See discussions, stats, and author profiles for this publication at: https://www.researchgate.

net/publication/340818221

Aeration System Design

Preprint · April 2020

DOI: 10.13140/RG.2.2.28903.80808

CITATIONS READS
0 524

1 author:

Johnny Lee
American Society of Civil Engineers
32 PUBLICATIONS   51 CITATIONS   

SEE PROFILE

Some of the authors of this publication are also working on these related projects:

wastewater treatment and other types of bioreactors View project

The Lee-Baillod model for submerged diffused bubble aeration View project

All content following this page was uploaded by Johnny Lee on 21 April 2020.

The user has requested enhancement of the downloaded file.

4/18/2020

Designing Aeration

Systems using Baseline

Mass Transfer

Coefficients:

for Water and Wastewater treatment

Johnny Lee
This book is dedicated to the memory of Dr. C. R. Baillod who first introduced to the author

the concept of a variable gas depletion rate. Dr. Baillod was a most sincere and diligent

scholar who thought of nothing but contributing to society. The author also wishes to thank

the reviewers for their painstaking review and their insights in this work.

1|Page
 Table of Contents

Preface .................................................................................................................................................................. 5

Chapter 1. Prologue ............................................................................................................................................. 11

References ...............................................................................................................................................................19

Chapter 2. Mass Transfer Coefficient and Gas Solubility ...................................................................................... 21

2.0 Introduction ...................................................................................................................................................21
2.1. THE TEMPERATURE CORRECTION MODEL FOR KLa ..................................................................................22
2.1.1. Basis for model development ..............................................................................................................22
2.1.2. Description of proposed model ...........................................................................................................25
2.1.3. Background ..........................................................................................................................................27
2.2. Theory .......................................................................................................................................................31
2.3. Materials & Methods ................................................................................................................................36
2.3.1. Hunter’s Experiment ............................................................................................................................38
2.3.2. Vogelaar et al.’s Experiment ................................................................................................................39
2.4. Results and Discussion ..............................................................................................................................39
2.4.1. Hunter’s data .......................................................................................................................................42
2.4.2. Vogelaar’s data ....................................................................................................................................46
2.4.3. Methodology for Temperature Correction ..........................................................................................48
2.5. THE SOLUBILITY MODEL ...........................................................................................................................50
2.6. Description of The Oxygen Solubility Model .............................................................................................52
2.7. Analysis .....................................................................................................................................................54
2.8. Conclusions ...............................................................................................................................................59
References ...............................................................................................................................................................62

Chapter 3. Development of a model to determine baseline mass transfer coefficients in aeration tanks ............ 65
3.0 Introduction ...................................................................................................................................................65
3.1 Model Development ......................................................................................................................................70
3.2 Material and Method ....................................................................................................................................76
3.3 Results and Discussions .................................................................................................................................78
3.3.1. Example calculation .............................................................................................................................78
3.3.2. Estimation of the effective depth ratio (e = de/Zd) .............................................................................82
3.3.3. Determination of the Standard Specific Baseline ................................................................................83
3.3.4. Relationship between the mass transfer coefficient and saturation concentration ...........................84
3.3.5. Relationship between the baseline and oxygen solubility...................................................................85
3.3.6. Relationship between the baseline and the gas flow rate...................................................................87
3.3.7. Relationship between the baseline and water temperature ...............................................................88
3.4 Discussion and Implications...........................................................................................................................89
3.4.1. Rating curves for aeration equipment .................................................................................................91
3.5 Potential for future applications ...................................................................................................................94
3.5.1. Scaling up .............................................................................................................................................94

2|Page
 3.5.2. Translation to in-process oxygen transfer ...........................................................................................95
3.6 Conclusions ....................................................................................................................................................96
References ...............................................................................................................................................................99

Chapter 4. The Lee-Baillod Equation .................................................................................................................. 102

4.0. Introduction to Derivation of the Lee-Baillod Model ..............................................................................102
4.1. Derivation of the Constant Bubble Volume Model .................................................................................103
4.1.1. CONSERVATION OF MASS IN THE GAS PHASE ...................................................................................103
4.1.2. CONSERVATION OF MASS IN THE LIQUID PHASE ..............................................................................110
4.1.3. DERIVATION OF THE DEPTH CORRECTION MODEL............................................................................113
4.1.4. THE HYPOTHESIS OF A CONSTANT BASELINE (KLa0) ...........................................................................117
4.1.5. DETERMINATION OF CALIBRATION PARAMETERS (n, m) FOR THE LEE- BAILLOD MODEL ................119
4.1.6. DETERMINATION OF THE EFFECTIVE DEPTH ‘de’ OR ‘Ze’ ...................................................................122
References .............................................................................................................................................................125

Chapter 5. Baseline Mass Transfer Coefficients and Interpretation of Non-steady State Submerged Bubble
Oxygen Transfer Data ........................................................................................................................................ 127
5.0. Introduction ............................................................................................................................................127
5.1. Theory .....................................................................................................................................................130
5.2. Methodology for depth correction .........................................................................................................133
5.3. Materials and Methods ..........................................................................................................................137
5.3.1. Case Study 1 - Super-oxygenation tests.............................................................................................137
5.3.2. Case Study 2 - ADS (Air Diffuser Systems) aeration tests. ................................................................151
5.3.3. Case Study 3 - FMC, Norton and Pentech Jet aeration shop tests.....................................................153
5.4. Example Calculations ..............................................................................................................................154
5.5. Discussion ...............................................................................................................................................159
5.6. Justification of the 5th power model over the ASCE method for temperature correction .......................162
5.7. Conclusion ...............................................................................................................................................164
5.8. Notation (major symbols) .......................................................................................................................168
References .............................................................................................................................................................170

Chapter 6. Is Oxygen Transfer Rate (OTR) in Submerged Bubble Aeration affected by the Oxygen Uptake Rate
(OUR)? ............................................................................................................................................................... 173
6.0 Introduction .................................................................................................................................................173
6.1 Theory ..........................................................................................................................................................178
6.1.1 Relationship between KLa and water characteristics .........................................................................178
6.1.2 Le Chatelier’s Principle applied to Gas Transfer ................................................................................182
6.1.3 The Hypothesis of a baseline KLa for wastewater ..............................................................................187
6.1.4 The application of the baseline KLa for wastewater ..........................................................................190
6.1.5 The Hypothesis of a microbial Gas Depletion Rate ............................................................................196
6.2 Materials and Methods ...............................................................................................................................197
6.2.1 Garcia et al.’s Experiment [2010].......................................................................................................197
6.2.2 Results and Discussions .....................................................................................................................200
6.2.3 Results from previous tests re-visited and Discussions .....................................................................204

3|Page
 6.3 Relationship between Alpha (α) and Apha'(α’) ...........................................................................................207
6.4 Measurement of Respiration Rate ..............................................................................................................209
6.5 Conclusions ..................................................................................................................................................217
6.6 Appendix ......................................................................................................................................................219
6.6.1 The Lee-Baillod Model in wastewater (speculative) ...............................................................................219
6.6.2 Determination of the Standard Specific Baseline for wastewater (speculative).....................................220
6.6.3 Mathematical Derivation for In-process Gas Transfer Model ................................................................222
6.6.4 A simple method to eliminate the impact of free surface oxygen transfer (speculative) .......................231
6.7 Notation ......................................................................................................................................................236
References .............................................................................................................................................................239

Chapter 7. Recommendation for further testing and research ........................................................................... 243

7.0. Introduction ............................................................................................................................................243
7.0.1. Testing for alpha in aeration tank ......................................................................................................244
7.0.2. Proposed Test Facility ........................................................................................................................248
7.0.3. Technical challenges ..........................................................................................................................250
7.0.4. Estimation of the effective depth ratio (e = de/Zd) ...........................................................................251
7.0.5. Standard Oxygen Transfer Rate .........................................................................................................255
7.0.6. Standard Oxygen Transfer Efficiency (SOTE) .....................................................................................256
7.1. Determination of Oxygen Transfer in wastewater .................................................................................258
7.1.1. BOD bottle Method ...........................................................................................................................259
7.1.2. Synthetic wastewater to determine actual oxygen uptake rate (Mines’ Method) ...........................260
7.1.3. Alternate Methods to be considered.................................................................................................261
7.1.4. Dilution Method.................................................................................................................................263
7.1.5. Procedure for determination of wastewater mass transfer coefficient KLaf and alpha (α) ...............267
7.2. Associated cost-benefit implications ......................................................................................................269
7.2.1. Upgrading of current Standards ........................................................................................................269
7.2.2. Translating Clean water test results to In-process water measurements .........................................271
7.2.3. Energy Conservation ..........................................................................................................................279
7.3. Appendix .................................................................................................................................................289

Chapter 8. Epilogue............................................................................................................................................ 300

References .............................................................................................................................................................318
END OF BOOK PAGE ..............................................................................................................................................319

4|Page
Preface
Nowadays, the way is totally open for an individual to revolutionise physics. Perhaps

someone will devise a new interpretation of a measurement problem, or show us how several

fundamental constants are really related, or ..... The same thing happens over and over again in

various fields; existing theories become entrenched, elaborations of them become increasingly

detailed and increasingly expensive, then someone produces a radically new theory that seems to

come out of nowhere (but actually doesn’t) and the cycle starts again.

Many people argue that people like Einstein and Newton took the low-hanging fruit and

that it’s no longer possible for an individual to make great advances working largely alone. I

disagree with this view, (as I have worked all alone for many years), even leaving aside the fact

that Newton and Einstein relied heavily on the work of others, as I have relied on other people’s

findings and data. What they came up with were paradigm shifts; not merely elaborations of

existing theories, but really new ways of looking at things. This happens very rarely. On the

contrary, by their very nature, it’s hard for teams to devise paradigm shifts. Teams have to set out

plausible-sounding grant applications, ones where they can expect to make some useful progress.

An individual, working in his or her spare time or within the protection of tenure (or in my case

my own savings), can sometimes afford to devote a large amount of time to working on something

that sounds a bit crazy but might actually be right.

While producing a theory that sounds crazy is no proof at all that the theory is correct (the

test of that is superior agreement with experimental evidence), paradigm-shifting theories do

always sound a bit crazy.

Newton faced criticism from people who objected that he had specified no mechanism for

gravity, which he liked to call “the hand of God”. Einstein faced opposition from people who were

5|Page
certain that time could not be relative and the luminiferous aether (a staple of physics at the time,

much talked about by the Michio Kakus of the day) must clearly exist. (I am facing opposition

because many people are certain that the mass transfer equation cannot contain double the amount

of respiration rate R, resulting in a mass transfer coefficient that is twice the value or so that would

be derived in a steady-state test. This is briefly explained in Chapter 1 below.)

Neither of them would have been taken seriously had proof of their theories not been

forthcoming. (My published papers [Lee 2018, 2019a, 2019b] contain case studies of previous

works by other researchers and mathematical proof of my theories---all it needs is some testing to

confirm and hence grants and fundings are required).

My new theory is based on the concept of a baseline for the mass transfer coefficient. As

it has never been used before, the discovery of this concept for designing aeration systems is

truly ground-breaking. But what is a baseline? A baseline is a fixed point of reference that is

used for comparison purposes. The baseline serves as the starting point against which all future

estimations are measured. A baseline can be any number that serves as a reasonable and defined

starting point for comparison purposes. It may be used to evaluate the effects of a change, track

the progress of an improvement project, or measure the difference between two periods of time.

One of the usefulness of a baseline mass transfer coefficient, in the context of wastewater

treatment, is to predict what will happen to the oxygen transfer in aeration systems if microbial

activity is present in the test water.

In line with Mahendraker's theory [Mahendraker V (2003)], the author believes that the

resistance to oxygen transfer is composed of two parts: one, a resistance by the reactor solution;

the other is the resistance due to the biological floc. [Mahendraker V., Mavinic, Donald S., and

Rabinowitz, B. (2005b).] And so if clean water testing is considered a baseline, the effect of cell

6|Page
concentrations can be superimposed onto the baseline to determine the oxygen transfer in the actual

mixed liquor of a treatment plant aeration basin, using the Principle of Superposition in physics.

This concept has been criticized by some experts. One university professor advised me that

“Consulting companies cannot benefit from investing in this research area. The only sector that I

can think of who may have an interest may be those that make aeration devices. However they

again would have to see a benefit from investing in your research. So currently they would do

standardized testing and have field data to verify their design calculations. Can investing in your

research help them improve their design or competitiveness leading to possible economic

benefits? If so, they are ones you may follow up with.”

“The other problem that I see is the cost and practicality of your research. Real life testing

is expensive enough and require an expensive infrastructure. I don’t see anyone funding that kind

of a setup only for this study. The facilities that offer services for standard water testing – I don’t

see them allowing the use to do testing with wastewater. It may be worth examining if there a

simpler and cheaper way of getting some data that will allow you to “validate” your model. Lower

cost of the study might make it easier to find an industrial partner. However again, they would

want to see a value to them from investing in the research.”

This raises the question "who is responsible for each aspect of aeration systems design?"

According to Stenstrom M.K. and Boyle W. (1998), "Some owners and consultants want to make

it entirely the responsibility of the manufacturer. Such attitudes are incorrect and produce

indifferent attitudes during the design process ("it is their job-why are we worrying about it?") and

wasteful litigation. Although manufacturers must be held responsible for the aspects of aeration

design that they control --- this means clean water transfer performance and mechanical integrity;

manufacturers have no control over wastewater characteristics, process design, or the way their

7|Page
equipment is operated and maintained. Design engineers must anticipate a range of operating

conditions and their effects on the aeration system. Alpha factors (RATIOS OF THE TRANSFER

COEFFICIENTS between tap water and wastewater) are strongly affected by process design and

operation and by system configuration, as well as wastewater characteristics.” [In this respect, my

finding differs from conventional thinking: in my opinion, alpha should only be dependent on the

reactor solution, not the process, in line with Mahendraker V. (2003)'s thinking.]

Another criticism comes from the American Oxygen Transfer Standards Committee:

“You're confusing oxygen transfer mechanisms with respiration. The respiration determines the

oxygen uptake rate (OUR) which is then matched by the oxygen transfer rate. There is no double

R term. R is NOT a part of the oxygen transfer mechanism.”

Stenstrom and Boyle (1998) continued: “Currently, Alpha factors cannot be specified by

the manufacturer: they must be determined by the design engineer for the range of operating

conditions anticipated by the owner. Alpha factors for design are not a single value but ranges of

values that occur for different process conditions, times, and locations within aeration tanks.

Owners must know and accept responsibility for their operating decisions; for example,

dramatically reducing sludge age decreases oxygen transfer efficiency in most aeration systems,

which should be considered before making process changes. Information on alpha must be

obtained through in-process testing experience or carefully documented data from the literature or

other credible sources. Small-scale testing such as laboratory testing has not been a reliable source

of data.” [Here again, my proposed model appears to be able to utilize such data to predict full-

scale performance.]

“Both owners and designers should not accept alpha factor claims by manufacturers. In

most cases it will be very difficult to hold manufacturers accountable for the alpha factors they

8|Page
might claim, because they cannot control process operation or wastewater characteristics.

Consultants who accept manufacturers' recommendations without verifying them are not

protecting their clients. Another important issue that remains the responsibility of the design

engineer is the compliance specification of the aeration equipment and the provision of any

required scale-up to the actual installation. Typically in the United States, clean water compliance

specifications are used. For process water, compliance specifications require considerations of

wastewater variability and process loading that lead to substantial uncertainty. If left to the

manufacturer, extremely conservative and costly systems will result for obvious reasons, because

the manufacturer has little knowledge of the wastewater and process operating conditions. In this

situation, it is incumbent on the designer to specify alpha and other process variables within the

specification. Another issue in compliance testing is scale-up. If shop tests are to be performed, it

is up to the designer to specify the shop test and to provide the necessary scale-up to field

conditions. Manufacturers may be consulted on issues, but it is ultimately the designers'

responsibility to ensure proper scale-up. To avoid this problem, some designers specify clean water

compliance testing in the field system."

From the discussion above, it would appear that approaching aeration device manufacturers

may not be a fruitful outcome. Maybe industrial partners such as paper and pulp companies may

be interested. Does anybody know of any? But my first choice would still be government bodies

and academic organizations and institutions, I reckon.

9|Page
References

Lee, J. (2018). “Development of a model to determine the baseline mass transfer coefficients in
aeration tanks”, Water Environ. Res., 90, (12), 2126 (2018).
Lee, J. (2019a). “Baseline Mass Transfer Coefficient and Interpretation of Non-steady State
Submerged Bubble Oxygen Transfer Data” 10.1061/(ASCE)EE.1943-7870.0001624.
Lee, J. (2019b). “Is Oxygen Transfer Rate (OTR) in Submerged Bubble Aeration affected by the
Oxygen Uptake Rate (OUR)?” 10.1061/(ASCE)EE.1943-7870.0001635.
Mahendraker V. (2003) “Development of a unified theory of oxygen transfer in activated sludge
processes – the concept of net respiration rate flux”, Department of Civil Engineering,
University of British Columbia.
Mahendraker, V., Mavinic, D.S., and Rabinowitz, B. (2005a). Comparison of oxygen transfer
test parameters from four testing methods in three activated sludge processes. Water Qual.
Res. J. Canada, 40(2).
Mahendraker, V. Mavinic, D.S., and Rabinowitz, B. (2005b). A Simple Method to Estimate the
Contribution of Biological Floc and Reactor-Solution to Mass Transfer of Oxygen in
Activated Sludge Processes. Wiley Periodicals, Inc. DOI: 10.1002/bit.20515.
Stenstrom M.K. and Boyle W. (1998): AERATION SYSTEMS-RESPONSIBILITIES OF
MANUFACTURER, DESIGNER, AND OWNER, Environmental Engineering Forum,
Journal of Environmental Engineering, May 1998 (398)

10 | P a g e
Chapter 1. Prologue
The US EPA in the 70's poured in substantial amount of money to fund fundamental

research, as they recognized the importance of the connection between clean water tests and

wastewater tests. Although they have made substantial progress, the fundamental question of

relating clean water and wastewater tests remains unresolved. [Mahendraker V., Mavinic, Donald

S., and Rabinowitz, B. (2005a).] A new revolutionary finding may revive their interest.

This book is focused primarily on submerged bubble aeration. In aeration systems, diffused

air is a simple concept which entails pumping (injecting) air through a pipe or tubing and releasing

this air through a diffuser below the water's surface. The submerged system has little visible pattern

on the surface, and is able to operate in depths up to and exceeding 12 m (40 ft). The best aerators

use quiet on-shore compressors that pump air to diffusers placed at a pond or tank bottom. From

stone diffusers to self-cleaning dome diffusers, they release oxygen throughout the water column

creating mass circulation that mixes bottom and top water layers, breaks up thermal stratification,

and replenishes dissolved oxygen through molecular oxygen mass transfer by means of gas

diffusion. Gas transfer is the exchange of gases between aqueous and gaseous phases. In a diffused

aeration, gas exchange takes place at the interface between submerged air bubbles and their

surrounding water. According to Lewis and whitman (1924), these bubbles are each wrapped with

two layers of films through which the gas must go through. The transfer rate is usually expressed

by a mass transfer coefficient symbolized by KLa.

No one has seen the two films around a bubble, let alone measuring the thicknesses of these

films based on which KLa can be quantified. The coefficient can only be determined by an indirect

method, such as the one used by the current ASCE standard (ASCE 2007). The transfer rate can

also be determined by mass balances---the gas depletion rate from the bubble must equal the

11 | P a g e
oxygen uptake rate in the liquid. This concept of gas-side oxygen depletion is not as readily

understood as it may seem:

The respiration determines Oxygen Uptake Rate (OUR) that equates to the Oxygen Transfer Rate

(OTR) at steady-state. The understanding that "The respiration determines the OUR which is then

matched by the oxygen transfer rate." concurs with my thesis in this book, and indeed is correct. But

in submerged aeration, there is the phenomenon known as gas-side oxygen depletion, so that the

oxygen transfer rate is affected by this effect and this effect (incorrectly) changes the value of KLa. To

make the correction, the OTR is therefore given by KLaf (C*∞f - c)-R (where f means “under field

conditions”) under the principle of superposition in physics (This concept is further explained in

Chapter 6). This is then matched by the oxygen transfer rate OTR at steady-state, which is equal to

respiration rate R.

Therefore,

KLaf (C*∞f -c)-R = R.

Although R is not part of the oxygen transfer mechanism, gas-side oxygen depletion is. If R is non-

variant within the test period, then it can be determined in a gas flow steady state, where R is matched

by the gas depletion rate in the bubbles which affects the value of the OTR. This has led to THE

ABOVE EQUATION when KLaf is understood to be (alpha.KLa) where alpha is a function of the

wastewater characteristics only. The current alpha as used in the conventional model treats it as a

lumped parameter that envelopes both effects (water characteristics and gas depletion), making it a

highly variable parameter that is indeterminate. The concept of gas-side depletion of oxygen from air

bubbles, at first glance, appears to be simple and straightforward, but is in fact less readily understood

than it may seem. In ordinary air bubble aeration, the OTE is typically 10 ~ 20%, since oxygen gas is

only slightly soluble in water. (In clean water, it can be as much as 40% depending on the aeration

device and the mixing intensity). This 10 ~ 20% by weight is the actual amount of oxygen successfully

being transferred to the liquid. This quantity is exactly equal to the quantity of gas depleted from the

air bubbles, since ‘oxygen transfer’ and ‘gas-side gas depletion’ are one and the same.

12 | P a g e
In fact, if in the absence of free surface gas transfer, gas-side gas depletion as the bubbles rise to the

free surface is the ONLY means of oxygen transfer, including any oxygen transfer at the bubble

formation stage. Therefore, any modeling of oxygen transfer into any liquid (tap water, sewage,

industrial wastes, etc.) must include the gas depletion effect, otherwise, the model cannot be valid.

In the review paper by Lars Uby (2019), in section 6.1, it was stated that "Among the CEN and DWA

standard test methods, no result dependence on initial conditions (supersaturation or depletion of

oxygen and nitrogen) has been detected (Wagner, 1991), but a rigorous uncertainty analysis is lacking.

This has been fully accepted in German and European practice (DWA, 2007; CEN, 2003). Gas side

depletion of oxygen from air bubbles has been shown to be a minor concern under common conditions

(Baillod and Brown, 1983; Jiang and Stenstrom, 2012), corroborating this approach [of Lars’ paper].

In the interest of standardization and uncertainty quantification, the difference should be quantified,

though experience speaks for at most a minor impact."

In my opinion, the above understanding by Lars (as well as the various standards) is absolutely

incorrect. It should be taken only in the context of the results of a clean water test, where the

parameters Cs (saturation value) and KLa (mass transfer coefficient) are estimated. The reason why

no result dependence on initial conditions (supersaturation or depletion of oxygen and nitrogen) has

been detected, is because these effects have already been absorbed in the Standard Model. In other

words, the calculated results of the two parameters have already included any dependence on these

effects, even though such dependence is not detected. In the application of clean water results to

sewage or other liquid, these effects will change in accordance with changes in the gas depletion rate

which is the same as changes in oxygen transfer rate under a changed environment. The bacterial

and other microbial composition and their metabolic functioning, in particular, constitute drastic

changes in the oxygen gas depletion rate which then drastically affects the value of the gas transfer

parameters, in particular the mass transfer coefficient KLa. Standardization and uncertainty analysis

by all means, but they will not significantly improve on the clean water test results. On the other hand,

if clean water test results are to be translated to other fluids with significant microbial cell content,

mixed liquor for example, then the principle of superposition must be applied to the Standard Model

13 | P a g e
to take into account this all-important gas depletion effect in diffused aeration, without which nothing

in terms of oxygen transfer happens. Therefore, gas-side gas depletion is not a minor impact, but in

fact is the ONLY impact in submerged diffused-air bubble aeration, the magnitude of which is a

function of a myraid of variables.

The amount of gas depleted from the bubble at any time not only depends on the films, but

also on the path taken by the bubble that follows a gas depletion curve which is a function of many

variables. This curve would vary with different heights and depths. Also, this gas depletion curve

in clean water is substantially different from that in wastewater. The loss rate of gas from the

bubble is the amount rate transferred at any time and place inside an aeration tank.

Given that the mass transfer coefficient (KLa) is a function of many variables, in order to

have a unified test result, it is necessary to create a baseline mass transfer coefficient, so that all

tests will have the same measured baseline. KLa is found to be an exponential function of this new

coefficient and is dependent on the height of the liquid column (Zd) through which the gas flow

stream passes. DeMoyer et al. (2003) and Schierholz et al. (2006) have conducted experiments to

show the effect of free surface transfer on diffused aeration systems, and it was shown that high

surface-transfer coefficients exist above the bubble plumes, especially when the air discharge (Qa)

is large. When coupled with large surface cross-sectional area and/or shallow depth, the oxygen

transfer mechanism becomes more akin to surface aeration where water entrainment with air from

the atmosphere becomes important. The water turbulence has a significant effect on oxygen

transfer. The alternative to a judicious choice of tank geometry and/or gas discharge, is perhaps

another mathematical model that could separate the effect of surface aeration from the actual

aeration under testing in the estimation of the mass transfer coefficient. This separate modelling

for surface aeration is not a topic in this book. Nevertheless, a simple graphical method to take this

14 | P a g e
effect into account in the establishing of the baseline coefficient is proposed in Chapter 6 Section

6.6.4.

In engineering, the mass transfer coefficient is a diffusion rate constant that relates the mass

transfer rate, mass transfer area, and concentration change as driving force, using the Standard

Model, typically stated in the form given by eq. 4-1 in Chapter 4. This can be used to quantify the

mass transfer between phases, immiscible and partially miscible fluid mixtures (or between a fluid

and a porous solid). Quantifying mass transfer allows for design and manufacture of separation

process equipment that can meet specified requirements and estimate what will happen in real life

situations (chemical spill, wastewater treatment, fermentation, and so forth) if the effect of other

factors, such as turbulence either due to the free surface exchange or due to mechanical mixing

within the water body, can be isolated, or eliminated or modelled separately.

Mass transfer coefficients can be estimated from many different theoretical equations,

correlations, and analogies that are functions of material properties, intensive properties and flow

regime (laminar or turbulent flow), all based on the Standard Model. Selection of the most

applicable model is dependent on the materials and the system, or environment, being studied.

This book is about the discovery of a new coefficient called the baseline mass transfer coefficient

(KLa0). The process of this discovery is described in Chapter 3. For open tank aeration, the author

defines it as the ordinary mass transfer coefficient (KLa) measured at the equilibrium pressure of

the standard sea-level atmospheric pressure (101.325 kPa). Since most testing is carried out in a

vessel of some physical height, the equilibrium pressure must exceed this baseline pressure of 1

atmosphere. If water is used for an aeration test in accordance with current standards [ASCE

2007][CEN 2003][DWA 2007], the system would attain a “super-saturated” state at equilibrium.

This super-saturated dissolved gas concentration (C*∞) would differ from the saturation

15 | P a g e
concentration that can be readily found from published data or any chemistry handbook on gas

solubility. The closest experiment that would yield a handbook solubility (CS) value (and the

corresponding baseline mass transfer coefficient) would be a laboratory-scale experiment.

In any other situations, KLa0 is not directly measurable. This book is about how the baseline

can be determined using the Standard Model for gas transfer, despite the many variables affecting

such transfer and KLa. Based on the various literature data cited, the baseline has proven to be a

valuable parameter (perhaps even more useful than KLa itself) that can be used to predict gas

transfer under different test conditions, such as different heights or liquid depths; perhaps even

different geometry. This is a revolutionary change as, up to now, it has not been possible to

correlate KLa from one test to another, even under ordinary testing circumstances. However, the

baseline, or more correctly the specific baseline upon normalizing with the gas flowrate, is a “true”

constant for every test. In the context of the meaning of “baseline”, the book is expected to be a

baseline itself for future upgrading when more data becomes available. People interested in this

book would certainly be scientists, engineers, researchers, treatment plant operators, and

manufacturers of aeration systems.

As mentioned, the mass transfer coefficient KLa is related to the air discharge and is found

to be dependent on the gas average flow rate (Qa) passing through the liquid column. Qa is

estimated from the gas mass flow rate (Qs), and is expressed in terms of actual volume of gas per

unit time, as distinct from Qs that is expressed as mass per unit time. For a uniform liquid

temperature (T) throughout the liquid column, Qa is calculated by Boyle’s Law, and taking the

arithmetic mean of the volumetric flow rates over the tank column. (Although the mass flow rate

Qs is sometimes also expressed as volume per unit time, it is not true volume because it is expressed

as standard conditions, which is equivalent to mass per unit time.) As such, Qa is a variable

16 | P a g e
dependent on temperature, pressure and volume, even when the gas supply Qs is fixed and non-

variant.

When an intensive property such as temperature is varied, KLa0 is directly proportional to

this mean gas flow rate (Qa) to a power q, where q is usually less than unity for water in a fixed

column height and a fixed gas supply rate at standard conditions, (Qs). However, KLa0 is not

proportional to Qs as the case studies presented in this book would show, although there may be

good correlation in some cases. When temperature is fixed, the same relationship holds for

different values of Qa, regardless of column height. This book provides theoretical development

and case studies that verify this baseline which can be standardized specifically to the average gas

flow rate as a new function (KLa0)/Qaq that is applicable to submerged bubble aeration testing. This

function is termed the specific baseline in this document, and is a constant quantity for any test

temperature T. This relationship between the baseline and gas flow can be determined by

experiments, as the case studies in Chapter 5 demonstrate, in which it was shown that the overhead

(or headspace) pressure is also an intensive property that, when varied, would give the same

baseline versus gas flowrate relationship. When the function is determined at standard conditions,

it is termed the standard specific baseline expressed as (KLa0)20/Qa20 q and is a constant independent

of tank height Zd and gas flow Qa.

Lastly, the suggested replacement of the temperature correction model for the mass transfer

coefficient that is based on the Arrhenius equation as stipulated by ASCE Standard 2-06 [ASCE

2007], with the new 5th power model, (see Chapter 2), may be controversial, because the former

equation is well known and is being used by the standard for a long time. This controversy is not

too important in this manuscript, as all the tests cited were conducted in the neighborhood of 20
0
C (within the range of 10 0C to 30 0C), and so there are only small differences in the calculations

17 | P a g e
of (KLa)20 or (KLa0)20 between the two models. Nevertheless, a discussion is in order since the new

model gives a slightly better correlation between the standard baseline and the gas flow rates in all

cases. As Dr. Stenstrom explained for the background: “In the first version of the standard, we

debated the value of theta (Ɵ) …and found that most of the literature data supported 1.020 to 1.028

with the diffused systems clustering toward the bottom of the range and the surface [aeration

systems] clustering toward the top of the range.”[Stenstrom and Lee, 2014].

From this, it can be inferred that there may be two different ranges of the temperature

correction factor Ɵ for the two aeration systems referred to by Stenstrom. Based on analyzing

literature data on diffused systems, the author found that the 5th power model fits more closely

with a theta (Ɵ) value in the range of 1.016~1.018 [Lee 2017][Chapter 2] which is closer to the

range for diffused systems. Furthermore, the ‘standard-recommended’ theta value of 1.024 is

probably based on tests on conventional treatment plants or shop tests of similar height that is

usually around 3 m (10 ft) to 4.5 m (15 ft). The 5th power model is suitable for ‘zero’ height since

most laboratory tests were carried out on a bench scale of very little height. Since the baseline

pertains to a mass transfer coefficient of an infinitesimally shallow tank, it would appear that this

new 5th power model is more suitable for correcting the baseline to the standard temperature. It

should be noted in passing that, temperature is an intensive property (i.e., independent of scale),

whereas KLa is a function of height and other variables, and it is dependent on scale; and so, it

cannot be accurately corrected by a single factor that summarily ignores changes in height and

other factors.

The book is divided into eight chapters. Chapter 2 below deals with the derivation of the

5th power model for temperature correction. Chapter 3 deals with the development of the model to

determine the baseline mass transfer coefficients in aeration tanks. Chapter 4 is dedicated to the

18 | P a g e
derivation and theoretical development of the Lee-Baillod model on which the subsequent depth

correction model is based. Chapter 5 illustrates the functionality of the Baseline Mass Transfer

Coefficient and Interpretation of Non-steady State Submerged Bubble Oxygen Transfer Data.

Chapter 6 asks the question: (Is Oxygen Transfer Rate (OTR) in Submerged Bubble Aeration

affected by the Oxygen Uptake Rate (OUR)?), concerning the use of the baseline for in-process

field working conditions; Chapter 7 recommends further research to elucidate the question posed

in Chapter 6, and Chapter 8 is the Epilogue that summarizes all the core findings. It is expected

that this book would serve practitioners in the designing of aeration systems, as well as serve as

Standard Guidelines for water and wastewater (both In-Process and non-In-Process) oxygen

transfer testing, enhancing the current standards and guidelines, ASCE 2-06 [ASCE 2007] and

ASCE-18-96 [ASCE 1997].

References

ASCE-2-06. (2007). “Measurement of Oxygen Transfer in Clean Water.” Standards

ASCE/EWRI. ISBN-10: 0-7844-0848-3, TD458.M42 2007

ASCE-18-96. (1997). ``Standard Guidelines for In-Process Oxygen Transfer Testing`` ASCE

Standard.ISBN-0-7844-0114-4, TD758.S73

CEN, 2003. EN 12255-15. Wastewater Treatment Plants – Part 15: Measurement of the Oxygen

Transfer in Clean Water in Aeration Tanks of Activated Sludge Plants. European

standard.

DeMoyer Connie D., Schierholz Erica L., Gulliver John S., Wilhelms Steven C. (2002). Impact

of bubble and free surface oxygen transfer on diffused aeration systems. Water Research

37 (2003) 1890-1904.

19 | P a g e
DWA, 2007. Merkblatt DWA-M 209. Messung der Sauerstoffzufuhr von Beluftung-

seinrichtungen in Belebensanlagen in Reinwasser und in belebtem Schlamm. (Formerly

ATV; Measurement of oxygen transfer of aeration equipment in biological treatment

plants in clean water and in activated sludge.)

Lee, J. (2017). Development of a model to determine mass transfer coefficient and oxygen

solubility in bioreactors. Heliyon 3(2): e00248.

Lewis, W.K., Whitman, W.G. (1924). “Principles of Gas Absorption”, Ind. Eng. Chem., 1924,
16 (12), pp 1215–1220 Publication Date: December 1924 (Article)
DOI:10.1021/ie50180a002
Mahendraker, V., Mavinic, D.S., and Rabinowitz, B. (2005a). Comparison of oxygen transfer
test parameters from four testing methods in three activated sludge processes. Water Qual.
Res. J. Canada, 40(2).
Schierholz Erica L., Gulliver John S., Wilhelms Steven C., Henneman Heather E. (2006). Gas

Transfer from air diffusers. Water Research 40 (2006) 1018-1026.

Stenstrom and Lee (2014). Private communication, email dated 2/7/2014.

Uby Lars (2019). “Next steps in clean water oxygen transfer testing --- A critical review of

current standards”. Water Research 157 (2019) 415 – 434.

20 | P a g e
Chapter 2. Mass Transfer Coefficient and Gas Solubility
2.0 Introduction

The main objective of this chapter is to develop a mechanistic model (based on

experimental results of two researchers, Hunter [1979] and Vogelaar et al. [2000]), to replace the

current empirical model in the evaluation of the standardized mass transfer coefficient (KLa)20

being used by the ASCE Standard 2-06 [ASCE 2007]. The topic is about gas transfer in water,

(how much and how fast), in response to changes in water temperature. This topic is important in

wastewater treatment, fermentation, and other types of bioreactors. The capacity to absorb gas into

liquid is usually expressed as solubility, Cs; whereas the mass transfer coefficient represents the

speed of transfer, KLa, (in addition to the concentration gradient between the gas phase and the

liquid phase which is not discussed here). These two factors, capacity, and speed, are related and

the manuscript advocates the hypothesis that they are inversely proportional to each other, i.e., the

higher the water temperature, the faster the transfer rate, but at the same time less gas will be

transferred.

This hypothesis was difficult to prove because there is not enough literature or

experimental data to support it. Some data [ASCE 1997], do support it, but they are approximate,

because some other factors skew the relationship, for example, concentration gradient; and the

hypothesis is only correct if these other factors are normalized or held constant.

This hypothesis may or may not be proved by theoretical principles, such as by means of

thermodynamic principles to find a relationship between equilibrium-concentration and mass

transfer coefficient, but such proof is beyond the expertise of the author. However, the hypothesis

can in fact be verified indirectly by means of experimental data that were originally used to find

the effects of temperature on these two parameters, solubility (Cs) and mass transfer coefficient

21 | P a g e
(KLa). Temperature affects both equilibrium values for oxygen concentration and the rate at which

transfer occurs. Equilibrium concentration values (Cs) have been established for water over a range

of temperature and salinity values, but similar work for the rate coefficient is less abundant.

This chapter uses the limited data available in the literature to formulate a practical model

for calculating the standardized mass transfer coefficient at 20 0C. The work proceeds with general

formulation of the model and its model validation using the reported experimental data. It is hoped

that this new model can give a better estimate of (KLa)20 than the current method.

2.1. THE TEMPERATURE CORRECTION MODEL FOR KLa

2.1.1. Basis for model development

Using the experimental data collected by two investigators Hunter [1979] [Vogelaar et al.

2000], data interpretation and analyses allowed the development of a mathematical model that

related KLa to temperature, advanced in this paper as a temperature correction model for KLa. The

new model is given as:

𝑬𝝆𝝈
(𝑲𝑳 𝒂)𝑻 = 𝑲 × 𝑻𝟓 × (𝟐 − 𝟏)
𝑷𝒔

where KLa = overall mass transfer coefficient (min-1); T = absolute temperature of liquid under

testing in Kelvin; the subscript T in the first term indicates KLa at the temperature of the liquid at

testing; and K = proportionality constant; E = modulus of elasticity of water at temperature T,

(kNm-2); ρ = density of water at temperature T, (kg m-3); σ = interfacial surface tension of water

at temperature T, (N m-1); Ps is the saturation pressure at the equilibrium position (atm). The

derivation is based on the following findings as described in Section 2-3.

The model was based on the two-film theory by Lewis and Whitman [1924], and the

subsequent experimental data by Haslam et al. [1924], whose finding was that the transfer

coefficient is proportional to the 4th power of temperature. Further studies by the subsequent

22 | P a g e
predecessors [Hunter 1979, Boogerd et al. 1990, Vogelaar et al. 2000] unveiled more relationships,

which when further analyzed by the author, resulted in a logical mathematical model that related

the transfer coefficient (how fast the gas is transferring when air is injected into the water) to the

5th order of temperature. Perhaps this is also a hypothesis, but it matches all the published data

sourced from literature.

Similarly, using the experimental data already published for saturation dissolved oxygen

concentrations, such as the USGS (United States Geological Survey) tables [Stewart Rounds

2011], Benson and Krause’s stochastic model [Benson and Krause 1984], etc., it was found that

solubility also bears a 5th order relationship with temperature.

So, there are actually three hypotheses. But are they hypotheses or are they in fact physical

laws that are beyond proof? For example, how does one prove Newton’s law? How does one prove

Boyle’s law, Charles’ law, or the Gay-Lussac’s law? They can be verified of course, but do not

lend themselves easily to mathematical derivation using basic principles. As mentioned, Prof.

Haslam found that the liquid film transfer coefficient varies with the 4th power of temperature, but

how does one prove it by first principles? The model just fits all the data that one can find although

it would be great if it can be proven theoretically. However, the correlation coefficients for (eq. 2-

1) are excellent as can be seen in the following sections.

The paper for this chapter is not a theory/modelling paper in the sense that a theory was

not derived based on first principles. Nor in fact is it an experimental/empirical paper since the

author did not perform any experiments. However, the research workers who did the experiments

did not recognize the correlation, and so they have missed the connection. This paper revealed that

these data can in fact support a new model that relates gas transfer rate to temperature that they

missed. They used their data for other purposes, and drew conclusions for their purposes.

23 | P a g e
 Further tests may therefore be required to justify these hypotheses. Although other people’s

data are accurate since they come from reputable sources, they are different from experiments

specifically designed for this model development purpose only. The novelty of the proposed model

is that it does not depend on a pre-determined value of theta (Ɵ) to apply a temperature correction

to a test data for KLa, if all other conditions affecting its value are held constant or convertible to

standard conditions.

The current model adopted by ASCE 2-06 is based on historical data and is given by the

following expression:

(𝐾𝐿 𝑎)20
= 1.024(20−𝑇) (2 − 2)
(𝐾𝐿 𝑎) 𝑇

In this equation, T is expressed in 0C and not in K (Kelvin) defined for (eq. 2-1). It has been widely

reported that this equation is not accurate, especially for temperatures above 20 0C. Current ASCE

2-06 employs the use of a theta (Ɵ) correction factor to adjust the test result for the mass transfer

coefficient to a standard temperature and pressure. The ratio of (KLa)T and (KLa)20 is known as the

dimensionless water temperature correction factor N, so that

(𝐾𝐿 𝑎)20
𝑁= (2 − 3)
(𝐾𝐿 𝑎) 𝑇

Current model is therefore given by:

𝑁 = 𝜃 (20−𝑇) (2 − 4)

where ϴ is the dimensionless temperature coefficient. This coefficient is based on historical

testing, and is purely empirical. Furthermore, the above equations indicate that the KLa water

temperature correction factor N is exclusively dependent on water temperature. This is definitely

not the case, as the correction factor is also dependent on turbulence, as well as the other properties

as shown in (eq. 2-1). Current wisdom is to assign different values of theta (θ) to suit different

24 | P a g e
experimental testing. While adjusting the theta (Ɵ) value for different temperatures may eventually

fit all the data, this may lead to controversies. Furthermore, it is necessarily limited to a prescribed

small range of testing temperatures.

2.1.2. Description of proposed model

The purpose of this chapter manuscript is to improve the temperature correction method

for KLa (the mass transfer coefficient) used on ASCE Standard 2-06 and to replace the current

standard model by (eq. 2-1).

The proposed model can also be expressed in terms of viscosity as described below.

Viscosity can be correlated to solubility. When a plot of oxygen solubility in water is made against

viscosity of water, a straight-line plot through the origin is obtained [IAPWS 2008]. When the

inverse of viscosity (fluidity) is plotted against the fourth power of temperature, the linear curve

as shown in Figure 2-1 below was obtained.

Water Fluidity and Temperature (0 to 50 C)

1.80
1.60
Riciprocal of viscosity (u = mPa.s)

y = 0.2409x - 0.7815
R² = 1
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00
0.00 2.00 4.00 6.00 8.00 10.00 12.00
T^4*1000 (T=degK/1000)

Figure 2-1. Reciprocal of Viscosity plotted against 4th power of temperature

25 | P a g e
 Therefore, viscosity happens to have a 4th order relationship with temperature, so that (Eq.

2-1) can be expressed in terms of viscosity and a first order of temperature, instead of using the 5th

order term. The concept of molecular attraction between molecules of water and the oxygen

molecule is important since changes in the degree of attraction would influence the equilibrium

state of oxygen saturation in the water system as well as its gas transfer rate. Although the above

plot (Figure 2-1) shows that the reciprocal of viscosity (fluidity) is linearly proportional to the 4th

order of absolute temperature, the line does not pass through the origin.

As viscosity is closely correlated to solubility, it is obvious that the molecular attraction

between water molecules that influences viscosity and the molecular attraction between water and

oxygen molecules are interrelated. This correlation does not establish that an alteration of water

viscosity, such as changes in the characteristics of the liquid, will have an impact on oxygen

solubility. However, it will certainly affect the mass transfer coefficient. Viscosity due to changes

in temperature is therefore an intensive property of the system, whereas viscosity due to changes

in the quality of water characteristics is an extensive property. The equation relating viscosity to

temperature is given by Fig. 2-1 as:

1 𝑇 4
= 0.2409 × 103 × ( ) − 0.7815 (2 − 5)
𝜇 1000

where µ = viscosity of water at temperature T, (mPa.s)

Rearranging the above equation, T4 can be expressed in terms of viscosity and therefore,

1
𝑇 4 = 𝐾 ′ × ( + 0.7815) (2 − 6)
𝜇

where K’ is a proportionality constant.

Substitute (eq. 2-6) into (eq. 2-1), therefore,

(𝐸𝜌𝜎) 𝑇 1
(𝐾𝐿 𝑎) 𝑇 = 𝐾 × × 𝐾 ′ × ( + 0.7815) × 𝑇 (2 − 7)
𝑃𝑆 𝜇

26 | P a g e
Grouping the constants therefore,

(𝑬𝝆𝝈)𝑻 𝟏
(𝑲𝑳 𝒂)𝑻 = 𝑲′′ × ( + 𝟎. 𝟕𝟖𝟏𝟓) × 𝑻 (𝟐 − 𝟖)
𝑷𝑺 𝝁

where K’’ is another proportionality constant.

Therefore, KLa can be expressed as either (eq. 2-8) or as (eq. 2-1). For the sake of easy

referencing to this model, this model shall be called the 5th power model.

2.1.3. Background

The universal understanding is that the mass transfer coefficient is more related to

diffusivity and its temperature dependence at a fundamental level on a microscopic scale. Although

Lewis and Whitman long ago advanced the two-film theory [Lewis and Whitman 1924] and

subsequent research postulated that the liquid film thickness is related to the fourth power of

temperature in K [Haslam et al. 1924], it was not thought that this relationship could be applied on

a macro scale. In a laboratory scale, Professor Haslam conducted an experiment to examine the

transfer coefficients in an apparatus, using sulphur dioxide and ammonia as the test solute. Based

on Lewis and Whitman’s finding that the molecular diffusivities of all solutes are identical, he

derived four general equations that link the various parameters affecting the transfer coefficients

which are dependent upon gas velocity, temperature, and the solute gas. He found that the absolute

temperature has a vastly different effect upon the two individual film coefficients. The gas film

coefficient decreases as the 1.4th power of absolute temperature, whereas the liquid film coefficient

increases as the fourth power of temperature. The discovery that the power relationship between

the liquid film coefficient and temperature can be applied to an even higher macroscopic level

where Cs is a function of depth, is based on a combination of seemingly unrelated events as

follows:

27 | P a g e
• Lee and Baillod [Lee 1978] [Baillod 1979] derived by theoretical and mathematical

development, a formula for the mass transfer coefficient (KLa) on a macro scale for a

bulk liquid treating the saturation concentration Cs as a dependent variable;

• The derived KLa mathematically relates to the “apparent KLa” [ASCE 2007] that is

defined in ASCE 2-06 standard;

• It was thought that KL (the overall liquid film coefficient) might perhaps be related to the

fourth power of temperature on a bulk scale similar to the same finding by Professor

Haslam on a laboratory scale, as described above;

• John Hunter [Hunter 1979] related KLa to viscosity via a turbulence index G;

• It was then thought that viscosity might be related to the fourth power temperature and a

plot of the inverse of absolute viscosity against the fourth power of temperature up to

near the boiling point of water gives a straight line;

• the interfacial area of bubbles per unit volume of bulk liquid under aeration is a function

of the gas supply volumetric flow rate which is in turn a first-order function of

temperature;

• It was then thought that KLa might be directly proportional to the 5th power of absolute

temperature and indeed so, as verified by Hunter’s data described in the following

Section 2.4.1 (Fig. 2-2); the relationship, however, was not exact because the data plot

deviates from a straight line at the lower temperature region;

• Adjustment of the initial equation based on observations of the behavior of certain other

intensive properties of water in relation to temperature improved the linear correlation

with a correlation coefficient of R2 = 0.9991 (Fig. 2-3);

28 | P a g e
• The relationship is based on fixing (holding constant) all the extensive factors affecting the

mass transfer mechanism. Specifically, KLa is dependent on the gas mass flow rate.

However, since Hunter’s data has slight variations in the gas mass flow rate over the

temperature tests, normalization to a fixed gas flow rate improves the accuracy for the

straight line passing through the origin with R2 = 0.9994 (Fig. 2-4).

20
18
16
14
KLa (hr^-1)

12
10
Kla
8
Kla(G)
6
4
2
0
0 10000 20000 30000 40000
T^5*1E-8 (T in degK)

Figure 2-2. KLa vs. 5th power of absolute Temperature

20.0
18.0
16.0
y = 3.6732x
R² = 0.9991
14.0
KLa (1/hr)

12.0
10.0
8.0 T= abs. temp
6.0 E= elasticity
4.0 ρ= sp. wt.
σ= surf. tension
2.0
0.0
0 1 2 3 4 5
T^5. E.ρ.σ

Figure 2-3. KLa vs. temperature, modulus of elasticity, density and surface tension

29 | P a g e
 20
18
y = 0.0336x
16 R² = 0.9994
14
12
KLa (hr^-1)

10
8
6
4
2
0
0 100 200 300 400 500 600
T^5.E.ρ.σ.Q^0.63

Figure 2-4. KLa vs. temperature, modulus of elasticity, density, surface tension, gas flow rate

Based on the above reasoning, data analysis as described in detail in the following sections

confirmed the validity of (eq. 2-1), but only for the special case where Ps is at or close to

atmospheric pressure (i.e. Ps =1 atm), assuming Hunter’s tests were carried out at 1 atm. The

experiments described in this paper have not proved that KLa is inversely related to Ps. The author

advances a hypothesis that KLa is inversely proportional to equilibrium concentration (Cs), which

can be related to pressure which therefore in turn is related to the depth of a column of water. Since

saturation concentration is directly proportional to pressure (Henry’s Law), therefore KLa must be

inversely proportional to pressure, if the reciprocity relationship between KLa and Cs is true. This

is discussed in another paper published by the author [Lee 2018] and in the following chapters

where relevant.

Furthermore, the concept of equilibrium pressure Ps and how to calculate Ps must be

clarified for a bulk column of liquid. (The details for the pressure adjustment are given in ASCE

2-06 Section 8.1 and ANNEX G) [ASCE 2007].) Insofar as the current temperature correction

30 | P a g e
model has not accounted for any changes in Ps due to temperature, this manuscript has assumed

that Ps is not a function of temperature for a fixed column height and therefore does not affect the

application of (eq. 2-1) for temperature correction.

2.2. Theory

The Liquid Film Coefficient (kl) can be related to the Overall Mass Transfer Coefficient (KL) for

a slightly soluble gas such as oxygen. For any gas-liquid interphase, Lewis and Whitman’s two-

film concept proved to be adequate to derive a relationship between the total flux across the

interface and the concentration gradient, given by:

𝑁0 = 𝐾𝐿 × (𝐶𝑆 − 𝐶) (2 − 9)

It can be proven mathematically that the bulk mass transfer coefficient is related to the respective

film coefficients by the following equation:

𝑘𝑔 𝑘𝑙
𝐾𝐿 = (2 − 10)
𝐻𝑘𝑙 + 𝑘𝑔

where kl and kg are mass transfer coefficients for the respective films that correspond directly to

their diffusivities and film thicknesses. H is the Henry’s Law constant.

When the liquid film controls, such as for the case of oxygen transfer or other gas transfer that has

low solubility in the liquid, the above equation is simplified to

𝐾𝐿 = 𝑘𝑙 (2 − 11)

This means that the gas transfer rate on a macro scale is the same as in a micro scale when the

liquid film is controlling the rate of transfer due to the fact that the liquid film resistance is

31 | P a g e
considerably greater than the gas film resistance. The four equations Prof Haslam developed are

given below:

𝑘𝑔 = 290 × 𝑀𝑈 0.8 𝑇 −1.4 (2 − 12)

0.8
𝑠 0.667
𝑘𝑔 = 0.72 × 𝑀𝑈 ( ) (2 − 13)
𝜇

𝑘𝑙 = 5.1 × 10−7 × 𝑇 4 (2 − 14)

𝑠 0.667
𝑘𝑙 = 37.5 ( ) (2 − 15)
𝜇

Equations (2-12) and (2-13) are not important, since any changes in the rate of transfer in the gas

film are insignificant compared to the changes in the liquid film for a slightly soluble gas such as

oxygen. Equation (2-15) relates the liquid film to two physical properties of water, density (s) and

viscosity (u). Equation (2-14) is most useful since it relates the mass transfer coefficient directly

to temperature, irrespective of the gas flow velocity (U) or the molecular weight (M), and appears

to be independent of Equation (2-15). Because the interphase concentrations are impossible to

determine experimentally, only the overall mass transfer coefficient KL can be observed in his

apparatus. However, by substituting the values of the film coefficients calculated using the above

equations into Equation (2-10), excellent agreement was found between the observed values of the

overall coefficients and those calculated. Because of Equation (2-14), it can be concluded that the

overall mass transfer coefficient in a bulk liquid is proportional to the fourth power of temperature,

given by:

𝐾𝐿 = 𝑘 ′ × 𝑇 4 (2 − 16)

32 | P a g e
where k’ is a proportionality constant.

For spherical bubbles, the interfacial area (a) is given by:

𝑄𝑎
× 6 𝜋𝑑𝑏 2
𝑎= 𝜋 3 × × 𝑡𝑐 (2 − 17)
𝑑𝑏 𝑉

where Qa = average gas volumetric flow rate (m3/min); db = average diameter of bubble (m,

mm); tc = contact time of bubble with liquid; V = tank volume.

The contact time is dependent upon the path of the bubble through the liquid and can be expressed

in terms of the average bubble velocity vb and the liquid depth Zd:

𝑍𝑑
𝑡𝑐 = (2 − 18)
𝑣𝑏

where, vb = average bubble velocity, (m s-1)

The area of bubble interface per unit of tank volume V is then

𝑄𝑎
𝑎 = 6× × 𝑍𝑑 (2 − 19)
𝑑𝑏 𝑣𝑏 𝑉

This shows that for a given tank depth, and a fixed aeration system, ‘a’ is proportional to the gas

flow rate Qa. The mass transfer coefficient is dependent on the volumetric gas flow rate which

changes with temperature and pressure----the higher the gas flow rate the faster is the transfer rate.

Average Volumetric Gas Fow Rate

The average gas flow rate is dependent on the test temperature of the bulk liquid. With this in

mind, Qa can be determined using the ASCE standard 2-06 [ASCE 2007] as follows:

Combining Eq. A-1b and Eq. A-2b in Section A.5.1 of Annex A where they were written as:

𝑇1 𝑃𝑃
𝑄1 = 𝑄𝑃 ( ) (2 − 20)
𝑇𝑃 𝑃1

33 | P a g e
 𝑄1 𝑇𝑆 𝑃1
𝑄𝑆 = (2 − 21)
𝑇1 𝑃𝑆

where,

Qs = gas flow rate given at standard conditions (i.e. the feed gas mass flow rate), (Nm3/min)

Q1 = gas flow at the gas supply system

QP = gas flow at the point of flow measurement (at the diffuser depth)

Ps = standard air pressure, 1.00 atm (101.3 kPa)

P1 = ambient (gas supply inlet) atmospheric pressure

PP = gas pressure at the point of flow measurement

Ts = standard air temperature (293 K for U.S. practice)

T1 = ambient (gas supply inlet) temperature, K (= 0C + 273)

TP = gas temperature at the point of flow measurement

By substituting (eq. 2-20) into (eq. 2-21), we have

𝑃𝑆 𝑇𝑃
𝑄𝑃 = 𝑄𝑆 ( ) ( ) (2 − 22)
𝑃𝑃 𝑇𝑆

Assuming the mass amount of gas is conserved, as the bubbles rise to the surface, Boyle’s Law

states that the volume is increased as the liquid pressure decreases, giving the following:

𝑃𝑃 𝑃𝑆 𝑇𝑃
𝑄𝑡𝑜𝑝 = ( ) 𝑄𝑆 ( ) ( ) (2 − 23)
𝑃𝑏 𝑃𝑃 𝑇𝑆

where Pb is the barometric pressure over the tank and Qtop is the volumetric flow rate at the top of

the tank. The average gas flow rate over the entire column is therefore obtained by averaging of

the gas flow rates given by eq. (2-22) and eq. (2-23) and is calculated by Qa =1/2(Qtop + QP) and

so,

34 | P a g e
 𝑄𝑆 𝑃𝑆 𝑇𝑃
2 1 1
𝑄𝑎 = ×( + ) (2 − 24)
𝑇𝑆 𝑃𝑃 𝑃𝑏

Since Ps = 1.01325 x 105 N/m2 and Ts = 293.15 K (20 0C),

Therefore, substituting the standard values into (eq. 2-24) yields the average gas flow rate in terms

of the standard gas flow rate as:

𝟏 𝟏
𝑸𝒂 = 𝑸𝑺 × 𝟏𝟕𝟐. 𝟖𝟐 × 𝑻𝑷 ( + ) (𝟐 − 𝟐𝟓)
𝑷𝑷 𝑷𝒃

Combining eq. (2-16), eq. (2-19) and eq. (2-25) yield:

1 1 𝑍𝑑
𝐾𝐿 𝑎 = 𝑘′𝑇 4 × 6𝑄𝑆 × 172.82 × 𝑇 × ( + ) (2 − 26)
𝑃𝑃 𝑃𝑏 𝑑𝑏 𝑣𝑏 𝑉

Grouping all the numerical constants together into one single term, we have

1 1 𝑍𝑑
𝐾𝐿 𝑎 = 𝑘′′𝑄𝑆 × 𝑇 5 × ( + ) (2 − 27)
𝑃𝑃 𝑃𝑏 𝑑𝑏 𝑣𝑏 𝑉

where k’’ is another proportionality constant. This equation (eq. 2-27) illustrates the 5th power

temperature correction relationship as shown in (eq. 2-1) for a fixed height Zd, volume V, and

assuming the pressures and the average bubble diameter (db) and velocity (vb) do not change

substantially over the temperature range tested.

As stated above, the response of KLa to temperature is affected by the behavior of the water

properties that are the other variables that also affect the 5th order temperature relationship. As the

temperature drops, the density of water (ρ) increases, and the maximum density is at about 4 0C.

Similarly, the surface tension (σ) also increases with the decrease of temperature. However, the

modulus of elasticity (E) decreases as the temperature decreases. This is because the modulus of

elasticity is proportional to the inverse of compressibility, which increases as the water approaches

the solid state. Compressibility of water is at a minimum at around 50 0C. Combining all the three

35 | P a g e
variables in response to temperature with the 5th order relationship would result in a curve that

resembles the error structure in Hunter’s experiment as described in Section 2-4 below. These

changes in water properties with respect to temperature are shown in Figs. 2-5, 2-6, and 2-7. The

variability of the compound parameter (Eρσ) with temperature is also shown in Fig. 2-7 for the

elasticity curve. Taking into account the changes in water properties in response to temperature,

(eq. 2-27) can be simplified to:

𝐸𝜌𝜎
(𝐾𝐿 𝑎) 𝑇 = 𝐾 × 𝑇 5 × (2 − 28)
𝑃𝑠

where the symbols are as defined in (eq. 2-1). The inverse relationship between (KLa)T and PS is

a hypothesis, based on the assumption that (KLa)T and CsT the solubility are inversely related.

2.3. Materials & Methods

To derive a temperature correction model, there are two ways. One is to use the solubility

law derived from the solubility table for water, (section 2.5), and the knowledge that KLa is

inversely proportional to Cs, under a reasonable temperature boundary range. The other method is

by use of examination and interpretation of actual data performed by numerous investigators, such

as Hunter’s data [Hunter 1979], on the relationship between KLa and temperature.

The new model for the correction number N as defined by (eq. 2-3), is based on the 5th

order proportionality. Numerous investigators have performed experiments of KLa determination

at different test water temperature, ranging from 0 0C to 55 0C. These data appear to support the

hypothesis that KLa is proportional to the 5th power of absolute temperature for a range of

temperatures close to 20 0C and higher. For temperatures close to 0 0C, however, the water

properties begin to change in anticipation of a change of physical state. (See Figs. 2-5, 2-6, 2-7

below).

36 | P a g e
 1.005

1.000

density (kg/m^3 x 10^-3)

0.995

0.990

0.985

0.980

0.975
0 10 20 30 40 50 60 70 80
Temperature in degC

Figure 2-5. Density vs. Temperature 0C

0.077
Surface Tension of water, N/m

0.076

0.075

0.074

0.073

0.072

0.071

0.070

0.069
0 10 20 30 40 50
Temperature in degC

Figure 2-6. Surface Tension vs. Temperature 0C

37 | P a g e
 2.30

2.25

2.20
E/10^6 (kN/m^2)
2.15

2.10 E
rho*E
2.05
rhoEsigma
2.00

1.95

1.90
0 10 20 30 40 50
Temperature in degC

Figure 2-7. Modulus of Elasticity vs. Temperature 0 C (Top Curve) (Note: E is modulus of
elasticity; rho is density of water; sigma is surface tension)

This change from a liquid state to a solid state at this low temperature is unique to water.

However, by incorporating these changes of the relevant properties into the KLa equation, as

described previously, it becomes possible to find a high degree of correlation for the data

interpretation.

The following paragraphs describe the derivation method to arrive at the proposed

temperature correction model by use of experimental data. This derivation is purely based on data

interpretation and data analysis using linear graphical verification, and is not derived theoretically.

2.3.1. Hunter’s Experiment

Hunter [1979] performed an experiment for the case of laboratory-scale submerged turbine

aeration systems. He derived an equation that relates KLa to the various extensive properties of the

system and to viscosity, and correlated his data for a temperature range of 0 – 40 0C. His method

is described in the paper cited in the manuscript and in his dissertation: Hunter, John S. “A Basis

38 | P a g e
for Aeration Design”. Doctor of Philosophy Dissertation, Department of Civil Engineering,

Colorado State University, Fort Collins CO, 1977.

2.3.2. Vogelaar et al.’s Experiment

The experiments performed by Vogelaar et al. [2000] consist of determining KLa using tap

water for a temperature range of 20 0C – 55 0C using a cylindrical bubble column with an effective

volume of 3 liters and subject to aeration flow rates of 0.15, 0.3, 0.45, and 0.56 vvm (volume air

volume liquid-1 min-1). The results for one particular volumetric air flow rate (0.3 vvm) among

all the data are given in Table 2-3 further below.

The following section describes how the data from these two research workers have been

used to develop the temperature correction equation for determining (KLa)20 for any clean water

test carried out in accordance with ASCE 2-06, and it is proposed that this new equation is to be

used to replace the current equation as stated in ASCE 2-06 Section 8.1 and the relevant sections

concerning the use of (Ɵ) in the calculation of this important parameter (KLa)20 – the standardized

KLa at standard conditions as defined in the ASCE Standard.

2.4. Results and Discussion

Hunter [1979] has suggested that turbulence can be related to viscosity as well as the

aeration intensity that created the turbulence. In surface aeration, aeration intensity can be the

power input to the water being aerated, while in subsurface diffused aeration, it is likely to be the

air bubbles flow rate. Therefore, for certain fixed power intensity, Hunter surmised that KLa is

only a function of viscosity which in turn is a function of temperature. He created a mathematical

model that related (KLa)T to viscosity at different temperatures from 0 0C to 40 0C. His results are

given below in Table 2-1, where KLa(G) are his modelled results. The model he used was

expressed as:

39 | P a g e
 𝐷 4
𝐾𝐿 𝑎(𝐺) = (4.04 + 0.00255𝐺 2 ∗ ( ) ) 𝑄 .63 (2 − 29)
𝑇

where D/T is a geometric function. [Note that T in his equation is NOT temperature], G2 = P/V/μ

where μ is viscosity, P is the power level (total power input into the water being aerated in ergs/s,

and V is the volume of tank in cm3). The term G was defined as the turbulence index. However,

just as in solubility, it is erroneous to consider G as a function of viscosity because viscosity is an

intensive property not extensive. Changing the viscosity would not increase turbulence, in the same

way turbulence does not affect viscosity for a fixed temperature. However, in his paper’s

attachment, he has theoretically derived a relationship between r, the rate of gas-liquid interfacial

surface renewal, and the turbulence index G, that they are equal. Since KL the liquid film

coefficient is related to r, it can be concluded that turbulence affects the mass transfer coefficient,

but this is not due to the apparent correlation between G and μ.

T(0C) Viscosity(poise) T(K) (T/1000)5x104 KLa(h-1) KLa(G)(h-1) Q(SCFH)*

0 0 0 0
0 0.01787 273.15 15.21 7.99 8.8 1.093
5 0.01519 278.15 16.65 9.12 9.64 1.1
10 0.01307 283.15 18.2 10.26 10.55 1.107
15 0.01139 288.15 19.87 11.39 11.51 1.113
20 0.01002 293.15 21.65 12.53 12.53 1.118
25 0.008904 298.15 23.56 13.66 13.6 1.124
30 0.007975 303.15 25.6 14.79 14.71 1.128
35 0.007194 308.15 27.79 15.93 15.87 1.132
40 0.006529 313.15 30.11 17.06 17.08 1.136

Table 2-1. Hunter’s Experimental data (*Note: The air flow rate Q is back calculated from
Hunter’s equation at D/T=0.35, P/V=2000)

In this table, the observed KLa results are given in column 5. His modelled results are given in

column 6. As one can see, his predicted results match up quite well with the true results for those

tests carried out at 20 0C and above. At the lower temperature range, however, his errors increase

40 | P a g e
progressively as the temperature drops to the water freezing point. His results can be seen from

the following plot in Figure 2-8 below:

18

16

14

12
KLa (1/hr)

10
Kla
8
Kla(G)
6

2
Temperature (deg C)
0
0 10 20 30 40 50

Figure 2-8. Hunter’s data of KLa plotted vs. temperature 0C

Hunter did not explain why the errors in terms of percent difference become more

pronounced toward the lower end of the temperature spectrum, since the turbulence index G has

already accounted for the increase of viscosity due to temperature, and so if turbulence was only a

function of viscosity, the changes due to viscosity to the mass transfer coefficient should have been

taken care of in his equation. However, in his attachment, he did derive an equation that relates

KLa not only to G, but also to other system variables which he had not defined. (Note: Hunter’s

formula did include the extensive properties as system variables in his experiment: geometry,

power level, volume, gas flow rate. But while the extensive properties are important factors

affecting KLa, it is found in this study that the relationship between KLa and the intensive properties

is always linear, and this linear relationship is independent of the extensive properties. The

intensive properties are all temperature dependent.) Hunter did not know of the 5th power model.

41 | P a g e
Had he plotted his KLa(G) values against the 5th power of absolute temperature, he would have

been astonished to see a perfect straight line as shown in Figure 2-2 before.

Hunter’s model is in fact correct if all the other system variables were fixed, so that KLa is

only a function of viscosity. The other system variables are in fact the other properties of water,

such as density, modulus of elasticity, and surface tension. As the liquid approaches its freezing

point, it is subject to all the changes in these properties in precedence to the anticipated changes of

physical states.

These changes in water properties can be seen by plotting the handbook values for these

parameters at different temperatures, as shown in Figs. 2-5, 2-6 and 2-7.

These changes in the other properties of water, explain why his data starts to deviate from

a straight line when the temperature drops below 20 0C. Figure 2-7 above includes showing what

happens when Hunter’s data is successively corrected for these changes. The final curve showing

the product ρ.E.σ (rhoEsigma) vs. T represents the correction by the product of density, elasticity,

as well as surface tension. The curve resembles the error structure in Hunter’s data (comparing

col.5 and col.6 in Table 2-1). And so, when the mass transfer coefficient data is plotted against T5

multiplied by the correction factor F which in this case is given by ρ.E.σ, a much better linear

relationship is obtained, as is shown in Figure 2-3 before.

Calculation

2.4.1. Hunter’s data

Data analysis based on Hunter’s experiments [Hunter 1979] has supported that KLa (the

oxygen mass transfer coefficient) needs to be corrected for surface tension in addition to E and ρ.

The effect of surface tension on KLa is more pronounced toward the lower temperature region

(below 20 0C and as it gets closer to the melting point (freezing point) of the solvent (Fig. 2-6)

42 | P a g e
where surface tension increases rapidly as the temperature decreases). From (eq. 2-8), it can be

seen that (KLa)20 can be calculated based on a single test data on KLa. It is important to note that

the temperature correction factor N should not be calculated as the ratio μ20 / μT , but as

(KLa)20/(KLa)T, therefore, at 20 0C

(T. E. ρσ)20 1
(𝐾𝐿 𝑎)20 = K’’ ×( + 0.7815) (2 − 30)
Ps 20 μ20

By eliminating K’’ and assuming Ps = Ps20 therefore,

[Eρσ × T × fn(u)]20
(𝐾𝐿 𝑎)20 = K L a. (2 − 31)
[Eρσ × T × fn(u)]T

or,

[Eρσ × T × fn(u)]20
N = (2 − 32)
[Eρσ × T × fn(u)]T

where fn(u) is given by (1/u + 0.7815)

Similarly, (eq. 2-1) for the 5th power model can be used to calculate (KLa)20 and result in the

following Table 2-2 and the following equation (eq. 2-33):

[EρσT 5 ]20
(𝐾𝐿 𝑎)20 = 𝐾𝐿 𝑎 × (2 − 33)
[EρσT 5 ]T

[EρσT 5 ]20
N = (2 − 34)
[EρσT 5 ]T

The correction number values are given in column 10 in Table 2-2. It should be noted that

even without including these additional variables E, ρ, σ, the 5th power model already gives a very

good fit to the experimental data. In fact, the fifth power model gives a slightly better fit than the

enhanced model for temperatures above 20 0C. The effects of these other physical properties seem

to wane as the temperature increases toward the boiling point region. This is apparent from

Hunter’s model as shown in Table 2-1 (comparing column 5 and column 6) where the prediction

43 | P a g e
error of his model becomes negligible when compared with the observed data when temperature

is above 20 0C. The enhanced model plot that is inclusive of the factors E, ρ, σ, is given as Figure

2-3 with a high degree of correlation (R2 = 0.9991).

1 2 3 4 5 6 7 8 9 10 11
*Corr. *Corr.
T T T5/1012 KLa ρ E/106 σ E.ρ.σ (KLa)20
No. F No. N
[degC] [K] * [1/hr] [kg/m3] [kN/m2] [N/m]
0 273.15 1.5206 7.99 999.8 1.98 0.0765 151.44 1.051 1.496 11.95
5 278.15 1.6649 9.12 1000.0 2.05 0.0749 153.55 1.036 1.348 12.29
10 283.15 1.8200 10.26 999.7 2.10 0.0742 155.77 1.022 1.215 12.47
15 288.15 1.9865 11.39 999.1 2.15 0.0735 157.88 1.008 1.099 12.51
20 293.15 2.1650 12.53 998.2 2.19 0.0728 159.15 1.000 1.000 12.53
25 298.15 2.3560 13.66 997.0 2.22 0.0720 159.36 0.999 0.918 12.54
30 303.15 2.5603 14.79 995.7 2.25 0.0712 159.51 0.998 0.844 12.48
35 308.15 2.7785 15.93 993.9 2.27 0.0704 158.48 1.004 0.782 12.46
40 313.15 3.0114 17.06 992.2 2.28 0.0696 157.45 1.011 0.727 12.40
* Note: F=(Eρσ)20/(Eρσ)T
N=F.(T20/T)5 or (KLa)20= KLa.N
T5 = (T/1000)5 x1000

Table 2-2. Simulated Results for the prediction of (KLa)20 by the 5th power model
Using the predicted (KLa)20 based on the 5th power model, and plotting the simulated results with

test temperatures, the following Figure 2-9 is obtained:

15
14
12.47 12.51 12.53 12.54 12.48 12.46 12.40
13 12.29
11.95
12
(KLa)20 (1/hr)

11
10
9
8
7
0 10 20 30 40 50
Test Temperature (0C)

Figure 2-9. Simulated results for the prediction of (KLa)20 (Hunter)

44 | P a g e
This shows that the variations in the prediction of (KLa)20 based on the various tests at different

temperatures are very small and in fact are much smaller than would be obtained from using the

current ASCE model. Figure 2-10 and Figure 2-11 below show the discrepancies between the

various models (Ɵ=1.024, Ɵ=1.018, and the 5th power model) even further.

13.50
13.00
12.50
(KLa)20 (/hr)

Kla20(T,u)
12.00
Kla20(T,u,ρ,E,σ)
11.50 θ=1.024

11.00 θ=1.018

10.50
10.00
0 20 40 60
Temperature in degC

Figure 2-10. Comparison of (KLa)20 as predicted (0 0C~40 0C) by various models (Hunter)

12.60 5th power model

12.55
12.50
(KLa)20 (1/hr)

12.45
12.40 theta 1.018

12.35
12.30
12.25
0 10 20 30 40
Temperature in degC

Figure 2-11. Comparison of (KLa)20 predicted by two close models between 10 0C and 30 0C

45 | P a g e
 It should be clear from these graphs that the 5th power model is superior to the current

model that uses the theta (Ɵ) correction factor, for temperatures between 10 0C and 30 0C, which

is the temperature range stipulated in ASCE 2-06.

The plot in Fig. 2-3 showing the linear relationship between the mass transfer coefficient

and the 5th power temperature function can be further improved if the KLa data are normalized to

the same gas flow rate (data given in Table 2-1 col. 7 for the flow rates). Hunter’s equation has

stipulated that the predicted KLa(G) is proportional to the value of Q0.63 and so plotting KLa against

the function T5. E.ρ.σ together with Q0.63 further improves the correlation as was shown in Figure

2-4. Therefore, based on Hunter’s experiment, and the good correlation results as shown in Figure

2-3 (R2 = 0.9991), and Figure 2-4 (R2 = 0.9994), it can be concluded that for a fixed mass gas flow

rate, the mass transfer coefficient under different test temperatures can be calculated by (eq. 2-1).

Therefore, the correction number N can be calculated by simple proportion as given by (eq. 2-34).

This equation has assumed that Ps remains constant at different temperatures.

2.4.2. Vogelaar’s data

Similarly, Vogelaar’s Experiment [Vogelaar et al. 2000] showed excellent correlation

between KLa and Cs for temperatures above 20 0C, and Vogelaar’s experimental result is given in

Table 2-3 below:

T (0C) T (K) (T/1000)5 *104 Cs(mg/L) 1/Cs KLa(h-1)

0 0 0 0

20 293.15 21.65 9.19 0.1088 22.4+/-0.4

30 303.15 25.60 7.43 0.1346 26.0+/-0.1

40 313.15 30.11 6.5 0.1538 30.6+/-0.2

55 328.15 38.05 5.15 0.1942 38.8+/-1.5

Table 2-3. Vogelaar’s experimental results

46 | P a g e
Figure 2-12 below shows a plot of KLa vs. T5.(Eρσ) and the correlation is excellent with R2 =

0.9975, assuming Ps = 1 atm. However, it is not as good as Hunter’s data using the same model.

At 55 0C, the deviation from the straight line is larger than the other data points. It is not clear why

this is so. It could be that the distribution of the experimental errors is not even, or that the gas

flow rate is not quite identical at this point. In any case, the prediction of (KLa)20 is still much better

than using θ = 1.024 or any other values except 1.016, as shown in Fig. 2-13.

45.0
40.0
35.0
y = 0.0656x
30.0 R² = 0.9975
KLa (1/hr)

25.0
20.0
15.0
10.0
5.0
0.0
0 100 200 300 400 500 600 700
Eρσ T^5 (Note: Temperature in K)

Figure 2-12. KLa against 5th power of temperature [Vogelaar et al. 2000]

At 55 0C, the discrepancy between the theta (Ɵ) model and the 5th power model is greater than

30%. When plotting the predicted (KLa)20 values using the various models (5th power, θ = 1.024,

θ = 1.018, θ = 1.016), the following graph is obtained (Fig. 2-13). As seen from this plot, the 5th

power model predicts a series of consistent values of (KLa)20, whereas the (θ) model using θ =

1.024 gives very poor results. Although using θ = 1.018 improves the prediction, it is still not as

good as the 5th power model. The difficulty of using the (θ) model is that the value of θ must be

pre-determined by testing which is the major disadvantage of this model.

47 | P a g e
 25.0
24.0
23.0
22.0
(KLa)20 (1/hr)
21.0 T^5 mod.
20.0 Ɵ=1.024

19.0 Ɵ=1.018

18.0 Ɵ=1.016

17.0
16.0
10 20 30 40 50 60
Temperature in 0C

Figure 2-13. Comparison of simulated results for (KLa)20 (Vogelaar)

2.4.3. Methodology for Temperature Correction

A new model to improve the temperature correction for KLa used in ASCE Standard 02

has been developed. Based on data analyses of two researchers’ work, it can be seen the new

model gives excellent simulated results for (KLa)20 based on series of tests at increasing water

temperatures, compared to the other models using the same data as seen in Fig. 2-10, and Fig. 2-

13. The major function of this model is to predict KLa for any changes in temperature so that

(KLa)20 can be predicted from any one single test at a specific temperature, and therefore would

replace the current model in ASCE 2-06 with a higher degree of accuracy. For a certain

equilibrium level (de) where the equilibrium pressure is at (Ps), the model is expressed by (eq. 2-

1), in which the proportionality constant K is dependent on the extensive properties of the

aeration system, such as gas flow rate, bubble size and other characteristics of the system. This

equation is not complete because temperature also affects the volumetric gas flowrate Qa.

Therefore, as a result of the foregoing analysis, the formula for estimating (KLa)20 based on any

given test at a test temperature T 0C is given by:

48 | P a g e
 (𝜌𝐸𝜎)20 𝑇20 + 273 5 𝑃𝑆 𝑇 𝑄𝑎20
〈𝐾𝐿 𝑎〉20 = (𝐾𝐿 𝑎)𝑇 ( )( ) ( )( ) (2 − 35)
(𝜌𝐸𝜎) 𝑇 (𝑇𝑇 + 273) 𝑃𝑠20 𝑄𝑎 𝑇

where T is expressed in 0C. For a series of tests under the same barometric pressure, as in Hunter’s

experiment, the change in (Ps) due to temperature is likely to be small, and so the ratio PsT/ Ps20

can be cancelled. Temperature affects the gas volumetric flow rate, even for a fixed mass gas flow

rate. But for a fixed gas flow rate and fixed pressure, and since the values of E, ρ and σ for water

are fixed, a table of correction factors can be compiled to make the application easy, as shown in

Table 2-4 below:

1 2 3 4 5 6 7 8
T T(K) ρ E/106 σ E.ρ.σ F T20 5
N = F. ( )
T
(0C) (kg/m3) (kN/m2) (N/m)
0 273.15 999.8 1.98 0.0765 151.44 1.051 1.496
5 278.15 1000.0 2.05 0.0749 153.55 1.036 1.348
10 283.15 999.7 2.10 0.0742 155.77 1.022 1.215
15 288.15 999.1 2.15 0.0735 157.88 1.008 1.099
20 293.15 998.2 2.19 0.0728 159.15 1.000 1.000
25 298.15 997.0 2.22 0.0720 159.36 0.999 0.918
30 303.15 995.7 2.25 0.0712 159.51 0.998 0.844
40 313.15 992.2 2.28 0.0696 157.45 1.011 0.727
50 323.15 988.0 2.29 0.0679 153.63 1.036 0.636
60 333.15 983.2 2.28 0.0662 148.40 1.072 0.566

Table 2-4. Table of Correction Factors for the temperature correction model (F, N)

Although in reality, (KLa)20 should be normalized to the same average gas volumetric flow

rate in order to be more precise, inasmuch as the current equation for correcting KLa in ASCE 2-

06 has not accounted for changes in gas flow rate nor any other effects such as (Ps), it is

recommended that, for the time being, it is sufficiently accurate to replace the current equation by

a simplified equation, in the effort to standardize the mass transfer coefficient to a standard

49 | P a g e
condition of 20 0C and standard atmospheric pressure. For a fixed mass gas flow rate, the equation

becomes:

(𝝆𝑬𝝈)𝟐𝟎 𝑻𝟐𝟎 + 𝟐𝟕𝟑 𝟓

(𝑲𝑳 𝒂)𝟐𝟎 = (𝑲𝑳 𝒂)𝑻 ( )( ) (𝟐 − 𝟑𝟔)
(𝝆𝑬𝝈)𝑻 (𝑻𝑻 + 𝟐𝟕𝟑)

where T is again expressed in terms of degree Celsius. Equation (2-36) is the proposed model for

temperature correction for KLa to be used on ASCE Standard 02, where F = (Eρσ)20/(Eρσ)T and

the correction number N would be given by N=F.(T20/T)5 where in the application of the

temperature correction model for (KLa)20, (KLa)20 is obtained by multiplying (KLa)T by the

correction number N in column 8 at the test temperature T. The correction factors can be plotted

against temperature for easy use, such as Fig. 2-13a below.

Temperature correction factors for Kla0

for oxygen transfer in water
1.6

1.4
corr. Factor N =Kla020/Kla0

1.2 y = 0.0002x2 - 0.0278x + 1.4807

1 R² = 0.9985

0.8

0.6

0.4

0.2

0
0 10 20 30 40 50 60 70
Temp in 0C

Fig. 2-13a. Temperature correction model in graphical form

(Note: for the meaning of Kla0, refer to Chapter 3)

2.5. THE SOLUBILITY MODEL

As mentioned in the introduction, the author advocates the hypothesis that solubility is

inversely proportional to KLa. The foregoing sections have established that KLa is related directly

50 | P a g e
to the 5th order of temperature. If this hypothesis is true, then one would expect the solubility also

bears a 5th order relationship with temperature, but in an inverse manner. The following sections

illustrate that solubility is indeed related to the 5th order of temperature using published scientific

data. This section is significant for 2 reasons:

First, a new physical law is discovered. By definition according to the Oxford English

dictionary, a physical law “is a theoretical principle deduced from particular facts, applicable to a

defined group or class of phenomena, and expressible by the statement that a particular

phenomenon always occurs if certain conditions be present.” The rationale behind the solubility

law is similar to the Universal Gas Law which is in fact an extension of Boyle’s Law or Charles’

Law. As Boyle’s Law states that for a fixed temperature, volume is inversely proportional to

pressure; so, the Universal Gas Law states that, for any pressure and temperature, volume is

inversely proportional to pressure, but directly proportional to temperature.

The solubility law relates oxygen solubility in water to the 5th power of temperature, and

also to certain properties of water. (eq. 2-37 below). This relationship has not appeared in any

literature until now and it is therefore accurate to claim that the 5th power inverse relationship is

hitherto unknown prior to this manuscript. The author believes that the reason this solubility law

has not been discovered earlier like the gas law is that, in the gas law all the parameters are first

order and can easily be verified experimentally. In the solubility law, the inverse 5th power

phenomenon is not directly observable. Furthermore, the solubility law deals with the interaction

of two phases and two species—solute gas and solvent liquid, whereas the gas law deals with only

a single gas phase. It is one thing to test a model once it has been discovered, but quite another to

find the physical law in the first place.

51 | P a g e
 Second, the law can be applied to real situations in wastewater treatment, and in many

bioreactor processes. One of the major applications is the prediction of oxygen transfer in water.

The topic discussed in this manuscript is about gas transfer in water, how much and how fast, in

response to changes in water temperature. The hypothesis is that KLa and Cs are in fact inversely

proportional to each other. This paper demonstrates how the discovered physical law for gas

solubility can be compared with the temperature correction model for KLa based on experimental

data [Hunter 1979] [Vogelaar et al. 2000] that will prove the hypothesis that KLa is inversely and

linearly proportional to Cs.

2.6. Description of The Oxygen Solubility Model

Oxygen solubility in water is affected by both temperature and pressure. The influence of

temperature on the solubility of gases is predictable. The Benson and Krause (1980, 1984) oxygen

solubility model is well known and is adopted by the USGS (United States Geological Survey)

and ASCE 2-06. This model however is only applicable for a special case where the atmospheric

pressure is at the standard pressure of 101.3 kN/m2. The model is empirical and based on data

collected for that pressure only.

Apart from temperature, pressure has a strong effect on the solubility of a gas. For a fixed

temperature, the relationship between solubility and pressure is governed by Henry’s Law.

Hitherto, however, equation has not existed that combines both effects into one single formula.

Henry’s Law, which states that the solubility (or saturation concentration) of a gas in a liquid is

directly proportional to the partial pressure of the gas if the temperature is constant, can be

explained by Le Chatelier’s principle in a body of water. The principle states that when a system

at equilibrium is placed under stress, the equilibrium shifts to relieve the stress. In the case of

saturated solution of a gas in a liquid, equilibrium exists whereby gas molecules enter and leave

52 | P a g e
the solution at the same rate. When the system is stressed by increasing the pressure of the gas,

more gas molecules go into solution to relieve that increase. This happens at the lower regions of

a body of water such as an aeration tank or a lake well-mixed. Conversely, when the pressure of

the gas is decreased, more gas molecules come out of solution to relieve the decrease and this

happens at the upper regions.

The solubility law proposed herewith is an extension of Henry’s Law. The proposed

solubility law states that for any temperature and pressure, solubility is directly proportional to

pressure, and inversely proportional to the fifth power of temperature in absolute, and inversely

proportional to density and modulus of elasticity of the solution, expressed as:

Ps
Cs = K × (2 − 37)
T 5 Eρ

where,

K = proportionality constant. For water of zero salinity, K has a value of approximately

43.4 kg −2 . N. degK. m−8 . atm−1 when the units of the parameters are defined as Cs = mg/L; Ps =

atm; T = K(Kelvin) x 10-3; E = (kN/m2) x 10-6; ρ = kg/m3.

Justification of this model, its derivation and verification, and the evaluation of the

importance of temperature-dependent properties of water including the bulk modulus of elasticity

of water and the density of water in the relationship will be presented later in Section 2.7 below.

(See also Section 5.7 in Chapter 5 for the justification for using the 5th power model for correcting

KLa). Since this solubility law is newly discovered in the scientific community, it should be given

a name such as the Law of Oxygen Solubility in Pure Water, and the constant K should be called

the Oxygen-Water Solubility Law Constant.

53 | P a g e
2.7. Analysis

There are many ways to confirm a physical law, once it has been discovered. For example,

thermodynamic data could be used such as enthalpy and entropy, and the Gibbs free energy may

be sufficient to verify the model. This method was used by Desmond Tromans [1998] in his

derivation of his model of oxygen solubility in pure water. In this manuscript, graphical methods

using linear proportions are used. Solubility data for other gases and other liquids are available, so

that it should be possible to test the law on other media in order to determine whether the law can

be applied to some other liquids. For example, solubilities of oxygen in water of different

chlorinities (salinities) are given by USGS, as well as in ASCE 2-06.

In the Office of Water Quality Technical Memorandum 2011.03 [Rounds 2011], it was

announced the equations that traditionally had been used by the U.S. Geological Survey (USGS)

to predict the solubility of dissolved oxygen (DO) in water result in slight discrepancies between

values predicted for DO solubility by USGS tables and computer programs compared with values

computed by following the methods listed in Standard Methods for the Examination of Water and

Wastewater (American Public Health Association, 2005) (Standard Methods). Subsequent

analysis resulted in a well-documented recommendation to replace the Weiss (1970) equations

with the equations developed by Benson and Krause (1980, 1984).

The Benson and Krause (1980, 1984) oxygen-solubility formulations (now adopted by

USGS) are documented in equations 1 and 7 through 11 of the Attachment to the Technical

Memorandum. The equations adopted by the USGS and now in line with the Standard Methods

are summarized in the following form:

DO = DOo x FS x FP where the dissolved oxygen (DO) concentration in mg/L is represented as a

baseline concentration in freshwater (DOo) multiplied by a salinity correction factor (FS) and a

pressure correction factor (FP). All three terms are a function of water temperature. In addition, the
54 | P a g e
salinity correction factor is a function of salinity and the pressure correction factor is a function of

barometric pressure. For freshwater (salinity = 0‰) and standard pressure (1 atm), the salinity and

pressure factors are equal to 1.0.

1.575701x105 6.642308x10 1.243800x10 8.621949x10

7 10 11

DO0 = exp (−139.3441 + - + – )

T T2 T3 T4

(2--38)

The salinity correction factor and the pressure correction factor are given by:

10.754 2140.7
𝐹𝑆 = exp [−S. (0.017674 − + )] (2 − 39)
𝑇 𝑇2

P − Pvt 1 − 𝜃0 . P
𝐹𝑃 = × (2 − 40)
1 − Pvt 1 − 𝜃0

where S is salinity in parts per thousand (‰) and T is temperature in Kelvin. P is the barometric

pressure in atmospheres, Pvt is the vapor pressure of water in atmospheres, and θo is related to the

second virial coefficient of oxygen. Using the above equations, it was possible to construct a

solubility table similar to the published Table CG-1 as given in ASCE 2-06. Such a constructed

table for zero salinity is given in Table 2-5 col. 2 [Metcalf & Eddy, 2nd Edition] below:

T Cs(T) ρ E/106 σ μx103 γx106 pv

(0C) (mg/L) (kg/m3) (kN/m2) (N/m) (N.s/m2) (m2/s) (kN/m2)
0 14.62 999.8 1.98 0.0765 1.787 1.785 0.61
5 12.77 1000 2.05 0.0749 1.518 1.519 0.87
10 11.29 999.7 2.10 0.0742 1.307 1.306 1.23
15 10.08 999.1 2.15 0.0735 1.139 1.139 1.70
20 9.09 998.2 2.19 0.0728 1.002 1.003 2.34
25 8.26 997.0 2.22 0.0720 0.890 0.893 3.17
30 7.56 995.7 2.25 0.0712 0.798 0.800 4.24
40 6.41 992.2 2.28 0.0696 0.653 0.658 7.38
50 5.49 988.0 2.29 0.0679 0.547 0.553 12.33
60 4.71 983.2 2.28 0.0662 0.466 0.474 19.92

Table 2-5. Physical Properties of water at various temperatures

55 | P a g e
The other data pertaining to the physical properties of water as shown in Table 2-5 is from the

standard handbook and textbook [ASCE 2007] [Benson and Krause 1984], which enabled

calculating:

• a temperature correction function: T5.E.ρ/Ps, and

• the reciprocal of solubility, or Insolubility

When the insolubility (1/Cs) is plotted against the temperature function at Ps= 1 atm, a straight

line passing through the origin is obtained with the correlation R2 = 0.9998. (Graph not shown).

The significance of this plot is that the extension of the linear plot passes through the point of

origin at zero K. This does not mean the absolute temperature could reach the zero point (the

molecular structure will have changed long before that), but such linear relationship offers a simple

means of calculating solubility at any physical parameters of the solvent, by simple ratios. Since

water changes from a liquid state to a solid state as the temperature approaches its melting point

(freezing point), once the temperature drops past the melting point (normally 0 0C at standard

pressure), the law no longer holds and any projection past the solid state is therefore purely

hypothetical. If the data of solubility is plotted against the inverse of the temperature correction

function affecting solubility, the straight-line linear plot would be as shown in Figure 2-14 below.

Therefore, the solubility law can be expressed either by the equation derived from plotting the

insolubility, or expressed by the equation from plotting the data as in Figure 2-14. In the former

method, the equation gives the insolubility of oxygen expressed by:

1 ρ
= 0.02302. T 5 × E × (2 − 41)
Cs Ps

where T is in K to the power 10-3.

56 | P a g e
 16.00
Cs = 43.457Ps/(E. ρ.T5)
14.00
R² = 0.9996
12.00

Solubility in mg/L 10.00

8.00

6.00

4.00

2.00

0.00
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
PS /(T^5.E.ρ)

Figure 2-14. Solubility Plot for water dissolving oxygen at Ps = 1 atm (1.013 bar)

In the latter case, the equation gives the solubility directly and is expressed by:

Ps
Cs = 43.457 × (2 − 42)
(T 5 𝐸𝜌)

1 bar 2 bar 4 bar

T(degC) T(K)
0 273.15 14.6 29.2 58.4
5 278.15 12.8 25.5 51.1
10 283.15 11.3 22.6 45.1
15 288.15 10.1 20.2 40.3
20 293.15 9.1 18.2 36.4
25 298.15 8.3 16.5 33.1
30 303.15 7.6 15.2 30.3
35 308.15 7.0 14.0 27.9
40 313.15 6.5 12.9 25.9
45 318.15 6.0 12.0 24.0
50 323.15 5.6 11.3 22.7

Table 2-6. Solubility of Oxygen in Fresh Water (Salinity ~ 0) at different pressures and
temperatures

Henry’s Law is applicable only to ideal solutions [Andrade 2013]; and for an imperfect liquid

subject to state changes at extreme temperatures, it is only approximate and limited to gases of

slight solubility in a dilute aqueous solution with any other dissolved solute concentrations not

57 | P a g e
more than 1 percent. At different pressures, the solubility will increase as shown in Table 2-

6:Similar plots (see Figure 2-15) for solubility at different pressures can be made using the data

[Engineering Toolbox, (2001)] obtained from downloading from the internet at

www.EngineeringToolBox.com as shown in Table 2-6 above. From Fig. 2-15, the solubility is

inversely proportional to the temperature function expressed in terms of T, E and ρ, so that

0.25

0.2 y = 0.023x
R² = 0.9995
1/Cs (Cs = mg/L)

0.15
1 bar
y = 0.0115x
0.1 R² = 0.9995 2 bar
4 bar
0.05
y = 0.0057x
R² = 0.9995
0
0 2 4 6 8 10
T^ 5.E.ρ (in appropriate units)

Figure 2-15. Comparison of oxygen solubility plots for various pressures

0.025
Insolubility/Temp ratio

0.02
y = 0.023x
R² = 1
0.015

0.01

0.005

0
0 0.2 0.4 0.6 0.8 1 1.2
1/Ps (reciprocal of pressure)

Figure 2-16. Proportionality Constant K plotted against Reciprocal of Pressure

58 | P a g e
 1
= K (T 5 . E. ρ) (2 − 43)
Cs

where K is a constant.

Plotting the K values (0.023 at 1 bar; 0.0115 at 2 bar; 0.0057 at 4 bar) from Figure 2-15 against

the reciprocals of pressures, the following graph shown in Figure 2-16 is obtained.

Therefore,

1
K = 0.023 ( ) (2 − 44)
Ps

Combining (Eq. 2-43) and (Eq. 2-44) we have

1 1
= 0.023 ( ) . ( T 5 . E. ρ) (2 − 45)
Cs Ps

or

Ps
Cs = 43.478 (2 − 46)
T 5 . E. ρ

(Eq. 2-46) is equivalent to (eq. 2-42) showing that solubility is indeed proportional to pressure, in

accordance with Henry’s Law. The slight discrepancy in the K value arises from the two different

sources of data, one from Bensen and Krause, and the other from Engineering ToolBox, (2001)

[online], available at: https://www.engineeringtoolbox.com. But it is likely that the former is more

accurate since the solubility data has two decimal places.

2.8. Conclusions

Based on the afore-mentioned literature review, the following conclusions are obtained:

2.8.1. The primary intent of this manuscript is to replace the geometric technique as used in ASCE

2-06 [ASCE 2007] The current method that uses an assigned theta (Ɵ) value for correcting the

effects of temperature on oxygen transfer coefficient (KLa)20 is empirical and attempts to lump all

59 | P a g e
possible factors, such as changes in viscosity, surface tension, diffusivity of oxygen, geometry,

rotating speed, type of aerators, etc. This empirical approach has produced a great variety of

correction factors for theta. Therefore, a wide range of temperature correction factors is reported

in the literature which has ranged from 1.008 to 1.047. ASCE 2-06 Commentary CG-3

recommends Ɵ to be 1.024 and clean water testing should be at temperatures close to 20 0C. When

a value different from 1.024 is proposed, it usually requires justification by an extensive array of

testing [Lee 1978] [Boogerd 1990], and preferably full scale for the range of testing temperatures

as required, under the same conditions from test to test. This may not be possible at all.

2.8.2. The 5th power model developed is mechanistic in nature. Unlike the conventional empirical

model, it does not require the selection of an uncertain parameter (a priori) value, such as theta (θ).

The correction number N, is independent on the extensive properties of an aeration system in the

estimation of (KLa)20; whereas the correction number for the Ɵ model cannot be applied

universally and pertains to the system that was used to obtain the parameter only. The new model

should prove to be valid for other similar testing especially in full-scale, because the resultant

(KLa)20 is dependent only on temperature and the other intensive properties of the fluid, if the

extensive properties are fixed.

2.8.3. For the temperature correction model, a formula is derived as defined by (eq. 2-36):

(𝝆𝑬𝝈)𝟐𝟎 𝑻𝟐𝟎 + 𝟐𝟕𝟑 𝟓

(𝑲𝑳 𝒂)𝟐𝟎 = (𝑲𝑳 𝒂)𝑻 ( )( ) (𝟐 − 𝟒𝟕)
(𝝆𝑬𝝈)𝑻 (𝑻𝑻 + 𝟐𝟕𝟑)

The improvement of this model relative to the old model as given by (eq. 2-2) is readily apparent

when plotting the simulated (KLa)20 for both models on a same plot, as shown in Figs. 2-8, 2-9, 2-

10, and 2-13. The prediction error is within 1% for the temperature range between 10 0C and 55
0
C. This is assuming that the measured (KLa)20 in the literature is correct, but there will be

experimental error associated with that measurement as well. The improvement over the existing

60 | P a g e
model can be as much as 30% since the error of the old model can be as much. It is recommended

that this equation replaces the current ASCE 2-06 model.

2.8.4. Although some extensive properties may change in response to a change in temperature in

the new model, such as the volumetric gas flow rate, bubble size, barometric pressure, etc., small

changes in these extensive properties can be easily normalized within a reasonable temperature

range, such as in the treatment of Hunter’s data, where the gas flow rates are normalized, resulting

in an improved correlation. However, this should be verified with testing before changing the

Standard. It may be difficult to normalize some extensive variables, such as the rotating speed of

an impeller-sparger type of aeration system. The effect of such extensive variables has not been

discussed in this manuscript, and if normalization is impractical, testing is required in the same

way that the theta model would require.

2.8.5. Hunter’s assertion that “equations do not exist… for full scale aeration systems that express

KLa as a function of G [the temporal velocity gradient which is dependent on viscosity]” is

incorrect. This is because even though turbulence affects KLa substantially, the new 5th power

model has excluded the turbulence effect due to temperature. Therefore, as long as such extensive

variables are fixed, any one test result can be extrapolated to estimate (KLa)20.

2.8.6. The discovery of a 5th order relationship between solubility and temperature leads to the

hypothesis that solubility (Cs) is related to (KLa), but in an inverse manner. Since solubility is

related to pressure as given by Henry’s Law, therefore KLa must be related to pressure as well.

Therefore, in a clean water test with a deep tank, the effect of pressure at the equilibrium level may

need to be considered in the use of (eq. 2-1) for the temperature correction model on KLa.

2.8.7. Apart from the advantage of a more accurate prediction of (KLa)20, the temperature

correction model has advantages in design. In a treatment process, the best design is usually when

61 | P a g e
the oxygen consumption balances the oxygen supply. This balance is needed not only to save

energy but also beneficial from the standpoint of the welfare of the microorganisms which are very

sensitive to water temperature. It is seldom practical to conduct full scale testing for a range of

water temperatures under process conditions. Therefore, a more accurate prediction of KLaT would

enhance designing the treatment process. This is certainly an enormous advantage in the

application of equation CG-1 in ASCE 2-06 for designing the oxygen transfer rate in process

condition. However, wastewater characteristics do need to be individually measured in the

application of the proposed model, since the properties of the fluid E, ρ and σ may all be different

from that of pure water or tap water.

References

Andrade Julia (2013) “Solubility Calculations for Hydraulic Gas Compressors” Mirarco Mining
Innovation Research Report
ASCE/EWRI 2-06. ``Measurement of Oxygen Transfer in Clean Water. `` ASCE Standard.
ISBN-13: 978-0-7844-0848-3, ISBN-10: 0-7844-0848-3, TD458.M42 2007
ASCE-18-96. ``Standard Guidelines for In-Process Oxygen Transfer Testing`` ASCE Standard.
ISBN-0-7844-0114-4, TD758.S73 1997
Baillod, C. R. (1979). “Review of oxygen transfer model refinements and data interpretation.”
Proc., Workshop toward an Oxygen Transfer Standard, U.S. EPA/600-9-78-021, W.C.
Boyle, ed., U.S. EPA, Cincinnati, 17-26.
Bruce B. Benson and Daniel Krause, Jr. (1984) “The concentration and isotopic fractionation of
oxygen dissolved in freshwater and seawater in equilibrium with the atmosphere”
Department of Physics, Amherst College, Amherst, Massachusetts 0 1002.
Boogerd F.C. et al. (1990). “Oxygen and Carbon Dioxide Mass Transfer and the Aerobic,
Autotrophic Cultivation of Moderate and Extreme Thermophiles: A Case Study Related
to the Microbial Desulfurization of Coal”, Biotechnology and Bioengineering, Vol. 35,
Pp. 1111-1119. DOI: 10.1002/bit.260351106.

62 | P a g e
Desmond Tromans (1998). “Temperature and pressure dependent solubility of oxygen in water:
a thermodynamic analysis”, University of British Columbia Department of Metals and
Materials Engineering, 6350 Stores Road, VancouÍer, British Columbia Canada, V6T
1Z4, Hydrometallurgy 48 _1998. 327–342
Hunter John S. III (1979). “Accounting for the Effects of Water Temperature in Aerator Test
Procedures.” EPA Proceedings Workshop Toward an Oxygen Transfer Standard EPA-
600/9-78-021
IAPWS The International Association for the Properties of Water and Steam, Berlin, Germany
September 2008, “Release on the IAPWS Formulation 2008 for the Viscosity of Ordinary
Water Substance”, 2008 International Association for the Properties of Water and Steam
Publication. (September 2008)
Lee J. (1978). “Interpretation of Non-steady State Submerged Bubble Oxygen Transfer Data”.
Independent study report in partial fulfillment of the requirements for the degree of
Master of Science (Civil and Environmental Engineering) at the University of Wisconsin-
Madison, 1978 [Unpublished]
Lee J. (2017). Development of a model to determine mass transfer coefficient and oxygen
solubility in bioreactors, Heliyon, Volume 3, Issue 2, February 2017, e00248, ISSN
2405-8440, http://doi.org/10.1016/j.heliyon.2017.e00248.
Lee, J. (2018). Development of a model to determine the baseline mass transfer coefficients in
aeration tanks, Water Environ. Res., 90, (12), 2126 (2018).
Metcalf & Eddy, Inc. second edition. “Wastewater Engineering: Treatment & Disposal” ISBN
0-07-041677-X
R. T. Haslam , R. L. Hershey , R. H. Kean. “Effect of Gas Velocity and Temperature on Rate of
Absorption”. Ind. Eng. Chem., 1924, 16 (12), pp 1224–1230 DOI: 10.1021/ie50180a004
Publication Date: December 1924
Solubility of oxygen in equilibration with air in fresh and sea (salt) water - pressures ranging 1 -
4 bar abs. Engineering ToolBox, (2001). [online] Available at:
https://www.engineeringtoolbox.com [Accessed Day Mo. Year].
Stewart Rounds (2011) Technical Memorandum, Office of Water Quality Technical
Memorandum 2011.03.” Change to Solubility Equations for Oxygen in Water”, USGS
Oregon Water Science Center.

63 | P a g e
Vogelaar et al. (2000). “Temperature Effects on the Oxygen Transfer Rate between 20 and 550C”
Water Res. Vol. 34, No.3, Elsevier Science Ltd.
W. K. Lewis, W. G. Whitman. “Principles of Gas Absorption”, Ind. Eng. Chem., 1924, 16 (12),
pp 1215–1220 Publication Date: December 1924 (Article) DOI: 10.1021/ie50180a002

64 | P a g e
Chapter 3. Development of a model to determine baseline mass transfer
coefficients in aeration tanks
3.0 Introduction

The term (KLa) has been widely used to mean the mass transfer coefficient of both micro-

scale and macro-scale aeration. On a macro-scale, the efficiency of porous fine-bubble diffusers

varies from 10 to 30 percent or more, depending on tank depth. Due to the uncertainty in predicting

the efficiency of an aeration system, the actual oxygen requirement cannot be accurately

determined, even though air use is a key parameter in sizing blowers, air piping, and the number

of diffuser plants in treatment plant aeration design. A safety factor as high as 2 is sometimes

assigned to compute the actual oxygen requirement in the sizing of blowers. (Metcalf and Eddy

1985). Furthermore, it has been observed by Eckenfelder (1952) and other researchers that the

Standard Oxygen Transfer Rate (SOTR) is constant for a given aeration system, and that at

different temperatures KLa and Cs (saturation concentration or oxygen solubility) adjust

accordingly, giving rise to the conjecture that these two parameters are inversely proportional to

each other under certain conditions, which has been proven valid for shallow tanks such as pilot

plants or experimental vessels in laboratories (Boogerd et al. 1990; Eckenfelder 1952; Hunter

1979; Vogelaar et al. 2000). This may not be true for deeper tanks.

The objective of this Chapter is to present an experimentally validated, mechanistic depth-

correction model for fine bubble aeration for different water depths under the same average

volumetric gas flow rate supply (Qa), and the same horizontal cross-sectional surface area, so

that it can be used to predict the mass transfer coefficient and the SOTE (Standard Oxygen Transfer

Efficiency) for aeration tanks. Furthermore, this manuscript shows that the proposed temperature

correction model (Lee 2017) as given in Chapter 2 can be applied to non-shallow tanks as well

when KLa is corrected to zero depth using the depth correction model. This depth correction model

65 | P a g e
deals with the changes in the mass transfer coefficient with depth inasmuch as the temperature

correction model deals with its changes with temperature. The proposed equations (Eq. [3-6] to

Eq. [3-10]) allow calculation of the baseline (KLa0) by solving them simultaneously for KLa0. The

baseline coefficient KLa0 is a hypothetical parameter which is defined in this manuscript as the

oxygen transfer rate coefficient at zero depth. The work presented here has shown that the standard

baseline (KLa0)20 (the standardized KLa0) determined from a single clean water test [ASCE 2007],

at any temperature, can predict (KLa)20 (the standardized bulk liquid apparent mass transfer

coefficient for clean water) for any other tank depth (if the gas flowrate Qa is kept constant or if

the baseline KLa0 can be normalized to Qa), using the proposed depth correction model and the

temperature correction model together. This manuscript shows that the variation of KLa with depth

is an exponential function with respect to the baseline parameter KLa0, and this relationship allows

a more precise determination of aeration system efficiencies.

Background

Because of the complexity of the subject, a dedicated Chapter 4 below seeks to clarify

about the progression of the development of the Lee-Baillod model and about the new things

beyond the Lee-Baillod model. Specifically, the development of the final model (a set of

equations) that would define the baseline KLa0 occurred in three distinct phases:

Phase 1: This happened in the 70’s. Lee and Baillod [Lee, J. 1978][Baillod, C.R. 1979] jointly

developed Eq. (4-1) to Eq. (4-16) when it was recognized that the saturation concentration C*∞ is

a variable rather than a constant assumed to be so in the oxygen transfer equation. The final

equation in this phase Eq. (4-16) was also attempted by Lakin and Salzman (1977), and more

recently by McGinnis et al (2002). This equation exists in a differential form. McGinnis et al.

66 | P a g e
managed to solve this equation by numerical integration, but their results appeared to have an error

of around 15% compared to measured data.

Phase 2: This pertains to Eq. (4-17) to Eq. (4-33). The final equation in this phase was derived by

Lee and Baillod [Lee, J. 1978][Baillod, C.R. 1979], as well as by Lakin and Salzman [1977], but

the latter’s equation contains an apparent error in sign [Lee 1978]. Even though Eq. (4-33) was

mathematically correct, the rising bubbles according to this model give an ever-increasing mole

fraction (it should be the other way round) and so this mole-fraction model was deemed to be

unrealistic, but it served as a conveyance equation for integrating this equation into a practical

oxygen transfer equation, allowing the researchers to conclude that the conventional oxygen

transfer equation describing the macroscopic transfer model in an aeration tank is valid, when the

KLa is interpreted as the apparent mass transfer coefficient. Dr. Baillod went on to develop a

parameter that he called the ‘true’ KLa. [EPA-600/2-83-102]

Phase 3: The remaining equations Eq. (4-34) to Eq. (4-76) pertain to this phase. The concept of

‘true’ KLa was not accepted as valid by the Standards Committee [ASCE 2007], and rightly so.

There are at least two reasons why this concept cannot be right:

First of all, the standard transfer equation given by dc/dt = KLa (C*∞ – c) is correct. In a majority

of non-steady state clean water tests, this model never fails to give a very good fit to the re-aeration

data;

Secondly, the calculated ‘true’ KLa is always higher than the apparent KLa. If the parameter

estimation has under-estimated KLa, then it must also have over-estimated C*∞, since the two are

co-related. Jiang P. and Stenstrom M. K. [2012] have monitored the off-gas content in non-steady

state clean water tests, and it was shown that the oxygen in the off-gas is depleted in the early part

of the test and then returns to 0.2095 mole fraction at the end of the test. This means there is no

67 | P a g e
net transfer when the system has reached the steady state. Had the saturation concentration been

over estimated, one would expect that there would still have been net negative transfer even at

steady state, and the exit mole fraction would have exceeded 0.2095 if the average saturation

concentration had been less than the bulk average DO concentration at equilibrium. This obviously

had not happened and would not have been logical. In other words, the measured C*∞ must be the

true saturation concentration and had not been over estimated. Since KLa and C*∞ are related

inversely to each other, the measured KLa must also be the true KLa and not under-estimated as

well.

So, what does this ‘true’ KLa parameter mean?

Consider the infinite series for any variable x:

𝑥2 𝑥3 𝑥4 𝑥𝑛
𝑒𝑥 = 1 + 𝑥 + + + + …..+ + …
2! 3! 4! 𝑛!

Recalling (see Chapter 4) that the proposed model is given by:

1 – exp (−𝐾𝑙𝑎0 Ф𝑍𝑑 )

𝐾𝐿𝑎 =
Ф𝑍𝑑
where

Ф = [HRST/Qa] (1-e)

Since, by expanding the exponential function into a series,

(−𝐾𝑙𝑎0 Ф𝑍𝑑 )2 (−𝐾𝑙𝑎0 Ф𝑍𝑑 )3

exp (−𝐾𝑙𝑎0 Ф𝑍𝑑 ) = 1 + (−𝐾𝑙𝑎0 Ф𝑍𝑑 ) + + + … ..
2 3!
Hence,

(−𝐾𝑙𝑎0 Ф𝑍𝑑 )2 (−𝐾𝑙𝑎0 Ф𝑍𝑑 )3

1 – [1 + (−𝐾𝑙𝑎0 Ф𝑍𝑑 ) + + + …..]
𝐾𝑙𝑎 = 2 3!
Ф𝑍𝑑
Therefore,

Kla = Kla 0 –Kla 0 2 Ф𝑍𝑑 /2 + Kla03 (Ф𝑍𝑑 )2/3! ….

thus, when Zd tends to zero, KLa tends to KLa0.

68 | P a g e
 Therefore, it was found by the author that the so-called ‘true’ KLa is a link between the

apparent KLa and the surface KLa, where the phenomenon of gas-side depletion is eliminated. By

adjusting the equations for both parameters (C*∞ and KLa) with calibration factors, based on the

effective depth (de) or the effective depth ratio (e), the proposed model linking KLa and KLa0

became valid for certain conditions, using existing data to verify. The model is further enhanced

by recognizing that, at saturation, the mole-fraction variation curve is a concave curve, so that

there is a minimum point in this curve that corresponds to a minimum oxygen mole fraction, at

which the absorption rate and the desorption rate are equalized at equilibrium. The derivative

(dy/dz) at this point must therefore equal zero.

The so-called ‘true’ KLa is in fact a parameter representing the mass transfer coefficient

when the gas depletion is absent. (i.e. tank height of zero.) This can be verified by plotting this

baseline KLa0 against the inverse of oxygen solubility in water (handbook values), (see Fig. 3-5),

and one would expect a straight line passing through the origin, since when the tank height is

infinitesimally small, KLa becomes KLa0 and C*∞ reduces to CS. In Chapter 2, it has been explained

that these two entities are inversely related to each other. This explains why, for surface aeration

in shallow tanks where oxygen is derived from atmospheric air rather than from diffused

submerged bubbles, the inverse relationship between KLa and Cs would hold because gas depletion

does not exist in such cases. It also explains why KLa0 is always higher in value than KLa as seen

in Fig. 3-8, since gas depletion hinders gas transfer. KLa0 without the depletion must therefore be

higher than that with depletion. This does not mean the hypothetical oxygen transfer rate at zero

depth would become higher, since at the surface, the saturation concentration is simultaneously

smaller than that in the bulk liquid. However, it is generally accepted that, the deeper the tank, the

higher the oxygen transfer efficiency, all things else being equal [Houck and Boon 1980] [Yunt et

69 | P a g e
al. 1988a, 1988b]. The suite of equations entailing this phase of the modeling also takes care of

the hydraulic pressure variations with respect to depth, so that rising bubbles would experience

volume changes, even without the gas depletion, due to this hydrostatic phenomenon.

3.1 Model Development

While Cs is proportional to pressure as stated by Henry’s Law, and the effect of hydrostatic

pressure on the dissolved oxygen (DO) saturation concentration is linear with changing water

depths; its effect on KLa is less certain. Boon (1979) found that the effect of immersion depth on

KLa is a general decline in the KLa versus increasing depths. This holds for different equipment

configuration and for different tank shapes. However, beyond a certain depth, this declining trend

ceases and the KLa value descends to a constant value.

The relationship between KLa and depth is not linear and is different from the relationship

between Cs (here CS is used in the context of equilibrium saturation concentration in a bulk liquid,

better known as C*∞) and depth, which is always an increasing function with depth. Houck et al.

(1980) found that the aeration efficiency should improve with increasing tank depth, but their data

showed no clear correlation between tank depth and oxygen efficiency at depths greater than 3.6

m (12 ft). Another interesting observation is that the variations of the KLa with depth is dependent

on the gas flow rates. Furthermore, their studies show that increases in blower efficiency can be

expected up to about 9.1 m (30 ft). Oxygen depletion beyond this depth clouds their analysis.

Therefore, it is not likely that the inverse proportionality between KLa and Cs still holds

for deep tanks, so that the previous findings of the researchers may be overturned for deep tank

aeration. In ASCE 18-96 Standard Guidelines (ASCE 1997) it is stated that the traditional

temperature correction coefficient (Ɵ) for KLa offsets the corresponding temperature correction

coefficient for the saturation concentration of DO in water. However, KLa is not only a non-linear

70 | P a g e
function of depth but also a function of a host of other factors (e.g., temperature, gas flow rate,

superficial velocity Ug ‒ the unit average gas flow rate over the cross-sectional area of the tank,

mixing intensity, etc.) (Metzger 1968). The data from the literature describe the general tendency

of the two variables (KLa and Cs) to move in different directions when temperature is changed,

but the product is not constant for non-shallow tanks. The statement in the Standard Guidelines is

questionable for deep tanks or any tank with a significant physical height. Lee (2017) showed that

for Ps = 1 atm, corresponding to a negligible tank depth, and for a constant average volumetric gas

flow rate Qa (m3/min), KLa is directly proportional to the water properties (this finding is also

supported by Daniil et al (1988)), as well as to the 5th power of temperature in absolute, as shown

in Chapter 2 Eq. (2-1) and again shown by eq. 3-1 below:

𝐸𝜌𝜎
(𝐾𝑙𝑎) 𝑇 = 𝐾 × 𝑇 5 × [3 − 1]
𝑃𝑠

where K is a proportionality constant; KLaT (min-1) is mass transfer coefficient at T; T is any

temperature in K (Kelvin); E is modulus of elasticity of water in (kN/m2) x 10-6; ρ is density of

water (kg/m3); σ is surface tension of water (N/m); Ps is the saturation pressure in atmospheres

(atm). Based on the experiments by Hunter (1979) and Vogelaar et al. (2000), Eq. [3-1] would

supersede the traditional Arrhenius equation for temperature correction because of the higher

accuracy, especially for water temperatures above 20 0C. Lee (2017) hypothesized that KLa is

inversely proportional to Ps, but this applies only to shallow tanks, according to those experiments.

The variation of the mass transfer coefficient KLa at different depths is due to the

phenomenon known as gas depletion. Gas-side depletion refers to the decrease in oxygen partial

pressure as the bubbles rise through the water column and is the major mechanism for oxygen

transfer in submerged bubble aeration. As the air bubbles rise, oxygen is transferred but no net

nitrogen is transferred. This occurs because the nitrogen concentration in the tank column is

71 | P a g e
constant, since there are no reactions that consume dissolved nitrogen [Stenstrom et al. 2001]. All

diffused aeration systems will experience higher gas-side depletion as the water depth increases

because of the longer contact times of the bubbles with water.

According to Stenstrom et al. (2001), more efficient systems encounter gas-side oxygen

depletion at shallow depths. Coarse bubble diffusers may not experience gas-side depletion until

15 m (50 feet) or more of depth. Fine pore systems experience gas side depletion at shallower

depths, but typically not less than 6m (20 feet). (This statement appears to be incorrect. Yunt’s

data [Yunt et al. 1988a] seems to suggest that gas-side depletion is significant even at 3 m (10 feet)

for diffused aeration. In fact, without gas depletion, there would be no gas transfer except any

transfer from the open atmosphere.) Fine pore diffuser systems using full floor coverage typically

have standard transfer efficiencies from 2 to 2.5% per cent per 0.3 m (1 foot) (SOTE/0.3m),

depending on the gas flow rate and diffuser density. For systems of high SOTE/0.3m, it is not

surprising that gas side depletion occurs at shallow depths. (Again, this usage of SOTE per unit

depth is not justified as the variations with depth is not linear.) However, at the baseline of zero

depth, there would be no gas depletion, and Eq. (3-1) for temperature correction should apply.

The model developed to calculate the baseline (KLa0) was based on fundamental gas-liquid

gas transfer principles and considered the oxygen mass balances on a rising bubble of a constant

volume, leading to the validation of the basic model for the non-steady state clean water test as

described in the ASCE 2-06 standard [ASCE 2007]. During the validation, two important models

have been found --- the depth correction model that gives a meaning to the apparent KLa in relation

to the liquid depth; and the Lee-Baillod model (as defined by the author herewith) that describes

the relationships between equilibrium concentration, depth and exit gas composition, based on an

oxygen mole fraction variation curve. The mathematical derivation of the Lee-Baillod model based

72 | P a g e
on a mass balance in the gas phase is given in Chapter 4. This gives rise to an expression Eq. [4-

33] as shown below:

𝐶 𝑌0 𝑃𝑑 𝐶
𝑦 = +( – ) exp(−𝑧)
𝐻𝑃 𝑃 𝐻𝑃

where y is the oxygen mole fraction at any point in the aeration tank as the bubble ascends to the

water free surface; C is the dissolved oxygen concentration in the bulk liquid; H is Henry’s Law

constant; Y0 is the mole fraction at the initial bubble release; Pd is the pressure at the diffuser; P is

the absolute pressure at the point corresponding to z; z is the depth measured from the bottom

(meter);  is a constant (see eq. 4-25). As for the liquid phase mass balance, the net accumulation

rate of dissolved gas in the liquid column is equal to the gas mass flow rate delivered by diffusion,

if there is no gas escape from the liquid column. The details of the derivation arising from mass

balance of the liquid phase is given in Section 4.1.2.

Therefore, from the mass balances in a non-steady state clean water test,

𝑑𝐶
= 𝐾1 (𝐾2 − 𝐶) [3 − 2]
𝑑𝑡

where

(1 – exp(−𝑍𝑑 ))
K1 = 𝐾𝐿 𝑎0 [3 − 3]
𝑍𝑑

and

K 2 = 𝐻𝑌0 𝑃𝑑 [3 − 4]

Thus, the basic transfer equation (the Standard Model) is proven mathematically, since K1 has the

same meaning as KLa, and K2 has the meaning of the saturation concentration C*∞. Eq. 3-3 is the

depth correction model for the KLa. However, as defined in Chapter 4, the Lee-Baillod model,

73 | P a g e
Eq. [4-33], that calculates the oxygen mole fraction y is not physically correct because of the

inherent assumptions, particularly the constant bubble volume assumption. However, it can be

amended to eq. [3-5] below by inserting two parameters n and m where appropriate, that is based

on a variable oxygen mole fraction curve versus the tank depth as can be seen in Fig. 3-1, (showing

the case when the DO is approaching saturation concentration C*∞). This equation is different

from other developed models such as the Downing-Boon’s model [Downing and Boon 1968] and

model developed by Jackson & Shen (1978), which are linear. Most models predict the equilibrium

level or the saturation level to be located at mid-depth (de/Zd = 0.5) which is unrealistic as both the

mole fraction and the bubble interfacial area change during the bubble rise to the surface, so that

de/Zd =< 0.5, where e is the effective depth ratio given by de/Zd, where de is the effective depth,

and Zd is the immersion depth of diffuser. After modification of the CBVM (Constant Bubble

Volume Model) with the calibration factors ‘n’ and ‘m’ for the Lee-Baillod model (Eq. [4-33]), to

account for the non-constant bubble volume in a deep tank, the following generalized equation is

obtained:

𝐶 𝑌0 𝑃𝑑 𝐶
𝑦 = +( – ) exp(−𝐻𝑘. 𝑚𝑧) [3 − 5]
𝑛𝐻𝑃 𝑃 𝑛𝐻𝑃

where m and n are calibration parameters for the curve; Hk = . (It can be shown that  = x. KLa0

where x = HRT/Ug where Ug is the height-averaged superficial gas velocity; R is specific gas

constant for oxygen; H is Henry’s constant; T is absolute temperature). Other symbols are as shown

in Fig. 3-1 below. This equation is equivalent to eq. 4-58 in Section 4.1.5 of Chapter 4.

This equation for the generalized Lee-Baillod model, can be differentiated by calculus with

respect to z, and then setting it to zero to obtain the minimum point. Another equation is thus

developed that gives the point along the curve at which the minimum mole fraction of oxygen

occurs. Similarly, the modified equation (eq. 3-5) can be subjected to mathematical integration

74 | P a g e
just like the previous case for the constant bubble volume model. All the resulting equations that

lend themselves to five simultaneous equations for solving the unknown parameters (n, m, KLa0,

ye, Ze) are summarized below:

[1 – exp(−𝑲𝑳 𝒂0 𝑥 (1 − 𝑒)𝑍𝑑 )]
𝑲𝑳 𝒂 = [3 − 6](eq. 4 − 48)
𝑥(1 − 𝑒)𝑍𝑑

𝐶 𝑌0 𝑃𝑑 𝐶
𝑦 = +( – ) exp(−𝑥𝑲𝑳 𝒂0 . 𝑚𝑧) [3 − 7](𝑒𝑞. 4 − 58)
𝑛𝐻𝑃 𝑃 𝑛𝐻𝑃

𝑃𝑎 – 𝑃𝑑 exp(−𝑚𝑥. 𝑲𝑳 𝒂0 . 𝑍𝑑 )
𝐶 ∗ ∞ = 𝑛𝐻 × 0.2095 × [3 − 8](𝑒𝑞. 4 − 63)
1 − exp(−𝑚𝑥. 𝑲𝑳 𝒂0 . 𝑍𝑑 )

1 – exp(−𝑚𝑥. 𝑲𝑳 𝒂0 . 𝑍𝑑 ) (𝑛 – 1)𝑲𝑳 𝒂0
𝑲𝑳 𝒂 = + [3 − 9](𝑒𝑞. 4 − 65)
𝑛𝑚𝑥. 𝑍𝑑 𝑛

1 𝑚𝑥𝑲𝑳 𝒂0 𝑛𝐻𝑌0 𝑃𝑑
𝑍𝑒 = {ln (𝑃𝑒 ) + ln ( ∗ − 1)} [3 − 10](𝑒𝑞. 4 − 74)
𝑚𝑥𝑲𝑳 𝒂0 𝑛𝑟𝑤 𝐶 ∞

free surface Cs = solubility at 1 atm

C*∞ = saturation
conc. at Pe
de
equilibrium
level at
z pressure Pe
Zd

bulk liquid
Ze

mole fraction of oxygen

ye assumed y0=0.2095
y

Fig.3-1. The MF(mole fraction) curve for the Lee-Baillod model (subscript e = equilibrium)

75 | P a g e
 Derivation of the above equations is given in Chapter 4. These equations are repeated in

the calculation sheet (Table 3-2) below. In the above suite of equations, Eq. [3-6] was uniquely

derived from the depth correction model, via an adjustment to the effective saturation depth ratio

(e) from e = 1 to e = de/Zd, in the same way the Lee-Baillod model was adjusted by the calibration

factors (n, m) to account for the constant bubble volume assumption.

After calculating the baseline mass transfer coefficient KLa0 at any test temperature T, the

standard baseline can be calculated by the use of the proposed 5th power temperature correction

model (Lee 2017). The use of the 5th power model as given by Eq. 3-1 is preferred to the current

ASCE model (ASCE 2007). The topic of temperature correction is discussed in Section 5.6 in

Chapter 5, where various temperature models are compared as shown in the bar graph of Fig. 5-

11. The discrepancies between these models in terms of standardizing (KLa0)T to (KLa0)20 in this

exercise are small, because all the tests cited were done within the narrow temperature range of

between 10 0C and 30 0C [ASCE 2007].

However, the use of a 5th power model appears to give the best regression analysis result

to yield the standard baseline (KLa0)20, but since the temperature effect compared to the depth

effect is small, other models will also give similar result, even though the prediction would not be

as accurate, especially for water temperatures above 20 0C (Lee 2017).

3.2 Material and Method

KLa0 can be calculated from a set of clean water test results. The following development is

based on data extracted from Yunt et al. (1988a). The test facility used for all tests was an all steel

rectangular aeration tank located at the Los Angeles County Sanitation Districts (LACSD) Joint

Water Pollution Control Plant. The dimensions of this tank are 6.1 m X 6.1 m X 7.6 m (20 ft X 20

ft X 25 ft) side water depth (SWD). As reported, more than 100 tests had been carried out on

various submerged aeration systems in the Control Plant. The test temperatures were reported to
76 | P a g e
be within the range of 16.2 0C to 25.2 0C. In the aeration tests, multiple diffusers were placed at a

submerged depth of 3.05-7.62 m with a tank water surface of 37.2 m2 and water volume of 113.2-

283.1 m3. Fine bubble diffusers were operated at air flow rates of 213.0-683.4 scmh (standard

cubic metres per hour). The FMC diffusers are a fine bubble tube diffuser system manufactured

by FMC Corporation. The testing configuration is given in Figure 10 of the LACSD report. The

diffuser media was a white porous modified acrylonitrile-styrene copolymer material. These tube

diffusers had a permeability of 23.7 L/s or 50 scfm (standard cubic feet per minute) at a headloss

of 25.4 mm (1 in.) of water. The diffuser air release point was 65 cm (25 in.) above the tank floor.

The tests were carried out at four different depths for a range of air flow rates for each depth. The

first group of tests were carried out from August 29, 1978 to September 29, 1978 with the last date

for one test only. This group is assumed to have a temperature of 25 0C for data interpretation and

analysis. This group entails the 3.05 m (10 ft) [Zd = 2.44 m] and 7.62 m (25 ft) [Zd=7.02m] tanks.

The second group was carried out from February 8, 1979 to February 9, 1979 and this group should

have a temperature of 16 0C. These tanks are 4.57 m (15 ft) [Zd = 4 m] and 6.10 m (20 ft) [Zd = 5.6

m]. Only standard values (KLa)20 and 𝐶 ∗ ∞ 20 were reported; the corresponding KLaT and C*∞T

were back-calculated from the formulae reportedly used in the conversion to standard conditions:

(𝐾𝐿 𝑎) 𝑇 = (𝐾𝐿 𝑎)20 Ɵ𝑇−20 [3 − 11]

𝐶 ∗ 0 = 8.96 + .271 𝑍𝑒𝑚𝑑 [3 − 12]

where the reported value of Ɵ used was 1.024 and Zemd is the effective depth similar to (de) in this

development. C*0 is identical to C*∞20. The actual measured saturation concentration C*∞T can

then be back-calculated by another equation using Zemd as an independent variable [Yunt et al.

1988a]. With these parameters thus determined, the baseline KLa (KLa0) for every test can then be

determined. For each water depth tested, the volumetric mass transfer coefficients can then be
77 | P a g e
plotted against the average flow rates for both the apparent and the baseline KLa values. The test

results are given in the LACSD report Table 5: “Summary of Exponential Method Results: FMC

Fine Bubble Tube Diffusers” and copied herewith as Table 3-1 below.

3.3 Results and Discussions

Tables 3-1 and 3-2 below are compiled based on data contained in the LACSD report for

FMC Fine Bubble Tube Diffusers. Table 3-1 shows all the raw data as given in the LACSD report.

Table 3-2 (Excel spreadsheet for estimating variables KLa0, n, m, de and ye) shows an example

calculation of the baseline mass transfer coefficient (KLa0)T using the Excel Solver with the model

equations (Eq. [3-6] to Eq. [3-10]) incorporated for tank 1 Run 1. Table 3-3 showcases calculation

of simulation tank for the 7.6m (25 ft) tank at a gas flow rate of 7.96 Nm3/min (281 scfm) using

the same specific baseline as calculated from Table 3-2, and using the same set of developed

equations, but with (KLa)20 as the unknown parameter to be solved for.

The simulated result from Table 3-3 gives a value of (KLa)20 = 0.1874 min-1 as compared

to the reported test value of (KLa)20 = 0.1853 min-1 which gives an error difference of around 1%

only comparing to the simulated value.

3.3.1. Example calculation

For an example, suppose the specific baseline (KLa0)20 has been established by a clean

water test to be 4.435x10-2 min-1 per Qa^0.82, where Qa is in m3/min. In customary units, it would

be 2.38 x 10-3 min-1 per Qa^0.82 (where Qa is in cfm). We want to estimate KLa for a 7.62 m (25

ft) tank with a diffuser submergence 0.6 m (2 ft) above the floor, (Zd = 7.01 m), at a gas supply

rate of 7.96 Nm3/min (281 scfm). The horizontal cross-sectional area of the tank is 37.2m2 (20 ft

x 20 ft). The pressure at the diffuser is given by:

78 | P a g e
 Standard
Rn Water Delivered Air-flow temperat apparent apparent saturation
Date Oxygen Transfer
No. Depth Z Power Density Rate Qs ure (KLa)20 (KLa)20 conc. C*∞20
Efficiency
(hp/1000
m (scmh) T (0C) * (1/hr) (1/min) (mg/L) (%)
ft^3)
Aug 29,78 1 3.05 2.02 700 25.2 17.46 0.2910 9.87 10.06
Aug 29,78 2 3.05 1.16 470 25.2 13.37 0.2228 9.99 11.68
Aug 29,78 3 3.05 0.54 241 25.2 7.63 0.1272 10.05 12.95
Aug 29,78 1 7.62 1.66 704 25.2 14.99 0.2498 11.23 23.93
Aug 29,78 2 7.62 1.07 478 25.2 11.12 0.1853 11.26 24.40
Aug 29,78 3 7.62 0.51 236 25.2 6.39 0.1065 11.54 31.71
Sep 29,78 1 3.05 1.19 472 25.2 13.39 0.2232 9.98 11.61
Feb 08,79 1 4.57 1.81 694 16.2 16.61 0.2768 10.50 15.34
Feb 08,79 2 4.57 1.05 449 16.2 11.90 0.1983 10.54 17.07
Feb 08,79 3 4.57 0.51 231 16.2 6.88 0.1147 10.63 19.87
Feb 08,79 1 6.10 1.74 709 16.2 16.73 0.2788 10.80 20.69
Feb 08,79 2 6.10 1.08 471 16.2 11.62 0.1937 11.05 22.17
Feb 08,79 3 6.10 0.49 224 16.2 6.10 0.1017 11.19 25.04
*Note: water temperature was deduced from the report statement: "The temperature range used in the study was 16.2 to 25.2
0C" Reported main data are given in bold; (K a) given in this table is based on the Arrhenius model using Ɵ=1.024 [ASCE 2007]
L 20
The 5th power temperature correction model [Lee 2017] to convert KLa0 estimated in Table 3-2 to (KLa0)20 and subsequently to
(𝐸𝜌𝜎)20 𝑇20 5
(KLa)20 is given by: (𝐾𝐿 𝑎)20 = 𝐾𝐿 𝑎 ( )
(𝐸𝜌𝜎)𝑇 𝑇
Note that there were some discrepancies in the reported data for the 7-m tank,
in that the data for the SOTE% were calculated by an equation in the Report and they did not match up for two points in the
report. These data were discarded and the calculated values using the Report’s equations were used in the above Table, but these
data are still suspect. The greater number of tests are done, the better would be the estimation of the unknown parameters.

Table 3-1. LACSD (Los Angeles County Sanitation District) Report Test Data (1978) for the FMC diffuser

78 | P a g e
Fixed parameters For Pe calc.: C*∞=Hye Pe SS error
diffuser depth Zd = 2.44 m saturation depth de (m) = 1.19
atm pressure Pa = 101325 N/m2 eff. Depth ratio e= 0.49
press at diff. Pd = 121964 N/m2 equil. Pressure Pe (N/m2) = 109758
x= HRST/Qa 0.1051 min/m Eq. I=(4-65) (Eq. I – KL a) = 9.951E-05 9.90288E-09
tank area S= 37.2 m^2 Eq. II=(4-63) (Eq. II – C*∞) = -1.275E-06 1.62751E-12
variables Eq. III=(4-74) (Eq. III – Ze) = 7.985E-05 6.37732E-09
min^-1 𝑲𝑳 𝒂 0 = 0.3349 Eq. IV=(4-51) (Eq. IV – KL a) = 5.377E-08 2.89175E-15
dimensionless n= 5.6629 sum= 1.62818E-08
Eq. I, II, III, IV, V given
dimensionless m= 3.0737
below
dimensionless ye= 0.2080 Eq. V=(4-58) offgas=y at exit 0.2095 checked
dimensionless yd= 0.2095
data 1 – exp(−𝑚𝑥. 𝐾𝑙𝑎0 . 𝑍𝑑) (𝑛 – 1)𝐾𝑙𝑎0
min^-1 KLa = 0.3276 𝑲𝑳 𝒂 = + (4 − 65)
𝑛𝑚𝑥. 𝑍𝑑 𝑛
mg/L C*∞ = 9.05
𝑃𝑎 – 𝑃𝑑 exp(−𝑚𝑥. 𝑲𝑳 𝒂0 . 𝑍𝑑)
mg/L/(N/m2) H= 3.96E-04 𝐶 ∗ ∞ = 𝑛𝐻 ∗ 0.2095 ∗ (4 − 63)
1 − exp(−𝑚𝑥. 𝑲𝑳 𝒂0 . 𝑍𝑑)
N/m3 rw = 9777
1 𝑚𝑥𝑲𝑳 𝒂0 𝑛𝐻𝑌0 𝑃𝑑
𝑍𝑒 = {ln (𝑃𝑒 ) + ln ( ∗ − 1)} (4 − 74)
𝑚𝑥 𝑲𝑳 𝒂0 𝑛𝑟𝑤 𝐶 ∞
[1 – exp(−𝑲𝑳 𝒂0 𝑥 (1 − 𝑒)𝑍𝑑)]
𝑲𝑳 𝒂 = (4 − 51)
𝑥(1 − 𝑒)𝑍𝑑
Vapor pressure Checking equation at system equilibrium:
Pvt = 3200
N/m2
(y=y0=0.2095; C=C*∞; z=Zd; P=Pa):
𝐶 𝑌0 𝑃𝑑 𝐶
𝑦 = +( – ) exp(−𝐻𝑘. 𝑚𝑧) (4 − 58)
𝑛𝐻𝑃 𝑃 𝑛𝐻𝑃

Table 3-2. Calculation of KLa0 for Run 1 for the 3.05 m (10 ft) Tank at T=25 0C

79 | P a g e
simulation of 7.6 m(25-ft) tank at
7.96 m3/min (281 scfm)
Fixed SS err
7.6 m Zd= 7.01 de(m)= 2.95
20 C Pa= 98992
Pd= 167613 V(m3)= 283.464 Pe= 127855
x= 0.1957 Qa(m3/m)= 6.35 Eq. I= 1.16E-03 1.344E-06
S= 37.2 Eq. II= -8.37E-06 7.005E-11
Variables 𝑲𝑳 𝒂 = 0.1874 (KLa)20= 11.25 Eq. III= 2.74E-04 7.499E-08
e= 0.42 sp.Kla0= 0.04434 Eq.IV= -9.04E-04 8.178E-07
n= 4.17 Eq.V= 2.28E-08 5.220E-16
m= 2.46 Eq. VI= -4.10E-05 1.679E-09
ye= 0.1958 Min (SS err) 2.236E-06
C*inf= 10.91 1 – exp(−𝑚𝑥. 𝐾𝑙𝑎0 . 𝑍𝑑) (𝑛 – 1)𝐾𝑙𝑎0
𝐾𝑙𝑎 = + (𝐸𝑞. 𝐼)
Data 𝑛𝑚𝑥. 𝑍𝑑 𝑛
𝑲 𝑳 𝒂 0= 0.2019 𝑃𝑎 – 𝑃𝑑 exp(−𝑚𝑥. 𝐾𝑙𝑎0 . 𝑍𝑑)
𝐶 ∗ ∞ = 𝑛𝐻 ∗ 0.2095 ∗ (𝐸𝑞. 𝐼𝐼)
1 − exp(−𝑚𝑥. 𝐾𝑙𝑎0 . 𝑍𝑑)
yd= 0.2095
4.383E- 1 𝑚𝑥𝐾𝑙𝑎0 𝑛𝐻𝑌0 𝑃𝑑
H= 𝑍𝑒 = {ln (𝑃𝑒 ) + ln ( ∗ − 1)} (𝐸𝑞. 𝐼𝐼𝐼)
04 𝑚𝑥𝐾𝑙𝑎0 𝑛𝑟𝑤 𝐶 ∞

rw= 9789 [1 – exp(−𝐾𝑙𝑎0 𝑥 (1 − 𝑒)𝑍𝑑)]

𝐾𝑙𝑎 = (𝐸𝑞. 𝐼𝑉)
𝑥(1 − 𝑒)𝑍𝑑
𝐶 𝑌0 𝑛𝑃𝑑 𝐶
𝑦 = +( – ) exp(−𝐻𝑘. 𝑚𝑧) (𝐸𝑞. 𝑉)
𝑛𝐻𝑃 𝑛𝑃 𝑛𝐻𝑃
pvt= 2333
at system equilibrium (y=y0=0.2095; C=C*∞; z=Zd; P=Pa)
𝐶 ∗ ∞ = (𝑟𝑤 ∗ 𝑑𝑒 + 𝑃𝑏 − 𝑃𝑣𝑡) ∗ 𝐶 ∗ 𝑠𝑡/(𝑃𝑠 − 𝑃𝑣𝑡) (𝐸𝑞. 𝑉𝐼)

Table 3-3. Calculation of (KLa)20 for Run 2 for the 7.62 m (25 ft) Tank at sp. (KLa0)20 = 0.04434

80 | P a g e
Pd = 101325 + 9789*7.01 = 169946 N/m2. Assuming vapor pressure has no effect on the

volumetric gas flowrate, the average gas flow rate is given by (Eq. 2-25) in Chapter 2, Qa =

172.82(293.15) (7.96) (1/101325+1/169946) = 6.35 m3/min (224 cfm)

The superficial velocity Ug = 6.35/(37.2) = 0.1707 m/min (0.56 ft/min)

Therefore, x = HRT/Ug = 4.382x10-4 (0.260) (293.15)/0.1707 = 0.1957 min/m

where R (specific gas constant for oxygen) is given as 0.260 KJ/kg-K; H is the handbook value

for Henry’s constant at 20 0C, given as 4.382E-4 (mg/L)/(N/m2). Assuming e = 0.45, Φ =

0.1957(1 - 0.45) = 0.1076 min/m (This assumption for ‘e’ is not needed in the spreadsheet

calculations) and,

From Eq. [3-6],

1 − exp(−𝛷𝑍𝑑 . 𝐾𝐿 𝑎0 )
𝐾𝐿 𝑎 =
𝛷𝑍𝑑

where Ø = 𝑥(1 − 𝑒)

Therefore, KLa = (1- exp (-0.1076 x 7.01 x 4.435x10-2 x 6.35^0.82))/0.1076/7.01 = 0.1873 min-1

= 11.23 hr-1. This compares with 11.12 hr-1 in the real test for this tank. This incurs an error of

about +2%

Furthermore, from ASCE 2-06 Eq. (F-1), the effective depth de is given as:

1 𝐶∗∞
𝑑𝑒 = [( ) (𝑃𝑠 – 𝑃𝑣𝑡) − 𝑃𝑏 + 𝑃𝑣𝑡] [3 − 13]
𝑟𝑤 𝐶 ∗ 𝑠𝑡

Rearranging gives,

(𝑟𝑤 𝑑𝑒 + 𝑃𝑏 – 𝑃𝑣𝑡 )𝐶 ∗ 𝑠𝑡
𝐶 ∗∞ = [3 − 14]
𝑃𝑠 – 𝑃𝑣𝑡

Therefore, C*∞ = (101325 + 9789 x 0.45 x 7.01 - 2340) x 9.09/ (101325 - 2340) =11.92 mg/L

The above equation (eq. 3-14) implicitly assumed that the mole fraction at the saturation

point is 0.21, but as the Excel Solver calculated, the true mole fraction at equilibrium (Ye) is

81 | P a g e
0.1958 (Table 3-3). Therefore, the corrected C*∞ will be given by: C*∞ = (0.1958/0.21) x 11.92

= 11.11 mg/L. This compares with the reported measured C*∞ of 11.26 mg/L. The percent error

is about -2%. The calculated SOTR (Standard Oxygen Transfer Rate) is given by (11.23) (11.11)

V=124.8V where V is the volume of tank, which compares well with the SOTR based on

reported values, of (11.12) (11.26) V = 125.2V. The percent error is practically insignificant.

3.3.2. Estimation of the effective depth ratio (e = de/Zd)

This paragraph should be read in conjunction with Chapter 4. Figure 3-2 below is a plot of

the effective depth ratios calculated from the test runs, the lower line showing the results based on

a constant equilibrium mole fraction at 0.21 similar to the equation in ASCE 2-06 Annex F [ASCE

2007], while the top line was based on the Depth Correction Model (Eq. 3-6); and the other

developed model equations (see Table 3-2), using Eqs. (4-46) for the effective depth ratio (e); Eq.

(4-74) or Eq. (3-10) for Ze and the minimum Y at Ye; Eq. (4-72) which is similar to eq. 3-13 above

given by ASCE 2-06 Annex F Eq. (F-1) but corrected for Ye for calculating de; Eq. (4-75) for Pe;

compare submergence between models

0.60

0.50
submerge ratio e

0.40

0.30
e(ye calc.)
0.20 e(Ye=0.21)
0.10

0.00
0 5 10 15
Run Number

Fig. 3-2. Comparison of submergence depth ratio (e) rigorous analysis vs. ASCE method

Eq. (4-76) for Pa (the atmospheric pressure N/m2). [Eq. 3-8, eq. 3-9] or [Eqs. (4-63) (4-65)] as

derived from the Lee-Baillod model (See Chapter 4) that describes the mole fraction variation

82 | P a g e
curve, and, after inserting boundary conditions, are for calculating C*∞ and KLa respectively which

lead to the calibration parameters, n and m; eq. 3-7 or Eq. (4-58) is used to double check the

calculations, as it should give a mole fraction at saturation of 0.2095 at exit.

It is important to note that the ASCE 2-06 Annex F Eq. (F-1) has treated Ye to be the same Y0 in

Eq. (4-72) which is not correct. As a result of this rigorous analysis, the top line in Fig. 3-2 gives

a more consistently uniform depth ratio of e.

3.3.3. Determination of the Standard Specific Baseline

Fig. 3-3 shows that the resulting KLa0 values are then adjusted for the standard temperature

by the temperature correction equation of the 5th power model (Lee 2017) and plotted against Qa20.

Amazingly, all curves fitted together after normalizing KLa0 values to 20 0C, as shown. The

exponent is 0.82.

(KLa0)20 vs. Qa20

0.3500

0.3000
y = 0.0444x0.82
0.2500 R² = 1
(KLa0)20 (min^-1)

0.2000

0.1500

0.1000

0.0500

0.0000
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Qa20 (m^3/min)

Fig. 3-3. Standard Baseline (KLa0)20 vs. Average standard gas flow rate Qa20 for various test

temperatures.

83 | P a g e
 The value obtained from the slope is 44.35 x 10 -3 (1/min) for all the gas rates normalized

to give the best NLLS (Non-Linear Least Squares) fit, bearing in mind that the KLa0 is assumed to

be related to the gas flowrate by a power curve with an exponent value [Stenstrom et al.

2006][Zhou et al. 2012]. The slope of the curve is defined as the standard specific baseline.

Therefore, the standard specific baseline (sp. KLa0)20 is calculated by the ratio of (KLa0)20

to Qa20^.82 or by the slope of the curve in Fig. 3-3. When the same information is compared with

a similar plot using the actual measured KLa values (plot not shown) it can be seen the correlation

was still quite good for the curve, but not as exactly as when the baseline values were plotted,

testifying the fact that the baseline mass transfer coefficient does represent a standardized

performance of the aeration system when the tank is reduced to zero depth (i.e. when the effect of

gas depletion in the fine bubble stream was eliminated.) As KLa is a local variable dependent on

the bubble’s tank location especially its height position, KLa0 represents the KLa at the surface, i.e.,

at the top of the tank, where the saturation concentration corresponds to the atmospheric pressure

(Ps = 1 atm).

3.3.4. Relationship between the mass transfer coefficient and saturation concentration

Fig. 3-4 below shows the apparent mass transfer coefficient KLaT as reportedly measured,

upon normalizing the (KLa)T values to the air flow rates, plotted against the inverse of the measured

saturation concentrations (C*∞T) for all the tests, and they give a linear correlation with R2 =

0.9859. This figure shows the inverse relationship between KLa and C*∞ for different temperatures;

but since the KLa data pertain to different gas flow rates (Qa), KLa must first be normalized to the

same Qa before it can be plotted against C*∞, otherwise it would be meaningless because KLa is

much more dependent on the air flow rate than the dissolved oxygen saturation concentration

would be [Hwang and Stenstrom 1985]. This normalization cannot occur until the relationship

84 | P a g e
between KLa and Qa is first determined (as shown in Fig. 3-3 above for the baseline plot). Since

the parameter estimation based on an assumed power function [Hwang and Stenstrom 1985] [Zhou

et al. 2012] has determined that it is a power function of Qa0.82, therefore, the relationship between

the specific mass transfer coefficient and the saturation concentration is given by K La/Qa0.82 =

0.4515(1/C*∞) as shown in the graph of Fig. 3-4.

KlaT [flow rate normalized] vs. 1/C*∞T

FMC diffusers
0.0600

0.0500
sp. KLaT/Qa^.82 (1/min)

0.0400 y = 0.4515x
R² = 0.9859
0.0300

0.0200

0.0100

0.0000
0 0.02 0.04 0.06 0.08 0.1 0.12
inverse of saturation concentration (1/C*∞T)

Fig. 3-4. specific KLaT vs. the inverse of measured saturation concentrations (C*∞T)

As will be seen later, using the same power exponent for tanks other than the baseline is

an approximation only, as R2 = 0.9859 is not as good as when the baseline values are used.

3.3.5. Relationship between the baseline and oxygen solubility

Fig. 3-5 shows the relationship between the baseline coefficients, (“air flow normalized"

base line coefficients, defined as the specific baseline), with the solubility of oxygen in water (Cs),

using handbook values [ASCE 2007] for the oxygen solubility. As expected, they bear an inverse

correlation, such that when KLa0 is plotted against the insolubility (1/Cs), a linear graph is obtained.

Since KLa0 represents the mass transfer coefficient at the surface and based on the hypothesis that

KLa and Cs are inversely proportional to each other [Lee 2017], the specific KLa0 for all the tanks

85 | P a g e
tested at temperature 16 0C would converge into a single point, and all the tanks tested at 25 0C

would focus to another single point, regardless of the individual tank depth and the gas flowrate

being applied.

(KLa0)T/QaT0.82 vs.1/Cs
0.06

0.05 16 0C
0.04
KLa0/Qa0.82

y = 0.4031x
0.03 R² = 0.9924
25 0C
0.02

0.01

0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Inverse of Cs---bassed on handbook values (1/Cs=L/mg)

Fig. 3-5. Sp. KLa0 vs. the inverse of surface saturation (solubility) concentration Cs

The specific baseline mass transfer coefficient at standard temperature (200C) would

become a single value, regardless of the tank depth (as shown later by the top curve in Fig. 3-8).

One would therefore expect that KLa0 would be proportional inversely to the surface saturation

value or the solubility when both parameters are varied with temperature [Lee 2017]. The

relationship between the specific baseline mass transfer coefficient and the gas solubility is given

by K La0/Qa0.82 = 0.4031(1/Cs) as shown in the graph of Fig. 3-5.

Although the slope of this curve given as K = 0.4031 indicates a linear proportionality, the

proportionality constant between the baseline and its solubility is not the same as the

proportionality constant between KLa0 and its corresponding temperature function, which is given

in Fig. 3-7 where the proportionality constant has a different value, K = 0.1284.

86 | P a g e
3.3.6. Relationship between the baseline and the gas flow rate

The equation for converting the gas mass flow rate (Qs) in col. 5 of Table 3-1, to the

average volumetric gas flowrate (Qa) is given by Eq. (2-25) in Chapter 2. Fig. 3-6 below shows

the resulting plot of (Kla0)T vs. QaT for the four different tank depths in this experiment.

(KLa0)T vs. QaT

0.4
0.35
25 C data depth
Baseline Kla0 (min^-1)

0.3
0.25 3.05 m

0.2 7.62 m
4.57 m
0.15
16 C data 6.09 m
0.1
0.05
0
0.00 2.00 4.00 6.00 8.00 10.00 12.00
average gas flowrate Qa (m3/min)

Fig. 3-6. Baseline KLa0 vs. Average gas flowrate Qa at various temperatures

Two distinct bands of curves were discovered: the 3.05 m (10 ft) [Zd = 2.44 m] tank and

the 7.62 m (25 ft) [Zd = 7.02 m] tank were carried out at 25 0C forming one band; and the 4.57 m

(15 ft) [Zd = 4 m] and the 6.09 m (20 ft) [Zd = 5.6 m] tanks form a different band at 16 0C. Here we

have the shallowest and the deepest tank curves almost coinciding, and similarly, for the other two

tanks, the curves are banded together as another group at 16 0C. The fact that all the (KLa0)T values

fitted together at a single temperature indicates that the baseline (KLa0)T is quite independent of

depth, for the different flow rates at each temperature. They tend to be best fitted by power curves,

with the average exponent of around 0.8. This phenomenon is equally pronounced when the

87 | P a g e
apparent KLa values are plotted (not shown), instead of the baselines KLa0 but with a lesser

correlation for each band, especially between the 3.05 m (10 ft) and 7.62 m (25 ft) tank.

3.3.7. Relationship between the baseline and water temperature

Fig. 3-7 below shows the relationship between the baseline mass transfer coefficient

(KLa0)T and the temperature function based on the 5th power model as given by Eq. (3-1) that was

developed in a previous manuscript (Lee 2017) and in Chapter 2.

KLa0 vs. Qa^.82*(EρσT^5)

0.4

0.35 y = 0.1284x
0.3
R² = 0.9958

0.25
KLa0 (1/min)

0.2

0.15

0.1

0.05

0
0 0.5 1 1.5 2 2.5 3
fn = T^5.Eρσ.Q^.82 (saturation pressure Ps=1 atm)

Fig. 3-7. Baseline (KLa0)T plotted against temperature function of 5th power model

Since Eq. [3-1] was based on a constant Qa, and the relation between KLa0 and Qa has been

established as power function of 0.82 as shown in Fig. 3-3, and Ps =1 atm for the baseline, the

correlation (KLa0 vs. Qa0.82(EρσT5)) shown in Fig. 3-7 has a value of R2 = 0.9958, testifying that

the 5th power model for temperature conversion is valid. The slope of this curve given by

K=0.1284 would be identical to the proportionality constant K specified in Eq. [3-1] for the

temperature correction model at Ps = 1 atm, since KLa0 pertains to this pressure. Therefore, for

this particular case where Qa is not constant, eq. 3-1 would become:

88 | P a g e
 (𝐸𝜌𝜎) 𝑇
𝐾𝐿 𝑎0 𝑇 = 0.1284 × 𝑇 5 × × 𝑄𝑇 0.82 [3 − 15]
𝑃𝑠

For comparison, the actual measured mass transfer coefficients KLa are plotted against the same

temperature function, and it can be seen that a good correlation can still be obtained, but the

correlation is less precise than when the baselines are plotted. Therefore, eq. 3-1 is still correct, for

the relationship (as shown in Fig. 3-7a) and the temperature correction model, as discussed in

Chapter 2. It is believed that the deeper the tank the further apart from linearity this plot will be,

but it appears that the temperature correction model holds for depths up to 7.6 m in this case.

KLaT vs. temperature for FMC diffusers

0.35

0.3

0.25 y = 0.1231x
R² = 0.992
KLa (1/min)

0.2

0.15

0.1

0.05

0
0 0.5 1 1.5 2 2.5 3
T^5.Eρσ .Q^0.82

Fig. 3-7a. (KLa)T plotted against temperature function of the 5th power model

3.4 Discussion and Implications

Modelling for the mass transfer coefficients through the use of image analysis (as is often

the case) of bubble size for diffused aeration is difficult and often system specific with ~15%

inaccuracies (McGinnis et al. 2002; Fayolle et al. 2007), especially if measuring in mixed liquor.

To avoid using bubble size as an input parameter; a mathematical, mechanistic, model has been

developed to predict mass transfer coefficients in deep water tanks when aeration is being

89 | P a g e
performed by the submerged diffused bubble-oxygen transfer mechanism. Using clean water tests

data created by Yunt et al. (1988a), the model formula precisely calculates a uniform value of KLa0

that is independent of tank depth for the standardized (KLa0)20 at 20 °C. This baseline value is

equivalent to the surface KLa that one would obtain from surface aeration, where the effect of gas

depletion is negligibly small.

This chapter has illustrated that, for a set of tanks of different heights subjected to a series

of gas flowrates under different temperatures, they yield different values of (KLa)20. However,

when all the (KLa)20 values are plotted against their respective flowrates Qa20, a good correlation

should be obtained regardless of what the tank heights are. On the other hand, though, when the

baseline (KLa0)20 is plotted against the same Qa20 values, an almost perfect power curve correlation

is obtained. The simulated results for the (KLa)20 for the various runs in the test are given in Table

3-3 below using the standard specific baseline (KLa0)20 of 44.35 x 10-3 (1/min) to predict the

standard mass transfer coefficients, (KLa)20. The results are then compared with the reported values

of the same, given in column 5 (reported) and column 7 (predicted) of the table. The new model

using the concept of a baseline KLa (KLa0) predicts oxygen transfer coefficients to within 1~3%

error compared to observed measurements and around the same for the standard oxygen transfer

efficiency (SOTE%), as shown in Table 3-3 (where p. stands for predicted values and rpt. means

the reported values). It must be remembered that the actual (KLa)20 and C*∞20 were never

measured at 20 0C. The conversion from the test temperature to the standard temperature of 20 0C

for KLa in the LACSD report was based on the Arrhenius model that assumed Ɵ = 1.024 which

may be the reason the error is larger for the data pertaining to 25 0C, since it is known that the

temperature model becomes more inaccurate for temperatures higher than 20 0C (Lee 2017).

Similarly, for the conversion of C*∞, the report relied on the ASCE (2007) method, but the

90 | P a g e
barometric pressure was not reported. The reported values of these standardized parameters may

therefore be imprecise and carry some inherent errors in the estimation of the baseline.

rpt.
p.
run rpt. rpt. p. SOTE p. %err %err
Zd (m)* T (oC) C*∞20
no. C*∞20 (KLa)20 (KLa)20 % SOTE% (SOTE) (KLa)20
(HyePe)
(eqt)
1 2.44 (8) 25 9.87 0.2910 9.68 0.2993 10.09 10.18 0.9 2.8
2 2.44 (8) 25 9.99 0.2228 9.68 0.2288 11.66 11.60 -0.5 2.6
3 2.44 (8) 25 10.05 0.1272 9.68 0.1305 13.03 12.87 -1.2 2.5
1 7.01 (23) 25 11.23 0.2498 11.56 0.2582 24.51 26.08 6.4** 3.2
2 7.01 (23) 25 11.26 0.1853 11.56 0.1915 26.88 28.52 6.1** 3.2
3 7.01 (23) 25 11.54 0.1065 11.56 0.1097 32.02 33.04 3.2 2.9
1 2.44 (8) 25 9.98 0.2232 9.68 0.2288 11.62 11.56 -0.5 2.5
1 3.96 (13) 16 10.50 0.2768 10.31 0.2759 15.46 15.12 -2.2 -0.4
2 3.96 (13) 16 10.54 0.1983 10.31 0.1976 17.18 16.73 -2.6 -0.4
3 3.96 (13) 16 10.63 0.1147 10.31 0.1141 19.47 18.79 -3.5 -0.5
1 5.49 (18) 16 10.80 0.2788 10.93 0.2787 20.90 21.15 1.2 -0.1
2 5.49 (18) 16 11.05 0.1937 10.93 0.1930 22.34 22.03 -1.4 -0.3
3 5.49 (18) 16 11.19 0.1017 10.93 0.1012 25.00 24.31 -2.8 -0.5
Notes: * numbers in brackets are in feet; diffuser depth Zd is two feet off the
tank floor; ** data error, see Table 3-1 footnote; assumptions: e = 0.48 and ye =
0.2; Pv = 2333 N/m2
Symbols: p. = predicted; rpt. = reported; eqt = equation in Yunt’s report.

Table 3-3. Comparison of Predicted and Reported Clean Water Tests Results

3.4.1. Rating curves for aeration equipment

The good prediction of KLa and the subsequent SOTE (standard oxygen transfer efficiency)

is a breakthrough since the correct prediction of the volumetric mass transfer coefficient (KLa) is

a crucial step in the design, operation and scale up of bioreactors including wastewater treatment

plant aeration tanks, and the equations developed allow doing so without resorting to multiple full-

scale testing for each individual tank under the same testing conditions for different tank heights

and temperatures. A family of rating curves for (KLa)20 with respect to depth can thus be

constructed for various gas flow rates applied, such as the one shown below (Fig. 3-8). In the chart,

the rating curves were constructed based on the three average gas flow rates, the individual flow

rates of which vary slightly for each tank depth (See Table 3-1). This has resulted in one tank (the

91 | P a g e
shallowest tank) having a specific KLa higher than the baseline value, but the error is negligibly

small. Although the rating curves in Fig. 3-8 show that the (KLa)20 values are always less than the

baseline (KLa0)20, it is generally accepted that, the deeper the tank, the higher the oxygen transfer

efficiency, all things else being equal. [Houck and Boon 1980][Yunt et al. 1988a, 1988b]

[EPA/625/1-89/023 (1989)]. This is simply because the dissolved oxygen saturation concentration

increases with depth, which offsets the loss in the transfer coefficient in a deep tank. The net result

is therefore still an increase in the overall aeration efficiency. Other clean water studies showed a

nearly linear correlation between oxygen transfer efficiency and depth up to at least 6.1 m (20 ft).

[Houck and Boon 1980]. The rating curves show that, in general, KLa decreases with depth at a

fixed average volumetric gas flowrate. For the gas flowrate of 3.3 m3/min, for example, the profile

is almost linear up to 6 m, which confirms Downing and Boon’s finding [Boon 1979] as mentioned

in the Section 3.1 for the model development.

Rating curves for sp. (KLa)20 vs. various tank depths

46.00
sp. (KLa0)20 and sp. (KLa)20 (min^-1*1000)

45.00

44.00

43.00
3.3 m^3/min
42.00
6.5 m^3/min
41.00
10 m^3/min
40.00 sp. Kla0
39.00

38.00
0.00 2.00 4.00 6.00 8.00 10.00
tank depth (m)

Fig. 3-8. Rating curves for the standard specific transfer coefficients (KLa0 and KLa)20 for various
tank depths and air flow rates

92 | P a g e
 It is interesting to observe that, for deeper tank depths, the trend is not always decreasing,

but actually starts to increase beyond a certain depth. This should be confirmed by further

exploratory testing.

The exponential functional relationship between the mass transfer coefficient and tank

depth presented herewith may explain this previously inexplicable phenomenon.

The predicted standard oxygen transfer efficiency (SOTE) using the simulation model

(Eq. 3-6) can be compared with the actual measured SOTE based on the reported values (Yunt et

al. 1988a). Fig. 3-9 below shows the compared results plotted in ascending order of the tank

depths. Within experimental errors and simulation errors, the results seem to match very well.

Comparison of predicted and reported efficiencies

7.62m
35.0
Aeration Efficiency in Percent

30.0 6.09m
25.0 4.57m
20.0
3.05 m
15.0 p. SOTE

10.0 rpt. SOTE

5.0
0.0
1 2 3 4 5 6 7 8 9 10 11 12 13
Run Numbers in ascending order of increasing Depth

Fig. 3-9. Comparison of the aeration efficiencies simulated vs. actual data

It would appear from the figure that the oxygen transfer efficiency is an increasing function of

depth, even though the gas flow rates were not exactly the same for all the tests.

93 | P a g e
3.5 Potential for future applications

3.5.1. Scaling up

In the application for scaling up, a clean water test must be performed. Most clean water

testing is performed by using the clean water standard developed by the American Society of Civil

Engineers (ASCE/EWRI 2-06). This standard uses a procedure that requires the test water (tap

water) to be deoxygenated and then reaerated with the test diffusers at the appropriate airflow rate.

The Standard was created for full-scale testing, not for small-scale testing to serve the purpose of

scaling up for a project. The Standard urges that similar geometries be used for testing and design;

however, the tank depth can be fixed at 3 m (10 feet) or 5 m (15 feet) or any other depth of choice.

Other differences exist because of the smaller scale. The data should be analyzed in strict

adherence to the Standard; the non-linear estimation procedure was used and the time to complete

the test (~ 4/KLa or 98% of equilibrium) will always be followed. It is preferable to repeat each

test several times to have a constant KLa, and the test is to be repeated for different applied gas

flow rates (average gas flow rate can be calculated from the standard gas flow rate), so that the

KLa vs. Qa relationship can be estimated. Once the baseline KLa is established, Eq. (3-6) can be

applied to find the transfer coefficient at another tank depth. The effect of tank depth on the OTE

is not just due to gas depletion, but also due to the natural volumetric expansion of the bubbles as

they rise to the surface. To solve these complex phenomena, an Excel spreadsheet using the built-

in software Solver is used to solve the simultaneous equations, using the established baseline

parameter KLa0, as well as the actual environmental conditions surrounding the second tank. (See

Table 3-3, but this time KLa0 is a data, and the KLa becomes a variable to be determined). In other

words, the same spreadsheet calculation method is used twice to calculate both KLa0 and KLa. The

94 | P a g e
proposed general procedure for estimating the specific baseline and the standard specific baseline

(KLa0)20 is shown in the flow chart (Fig. 5-1) in Chapter 5.

3.5.2. Translation to in-process oxygen transfer

In the application for wastewater treatment, using the transfer of oxygen to clean water as

the datum, it may then be possible to determine the equivalent bench-scale oxygen transfer

coefficient (KLaf0) for a wastewater system, and the ratio of the two coefficients can then be used

as a correction factor to be applied to fluidized systems treating wastewaters via aerobic biological

oxidation, where microbial respiration has a significantly different contribution to gas depletion

compared to clean water. However, before any mass balance equations can be used to

evaluate this difference in the gas depletion rates, it is paramount to determine alpha (α) where

alpha is the correction factor (Stenstrom et al. 2006) given by:

𝐾𝐿 𝑎𝑓 0 𝐾𝐿 𝑎𝑓
𝛼= ≈ [3 − 16]
𝐾𝐿 𝑎0 𝐾𝐿 𝑎

It is postulated that this correction factor (α) can be determined by bench scale experiments.

KLaf Vs ØZd for KLa0 = 1

1.2
KLB = [1- exp(-αKlao ØZd)]/ØZd
1
KLB = KLa or KLaf

0.8 KLB for α = 1

KLB for α = 0.8
0.6
KLB for α = 0.6

0.4 KLB for α = 0.4

KLB for α = 0.3
0.2 KLB for α = 0.2

0
0 2 4 6 8 10 12
ØZd

Fig. 3-10. The apparent KLaf plotted against ØZd for KLa0 = 1

95 | P a g e
It is hypothesized that this alpha value is not dependent on the liquid depth and geometry of the

aeration basin and the model developed that relates KLa to depth then allows the alpha value to be

used for any other depths and geometry of the aeration basin.

Therefore, using Eq. (3-6), after incorporating α into the mass transfer coefficient for in-process

water, the mass transfer coefficient in in-process water KLaf would be given by:

1 − exp(−𝛷𝑍𝑑 . 𝜶𝐾𝐿 𝑎0 )
KLB = 𝐾𝐿 𝑎𝑓 = [3 − 17]
𝛷𝑍𝑑

(where KLB = KLa for water (α = 1) or KLaf for wastewater (α < 1))

This equation can be plotted for KLaf against the function ØZd for when the baseline is unity, for

various α values, as shown in Fig. 3-10 above.

The use of this equation for in-process water parameter estimations will be the subject of

another paper to be submitted, pending further investigations. However, this subject is discussed

in great length in Chapter 6. This graph of Fig. 3-10 shows exactly what Boon (1979) has found

in his experiments, that KLaf is a declining trend with respect to increasing depth of the immersion

vehicle of gas supply.

3.6 Conclusions

The objective of this paper is to introduce a baseline oxygen mass transfer coefficient

(KLa0), a hypothetical parameter defined as the oxygen transfer rate coefficient at zero depth, and

to develop new models relating KLa to the baseline KLa0 as a function of temperature, system

characteristics (e.g., the gas flow rate, the diffuser depth Zd), and the oxygen solubility (Cs).

Results of this study indicate that a uniform value of KLa0 that is independent of tank depth can be

obtained experimentally. This new mass transfer coefficient, KLa0 is introduced for the first time

in the literature and is defined as the baseline volumetric transfer coefficient to signify a baseline.

96 | P a g e
This baseline, KLa0, has proven to be universal for tanks of any depth when normalized to the same

test conditions, including the gas flow rate Ug, (commonly known as the superficial velocity when

the surface tank area is constant). The baseline KLa0 can be determined by simple means, such as

a clean water test as stipulated in ASCE 2-06.

The developed equation relating the apparent volumetric transfer coefficient (KLa) to the

baseline (KLa0) is expressed by Eq. (3-6).

The standard baseline (KLa0)20 when normalized to the same gas flowrate is a constant

value regardless of tank depth. This baseline value can be expressed as a specific standard baseline

when the relationship between (KLa0)20 and the average volumetric gas flow rate Qa20 is known.

Therefore, the standard baseline (KLa0)20 determined from a single test tank is a valuable parameter

that can be used to predict the (KLa)20 value for any other tank depth and gas flowrate (or Ug

(height-averaged superficial gas velocity)) by using Eq. (3-6) and the other developed equations,

provided the tank horizontal cross-sectional area remains constant and uniform as the bubbles rise

to the surface. The effective depth ‘de’ can be determined by solving a set of simultaneous

equations using a spreadsheet Solver, but, in the absence of more complete data, ‘e’ can be

assumed to be between 0.4 to 0.5 (Eckenfelder 1970).

Therefore, (KLa0)20 can be used to evaluate the KLa in a full-size aeration tank (e.g., an

oxidation ditch with a closed loop flow condition) without having to measure or estimate

numerically the bubble size needed to estimate the KLa for such simulation. However, the proposed

method herewith may require multiple testing under various gas flowrates, and preferably with

testing under various water depths as well, so that the model can be verified for a system. Using

the baseline, a family of rating curves for (KLa)20 (the standardized KLa at 20 oC) can be

constructed for various gas flow rates applied to various tank depths. The new model relating KLa

97 | P a g e
to the baseline KLa0 is an exponential function, and (KLa0)T is found to be inversely proportional

to the oxygen solubility (Cs)T in water to a high degree of correlation. Using a pre-determined

baseline KLa0, the new model predicts oxygen transfer coefficients (KLa)20 for any tank depths to

within 1~3% error compared to observed measurements and similarly for the standard oxygen

transfer efficiency (SOTE%).

Hopefully, the problem with energy wastage due to inaccurate supply of air is ameliorated

and the current energy consumption practice could be improved by applying the models to estimate

the mass transfer coefficient (KLa) correctly for different tank depths at the design stage. As a side,

this analysis appears to support the temperature correction model [Lee 2017] as shown by Fig. 3-

7, showing the excellent regression correlations when the baseline is used in conjunction with the

temperature model. Using the baseline KLa0 is tantamount to using a shallow tank, which is the

fundamental basis for the 5th power temperature model.

Although the mass transfer of oxygen in clean water is well researched and documented in

the literature, its application in wastewater process conditions is not well understood. The

development of Eq. [3-16] and Eq. [3-17] may lead to better relationships between clean water and

process water in the attempt to elucidate the alpha factor (α) which currently appears to be a

complicated function of process variables. The discovery of a standard baseline (KLa0)20 that may

be determined from shop tests for predicting the (KLa)20 value for any other aeration tank depth

and gas flowrate, and even for in-process water with an alpha (α) factor incorporated into the

equation as Eq. [3-17], is important. This finding may be utilized in the development of energy

consumption optimization strategies for wastewater treatment plants. This work may also improve

the accuracy of aeration models used for aeration system evaluations.

98 | P a g e
References

ASCE-2-06. (2007). “Measurement of Oxygen Transfer in Clean Water.” Standards

ASCE/EWRI. ISBN-10: 0-7844-0848-3, TD458.M42 2007

ASCE-18-96. (1997). ``Standard Guidelines for In-Process Oxygen Transfer Testing`` ASCE
Standard.ISBN-0-7844-0114-4, TD758.S73

Baillod, C. R. (1979). Review of oxygen transfer model refinements and data interpretation.
Proc., Workshop toward an Oxygen Transfer Standard, U.S. EPA/600-9-78-021, W.C.
Boyle, ed., U.S. EPA, Cincinnati, 17-26.

Boogerd, F.C., Bos, P., Kuenen, J.G., Heijnen, J.J. and Van der Lans, R.G.J.M., (1990).
Oxygen and carbon dioxide mass transfer and the aerobic, autotrophic cultivation of
moderate and extreme thermophiles: a case study related to the microbial desulfurization
of coal. Biotechnology and bioengineering, 35(11), pp.1111-1119.

Boon A. G. (1979). “Oxygen Transfer in the Activated-Sludge Process” Water Research

Centre, Stevenage Laboratory, England, United Kingdom.

Daniil E. I., Gulliver J.S. (1988). “Temperature Dependence of Liquid Film Coefficient for
Gas Transfer” Journal of Environmental Engineering Oct 1988, 114(5): 1224-1229

Downing, A.A., A.G. Boon. (1968). “Oxygen Transfer in the Activated Sludge Process”, In:
Advances in Biological Waste Treatment, Ed. by W. W. Eckenfelder, Jr. and B.J.
McCabe, MacMillian Co., NY, p. 131

Eckenfelder, W.W., (1952). Aeration efficiency and design: i. measurement of oxygen

transfer efficiency. Sewage and Industrial Wastes, pp.1221-1228.

Eckenfelder (1970). “Water Pollution Control. Experimental procedures for process design.”
The Pemberton press, Jenkins publishing company, Austin and New York.

EPA/600/2-83-102 (1983). “Development of Standard Procedures for Evaluating Oxygen

Transfer Devices” Municipal Environmental Research Laboratory Office of Research
and Development, US Environmental Protection Agency Cincinnati, OH 45268

99 | P a g e
EPA/600/S2-88/022 (1988). “Project Summary – Aeration Equipment Evaluation: Phase I –
Clean Water Test Results” Water Engineering Research Laboratory Cincinnati OH 45268

EPA/625/1-89/023 (1989). “Fine pore aeration systems”, U.S. Environmental Protection

Agency, Office of Research and Development, Center for Environmental Research
Information: Risk Reduction Engineering Laboratory.

Fayolle, Y., Cockx, A., Gillot, S., Roustan, M., & Héduit, A. (2007). Oxygen transfer
prediction in aeration tanks using CFD. Chemical Engineering Science, 62(24), 7163-
7171.

Houck, D.H. and Boon, A.G. (1980). “Survey and Evaluation of Fine Bubble Dome Diffuser
Aeration Equipment”, EPA/MERL Grant No. R806990, September, 1980.

Hunter III, J.S., (1979). Accounting for the effects of water temperature in aerator test
procedures. Proceedings: Workshop Toward an Oxygen Transfer Standard, EPA-600/9-
78 (Vol. 21, pp. 85-9).

Hwang and Stenstrom (1985). Evaluation of fine-bubble alpha factors in near full-scale
equipment. H. J. Hwang, M. K. Stenstrom, Journal WPCF, Volume 57, Number 12,
U.S.A

Jackson, M. L., and Shen, C-C., "Aeration and Mixing in Deep Tank Fermentation
Systems," J. AIChE, 24, l, 63 (1978).

Jiang P. and Stenstrom M. K., (2012). Oxygen Transfer Parameter Estimation: Impact of
Methodology. Journal of Environmental Engineering 138(2):137-142 · February 2012
DOI: 10.1061/(ASCE)EE.1943-7870.0000456

Lakin, M.B., and R.N. Salzman. (1977). “Subsurface Aeration Evaluation.” Paper presented
at the 50th Annual Conference, Water Pollution Control Federation, Philadelphia, 1977.

Lee J. (1978). “Interpretation of Non-steady State Submerged Bubble Oxygen Transfer

Data”. Independent study report in partial fulfillment of the requirements for the degree

100 | P a g e
 of Master of Science (Civil and Environmental Engineering) at the University of
Wisconsin, 1978 [Unpublished]

Lee J. (2017). Development of a model to determine mass transfer coefficient and oxygen
solubility in bioreactors, Heliyon, Volume 3, Issue 2, February 2017, e00248, ISSN
2405-8440, http://doi.org/10.1016/j.heliyon.2017.e00248.

McGinnis, D.F. and Little, J.C., (2002). Predicting diffused-bubble oxygen transfer rate using
the discrete-bubble model. Water research, 36(18), pp.4627-4635.

Metcalf & Eddy, Inc. second edition. (1985) “Wastewater Engineering: Treatment &
Disposal” ISBN 0-07-041677-X

Metzger, I. (1968). Effects of temperature on stream aeration. Journal of the Sanitary

Engineering Division, 94(6), 1153-1160.

Stenstrom et al (2001). “2001” OXYGEN TRANSFER REPORT: CLEAN WATER

TESTING (In accordance with latest ASCE standards) for AIR DIFFUSION SYSTEMS
SUBMERGED FINE BUBBLE DIFFUSERS on MARCH 7, 8, 9, & 10 in 2001”

Stenstrom et al (2006). “Alpha Factors in Full-scale Wastewater Aeration Systems.” 2006

Water Environment Foundation.

Vogelaar, J.C.T., KLapwijk, A., Van Lier, J.B. and Rulkens, W.H., (2000). Temperature
effects on the oxygen transfer rate between 20 and 55 C. Water research, 34(3), pp.1037-
1041.

Yunt F. et al. (1988a). Aeration Equipment Evaluation- Phase 1 Clean Water Test Results.
Los Angeles County Sanitation Districts, Los Angeles, California 90607. Municipal
Environmental Research Laboratory Office of Research and Development.
USEPA, Cincinnati, Ohio 45268.

Zhou, Xiaohong et al. (2012). “Evaluation of oxygen transfer parameters of fine-bubble

aeration system in plug flow aeration tank of wastewater treatment plant” Journal of
Environmental Sciences 2013, 25(2) ISSN 1001-0742 CN

101 | P a g e
Chapter 4. The Lee-Baillod Equation
4.0. Introduction to Derivation of the Lee-Baillod Model

Conceptually, before reaching the saturation state in a non-steady state test, since the

oxygen concentration in the water is less than would be dictated by the oxygen content of the

bubble, Le Chatelier’s principle requires that the process in the context of a bubble containing

oxygen and rising through water with a dissolved-oxygen deficit, relative to the composition of

the bubble, would seek an equilibrium via the net transfer of oxygen from the bubble to the water

[Mott H. 2013]. In this scenario, even for the ultimate steady-state, oxygen goes in and out of the

gas stream depending on position and time of the bubble of the unsteady state test. In clean water,

one can view the mass balances as having two sinks---one by diffusion into water; and the other

by diffusion from water back to the gas stream which serves as the other sink. Whichever is the

greater depends on the driving force one way or the other. At system equilibrium, these two rates

are the same at the equilibrium point of the bulk liquid, the equilibrium point being defined by ‘de’

in ASCE 2-06 [ASCE 2006]. At steady state, the entire system is then at a dynamic equilibrium,

with gas depletion at the lower half of the tank below the ‘de’ level, and gas absorption back to the

gas phase above de; the two movements balancing each other out. The expression applicable to a

stream of gas bubbles undergoing gas transfer in a tank is given by Eq. [4-1]. The standard mass

transfer equation (the Standard Model) is usually written as:

𝑑𝐶
[4 − 1] = 𝐾𝐿 𝑎 (𝐶 ∗ ∞ − 𝐶)
𝑑𝑡

where KLa is defined in the ASCE 2-06 standard as the apparent volumetric mass transfer

coefficient; 𝐶 ∗ ∞ is the determination point of the steady-state DO (dissolved oxygen) saturation

concentration as time approaches infinity. This standard model can be derived by using the

principle of conservation of mass; when C is the dissolved oxygen concentration (mg/L) at time t

102 | P a g e
(min). The mathematical method employs the concept of substantial derivative or the “derivative

following the motion” where the transfer is being observed as the bubble ascends to the surface.

In considering an oxygen balance on a rising bubble, the transfer rate is given by the oxygen flux

integrated over the bubble surface.

4.1. Derivation of the Constant Bubble Volume Model

4.1.1. CONSERVATION OF MASS IN THE GAS PHASE

Hence, the total rate of mass increase is related to the transfer rate by:

𝑑𝑀
= − ∫ 𝐽 . 𝑑𝐴 (4 − 2)
𝑑𝑡 𝐴

where M is the mass of oxygen inside the bubble. 𝐽 is the f1ux. The bar indicates it is a vector

quantity. ‘A’ is the overall interfacial area of the bubble.

The flux N is given by

𝐽 = − 𝐾𝐺 (𝑃∗ − 𝑃𝐺 ) (4-3)

where P* is the saturation gas content corresponding to the dissolved oxygen concentration C

and PG is the partial pressure of oxygen in the gas phase; KG is the overall mass transfer

coefficient in the gas phase.

The flux equation can be expressed based on Dalton’s Law as:

𝐽 = −𝐾𝐺 (𝐶/𝐻 – 𝑦𝑃) (4-4)

where H is Henry’s Law constant and y is the oxygen mole fraction and P is the total pressure.

Therefore, the mass balance equation (eq. 4-2) becomes

𝑑𝑀 𝐶
= − ∫ 𝐾𝐺 (𝑦𝑃 – ) 𝑑𝐴 (4 − 5)
𝑑𝑡 𝐴 𝐻

where M is given by the universal gas law as:

𝑦𝑃𝑉𝐵
𝑀 = (4 − 6)
𝑅𝑇

103 | P a g e
where VB is bubble volume and R is specific gas constant of oxygen.

Applying the concept of substantial derivative, and assuming the flux to be constant over the

bubble area, we have,

𝑑𝑀   𝐶
= (𝑀) + 𝑣𝑏 (𝑀) = −𝐾𝐺 (𝑦𝑃 – ) (𝑎’𝑉𝐵 ) (4 − 7)
𝑑𝑡 𝑡 𝑧 𝐻

where z is the vertical ordinate of the bubble in the tank or the distance through which the bubble

has traveled; a’ is the interfacial area per unit bubble volume, and is assumed constant; and vb is

the velocity of bubble, which can be assumed constant when the radius of the bubble falls within

the following range [McGinnis et al. 2002]:

7.0 x 10-4 < r < 5.1 x 10-3 (4 -- 8)

where r is the radius of bubble in meters. Conceptually the first term on the right in Eq. (4-7)

represents the local time rate of change of mass. This can be neglected because the gas flow rate

is rapid in the liquid column, so that the oxygen content in the gas phase instantaneously present

in the aeration liquid column is small, compared to the total amount of oxygen passing through the

liquid column. In other words, if the bulk aqueous-phase concentration does not change

significantly during the time a bubble takes to rise through the tank, the pseudo-steady-state

assumption may be invoked. The overall gas phase mass transfer coefficient is related to the

individual mass transfer coefficients as

1/𝐾𝐺 = 1/𝑘𝐺 + 1/𝐻 (1/𝑘𝐿 ) (4 -- 9)

where kG and kL are mass transfer coefficients for the gas film and liquid film respectively, in

accordance with the two-film theory [Lewis and Whitman 1924].

Note that for a very soluble gas, H is large and the second term becomes negligible so that

1 1
≈ (4 − 10)
𝐾𝐺 𝑘𝐺

104 | P a g e
Similarly, the overall liquid phase mass transfer coefficient can be related to the individual mass

transfer coefficients as follows:

1 1 1
= + 𝐻 ( ) (4 − 11)
𝐾𝐿 𝑘𝐿 𝑘𝐺

For a slightly soluble gas such as oxygen, H is very small and therefore the second term on the

right-hand side of the equation becomes negligible, so that

1/𝐾𝐿 ≈ 1/𝑘𝐿 (4 -- 12)

When the liquid film controls, kG >>>kL, so that

𝐾𝐺 ≈ 𝐻. 𝐾𝐿 (4 -- 13)

Neglecting the first term and substituting (eq. 4 -- 13) into (eq. 4 -- 7), the equation (eq. 4 -- 7)

reduces to:

𝑑 𝐶
𝑣𝑏. (𝑀) = − 𝐻𝐾𝐿 (𝑦𝑃 – ) (𝑎’𝑉𝐵 ) (4 − 14)
𝑑𝑧 𝐻

Since a’=6/d, where d is the diameter of bubble, and

VB = 4/3πr3 where r is the radius of bubble, therefore,

𝑑
𝑣𝑏 (𝑀) = −𝐾𝐿 (𝐻𝑦𝑃 – 𝐶)(4𝜋𝑟 2 ) (4 − 15)
𝑑𝑧

If M is expressed as the molar flow rate of gas instead of mass, and N is the number flux of

bubbles entering the tank, the above equation can be written as

𝑑 (4𝜋𝑟 2 )𝑁
(𝑀’) = −𝐾𝐿 (𝐻𝑦𝑃 – 𝐶) (4 − 16)
𝑑𝑧 𝑣𝑏

where M’ is the molar gas flow rate of oxygen.

McGinnis, Little et al. (2002) derived a similar equation. [McGinnis et al. 2002]

The relationship between M and M’ is therefore given by:

M’ = dM/dt (4 -- 17)

105 | P a g e
where N is given by:

𝑁 = 𝑄0 /𝑉0 (4 -- 18)

where V0 is the initial bubble volume and Q0 is the actual volumetric gas flow rate at the

diffuser.

Therefore,

𝑄0
𝑀′0 = 𝑦0 𝑃0 (4 − 19)
𝑅𝑇0

where M’0 is the initial molar gas flow rate of gaseous oxygen or nitrogen; y0 is the initial mole

fraction of the gas, P0 is the standard pressure, Q0 is the gas flow rate at standard temperature and

pressure (00 C and 1 bar), R is the ideal gas constant for oxygen, and T0 is the standard

temperature.

and when M’ is a variable,

𝑀’ = 𝑦𝑃𝑄/(𝑅𝑇) (4 -- 20)

1 𝑑 (𝐻𝑦𝑃 – 𝐶)(4𝜋𝑟 2 )𝑁
( ) . (𝑦𝑃𝑄) = −𝐾𝐿 (4 − 21)
𝑅𝑇 𝑑𝑧 𝑣𝑏

Assuming a uniform cross-sectional area of the tank S, then

𝑈𝑔 𝑑(𝑦𝑃) 4𝜋𝑟 2 𝑁 𝐶
𝑆. . = − . 𝐾𝐿 . 𝐻. 𝑃 (𝑦 – ) (4 − 22)
𝑅𝑇 𝑑𝑧 𝑣𝑏 𝐻𝑃

where Ug is the average gas superficial velocity over the tank cross-sectional area S given by Ug =

Qa/S. Here, an important assumption is made: that the bubble volume remains constant as it rises

to the surface, so that r is constant. This assumption is made because of the limitations of the state

of the art of solving calculus, without which the differential equation Eq. (4 -- 22) cannot be solved.

McGinnis and Little [2002] uses numerical integration to solve for both oxygen and nitrogen, in

order to obtain the change in the molar flow rate while the gas bubble is in contact with the water

in the aeration tank. They assumed that vb and KL are functions of r. However, as mentioned

106 | P a g e
before, vb can be assumed constant for a certain range of r, but the assumption that r is constant

with respect to depth is more difficult to justify. This is because the bubble radius increases in

response to decreasing hydrostatic pressure as well as the amount of oxygen and nitrogen

transferred between the bubble and the water. While the latter gas-exchange effect can be ignored

because of the pseudo-steady-state assumption, the first effect from the changes in hydrostatic

pressure cannot be ignored because of Boyle’s Law. The assumption of a constant r is therefore

purely for facilitating the solution of the differential equation to solve the equation mechanistically

instead of by numerical integration which required that the incremental results in the changes of

partial pressure of oxygen and nitrogen within the bubble to be recalculated at incremental steps

as the bubble rises through the tank. However, this assumption of r is applicable to a stream of gas

bubbles of approximately equal bubble volumes in a shallow tank.

First Order Linear Differential Equation

The mathematical derivation of the Constant Bubble Volume model is based on the first order

type of a differential equation,

dy/dx + f(x).y = g(x) (i)

First, let f(x) = a, where a is a given constant, so that

dy/dx +ay = g(x) (ii)

The left side is not an exact differential but can be made so by multiplying by an integrating

factor. If g(x) = 0, equation (ii) has the solution y = Ae-ax, or yeax = A, i.e. d(yeax) = 0. This

suggests that the left side of equation (ii) can be made an exact differential by multiplication by

eax.

Multiplying both sides of equation (ii) by eax (≠ 0) gives

eax dy/dx + aeax y = eax g(x)

107 | P a g e
or

d/dx (yeax) = eax g(x)

Integrating,

yeax = ∫ eax g(x) dx + k

or

y = e-ax ∫ eax g(x) dx + ke-ax (iii)

where k is an arbitrary constant.

Next, consider the more general equation (i). This can be made exact by multiplication by an

integrating factor q(x) to be determined. For let f(x) = dq/dx, or q = ∫f(x)dx, then equation (i)

becomes

dy/dx + dq/dx y = g(x)

Multiplication by eq gives

d/dx (eq y) = eq g(x),

from which on integration,

eq y = ∫ eq g(x) dx + k’ (iv)

where k’ is an arbitrary constant.

Since q = ∫ f(x) dx

is known, y is given in terms of x. Hence, multiplying equation (i) by the integrating factor exp

(∫f(x)dx) makes the left side of equation (i) exactly integrable. Therefore,

From (eq. 4-22),

𝑈𝑔 𝑑(𝑦𝑃) 4𝜋𝑟 2 𝑁 𝐶
𝑆. . = − . 𝐾𝐿 . 𝐻. 𝑃 (𝑦 – ) (4 − 23)
𝑅𝑇 𝑑𝑧 𝑣𝑏 𝐻𝑃

Rearranging,

108 | P a g e
 𝑑(𝑦𝑃) 4𝜋𝑟 2 𝑁 𝑅𝑇 𝐶
= − . 𝐾𝐿 . 𝐻 . 𝑃 (𝑦 – ) (4 − 24)
𝑑𝑧 𝑣𝑏. 𝑈𝑔 𝑆 𝐻𝑃

letting

4𝜋𝑟 2 𝑁 𝑅𝑇
𝛺 = . 𝐾𝐿 . 𝐻 (4 − 25)
𝑣𝑏. 𝑈𝑔 𝑆

Therefore,

𝑑(𝑦𝑃) 𝛺𝐶
+ 𝛺 𝑃𝑦 = (4 − 26)
𝑑𝑧 𝐻

Since both y and P are functions of z, the equation can be expanded as,

𝑦𝑑𝑃 𝑃𝑑𝑦 𝛺𝐶
+ + 𝛺𝑃𝑦 = (4 − 27)
𝑑𝑧 𝑑𝑧 𝐻

Hence,

𝑑𝑦 𝑑(𝑙𝑛𝑃) 𝛺𝐶
+ 𝑦 (𝛺 + )= (4 − 28)
𝑑𝑧 𝑑𝑧 𝐻𝑃

Assuming vb to be constant, this is a first order linear differential equation with non-constant

coefficients just like equation (iv), and letting

𝑑(𝑙𝑛𝑃)
𝑞 = ∫ (𝛺 + ) 𝑑𝑧 (4 − 29)
𝑑𝑧

integrating gives:

𝑞 = 𝛺𝑧 + ln 𝑃 (4 − 30)

letting x = z and g(x) = ΩC/(HP), equation (iv) becomes

𝛺𝐶
𝑦 = exp(−(𝛺𝑧 + ln 𝑃)) [∫ . exp(𝛺𝑧 + 𝑙𝑛𝑃) . 𝑑𝑧 + 𝑘’] (4 − 31)
𝐻𝑃

Simplifying,

𝐶
𝑦𝑃 exp( 𝑧) = exp( 𝑧) + 𝑘’ (4 − 32)
𝐻

where k’ is an integration constant.

109 | P a g e
The boundary condition is that at z = 0, y = Y0, and P = Pd with the datum of origin at the level of

the diffuser orifice. Y0 is the initial oxygen mole fraction at the diffuser depth usually assumed to

be 0.21; Pd is the hydrostatic pressure at the diffuser depth. Using the boundary values, the definite

integral becomes:

𝐶 𝑌0 𝑃𝑑 𝐶
𝑦 = +( – ) exp(−𝑧) (4 − 33)
𝐻𝑃 𝑃 𝐻𝑃

The above equation represents the oxygen mole fraction at any depth z measured from the bottom.

When y is plotted against z, an oxygen mole fraction variation curve is obtained. In the CBVM

(Constant Bubble Volume Model), this mole fraction variation curve is a special case, where the

inert gas in the bubble is continually being vented in such a way that the expansion of the bubble

as it rises to the surface is compensated by the loss of the inert gas, so that the bubble volume

remains constant. The derivative dy/dz in this equation is always positive, indicating that the

equation is an ever-increasing function and it makes sense, since the ever depleting of the inert gas

must correspond to an ever-increasing oxygen mole fraction. For the general case where the inert

gas mole fraction is constant within the bubble, the equation can be adjusted by calibration factors

introduced to the two ‘independent’ variables P and z.

4.1.2. CONSERVATION OF MASS IN THE LIQUID PHASE

For the bulk liquid, the net accumulation rate of dissolved gas in the liquid column is equal

to the gas mass flow rate delivered by diffusion, if there is no gas escape from the liquid column.

(This assumption is not entirely satisfactory because the liquid column may be supersaturated with

the dissolved gas with respect to any gas phase outside the liquid column as C approaches C*∞. In

the case of air aeration, dissolved oxygen from the bulk liquid will flow out to the atmosphere in

addition to the gas stream exit.) The diffusional flux is again given by the two-film theory as:

110 | P a g e
N = KL (C* - C) where C* is the dissolved gas saturation concentration in the liquid phase; KL is

the overall mass transfer coefficient in the liquid phase.

or,

𝑑𝑤
= ∫ 𝐾𝐿 𝑎 (𝐻𝑦𝑃 – 𝐶) 𝑑𝑉 (4 − 34)
𝑑𝑡

but since w = CV, the differential equation becomes

𝑑𝐶 𝐾𝐿 𝑎 𝑍𝑑
= ∫ (𝐻𝑦𝑃 – 𝐶) 𝑑𝑧 (4 − 35)
𝑑𝑡 𝑍𝑑 0

integral being applied from 0 to Zd where Zd is the total depth.

Therefore,

𝐾𝐿 𝑎 𝑡 𝑍𝑑
𝐶 = ∫ ∫ (𝐻𝑦𝑃 – 𝐶) 𝑑𝑧 𝑑𝑡 (4 − 36)
𝑍𝑑 0 0

Substituting y by Eq. (4-33) as derived previously, the first integral with respect to z can be

solved by integration, recognizing that P is also a function of z, resulting in the following:

𝑍𝑑 (𝐻𝑌0 𝑃𝑑 – 𝐶)(1 – exp(−𝑍𝑑 ))
∫ (𝐻𝑦𝑃 – 𝐶) 𝑑𝑧 = (4 − 37)
0 𝑍𝑑

Therefore, substituting Eq. (4-37) into Eq. (4-36), we have

𝑡 (1 – exp(−𝑍𝑑 ))
𝐶 = 𝐾𝐿 𝑎 ∫ (𝐻𝑌0 𝑃𝑑 – 𝐶) 𝑑𝑡. (4 − 38)
0 𝑍𝑑

As explained before, this equation is valid only when the bubble size is constant with depth and

time. In other words, KLa was assumed to be constant with depth and time. In reality, the oxygen

transfer film is affected by several factors, notably changes in pressure, and gas depletion both of

which are functions of depth and time. Taking the parameter KLa out of the integral in eq. 4-35 is

an approximate mathematical treatment only. However, the equation Eq. (4 -- 38) would be

111 | P a g e
approximately true if the aeration tank is shallow, so that the changes in hydrostatic pressure is

small. Defining a baseline KLa as:

lim 𝐾𝐿 𝑎 = 𝐾𝐿 𝑎0 (4 -- 39)
𝑍𝑑→0

The equation can be written as

(1 – exp(−𝑍𝑑 )) 𝑡
𝐶 = 𝐾𝐿 𝑎0 ∫ (𝐻𝑌0 𝑃𝑑 – 𝐶) 𝑑𝑡 (4 − 40)
𝑍𝑑 0

In the differential form, the equation would become

𝑑𝐶 (1 – exp(−𝑍𝑑 ))
= 𝐾𝐿 𝑎0 (𝐻𝑌0 𝑃𝑑 – 𝐶) (4 − 41)
𝑑𝑡 𝑍𝑑

Since in a non-steady state clean water test, the standard model is as stated by eq. 4-1,

𝑑𝐶
= 𝐾𝐿 𝑎 (𝐶 ∗ ∞ − 𝐶) (4 − 42)
𝑑𝑡

Comparing Eq. (4-41) and Eq. (4-42), it is obvious that the two equations would match if

𝐶 ∗ ∞ = 𝐻𝑌0 . 𝑃𝑑 (4 -- 43)

(1 – 𝑒𝑥𝑝(−𝑍𝑑 ))
𝐾𝐿 𝑎 = 𝐾𝐿 𝑎0 (4 − 44)
𝑍𝑑

The above set of equations (eq. 4-43 and eq. 4-44) represents the CBVM (Constant Bubble Volume

Model). The model can be improved if it is recognized that the equilibrium concentration C*∞ does

not saturate at the bottom of the tank at the diffuser depth. Both the equilibrium pressure and the

equilibrium mole fraction of oxygen are different from Eq. (4-43) and so a calibration is required.

Furthermore, Eq. (4-43) can be written as,

𝐶 ∗ ∞ = 𝐻 𝑌0 . (𝜌𝑤 . 𝑔. 𝑍𝑑 + 𝑃𝑏 – 𝑃𝑣𝑡) (4 -- 45)

where ρw is the density of water; g is the gravitational constant, and Pb is the barometric pressure

and Pvt is the vapor pressure at the free surface.

112 | P a g e
4.1.3. DERIVATION OF THE DEPTH CORRECTION MODEL

The CBVM recognized that both parameters (KLa) and (C*∞) are functions of Zd.

Comparing Eq. (4-41) and Eq.(4-42), since they are one and the same equation, it is easy to see

that if it is required to apply a correction factor similar to what Downing and Boon [Downing and

Boon 1968] did to their equation for C*∞ to match with reality, then a corresponding correction

for KLa is similarly required, and the common linkage between these two equations must be the

submergence depth Zd.

The equation for C*∞ (eq. (4-45)) appears to suggest that the saturation equilibrium level

has occurred at the submergence depth Zd. As the submergence depth is not the equilibrium level,

which should occur at an effective depth de, an adjustment must be made in order to correctly

predict the C*∞ value. Since C*∞ and KLa are co-related (approximately inversely proportional to

each other), an adjustment to C*∞ must have a corresponding adjustment to KLa. Bearing in mind

that C*∞ is measured from the surface downward toward the bottom, whereas KLa was derived

based on the travel distance of bubble measured from the bottom to the top, the point of origin of

the parameters’ individual reference frame is not the same but opposite to each other. If the

adjustment to C*∞ is e.Zd, the corresponding adjustment to the submergence depth for KLa must

be (1 – e) where e is the saturation depth correction for Eq. (4-45), given as

𝑑𝑒
𝒆 = (4 − 46)
𝑍𝑑

and Pb is the overburden atmospheric pressure on the liquid column. Therefore, (eq. 4-45)

becomes:

𝐶 ∗ ∞ = 𝐻 𝑌0 (𝜌𝑤 𝑔. 𝒆 𝑍𝑑 + 𝑃𝑏 − 𝑃𝑣𝑡) (4 -- 47)

113 | P a g e
 This equation assumed that the oxygen mole fraction at the equilibrium level is the same

as the initial mole fraction at the bottom Y0, which is 0.21. In reality, as seen in Figure 3-1 in

Chapter 3, the oxygen mole fraction at equilibrium is slightly less than Y0 because of gas depletion

prior to reaching this level, and so to compensate, the equilibrium mole fraction ye must be slightly

smaller than y0 of 0.21 at the true equilibrium level as illustrated by Fig. 3-1. However, for the

purpose of calibrating the model, this error in Ye is often deemed acceptable and Eq. (4-47) is

deemed to be valid [ASCE 2007]. This equation has been used in the current ASCE Standards for

determining ‘de’ based on clean water test results for C*∞.

Another argument is that, since the correction factor ‘e’ is applied to Zd for the calculation

of C*∞, by mathematical induction or de facto implied by the analytic proof (eq. 4-2 to eq. 4-44)

that the standard model (eq. 4-1) is valid for the general case as well, the same correction factor is

applied to Zd in the mathematical equation for KLa because of the hypothesis that KLa0 and Cs are

inversely proportional to each other [Lee 2017]; but the correction factor is (1-e) because of the

point of origin being fixed at the bottom. Therefore, the final model for KLa vs. KLa0 can be

expressed as a variation of the special case of constant bubble equation (Eq. 4-44), as shown below

by eq. 4-48:

(1 – 𝑒𝑥𝑝(− (𝟏 − 𝒆)𝑍𝑑 ))
𝐾𝐿 𝑎 = 𝐾𝐿 𝑎0 (4 − 48)
 (𝟏 − 𝒆)𝑍𝑑

 given by Eq. (4-25) can be expressed in terms of KLa0 by recognizing that a’ = 6/db where db is

the diameter of a spherical bubble (assumed constant), and vb=Ug (a’/a), as well as that

𝑁 = 3𝑄/4𝜋𝑟 3 (4 -- 48a)

Therefore,

114 | P a g e
 𝐻𝑅𝑆𝑇
 = 𝐾𝐿 𝑎0 (4 − 49)
𝑄

where a0 is the interfacial area per unit of liquid volume V (at constant db) given by:

6 𝑍𝑑
𝑎0 = 𝑄 (4 − 50)
𝑑𝑏 𝑣𝑏. 𝑉

where Q is the average gas flow rate of air or Qa, V is the volume of the bulk liquid, and

therefore Eq. (4-48) can be written as

[1 – exp(−𝐾𝐿 𝑎0 𝑥 (1 − 𝑒)𝑍𝑑 )]
𝐾𝐿 𝑎 = (4 − 51)
𝑥(1 − 𝑒)𝑍𝑑

where


𝑥 = (4 − 52)
𝐾𝐿 𝑎0

and is given by

𝑆𝑇
𝑥 = 𝐻𝑅 (4 − 53)
𝑄

where S is the horizontal cross-sectional area assumed to be uniform throughout the liquid

column, R is the specific gas constant for the oxygen gas under transfer. The parameter x is

hereby defined as the gas-flow constant, when the average volumetric gas flow rate is fixed, for a

specific temperature T.

The important assumptions for the derivation of this model are as follows:

• Only one gas is under transfer, all other gases inside bubbles are inert;

• Transferred gas is only slightly soluble so that the liquid film controls;

• the liquid column within the confine of its boundary is well mixed so that the dissolved

gas concentration is uniform;

115 | P a g e
 • The bubbles assumed rising in a uniform column with uniform bubble size and velocity

are corrected to an effective depth (de) by a depth ratio (e);

• Uniform horizontal cross-sectional area in the liquid column;

• The effects of gas hold-up, coalescence of bubbles or breaking up of bubbles, and gas

transfer during the bubble formation stage are ignored. Any gas transfer at the surface of

the liquid column has been ignored.

The effective saturation depth, de, represents the depth of water under which the total pressure

(hydrostatic plus atmospheric) would produce a saturation concentration equal to C*∞ for water in

contact with air at 100% relative humidity. In a clean water test, it is calculated based on ASCE 2-

06 (Section 8.1 and Annex F) [ASCE 2007]. The method given in the ASCE standard, however,

is only an approximation.

Eq. (4-51) can also be written as:

1 – exp(−Ø𝑍𝑑 . 𝐾𝐿 𝑎0 )
𝐾𝐿 𝑎 = (4 − 54)
Ø𝑍𝑑
where

Ø = 𝑥(1 − 𝑒) (4 − 55)

Eq. (4-54) is herewith defined as the Depth Correction Model. Since the parameter Ø is an

adjustment to the gas-flow constant x, it can be defined as the effective gas-flow constant. The

compound parameter Ø𝑍𝑑 is defined as the characteristic depth of the diffuser or the immersion

depth of the aeration device. By rearranging Eq. (4-54), KLa0 can be obtained as:

𝐾𝐿 𝑎0 = −1/(Ø𝑍𝑑 ) {ln(1 – 𝐾𝐿 𝑎(Ø𝑍𝑑 )} (4 − 56)

where KLa (the apparent KLa) can be obtained from a curve fitting to clean water testing data using

the ASCE 2-06 equation as shown here:

𝐶 = 𝐶 ∗ ∞ – (𝐶 ∗ ∞ – 𝐶0 )𝑒𝑥𝑝 (−𝐾𝐿 𝑎 𝑡) (4 − 57)

116 | P a g e
The implicit function (e) in Eq. (4-51) can be solved in a spreadsheet. Solving for (e) depends on

solving the depth of equilibrium level (de). Eq. (4-54) and (4-55) constitute the proposed Depth

Correction Model. It is envisaged that the baseline KLa0 as calculated by Eq. (4-56) will not change

for any depth of tank under a fixed average volumetric gas flow rate (Qa) for a system under testing.

4.1.4. THE HYPOTHESIS OF A CONSTANT BASELINE (KLa0)

As mentioned before, this thesis advances the concept of a constant baseline for the mass

transfer coefficient. For every tank tested using the non-steady state testing method, there is a

baseline mass transfer coefficient that would be constant regardless of the tank height, as long as

it was tested under the same average volumetric gas flow rate Qa.

It would appear from the above derivation that the baseline mass transfer coefficient (KLa0)

is indeed only dependent on the volumetric gas flow rate Qa, although from (eq. 4 -- 44) the

apparent mass transfer coefficient is not only a function of Qa but also a function of depth Zd.

Based on the above calculations, the author advances the hypothesis that, for the same volumetric

average gas flow rate, and the same water temperature and barometric pressure, the baseline mass

transfer coefficient as calculated by (eq. 4-44) would be a constant for any tank depth or diffuser

depth Zd. In other words, a clean water test carried out on a 3.05 m (10-ft) tank would give the

same KLa0 for the test carried out in a 4.57 m (15-ft) tank, or in a 6.09 m (20-ft) tank, or in a 7.62

m (25-ft) tank, as long as the volumetric gas flowrate is kept constant in all tests. A corollary of

this finding is that, given a system of known depth of aeration, and given a supplied gas flow rate,

it would be possible to estimate the mass transfer coefficient (KLa) for any tank depth, without

having to conduct a full-scale clean water test, if the baseline KLa0 is known. However, the

determination of the baseline relies on an accurate determination of the equilibrium point of the

117 | P a g e
saturation concentration, previously assumed to be at mid-depth [Eckenfelder 1970] but can be

more accurately determined by the mole variation equation (Lee-Baillod model) over a tank height.

The point at which the curve gives a minimum mole fraction is the equilibrium point. This can be

determined by differentiation of the equation and at dy/dx = 0, the minimum point is obtained.

The model for the C*∞ equation relies on the estimation of “e”, for a correction term. This value

of “e” estimated in one tank may not be used in a tank of another depth, since no test results have

been presented to substantiate that “e” is more or less constant for tanks of different heights.

Even though the current ASCE clean water standard has mandated an effective-depth

correction, this ASCE correction can only be applied to each individual tank under testing. This

correction varies among different tank sizes, diffuser densities and layouts, and diffuser types, not

to mention different tank depths. This chapter attempts to prove that, for all other conditions having

been fixed, there is a unique relationship between KLa and tank depth, and that for as long as these

conditions are fixed, the effective depth ratio (e) can be proven to be quite constant, as can be seen

in Fig. 3-2 in the last chapter. Since a model has been developed based on the mole fraction

variation along the tank height (Lee-Baillod model), this constant e value can be exploited for

scale-up purposes as approximate solutions. Even though its usefulness in scaling up requires

further study, the constant baseline KLa0 would be useful as a function of comparing aeration

equipment tested under different tank depths, and for evaluation of performance for the purpose

of rating aeration equipment. Furthermore, the baseline mass transfer coefficient can be used to

determine the oxygen transfer coefficient in a wastewater aeration tank, using Eq. (3-17) given in

Chapter 3 by incorporating the alpha factor (α) into the clean water depth correction model (eq. 3-

6), assuming that the in-process water mass transfer coefficient (KLaf0) can be determined on a

bench scale. For in-process water, the model will depend on modifying the clean water gas transfer

118 | P a g e
equations to include the additional gas depletion due to the microbial respiration as explained in

Chapter 6. The application of alpha (α) so determined by bench-scale tests to determine the

corresponding oxygen transfer coefficient in a full-scale situation is discussed in Chapter 5 and

Chapter 6.

4.1.5. DETERMINATION OF CALIBRATION PARAMETERS (n, m) FOR THE LEE-

BAILLOD MODEL

There are two methods to adjust the original equations. One method, by the correction

factor ‘e’, has already been explained. For the other method, since the assumption of constant

bubble volume and therefore constant interfacial area seriously compromises the Lee-Baillod

model equation (Eq. 4-33) for the real case, the variables z and P need to be adjusted. Lee [1978]

and Baillod [1979] postulated that the mole fraction variation curve is not linear, and certainly not

an ever-increasing function as the CBVM (Constant Bubble Volume Model) has predicted. By

assigning a parameter n to all the pressure values, and another parameter m to z, the equation can

be written as:

𝐶 𝑌0 𝑛𝑃𝑑 𝐶
𝑦 = +( – ) exp(−𝐻𝑘. 𝑚𝑧) (4 − 58)
𝑛𝐻𝑃 𝑛𝑃 𝑛𝐻𝑃

where Hk = Ω. Therefore, k = KLa0 RST/Qa or k = KLa0 RT/Ug. The above equation is hereby

defined as the generalized Lee-Baillod Model and k can be defined as a specific baseline constant.

Rearranging the equation gives

𝐶 𝐶
𝑦𝑃 = + (𝑌0 𝑃𝑑 – ) exp(−𝐻𝑘. 𝑚𝑧) (4 − 59)
𝑛𝐻 𝑛𝐻

This equation allows the partial pressure of oxygen in the bubble at any location and at any time

to be expressed. Rearranging Eq. (4 -- 59) to solve for k as:

119 | P a g e
 1 𝑛𝐻𝑌0 𝑃𝑑 – 𝑐
𝐻𝑘 = [ln { }] (4 − 60)
𝑚𝑧 𝑛 𝐻𝑌𝑃 – 𝑐

or,

𝑄𝑎 𝑛𝐻 𝑌0 𝑃𝑑 – 𝑐
𝐾𝐿 𝑎0 = /(𝑚𝑧) [ln { }] (4 − 61)
𝐻𝑅𝑆𝑇 𝑛 𝐻𝑌𝑃 – 𝑐

At the exit gas when t = ∞, c = C*∞, Y (exit gas mole fraction) = Y0, P = Pa (the atmospheric

pressure) and Z = Zd, then

1 𝑛𝐻 𝑌0 𝑃𝑑 – 𝐶 ∗ ∞
𝐾𝐿 𝑎0 = ( ) /(𝑚𝑍𝑑 ) [ln { }] (4 − 62)
𝑥 𝑛𝐻 𝑌0 𝑃𝑎 – 𝐶 ∗ ∞

where x = HRST/Qa; Pa = Pb – Pvt

Rearranging Eq. (4-62) gives

𝑃𝑎 – 𝑃𝑑 exp(−𝑚𝑥. 𝐾𝐿 𝑎0 . 𝑍𝑑 )
𝐶 ∗ ∞ = 𝑛𝐻 × 𝑌0 × (4 − 63)
1 − exp(−𝑚𝑥. 𝐾𝐿 𝑎0 . 𝑍𝑑 )

Eq. (4-58) is similar to Eq. (4-33) and after substituting y by Eq. (4-58) as derived previously into

Eq. (4-36), the first integral with respect to z can be solved by integration w.r.t. z to give,

1 – exp(−𝑚 𝐻𝑘 𝑍𝑑 )
𝐶 ∗ ∞ = 𝐻 𝑌0 𝑃𝑑 𝑛 (4 − 64)
(1 – exp(−𝑚 𝐻𝑘 𝑍𝑑 ) + (𝑛 – 1)𝑚𝐻𝑘 𝑍𝑑
and,

1 – exp(−𝑚𝑥. 𝐾𝐿 𝑎0 . 𝑍𝑑 ) (𝑛 – 1)𝐾𝐿 𝑎0
𝐾𝐿 𝑎 = + (4 − 65)
𝑛𝑚𝑥. 𝑍𝑑 𝑛

In the application of the above equations, one equation (eq. (4-64)) is not valid because of gas

super-saturation at the free surface, where in the derivation any dissolved gas escaping into the

atmosphere was ignored. This will tend to over-estimate C*∞, but the effect of supersaturation on

the mass transfer coefficient is assumed to be small, and so Eq. (4-65) is considered valid, as this

120 | P a g e
parameter is less sensitive to the DO concentration as C approaches C*∞. The equations that can

be used to find the unknown parameters are:

1) oxygen mole fraction variations with depth: eq. (4-58)

2) boundary conditions at the exit: eq. (4-63)

3) integrated form of the mass transfer equation Eq. (4-65)

Therefore, we have basically three equations to solve three unknowns, n, m, and KLa0. However,

the solution does not include the effective depth de, which corresponds to the minimum mole

fraction of the MF (mole fraction) curve at equilibrium. This can be achieved in two ways: plotting

the MF curves using the MF equation (Eq. (4-58)) for a series of DO values upon knowing the n

and m values; or, differentiating this equation and set dy/dz to zero to find the minimum point. In

using the first method, a series of MF curves can be obtained as shown in the following example

[EPA-600/2-83-102], where the tank height is 5.55 m (18.2 ft) and the saturation concentration

was measured to be 11.43 mg/L as shown in Figure 4-2:

20
18
Tank Height from Bottom (ft)

16
14
C=2
12
c=3
10
c=5
8
c=0
6
c=11.43
4
c=9
2
Mole Fraction of Oxygen in gas stream, y
0
0.1750

0.1800

0.1850

0.1900

0.1950

0.2000

0.2050

0.2100

0.2150

Figure 4-2. Off-gas mole fraction for a 5.55 m (18.2-ft) tank and the MF distribution curves

121 | P a g e
From the MF curve at saturation, the minimum mole fraction corresponding to the effective

depth can be determined by graphical interpretation.

The parameters ‘n’ and ‘m’ can be considered as the characteristics of a particular aeration

system at a fixed gas flow rate and can be determined by using a clean water test. The parameters

would also serve as a correction for the underlying assumptions necessary for the model

development, one of which is that any transfer of other gases from the bubbles, particularly

nitrogen, was not incorporated. Bearing in mind that k is a function of the baseline KLa0, this

equation (Eq. (4-58)) will eventually yield the result of the KLa0 in a clean water test as given by

Eq. (4-62).

4.1.6. DETERMINATION OF THE EFFECTIVE DEPTH ‘de’ OR ‘Ze’

From Eq. (4-47)

𝐶 ∗ ∞ = 𝐻 𝑌0 (𝜌𝑤 𝑔. 𝒆 𝑍𝑑 + 𝑃𝑏 − 𝑃𝑣𝑡) (4 -- 66)

This equation assumed that the oxygen mole fraction at the equilibrium level is the same

as the initial mole fraction at the bottom Y0, which is 0.21. In reality, as seen in Chapter 3 Figure

3-1, the oxygen mole fraction at equilibrium is slightly less than Y0 because of gas depletion prior

to reaching this level, and so to compensate, ye must be slightly smaller than 0.21 to substitute for

Y0 in the above equation, at the true equilibrium level. This means e would be slightly bigger than

the value calculated by (Eq. 4-66) when C*∞ is known.

It was mentioned before that there are two methods to measure the system parameters, KLa

and C*∞. Both methods would require the determination of the effective depth de. In EPA/625/1-

89/023 as well as in ASCE 2-06 [ASCE 2007], de was determined based on Eq. (4-66) which

122 | P a g e
assumed a constant mole fraction when steady state is reached. This assumption is an

approximation that can be corrected by the following method with the following assumptions:

• mass balances of the gas absorption and desorption of a rising gas stream give a non-linear

mole fraction curve (Eq. (4-58));

• the driving force is zero at this equilibrium level, so that the determination point C*∞ is the

saturation concentration at this level;

• The mole fraction at this level is at the minimum of the non-linear mole fraction variation

curve which can be determined by differentiating Eq. (4-58) and setting it to zero.

Rigorous mathematical derivation showed that de is a function of environmental constants

(barometric pressure, temperature, density of water), depth of tank, gas flux, KLa, as well as C*∞,

and most importantly the aeration system characteristics.

The following derived equation can be used to demonstrate to the European engineers who

like to adopt a mid-depth correction, that de is in fact a variable, and not necessarily mid-depth.

This equation does not confirm that the saturation level is necessarily less than mid-depth----

caution must be advised against jumping to the conclusion that saturation must occur less than

mid-depth, due to the inherent definition of de. The derivation of the equation to calculate the

minimum mole fraction ye that corresponds to de is given below.

Recalling from Eq. (4-58) that the partial pressure of oxygen in a bubble is related to the

depth of bubble measured as its distance from the diffuser depth, Z, the equation can be

differentiated with respect to Z to find the minimum mole fraction, thus:

Eq. (4-58)

𝐶 𝑌0 𝑛𝑃𝑑 𝐶
𝑦 = +( – ) exp(−𝐻𝑘. 𝑚𝑧) (4 − 67)
𝑛𝐻𝑃 𝑛𝑃 𝑛𝐻𝑃

or

123 | P a g e
 𝐶 𝐶
𝑦𝑃 = + (𝑌0 𝑃𝑑 – ) exp(−𝛺𝑚𝑧) (4 − 68)
𝑛𝐻 𝑛𝐻

where Ω = Hk, then

𝑛𝐻 𝑦𝑃 = 𝐶 + ((𝑛𝐻 𝑌0 𝑃𝑑 – 𝐶)𝑒𝑥𝑝(−𝑚𝑥𝐾𝐿 𝑎0 𝑧) (4 − 69)

𝑑𝑃 𝑑𝑦
𝑛𝐻 (𝑦 + 𝑃 ) = (𝑛𝐻 𝑌0 𝑃𝑑 – 𝐶)(−𝑚𝑥 𝐾𝐿 𝑎) exp(−𝑚𝑥𝐾𝐿 𝑎0 𝑧) (4 − 70)
𝑑𝑧 𝑑𝑧

For the Boundary conditions at the minimum point: dy/dz = 0 when P = Pe, z = Ze, y = Ye, C =

C*∞

(where Ye and Pe are the equilibrium mole fraction and the equilibrium total pressure

respectively.)

Therefore,

𝐾𝐿 𝑎0 𝑚𝑥
𝑌𝑒 = (𝑛𝐻𝑌0 𝑃𝑑 – 𝐶 ∗ ∞ ) exp(−𝑚𝑥𝐾𝐿 𝑎0 𝑍𝑒) (4 − 71)
𝑛𝑟𝑤 𝐻

At equilibrium, according to Dalton’s Law, the saturation concentration in the liquid phase C*∞

is a function of the partial pressure in the gas phase given by:

𝐶 ∗ ∞ = 𝐻 𝑌𝑒 𝑃𝑒 (4 -- 72)

where

H is Henry’s Law Constant. Substituting Ye in Eq. (4-71) into Eq. (4-72), we have

𝑚𝑥𝐾𝐿 𝑎0
𝐶 ∗∞ = (𝑛𝐻𝑌0 𝑃𝑑 – 𝐶 ∗ ∞ ) exp(−𝑚𝑥𝐾𝐿 𝑎0 𝑍𝑒) (𝑃𝑒) (4 − 73)
𝑛𝑟𝑤

Rearranging terms, we have

1 𝑚𝑥𝐾𝐿 𝑎0 𝑛𝐻𝑌0 𝑃𝑑
𝑍𝑒 = {ln (𝑃𝑒 ) + ln ( ∗ − 1)} (4 − 74)
𝑚𝑥𝐾𝐿 𝑎0 𝑛𝑟𝑤 𝐶 ∞

where Pe is given by:

𝑃𝑒 = 𝑃𝑎 + 𝑟𝑤(𝑍𝑑 – 𝑍𝑒) (4 − 75)

124 | P a g e
where

Pa is the net pressure at the free surface given by:

𝑃𝑎 = 𝑃𝑏 − 𝑃𝑣𝑡 (4 − 76)

Ye can then be determined from eq. (4-72) when C*∞ is determined by a clean water test.

Eq. (4-74) gives one more equation to determine the five parameters m, n, KLa0, Ye and Ze.

In the next Chapter, examples are given to show how the KLa0 can be calculated for a set of clean

water test result based on case studies on several experiments carried out by various

investigators.

References

ASCE-2-06. (2007). “Measurement of Oxygen Transfer in Clean Water.” Standards

ASCE/EWRI. ISBN-10: 0-7844-0848-3, TD458.M42 2007
Baillod, C. R. (1979). Review of oxygen transfer model refinements and data interpretation.
Proc., Workshop toward an Oxygen Transfer Standard, U.S. EPA/600-9-78-021, W.C.
Boyle, ed., U.S. EPA, Cincinnati, 17-26.
Eckenfelder (1970). “Water Pollution Control. Experimental procedures for process design.”
The Pemberton press, Jenkins publishing company, Austin and New York.
EPA/600/2-83-102 (1983). “Development of Standard Procedures for Evaluating Oxygen
Transfer Devices” Municipal Environmental Research Laboratory Office of Research
and Development, US Environmental Protection Agency Cincinnati, OH 45268
EPA/625/1-89/023 (1989). “Fine pore aeration systems”, U.S. Environmental Protection
Agency, Office of Research and Development, Center for Environmental Research
Information: Risk Reduction Engineering Laboratory.
Lee J. (1978). “Interpretation of Non-steady State Submerged Bubble Oxygen Transfer
Data”. Independent study report in partial fulfillment of the requirements for the degree
of Master of Science (Civil and Environmental Engineering) at the University of
Wisconsin, 1978 [Unpublished]

125 | P a g e
Lee J. (2017). Development of a model to determine mass transfer coefficient and oxygen
solubility in bioreactors, Heliyon, Volume 3, Issue 2, February 2017, e00248, ISSN 2405-
8440, http://doi.org/10.1016/j.heliyon.2017.e00248.
Lewis W. K., Whitman W. G. (1924). “Principles of Gas Absorption”, Ind. Eng. Chem.,
1924, 16 (12), pp 1215–1220 Publication Date: December 1924 (Article) DOI:
10.1021/ie50180a002
McGinnis, D.F. and Little, J.C., (2002). Predicting diffused-bubble oxygen transfer rate using
the discrete-bubble model. Water research, 36(18), pp.4627-4635.
Mott H.V. (2013). Environmental Process Analysis: Principles and Modelling. John Wiley &
Sons (2013).

126 | P a g e
Chapter 5. Baseline Mass Transfer Coefficients and Interpretation of
Non-steady State Submerged Bubble Oxygen Transfer Data
5.0. Introduction

The clean water standard 2-06 (ASCE 2007) was originally published to provide the

industry with a tool that ensures all manufacturers provide data using the same methodology. The

standard has been used successfully since it was first published in 1992. It has recently been revised

and re-balloted through the consensus standards process to provide an updated standard, which is

expected to be published soon. While variations in equipment in terms of individual performances

are important, these have less adverse impact on parameter estimation than variations in the other

factors, such as, in the past, variations in the test results obtained between test methods, as well as

between different analyses of variance methods for the data. The emphasis on the non-linear least

squares (NLLS) regression analysis method will have greatly re-assured manufacturers to the

standard. The "Log Deficit" method which requires a priori estimation of the equilibrium oxygen

concentration (𝐶 ∗ ∞ ), is expected to be removed from the standard. This deletion will bring data

interpretation and analysis to an even higher degree of accuracy and consistency [Jiang and

Stenstrom, 2012].

However, the effects of other variables, such as temperature, pressure, geometry, etc., are

still requiring deliberations in order to obtain reproducible results and more importantly, in the

application of clean water results to field conditions which are subject to additional whole hosts of

variables affecting oxygen transfer. Therefore, a systematic and progressive elimination of these

effects is the way forward, and the proposals made in this paper appear to be a first step in this

direction.

Page | 127
 Although the clean water standard has fulfilled the purpose of setting a standard of

conformance for manufacturers to use in the measurement of oxygen transfer, especially in the

area of test methods and data interpretation, it falls short in the purpose of compliance testing,

which is in fact the main purpose of the standard. The mass transfer coefficient (KLa or KLaf) is

one of the most important parameters in the water and wastewater treatment technology. [ASCE

1997] [ASCE 2007] (The subscript f stands for any field obtained measurement.) Testing in clean

water eludes the many factors that affect the use of this parameter. One factor particularly for

submerged diffused aeration, for example, is the varying gas-phase gas depletion rate. In in-process

water, care must be taken to ensure that the parameter is not a function of dissolved oxygen

concentration. This dependency can occur where air is blown through diffusers on the bottom of

activated sludge tanks, where rising air bubbles are significantly depleted of oxygen as they ascend

to the water surface. The extent of oxygen depletion is a function of the oxygen concentration in

the activated sludge mixed liquor [Ahlert 1997] [Rosso and Stenstrom 2006].

The dissolved oxygen concentration C in a well-mixed tank is a function of the biological

uptake rate R. Since the effect of this uptake rate on oxygen transfer is additive (negative in the

context of the basic mass transfer model), the attendant gas depletion rate effect on the oxygen

transfer must also be additive. This effect is not associative as a scaling quantity of KLa, with the

use of a scalar factor alpha (α) as is the current practice [Rosso and Stenstrom 2006]. Since the

microbial gas depletion rate arising from microbial respiration comes only from the presence of

microbes, the respiration rate R must equal the gas depletion rate, other minor factors impacting

on the transfer rate notwithstanding. Hence, in a batch process, in order to utilize the clean water

mass transfer coefficient, the gas depletion rate (gdp) due to the microbes must be accounted for;

𝑑𝐶
that the author believes should result in an equation = 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 – 𝐶) − 𝑅 − 𝑔𝑑𝑝 which,
𝑑𝑡

Page | 128
 𝑑𝐶
when equating R with gdp, gives: = 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 – 𝐶) − 2𝑅. This is the only way to take
𝑑𝑡

into account the microbial gas depletion rate effect on the oxygen transfer in the in-process water.

Simply multiplying an alpha factor associative to the mass transfer coefficient does not fully

account for this effect. This is described in Chapter 6 and is the subject of another paper being

published.

Similarly, gas depletion comes from other sources as well, such as tank height and its

associated water depths. In a non-steady state test as described in the standard, the driving force as

derived from the concentration gradient gradually diminishes as the dissolved oxygen increases

until it reaches saturation, and so the gas-side gas depletion rate varies throughout the test [Baillod

1979] [DeMoyer et al. 2002] [McGinnis and Little, 2002] [Schierholz et al. 2006] [Lee 2018].

Naturally, this spectrum of gas depletion would be dependent on the tank height, so that deeper

tanks will have a higher overall gas depletion. Given that KLa is a function of many variables,

including the water depth under test, in order to have a unified test result, it is necessary to create

a baseline mass transfer coefficient, so that all tests will have the same measured baseline. A paper

published by the author [Lee 2018] examines the depth effect by the introduction of this baseline

mass transfer coefficient (KLa0). The baseline would be independent of tank height, since it

measures KLa for a tank of virtually zero height. Manufacturers who can calculate the baseline

based on a series of testing and measurements with different gas velocities should be expected to

produce a uniform constant value of the specific baseline mass transfer coefficient regardless of

the tank heights they use. In this context, unlike the effect of microbes, gas depletion due to this

source (tank height) is eliminated indirectly not so much by incorporating this effect into the

transfer equation, but by changing the evaluation of a mass transfer coefficient to one of zero-

depth tank. The previous paper has explained how this is done based on theoretical development

Page | 129
and numerous testing of data reported in the literature widely available to the public. This has led

to the discovery of several physical mathematical models in nature applicable to the calculation of

gas-phase oxygen mass transfer in water for submerged bubble aeration [Lee, 2017] [Lee, 2018].

In this chapter, additional previously published aeration data by others were re-analyzed by

conducting regression analyses to determine and to verify this concept of a standardized specific

baseline mass transfer coefficient (KLa0)/Qaq so that it can be used to offer a standardized practice

of measurement of oxygen transfer. Only when all the negative effects impacting oxygen transfer

are eliminated can it be confidently proclaimed that the standard is successful, especially in terms

of compliance testing. A baseline is invaluable in this regard.

5.1. Theory

Reports on aeration equipment rely on the basic transfer equation. Of the two main

parameters (KLa and C*∞) pertaining to the standard transfer equation, changing variables affect

both the equilibrium values for oxygen concentration and the rate at which transfer occurs. The

former has been studied extensively over a range of variables, but similar work for the rate

coefficient KLa is less abundant. The more relevant papers on diffused bubble aeration include

McGinnis et al. (2002), whose discrete bubble model forms the basis for the models developed by

the author [Lee 2018]. Other similar work includes McWhirter and Hutter (A.I.Ch.E. J. 35(9)

(1989)) which is the basis for subsequent development by DeMoyer et al. (2003) and Schierholz

et al. (2006) that are now cited in this manuscript in the Discussion section. Works involving this

parameter almost invariably focus on the oxygen transfer rate (OTR) which includes both the

equilibrium concentration and the transfer coefficient together. Ashley et al. (2009) looked at the

effect of air flow rate, depth of air injection, among other things, on the oxygen transfer. Graphs

were plotted to show that KLa is a function of air flow rate, but each depth tested would give a

Page | 130
different unique graph. There is no correlation between graphs of different depths because KLa

would depend on the gas depletion rate which varies with different depths. While 𝐶 ∗ ∞ varies with

depths as governed by Henry’s Law, what is the physical law governing how KLa varies with depth

seems quite unknown. If the author is not mistaken, this is the first time a mechanistic model based

on first principles was ever derived, and it is an exponential function given by eq. 3-6, restated

below as eq. 5-2.

Analysis of bubble aeration depends on average values. Oxygen transfer rate depends on

the average surface area of the bubbles and thus on the mean bubble diameter db. Eckenfelder

[1966] used this to relate to the average gas flow rate [Schroeder 1977]. Since db depends on

temperature and pressure, Qa would require adjustment to temperature and pressure as well,

otherwise the basic transfer model cannot be used correctly. The mass transfer coefficient KLa is

dependent on the gas average volumetric flow rate (Qa) passing through the liquid column [Hwang

and Stenstrom 1985]. Qa is expressed in terms of volume of gas per unit time and is calculated by

the universal gas law, or Boyle’s Law if the liquid temperature is uniform throughout the liquid

column, and taking the arithmetic mean of the flow rates over the tank column. KLa is directly

proportional to this averaged gas flow rate to power q, where q is usually less than unity (for fine

bubble aeration) for water in a fixed column height and a fixed gas supply rate at standard

conditions. This average gas flowrate Qa is determined from the given gas flow at standard

conditions Qs. (The subscript s represents standard). The salient equation to convert Qs to Qa is

given by Lee(2017) and in Chapter 2, eq. 2-25, and restated here as eq. 5-1 below:

1 1
𝑄𝑎 = 172.82 × 𝑄𝑆 × 𝑇𝑃 [ + ] (5 − 1)
𝑃𝑃 𝑃𝑏

where 𝑇𝑃 is the gas temperature at the point of flow measurement, in Kelvin, assumed to be equal

to the water temperature; Pp and Pb are the corresponding gas pressure and the barometric pressure

Page | 131
respectively. (Units are in Pa). This equation has assumed the standard air temperature to be 20 0C

or 293 K, and a standard air pressure of 1.00 atm (101.3 kPa).

The effect of changing depth on the transfer rate coefficient was explored by Yunt [1988a], and,

building on his research, the new depth-correction model relating KLa to the baseline KLa0 [Lee,

2018] is expressed by:

1 − exp(−𝛷𝑍𝑑 . 𝐾𝐿 𝑎0 )
𝐾𝐿 𝑎 = (5 − 2)
𝛷𝑍𝑑

where Φ is a constant dependent on the aeration system characteristics x.(1 – e) and Zd is the

immersion depth of the diffusers, where x and e are defined in Chapter 4 (Eq. 4-53 to Eq. 4-55) as:

x = HR0T/Ug where Ug is the height-averaged superficial gas velocity); R0 is the specific gas

constant for oxygen (note: a different symbol is used to distinguish it from the respiration rate R);

T is the water temperature; e is the effective depth ratio e = de/Zd. Therefore, KLa is an exponential

function of this new coefficient KLa0 and their relationship is given by eq. (5-2), where KLa is

dependent on the height of the liquid column Zd through which the gas flow stream passes. The

error value of KLa was obtained by comparison of the numerical results from model solution and

the experimental data for dissolved oxygen concentration, and it was found that the error is around

1~3% [Lee, 2018]. The full suite of equations [Lee, 2018] derived following from this basic model

is reiterated below as:

[1 – exp(−𝐾𝐿 𝑎0 𝑥 (1 − 𝑒)𝑍𝑑 )]
𝐾𝐿 𝑎 = (5 − 3)
𝑥(1 − 𝑒)𝑍𝑑

𝐶 𝑌0 𝑃𝑑 𝐶
𝑦 = +( – ) exp(−𝑥𝐾𝐿 𝑎0 . 𝑚𝑧) (5 − 4)
𝑛𝐻𝑃 𝑃 𝑛𝐻𝑃

𝑃𝑎 – 𝑃𝑑 exp(−𝑚𝑥. 𝐾𝐿 𝑎0 . 𝑍𝑑 )
𝐶 ∗ ∞ = 𝑛𝐻 𝑌0 (5 − 5)
1 − exp(−𝑚𝑥. 𝐾𝐿 𝑎0 . 𝑍𝑑 )

Page | 132
 1 – exp(−𝑚𝑥. 𝐾𝐿 𝑎0 . 𝑍𝑑 ) (𝑛 – 1)𝐾𝐿 𝑎0
𝐾𝐿 𝑎 = + (5 − 6)
𝑛𝑚𝑥. 𝑍𝑑 𝑛

1 𝑚𝑥𝐾𝐿 𝑎0 𝑛𝐻𝑌0 𝑃𝑑
𝑍𝑒 = {ln (𝑃𝑒 ) + ln ( ∗ − 1)} (5 − 7)
𝑚𝑥𝐾𝐿 𝑎0 𝑛𝑟𝑤 𝐶 ∞

Finally, temperature has an effect on the value of KLa, and the solution for temperature

correction to standard conditions is given by Lee [2017] and in Chapter 2 eq. 2-35 as:

(𝐸𝜌𝜎)20 𝑇20 5 𝑃𝑎𝑇

(𝐾𝐿 𝑎)20 = 𝐾𝐿 𝑎 ( ) (5 − 8)
(𝐸𝜌𝜎) 𝑇 𝑇 𝑃𝑎 20

where E, ρ, σ are properties of the water under aeration. The model has its most precise application

when used for shallow tanks or bench-scale experiments, at atmospheric pressures. All symbols

are given in the Notation and also in previous paper [Lee 2018] or Chapter 2.

5.2. Methodology for depth correction

The following steps lay out the procedure for calculating the specific baseline, and the

flowchart for this procedure is as shown in Fig. 5-1 below:

(i) First, clean water tests (CWT) are to be done for 2 ~ 3 temperatures, preferably one

below 20 0C, one at 20 0C and one above. CWT is also required for 2 different gas flow

rates, so that altogether a minimum of 4 tests are recommended for a tank of adequate

size; and the tank water depth is suggested to be fixed at 3 m (10 feet) or 5 m (15 feet)

up to 7.6 m (25 ft). Tests will be repeated several times (minimum 3 as per ASCE

standard) to have a constant KLa, for each temperature, and the test is to be repeated

for different applied gas flow rates (minimum number of gas flow rates is 2, since the

point of origin constitutes a valid data point) so that the KLa vs. Qa relationship can be

estimated.

Page | 133
 (ii) All diffused aeration systems will experience gas-side depletion as the water depth

increases. This changes in gas-side depletion is dealt with by the Lee-Baillod model

[Lee 2018], allowing calculation of the baseline (KLa0) by solving 5 simultaneous

equations (eq 5-3 to eq 5-7) (using the Microsoft Excel Solver or similar) where KLa0

is a variable to be determined, with the measured KLa and C*∞ as the independent

variables.

(iii) Once the baseline KLa0 for every test is established, a specific baseline can be

determined using the KLa0 versus Qa relationship. For the tests not done at 20 0C and 1

atmosphere (atm) pressure, a temperature correction model to convert the specific

baseline to standard conditions is required. The temperature correction model of choice

is the fifth power model (Lee 2017) as advocated by the author. This standardized value

can be used to find the transfer coefficient at another tank depth. By solving the

simultaneous equations again, but using the established specific baseline parameter

(KLa0)20 as the independent variable, together with the actual environmental conditions

surrounding the scaled-up tank, the KLa for the simulated tank can be found. Hence,

the same set of equations are used twice, to calculate both the KLa0 and the KLa.

(iv) All the measured apparent KLa values can be used to formulate the relationship between

KLa and Qa, but the resultant slopes may have some differences. This should be

compared with the plot of (KLa0)T vs. QaT and also to compare with (KLa0)20 vs. Qa20.

The latter curve should give the best correlation. Likewise, all (KLa0)T values are to be

plotted against their respective handbook solubilities (CS)T.

The specific baseline (KLa0)20/Qa20^q is expected to be constant for all the tanks tested.

From the standardized baseline (KLa0)20 at 20 °C, a family of rating curves for the standard mass

Page | 134
 KLa, C*∞ determined as Calc. baseline (KLa0)T
CWT for 2 variables:
per ASCE 2-06 for each for all the tests by Lee-
T1, T2 and/or (P1, P2) temperature, fixing Qa Baillod model (eq. 5-3
to eq. 5-7)

KLa, C*∞ determined

CWT for 2 flowrates: as per ASCE 2-06 for
each air flowrate
Qs1 Qs2 fixing T or P

Calc. QaT and Qa20

Plot (KLa0)20 Calc. (KLa0)20
(eq. 5-1)
vs. Qa20 using 5th power
model (eq. 5-8)

Plot KLaT and (KLa0)T vs.

QaT for each temp or
pressure tested
Determine slope of plot (q)
and calc. the standard
specific baseline given by:
Determine slopes of sp. (KLa0)20
plots q and calc. =KLa0)20/Qa20^q
average q

Plot (KLa0)T vs.

(Qa^q) (Eρσ)T^5
Normalize KLa
values to Qa^q

Plot normalized KLa

values vs. 1/C*∞ Design for KLa and C*∞ for
any other tank depth, air
flowrate, temperature or
pressure using the standard
Plot standard specific specific baseline and the
Calc. specific KLa = baseline vs. 1/Cs to depth correction model (eq.
KLaT/QaT^q and std. specific confirm validity 5-2)
baseline= (KLa0)20/Qa20^q

Fig. 5-1. Flowchart for calculating the standard specific baseline (KLa0)20

Page | 135
transfer coefficient (KLa)20 can thus be constructed for various gas flow rates applied to various

tank depths using eq. (5-2). Note that the standard specific baseline can also be expressed in terms

of the superficial gas velocity Ug by simply dividing the average gas flowrate Qa by the cross-

section area S of the aeration column. It is difficult to describe a required geometry or placement

for testing conducted in tanks other than the full-scale field facility. According to the ASCE

Standard, appropriate configurations for shop tests should simulate the field conditions as closely

as possible. For example, width-to-depth or length-to-width ratios should be similar. Potential

interference resulting from wall effects and any extraneous piping or other materials in the tank

should be minimized.

The density of the aerator placement, air flow per unit volume (or area), and power input

per unit volume are examples of parameters that can be used to assist in making comparative

evaluations. Notwithstanding these difficulties, the work here is to prove that, for the same

configurations of aerator placement and tank dimensions, the model is able to predict oxygen

transfer efficiency for a range of tank water depths and/or a range of other testing conditions, using

a universal standard specific baseline mass transfer coefficient (KLa0)20. Conventional modeling

uses the gas flow rate at standard conditions (Qs) [Hwang and Stenstrom 1985], but since Qs is in

fact a mass flow rate rather than volumetric, KLa may not correlate well with Qs. For any fixed gas

supply rate of Qs, KLa can be highly variable, such as in Case Study No. 1 described below, where

the major variable on which KLa is dependent is the overhead pressure. There is no relationship

between KLa and Qs, but a strong correlation can be found between the average volumetric gas

flow rate, Qa, as calculated by eq. 5-1, and KLa. The hypothesis advocated is that the baseline KLa0

can be correlated with the volumetric gas flow rate Qa as well. The fact that the mass transfer

coefficient is an exponential function of the baseline means that the relationship between the

Page | 136
former and gas flow rate would be different from that between the baseline and gas flow rate. The

hypothesis in this manuscript is that the latter would constitute a better correlation, as can be seen

in the case studies below, and is a power function.

5.3. Materials and Methods

5.3.1. Case Study 1 - Super-oxygenation tests

1. This research [Barber, 2014] aimed to determine oxygen transfer rates, mass transfer

coefficients, and saturation concentrations in clean water at different overhead pressures for a

sealed aeration column, using air as well as high purity oxygen as the sources of oxygen. The

experimental groups were designed to increase headspace pressures incrementally by 0.5 atm

intervals, up to a pressure of 3 atmospheres, as shown in Table 5-1 below. (Note that LPM

stands for litres per minute.) The aeration apparatus was a clear acrylic tubular column

totaling 5.64 m in height and 238 mm in diameter of a circular cross-section. The horizontal

area is therefore given by S = 0.0445 m2. The column was fitted with a lid and O-ring to

create an air-tight seal. After filling up with the test fluid but leaving a 0.3 m gap at the

headspace, the column was pressurized to a desired pressure at the headspace. Sealed through

the lid were three dissolved oxygen measuring probes---one near the surface of the water, one

at mid-depth, and the third at near the bottom.

Headspace Gas type Gas No. of No. of Total

Pressure flowrate experiments Replicates
(atm) (LPM)
1 Air/Oxygen 4 2 4 8
1.5 Air/Oxygen 4 2 4 8
2 Air/Oxygen 4 2 4 8
2.5 Air/Oxygen 4 2 4 8
3 Air/Oxygen 4 2 4 8
Total 40

Table 5-1. Summary of the experimental design

Page | 137
A temperature probe was fitted at the mid-depth level to measure the water temperature during

each test. For the air/oxygen supply, two 140 micron-air diffusers placed at the bottom and

connected by air-hose flexible pipe 6.4 mm dia. that runs to the top of the column are connected

via a drilled hole in the lid to the aeration feed-gas supply system.

5.3.1.1. Test Result for air aeration tests

The aeration tests were carried out in accordance with The American Society of Civil

Engineers Standard 2-06 [ASCE 2007] that requires a minimum of 3 replicates for non-steady

state reaeration tests. However, in this experiment 4 replicates were provided for each probe, 12

replicates for each test pressure. The reported values for (KLa)20 and C*∞20 are given in column

3 and 5 as shown in Table 5-2. (Note that N is the number of replicate tests). Experiments were

conducted as close to the standard temperature of 20 0C as possible, and the Arrhenius equation

was used to correct to standard conditions whenever an experiment was not conducted at that

temperature. The actual temperatures were not reported, but given the range of temperatures

reported as 100C ~ 200C for the air tests, and 150C ~ 210C for the HPO (high purity oxygen)

tests, the Arrhenius model should be quite accurate; in any case, as the effect of pressures must

be greater than the effect from temperature discrepancies. In Table 5-3, Ye is the oxygen mole

fraction at the equilibrium position de, which is defined in ASCE 2-06 Standard as the effective

depth; e is the ratio of the effective depth to the diffuser depth; and Pe is the equilibrium pressure

at the effective depth. Note that Ye can exceed 0.2095 at higher pressures, and the baseline

becomes closer to the measured KLa simply because the overhead pressure dominates the effect

on KLa rather that the water depth. The increase in Pe comes mainly from the overhead pressure

although the hydrostatic pressure also contributes. In the above suite of equations (eq. 5-3 to eq.

5-7), the initial oxygen mole fraction Y0 is 0.2095 for air, and 0.80 for the HPO tests.

Page | 138
 Zd Probe (KLa)20 (KLa)20 C*∞20 SOTR SAE SOTE N
5.33 m No. (1/hr) (1/min) (mg/L) (kg/hr) (kg/Kwh) (%)
1 atm 1(bot) 6.94(0.2) 0.1157 11.62(0.07) 19.48 1079 26.81 4
2(mid) 6.55(0.1) 0.1092 11.38(0.08) 18.00 996 24.76 4
3(surf) 6.49(0.16) 0.1082 11.13(0.08) 17.45 966 24.01 4
Average 6.66(0.1) 0.1110 11.38(0.07) 18.31 1014 25.19 12

1.5 atm 1(bot) 4.84(0.41) 0.0807 19.2(1.31) 22.11 1078 30.83 4

2(mid) 5.42(0.29) 0.0903 16.82(0.36) 22.01 1073 30.68 4
3(surf) 4.92(0.43) 0.0820 16.17(0.63) 19.36 943 26.95 4
Average 5.06(0.21) 0.0843 17.39(0.60) 21.16 1031 29.49 12

2 atm 1(bot) 4.39(0.30) 0.0732 24.06(1.97) 25.25 1122 35.61 4

2(mid) 4.81(0.34) 0.0802 21.2(0.32) 24.64 1095 34.75 4
3(surf) 3.93(0.49) 0.0655 19.49(1.09) 18.24 811 25.74 4
Average 4.38(0.23) 0.0730 21.58(0.89) 22.71 1009 32.03 12

2.5 atm 1(bot) 4.35(0.08) 0.0725 30.36(0.72) 31.57 1288 44.37 4

2(mid) 3.38(0.50) 0.0563 31.43(2.42) 24.58 1003 34.55 4
3(surf) 2.83(0.15) 0.0472 28.71(1.44) 19.25 786 27.06 4
Average 3.52(0.25) 0.0587 30.17(0.94) 25.13 1026 35.33 12

3 atm 1(bot) 3.59(0.19) 0.0598 34.56(1.13) 29.85 1156 42.67 4

2(mid) 3.17(0.71) 0.0528 29.98(0.65) 22.99 888 32.76 4
3(surf) 3.46(0.42) 0.0577 36.98(3.44) 31.64 1232 45.45 4
Average 3.41(0.26) 0.0568 33.84(1.41) 28.16 1092 40.29 12

Table 5-2. Data of the test results. (Number in parenthesis is +/- standard errors of the mean)
(symbols: bot = bottom; mid = mid-depth; surf = surface)

The baseline values were calculated by solving the simultaneous equations (eq. 5-3 to eq. 5-7).

The calculated values of (KLa0)20 are given in column 4 of Table 5-3. At higher pressures, it is

Page | 139
not surprising that nitrogen gas is no longer inert (i.e., it participates in gas exchange) even

though the water may already contain much nitrogen.

The Data Analysis Result for the Air aeration is given in Table 5-3 below:

Handbook
Test Qs Qa K La 0 Pe
solubility Ye e=de/Zd
Pressure (L/min) (m3/min) (1/min) (N/m^2)
CS (mg/L)
1 atm 4 0.0033 0.1254 9.09 0.2096 0.53
0.1161 9.09 0.2073 0.50
0.1155 9.09 0.2047 0.48
Average 0.1190 9.09 0.2072 0.50 125281

1.5 atm 4 0.0023 0.0898 13.64 0.2095 0.56

0.0911 13.64 0.2136 0.58
0.0890 13.64 0.2093 0.51
Average 0.0900 13.64 0.2108 0.55 181875

2 atm 4 0.0018 0.0724 18.18 0.2259 0.82

0.0808 18.18 0.2114 0.55
0.0750 18.18 0.2025 0.37
Average 0.0761 18.18 0.2133 0.58 230478

2.5 atm 4 0.0015 0.0657 22.73 0.2297 0.97

0.0630 22.73 0.2342 1.06
0.0611 22.73 0.2230 0.82
Average 0.0633 22.73 0.2290 0.95 297622

3 atm 4 0.0012 0.0548 27.27 0.2251 0.93

0.0546 27.27 0.2090 0.49
0.0597 27.27 0.2362 1.07
Average 0.0564 27.27 0.2234 0.83 344773

Table 5-3. Results of calculations of KLa0 (note that S = 0.0445 m2)

This dissolution of nitrogen has an effect on Ye, as the calibration factors n and m will

change in response to nitrogen depletion in the gas stream. In fact, the Constant Bubble Volume

Model (CBVM) was derived based on a particular scenario that the nitrogen gas in the bubble is

being depleted as it rises to the surface, in such a way that the expansion of the bubble volume

Page | 140
due to decreasing pressure is balanced by the reduction in volume due to loss of the nitrogen, so

that the volume remains constant, and the calibration factors (n, m) are unity [Lee 2018]. Fig. 5-2

shows the resulting KLa0 values (assumed to be at 20 0C) plotted against the superficial gas

velocity (also assumed to be at 20 0C), from which the standard specific baseline (KLa0)20/Ug20q

is determined. The value obtained from the slope is 0.840 min-1/(m/min)0.75 for the air aeration.

This standard specific baseline value so determined can then be used to simulate aeration tests

for other conditions as described by Lee (2018). The oxygen-in-air solubility value (CS) in Table

5-3, as stated in column 5, is obtained from Table 2-5 from Chapter 2. Previously, Lee (2017) has

established that there is a definite inverse linear relationship between KLa and C*∞, provided that

the test is done on a very shallow tank. The use of the baseline is tantamount to testing on a very

shallow tank, and so, the same relationship would be expected to hold between KLa0 and

solubility CS. Since the relationship between the baseline and the gas flow rate is a power curve

(Fig. 5-2), the relationship between the baseline and the inverse of solubility (or any other related

function, if it exists, such as pressure) would also be a power curve, as shown by Fig. 5-3,

because the baseline depends on the gas flow rate and varies with it as a power function. (For the

sake of comparison, this plot also includes plotting the mass transfer coefficients KLa against the

same insolubility values and it can be seen the relationship is not as good as the baseline plot

relationship.) The gas flow rate is, in turn, dependent on the hydrostatic pressure at the point of

flow measurement PP (see Eq. 5-1), which is dependent on tank height. Therefore, the specific

baseline (i.e., KLa0 normalized to gas flow rate and pressure) versus the inverse of solubility, Cs,

would form a straight line passing through the origin (as shown in Fig. 5-3a), as both parameters

are then independent of tank height, and so, the inverse law between these two parameters (KLa0

vs. Cs) would be obeyed (Lee 2017). Note that KLa0 is defined at 1 atm pressure only. (Example,

Page | 141
from Table 5-3, KLa0(n) = 0.0611*(.0033/.0015)^.75*(1/2.5) = .044 min^-1 corresponding to CS

=22.73 mg/L or 1/Cs = .044 L/mg).

Fig. 5-3b shows what happens when the saturation concentration is plotted against the

overhead pressure. Barber (2014) also made the same plot as shown in Figure 3.2 in his report. At

the same time, he calculated the saturation concentration using Henry’s Law constant and the

partial pressure at mid-depth, and found that the relationship is not linear. However, because of

gas-sde oxygen depletion, the equilibrium pressure may not be at mid-depth. When C*∞20 is

plotted against Pe the correlation improves as shown in the second curve from top. Furthermore,

Henry’s Law only applies to gas solubility, not to saturation concentration. The bulk liquid is

actually “super-saturated” in aeration, and this saturation value may not relate to pressure in

accordance with Henry’s Law. When the oxygen solubility is plotted against the overhead

pressure, then Henry’s Law would apply and this is shown as a perfect straight in the bottom curve

in Fig. 5-3b. Similar to Figure 3.1 in Barber’s report, the standard mass transfer coefficient is

plotted against the pressure and obtains the similar finding as shown in Fig. 5-3c. However, when

the baseline plot is superimposed on this graph, it can be seen that the correlation between the mass

transfer coefficient (baseline) and the headspace pressure becomes much better. When it is plotted

against the inverse of the pressures, it can be seen that a power curve is obtained as shown in Fig.

5-3d. This confirms the hypothesis of eq. 2-1 in Chapter 2 where it was postulated that KLa is

inversely proportional to Ps but only for shallow tanks, which is equivalent to plotting KLa0 against

the inverse of pressure Ps. When the baseline is normalized to the gas flow rate to power q, a

straight line linear relationship would be obtained for such correlation, (plot not shown but would

be similar to Fig. 5-3a for solubility). Similar findings on the inverse relationship between the mass

transfer coefficient (KLa) and saturation concentration (C*∞), as well as the inverse relationship

Page | 142
between the baseline mass transfer coefficient (KLa0) and pressure (PS), can be found for high

purity oxygen aeration, as shown in Figs. 5-5, 5-5a, and 5-5b.

KLa0 vs. Ug
AIR AERATION
0.1400
baseline mass transfer coeff KLa0 (1/min)

0.1200

0.1000 KLa0 = 0.8404Ug0.753

R² = 0.9951
0.0800

0.0600

0.0400 sp. KLa0 = 0.840 min-1/(m/min)0.75

0.0200

0.0000
0.0000 0.0100 0.0200 0.0300 0.0400 0.0500 0.0600 0.0700 0.0800
superficial gas velocity Ug (m/min)

Fig. 5-2. Standard baseline (KLa0)20 versus average standard gas velocity Ug
for various test pressures

KLa0 vs. 1/Cs

AIR AERATION
Baseline mass transfer coeff KLa0 (1/min)

0.1400

0.1200
KLa0 = 0.5493(1/Cs)0.6905
0.1000 R² = 0.9991
0.0800

0.0600 KLa0
Kla
0.0400

0.0200

0.0000
0 0.02 0.04 0.06 0.08 0.1 0.12
Insolubility 1/Cs (L/mg)

Fig. 5-3. Standard baseline (KLa0)20 versus inverse of solubility 1/Cs for various pressures

Page | 143
 KLa0(np,nQ) vs. 1/Cs
AIR AERATION
0.1400
Baseline mass transfer coeff Kla0(n) (min^-1)
0.1200

0.1000
y = 1.08x
0.0800 R² = 0.9932

0.0600

0.0400

0.0200

0.0000
0 0.02 0.04 0.06 0.08 0.1 0.12
Insolubility (L/mg)

Fig. 5-3a. Standard baseline (KLa0)20 normalized to P and Q versus inverse of solubility
1/Cs for various pressures

saturation concentration vs. pressure

(AIR AERATION C*∞ vs. Pe)
40

35
saturation concentration (mg/L)

y = 0.113x
30 y = 0.0977x
R² = 0.9943
R² = 0.9962
25

20 C*inf v Pa
C*inf v Pe
15
y = 0.0897x
Cs vs Pa
R² = 1
10

0
0 50 100 150 200 250 300 350 400
pressure (kPa)

Fig. 5-3b. Saturation concentration for air for various pressures (Note that Henry’s Law is
given by the third curve from the top)

Page | 144
 Average (KLa)20 vs. overhead pressure Ps
AIR AERATION
8.00

mass transfer coefficient (1/hr) 7.00

6.00 R² = 0.9659

5.00

4.00
Kla20
3.00
(Kla0)20
2.00

1.00

0.00
0 50 100 150 200 250 300 350
overhead pressure (kPa)

Fig. 5-3c. Standard baseline (KLa0)20 compared with mass transfer coefficient (KLa)20 for
various pressures

KLa0 vs. 1/Ps

AIR AERATION
8.00
Baseline mass transfer coefficient (1/hr)

7.00

6.00

5.00

4.00 y = 7.1767x0.69
3.00
R² = 0.987

2.00

1.00

0.00
0 0.2 0.4 0.6 0.8 1 1.2
reciprocal of
pressure in atm.

Fig. 5-3d. Standard baseline (KLa0)20 versus inverse of pressures 1/Ps

Page | 145
5.3.1.2. High Purity Oxygen (HPO) aeration test Result

The reported HPO test values for (KLa)20 and C*∞20 are given in column 3 and 5 as shown

in Table 5-4 below. Table 5-5 shows the analysis result of the HPO tests. Fig. 5-4 shows the

resulting KLa0 values (assumed to be at 20 0C) plotted against the superficial gas velocity (also

assumed to be at 20 0C), from which the standard specific baseline KLao20/Ug20q is determined. The

value obtained from the slope is 1.511 min-1/(m/min)0.92 which is the specific baseline for the HPO

aeration. This value can then be used to simulate aeration tests for other conditions as described

by Lee (2018). As mentioned before, since the relation between the mass transfer coefficient and

the gas flow rate is a power curve, and since the baseline KLa is inversely proportional to oxygen

solubility Cs, the plot of the baseline versus the inverse of solubility would also give a power curve,

as shown in Fig. 5-5. Pure oxygen solubility values in water is given in col. 5 in Table 5-5. Similar

argument can be applied to the HPO tests as to the air aeration tests, giving, therefore, the plots as

shown in Fig. 5-4 and Fig. 5-5, showing the excellent correlation between the baseline and the

reciprocal of oxygen solubility in the latter plot. The relationship is a power curve, as is the

relationship between the baseline and the gas flow rate (Fig. 5-4). As would be expected, the

specific baseline (i.e., KLa0 normalized to gas flow rate and pressure) versus the inverse of

solubility, Cs, would form a straight line passing through the origin (plot not shown), as both

parameters are then independent of tank height, and so, the inverse law between these two

parameters would be obeyed (Lee 2017). For comparison the measured apparent mass transfer

coefficients are also plotted against the inverse of solubility in Fig. 5-5, and it can be seen that the

correlation is not as good compared to the baseline plot, further confirming the model validity for

the baseline.

Page | 146
 Zd Probe (KLa)20 (KLa)20 C*∞20 SOTR SAE SOTE N
5.33 m No. (1/hr) (1/min) (mg/L) (kg/hr) (kg/Kwh) (%)
1 atm 1(bot) 7.85(0.50) 0.1308 51.37(0.32) 97.75 241.0 35.12 4
2(mid) 8.13(0.28) 0.1355 49.63(0.42) 97.89 241.3 35.17 4
3(surf) 8.27(0.30) 0.1378 49.17(0.33) 98.62 243.1 35.43 4
Avg. 8.08(0.20) 0.1347 50.06(0.34) 98.09 241.8 35.24 12

1.5 atm 1(bot) 5.51(0.29) 0.0918 75.30(2.19) 99.67 223.5 38.44 4

2(mid) 5.44(0.55) 0.0907 74.15(1.33) 97.15 218.2 37.40 4
3(surf) 5.52(0.43) 0.0920 71.07(0.65) 94.68 212.6 36.45 4
avg. 5.49(0.23) 0.0915 73.51(0.96) 97.17 218.1 37.43 12

2 atm 1(bot) 4.51(0.28) 0.0752 91.84(1.56) 97.88 204.0 39.97 4

2(mid) 5.38(0.21) 0.0897 87.14(2.33) 110.80 231.0 45.38 4
3(surf) 5.26(0.25) 0.0877 81.05(2.54) 100.60 209.6 41.20 4
5.05(0.17) 0.0842 86.68(1.75) 103.10 214.9 42.18 12

2.5 atm 1(bot) 3.07(0.22) 0.0512 106.0(5.73) 77.76 152.9 32.68 4

2(mid) 4.70(0.26) 0.0783 94.70(5.71) 106.40 208.8 44.57 4
3(surf) 4.53(0.69) 0.0755 91.73(1.99) 100.20 197.7 42.15 4
avg. 4.10(0.26) 0.0683 97.46(2.17) 94.75 186.5 39.80 12

3 atm 1(bot) 4.07(0.07) 0.0678 119.5(3.10) 115.10 217.8 45.59 4

2(mid) 3.02(0.54) 0.0503 123.7(6.78) 85.22 161.6 35.49 4
3(surf) 3.68(0.17) 0.0613 106.3(2.15) 92.58 175.5 36.72 4
Avg. 3.59(0.18) 0.0598 116.5(2.90) 97.63 185.0 39.27 12

Table. 5-4. Data of the pure oxygen test results. (Number in parenthesis is +/- standard errors of

the mean) (note that S = 0.0445 m2)

Page | 147
 Handbook
Test Qs Qa K La 0 Pe
solubility Ye e=de/Zd
Pressure (L/min) (m3/min) (1/min) (N/m^2)
Cs (mg/L)
1 atm 4 0.0033 0.1423 34.71 0.7773 0.99
0.1429 34.71 0.7509 0.99
0.1431 34.71 0.7440 0.99
Avg. 0.1428 34.71 0.7574 0.99 152979

1.5 atm 4 0.0023 0.0987 52.09 0.8534 0.99

0.0989 52.09 0.8404 0.99
0.0995 52.09 0.8055 0.99
Avg. 0.0990 52.09 0.8331 0.99 203641

2 atm 4 0.0018 0.0774 69.42 0.8320 0.99

0.0780 69.42 0.7895 0.99
0.0788 69.42 0.7343 0.99
Avg. 0.0781 69.42 0.7853 0.99 254304

2.5 atm 4 0.0015 0.0693 86.80 0.7999 0.99

0.0658 86.80 0.7146 0.99
0.0661 86.80 0.6922 0.99
Avg. 0.0671 86.80 0.7356 0.99 304966

3 atm 4 0.0012 0.0554 104.13 0.7726 0.98

0.0551 104.13 0.7998 0.98
0.0596 104.13 0.6873 0.98
Avg. 0.0567 104.13 0.7532 0.98 355107

Table 5-5. Results of calculations of KLa0 for the HPO tests (note that S = 0.0445 m2)

In Fig. 5-5a, the mass transfer coefficients are plotted against the inverse of C*∞, which

gives a reasonably good fit, but always inferior to the baseline plot given in Fig. 5-5. Similarly,

when the baseline is plotted against the inverse of pressure, an excellent correlation is obtained, as

shown in Fig. 5-5b. This curve is similar to Fig. 5-3d for the air aeration, thus validating the

proposed model in Chapter 2, as stated by eq. 2-35 for the pressure correction, and also by eq. 5-8

above.

Page | 148
 KLa0 vs. Ug
HPO AERATION
0.1600
Baseline mass transfer coeff KLa0 (1/min)
0.1400

0.1200
KLa0 = 1.511Ug0.9164
0.1000 R² = 0.9926

0.0800 sp. KLa0 = 1.511 min-1/(m/min)0.92

0.0600

0.0400

0.0200

0.0000
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080
Superficial gas velocity Ug (m/min)

Fig. 5-4. Standard baseline (KLa0)20 versus average standard gas velocity Ug

(KLa0)20 vs. 1/Cs

HPO AERATION
0.1600
Baseline mass transfer coeff KLa0 (1/min)

0.1400

0.1200
KLa0 = 2.7029(1/Cs)0.8326
R² = 0.996
0.1000

0.0800
Kla0
0.0600 Kla
0.0400

0.0200

0.0000
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
Insolubility 1/Cs (L/mg)

Fig. 5-5. The Inverse relationship between baseline KLa0 and Solubility 1/Cs for various
pressures

Page | 149
 (KLa)20 vs. 1/C*∞
HPO AERATION
0.1400

0.1200
mass transfer coefficient (1/min)

0.1000

0.0800

0.0600

y = 0.4824x0.6129
0.0400
R² = 0.8557
0.0200

0.0000
0 0.02 0.04 0.06 0.08 0.1
reciprocal of saturation concentration (L/mg)

Fig. 5-5a. Standard mass transfer coefficient (KLa)20 versus inverse of saturation
concentration 1/C*∞ for various pressures

(KLa0)20 vs. 1/Ps

HPO AERATION
0.1600
baseline mass transfer coef (1/min)

0.1400

0.1200 y = 4.5521x0.7542
0.1000 R² = 0.9744
0.0800

0.0600

0.0400

0.0200

0.0000
0 0.002 0.004 0.006 0.008 0.01 0.012
reciprocal of pressure Ps in kPa

Fig. 5-5b. Standard baseline (KLa0)20 versus inverse of pressure 1/PS

Page | 150
5.3.2. Case Study 2 - ADS (Air Diffuser Systems) aeration tests.

Testing was performed by Stenstrom (2001) in a clear acrylic column with a 222 mm (8.75

in.) internal diameter by 4.88 m (16 ft) maximum depth. The depth of the column was varied by

filling it to different heights with tap water to the appropriate depth. Small sections of aeration

tubing with slits were placed at the bottom to create orifices for aeration. The reported test results

for the standard mass transfer coefficient and the saturation concentration are given in Table 5-6,

with the data analysis results and the calculated baseline values for the mass transfer coefficients.

The test water came from Lake Bluff tap water, and since the tests were done in early March, is

expected to be below 20 0C. The tests were performed and the data analyzed in strict adherence to

the ASCE standard, and, therefore, the Arrhenius temperature-correction model was used. The

correction is deemed to be accurate because the model is known to work well for water

temperatures below 200C within the range of 100C and 300C stipulated by the standard.

Two types of ADS Aeration Tubing were tested, one has 6 slits and the other had 14 slits.

Tests were done at three depths: 1.52m (5 feet), 3.05m (10 feet), and 4.57m (15 ft). Three probes

were deployed in the last two depths. Three tests were performed at the design flow rate for each

orifice configuration in order to provide a measure of the precision that is required by the Standard.

Fig. 5-6 shows the resulting KLa0 values (assumed to be at 20 0C) plotted against the superficial

gas velocity (also assumed to be at 20 0C), from which the standard specific baseline KLao20/Ug20q

is determined. The value obtained from the slope is 3.256 hr-1/(m/hr)0.71 or 0.994 min-1/(m/min)0.71

which is the specific baseline for the air aeration. This value can then be used to simulate aeration

tests for other conditions as described by Lee (2018).

Page | 151
 Run water AFR C*∞20 Qa Ug (𝐾𝐿 𝑎0)20 (KL a)20 ye sp.
No. dep(m) (scmh) (mg/L) (m3/h) (1/h) (1/h) (Kla0)20
(m/h)
4 4.57 0.0401 10.61 0.034 0.89 3.16 2.73 0.1979 3.44
1 4.57 0.0802 10.67 0.069 1.77 6.26 5.52 0.1988 4.17
2 4.57 0.0802 10.67 0.069 1.77 6.27 5.53 0.1988 4.18
3 4.57 0.0802 10.79 0.069 1.77 6.20 5.49 0.2002 4.13
5 4.57 0.1604 10.75 0.138 3.55 7.39 6.76 0.1998 3.01
12 3.05 0.0401 10.04 0.036 0.93 3.16 2.86 0.2012 3.34
9 3.05 0.0802 10.08 0.072 1.85 4.69 4.35 0.2022 3.03
10 3.05 0.0802 10.06 0.072 1.85 4.67 4.34 0.2020 3.02
11 3.05 0.0802 10.06 0.072 1.85 4.58 4.27 0.2020 2.96
13 3.05 0.1604 10.13 0.144 3.70 6.70 6.33 0.2029 2.64
23 1.52 0.0401 9.56 0.038 0.97 3.64 3.38 0.2096 3.71
20 1.52 0.0802 9.489 0.076 1.95 5.55 5.27 0.2094 3.46
21 1.52 0.0802 9.473 0.076 1.95 5.52 5.24 0.2092 3.44
22 1.52 0.0802 9.526 0.076 1.95 5.35 5.08 0.2100 3.33
24 1.52 0.1604 9.525 0.151 3.89 7.75 7.45 0.2104 2.95
6 4.57 0.0344 10.94 0.030 0.76 2.31 2.01 0.2019 2.81
7 4.57 0.0344 10.93 0.030 0.76 2.60 2.23 0.2018 3.15
8 4.57 0.0344 10.9 0.030 0.76 2.45 2.11 0.2014 2.97
14 3.05 0.0344 10.18 0.031 0.79 2.69 2.42 0.2032 3.17
15 3.05 0.0344 10.16 0.031 0.79 2.62 2.35 0.2030 3.08
16 3.05 0.0344 10.17 0.031 0.79 2.63 2.37 0.2031 3.10
17 1.52 0.0344 9.504 0.032 0.83 2.81 2.64 0.2091 3.20
18 1.52 0.0344 9.507 0.032 0.83 2.78 2.61 0.2092 3.16
19 1.52 0.0344 9.463 0.032 0.83 2.78 2.61 0.2086 3.16

Table 5-6. Results of calculations of KLa0 (note that S=0.0387 m2)

Since the airflow rate for the 14-slit tubing was 0.0802 scmh (0.0472 scfm), and that for the 6-slit

was 0.0344 scmh (0.0203 scfm), where scmh stands for standard cubic metres per hour, and scfm

stands for standard cubic feet per minute, with the 14-slit additionally tested at 50% and 200% of

the design flow rates to determine the impact of airflow rate on oxygen transfer efficiency, there

may be some discrepancies in the baseline calculations, especially at the higher gas flow rate values.

This probably explains why the data at the top end of the graph shows some anomalies.

Page | 152
 (KLa0)20 vs. Ug20
all slits
9

8 KLa0 = 3.256Ug0.7061
7 R² = 0.9787
Baseline Kla020 (1/hr)

3 sp. KLa0 = 3.256 hr-1/(m/hr)0.706

2

0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
superficial gas velocity Ug (m/hr)

Fig. 5-6. Standard baseline (KLa0)20 versus average standard gas velocity Ug

5.3.3. Case Study 3 - FMC, Norton and Pentech Jet aeration shop tests

The test facility used by Yunt et al. (1988a, 1988b) for all tests was an all steel rectangular

aeration tank located at the Los Angeles County Sanitation Districts (LACSD) Joint Water

Pollution Control Plant. The details of the test have already been described in Chapter 3 sec. 3.3

for the FMC diffusers. This case study is included in this chapter for the purpose of completeness.

The Norton diffusers are fine bubble dome type, and consist of 126 ceramic dome diffusers

mounted on PVC headers. The test results are given in the LACSD report Table 5: “Summary of

Exponential Method Results: FMC Fine Bubble Tube Diffusers” and copied herewith as Table 5-

7 (which is the same as Table 3-1) below. Similarly, Table 5-8 gives the test results for the Norton

diffusers. Table 5-7 and Table 5-8 below are compiled based on data contained in the LACSD

report for FMC and Norton Fine Bubble Diffusers. (A similar table for the Pentech Jet test data

has also been compiled but not shown here). These tables show all the raw data as given in the

Page | 153
LACSD report. The calculations for estimating the variables KLa0, n, m, de and ye are not shown

in this chapter, but the reader is referred to the example calculation of the baseline mass transfer

coefficient (KLa0)T using the model equations in Chapter 3. The simulated result for the FMC

diffusers in this example gives a value of (KLa)20 = 0.1874 min-1 for a typical run test as compared

to the test-reported value of (KLa)20 = 0.1853 min-1 which gives an error difference of around 1%

only comparing to the simulated value.

5.4. Example Calculations

An example has been given in Chapter 3 sec. 3.4 for the FMC diffusers. The test data are

as shown in Table 5-7. Fig. 5-7 (which is the same as Fig. 3-3) shows that the resulting KLa0 values

are adjusted to the standard temperature by the temperature correction equation of the 5th power

model (Lee 2017) and plotted against Qa20. These curves relating KLa0 with Qa for each tank depth

all fitted together after normalizing KLa0 values to 20 0C, as shown in the graph, to form one single

curve. The exponent determined is 0.82. The value obtained from the slope is 0.044 min-
1/(m3/min)0.82 or 0.861 min-1/(m/min)0.82 for all the gas rates normalized to give the best NLLS

(Non-Linear Least Squares) fit, bearing in mind that the KLa0 is assumed to be related to the gas

flowrate by a power curve with an exponent value [Rosso and Stenstrom 2006][Zhou et al. 2012].

The slope of the curve is defined as the standard specific baseline. Therefore, the standard specific

baseline (sp. KLa0)20 is calculated by the ratio of (KLa0)20 to Qa20^.82 or by the slope of the curve

in Fig. 5-7. The graph output for the Pentech Jet is shown in Fig. 5-8a. Again, a standard specific

baseline can be obtained for all the gas rates normalized to give the best NLLS (Non-Linear Least

Squares) fit, since the KLa0 value is related to the gas flowrate by a power curve with an exponent

q [Hwang and Stenstrom 1985][Stenstrom et al. 2006][Zhou et al. 2012].

Page | 154
 FMC diffusers (KLa0)20 vs. Qa20
0.3500

0.3000
y = 0.0444x0.82
0.2500
R² = 1
(KLa0)20 (min^-1)

0.2000

0.1500

0.1000

0.0500

0.0000
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Qa20 (m^3/min)

Fig. 5-7. Calculation of the Standard specific baseline (KLa0)20/Qa200.82 for various test
temperatures and water depths (FMC diffusers)

For the Pentech jets, the resulting baseline (with one outlier removed) is given as 0.0515 min-
1/(m3/min)0.728 or 0.716 min-1/(m/min)0.71. The graph for all the data including the outlier is

given in Fig. 5-8b below. Therefore, it would seem that the methodology for calculating the

baselines has the added benefit of spotting outliers in the testing data, as it seems obvious that the

last data shown in the graph is an outlier. At the very least, it serves as a red flag that this last

data point is questionable, enabling the researcher to re-visit this particular test and perhaps carry

out the test once more to confirm its validity. For the Norton diffusers, the baseline is measured

to be 0.072 min-1/(m3/min)0.80 or 1.305 min-1/(m/min)0.80 as shown in Fig. 5-8.

Page | 155
 Air- Standard
Delivered
Water flow temper Oxygen
Date # Power (KLa)20 (KLa)20 C*∞20
Depth Rate ature Transfer
Density
Qs Efficiency
(hp/1000 T (0C)
m scmh 1/hr 1/min mg/L (%)
ft^3) *
Aug
1 3.05 2.02 700 25.2 17.46 0.2910 9.87 10.06
29,78
Aug
2 3.05 1.16 470 25.2 13.37 0.2228 9.99 11.68
29,78
Aug
3 3.05 0.54 241 25.2 7.63 0.1272 10.05 12.95
29,78
Aug
1 7.62 1.66 704 25.2 14.99 0.2498 11.23 23.93
29,78
Aug
2 7.62 1.07 478 25.2 11.12 0.1853 11.26 24.40
29,78
Aug
3 7.62 0.51 236 25.2 6.39 0.1065 11.54 31.71
29,78
Sep
1 3.05 1.19 472 25.2 13.39 0.2232 9.98 11.61
29,78
Feb
1 4.57 1.81 694 16.2 16.61 0.2768 10.50 15.34
08,79
Feb
2 4.57 1.05 449 16.2 11.90 0.1983 10.54 17.07
08,79
Feb
3 4.57 0.51 231 16.2 6.88 0.1147 10.63 19.87
08,79
Feb
1 6.10 1.74 709 16.2 16.73 0.2788 10.80 20.69
08,79
Feb
2 6.10 1.08 471 16.2 11.62 0.1937 11.05 22.17
08,79
Feb
3 6.10 0.49 224 16.2 6.10 0.1017 11.19 25.04
08,79
*Note: water temperature was deduced from the report statement: “The temperature range used in
the study was 16.2 to 25.2 0C” Reported main data are given in bold; (KLa)20 given in this table is
based on the Arrhenius model using Ɵ=1.024 [ASCE 2007] The 5th power temperature correction
model [Lee 2017] to convert KLa0 estimated to (KLa0)20 and subsequently to (KLa)20 is given by:
(𝐸𝜌𝜎)20 𝑇20 5
(𝐾𝐿 𝑎)20 = 𝐾𝐿 𝑎 ( )
(𝐸𝜌𝜎)𝑇 𝑇
Note that there were some discrepancies in the reported data for the
7-m tank, in that the data for the SOTE% were calculated by an equation in the Report and they did
not match up for two points in the report. These data were discarded and the calculated values using
the LACSD Report’s equations were used, but these data are still suspect. The greater number of tests
are done, the better would be the estimation of the unknown parameters.

Table 5-7. Los Angeles County Sanitation District Report test data (1978) FMC Diffusers

Page | 156
 Air- Standard
Delivered
Water flow temper Oxygen
Date # Power (KLa)20 (KLa)20 C*∞20
Depth Rate ature Transfer
Density
Qs Efficiency
(hp/1000 T (0C)
m scmh 1/hr 1/min mg/L (%)
ft^3) *
Mar
1 7.62 0.28 125 20 5.34 0.0890 11.42 49.48
24,78
Apr
1 3.05 0.57 214 20 11.31 0.1885 9.81 21.30
21,78
Apr
1 3.05 0.32 126 20 7.17 0.1195 9.88 23.20
24,78
Apr
1 4.57 0.31 127 20 6.41 0.1068 10.24 32.03
25,78
Apr
1 4.57 0.54 214 20 9.87 0.1645 10.45 29.71
26,78
Apr
1 4.57 1.24 430 20 17.66 0.2943 10.6 26.61
27,78
May
1 6.10 0.51 216 20 9.47 0.1578 11.12 39.81
04,78
May
1 6.10 1.15 435 20 16.39 0.2732 11.02 33.80
05,78
May
1 7.62 1.16 463 20 14.61 0.2435 11.67 37.16
08,78
May
1 7.62 0.50 217 20 8.54 0.1423 11.65 46.69
09,78
May
1 6.10 0.30 130 20 6.07 0.1012 11.33 43.55
10,78
May
1 3.05 1.37 422 20 19.3 0.3217 10.17 19.14
15,78
May
1 6.10 0.30 127 20 5.82 0.0970 11.44 42.85
16,78
*Note: water temperature was assumed to be 20 C, actual temperatures not reported.
Reported main data are given in bold; (KLa)20 given in this table is based on the Arrhenius model
using Ɵ=1.024 [ASCE 2007]. Since the tests were carried out from March to May, the water
temperature is likely to be 20 C or less, so that the Arrhenius model is likely to be quite accurate.
The greater number of tests are done, the better would be the estimation of the unknown
parameters. In the absence of more data, all the tests are assumed to have been carried out under
standard conditions.

Table 5-8. Los Angeles County Sanitation District Report test data (1978) Norton Diffusers

Page | 157
 Norton diffusers
(KLa0)20 vs. Qa20
0.4

Standardized Baseline (Kla0)20

0.35
0.3 y = 0.0723x0.8017
0.25 R² = 0.9982
0.2
0.15
0.1
0.05
0
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00
Height-weighted gas flow rate at 20 C
(m3/min)

Fig. 5-8. Calculation of the Standard specific baseline (KLa0)20/Qa200.80 for various water
depths (Norton diffusers)

Pentech Jet (EMJA)

KLa0 vs. Qa
0.25

0.2
y = 0.0515x0.7228
R² = 0.9993
Kla0 (1/min)

0.15

0.1

0.05

0
0 1 2 3 4 5 6 7 8 9
Qa20 (m3/min)

Fig. 5-8a. Calculation of the Standard specific baseline (KLa0)20/Qa200.72 for various test
temperatures and water depths (Pentech Jets) w/ outlier removed

Page | 158
 Pentec Jet (EMJA)
KLa0 vs. Qa
0.2500

0.2000 y = 0.0505x0.7128
R² = 0.9971
KLa0 (1/min)

0.1500

0.1000

0.0500

0.0000
0 1 2 3 4 5 6 7 8 9
Qa20 (m3/min)

Fig. 5-8b. Calculation of the Standard specific baseline (KLa0)20/Qa200.72 for various test
temperatures and water depths (Pentech Jets) w/ all data

5.5. Discussion

Rating curves for aeration equipment

The good prediction of the baseline mass transfer coefficient is a breakthrough since the correct

prediction of the volumetric mass transfer coefficient (KLa) using the baseline is a crucial step in

the design, operation and scale up of bioreactors including wastewater treatment plant aeration

tanks, and the equation developed allows doing so without resorting to multiple full-scale testing

for each individual tank under the same testing condition for different tank heights and

temperatures. As mentioned in the Methodology section, a family of rating curves for (KLa)20

with respect to depth can thus be constructed for various gas flow rates applied, such as the one

shown in Fig. 5-9 and 5-10 below.

Page | 159
 Rating curves for FMCdiffusers
sp. (KLa)20 vs. various tank depths
45

sp. (Kla0)20 and sp. (KLa)20 (min^-1*1000)

44.5
44
43.5
Qa
43
42.5 3.4 m3/min
42 6.7 m3/min
41.5
10.1 m3/min
41
40.5 sp. (Kla0)20
40
39.5
0 2 4 6 8 10
tank depth (m)

Fig. 5-9. Rating curves for the standard specific transfer coefficients (KLa0 and KLa)20 for
various tank depths and air flow rates (FMC Diffusers)

In Fig 5-9, the rating curves were constructed based on the three average gas flow rates,

which vary slightly for each tank depth, but the sp. KLa0 has been normalized to a constant gas

flow rate for each curve. (Compare this graph with Fig. 3.8 which has used the test gas flow as is

without any normalization, showing that the shape of the curves is sensitive to gas flow rate.)

As mentioned in Chapter 3, although the rating curves show that the (KLa)20 values are

always less than the baseline (KLa0)20, it is generally accepted that, the deeper the tank, the higher

the oxygen transfer efficiency, all things else being equal [Houck and Boon 1980] [Yunt et al.

1988a, 1988b]. This is simply because the dissolved oxygen saturation concentration increases

with depth, which offsets the loss in the transfer coefficient in a deep tank. The net result is

therefore still an increase in the overall aeration efficiency.

Page | 160
 Rating curves for Norton diffusers
sp. (KLa)20 vs. tank depths
0.072

sp. (KLa0)20 and sp. (KLa)20 (1/min) 0.07

Qa
0.068
~2 m3/min
0.066
~3 m3/min
0.064 ~6 m3/min

0.062 sp.(Kla0)20

0.06
0 2 4 6 8
tank depth (m)

Fig. 5-10. Rating curves for the standard specific transfer coefficients (KLa0 and KLa)20
for various tank depths and air flow rates (Norton Diffusers)

Other clean water studies showed a nearly linear correlation between oxygen transfer

efficiency and depth up to at least 6.1 m (20 ft) [Houck and Boon 1980]. The rating curves show

that, in general, KLa decreases with depth for a fixed average volumetric gas flowrate. For the gas

flowrates in Fig. 5-9, for example, the profile is almost linear up to 4 m, which confirms Downing

and Boon’s finding [Boon 1979]. For Norton diffusers, the curves are linear up to 7 m as shown in

Fig. 5-10.

DeMoyer et al. (2003) and Schierholz et al. (2006) have conducted experiments to show

the effect of free surface transfer on diffused aeration systems, and it was shown that high surface-

transfer coefficients exist above the bubble plumes, especially when the air discharge rate (Qa) is

large. In the establishment of the baselines, care must be taken in selecting the test gas flow rates

and tank geometry such that other such effects would not render the simulation model invalid. The

simulation model has ignored any free surface effects [Lee 2018]. When coupled with large surface

cross-sectional area and/or shallow depth, the oxygen transfer mechanism becomes more akin to

Page | 161
surface aeration where air entrainment from the atmosphere becomes important. In order to make

the model valid, the alternative to a judicious choice of tank geometry and/or gas discharge, is

perhaps another mathematical model that could separate the effect of surface aeration from the

actual aeration under testing in the estimation of the baseline coefficient [DeMoyer et al. 2003]

[Schierholz et al. 2006]. This topic would be the subject of another paper and is briefly discussed

in Chapter 6 Section 6.5.4.

5.6. Justification of the 5th power model over the ASCE method for temperature correction

The advantage of the 5th power model for temperature correction is that it is a base model,

around which other effects can be built on, such as the stirrer speed or rotation speed of impeller,

gas flow rates, geometry, dissolved solids, liquid characteristics, and so forth, which can be

accounted for provided these effects’ relationships with temperature are individually known. Since

the effect of tank height has been accounted for in the study, it was thought that this model would

give a more reliable estimate of (KLa0)20 than the Arrhenius equation. By comparison, the

Arrhenius relation, derived for the temperature dependence of the equilibrium constants of ideal

gas mixtures and shown to fit data for the temperature relationship of many reaction rate constants,

is used empirically for gas mass transfer. However, gas transfer is a diffusion process, not a

chemical reaction. KL is not a reaction rate coefficient and so the Arrhenius equation is not

theoretically based. Therefore, the Arrhenius equation used in this context is purely empirical.

Daniil and Gulliver et al. (1988) have made a comparison between various temperature correction

models, and concluded that the one using properties of water and derived using dimensional

analysis incorporating the Schmidt number and others, has the best similarity with the Ɵ model of

1.0241. Although they recommended replacing the Arrhenius equation with this dimensional

Page | 162
equation, it was also recognized that their favored equation is not universal either, since the

equation has not accounted for the turbulence effect and the effect of KL itself on the value of Ɵ.

Since the test temperature range falls within the ASCE prescribed temperature range (10
0
C ~ 30 0C), the Arrhenius equation using Ɵ=1.024 is also approximately valid. (See Fig. 5-11 for

comparison of various temperature correction models for the FMC diffusers.) However, as the

purpose of this manuscript is to advance a depth correction model, so that one test carried out at a

certain tank depth can be translated to test results for other tank depths, the use of a 5th power

model appears to give the best regression analysis yet to yield the standard baseline (KLa0)20.

The Ɵ parameter attempts to lump all the effects together, and therefore it does not allow

modifications to include other effects, except doing more experiments to suit each case, and

altering the Ɵ value altogether based on these experiments (Lee 2017). The Ɵ parameter is an ‘all-

in’ function of many effects, sometimes including temperature itself as explained in this paper

“Temperature Effects in Treatment Wetlands” [Kadlec and Reddy 2001]. As shown in Fig. 5-11

for the FMC diffusers, the discrepancies between these models in terms of standardizing (KLa0)T

to (KLa0)20 are very small. Table 2-3 in Chapter 2 showed Vogelaar et al.’s data [Vogelaar et al.

2000] and the relationship between the (KLa)T and the inverse of C*∞T is plotted in Fig. 7-1 in

Chapter 7 clearly illustrating the linear relationship between these two parameters for different

temperatures. It should be noted in passing that, Fig. 3-4 in Chapter 3 is irrelevant insofar as the

main theme of the manuscript is concerned; its purpose is to only show that the relationship of the

measured KLa (normalized to gas flow rate) with saturation concentration, although such

relationship exists, is not as good as the same relationship using the baseline KLa (KLa0) instead,

as given in Fig.3-5, which has a higher correlation coefficient R2 = 0.9924 compared to R2 =

Page | 163
0.9859. (Note that when the baseline KLa0 is used, the corresponding saturation concentration is

reduced to surface saturation.)

0.3500

0.3000

0.2500
p. (KLa0)20 (1/min)

0.2000
θ=1.024
0.1500
5th mod
0.1000 θ=1.017

0.0500

0.0000
1 2 3 4 5 6 7 8 9 10 11 12 13
Run No. (consecutive)

Fig. 5-11. Comparison of the standard baseline (KLa0)20 using various temperature models

The farther the tank departs from a shallow tank, the more the deviation is for the mutual

linear correlation between the two parameters (KLa vs. C*∞), according to the model as given by

Eq. (3-6) for the depth correction, shown in Fig. 3-10, which is an exponential function; as contrary

to the solubility (Cs) or the saturation concentration (C*∞) variation model with depth, which is

linear in accordance with Henry’s Law.

5.7. Conclusion

Oxygen is only slightly soluble in water. Therefore, the mass transfer coefficient KLa is

extremely sensitive to the gas depletion rate in a bubble which in turn is highly sensitive to changes

in the factors affecting its depletion. By citing several case studies, this paper has illustrated that,

for each specific submerged bubble aeration equipment, the standard baseline (KLa0)20 at the

standard temperature of 20 0C and standard pressure of 1 atm for a specific feed-gas composition,

when normalized to the same gas flowrate Qa is a constant value regardless of tank depth, test

Page | 164
water temperature and overhead test pressure. This baseline value can be expressed as a specific

standard baseline when the relationship between (KLa0)20 and the average volumetric gas flow rate

Qa20 is known. The model has been tested, based on the experimental reports by several researchers,

for a range of superficial gas flow rates (0.08 m/min for the pilot scale tests as in Study Cases 1

and 2 to 0.18 m/min for the full-scale shop tests as in Case Study No. 3), with the shop tests tank

depths ranging from 3 m (10 ft) to 7.6 m (25 ft) under identical diffuser placement density. The

model should be valid for these ranges of gas flow rates and tank heights. Therefore, the standard

baseline (KLa0)20 determined from a single test tank is a valuable parameter that can be used to

predict the (KLa)20 value for any other test variables and gas flowrate (or height-averaged

superficial gas velocity, Ug) by using the proposed model equations, provided the tank horizontal

cross-sectional area remains constant and uniform as the bubbles rise to the surface.

Case Major Sp. Exponent

Gas type Diffuser Type
No. Variables KLa020 q
Pressure, Qa,
140-micron
1 Air gas 0.840 0.75
diffuser
composition
Pressure, Qa,
140-micron
Oxygen gas 1.511 0.92
diffuser
composition
Slitted ADS
2 Air Height, Qa 0.994 0.71
tubing
FMC fine pore Temperature,
3 Air 0.861* 0.82
tubing height, Qa
Norton fine Temperature,
Air 1.305 0.80
dome height, Qa
Pentech Temperature,
Air 0.716 0.73
Jet height, Qa
*Note: this calculated value was obtained from Chapter 3

Table 5-9. Summary of the test results for standardized specific baseline mass transfer
coefficient (KLa0)/Ugq

Page | 165
 All the test results from the three experimenters are summarized in Table 5-9 below. The

effective depth ‘de’ can be determined by solving a set of simultaneous equations but, in the

absence of more complete data, ‘e’ can be assumed to be between 0.4 to 0.5 (Eckenfelder 1970)

for ordinary air feed gas which has an oxygen mole fraction of 21%. For HPO, e can approach

unity due to the higher driving force at bubble release as can be seen in Table 5-5 in Case Study

No. 1. Yunt’s experiments, which were carried out in large tanks that resemble full-scale, further

demonstrate that simulation and translation from one test to another is possible, with an error of

not more than 3% in the estimation of KLa. The examples provided in this paper proved that the

concept of a constant baseline for an aeration equipment is true for the range of gas flows and

water depths tested. However, the tests were carried out with the tank horizontal cross-sectional

area remaining constant and uniform as the bubbles rise to the surface for each test. Further testing

is required on whether the same is true for tanks of different cross-sections for the same aerator,

i.e., whether pilot test results can be translated to shop tests and to full-scale. For example, it would

be interesting to find out if the specific baseline determined from Case 1 or 2 can extrapolate to

the same results on Case 3 using the same aerator. The comparison graph in Fig. 5-12 below shows

that the Norton diffuser is obviously superior since it has the highest baseline. However, when

comparing the Pentech Jet and the FMC fine pore diffusers, it can be concluded that there is little

to choose between the two for gas flow rates below 4.3 m3/min, but the FMC would become more

superior if the design gas flow is beyond this gas flow rate. (Care must be taken in selecting the

test gas flow rates and tank geometry such that other effects would not render the simulation model

invalid. DeMoyer et al. (2003) and Schierholz et al. (2006) have conducted experiments to show

the effect of free surface transfer on diffused aeration systems, and it was shown that high surface-

transfer coefficients exist above the bubble plumes, especially when the air discharge (Qa) is large.)

Page | 166
 (Kla0)20 vs. Qa20
Norton vs. Pentec vs. FMC
Standardized Baseline (KLa0)20 0.4

0.35 y = 0.045x0.8164
R² = 0.9984
0.3 y = 0.0728x0.8065
R² = 0.9991
0.25

0.2 Norton

0.15 Pentec
y = 0.0515x0.7228
0.1 R² = 0.9993 FMC

0.05

0
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Qa20 (Height-weighted gas flow rate at 20 C) (m3/min)

Fig. 5-12. Baselines between Norton, Pentech jet, and FMC diffusers, data Yunt [1988a]

This good accuracy for estimating the baseline enables the production of rating curves for

the aeration equipment under various operating conditions that include tank depths, pressures,

temperatures as well as different gas flow rates. Therefore, given its importance, (KLa0)20 should

be expressed as an important parameter estimation to comply with the current standard. It can be

used to evaluate the KLa in a full-size aeration tank along with the standard procedures covering

the measurement of the oxygen transfer rate by any such submerged systems. The use of this

parameter to determine KLaf, and whether the biological uptake rate R should be incorporated into

the mass transfer equation as postulated in the Introduction, requires further investigation and is

discussed in the next chapter in Chapter 6.

Page | 167
5.8. Notation (major symbols)

𝐶 ∗∞ oxygen saturation concentration in an aeration tank (mg/L)

𝐶 ∗∞f oxygen saturation concentration in an aeration tank under field conditions (mg/L)
𝐶 ∗ ∞ 20 oxygen saturation concentration in an aeration tank at 20 0C (mg/L)

Cs oxygen saturation concentration in an infinitesimally shallow tank;

also known as oxygen solubility in water at test water temperature and
barometric/headspace pressure; for open tank, also known as DO surface
saturation concentration---handbook tabular values as a function of barometric
pressure and water temperature (mg/L)
C dissolved oxygen (DO) concentration in a fully mixed aeration tank (mg/L)
DO dissolved oxygen in water (mg/L)
E modulus of elasticity of water at atmospheric pressure, E/106 (kN/m2)
ρ density of water (kg/m3)
σ surface tension in contact with air (N/m)
R microbial oxygen respiration rate (mg/L/(m3-hr))
R0 specific gas constant for oxygen (kJ/(kg-K))
R2 curve fitting correlation coefficient
KLa volumetric mass transfer coefficient also known as the apparent volumetric mass
transfer coefficient (min-1 or hr-1)
(KLa)20 standard volumetric mass transfer coefficient (min-1 or hr-1)
KLaf mass transfer coefficient as measured in the field (min-1 or hr-1)
KLa0 baseline mass transfer coefficient, equivalent to that in an infinitesimally shallow
tank with no gas side oxygen depletion (min-1 or hr-1)
(KLa0)20 standard baseline mass transfer coefficient, at zero gas side oxygen depletion
(min-1 or hr-1)
z depth of water at any point in the tank measured from bottom (m)
Zd submergence depth of the diffuser plant in an aeration tank (m)
Ze equilibrium depth at saturation measured from bottom (m)
Pa, Pb atmospheric pressure or barometric pressure at time of testing (kPa; Pa)

Page | 168
Pe equilibrium pressure of the bulk liquid of an aeration tank defined such
that: Pe = Pa + rw de -Pvt where Pvt is the vapor pressure and rw is the specific
weight of water in kN/m3 or N/m3 (kPa; Pa)
de effective saturation depth at infinite time (m)
e effective depth ratio (e = de/Zd)
Ye oxygen mole fraction at the effective saturation depth at infinite time
Y0, Yd initial oxygen mole fraction at diffuser depth, Zd, also equal to exit gas mole
fraction at saturation of the bulk liquid in the aeration tank, Y0 = 0.2095 for air
aeration, Y0 = 0.80 for HPO (high purity oxygen) aeration
H Henry’s Law constant (mg/L/Pa) defined such that:
𝐶 ∗ ∞ = HYePe or Cs = HY0Pa
Yex exit gas or the off-gas oxygen mole fraction at any time
y oxygen mole fraction at any time and space in an aeration tank defined by an
oxygen mole fraction variation curve
Qa height-averaged volumetric air flow rate (m3/min or m3/hr)
Qa20 height-averaged volumetric air flow rate at 200C (m3/min or m3/hr)
Qs, AFR gas (air) flow rate at standard conditions (20°C for US practice and 0°C for
European practice), in (std ft3/min or Nm3/hr)
S cross-sectional area of aeration tank (m2)
Ug superficial gas velocity given by Qa/S (m/min; m/hr)
sp. KLa0 specific baseline mass transfer coefficient, KLa0/Ugq also expressed as KLa0/Qaq
sp. (KLa0)20 standard specific baseline mass transfer coefficient, KLa0/Ugq also expressed as
(KLa0)20/Qaq
V volume of aeration tank given by S.Zd (m3)
𝑇𝑃 gas temperature at the point of flow measurement, Kelvin, assumed to be equal to
the water temperature
Pp and Pb the corresponding gas pressure and the barometric/headspace pressure
respectively to 𝑇𝑃 (Pa)
T Test water temperature in degree Celsius or in Kelvin
Ts standard air temperature 20 0C or 293 K
Ps standard air pressure of 1.00 atm (101.3 kPa); or overhead pressure

Page | 169
Pd Total pressure at diffuser depth (kPa)
n, m calibration factors for the Lee-Baillod model equation for the oxygen mole
fraction variation curve
Ø, x system variables for eq. 5-2 and eq. 5-3 [x = HR0T/Ug; Ø = x.(1 – e)]
Ɵ.𝜏. ß. 𝛺 temperature, solubility, salinity, pressure correction factors as defined by the
standard [ASCE 2007]

References

Ahlert, R. C. 1997. Process dynamics in environmental systems.Edited by J. W. Walter, Jr. and A.

D. Francis, 1-943. New York: Wiley. https://doi.org/10.1002/ep.3300160107.
American Society of Civil Engineers (2007). Standard 2-06: Measurement of oxygen transfer in
clean water. Reston, VA.
American Society of Civil Engineers (1997). Standard guidelines for in-process oxygen transfer
testing, 18-96. Reston, VA.
Ashley Ken I., Mavinic Donald S., Hall K.J. (first name n/a) (2009) Effect of orifice diameter,
depth of air injection, and air flow rate on oxygen transfer in a pilot-scale, full lift,
hypolimnetic aerator. Can. J. Civ Eng. 36: 137-147, NRC Research Press.
Baillod, C. R. (1979). Review of oxygen transfer model refinements and data interpretation.
Proc., Workshop toward an Oxygen Transfer Standard, U.S. EPA/600-9-78-021, W.C.
Boyle, ed., U.S. EPA, Cincinnati, 17-26.
Barber Tyler W. (2014). Superoxygenation: Analysis of oxygen transfer design parameters using
high purity oxygen and a pressurized aeration column. University of British Columbia.
Boon, Arthur G. (1979). Oxygen transfer in the activated-sludge process. Water Research Centre,
Stevenage Laboratory, U.K.
Daniil E. I., Gulliver J.S. (1988). “Temperature Dependence of Liquid Film Coefficient for
Gas Transfer” Journal of Environmental Engineering Oct 1988, 114(5): 1224-1229
DeMoyer Connie D., Schierholz Erica L., Gulliver John S., Wilhelms Steven C. (2002). Impact
of bubble and free surface oxygen transfer on diffused aeration systems. Water Research
37 (2003) 1890-1904.
Eckenfelder, Wesley W., Jr. (1966). Industrial water pollution control (New York: McGraw-Hill).

Page | 170
Eckenfelder, Wesley W. (1970). Water pollution control: experimental procedures for process
design (Austin: Pemberton Press).
EPA/600/S2-88/022 (1988). Project summary: aeration equipment evaluation. Phase I: Clean
water test results. Water Engineering Research Laboratory, Cincinnati, OH.
Houck, D. H., Boon, Arthur G. (1980). Survey and evaluation of fine bubble dome diffuser
aeration equipment. EPA/MERL Grant No. R806990.
Hwang, Hyung J., Stenstrom, Michael K. (1985). Evaluation of fine-bubble alpha factors in near
full-scale equipment. Journal Water Pollution Control Federation 57(12).
Jiang, Pang; Stenstrom, Michael K. (2012). Oxygen transfer parameter estimation: impact of
methodology. Journal of Environmental Engineering 138(2):137-142.
Kadlec, R. H., & Reddy, K. R. (2001). Temperature effects in treatment wetlands. Water
Environment Research, 73(5), 543-557.
Lee, Johnny (2017). Development of a model to determine mass transfer coefficient and oxygen
solubility in bioreactors. Heliyon 3(2): e00248.
Lee, Johnny (2018). Development of a model to determine the baseline mass transfer coefficient
in bioreactors (Aeration Tanks). Water Environment Res., 90, vol. 12, 2126
McGinnis, Daniel F., Little, John C. (2002). Predicting diffused-bubble oxygen transfer rate
using the discrete-bubble model. Water Research 36(18):4627-4635.
McWhirter John R., Hutter Joseph C. (1989). Improved oxygen mass transfer modeling for
diffused/subsurface aeration systems. AIChE J 1989;35(9):1527-34
https://doi.org/10.1002/aic.690350913
Rosso D. & Stenstrom M. (2006). Alpha Factors in Full-scale wastewater aeration systems.
Water Environment Foundation. WEFTEC 06.
Schroeder, E. D. (1977). Water and wastewater treatment (Tokyo: McGraw-Hill).
Schierholz Erica L., Gulliver John S., Wilhelms Steven C., Henneman Heather E. (2006). Gas
Transfer from air diffusers. Water Research 40 (2006) 1018-1026.
Stenstrom Michael K. (2001). Oxygen transfer report: clean water testing (in accordance with
latest ASCE standards) for Air Diffusion Systems submerged fine bubble diffusers on
March 7, 8, 9, & 10 in 2001. http://www.aqua-sierra.com/wp-content/uploads/ads-full-
oxygen-report.pdf

Page | 171
Vogelaar, J.C.T., KLapwijk, A., Van Lier, J.B. and Rulkens, W.H., (2000). Temperature effects
on the oxygen transfer rate between 20 and 55 C. Water research, 34(3), pp.1037- 1041.
Yunt Fred W., Hancuff Tim O., Brenner Richard C. (1988a). Aeration equipment evaluation.
Phase 1: Clean water test results. Los Angeles County Sanitation District, Los Angeles,
CA. Municipal Environmental Research Laboratory Office of Research and
Development, U.S. EPA, Cincinnati, OH.
Yunt Fred W., Hancuff Tim O. (1988b). EPA/600/S2-88/022. Project summary: aeration
equipment evaluation. Phase I: Clean water test results. Water Engineering Research
Laboratory, Cincinnati, OH.
Zhou Xiaohong, Wu Yuanyuan, Shi Hanchang, Song Yanqing (2012). Evaluation of oxygen
transfer parameters of fine-bubble aeration system in plug flow aeration tank of
wastewater treatment plant. Journal of Environmental Sciences 25(2).

Page | 172
Chapter 6. Is Oxygen Transfer Rate (OTR) in Submerged Bubble Aeration
affected by the Oxygen Uptake Rate (OUR)?
6.0 Introduction

Although the oxygen transfer rate and the oxygen uptake rate are two sides of the same

coin, i.e., OTR = OUR, where the accumulation term in the bulk liquid is included in the OUR, the

amount of evidence pointing to errors in the estimating of the oxygen mass transfer coefficient

(KLaf) for diffused aeration is overwhelming, (the subscript for KLa denotes mass transfer

coefficient in the field) such as described in McCarthy, J. (1982) where in section 3 under

“Methods of Aeration Equipment Testing”, it was stated that: “The need to accurately correlate

clean water and wastewater test results…has been recognized by the U.S. Environmental

Protection Agency (EPA) as an important area of research.” Some literature has offered

explanations, such as that the response time of probes caused the errors; or that the OUR

measurement technique is faulty; etc., but none of these explanations are convincing enough to

explain the non-correlation between the clean water and the wastewater mass transfer coefficients.

In fact, as early as 1979, experiences have indicated that OUR under 60 mg/L/hr can be measured

with a minimum of error (McKinney and Stukenberg 1979). As for the probes, the lag times for

modern fast-response probes have been drastically reduced (Baquero-Rodriguez et al., 2016, 2018)

(ASCE 2007). No one has ever considered that the equations for the mass balancing might be

incomplete.

The purpose of this manuscript is to address the anomaly in KLaf estimation (KLa vs.

KLaf) by re-examining the equations as given in Section 2 and Section 3 of the ASCE Standard

Guidelines for In-Process Oxygen Transfer Testing (ASCE 1997). For simplicity, the following

arguments pertain to a completely-stirred batch process only, with a tank/vessel volume of unity.

The author postulates that inconsistency in the evaluation of KLaf between the Non-steady State

Page | 173
Methods (Changing Power levels (CPL) or Hydrogen Peroxide Addition (HPA)) and the Steady

State Methods (Oxygen Uptake Rate (OUR) or Off-gas (OG)), for submerged diffused aeration is

caused by the different gas depletion rates (Gas depletion is defined as the difference between

the oxygen content of the feed and exit gas due to the loss of oxygen partial pressure as the

bubbles rise to the surface) during testing between the two broad categories of methods under the

same mass gas flow rate and substrate loading conditions prior to testing. This difference in gas

depletion rates (gdp) must be accounted for in the mass balancing equations. In in-process water,

care must be taken to ensure that the parameter KLaf is not a function of dissolved oxygen

concentration. This dependency can occur where air is injected through diffusers on the bottom

of activated sludge tanks or fermentation bioreactor vessels, where rising air bubbles are

significantly depleted of oxygen as they ascend to the water surface [CEE 453, 2003] [Rosso and

Stenstrom 2006a]. The extent of oxygen depletion is a function of the oxygen concentration in

the activated sludge mixed liquor.

KLa, by definition, is the product of the liquid film coefficient KL and the interfacial area

of the gas-liquid interface. (The theory of oxygen transfer is given in Chapter 4, and is based on a

mole fraction variation curve as shown in Fig 6-1.) Why would KLa be dependent on anything

else? Although the oxygen transfer rate is affected by the microbial cells of the activated sludge,

KLa itself should not, since in principle, it is not physically or chemically or biologically connected

to the microbial interactions of any live microorganisms. The availability of oxygen has nothing

to do with this parameter, although the oxygen availability in terms of the dissolved oxygen

concentration may be a critical parameter for ethanol production (in fermentation) or biomass

production.

Page | 174
 A study with a control chemostat in parallel, but with the cells killed off either by UV or

by chemicals such as sulfamic acid-copper sulphate may illustrate this point. The characteristics

of the liquid interface would indeed affect diffusivity and hence KL, and the interfacial area may

be affected by many factors such as the air supply system. Looking at all the literature, there is no

evidence that KLa would be affected by live organisms. The non-steady state method (gassing-in

method in the language of fermentation literature) should give a more reliable value of KLa, since

it is free from any interference from any live organisms if first destroyed, and it should compare

well with the steady-state method. Unfortunately, they usually differ by as much as 50%.

Shraddha et al. (2018) have postulated that one must assume that the KLa measured in cell-

free medium persists in the presence of growing cells. However, according to Shraddha, this

assumption is not tenable in their experiments because the rheology of the culture changes with

the operating conditions — the culture is viscous and foams during dual limited (substrate-limited

and oxygen-limited) growth. However, when comparing two cultures, one with living cells and

the other without, the media can be made the same at a certain fixed set of operating conditions,

but with the control devoid of the cells. The control can be used to measure KLa using the non-

steady state method, while the other one's KLa can be determined by the steady-state method. In

principle, both values should be the same. Garcia's experiments [Garcia-Ochoa, F. et al. 2009]

[Garcia-Ochoa, F. et al. 2010] [Santos et al. 2006] using Xanthomonas compestris culture in one

test, and rhodococcus erythropolis culture in another, was illustrated by the author in detail (Fig.

6-5, Fig. 6-6 and Fig. 6-7). These experiments showed a discrepancy between the steady-state and

non-steady state tests of around 50%. If one modifies the transfer equation to include the gas

depletion (gdp) effect, as the author had tried with Garcia's data, the author found that the same

value of KLa is arrived at in both methods. The gas depletion rate is no more than the respiration

Page | 175
rate R, and the transfer equation becomes dC/dt= KLa (Cs-C) - R-gdp which when equating R with

gdp gives dC/dt=KLa (Cs-C)-2R.

Therefore, the author suspects this discrepancy in the conventional model is due to the

mass balance equations missing some important factors, such as the effect of gas-phase gas

depletion during aeration, that is different between the two cases (steady-state versus non-steady

state), making the former 50% less than the latter. If this is accounted for, the two values should

be similar. Unfortunately, few people have done such experiments. The nearest such experiments

are given by Garcia-Ochoa F., et al. (2010) "oxygen uptake rate in microbial processes..." as stated

before.

In the author’s opinion, only when the oxygen transfer coefficient is accurately measured

can the oxygen uptake rate be accurately determined, since, in a steady state, the OTR and the

OURf are two sides of the same coin, and the former is dependent on KLa. There is no accumulation

term in a steady state. Therefore, at steady state, the oxygen uptake rate OURf is the microbial

respiration rate R. If KLaf is not estimated correctly, then these two terms (OTR, OURf) will not

balance each other, with the usual culprit blamed being the OURf as mentioned in the first

paragraph. This happens when a method such as the off-gas steady-state method is chosen to

estimate KLaf, leading to an erroneous estimation of the OTRf. On the other hand, if KLaf is

measured correctly, such as by the non-steady state method, or by Garcia’s gassing-in method,

then any independent separate measurement of the OURf will not give the same KLaf, leading to

doubts about the OURf method of measurement and/or the steady-state method itself [ASCE

1997].

In an aeration tank of a wastewater treatment plant, when a steady state is reached (i.e.,

the oxygen supply meets the oxygen demand from the biological system), the mole fraction of

Page | 176
oxygen in the gas phase would decrease as the depth decreases, so that the exit gas has a smaller

mole fraction than the feed gas. (See Fig. 6-1 in Section 6.1.2). There is evidence that the gas

depletion rate or the oxygen transfer rate is affected by any biochemical reactions such as the

respiration rate of any microorganisms occurring within the liquid, as shown in Fig. 6-3 and Fig.

6-4. The hypothesis presented in this manuscript is that, for the same gas supply rate, the effect

of such reactions is a negative impact on gas depletion, so that the higher the reaction rate, the

smaller is the gas depletion rate, and therefore less gas will be transported or transferred into the

liquid under aeration. In mathematical terms, F1 – F2 = R, where F1 is the gas depletion rate

unaffected by any biochemical reactions; F2 is the gas depletion rate in the presence of

biochemical reactions in the liquid, and R is the reaction rate or the microbial respiration rate or

the microbial oxygen uptake rate (steady-state OURf). This effect of changes in the gas depletion

rate with respect to changes in the mixed liquor suspended solids (MLSS) under a constant gas

flow rate is illustrated by Hu J. (2006), as shown in Fig. 6-3.

This chapter presents mass balance equations that would include the gas depletion effect,

so that the testing methods give consistent results, and that the measured mass transfer

coefficients in the field can be related to clean water KLa. Based on data on tests extracted from

literature, the proposed revised equations for the American Society of Civil Engineers (ASCE

1997) testing methods are shown to result in a consistent estimation of the mass transfer

coefficient (KLaf), where previously the estimation among the methods generally has a

discrepancy of around 40 to 50%.

Page | 177
6.1 Theory

6.1.1 Relationship between KLa and water characteristics

In oxygen transfer, Lewis and Whitman (1924) advanced the two-film theory as a classical

theory of aeration. Using this theory, most models simulate the movement of gas into the water,

but not the other way around. In fact, gas transfer is a two-way street, because the gas dissolves

into the water as well as flows back to the gas stream from the water (Baillod 1979) (Jiang et al.

2012). To make a truly accurate model, one must simulate the dynamic movement.

After so many years of research since the inception of the activated sludge technology, the

author believes that it is a universally established acceptance that the parameter KLa, whether it be

for clean water or wastewater, or mixed liquor, is a function ONLY of the physical characteristics

of the water involved, so long as the external variables such as temperature, pressure, tank

geometry, diffuser plant, solute gas, gas flow rate, turbulence, etc., are not changed. If this sole

dependence is not accepted, it will be necessary to thoroughly review the fundamentals of oxygen

transfer, the two-film theory, etc., and re-visit all the researches carried out so far in the literature

world on this topic. This accepted understanding has led to the standard oxygen transfer equation

given by ASCE 2-06 [2007]; Metcalf and Eddy [1985]; etc.:

𝑑𝐶
= 𝐾𝐿 𝑎 (𝐶 ∗ − 𝐶) [6 − 1]
𝑑𝑡

where, although the meanings and definitions of the symbols may change with respect to the

applications being applied to, the general form of the equation appears to hold for any systems of

oxygen transfer in liquid water under any conditions, no matter whether it is clean water or dirty

water. C* can have different meanings within the context of each application, but it always pertains

to an equilibrium concentration value.

Page | 178
 For the main issue that the manuscript aims at solving (the gap between gas transfer rate

under clean water and wastewater conditions), conventional method is that the consumption of

oxygen due to biological reactions is dealt with using a coupled mass balance equation while the

mass transfer equation is well maintained as its current, widely accepted form. In other words, the

right-hand side of equation (eq. 6-1) can still be used to describe the mass transfer rate, however,

the change in oxygen concentration in the liquid phase will be influenced by mass transfer and

biological consumption R. The salient equation for this scenario is:

𝑑𝐶 ∗
= 𝐾𝐿 𝑎𝑓 (𝐶∞𝑓 − 𝐶) − 𝑅 [6 − 2]
𝑑𝑡

This equation has recognized that R is an additive quantity and not a scalar quantity

associated with KLa but is still flawed because, granted that such a unique function of variability

of KLa is an accepted fact, then the water characteristics may be changed by outside factors, such

as the quantity and character of suspended solids in the water. Suspended solids concentrations,

perhaps along with other constituents, change the viscosity and density of the liquid and hence

affect KLa. Lee (2017) has shown, using water temperature as the independent variable, that KLa

in fact is related to the above water properties, as well as surface tension. When comparing

wastewater with these altered characteristics to clean water, the KLa value usually becomes slightly

smaller (Tchobanoglous et al. 2003). In England, the Water Research Centre (WRc) had used

detergent added to clean water to mimic municipal wastewater, so that the measured KLa would

be representative of the field KLa without having to measure the in-situ KLa in a treatment plant

[Boon 1979] [ASCE 2007].

Now, if the conventional model ASCE 18-96 equation 2 [ASCE 1997] is correct, so long

as the water characteristics remain constant, the measured KLa should be a constant, but it is not;

since the value of KLa in the field is highly variable with respect to the microbial population that

Page | 179
can be represented by the oxygen uptake rate R. For the same oxygen supply rate, the higher the

value of R, the lower is the value of the measured KLa. Hwang and Stenstrom (1985) plotted the

‘mass transfer coefficient’ versus ‘oxygen uptake rate’ in decaying OUR tests and showed a linear

declining trend with increasing R. Why?

There are only two possible explanations: one is that KLa is in fact a function of R. The

other explanation is that the mass balancing equation is wrong or incomplete for diffused aeration

because it hasn't taken into account the changes in the gas depletions due to microbial respiration.

(Gas depletion is defined as the difference between the oxygen content of the feed and exit gas due

to the loss of oxygen partial pressure as the bubbles rise to the surface.) If the first explanation is

correct, logically it overturns the established concept that KLa is only dependent on the water

characteristics and properties. However, it can be argued that the nature of the water may change

due to the presence of microorganisms. Indeed, Hwang and Stenstrom showed in another graph,

that as R increases, the surface tension (as measured by the Du Nouy ring method) decreases. But

since previous plotting has showed a linear declining trend of KLa in relation to R, it can be argued

that the altered water characteristics due to the microbes exert a resistance to oxygen transfer and

therefore this resistance decreases the KLa. This then conforms with the concept that KLa is only a

function of water characteristics and not a function of R.

But where does this additional alteration of the characteristics come from? Why would the

microbial respiration alter the surface tension of the water in question? This is obviously something

for future research. But for now, it can be argued that the presence of the living microbes or the

biochemical reactions associated with their metabolism provide the alteration, and so the change

in the corresponding gas depletion rate that changes the KLa must come from the microbial

respiration itself. There is no other factor that may have caused that change. However, unlike water

Page | 180
characteristics, the effect on the gas depletion rate is additive, not associative. While one can use

a partial factor, alpha (α), on KLa to account for the changes in water characteristics, whose

property in intensive (i.e., not dependent on scale as long as the fluid is well-mixed), one cannot

do the same for the gas depletion rate which is reactive and consecutive to the respiration rate.

Hence, changes in gas depletion rate due to the microbes is an extensive property (i.e. the oxygen

uptake rate of the microbes can be changed substantially without effecting a substantial change in

the water characteristics and properties, even though some changes are inevitable, such as surface

tension). The consequence of R is therefore mostly additive in the mass balance equation.

If the second explanation is correct, and since KLaf should be constant when the water

characteristics is constant, then the oxygen transfer rate cannot be just given by the ASCE equation

2, which has not accounted for that additional change in the gas depletion rate due to the microbial

respiration that changes the surface tension. Based on the above argument, the first explanation is

not entirely incorrect. However, the variation in water characteristics due to the microbes is much

less than the variation in the gdp, because R is highly variable. This is especially true when the

situation is close to endogenous where the DO is very low, and the gdp is at a minimum due to the

high biological activity and the high resultant resistance. At the same time, it is also at a maximum

because of the high driving force. The conventional model adopts a holistic approach, in which

KLaf is corrected by alpha (α) associated with the clean water KLa that does not adequately account

for its changes due to the gas depletion effect. Using alpha purely to correct for the water

characteristics would give a much more consistent value of KLaf. This manuscript introduces the

hypothetical concept that the difference in the respective gas depletion rates is precisely equivalent

to the microbial respiration rate R. If this is true, it has explained why, in the proposed equation

Page | 181
by the author, the 2R is required instead of just a single R which is only correct for surface aerators

where the phenomenon of gas depletion is not existing.

6.1.2 Le Chatelier’s Principle applied to Gas Transfer

Conceptually, before reaching the saturation state in a non-steady state test, since the

oxygen concentration in the water is less than would be dictated by the oxygen content of the

bubble, Le Chatelier's principle requires that the process in the context of a bubble containing

oxygen and rising through water with a dissolved-oxygen deficit, relative to the composition of

the bubble, would seek an equilibrium via the net transfer of oxygen from the bubble to the water

[Mott, 2013]. In this scenario, even for the ultimate steady-state, oxygen goes in and out of the gas

stream depending on position and time of the bubble of the unsteady state test. In clean water, one

can view the mass balances as having two sinks---one by diffusion into water; and the other by

diffusion from water back to the gas stream which serves as the other sink. Whichever is the greater

depends on the driving force one way or the other. At system equilibrium, these two rates are the

same at the equilibrium point of the bulk liquid, the equilibrium point being defined by the

effective depth 'de' in ASCE 2-06 [ASCE 2007]. At steady state, the entire system is then in a

dynamic equilibrium, with gas depletion at the lower half of the tank below the 'de' level, and gas

absorption back to the gas phase above de; the two movements balancing each other out. Therefore,

the general understanding that: "The overall mass transfer coefficient “KLa” incorporates the mass

transfer through the gaseous and liquid films at equilibrium", is applicable to clean water only, if

equilibrium is being defined as a state where the gas flowing into the bulk liquid equals the gas

flowing out of the bulk liquid. In the author’s opinion, only clean water tests can achieve

equilibrium where the potential to transfer (fugacity) is fully utilized. At equilibrium, which is also

steady state, the inlet feed gas mass flow rate will be equal to the exit gas mass flow rate. On the

contrary, in in-process wastewater, only steady state can be achieved, as the fugacity may not be
Page | 182
fully utilized. When the DO changes, or the OUR changes, the potential to transfer may change

accordingly. This can be understood by examining the gas side oxygen depletion curve, where the

exit gas oxygen mole fraction is lower than the feed gas mole fraction at steady state for process

water. The steady state DO concentration (CR) is only an “apparent” saturation concentration that

is not stable as opposed to the saturation concentration in clean water when steady state is achieved.

However, the effect of gas depletion must be considered in both cases, as this manuscript explains

below. Equilibrium means the fugacity of the oxygen in the gas phase is equal to the fugacity of

the oxygen in the liquid phase and LeChatelier’s principle applies to equilibrium.

tank wall

process water clean water

curve at SS curve at SS free surface

equilibrium
level at de
pressure Pe

z
Zd
bulk mixed liquor
DO = C*∞f Ze

initial mole fraction of

ye oxygen assumed Y0 = 0.2095
for air

y
Fig. 6-1. Oxygen Mole fraction curves at saturation for both in-process water and clean water
based on the Lee-Baillod model (e = equilibrium) [Lee 2018]

If the system is at equilibrium, then it is at steady state, as shown in Figure 6-1 at clean water

saturation (also shown in Fig. 3-1 of Chapter 3). As mentioned, the mole fraction variation curve

Page | 183
for any in-process water will not reach equilibrium even at steady state (SS), as shown by the

other curve. The standard mass transfer model for a bulk liquid aeration under constant gas flow

rate has been theoretically derived in Chapter 4, given by eq 4-1 repeated herewith as eq. 6-3:

𝑑𝐶
= 𝐾𝐿 𝑎 (𝐶 ∗ ∞ − 𝐶) [6 − 3]
𝑑𝑡

For the general case, the equation for K1 in eq 4-44 in the previous Chapter 4 (K1 is

defined in Chapter 3 eq. 3-3 and in Chapter 4 eq. 4-44) can be modified to (eq. 6-4) below [Lee

2018], and the generalized Lee-Baillod equation (Eq. 6-5) can be subjected to mathematical

integration to yield eq. 6-6 and eq. 6-7, just like the previous case for the CBVM (constant

bubble volume model) [Lee 2018]. All the resulting equations that lend themselves to five

simultaneous equations for solving the unknown parameters (n, m, KLa0, ye, Ze) are reiterated

and summarized below:

[1 – exp(−𝐾𝐿 𝑎0 𝑥 (1 − 𝑒)𝑍𝑑 )]
𝐾𝐿 𝑎 = [6 − 4]
𝑥(1 − 𝑒)𝑍𝑑

𝐶 𝑌0 𝑃𝑑 𝐶
𝑦 = +( – ) exp(−𝑥𝐾𝐿 𝑎0 . 𝑚𝑧) [6 − 5]
𝑛𝐻𝑃 𝑃 𝑛𝐻𝑃

𝑃𝑎 – 𝑃𝑑 exp(−𝑚𝑥. 𝐾𝐿 𝑎0 . 𝑍𝑑 )
𝐶 ∗ ∞ = 𝑛𝐻(0.2095) [6 − 6]
1 − exp(−𝑚𝑥. 𝐾𝐿 𝑎0 . 𝑍𝑑 )

1 – exp(−𝑚𝑥. 𝐾𝐿 𝑎0 . 𝑍𝑑 ) (𝑛 – 1)𝐾𝐿 𝑎0
𝐾𝐿 𝑎 = + [6 − 7]
𝑛𝑚𝑥. 𝑍𝑑 𝑛

1 𝑚𝑥𝐾𝐿 𝑎0 𝑛𝐻𝑌0 𝑃𝑑
𝑍𝑒 = {ln (𝑃𝑒 ) + ln ( ∗ − 1)} [6 − 8]
𝑚𝑥𝐾𝐿 𝑎0 𝑛𝑟𝑤 𝐶 ∞

(where x = HR0T/Ug where Ug is the height-averaged superficial gas velocity); R0 is the specific

gas constant of oxygen (note: a different symbol is used to distinguish it from the respiration rate

R); T is the water temperature; e is the effective depth ratio e=de/Zd.) Hence, the basic transfer

equation for the non-steady state clean water test as given by eq 6-3, is proven for the general

Page | 184
case (non-constant bubble volume) as well, where 𝐾𝐿 𝑎 and 𝐶 ∗ ∞ are obtainable by solving the

above set of equations when the baseline KLa0 is known. Based on the above derivation, by the

principle of mathematical induction, it can be argued that, for very shallow tank (Zd ≈ 0), the

basic transfer equation is again applicable. Hence, the following equation would apply:

𝑑𝐶
= 𝐾𝐿 𝑎0 (𝐶𝑆 − 𝐶) [6 − 9]
𝑑𝑡

where CS is the handbook solubility value at the atmospheric pressure and water temperature at

testing. Comparing Eq. (6-1) with Eq. (6-9), the two mass transfer coefficients are not the same,

since the former has incorporated the effect of gas depletion as seen in the derivation (see

Chapter 4), whereas in the latter equation, gas depletion is non-existent because of the zero

depth. However, for tank aeration with gas depletion, Eq. (6-3) can be modified to:

𝑑𝐶
= 𝐾𝐿 𝑎0 (𝐶 ∗ ∞0 − 𝐶) − 𝑔𝑑𝑝𝑐𝑤 [6 − 10]
𝑑𝑡

where 𝐾𝐿 𝑎0 is as calculated by eq 6-4 from a known value of KLa. The parameter 𝐶 ∗ ∞0 is the

saturation concentration that would have existed without the gas depletion (note that 𝐶 ∗ ∞0 is not

CS), and 𝑔𝑑𝑝𝑐𝑤 is the overall gas depletion rate during a clean water test. This equation is based

on the Principle of Superposition in physics where the transfer rate is given by the vector sum of

the transfer rate as if gdp (gas depletion rate) does not exist, and the actual gas depletion rate

which is a negative quantity. 𝐶 ∗ ∞0 cannot be the same as Cs because the latter is the oxygen

solubility under the condition of 1 atmosphere pressure only, while 𝐶 ∗ ∞0 should correspond to

the saturation concentration of the bulk liquid under the bulk liquid equilibrium pressure, but

deducting the gas depletion (this of course cannot happen, since without gas depletion there can

be no oxygen transfer). The hypothetical 𝐶 ∗ ∞0 must therefore be greater than 𝐶 ∗ ∞ which in turn

is greater than Cs since the former corresponds to a pressure of Pe while the latter corresponds to

Page | 185
the free surface pressure Pa. This method of reasoning allows solving for the transfer from the

baseline mass transfer coefficient as shown in eq 6-10. Since KLa is a function of gas depletion,

and since every test tank may have different water depths and different environmental

conditions, their gas depletion rates are not the same; hence, they cannot be compared without a

baseline [Lee 2018]. Furthermore, by introducing the term gdpcw, the oxygen transfer rate based

on the fundamental gas transfer mechanism (the two-film theory) can be separated from the

effects of gas depletions on KLa. This gas depletion rate cannot be determined experimentally,

since gdp varies with time throughout the test. Jiang and Stenstrom (2012) have demonstrated

the varying nature of the exit gas during a non-steady state clean water test. Therefore, the only

equation that can be used to estimate the parameters is eq. 6-1 (where C*= 𝐶 ∗ ∞ in the transfer

equation) giving eq. 6-3 above:

𝑑𝐶
= 𝐾𝐿 𝑎 (𝐶 ∗ ∞ − 𝐶)
𝑑𝑡

eq. 6-3 is essentially equivalent to eq. 6-10 but expressed differently (KLa vs. KLa0). Therefore,

by the same token using the Principle of Superposition, for in-process water without any

microbes, eq 6-10 would become eq. 6-11:

𝑑𝐶
= 𝐾𝐿 𝑎0𝑓 (𝐶 ∗ ∞0𝑓 − 𝐶) − 𝑔𝑑𝑝𝑤𝑤 [6 − 11]
𝑑𝑡

giving, in the presence of microbes:

𝑑𝐶
= 𝐾𝐿 𝑎0𝑓 (𝐶 ∗ ∞0𝑓 − 𝐶) − 𝑔𝑑𝑝𝑤𝑤 − 𝑔𝑑𝑝𝑓 − R [6 − 12]
𝑑𝑡

or when expressed differently using the measurable parameters:

𝑑𝐶
= 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 − 𝐶) − 𝑔𝑑𝑝𝑓 − R [6 − 13]
𝑑𝑡

Page | 186
where gdpf is the gas depletion rate due to the microbial respiration. Note that in this equation,

when dC/dt = 0, gdpf would be given by 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 − 𝐶) − 𝑅, where C becomes a constant,

usually denoted by CR as the apparent saturation concentration at steady state.

6.1.3 The Hypothesis of a baseline KLa for wastewater

When the system has reached a steady state in the presence of microbes, the gas depletion

rate is a constant, and so it would be possible to calculate the microbial gdp by the same equation

and by incorporating R as well when dC/dt = 0 and C = CR. In the presence of microbes, the

advocated hypothesis is that this gdpf due to the microbes is the same as the reaction rate R and so

dC/dt = KLaf (C*f-c)-R-R, compared to clean water where the microbial gdp = 0. In other words,

if F1 is the gas depletion rate for clean water, and F2 is the gas depletion rate in process water, then

F1 – F2 = R. It should be noted that, as mentioned before, the basic mass transfer equation is

universal, its general form given by Eq. (6-1). Therefore, in a non-steady state test for in-process

water for a batch test, the transfer equation is given by:

𝑑𝐶
= 𝐾𝐿 𝑎𝑓 (𝐶𝑅 − 𝐶) [6 − 14]
𝑑𝑡

where CR is the “apparent” saturation concentration or the “pseudo” steady-state DO value in

the test tank at the in-situ oxygen uptake rate, R. But the transfer equation is also given by

dC/dt= KLaf (𝐶 ∗ ∞𝑓 -c)-R-R. Equating the two gives,

𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 – 𝐶) − 𝑅 − 𝑅 = 𝐾𝐿 𝑎𝑓 (𝐶𝑅 − 𝐶) [6 − 15]

which gives:

2𝑅
𝐾𝐿 𝑎𝑓 = [6 − 16]
(𝐶 ∗ ∞𝑓 − 𝐶𝑅 )

Note that in this equation, C is cancelled out, so that the above equation is valid for any value of

C, at any state, so long as dC/dt > 0 and C < CR. Most models did not simulate the gas phase, and

Page | 187
so is missing this important element in their balancing equations. This 𝐾𝐿 𝑎𝑓 can then be related to

the clean water KLa which serves as a baseline for extrapolating the clean water test results to

wastewater.

ASCE Guidelines 18-96 [ASCE 1997] reported that “in an EPA Cooperative Agreement

research program, side-by-side comparisons were made of process water oxygen transfer test

procedures (Mueller and Boyle, 1988). Based on these test results on the estimation of KLaf it was

concluded that steady-state testing using oxygen uptake rates, although the easiest procedure to

conduct, is not recommended, because it may significantly overestimate or underestimate the real

oxygen transfer rate. Overestimates are detected in low DO systems. Underestimates appear to be

caused by the presence of a readily available exogenous food source that is rapidly consumed, and,

therefore, is not effectively measured (as uptake) in samples removed from the basin.”

Mahendraker et al. (2005a) compared oxygen transfer test parameters from four testing

methods in three activated sludge processes and found different kinds of discrepancies from the

ASCE Guidelines. The concept of an additional resistance is advanced by Mahendraker (2003)

with the advocacy of a net respiration flux, and Mahendraker et al. (2005b) who demonstrated the

different mass transfer coefficients taking into account the gas depletion effect by the floc.

It is notable that the conclusion reached by Mahendraker V. et al. (2005a) about the

methods is completely opposite to the ASCE Guidelines. In their paper, it was the non-steady

method that was considered suspect, and the steady-state methods Oxygen Uptake Rate (OUR) or

Off-gas (OG) were considered correct, because of the close agreement between these two in their

estimation of KLaf. (In fact, these two methods both give an erroneous estimation of KLaf but by

the same amount, leading to the mistaken conclusion that they were better methods than the non-

steady state method.) The author postulates that all methods will be correct if the mass balance

Page | 188
equation has included the effect of gdp, notwithstanding the various legitimate defects for each

individual method of testing.

Paradoxically, Boyle et al. (1984) appear to agree with Mahendraker’s overall conclusion

that the steady-state method is valid, as can be seen from Table 6 of their report, where the OTE

(oxygen transfer efficiencies) are compared between the off-gas method and the steady-state

method in testing on a municipal wastewater treatment plant. Furthermore, they found that

excellent agreement between the gas tracer method which is considered as a referee method (ASCE

1997), and the off-gas method was achieved in another experiment, therefore suggesting that the

steady-state method is accurate as well. Unfortunately, unlike in Mahendraker’s experiment, the

non-steady state method (NSS) was not compared. Had it been done, they would have found an

anomaly in the KLaf estimates as Mahendraker et al. have found between the NSS method and the

SS method. It should be noted in passing that, during the development of the inert gas radiotracer

procedure for oxygen transfer measurement, the ratio KKr/KLa where KKr is the volumetric gas

transfer rate coefficient for krypton-85, was determined experimentally in laboratory studies using

surface aeration apparatus. The value obtained, 0.83, has been proven accurate for surface transfer

systems only. In sub-surface aeration, the effect of gas depletion must be considered, as can be

seen in eq. 6-12 above. The gas depletion not only comes from the depth of aeration, but also from

the microbial oxygen uptake rate R, which according to the hypothesis in this manuscript is

equivalent to the attendant gas depletion rate coming from the resistance of biological floc, and is

an associative-additive quantity in the oxygen transfer Standard Model. Therefore, the ratio

KKr/KLa may not be 0.83 for a sub-surface system under process conditions. It is also important to

note that, the krypton method only gives KLa estimation, whereas the off-gas method gives

estimation of OTE, necessitating calculation for the KLa from the OTR test results, based on the

Page | 189
standard model. If the calculation has not included the gas depletion effect, the good match

between the two methods is only a coincidence, since both methods have neglected to take into

account of gas depletion in their estimation of KLaf. The same argument goes with the oxidation

ditch tests carried out by Boyle et al. (1989).

6.1.4 The application of the baseline KLa for wastewater

However, it appears that all these discrepancies can be explained by bearing in mind that

in the equation KLaf = α(KLa) where KLa is a baseline based on a clean water test, α represents a

contamination partial factor dependent only on the water characteristics. (α is around 0.8 for

domestic wastewater usually.)

To distinguish this ratio for the baseline case from the other case where α is directly

measured from the field, it would be better to use a different symbol, such as α’. (Mahendraker

used the symbol αe to represent the same parameter.) Based on this modified equation, since dC/dt

= OTRf – R, then OTRf = α’KLa (C*-c) - R which says OTR is affected negatively by the OUR.

The higher the value of R, the lower is the transfer rate. This hypothesis concurs with Hwang and

Stenstrom [1985]’s findings. According to their finding, the degree of reduction of KLaf due to the

microbial respiration R is dependent on tank depth, and the air flow rate. Since the gas depletion

rate as seen in the Lee-Baillod model [Lee 2018] is also dependent on tank depth and the air flow

rate, it can be demonstrated that, by using the data from the literature [Mahendraker 2003], the

microbial gas depletion rate and the respiration rate are the same. This makes the translation of the

clean water baseline KLa to dirty water KLaf mathematically possible. (Note: V = 1 in the

discussion).

The main problem in modeling is the issue of scale. In a full-scale plant, many factors come

into play that would affect the mass transfer coefficient, so that the parameter α becomes a variable.

One of such factors is the gas depletion rate in a diffused submerged bubble aeration system as
Page | 190
mentioned before. However, the alpha’ (α’) factor, which is the ratio of mass transfer coefficients

between dirty water and clean water, is an intensive property if the physical characteristics of the

wastewater is constant, and so can be established by bench-scale experiments [Eckenfelder 1970].

Due to gas depletion, the author has previously developed an equation that would give a more

realistic KLaf value in full-scale, based on their different tank heights [Lee 2018]. The derived

equation is given below as:

1 − exp(−𝛷𝑍𝑑 . 𝛼𝐾𝐿 𝑎0 )
𝐾𝐿 𝑎𝑓 = [6 − 17]
𝛷𝑍𝑑

where 𝐾𝐿 𝑎0 is again a baseline in the context of eliminating gas depletion effects due to tank

height, similar to Eq. (6-4), and α = α’.

KLaf Vs ØZd for KLa0=1

1.2
KLB = [1- exp(-αKLao ØZd)]/ØZd
1
KLB =KLa or KLaf

0.8
KLB for α = 1

KLB for α = 0.8

0.6
KLB for α = 0.6

0.4 KLB for α = 0.4

KLB for α = 0.3

0.2 KLB for α = 0.2

0
0 2 4 6 8 10 12
ØZd

Fig. 6-2. Apparent mass transfer coefficient vs. function of ØZd

Fig. 6-2 (reproduced from Fig. 3-10) shows the effect on the parameter estimation for

different values of α, assuming 𝐾𝐿 𝑎0 = 1. Notice that the value of Φ given by Φ = x.(1-e) may

change when applied to wastewater.

Page | 191
 Eq. (6-17) then accounts for the gas depletion effect and allows translation from the

baseline to full-scale KLaf. Hence, for the mass balance as shown by the ASCE equation 2, for the

case where continuous wastewater flow is absent, although the author agrees that the Respiration

rate (R) must equal transfer rate minus any accumulation rate that is occurring via a changing

dissolved oxygen (DO) level, such that at steady state the OTR must equal the respiration rate R

or the OUR, the author challenges the conventional thinking for the expression of the transfer rate

using the transfer coefficient KLaf that has not included gas depletion. The question presented in

this manuscript is:

“In submerged aeration, should the oxygen transfer rate (OTR) be given by KLaf. (C*∞f -

C) or should it be KLaf. (C*∞f - C) - R?” Using the baseline KLa for clean water, it would appear

that the latter is correct because of the different gas depletion rates between clean water (or non-

respiring water) and in-process water where microbial cells are active with a respiration rate of

R. Below is an example of calculating the baseline using a typical case [Baillod, 1979]:

Suppose the following results are used/obtained by a clean water test:

Zd = 3.05 m (10 ft)

T = 19.5 0C

H = 4.4248 x 10-4 mg/L/(kPa)

Cs = 9.2 mg/L

C = 2 mg/L (measured or calculated)

Yd = 0.2095

Yex = 0.19 (measured or calculated)

ρa Qa/A = 197 g/min-m2 or 11.82 kg/hr/m2

(the product ρa Qa is the mass flow of air at standard conditions; A is the horizontal area of tank.

Page | 192
This gives Qa per unit area as 0.1636 m3/min/m2 when ρa = 1.204 kg/m3 at 20 0C)

KLa = 0.15 min-1 (measured)

C*∞ = 10.0 mg/L (measured)

From re-arranging eq 6-5, we have,

𝑃𝑦
( 𝑌 − 𝑃𝑑 exp(−𝑥𝐾𝐿 𝑎0 . 𝑚𝑧))
0
𝐶 = 𝑛𝐻𝑌0 [6 − 18]
(1 − 𝑒𝑥𝑝(−𝑥𝐾𝐿 𝑎0 . 𝑚𝑧))

At z= Zd, y=Yex, Y0 = Yd = .2095, P = Pa, C can be calculated to be 2 mg/L which confirmed the

measured value. (Note that if C was measured instead, the same equation can be used to calculate

the exit gas mole fraction Yex). The calibration factors (n, m) and other variables, including the

baseline KLa0 [Lee 2018], can be found by using the Excel Solver as shown in Table 6-1,

assuming the exit gas mole fraction is now 0.1900. Assuming pseudo-steady state (i.e. the mass

flow rate has only negligible changes during the transit from tank bottom to top), the gas

depletion rate (gdp) is approximately given by:

𝑔𝑑𝑝 = 𝜌𝑎 𝑄𝑎 (𝑌0 − 𝑌𝑒𝑥 ) [6 − 19]

Hence,

gdp = 1.204 kg/m3 x 0.1636 m3/min/m2 (.2095-.1900) x 60 min/hr = .23 kg/hr/m2

On the other hand, the oxygen transfer rate OTR at C=2 mg/L is also given by the liquid phase

mass balance as:

OTRC=2 = KLa min-1 (C*∞ - 2) x .001 kg/m3 x 60 min/hr x 3 m = 0.21 kg/hr/m2

which is close to the gdp as calculated by the gas phase mass balance. Next, we consider the case

of a mass balance in wastewater where microbes with a respiration rate is present (according to

Mahendraker (2005b), the effect of the microbes in the floc is understood to be an increase in

resistance to oxygen transfer); in this case, since the resistance is increased, the exit gas Yex

Page | 193
 Environmental data
water 0
T= 19.5 C
temp.
atm press. P a= 101325 N/m2
tank area S= 1 m2 Error Analysis
calc. variables Eq. I= 1.79E-05 3.20E-10

diff press. Pd= 128359 N/m2 Eq. II= 2.98E-06 8.88E-12

x=HR’ST/Q x= 0.2058 min/m Eq. III= 4.25E-07 1.81E-13

Solver baseline
KLa 0= 0.1516 1/min 9.10(h-1) Eq.V= -5.60E-09 3.13E-17
soln. KLa
n= 3.62 - SS(sum of squares)= 3.29E-10

m= 3.78 -
check
equil. mole fraction Y e= 0.2028 - Eq.IV= 0.1476 min-1
KLa
diff mole fraction Yd= 0.2095 - exit gas Yex = 0.1900 -

eff. depth de = 1.27 m DO C= 2.00 mg/L

Check
depth ratio e= 0.42 - C*∞= 10.00 mg/L
C*inf
press. at de P e= 111422 N/m2
Qa
Input Zd 3 KLa C*∞ rw Pvt
(m /min H(mg/L/kPa)
Data (m) (1/min) (mg/L) (N/m3) (N/m2)
)
3.00 0.1636 0.1450 10.00 9789 4.425E-04 2333
equations used
Eq 6-7 Eq. I: Kla = (1 -exp (-mxKla0 Zd))/n/m/x/Zd +(n-1) Kla0/n
Eq 6-6 Eq. II: C*∞ = nH*0.2095*(Pa-Pvt-Pd exp (-mxKla0 Zd))/(1-exp(-mxKla0 Zd))
Eq 6-8 Eq. III: ln (Pe mx Kla0/n/rw) +ln (nHYdPd/C*∞ - 1) = mx Kla0 Ze
Eq 6-4 Eq. IV: Kla=(1-exp(-x(1-e) Zd Kla0))/(x(1-e) Zd)
Eq 6-5 Eq. V: y0=C*∞/(nH(pa-pvt)) +(0.2095nPd/(n(Pa-Pvt)-C*∞/(nH(pa-pvt)) exp(-HkmZd)
note: de=1/rw(C*inf/(Hye)-Pa+pvt)

Table 6-1. Calculation of the baseline mass transfer coefficient [Lee 2018]

would be increased at C=2 mg/L, hence, gdpf = 1.204 x.1636 (.2095-.2025) x 60 = 0.083 kg/hr/m2,

hypothetically assuming Yex = 0.2025 when the DO value is very close to zero. If the system is at

Page | 194
steady state, the gas depletion rate in the air stream must be equal to the respiration rate RV, and,

based on eq. 6-16, the in-process mass transfer coefficient KLaf is calculated by:

KLaf = (.083 + 0.083) (kg/hr/m2)/ (0.99 x 10 – 2)/.001 (kg/m3)/60 (min)/3 (m3/m2) =

0.1167 min-1

Therefore,

alpha (α) = KLaf/KLa = .1167/.1450 = 0.80

R = 0.0830/(3x1) x1000 = 27.7 mg/L/hr

As can be seen, alpha (α) is very sensitive to the exit gas oxygen mole fraction Yex so that

the off-gas method must be carried out with extreme care in order to obtain a credible alpha value,

especially when the exit gas is close to the feed gas mole fraction as can be seen in Hu (2006)’s

experiment, shown in Fig. 6-3 based on his test results.

OFFGAS vs. MLSS for membrane diffuser @ 0.028

m3/min (1 scfm)
0.2050
0.2040
Off gas Oxygen mole fraction yex

0.2030
0.2020
0.2010
0.2000
0.1990
0.1980
0.1970
0.1960
0.1950
0 5000 10000 15000 20000
MLSS concentration (mg/L)

Figure 6-3. Off-gas mole fraction vs. MLSS concentrations [Hu 2006]

Small changes in the measurement of the off-gas can give a large error in the estimate of KLaf.

Therefore, from this example, it can be seen that with the concept of a baseline KLa applied to dirty

Page | 195
water, a more realistic alpha (α) value can be obtained, bearing in mind that α = α’ in the context

of this estimation.

6.1.5 The Hypothesis of a microbial Gas Depletion Rate

In the ASCE equation, [ASCE 1997] [ASCE 2007], the parameter KLaf is used to serve a

dual purpose, one to account for the changes in wastewater physical characteristics from clean

water to dirty water, but also to account for the variations in the gas depletion rates due to the

presence of respiring cells. This equation then makes KLaf a variable, dependent on the value of

R [Hwang and Stenstrom 1985] because different values of R produces different values of gdp.

αF Factors (Jet vs. Ceramic disk)

1
0.9
0.8
αF Factor (Klaf/Kla)

0.7
0.6
0.5
disk
0.4
Jet
0.3
0.2
0.1
0
7.6 22.9 38.1 53.3 68.6 83.8
Tank distance from Headwork (m)

Fig. 6-4. Comparison of alpha (α) factors for two different microbial respirations [Yunt 1988c]

As an example, Fig. 6-4 shows the anomaly in the traditional method of determining the

ratio of the mass transfer coefficients between in-process water and clean water [Yunt 1988c].

Although two different aeration equipment and two different wastewater flows were used in the

experiments, the ratios should not be so dramatically different, since the clean water tests were

carried out comparable in all respects to the process water tests.

Page | 196
 Because off-gas measurements in the field tests are reported as OTEf, it was necessary to

translate this value to KLaf. If the equation used was that of the ASCE 18-96 section 5, given by

KLaf = (OTEf WO2 x 103)/ (C*∞f - C) V (where WO2 is the mass flow of oxygen in air stream),

then this equation has not included the effect of gas depletion which is dependent on R.

The author suspects this difference in αF is more due to the different R values in the field tests. On

the other hand, with the new equation (eq. 6-17), KLaf will have only one meaning, which reflects

the characteristics of the dirty water only [Eckenfelder 1970] [Stenstrom et al. 1981] [Bewtra et

al. 1982] [Stenstrom et al. 2006], and independent of R. Therefore, the wastewater mass transfer

coefficient should be given by: KLaf = [(OTEf WO2 x 103) + RV]/ (C*∞f - C) V to conform to the

hypothesis that the microbial gdp is the same as the respiration rate.

6.2 Materials and Methods

6.2.1 Garcia et al.’s Experiment [2010]

Using the modified equations and based on the test data by Mahendraker (2003), Garcia et

al. (2010) and Hu (2006), it was found that all the testing methods within the ASCE document

[ASCE 1997] are valid, as they produce similar values for the mass transfer coefficient KLaf. In

particular, Garcia et al. compared two determination methods for the oxygen uptake rate R, namely

the dynamic method and the oxygen profile data method for a fermentation broth. In terms of

estimating the OUR and KLa, the methods are similar to the steady-state method in the

measurement of OUR, and the non-steady state method in the measurement of KLa respectively

[ASCE 1997]. In the dynamic method example, as described by Garcia and as shown in Fig. 6-5,

the airflow inlet to the fermentation broth is interrupted for a few minutes so that a decrease of DO

concentration can be observed. When the DO has dropped to an acceptable level, air is turned back

on under the same operational conditions until it reaches the same steady state as before. The OUR

Page | 197
is determined from the depletion slope from after the stopping of the air flow, and the procedure

repeated several times for precision.

60
Cr
50
air off re-aeration curve
of wastewater test
DO concentration (%)

40

30
Re-aeration

20 OUR

10 air on
@C0
0
0 200 400 600 800 1000
Time t (s)

Fig. 6-5. Dynamic measurement of OUR and KLa [Garcia et al. 2010]

The second part of the dynamic method is actually identical to the oxygen profile data

method that Garcia described, in that both methods require generating an oxygen profile curve.

In the dynamic method, the re-aeration is made following the de-aeration by the microbes upon

stopping the air supply. (This curve allows the KLa to be calculated.) However, the dynamic

method requires the OUR to be separately determined, as mentioned in the first part of the test,

to be substituted into the oxygenation curve equation to determine KLaf. Contrasting with the

profile data method where the KLa is pre-determined by other means, including the re-aeration

curve, the OUR can be calculated directly from the basic oxygen transfer equation, similar to

ASCE Guidelines’ equation 2. Therefore, in Garcia’s example, since the oxygenation profile is

created by re-aerating back to the original DO level, which is similar to the ASCE non-steady

state method, which is similar to the profile method, the OUR so determined should be the same

Page | 198
as the dynamic method in this example. But it is not (See section 6.2.2 below). The calculation

shows that the two R values differ by 50% when comparing the uptake test with the re-aeration

test. In addition, Garcia cited experiments by Santos et al. (2006) that showed that, in a bio

desulphurization microbiological system with the dynamic method employed to measure the

OUR, the method differs in the value of OUR dramatically when compared to fitting a metabolic

kinetic model to experimental values of oxygen concentration with time. Fig. 6-6 and Fig. 6-7

compare experimental OUR values obtained from the oxygen concentration profile data (OURp)

and those obtained using the dynamic method (OURd) for two bioprocesses, Xanthomonas

campestris and Rhodococcus erythropolis IGTS8 cultures.

steady-state vs. non-steady state tests

Xanthomonas compestris culture [Santos et al., 2006]
non-steady state reaeration OURp (mol/m3.s)

10
9
8
7
6
OURp
5 y = 2.04x
theotical
4
all rpm
3
2 Linear (OURp)

1
0
0 2 4 6 8 10
steady-state respiration rate OURd (mol/m3.s)

Fig. 6-6. OURp vs. OURd (mol O2/m3 s x10-4) [Garcia et al. 2009].

As shown in Fig. 6-6, the theoretical relationship between the steady-state test and the non-

steady state test should be given by y = x if both tests give the same answer for the mass transfer

coefficient, but the actual measurements differ by 50% as seen from the linear relationship y =

2.04x. Even though the rotational speed of the magnetic stirrer may differ from test to test, the

Page | 199
discrepancy should not be so dramatic. The same holds for the Rhodococcus culture, even though

the speeds range from N = 150 rpm to N = 550 rpm. Speed of rotation does affect the KLa and

hence in the prediction of the oxygen uptake rate (OUR), and it can be seen that at higher speeds,

the mass transfer coefficient increases dramatically beyond a certain speed. The linear relationship

becomes y = 3.1x at 550 rpm, as opposed to y = 2.3x at N = 150 rpm ~ 400 rpm. This effect of

speed may be modelled separately, perhaps by adding a scaling factor to KLa pending further

experimental investigations. But the effect of gas depletion rate is clear from these experiments,

and the effect is additive to the transfer equation as discussed previously, and further below.

steady-state vs. non-steady state tests

Rhodococcus erythropolis culture [Santos et al., 2006]
20
non-steady state reaeration OURp (mol/m3.s)

18 y=x
16 N=150
N=250
14
N=400
12
N=550
10
8 y = 2.3x
(N=150)
6
y = 2.3x
4 (N=250)

2 y = 2.2x
(N=400)
0
-1 1 3 5 7 9 11 13 15 y = 3.1x
(N=550)
steady state respiration rate OURd (mol/m3.s)

Fig. 6-7. OURp vs. OURd (mol O2/m3 s x10-4) [Garcia et al. 2009].

6.2.2 Results and Discussions

As can be seen from Fig. 6-6 and Fig. 6-7, the experimental OUR values obtained from the

DO concentration profile are higher than those by the dynamic method, in fact as much as 100%,

depending on the stirring speed (N) of the mixer impeller. The author believes this is due to the

Page | 200
mass balance equation in the profile method neglecting the gas side oxygen depletion, thus giving

an erroneous OUR value that is twice the value obtained in an in-situ oxygen uptake experiment

for the same KLa. When the OUR = R, based on the premise that the OTRf = KLaf. (C*∞f – C) – R,

this concept leads to modification of ASCE equation 2 for a batch test.

Therefore, to illustrate the anomaly, using Garcia’s equation for the re-aeration in Fig. 6-5,

dC/dt = KLa.(C* - C) - OUR

C = Cr - (Cr - C0) exp(-KLa.t)

From Fig. 6-5, it can be read the following:

Cr = 59.5%; C0 = 14.7%, therefore,

C ≈ 59.5 - (59.5 - 14.7) exp (-0. 0052.t)

where KLa is found to be 0.0052 s-1 by fitting the read data to the model by the non-linear least

square (NLLS) method, using the Excel solver as shown in the table below:

time
Cr, Kla,
start from duration c(model error
c (%) C0 SS
time start (s) fit) (c-c(m))
paramtrs
(s)
330 330 0 15 59.49205 14.68 0.322 0.103
420 90 30 0.005168 31.35 -1.347 1.815
480 150 40 14.67843 38.85 1.149 1.319
630 300 50 49.99 0.015 0.000
900 570 57 57.14 -0.137 0.019
min.
3.257
sum(SS)

Table 6-2. Reproduced data from Garcia et al. [2009]

Garcia’s own calculation in their report gave: KLaf = 0.0057 s-1

If the steady state concentration is taken to be around 55%, then R would be calculated as:

R = 0.0057.(100 – 55)/100 = 0.256% s-1

Page | 201
From the oxygen uptake rate test,

dC/dt = -R

R ≈ (55-13)/335*100 = 0.125% s-1 which is only half the value calculated by the profile method.

If this measured uptake rate is inserted in the above equation as a known quantity, the value of

KLaf obtained is only one-half of that obtained by the NLLS method (the profile method), and

will not be correct, since Garcia’s formula did not include the effect of gas depletion.

This concept of accounting for gas depletion, based on the depth correction model eq. 6-

17 above, is at first seemingly counter-intuitive, as one of previous reviewers has mentioned: “In

the text is indicated that "The higher the reaction rate, the smaller the gas depletion rate". This

phrase is difficult to understand because in an aerobic process if reaction rate increased, the

oxygen consumption will be higher (the oxygen is needed to degrade organic matter) and gas

depletion should be higher”, but can be readily understood when a gas phase mass balance of

oxygen is taken for a liquid volume when the system is at steady state. The difference between the

feed gas rate and the exit gas rate must be the oxygen transfer rate (OTR), which is equal to R; but

the OTR is also the gas depletion rate, and so the microbial gdp must also equal to R.

The text simply means that, for the same gas supply rate (therefore constant KLaf during

the duration of the study), an increase of R such as an organic shock load, adds an additional

resistance and so the microbial gdp would increase, but the overall gdp or OTR would decrease

(see Fig. 6-3), requiring the system to adjust to a new steady-state by lowering the steady state DO

concentration CR, thereby increasing the driving force so that the OTRf would match the new

oxygen demand. However, if CR becomes too low, the blowers might then need to work doubly

hard, not only to constantly provide enough air to maintain the oxygen being consumed by the

biomass (oxygen uptake rate OURf = R), but also to maintain a stable ‘spare’ DO level required to

Page | 202
overcome the additional resistance. In this case, the gdp would obviously increase to counteract

the increase in R, but the gas flow rate Qa would also be different and then it would violate the

limitations of the test [ASCE 1997], as change in the gas flow rate means that KLaf is no longer

constant. In the experiment on the performance of a membrane bioreactor (MBR) treating high

strength municipal wastewater, conducted by Birima et al. (2009), the results of dissolved oxygen

(DO) and aeration rate show that the effect of the organic loading rate (OLR) on aeration rate and

DO concentration was very significant. For instance, comparing the results of a trial with low OLR

with those of another trial with high OLR shows that the aeration rate in the first trial was 20 L/min

corresponding to DO of above 4 mg/L, whereas, the rate of aeration in the second trial increased

rapidly till 60 L/min but corresponding to a DO of below 2 mg/L. Similarly for other trials, it was

noted that the higher the organic loading rate, the higher would be the aeration rate and

correspondingly the lower the DO concentration. This observation appears to support the

hypothesis of a higher resistance to oxygen transfer when the demand for oxygen has increased,

even though the driving force has increased because of the lowered DO. This implies that for

higher organic load, a higher rate of aeration is required to obtain the same DO. This means that

operating the MBR with a high organic load means that more energy is required. Generally, the

results of their study showed that for the low OLR trials the aeration rate varied from 6 to 12 m3

/m2 membrane area per hour and the DO varied from 3.7 to 5.7 mg/L, whereas for the high OLR

trials the aeration rate and the DO varied from 6 to 18 m3 /m2 membrane area per hour and 0.9 to

4.4 mg/L, respectively. This depends on the concentration of MLSS in the reactor that in turn

directly affects the respiration rate of the microbial communities.

Page | 203
6.2.3 Results from previous tests re-visited and Discussions

6.2.3.1. Yunt et al. (1988a)’s reported data

Experimental verification to justify the depth correction model for a batch mode in clean

water is given by Lee [2018] and Yunt’s experiments have been described in Chapter 3 section

3.3. The test results are given in the LACSD report Table 5: “Summary of Exponential Method

Results: FMC Fine Bubble Tube Diffusers” and copied over as shown by Table 5-8 in Chapter 5

as well as Table 3-1 in Chapter 3, where the calculations for the baseline KLa0 is shown by Table

3-2. The calculation spreadsheet for estimating the variables KLa0, n, m, de and ye is not repeated

in this Chapter. Using the standardized baseline, (KLa0)20, the simulated result for a typical run test

for the FMC diffusers gives a value of (KLa)20 = 0.1874 min-1 as compared to the test-reported

value of (KLa)20 = 0.1853 min-1 which gives an error difference of around 1% only comparing to

the simulated value [Lee 2018]. The compared results of the aeration efficiencies plotted in

ascending order of the tank depths is shown herewith in Fig. 6-8 (which is identical to Fig. 3-9 in

Chapter 3). Within experimental errors and simulation errors, the results seem to match very well,

with the aeration efficiency slightly over-predicted at the higher depths. (This is probably because

in the development of the model, any free water surface oxygen transfer has been ignored. This

effect is not uniform for all tanks – more important for shallower tanks than for the deeper ones

[DeMoyer et al. 2002], and so the actual baseline would have been slightly smaller for the deeper

tanks if the effect of surfacing bubble plume had been considered, matching the report values.) It

would appear from the graph that the oxygen transfer efficiency is an increasing function of depth,

even though the gas flow rates were not exactly the same for all the tests. These results, along with

other tests, clearly show that the Lee-Baillod model [Lee 2018] is valid in considering the effect

of oxygen gas depletion for the purpose of predicting oxygen transfer.

Page | 204
 Comparison of predicted and reported efficiencies

Aeration Efficiency in Percent

35.0

30.0

25.0

20.0

15.0 p. SOTE

10.0 rpt. SOTE

5.0

0.0
1 2 3 4 5 6 7 8 9 10 11 12 13
Run Numbers in order of increasing Depth (Note: 1-4 for 3.05m, 5-7
for 4.57m, 8-10 for 6.09m, 11-13 for 7.62m)

Fig. 6-8. Simulated efficiencies for FMC Diffusers [Lee 2018]

As shown in Fig. 6-2 for the depth correction model given in Section 6.1.3, KLaf is a

declining trend with respect to increasing depth of the immersion vehicle of gas supply, similar to

what Boon (1979) has found in his experiments. The detailed analysis and the derivation of the

model equation (eq 6-17) is given by Lee [2018] and also in Chapter 4. For similar tests in in-

process water, the mass transfer coefficient so determined by the depth correction model while

incorporating the alpha factor, should match the result of field testing by following the ASCE

Standard Guidelines for In-Process Oxygen Transfer Testing methods [ASCE 1997]. The

verification of validity for this equation would require testing full-scale in the field, using the

alpha’ value (α’) derived by bench-scale testing or shop tests, with solids filtered out or the cells

destroyed prior to testing.

6.2.3.2. Mahendraker et al. (2003)’s data

Based on re-analyzing the data from Mahendraker’s dissertation [Mahendraker 2003], the

average value of α’ based on all the test results is about 0.82. The new equations after incorporating

Page | 205
the effect of gdp, give an overall discrepancy for KLaf of only around 12% between the various

tests, notwithstanding the various inaccuracies of the individual methods stated in the literature

[ASCE 1997]. The calculations are not shown in this paper, but readers can satisfy themselves by

verifying the results and by examining their paper and re-analyzing the experimental data. It should

be noted that, in calculating for KLaf in their steady-state tests, the new equation eq. 6-16 as

postulated by the author as repeated below was used instead of the conventional model:

2𝑅
𝐾𝐿 𝑎𝑓 =
(𝐶 ∗ ∞𝑓 − 𝐶𝑅 )

In this equation, although R was determined by the respirometer method as described in

Mahendraker’s dessertation, the author considers this method as incorrect in that the dissolved

oxygen content in the sample was artificially aerated to a higher level, thus in the author’s opinion,

artificially making the estimation of the microbial respiration rate twice as much as would be in

the actual treatment system. R must therefore be reduced by 50% in the mass balancing equations

for calculating the mass transfer coefficients in these tests.

Furthermore, Mahendraker et al. (2005b) postulated that the resistance to oxygen transfer

is composed of two elements: the resistance due to the reactor’s solution, and the resistance due to

the biological floc. They formulated the relationship between these resistances as:

1 1 1
= ′ + (6 − 20)
∝ 𝐾𝐿 𝑎 ∝ 𝐾𝐿 𝑎 𝐾𝐿 𝑎𝑏𝑓

in which the scaling factor for the reactor solution was given in their equation as ∝𝑒 , instead of

the symbol α’ and the subscript for the second resistance bf represents that due to the biological

floc. Rearranging eq. 6-20 gives:

∝′ 𝐾𝐿 𝑎 × 𝐾𝐿 𝑎𝑏𝑓
∝ 𝐾𝐿 𝑎 = ′ (6 − 21)
∝ 𝐾𝐿 𝑎 + 𝐾𝐿 𝑎𝑏𝑓

Page | 206
Substituting eq. 6-21 into eq. 6-2 which is identical to ASCE 18-96 equation 2, in a steady state

and for a batch reactor, and solving for 𝐾𝐿 𝑎𝑏𝑓 we have:

∝′ 𝐾𝐿 𝑎 × 𝑅
𝐾𝐿 𝑎𝑏𝑓 = (6 − 22)
∝′ 𝐾𝐿 𝑎(𝐶 ∗ ∞𝑓 − 𝐶𝑅 ) − 𝑅

If it is assumed that the resistance of the biological floc is the same as the resistance from the

reactor solution, (this is similar to assuming that the gdp due to the microbes is due to the

resistance in the bioreactor solution KLaf), then

𝐾𝐿 𝑎𝑏𝑓 =∝′ 𝐾𝐿 𝑎 (6 − 23)

Substituting eq. 6-23 into eq. 6-22 and solving for ∝′ 𝐾𝐿 𝑎,

2𝑅
∝′ 𝐾𝐿 𝑎 = (6 − 24)
(𝐶 ∗ ∞𝑓 − 𝐶𝑅 )

This equation is similar to eq. 6-16 previously derived. However, this requires an assumption for

the biological floc resistance which may not be true, as well as the use of the ASCE equation that

the author has disputed its validity. Mahendraker’s concept requires further investigation. The

logical explanation may be that the biological floc resistance results in a change of the gas

depletion rate, which coincides with the microbial respiration rate at steady state. This explanation

then unifies the two concepts together, and result in the same conclusion as stated by eq. 6-24.

6.3 Relationship between Alpha (α) and Apha'(α’)

As mentioned in Section 6.1.3, the concept of a baseline mass transfer coefficient for wastewater

requires the employment of an additional correction factor (α’) for the clean water KLa when

applying the standard model to wastewater. However, alpha (α) and alpha’(α’) are inter-

convertible. From eq 6-13, we have,

𝑑𝐶
= 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 − 𝐶) − 𝑔𝑑𝑝𝑓 − R
𝑑𝑡

Page | 207
Therefore, substituting 𝛼′𝐾𝐿 𝑎 for 𝐾𝐿 𝑎𝑓 , we have,

𝑑𝐶
= 𝛼′𝐾𝐿 𝑎 (𝐶 ∗ ∞𝑓 − 𝐶) − 𝑔𝑑𝑝𝑓 − R (6 − 25)
𝑑𝑡

Therefore, if the microbial gas depletion rate is the same as the respiration rate, we have

𝑂𝑇𝑅𝑓 = 𝐾𝐿 𝑎. 𝛼’ (𝑑𝑒𝑓𝑖𝑐𝑖𝑡)– 𝑅 (6 − 26)

where deficit = 𝐶 ∗ ∞𝑓 − 𝐶

For the case where alpha is directly determined in the field by comparing in-process wastewater

to clean water tests simultaneously, as described in the previous sections,

𝑂𝑇𝑅𝑓 = 𝐾𝐿 𝑎. 𝛼 (𝑑𝑒𝑓𝑖𝑐𝑖𝑡) (6 − 27)

Equating Eq. (6-26) and Eq. (6-27), alpha and alpha’ can be inter-related as:

𝑅
𝛼’ = 𝛼 + (6 − 28)
𝐾𝐿 𝑎(𝑑𝑒𝑓𝑖𝑐𝑖𝑡)

Both Eq. (6-26) and Eq. (6-27) can be used to determine the oxygen transfer rate under process

conditions. However, Eq. (6-27) has two degrees of freedom, with both variables (water

characteristics and cell respiration) incorporated into this one single value for alpha. Alpha values

can vary from a small value to a large number, depending on the initial cell content and the degree

of treatment which directly affects the value of R. It is also dependent on the organic loading rate

(oxygen demand) to which the metabolic rate (consumption) directly responds.

Alpha’, on the other hand, depends only on the nature of the wastewater, which can be much more

constant. The two equations are easily reconciled by substituting eq. 6-28 into eq. 6-25, giving

𝑑𝐶 𝑅
= (𝛼 + ) 𝐾 𝑎(𝑑𝑒𝑓𝑖𝑐𝑖𝑡) − 𝑔𝑑𝑝𝑓 − R (6 − 29)
𝑑𝑡 𝐾𝐿 𝑎(𝑑𝑒𝑓𝑖𝑐𝑖𝑡) 𝐿

or

𝑑𝐶
= 𝛼𝐾𝐿 𝑎 (𝐶 ∗ ∞𝑓 − 𝐶) − 𝑔𝑑𝑝𝑓 (6 − 30)
𝑑𝑡

Page | 208
Again, if gdp is equal to R, we have

𝑑𝐶
= 𝛼𝐾𝐿 𝑎 (𝐶 ∗ ∞𝑓 − 𝐶) − R (6 − 31)
𝑑𝑡

which is identical to ASCE equation 2 in the guidelines [ASCE 1997], for a batch process.

In the application of Eq. (6-26) or Eq. (6-27), both equations should give similar results if the

objective is to find the oxygen transfer rate under process conditions, given clean water test results

for 𝐾𝐿 𝑎 and C*∞ (note that C*∞ is required to determine the deficit). Eq. (6-26) would additionally

require the determination of alpha’, as well as determination of R which can be done in accordance

with the ASCE 18-96 methods. On the other hand, Eq. (6-27) does not contain an R term, therefore,

Eq. (6-27) is the more popular choice when the objective is to determine the OTRf directly because

of its simplicity.

However, if the objective is to find R, the oxygen uptake rate commonly known as the

OUR, Eq. (6-27) cannot be used even though all the other parameters are known, since alpha has

two degrees of freedom, and so it is impossible to determine how much of the transfer is attributed

to R and how much is due to the gas-liquid transfer mechanism that is affected by the water

characteristics. Additionally, the intensive variable of the difference in gas depletion rates is also

unknown. With three unknowns, R cannot be calculated even if all the other parametric values are

known in (Eq. 6-27). R must therefore be independently measured.

6.4 Measurement of Respiration Rate

According to ASCE guidelines (ASCE 1997, 2018), measuring the OUR under oxygen limiting

conditions is extremely difficult, although the principle of respirometry is remarkably simple—-

the slope of the decline curve of DO vs. time must be the respiration rate. The problem is not so

much the method as the methodology commonly employed to make the sample measurable. When

the oxygen level in the sample is low, say, at 2 mg/L, it would need to be artificially aerated to a

Page | 209
higher level, say, 5 mg/L, before a meaningful curve (usually a straight line if the sample is not

substrate-limiting as well) for calculating the slope can be obtained. This boosting of the DO

concentration may make the sample measurement artificially high, and so the true uptake rate in

the aeration tank is not measured correctly. How do we correct this error?

To estimate the respiration rate R, ASCE (2018) recommends the off-gas column steady-state test.

In the recommendation, an acrylic or fiberglass reinforced tank is used, such as a 30 in. (760 mm)

diameter by 11 ft (3.4 m) deep column. The column depth was selected based on work done at the

University of Wisconsin (Doyle, 1981) where it was found that alpha decreased as the liquid depth

increased over a range of two to ten feet (3.05 m); however, the decrease was relatively small

above eight feet (2.4 m). Mixed liquor is continuously pumped to the test column from a position

within the existing aeration tank using a submersible pump. The liquid detention time in the

column is typically maintained between 10 and 15 min. The mixed liquor should be aerated using

a fine pore (fine bubble) diffuser identical to the type used in the tank. The oxygen transfer

efficiency of the diffuser used in the column using process mixed liquor is measured using the off-

gas techniques described in Section 3.0 of the Guideline. The airflow rate to the test diffuser is

adjusted so that the DO concentration in the steady state column is maintained in the range of those

found in the test section of the aeration basin. A schematic of the column test system is given in

Figure D-1 of the ASCE Guidelines. An example is given in the Guidelines as shown below:

Oxygen uptake rate is determined by a mass balance of oxygen around the column system as:

oxygen uptake rate = (oxygen transfer rate - net change in DO)/column volume,

or R = (OTRf - (DOo - DOi)Qi)/V

For an example, an ex situ column test is performed at a test section of the aeration basin. The

following data are collected:

Page | 210
DOi (at transfer pump) = 0.55 mg/L

DOo (in test column) = 0.80 mg/L

qi (airflow rate to column) = 1.07 NL/s (normal litres per second)

OTEf (measured in column) = 0.130 mg O2 transferred / mg O2 supplied

Qi (mixed liquor pump rate) = 2.16 L/s

V (column volume) = 1,460 L

OTRf = 1.07 NL air/s × 299.3 mg O2/ NL air × 0.13 mg O2 transferred / mg O2 supplied = 41.6

mg O2/s

R = [41.6-(0.80-0.55) × 2.16]/1,460 = 0.0278 mg/L•s × 3600 s/hr = 101 mg/L/hr

Although, unfortunately, there is no comparable data using the other methods such as the BOD

bottle method, Chisea S.C. et al. [1990] conducted a series of bench‐ and pilot‐scale experiments

to evaluate the ability of biochemical oxygen demand (BOD) bottle‐based oxygen uptake rate

(OUR) analyses to represent accurately in-situ OUR in complete mix‐activated sludge systems.

Aeration basin off‐gas analyses indicated that, depending on system operating conditions, BOD

bottle‐based analyses could either underestimate in-situ OUR rates by as much as 58%

or overestimate in-situ rates by up to 285%. A continuous flow respirometer system was used to

verify the off‐gas analysis observations and assessed better the rate of change in OUR after

mixed liquor samples were suddenly isolated from their normally continuous source of feed.

OUR rates for sludge samples maintained in the completely mixed bench‐scale respirometer

decreased by as much as 42% in less than two minutes after feeding was stopped. Based on these

results, BOD bottle‐based OUR results should not be used in any complete mix‐activated sludge

process operational control strategy, process mass balance, or system evaluation procedure

requiring absolute accuracy of OUR values. This echos the author’s previous suspicion about

Page | 211
Eckenfelder's experiment (Eckenfelder, 1952) for calculating the oxygen transfer efficiency

based on sample testing of the respiration rate which required artificial boosting of the DO

concentration in a BOD-bottle, as described below.

There is certainly a need to get to the bottom of this. According to the published article,

"Aeration Efficiency and Design", in which Eckenfelder (1952) described two methods of testing

for the microbial respiration rate, both the steady-state method and the non-steady state method

were used to "validate" that these two methods are compatible with each other. The results were

given in a table as reproduced and summarized below (Table 6-3). In this table was shown the

test results for 6 runs, for an aeration tank of 33 inches (838 mm) tall, and aerated at different

flow rates from 33 cu ft/hr (0.016 m3/min) to 92 cu ft/hr (0.043 m3/min). In terms of SI units, the

air flow rate (AFR) and the height-averaged air flow rate would essentially be the same given the

small height. Eckenfelder used the log-deficit method to calculate the mass transfer coefficients,

corresponding to each AFR, for the nonsteady state (NSS) test results, shown in red in col. 5.

AFR NSS log-def SS rpt

calc. 𝐾𝐿 𝑎
Q Qa rpt 𝐾𝐿 𝑎 R
Q(cfh) 𝐾𝐿 𝑎 (ASCE)
(m3/min) (m3/min) 1/h (mg/l/h)
1/h 1/h
0 0
33 0.0156 0.0147 8.03 8.20 8.50 27.5
39 0.0184 0.0173 11.31 9.80 9.37 30.5
52 0.0245 0.0231 12.21 12.00 11.96 31.5
64 0.0302 0.0284 16.01 15.00 15.89 34.4
77 0.0363 0.0342 18.99 18.20 18.78 33.4
92 0.0434 0.0409 24.87 23.50 22.70 32.2

Table 6-3. Laboratory test results by Eckenfelder (1952)

His data was reproduced and converted and then the non-linear regression analysis (NLLS) was

used as recommended in the standard (ASCE 2-06) to re-calculate the 𝐾𝐿 𝑎 's, shown in col. 4.

Page | 212
The plot of the re-aeration curves is as shown in Fig.6-9. These data were derived from an

experimental aeration tank employing an agitator and air sparger ring. Next, the author used the

steady-state (SS) method to again calculate 𝐾𝐿 𝑎, after using the measurements of the individual

respiration rates from the BOD bottle method, and equating those with the oxygen transfer rate

as 𝐾𝐿 𝑎 (C*-C), and the results are similar to those of the non-steady state method, ostensibly

proving the validity of both test methods. Unfortunately, clean water tests were not performed,

and so there is no way to estimate alpha. However, the respiration tests were done at 2 p.p.m. as

reported in the article by Eckenfelder, and so these samples must have been re-aerated by

vigorous shaking to at least twice the value of the in-situ dissolved oxygen concentration.

Eckenfelder's non-steady state tests

9
8
DO concentration (mg/L)

7
6 C1

5 C2
4 C3
3 C4
2
C5
1
C6
0
0 2 4 6 8 10 12 14 16
time (min)

Fig. 6-9. Illustrative problem in oxygen transfer measurement reproduced from Eckenfelder (1952)

If the author’s hypothesis is correct, i.e., that the microbial oxygen uptake rate is linearly

proportional to the oxygen availability, then the measured R values must have been at least twice

the actual values in the aeration basin where the samples were withdrawn. The resultant 𝐾𝐿 𝑎 values

would then be half the actual values measured by the non-steady state method. This experiment

Page | 213
then in fact did not prove the validity of either one or the other, but instead proved that these two

methods give results of 𝐾𝐿 𝑎 that are off by 50% using the ASCE methods.

According to the author’s thesis, the equation for the steady-state method should come out

to be 𝐾𝐿 𝑎f = 2R/(C* -Cr) instead of a single R as conventionally used, because of the gas depletion

effect in the air bubbles. With this modification, this would then give the exact results of the 𝐾𝐿 𝑎f

as before using the ASCE non-steady state method, if it is reckoned that the BOD method of

measuring the OUR is incorrect (over-estimation) because of the additional aeration. It is therefore

vital, to prove one way or another, that an in-situ oxygen uptake rate test be performed similar to

that described in the Guideline ASCE 18-96 (or more recently ASCE 18-18) for the steady-state

column test using the off-gas measurement techniques. This may confirm, once and for all, whether

oxygen availability has an effect on the oxygen uptake rate in a sample upon re-aeration.

The beauty of this off-gas method in measuring the respiration rate is that it does not require

artificially aerating the sample to a higher DO level, and 𝐾𝐿 𝑎 doesn’t come into the picture as well.

The author suggests that an experiment be done in one treatment plant with an acrylic column 3 m

or so high; and then comparing the result with the traditional BOD bottle method and observing

the difference, especially for oxygen-limiting low DO conditions. The key element of success is

the off-gas analyzer that must measure the offgas accurately, since the OTE is highly sensitive to

changes in the gas composition. (See Chapter 5 above.)

Measuring OUR under oxygen limiting conditions is difficult, and it was found that OUR

measurements in general can be impacted by how the test is run.[Doyle 1981][Private

communication Doyle and Lee]. When one withdraws a sample of mixed liquor and runs an OUR

the typical way (aerate to high DO in a BOD bottle, stop aerating and then track the depletion of

DO over time) one changes the conditions in the sample compared to the conditions in the aeration
Page | 214
tank. Of course, that method is not at all appropriate for an aeration basin near zero DO (ie, the

tank OUR is limited by the oxygen transfer rate). When given more DO the bacteria will increase

the OUR compared to the oxygen limited OUR in the aeration tank. But even with sufficient

aeration basin DO, one can get different OUR values depending on the measurement method. For

example, shaking the sample to aerate can break up the floc, making the substrate and DO more

available to the bacteria. Doyle noticed in one of his thesis work that a BOD bottle OUR did not

agree with a respirometer OUR. When using the old Arthur respirometer, it consistently gave

higher OUR measurements compared to the BOD bottle/DO depletion method. He attributed it to

the intensive aeration in the respirometer which was rather violent and may have broken up the

floc causing increased delivery of DO and substrate to the floc. [Private communication Doyle and

Lee].

In Garcia’s experiment (see section 6.2.2), Garcia tried to explain this anomaly by the

"cellular economy principle" that, during the time oxygen is not transferred, (i.e. during the

shutting off the gas supply in the microbial desorption test), microbial cells consume oxygen at a

lower rate. There are four reasons that this claim is wrong:

i. the desorption test is done in-situ, there is no time lag between DO (dissolved

oxygen) analysis and sample collection;

ii. The desorption curve is linear which means the decrease in DO content is

uniform. (See Figure 6-5). This in turn means the microbes are consuming the

oxygen at a uniform rate. If the microbes had been using oxygen at a declining

rate, the curve would have been concave in shape;

iii. During desorption, there is still plenty of oxygen in the liquid phase, beginning at

55% saturation. Bacteria are not so smart that they could sense a continual

Page | 215
 diminishing of oxygen that they would start economizing from the start. Only

when the DO has reached the endogenous zone would this occur. Even if the

consumption rate has decreased it could never by as much as 50%.

iv. There is no deficiency in soluble substrate and so the respiration rate would not be

affected by soluble substrate uptake depletion, i.e. the system was not at substrate-

limiting condition before or during the test.

According to Doyle, there is no data to show that isn’t the case that the microbes change

their respiration rate at low DO levels, but it seems to Doyle that oxygen is used as fast as it can

be delivered. Any reduction is due to physical/chemical limitations (diffusion limitations, and

concentration gradients and transport across the cell membrane). It would be difficult to determine

whether the microbes are doing this voluntarily since to test the theory one has to limit the DO,

which impacts the other factors, such as an increase in the biological floc respiration rate due to

the breaking up of the floc by violent agitation in order to bring the DO level back up to a

measurable level for testing.

In summary, the author reckons there are only three possible explanations for an over-

estimation of the respiration rate. Firstly, when the sample is agitated by vigorous shaking, the

activity level of the microbes might increase. Like any living organisms under stress, they respire

more. Secondly, the oxygen level in the sample may be limiting (although at 2 ppm, it shouldn't

be); thirdly, Garcia-Ochoa [Garcia et al.] suggested a cell economy principle by which the

microbes voluntarily reduce the respiration level at low DO and changes the respiration rate at

elevated level due to increased oxygen availability. There have been many literature on this but

none seems to have given a definite answer.

Page | 216
 However, Doyle’s article [Doyle 1981] suggested an interesting method of testing for

alpha. It seems that it may be possible to use a dilution method to test out the determination. By

first aerating a tank of pure water to an elevated DO, say 7 ppm, and then pouring the activated

sludge mixed liquor into the tank, and then gently mixing them together, it may be possible to

measure the slope of the DO decline curve at quiescent conditions, thereby eliminating the first

possible explanation for the cause of increased OUR measurement. If the sample has been diluted

to 50%, the resultant slope should then be multiplied by 2 to get the true OUR. This should then

be compared with a corresponding steady-state column test with an insitu measurement.

6.5 Conclusions

In this manuscript, the author postulates that the true OTRf is given by KLaf.(C*f -C) -R,

since the transfer rate is affected by biochemical reactions in the cells [Hwang and Stenstrom

1985], which changes not only the water characteristics but also changes the gdp.

Based on the studies so far cited in this manuscript, it is concluded that:

1. The oxygen transfer efficiency based on the oxygen transfer rate by a prescribed CWT

(Clean Water Test) for a fixed gas supply is a property of an aeration equipment, and so

will not be affected by external factors (i.e. clean water test data are reproducible)[ASCE

2007] and would be uniquely defined by a standard specific baseline value 𝐾𝐿 𝑎0;

2. A new mathematical model for gdp has been derived (Eq. 6-17) and is verified by testing

under a variety of water depths for clean water (Yunt et al. 1988a). This model is shown to

be applied to wastewater through a revised alpha (α’) that pertains specifically to

wastewater characteristics;

Page | 217
3. The respiration rate produces additional resistance to oxygen transfer in the system

resulting in a loss of gdp, and therefore must be accounted for in the mass balancing

equations. In this sense, the OTRf in the system (as opposed to the aeration efficiency of

the device definable by the CWT) is indeed affected negatively by the OUR;

4. The difference in the gas depletion rates due to the microbial cells that affect oxygen

transfer is precisely the respiration rate itself, based on all the test results, and therefore is

important for the revising of the ASCE equations [Mahendraker et al. 2003, 2005a];

5. The present equations used in the ASCE Guidelines [1997] are not correct for submerged

aeration. This has resulted in discrepancies of around 40 ~ 50% in the estimation of KLaf

for batch test analyses (i.e. the steady-state test results are lower than the non-steady state

tests by such). The new equations after incorporating the effect of gdp, give an overall

discrepancy of only around 12%, notwithstanding the various inaccuracies of the individual

methods stated in the literature [ASCE 1997] [Mahendraker et al. 2003] and for the

continuous water flow testing;

6. KLaf is dependent on the AFR (air flow rate), but since the AFR also affects the

corresponding KLa in clean waters, the resultant effect on α is not overly significant when

α is defined as α’. The average value of α’ based on all the test results is about 0.82

according to Mahendraker’s data. This value of α’ is in line with the traditional design

value of 0.8 used in many treatment plant’s designs. However, using this value would now

require revising the design equations to include the gas depletion effect as explained in this

manuscript, otherwise, α would become highly variable [Rosso and Stenstrom 2006a,

2006b] [Rosso et al. 2017].

Page | 218
 As a consequence of including the gas depletion effect in the application of Clean Water

Test Results to estimate Oxygen Transfer Rates in Process Water at Process DO levels, for the

same oxygenation system, the following amendment is applicable to Eq. CG-1 in ASCE 2-06

[ASCE 2007]:

1
[6 − 32] 𝑂𝑇𝑅𝑓 = ( ∗ ) [ 𝛼 (𝑆𝑂𝑇𝑅)Ɵ𝑇−20 ]( 𝜏. ß. 𝛺. 𝐶 ∗ ∞ 20 − 𝐶) – 𝑅𝑉
𝐶 ∞ 20

where α = α’ in the context of this submitted document, with all symbols referring to the ASCE

(2007) Standard. The implication of this equation is that the oxygen transfer in the field of an

aeration equipment can be closely determined by clean water tests by applying relevant correction

factors to the clean water measured parameters, together with accurate measurements of the

respiration rate in the field. To complete the equation, the effects of temperature in the selection

of a proper value for the temperature correction parameter Ɵ, and the effect on KLa due to geometry

have been discussed in previous chapters and manuscripts [Lee 2017] [Lee 2018].

6.6 Appendix
6.6.1 The Lee-Baillod Model in wastewater (speculative)

If the clean water KLa is to be applied to in-process water, this discrepancy in the gas

depletion rate between the two systems must be accounted for in any mathematical model

describing oxygen transfer in in-process water, so that the meaning of the mass transfer coefficient

is consistent with both systems. It is thought that the mathematical model, starting with clean water

at the equilibrium state where the parameters (KLa and C*∞) describe the oxygenation curve that

can be determined in a clean water test, might be applied to wastewater. The derivation of such a

model for clean water has been published in the WERF Journal [Lee 2018], and also explained in

detail in previous Chapter 4. It is envisaged that the standard specific baseline for wastewater can

be similarly determined as for clean water, pending further investigations.

Page | 219
6.6.2 Determination of the Standard Specific Baseline for wastewater (speculative)

FMC diffusers
(KLa0)20 vs. Qa20

Baseline mass transfer coefficient

20
y = 2.6732x0.8167
KLa020 (1/h) 15 R² = 1
10 3.05 m
4.57 m
5
6.10 m
7.62 m
0
0 2 4 6 8 10 12
Height-averaged Air flow Rate
Qa (m3/min)

Fig. 6-10. Effect of diffuser submergence and airflow rate on the baseline transfer coefficient

Revisiting Yunt’s experiment [Yunt et al. 1988a] in a shop test, Fig. 6-10 shows that the

resulting KLa0 values [Lee 2018] obtained for various tests are adjusted to the standard temperature

by the temperature correction equation of the 5th power model (Lee 2017) and plotted against

Qa20.

This curve is identical to Fig. 5-7 in the previous chapter, except that the unit for the

baseline mass transfer coefficient is in 1/hr where previously it was in 1/min. Remarkably, all

curves fitted together after normalizing KLa0 values to 20 0C, as shown by the data points

(represented by different symbols) on the different depths. The exponent determined is 0.82. The

value obtained from the slope is 44.35 x 10-3 (1/min) for all the gas rates normalized to give the

best NLLS (Non-Linear Least Squares) fit, bearing in mind that the KLa0 is assumed to be related

to the gas flowrate by a power curve with an exponent value [Stenstrom et al. 2006] [Zhou et al.

2012]. The slope of the curve is defined as the standard specific baseline. Therefore, the standard

Page | 220
specific baseline (sp. KLa0)20 is calculated by the ratio of (KLa0)20 to Qa20^.82 or by the slope of

the curve in Fig. 6-10.

FMC diffusers
(KLa)20 vs. Qa20
20
volumetric mass transfer coefficient

18
y = 2.7677x0.7841
16
14 y = 2.5784x0.8057
12
10 y = 2.2428x0.8726 3.05 m
KLa (1/h)

8 4.57 m
6 y = 2.5777x0.781
6.10 m
4
2 7.62 m
0
0 2 4 6 8 10 12
Height-averaged Air Flow Rate
Qa (m3/min)

Fig. 6-11. Effect of diffuser submergence and airflow rate on the mass transfer coefficient

When the same information is compared with a similar plot using the actual measured KLa

values (plot shown as Fig. 6-11) it can be seen the correlation was still quite good for the curve,

but not as exactly as when the baseline values were plotted, testifying the fact that the baseline

mass transfer coefficient does represent a standardized performance of the aeration system when

the tank is of zero depth (i.e. when the effect of gas depletion in the fine bubble stream was

eliminated). As KLa is a local variable dependent on the bubble’s location especially its height

position, KLa0 represents the KLa at the water surface, i.e., at the top of the tank with no gas

depletion, where the saturation concentration corresponds to the atmospheric pressure (Ps = 1 atm).

By the same token, it is speculated that the same correlation would exist for wastewater,

so that a standard specific field baseline (KLa0f)20 per Qa20 q can be equally established by

Page | 221
similar testing on wastewater, provided that any gas depletion effect from microbial respiration

is avoided. The wastewater mass transfer coefficient KLaf can then be calculated by eq. 6-17,

after the baseline KLa0f has been determined by such testing or by laboratory bench-scale testing.

If the Lee-Baillod model (eq. 6-5) is applied, then the water characteristics, such as E (modulus of

elasticity), ρ (density of the wastewater), and σ (surface tension of the wastewater) must be

separately determined in order for the model to be applicable in the temperature correction model

(eq. 2-1). Also, the Henry’s Law constant for the wastewater would be different from that of clean

water, so that the parameter x given by x = HR’T/Ug needs to be adjusted accordingly. (x has been

defined as the gas flow constant in Chapter 4.) The solutions for KLaf by the simulation model

should match actual field-testing results using the methods described in the ASCE Guidelines,

provided the OTRf and KLaf are calculated by eq. 6-32 that would include the respiration rate

separately determined in the field. The concept of additional resistance from the microbes can be

further illustrated by the following mathematical derivation.

6.6.3 Mathematical Derivation for In-process Gas Transfer Model

The various cases are examined as below:

(i) Baseline (R = 0)

First consider the baseline case. For the simple case where oxygen uptake rate is zero, ASCE

(Eq. 2-2) or eq. 6-2 based on a mass balance on the liquid phase gives:

𝑑𝐶
= 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞ 𝑓 – 𝐶) (6 − 33)
𝑑𝑡

Based on a mass balance on the gas phase gives:

𝐹 = 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞ 𝑓 – 𝐶) (6 − 34)

where F is the gas depletion rate per unit volume given by Figure 6-12,

Page | 222
 where,

F.V = 𝜌𝑖 𝑞𝑖 𝑌𝑖 − 𝜌𝑒 𝑞𝑒 𝑌𝑒

where ρ is density of the gas; q is the gas flow rate; subscripts i and e are inlet and exit.

ρe, qe, Ye

EXIT GAS

Aerated liquid volume,

V

Dissolved oxygen
concentration, C

INLET GAS ρi, qi, Yi

Fig. 6-12. Mass Balance on the gas phase

F (t=0) = 𝐾𝐿 𝑎𝑓 . 𝐶 ∗ ∞ 𝑓

𝐹 = 𝐾𝐿 𝑎𝑓 . 𝐶 ∗ ∞ 𝑓 . 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡)

Figure 6-13. Gas Depletion Time Variation (R = 0)

Simplifying the case by assuming the test starts at zero DO, and

integrating Eq. (6-33) gives:

Page | 223
 𝐶 = 𝐶 ∗ ∞𝑓 (1 – 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡)) (6 − 35)

Substituting C from the above expression into Eq. (6-33) gives:

𝑑𝐶
= 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞ 𝑓 − 𝐶 ∗ ∞ 𝑓 (1 – 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡))) (6 − 36)
𝑑𝑡
Hence,

𝑑𝐶
= 𝐾𝐿 𝑎𝑓 . 𝐶 ∗ ∞𝑓 . 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡) (6 − 37)
𝑑𝑡

Since Eq. (6-33) and Eq. (6-34) are the same, (dC/dt = F), therefore,

𝐹 = 𝐾𝐿 𝑎𝑓 . 𝐶 ∗ ∞ 𝑓 . 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡) (6 − 38)
Therefore,

𝐹 (𝑡 = 0) = 𝐾𝐿 𝑎𝑓 . 𝐶 ∗ ∞ 𝑓 (6 − 39)

and, F (t=∞) = 0

Plotting this function F for gas depletion would give an exponential curve as shown in Figure 6-

13. This is the baseline case plot. Without the action of microbial respiration, the oxygenation

capacity of the aeration system is fully utilized. Eventually, the system will balance itself so that

the tank becomes saturated, and the gas transfer is complete. Further continual supply of gas would

not increase the oxygen content in the tank, and the system is said to be in a steady state, as the

feed gas is balanced by the exit gas, and there is no gas depletion at steady state.

(ii) ASCE model for R > 0

In the presence of cell respiration, according to current ASCE 18-96, Eq. (3-1), the gas depletion

rate remains the same under the influence of R, but ASCE Eq. (2-2) now becomes:

𝑑𝐶
= 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 – 𝐶) – 𝑅 (6 − 40)
𝑑𝑡

Integrating Eq. (6-40) with respect to time, gives

Page | 224
 𝐾𝐿 𝑎𝑓 . (𝐶 ∗ ∞𝑓 – 𝐶) − 𝑅
= 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡) (6 − 41)
𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞ 𝑓 – 𝐶0 ) – 𝑅

Again, assuming C0 = 0, at time t = 0, and re-arranging terms,

𝐶 = 𝐶𝑅 (1 – 𝑒𝑥𝑝 (−𝐾𝐿 𝑎𝑓 . 𝑡)) (6 − 42)

where

From eq. (6-40), dC/dt = 0 at steady-state, and C = CR, then

𝑹
𝑪𝑹 = 𝑪∗ ∞ 𝒇 – (𝟔 − 𝟒𝟑)
𝑲𝑳 𝒂𝒇

Substituting Eq. (6-42) into Eq. (6-40) for C,

𝑑𝐶
= 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞ 𝑓 – 𝐶𝑅 (1 – 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡))) – 𝑅 (6 − 44)
𝑑𝑡

Simplifying,

𝑑𝐶
= 𝐾𝐿 𝑎𝑓 . 𝐶𝑅 . 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡) (6 − 45)
𝑑𝑡

From ASCE Eq. (3-1), or eq. 6-34, where ASCE has assumed to be same as the baseline case,

𝐹 = 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 – 𝐶) (6 − 46)

Differentiating w.r.t. t,

𝑑𝐹 𝑑𝐶
= −𝐾𝐿 𝑎𝑓 . (6 − 47)
𝑑𝑡 𝑑𝑡
Substituting (6-45) into (6-47),

𝑑𝐹
= −𝐾𝐿 𝑎𝑓 . 𝐾𝐿 𝑎𝑓 . 𝐶𝑅 . 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡) (6 − 48)
𝑑𝑡

integrating,

𝐹 = 𝐾𝐿 𝑎𝑓 . 𝐶𝑅 . 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡) + 𝐾 (6 − 49)

where K is an integration constant.

The boundary condition is that when t→∞, F→R, and therefore K = R

Page | 225
 Hence,

𝐹 = 𝑅 + 𝐾𝐿 𝑎𝑓 . 𝐶𝑅 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡) (6 − 50)

But since from eq. 6-43,

𝑅
𝐶𝑅 = 𝐶 ∗ ∞𝑓 –
𝐾𝐿 𝑎𝑓

Therefore,

𝐹 = 𝑅 + (𝐾𝐿 𝑎𝑓 . 𝐶 ∗ ∞𝑓 – 𝑅) 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡) (6 − 51)

at t = 0, therefore,

F = 𝐾𝐿 𝑎𝑓 . 𝐶 ∗ ∞ 𝑓

at t = ∞, F = R, the following plot is obtained as shown in Fig. 6-14. The plot as shown in Figure

6-14 is similar to the baseline plot, except that the final steady state gas depletion rate at infinite

time is not zero, but is given by the fixed respiration rate R. At steady state, therefore, the

respiration rate equals the gas depletion rate which is concurrent with the thesis of this paper.

F (t=0) = Klaf. C*∞f 𝐹 = 𝑅 + (𝐾𝐿 𝑎𝑓 . 𝐶 ∗ ∞𝑓 – 𝑅) 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡)

F=R

Fig. 6-14. Gas Depletion Time Variation (R > 0) [ASCE model]

Page | 226
 However, this plot as given by Figure 6-14 shows that the gas depletion is not impaired at

the beginning in the presence of R. Like the previous plot for the case where cells are absent, the

oxygenation capacity is fully utilized at time t = 0. Experiments have shown that this is not the

case, and it is really not logical, since R must affect the gas depletion rate, no matter whether it is

at the beginning, during, or at the end of the test. Mancy and Barlage (1968) described the

phenomenon where long chain charged molecules attach to the gas bubble interfaces and impede

the diffusion of oxygen to bulk solution. The longer the bubbles are in transit to the surface the

more of these materials are attached to the bubbles resulting in a greater resistance to oxygen

transfer and a reduction in alpha (α). Rosso and Stenstrom (2006) have found that bubble surface

contamination equilibrates even before detachment, so that after bubble detachment and during the

transit of bubbles through the liquid, the liquid-side gas transfer coefficient KL is reduced to a

steady-state process value, always lower than the gas transfer coefficient in pure water. This

means that the gas depletion must occur almost immediately upon detachment, and if the cells

exert a transfer resistance, then the reduction of the gas-side depletion rate must start upon

detachment at time t = 0, neglecting the bubble formation stage which is small compared to the

time taken for the bubble transit to the surface. Therefore, the gdp at t = 0 must be smaller than

the baseline case at t = 0. They should not be the same. This graph based on the ASCE model must

therefore be incorrect.

(iii) Proposed Model for R > 0

Going through the same process, but with the ASCE Eq. (2-2) or eq. 6-2 proposed to be changed
to:

𝑑𝐶
= 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞ 𝑓 – 𝐶) – 2𝑅 (6 − 52)
𝑑𝑡

and the gas phase mass balanced is changed to:

Page | 227
 𝐹 = 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 – 𝐶) − 𝑅 (6 − 53)

the same expression for the gas depletion function is obtained, i.e.,

𝐹 = 𝑅 + 𝐾𝐿 𝑎𝑓 . 𝐶𝑅 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡) (6 − 54)

This is similar in expression as Eq. (6-43) above, but with CR modified to:

𝟐𝑹
𝑪𝑹 = 𝑪∗ ∞ 𝒇 – (𝟔 − 𝟓𝟓)
𝑲𝑳 𝒂𝒇

Therefore,

𝐹 = 𝑅 + (𝐾𝐿 𝑎𝑓 . 𝐶 ∗ ∞ 𝑓 – 2𝑅) 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡) (6 − 56)

at t = 0, therefore,

𝐹 = 𝐾𝐿 𝑎𝑓 . 𝐶 ∗ ∞𝑓 – 𝑅 (6 − 57)

at t=∞, F = R

The plot then becomes as shown in Figure 6-15.

F = 𝑅 + (𝐾𝐿 𝑎𝑓 . 𝐶 ∗ ∞ 𝑓 – 2𝑅) 𝑒𝑥𝑝(−𝐾𝐿 𝑎𝑓 . 𝑡)

F (t=0) = Klaf. C*∞f -R

Fig. 6-15. Gas Depletion Time Variation (R > 0) Modified model

Page | 228
 This plot shows that the initial depletion rate is reduced by an amount equal to the

respiration rate R. Experiments have borne out the fact that, when respiring cells are present, the

initial gas depletion must be smaller than when cells are absent, as evidenced by the higher off-

gas content compared with the non-cell test. Furthermore, the non-cell condition would give a zero

depletion rate at the end when the off-gas is equal to the feed gas content; whereas, in the case of

the oxygen uptake rate (OURf) R achieving a steady state, the off-gas mole fraction becomes

constant at a lower value than 0.2095, and F at steady state equates to the respiration rate R.

Since the gas depletion represents the net oxygen transfer, the OTRf therefore equates to

the consumption by the microbes, as is expected if a steady state is reached under the influence of

the respiring cells. Using the principle of superposition, the total oxygen transfer rate remains

given by KLaf. ( 𝐶 ∗ ∞𝑓 – C) as if the cells are not present (the baseline case), and KLaf is then a

fixed constant independent of R and gas depletion. This plot is therefore more correct for a

consistent interpretation of the mass transfer coefficient KLaf.

The conclusion of this exercise is that, for submerged aeration where gas loss rate from the

system is significant, the rate of transfer under the action of microbial respiration should be given

by Eq. (6-52) reproduced below:

𝒅𝑪
= 𝑲𝑳 𝒂𝒇 (𝑪∗ ∞ 𝒇 – 𝑪) – 𝟐𝑹
𝒅𝒕

This equation should then replace Eq. (2-2) in the ASCE 18-96 Guidelines. Experimental data does

not exist to verify the gas depletion model as shown in Figure 6-15, since no data on direct

comparison of a baseline case and a real case (R > 0) is available. However, since it is illogical to

assume that R does not affect the gas depletion in the beginning of the test but does affect it at the

end, it is likely the current ASCE model is incorrect.

Page | 229
Furthermore, if the same gas flow rate is applied, successive tests for estimating alpha using

increasing MLSS (hence increasing steady state uptake rate) will indicate whether the initial gas

depletion rate should be diminished by the uptake rate R.

Alpha vs. MLSS

3.50
3.00
alpha (Klaf/Kla) x10

2.50
2.00
1.50
1.00
0.50
0.00
0 5000 10000 15000 20000
MLSS (mg/L)

Fig. 6-16. Relationship between alpha and MLSS for the membrane diffuser at 0.0283 m3/min (1
SCFM)

OFFGAS @ (.0283 m3/min) 1 scfm

0.2050
Off gas Oxygen mole fraction ye

0.2040
0.2030
0.2020
0.2010
0.2000
0.1990
0.1980
0.1970
0.1960
0.1950
0 5000 10000 15000 20000
MLSS concentration (mg/L)

Fig. 6-17. Relationship between offgas and MLSS for the membrane diffuser at 1 SCFM

Page | 230
This is indeed the case by examining Jing Hu’s data [Hu 2006]. His data on the measurements of

alpha and offgas values are plotted as shown in Figure 6-16 and Figure 6-17 (which is the same as

Fig. 6-3). Figure 6-16 shows that there is a general trend of decreasing alpha, hence in Figure 6-

17, a decreasing gas depletion rate or increasing off-gas emission rate is obtained, for increasing

MLSS or increasing R. In other words, the effect of R is a suppression of the gas depletion or a

suppression of the net oxygen transfer rate in the system.

This phenomenon then agrees with the model that the gas depletion is given by eq. 6-53 above:

𝑭 = 𝑲𝑳 𝒂𝒇 (𝑪∗ ∞ 𝒇 – 𝑪) − 𝑹

instead of the current ASCE 18-96 model for gas depletion rate given by ASCE’s Eq. (3-1) as F =

KLaf (𝐶∗ ∞ 𝑓 – C). Similarly, the gas transfer rate on the liquid phase would then be given by Eq.

(6-52) above. It should be noted in passing that the above plot as shown in Figure 6-17 is obtained

when the offgas data for the same test is plotted against the MLSS [Hu J. 2006].

6.6.4 A simple method to eliminate the impact of free surface oxygen transfer (speculative)

McWhirter et al. (1989) points out that the mass transfer analysis of the oxygen transfer

performance of diffused air or subsurface mechanical aeration systems has progressed very little

over the past decades and is still true today. The recently‐developed ASCE Standard [ASCE 2007]

method for determination of the oxygen mass transfer performance of diffused or subsurface

aeration systems is based on a greatly over‐simplified mass transfer model. Although the ASCE

Standard can be used to empirically evaluate point performance conditions, it does not provide a

meaningful representation of the actual mass transfer process and is not capable of accurately

assessing or predicting performance under changing operating or environmental conditions.

Page | 231
According to DeMoyer et al. (2003), the standard testing methodology for oxygen transfer makes

adjustment of measured values to other depths intangible.

Although they now have a model that separates the bubble mass transfer coefficient (KLab)

and the surface transfer coefficient (KLas), their method only gives insight into the relative

importance of transfer across the free water surface versus bubble surface. They rightly point out

that bubble gas-water transfer is the dominant means of oxygen transfer. Their model results

indicate that the surface transfer coefficient in a 9.25 m water depth circular tank of a diameter 7.6

m with an air flow rate of 51 to 76 scmh (Ug = .019 m/min to .028 m/min) is 59-85% of the bubble

transfer coefficient. However, because of the hydrostatic pressure, the driving force inside a tank

is higher than that on the surface, and so, the deeper the tank, the less is the relative importance of

surface transfer. Also, the coefficients would depend on the gas flow rates.

The approach taken by this book is different from DeMoyer’s approach in that the baseline

mass transfer coefficient (KLa0) is a lumped parameter that includes both effects. The model was

developed based on first assuming that the surface transfer has no effect along with other

assumptions such as constant bubble volume, and then later on adjusting the model by introducing

calibration factors--- n, m for the Lee-Baillod model; and e and (1 – e) for the depth correction

model. This approach has proven to be successful for translating the mass transfer coefficient KLa

from one depth to another via the baseline KLa0, within the cited range of gas flow rates tested, but

is not expected to simulate well for high gas flow rate discharge and/or shallow tanks where gas

transfer over the surface is expected to adopt a more prominent influence than the bubble transfer

on which the theoretical development was based. This can be illustrated by citing some examples

using the data obtained by GSEE, Inc. (available online at www.canadianpond.ca) who performed

clean water tests at various air flow rates on a 6.5 m dia. circular tank and 9.45 m deep, using

Page | 232
bubble tubings and OctoAir-10 aeration systems. Tests were carried out at 1.52 m (5 ft), 3.05 m

(10 ft), 4.57 m (15 ft), and 6.10 m (20 ft) for the tubings. For the OctoAir, the depths range from

1.32 m, 2.84 m, 4.37 m to 5.89 m inclusive.

Using the method proposed in this book, the baseline coefficients were calculated and

plotted against the average gas flowrates, as shown in Fig.6-17 below. As can be seen, the curves

are not equal, indicating that other factors are affecting the baseline curve. However, by assuming

that the deepest tank has a baseline that is almost free from interference or suffering the least from

such interference due to surface effect, the other tanks are then normalized to this tank with Zd =

4.57 m. On examining the data, it can be shown that there is a definite correlation between the

depth ratio of various tanks and the baseline coefficient and the correlation is a power function.

The exponent for the depth ratio for the adjustment factor is found by minimizing the

standard error between the predicted and measured baseline values, which will then give the best-

fit values of the normalized baseline coefficients KLa0(N) as shown in Fig. 6-18.

The graphs (Fig. 6-18) show that the adjustment factor (z/z).5 appears to give a better correlation

than (z/z).3, therefore, where Zd = 4.57 m. The normalized baseline KLa0(N) for any average gas

flow rate and tank height zd would be given by eq. 6-58 as stated below.

Similarly, for the OctoAir aeration device, it would appear that the baseline should be

normalized to 5.89 m at a normalization factor of (z/z)0.33, resulting in the graph shown in Fig. 6-

19. (The last three points are for 4.37 m, 2.84 m, and 1.32 m respectively.)

𝐾𝐿 a0 (N) = 𝐾𝐿 a0 x sqrt(zd/Zd) (6 − 58)

The contribution of the water surface transfer amounts to 9%, 21% and 39% respectively. Once

the normalization factor is determined, the baseline at any gas flow rate can be determined by eq.

6-58, but with (z/Zd)^0.33 as the correction factor instead of (z/Zd)^0.5, and the normalized

Page | 233
baseline can then be used to simulate the mass transfer coefficient (KLa) for any other conditions

just like the methodology used before for the other tests. It should be understood that the

Bubble Tubing 3/4"

baseline KLa for various depths
0.0180
0.0160
y = 0.036x0.5621 y = 0.0278x0.5715
0.0140 R² = 0.9977
R² = 1
0.0120
Kla0 (1/min)

0.0100
Zd=1.52
0.0080
Zd=3.05
0.0060
y = 0.0213x0.5615
R² = 0.9405 Zd=4.57
0.0040
0.0020
0.0000
0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000
avg. gas flowrate Qa (m3/min)

Fig. 6-17. Relationship of baseline and gas flow rate before normalization

Bubble Tubing 3/4"

Kla0(N) [normalized to 4.57 m]
0.0140
Klao(N)=Kla0 x sqrt(zd/Zd) (1/min)

0.0120 y = 0.0236x0.5672
R² = 0.9374
0.0100

0.0080

0.0060 (z/z)^.5
y = 0.0216x0.5646 (z/z)^.3
0.0040 R² = 0.9983

0.0020

0.0000
0.000 0.050 0.100 0.150 0.200 0.250 0.300
Qa (m3/min)

Fig. 6-18. Comparison of two different normalization factors

Page | 234
 Octo air 10
(Baseline normalized to 5.89 m at (z/z)^.33)

0.0250

0.0200
(KLa0)20 (1/min)

y = 0.0295x0.7378
0.0150 R² = 0.9994
KLa0
0.0100
KLa0(n)

0.0050

0.0000
0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450
Qa20 (m3/min)

Fig. 6-19. Relationship between the baseline and gas flow rate after normalization

normalized baseline represents that which is free from the surface effect; so that where the gas

discharge is large, or the tank is shallow, therefore, any significant surface transfer must be added

to such simulation either afterwards, using an additional model such as postulated by DeMoyer et

al. (2002), as a percentage of the estimated KLa, or by pro rata using eq. 6-58 to calculate the true

(surface + bubble) KLa0 prior to the simulation. As an example, to calculate the standard baseline

for the 1.32 m tank at a gas flow rate of 0.23 m3/min, the reading from the graph gives a value of

0.01 min-1, and so the true baseline mass transfer coefficient would be given by (0.01) x

(5.89/1.32)0.33 which equals to 0.0164 min-1 for the baseline value. This value should then be used

for any simulation on this tank for any gas flowrate and environmental conditions in order to

calculate the standard mass transfer coefficient (KLa)20.

Page | 235
6.7 Notation

α wastewater correction factor, ratio of process water α.KLa to clean water KLa

α·F ratio of the process water (α.KLa)20 of fouled diffusers to the clean water (KLa)20
of clean diffusers at equivalent conditions (i.e., diffuser airflow, temperature,
diffuser density, geometry, mixing, etc.), and assuming(α.KLa)20/(KLa)20
≈(α.KLa)/(KLa)
α’ wastewater correction factor, ratio of process water (with no microbes) KLaf to
clean water KLa
β correction factor for salinity and dissolved solids, ratio of 𝐶 ∗ ∞ in wastewater to
tap water
𝐶 ∗∞ oxygen saturation concentration in an aeration tank (mg/L)
𝐶 ∗∞f oxygen saturation concentration in an aeration tank under field conditions (mg/L)
𝐶 ∗ ∞ 20 oxygen saturation concentration in an aeration tank at 20 0C (mg/L)

𝐶 ∗∞0 hypothetical oxygen saturation concentration in an aeration tank at zero gas

depletion rate (mg/L)
Cs oxygen saturation concentration in an infinitesimally shallow tank, also known as
oxygen solubility in water at test temperature and barometric pressure (mg/L)
C dissolved oxygen concentration in a fully mixed aeration tank (mg/L)
C* a broad term for any saturation concentration in equilibrium with the partial
pressure of oxygen in the gas phase, microscopically or macroscopically
CR the “apparent” or steady-state saturation concentration in the field (mg/L)
DO Dissolved oxygen or dissolved oxygen concentration (mg/L)
KLa volumetric mass transfer coefficient also known as the apparent volumetric mass
transfer coefficient (min-1 or hr-1)
(KLa)20 volumetric mass transfer coefficient also known as the apparent volumetric mass
transfer coefficient at standard conditions (min-1 or hr-1)
KLaf mass transfer coefficient as measured in the field, equals α’.KLa (min-1 or hr-1)
KLa0 baseline mass transfer coefficient, equivalent to that in an infinitesimally shallow
tank with no gas side oxygen depletion (min-1 or hr-1)
(KLa0)20 clean water baseline mass transfer coefficient at standard conditions, equivalent to
that in an infinitesimally shallow tank with no gas side oxygen depletion at
standard conditions (min-1 or hr-1)

Page | 236
KLa0f wastewater baseline mass transfer coefficient, equivalent to that in an
infinitesimally shallow tank with no gas side oxygen depletion (min-1 or hr-1)
(KLa0f)20 wastewater baseline mass transfer coefficient at standard conditions, equivalent to
that in an infinitesimally shallow tank with no gas side oxygen depletion (min-1 or
hr-1)
Ɵ.𝜏. ß. 𝛺 temperature, solubility, pressure correction factors as defined in ASCE 2007
MLSS mixed liquor suspended solids (mg/L or g/L)
OTE oxygen transfer efficiency (%)
p. SOTE predicted standard oxygen transfer efficiency (%)
rpt. SOTE reported standard oxygen transfer efficiency (%)
OTR oxygen transfer rate (kg O2/hr)
OTRf oxygen transfer rate in the field under process conditions, equals:
KLaf (C*∞f – C) V -RV (for batch process) (kg O2/hr)
OTRcw oxygen transfer rate under non-steady state oxygenation in clean water
(kg O2/hr)
OTRww oxygen transfer rate under non-steady state oxygenation in wastewater
without any microbial cell respiration (kg O2/hr)
SOTR standard oxygen transfer rate in clean water as defined in ASCE 2007 (kg O2/hr)
OUR oxygen uptake rate (kg O2/hr)
OURf oxygen uptake rate in the field, measurable by the off-gas method (kg O2/hr)
R microbial respiration rate, also known as the oxygen uptake rate (OUR) (kg
O2/m3/hr usually expressed as mg/L/hr)
R0 specific gas constant for oxygen (kJ/kg-K)
V volume of aeration tank (m3)
z depth of water at any point in the tank measured from bottom, see Fig. 1 (m)
Zd submergence depth of the diffuser plant in an aeration tank (m)
Ze equilibrium depth at saturation measured from bottom, see Fig. 1 (m)
Pa, Pb atmospheric pressure or barometric pressure at time of testing (kPa)
Pe equilibrium pressure of the bulk liquid of an aeration tank (kPa) defined such
that: Pe = Pa + rw de -Pvt where Pvt is the vapor pressure and rw is the specific
weight of water in N/m3 (kPa)

Page | 237
de effective saturation depth at infinite time (m)
Ye oxygen mole fraction at the effective saturation depth at infinite time
Y0 initial oxygen mole fraction at diffuser depth, Yd, also equal to exit gas mole
fraction at saturation of the bulk liquid in the aeration tank, Y0 = 0.2095 for air
aeration
H Henry’s Law constant (mg/L/kPa) defined such that:
𝐶 ∗ ∞ = HYePe or Cs = HY0Pa
Yex exit gas or the off-gas oxygen mole fraction at any time
y oxygen mole fraction at any time and space in an aeration tank defined by an
oxygen mole fraction variation curve
gdp gas-side oxygen depletion rate (equals OTR; equals zero at steady state for clean
water; equals OTRf in process water; equals gdpf at steady state) (kg O2/hr)
𝑔𝑑𝑝𝑐𝑤 gas-side oxygen depletion rate in clean water (equals OTRcw) (kg O2/hr)
𝑔𝑑𝑝𝑤𝑤 gas-side oxygen depletion rate in wastewater (equals OTRww) (kg O2/hr)
𝑔𝑑𝑝𝑓 specific gas-side oxygen depletion rate due to microbes-induced resistance
(equals OTRf at steady state); equals the microbial respiration rate R in process
water (kg O2/hr)
Qa height-averaged volumetric air flow rate (m3/min or m3/hr)
Qs, AFR gas (air) flow rate at standard conditions (20°C for US practice and 0°C for
European practice), in (std ft3/min or Nm3/h)
n, m calibration factors for the Lee-Baillod model equation for the oxygen mole
fraction variation curve
WO2 mass flow of oxygen in air stream (kg/h)

Page | 238
References

American Society of Civil Engineers (1997). Standard guidelines for in-process oxygen transfer
testing, 18-96. Reston, VA.
American Society of Civil Engineers (2018). Standard Guidelines for In-Process Oxygen
Transfer Testing ASCE/EWRI 18-18. Reston, VA.
American Society of Civil Engineers (2007). Standard 2-06: Measurement of oxygen transfer in
clean water. Reston, VA.
Baillod Robert, C. (1979). Review of oxygen transfer model refinements and data interpretation.
Proc., Workshop toward an Oxygen Transfer Standard, U.S. EPA/600-9-78-021, W. C.
Boyle, ed., U.S. EPA, Cincinnati, 17-26.
Baquero-Rodríguez, G. A.; Lara-Borrero, J. (2016) The Influence of Optic and Polarographic
Dissolved Oxygen Sensors Estimating the Volumetric Oxygen Mass Transfer Coefficient
(KLa). Mod. Appl. Sci., 10 (8), 142–151
Baquero-Rodriguez Gustavo A., Lara-Berrero Jaime A., Nolasco Daniel, Rosso Diego (2018). A
Critical Review of the Factors Affecting Modeling Oxygen Transfer by Fine-Pore
Diffusers in Activated Sludge. Water Environment Research, Volume 90, Number 5, pp.
431-441(11).
Bewtra, J., Tewari, P. (1982). Alpha and Beta Factors for Domestic Wastewater. Journal (Water
Pollution Control Federation), 54(9), 1281-1287. Retrieved from
http://www.jstor.org/stable/25041683
Birima, A. H., Noor, M. J. M. M., Mohammed T. A., Idris, A., Muyibi, S. A., Nagaoka, H.,
Ahmed, J., Ghani, L. A. A. (2009). The effects of SRT, OLR and feed temperature on the
performance of membrane bioreactor treating high strength municipal wastewater,
Desalination and Water Treatment, 7:1-3, 275-284.
Boon, Arthur G. (1979). Oxygen transfer in the activated-sludge process. Water Research
Centre, Stevenage Laboratory, U.K. Proc., Workshop toward an Oxygen Transfer
Standard, U.S. EPA/600-9-78-021, W. C. Boyle, ed., U.S. EPA, Cincinnati, 17-26.
Boyle, W.C., Campbell, H.J. Jr. (1984) “Experiences with oxygen transfer testing of diffused air
systems under process conditions”, Wat. Sci. Tech. Vol. 16, Vienna, pp. 91-106.

Page | 239
Boyle, W.C., Stenstrom, M.K., Campbell H.J. Jr., Brenner R.C. (1989) “Oxygen Transfer in
Clean and process Water for Draft Tube Turbine Aerators in Total Barrier Oxidation
Ditches”, Journal of the Water Pollution Control Federation, Vol. 61, No. 8, (Aug. 1989)
CEE 453: (2003). Laboratory Research in Environmental Engineering
http://ceeserver.cee.cornell.edu/mw24/Archive/03/cee453/Lab_Manual/pdf/Gas%20transf
er.pdf
Chiesa, S.C., Rieth, M.G., and Ching T. (1990). Evaluation of activated sludge oxygen uptake
rate test procedures. J. Environ. Engr. Div., ASCE, 116:472.
DeMoyer, Connie D., Schierholz Erica L., Gulliver John S., Wilhelms Steven C. (2002). Impact
of bubble and free surface oxygen transfer on diffused aeration systems. Water Research
37 (2003) 1890-1904.
Doyle, M. 1981. Small scale determination of alpha in a fine bubble diffuser system. Madison,
WI: Department of Civil and Environmental Engineering, University of Wisconsin.
Eckenfelder, W.W. (1952). Aeration Efficiency and Design: I. Measurement of Oxygen Transfer
Efficiency. Sewage and Industrial Wastes, 24(10), 1221-1228.
Eckenfelder, W. W. (1970). Water pollution control: experimental procedures for process design
(Austin: Pemberton Press).
Garcia-Ochoa, F., & Gomez, E. (2009). Bioreactor scale-up and oxygen transfer rate in microbial
processes: an overview. Biotechnology advances, 27(2), 153-176.
Garcia-Ochoa, F., E Gomez, VE Santos, JC Merchuk (2010). Oxygen uptake rate in microbial
processes: An overview. Biochemical Engineering Journal 49 (3), 289-307.
Hu, J. (2006). Evaluation of parameters influencing oxygen transfer efficiency in a membrane
bioreactor (Doctoral dissertation).
Hwang, H. J., Stenstrom, M. K. (1985). Evaluation of fine-bubble alpha factors in near full-scale
equipment. Journal (Water Pollution Control Federation), Volume 57, Number 12.
Jiang, P. & Stenstrom, M. K. (2012). Oxygen transfer parameter estimation: impact of
methodology. Journal of Environmental Engineering, 138(2), 137-142.
Lee, J. (2017). Development of a model to determine mass transfer coefficient and oxygen
solubility in bioreactors. Heliyon 3(2): e00248.
Lee, J. (2018). Development of a model to determine the baseline mass transfer coefficients in
aeration tanks, Water Environ. Res., 90, (12), 2126 (2018).

Page | 240
Lewis, W. K., Whitman, W. G. (1924). Principles of gas absorption. Ind. Eng. Chem.
16(12):1215-1220.
Mancy, K. H., and Barlage, W. E., Jr. (1968) Mechanism of Interference of Surface Active
Agents with Gas Transfer in Aeration Systems, Advances in water quality, F. Gloyna and
W W Eckenfelder Jr. (eds.), University of Texas Press, Austin, TX, 262-286
Mahendraker V. (2003) “Development of a unified theory of oxygen transfer in activated sludge
processes – the concept of net respiration rate flux”, Department of Civil Engineering,
University of British Columbia.
Mahendraker, V., Mavinic, D.S., and Rabinowitz, B. (2005a). Comparison of oxygen transfer
test parameters from four testing methods in three activated sludge processes. Water Qual.
Res. J. Canada, 40(2).
Mahendraker, V. Mavinic, D.S., and Rabinowitz, B. (2005b). A Simple Method to Estimate the
Contribution of Biological Floc and Reactor-Solution to Mass Transfer of Oxygen in
Activated Sludge Processes. Wiley Periodicals, Inc. DOI: 10.1002/bit.20515.
McKinney and Stukenberg (1979). On-site Evaluation: Steady State vs. Non-Steady State
Testing. Proc., Workshop toward an Oxygen Transfer Standard, U.S. EPA/600-9-78-021,
W. C. Boyle, ed., U.S. EPA, Cincinnati, 17-26.
McCarthy, J. (1982). TECHNOLOGY ASSESSMENT OF FINE BUBBLE AERATORS. U.S.
Environmental Protection Agency, Washington, D.C., EPA/600/2-82/003 (NTIS
PB82237751).
McWhirter John R., Hutter Joseph C. (1989). Improved oxygen mass transfer modeling for
diffused/subsurface aeration systems. AIChE J 1989;35(9):1527-34
https://doi.org/10.1002/aic.690350913
Metcalf and Eddy. (1985) Wastewater engineering: treatment and disposal, 2nd ed. (New York:
McGraw-Hill).
Mott, H. V. (2013). Environmental process analysis: principles and modelling (John Wiley &
Sons).
Mueller, J. A. and Boyle, W. C. 1988. Oxygen transfer under process conditions. J. Water Pollut.
Cont. Fed., 60:332–341.
Rosso D. & Stenstrom M. (2006a). Alpha Factors in Full-scale wastewater aeration systems.
Water Environment Foundation. WEFTEC 06.

Page | 241
Rosso D. & Stenstrom M. (2006b). Surfactant effects on α-factors in aeration systems. Water
Research 40(7): 1397-1404, Elsevier Ltd.
Rosso et al. (2017). Modelling oxygen transfer using dynamic alpha factors. Water Research,
124 (2017) 139-148.
Santos V.E. et al. (2006) Oxygen uptake rate measurements both by the dynamic method and
during the process growth of Rhodococcus erythropolis IGTS8: modelling and difference
in results, Biochem. Eng. J. 32 (2006) 198-204.
Shraddha M., Narang Atul (2018). Quantifying the parametric sensitivity of ethanol production
by Scheffersomyces (Pichia) stipitus: development and verification of a method based on
the principles of growth on mixtures of complementary substrates. Microbiology. 164.
10.1099/mic. 0.000719.
Stenstrom et al. (1981). Effects of alpha, beta and theta factor upon the design, specification and
operation of aeration systems. Water Research, 15(6), 643-654.
Tchobanoglous, G., Burton, F.L., Stensel, H.D., Eddy, M. (2003). Wastewater engineering:
Treatment and Reuse, McGraw-Hill Series in Civil and Environmental Engineering,
Fourth ed. McGraw-Hill New York, NY.
Yunt Fred W., Hancuff Tim O., Brenner Richard C. (1988a). EPA/600/2-88/022. Aeration
equipment evaluation. Phase 1: Clean water test results. Contract No. 14-12-150. Los
Angeles County Sanitation District, Los Angeles, CA. Municipal Environmental Research
Laboratory Office of Research and Development, U.S. EPA, Cincinnati, OH.
Yunt Fred W., Hancuff Tim O. (1988b). EPA/600/S2-88/022. Project summary: aeration
equipment evaluation. Phase I: Clean water test results. Water Engineering Research
Laboratory, Cincinnati, OH.
Yunt, F. et al. (1988c). Aeration equipment evaluation – Phase II Process Water Test Results.
Contract No. 68-03-2906. (undated report). Los Angeles County Sanitation Districts, Risk
Reduction Engineering Laboratory Office of Research and Development, U. S. EPA,
Cincinnati, OH.
Zhou, Xiaohong, et al. (2012). Evaluation of oxygen transfer parameters of fine-bubble aeration
system in plug flow aeration tank of wastewater treatment plant. Journal of Environmental
Sciences 25(2).

Page | 242
Chapter 7. Recommendation for further testing and research

7.0. Introduction

The alpha factor (α) represents the ratio of the mass transfer coefficient in process water

KLaf to KLa in clean water at equivalent test conditions and this ratio can range from approximately

0.1 to greater than 1.0 (ASCE 2007). The subscript f signifies field conditions. This wide range

makes it very difficult to design an aeration tank for a wastewater treatment plant. An opportunity

exists for testing for relationships between alpha (α) and tank volume. The Water Research

Foundation (WRF) has accepted a pre-proposal for initiating such a project, based on new model

discoveries in recently published papers [Lee, 2017][Lee, 2018]. If accepted, the foundation would

sponsor $75,000. The budgeted project would require the erection of a tank, say 4.6 m (15 ft) to

6.1 m (20 ft) tall, and with a tentative plan area of 2 m by 3 m. The proposal applies to diffused

bubble aeration only.

Other investigators have concluded that it is not alpha versus tank volume that matters, but alpha

versus diffuser submergence [Boon and Lister, 1973, 1975][Doyle et al., 1983][Groves et al.,

1992] [Mike Stenstrom et al., 2006a, 2006b]. In general, these researchers found that as diffuser

submergence is increased, alpha is reduced. Mancy and Barladge (1968) described the

phenomenon where long chain charged molecules attach to the gas bubble interfaces and impede

the diffusion of oxygen to the bulk aqueous solution as dissolved oxygen (DO). The longer the

bubbles are in transit to the free water surface the more of these materials are attached to the

bubbles resulting in a greater resistance to oxygen transfer and a reduction in alpha. However, Keil

and Russell (1987) developed the bubble recirculation cell to model sparged aeration tank. The

method regards each sparger (or diffuser) to be substantially independent of neighbouring spargers,

Page | 243
and hence the aerated liquid can be divided into cells, each cell corresponding to the space around

one sparger. This approach means that the behaviour of a single sparger can be studied and the

results then can be applied to the whole aerated volume if the bulk liquid is completely mixed.

This is especially true if the diffuser plants are arranged as full-floor coverage.

In clean water aeration, the proposed new model called the Lee-Baillod model allows plotting the

mole fraction curve along the tank height starting from the diffuser submergence depth Zd. At

equilibrium, the mole fraction is at a minimum ye which occurs at an effective depth de [ASCE

2007], as shown in Fig. 7-1 below.

Furthermore, another proposed model, called the depth correction model [Lee 2018], then allows

plotting KLa with depths based on a baseline KLa that is defined as the mass transfer coefficient at

zero depth, essentially meaning zero gas depletion. The Proposer postulates that only then can one

compare clean water KLa with in-process water KLaf that would make alpha invariant. Since the

model depends on the superficial gas flow rate (Ug) that depends on the horizontal cross-sectional

area of the tank for a fixed gas supply, the volume does come into play.

7.0.1. Testing for alpha in aeration tank

After building a tank of an appropriate size it may then be possible to see if a bench scale

experiment can actually predict the KLa in a higher scale. The work so far has agreed that KLa

would decrease with depth which is why the baseline KLa is always the maximum value. By testing

at different depths and different flow rates, it may, according to the proposed thesis, yield a

constant baseline value. If this is true, then rating curves may be readily produced for aeration

systems for various gas flow rates, depths, and other environmental conditions. Some researchers

believe that the baseline KLa at zero depth is a hypothetical concept that has no practical value,

and have suggested that study must be carried out on full-scale systems. Then there

Page | 244
 tank wall clean water curve barometric
at SS presssure Pa or Pb
process water
curve at SS free surface

equilibrium
level at de
pressure Pe

z bulk mixed liquor Zd

DO = C*∞f (R = 0) Ze
DO = CR (R > 1)

initial mole fraction of

ye oxygen assumed Y0 = 0.2095
y
for air at pressure Pd

Fig. 7-1. Oxygen Mole fraction curves at saturation for in-process water compared to clean water
based on the Lee-Baillod model (e = equilibrium; SS = steady state) [Lee 2018]

is no need to worry about scale up. According to them, in real systems there are so many things that

impact alpha, including sludge retention time (SRT) and organic loading among others, which are

so highly variable that generating meaningful rating curves is not deemed possible. The proposer

disagrees with the above argument that the baseline has no practical value. This proposal has

included a worked example of how to use the baseline to predict or simulate mass transfer

coefficients for full scale plants. The proposal is that these other ‘things’ can be separately

modelled, so that alpha pertains only to the waste characteristics as an intensive property, while

the mass transfer equations are modified to include the other things as extensive properties. The

baseline KLa is a hypothetical parameter defined mathematically as lim 𝐾𝐿 𝑎 = 𝐾𝐿 𝑎0 , where KLa0

𝑍𝑑→0

represents the baseline. Since every tank under aeration has gas depletion in submerged aeration,

Page | 245
it is not possible to relate tanks of different height or diffuser submergence (Zd), unless they are all

reduced to zero depth or to a very, very shallow depth as to be infinitesimal. The proposed Lee-

Baillod model relates KLa to depths based on this hypothetical parameter. For every tank, it is

possible to back calculate from the measured KLa to this baseline where the gas depletion becomes

zero or approaching zero. Baillod [1979] called this the "true" KLa. [See Appendix].

Since every tank no matter how shallow has some physical height, there is no such thing as "true"

KLa in any clean water test. However, when extrapolating to zero depth, it becomes a baseline,

from which other depths can be contemplated. Based on Yunt's experiments [Yunt et al.

1980][Yunt 1988], tanks of 3.05 m (10 ft), 4.57 m (15 ft), 6.09 m (20 ft) and 7.62 m (25 ft) were

measured. The proposer found that each of these tanks can be back calculated to find the baseline

and the baseline is a constant no matter what the tank depth is, when the gas flow rate is normalized

to the same "height-averaged" volumetric flow rate. The error of estimation is around 1 ~ 3%.

Therefore, it is hypothesized that the same model would apply to wastewater under aeration (as

shown by the second curve, for process water, in Fig. 7-1). At steady state, the initial mole fraction

curve is shifted to the left due to the effect of microbial activities. How much it is shifted would

depend on the respiration rate of the microbes. The proposer postulates that this amount of shifting

of the oxygen mole fraction curve can be estimated by the principle of superposition [Lee

2018][Lee 2019b] by first assuming that the microbiological effect is negligible, and then

vectorially adding the effect to the mass balance equation. If one can determine KLaf in bench scale

or pilot scale, it should be possible to calculate the corresponding KLaf in full scale, based on the

depth correction model verified by clean water tests. This is the hypothesis underlying this

proposal. The parameter alpha (α) is normally measured in the field as a common practice. Such

measurement necessarily includes all the effects of field variables on the mass transfer coefficient.

Page | 246
Such an approach confound the meaning of alpha with too many variables. When Eckenfelder

(1952) first designed his test, the biological solids was filtered out to exclude the biological

interference, so that his alpha value is dependent only on the wastewater characteristics which is,

for all intents and purposes, an intensive property (ie., independent of scale) of the mixed liquor.

When the wastewater characteristics is constant, alpha is constant.

It is possible that the magnitude of KLa decreases in systems where there is biological

growth. According to some researchers, that could be because, for instance, a high rate of O2

utilization means a lot of suspended bacteria in the water, and that affects the aqueous diffusivity

of O2 in the water. (Exactly how is not known, since suspended bacteria only constitute a tiny

fraction of the bulk mass of the liquid.) Hence, the “individual” mass-transfer coefficients kL, kG

could be affected. Since (1/KL) = (1/kL) + (1/H*kG), so if the chemistry or physics of the water

decreases kL, then KL will also be decreased. This could explain why mass transfer is lower than

expected in systems with biological activity. But this is still because of the changes in wastewater

characteristics (by whatever means) that changes the diffusivity resulting in a change of resistance

to gas transfer. Therefore, by relating alpha only to the wastewater characteristics, it could at least

eliminate one variable, the biological activities affecting alpha, that leaves the other variables to

be elucidated one by one by other means. This alpha should be distinguished by a different symbol

such as α’ to avoid confusion with the current understanding of alpha.

Mahendraker et al. (2005) postulated that the overall resistance in the mixed liquor is the sum of

the liquid’s resistance (i.e. that due to the water characteristics alone) and an additional resistance

due to the biological floc. This concept leads to the equation that 1/αKLa =1/α’KLa + 1/KLabf where

the subscript bf denotes biological floc which is a function of the respiration rate R. In their paper,

the authors did not correlate the biological floc with the respiring rate R, but it can be seen that the

Page | 247
parameter KLabf has the same form as in the basic transfer equation. From this, it can be shown

that α’ = 2α which is similar to the hypothesis proposed by the Proposer whose proposed

undertanding of alpha is α’. This has been the subject of another paper [Lee 2019b]. If this concept

is true, Eq. CG-1 in ASCE 2-06 [ASCE 2007] would become modified to the following:

1
𝑂𝑇𝑅𝑓 = ( ∗ ) [ 𝛼 (𝑆𝑂𝑇𝑅)Ɵ𝑇−20 ]( 𝜏. ß. 𝛺. 𝐶 ∗ ∞ 20 − 𝐶) – 𝑅𝑉 (7 − 1)
𝐶 ∞ 20

where α = α’ and other symbols are as defined in the Standard [ASCE 2007]. This equation would

require determination of the respiration rate R as opposed to the original equation where the effect

of biological activities is included in alpha [Lee 2019b].

7.0.2. Proposed Test Facility

Most clean water testing is performed using the clean water standard developed by the American

Society of Civil Engineers (ASCE 2007). It is proposed that an outdoor, all-steel, rectangular

aeration tank with dimensions of 6.1 m x 6.1 m x 7.6 m sidewater depth (SWD) to be used for all

tests. Depending on budget, the plan size of the tank may be reduced, perhaps to half-scale. The

test tank for use in this project is proposed to be similar to that shown in Fig. 7-4 and Fig. 7-5

below extracted from the literature [Wagner et al., 2008]. To study the wall effect, a small scale

test column to the same maximum height can be separately erected to serve as a control. Potable

water is to be used in all clean water tests for this study. The air delivery system and other

equipment to be used will be similar to those used in Yunt’s test program [Yunt, 1980], and tests

are to be performed in accordance with the ASCE standard as far as possible. The following

flowchart procedure lay out the steps:

Page | 248
• First, clean water tests (CWT) are to be done for 2~ 3 temperatures, preferably one

below 20 0C, one at 20 0C and one above. CWT is also required for 2 different gas flow

rates, so that altogether a minimum of 4 tests are recommended for a tank of adequate

size; and the tank water depth is suggested to be fixed at 3 m (10 feet) or 5 m (15 feet)

or any other depth of choice. Tests will be repeated several times to have a constant KLa,

for each temperature, and the test is to be repeated for different applied gas flow rates

so that the KLa vs. Qa relationship can be estimated.

• All diffused aeration systems will experience gas-side depletion as the water depth

increases. This changes in gas-side depletion is dealt with by the Lee-Baillod model,

allowing calculation of the baseline (KLa0) using the microsoft Excel Solver or similar

where KLa0 is a variable to be determined, with the measured KLa and C*∞ as the

independent variables.

• Once the baseline Kla0 is established, a specific standard baseline can be determined

using the temperature correction model and the established Kla0 vs. Qa relationship, and

this value can be used to find the transfer coefficient at another tank depth. The Excel

Solver or similar is used to solve the simultaneous equations, using the established

baseline parameter KLa0, as well as the actual environmental conditions surrounding the

scaled-up tank. Hence, the same Solver method is used twice, to calculate both the KLa0

and the KLa. The temperature correction model of choice is the 5th power model [Lee

2017] as advocated by the proposer.

• All the measured apparent KLa values can be used to formulate the relationship between

KLa and Qa, but the resultant slopes may have some differences. These should be

compared to the plot of (KLa0)T vs. QaT and also to be compared with (KLa0)20 vs. Qa20.

Page | 249
 The latter curve should give the best correlation. Likewise, all (KLa0)T values are to be

plotted against their respective handbook solubilities (CS)T.

• The specific baseline (KLa0)20/Qa20^q [Zhou et al. 2012] [Lee2018] is expected to be

constant for all the tanks tested. From the standardized baseline (KLa0)20 at 20 °C, a

family of rating curves for the standard mass transfer coefficient (KLa)20 can thus be

constructed for various gas flow rates applied to various tank depths using Eqs. (3-6 to

3-10) as stated in Chapter 3.

7.0.3. Technical challenges

Presumably, changes in tank shapes and sizes, diffuser layouts and tank depths will affect the

oxygen transfer film and the KL value of the scaled-up tank. Since the model is a holistic approach

on the overall change in both parameters KL and ‘a’, one of the challenges is in determining the

limitations and the validity boundaries of this model. A trial-and-error approach to establish the

boundaries may be necessary. The proposed model appears correct and valid for the set of tanks

cited in the paper [Lee 2018][Yunt et al. 1980], but the goal of scaling up test data from a smaller

tank size to a larger tank size may depend on external factors that may confound the new model.

Transfer devices typically produce irregularly sized bubbles that often swarm in various

hydrodynamic patterns, e.g. spiral roll devices vs full-floor coverage. So, scale-up changes that

affect the size and shape and depth and roll patterns in a tank effect on oxygen transfer

performance. Small tanks are notorious for wall effects. The second challenge would be in the

selection of suitable size and shape of the test tank that would give the best geometric similitudes

between the various tests. It would also be important to select the aeration system that would not

produce excessive plume entrainment of air during the testing [DeMoyer et al. 2002].

Page | 250
It is difficult to describe a required geometry or placement for testing conducted in tanks other

than the full-scale field facility. According to the ASCE Standard, appropriate configurations for

shop tests should simulate the field conditions as closely as possible. For example, width-to-depth

or length-to-width ratios should be similar. Potential interference resulting from wall effects and

any extraneous piping or other materials in the tank should be minimized. The density of the

aerator placement, air flow per unit volume, or area and power input per unit volume are examples

of parameters that can be used to assist in making comparative evaluations. However, the work

here is to prove that, for the same configurations of aerator placement and tank dimensions, the

model is able to predict oxygen transfer efficiency for a range of tank water depths using a

universal standard specific baseline mass transfer coefficient (KLa0)20.

compare submergence between models

0.60

0.50
submerge ratio e

0.40

0.30
e(ye calc.)
0.20 e(Ye=0.21)

0.10

0.00
0 5 10 15
Run Number

Fig. 7-2. Comparison of effective depth ratio: rigorous analysis versus ASCE method

7.0.4. Estimation of the effective depth ratio (e = de/Zd)

Figure 7-2 (also Fig. 3-2) is a plot of the effective depth ratios calculated from the test runs for the

FMC diffuser tests, the lower line showing the results based on a constant equilibrium mole

fraction at 0.21 similar to the equation in ASCE 2-06 Annex F [ASCE 2007]; while the top line

Page | 251
was based on the developed model equations which describe the mole fraction variation curve and

the calibration parameters, n and m. It is important to note that the ASCE 2-06 Annex F Eq. (F-1)

has treated Ye to be the same Y0 which is not correct. As a result of this rigorous analysis, the top

line in Fig. 7-2 gives a more consistently uniform depth ratio of e = de/Zd.

Note for the development of the effective depth based on ye

The English chemist William Henry, who studied the topic of gas solubility in the early 19th

century, in his publication about the quantity of gases absorbed by water, described the results of

his experiments:

“… water takes up, of gas condensed by one, two, or more additional atmospheres, a quantity

which, ordinarily compressed, would be equal to twice, thrice, &c. the volume absorbed under the

common pressure of the atmosphere.”

Unfortunately, Henry killed himself in the end, perhaps because his brilliancy was not fully

appreciated. If he hadn't died young, he might have discovered more, such as:

Dalton's Law says that in a mixture of gases within a vessel, the total pressure is the sum of the

individual partial pressures of the gases inside the vessel.

Combining the two physical laws, we have

Cst = H.(Y.P) (7—2)

where H is Henry's law constant; Cst is the gas solubility. In the standard, Cst = tabular value DO

surface saturation concentration at test temperature, standard total pressure of 1.00 atm (101.3 kPa)

and 100% relative humidity, in mg/L; Y = oxygen mole fraction in the gas phase; the product Y.P

is the partial pressure in equilibrium with the liquid phase, where P is the total pressure.

Page | 252
Henry's law can be applied to any point within a bulk liquid. When applied to the surfical situation

under atmospheric pressure Pa, the partial pressure of oxygen is 0.21.Pa, hence the law constant

can be determined by re-arranging the above equation to form:

H = Cst /(0.21.Ps) where Ps is standard barometric pressure, 101.3 kPa , in kPa (atm).

Now, the vapor pressure has an effect on solubility so that a correction needs to be made to the

surfical pressure so that

H = Cst /(0.21.(Ps - Pvt)) (7—3)

Now, Henry's law can also be applied to the equilibrium point for a bulk liquid under aeration, so

that

C*∞ = H.(Ye.Pe) (7—4)

Substituting H by the previous equation, we have

C*∞ = [Cst/(0.21.(Ps - Pvt))] (Ye.Pe) (7—5)

It is unfortunate that the ASCE standard has assumed Ye = 0.21 as well, so that in the standard, the

mole fraction is cancelled out, giving:

C*∞ = Cst/(PS - Pvt).Pe (7—6)

The equilibrium pressure, if we assume vapor pressure has similar effect, is given by:

Pe = Pa + rw.de - Pvt where Pa is at test condition, also symbolized by Pb the barometric pressure.

Therefore,

C*∞ = Cst/(Ps - Pvt).(Pa + rw.de - Pvt) (7—7)

Hence,

de = 1/rw [(C*∞/Cst).(Ps - Pvt) - Pa + Pvt] (7—8)

which is identical to the ASCE 2007 Annex F equation for the estimation of de.

Page | 253
Unfortunately, this equation is wrong or approximate only--- the mole fraction of oxygen in the

bubble at the equilibrium point cannot be ignored! Although oxygen is only slightly soluble in

water, the mole fraction of oxygen in the bubble at the equilibrium point is calculated by the gas-

side gas depletion following a mole fraction variation curve, and at the equilibrium point the mole

fraction is different from 0.21. It could be more, it could be less, depending on the initial gas

composition at the point of release from the diffuser and the other factors such as gas depletion. In

ordinary circumstances, it is usually slightly less, so that Ye < 0.21. Therefore, C*∞ = [Cst

/(0.21.(Ps - Pvt))].Ye.Pe

Rearranging gives

de = 1/rw [(C*∞/Cst).(Ps - Pvt).0.2095/Ye - Pa + Pvt] (7—9)

This gives one more equation to the original developed five equations (eq. 3-6 ~eq. 3-10) relating

the effective depth, de, to the oxygen mole fraction at equilibrium, Ye, for simulation giving a total

of six equations with six unknowns (Ye, n, m, de, KLa, C*∞) when the baseline KLa0 is an

independent input variable pre-determined by clean water tests.

KLaf Vs ØZd for KLa0=1

1.2
KLB = [1- exp(-αKLao ØZd)]/ØZd
1
KLB = KLa or KLaf

0.8 KLB for α = 1

KLB for α = 0.8

0.6
KLB for α = 0.6
0.4 KLB for α = 0.4

KLB for α = 0.3

0.2
KLB for α = 0.2

0
0 5 10
ØZd

Fig. 7-3. Apparent mass transfer coefficient vs. function of Zd

Page | 254
It is postulated that this correction factor (α) can be determined by bench scale experiments. Eq.

(3-6) stated in Chapter 3 for the main model can be plotted for KLaf against the function ØZd for

when the baseline is unity, for various α values, as shown in Fig. 7-3 (also Fig. 3-10 and Fig. 6-2).

This graph shows exactly what Boon (1979) has found in his experiments, that KLaf is a declining

trend with respect to increasing depth of the immersion vehicle of gas supply. The generic term

for the mass transfer coefficient is symbolized as KLB.

7.0.5. Standard Oxygen Transfer Rate

In accordance with the ASCE 2-06 standard [ASCE 2007], the average value of SOTR shall be

calculated by averaging the values at each of the n determination points by

𝑉
𝑆𝑂𝑇𝑅 = ∑𝑛𝑖=1 𝐾𝐿 𝐵20𝑖 𝐶∞20𝑖
∗
(7--10)
𝑛

1
𝑆𝑂𝑇𝑅 = 𝑛 ∑𝑛𝑖=1 𝑆𝑂𝑇𝑅𝑖 (7--11)

where
∗
𝑆𝑂𝑇𝑅𝑖 = 𝐾𝐿 𝐵20𝑖 𝐶∞20𝑖 𝑉 (7--12)

where KLB = KLa for clean water (α = 1) and KLB = KLaf for wastewater (α < 1). Note that

when α < 1, and in the presence of microbes, the biological floc exerts a resistance force (Fbf)

to the gas transfer, so that by the principle of superposition, the net transfer in the field

would be given by OTRf where OTRf = SOTR – Fbf. This resistance force is equivalent to the

additional gas-side gas depletion rate in the field (gdpf) and can be measured by the off-gas

method applied to an aeration basin. At steady state, this gas depletion is the same as the

respiration rate R, and so OTRf would be given by SOTR calculated by eq. 7-11 and eq. 7-12

and re-converting to field conditions, and then subtracting the respiration rate over the

entire volume (RV) as shown by Eq. 7-1 [Lee 2019b].

Page | 255
The following photographs (Fig. 7-4, Fig. 7-5) extracted from the literature are an example of a

test tank suitable for the purpose of this proposal :

Fig. 7-4. Schematic of Proposed Test Tank [Wagner 2008]

7.0.6. Standard Oxygen Transfer Efficiency (SOTE)

Oxygen transfer efficiency refers to the fraction of oxygen in an injected gas stream dissolved

under given conditions. The standard oxygen transfer efficiency (SOTE), which refers to the OTE

Page | 256
at a given gas rate (see ASCE 2007 Annex A), water temperature of 20°C, and barometric pressure

of 1.00 atm (101.3 kPa), may be calculated for a given flow rate of air by:

SOTE = SOTR/W02 (7--13)

where

SOTE = standard transfer efficiency as a fraction; and

W02 = mass flow of oxygen in air stream, lb/min (kg/hr)

For subsurface gas injections systems, the value of SOTE should be reported as per ASCE 2007
∗
Section 8.4. If possible, the standard deviations of the parameter estimates, KLa, 𝐶∞ , and standard

error of estimate should also be reported. The above applies to clean water. Under process

conditions, the principle of conservation of mass must be applied, so that the OTEf is given by

OTRf /WO2 with the same units as for clean water test, where OTRf is given by Eq. 7-1, and the

Standard Model would be written as:

𝑑𝐶
𝑂𝑇𝑅𝑓 = 𝑉 + 𝑅𝑉 (7 − 14)
𝑑𝑡

where OTRf is given by Eq. 7-1 which is based on the principle of superposition equivalent to:

𝑂𝑇𝑅𝑓 = 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 − 𝐶)𝑉 – 𝑅𝑉 (7 − 15)

Equating the two equations for the OTRf gives :

𝑑𝐶
2𝑅 +
𝐾𝐿 𝑎𝑓 = 𝑑𝑡 (7 − 16)
𝐶 ∗ ∞𝑓 − 𝐶

Page | 257
Therefore, the above equations show that the mass transfer coefficient can be determined in the

field provided that the respiration rate R is known.

Fig. 7-5. Proposed diffused aeration configuration (diffusers can be soaker hose pipes or similar)

7.1. Determination of Oxygen Transfer in wastewater

Procedure for determination of dissolved oxygen uptake rate and oxygen transfer rate

In principle, the determination of the oxygen transfer rate (OTR) in an aerobic bioreactor is

remarkably simple. At any point in time during fermentation or wastewater treatment, the oxygen

uptake rate (OUR) must equal the transfer rate when the process is at a steady state. At steady

state, there is no change of dissolved oxygen (DO) concentration with time so that all the oxygen

transferred is uptaken by the microbes within the bioreactor, provided that the content is well-

Page | 258
mixed and both the microbes and the DO are uniformly and spatially distributed within the reactor

vessel. At steady state, there is no oxygen uptake by the aqueous solution or the reactor solution,

provided no chemical oxidation processes are occurring. Therefore, the mass balance equation is

simply given by:

OTR = OUR (7--17)

The difficulty is that there is no direct method to measure either the oxygen transfer rate or the

oxygen uptake rate, resulting in only an approximate method of evaluation.

Traditionally, the OTR determination relies on the mass transfer coefficient (KLa) which is an

uncertain parameter. The determination is based on a non-steady state method as per ASCE 2-06

standard or similar, which is an indirect method. On the other hand, the OUR is also subject to

many uncertainties, so that discrepancies can be as much as 50% even under the best circumstances

when attempts are made to equate OUR with OTR. The various methods to be investigated for

determining the OUR are described below:

7.1.1. BOD bottle Method

The proposer reckons there are only three possible explanations for the increase of the OUR using

this method. Firstly, when the sample is agitated by vigorous shaking, the activity level of the

microbes might increase and so they respire more. Secondly, the oxygen level in the sample may

be limiting (although at 2 p.p.m., it shouldn't be) so the respiration rate is not as high as in a normal

DO level; thirdly, Garcia-Ochoa et al. (2010) suggested a cell economy principle by which the

microbes voluntarily reduce the respiration level at low DO or at declining DO from a high level,

changing the respiration rate at elevated level due to increased oxygen availability and conversely,

reducing the respiration rate at decreasing DO concentration. There have been many literatures on

this but none seems to have given a definite answer. Chisea S.C. et al. [1990] conducted a series

Page | 259
of bench‐ and pilot‐scale experiments to evaluate the ability of biochemical oxygen demand (BOD)

bottle‐based oxygen uptake rate (OUR) analyses to represent accurately in-situ OUR in complete

mix‐activated sludge systems. Aeration basin off‐gas analyses indicated that, depending on system

operating conditions, BOD bottle‐based analyses could either underestimate in-situ OUR rates by

as much as 58% or overestimate in-situ rates by up to 285%. A continuous flow respirometer

system was used to verify the off‐gas analysis observations and assessed better the rate of change

in OUR after mixed liquor samples were suddenly isolated from their normally continuous source

of feed. OUR rates for sludge samples maintained in the completely mixed bench‐scale

respirometer decreased by as much as 42% in less than two minutes after feeding was stopped.

Based on these results, BOD bottle‐based OUR results should not be used in any complete mix‐

activated sludge process operational control strategy, process mass balance, or system evaluation

procedure requiring absolute accuracy of OUR values. It should be noted also that the intensive

aeration in the respirometer which was rather violent and may have broken up the floc caused

increased delivery of DO and substrate to the floc, so that the respirometer method is not

considered reliable as well.[Doyle et al., 1983]

7.1.2. Synthetic wastewater to determine actual oxygen uptake rate (Mines’ Method)

Mines et al. (2016) indicated that the proper design of aeration systems for bioreactors is critical

since it can represent up to 50% of the operational and capital cost at water reclamation facilities.

Transferring the actual amount of oxygen needed to meet the oxygen demand of the wastewater

requires α- and β-factors, which are used for calculating the actual oxygen transfer rate (AOTR)

under process conditions based on the standard oxygen transfer rate (SOTR). In their experiment,

the SOTR is measured in tap water at 20°C, 1 atmospheric pressure, and 0 mg L-1 of dissolved

oxygen (DO). In their investigation, two 11.4-L bench-scale completely mixed activated process

Page | 260
(CMAS) reactors were operated at various solid retention times (SRTs) to ascertain the relationship

between the α-factor and SRT, and between the β-factor and SRT. The second goal was to

determine if actual oxygen uptake rates (AOURs) are equal to calculated oxygen uptake rates

(COURs) based on mass balances. Each reactor was supplied with 0.84 L m-1 of air resulting in

SOTRs of 14.3 and 11.5 g O2 d-1 for Reactor 1 (R-1) and Reactor 2 (R-2), respectively. The

estimated theoretical oxygen demands of the synthetic feed to R-1 and R-2 were 6.3 and 21.9 g

O2 d-1, respectively. R-2 was primarily operated under a dissolved oxygen (DO) limitation and

high nitrogen loading to determine if nitrification would be inhibited from a nitrite buildup and if

this would impact the α-factor. Nitrite accumulated in R-2 at DO concentrations ranging from 0.50

to 7.35 mg L-1 and at free ammonia (FA) concentrations ranging from 1.34 to 7.19 mg L-1.

Nonsteady-state reaeration tests performed on the effluent from each reactor and on tap water

indicated that the α-factor increased as SRT increased. A simple statistical analysis (paired t-test)

between AOURs and COURs indicated that there was a statistically significant difference at 0.05

level of significance for both reactors. Mines concluded that the ex situ BOD bottle method for

estimating AOUR appears to be invalid in bioreactors operated at low DO concentrations

(<1.0 mg L-1).

7.1.3. Alternate Methods to be considered

In enzymes technology, bacterial oxidation is hugely related to enzymes and delta G (∆G), where

G is the Gibbs free energy, in relation to their different growth phases, in particular the log-growth

phase and the endogenous phase. Since bacteria cells are extremely good model systems, as such,

the energy balance of the cells may be used to calculate the oxygen demand, together with

techniques such as FISH, 16S RNA and so forth, to estimate the cell numbers.

Page | 261
This will then be an alternative method of estimating the OUR (Oxygen Uptake Rate) in a

treatment plant. The OUR is an overriding important parameter that governs the treatment

performance. However, even though the idea is laboratory proven and reproducible, in the field,

in open vessels with hundreds if not thousands of types of bacteria, with hundreds if not thousands

of substrates in wastewater, we do not have the tools to determine OUR in the way proposed.

Therefore, the proposer is trying to break through the traditional "black box" mentality by finding

an "out of the box" way to estimate oxygen transfer, and he speculated that perhaps Specific

Oxygen Uptake Rate (SOUR) would be a tool. Under the right conditions, one may get a very high

correlation coefficient between substrate consumption, theoretical oxygen demand, and oxygen

transfer, provided that the microbial community is definable in terms of oxygen respiration rates.

However, when looking at degradation potential for a given substrate, one generally looks at both

makeup and structure to understand the availability of the carbon and nitrogen to the organisms.

Single bonds more available than double bonds; double bonds more available than triple bonds;

the number of chlorines clustered around a carbon molecule is a strong influence on kinetics and

on whether the reaction will go; the number of carbon in a ring; and so forth. All of these things

influence the availability of the molecule. As many researchers point out, if we cannot control the

substrate, the correlation between uptake rate and oxygen transfer becomes more difficult to sort

out. This becomes a research activity rather than an oxygen transfer estimation technique.

However, one might be able to make this work in the same way we do a clean water test by

dictating the substrate and the other physical parameters even though one could not make this work

in a generic wastewater. Instead of treating the ecosystem as a black box, it is proposed to look at

the microorganisms at the molecular level. It is proposed that this work is to be coordinated by the

ASCE/EWRI Oxygen Transfer Standards Committee (OTSC) in conjunction with the USEPA in

Page | 262
an effort to upgrade relevant guidelines and standards. By considering the ADP-ATP cellular-

energy transfer system; the heterotrophic bacterial metabolism; chemosynthetic autotrophic

bacterial metabolism; photosynthetic autotrophic bacterial metabolism, and so forth, it may be

possible to play the zero-sum game in the matter of oxygen supply and demand. This may then

allow development of a reference standard method for measurement against which all other

methods can be judged.

This would then allow sensible methods to be devised, and calibrated against the reference

standard method. If a baseline mass transfer coefficient can be established, the mass transfer can

be calculated at a baseline scale. Similarly, by knowing the SOUR, using a reference substrate and

a reference microbial make-up, it might be possible to identify valid procedures and methodologies

to correlate OTR with OUR. However, at this stage the method is primarily used to validate Eq.

(1) that states that the oxygen transfer rate (OTR) is indeed related to the respiration rate if such is

true, and is a valid method to confirm the other methods such as the dilution method explained

below.

7.1.4. Dilution Method

In the article by Doyle (1981), an interesting method of testing for alpha was suggested. It appears

that it may be possible to use the dilution method as suggested by the article to test out the

determination. By first aerating a tank of pure water (as shown in Fig. 7-6 below) to an elevated

DO, say to 7 p.p.m., and then, upon stopping the aeration, gradually and carefully pouring a sample

of known volume of the activated sludge mixed liquor into the tank, and then gently mixing them

together, it may be possible to measure the slope of the DO decline curve at quiescent conditions

(Fig. 7-6a), thereby eliminating the first possible explanation for the cause of increased OUR

measurement. If the sample has been diluted to 50% by the tank, the resultant slope should then

Page | 263
be multiplied by 2 to get the true OUR. This should then be compared with the steady-state column

test with an in-situ measurement as recommended by the ASCE Guidelines (ASCE 1997).

Alternatively, mixed liquor can be continuously pumped to the test tank from a position within the

existing aeration basin using a displacement pump until a set known volume is withdrawn. This

should give the same oxygen depletion curve as the steady-state test, allowing the measurement of

the slope of the curve as a measurement of the microbial oxygen uptake rate. To avoid substrate

limiting effect, the test should be done in-situ as quickly as possible just like the off-gas column

test. A schematic of the method is depicted in the diagrams below (Fig. 7-6b, Fig. 7-6c).

Fig. 7-6. A Typical bench-scale aeration unit (image from Armfield Ltd.)

Page | 264
DO (mg/L) 𝑎+𝑏
𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝐷𝑂 =
2

Time (min)
Fig. 7-6a. Plot of DO (mg/L) versus time (minutes) t

mixed liquor sample

DO = a (mg/L)

Fig. 7-6b. Sample of mixed liquor of a known volume in a container

Page | 265
Steps for the dilution method:

1/ Aerate a tank of pure water of 4 to 6L volume (as shown in Fig. 7-6) to an elevated DO, say to

7 p.p.m.

2/ Collect a sample of mixed liquor

to be evaluated into a container of

approximately 4 to 6 L

and then, pure water aerated

DO = b (mg/L)
3/ upon stopping the aeration, R = (a + b) x
60/T(mg/L/hr)
gradually and carefully

pour the sample of known

volume of the activated sludge

mixed liquor into the tank, and

4/ then gently mix them together

by a mechanical stirrer.

5/ Immediately measure the DO

Fig. 7-6c. Aeration tank of same volume of pure water

by the Winkler Method or by

using a calibrated fast-response DO probe with probe time constants less than 0.02/KLa.

6/ Plot the DO versus time on a graph (as shown in Fig. 7-6a) and calculate R using a linear

least-squares regression to fit a straight line through the data points. Since the mixed liquor is

diluted to 50% its original concentration, the resultant slope of the line is multiplied by 2 to

obtain the in-situ R value. The time lapse between sample collection and uptake rate

measurement is critical in this ex-situ procedure. The entire process from colllection of sample to

Page | 266
start of DO monitoring should take less than 2 min. [ASCE 18-18]. The procedure should be

replicated at least three times at any given sampling point. A worked example is given below.

Suppose the treatment plant in-situ aeration tank has a DO level of 2 mg/L and the bench-scale

aeration unit is aerated to 7 mg/L, then the mixture will have a DO concentration of:

(a + b)/2 = (2 + 7)/2 = 4.5 mg/L

Suppose the DO level dropped to zero in 7 mins, then the slope of the decline curve would be:

4.5/7 x 60 = 38.57 mg/L/hr

The actual oxygen uptake rate in-situ is therefore given by twice this value or

2 x 38.57 = 77 mg/L/hr.

7.1.5. Procedure for determination of wastewater mass transfer coefficient KLaf and alpha (α)

The following procedure can be used in the bench-scale determination of the oxygen transfer

coefficient, α:

1. A vessel similar to those shown in Fig. 7-7 can be used. For the diffused system, where the

gas-side gas depletion effect is significant, an air measuring rotameter can be installed for

recording the gas flow rate.

2. The container is filled with a defined volume of tap water or wastewater as the case may

be, and the water temperature and air barometric pressure are recorded. The water is then

deoxygenated as per ASCE standard 2-06 either chemically or physically stripping the oxygen

from solution with an inert gas such as nitrogen. In the laboratory unit, the latter is preferred in

order to eliminate any possible chemical interferences.

3. Once the contents have been deoxygenated, the water is re-aerated at a controlled diffused

air flow together with any mechanical rotational speed if used to increase the transfer rate. If the

Page | 267
latter is employed, care must be taken to ensure that the device simulates the actual device used in

the field (this would require separate modeling). The re-aeration curve can be plotted and the

important parameters KLa and C* ∞ can then be estimated using the non-linear least squares

(NLLS) method as described in the standard. This step can be repeated for various gas flow rates,

mixing intensities, temperatures and pressures.

4. If wastewater is used in the test, the procedure should come after the clean water test, and

should be repeated using the same volume of wastewater as the previous cleanwater. As the test

results would depend on the microbial respiration rate, as previously described, every effort should

be made to eliminate the influence of the microbial respiration.

Fig. 7-7. Laboratory apparatus for determination of α [Eckenfelder 1970]

This can be achieved by killing off all live microbes, such as using a sulphamic acid-copper

sulphate solution, or if a bench scale biological reactor has been used, the effluent from the reactor

should be used for the test when a completely mixed system is contemplated either in the entire

aeration basin, or a section of the basin where completely-mixed is envisaged. If the latter method
Page | 268
is used, the biological solids should be filtered out to prevent any microbial oxygen uptake that

remains during the test period. The respective KLa values as determined from the two tests (clean

water vs. wastewater) can then be compared at the same temperature, pressure and mixing

conditions, and α can be calculated in accordance with the following equation:

α = (KLa wastwater)/( KLa clean water) (7--18)

Should all the methods described above give the same alpha value and the same in-process mass

transfer coefficient at full scale, this proposal would be a success and the standards and guidelines

for oxygen transfer measurement should be amended accordingly.

7.2. Associated cost-benefit implications

7.2.1. Upgrading of current Standards

The validity of this proposal is based on two concepts:

1/ the oxygen transfer rate in the field under process condition is affected by the microbial oxygen

uptake rate for a diffused aeration system;

2/ the alpha-factor (α) is only dependent on the wastewater characteristics.

If these concepts are correct, then the oxygen transfer rate calculations must take into account of

the respiration rate at the time of measurement. This would result in replacing Eq. CG-1 in the

clean water measurement standard ASCE 2-06 by eq. (7—1) recalling as follows:

1
𝑂𝑇𝑅𝑓 = ( ∗ ) [ 𝛼 (𝑆𝑂𝑇𝑅)Ɵ𝑇−20 ]( 𝜏. ß. 𝛺. 𝐶 ∗ ∞ 20 − 𝐶) – 𝑅𝑉
𝐶 ∞ 20

where SOTR is determined as per eq. 7-10 above; C is the DO concentration at any process DO

level; R is the respiration rate at this process level. Both the parameters SOTR and the oxygen

Page | 269
saturation concentration C*∞ are based on a series of clean water tests at pilot scale or shop tests

as described in the section Technical Approach. It is however required to determine the baseline

clean water mass transfer coefficient KLa0 in order to extrapolate this coefficient to full scale.

Alpha is determined in bench scale as shown by Fig. 7-7 and by using eq. 7-18 and eq. 7-20 below.

The mass transfer coefficient at full scale under process condition is estimated by eq. 7-19 at any

determined alpha value in the laboratory at bench scale. The other correction factors are as per the

current standard. The cost-benefit implication is that there is no longer any need to determine the

essential parameters at full scale, since at the outset of a design project, full scale aeration tanks

are not available prior to design. With the current standard, the discrepancies between anticipated

and actual performance are often sufficiently large to warrant substantial field modifications to the

aeration equipment furnished. The costs of performing such modifications and the ill will

generated testify to the need for improved oxygen transfer design and testing procedures. Of the

other correction factors, the temperature correction factor Ɵ should be based on the 5th power

model as advanced by Lee (2017) and not just using Ɵ = 1.024 as per the standard that has many

fallacies.

The analysis in Yunt’s experiment [Yunt et al. 1980] based on the FMC diffusers, appears to

support the temperature correction model [Lee 2017] as shown by Fig. 7-8 (also Fig. 3-7a),

showing the excellent regression correlations when the baseline is used with the model. Using the

baseline KLa0 is tantamount to using a shallow tank, which is the fundamental basis for the 5th

power temperature model. However, the temperature model used in conjuction with the depth

correction model, allows scaling up to a higher water depth tank. In biotechnology, temperature

correction is very important, as the standard and current guidelines have pointed out in various

sections. Currently, reported values for Ɵ range from 1.008 to 1.047. Because it is a geometric

Page | 270
function, large error can result if an incorrect value of Ɵ is used. While the Arrhenius equation is

used substantially for the wastewater and other industries, and is applicable to cooler and temperate

climates, it should be used with caution at higher temperatures. At temperatures below 20 C, it is

not important which equation is used, but in countries with hot climates, and especially those with

high humidity, where the effect of evaporative cooling is reduced, the Arrhenius equation may not

be applicable and could result in systems that do not perform to design specifications. Furthermore,

temperature correction is affected by size of bioreactors, aeration tanks, contaminants, and mixing

intensities. Using the proper temperature model would enhance the accuracy of the calculation of

the standard specific baseline as demonstrated in Fig. 5-7 for the FMC diffusers, whereas the

Arrhenius formula may be less exact, as shown in Fig. 5-8 for the Norton diffusers.

Therefore, this project has the additional benefit of confirming the validity of the 5th power

temperature correction model as well as the other models concerning the effect of geometry on the

mass transfer coefficient. For the upgrading of the standard, the research should additionally be

done under the umbrella of a university or similar institution, and subjected to peer review. If the

results are believed to be applicable for a standard, it could then be submitted to the Standards

Committee for consideration and further verification.

7.2.2. Translating Clean water test results to In-process water measurements

In the application to wastewater treatment, using the transfer of oxygen to clean water as

the datum, it may then be possible to determine the equivalent bench-scale oxygen transfer

coefficient (KLa0f) for a wastewater system, where the subscript f denotes in-field process water;

and the ratio of the two coefficients can then be used as a correction factor to be applied to fluidized

systems treating wastewaters via aerobic biological oxidation. It is paramount to determine alpha

(α) where alpha is the correction factor (Stenstrom et al. 2006) given by:

Page | 271
 𝐾𝐿 𝑎0𝑓 𝐾𝐿 𝑎𝑓
𝛼= ≈ (7 − 19)
𝐾𝐿 𝑎0 𝐾𝐿 𝑎

It is postulated that this correction factor (α) can be determined by bench scale experiments. It is

hypothesized that this alpha value is not dependent on the liquid depth and geometry of the aeration

basin and the model developed that relates KLa to depth then allows the α value to be used for any

other depths and geometry of the aeration basin. Therefore, after incorporating α into the baseline

mass transfer coefficient for clean water, the mass transfer coefficient in in-process water KLaf

would be given by eq. 3-6 and repeated here upon incorporating the alpha-factor as:

1 − exp(−𝛷𝑍𝑑 . 𝛼𝐾𝐿 𝑎0 )
𝐾𝐿 𝑎𝑓 = (7 − 20)
𝛷𝑍𝑑

The mass transfer coefficient so determined should match the result of field testing by following

the ASCE Standard Guidelines for In-Process Oxygen Transfer Testing methods [ASCE 1997].

16.00
Cs = 43.457Ps/(E. ρ.T5)
14.00
R² = 0.9996
12.00
Solubility in mg/L

10.00

8.00

6.00

4.00

2.00

0.00
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
PS /(T^5.E.ρ)

Fig. 7-8. Solubility Plot for water dissolving oxygen at Ps = 1 atm (1.013 bar)

The outcome of this project, if successful, would challenge the current concept that there is

absolutely no way to relate alpha with any of these affecting variables other than full scale testing

Page | 272
for every condition [ASCE/EWRI 2-06] because of the difficulties in calculating alpha

[Mahendraker et al., 2005].

If the testing were done on an old diffuser, then α would be replaced by α.F where F is a fouling

factor as defined in the standard. It should be noted in passing that, in the above equation, the

parameter Ø depends on Henry’s Law constant.

Therefore, this equation is equivalent to eq. 3-6, where Ø is x(1-e). If the handbook data of oxygen

solubility is plotted against the inverse of the temperature correction function affecting solubility,

the straight-line linear plot would be as shown in Figure 7-8 below.

Therefore, the solubility law [Lee 2017] can be expressed either by the equation derived from

plotting the insolubility, or expressed by the equation from plotting the data as in Figure 7-8. In

the former method, the equation gives the insolubility of oxygen expressed by:

1 𝜌
( = 0.02302. 𝑇 5 × 𝐸 × ) (7 − 21)
𝐶𝑠 𝑃𝑠

where T is in K(Kelvin) to the power 10-3.

In the latter case, the equation gives the solubility directly and is expressed by:

𝑃𝑠
𝐶𝑠 = 43.457 × (7 − 22)
(𝑇 5 𝐸𝜌)

Henry’s Law is applicable only to ideal solutions [Andrade 2013]; and for an imperfect liquid

subject to changes in physical state, at extreme temperatures between 273 K and 373 K, it is only

approximate and limited to gases of slight solubility in a dilute aqueous solution with any other

dissolved solute concentrations not more than 1 percent. Since in Henry’s Law, the solubility Cs

is proportional to the partial pressure, the Henry’s Law constant would be given by H =

43.457/(𝑇 5 𝐸𝜌)/Y0, where Y0 = 0.2095 for air. In a mixed liquor, the liquid density may depart from

Page | 273
that of clean water significantly, and so the sample should be measured for its density using a

hygrometer to measure the humidity and a hydrometer to measure the relative density.

The discovery of a standard baseline (KLa0)20 that may be determined from shop tests for predicting

the (KLa)20 value for any other aeration tank depth and gas flowrate, and even for in-process water

with an alpha (α) factor incorporated into the equations, is important not only in the development

of energy conservation for wastewater treatment plants, but also in the prediction of field in-

process performance of an aeration tank in a utility setting. Such prediction would depend on the

veracity and validity of Eqs. (7-1, 7-19 and 7-20) on which this proposal is based and on the

assumption that the respiration rate R can be accurately measured.

Worked Example

The proposed method of calculating the oxygen transfer rate in the field (OTRf) is best illustrated

by a mock example as shown below. Using the test results for the Norton diffusers as shown in

Table 5-8, the derived baseline mass transfer coefficient can be obtained to be 0.1076 min-1 as

shown in Table 7-1 below, for a gas flowrate of 127 scmh (standard cubic meters per hour). This

baseline value is used to design an aeration basin of the same surface area but 9 m deep, based on

the following hypothetical test conditions:

V = 37.2 m2 x 9 m = 335 m3; C (the mixed liquor DO concentration) = 2 mg/L; measured average

oxygen uptake rate R = 20 mg/L; mixed liquor temperature T = 20 0C; barometric pressure Pa =

101.3 kPa; alpha-factor (α’) = 0.6 derived by testing devoid of living cells as descibed in section

7.1.5 (see Fig. 7-7). The calculation steps are given below:

(i) Determine KLaf for the 9m tank

In the application of eq. 7-20 for the full-scale mass transfer coefficients, the first step

is to calculate Ø.Zd. As given by eq. 7-22, Henry’s Law constant is calculated by H =

Page | 274
 43.457/(𝑇 5 𝐸𝜌) but since the bioreactor is only lightly fed, the modulus of elasticity

and density of the mixed liquor are assumed to be same as pure water, and so it can be

read from a chemistry handbook to give H = 4.382 x 10-4 (mg/L)/(N/m2) compared

with 4.383 x 10-4 using the formula; Therefore, the hydrostatic pressure at diffuser

depth is given by,

Pd = 101325 + 9789 x 9 – 2333 = 187093 Pa; and the hydrostatic pressure at

equilibrium depth, assuming mid-depth (e = 0.5), is given by:

Pe = 101325 + 9789 x 9/2 – 2333 = 143043 Pa;

the height-averaged gas flowrate is given by Lee (2017) and also given in Chapter 2

eq. 2-25 as Qa = Qs x 172.82 x 293.15 x (1/101325 + 1/187093) = 0.77Qs where Qs is

given in Table 7-2 as 127 scmh. Therefore Qa can be estimated to be 0.77 x 127/60 =

1.63 m3/min. Therefore, Ø = x(1-e) where x = HR’ST/Qa = 4.382E-4(0.260 kJ/kg-K)

(37.2 m2)(293.15 K)/1.63 m3/min = 0.7622 min/m and Φ = 0.7622( 1 – 0.5) = 0.3811

min/m. This gives ØZd = 3.43 min.

Hence, from eq. 7-20, the mass transfer coefficient in the field for this scaled-up tank

would be: KLaf = (1 – exp(-α.KLa0 x ØZd))/ ØZd calculated as follows:

KLaf = (1 – exp(-0.6 x 0.1076 x 3.43))/3.43 = 0.0579 min-1 = 3.48 hr-1. Assuming the

equilibrium mole fraction Ye is about 0.2000, C*∞f = HYePe = 4.382E-4 (0.2000)

(143043) = 12.54 mg/L (This value should be corrected for β as per ASCE 2007.

Assuming β = 0.966, this gives C*∞f = 0.966 x 12.54 = 12.12 mg/L).

(ii) Determine KLa for the 9m tank

Page | 275
 The clean water mass transfer coefficient for this 9 m tank can be calculated by eq. 7-

20 again with α = 1 as follows: KLa = (1 – exp(-KLa0 x ØZd))/ ØZd. Hence KLa = (1-

exp(-0.1076 x 3.43))/3.43 = 0.08998 ~ 0.0900 min-1 = 5.40 hr-1.

(iii) Determine α’

Therefore the calculated alpha value (α’) would be given by α’ = 0.0579/0.0900 = 0.64

which is quite close to the experimental value of 0.6, thus verifying that eq. 7-19 is

practically valid. The above calculations also proves that the hypothesis α’ = 2α is

correct, since α’ = 2 x 0.34 = 0.68 which is close to the α’ of 0.64 as calculated by the

proposed model.

(iv) Determine OTEf and α

Therefore, from eq. 7-15, written as 𝑂𝑇𝑅𝑓 = 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 − 𝐶)𝑉 – 𝑅𝑉

OTRf = [3.48(12.12 – 2)335 – 20 x 335] x 10-3 = 5.097 kg/hr at a mass gas flowrate

of 127 scmh. The OTEf is given by 5.097/(127 x 1.20 x 0.23) = 0.1457 or 14.57%. This

efficiency should correspond to the field test such as by the off-gas method. From Table

5-8, the Norton clean water standard OTE was calculated to be 42.85% for the 6-m test

tank, therefore the ratio of efficiencies would be alpha (α) = 14.57/42.85 = 0.34

approximately if the cleanwater efficiency holds for the 9-m tank as well. A more

precise method is to use the spreadsheet to simulate the parameters by solving all the

model equations for the scale-up tank (eqs. 3-6 ~ 3-10) as shown in the following

calculation sheet (Table 7-2) below, where the estimated result for the clean water mass

transfer coefficient is KLa = 5.33 hr-1 and the equilibrium DO concentration would be

C*∞ = 11.59 mg/L. Therefore, the clean water OTR at 2 mg/L would be given as 5.33

(11.59 – 2) x 335 x 10-3 = 17.1 kg/hr. Since WO2 = 127 x 1.20 x 0.23 = 35.05 kg/hr

Page | 276
 [ASCE 2007], the OTE = 17.1/35.05 x 100 = 48.8% which is higher than the previously

assumed value of 42.85%. Since β = 0.97, C*∞f would become 0.97 x 11.59 = 11.24

mg/L.

Therefore, from eq. 7-15, OTRf = KLaf (C*∞f – C)V – RV = [3.48(11.24 – 2)335 – 20

x 335] x 10-3 = 4.072 kg/hr at a mass gas flowrate of 127 scmh. Then OTEf becomes

4.072/35.05 = 11.6 %. Therefore alpha would be given by: α = 11.6/48.8 = 0.24.

(v) Determine the respiration rate R and compare with measured OUR

Comparing this estimated α with the model result that gives α’ = 3.48/5.33 = 0.65 as

determined in step (iii), which, although is quite close to the meausred α’ of 0.60, is at

least more than twice the value of alpha (α) of 0.24. If it is desired to balance the two

estimations assuming α’ = 2α (for the steady state), then the measured OURf should be

around 16.8 mg/L/hr instead of 20 mg/L/hr.

In a previous paper, the proposer derived an equation that links up the two parameters

alpha (α) and alpha’ (α’) at any value of DO concentration as follows [Lee 2019b] and

also given by eq. 6-28 in Chapter 6:

𝑅
𝛼’ = 𝛼 +
𝐾𝐿 𝑎(𝑑𝑒𝑓𝑖𝑐𝑖𝑡)

Rearranging this equation gives: R = (α’ – α) KLa (β C*∞ - C)

Therefore, R = (0.65 – 0.24) 5.33 (11.24 – 2) = 20 mg/L/hr which is greater than 16.8

mg/L, indicating that the test was not done at steady state.

The only way to verify this result is by means of an actual testing of wastewater with

the appropriate composition, and aerating it inside a 9-m column or tank using the same

Norton diffuser to observe the resultant KLaf and compare it with the clean water test

Page | 277
 Test Date 5/16/1978 Norton diffuser
Environmental data
water temp. T= 20 degC
atm press. Pa= 101325 N/m^2
tank area S= 37.2 m^2 Error Analysis
calc. variables Eq. I= -2.21E-04 4.89E-08
diff press. Pd= 152696 N/m^2 Eq. II= -9.97E-04 9.94E-07
x=HRST/Qa x= 0.7182 min/m Eq. III= -5.09E-05 2.59E-09
Solver Eq.IV=
baseline Kla Kla0= 0.1076 1/min(hr) 6.45 1.13E-04 1.27E-08
soln.
n= 2.77 - SS(sum of squares)= 1.05832E-06
m= 1.63 - offgas
equil. mole fraction ye= 0.2067 - Eq.V= 0.2095 checked
diff mole fraction yd= 0.2095 -
eff. depth de= 2.79 m
depth ratio e= 0.51 -

press. at de Pe= 126324 N/m^2

Input Zd Qa Kla C*inf rw H pvt

Data (m) (m^3/min) (1/min) (mg/L) (N/m^3) (mg/L/(N/m^2) (N/m^2)
5.49 1.73 0.0970 11.44 9789 4.382E-04 2333

Table 7-1. Example calculation for the baseline Kla0 based on Norton Diffusers data (Eqs. I to V are identical to Eqs. 3-6 to 3-10)
 Fixed SS error
Qa
T (deg C) Zd= 9.00 (m3/min) 1.63 de= 4.0979
20 Pa= 101325 T (K) 293.15 Pe= 139106
Pd= 187093 Qs = 127 scmh Eq. 1 5.9024E-04 3.48389E-07
-3.3022E-
x= 0.7624 Eq. 2 06 1.09046E-11
S= 37.2 Eq. 3 1.8525E-05 3.43165E-10
-3.6811E-
Variables Eq. 4 04 1.35503E-07
Kla= 0.0890 5.33 hr-1 Eq. 5 2.1676E-08 4.69838E-16
e= 0.46 Eq. 6 3.0261E-06 9.15723E-12
n= 2.33 4.84255E-07
m= 1.47 Eq. I,II,III, IV, V, VI given by:
Ye= 0.1901 (1) Kla = (1 -exp(-mxKla0 Zd))/n/m/x/Zd +(n-1) Kla0/n
C*inf 11.59 (2) C*∞ = nH*Yd*(Pa-Pvt-Pd exp(-mxKla0 Zd))/(1-exp(-mxKla0
Data Kla0= 0.1076 Zd))
Yd= 0.2095 (3) ln(Pe mx Kla0/n/rw) +ln(nHYdPd/C*∞ - 1) = mx Kla0 Ze
H= 4.383E-04 (4) Kla=(1-exp(-x(1-e)Zd Kla0))/(x(1-e)Zd)
rw= 9789 (5) y0=C*∞/n/H/(pa-pvt)+((Yd.Pd/(Pa-Pvt)-C*∞/(nH(pa-
pvt)))exp(-mx.Kla0Zd)
(6) C*inf = HYePe
pvt= 2333
Ps= 98992

Table 7-2. Calculation sheet for 9-m tank using Norton diffuser at 1.63 m3/min (127 scmh) for

clean water aeration at Pb = 1 atm (1.013 bar) and standard temperature of 20 0C

KLa using the same column. The respiration rate should be independently measured

by a suitable method such as the dilution method as described in section 7.1.4, or the

off-gas Column Test as given by ASCE 18-18 Appendix D.

7.2.3. Energy Conservation

About 50 to 85 percent of the total energy consumed in a biological wastewater treatment plant is

in aeration. The activated sludge process, the most common process, is performed in large aeration

basins and an excess factor of safety is used in designing for the air supply to meet sustained peak

Page | 279
organic loading and to avoid endogenous situations. This may lead to unsatisfactory treatment

performance and even plant failure.

In terms of the issues of environmental and economic significance of the research, the current

improper sizing of the aeration system is primarily due to the inability to estimate the mass transfer

coefficient (KLa) correctly for different tank depths, among other things, leading to improper

blower design and to inappropriate operation. The new findings have potential to help against the

wasteful energy practice in WWTP (Wastewater Treatment Plants) - supplying the air in the

excess, which has also adverse impact on the regime of treatment.

The objective of this proposal is to introduce a baseline oxygen mass transfer coefficient (KLa0), a

hypothetical parameter defined as the oxygen transfer rate coefficient at zero depth, and to develop

new models relating KLa to the baseline KLa0 as a function of temperature, system characteristics

(e.g., the gas flow rate, the diffuser depth Zd), and the oxygen solubility (Cs). Results of this study

indicate that a uniform value of KLa0 that is independent of tank depth can be obtained

experimentally. This new mass transfer coefficient, KLa0 is introduced for the first time in the

literature and is defined as the baseline volumetric transfer coefficient to signify a baseline. This

baseline, KLa0 has proven to be universal for tanks of any depth when normalized to the same test

conditions, including the gas flow rate Ug, (commonly known as the superficial velocity when the

surface tank area is constant). The baseline KLa0 can be determined by simple means, such as a

clean water test as stipulated in ASCE 2-06. The developed equation relating the apparent

volumetric transfer coefficient (KLa) to the baseline (KLa0) is mainly expressed by Eq. (3-6).

The standard baseline (KLa0)20 when normalized to the same gas flowrate is a constant value

regardless of tank depth. This baseline value can be expressed as a specific standard baseline when

the relationship between (KLa0)20 and the average volumetric gas flow rate Qa20 is known.

Page | 280
Therefore, the standard baseline (KLa0)20 determined from a single test tank is a valuable parameter

that can be used to predict the (KLa)20 value for any other tank depth and gas flowrate (or Ug

(height-averaged superficial gas velocity)) by using Eqs. (3-6 ~ 3-10) and the other developed

equations, provided the tank horizontal cross-sectional area remains constant and uniform as the

bubbles rise to the surface. The effective depth ‘de’ can be determined by solving a set of

simultaneous equations using the Excel Solver or similar, but, in the absence of more complete

data, ‘e’ can be assumed to be between 0.4 to 0.5 (Eckenfelder 1970) for conventional aeration.

Therefore, (KLa0)20 can be used to evaluate the KLa in a full-size aeration tank (e.g., an oxidation

ditch with a closed loop flow condition) without having to measure or estimate numerically the

bubble size needed to estimate the KLa for such simulation. However, the proposed method

herewith may require multiple testing under various gas flowrates, and preferably with testing

under various water depths as well, so that the model can be verified for a system. Using the

baseline, a family of rating curves for (KLa)20 (the standardized KLa at 20 oC) can be constructed

for various gas flow rates applied to various tank depths. The new model relating KLa to the

baseline KLa0 is an exponential function, and (KLa0)T is found to be inversely proportional to the

oxygen solubility (Cs)T in water to a high degree of correlation. Using a pre-determined baseline

KLa0, the new model predicts oxygen transfer coefficients (KLa)20 for any tank depths to within

1~3% error compared to observed measurements and similarly for the standard oxygen transfer

efficiency (SOTE%). The discovery of a standard baseline (KLa0)20 determinable from shop tests

is important for predicting the (KLa)20 value for any other aeration tank depth and gas flowrate,

and this finding is expected to be utilized in the development of energy optimization strategies for

wastewater treatment plants and also to improve the accuracy of contemporary aeration models

used for aeration system evaluations. Hopefully, the problem with energy wastage due to

Page | 281
inaccurate supply of air is ameliorated and the current energy consumption practice could be

improved by applying the models to estimate the mass transfer coefficient (KLa) correctly for

different tank depths at the design stage.

Notation

α wastewater correction factor, ratio of process water α.KLa to clean water

KLa

α·F ratio of the process water (α.KLa)20 of fouled diffusers to the clean water

(KLa)20 of clean diffusers at equivalent conditions (i.e., diffuser airflow,

temperature, diffuser density, geometry, mixing, etc.), and

assuming(α.KLa)20/(KLa)20 ≈(α.KLa)/(KLa)

α’ wastewater correction factor, ratio of process water (with no microbes)

KLaf to clean water KLa

β correction factor for salinity and dissolved solids, ratio of 𝐶 ∗ ∞ in

wastewater to tap water

𝐶 ∗∞ oxygen saturation concentration (clean water) in an aeration tank (mg/L)

𝐶 ∗∞f oxygen saturation concentration in an aeration tank under field

conditions and R = 0 (mg/L)

𝐶 ∗ ∞ 20 oxygen saturation concentration (clean water) in an aeration tank at 20 0C

(mg/L)

Page | 282
𝐶 ∗∞0 hypothetical oxygen saturation concentration (clean water) in an aeration

tank at zero gas depletion rate (mg/L)

Cs oxygen saturation concentration (clean water) in an infinitesimally

shallow tank, also known as oxygen solubility in clean water at test

temperature and barometric pressure (mg/L)

C dissolved oxygen concentration in a fully mixed aeration tank (mg/L)

C* a broad term for any saturation concentration in equilibrium with the

partial pressure of oxygen in the gas phase, microscopically or

macroscopically

CR the “apparent” or steady-state saturation concentration in the field

(mg/L)

E Modulus of elasticity of water (kN/m2 x 106)

Fbf resistance force by the biological floc against oxygen transfer (mg/L/hr)

DO Dissolved oxygen or dissolved oxygen concentration (mg/L)

kL mass transfer coefficient in the liquid film (m/min)

kG mass transfer coefficient in the gas film (m/min)

KL overall mass transfer coefficient based on the liquid phase (m/min)

KLB generic term for the apparent mass transfer coefficient measured either

for clean water or wastewater (min-1 or hr-1)

Page | 283
KLa volumetric mass transfer coefficient (clean water) also known as the

apparent volumetric mass transfer coefficient (min-1 or hr-1)

(KLa)20 volumetric mass transfer coefficient (clean water) also known as the

apparent volumetric mass transfer coefficient at standard conditions

(min-1 or hr-1)

KLaf mass transfer coefficient as measured in the field, equals α’.KLa (min-1 or

hr-1)

KLa0 baseline mass transfer coefficient (clean water), equivalent to that in an

infinitesimally shallow tank with no gas side oxygen depletion (min-1 or

hr-1)

(KLa0)20 baseline mass transfer coefficient (clean water) at standard conditions,

equivalent to that in an infinitesimally shallow tank with no gas side

oxygen depletion at standard conditions (min-1 or hr-1)

KLa0f wastewater baseline mass transfer coefficient;

equivalent to that in an infinitesimally shallow tank without gas side oxygen

depletion (min-1 or hr-1)

(KLa0f)20 wastewater baseline mass transfer coefficient at standard conditions,

equivalent to that in an infinitesimally shallow tank with no gas side

oxygen depletion (min-1 or hr-1)

Ɵ.𝜏. ß. 𝛺 temperature, solubility, pressure correction factors as defined in ASCE

2007

Page | 284
MLSS mixed liquor suspended solids (mg/L or g/L)

OTE oxygen transfer efficiency (%)

p. SOTE predicted standard oxygen transfer efficiency (%)

rpt. SOTE reported standard oxygen transfer efficiency (%)

OTR oxygen transfer rate (kg O2/hr)

OTRf oxygen transfer rate in the field under process conditions, equals:

KLaf (C*∞f – C) V -RV (for batch process) (kg O2/hr)

OTRcw oxygen transfer rate under non-steady state oxygenation in clean water

(kg O2/hr)

OTRww oxygen transfer rate under non-steady state oxygenation in wastewater

without any microbial cell respiration (kg O2/hr)

SOTR standard oxygen transfer rate in clean water/wastewater as (Fig. 7-3 and eq. 7—

12) also defined in ASCE 2007 standard for clean water (kg O2/hr)

OUR oxygen uptake rate (kg O2/hr)

OURf oxygen uptake rate in the field, measurable by the off-gas method (kg

O2/hr)

R microbial respiration rate, also known as the oxygen uptake rate (OUR)

(kg O2/m3/hr usually expressed as mg/L/hr)

R0 specific gas constant for oxygen (kJ/kg-K)

Page | 285
ρ density of water (kg/m3)

σ surface tension of water (N/m)

z depth of water at any point in the tank measured from bottom, see Fig. 7-1

(m)

Zd submergence depth of the diffuser plant in an aeration tank (m)

Ze equilibrium depth at saturation measured from bottom, see Fig. 7-1 (m)

Ps net surface atmospheric pressure, Pa – Pvt (kPa)

Pa, Pb atmospheric pressure or barometric pressure at time of testing (kPa)

Pe equilibrium pressure of the bulk liquid of an aeration tank (kPa) defined

such that: Pe = Pa + rw de -Pvt where

Pvt the vapor pressure and

rw specific weight of water in kN/m3 (at test temperature)

de effective saturation depth at infinite time (m)

Ye oxygen mole fraction at the effective saturation depth at infinite time

Y0, Yd initial oxygen mole fraction at diffuser depth, Zd, also equal to exit gas

mole fraction at saturation of the bulk liquid in the aeration tank, Y0 =

0.2095 for air aeration

H Henry’s Law constant (mg/L/kPa) defined such that:

𝐶 ∗ ∞ = HYePe or Cs = HY0 (Ps)

Page | 286
Yex exit gas or the off-gas oxygen mole fraction at any time

y oxygen mole fraction at any time and space in an aeration tank defined by

an oxygen mole fraction variation curve

gdp gas-side oxygen depletion rate (equals OTR; equals zero at steady state

for clean water; equals OTRf in process water; equals gdpf at steady state)

(kg O2/hr)

𝑔𝑑𝑝𝑐𝑤 gas-side oxygen depletion rate in clean water (equals OTRcw) (kg O2/hr)

𝑔𝑑𝑝𝑤𝑤 gas-side oxygen depletion rate in wastewater (equals OTRww) (kg O2/hr)

𝑔𝑑𝑝𝑓 specific gas-side oxygen depletion rate due to microbes-induced

resistance (equals OTRf at steady state); equals the microbial respiration

rate R in process water (kg O2/hr)

Qa height-averaged volumetric air flow rate (m3/min or m3/hr)

S horizontal cross-sectional area of an aeration basin or tank (m2)

T absolute temperature in Kelvin (K) or in 0C

Ug superficial gas velocity (m/min)

V volume of aeration tank (m3)

Qs, AFR gas (air) flow rate at standard conditions (20°C for US practice and 0°C for

European practice), in (std ft3/min or Nm3/h for scmh)

n, m calibration factors for the Lee-Baillod model equation for the oxygen mole

fraction variation curve

Page | 287
q exponent in the mass transfer coefficient vs. gas flowrate curve

WO2 mass flow of oxygen in air stream (kg/h)

Page | 288
7.3. Appendix

The differentiation between the mass transfer coefficient and the baseline has been described in

Chapter 3, under the section ‘Background’. Here the author gives a further elaboration on one of

the many myths that has hindered, for many years, the development of new techniques in the

clean water measurement and dirty water applications of this technology.

The Myth of ‘true’ KLa

In the 70’s and 80’s it had been correctly recognized that the saturation concentration C* is

a function of both space and time for bubble aeration [Baillod 1979][Boon and Lister

1973][Downing and Boon 1968]. What was neglected is that the mass transfer coefficient KLa is

also a function of both space and time since KLa and C* are the two sides of a same coin [Lee

2017][Lee 2018]. Therefore, in the standard model, the accumulation term would be given by:

dC/dt = KLa (C* - C) (A1)

where both KLa and C* are variables (the symbol indicates a point in space and time), even when

C is uniform at any point in time in a well-mixed tank. By treating KLa as a constant with respect

to space and time however, the following equation is obtained:

dC/dt = KLa (C*- C) (A2)

where the bar indicates an average value over the height of the bulk volume. Eq. A2 was then

equated with the standard model applied to a bulk liquid for a bulk aeration system that is described

by the terms (bulk-averaged saturation concentration C*∞) and (the apparent bulk mass transfer

coefficient KLa’), both of which can be obtained by a clean water test, giving the following equality

equation:

Page | 289
dC/dt = KLa’ (C*∞ - C) (A3)

KLa is then termed the ‘true’ mass transfer coefficient, as opposed to the ‘apparent’ mass transfer

coefficient KLa’. Conceptually, both coefficients are ‘true’ KLa but applied to different scenarios.

The former equation (eq. A2) was deemed to apply to an average saturation concentration over the

tank height at a unique point in time during the reaeration test (not a unique point in space), but

this equation cannot be valid since the mass transfer coefficient also varies with space and time as

the bubble composition changes throughout the test. Apart from the composition, the bubble

volume is also a function of time and space, firstly due to the varying oxygen mole fraction, and

secondly due to the hydrostatic pressure. Both the liquid film coefficient KL and the interfacial

bubble-surface area ‘a’ are functions of space and time – any changes in the bubble radius will

affect the bubble film thickness and the surface tension, changing the value of KL; and the

interfacial area is also changing as well during the bubble ascent to the surface. In fact, in a bulk

liquid volume, the bubble size distribution within the bulk volume becomes important to the bulk

overall mass transfer coefficient. Adjusting a point oxygen saturation concentration C* to a height-

averaged value C* without doing the same for KLa is fundamentally flawed. Treating KLa as a

constant is therefore errorneous in eq. A2, and so the two equations (eq. A2 and eq. A3) cannot be

equated to each other unless both parameters are adjusted with respect to height, simultaneously.

Hence, the concept of a ‘true’ KLa is totally unjustified. Similarly, the term ‘apparent’ KLa makes

no sense at all.

Page | 290
Principle of Superposition and the concept of a baseline KLa

"The superposition principle, also known as superposition property, states that, for all linear

systems, the net response caused by two or more stimuli is the sum of the responses that would

have been caused by each stimulus individually. [Wikepedia]"

One of the major conflicts between the ASCE standard and the European standards is the

mathematical treatment on the oxygen saturation concentration in the CWT. This is because the

relationship between KLa and Cs is not fully understood by any of the standards. These two

functions are in fact the two sides of a same coin. When the water molecules are bonded as a liquid

medium, the inter-molecular forces can change according to environmental conditions. When these

bonds are weakened, gas can enter the liquid easier and so the rate of gas transfer (as exemplified

by the KLa parameter) increases. At the same time weakened forces between the water molecules

means that the water cannot hold as much molecular gas entering the system, and so the solubility

of the gas decreases. A simple experiment with a beaker of water would illustrate the fact. If the

beaker is initially devoid of dissolved oxygen when subjected to oxygen dissolution by diffusion,

the amount of oxygen transfer is the product of KLa and Cs over the beaker volume. This product

is a constant no matter how the inter-molecular forces change, as can be demonstrated by changing

the temperature (within the normal working range) that would affect the kinetic energy that would

change the forces. [Eckenfelder 1952][ASCE 1997][ASCE 2006][Vogelaar 2001]. The initial rate

of transfer is therefore given by dC/dt = Cs. KLa. This is the standard model of oxygen transfer in

its most basic form.

Whereas the saturation concentration is related to Henry’s law that governs solubility of the gas

into liquid, the mass transfer coefficient is not related to Henry’s Law. Both parameters however,

Page | 291
are related to the molecular attraction between the water molecules. When the attraction force is

strengthened (such as when the temperature is reduced), the capacity to hold more oxygen

molecules increases, and so the solubility increases. Conversely, when the attraction force is

decreased (such as by an increase in temperature), and the inter-molecular bonding is weakened,

oxygen molecules enter at a faster rate, but the capacity to hold them becomes smaller. Since the

mass transfer coefficient is related to how fast dissolution occurs, the two parameters are inversely

correlated when the liquid is disturbed by an external force. In Chapter 2, a detailed dichotomy on

the effect of temperature on these parameters is already given.

Like any physics problem, the principle of superposition is a powerful tool in tackling such

problems as oxygen transfer in an aqueous solution where many different forces are at play

simultaneously. As the dissolved oxygen content builds up in an aeration basin, this build-up of

gas dissolution in the aqueous solution exerts a counter-force for diffusion from the liquid to the

gas phases. When the principle of superposition is applied to the system, the net rate of oxygen

transfer between the aqueous phase and the gas phase is the vector sum of the two opposing forces,

depending on which force is more superior. In mathematical form, therefore, the net transfer is

given by dC/dt = ∑𝐶𝑠

𝐶 (𝐶. 𝐾𝐿 𝑎), where ‘C’ ranges from zero to Cs. This can be written as dC/dt =

KLa (Cs – C) when we make the assumption that the overall liquid film coefficient KLa does not

change regardless of whether the gas molecules are going into or out of the bubble. This

assumption is only correct when the height of the aeration basin is small. In the attempt to delineate

the effect of height on the mass transfer coefficient, eq. A2 makes a lot of sense when considering

the KLa at zero height as a baseline case. The author has now termed KLa0 as a special case for a

tank of infinitesimal height, and, since the height is now zero, the corresponding saturation

Page | 292
concentration must be identical to its surface solubility, and so the universal equation for this

baseline case can be written as:

dC/dt = KLa0 (Cs – C) (A4)

where Cs is the surface DO saturation concentration that can be read from any chemistry handbook

table on solubility (ASCE 2007 considered the table provided by Benson and Krause (1984) as

most accurate). This equation, however, still cannot be equated to the bulk transfer equation as

determined by a clean water test because these two cases have different gas-side gas depletion

rates. (In theory, eq. A4 has zero bubble gas depletion). However, for tank aeration with gas

depletion, eq. (A3) can be modified to:

𝑑𝐶
= 𝐾𝐿 𝑎0 (𝐶 ∗ ∞0 − 𝐶) − 𝑔𝑑𝑝𝑐𝑤 (A5)
𝑑𝑡

where 𝐾𝐿 𝑎0 is calculated by re-arranging the depth correction model (eq. 3-6) from a known value

of KLa, the ‘apparent’ KLa determined by an experiment, as follows:

𝑙𝑛(1 – 𝐾𝐿𝑎(𝛷𝑍𝑑 ))
𝐾𝐿 𝑎0 = − (𝐴6)
(𝛷𝑍𝑑 )

The parameter 𝐶 ∗ ∞0 is the saturation concentration that would have existed without the gas

depletion (note that 𝐶 ∗ ∞0 is not Cs), and 𝑔𝑑𝑝𝑐𝑤 is the overall bubble gas depletion rate during a

clean water test. This equation is again based on the Principle of Superposition in physics where

the mass transfer rate is given by the vector sum of the transfer rate as if gdp (gas depletion rate)

does not exist, and the actual gas depletion rate which is a negative quantity. The saturation

concentration 𝐶 ∗ ∞0 cannot be the same as Cs because the latter is the oxygen solubility under the

condition of pressure of 1 atmosphere only, while 𝐶 ∗ ∞0 corresponds to the saturation

concentration of the bulk liquid under the bulk liquid equilibrium pressure, but deducting the gas

Page | 293
depletion (this of course cannot happen, since without gas depletion there can be no oxygen

transfer). The hypothetical 𝐶 ∗ ∞0 must therefore be greater than 𝐶 ∗ ∞ which in turn is greater than

Cs since the former corresponds to a pressure of Pe (see Fig. 7-1 in the main text) while the latter

corresponds to the surfical pressure Pa, the atmospheric pressure or the barometric pressure at the

time of testing. This method of reasoning allows solving for the transfer rate from the baseline

mass transfer coefficient.

Since KLa is a function of gas depletion, and since every test tank may have different

water depths and different environmental conditions, their gas depletion rates are not the same;

hence, they cannot be compared without a baseline [Lee 2018]. Furthermore, by introducing the

term gdpcw, the oxygen transfer rate based on the fundamental gas transfer mechanism (the two-

film theory) can be separated from the effects of gas depletions on KLa. This gas depletion rate

cannot be determined experimentally, since gdpcw varies with time throughout the test. Jiang and

Stenstrom (2012) have demonstrated the varying nature of the exit gas during a non-steady state

clean water test. Therefore, the only equation that can be used to estimate the parameters is still

the basic transfer equation given by eq.A3:

𝑑𝐶
= 𝐾𝐿 𝑎 (𝐶 ∗ ∞ − 𝐶)
𝑑𝑡

Note that the use of the ‘apparent’ term is no longer necessary, since the concept of ‘true’ KLa is

not acceptable (as explained above). Eqs. A3 and A5 are essentially equivalent to each other but

expressed differently (KLa vs. KLa0). Therefore, by the same token using the Principle of

Superposition and the Principle of Mathematical induction (ie. if the phenonmenon is true for

clean water it must be true for wastewater), for in-process water without any microbes, eq A3

would become:

Page | 294
𝑑𝐶
= 𝐾𝐿 𝑎0𝑓 (𝐶 ∗ ∞0𝑓 − 𝐶) − 𝑔𝑑𝑝𝑤𝑤 (A7)
𝑑𝑡

It is postulated that the biological floc exerts biological-chemical reactions that produce a

resistance equal to gdpf giving, in the presence of microbes therefore:

𝑑𝐶
= 𝐾𝐿 𝑎0𝑓 (𝐶 ∗ ∞0𝑓 − 𝐶) − 𝑔𝑑𝑝𝑤𝑤 − 𝑔𝑑𝑝𝑓 − R (A8)
𝑑𝑡

Yex EXIT GAS

In an aerobic reactor, the aerated liquid volume

V, (for V = 1), has, under a fixed gas flowrate:

1) microbial population with a collective

respiration rate R
biological reactor 2) Oxygen transfer rate given by:
OTRf = 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 − 𝐶) − 𝑔𝑑𝑝𝑓
3) Dissolved oxygen concentration, C,
changing at a rate ± dC/dt and mass
balance,
INLET GAS 4) OTRf = ±dC/dt + R

Fig. A1. Mass Balance on the gas phase

Note that the R term in this equation is based on the Principle of Conservation of Mass or a material

balance, not the Principle of Superposition, as illustrated by Fig. A1 below. [Note: The law of

conservation of mass or principle of mass conservation states that for any system closed to all

transfers of matter and energy, the mass of the system must remain constant over time, as the

system's mass cannot change, so quantity can neither be added nor be removed.] Expressed

differently using the measurable parameters, the equation can be written as:

𝑑𝐶
= 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 − 𝐶) − 𝑔𝑑𝑝𝑓 − R (A9)
𝑑𝑡

Page | 295
where in the above equations, gdpww is the gas depletion rate for bubbles in wastewater, gdpf is the

gas depletion rate due to the microbial respiration, and R is the microbial respiration rate.

The subscript f refers to field conditions for all the parameters, and that in this last equation, when
dC/dt = 0, gdpf would be given by 𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 − 𝐶) − 𝑅, where C becomes a constant, usually

denoted by a symbol CR representing the saturation concentration at steady state under process

conditions, where Yex is the off-gas mole fraction that can be measured by an off-gas test [ASCE

1997]. When the system has reached a steady state in the presence of microbes, the gas depletion

rate is a constant, and so it would be possible to calculate the microbial gdp by the same equation
and by incorporating R as well when dC/dt = 0 and C = CR. In the presence of microbes, the

advocated hypothesis is that this gdpf due to the microbes is the same as the reaction rate R and so
dC/dt = KLaf (C*∞f - C)-R-R, compared to clean water where the microbial gdp = 0. In other words,

if F1 is the gas depletion rate for clean water, and F2 is the gas depletion rate in process water, then

F1 – F2 = R. It should be noted that, as mentioned before, the basic mass transfer equation is

universal, its general form given by the standard model. Therefore, in a non-steady state test for

in-process water for a batch test, the transfer equation is given by:
𝑑𝐶
= 𝐾𝐿 𝑎𝑓 (𝐶𝑅 − 𝐶) (𝐴10)
𝑑𝑡

where CR is the steady-state DO concentration value attained in the test tank at the in-situ oxygen

uptake rate, R, under a constant gas supply. But the transfer equation is also given by dC/dt=

KLaf (𝐶 ∗ ∞𝑓 -c)-R-R as mentioned above. Equating the two gives,

𝐾𝐿 𝑎𝑓 (𝐶 ∗ ∞𝑓 – 𝐶) − 𝑅 − 𝑅 = 𝐾𝐿 𝑎𝑓 (𝐶𝑅 − 𝐶) (𝐴11)

which gives:

2𝑅
𝐾𝐿 𝑎𝑓 = (𝐴12)
(𝐶 ∗ ∞𝑓 − 𝐶𝑅 )

Page | 296
Note that in this equation, C is cancelled out, so that the above equation is valid for any value of

C, at any state, so long as dC/dt ≥ 0 and C < CR. Most models do not simulate the gas phase, and

so are missing this important element in their balancing equations. This 𝐾𝐿 𝑎𝑓 can then be related

to the clean water KLa which serves as a baseline for extrapolating the clean water test results to

wastewater by using the correction factors α and β applied to KLa and C*∞ respectively. The correct

transfer equation under process conditions at any DO level of C would then by given by eq. A9

where the gdpf is given by the microbial respiration rate at a steady state when C = CR, provided

that R does not change drastically under another process DO level for the same gas flowrate

supplied to the system under test. The estimated value from eq. A12 (that relies on a measurement

of R) for the mass transfer coefficient KLaf should then be compared with that determined by a

bench-scale or pilot-scale test as depicted in Fig. 7-7 that does not depend on any measurement of

R. These two estimations of KLaf should give similar results to each other.

Page | 297
References

1. Andrade Julia (2013) “Solubility Calculations for Hydraulic Gas

Compressors” Mirarco Mining Innovation Research Report
2. ASCE 1997. ASCE-18-96. Standard Guidelines for In-Process Oxygen
Transfer Testing. ASCE Standard. ISBN-0-7844-0114-4, TD758.S73 1997
3. ASCE 2007. ASCE/EWRI 2-06. Measurement of Oxygen Transfer in Clean
Water. ASCE Standard. ISBN-13: 978-0-7844-0848-3, ISBN-10: 0-7844-
0848-3, TD458.M42 2007
4. ASCE 2018. ASCE/EWRI 18-18 (2018). American Society of Civil Engineers
Standard Guidelines for In-Process Oxygen Transfer Testing. Reston,
VA.
5. Baillod, C. R. (1979). Review of oxygen transfer model refinements and
data interpretation. Proc., Workshop toward an Oxygen Transfer
Standard, U.S. EPA/600-9-78-021, W.C. Boyle, ed., U.S. EPA,
Cincinnati, 17-26.
6. Benson, B. B., and Krause, D., Jr. (1984). “The Concentration and
Isotopic Fractionation of Oxygen Dissolved in Fresh Water and Sea-
water in Equilibrium with the Atmosphere.” Limnology and Oceanography,
29, 620–632
7. Boon, A. G. and Lister, A. R. (1973): Aeration in deep tanks: An
evaluation of a fine bubble diffused-air system”. Journal of the
Institute of Water Pollution Control, No. 5, pp. 3 – 18, 1973.
8. Boon, A. G. and Lister, A. R. (1975): Formation of sulphide in a rising
main sewer and its prevention by injection of oxygen. Progress in
Water Technology.7, pp. 289-300
9. Boon A. G. (1979). “Oxygen Transfer in the Activated-Sludge Process”
Water Research Centre, Stevenage Laboratory, England, United Kingdom.
10. DeMoyer, Connie D., Schierholz Erica L., Gulliver John S., Wilhelms
Steven C. (2002). Impact of bubble and free surface oxygen transfer on
diffused aeration systems. Water Research 37 (2003) 1890-1904.
11. Downing, A.A., A.G. Boon. (1968).“Oxygen Transfer in the Activated
Sludge Process”, In: Advances in Biological Waste Treatment, Ed. by W.
W. Eckenfelder, Jr. and B.J. McCabe, MacMillian Co., NY, p. 131.
12. Doyle, M., Boyle, W. C., (1981). Translation of Clean to Dirty Water
Oxygen Transfer Rates. University of Wisconsin, Madison, Wisconsin
(Unpublished).
13. Doyle, M. L., Rooney, T., & Huibregtse, G. (1983). Pilot Plant
Determination of Oxygen Transfer in Fine Bubble Aeration. Journal
(Water Pollution Control Federation), 55(12), 1435-1440. Retrieved
from http://www.jstor.org/stable/25042126
14. Eckenfelder, W.W., (1952). Aeration efficiency and design: i.
measurement of oxygen transfer efficiency. Sewage and Industrial
Wastes, pp.1221-1228
15. Eckenfelder (1970). “Water Pollution Control. Experimental procedures
for process design.” The Pemberton press, Jenkins publishing company,
Austin and New York.EPA/600/S2-88/022 (1988)
16. Garcia-Ochoa, F., E Gomez, VE Santos, JC Merchuk (2010). Oxygen uptake
rate in microbial processes: An overview. Biochemical Engineering
Journal 49 (3), 289-307.
17. Groves, K., Daigger, G., Simpkin, T., Redmon, D., & Ewing, L. (1992).
Evaluation of Oxygen Transfer Efficiency and Alpha-Factor on a Variety
of Diffused Aeration Systems. Water Environment Research, 64(5), 691-
698. Retrieved from http://www.jstor.org/stable/25044209

Page | 298
18. Houck, D.H. and Boon, A.G. (1980). “Survey and Evaluation of Fine
Bubble Dome Diffuser Aeration Equipment”, EPA/MERL Grant No. R806990,
September, 1980
19. Hwang and Stenstrom (1985). Evaluation of fine-bubble alpha factors in
near full-scale equipment H. J. Hwang, M. K. Stenstrom, Journal WPCF,
Volume 57, Number 12, U.S.A
20. Keil Z,Otero and Russel T W F (1987). Design of Commercial Scale Gas-
Liquid Contactors. A.I.Ch. E. J. 33(3) March, pp 488-496.
21. Lee, J.(2017). Development of a model to determine mass transfer
coefficient and oxygen solubility in bioreactors, Heliyon, Volume 3,
Issue 2, February 2017, e00248, ISSN 2405-8440,
http://doi.org/10.1016/j.heliyon.2017.e00248
22. Lee, J. (2018). Development of a model to determine the baseline mass
transfer coefficients in aeration tanks, Water Environ. Res., 90,
(12), 2126 (2018).
23. Lee, J. (2019a). Baseline Mass-Transfer Coefficient and Interpretation
of Nonsteady State Submerged Bubble-Oxygen Transfer Data. J. Environ.
Eng., 2020, 146(1): 04019102
24. Lee J. (2019b). Forum: Oxygen Transfer Rate and Oxygen Uptake Rate in
Subsurface Bubble Aeration Systems. J. Environ. Eng., 2020, 146(1):
02519003
25. Mahendraker, V. Mavinic, D.S., and Rabinowitz, B. (2005). A Simple
Method to Estimate the Contribution of Biological Floc and Reactor-
Solution to Mass Transfer of Oxygen in Activated Sludge Processes.
Wiley Periodicals, Inc. DOI: 10.1002/bit.20515
26. Mancy and Barlage (1968). Mechanisms of Interference of surface active
agents in aeration systems. Advances in water quality improvement.
E.F. Gloyna and W.w. Eckenfelder, Jr. Univ of Texas Press
27. Mines RO, Callier MC, Drabek BJ, Butler AJ (2016).Comparison of
oxygen transfer parameters and oxygen demands in bioreactors operated
at low and high dissolved oxygen levels. J Environ Sci Health A Tox
Hazard Subst Environ Eng. 2017 Mar 21;52(4):341-349. doi:
10.1080/10934529.2016.1258871. Epub 2016 Dec 7.
28. Stenstrom et al.(2006a). “Surfactant effects on α-factors in aeration
systems” Water Research 40, Elsevier Ltd.
29. Stenstrom et al (2006b). “Alpha Factors in Full-scale Wastewater
Aeration Systems.” 2006 Water Environment Foundation
30. Wagner et al. (2008). Oxygen Transfer Tests in. Clean Water in a Glass
Test Tank. Client: Innowater BV / Boxtel, The Netherlands.
Certificate
31. Yunt F. et al. (1980). Aeration Equipment Evaluation- Phase 1 Clean
Water Test Results. Los Angeles County Sanitation Districts, Los
Angeles, California 90607. Municipal Environmental Research Laboratory
Office of Research and Development. USEPA, Cincinnati, Ohio 45268
32. Yunt (1988). “Project Summary – Aeration Equipment Evaluation: Phase I
– Clean Water Test Results” Water Engineering Research Laboratory
Cincinnati OH 45268
33. Zhou, Xiaohong et al. (2012). “Evaluation of oxygen transfer
parameters of fine-bubble aeration system in plug flow aeration tank
of wastewater treatment plant” Journal of Environmental Sciences 2013,
25(2) ISSN 1001-0742 CN

Page | 299
 Chapter 8. Epilogue
The important findings that have been described in this book are summarized in this chapter as

listed below.

The Standard Model

For many years, attempts have been made to develop correlations between KLa values in

particular wastes and in pure water, using the Standard Model for bubble-oxygen transfer. The

standard model is given in eq. 4-1 in Chapter 4 as:

𝑑𝐶
= 𝐾𝐿 𝑎 (𝐶 ∗ ∞ − 𝐶)
𝑑𝑡

Even though the standard model has been in existence since 1924 [Lewis and Whitman 1924], the

fundamental principles of the model applied to a bulk liquid under aeration have not been fully

understood. By definition, the term KLa must necessarily envelop all of the aerator characteristics

associated with oxygenation in a water basin. Hence, the characteristic bubble size, relative

velocity, retention time and convective flow patterns are all lumped into this single mass transfer

parameter. By applying the principles of superposition and mathematical induction, this book has

shown that it is possible to mathematically and theoretically derive the equation for the standard

model from first principles. The various findings regarding the standard model applied to a bulk

liquid are summarized below:

Effect of height of liquid column

As a result of this rigorous analysis effort, a mathematical model to determine a baseline

mass transfer coefficient (KLa0) that is independent of the liquid depth, so that all the baseline

Page | 300
values determined with any tank height are equal, has been advanced. This method does not require

any measurement of bubble size, but does require a typical clean water testing using the

ASCE/EWRI 2-06 standard method (ASCE 2007). Conceptually, the baseline coefficient is that

which occurs when the tank depth tends to zero (i.e. where the tank height is virtually non-existent

or very small). The mathematical definition of this baseline transfer coefficient is given by:

𝑙𝑖𝑚 𝐾𝐿𝑎 = 𝐾𝐿𝑎0 (8 − 1)

𝑍𝑑 →0

where Zd = diffuser depth; KLa = apparent oxygen mass transfer coefficient; KLa0 = baseline

oxygen mass transfer coefficient. The baseline mass transfer coefficient (KLa0) is defined as the

hypothetical mass transfer coefficient when the measured KLa in a typical clean water test is

converted to that of an aeration tank of infinitesimal height, so that the diffuser immersion depth

Zd tends to zero. This physical meaning is equivalent to having no gas depletion (noting that in a

gas bubble, Dalton’s law states that the total pressure of a mixture of ideal gases is equal to the

sum of the partial pressures of the constituent gases. A corollary of this law states that the partial

pressure is given by the product of the mole fraction and the total pressure. Under this law, the

result of both gradual decreases in hydrostatic pressure and the mole fraction of oxygen as the

bubble rises, is a gradual decrease in oxygen partial pressure. The latter effect, manifested by the

difference between the oxygen content of the feed and exit gases, is termed “exit gas depletion” or

simply “gas depletion”) during gas transfer, similar to surface aeration. The depth correction

model as given by Eq. (4-51), is developed based on mass balances in both the liquid phase and

the gas phase as given in the Chapter 4, together with the derivation of the mathematical intricacies.

The effect of changing depth on the transfer rate coefficient KLa has been explored in this

book. As shown in Chapter 4 (Eqs. 4-53 and 4-54), the new model relating KLa for any tank

depth to the baseline KLa0 is expressed by:

Page | 301
 1 − exp(−𝛷𝑍𝑑 . 𝐾L𝑎0 )
𝐾 L𝑎 = (8 − 2)
𝛷𝑍𝑑

where Φ is a constant dependent on the aeration system characteristics. This equation is of the

same form as Eq. (6-17) given in Chapter 6 for wastewater, and is similar to eq. 4-44 in Chapter 4

for clean water. By expanding the exponential function into a series, KLa = KLa0 – KLa02Ф𝑍𝑑/2 +

KLa03(Ф𝑍𝑑 )3/3! ….it can be seen that, when Zd tends to zero, KLa tends to KLa0.

The real example given in Chapter 3 (data shown in Table 3-2), [Yunt et al. 1988a], using

tanks of 4 different heights---3.05 m (10 ft), 4.57 m (15 ft), 6.09 m (20 ft) and 7.62 m (25 ft) ---

illustrates the validity of the developed model. All the tank heights with the same average

volumetric gas flow rate yield an identical baseline mass transfer coefficient standardized to 20
0
C. This baseline coefficient (symbolized as KLa0) is conveniently expressed as (KLa0)20 where the

subscripts refer to baseline and the standard temperature of 20 0C respectively.

Effect of gas solubility

Furthermore, it was found that for any tank depth and temperature (subscript T refers), the

mass transfer coefficient, (KLa)T, is inversely related to the equilibrium saturation concentration,

(C* ∞ )T, with R2 = 0.9859 (Fig. 3-4); in the same way that the baseline (KLa0)T is inversely

proportional to the oxygen solubility (Cs)T (Fig. 3-5, R2 =0.9924) in water, that has already been

envisaged in Chapter 3 and in a previous manuscript (Lee 2017). The baseline KLa0 can be

determined by re-arranging Eq. (8-2) to obtain the following:

𝑙𝑛(1 – 𝐾𝐿𝑎(𝛷𝑍𝑑 ))
𝐾𝐿 𝑎0 = − (8 − 3)
(𝛷𝑍𝑑 )

The solution for the baseline involves solving Eq. (8-3), together with a set of simultaneous

equations as given by Eqs. 4-63, 4-65, 4-68, 4-72 and 4-74. Note that Eq. (4-72) is similar to Eq.

Page | 302
(4-66) but modified to replace y0 by ye in Eq. (4-66) to become the following equation to be used

in the spreadsheet:

𝐶 ∗∞ = 𝐻 𝑌𝑒 (𝜌𝑤 𝑔. 𝑒 𝑍𝑑 + 𝑃𝑏 − 𝑃𝑣𝑡) (8 − 4)

where Ye is the mole fraction at the equilibrium level (Fig. 3-1 refers); ρw g is the specific weight

of water; Pb is the barometric pressure; Pvt is the vapor pressure at the time of test. The main theme

of this manuscript is that, there is an inverse relationship between the mass transfer coefficient and

the dissolved oxygen saturation concentration, but the inverse linear relationship between KLa and

C*∞ holds only for shallow tanks, in which cases C*∞ approaches Cs which is the solubility of

oxygen in water as given by handbook values for all temperatures within the ASCE prescribed

range of 10 0C to 30 0C. (Both Hunter’s data and Vogelaar’s data in Chapter 2 have shown that

this temperature range can be more extensive in fact.) In fact, the relationship between KLa and

C*∞ will have its most precise application when Zd approaches 0, so that the saturation pressure

Ps approaches 1 atm, and the saturation concentration C*∞ approaches Cs or the oxygen

solubility, as shown by Fig. 3-5, even though, in the example cited [Yunt et al. 1988a], we only

have two temperature data to go by (three if the point of origin is also counted). The author suggests

that more temperatures are used to do testing in the future to verify the KLa vs. C*∞ relationship

for different tank depths, and in order to alleviate the concern about bunching up of the data at the

right-hand corner of Fig. 3-4 and Fig. 3-5. In this present exercise, each tank has only one single

temperature (either 16 0C or 25 0C) and so the author agrees that the data may not be sufficient to

affirm the definite relationship between KLa and C*∞ for different temperature, and for shallow

tank depths only. The definition of shallow requires further investigations. However, the effect of

gas solubility on the mass transfer coefficient can be equally examined by varying the overhead

pressure rather than by varying the temperature. The case studies as given in Chapter 5 based on

Page | 303
changing the headspace pressure from 1 atm to 3 atm dramatically illustrate the precise relationship

between the baseline mass transfer coefficient and the gas solubility. The gas solubility model for

oxygen as derived in Chapter 2 is thus verified.

Effect of gas flow rate

The standard mass transfer coefficient (KLa)20 is chiefly dependent on the average gas

flowrate (Qa)20 and to a lesser extent also on the tank depth. Qa is governed by another developed

model as given in Chapter 2 (Eq. 2-25):

1 1
𝑄𝑎 = 𝑄𝑆 × 172.82 × 𝑇𝑃 × [ + ] (8 − 5)
𝑃𝑃 𝑃𝑏

It should be noted in passing that, in the use of the baseline model given by Eq. (7-3), the term Zd

is the diffuser depth as opposed to tank depth which is usually about 0.6 m (2 feet) above the tank

bottom. The model requires an estimation of the submergence ratio (e) which can be determined

by the following equation:

e = de/ Zd (8--6)

The standard baseline (KLa0)20 has a relationship with the average gas flowrate (Qa)20 in water,

and the relationship can be found via the following equation:

(KLa0)20/(Qa20)q = A (8--7)

where A is a numeric constant and q is an exponent of the gas flowrate. This constant ‘A’ has

been defined as the standard specific mass transfer coefficient in this manuscript when (KLa)20 is

used in this equation. Otherwise, it is the standard specific baseline mass transfer coefficient, if

the baseline is used, as shown in eq. 8-7.

The baseline (KLa0)T has a relationship with the oxygen solubility (Cs)T in water, and the

relationship can be found via the following equation:

(KLa0)T/(QaT)q = B x (1/CsT) (8--7a)

Page | 304
where B is another numeric constant. For the data of Yunt et al. (1988a), q = 0.82

(dimensionless) and A = 0.0444 (see Fig. 3-3); B = 0.4031 (see Fig. 3-5). Eq. 8-7a effectively

stipulates that the specific baseline mass transfer coefficient at any temperature T is directly

proportional to the inverse of oxygen solubility at the same temperature.

Effect of mixing

It should be emphasized that the above derivation assumes that bubbles rise in plug flow

through a tank of well-mixed water. The initial bubble-size distribution and the rate of bubble

formation are assumed to be constant, and that they depend only on the specific aeration

equipment. Bubble coalescence and mass transfer of gases other than nitrogen and oxygen are

considered negligible. The water temperature and the ambient air temperature as well as the bubble

feed gas temperature are assumed to be equal and constant. Mass transfer through the water surface

at the top of the tank is neglected.

The above assumptions effectively ignore the fact that transfer devices typically produce

irregularly sized bubbles that often swarm in various hydrodynamic patterns, e.g. spiral roll

devices vs. full-floor coverage. In addition, oxygen transfer takes place during bubble formation,

bubble retention and bubble exit at the surface. (In ordinary aeration tanks, the average hydraulic

retention time for a fine bubble is in the order of 20-80 s. This reduces the time of bubble formation

to a negligible fraction of the total bubble residence time. Therefore, the gas transferred in the

bubble formation process is a negligible fraction of the total gas transferred [Stenstrom et al.

2006]). Therefore, within reasonable boundaries to be established by future research and testing

on tanks of different heights, it is reasonable to assume that, statistically, these effects are more

intensive in nature than extensive (i.e. these effects can be considered to be less dependent on scale

than the mass transfer coefficient will be), so that they can be controlled by similitude. Calibration

Page | 305
factors have been incorporated into the equations to account for such variables. However, these

assumptions cannot be all correct, especially for the fact that the impeller speed in a sparger system

may have a significant impact on the KLa value. It is assumed that, in this exercise, the rotating

speed, if any, is low to moderate sufficient to maintain a continuously stirred CSTR (completely

stirred and mixed tank reactor) and not affecting the KLa value from tank to tank, and so the transfer

rate is reasonably uniform, so that the instantaneous DO concentration throughout the tank is

deemed distributed uniformly and constant.

Effect of liquid temperature

If Vogelaar’s [Vogelaar et al. 2000] data is re-examined, which is based on testing in a 3-

litre bottle, the exact linear relationship between KLa and the inverse of C*∞ is confirmed ,

(within tolerances of the experimental errors) as shown in Table (8-1) below (which is the same

as Table 2-3). When KLa is plotted against the reciprocal of saturation concentration (which is

similar to solubility because of its small scale as it is only a 3-litre bottle), the correlation is

R2=0.9923 (see Fig. 8-1), which is similar in terms of correlation to Fig. 3-5 in Chapter 3 using

the baseline KLa0.

T (0C) T (K) (T/1000)5 *104 C*∞(mg/L) 1/C*∞ KLa (h-1)

0 0 0 0

20 293.15 21.65 9.19 0.1088 22.4+/-0.4

30 303.15 25.60 7.43 0.1346 26.0+/-0.1

40 313.15 30.11 6.5 0.1538 30.6+/-0.2

55 328.15 38.05 5.15 0.1942 38.8+/-1.5

Table 8-1. Vogelaar et al.’s test data of KLa and C*∞ at different temperatures at a fixed gas rate

(3 vvm)

Page | 306
 Vogelaar's data
KLa vs. 1/C*∞
45
40 y = 195.98x + 0.4741
R² = 0.9923
35
30
KLa (h^-1)

25
20
15
10
5
0
0 0.05 0.1 0.15 0.2 0.25
Reciprocal of C*∞ (L/mg)

Fig. 8-1. Plot of KLa vs. 1/C*∞ for Vogelaar’s data

The y-intercept is so small it can be considered zero for all intents and purposes. This shows

that a laboratory-scale test can give a similar value as the baseline calculated from a higher-scale

tank. The author believes that, owing to its shallow depth, the measured KLa in this experiment is

not much different from the baseline, KLa0, so that, for all intents and purposes, this measurement

of KLa can be regarded as a true measurement of the baseline KLa0. This proves that for very

shallow tanks, where C*∞ approaches CS, there is a definite linear correlation between the two

parameters KLa and C*∞, for varying temperatures and under a constant volumetric gas flow rate

(or average volumetric gas flow rate if the tank height becomes significant, as calculated by Eq.

(8-5)). This relationship becomes less precise the further the tank height departs from the “shallow”

tank criterion. This good-fit relationship cannot be simulated by the Arrhenius equation using Ɵ =

1.024 [Lee 2017] because the temperature range is much wider in this case (0 to 55 0C) than what

the Ɵ model can handle; however, using Ɵ = 1.016 would give a good fit as well. But, for the theta

Page | 307
model, this is a ‘chicken or egg’ problem. Without the experimental data in a simulation, one

would not know which Ɵ value to use. The 5th power model does not have this problem, and the

model is good for such temperature range [Lee 2017]. With this model, any one single set of data

(KLa and C*∞)T, would be sufficient to predict any other set of data within this temperature range

of 00C to 550C. The 5th power model was derived in Chapter 2 and is stated by eq. 2-47 as below:

(𝝆𝑬𝝈)𝟐𝟎 𝑻𝟐𝟎 + 𝟐𝟕𝟑 𝟓

(𝑲𝑳 𝒂)𝟐𝟎 = (𝑲𝑳 𝒂)𝑻 ( )( )
(𝝆𝑬𝝈)𝑻 (𝑻𝑻 + 𝟐𝟕𝟑)

where the symbols are as defined in Chapter 2. This equation is applicable to a tank of small height.

The sensitivity of this model to tank height requires further investigations.

The specific standard baseline (KLa0)20/Qa20q

As for the normalization to standard temperature and pressure, on the other hand, the

relationship between (KLa0)20 and Qa20 is a power curve such as is given by Fig. 3-3, where the

exponent is found to be 0.82, bearing in mind that Qa is calculated by Eq. (8-5), and the baseline

KLa0 is calculated from the measured KLa using the depth correction model. Here the relationship

between the baseline and the gas flowrate is not linear and not known until the best fit curve is

plotted in Fig. 3-3 with an R2 = 1, giving the value of the exponent as 0.82. It is not a circular logic

because, previously, the value of the exponent is unknown; it can only be determined by plotting

the data to find the best correlation using curve fitting, because, unlike temperature, gas flow rate

is an extensive property that is dependent on both the tank height and the gas supply Qs. The plot

can be determined by using the Excel Solver or similar to solve for q, and minimizing the sum of

squares error. This plot is expected to be true for all tank heights, but, if the mass transfer

coefficient rather than the baseline is plotted against the gas flow rate, each tank height would

yield a different curve.

Page | 308
 The power curve relationship between KLa and Qa is supported by many researchers in the

literature [Hwang and Stenstrom 1985][Jackson and Shen 1978]. Jackson and Hoech [1977]

related KLa value to a power function of the superficial air velocity, and found that the exponent q

varied from 1.08 to 1.13. King [1955] showed that the rate of oxygen absorption varied from

0.825th to 0.86th power of air flow rate depending on liquid depth and geometry. The exponent is

dependent on the diffuser type, and therefore it must be established by testing, as recommended in

Chapter 5 (See Fig. 5-1 for the flow chart procedure). Zhou X. et al. [2012] performed testing on

a full-scale wastewater treatment plant in Wuxi, China, and found that (KLa)20 is proportional to

Q^0.877 for fine bubble aerators. Long ago, Eckenfelder [1966] reported that the typical bubble

diameter db is related to the air volumetric flow rate as:

db α Qa^q’ (8--8)

where q’ is an empirical coefficient ranging from 0.2 to 1.0. Since the interfacial area per unit of

tank volume V is given by eq. 2-19 in Chapter 2:

a = 6 Qa. 𝑍𝑑 / (db vb V) (8--9)

and since it is assumed that db is a pure function of Qa as given by eq. 8-8, the interfacial area is a

pure function of Qa, also. Therefore, KLa must be a pure function of Qa also, (since it is a product

of KL and a), raised to some power q = (1 - q’) and assuming the bubble velocity is constant for

fine bubble diffusers for bubble size between 0.7 mm to 5 mm (eq. 4-8 in Chapter 4). In light of

the great variety of the exponent value in practice, the exponent is best determined by curve fitting;

and in the case of the FMC diffusers, the exponent that gives the best fit to the data is 0.82.

However, it is proven in this book that the exponent is independent of depth, provided that other

variables are held constant. The established best-fit exponent value is then used to normalize KLa

Page | 309
in order to produce the plots given in the main manuscript. The constancy of the exponent requires

further investigation.

Clean water compliance testing

This book has explained that it is possible to design for a system of air aeration or oxygen

aeration based on testing in clean water in accordance with the ASCE 2-06 standard together with

the concept of a baseline, and therefore the usefulness of the standard has been augmented

enormously. It is hoped that this book would serve as a standard guideline for professional

practitioners----engineers, owners, and manufacturers alike in evaluating the performance of

aeration devices operating at full-scale and under process conditions. The methods presented in

this book are intended for compliance testing of such, even though performance under process

conditions is affected by a large number of process variables and wastewater characteristics that

are not easily controlled.

Different from other textbooks, this book aims to solve a pressing engineering problem

using the fundamental theories and validated by experimental results extracted from the literature.

There are several major discoveries in the book:

1. It is the height-averaged volumetric gas flow rate that is proportional to the mass

transfer coefficient, not the standard mass gas flow rate;

2. The proportionality function between mass transfer coefficient and gas flow rate

is a power function but it is the baseline mass transfer coefficient that is exactly

correlated to the mean gas flow rate, not the mass transfer coefficient itself;

3. It is the baseline mass transfer coefficient that bears an exact correlation to the

oxygen solubility in water, not the mass transfer coefficient itself;

Page | 310
 4. The proportionality between the baseline and solubility is an inverse linear

function, and is in concurrence with the gas solubility law as explained in Chapter 2;

5. The mass transfer coefficient can be correlated to the equilibrium saturation

concentration but only approximately, when testing under barometric pressure at the top

surface and would be less accurate the further the depth increases;

6. Henry’s Law is verified, and extended to create a new general gas solubility law

(see Chapter 2), much like Boyle’s Law is extended to form the universal gas law;

7. The new findings can be logically applied to real situations (although not yet

verified by testing), such as aeration performance compliance testing in a full-scale

facility in in-process wastewater treatment.

Wastewater compliance testing and design

Chapter 6 presents a brand-new concept about oxygen transfer in the field, and Chapter 7

gives recommendations for further research requirements on this new concept. The novel

hypothesis is that the net oxygen transfer rate (OTR) is in fact affected by the microbial respiration

rate R, because of the additional resistance, produced and influenced by the microbes, that happens

to be the oxygen uptake rate (OUR) at steady-state. The salient equation for calculating the OTRf

is given by eq. 6-25 which develops into eq. 6-32. The hypothesis was based on the argument that,

the oxygen transfer capacity (OTRww) of an aeration system is not affected by the respiration rate

of micro-organisms present in the water, so that the mass transfer coefficient KLaf is constant

relative to the amount of microbes, and only varies relative to the wastewater characteristics.

(OTRww) gives rise to wastewater or in-reactor solution mass transfer as opposed to the net transfer

of the oxygen transfer rate in in-process water (OTRpw). Based on the premise that the net oxygen

Page | 311
transfer rate is now affected by the microbial oxygen uptake rate, it must be the vector sum of the

aeration-system transfer rate and the respiration rate which is a negative quantity because of the

resistance of the floc. If this resistance is not dependent on whether the system is at steady state or

not, then it can be determined by a mass balance at steady state such as by adjusting the gas

flowrate until a steady state is reached; since this resistance must be equal to the respiration rate at

steady state. Therefore, by the principle of superposition, (OTRww) – (OTRpw) = R. However, when

the gas flowrate is altered, the DO level changes as well, and it must be assumed that this alteration

of the DO would not affect the respiration rate or any other factors that may affect the value of the

mass transfer coefficient, such as the mixing intensity. The logical steps for the derivation of eq.

6-32 can be summarized as follows:

In any closed system of bulk liquid under aeration, the oxygen uptake rate by the bulk

volume must be equal to the oxygen transfer rate to the bulk volume, if there are no

chemical reactions involved. Therefore, considering a unit bulk volume,

OUR = OTR (8-10)

But OUR has two components, the uptake by the bulk liquid through the process of

diffusion and dissolution, and the uptake by the microbial communities, therefore, by the

Principle of Conservation of Mass,

𝑑𝐶
𝑂𝑈𝑅 = +𝑅 (8 − 11)
𝑑𝑡

where dC/dt is the accumulation rate. Substituting eq. 8-11 into eq. 8-10, therefore,

𝑑𝐶
𝑂𝑇𝑅 = +𝑅 (8 − 12)
𝑑𝑡

But the effective oxygen transfer in in-process water is affected by the respiration rate, so

that

Page | 312
 𝑂𝑇𝑅𝑝𝑤 = 𝑂𝑇𝑅𝑤𝑤 − 𝑅𝑏𝑓 (8 − 13)

where OTRpw is the oxygen transfer rate in process water; OTRww is the oxygen transfer

rate in wastewater; Rbf is the resistance to transfer by the biological floc that provides a

negative driving force. (This resistance can be determined by the off-gas method that

measures the differences in the off-gas oxygen mole fraction arising from the changes in

the gdp.) The oxygen transfer rate in the wastewater is a function of the mass transfer

coefficient and the positive driving force given by the concentration gradient between the

saturation DO concentration and the actual DO in the water. Therefore,

∗
𝑂𝑇𝑅𝑤𝑤 = 𝐾𝐿 𝑎𝑓 (𝐶∞𝑓 − 𝐶) (8 − 14)

Based on the assumption that the resistance is the same as the respiration rate, therefore,

from eq. 8-13,

∗
𝑂𝑇𝑅𝑝𝑤 = 𝐾𝐿 𝑎𝑓 (𝐶∞𝑓 − 𝐶) − 𝑅𝑏𝑓 (8 − 15)

and

∗
𝑂𝑇𝑅𝑝𝑤 = 𝐾𝐿 𝑎𝑓 (𝐶∞𝑓 − 𝐶) − 𝑅 (8 − 16)

This equation is equivalent to eq. 6-25 or eq. 6-32. Substituting this equation into eq. 8-12,

therefore,

𝑑𝐶 ∗
= 𝐾𝐿 𝑎𝑓 (𝐶∞𝑓 − 𝐶) − 𝑅 − 𝑅 (8 − 17)
𝑑𝑡

This equation is equivalent to eq. 6-52. Hence, it is proven that the accumulation rate in

the bulk liquid is given by the transfer rate minus twice the respiration rate of the cells

within the liquid. This equation differs from the ASCE -18-96 Guidelines by a single R in

the Equation 2 of the Guidelines which needs to be corrected for diffused aeration. The

above argument is based on a batch process, and so for a continuous process, the equation

eq. 8-17 would need to be amended to include the wastewater flow rate and the DO

Page | 313
 concentration in the influent in such cases. The takeaway from the arguments presented in

this book is that the mass transfer coefficient is affected by two major effects in in-process

water oxygen transfer, namely, the wastewater characteristics; and, the gas-side gas

depletion rate gdp accompanying the biological floc resistance due to the microbes. All the

effects are associative in nature, but the first effect is associative by scale, while the other

effect is associative by addition (superposition). The current ASCE equation has treated

both the effects as by scale, so that the oxygen transfer rate is given as:

∗
𝑂𝑇𝑅𝑓 =∝ 𝐾𝐿 𝑎(𝐶∞𝑓 − 𝐶) (8 − 18)

The parameter ∝ is a lumped parameter that includes both effects together bound into this

one single parameter. The transfer equation advocated by the author is written as:

∗
𝑂𝑇𝑅𝑓 =∝′ 𝐾𝐿 𝑎(𝐶∞𝑓 − 𝐶) − 𝑅 (8 − 19)

where ∝′ is associated with the water characteristics only. Given that the oxygen

accumulation rate (or the liquid phase uptake rate) in the bulk liquid is the vector

mathematical sum of the oxygen transfer rate and the microbial uptake rate, this concept

has resulted in a mathematical relationship given by eq. 8-17 above.

It must be remembered that the mass transfer coefficient in clean water is determined by a non-

steady state test. It cannot be determined by any steady-state test. On the other hand, when

determining the mass transfer coefficient for in-process water, a steady-state or quasi-steady state

test is required. Even if a non-steady state method is used, a pseudo-steady state is still required

for testing in-process water. Hence, the gas depletion rates between the two types of test must be

different. This difference must be reconciled if it is the intention to use the clean water coefficient

as a baseline for the in-process coefficient. The book has suggested that further testing is needed

to validate this mass balance equation (eq. 8-17), as explained in detail in Chapter 7.

Page | 314
 The current use of a single constant value to represent the α-factor as exhibited in eq. 8-18

has a tremendous flaw as is recognized by Baquero-Rodriguez et al. (2018). Their proposed

solution is to change the current practice of a constant alpha (α) to using a dynamic α-factor, and

to use a dynamic model to describe aeration energy demand, both in 24-hour periods with organic

load variations and α-factor changes. It would be interesting to compare their results, when such a

dynamic model becomes available, to the results based on eq. 8-19 that uses the approach

recommended in this book, which is to separate the dual effects of respiration rate and wastewater

characteristics, and as more data with regard to both approaches are gathered. The respiration rate

is a function of the organic loading rate and the amount of bacterial biomass (the respiring cells)

present in the mixed liquor which is a function of the MLSS concentration. In the author’s opinion,

the respiration rate R is easily measurable, such as by means of the ASCE method (ASCE 1997)

for in-situ oxygen uptake rate measurement or other methods as described in Chapter 7. However,

the Guidelines ASCE 18-96 have not given an assessment of the recommended steady-state

column method in detail and how well this method compares with the ex-situ methods which,

according to the Guidelines, depend much on the time lapse between sample collection and uptake

rate measurement. This book has recommended a dilution method that may give a better estimate

of the in-situ respiration rate, as long as the test is carried out as soon as the sample is withdrawn.

The author believes the depletion of substrate in the sample is much slower than the depletion rate

of the dissolved oxygen so that the rate of DO decline without aeration should give a good estimate

of the microbial oxygen uptake rate. An accurate measurement of R is critical to the approach

using eq. 8-19. The conventional BOD bottle method is not recommended as it tends to over-

estimate the in-situ respiration rate. The proposed concept of separating out the effect of respiration

Page | 315
from other effects has resulted in a different α-factor symbolized as alpha’ (α’) that is to be used

in this bubble-oxygen transfer rate equation (eq. 8-19).

The main breakthrough

The discovery of a Standard Specific Baseline Mass Transfer Coefficient represents a

revolutionary change in the understanding, designing, operation and maintenance of the aeration

equipment, as well as in providing the baseline for future research, development and design.

Compliance testing means that, subject to certain achievable constraints, all measurements of

oxygen transfer in clean water in accordance with the standard ASCE 2-06 should yield the same

standardized specific baseline. Simulation means that such baseline as measured is used for scaling

up and predicting performances in raw wastewater aeration through a constant correction factor

alpha’ (α’) for the parameter KLa0 and through knowledge of the respiration rate R. In this book,

α’ is treated as dependent only on the characteristics and nature of the waste. This is substantially

different from the classical method of designing the in-process mass transfer coefficient KLaf

where the parameter alpha (α) must be designed as a range. Using the transfer of oxygen to tap

water as the datum, the new approach of calculating via the use of a baseline can now be used to

relate the overall mass transfer coefficient of the wastewater to that of tap water. This also means

that a bench-scale determination will become meaningful for translating such test results to full-

scale, pending further testing and validation.

As stated in Chapter 3, this finding, if proven to be correct, may be utilized in the

development of energy consumption optimization strategies for wastewater treatment plants and

may also improve the accuracy of aeration models used for aeration system evaluations. The major

achievement of this book was showing that gas transfer is a consistent relativistic theory of

Page | 316
molecular interactions based on the Standard Model, and matched the predictions observed in

experiment. The standard model is really the universal model that everybody is looking for,

recognizing that the parameter estimation for KLa in a typical set of non-steady state, clean water

oxygen-transfer data is not the real KLa for all types of aeration equipment. Therefore, solving for

the real KLa is almost like an impossible task. Instead, all we can do is make some assumptions

and either tease out some higher-order approximate terms or to examine the specific form of a

problem and attempt to solve it either numerically such as that carried out by McGinnis et al.

(2002), or mechanistically using all the physical laws, theories, and mathematics available, such

as carried out in this book so that the problem becomes solvable, such as by the assumption of a

constant bubble volume and constant bubble velocity.

We can then extract how the behavior of a solvable system differs from the general system

in real life and find corrections by identifying the important variables in the solvable system that

can be calibrated against real situations, and then apply those corrections to a more complicated

system that perhaps we cannot solve. The corrected calibrated model can then form a baseline from

which the standard model can be adapted to this baseline model that would yield a baseline KLa

that would be true for all types of aeration equipment. (See case studies presented in Chapter 5).

Hitherto, the primary challenge was the appearance of divergences in the mass transfer

coefficient calculations and estimations. The whole procedure of renormalization to a baseline and

to a depth-averaged gas flow rate was a great important achievement even if it had been another

three decades before it was properly understood (the author first postulated the concept of gas-

phase gas depletion in 1978) — these theories could have been thrown away for that interim period,

possibly delaying physics for 30 to 50 years.

Page | 317
References

ASCE-2-06. (2007). “Measurement of Oxygen Transfer in Clean Water.” Standards

ASCE/EWRI. ISBN-10: 0-7844-0848-3, TD458.M42 2007

Baquero-Rodriguez Gustavo A., Lara-Berrero Jaime A., Nolasco Daniel, Rosso Diego (2018). A
Critical Review of the Factors Affecting Modeling Oxygen Transfer by Fine-Pore
Diffusers in Activated Sludge. Water Environment Research, Volume 90, Number 5, pp.
431-441(11).

Hwang H. J., Stenstrom M. K., (1985). Evaluation of fine-bubble alpha factors in near full-
scale equipment. Journal WPCF, Volume 57, Number 12, U.S.A

Jackson, M. L., and Shen, C-C., "Aeration and Mixing in Deep Tank Fermentation Systems,"
J. AIChE, 24, l, 63 (1978).

Jackson, M. L., and Hoech, G. W. (1977). "A Comparison of Nine Aeration Devices in a 43-
Foot Deep Tank," A Report to the Northwest Pulp and Paper Association, Univ. of Idaho,
Moscow (1977)

King, H. R., "Mechanics of Oxygen Absorption in Spiral Flow Aeration Tanks: I . Derivation
of Formulas," Sew. Ind. Wastes, 27, 894 (1955).

King, H. R., "Mechanics of Oxygen Absorption in Spiral Flow Aeration Tanks: II.
Experimental Work," Sew. Ind. Wastes, 27, 1007 (1955).

Lee J. (2017). Development of a model to determine mass transfer coefficient and oxygen
solubility in bioreactors, Heliyon, Volume 3, Issue 2, February 2017, e00248, ISSN 2405-
8440, http://doi.org/10.1016/j.heliyon.2017.e00248.

Lewis, W.K., Whitman, W.G. (1924). “Principles of Gas Absorption”, Ind. Eng. Chem., 1924,
16 (12), pp 1215–1220 Publication Date: December 1924 (Article)
DOI:10.1021/ie50180a002

Stenstrom et al. (2006). “Alpha Factors in Full-scale Wastewater Aeration Systems.” 2006
Water Environment Foundation.

Page | 318
 Vogelaar, J.C.T., KLapwijk, A., Van Lier, J.B. and Rulkens, W.H., (2000). Temperature effects
on the oxygen transfer rate between 20 and 55 C. Water research, 34(3), pp.1037- 1041.

Yunt Fred W., Hancuff Tim O., Brenner Richard C. (1988a). Aeration equipment evaluation.
Phase 1: Clean water test results. Los Angeles County Sanitation District, Los Angeles,
CA. Municipal Environmental Research Laboratory Office of Research and
Development, U.S. EPA, Cincinnati, OH.

END OF BOOK PAGE

Page | 319

View publication stats