Desalination and Water Purification Research

and Development Program Report No. 115

Plant-Scale Engineering Design of

Membrane Processes from
Bench-Scale Measurements,
Version 1

U.S. Department of the Interior

Bureau of Reclamation
Technical Service Center
Denver, Colorado April 2006
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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER

03-FC-81-0918
Plant-Scale Engineering Design of Membrane Processes from Bench-Scale
Measurements, Version 1 5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S) 5d. PROJECT NUMBER

John Pellegrino and Kohn Phu
5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT

Department of Civil, Environmental, and Architectural Engineering, 428 UCB NUMBER
University of Colorado – Boulder
Boulder, CO 80309-0428

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S)

Bureau of Reclamation, Denver Federal Center,
P.O. Box 25007, Denver, CO 80225
11. SPONSOR/MONITOR'S REPORT
NUMBER(S)
DWPR Report No. 115

12. DISTRIBUTION/AVAILABILITY STATEMENT

Available from National Technical Information Service, Operations Division,
5285 Port Royal Road, Springfield, VA 22161

13. SUPPLEMENTARY NOTES

Final Report can be downloaded from Reclamation website: www.usbr.gov/pmts/water/reports.html

14. ABSTRACT
The initial components of a self-contained software application have been created to provide material balances (that is, the
permeate and retentate flowrates and compositions) and the pump energy for plant scale reverse osmosis (RO), nanofiltration
(NF), ultrafiltration (UF), and microfiltration (MF) membrane unit operations. The application modeling allows some input
parameters to come from the data normally reported from bench-scale measurements performed in standard stirred-cell,
crossflow (aka swatch) apparatus, or small hollow-fiber modules. The software application was developed using the LabView®
programming environment.
15. SUBJECT TERMS
membranes, water treatment processes, modeling

16. SECURITY CLASSIFICATION OF: 17. 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON
LIMITATION OF OF PAGES Saied Delagah
ABSTRACT
a. REPORT b. ABSTRACT a. THIS PAGE 19b. TELEPHONE NUMBER (Include area code)
U U U 303-445-2248

Standard Form 298 (Rev. 8/98)

Prescribed by ANSI Std. Z39.18
Desalination and Water Purification Research
and Development Program Report No. 115

Plant-Scale Engineering Design of

Membrane Processes from
Bench-Scale Measurements,
Version 1

Prepared for the Bureau of Reclamation Under Agreement

No. 03-FC-81-0918
by

John Pellegrino and Kohn Phu

University of Colorado
Civil, Environmental, and Architectural Engineering Department

U.S. Department of the Interior

Bureau of Reclamation
Technical Service Center
Denver, Colorado April 2006
MISSION STATEMENTS

The U.S. Department of the Interior protects America’s natural resources and
heritage, honors our cultures and tribal communities, and supplies the energy to
power our future.

The mission of the Bureau of Reclamation is to manage, develop, and protect

water and related resources in an environmentally and economically sound
manner in the interest of the American public.

Disclaimer
The views, analysis, recommendations, and conclusions in this report are those of
the authors and do not represent official or unofficial policies or opinions of the
United States Government, and the United States takes no position with regard to
any findings, conclusions, or recommendations made. As such, mention of trade
names or commercial products does not constitute their endorsement by the
United States Government.

Acknowledgments
We wish to recognize the major contributions made to this project by Kate
Worster and Laura Richards. Kate Worster undertook the original ill-fated, thrust
of developing this application as an Excel® Workbook, and then provided the
initial LabView® VIs that evolved into the final product. Laura Richards
undertook the thankless task of transcribing and creating fitting correlations for
the aqueous electrolyte properties that were used in the subroutines. In addition,
the Technical Project Manager, Mr. Saied Delagah, has continuously provided
both intellectual input and moral support that allowed us to bring the effort to its
current conclusion. This research was sponsored by the Desalination and Water
Purification Research and Development Program of the Bureau of Reclamation.
CONTENTS
Page
Executive Summary ................................................................................................ 1
Background ............................................................................................................. 2
Review of Membrane Process Modeling .......................................................... 2
Conclusions and Recommendations ....................................................................... 5
Results…. ................................................................................................................ 5
Overview ........................................................................................................... 5
Excel® Workbook Modeling Code Problem .............................................. 5
Current Approach........................................................................................ 6
Subprograms (vi’s)...................................................................................... 7
Benchmarking Data .................................................................................... 7
Vendor Software ......................................................................................... 7
Physical Property Data ................................................................................ 7
Modeling Approach .......................................................................................... 8
Membrane Element Geometric Model........................................................ 8
Net driving pressure .................................................................................. 10
Concentration ............................................................................................ 10
Mass transfer coefficient ........................................................................... 10
Code Structure (current programming flowsheet) .......................................... 11
Current Overall Model .......................................................................................... 15
Appendices ............................................................................................................ 17
Appendix 1: Description of Some of the Working Subprogram vi’s ............ 17
Appendix 2: Benchmarking Analysis of Vendor Specifications
(Hydranautics SWC-2521).................................................................. 23
Appendix 3. Vendor Design Software: Program Comparisons ...................... 8
ROSA 5.3 .................................................................................................... 8
Appendix 4: Physical and Chemical Property Data Accumulation and
Correlation ............................................................................................ 9
References ............................................................................................................. 13

Figures
Page
Figure 1. Unrolling a spiral-wound element. ..........................................................8
Figure 2. Division of membrane into sections (calculation-elements). ..................9
Figure 3. An individual area element; top view (left), side view (right). ...............9
Figure 4. Flowchart color code. ............................................................................11
Figure 5. Code flowchart of steady state problem. ...............................................12
Figure 6. Outline of subfunction for calculation of intrinsic property
parameters of the membrane and the feed stream. ...............................12
Figure 7. Outline of subfunction for stream composition. ....................................12

iii
Figure 8. Outline of subfunction for specifying problem criteria. ........................12
Figure 9. Flowchart for specifying the properties of the membrane
elements. ..............................................................................................13
Figure 10. Flowchart for shortcut analysis to develop the initial starting
guesses. ................................................................................................13
Figure 11. Flowchart for the pseudo-steady-state problem. .................................14
Figure 12. Flowchart for subfunctions to calculate the concentration
polarization of each solute (left) and the solute material balance
in each step (right). ..............................................................................14
Figure 13. Outline of next step in the problem. ....................................................14

Figure A-1. Schematic of the generic membrane process. .................................. 23

Figure A-2. Side view of spiral element. ............................................................... 3
Figure A-3. Nomenclatue applied to the SWC-2521 membrane. .......................... 3

Tables
Page
Table A-1. Bulk properties of aqueous NaCl solution at 293 K ............................ 6
Table A-2. Aqueous solutions properties at 293.15 K transcribed thus far ........... 9
Table A-3. Molarity (g-mol/L) as a function of mass percent salt ...................... 10
Table A-4. Relative density (kg/L) at 293 K as a function of mass percent salt . 10
Table A-5. Viscosity (Pa·s or kg·s·m–1) as a function of mass percent salt ........ 11
Table A-6. Conductance (mS/cm) as a function of mass percent salt ................. 12

iv
SYMBOLS, ACRONYMS, AND ABBREVIATIONS
d diameter of pipe
D diffusivity
f (as subscript) feed water
g-mol gram-mole
k boundary layer coefficient
L liter
L length
M molar concentration
MF microfiltration
mol mole
NDP net driving pressure
NF nanofiltration
P pressure
p (as subscript) permeate
R the ideal gas constant
Re Reynolds number
RO reverse osmosis
Sc Schmidt number
Sh Sherwood number
T temperature
UF ultrafiltration
vi any of various subroutines of the LabView® program
V velocity
MPa megapascals
r (as subscript) retentate, concentrate, or reject
V volumetric flowrate
C concentration
K kelvins
ΔP applied transmembrane pressure
μ fluid viscosity
Π osmotic pressure
ρ fluid density

v
EXECUTIVE SUMMARY
The initial components of a self-contained software application have been created
to provide material balances (that is, the permeate and retentate flowrates and
compositions) and the pump energy for plant scale reverse osmosis (RO),
nanofiltration (NF), ultrafiltration (UF), and microfiltration (MF) membrane unit
operations. The application modeling allows some input parameters to come from
the data normally reported from bench-scale measurements performed in standard
stirred-cell, crossflow (aka swatch) apparatus, or small hollow-fiber modules.

The software application was developed using the LabView® programming

environment. Though normally a data acquisition and process control software,
LabView® provides many library routines (called VIs) that perform sophisticated
mathematical functions. In addition, LabView® provides a built-in, graphical
user interface for both input and output; can be used on Unix, MacOS, and
Windows operating systems; applications can be distributed as compiled, run-only
programs that do not require ownership of the LabView® software; and the
application routines can be directly used to create an unlimited number of process
“flow diagrams”.

The purpose of the modeling software is to provide researchers, designers, and

project managers with a tool to quantitatively assess the significance of technical
advances that are often first described by reports of bench-scale measurements.
This modeling tool can: i) enhance the accuracy and capability of sensitivity
analyses done to focus research efforts, ii) reduce the time (and cost) to
implement new technology and applications of existing technology, and iii)
facilitate optimizing cost and/or design for different operating conditions. The
proposed software should, eventually, be integrated with the Reclamation-
sponsored WTCost© software.

The modeling software has several separate components including fitting

correlations for the density, viscosity, and osmotic pressure of aqueous
electrolytes. Mixtures are currently represented as perfect solutions. We have not
incorporated any phase equilibrium considerations in this initial application. The
model is primarily able to reduce experimental data in terms of an “effective
medium” that represents the complex (or simple) mixture that the researcher used.
That is, there will be one or two “key” solutes whose transport properties through
the membrane fit the reported data based on the conditions that the experimenter
used—for instance, this is similar to representing a mixture of natural organic
matter (NOM) as a single molecular mass component. Importantly, the test
solution can also have an “effective” flux decline character that will be defined in
terms of empirical parameters. The membrane’s solvent flux permeability can be
defined based on the experimental results. These calculated parameters and
properties will then be integrated (using a general shell balance) through a user-
specified membrane system with a desired capacity and configuration to calculate

1
the local concentrations and pressure required over the course of time. The
ultimate output will be the permeate and retentate (reject) flowrates and
compositions, and the increase in pumping pressure over time that is required to
keep the specified production rate. Temperature changes are not be included in
this Version 1.

BACKGROUND
If we accept the conclusions of the Desalination Roadmap (2003), the
development of viable, cost-effective, and technically efficient water supplies will
include membrane processes for the foreseeable future. Membrane-based
separations are an embodiment of the ideal separation processes. Accepting that
premise, then it is clear that anything that reduces development time and cost will
hasten the installation of viable water reclamation process equipment and will
decrease the cost of the water produced. Better process design will not only lower
the development costs incurred, but will also lowering the cost of capital and
increase system reliability. In addition, process design tools that provide the
means to assess the value of technological innovations, will help focus limited
resources into the areas that will provide the greatest benefit and/or leveraging.

At present, the development of plant scale membrane installations are often

slowed by the “cherished belief” that every new combination of water
composition (quality) and membrane material and/or module requires extensive
testing and validation through pilot scale. The belief that every new water
recovery application needs to follow the exact same piloting processes often
limits the number of membrane materials and process operating conditions that
will be evaluated.

Process-modeling tools help create the means for extending lower-cost bench-
scale measurements into “virtual pilot-scale evaluations.” Many more bench scale
measurements can be done in a shorter period of time, which is desireable
following Allgeier and Summers (1995), and therefore fewer pilot-scale
evaluations need be performed with the optimum materials and under suitable
process conditions to validate the design premises. This software model can be
made widely accessible to maximize its impact on the development of new
membrane technology—the scale-up modeling approach does not need to change
as new membrane technology is developed, as long as the new technology is
embodied in the bench scale tests.

Review of Membrane Process Modeling

Academic and industrial researchers have produced many outstanding
fundamental modeling efforts. More specifically, in recent years very detailed
numerical studies have been executed to predict the membrane filtration behavior

2
of more and more complicated systems, including such fundamental aspects as
solute and membrane pore-size distributions, fluid flow hydrodynamics, module
non-uniformities, and variable fluid properties (for example, viscosity gradients).
In addition, membrane manufacturers have their own proprietary, black-box
engineering models that are used to recommend their membrane systems to
customers. These prior contributions and resources will inform the proposed
project, but have been developed for different goals and thus manipulate physical
models that are not completely appropriate for the current purpose.

The engineering design of a membrane process includes at least three (3) levels of
transport phenomena:
1. The first level is the transport of solvent and other species through the
membrane (Mulder, 1992). This is the most fundamental level of membrane
science and technology and has been predominately studied by academia in
the current era (the last 40–50 years) of synthetic membrane development.
The variety of models include a spectrum of approaches that incorporate
solution-diffusion (Adam et al, 1983; Theil, 1990; Kataoka et al, 1991),
frictional flow (Sourirajan and Matsuura, 1986), and hindered
partitioning/diffusion (Deen, 1989; Schaep et al, 1999) which will cover the
range of reverse osmosis, nanofiltration, ultrafiltration, and microfiltration.
In fact, the most complete, and fundamental, form was presented by Mason
and Lonsdale (1990), with which they show how all other membrane
transport models directly result by making appropriate assumptions and
neglecting terms. Nonetheless, for the modeling of this proposed project,
we choose to use the “black box”, non-equilibrium transport model initially
developed by Kedem and Katchalsky (1958) and Spiegler and Kedem
(1966). This modeling approach has been widely accepted especially when
the parameters are experimentally measured (or estimated with minimum
assumptions.)
2. The next level of transport concern is in the mass transfer that occurs in the
fluid phase that is next to the membrane. It is well recognized that
concentration polarization, colloid (or particle) deposition, and adsorption
phenomena influence flux decline (fouling) as well as the solute and solvent
transport through the membrane—and vice-versa. Unlike membrane
transport, the study, prediction, and measurement of mass transfer (and
prevention of flux decline) in membrane filtration has been the major pre-
occupation of membrane technologists and researchers from, not only,
academia but also industry and government in the last 20 years. One of the
earliest reviews of this subject was presented by Brian (1967) (for RO) and
one of the most recent by Bowen (1995) is more broadly applicable. In
general, a robust description of the concentration of solutes at the membrane
interface is determined by solving a form of the advection-diffusion
equation in the channel next to the membrane. Alternatively, a less detailed
approach can be taken in which a “lumped” boundary-layer mass transfer
coefficient is used to describe the change in solute concentration between
the bulk fluid and the membrane interface, as done by Pradanos et al (1995).

3
 In this case, a correlation is used to calculate the effective mass transfer
coefficient. The description of the hydrodynamic environment is critical to
this type of model and the effect of turbulence promoters must be
adequately included, as shown by DaCosta (1993).
3. The final level of transport concern is the extension of the above mass
transfer analysis to integration over the entire channel (or tube) length.
When this is done, the prediction of the flowrates and compositions from
complete elements, series of elements, and entire membrane process trains
can be made. Many academic and industrial studies have been published on
this topic. In the context of RO, the usefulness of short cut models (Sirkar
et al, 1982, 1983; Evanelista 1985) have been recognized and successfully
applied in proprietary software (Dow/FilmTec) and in academic studies to
optimize process configuration (Evangelista, 1989). More complex models
that include pressure drop in the module and variations in 10 geometry and
mass transfer coefficient, and that require numerical solutions, have been
developed and applied to RO (Wiley et al, 1985; El-Halwagi et al, 1996).
Considering NF and UF, a variety of numerical modeling developments that
include concentration polarization and particle deposition over an integrated
channel geometry have been reported (Bhattachajee et al., 2001; Bacchin et
al., 2002), with the recent NF-PROJECT (Noronha et al., 2002) being the
most closely aligned with the objectives of this proposal.

Norona et al. (2002, 2003) have developed a computer simulation (NF-

PROJECT) which uses as its input parameters the experimentally-obtained data
from measurements on 2.5” × 40” elements. They fit the data from these
measurements to the parameters for an irreversible thermodynamic model
(Spiegler and Kedem, 1966) and, with several other phenomenological equations,
perform an isothermal, steady state analysis of a process train of membrane
elements. They have reported the use of this simulator to optimize the energy
cost, permeate quality, and product flow The authors don’t describe their actual
algorithm except that it is a numerical calculation (versus) analytical.

The current project builds very closely off the work of Norona et al. (2002, 2003),
with the additions that: i) we wish to incorporate flux decline mechanisms due to
adsorption and deposition, as well as, concentration polarization; ii) we will create
an interface that can work with bench-scale (“swatch”) measurements collected in
a variety of ways; iii) we will facilitate the adaptation of the software to all the
pressure-driven membrane filtration processes; iv) we will build in a framework
to extend the simulation to non-isothermal and unsteady-state operation (i.e.,
cleaning cycles); and, most importantly, v) the simulation tool will be directly
available for Reclamation and their constituency to use.

4
CONCLUSIONS AND RECOMMENDATIONS
The development of this software application needs to be continued. There is still
significant programming needed to complete even the simple RO design case
without fouling. The major items include finishing the iterative approach of
linking together elements (and the steps within elements), as well as the use of the
mixed solutes properties. The next step after that would be to develop a way to
link together multiple elements into a process train. The most straightforward
approach is to define a fixed number of possible configurations and allow the user
to choose among these.

In addition, we need to incorporate the fouling modeling, which is a very

significant amount of further programming development because of the unsteady-
state aspects.

The full program structure developed for the RO will be directly transferable to
MF, UF, and NF. Only the transport equations and input parameters would need
to be modified.

Comparison between the model application created in this project and the
commercial offerings from membrane vendors should also be part of a future
project.

RESULTS
Overview
As requested by Reclamation at the contract award, we had focused on
developing an Excel® workbook environment that facilitates partitioning full
membrane elements into smaller, discrete mass transfer units and solving the
conservation equations over these spatial elements in a step-wise and iterative
fashion for each time step.

Excel® Workbook Modeling Code Problem

Aside from a slightly different method of solving for the final variables for one
element, the integration approach (summation of multiple divisions of one
membrane element versus solving the entire membrane element as one mass
transfer step) was not as successful when computing the equations with the
macros. The main deterrents were the generation of several errors in Microsoft
Excel®. These unsolved errors often caused the program to crash. After much
deliberation and repeated attempts to correct these errors, it was concluded that
there may be inherent limitations to using the Microsoft Excel® Solver Add-In as
part of a set of nested calculations. The inter-relationships between the variables

5
overwhelms Excel’s ability to compute the final iterations using macros and
display them in a single output column.

Current Approach
We switched to a more suitable programming environment that is as user friendly
as Microsoft Excel®, and also meets the program interface criteria. It is National
Instrument’s LabView®. We followed the same algorithm structure that is
described in Appendix 1.

Some of the advantages to using LabView® are that graphical user interface is
more user friendly, the program is more capable of handling large loop and
iterative calculations, and the results can be outputed in other programming
languages or programs such as C+ and Microsoft Excel®. An application file
with the model can be created and distributed quite easily (across multiple
platforms), and does not require the expense of buying the full LabView®
Development Environment. The application file will also be resistant to user
tampering and unintentional changes. Reclamation will have the Full
Development version of the model and can make any changes in future versions.

Several initial LabView® subroutines (called vi’s) have been created (these are
described in Appendix 1), that calculate the previously outlined equations for an
element with multiple sub-sections—which had caused many problems in the
Excel® environment—and, as of this time, the current version has no errors. The
immediate benefit to using the LabView® programming environment is the
production of a program that not only works but is also stable and consistent,
which Microsoft Excel® has not been. In addition, it will be easier to implement
the use of collocation methods, to solve the entire mass and pressure balance in
membrane elements on a global basis, in the LabView® environment. This
collocation approach will take a future new project to develop and has some
uncertainties, but will probably be a more robust software tool if successful.

The following list are the primary tasks and their current status:
• Task 1a. Develop the time-dependent, shell-balance, engineering design
equations that include the pertinent macroscopic phenomena at a differential
level. These include key solute component(s) permeability (1 component
done); the solvent permeability (done); the hydrodynamic mass transfer
coefficient (done); and the three main modes of flux decline (not started).
• Task 1b. Define the step size (for both membrane area and time integration)
approaches for various types of membrane elements and module
configurations, including consideration of overall plant size (done for spiral
wound elements, not started for hollow fiber units).
• Task 1c. Develop the models for defining: i) the feed side mass transfer
coefficient and ii) the pressure drop through an element/module (done).

6
 • Task 1d. Define the integration algorithm for performing the summation of
the material balance through each single element in a series and checking its
convergence accuracy (simple forward difference method done).
• Task 1e. Develop a set of experimental data to input into the model for
benchmark testing (done).
• Task 1f. Create a complete set of default parameters for all the model input
variables (in progress).
• Task 1g. Create the Excel® workbook application using the specifications
from subtasks 1a-f (effort using Excel® workbook has been cancelled in
favor of using LabView® visual programming environment—this work is in
progress).
• Task 1h. Beta test the Excel® workbook application with a selected data set
that includes both bench-scale and pilot-scale measurements over extended
periods of time (not begun).

Subprograms (vi’s)
Labview® calls its subprograms vi’s. Appendix 1 contains a detailed description
of the vi’s including the equations, definitions of the variables, and how the
subprogram executes.

Benchmarking Data
We illustrate a reverse-engineering analysis of an arbitrarily chosen vendor’s
(Hydranautics SWC-2521 RO) specifications for a membrane element in
Appendix 2. This was done in order to make comparisons with bench-scale data
and the modeling algorithm results—and to facilitate extrapolation of that
membrane to other process conditions.

Vendor Software
We need to compare and contrast our design modeling development with the
vendor software tools available for RO systems. We have started with ROSA 5.3
(Dow- FilmTec). The initial results are shown in Appendix 3.

Physical Property Data

We are transcribing literature data of physical properties for representative species
(salts, water, macromolecules, and solutions thereof) into electronic (spreadsheet)
formats and fitting correlating equations to be used in process design models.
These data include transport and volumetric properties of water as a function of
temperature and pressure; density and activity coefficients for representative
electrolytes and organic molecules in order to estimate osmotic pressures as a

7
function of recovery. We have included the correlations in Appendix 4 and are
included as subprogram vi’s.

Modeling Approach
This section presents the overall algorithm and general modeling equations for
performing stepwise material and energy (pressure) balances across a spiral-
wound membrane element.

Membrane Element Geometric Model

We first consider unrolling a spiral-wound element as depicted in Figure 1. The
overall flows are depicted and we consider discretizing the membrane into n
sequential, differential area elements (a similar approach to Finite Element
Methods) for further analysis as in Figure 2. Initially, we ignore any edge effects
and the permeate side’s composition effect on the mass transfer from the feed side
of a semi-permeable membrane—though this assumption is easily relaxed. For
the initial calculation scheme we make the pseudo-steady-state approximation for
each time step.

Figure 1. Unrolling a spiral-wound element.

8
Figure 2. Division of membrane into sections (calculation-elements).

Figure 3. An individual area element; top view (left), side view (right).

Referring to Figure 3 we see that the differential membrane (or mass transfer)
area is given by the following equation:
ΔA = wΔL (1)
where ΔL is determined by the degree that we discretize the overall element
length, L. (N.B., We will need to evaluate the process design result’s sensitivity
to the choice of ΔL but, initially, we will use ~15 cm as a starting value. Thus,
one would need to increase the size of the problem (that is, the number of
calculation area elements) as the length of the full membrane element increases.
For example, the number of differential area elements would be n = L/15, where L
is given in centimeters.)

The mass balance is depicted in the RHS of Figure 3, where Q is the flow in the
feed channel and W1 and W2 are the mass flows of water and solute (for example,
NaCl) that permeate the membrane. The mass flows through the membrane are
given (at any time t) by the specific flux equations and are coupled to the overall
mass balance—in an iterative sense—by the compositions.

9
The Reynolds number (Re) will be calculated using Eq. 2. The cross-sectional
area for flow is Ax = h·w (where h is the height of the feed channel —determined
by the spacer thickness). The superficial velocity in each area element is given by
v = (Qi + Qi+1)/(2Ax) and will be coupled with the fluid’s density (ρ) and viscosity
(μ).
𝑣𝑣𝑣𝑣ℎ 𝜌𝜌
Re = 𝜇𝜇
(2)

The Re will be used with the appropriate correlations to determine the mass
transfer coefficient and the frictional component of hydraulic pressure drop in the
area element.

Net driving pressure

After calculating the mass transfer coefficient and frictional pressure loss, the Net
Driving Pressure (NDP) must be applied to the differential area element to
determine the permeate fluxes. The NDP is found by subtracting the pressure
head of the permeate (Pp) and the osmotic pressure of the feed (Πf) from the
average of the incoming and outgoing feed pressure (Pf) and the osmotic pressure
of the permeate (Πp).
NDP = Pf + Πp – Pb – Πf (3)
The Van’t Hoff equation will be used for osmotic pressure of larger solutes
present as dilute species.
𝐶𝐶𝐶𝐶𝐶𝐶
Π= 𝑀𝑀
(4)

The osmotic pressure Π is in pascals and is calculated from the solute

concentration (C) in moles per liter, temperature (T) in kelvins, the ideal gas
constant (R), and the molecular mass of the solute (M) in grams per mole. For the
electrolytes the correlations described in Appendix 4 are used.

Concentration
The concentrations of both the solvent and solute were required for all major
calculations and are therefore important items to calculate correctly when
converting units. The best estimates of real solution densities are being
incorporated in order to make certain that mass and molar balances are done
correctly. Even though fluxes are reported on a volumetric basis, the transport
equations are really only accurate on a molar basis.

Mass transfer coefficient

The correlation approach described by DeCosta (1993) is incorporated employed
to calculate the mass transfer coefficients in channels and the frictional pressure
losses of the system. In conjunction with dimensional analysis, the Sherwood

10
(Sh), Schmidt (Sc), and Reynolds (Re) numbers were used to compute the mass
transfer coefficient (km), which is the rate of mass transfer per unit area per unit
concentration.

𝑘𝑘𝑘𝑘ℎ
𝑆𝑆ℎ = (5)
𝐷𝐷

𝜇𝜇𝜇𝜇
𝑆𝑆𝑐𝑐 = 𝜌𝜌
(6)

𝑚𝑚̇
𝑘𝑘𝑚𝑚 = 𝐴𝐴 (7)
𝑚𝑚 (𝐶𝐶𝑖𝑖 −𝐶𝐶𝑖𝑖+1 )

The equation for the mass transfer coefficient consists of ṁ, which is the mass
flow rate in units of kilograms per second; (Ci –Ci+1), which is the concentration
of Am in moles per cubic meter; and Am, which is the area of the membrane section
in square meters, thus giving the mass transfer coefficient units of meters per
second.

Code Structure (current programming flowsheet)

A preliminary outline of the problem was used to create a base structure of the
necessary program code. This base structure’s framework was laid out in a
flowchart format using Microsoft PowerPoint. In addition to following the
number references from the text version of the code outline, all of the flow charts
utilize the same color coding for their boxes defining the various different types
and parts of the code (Figure 4).

Figure 4. Flowchart color code.

The ideal case of steady state example was first modeled and broken down into
sections that required sub-functions, user inputs, and program defined constants
Figure 5.

Each of the boxes designated as containing subfunctions (green) are also outlined
in the following flowcharts Figures 6 to 10.

11
Figure 5. Code flowchart of steady state problem.

Figure 6. Outline of subfunction for calculation of intrinsic

property parameters of the membrane and the feed stream.

Figure 7. Outline of
subfunction for Figure 8. Outline of subfunction for specifying problem
stream composition. criteria.

12
Figure 9. Flowchart for specifying the properties of the membrane elements.

Figure 10. Flowchart

for shortcut analysis to
develop the initial
starting guesses.

Once an ideal (perfectly mixed) steady state case and its respective subfunctions
are modeled, a stepwise modeling (using the pseudo-steady-state approach) using
the same methodology, numbering conventions, and color codes as described
above for the steady state case is done. The time-dependent (pseudo-steady state
approach) flowchart for an element are shown below as Figures 11 and 12.

For a time dependent problem, a pre-processor (in the problem input) will define
the way to proceed in doing multiple time-steps. But, in general, each time-step
will require solution of the pseudo-steady problem with changing initial
conditions, until either a convergence criteria is reached, or a specified amount of
time has passed.

13
 Figure 11. Flowchart for the pseudo-steady-state problem.

Figure 12. Flowchart for subfunctions to calculate the concentration polarization of each solute
(left) and the solute material balance in each step (right).

Figure 13. Outline of next step in

the problem.

14
CURRENT OVERALL MODEL
The following “screenshot” presents the current topmost level. It is called the
“Differential Unit.vi”. The values in the input boxes are the current defaults.

This subprogram does the material and pressure balance on the inflows and
outflows for an element. It needs significant further refinement, but right now
does the calculations as described in Appendices 3 and 4 for a single element.
The input is in the four leftmost sections and the output is in the rightmost section.
To link together multiple elements the output from one element becomes the input
to the next. A higher level subprogram (vi) is needed to do this and there are a
variety of ways to approach this.

Feed Properties: The feed flow rate, composition, and pressure are the inputs.

Right now this subprogram (vi) works with a single “real” solute (and its
properties) as the feed. This single “real” solute can be replaced by a “virtual”
solute using the mixture properties as output from the Solution Properties.vi
subprogram (see the following screenshot). This subprogram currently is set up
for 3 components and uses a perfect mixture model (properties combine in
proportion to their mole fractions). The properties (viscosity, density, osmotic

15
pressure, and the conversion between mass fractions and molarity) are
recalculated throughout the modeling in order to correctly account for the
composition change. They are calculated at a constant temperature (20 °C), but
that can (and will) be modified to use a general temperature dependency
correlation based on perfect mixture theory.

Solution Properties.vi screenshot:

Permeate Properties: only the permeate pressure needs to be specified at this

point because we are assuming that the permeate from each membrane element is
collected separately (a good assumption with small spiral elements.)

Membrane Properties: these include the differential element’s dimensions; the

water permeability; the assumed salt (solute) rejection; the mass transfer model to
use for calculating the boundary layer concentration; and the solute’s bulk
diffusion coefficient. Right now we assume a pressure drop along the element, in
the future we will include a correlation to calculate it, and/or allow the user to
input it as a variable.

General Settings: this is where items that relate to the calculation algorithm are
input. The tolerance on iterating for the flux is the only operational value at
present. The number of differential units is not used yet. It will be used to
control the step size when working with longer elements.

16
APPENDICES
Appendix 1: Description of Some of the Working
Subprogram vi’s
Average.vi
Purpose: Computes average of up to five numbers.
Inputs: Number of Values, Value 1–5.
Outputs: Average.
Notes: vi assumes number of variables is correct.

Equations:
∑𝑖𝑖 𝑉𝑉𝑖𝑖
𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎 =
𝑖𝑖
Vavg : Average value
Vi : ith value
i : number of values

Boundary_layer_coeff2.vi
Purpose: Calculates the boundary layer coefficient in many different flow
regimes/using different methods (SW Eriksson, Laminar Flow Between
Parallel Plates, Laminar Flow in Round Tubes, Turbulent Flow in Stirred
Batch Vessels, Turbulent Flow in Tubes 1 and 2).
Inputs: Diffusivity, viscosity, length, shear viscosity, diameter, avg. velocity,
Reynold’s number, stirrer speed, density, Schmidt number, height.
Outputs: Boundary layer coefficient.
Notes: Not all inputs have to be used for every model, vi does not check for this.
User must make sure all required inputs have values.

Equations:

SW Eriksson

𝐷𝐷 ∙ Re0.54 ∙ Sc 0.33
𝑘𝑘 =
2 ∙ 𝑑𝑑
k: boundary layer coefficient
D: diffusivity
Re: Reynolds number
Sc: Schmidt number
d: diameter of pipe

17
Laminar Flow Between Parallel Plates

𝑈𝑈𝑐𝑐 ∙ 𝐷𝐷2
3
𝑘𝑘 = 1.177 ∙ �
ℎ ∙ 𝐿𝐿

k: boundary layer coefficient

Uc: average velocity
h: distance between plates
L: length of plates

Laminar Flow in Round Tubes

3 2 ∙ 𝑈𝑈 ∙ 𝐷𝐷 2
𝑐𝑐
𝑘𝑘 = 1.295 ∙ �
𝑑𝑑 ∙ 𝐿𝐿

d: diameter of tube
L: length of tube

Turbulent Flow in Stirred Batch Vessels

1 3
𝐷𝐷 𝜌𝜌 3 𝜔𝜔 ∙ 𝜇𝜇 ∙ 𝑑𝑑 2 4
𝑘𝑘 = 0.0443 ∙ ∙ � � ∙� �
𝑑𝑑 𝜇𝜇 ∙ 𝐷𝐷 𝜌𝜌

ρ: fluid density
μ: fluid viscosity

Turbulent Flow in Tubes 1

0.0791 −1 −2
𝑘𝑘 = 𝑈𝑈𝑐𝑐 ∙ ∙ 𝑅𝑅𝑅𝑅 4 ∙ 𝑆𝑆𝑆𝑆 3
2
Generally, to use this equation the Reynold’s number (Re) must be greater than
20,000.

Turbulent Flow in Tubes 2

𝐷𝐷 1
𝑘𝑘 = 0.023 ∙ ∙ 𝑅𝑅𝑅𝑅 0.83 ∙ 𝑆𝑆𝑆𝑆 3
𝑑𝑑

18
Program Flowsheet:

concentration_permeate.vi
Purpose: Calculates the concentration of the permeate using a very simplistic
model.
Inputs: Retentate concentration, intrinsic rejection, solution flux, boundary layer
coefficient.
Outputs: Permeate concentration.
Notes: N/A

Equations:
𝐽𝐽𝑣𝑣
𝐶𝐶𝑟𝑟 ∙ (1 − 𝑅𝑅𝑜𝑜 ) ∙ 𝑒𝑒 𝑘𝑘
𝐶𝐶𝑝𝑝 = 𝐽𝐽𝑣𝑣
𝑅𝑅𝑜𝑜 + (1 − 𝑅𝑅𝑜𝑜 ) ∙ 𝑒𝑒 𝑘𝑘
Cp: permeate concentration
Cr: concentration retentate
Ro: membrane’s intrinsic rejection
Jv: flux through membrane
k: boundary layer coefficient

19
concentration_retentate.vi
Purpose: Calculates the concentration of the retentate using a very simplistic
model.
Inputs: Boundary layer coefficient, intrinsic rejection, feed concentration,
solution flux, recovery.
Outputs: Retentate concentration.
Notes: N/A

Equations:
𝐶𝐶𝑓𝑓
𝐶𝐶𝑟𝑟 = 𝐽𝐽𝑣𝑣
𝜃𝜃 ∙ (1 − 𝑅𝑅𝑜𝑜 ) ∙ 𝑒𝑒 𝑘𝑘
(1 − 𝜃𝜃) + 𝐽𝐽𝑣𝑣
𝑅𝑅𝑜𝑜 + (1 − 𝑅𝑅𝑜𝑜 ) ∙ 𝑒𝑒 𝑘𝑘

Cf: feed concentration

θ: recovery

concentration_wall.vi
Purpose: Calculates the concentration at the membrane wall.
Inputs: Boundary layer coefficient, intrinsic rejection, bulk concentration, solution
flux.
Outputs: Membrane wall concentration.
Notes: N/A

Equations:
𝐽𝐽𝑣𝑣
𝐶𝐶𝑏𝑏 ∙ 𝑒𝑒 𝑘𝑘
𝐶𝐶𝑤𝑤 = 𝐽𝐽𝑣𝑣
𝑅𝑅𝑜𝑜 + (1 − 𝑅𝑅𝑜𝑜 ) ∙ 𝑒𝑒 𝑘𝑘
Cw: wall concentration
C : bulk concentration

differential_unit_guess.vi
Purpose: Calculates initial guesses for multiple conditions which are not given.
This vi provides a starting point for differential_unit.vi. The values found by
this vi are, for the most part, inaccurate as they are only guesses.
Inputs: Feed flowrate, salt, salt concentration, minimum salt rejection, intrinsic
water permeability, feed pressure, permeate pressure, differential membrane
permeation area, number of differential units.

20
Outputs: Retentate concentration, permeate concentration, retentate volumetric
flowrate, retentate pressure, Jv, recovery.
Sub vi’s: Retentate_pressure.vi, Jv_Ro.vi.
Strategy: The vi first solves for the permeate concentration using the assumption
that volume is additive (bad assumption unless fluid is incompressible). The
vi then uses Retentate_pressure.vi which is a guess for the pressure drop along
the membrane. After, the vi uses Jv_Ro.vi to solve for the flux. This flux will
be off due to the fact that the initial guesses are off. The flux is then used to
solve for permeate flowrate, which is in turn used to solve for retentate
flowrate (doing a volumetric balance, assuming volume is additive once
again). Retentate flowrate can then be used to solve for recovery. This then
leads to retentate concentration being solvable.

Equations:

Volume Balance (assuming constant density)

Cp = Cf ∙(1 – Ro)

Cp: permeate concentration

Cf : feed concentration
Ro: intrinsic rejection

Estimate for Permeate Flowrate

Vp = Jv∙dAxp

Vp: permeate volumetric flowrate

Jv: flux through membrane
dAxp: differential membrane cross sectional area

1st Estimate for Retentate Flowrate (assuming constant density)

Vr = Vf – Vp

Vf : feed volumetric flowrate

1st Estimate for Recovery

𝑉𝑉𝑝𝑝
𝜃𝜃 =
𝑉𝑉𝑓𝑓

1st Estimate for Retentate Concentration

𝑉𝑉𝑓𝑓 ∙ 𝐶𝐶𝑓𝑓 − 𝑉𝑉𝑝𝑝 ∙ 𝐶𝐶𝑝𝑝
𝐶𝐶𝑟𝑟 =
𝑉𝑉𝑟𝑟
Vr : retentate volumetric flowrate

21
Program Flowsheet:

Differential Unit.vi
Purpose: Puts together all the basic vi’s

22
Appendix 2: Benchmarking Analysis of Vendor
Specifications (Hydranautics SWC-2521)
Pure water permeability is the first figure-of-merit determined when performing
bench scale tests. This should be equivalent to the membrane’s nominal intrinsic
water permeability obtained when performing measurements containing solutes
that are rejected by the membrane (at constant temperature). Temperature
corrections would need to be made based on the change in viscosity and solute
activity. Manufacturers typically publish data for an element test performed
under specified conditions. In the following, we “deconstruct” a set of published
data to show the algorithm that will be followed for translating (and verifying)
bench scale data into parameters that will be forward-integrated in the modeling
process.

We use Figure A-1 to define the design

problem schematic. The variables (f =
feed; r = retentate, concentrate, or reject;
and p = permeate) are:
Vi = volumetric flowrate of stream i, (L/s)
Ci = concentration of stream i, (mol/L)
ρi = density of stream i, (kg/L)
ΔP = applied transmembrane pressure
(psi)
ΔPf = frictional pressure loss at flow
Figure A-1. Schematic of the conditions (psi)
generic membrane process. Jv = water flux through the membrane
(m/s)
Ax,P = membrane area for permeation (m2)
From the vendor specification sheet we use the following values for the
parameters:
Cf = 32,000 ppm NaCl 10% permeate recovery
ΔP = 800 psi (~5.5 MPa) Ax,P = 12 square feet (~1.115 m2)
T = 25 °C Vp = 225 gallons per day (~0.9 m3/d)

1) Calculate all flowrates:

Vp = 1.0417 × 10–5 m3 0.9 m3 d h

𝑉𝑉𝑝𝑝 = ∙ ∙
d 24 h 3600 s
Vf = 1.0417 × 10–4 m3/s Based on 10% recovery and assuming it was calculated on
a volumetric basis and all flows were measured at
atmospheric pressure and the same temperature (298 K).
𝑉𝑉𝑝𝑝
𝑉𝑉𝑓𝑓 =
recovery

23
 Vr = 0.9375 × 10–4 m3/s Using volumes to do the overall mass balance is the first
assumption which can be corrected after calculating
compositions. In the absence of certain knowledge of what
the manufacturer used for their specifications, it’s the best
we can do to get started.
Vr = Vf – Vp

2) Calculate all compositions:

As presented in the previous quarterly report, the physical properties of aqueous
NaCl solutions were tabulated for our use. The densities in the following are
based on that data. (See Table A-1, at end of this appendix).

Cf = 0.5585 mol/L The 32,000 ppm is assumed to be based on mass, which is

0.032 g NaCl/g solution. The density of such a solution is
~ ρf = 1.021 kg/L. Thus:
0.032 kg NaCl 1.021 kg solution mol NaCl
𝐶𝐶𝑓𝑓 = ∙ ∙
kg solution L solution 0.0585 kg NaCl
Cp = 5.585 × 10–3 mol/L Minimum salt rejection is specified as 99% and we assume
that is the observed rejection. We use the molar
concentration directly in the absence of better information.
Cp = Cf ∙ (1–0.99) = 0.5585 ∙ (0.01)M
Cr = 0.5812 mol/L The overall molar balance on the salt gives us Cr.
𝑉𝑉𝑓𝑓 ∙ 𝐶𝐶𝑓𝑓 − 𝑉𝑉𝑝𝑝 ∙ 𝐶𝐶𝑝𝑝
𝐶𝐶𝑟𝑟 =
𝑉𝑉𝑟𝑟
(1.0417×10–4 ∙ 0.5585) – (1.0417×10–5 ∙ 0.5585×10–3 )
Cr =
0.9373×10–4

3) Calculate average concentration in the bulk in the element:

Cb = 0.5699 mol/L 𝐶𝐶𝑟𝑟 + 𝐶𝐶𝑓𝑓 0.5812 + 0.5585

𝐶𝐶𝑏𝑏̅ = =
2 2

4) Calculate the spacer dimension and open area for flow in the element:
We need to determine the average fluid velocity in the feed channel in order to
determine the mass transfer coefficient to be used in calculating the wall
concentration during the manufacturer’s test. To do this, we need the actual
cross-sectional area for flow.

2
Figure A-2. Side view of spiral element.

The manufacturer’s specifications are:

D = 0.061 m A = 0.5334 m
d = 0.0191 m c = 0.0305 m

We need to estimate the overall length of the spiral in order to determine the
cross-sectional area for the feed flow to be passing through. Refer to the
following Figure A-3 for our nomenclature and note that is somewhat different
than that used in the report body when we are defining our flux modeling. This
change in nomenclature is simply an immediate convenience.

Figure A-3. Nomenclature applied to the SWC-2521 membrane.

We will use the following variable definitions:

L= the unwound length of a spiral envelope. A spiral envelope is two
membranes enclosing a permeate carrier.
w= the width of membrane (spiral envelope).
tm = the overall “compressed” thickness of the spiral envelope.

3
ts = the thickness of the feed channel (presumed equal to the feed spacer
thickness.)
D = the outer diameter of the spiral.
d= the core diameter for the permeate header.
c= the length of the core extensions.
n= the number of envelopes (and recognizing that each envelope has two
membranes.)
Ax,T = the total projected, cross-sectional area presented by the element to the
feed.
Ax,M = the total projected, cross-sectional area presented by the membrane
envelope to the feed.
Ax,F = the total projected, cross-sectional area available for feed flow (including
the feed spacer).

We assume that the nominal 2.5” diameter element is made from one envelope.

w = 0.4572 m We assume that some of the dimension A is used for the brine seals
and glue lines (~0.6 inches or 0.01524 m). Thus
w = 0.5334 – 2·0.0305 – 0.01524

The formula for the length of a spiral whose thickness is tm + ts is given by:
𝐷𝐷 − 𝑑𝑑 𝐷𝐷 + 𝑑𝑑
𝐿𝐿 = 𝜋𝜋 ∙ ∙
2 ∙ (𝑡𝑡𝑚𝑚 + 𝑡𝑡𝑠𝑠 ) 2

The membrane area for permeation must be equal to L·w. Thus:

𝐴𝐴𝑥𝑥,𝑃𝑃 𝜋𝜋 𝐷𝐷 − 𝑑𝑑 𝐷𝐷 + 𝑑𝑑
= 𝑤𝑤 ∙ ∙ ∙
2 ∙ 𝑛𝑛 2 (𝑛𝑛 ∙ 𝑡𝑡𝑚𝑚 + 𝑡𝑡𝑠𝑠 ) 2
where 2·n is included because there are two membranes per envelope. We
rearrange this to estimate tm+ts. And, we must assume something about the
spacer. We assume a standard 28-mil (0.028”) diamond-shaped spacer (personal
commun., Peter Eriksson, Osmonics-Desal, 1999). Thus, ts = 7.112×10–4 m.

tm+ts = 2.16×10–3 m 𝜋𝜋 2 ∙ 𝑛𝑛 𝐷𝐷2 − 𝑑𝑑2

𝑛𝑛 ∙ 𝑡𝑡𝑚𝑚 + 𝑡𝑡𝑠𝑠 = 𝑤𝑤 ∙ ∙ ∙
ts = 7.112×10–4 m 2 𝐴𝐴𝑥𝑥,𝑃𝑃 2
tm = 1.45×10–3 m if we assume n = 1
𝜋𝜋 2
𝑡𝑡𝑚𝑚 + 𝑡𝑡𝑠𝑠 = 0.457 ∙ ∙ ∙ 0.0612 − 0.01912
4 1.115
Ax,T = 2.636×10–3 m2 𝜋𝜋
𝐴𝐴𝑥𝑥,𝑇𝑇 = ∙ (𝐷𝐷 2 − 𝑑𝑑2 )
4
L = 1.2195 m
Ax,M = L∙tm
Ax,M = 1.768×10–3 m2
Ax,F = 0.868×10–3 m2 Ax,F = Ax,T – Ax,M

4
5) Calculate the superficial velocity through the feed channel and the bulk
fluid mass transfer coefficient:
v = 0.114 m/s v = the superficial velocity in the feed channel of the element
𝑉𝑉𝑓𝑓 + 𝑉𝑉𝑟𝑟 1 1.0417 + 0.9375 10−4
𝑣𝑣 = ∙ = ∙ −4
2 𝐴𝐴𝑥𝑥,𝐹𝐹 2 ∙ 8.68 10
Re = 78.33 We are using the properties for a 0.613 M (35,000 ppm)
solution; rigorous evaluation requires a trial-and-error approach
𝑣𝑣 ∙ 𝑑𝑑ℎ ∙ 𝜌𝜌
𝑅𝑅𝑅𝑅 =
𝜇𝜇
dh = ts = 7.112×10–4 m
ρ = density of the bulk fluid, 1.025×103 kg/m3
μ = viscosity of the bulk fluid, 1.061×10–3 kg·m–1·s–1
Sc = 1,293.9 𝜇𝜇
Sc =
𝜌𝜌 ∙ 𝐷𝐷
D = diffusion coefficient of salt in concentrated salt solution,
0.8×10–9 m2/s
k = 2.556×10–3 m/s we apply the mass transfer correlation presented by Eriksson
(1999)
𝐷𝐷
𝑘𝑘 = ∙ 𝑎𝑎 ∙ Re𝑏𝑏 ∙ Sc 0.33
𝑑𝑑ℎ
a = 0.5 and b = 0.54

6) Calculate the average salt concentration at the membrane interface and

the average osmotic pressure differences:
C̅w = 0.5720 mol/L Using the film theory approach:
exp(𝐽𝐽𝑣𝑣 ⁄𝑘𝑘 )
̅ = 𝐶𝐶𝑏𝑏̅ ∙
𝐶𝐶𝑤𝑤
𝑅𝑅𝑜𝑜 + (1 − 𝑅𝑅𝑜𝑜 ) ∙ exp(𝐽𝐽𝑣𝑣 ⁄𝑘𝑘 )
𝑉𝑉𝑝𝑝 1.0417 × 10−5 m3 ⁄s
𝑗𝑗𝑣𝑣 = = = 9.34 × 10−6 m⁄𝑠𝑠
𝐴𝐴𝑥𝑥,𝑃𝑃 1.115 m2
𝐽𝐽𝑣𝑣 9.34 × 10−6 m⁄s
= = 3.66 × 10−3
𝑘𝑘 2.556 × 10−6 m⁄s
Ro = 0.99
Πw = 2.421 MPa The fit for the osmotic pressure (MPa) of NaCl solutions with
Πp = 0.022 MPa concentration expressed as mol/L is:
Π = 3.8954 ∙ c + 0.5911 ∙ c2
Jv = Kw (ΔP – ΔΠ)
𝐽𝐽𝑣𝑣 9.34 × 10−6 m⁄𝑠𝑠
𝐾𝐾𝑤𝑤 = =
(∆𝑃𝑃 − ∆Π) 5.514 − (2.421 − 0.022) MPa
5) Calculate the membrane’s nominal intrinsic water permeability:

5
 Kw = 3×10–6 m·MPa–1·s–1 The phenomenological flux equation is:
Jv = Kw (ΔP – ΔΠ)
the applied transmembrane pressure was set at 5.514 MPa
𝐽𝐽𝑣𝑣 9.34 × 10−6 m⁄𝑠𝑠
𝐾𝐾𝑤𝑤 = =
(∆𝑃𝑃 − ∆Π) 5.514 − (2.421 − 0.022) MPa

The following Table A-1 is extracted from the CRC Handbook of Chemistry and
Physics, 71st Ed.

Table A-1. Bulk properties of aqueous NaCl solution at 293 K

Relative Specific Salt con- Water Water
Mass Molarity, Viscosity
density gravity centration concen- displaced
%, A M (×103)
@ 293 K (ρH2O= (anhyd) tration by salt
(g/100 g) (g-mol/L) (kg/m/s)
(kg/L) 0.99823) (g/L) (g/L) (g/L)
0.1 0.9989 1.0007 1 0.017 997.9 0.3 1.005
0.5 1.0018 1.0036 5 0.086 996.8 1.5 1.012
1 1.0053 1.0071 10.1 0.172 995.3 3 1.021
1.5 1.0089 1.0107 15.1 0.259 993.8 4.5 1.029
2 1.0125 1.0143 20.2 0.346 992.2 6 1.037
3 1.0196 1.0214 30.6 0.523 989 9.2 1.053
3.5 1.0232 1.025 35.8 0.613 987.4 10.8 1.061
7.2 1.0500 1.0519 75.6 1.294 974.4 23.8 1.129
9.4 1.0662 1.0681 100.2 1.715 966 32.2 1.179
11.5 1.0819 1.0838 124.4 2.129 957 40.8 1.238
14 1.1008 1.1028 154.1 2.637 946.7 51.5 1.318
20 1.1478 1.1498 229.6 3.928 918.2 80 1.559
23 1.1721 1.1742 269 4.613 902 95.7 1.747
26 1.1972 1.1993 311.3 5.326 885.9 112.3 1.992

We have used this data to develop a density correlation which is needed to change
mole or mass fractions to volumetric flow rates. The correlation is:

𝜌𝜌� = 55.426 − 0.44258𝑥𝑥2 − 95.156𝑥𝑥22 , where ρ̂ is in gram-moles per liter and x2

is the NaCl mole fraction.

6
7
Appendix 3. Vendor Design Software: Program
Comparisons
There are several reverse osmosis process simulators offered by equipment
vendors. They include the Dow/Filmtec ROSA (a computer design program for
designing plants with FILMTEC reverse osmosis membranes); GE Osmonics
WINFLOWSTM (a computer design program for complex system configurations
such as designs with feed bypass, recycle, two-pass, and two-stage configurations);
and IMSDesign (a comprehensive membrane software design package that allows
the user to design a membrane system using Hydranautics membranes).

ROSA 5.3
The ROSA version 5.3 commercial program was also run to compare with the
results of the calculations from the Microsoft Excel® code. The following results
were run for a reverse osmosis system under the same conditions as defined in the
Microsoft Excel® code.

8
As displayed above, the results are similar to the Microsoft Excel® results for a
single element with zero sub-sections.

Appendix 4: Physical and Chemical Property Data

Accumulation and Correlation
Physical properties of single component electrolyte solutions are being
transcribed into digital format (spreadsheets) for several solutes. These databases
will allow us to create correlation equations for density, viscosity, and osmotic
pressure for the solutions formed during the membrane filtration process,
including the concentration polarization effects. Thermodynamically consistent
mixing rules will be applied to use this data for mixtures of electrolytes and/or
electrolytes and organic species. Data for the following solutes have been added
to our digital database. Thus far, we have used two main sources for the data, the
CRC Handbook (1982) and Hamer and Wu (1972). The osmotic pressure data is
currently being collected and verified for consistency.

Table A-2. Aqueous solutions properties at 293.15 K transcribed thus far

Concentration
Solute Density Viscosity Conductance
(anhyd g/L)
BaCl2 5 – 332 Yes Yes Yes
CaCl2 5 – 558.3 Yes Yes Yes
CdCl2 10 – 1015.4 Yes Yes Yes
K2CO3 5 – 770.2 Yes Yes Yes
K2SO4 5 – 108.1 Yes Yes Yes
KCl 5 – 278.9 Yes Yes Yes
KHCO3 5 – 280.4 Yes Yes Yes
LiCl 5 – 353.7 Yes Yes Yes
MgCl2 5 – 382.9 Yes Yes Yes
MgSO4 5 – 337.0 Yes Yes Yes
Na2CO3 5 – 173.6 Yes Yes Yes
NaCl 1 – 311.3 Yes Yes Yes
Seawater 5 – 166.8 Yes Yes Yes
ZnSO4 5 – 188.9 Yes Yes Yes

9
 Table A-3. Molarity (g-mol/L) as a function of mass percent salt
[Molarity = A×(Mass%)2 + B×(Mass%)]
Salt A B
Ammonia –0.0019 0.5814
Ammonium chloride 0.0005 0.187
Barium chloride 0.0006 0.0468
Cadmium chloride 0.0009 0.0465
Calcium chloride 0.0009 0.0876
Ethanol –0.0005 0.2231
Lead nitrate 0.0004 0.0288
Lithium chloride 0.0015 0.2342
Magnesium chloride 0.001 0.1033
Magnesium sulfate 0.001 0.0817
Potassium bicarbonate 0.0007 0.099
Potassium carbonate 0.0008 0.0695
Potassium chloride 0.0009 0.1333
Potassium hydroxide 0.0019 0.1733
Potassium nitrate 0.0007 0.0981
Potassium sulfate 0.0005 0.0572
Sea water ---- ----
Sodium carbonate 0.001 0.094
Sodium chloride 0.0013 0.1697
Sodium hydroxide 0.0027 0.251
Zinc sulfate 0.007 0.0613

This correlation simplifies performing process design material balances based on

concentration units. Since molarity requires the solution density, this correlation
implicitly includes the density variations for the different salt solutions, as well as
their different molecular masses.

Table A-4. Relative density (kg/L) at 293 K as a function of mass percent salt
[Density = A×(Mass%)2 + B×(Mass%) + C]
Salt A B C
Ammonia 3.00×10–5 –0.0043 0.99823
Ammonium chloride –1.00×10–5 0.0032 0.99823
Barium chloride 9.00×10–5 0.0085 0.99823
Cadmium chloride 0.0001 0.0074 0.99823
Calcium chloride 5.00×10–5 0.008 0.99823
Ethanol –8.00×10–6 –0.0013 0.99823
Lead nitrate 0.0001 0.0082 0.99823
Lithium chloride 1.00×10–5 0.0056 0.99823

10
 Salt A B C
Magnesium chloride 4.00×10–5 0.008 0.99823
Magnesium sulfate 6.00×10–5 0.0099 0.99823
Potassium bicarbonate 3.00×10–5 0.0064 0.99823
Potassium carbonate 4.00×10–5 0.0088 0.99823
Potassium chloride 2.00×10–5 0.0063 0.99823
Potassium hydroxide 3.00×10–5 0.0086 0.99823
Potassium nitrate 3.00×10–5 0.0062 0.99823
Potassium sulfate 3.00×10–5 0.008 0.99823
Sea water 8.00×10–6 0.0075 0.99823
Sodium carbonate 2.00×10–5 0.0103 0.99823
Sodium chloride 3.00×10–5 0.007 0.99823
Sodium hydroxide –9.00×10–6 0.0112 0.99823
Zinc sulfate 8.00×10–5 0.0101 0.99823

Table A-5. Viscosity (Pa·s or kg·s·m–1) as a function of mass percent salt

[Viscosity = A×(Mass%)4 + B×(Mass%)3 + C×(Mass%)2 + D×(Mass%) + E]
Salt A B C D E
Ammonia ---- –1.00×10–8 2.00×10–7 1.00×10–5 0.001002
Ammonium chloride ---- 2.00×10–9 1.00×10–7 –5.00×10–6 0.001002
Barium chloride ---- 1.00×10–8 –5.00×10–8 1.00×10–5 0.001002
Cadmium chloride 8.00×10–9 –6.00×10–7 1.00×10–5 –8.00×10–5 0.001002
Calcium chloride 9.00×10–9 –4.00×10–7 7.00×10–6 –6.00×10–6 0.001002
Ethanol 2.00×10–10 –3.00×10–8 8.00×10–7 5.00×10–5 0.001002
Lead nitrate ---- ---- 3.00×10–7 3.00×10–6 0.001002
Lithium chloride 3.00×10–9 –9.00×10–8 2.00×10–6 3.00×10–5 0.001002
Magnesium chloride 1.00×10–8 –4.00×10–7 6.00×10–6 2.00×10–5 0.001002
Magnesium sulfate 2.00×10–8 –3.00×10–7 4.00×10–6 5.00×10–5 0.001002
Potassium bicarbonate ---- ---- 4.00×10–7 1.00×10–5 0.001002
Potassium carbonate 4.00×10–9 –2.00×10–7 5.00×10–6 –4.00×10–6 0.001002
Potassium chloride –3.00×10–10 2.00×10–8 –2.00×10–7 –6.00×10–7 0.001002
Potassium hydroxide 3.00×10–9 –1.00×10–7 3.00×10–6 1.00×10–6 0.001002
Potassium nitrate ---- ---- 2.00×10–7 –6.00×10–6 0.001002
Potassium sulfate ---- ---- 4.00×10–7 1.00×10–5 0.001002
Sea water 1.00×10–7 –1.00×10–6 5.00×10–6 1.00×10–5 0.001002
Sodium carbonate ---- ---- 5.00×10–6 3.00×10–5 0.001002
Sodium chloride ---- 4.00×10–6 1.20×10–3 1.71×10–1 0.001002
Sodium hydroxide 1.00×10–8 –5.00×10–8 2.00×10–6 6.00×10–5 0.001002
Zinc sulfate 2.00×10–8 –5.00×10–7 5.00×10–6 3.00×10–5 0.001002

11
Table A-6. Conductance (mS/cm) as a function of mass percent salt
[Conductance = A×(Mass%)5 + B×(Mass%)4 + C×(Mass%)3 + D×(Mass%)2 + E×(Mass%)]
Salt A B C D E
Ammonia 5.00×10–6 –4.00×10–4 9.50×10–3 –1.19×10–1 0.6406
Ammonium chloride --- --- --- –1.43×10–1 19.691
Barium chloride --- --- --- –8.13×10–2 8.4849
Cadmium chloride 1.00×10–6 –2.00×10–4 9.20×10–3 –2.83×10–1 4.3384
Calcium chloride --- 9.00×10–6 –1.00×10–3 –2.76×10–1 14.636
Lead nitrate --- --- 1.20×10–3 –1.02×10–1 4.3442
Lithium chloride --- --- 3.80×10–3 –5.43×10–1 17.794
Magnesium chloride --- --- 4.20×10–3 –5.49×10–1 15.978
Magnesium sulfate --- --- 9.00×10–4 –2.07×10–1 6.4111
Potassium bicarbonate --- --- --- –8.68×10–2 8.1201
Potassium carbonate --- --- --- –1.48×10–1 12.358
Potassium chloride --- --- --- –5.89×10–2 14.839
Potassium hydroxide --- --- --- –9.37×10–1 39.832
Potassium nitrate --- --- --- –9.79×10–2 9.7801
Potassium sulfate --- --- --- –1.68×10–1 10.491
Sea water --- --- --- –4.88×10–1 15.12
Sodium carbonate --- --- --- –3.54×10–1 11.097
Sodium chloride --- --- --- –2.50×10–1 15.165
Sodium hydroxide --- --- --- –1.76 49.943
Zinc sulfate --- --- --- –1.18×10–1 4.6183

12
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