Accepted Manuscript

Title: A new practical CFD-based methodology to calculate

the evaporation rate in indoor swimming pools

Authors: Juan Luis Foncubierta Blázquez, Ismael R. Maestre,

Francisco Javier González Gallero, Pascual Álvarez Gómez

PII: S0378-7788(16)31312-3
DOI: http://dx.doi.org/doi:10.1016/j.enbuild.2017.05.023
Reference: ENB 7604

To appear in: ENB

Received date: 26-10-2016

Revised date: 12-4-2017
Accepted date: 11-5-2017

Please cite this article as: Juan Luis Foncubierta Blázquez, Ismael R.Maestre, Francisco
Javier González Gallero, Pascual Álvarez Gómez, A new practical CFD-based
methodology to calculate the evaporation rate in indoor swimming pools, Energy and
Buildingshttp://dx.doi.org/10.1016/j.enbuild.2017.05.023

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A new practical CFD-based methodology to calculate the evaporation rate in indoor swimming

pools

Juan Luis Foncubierta Blázqueza, Ismael R. Maestrea, Francisco Javier González Galleroa, Pascual

Álvarez Gómeza

a
Escuela Politécnica Superior de Algeciras, University of Cádiz. Avenida Ramón Puyol, s/n.

Algeciras 11202 (Spain)

*Corresponding author. Tel.: +34 956 028000; fax: +34 956 028001. E-mail address:

javier.gallero@uca.es

Highlights

1.- A new CFD-based method to calculate evaporation rate.

2.- Main hypotheses set no shear stress at the air-water interface.
3.- The model was experimentally validated with data from test chambers and a real swimminng-
pool.
4.- Results were quite satisfactory with low relative errors.

Abstract

This paper presents a new methodology in which a computational fluid dynamics model is

applied to estimate water evaporation rate in indoor swimming pools. This rate is needed to

achieve a suitable energy performance of ventilation and dehumidification systems. The main

hypotheses of the model set the following boundary conditions at the air-water interface: air

temperature equal to water temperature, water vapour concentration equal to saturation

humidity of air at water temperature and free slip wall condition (no shear stress). This last

condition can be justified by the fact that Prandtl and Schmidt turbulent numbers are usually

less than one in this kind of flows. Consequently, the dynamic boundary layer depth will be
smaller than the thickness of the thermal and humidity boundary layers. The model was

experimentally validated by using data from three different test chambers and from a real

swimming-pool. A total of 233 different flow conditions were simulated. The results were quite

satisfactory, with a relative error of only 3% in the simulations of the real swimming-pool and a

total average relative error smaller than 9%.

Keywords: Evaporation rate, indoor swimming-pool, CFD.

Nomenclature

Semi-Implicit Method for

A area, m2 SIMPLE
Pressure Linked Equations

dA differential area, m2 SRT Statistical Rate Theory

E evaporation rate, kg/(m2s) SST Shear-Stress Transport

g gravitational acceleration, m/s2 Greek symbols

Gr Grashof number (-) δ relative percentage error

L characteristic length (m)  density, kg/m3

𝑚̇e mass flow rate of evaporated water, kg/s  shear stress, Pa

n unit normal vector µ dynamic viscosity, Pa·s

P vapour pressure, Pa  free stream

Re Reynolds number (-) Subscripts

Sc Schmidt number a air

t time, s e evaporation

T temperature exp experimental

v velocity vector, m/s i time index

V volume, m3 in inlet

Y water vapour mass fraction out outlet

 y+ dimensionless distance prop proposed model

Acronyms ref reference model

CFD Computational Fluid Dynamics s saturation

IEA International Energy Agency t turbulent

saturated air at water

KTG Kinetic Theory of Gases w
temperature, water

1. Introduction

According to the International Energy Agency (IEA) [1], building energy consumption of heating

and cooling systems is about 50% of energy use in cold climate countries and 16% in moderate

and warm climate countries. Specifically, in buildings with indoor swimming pools, in which

water evaporation can lead to inappropriate humidity and comfort conditions, some energy

studies [2] estimate that energy consumption of heating and cooling systems, used both for air

dehumidification and water heating, is about 60% of the total consumption of thermal facilities.

Although there may be other factors, the main source of moisture in buildings with indoor

swimming-pools is the evaporation of water in the pool, called evaporation rate, that is usually

expressed in terms of mass flow per unit surface of pool. Evaporation rate is a determining factor

in the selection process of the dehumidification equipment and its estimation has been the

subject of many studies on a theoretical and practical basis [3].

From a theoretical point of view, evaporation process has been analysed for decades at

molecular level using the well-known Kinetic Theory of Gases (KTG). This theory provides an

expression (Hertz-Knudsen-Schrage equation) to estimate evaporation rate in terms of the

interfacial temperatures, interfacial pressures, and condensation and evaporation coefficients.

Many experiments have been made to determine the values of these coefficients for water, but

the results obtained by different authors show a large span [4]. This uncertainty and the lack of

a suitable model to predict the evaporation rate have caused some authors [5] to develop other
theoretical models, based on the transition probability concept (statistical rate theory, SRT).

These models do not need fitting parameters and their results appear to be better than those

obtained by the KTG when compared with experimental observations.

<Nomenclature>

Other authors propose the use of experimental correlations, in which evaporation rate is related

with average values of the thermal conditions of air and water in the swimming-pool [6]. These

correlations can be classified according to the nature, natural or forced, of the convection

process. In general, forced convection correlations relate evaporation rate with average air

velocity and the difference between water vapour partial pressure at air temperature and at

water temperature [7]. Analogously, for natural convection evaporation rate seems to be a

function of the density and vapour pressure differentials. The experiments carried out to obtain

correlations usually involve the measure of the evaporated water during a certain period of time,

changing thermal conditions of air and water in a swimming-pool with a fixed geometrical

configuration. Although the results achieved by using correlations can be accurate, significant

discrepancies of up to 20% have been reported when compared with experimental observations

[8]. Furthermore, their scope of application is limited to the topology and to the conditions they

were obtained from [9]. Then, the accuracy of the results extrapolated to other geometries is

not guaranteed. There is no general consensus on how and where conditions must be measured.

For instance, according to Sartori [10], velocity should be measured at a place between 0.3 and

2 meters over the water surface, depending on the correlation used. Also, in this study a

comparison between the most common correlations is carried out, showing differences of up to

80%, approximately, under the same operating conditions. Despite these limitations, the use of

correlations is widespread in the absence of more accurate methods [6].

For the estimation of evaporation rate and humidity balance in a room such as an indoor

swimming-pool, it is necessary to know velocity, temperature and vapour pressure close to the

water surface. Their local values will depend on the distribution of velocity, temperature and
relative humidity within the room. Thus, concentration, linear momentum and energy equations

must be solved simultaneously. This procedure to calculate the evaporation rate, although more

complex than others, has the advantage of providing thermal-hygrometric comfort conditions

and it also allows the control of heating and cooling systems. As an example of this coupling and

interaction among the different transport phenomena, some authors (Kolokotroni et. al. [11])

show how to get a significant decrease of moisture (up to 50%, approximately) in some zones of

a building, by using different options of air distribution under thermal comfort conditions.

In this context, Li and Heiselberg [12] developed a study about the influence of the evaporation

on indoor air conditions in an indoor swimming-pool by using a CFD model. However, authors

imposed the evaporation rate as a boundary condition on water surface, which was calculated

through experimental correlations. Furthermore, numerical results were not compared with

experimental data. In contrast, Liu et. al. [13] provided an experimental validation in a test

chamber of the air temperature and humidity CFD numerical distributions, although again,

evaporation mass flow was imposed as a boundary condition. No experimental validations of

CFD models to simulate air flow in real indoor swimming-pools have been found in the published

literature.

Although it is possible to predict air movement and thermal conditions by using traditional CFD

techniques, the direct estimation of evaporation rate is complex unless experimental

correlations or coefficients which depend on bulk flow properties are used. Most studies on this

matter assume laminar flow conditions [14] [15] [16]. However, laminar flow hypothesis cannot

usually be assured in the field of air conditioning, where turbulent flow is the prevailing flow in

some situations. For instance, the evaporation of deposited liquid is sometimes affected by an

external air stream which under most practical circumstances is fully turbulent. Thus, Raimundo

et. al. [17] used an own CFD turbulent model to calculate evaporation rate directly. Authors used

and extended k-ε model, with wall-functions to solve both velocity and water vapour

concentration profiles in the vicinity of boundaries. These functions were adapted to get the
best possible results of the evaporation rate at the air-water interface. For the calculation of

mass flows and considering the predominance of advection, variable turbulent Schmidt numbers

dependent on air velocity values were taken into account.

The present work shows a new methodology to obtain the water evaporation rate in indoor

swimming-pools. It is based on the stability of the saturated vapour layer that forms just over

the water surface. This leads to set the following boundary conditions at the air-water interface:

air temperature equal to water temperature, water vapour concentration equal to saturation

humidity of air at water temperature and free slip wall condition (no shear stress). These

boundary conditions on the air-water interface do not depend on any experimental parameter

fitting such that no experimental correlations for evaporation rate are needed. These

hypotheses are easy to implement and they can be used in most commercial CFD software

programs [18] [19], which allow the complete determination of air flow for different conditions

and flow regimes. ANSYS FLUENT 17.1 has been used in the present work.

First, the methodology proposed is shown together with its experimental validation through

four experimental tests with different geometries, scales and air conditions. Finally, the results

obtained are shown and discussed.

2. Proposed Methodology

2.1 Model description and hypotheses

In the area of air conditioning, it can be assumed that humid air is a mixture of two gases: dry

air and water vapour. In enclosed spaces with swimming-pools, experimental studies confirm

that, due to evaporation, a stable air thin layer develops near the water surface which becomes

saturated [5], that is, the layer has the maximum possible water content in vapour state. Water

losses in this layer, which take place through diffusion and advection mechanisms to the space

air, are replaced by evaporation of pool water. Thus, evaporation is able to keep this layer

unchanged such that its modelling can be avoided (Fig. 1).

Then, one could consider the control volume to be the volume of humid air within the space,

excluding the former saturated layer (Fig. 1).

Once control volume is defined (Fig. 1), and under the conditions described above, it is possible

to determine air behaviour, distribution and thermal state by using a CFD model. Thus, the

characteristic boundary conditions at the boundary with the saturated layer are the following:

1.- Imposed air temperature equal to water temperature,

2.- Imposed water concentration equal to saturated vapour at water temperature; and

3.- Free slip wall, τx = τy = τz = 0.

As it can be seen, the usual no-slip wall condition is not applied at the lower surface of the

control volume, because air velocity at the upper surface of the saturated layer may not be zero

in general. Air velocities in indoor swimming-pools are usually limited by regulatory framework

and they are commonly low (0.05~0.15 m/s) [20] for reasons of comfort. These low values of air

velocities together with the small value of air viscosity, make laminar viscous stresses not

significant. Furthermore, as the values of the Schmidt turbulent number near the air-water

interface are smaller or near one [17], the thickness of the dynamic boundary layer will be similar

or smaller than the thickness of the saturated water vapour layer. Thus, there will be no

significant velocity gradients above the saturated layer and the no shear stress condition could

be considered plausible.

<Figure 1>

2.2 CFD solution

Considering the former hypotheses, liquid water in the pool and water-air interface can be

excluded from the domain for the calculation of the evaporation rate. CFD is a possible

technique that can be used for such calculation. In this work, ANSYS Fluent 17.1, a CFD program,

has been selected. This section describes the general process followed for performing the CFD

analysis in detail.
An unstructured tetrahedral mesh has been used with inflation layers (prismatic layers of

increasing thickness) in the proximity to the water-air interface in order to capture gradients

within the humidity boundary layer. Specifically, for this near-interface mesh, a value for the

interface dimensionless distance (y+) lower than one has been set.

Several runs of the initial CFD model were performed on different size meshes and their results

were compared to analyse the grid independence of the solution [21]. Mesh refinement was

repeated until the absolute residual for the mass conservation equation was at least two orders

of magnitude smaller than the calculated evaporation rate.

In addition to the boundary conditions at the interface described in section 2.1, constant air

speed, temperature and water vapour concentration at the inlets have been considered.

Constant atmospheric pressure has been set at the outlet boundaries. Adiabatic and no-slip wall

condition was applied to the rest of boundaries in the control volume.

The model used for the simulation of humid air has been the multi-component flow model [19]

(species transport, ANSYS Fluent), in which a mixture of ideal gases (dry air and water vapour) is

considered. Thermodynamic properties of each component and the diffusivity of water vapour

in air are considered temperature-dependent, as shown in different references [22]. Thus,

assuming ideal gas behaviour, kinetic theory shows that mass diffusivity depends on

temperature (T, K) and pressure (P, atm) as follows:

𝐷~𝑃−1 𝑇 3/2

Furthermore, for the evaluation of saturation vapour pressure (𝑃𝑣,𝑠 ) as a function of

temperature the following expression has been used [23]:

7235.46
𝑃𝑣,𝑠 = 105 𝑒𝑥𝑝 [65.832 − 8.2 ln(𝑇𝑠 ) + 5.717 · 10−3 𝑇𝑠 − ]
𝑇𝑠

Buoyancy effects have been modelled by using Boussinesq approximation. The turbulent

Schmidt number (Sct) is considered constant and equal to 0.7.

In order to minimise numerical error, mass conservation equation is solved for water vapour,

and dry air mass fraction is calculated as 1-Y, where Y is water vapour mass fraction. Shear-Stress
Transport (SST) model has been used to model turbulence, due to its high levels of accuracy and

reliability in many flow typologies [24].

Conservation equations are solved by using SIMPLE (Second Order Upwind Numerical Scheme)

with a scaled error smaller than 10-4.

Evaporation rate or evaporated water vapour flow in each time step ti, can be estimated as:

  YdV    YdV
  V ti V ti 1 (1)
m e ( t )    Y v · n dA    Y v · n dA 
t
Aout ti Ain ti

Particularly, for stationary conditions it can be expressed as follows:

 
m e ( t )    Y v · n dA    Y v · n dA (2)
Aout Ain

Where  is the density of humid air ( kg / m 3 ), Y is the water vapour mass fraction (kg of water

vapour / kg of humid air), v is fluid velocity and n̂ is the unit normal vector to inlet (Ain) and

outlet (Aout) surfaces of the control volume. Thus, surface integrals represent convective fluxes

of water vapour through the former surfaces, while volume integrals represent the mass of

water vapour at each time step.

The methodology proposed here allows the calculation of the evaporation rate both in transient

and steady state conditions.

3. Validation

In the present work, the proposed method has been validated by comparing the numerical

results of the evaporation rate with those obtained in four experimental tests with different

geometries and scales in stationary conditions. These experiments (tests A, B, C and D)

correspond to those developed by Asdrubali et. al. [25], Jodat et. al. [7], Raimundo et. al. [17]

and Hyldgård [26], respectively.

Considering symmetry and experimental conditions, a two-dimensional (2D) model was used for

tests A and B. For tests C and D (real swimming-pool), the three-dimensional (3D) model is a

faithful reproduction of the real case.

3.1 Test A

In this case, the experiment developed by Asdrubali et al. [25], in which a pool scale model is

tested in stationary conditions, is simulated numerically. The model is inserted in an

environmental test chamber with internal dimensions of 700 mm × 660 mm × 680 mm. There is

an aluminium container with water in the lower part (250 mm × 150 mm × 70 mm), which is

insulated with polyurethane such that heat exchange occurs mainly through the free water

surface. Air was blown above the water surface with velocities that were set to 0.05 m/s, 0.08

m/s and 0.17 m/s. Velocity is measured at about 10 mm from water surface in five different

positions. Air temperature varies from 22ºC to 32ºC, while water temperature is two degrees

smaller. Relative humidity values were 50%, 60% and 70%.

Fig. 2 shows a scatter plot of the numerical results obtained by the proposed method against

the experimental data. Dashed lines show the margin of error of experimental data. It can be

seen how numerical method gets satisfactory results in most cases, obtaining an average total

error of 9.95%.

<Figure 2>

3.2 Test B

In this case (Jodat et. al. [7]), an experimental test chamber with a square section of 1m2 and 1.5

m in length is used. Water layer spreads over all the lower part of the chamber in order to reduce

convective edge effects. Tests were developed in steady state, with measurements that are

carried out over a wide range of water temperatures (from 20 to 55ºC) and air velocities (from

0.05 to 5 m/s).

Fig. 3 and Fig. 4 show the results obtained for velocity values of 1 m/s and 2 m/s, respectively.

The proposed method gives predictions in close agreement with measurements, with an
average relative error of 8.4%, approximately. A better fitting than in test A is found, even with

greater velocity values. This may be due to the fact that water surface in test B occupies almost

all the lower part of the test chamber and the development of a stable dynamic boundary layer

is more likely to happen.

<Figure 3>

<Figure 4>

3.3 Test C

In this case, tests were performed in a chamber that consisted of a low velocity wind tunnel and

a container of water (Raimundo et. al. [17]). The authors compared the experimental

measurements of evaporation rate with the numerical results obtained by using an own CFD

program. The tunnel had a length of 3.3 m and a square cross-section with a side of 0.4 m. Water

surface was at 1.8 m from the inlet with dimensions of 0.15m × 0.15 m. Experimental conditions

of air velocity, temperature and relative humidity and water temperature, are shown in Table 1,

together with a comparison among the experimental and numerical results given by Raimundo

et al. [17] and those obtained by the numerical method proposed here.

<Table 1>

As it is shown in Fig. 5, the results obtained by both numerical methods are quite similar to

experimental data, with average evaporation rates of 0.25 and 0.26 g/(s·m2), and standard

deviations of 0.13 and 0.14 g/(s·m2), for reference and proposed methods, respectively. Average

relative percentage errors with respect to the experimental values are 7.6% and 12.4%,

respectively.

<Figure 5>

3.4 Test D

Although numerical results fit well to experimental data in the former cases, it would be

interesting to analyse the performance of our method in a full-scale model of an indoor

swimming-pool. Thus, this test (Hyldgård [26]) is developed in a room with internal dimensions
of 5.43 m × 3.60m × 2.42 m, and a swimming-pool with dimensions of 3.69 m × 1.90 m. The room

had four air inlets uniformly distributed and located on the top of the room and two outlets on

the lower part of one of the side walls. The experimental tests were developed in steady

conditions, one group of the experiments with average velocities of 0.15 m/s and other one with

velocities smaller than 0.1 m/s. Velocity was measured 20 cm above surface water. Air

temperatures ranged from 24ºC to 32ºC, with relative humidity of 40%, 50%, 60% and 70%.

Water temperatures were 24ºC, 26ºC and 28ºC.

Since Hyldgård [26] does not specify exact values of the air velocity values smaller than 0.1 m/s,

the simulation run in the present work has focussed on the cases of average velocity equal to

0.15 m/s. The results obtained are shown in Fig. 6. Fitting is quite satisfactory, with an average

relative error of 3.34%, and a maximum value of 9.3%.

Fig. 7 and Fig.8 show the spatial distribution of water vapour mass fraction and air temperature

for test D in different parts of the domain, for the following conditions: air and water at 26C

and 24C, respectively, and 40% of relative humidity. Obviously, the distribution of all physical

properties will depend on the room configuration and boundary conditions. However, humidity

and temperature profiles show an expected behaviour, with significant gradients near the

water-air interface and a practically uniform distribution above. As it can be seen (Fig. 8), these

gradients are reduced due to the air jets from the inlets.

Table 2 summarizes the results obtained in all tests considered herein. 233 different flow

conditions have been simulated, 205 of them in pool scale models and 28 in a full-scale

swimming-pool. Results of the average, maximum and minimum relative and absolute error

values for each case can be compared in Table 2. Relative errors range from 3.3% (test D, real

swimming-pool) to 12.4% (test C). Total average relative error is only 8.9%, with a standard

deviation of 6.8%. On average, 48.5% of the numerical results are within the error margin of the

experimental data.

<Figure 6>
<Table 2>

<Figure 7>

<Figure 8>

3.5 Dependence of the error in simulated evaporation rate on convection processes

Dimensional analysis can be used to assess the dependence of the simulated evaporation rate

and its error on the different convection processes. Thus, buoyancy and inertial forces can be

represented by the dimensionless product Gr/Re2 [27], where Re is the Reynolds number and

Gr is the mass Grashof number, defined as:

𝜌(𝜌∞ − 𝜌𝑤 )𝑔𝐿3
𝐺𝑟 =
𝜇2

Here,  is the free stream air density, w is air density at the interface at 100% saturation, L is

the characteristic length of the evaporation surface and µ the dynamic viscosity of air.

The experiments considered in the present work cover the range 0.009 < Gr/Re2 < 15. As pointed

out by Pauken in a similar study [27], for Gr/Re2 > 5, the contribution of forced convection to the

total evaporation rate was less than 10%. Furthermore, this author showed that 30% of

evaporation rate was contributed by the free convection component at Gr/Re2 = 0.1. Then, the

following limits have been considered here in order to study how the proposed model estimates

the evaporation rate under different convection processes: Forced convection (Gr/Re2 < 0.1),

mixed convection (0.1  Gr/Re2 < 5) and free convection (Gr/Re2  5).

Relative errors of the evaporation rate with respect to experimental values and the ratios of

numerical and experimental values have been shown as a function of the dimensionless number

Gr/Re2 (Fig. 9, Fig. 10 and Fig. 11). Also, a brief statistical analysis is shown in Table 3. Most of

the cases covered (73%) are within the mixed convection range. As it can be seen, relative error

values are smaller for mixed convection conditions (8.16.6%) with a nearly constant trend also

for the numerical/experimental ratios within this range. Higher values are found for both forced

and free convection conditions with relative errors that decrease as Gr/Re2 increases. It seems
that there is also a greater dispersion of error values for forced convection and mixed convection

conditions in which forced convection is dominant (Fig. 10, 0.1  Gr/Re2 < 1).

<Figure 9>

<Figure 10>

<Figure 11>

<Table 3>

4. Conclusions

A new model to obtain the water evaporation rate in indoor swimming-pools is presented. The

boundary conditions set on the air-water interface, which are based on the stability of the

saturated vapour layer that forms just over the water surface, avoid the use of any experimental

correlations for evaporation rate. The proposed model may be very interesting due to the

simplicity of its hypotheses and ease of implementation in most common CFD programs,

allowing the estimation of the distribution of air flow in turbulent flow regime and its coupling

with heating and cooling systems.

The model has been validated with four experimental studies, under different conditions and

combinations of air temperature, air humidity and air inlet velocity. Three of the four

experiments were carried out on scale models by using test chambers, while the last one was

developed in a real swimming-pool. A total of 233 experimental conditions were simulated, with

an average relative error of 8.9% with respect to experimental data. Numerical simulations of

the real swimming-pool, in which air conditioning system maintains small air velocities, achieve

very good results, with an average relative error of 3.3% and a maximum value of 9.3%.

Finally, the dependence of the error in the estimated values of evaporation rates on the different

convection mechanisms is studied in terms of the dimensionless number Gr/Re2, which

approximates the ratio of the buoyancy and inertial forces. Relative errors of the evaporation

rate with respect to experimental values are smaller for mixed convection conditions for which
the numerical/experimental ratio is 1.010.10. The highest dispersion of errors is found for

conditions in which forced convection dominates.

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Figure captions

Fig. 1. Fluid domain and boundary conditions at the air-water interface in the proposed model.

Fig. 2. Numerical and experimental evaporation rate values. Test A.

Fig. 3. Numerical and experimental evaporation rate values. Test B (v=1m/s).

Fig. 4. Numerical and experimental evaporation rate values. Test B (v=2m/s).

Fig. 5. Numerical and experimental evaporation rate values. Test C.

Fig. 6. Numerical and experimental evaporation rate values. Test D (v=0.15m/s).

Fig. 7. Spatial distribution of water vapour mass fraction. Case Hyldgård Tair = 299 K, Twater = 297

K, HR = 40%.

Fig. 8. Spatial distribution of temperature and stream lines. Case Hyldgård Tair = 299 K, Twater =

297 K, HR = 40%.
Fig. 9. Relative errors and numerical/experimental ratios for the simulated evaporation rate

under forced convection conditions (Gr/Re2 < 0.1).

Fig. 10. Relative errors and numerical/experimental ratios for the simulated evaporation rate

under mixed convection conditions (0.1  Gr/Re2 < 5).

Fig. 11. Relative errors and numerical/experimental ratios for the simulated evaporation rate

under free convection conditions (Gr/Re2  5).

Table 1. Description of velocity, temperature and humidity conditions in Test C. Experimental

and numerical results for evaporation rate from reference (ref, Raimundo et al [17]) and

proposed (prop) numerical method.

Eexp Eprop
Vin Tin Tw Pw-Pa Eref δref δprop
[g/(s [g/(s
[m/s] [K] [K] [Pa] [g/(s m²)] [%] [%]
m²)] m²)]

293.3 301.9 2051 0.059 0.056 5.3 0.059 1.4

294.6 307.9 4026 0.136 0.119 12.6 0.126 7.4

0.101 293.9 315.3 6835 0.27 0.256 5.2 0.244 9.5

293.1 317.7 7968 0.342 0.33 3.7 0.296 13.3

295.5 301.3 2007 0.081 0.055 31.6 0.053 34.7

293.6 309.1 4429 0.182 0.166 8.8 0.151 16.7

294 315.2 6786 0.287 0.285 0.7 0.240 16.5

0.194
295 319 8265 0.355 0.369 3.9 0.300 16.4

295.1 301.1 1850 0.087 0.075 13.2 0.060 30.5

294.9 309.8 4201 0.208 0.188 9.6 0.177 15.0

295.4 314.2 6076 0.294 0.276 6.4 0.256 13.0

0.308
295.5 319.3 8341 0.401 0.381 5 0.352 12.2

295.1 300.8 1976 0.106 0.094 11 0.083 21.4

295.9 308.2 4141 0.206 0.196 4.7 0.206 0.01

295.3 314.2 5910 0.301 0.284 5.5 0.293 2.7

0.406
297.5 319.4 8412 0.4 0.412 3 0.413 3.5

296.1 302.5 2105 0.114 0.108 5.1 0.105 7.5

0.497 294.9 309.9 4238 0.244 0.217 11.4 0.240 1.4

 297 314.9 6226 0.332 0.316 4.9 0.350 5.1

291.6 319.3 8405 0.449 0.425 5.5 0.475 5.6

294.4 301.7 2147 0.152 0.124 18.9 0.124 18.6

296.8 309.7 4240 0.257 0.244 5.1 0.269 4.8

294 313.7 5767 0.328 0.327 0.3 0.365 11.1

0.596
296.8 319.8 8751 0.46 0.48 4.5 0.548 19.1

296.2 301.5 2059 0.145 0.127 12.2 0.133 8.3

295.9 308.7 3866 0.239 0.247 3.5 0.272 13.9

0.697 297.7 314.7 6061 0.359 0.372 3.5 0.422 17.5

296.6 319.1 8295 0.471 0.508 7.8 0.573 21.6

Table 2. Relative and absolute errors obtained from numerical simulations of all the

experimental tests considered herein.

Number of Relative Errors Absolute Errors

Experiments [%] [kg/(h·m²)] ·10-3

Test Minimum Average Maximum Minimum Average Maximum

A 99 0.5 9.9 22.9 0.4 7.3 20.8

B 78 0.0 8.4 31.9 0.0 57.0 150.0

C 28 0.0 12.4 34.7 0.0 56.0 133.0

D 28 0.0 3.3 9.3 0.0 2.8 7.1

Table 3. Relative errors and ratios of numerical and experimental values of the evaporation rate

under different convection conditions (overall tests A, B, C and D).

Gr/Re2 < 0.1 0.1  Gr/Re2 < 5 Gr/Re2  5

Rel. Error Num/Exp Rel. Error Num/Exp Rel. Error Num/Exp

% Ratio % Ratio % Ratio

Maximum 31.94 1.47 30.58 1.23 34.72 1.07

Average 12.14 1.12 8.10 1.01 10.42 0.92

Minimum 1.25 0.92 0.01 0.81 1.41 0.80

Std. Deviation 8.53 0.16 6.56 0.10 6.01 0.05