Effects of mass transfer on heat and mass transfer ...

International Journal of Heat and Mass Transfer 122 (2018) 1093–1102

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Effects of mass transfer on heat and mass transfer characteristics between water surface and airstream L.D. Gu, J.C. Min ⇑, Y.C. Tang Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

a r t i c l e

i n f o

Article history: Received 20 October 2017 Received in revised form 19 January 2018 Accepted 15 February 2018

Keywords: Convective heat and mass transfer Chilton-Colburn analogy Water evaporation Variable air property High mass flux

a b s t r a c t A numerical study has been conducted to simulate the convective heat and mass transfer between water surface and air fluid flowing over it. The airflow is laminar and steady, and has a temperature much higher than the water, causing a combined heat and mass transfer accompanied with water evaporation into the airstream. Calculations are performed to investigate the effects of mass flux on heat and mass transfer coefficients and the applicability of the Chilton-Colburn analogy for 200 °C air temperature, 1–10 m/s air velocities, and 10–90 °C water temperatures. Calculations are implemented with and without consideration of the air property variations caused by the air temperature and humidity changes near the water surface and in the airflow direction. The results show that the heat and mass transfer coefficients both decrease with increasing water surface temperature, i.e. increasing mass flux. The ChiltonColburn analogy holds only for low water temperature case, the deviation of the heat to mass transfer coefficient ratio given by the Chilton-Colburn analogy relative to that by the numerical simulation is less than 5% when the water surface temperature is below 60 °C. The air property variability has a notable and complex effect on heat transfer coefficient but an inconspicuous effect on mass transfer coefficient. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Water evaporation into air fluid is a process often seen in many areas such as drying, concentration, and water desalination [1,2]. In such a process, the heat and mass transfer coexist, they affect each other, making it difficult to accurately predict the heat and mass transfer coefficients. When the mass flux is low, the mass transfer coefficient can be obtained from the heat transfer coefficient based on the Colburn-Chilton analogy [3–5]. When the mass flux is larger, however, the analogy may become inapplicable because of the significant influence of mass transfer on heat transfer. Also, the properties of moist air, which were often treated as constant in most previous investigations, may vary significantly when the air temperature and humidity change considerably. Many previous studies involved the problem of simultaneous heat and mass transfer between water surface or wetted solid surface and air fluid. Chow and Chung [6] numerically investigated the vaporization rate of water evaporation between water surface and humid air or superheated steam in a laminar boundary layer flow. They reported that the fluid property variability had little influence on the vaporization rate, but they gave no information on the effects of property variability on heat transfer. Yuan et al. [7] ⇑ Corresponding author. E-mail address: [email protected] (J.C. Min). https://doi.org/10.1016/j.ijheatmasstransfer.2018.02.061 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

numerically studied the heat and mass transfer characteristics of water evaporation in a laminar boundary layer flow and found that the sensible heat transfer at the interface degraded obviously due to the evaporation: it degraded 11% over the parameter range of their study. Tang and Etzion [8] experimentally investigated the water evaporation rates from a wetted surface and from a free water surface. Boukadida and Nasrallah [9] numerically analyzed the mechanism of heat and mass transfer during water evaporation into a two-dimensional steady laminar flow of dry or humid air in a horizontal channel, they found that the Chilton-Colburn analogy was valid only at low free stream temperatures and vapor concentrations, but they provided no range for which the Chilton-Colburn analogy applies. Stegou-Sagia [10] performed numerical and experimental studies of air humidification process involving simultaneous heat and mass transport in a nearly horizontal tube, and gave the profiles of air velocity, temperature and humidity along the tube. Talukdar et al. [11] conducted CFD simulations for convective heat and mass transfer between water surface and humid air flowing in a horizontal 3D rectangular duct, they found that introducing a heat source/sink at the water surface could cause negative Nusselt and Sherwood numbers. Raimundo et al. [12] investigated the relationship between evaporation from heated water surfaces and mean aerothermal properties of a forced airflow. Their results showed that the rate of evaporation was affected by the air velocity, humidity, and water-air temperature difference.

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L.D. Gu et al. / International Journal of Heat and Mass Transfer 122 (2018) 1093–1102

Nomenclature Cp D h hm i J Le l M Nu P Pc Pr q Re Rg Sc Sh T

specific heat, J/kgK diffusivity, m2/s heat transfer coefficient, kg/m2K mass transfer coefficient, kg/m2s enthalpy, J/kg mass flux, kg/m2s Lewis number (=Sc/Pr) length of water surface, m molar mass Nusselt number (=hl/k) pressure, Pa or atm critical pressure, atm Prandtl number (=lCp/k) heat flux, W/m2 Reynolds number (=qul/l) gas constant, J/(kgK) Schmidt number (=l/qD) Sherwood number (=hml/qD) temperature, °C

Poós and Varju [13] experimentally examined the rate of evaporation from free water surface to a tubular artificial flow and purposed correlations for calculating the Sherwood numbers at different Reynolds numbers. However, the mass transfer rate in their investigations was low. Iskra and Simonson [14] performed experiments to determine the convective mass transfer coefficient for evaporation in a horizontal rectangular duct. Wei et al. [15] proposed a simplified CFD-based model to analyze the heat and mass transfer in air-water direct contact through the water surface in a rectangular duct. Kumar et al. [16] investigated the effect of roughness on evaporation rates under varying conditions of air velocities and water temperatures and reported that the increase in the roughness caused an increase in the evaporation rate. Schwartze and Brocker [17] and Volchkov et al. [18] theoretically studied the evaporation of water into air-steam mixture and pure superheated steam, they paid a special attention on the phenomena of inversion temperature, above which the rate of evaporation into pure superheated steam became higher than that into dry air. There are also many investigations on simultaneous heat and mass transfer characteristics for falling film evaporation or condensation in vertical channels or inclined planes [19–32]. Jang et al. [19] numerically investigated the mixed convective heat and mass transfer with film evaporation in inclined square ducts, while Huang et al. [20] numerically simulated the mixed convective heat and mass transfer with film evaporation and condensation in vertical ducts, they both reported that the latent heat transport with film evaporation tremendously augmented the heat transfer rate and claimed better heat and mass transfer rates related with film evaporation for the case with a higher wetted wall temperature. Yan and Lin [21] numerically studied the natural convective heat and mass transfer with film evaporation and condensation in vertical concentric annular ducts, they also found the tremendous enhancement in heat transfer due to the exchange of latent heat associated with film evaporation and condensation. The extent of augmentation of heat transfer due to mass transfer was more significant for a system with a higher wetted wall temperature. Nasr [22,23] and Terzi et al. [24,25] studied the heat and mass transfer of evaporation and condensation of falling film with porous layer inside, their results supported that use of the porous layer could promote the heat and mass transfer. Cherif et al. [26] experimentally and numerically investigated the effects of film evaporation on mixed convective heat and mass transfer in a ver-

Tc u, v w

critical temperature, K velocity, m/s specific humidity, kg/kg

Greek symbol h correction factor q density, kg/m3 l viscosity, Pas k thermal conductivity, W/mK Subscript 1 a h hm v s x

free airstream air heat transfer mass transfer water vapor saturation state or water-air interface local

tical rectangular channel as well as those of liquid film temperature, evaporated flow rate, and upward airflow rate on the heat and mass transfer. Charef et al. [27] numerically studied the liquid film condensation from vapor-gas mixture in a vertical tube with constant temperature or uniform heat flux, and reported that the increases of relative humidity and inlet-to-wall temperature difference acted to enhance the condensation process. For a fixed heat flux, increasing the inlet temperature substantially increased the accumulated condensation rate. Wan et al. [28,29] numerically analyzed the combined heat and mass transfer characteristics in vertical plate channels with falling film evaporation, and proposed correlations for predicting the heat and mass transfer coefficients. Tang and Min [30] theoretically studied the transient evaporation characteristics of water film attaching to an adiabatic solid wall, they [31] also analyzed the evaporation characteristics of water film on a thermally conductive spherical solid particle and found that the water film transient evaporation characteristics were affected more by the heat capacity than by the thermal conductivity of the solid particle. Although there are many investigations on heat and mass transfer associated with water evaporation and simultaneous heat and mass transfer, few studies focused on the effects of mass flux and fluid property variability on the heat and mass transfer characteristics. In this research, a numerical study is carried out to analyze the coupled heat and mass transfer between water surface and laminar airflow. The specific objectives of this research are to investigate the effects of mass flux and air property variability on heat and mass transfer coefficients and discuss the applicability of the Chilton-Colburn analogy principle.

2. Computational details 2.1. Physical model Fig. 1 illustrates the convective heat and mass transfer between water surface and air fluid flowing over it. The free airstream has a zero specific humidity and a temperature much higher than that of the water surface, leading to a combined heat and mass transfer accompanied with water evaporation into the airstream. To simplify the problem and stress the focal point, the following assumptions are adopted:

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At the air-water interface, the normal velocity mined by the mass diffusion rate, i.e.

ð1 ws Þqv s ðxÞ ¼ qD

Fig. 1. Convective heat and mass transfer between water surface and airstream.

(1) The airflow is a laminar boundary layer flow at a steady state. (2) The water surface is stationary and has a uniform temperature. (3) The air at the air-water interface is saturated and the specific humidity there can be evaluated at the water surface temperature. The investigation is implemented for the following conditions: the free airstream is at the atmospheric pressure of 101.325 kPa, its temperature is T1 = 200 °C, specific humidity is w1 = 0, and velocity ranges u1 = 1–10 m/s, while the water surface temperature ranges Ts = 10–90 °C. 2.2. Governing equations The process involves the exchanges of momentum, energy and mass, and the equations governing such exchanges can be summarized as follows.

@ qu @ qv þ ¼0 @x @y

ð1Þ

Momentum equation:

@u l @y

ð2Þ

Energy equation:

! @ qui @ qv i @ @T @ X @wj þ k cj ij ¼ 0 @x @y @y @y @y @y j

ð3Þ

Mass diffusion equation:

@ quwj @ qv wj @ @w c j þ @y j @y @x @y

is deter-

ð11Þ

2.4. Dependence of moist air properties on temperature and specific humidity Moist air may show significant variations in its properties if its temperature and specific humidity change considerably. The density, specific heat, viscosity, thermal conductivity and diffusion of moist air can be calculated using the equations listed below. (1) Density [33]: The ideal gas model can be used to calculate the density of moist air

p Tðwa Rga þ wv Rgv Þ

q¼

ð12Þ

where wa and wv are the mass fractions of air and water vapor in moist air, and wa = 1 w and wv = w, Rga is the gas constant of air whose value is 287 J/(kgK), Rgv is the gas constant of water vapor whose value is 462 J/(kgK), P is the pressure, and T is the temperature. (2) Specific heat [33]:

ð13Þ

where Cp, Cpa and Cpv are the specific heats of moist air, dry air and water vapor, respectively. (3) Viscosity [33,34]:

ðx > 0; y ¼ 0Þ

C p ¼ wa C pa þ wv C pv

Continuity equation:

@ quu @ qv u @ þ ¼ @x @y @y

@w @y

vs(x)

¼0

ð4Þ

xa l a xv l v þ xa þ ð1 xa ÞUav xv þ ð1 xv ÞUv a

l¼

ð14Þ

in which la and lv are the viscosities of air and water vapor, and xa and xv are the molar fractions of air and water vapor. For an ideal gas mixture, the molar fractions can be calculated by

xa ¼

wa Mv wa Mv þ wv Ma

ð15Þ

xv ¼

wv M a wv M a þ wa M v

ð16Þ

The boundary conditions can be expressed as below.

where Ma is the molar mass of air whose value is 28.97 103 kg/mol, and Mv is the molar mass of water vapor whose value is 18.01 103 kg/mol, while Uav and Uva are given by

Free flow region (x = 0, y > 0; x > 0, y ? 1):

2.3. Boundary conditions

u ¼ u1 ;

v ¼0

Uav

1=2 " 1=2 1=4 #2 1 Ma la Mv ¼ pffiffiffi 1 þ 1þ M l Ma v 8 v

ð17Þ

Uva

1=2 " 1=2 1=4 #2 1 Mv lv Ma ¼ pffiffiffi 1 þ 1þ Ma la Mv 8

ð18Þ

ð5Þ

T ¼ T1

ð6Þ

w¼0

ð7Þ

(4) Thermal conductivity [33,35]:

Air-water interface (x > 0, y = 0):

u ¼ 0;

v ¼ v s ðxÞ

ð8Þ

T ¼ Ts

ð9Þ

w ¼ ws

ð10Þ

k¼

xa ka xv kv þ xa þ ð1 xa ÞUav xv þ ð1 xv ÞUv a

ð19Þ

where ka and kv are the thermal conductivities of air and water vapor, while Uav and Uva are also given by Eqs. (17) and (18).

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(5) Diffusivity:

h¼

The diffusion of water vapor in air or air in water vapor is irrelevant with humidity, it varies only with the temperature and pressure [36]. The diffusivity can be calculated from [37]:

pD ðpca pcv Þ1=3 ðT ca T cv Þ5=12 ð1=M a þ 1=M v Þ1=2

b T ¼ a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ca T cv

ð20Þ

in which the diffusivity D has a unit of cm2/s, the pressure has a unit of atm, and the temperature has a unit of K, a and b are constants whose values are a = 3.640 10 and b = 2.334 for moist air, Pca and Pcv are the critical pressures of air and water vapor whose values are Pca = 36.4 atm and Pcv = 218 atm, and Tca and Tcv are the critical temperatures of air and water vapor whose values are Tca = 132 K and Tcv = 647.3 K. The properties of dry air and water vapor can be obtained from Tsilingiris [33] and some other references [38–41]. Based on the equations given above, the properties of moist air can be calculated as a function of temperature and specific humidity. Fig. 2 illustrates the dependences of moist air properties and water vapor diffusivity in air on air temperature and specific humidity. The Fig. 2 data are used in the numerical calculations of this research. 2.5. Numerical method and grid independence The computational domain includes an airflow area that has a length of 0.5 m in the airflow direction and a width of 0.064 m in the perpendicular direction, it covers the region of 0 x 0.5 m and 0 y 0.064 m as shown in Fig. 1, constant temperature boundary condition is used for the air-water interface (y = 0). The grids are uniform in the airflow direction (x direction) but nonuniform in the perpendicular direction (y direction), with finer grids being used for the region near the water surface. The control-volume method is used to discretize the governing equations. The first order upwind scheme is employed for the convective terms, with the linear equations formed by discretizing the governing equations being solved using the TDMA method. A grid independence test is done for four grid systems of 125 25, 250 50, 500 100 and 1000 200. Calculations are conducted for T1 = 200 °C free airstream temperature, w1 = 0 specific humidity, u1 = 1 m/s inlet velocity, and Ts = 10 °C water surface temperature. The obtained results are presented in Fig. 3, which shows the variations of overall heat and mass transfer coefficients with number of grids. Only small changes are observed in both heat and mass transfer coefficients when the number of grids exceeds 250 50, and the difference between the schemes 500 100 and 1000 200 is less than 1.5%, the grid system of 500 100 is therefore chosen for the simulations. Note that the number of 500 is for the airflow direction and that of 100 is for the perpendicular direction. 2.6. Data reduction The local heat and mass transfer coefficients can be calculated from the temperature and concentration gradients near the water surface, i.e.

@T ðT s T 1 Þ hx ¼ k @y s hm;x

@w ðws w1 Þ ¼ qD @y s

ð21Þ

Z

1 l

hx dx

ð23Þ

0

1 l

hm ¼

l

Z

l

hm;x dx

ð24Þ

0

where l is the length of water surface. The Nusselt and Sherwood numbers corresponding to the overall heat and mass transfer coefficients can be computed from

Nu ¼

hl k

ð25Þ

Sh ¼

hm l qD

ð26Þ

The mass flux of water vapor across the water surface is equal to the sum of the mass flux by diffusion and that by convection, i.e.

J ¼ qD

@w þ qwv s @y y¼0

ð27Þ

where vs is the velocity pointing to the y direction at the air-water interface. For simulations that treat the air properties as constant, the air properties are evaluated at the mean temperature and specific humidity as

Tf ¼

Ts þ T1 2

ð28Þ

wf ¼

ws þ w1 2

ð29Þ

where Ts and ws are the air temperature and specific humidity at the air-water interface, while T1 and w1 are those of the free airstream. To analyze the variations of heat and mass transfer coefficients with mass flux, the following two correction factors are defined, i.e. the heat transfer coefficient correction factor

hh ¼

h h0

ð30Þ

and the mass transfer coefficient correction factor

hhm ¼

hm hm0

ð31Þ

in which h0 and hm0 are the heat and mass transfer coefficients with mass flux approaching zero. 3. Results and discussion Calculations are implemented for T1 = 200 °C free airstream temperature, w1 = 0 specific humidity, u = 1–10 m/s velocities, and Ts = 10–90 °C water surface temperatures. The air at the airwater interface is saturated and has a specific humidity evaluated at the water surface temperature at atmospheric pressure, which can be expressed as

ws ¼ expð8:407777e 5 T 2s þ 6:252234e 2 T s 5:458808Þ ð22Þ

The overall heat and mass transfer coefficients can be obtained by integrating the local heat and mass transfer coefficients over the length of water surface along the airflow direction:

ð32Þ based on the Ref. [42] data. Eq. (32) is valid for 0–100 °C. Note that the air velocity range of u = 1–10 m/s and the water surface length of l = 0.5 m (see Section 2.5) guarantee the airflow to be a laminar boundary layer flow over a plane surface.


(a) Moist air density

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(b) Moist air thermal conductivity

(c) Moist air specific heat

(d) Moist air viscosity

(d) Water vapor diffusivity in air Fig. 2. Dependences of moist air properties and water vapor diffusivity in air on air temperature and specific humidity.

3.1. Model validation

hm;x ¼ 0:332

The local heat transfer coefficient for a laminar boundary layer flow over a flat plate can be represented by [43]

k hx ¼ 0:332 Rex1=2 Pr 1=3 x

ð33Þ

where x is the distance to the leading edge, Rex is the Reynolds number based on x, and Pr is the Prandtl number. When the mass flux is low, the local mass transfer coefficient for a laminar boundary layer flow over a flat plate can be achieved from the Chilton-Colburn analogy as [43]

qD x

1=3 Re1=2 x Sc

ð34Þ

where D is the diffusivity and Sc is the Schmidt number. Fig. 4 compares the local heat and mass transfer coefficients generated by the numerical simulations and those given by Eqs. (33) and (34). The simulations use T1 = 200 °C free airstream temperature, w1 = 0 specific humidity, u = 5 m/s velocity, Ts = 10 °C water surface temperature, and the constant air properties evaluated at the mean temperature and mean specific humidity given by Eqs. (28) and (29). The saturation specific humidity corresponding to Ts = 10 °C is 0.0076 kg/kg, this value guarantees a low mass

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Fig. 3. Variations of overall heat and mass transfer coefficients with number of grids.

Fig. 4. Comparison of local heat and mass transfer coefficients obtained from the numerical simulations and available equations.

Fig. 5. Variations of mass flux with water surface temperature for different air velocities.

Fig. 6. Variations of overall heat transfer coefficient with water surface temperature for different air velocities.

flux across the water-air interface, making the Chilton-Colburn analogy applicable. Fig. 4 shows that the two sets of data agree well, supporting the reliability of our numerical procedures. 3.2. Overall heat and mass transfer characteristics Fig. 5 depicts the variations of mass flux with water surface temperature for different air velocities. The mass flux, J, is obtained from Eq. (27), which suggests that J is the sum of the mass flux by diffusion and that by convection. The calculations used the constant air properties evaluated at the mean air temperature and specific humidity given by Eqs. (28) and (29). The mass flux increases with increasing water surface temperature because of the increase of the saturation specific humidity. Also, the mass flux increases with increasing air velocity. Figs. 6 and 7 show the variations of overall heat and mass transfer coefficients with water surface temperature for different air velocities. As the water temperature rises, which causes an increase of mass flux, the overall heat and mass transfer coefficients both decrease. They decrease gently for low water temperatures but more rapidly for higher temperatures. Further, the overall mass transfer coefficient decreases more conspicuously than the overall heat transfer coefficient. As the air velocity increases, the heat and mass transfer coefficient both increase considerably.

Fig. 7. Variations of overall mass transfer coefficient with water surface temperature for different air velocities.

Figs. 8 and 9 present the variations of Nusselt and Sherwood numbers with Reynolds number for different water surface temperatures. The figures show that the Nusselt and Sherwood num-


Fig. 8. Variations of Nusselt number with Reynolds number for different water surface temperatures.

Fig. 9. Variations of Sherwood number with Reynolds number for different water surface temperatures.

bers both increase with increasing Reynolds number but decrease with increasing water surface temperature, and they increase more obviously for a smaller Reynolds number but decrease more conspicuously for a higher water surface temperature. 3.3. Effects of mass flux on overall heat and mass transfer coefficients Figs. 10 and 11 illustrate the variations of heat and mass transfer coefficient correction factors with water surface temperature for different air velocities. Such factors are the ratios of the overall heat and mass transfer coefficients to those with mass flux approaching zero, as defined by Eqs. (30) and (31), they reflect the effects of mass flux on overall heat and mass transfer coefficients. As observed in the figures, both factors begin with unity and decrease with increasing water surface temperature, which can be attributed to the increase of mass flux with water temperature as indicated by Fig. 5. A higher mass flux yields a more obvious effect of mass transfer on the heat and mass transfer coefficients. In contrast, the air velocity has little influence on the heat and mass transfer coefficient correction factors. This can be explained as below. The correction factors reflect the effect of mass flux on heat and mass transfer coefficients, which is also the effect of mass flux on profiles of temperature and humidity. For a bound-

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Fig. 10. Variations of heat transfer coefficient correction factor with water surface temperature for different air velocities.

Fig. 11. Variations of mass transfer coefficient correction factor with water surface temperature for different air velocities.

ary layer flow, the dimensionless profiles of velocity, temperature and humidity in the boundary layer are independent of the inlet air velocity except that the boundary layer thickness decreases with increasing air velocity. The correction factors are independent of the inlet air velocity means that the effects of mass flux on the dimensionless profiles of velocity, temperature and humidity are independent of the inlet air velocity There are some existing theories and models for calculating the correction factors, of which the film theory [44,45] and permeation model [36,46] are most popular, they both consider the effects of mass flux on heat and mass transfer coefficients. The film theory assumes the boundary layer film to be steady and vary only with the fluid velocity. The solute diffuses into the free stream from the film, with the correction factor being calculated by

hh ¼

lnð1 þ Rh Þ Rh

hhm ¼

lnð1 þ Rm Þ Rm

ð35Þ

ð36Þ

where hh is the heat transfer coefficient correction factor and hhm is the mass transfer coefficient correction factor. For the heat and mass transfer process shown in Fig. 1,

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i1 is qs

Rh ¼

ð37Þ

for the heat transfer, where i1 and is are the air enthalpy of free airstream and that at the air-water interface, and qs is the heat flux across the interface, and

Rm ¼

w1 ws ws 1

ð38Þ

for the mass transfer, where w1 and ws are the specific humidity of free airstream and that at the air-water interface. The permeation model is an unsteady diffusion model based on the film theory, it considers the variation of film thickness caused by mass diffusion. The heat transfer coefficient correction factor in the permeation model is calculated by the following equations:

hh ¼

expð/2h =pÞ pffiffiffiffi 1 þ erf ð/h = pÞ

ð39Þ

! /h /2h p ffiffiffi ffi Rh ¼ 1 þ erf /h exp

p

p

ð40Þ

while the mass transfer coefficient correction factor in the permeation model is calculated by:

hhm ¼

expð/2m =pÞ pffiffiffiffi 1 þ erf ð/m = pÞ

ð41Þ

! / /2m ffiffiffiffi /m exp Rh ¼ 1 þ erf pm

p

p

ð42Þ

Fig. 13. Comparison of mass transfer coefficient correction factors obtained from the film theory, permeation model, and numerical simulations.

the correction factors given by the film theory and permeation model are very close to those by the numerical simulations, for a higher water temperature, however, the film theory and permeation model tend to over-predict the correction factor, with the former yielding a more serious over-prediction than the latter, because the latter considers the variation of film thickness caused by the mass diffusion while the former ignores it. The deviation ranges 0–24% for the film theory and 0–11% for the permeation model, with a higher water temperature giving a larger deviation.

in which Rh can still be computed from Eq. (37) for the heat transfer and Eq. (38) for the mass transfer. Figs. 12 and 13 compare the heat and mass transfer coefficient correction factors obtained from the film theory, permeation model and numerical simulations. No air velocities are given here because hh and hhm are unaffected by the air velocity, as seen in Figs. 10 and 11. In other words, they are applicable for all air velocities of u = 1, 5 and 10 m/s. Fig. 12 shows that the permeation model yields a correction factor very close to that generated by the numerical simulations, whereas the film theory overestimates the correction factor, with a higher water temperature producing a more serious overestimation. The maximum deviation is 4.8%, which occurs at Ts = 90 °C water temperature. Fig. 13 shows that for low water temperatures,

for the Chilton-Colburn analogy, in which Le is the Lewis number. The figure applies for all air velocities because the result is unaffected by the air velocity. The deviation of the ratio generated by the analogy relative to that by the simulations ranges 0–27%, with a higher water temperature yielding a larger deviation. It is less than 5% when the water surface temperature is below 60 °C. So,

Fig. 12. Comparison of heat transfer coefficient correction factors obtained from the film theory, permeation model, and numerical simulations.

Fig. 14. Comparison of heat to mass transfer coefficient ratios obtained from the Chilton-Colburn analogy and numerical simulations.

3.4. Validity of Chilton-Colburn analogy Fig. 14 compares the ratios of heat to mass transfer coefficient obtained from the Chilton-Colburn analogy and numerical simulations. Note that such a ratio is calculated by

h ¼ C p Le2=3 hm

ð43Þ


to be able to obtain an accurate result, the Chilton-Colburn analogy principle should not be used if the mass flux is high. 3.5. Effects of air property variability on overall heat and mass transfer characteristics When the temperature and humidity of moist air change considerably, its properties may vary significantly. Figs. 15 and 16 compare the overall heat and mass transfer coefficients calculated with and without consideration of air property variability for u = 5 m/s air velocity. Noting that the variable properties are based on the Fig. 2 data while the constant properties are evaluated at the mean air temperature and specific humidity given by Eqs. (28) and (29). It is seen from Fig. 15 that large differences exist between the two sets of the data. For water temperatures below 50 °C, the heat transfer coefficient for variable properties is larger than those for constant properties. When the water temperature exceeds 50 °C, the results are opposite. The maximum deviation is 35%, which occurs at Ts = 90 °C water temperature. So, the effects of the air property variability on the heat transfer coefficient are conspicuous and complex, ignorance of air property variability may either overestimate or underestimate the heat transfer coefficient. A pos-

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sible explanation is that the heat transfer relating properties, such as the thermal conductivity, specific heat and viscosity, are closely related with the temperature and specific humidity, which complicates the effect of property variability on heat transfer coefficient. It is seen from Fig. 16 that the differences between the constant and variable property results are quite small, and the deviation between the two sets of data is less than 3%. The mass transfer is affected primarily by the water vapor diffusivity, which changes moderately with temperature and is unrelated with specific humidity. 4. Conclusions (1) The heat and mass transfer coefficients both decrease with increasing water surface temperature, i.e. increasing mass flux. (2) The correction factors for heat and mass transfer coefficients, which reflect the effects of mass flux on heat and mass transfer coefficients, both decrease with increasing water surface temperature, i.e. increasing mass flux, but they are unaffected by the air velocity. (3) The correction factors for heat transfer coefficient predicted by the film theory and permeation model agree reasonably with the numerical results, while those for mass transfer coefficient predicted by the film theory and permeation model differ considerably from the numerical results. (4) The Chilton-Colburn analogy holds only for low water temperature i.e. low mass flux case. The deviation of the heat to mass transfer coefficient ratio given by the ChiltonColburn analogy relative to that by the numerical simulation is less than 5% when the water surface temperature is below 60 °C. (5) The air property variability has a conspicuous and complex effect on heat transfer coefficient but an unobvious effect on mass transfer coefficient. Conflict of interest statement We declare no any interest conflict. Acknowledgments

Fig. 15. Comparison of overall heat transfer coefficients calculated with and without consideration of property variability for u = 5 m/s air velocity.

This research is funded by National Natural Science Foundation of China (51376103). References

Fig. 16. Comparison of overall mass transfer coefficients calculated with and without consideration of property variability for u = 5 m/s air velocity.

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