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lnternational journal of Heat and Mass Transfer 122 (2018 ) 1093-1102

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International journal of Heat and Mass Transfer jo u rn a l h o m e pa g e: ww w. e l sev i e r.co m / l oca t e/ i j h m t

ELSEVIER

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 ºC air temperature, 110 m/s air velocities, and 10-90 ºC water temperatures . Calculations are implemented with and with- out 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 coeffi- cients both decrease with increasi ng water surface temperature, i .e. increasing mass flux. The Chilton- Colburn 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 ºC. The air property variability has a notable and complex effect on heat transfer coefficient but an inconspicuous effect on mass transfer coefficient. © 2018 Elsevier Ltd. Ali 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 Colbum-Chilton analogy [3-5). When the mass flux is larger, however, the analogy may become inapplicable beca use 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 !ayer 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] Q.C. Min).

https://doi.org/ 10.1016/j.ijheatmasstransfer.2018 .02.061 0017-9310 /© 2018 Elsevier Ltd. Ali rights reserved .

numerically studied the heat and mass transfer characteristics of water evaporation in a laminar boundary !ayer 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-Colbum analogy was valid only at low free stream temperatures and vapor concentrations, but they provided no range for which the Chilton-Colbum 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.

L.D. Gu et al./ lntemational joumal of Heat and Mass Transfer 122 (2018) 1093-1102

1094

Nomenclature CP D h hm

J Le l M

Nu p

Pe Pr q

Re Rg Se Sh T

specific heat, J/kg- K diffusivity, m2 /s heat transfer coefficient, kg/m 2 - K mass transfer coefficient, kg/m2 -s enthalpy, J/kg mass flux, kg/m 2 -s Lewis number ( =Sc/Pr) length of water surface, m molar mass Nusselt number ( =hl/Jc ) pressure, Pa or atm critica( pressure, atm Prandtl number (=µCpf J,) heat flux, W/m 2 Reynolds number ( =pul/µ ) gas constant, J/(kg- K) Schmidt number ( =µ/ pD ) Sherwood number (=hml/ pD ) temperature, ºC

Poós and Varju [13) experimentally examined the rate of evaporation from free water surface to a tubular artificial flow and purposed correlations far 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 far 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 far 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 far 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 far 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-

Te U, V

w

critica( temperature, K velocity, m/s specific humidity, kg/kg

Greek symbol

e p µ Je

correction factor density, kg/m3 viscosity, Pa-s thermal conductivity, W/m - K

Subscript 00

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. Far 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 far predicting the heat and mass transfer coefficients. Tang and Min [30) theoretically studied the transient evaporation characteristics of water film attaching to an adiabatic salid wall, they [31) also analyzed the evaporation characteristics of water film on a thermally conductive spherical salid particle and found that the water film transient evaporation characteristics were affected more by the heat capacity than by the thermal conductivity of the salid 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 investigare the effects of mass flux and air property variability on heat and mass transfer coefficients and discuss the applicability of the Chilton-Colburn analogy principie.

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:

L.D. Gu et al./ lntemational joumal of Heat and Mass Transfer 122 (2018) 1093-1102

1095

At the air-water interface, the normal velocity v5(x ) is determined by the mass diffusion rate, i.e.

U =Ua:J

Air tlow

()w

W=W 00

(1 - w,)pv 5 (x ) = -pD ()y

Yot- , x"----T ---=--T-',-,-w ---=--w-,''------"_,...., " '...,. - -A--ir--w - ater interface Fig. t. Convective heat and mass transfer between water surface and airstream.

(1) The airflow is a laminar boundary !ayer 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 T00 = 200 ºC. specific humidity is w00 = O, and velocity ranges u 00 = 1-1O m/s, while the water surface temperature ranges T5 = 10-90 ºC. 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.

(x > O, y = O)

(11)

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

p

= T ( wa Rga

(12)

+ Wv Rgv )

where w0 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/( kg-K), Rgv is the gas constant of water vapor whose value is 462 J/(kg-K), P is the pressure, and T is the temperature. (2) Specific heat [33): (13)

Continuity equation: fJpu ax

+ fJpv = 0 ()y

(1)

where Cp, Cpa and Cpv are the specific heats of moist air, dry air and water vapor, respectively.

Momentum equation:

(3) Viscosity [33,34): (2)

Energy equation: (3)

µ

=

Xaµa Xa

+ (1 - Xa )av

+ --X-'v-µ'-v--"--X v + (1 - X v ) va

(14)

in which µª and µv 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 WaM v

Mass diffusion equation:

Xa =

Wa M v

(4)

(15)

+ WvM a

WvM a

(16)

Xv = Wv M a + waM v

where M0 is the molar mass of air whose val ue is 28.97 x 10-3 kg/mol, and Mv is the molar mass of water vapor whose value is 18.01 x 10-3 kg/mol, while <Pav and
2.3. Boundary conditions The boundary conditions can be expressed as below.

_ 1 ( Free flow region (x = O, y > O; x > O, y --. oo): U = U'""

V =0

(5) (6)

av -

W =0

(7)

1 +-

J8

_ 1 ( va -

2 Ma )-1/2 [ (µª)1/2 (Mv)1/4]

J8

Mv

1+

µv

Ma

Mv)-1 /2 [ (µv)1/2 (Mª)1/4]

1 +Ma

1+

-

-

µa

Mv

2

(17)

(18)

(4) Thermal conductivity [33,35):

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

T = T5 W = Ws

V = V5 (X )

(8)

(19)

(9) (10)

where Aa and Av are the thermal conductivities of air and water vapor, while <Pav and
L.D. Gu et al./ lntemational joumal of Heat and Mass Transfer 122 (2018) 1093-1102

1096

(5) Diffusivity:

(23)

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 ( 12 (P ea · P cv) 1 13 ( Tca · Tcv) 5112 ( 1/ Ma + 1/Mv)/ = a

)b

T

(20)

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 x 10 and b = 2.334 for moist air, Pea and Pcv are the critica! pressures of air and water vapor whose values are Pea = 36.4 atm and Pcv = 218 atm, and Tea and Tcv are the critica! temperatures of air and water vapor whose values are Tea = 132 K and Tcv = 647.3 K. The properties of dry air and water vapor can be obtained from Tsilingiris [33) and sorne 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 O ::; x ::; 0.5 m and O ::; y ::; 0.064 m as shown in Fig. 1, constant temperature boundary condition is used for the air-water interface (y = O). 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 arder upwind scheme is employed for the convective terms, with the linear equations formed by discretizing the governing equations being solved using the TOMA method. A grid independence test is done for four grid systems of 125 x 25, 250 x 50, 500 x 100 and 1000 x 200. Calculations are conducted for T = 200 ºC free airstream temperature, w = O specific humidity, u = 1 m/s inlet velocity, and T5 = 1O ºC 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 x 50, and the difference between the schemes 500 x 100 and 1000 x 200 is less than 1.5%, the grid system of 500 x 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. 00

(24) 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 = !!.!_ ;,

(25)

Sh = hml pD

(26)

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=

-pD °:I

y y= O

+

pwv

(27)

5

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 T¡ = Ts + T oo 2

(28)

W5 + woo

W¡ =

(29)

2

where T5 and w5 are the air temperature and specific humidity at the air-water interface, while T and w 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 00

00

(30)

00

00

2.6. Data reduction

and the mass transfer coefficient correction factor (31) in which h0 and hmo are the heat and mass transfer coefficients with mass flux approaching zero. 3. Results and discussion Calculations are implemented for T = 200 ºC free airstream temperature, w = O specific humidity, u = 1-1O m/s velocities, and T5 = 10-90 ºC 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 00

The local heat and mass transfer coefficients can be calculated from the temperature and concentration gradients near the water surface, i.e.

00

W5 = exp(-8.407777e - 5 ·

(21)

r;+ 6.252234e - 2 ·T

5-

5.458808) (32)

(22) 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:

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

L.D. Gu et al./ lntemational joumal of Heat and Mass Transfer 122 (2018) 1093-1102

l.3

0.040

1.2

0.038

l.l

0.036

1.0

0.034

E 0.9

0.032

,:;--

1li

0.8

1097

._, 0.030

c:i. 0.7

0.028

0.6

0.026

0.5 0.024 0.4

o

50

100

150

o

200

100

50

T (°C )

150

20 0

T CC)

(a) Moist air density

(b) Moist air thermal conductivity

3000

2.8E-5 2.6E-5

2500 ,....._ :,¿

2.4E-5 2.2E-5

2000

6n -"'

oü"'

'V>' 2.0E-5 O¡ .c_..,

l.8E-5

1500

::::.. l .6E-5

1000

l .4E-5

w=O

l .2E-5 500

o

100

50

o

200

150

100

50

T ('C )

150

200

T ('C )

(e) Moist air specific heat l.I E-5

(d) Moist air viscosity

-

l .OE-5 9.0E-6

,....._

<J>

H.OE-6

º....

7.0E-6

Cl

6.0E-6

_§,

w=0-1.0 5.0E-6 4.0E-6 3.0E-6

o

50

100

150

200

T (°C )

(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

(34)

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

=

0.332 Re;l2 Pr 1!3

(33)

X

where D is the diffusivity and Se 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 T = 200 ºC free airstream temperature, w = O specific humidity, u = 5 m/s velocity, T5 = 10 ºC 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 T5 = 1O ºC is 0.0076 kg/kg, this value guarantees a low mass 00

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)

00

L.D. Gu et al./ lntemational joumal of Heat and Mass Transfer 122 (2018) 1093-1102

1098

15

4.5E-3

N

E

--D- u=l m/s

4.0E-3

,.-.,

10

-o- u=S mis u=I O m/s

3.5E-3 3.0E-3

'-'

5

D

,.-.,

D

"'a2.5E-3 2.0E-3 bo

0.003

o ....,

,.-.,

o 0.002

l.OE-3

o

•

'-'

l .5E-3

5.0E-4

o

o

O.O

o 125x25

250 x50

500x l00

10

20

30

40

1 000x200

20

80

90

100

-o-u= 1 m/s --O- u=S m/s u=l O mis

-o-Numerical -o- Eq . (33)

100

70

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

1000 ,.-.,

60

TS CC )

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

50

16

N

a

10 12

'-'

' ,.-.,

1 1---

-'---

-'-

0.1

----1

-o-o-Numerical Eq. (34)

N

-E-

8

'-'

'-'

4

0.01

..s:;: o

0.001 0.001

0.01

0.1 X

..

o

..

10

.._.._.._.._

20

30

(m)

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

50

60

70

80

90

100

T, (°C) Fig. 6. Variations of overall heat transfer coefficient with water surface temperature far 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.

0.006

--o- u=l mis -o- u=S m/s u=lO m/s

0.005

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,], is obtained from Eq. (27), which suggests that ]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.

40

,.-.,

0.004

0

lf

0.003

• 0.002

'-'

0.001

o

10

20

30

40

50

60

70

80

90

100

T, ('C ) Fig. 7. Variations of overall mass transfer coefficient with water surface temperature far different air velociti es.

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-

L.D. Gu et al./ lntemational joumal of Heat and Mass Transfer 122 (2018) 1093-1102

1.05

300

1099

-

-o- J O ºC --0-- 20 ºC 30 ºC -v-40 ºC ---<>--- 50 ºC ---4--- 60 ºC --t>-- 70 ºC --0-- 80 ºC -*- 90 ºC

250

200

1 50

-o- u=l m/s -O- u=S m/s ---1:.- u=l O m/s

1.00 0.95 0.90 0.85 0.80

100

0.75 50

o

50000

100000

150000

o

10

20

30

40

200000

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

50

60

70

80

90

1 00

T, (C) Fig. to. Variations of heat transfer coefficient correction factor with water surface temperature far different air velocities.

1.1

500

-o- J O ºC --0-- 20 ºC 30 ºC

400

300

-o- u=l mis -o-u=S m/s ---1:.- u=I O mis

1.0

-v-40 ºC ---<>--- 50 ºC ---4--- 60 ºC --t>-- 70 ºC

0.9 0.8 ¡¡

"" 0.7 0.6

200

0.5 100

0.4

o

50000

100000

150000

200000

Re

o

10

20

30

40

50

60

70

80

90

100

T, (°C )

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

Fig. 11. Variations of mass transfer coefficient correction factor with water surface temperature far different ai r velocities.

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.

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 sorne 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 salute diffuses into the free stream from the film, with the correction factor being calculated by

3.3. E.ffects of mass flux on overa!! heat and mass transfer coe.f.ficients Figs. 1O 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-

(35)

(36) where lh is the heat transfer coefficient correction factor and lhm is the mass transfer coefficient correction factor. For the heat and mass transfer process shown in Fig. 1,

L.D. Gu et al./ lntemational joumal of Heat and Mass Transfer 122 (2018) 1093-1102

1100

Rh = i00 -

(37)

is

1.0

qs for the heat transfer, where i 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 R - Woo - Ws (38) m - Ws - 1 00

0.9 0.8 <:):)J.

for the mass transfer, where w 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:

0.7

--o-- Film theory

00

(J _ exp( - / n) h - 1 + erf (h / Vn)

-o-Permeation model 0.6

---b--

Numerical

0.5

o

1o

20

30

(39)

40

50

60

70

80

90

100

T,('C ) Fig. 13. Comparison of mass transfer coefficient correction factors obtained from the film theory, permeation model, and numerical simulations.

(40)

while the mass transfer coefficient correction factor in the permeation model is calculated by: exp(-/ n) (Jhm = _1_+_erf_(_m!-v_n_)

(41)

the correction factors given by the film theory and permeation model are very clase 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.

(42)

3.4. Validity of Chilton-Colburn analogy

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 (Jh and (Jhm are unaffected by the air velocity, as seen in Figs. 1O and 11. In other words, they are applicable for ali air velocities of u = 1, 5 and 10 m/s. Fig. 12 shows that the permeation model yields a correction factor very clase 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 ºC water temperature. Fig. 13 shows that for low water temperatures,

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

..!!.._ = CpLe2l3

for the Chilton-Colburn analogy, in which Le is the Lewis number. The figure applies for ali 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 ºC. So,

1.00

4800

0.95

4400 4000

0.90 <:):)""

(43)

hm

-.._ ,,

-a- Chilton-Colurn analogy --O- Numerical

the film theory, permeation model, and numerical simulations.

0.85

--o-- Film thcory

-o- Permeation model 0.80

o

---b--

Numerical

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Ts (°C ) Fig. 12. Comparison of heat transfer coefficient correction factors obtained from

..:::." 3600

L.D. Gu et al./ lntemational joumal of Heat and Mass Transfer 122 (2018) 1093-1102

3200 2800 L---1 -L.....1-.L. J-L-'----'-¡_J o lO 20 30 40 50 60 70 80 90 100

T (°C ) 5

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

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L.D. Gu et al./ lntemational joumal of Heat and Mass Transfer 122 (2018) 1093-1102

1100

to be able to obtain an accurate result, the Chilton-Colbum analogy principie should not be used if the mass flux is high.

3.5. Effects of air property variability on overa!! 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 ºC, the heat transfer coefficient for variable properties is larger than those for constant properties. When the water temperature exceeds 50 ºC. the results are opposite. The maximum deviation is 35%, which occurs at T5 = 90 ºC 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-

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

(2)

(3)

13

-a-Property constant

(4)

-O- Property variable

12 11 ,.-.._

".'.E.._ "-'

10

(5) 9

..s::

increasing water surface temperature, i.e. increasing mass flux. 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. 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. The Chilton-Colbum 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 ChiltonColbum analogy relative to that by the numerical simulation is less than 5% when the water surface temperature is below 60 ºC. The air property variability has a conspicuous and complex effect on heat transfer coefficient but an unobvious effect on mass transfer coefficient.

8

Conflict of interest statement 7

We declare no any interest conflict.

o

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Acknowledgments

Ts (°C) Fig. 15. Comparison of overall heat transfer coefficients calculated with and without consideration of property variability far u 5 m/ s air velocity.

References

4.5E-3 -

-o- Propcrty constant

-o-Property variable

4.0E-3 3.5E-3

2.0E-3 l.5E-3

o

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

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T1 (°C) Fig. 16. Comparison of overall mass transfer coefficients calculated with and without consideration of property variability far u 5 m/ s air velocity.

[1] T. Yoshida, T. Hyodo, Evaporation of water in air, humid air, and superheated steam, lnd. Eng. Chem. Process Des. Dev. 9 (2) (1970) 207-214. [2] M. Al-Shammiri, Evaporation rate as a function of water salinity, Desalination 150 (2) (2002) 189-203. [3] J.C. Min, j.F. Duan, Membrane-type total heat exchanger performance with heat and moisture transferring in different directions across membranes, Appl. Therm. Eng. 91 (2015) 1040-1047. [4] J.C. Min, M. Su, Performance analysis of a membrane-based energy recovery ventilator: effects of outdoor air state, Appl. Therm. Eng. 31 (2011) 4036-4043 . [5] j.L. Niu, L.Z. Zhang, Membrane based enthalpy exchanger: Material considerations and clarification of moisture resistance, J. Membr. Sci. 3 (189) (2001) 179-191. [6] L.C. Chow, j.N. Chung, Evaporation of water into a laminar stream of air and superheated steam, lnt. J. Heat Mass Transf. 26 (3) (1983) 373-380. [7] Z.X. Yuan, X.T. Yan, C.F. Ma, A study of coupled convective heat and mass transfer from thin water film to moist air flow, lnt. Commun. Heat Mass Transf. 31 (2) (2004) 291-301. [8] R. Tang, Y. Etzion, Comparative studies on the water evaporation rate from a wetted surface and that from a free water surface, Build. Environ. 39 (1) (2004) 77-86. [9] N . Boukadida, S.B. Nasrallah, Mass and heat transfer during water evaporation in laminar flow inside a rectangular channel-validity of heat and mass transfer analogy, lnt. J. Therm. Sci. 40 (1) (2001) 67-81.

L.D. Gu et al./ lntemational joumal of Heat and Mass Transfer 122 (2018) 1093-1102

[10] A. Stegou-Sagia, An experimental study and a computer simulation of heat and mass transfer far three-dimensional humidification processes, lnt. J. Numer. Methods Biomed . Eng. 12 (11) (1996) 719-729. [11] P. Talukdar, C.R. lskra, C.j. Simonson, Combined heat and mass transfer far laminar flow of moist air in a 30 rectangular duct: CFD simulation and validation with experimental data, lnt. J. Heat Mass Transf. Sl (11) (2008) 3091-3102 . [12] A.M. Raimundo, A.R. Gaspar, A.V.M. Oliveira, O.A. Quintela, Wind tunnel measurements and numerical simulations of water evaporation in forced convection airflow, lnt. J. Therm. Sci. 86 (2014) 28-40. [13] T. Poós, E. Varju, Dimensionless evaporation rate from free water surface at tubular artificial flow, Energy Procedia 112 (2017) 366-373. [14] C.R. lskra, C.j. Simonson, Convective mass transfer coefficient far a hydrodynamically developed airflow in a short rectangular duct, lnt. J. Heat Mass Transf. SO ( 11-12) (2007) 2376-2393. [lS] X. Wei, B. Duan, X. Zhang, Y. Zhao, M. Yu, Y. Zheng, Numerical simulation of heat and mass transfer in air-water direct contact using computational fluid dynamics, Procedia Eng. 20S (Supplement C) (2017) 2S37-2S44 . [16] M. Kumar, C.S.P. Ojha, j.S. Saini, lnvestigation of evaporative mass transfer with turbulent-forced convection air flow over roughness elements, J. Hydrol. Eng. 19 (11) (2014) 06014004. [17] j.P. Schwartze, S. Brocker, The evaporation of water into air of different humidities and the inversion temperature phenomenon, lnt. J. Heat Mass Transf. 43 (10) (2000) 1791-1800. [18] E.P. Volchkov, A.1. Leontiev, S.N. Makarova, Finding the inversion temperature far water evaporation into an air-steam mixture, lnt. J. Heat Mass Transf. SO (11) (2007) 2101-2106 . [19] j.H . jang, W.M. Yan, C.C. huang, Mixed convection heat transfer enhancement through film evaporation in inclined square ducts, lnt. J. Heat Mass Transf . 48 (11) (200S) 2117-212S. [20] C.C. Huang, W.M. Yan, j.H . jang, Laminar mixed convection heat and mass transfer in vertical rectangular ducts with film evaporation and condensation, lnt. J. Heat Mass Transf. 48 (9) (200S) 1772-1784. [21] W.M. Yan, D. Lin, Natural convection heat and mass transfer in vertical annuli with film evaporation and condensation, lnt. J. Heat Mass Transf. 44 (6) (2001) 1143-1lSl. [22] A. Nasr, A.S. Al-Ghamdi, Numerical study of evaporation of falling liquid film on one of two vertical plates covered with a thin porous !ayer by free convection, lnt. J. Therm. Sci. 112 (2017) 33S-344. [23] A. Nasr, Heat and mass transfer far liquid film condensation along a vertical channel covered with a thin porous !ayer, lnt. J. Therm. Sci. 124 (2018) 288299. [24] A. Terzi, W. Foudhil, S. Harmand, S.B. jabrallah, Liquid film evaporation inside an inclined channel: effect of the presence of a porous !ayer, lnt. J. Therm . Sci. 109 (2016) 136-147. [2S] A. Terzi, W. Foudhil, S. Harmand, S. Ben jabrallah, Experimental investigation on the evaporation of a wet porous !ayer inside a vertical channel with resolution of the heat equation by inverse method, Energy Convers. Manage. 126 (2016) 1S8-167. [26] A.S. Cherif, M.A. Kassim, B. Benhamou, S. Harmand, j.P. Corriou, S. Ben jabrallah, Experimental and numerical study of mixed convection heat and

1101

mass transfer in a vertical channel with film evaporation, lnt. J. Therm. Sci. SO (6) (2011) 942-9S3. [27] A. Charef, M.B. Feddaoui, M. Najim, H. Meftah, Liquid film condensation from water vapour flowing downward along a vertical tube, Desalination 409 (2017) 21-31. [28] Y. Wan, C. Ren, L. Xing, Y. Yang, Analysis of heat and mass transfer characteristics in vertical plate channels with falling film evaporation under uniform heat flux/uniform wall temperature boundary conditions, lnt. J. Heat Mass Transf. 108 (2017) 1279-1284. [29] Y. Wan, C. Ren, Y. Yang, L. Xing, Study on average Nusselt and Sherwood numbers in vertical plate channels with falling water film evaporation, lnt. J. Heat Mass Transf. 110 (2017) 783-788. [30] J.C. Min, Y.C. Tang, Theoretical analysis of water film evaporation characteristics on an adiabatic salid wall, lnt. J. Refrig. S3 (201S) SS-61. [31] Y.C. Tang, J.C. Min, Evaporation characteristics analysis of water film on a spherical salid particle, Appl. Therm. Eng. 102 (2016) S39-S47. [32] Y. Azizi, R. Bennacer, B. Benhamou, N. Galanis, M. El-Ganaoui, Buoyancy effects on upward and downward laminar mixed convection heat and mass transfer in a vertical channel, lnt. J. Numer. Meth. Heat Fluid Flow 17 (3) (2007) 3333S3. [33] P.T. Tsilingiris, Thermophysical and transport properties of humid air at temperature range between O and lOOºC, Energy Convers. Manage. 49 (S) (2008) 1098-1110. [34] C.R. Wilke, A viscosity equation far gas mixtures, J. Chem. Phys. 18 (4) ( 19SO) S17-S19. [3S] E.A. Masan, S.C. Saxena, Approximate formula far the thermal conductivity of gas mixtures, Phys. Fluids 1 (S) (19S8) 361-369. [36] R. Bird, W.E. Stewart, E.B. Lightfoot, Transport Phenomena, second ed., john Wiley & Sons, New York, 2002. [37] J.C. Slattery, R.B. Bird, Calculation of the diffusion coefficient of dilute gases and of the self-diffusion coefficient of dense gases, AIChE J. 4 (2) (19S8) 137142. [38] W.M. Rohsenow, j.P . Hartnett, Y.l. Cho, Handbook of Heat Transfer, 3rd ed., McGraw-Hill, 1998. [39] T.F. lrvine, P. Liley, Steam and Gas Tables with Computer Equations, Academic Press, San Diego, 1984. [40] Touloukian YS, Powell RW, Ho CY, Clemend PG. Thermophysical Properties of Matter, vol. 1. NY, 1970. [41] C.A. Meyer, R.B. McClintock, G.j. Silvestri, R.C. Spencer, ASME Steam Tables, New York, 1967. [42] W.M. Kays, M.E. Crawford, B. Weigand, Convective Heat and Mass Transfer, fourth ed., Tata McGraw-Hill Education, 2012. [43] F.P. lncropera, O.P. DeWitt, Fundamentals of Heat and Mass Transfer, fourth ed., john Wiley & Sons, New York, 1996. [44] W.K. Lewis, K.C. Chang, Applied the film theory to the interphase transfer of both species in a binary mixture at high mass-transfer rates, Trans. AIChE 21 (1928) 127-136. [4S] A.P. Colburn, T.B. Drew, The condensation mixed vapors, Trans. Am. lnst. Chem. 33 (1937) 197-212. [46] R. Higbie, The rate of absorption of a pure gas into a still liquid during short periods of exposure, Trans. AIChE 31 (193S) 36S-389.