Laminar Forced Convection Heat and Mass Transfer of Humid Air across a Vertical Plate with Condensation

FLUID FLOW AND TRANSPORT PHENOMENA Chinese Journal of Chemical Engineering, 19(6) 944ü954 (2011)

Laminar Forced Convection Heat and Mass Transfer of Humid Air across a Vertical Plate with Condensation* LI Cheng (हю) and LI Junming (हࢎੜ)** Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China Abstract Condensation of humid air along a vertical plate was numerically investigated, with the mathematical model built on the full boundary layer equations and the film-wise condensation assumption. The velocity, heat and mass transfer characteristics at the gas-liquid interface were numerical analyzed and the results indicated that it was not reasonable to neglect the condensate film from the point of its thickness only. The condensate film thickness, interface temperature drop and the interface tangential velocity affect the physical fields weakly. However, the subcooling and the interface normal velocity were important factors to be considered before the simplification was made. For higher wall temperature, the advective mass transfer contributed much to the total mass transfer. Therefore, the boundary conditions were the key to judge the rationality of neglecting the condensate film for numerical solutions. The numerical results were checked by comparing with experiments and correlations. Keywords condensation, binary mixture, convective heat and mass transfer

1

INTRODUCTION

When humid air encounters a cold surface, heat and mass transfer can occur simultaneously if the cold wall temperature is below the dew point [1]. Humid air condensation [24] is a common phenomenon in many industrial processes used to speed up the heat removal or dehumidification [5, 6], such as containment safety system for nuclear power plants, air-conditioning applications, cooling towers and heat recovery of combusting exhaust. Regardless of condensation being desired or to be avoided, all natural air used in industrial fields includes water vapor. Condensation of vapor mixed with a noncondensable gas has been extensively studied since the boundary layer approximations were derived by Sparrow et al [7]. There have been many numerical analyses that describe condensation processes in forms of velocity, temperature and concentration profiles. However, robust models are difficult to develop and predictions often do not match experiments. Then, the Reynolds analogy is often paid attention to due to its simplicity when the Lewis number (Le) is close to unity. Extensive numerical results have showed that the Reynolds analogy is valid if the vapor fraction is small enough, but the assumption of Le 1 is not reasonable for high vapor fractions. However, the numerical results of Volchkov et al. [8] indicated that even for low vapor mass fractions of less than 0.01, the convection heat transfer fractions in numerical simulation and experiment differ by threefold. They did not give a specific explanation from physical mechanism or phenomena observation, but assumed that the differences were due to the wavy condensate films. Brdlik et al. [9] showed that even for natural convection the Reynolds analogy did not ap-

pear to hold in an early experimental analysis. Opposite to their conclusions, the numerical study of the heat and mass transfer by Desrayand and Lauriat [10] indicated that the Sherwood number and Nusselt number were similar. Their formulation of the heat and mass transfer analogy held well for both low and high mass condensation rates. Thus, more studies are needed to understand the effects of velocity, temperature and concentrations to elucidate these discrepancies. Many studies neglected the effects of the liquid film on the humid air phase temperature and concentration distributions [2, 8, 10] for forced convection, natural convection or thin film evaporation in their simplified models. If the interface conditions are neglected, the numerical analysis will be much simpler and some commercial software [11] such as CFX can get solution easily. Then, the total heat rate can be obtained from the relationship of the heat and mass transportation analogy. But if the liquid film is not neglected, the phase interface can not be confirmed clearly and these software can not achieve the correct prediction. Yan [12] analyzed the effect of the liquid film thickness and concluded that it could be neglected when the liquid mass flow was small. Volchkov et al. [8] pointed out that the condensate film on the cold wall was thin and the thermal resistance was small, so the interface temperature was almost the same as the cold wall temperature, and the condensation could be modeled by setting a velocity component normal to the interface. Ren and Gu [2] on the other hand, simply modeled the vapor condensation as a chemical reaction and the effects of condensate film on the velocity, temperature and concentration boundary layers were totally neglected. Thus, more studies are needed to evaluate the effects of the condensate film on the heat and mass transfer. The present work starts from the full laminar

Received 2010-09-17, accepted 2011-06-16. * Supported by the National Basic Research Program of China (2011CB706904) and Beijing Natural Science Foundation (3071001). ** To whom correspondence should be addressed. E-mail: [email protected]

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boundary layer with the actual phase change interface conditions, analyzes the condensate film, interface velocity and characteristics of heat and mass transfer to better understand the condensation of humid air flowing over a vertical plate, and provides a theoretical and quantitative basis for evaluating the simplified analyses previously referred. The thermal physical properties changing with temperature and concentration are included in the momentum, energy and concentration equations. Experimental observation is also made to analyze the discrepancy between the experiments and the numerical solutions of the heat and mass transfer analogy. 2

MODELS AND ANALYSES

A 2-dimensional (2-D) model was adopted, based on the characteristics of the film-wise condensation, the parallel flow and gravity direction. The schematic of the 2-D model with the coordinate system is shown in Fig. 1. The computational domain, of 0.1 m in x direction and around 0.05 m in y direction, is split into the liquid film region and the humid air region with each region analyzed separately. The domain size is chosen so wide that the external flow of gas mixture is not influenced by the domain width. The interface boundary conditions between the two regions are used to relate the two solutions during iterative calculation.

Figure 1 Condensation of humid air flowing along a vertical cold wall

2.1

wul wvl  wx wy

(1)

0

Momentum conservation in the liquid film: w  ul ul wx

 Ul

w  ul vl wy

w § wul P wy ¨© wy

wp · ¸  g ˜ Ul  wx (2) ¹

The term (wp / wx) in Eq. (2) is approximately replaced by Ui g , which is calculated from the momentum conservation solution for the humid air phase at the interface, with the total pressure in the whole

w  c p ,l Tl

w  c p ,l Tl

w § wTl · (3) wx wy wy ¨© wy ¸¹ The thermal physical properties in Eqs. (1)(3) are assumed to be constant, calculated using the reference temperature Tref,l as recommended by Poots and Miles [13]: 1 Tref ,l Tw  Tm,i  Tw (4) 3 Stephan [14] has shown that the temperature difference could reach 19.78 °C, so the transport properties can change much. Thus, the correct Tref,l leads to better results, especially for high condensation rates.

Ul ul

2.2

 Ul vl

kl

Humid air boundary layer model

Mass conservation in the gas-vapor region: wU m um wU m vm 0  wx wy

(5)

Momentum conservation in the humid air region: w w  Um um um   Um um vm wx wy wu · w § Pm m ¸  g  U m  U m,f (6) ¨ wy © wy ¹ where term g ( U  Uf ) represents the effects of buoyancy, due to the density difference caused by temperature and concentration gradients between the bulk flow and the humid air boundary layer. Liao et al. [15] concluded that buoyancy effects must be included, as they numerically showed that the bulk flow blew away some non-condensable gas previously accumulated near the interface, which enhanced the condensation heat transfer with the heat transfer coefficient increased by up to 45%. Thus, buoyancy effects must be included. Energy conservation in the humid air region:

U m um

Liquid film model Mass conservation in the liquid film:

Ul

gas region assumed to be constant . Energy conservation in the liquid film:

w w c p ,mTm  U m vm  c p ,mTm  wx wy

wW w § wTm · w ª º U m D  c p ,g  c p ,v m Tm »  km wy ¨© wy ¸¹ wy «¬ wy ¼ (7) where the last term indicates the effect of mass diffusion on heat transfer. The effects of the thermal diffusion on the energy transport are negligible. Species conservation in the humid air region: wWv · w § U m Dm (8) ¨ wy © wy ¸¹ Thermal physical properties of gas-vapor mixture are a function of the concentration as well as temperature,

U m um

wWv wWv  U m vm wx wy

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so they are quite different from those for a pure gas. Thus, the thermal physical properties of the gas-vapor mixture are all taken to be variable. 2.3

Boundary conditions

At the wall surface y ul or x

vl

0:

0 and Tl

Tw

(9)

At the free humid air stream or the inlet: y o f 0

um

uf , vm

0 and Tm

Tf

(10)

They concluded that the mass transfer error, if the advection is neglected, would result in large errors for high condensation mass rate. Hammou et al. [18] found that vm,i was the key factor to determine whether it was condensation or evaporation. As noted earlier, Ren and Gu [2] entirely neglected the effect of the condensate film on the heat and mass transfer. Thus, different from the considerations in former studies, the interfacial velocity component, vm,i, here is considered as shown in Eq. (18). The condensation rate should abide by both the Fick’s law and the advective mass transfer at the interface, in addition to the mass balance:

At the phase interface, no slip is assumed with the shear stress for a Newtonian fluid: ul ,i

Pl

wul ,i wy

(11)

um,i

Pm

2.4

wum,i

(12)

wy

Equations. (11) and (12) indicate that the tangential velocity component up is a function of the velocity gradients in both the humid air and the liquid. up makes the condensation air-vapor field boundary layer different from dry conditions. Therefore, neglecting condensate film [2, 8, 10] means that up is not included. Temperature jumps at the interface [16], as the partial vapor pressure at the interface is higher than its saturated pressure at temperature, Tm,i. Therefore, the interfacial thermal resistance exists from Eq. (13). N 1 m (13) ª pv  psat (Tl ,i ) ¼º 1  0.5N 2SRTl ,i ¬ where, the saturated water vapor pressure, psat, is related to the interfacial temperature [17]: 5965.6 · § psat (T ) pt ˜ exp ¨ 18.79  0.0075T  ¸ (14) © T ¹ At the humid air-liquid interface, both the thermal balance and the mass balance require: (15)

dG dG  Ul vl U m um  U m vm (16) dx dx where hf,g for water is the function of temperature. It is quadratic fitted as m

Ul ul

hf ,g

2.7554 u 106  2.464Tl 2,i

(17)

The velocity component normal to the interface, vm,i due to the advective condensation from air-vapor mixture to liquid is deduced by Comini and Savino [1] as vm,i



D wW 1  Wv,i wn

(19)

(18) i

Thermal properties for the humid air

The specific heat at constant pressure is a function of components and their mass fractions:

c p ,m

Wv c p ,v  Wa c p ,a

(20)

where Wv can be obtained from the water vapor partial pressure and Wa equals ( 1  Wv ). The total pressure, pt, here is 1.01325×105 Pa. The density of the gas-vapor based on temperature and concentration is given by

Um

1 c· § pt ¨ 0.003484  0.00134 ¸ © T T¹

(21)

The thermal conductivity and viscosity of the humid air are also functions of mass fractions:

2.5

km

(1  W )ka  WK v

(22)

Pm

(1  W ) Pa  W P v

(23)

Solution method

The analyses were made, based on the geometry in Fig. 1. The coordinate transform in terms of

x

L F and y G ( x)K

(24)

was used to manually transform all the conservation equations and boundary conditions onto the new coordinate system. In the original x-y coordinate system, the transverse dimension of condensate film was a variable in the x direction. While, the dimensionless condensate film thickness in the new coordinate system Ȥ-Ș was always the same along the air flow direction and it was convenient to capture the dimensionless interface position accurately. These conversation equations in the original coordinate and in the new boundary-fitted coordinate systems should satisfy the following conversion relations, simplified by using Eq. (24) [19]:

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w wx

wF w wK w ˜  ˜ wx wF wx wK w wy

1 w K dG w (25) ˜  ˜ ˜ L wF G ˜ L dF wK

wK w wF w  wy wK wy wF

1

w

˜

(26)

G wK

To realize the meshes within the humid air domain as well as in the condensate film domain, the width of the whole numerical domain H, was set to be the function of condensate film thickness or Ȥ

G

ª¬0.05  G F 1 º¼ GF

H

G

(27)

1

where H means the meshes along y direction. į is the local condensate film thickness. The control equations in the new coordinate system were then obtained by putting Eqs. (25) and (26) into the original control equations, Eqs. (1)(3), (5)(8). For example, Eqs. (1)(3) are transformed to 1 wu K dG wu 1 wv (28) 0 ˜  ˜ ˜  ˜ L wF G ˜ L dF wK G wK

Ul w  ul ul UlK dG w  ul ul ˜  ˜ ˜  L wF wK G ˜ L dF Ul w  ul vl Pl w 2 ul ˜  2 ˜ 2  g  U l  Ui 0 wK G G wK

from the grids next to the leading edge in Ȥ direction. Since condensation only occurs at the interface, it was obviously important to capture the phase interface, where the interface physical conditions must be enforced with good accuracy. The finite difference method was used here, considering the clear phase interface existing between the two numerical regions. The Tri-diagonal Matrix Algorithm method for matrix solutions and the unsteady state solution method for the control equations were used for robust convergence. The Alternative Direction Implicit method was applied to solve the matrix equations simultaneously. The computer code was developed in our lab and compiled in Fortran. The solution was first to assume an initial film thickness and then to solve the momentum equations and continuity equations, then updated interfacial flow conditions. An iterative solution was needed to correct the shear stress and no-slip conditions at the interface. Then, the energy equations and the species equation were solved. The last step was to update all the interfacial conditions, the thermal physical properties and to calculate the local condensate film thickness į. The condensation mass rate m can be obtained by Eqs. (16) and (19). Meanwhile, į could be calculated, based on the local mass conservation at interface in the original x-y coordinate:

(29)

m( x )

0

0 and Tl

vl

At the free humid air stream K inlet F 0 : um

uf vm

0 and Tm

Tf

0

Ul ul dy



'x

(30)

(31)

Tw

G ( x)

j

The structured meshes were obtained in the new coordinate system Ȥ-Ș, with the corresponding boundary conditions. At the wall surface K 0 : ul

³

¦ 'G j ª¬ Ul ul x 'x   Ul ul x º¼

Ul ul w  c p ,l Tl Ul ulK dG w  c p ,l Tl ˜  ˜ ˜  L wF wK G ˜ L dF Ul vl w  c p ,l T kl w 2Tl ˜  2˜ 2 wK G G wK

d dx

(36) x

The iterative procedure continued for all field variables until the maximum relative error of the interface temperature reached 109. Mesh independence was carried out to make sure that the simulation results were independent of numerical grids, see Fig. 2. For both the humid air and the condensate regions, the suitable meshes were the 100×40 structured grids respectively for Ȥ and Ș directions for all subsequent simulations.

H or at the (32)

At the phase interface K 1 : ul ,i ul

wul ,i wK

um,i

Pm

wum,i wK

(33) (34) (35)

Therefore, the structured grids were achieved within the two numerical domains in new coordinate system. At the leading edge of the vertical wall F 0 , water vapor condensation occurred strongly but condensate film was set to be zero and numerical solution began

Figure 2 Effect of meshes on the calculated local condensation rate (Tw 280 K, TinTw 20 K, uin 2.0 m·s1, x 0.05 m)

3

RESULTS AND DISCUSSION

When the iterative procedure described in Section

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2.5 is converged, the physical fields of velocity, temperature and concentration can be obtained. Here, the influencing factors of the condensate film will be discussed based on the numerical solutions. 3.1

Physical fields within condensate

Characteristics of the local condensate film thickness, the temperature and velocity component, u, distributions within the condensate film along x direction with different Win are shown in Figs. 3 and 4.

Figure 3 Local condensate thickness for different Win (uin 2.0 m·s1, Tin 300 K, Tw 280 K) Win: 0.010; 0.013; 0.016; 0.020

It is seen from Fig. 4 that differences of temperature T and velocity component u within the condensate film are very small. Illustrations in Fig. 4 (a) indicate that except the inlet domain, T increases almost linearly in y direction, which means that the heat conduction is the dominant heat transfer model within condensate film. In essence, although u in Fig. 4 (b) displays a non-linear increase in y direction, its absolute value is very small and these data are all within the same magnitude. Thus, it is reasonable to recognize the constant u gradient in y direction from the characteristics of their distributions. The humid air field with condensation shown above was thought to be similar to that of dry air, but they are essentially different from each other as the

(a) Temperature distribution Figure 4

condensate film, with the interface velocity, the subcooling and the interface water vapor mass fraction makes the interface boundary condition complicated and dramatically different from the dry air heat transfer. These differences will be discussed below in order to comprehensively conclude reasonableness of neglecting the condensate film. 3.2 Local condensation mass rate and local condensate film thickness

The humid air condensation is the main reason that the physical fields in Section 3.1 are different from dry ones. With condensation occurring, the forced convection of humid air makes water vapor flow directly to and cross the interface. So the diffusion and the advective mass transfer both contribute to condensation. Here, the total condensation mass rate, m, for different wall temperature is shown in Fig. 5. It is obvious that the local m shows larger for high uin and high Win. The effect of uin on m becomes significant for higher Win, and vice versa. When condensation continually occurs, the condensate film will form on the wall and flows down, see Fig. 3. Therefore, the local condensate film thickness į depends primarily on the condensation rate by considering Figs. 5 and 6 together. Intuitively, the higher uin could drag the film thinner for no slip interface conditions. However, the effect of uin on the interface tangential velocity component up, is weak during present study, as the viscosity ratio of the liquid and air-vapor mixture is around 60100 and subsequently the direct contribution of uin to up can be neglected. But it is obvious from Fig. 5 that uin shows positive influence on m, so the higher uin devotes much to thicker į due to m increase (see Fig. 6). It also can be seen that į shows similar trend with the condensation mass rate m. Threlkeld [20] stated the condensate film thickness of around 0.1 mm in the finned-tube heat exchanger design for wet conditions,

(b) Velocity component (u/m·s1) distribution

Characteristics of physical fields within the condensate film (uin 2.0 m·s1, Tin 300 K, Tw

280 K, Win 0.010)

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(a) Tw 280 K Win:Ƶ0.010;ƽ0.013;Ʒ0.016;ͩ0.020

(b) Tw 320 K Win:Ƶ0.07;ƽ0.08;Ʒ0.09;ͩ0.10; Ż 0.11; Ź 0.12

Figure 5 Effects of uin and Win on local condensation rate (TinTw

(a) Tw 280 K Win:Ƶ0.010;ƽ0.013;Ʒ0.016;ͩ0.020

20 K, x 0.05 m)

(b) Tw 320 K Win:Ƶ0.07;ƽ0.08;Ʒ0.09;ͩ0.10; Ż 0.11; Ź 0.12

Figure 6 Effects of uin and Win on condensate film thickness (TinTw

(a) Tw 280 K Win:Ƶ0.010;ƽ0.013;Ʒ0.016;ͩ0.020

(b) Tw 320 K Win:Ƶ0.07;ƽ0.08;Ʒ0.09;ͩ0.10; Ż 0.11; Ź 0.12

Figure 7 Effects of uin and Win on interfacial temperature drop (TinTw

and his proposal is approximate to the value in Fig. 6 (b) for Win 0.11 and uin 2.5 m·s1. 3.3

Temperature drop at phase interface

The interfacial thermal resistance appears when there exists water vapor passing through the phase interface into condensate, so the temperature drop exists for humid air condensation, similar to the case of

20 K, x 0.05 m)

20 K, x 0.05 m)

pure vapor condensation. The local temperature drop across the phase interface increases with both increasing uin and Win, see Fig. 7. For two typical cold wall temperatures the interface temperature drops are both less than 0.0003 K, which are smaller than the precision of equipment for temperature measurement and the temperature drop is in contrast to that of 0.10.5 K for pure vapor condensation concluded by Rose [21]. Thus, the interfacial thermal resistance due to phase change is quite small

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as justified in [2124], who ignored the effects of the interfacial thermal resistance. 3.4

Local subcooling of the condensate film

The condensate film has been neglected in previous numerical studies [2, 8, 10], presuming that the condensate film was thin and the thermal resistance was small. Here, the local subcooling of the condensate film for two cold wall temperatures is shown in Fig. 8.

3.5

Local interface tangential velocity component

As discussed in Section 3.2, uin shows little influence on up. Assume that the dynamic viscosity ratio of the water and humid air at phase interface is 100 and Eq. (12) can be simplified to wum,i wy

(37)

The velocity component distribution in Fig. 4 (b) indicates that the u within the condensate film varies slightly and within the same magnitude in y direction, so the partial differential term on the right side of Eq. (37) can be expressed linearly for local į. For given step size 'ym ( 'ym | 0.001 for 40 meshes) next to the phase interface, up can be expressed as: up

ui

um,i |

um,k 100 'y  1 G ( x) m

(38)

where um,k is the longitudinal velocity component of humid air next to the phase interface um,i with the numerical step size of 'ym . Figure 6 shows that even for higher m, the local condensate film thickness (x=0.05) is no more than 1.2×104 m, then Eq. (38) can be rewritten as

(a) Tw 280 K Win:Ƶ0.010;ƽ0.013;Ʒ0.016;ͩ0.020

up |

(b) Tw 320 K Win:Ƶ0.07;ƽ0.08;Ʒ0.09;ͩ0.10; Ż 0.11; Ź 0.12 Figure 8 Local subcooling of condensate film (TinTw K, x 0.05 m)

wul ,i Pl wul ,i | 100 Pm wy wy

20

The local subcooling shows linear increase with uin. The subcooling for low Win is less than 0.06 K, as shown in Fig. 8 (a). However, for higher Win the subcooling can be larger than 0.56 K, around 2.8% of the total temperature difference between the inlet air and the cold wall. Such a large subcooling will further influence the condensate temperature and mass transfer rate, because the interface should obey Eq. (14). Thus, the temperature drop across the liquid layer is significant for high Win and high uin, both of which influence nearly linearly the subcooling. Hence, the effects of the subcooling on heat transfer should be considered for relatively high Win and uin, which are common conditions in containment safety analysis, cooling towers and the heat recovery of the combusting exhausts.

um,k | 1.2 u 103 um,k (39) 100 3 1.0 u 10  1 1.2 u 104

Equation (39) indicates that the up will not increase dramatically, by directly increasing uin. Although it is concluded that uin influences up by increasing condensation rate, m, as shown in Fig. 5 slopes of thickness become smaller by increasing uin, which means that the magnitude of į is around 104 m. Hence, up will change small for humid air condensation on the vertical wall. The conclusion is proved by numerical results shown in Fig. 9. 3.6

Local interface normal velocity component

The phase interface is permeable to water vapor during condensation, but is impermeable to air. Thus, there is velocity component vn normal to the phase interface due to the advection and sucking effects of water vapor condensation, as shown in Fig. 10. It can be seen that low condensation rate makes vn in Fig. 10 (a) very low. But when the condensation rate is large, the velocity can hit 0.001 m·s1 and its effect does not lie in the momentum equation, but the latent heat from condensation. For example, when uin 3.0 m·s1 and Win 0.12, the latent heat flux, qad, is about 180 W·m2 as calculated from qad

U m,iWi vn hf ,g

(40)

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(a) Tw 280 K Win:Ƶ0.010;ƽ0.013;Ʒ0.016;ͩ0.020 Figure 9

Interfacial velocity component parallel to vertical wall (TinTw 20 K, x

(a) Tw 280 K Win:Ƶ0.010;ƽ0.013;Ʒ0.016;ͩ0.020 Figure 10

(b) Tw 320 K Win:Ƶ0.07;ƽ0.08;Ʒ0.09;ͩ0.10; Ż 0.11; Ź 0.12 0.05 m)

(b) Tw 320 K Win:Ƶ0.07;ƽ0.08;Ʒ0.09;ͩ0.10; Ż 0.11; Ź 0.12

Interface velocity component of gas mixture normal to vertical wall (TinTw 20 K, x 0.05 m)

Therefore, the advective mass rate is important if vn is large enough. The advective mass fraction can be expressed by Eqs. (18) and (19): mod U mWi vn (41) Wi f Tm,i U D wW mt 1  Wv,i wn i

Equation (41) indicates that the advective mass fraction is the single function of phase interface temperature Tm,i, independent of Win and uin. So according to the analysis on subcooling, if the subcooling is large enough, the interface advective mass transfer will show much difference from the cold wall. It can be seen from Fig. 11 that the advective mass transfer fraction will grow faster by increasing the phase interface temperature. Therefore, high wall temperature and large condensation rate make the assumption of neglecting the condensate film unreasonable. 4 COMPARISONS AND EXPERIMENTAL OBSERVATION

The present study is compared with previous work for the ratio of the condensation heat transfer to the total heat transfer, as shown in Fig. 12. The present results agree well with data from

Figure 11 Effect of wall temperature on advective mass transfer fraction (Tin 345 K, Win 0.10, uin 2.0 m·s1)

Volchkov et al. [8], which were also based on the film-wise condensation, but there exists a little difference from model of Comini and Savino [1], which is a theoretical model based on Tw:

qc qt

2 ª º hf ,h Tw W  Ww 3 «1  Le ˜ f » 1  Ww c p Tin  Tw »¼ «¬

1

(42)

However, these numerical and analytical results do not agree well with the experimental data from Takarada et al. [25] for low value of (W’Ww). Previous studies by Volchkov et al. [8] also found such

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Figure 12 Comparison of the condensation heat flux among different work (Tin 313.15 K, Tw 294.15 K) üƻü  present numerical results;ͩexperimental data [25]; ƽ numerical results [8]; Comini and Savino model [1]

discrepancy and they attributed it to the waviness of film flow. However, the difference between Win and the saturated humid air Ww at wall temperature is less than 0.022, and it is impossible to result in wavy flow or turbulent flow for such condensation. Here, Ww is often assumed to be the saturated humid air at Tw, as the phase interface temperature is an important parameter for condensation, but is hard to obtain form experimental observation. Therefore, observation was carried out on the vertical cold wall, with Win 0.02045 and Ww 0.00575. The condensate on the wall (in Fig. 13) shows irregular shape, which is very complex [26] and different from the numerical film assumption.

the numerical or theoretical results from the experimental conclusion. As there is lack of experimental research on humid air, there has been no experiment with vertical wall length of 0.1 m. The comparison between the present numerical solution and early experimental data by Lebedev et al. [27] was carried out on the total convective heat transfer coefficient ht. According to the experiment [27], the length of the cold wall is 0.6 m. Here, the numerical length and meshes are both increased to match the same condition with experiment. ht was defined as [27]: q (43) ht Tin  Tw The results independent of meshes was achieved with the grids of (600×100 for the humid air numerical region), as described in Section 3.2. Unfortunately, experimental data by Lebedev et al. [27] uin were given by a small range, but not specific values. So the experimental average uin is adopted here for present numerical solutions. The comparison is based on the relative humidity and these results fit well, as shown in Fig. 14.

Figure 14 Comparison of ht between numerical and experimental results (L 0.6 m, H 0.1 m, Tw 278.15 K) uin (present, Tin 331.65 K)/m·s1:Ƶ3.75;ƽ2.15;Ʒ0.70 uin (exp. [27], Tin 329.15334.15 K)/m·s1:ƶ3.63.9; ƻ 2.02.3;Ƹ0.70

Figure 13 Observation of the condensate film on the vertical cold wall (Surface material: hydrophilic aluminum foil surface with static contact angle 34.2°)

There are various partial rivulets, film and droplets that adhered to the cold wall. The irregular condensate shape augments the effective surface for heat and mass transfer, so the heat and mass transfer are both enhanced. On the other hand, according to the Yang-Laplace Equation, curvature of the irregular small shape restrains mass transfer due to water vapor partial pressure drop significant for large curvature. Therefore, the mass transfer driving force ǻW becomes small and the enhancement of irregular phase interface on mass transfer is not as significant as on heat transfer. This may explain the difference between

5

CONCLUSIONS

Condensation of humid air flowing over a vertical plate was analyzed numerically. The local condensation rates, condensate film thickness, characteristic temperature drop of the phase interface, subcooling, and tangential and normal velocities of humid air were numerically studied for various cold wall temperatures and inlet conditions at x 0.05 m. The results showed that: The condensation mass rate, m, almost increases linearly with uin and higher Win contributes much to the effect of uin on m. The influence of uin and Win on the local condensate film thickness, į, is similar with their influence on m, but į is less than 104 m, with the condensate temperature linearly increase along y direction, except the inlet domain. The phase interface temperature drop is very small, no matter m is high or

Chin. J. Chem. Eng., Vol. 19, No. 6, December 2011

low. However, the subcooling of condensate film changes significantly with uin and Win, with the local subcooling larger than 0.56 °C for higher m. The interface tangential velocity component of humid air up, changes little even for high uin or high m, as it is mainly affected by the liquid side. But the interface normal velocity component of humid air, vn, can dramatically affect the heat rate, as the latent heat, hf,g, is relative enormous. Higher m and higher Tw devote much to higher vn, and relative higher advective mass transfer. Therefore, the interface temperature drop, the up and the condensate film thickness show little contribution to the physical fields of air flow, even for relative higher m. The assumption of neglecting the condensate film thickness should be based on the subcooling, interface normal velocity component, vn, for higher m and should be based on vn for high Tw. NOMENCLATURE c cp D d H hf,g h g K L m p pt T u v W ǻW į Ș ț ȝ ȡ Ȥ

water vapor molar fraction, kJ·kg1·K1 specific heat capacity, kJ·kg1·K1 diffusion coefficient, D 2.16×105 (T/273.18)1.80 m2·s1 humidity ratio, g·kg1 height of the numerical region, m latent heat, J·kg1 total heat transfer coefficient, W·m2·K1 gravitational acceleration, m2·s1 thermal conductivity, W·m1·K1 length of the vertical wall, m condensation mass rate, kg·m2·s1 pressure, Pa total pressure, Pa temperature, K velocity component in x direction, m·s1 velocity component in y direction, m·s1 mass fraction mass fraction difference (Wv,inWv,i) condensate film thickness, m coordinate transformation from y condensation coefficient viscosity, kg·m1·s1 density, kg·m3 coordinate transformation from x

Subscripts a ad c i in j k l m n p ref sat t v

dry air advective condensation interface humid air inlet grid node of condensate film in y direction grid node of humid air in y direction next to the phase interface liquid gas and water vapor mixture normal to the interface parallel to the interface reference parameter water vapor saturated state total water vapor

w f

953

cold wall bulk flow

REFERENCES 1

2

3

4

5

6

7

8

9

10

11

12 13

14

15

16 17

Comini, G., Savino, S., “Latent and sensible heat transfer in air-cooling applications”, Int. J. Num. Methods Heat Fluid Flow, 17 (6), 608617 (2007) Ren, N., Gu, B., “Experimental study and numerical simulation of heat and mass transfer on plain fin in wet conditions”, J. Chem. Ind. Eng. (China), 58 (7), 16261631 (2007). (in Chinese) Warmuzinski, K., Bartelmus, G., Berezowski, M., “Modeling of countercurrent partial condenser for light hydrocarbon streams containing hydrogen”, Chem. Eng. Process., 37 (5), 389403 (1998). Kang, H.C., Kim, M.H., “Characteristics of film condensation of supersaturated steam-air mixture on a flat plate”, Int. J. Multiphase Flow, 25 (8), 16011618 (1999). Cheng, H.G., Wang, S.C., “Modelling and experimental investigation of humidification- dehumidification desalination using a carbonfilled-plastics shell-tube column”, Chin. J. Chem. Eng., 15 (4) 478475 (2007). Quan, Z.H., Chen, Y.C., Ma, C.F., “Experimental study of fouling on heat transfer surface during forced convective heat transfer” Chin. J. Chem. Eng., 16 (4), 535540 (2008). Koh, J.C.Y., Sparrow, E.M., Hartnett, J.P., “The two phase boundary layer in laminar film condensation”, Int. J. Heat Mass Transfer, 2 (1-2), 6982 (1961). Volchkov, E.P., Terekhov, V., Terekhov, V., “A numerical study of boundary-layer heat and mass transfer in a forced flow of humid air with surface steam condensation”, Int. J. Heat Mass Transfer, 47 (6-7), 14731481 (2004). Brdlik, P.M., Kozhinov, I.A., Petrov, N.G., “An experimental heatand mass-transfer study of humid-air flows with steam condensation on a vertical surface under national-convection conditions”, Journal of Engineering Physics and Thermophysics (Russia), 8 (2), 243246 (1965). Desrayand, G., Lauriat, G., “Heat and mass transfer analogy for condensation of humid air in a vertical channel”, Heat Mass Transfer, 37, 6776 (2001). Martín-Valdepeñas, J.M., Jiménez, M.A., Martín-Fuertes, F., Fernández, J.A., “Comparison of film condensation models in presence of non-condensable gases implemented in a CFD Code”, Heat Mass Transfer, 41 (11), 961976 (2004). Yan, W.M., “Mixed convection heat transfer in a vertical channel with film evaporation”, Can. J. Chem. Eng., 71, 5462 (1993). Poots, G., Miles, R.G., “Effects of variable properties on laminar film condensation of saturated steam on a vertical flat plate”, Int. J. Heat Mass Transfer, 10 (12), 16771692 (1967). Stephan, K., “Interface temperature and heat transfer in forced convection laminar film condensation of binary mixtures”, Int. J. Heat Mass Transfer, 49 (3-4), 805809 (2006). Liao, Y., Vierow, K., Dehbi, A., Guentay, S., “Transition from natural to mixed convection for steam-gas flow condensing along a vertical plate”, Int. J. Heat Mass Transfer, 52 (1-2), 366375 (2009). Ghiaasiaan, S.M., Two-phase Flow, Boiling and Condensation, Cambridge University Press, New York, 2159 (2008). Srzic, V., Soliman, H.M., Orimiston, S.J., “Analysis of laminar mixed-convection condensation on isothermal plates using the full boundary-layer equations: mixtures of a vapor and a lighter gas”, Int. J. Heat Mass Transfer, 42 (4), 685695 (1999).

954

18

19 20 21 22

23

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Hammou, Z.A., Benhamou, B., Galanis, N., Orfi, J., “Laminar mixed of humid air in a vertical channel with evaporation or condensation at the wall”, Int. J. Therm. Sci., 43 (6), 531539 (2004). Tao, W.Q., Numerical Heat Transfer, 2nd edition, Xi’an Jiao Tong University Press, Xi’an, 442458 (2001). (in Chinese) Threlkeld, J.L., Thermal Environmental Engineering, 2nd edition, Prentice-Hall, Englewood Cliffs, New Jersey, 263267 (1970). Rose, J.W., “Condensation heat transfer fundamentals”, Chem. Eng. Res. Des., 76 (2), 143152 (1998). Dharma, R.V., Murali, K.V., Sharma, K.V., Mohana-Rao, P.J.V., “Convection condensation of vapor in the presence of a non-condensable gas of high concentration in laminar flow in a vertical pipe”, Int. J. Heat Mass Transfer, 51 (25-26), 60906101 (2008). Siow, E.C., Ormiston, S.J., Soliman, H.M., “Two-phase modeling of laminar film condensation from vapor-gas mixtures in declining par-

24

25

26

27

allel-plate channels”, Int. J. Therm. Sci., 46 (5), 458466 (2007). Rosa, J.C., Escrivá, A., Herranz, L.E., Cicero, T., Muñoz-Cobo, J.L., “Review on condensation on the containment structures”, Progr. Nucl. Energy, 51 (1), 3266 (2009). Takarada, M., Ikeda, S., Izumi, M., “Forced convection, and heat and mass transfer from humid air under condensation conditions”, Proceedings of Heat Transfer, Fluid Mech. Thermodyn., 2, 11031116 (1997). Wang, X.D., Peng, X.F., Duan, Y.Y., Wang, B.X., “Dynamics of spreading of liquid on solid surface”, Chin. J. Chem. Eng., 15 (5), 730737 (2007). Lebedev, P.D., Baklastov, A.M., Sergazin, Z.F., “Aerodynamics, heat and mass transfer in vapor condensation from humid air on a flat plate in a longitudinal flow in asymmetrically cooed slot”, Int. J. Heat Mass Transfer, 12, 833842 (1969).