Abstract: This research consists of a numerical investigation of coupled heat and mass transfers by natural convection during water evaporation in a vertical channel. The two channel walls were symmetrically heated by a uniform flux density. One wall is partially wetted by an extremely thin water film and the other is dry. The partially humid plate is divided into 2N with equal lengths being alternatively wet and dry zones. The results are reported in terms of local Sherwood number, the inlet velocity and evaporative rate for different wet zone position and for different wet number zones. However, the mass transfer is extremely influenced by the number of the wetted zones and their positions. The evaporative rate is more intense when the wetted zone is situated at the channel exit. Finally, it is observed that the evaporation is intensified by increasing the number of wetted zones.

INTRODUCTION

The free convection duct flows with coupled heat and mass transfer in a flowing gas mixture can be significantly affected by the combined buoyancy forces due to the existence of temperature and concentration variations. The understanding of the modification of flow structure in a duct is important in various thermal systems, such as the desalination, solar energy collectors, design of heat exchangers, geothermal energy systems, cooling of the nuclear reactor. Nawayseh et al. (1999) studied the performance of desalination systems and insisted on the accurate evaluation of the heat and mass transfer coefficients before the optimization study. The effects of combined buoyancy forces of heat and mass diffusion on laminar free convection heat transfer in vertical and horizontal rectangular ducts were studied extensively (Aung and Worku, 1986a; Dalbert et al., 1981; Prakash and Liu, 1985). The effects of mass diffusion on natural thermal convection flow have been widely investigated for vertical, horizontal and recently inclized flat plates (Gebhart and Pera, 1971; Yan and Lin, 1990; Mammou et al., 1992). Numerically studied the evaporation of water vapor along an inclined heated plate. The influences of the inclined angle, the wall heating flux, the inlet film thickness and the free stream velocity on the momentum, heat and mass transfer in the system are clarified. Mammou et al. (1992) presented a numerical study of the laminar heat and mass transfer from an inclined flat plate with a dry zone inserted between two wet zones. They concluded that the inclination angle has a small influence on the local Nusselt and Sherwood numbers. Tsay et al. (1990) and Chang et al. (1986) have treated the evaporation of liquids by free convection driven by thermal and mass buoyancy forces into air. Yan and Lin (1990) presented a numerical analysis to investigate the effects of the latent heat transfer, in association with the evaporation into air of a finite liquid film on the channel wall, on the free convective heat and mass transfer.

Debbissi et al. (2001, 2003) analyzed the evaporation of water by free into humid air and superheated steam. In their research, particular attention is paid to study the effect of the ambient conditions on evaporation rate of water and the inversion temperature of the phenomenon in the condition of free and mixed convection. The effect of ambient conditions, channel width and walls radiation are analysed in this study. Recently, Orfi et al. (2004) have studied the evaporation of water by natural convection, of a thin liquid film down on the internal face of one plate of a vertical heated channel.

To our knowledge, the heat and mass transfer by free convection along a partially wetted plate, which is composed, respectively by an alternation of humid and dry zones is not studied. The main objective of this research is to study the evaporation of water into natural and mixed convection flow of humid air in a partially wetted channel. A particular attention will be addressed to the effect of the number N of wetted zones on the evaporation rate.

MATERIALS AND METHODS

This study presents, a numerical analysis of heat and mass transfer during water evaporation by free convection in a finite vertical channel. The studied channel is made up of two parallel plates symmetrically heated by uniform density fluxes. The left plate is made of a 2N equals zones alternately wet and dry. The second plate (y = d) is dry. This geometric configuration can represent a drying system. The role of the dry and heated zone is to generate an upward natural flow in the channel, through the thermal buoyancy forces. The heating imposed flux can be assured by the solar radiation.

At the channel entrances, the moist air flows upwards with the ambient conditions of temperature T₀, Pressure p₀ and mass concentration c₀. The geometry of the problem under consideration (for N = 1) is shown in Fig. 1a, b. The left plate is divided into two regions with equal lengths (H/2) being alternatively wet and dry zones. Two configurations were considered in this study, in the first case (wet exit), the wetted zone is at the channel exit and the part of the plate is dry. In the second case (dry exit) the configuration is reversed.

In order to set the partial differential system equations describing momentum, heat and mass transfers and some simplifying assumptions are taken into consideration. The boundary layer approximations are generally used. The moist air in the channel is considered as an ideal gas with variable thermo-physical properties. The viscous dissipation and the pressure work are negligible. For wet zones, the moist air is assumed to be at thermodynamic equilibrium so that the wall temperature and water concentration can be related through the saturated vapor pressure (Debbissi et al., 2001, 2003). One can note that these humid zones can be modeled by considering an extremely thin liquid film. Thus, transport in the liquid film can be replaced by approximate boundary conditions for gas flow (Debbissi et al., 2001, 2003). Some other classic assumptions are used such as steady state flow, the negligible Dufour and Soret effects and radiative transfer.

From the above assumptions, the bi-dimensional flow of a gas mixture is described in the (x, y) coordinate system by the continuity equation and the balances for momentum, heat and species concentration (Kays and Crawford, 1980; Schlichting, 1979):

Fig. 1:	Schematic diagram of the physical system, a): N = 1 and b): N = 2

(1)

(2)

(3)

(4)

where, (βg(T-T₀) + β^*g(C-C₀)) represents the momentum transfer caused by the combined buoyancy forces. The thermo-physical properties of gas mixture are considered as variable with temperature and composition.

In this study, the overall mass balance described by the following equation should be satisfied at every axial location:

(5)

The boundary conditions for the problem are as follows:

(6)

The transverse gas velocity is deduced by assuming that the air-water interface is semipermeable:

(7a)

The value of ε is zero for the case of dry zone and unity for the case of wetted zone.

The energy balance at the interface (y = 0) is evaluated by:

(7b)

It is clear that the imposed heat flux q_w is the sum of a sensible (q_s) and a latent (q_l) component.

According to Dalton's law and by assuming the interface to be at thermodynamic equilibrium and the air vapor mixture is an ideal gas mixture, the concentration of vapor can be evaluated by:

(7c)

p_vs is the equilibrium pressure of vapor given by the Eq. 7d (Vachon, 1989):

(7d)

The impermeability of the dry plate (y = d) to the water vapor can be described by:

(7e)

In order to describe the mass and energy transfers between the channel walls and moist air, the following dimensionless coefficients are used (Shah and London, 1978):

The local Nusselt number is defined as:

(8a)

where, h_x is the local heat transfer coefficient. T_m is the fluid bulk temperature at a cross section:

(8b)

The mean Nusselt number is:

(8c)

The local Sherwood number is defined as:

(9a)

C_m is the fluid bulk concentration at a cross section:

(9b)

The mean Sherwood number is:

(9c)

The local evaporated mass flux is given by:

(10)

Solution method: The system of Eq. 1-5 is solved numerically using a finite difference method. The flow area is divided into a regular mesh placed in axial and transverse direction. A fully implicit marching scheme, where the axial convection terms were approximated by the upstream difference and the transverse convection and diffusion terms by the central difference is employed to transform the governing equation into finite difference equations. The resolution of the obtained algebraic equations was marched in a downstream direction since flow under consideration is a boundary-layer type. The discrete equations are resolved line by line from the inlet to the outlet of the channel.

For a given thermal and mass boundary conditions, the resolution procedure is described as follows:

If this criteria is not satisfied, return to Eq. 2 and modify the wetted wall temperature by using the bisection method

Test if the exit dynamic pressure is zero, else return to step 1 and modify the inlet velocity by using bisection method

To ensure that results were grid independent, the solution was obtained for different grid sizes for typical case program test. Table 1 shows that the differences in the evaporative rate obtained using 71-71 and 101-101 grids are always <1%. A (71-71) grid is retained for the present computations.

Table 1:	Comparison of the total evaporative rate of water at the exit (case when the left plate (y = 0) is entirely wetted) for various grid arrangement (T0 = 298.15 K; qw = 500 W m-2; p0 = 1; d/H = 0.015; C0 = 0.005)

RESULTS AND DISCUSSION

To validate the numerical scheme adopted in the present study, different limiting cases for laminar mixed and free convection have been considered. The results for the case of mixed convective heat and mass transfers inside a channel were obtained. The plates of the channel are maintained isothermal.

The first plate (y = 0) is wetted by an extremely thin water film and the second one is dry and kept at the ambient temperature T₀. The procedure has been tested by comparing the present results for the mass transfer coefficient to those of Shah and London (1978).

Figure 2a shows a good agreement between the result and those obtained by Shah and London (1978). Furthermore, the numerical code has been tested successfully by comparing the present results for mean Nusselt number (Nu_m) (Fig. 2b) at the isothermal and dry wall to the analytical solution obtained by Shah and London (1978).

Finally, results of evaporation by natural, forced and mixed convection in a heated channel were validated informer paper Dalbert et al. (1981, 2001), Prakash and Liu (1985).

Fig. 2:	Axial evolution of a): Mass transfer abd b): Heat transfer coefficient

Fig. 3:	Effect of the heat flux qw on the inlet velocity for a): Wet exit, N = 1 (T0 = 298.15 and b): Dry exit K; p0 = 1; d/H = 0.015; C0 = 0.005)

Fig. 4:	Effect of the inlet concentration on the local Sherwood number for a): Wet exit and b): Dry exit N = 1 (T0 = 298.15K; qw = 250 W m-2; p0 = 1; d/H = 0.015)

Through, these program tests, the present numerical code is considered to be suitable for the present investigation.

All the results of this study have been cared out for a channel placed into a upward flow of humid air with the ambient conditions: C₀ = 0.005; T₀ = 298.15K; the imposed walls heat flux density q_w = 250 W m^-2, the geometrical ratio is d/H = 0.015. The left plate (x = 0) is divided along the channel into 2N equally wet and dry zones.

The second plate is dry and subjected to a uniform same heat flux density q_w. This study includes two configuration, for the first case, the dry zone is located in the first half of the plate (x = 0) and the wetted zone is located at the channel exit. For the second (case 2), situation is inverted.

In this study, attention was paid to the evaporation of water by natural convection driven by the simultaneous presence of combined buoyancy effects of heat and mass diffusion. It is clear in Fig. 3, when the imposed heat flux density q_w increases, the inlet velocity increases.

The effect of the ambiance conditions and of the imposed heat flux on the mass transfer is shown in Fig. 4-6.

Figure 4 shows that the Sherwood number Sh_x is practically not influenced by the inlet concentration and especially, in the case 1.

Figure 5 shows the effect of the inlet temperature on the Sherwood number Sh_x. It is clearly seen, when T₀ increases, the Sherwood number Sh_x along the wet zone decreases.

Fig. 5:	Effect of the inlet temperature on the local Sherwood number for a): Wet exit and b): Dry exitN = 1 (C0 = 0.005; qw = 250 W m-2; p0 = 1; d/H = 0.015)

Fig. 6:	Effect of the heat flux on the local sherwood number for a): Wet exit and b): Dry exit N = 1 (C0 = 0.005; T0 = 298.15K; p0 = 1; d/H = 0.015)

As shown in Fig. 6 and along the wetted zone, the Sherwood number Sh_x is influenced by the heat flux density applied to the wall. Thus, it is noted that this number increases when the heat flux density applied to the wall increases. This can be allotted to the fact that one increases the imposed heat flux, the evaporation is increasing. Figure 4-6 show that along the dry zone, for which there is practically no mass transfer, the Sherwood number Sh_x is essentially zero, because there is no evaporation process in this zone.

As shown in Fig. 7, a increasing in the heat flux density increases the interfacial mass flux. It is clearly observed in Fig. 8, that the interfacial mass rate increases with an decrease in ambient pressure.

Figure 9 presents the interfacial mass flux at the channel exit as the function of inlet temperature T₀ for various ambient humidity. It is clearly observed that the interfacial mass rate increases with an increase in inlet temperature. As shown in Fig. 10 that the interfacial mass flux at the channel exit increases with an decrease in inlet concentration.

For different configurations, the local evaporative rate for the case of two wetted zones position is plotted also in Fig. 11.

Figure 11 shows that the evaporative rate in the first case (wet exit) is more important than that of the second case (dry exit). This result can be justified by the higher temperature of the fluid arriving to the entry of the second humid zone for the first case.

To provide further perspective about the role of the number of humid zones, for the same wetted length, Fig. 12 displays the result of local evaporative rate for different numbers of the wetted zones (N = 1-4). In all these studied cases, the channel inlet is occupied by a

Fig. 7:	Effect of the heat flux on the local evaporating rate for a): Wet exit and b): Dry exit N = 1 (C0 = 0.005; T0 = 298.15 K; p0 = 1; d/H = 0.015)

Fig. 8:	Effect of the ambient pressure on the local evaporating rate for a): Wet exit and b): Dry exit, N = 1 (C0 = 0.005, T0 = 298. 15K, qw = 250 W m-2, d/H = 0.015)

Fig. 9:	Effect of the ambient temperature on the local evaporating rate for a): Wet exit and b): Dry exit, N = 1 (P0 = 1atm, qw = 250 W m-2, d/H = 0.015)

Fig. 10:	Effect of the ambient concentration on the local evaporating rate for a): Wet exit and b): Dry exit, N = 1 (P0 = 1 atm, qw = 250 W m-2, d/H = 0.015)

Fig. 11:	Effect of the wetted zone position on the local evaporative rate for N = 1 (C0 = 0.005; T0 = 298.15K; qw = 250 W m-2; d/H = 0.015)

dry zone. This choice is approved by the previous results. One notices that at the channel exit, the evaporation is intensified by increasing the number N of the humid zones. This result can be justified by the fact that the fluid temperature reaching every humid zone increases when, one increases the number of these zones. Figure 12 also shows that by passing from N = 1-4, the relative gap of the evaporative rate value can exceed 46%.

Fig. 12:	Effect of the number N of the humid zones on the local evaporative rate (C0 = 0.005; T0 = 298.15 K; qw = 250 W m-2; d/H = 0.015)

CONCLUSION

The evaporation by free convection in a partially wetted channel has been numerically studied for an air-water system. The studied channel is made up of two parallel plates. The partially wetted plate is heated by uniform density flux, while the second is dry and isothermal. The effect of the humid zones position and the number of wetted zones N on the characteristics of the heat and mass transfers has been analyzed.

The major results are briefly summarized as follows: