Three-Dimensional Simulation of Mixing Flow in a Porous Medium with Heat and Mass Transfer in a Moisture Recovery System

Transcription

1 12 th Fluid Dynamics Conference, Babol Noshirvani University of Technology, April 2009 Three-Dimensional Simulation of Mixing Flow in a Porous Medium with Heat and Mass Transfer in a Moisture Recovery System Amir Hossein Sharafian Ardakani Mohammad Hadi Akbari * M.Sc. student Assistant Professor School of Engineering Faculty of Mechanical Engineering Shiraz University, Shiraz, , Iran Abstract: In this article, simulation results of steady three-dimensional heat and mass transfer between two parallel channels separated by a porous medium are presented. The two fluid streams flowing in the two channels have different compositions and are at different temperatures. The governing equations at the present phase include conservations of mass, momentum, energy, and species. The flow is laminar since the Reynolds number based on the hydraulic diameter of the channel is less than 500 in all cases. Density is variable and is evaluated based on an ideal gas mixture. Mass diffusivity is calculated based on binary diffusion coefficients for species pairs. Comparison of the results indicate that a counter flow arrangement results in a higher performance in terms of both heat transfer and moisture recovery compared to a parallel flow case. In fact, for the particular dimensions considered in this study, an increase of about 50% in heat transfer, and an increase of about 24% in moisture recovery are observed. Keywords: Heat and mass transfer, porous media, moisture recovery, CFD modeling, Simple algorithm Nomenclature a Constant for gas pairs in binary diffusion coefficient formula b Constant for gas pairs in binary diffusion coefficient formula C eff Effective heat capacity, J/kg.K d Hydraulic diameter, m D Diffusion coefficient, mm 2 /s K Permeability, m 2 K eff Effective thermal conductivity, W/m.K L Height of duct, m M Molar mass, gr/mol n Normal direction to wall N Number of species P Pressure, Pa P c Critical pressure, Pa R Ideal gas constant, J/mol.K T Temperature, K T c Critical temperature, K u Velocity component in the x direction, m/s v Velocity component in the y direction, m/s Velocity vector, m/s w Velocity component in the z direction, m/s W Mass fraction Y Mole fraction * Corresponding author: address: h-akbari@shirazu.ac.ir (M.H. Akbari) Tel.:

2 Greek symbol Porosity Relative humidity Viscosity, kg/m.s Mixture density, kg/m 3 Tortuosity Subscripts H 2 O Water vapor j Denotes gas species k Denotes gas species mix Gas mixture max Maximum min Minimum N 2 Nitrogen Oxygen O 2 Superscripts * Dimensionless parameter 1. Introduction Nowadays, optimum consumption of energy is of great consideration due to its increasing cost. At the same time, depletion of the fossil fuels on the one hand, and their environmental impact on the other, are pushing us to use renewable energy resources. In such a "green energy" scenario, hydrogen is considered as a very suitable energy carrier. One of the main applications of hydrogen is in different types of fuel cells, such as a polymer electrolyte membrane fuel cell (PEMFC). The initial and operational costs of energy production associated with a PEMFC is higher compared to other, more conventional, technologies specially those that are powered directly by fossil fuels. However, higher efficiency and environmental consistency of PEMFCs make this technology a viable option for the medium to long term planning of energy production. Consequently, every little opportunity to increase the efficiency and reduce the energy cost of a PEMFC system must be seriously taken into consideration. One such opportunity lays in using a heat and moisture recovery subsystem in a PEMFC system. PEMFC systems perform in C temperature range. The sole product of the electrochemical reaction occurring in a PEMFC is water. The exhausted gas stream from the cathode is at a higher temperature and relative humidity than the air flowing into the cathode. Therefore, a simple device for heat and moisture exchange can be utilized to recover some of the heat and moisture content of the cathode exhaust. It must be noted that for a PEMFC to operate efficiently, the membrane must be maintained at a state of high moisture content. Therefore, moisturizing the inlet air to the cathode is essential in safe and efficient operation of a PEMFC system. Heat transfer in channels partially filled with porous media has received considerable attention and was the focus of several investigations, such as Al-Nimr and Alkam [1], Chikh et al. [2], Vafai and Kim [3], and Poulikakos and Kazmierczak [4]. In general, most analytical studies of fluid flow and heat transfer in a porous medium adopt Darcy s law. Carman [5] and Collins [6] have investigated the fluid flow through porous material using Darcy's law. In recent researches conducted by Al-Nimr and Alkam [7,8], very limited effort have been placed on the problem of forced convection in composite fluid and porous layers. Beavers and Joseph [9] first investigated the fluid mechanics at the interface between a fluid layer and a porous medium over a flat plate. Vafai and Thiyagaraja [10] obtained an analytical approximate solution for the same problem based on matched asymptotic expansions for the velocity and temperature distributions. Later on, Vafai and Kim [11] presented an exact solution for the same problem. Using Darcy-Brinkman-Forchheimer model, the problem of forced convection in channels partially filled with porous media was numerically investigated by Jang and Chen [12]. Nield [13] discussed the limitation of the Brinkman-Forchheimer model in porous media and at the interface, between the clear fluid and porous region. In this article, simulation results of steady three-dimensional heat and mass transfer between two parallel channels separated by a porous medium are presented. The two fluid streams in the two channels have different compositions and different temperatures. Fig (1) shows the schematic of this system. This simulation attempt is the first step of a more inclusive program to perform a three-dimensional modeling of a PEMFC with two-phase flow in the porous diffusion layers and distribution channels. In this first step, we investigate heat and mass transfer between two channels through a porous layer separating the two. Moist and warmer air flows in one 2

3 channel, while air at lower humidity and temperature flows in the other. Such a system forms the basis of a moisture recovery device that is used in PEMFC systems. 2. Governing equations Continuity, momentum, energy and mass transfer equations are solved. All of fluid properties, i.e., density, viscosity, thermal conductivity and specific heat, are variable. Gas behavior follows the ideal gas mixture law. Flow field inside the channels is obtained by solving steady state continuity and Navier-Stokes equations [14]:. 0 (2.1). (2.2). (2.3). (2.4) In the porous medium, continuity equation is similar to Eq. (2.1). Momentum equations are modified as follows [15]: In these equations, (2.5) (2.6) (2.7) (2.8) (2.9) In the construction of a continuum model for a porous medium, there are two ways to perform the variables averaging: spatial and statistical. In the spatial approach, a macroscopic variable is defined as an appropriate mean over a sufficiently large representative elementary volume (r.e.v.); this operation yields the value of that variable at the centroid of the r.e.v. It is assumed that the result is independent of the size of the representative elementary volume. The length scale of the r.e.v. is much larger than the pore scale, but considerably smaller than the length scale of the macroscopic flow domain. In the statistical approach, the averaging is over an ensemble of possible pore structures that are macroscopically equivalent. A difficulty is that usually the statistical information about the ensemble has to be based on a single sample, and this is possible only if statistical homogeneity (stationarity) is assumed [15]. We construct a continuum model for a porous medium, based on the r.e.v. concept and use Darcy velocity. Eq. (2.8) is the Karman-Kozeny relation and Eq. (2.9) is the modified Ergun relation. The term is Darcy term and is Ergun inertia term. In this article; we use 0.375, 2.0 and = m [16]. The species mass fractions are obtained by solving mass transfer equations [17]:... (2.10)... (2.11) 1 (2.12) The diffusion coefficients for multi-component flow are calculated from Eq. (2.13) that is based on a simplification of the Stefan-Maxwell equations [18]. The binary diffusion for each of mixture components are given by Eq. (2.14), [19]. 3

4 1 (2.13) where, 1 1 : , 1.823, : , (2.14) Gas diffusion in porous medium is calculated by the following [17]: 1. (2.15) Density of fluid is given by the ideal gas mixture law (2.16) Viscosity of fluid is obtained by (2.17) where, (2.18) The energy equation is given by [16].. (2.19) where, (2.20) (2.21) The mole fraction and mass fraction of a species are related by (2.22) We also use some dimensionless parameters as follows (2.23) 4

5 (2.24) (2.25) 3. Boundary conditions The hydrodynamic boundary conditions at the inlets of the channels, at the outlets and at the walls are: At x = 0, 0 < y < L 2 and 0 < z < L 3 In cold and low moist channel: In warm and moist channel: u=u 01, v = w = 0, T = T 1, φ = 30%,,, u=u 02, v = w = 0, T = T 2, φ = 100%,,, At x = L 1, 0 < y < L 2 and 0 < z < L 3 In cold and low moist channel: P = P 1 In warm and moist channel: P = P 2 At the walls u = v = w = 0, = 0 Boundary conditions are shown in Fig. (2) for parallel flow channels. Inlet and outlet conditions are described in Table (1). 4. Numerical method In this study, all equations are solved simultaneously in a 3-D single domain. The full computational domain consists of two gas flow channels and porous medium between them, as shown in Fig. (3). The main unknowns in this model are: u, v, w, P, T, 2, 2, 2. The governing equations are discretized using a finite volume method. The convective terms in the momentum and mass transfer equations are discretized by the power law scheme, and the diffusion terms in these equations are modeled by a central difference scheme. For coupling pressure to velocity the SIMPLE algorithm is applied [20,21]. To obtain the flow field, the continuity and momentum equations are calculated first. Then, mass transfer equations, and finally energy equation are solved. Fluid properties are updated in every iteration. The convergence criteria are checked, and if convergence is not reached calculations are repeated in the next iteration [22]. 5. Results Simulation results are presented for a heat and mass recovery system for two arrangements: 1) parallel flow 2) counter flow. Results for these two arrangements are compared with each other. The overall dimensions of the domain are 3021 cm, which consists of 75,000 computational cells. In order to validate the present code, first a comparison is made in Fig. (4) between the predicted fully developed axial velocity profile and the corresponding analytical solution given by Poulikakos et al. [4]. The discrepancy between the numerical and analytical solutions is mainly due to the simplifying assumptions made in the analytical solution. Velocity, species mole fractions, temperature, density and viscosity distributions for the parallel flow arrangement are presented in Figs. (5) to (14). The velocity distribution in the x direction is shown in Fig. (5). This velocity component is generally higher in the warm and moist channel than that in the cold and low humidity channel. Velocity distributions in the y and z directions are shown in Figs. (6) and (7). These figures indicate that the velocity variations in the y and z directions are not considerable. Pressure contours inside the channels are shown in Fig. (8). It shows that pressure decreases in the longitudinal direction. Temperature distribution is shown in Fig. (9). Temperature increases in cold side; the increase in the dimensionless temperature T * is more than 16% in this channel. Mole fraction of nitrogen is shown in Fig. (10). The total mass flow rate of nitrogen is the same in two sides. However, the difference in the concentration of nitrogen in two sides causes nitrogen to diffuse from the cold side toward the warm side. The reason for this decrease in the nitrogen mole fraction is mainly due to the increase in the moisture content of the warm stream coming out of the fuel cell. Although about half of the oxygen content is consumed in the fuel cell, but due to the addition of the product water to the air stream in the cathode, the overall concentration of nitrogen will decrease at the fuel cell exit (the warm stream in the moisture 5

6 recovery device). Mole fraction distribution of oxygen is shown in Fig. (11). Oxygen also tends to diffuse toward the warm channel because it has higher concentration in the cold channel. Mole fraction contours of water vapor are plotted in Fig. (12). Relative humidity of air at the inlet of the warm channel is 100%; thus water vapor moves toward the cold and low humidity channel where the relative humidity is 30% at its inlet. As a result, the amount of moisture increases at the outlet of the cold and low moist channel. Variation of the density inside the channels is shown in Fig. (13). Density at the inlet of the warm and moist channel is nearly half of that at the inlet of cold and low humidity channel. However, the outlet density increases in the warm and moist channel, and decreases in cold and low humidity channel. Viscosity distribution is shown in Fig. (14). It shows the viscosity of the fluid changes inside the channels and thus this effect can not be neglected. Velocity, species mole fractions, temperature, density and viscosity distributions for the counter flow arrangement are plotted in Figs. (15) to (24). The velocity distribution in the x direction inside the counter flow channels is shown in Fig. (15). The flow field is different than that in the previous case, and the mixing rate of species are higher than previous case. Velocity variations in the y and z directions are shown in Figs. (16) and (17). Pressure contours are shown in Fig. (18). It is seen that in a part of the porous medium the amounts of pressure are equal in two sides, thus there is no flow in this part. Temperature distribution is shown in Fig. (19). Temperature variations are more than that in the previous case; this has increased to about 25% in this case. Mole fraction contours of nitrogen are shown in Fig. (20).Nitrogen tends to diffuse from the cold side to the warm side in this case also, for the same reason mentioned in the parallel flow case. Contours of oxygen mole fraction are shown in Fig. (21). In this arrangement, more oxygen diffuses toward the warm and moist channel compared to the previous case; but this is not considerable. However, the amount of water vapor diffusion increases noticeably in this arrangement compared to the previous case; that is shown in Fig. (22). Distribution of density is shown in Fig. (23) for this arrangement. Also, viscosity distribution is shown in Fig. (24) where it is seen that viscosity increases along the cold and low humidity channel, and decreases along the warm and moist channel. Comparing the results for the two cases, it is observed that the mixing rates in the counter flow arrangement are more than those in the parallel flow arrangement. The velocity vectors in the porous medium are plotted in Fig. (25). The rate of diffusivity of fluid decreases in the longitudinal direction. Thus we must consider a suitable length to allow the desired mass transfer rate. The mean exit temperatures and mole fractions for different species in the parallel flow and counter flow arrangements are summarized in Tables (2) and (3). By comparing these results, we can observe that the counter flow arrangement causes a better heat transfer than the parallel flow arrangement, as the dimensionless temperature rise in the former case is more than 50% higher than that in the latter case. More specifically, T * is 0.25 at the exit in the counter flow case, while it is in the parallel flow case. Moreover, the results also indicate that the amount of moisture recovery is about 24% higher in the counter flow arrangement compared to that in the parallel flow case. It is also noted that the oxygen mole fraction at the outlet of the cold channel decreases more in the counter flow arrangement compared to the parallel flow case. However, this cannot be considered as a problem since more than the "stoichiometric air" is normally pumped into a PEMFC, and the slight decrease in the oxygen mole fraction will not cause any problem in the operation of the fuel cell. This is while the considerable increase in the humidity content of the inlet air (the cold channel) will have a significant desirable impact on the operation of the PEMFC. Having stated all the above simulation results, we can conclude that the counter flow arrangement is the suitable choice in a heat and moisture recover device for a PEMFC system in order to save energy consumption and to increase the overall efficiency of the system. 6. Conclusion In this article, steady three-dimensional heat and mass transfer between two parallel channels separated by a porous medium are simulated. The governing equations that include conservations of mass, momentum, energy, and species, are solved using a finite volume method based on the SIMPLE algorithm. The CFD code developed is first validated against an analytical solution for a simple case. The simulation results are then compared for two arrangements for the main problem at this stage. These results indicate that a counter flow arrangement in the heat and moisture recovery system has better overall performance: the heat transfer rise in the counter flow case is higher than that in the parallel flow case by about 50%; also the increase in the water vapor mole fraction is about 24% more than that in the parallel flow case. Thus by using a counter flow arrangement in a heat and moisture recovery system, one can reduce energy consumption and increase the overall efficiency of a PEMFC system. 6

7 References [1] Al-Nimr, M.A. and Alkam, M.K. (1997) Unsteady non-darcian forced convection analysis in an annuls partially filled with a porous material, ASME J. Heat Transfer 119, [2] Chikh, S., Boumedien, A., Bouhadef, K. and Lauriat, G. (1995) Analytical solution of non-darcian forced convection in an annular duct partially filled with a porous medium, Int. J. Heat Mass Transfer 38, [3] Vafai, K. and Kim, S.J. (1990) Fluid mechanics of interface region between a porous medium and fluid layer - an exact solution, Int. J. Heat Mass Transfer 11, [4] Poulikakos, D. and Kazmierczak, M. (1987) Forced convection in a duct partially filled with a porous material, ASME J. Heat Transfer 109, [5] Carman, P.C. (1956) Flow of Gases Through Porous Material, Academic Press, New York. [6] Collins, R.E. (1961) Flow of Fluids Through Porous Material, Reinhold, New York. [7] Al-Nimr, M.A. and Alkam, M.K. (1998) Unsteady non-darcian fluid flow in parallel channels partially filled with porous materials, Heat Mass Transfer 33, [8] Alkam, M.K., Al-Nimr, M.A. and Mousa, Z.N. (1998) Transient forced convection of non-newtonian fluid in the entrance region of porous concentric annuli, Int. Num. Meth. Heat Fluid Flow 8 (5), [9] Beavers, G.S. and Joseph, D.D. (1967) Boundary conditions at naturally permeable wall, J. Fluid Mech. 13, [10] Vafai, K. and Thiyagaraja, R. (1987) Analysis of flow and heat transfer at the interface region of a porous medium, Int. J. Heat Mass Transfer 30, [11] Vafai, K. and Kim, S.J. (1995) On the limitations of the Brinkman-Forchheimer-extended Darcy equation, Int. J. Heat Mass Transfer 16, [12] Jang, J.Y. and Chen, J.L. (1992) Forced convection in a parallel plate channel partially filled with a high porosity medium, Int. Com. Heat Mass Transfer 19, [13] Nield, D.A. (1991) The limitation of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface, Int. J. Heat Mass Transfer 12, [14] Streeter, V.L., Wylie, E.B. and Bedford, K.W. (1998) Fluid Mechanics, ISBN: , pp [15] Nield, D.A. and Bejan, A. (2006) Convection in Porous Media, ISBN: , pp [16] Kaviany, M. (1995) Principles of Heat Transfer in Porous Medium, ISBN: , pp [17] Siegel, N.P. (2003) Development and validation of a computational model for a proton exchange membrane fuel cell, PhD. Thesis, Virginia Polytechnic Institute and State University, pp [18] Gurau, V., Liu, H. and Kakac, S. (1998) Two dimensional model for proton exchange membrane fuel cells, J. AIChE Vol. 44, pp [19] Bird, R.B., Steward, W.E. and Lightfoot, E.N. (2002) Transport Phenomena, ISBN: , pp [20] Patankar, S.V. (1980) Numerical Heat and Fluid Flow, ISBN: [21] Hirsch, C. (2007) Numerical Computational of Internal and External Flows, ISBN: [22] Versteeg, H.K. and Malalaskerea, W. (1995) An Introduction to Computational Fluid Dynamics: The Finite Volume Method, ISBN:

8 Figure (1) Schematic diagram of the problem. Figure (2) Boundary conditions of the computational domain for parallel flow arrangement. Figure (3) Computational grid. 8

9 Figure (4) - Comparison of the present results with the analytical solution by Poulikakos et al. [4]. Figure (5) x-velocity component distribution for the parallel flow arrangement. 9

10 Figure (6) y-velocity component distribution for the parallel flow arrangement. Figure (7) z-velocity component distribution for the parallel flow arrangement. Figure (8) Pressure distribution for the parallel flow arrangement. 10

11 Figure (9) Temperature distribution for the parallel flow arrangement. Figure (10) Nitrogen mole fraction distribution for the parallel flow arrangement. Figure (11) Oxygen mole fraction distribution for the parallel flow arrangement. 11

12 Figure (12) Water vapor mole fraction distribution for the parallel flow arrangement. Figure (13) Density distribution for the parallel flow arrangement. Figure (14) Viscosity distribution for the parallel flow arrangement. 12

13 Figure (15) x-velocity component distribution for the counter flow arrangement. Figure (16) y-velocity component distribution for the counter flow arrangement. Figure (17) z-velocity component distribution for the counter flow arrangement. 13

14 Figure (18) Pressure distribution for the counter flow arrangement. Figure (19) Temperature distribution for the counter flow arrangement. Figure (20) Nitrogen mole fraction distribution for the counter flow arrangement. 14

15 Figure (21) Oxygen mole fraction distribution for the counter flow arrangement. Figure (22) Water vapor mole fraction distribution for the counter flow arrangement. Figure (23) Density distribution for the counter flow arrangement. 15

16 Figure (24) Viscosity distribution for the counter flow arrangement. Figure (25) Velocity vectors in the porous medium for the parallel flow arrangement. Warm and Moist Side Inlet condition Outlet condition Cold and Low Moist Side Inlet condition Outlet condition Table (1) - Inlet and Outlet boundary conditions. u (m/s) v (m/s) w (m/s) P (pag) T (K) P (pag) 10,000 u (m/s) v (m/s) w (m/s) P (pag) T (K) P (pag) 10,000 Table (2) Outlet parameters for the parallel flow arrangement. Warm and Moist Side T (K) Cold and Low Moist Side T (K) Table (3) Outlet parameters for the counter flow arrangement. Warm and Moist Side T (K) Cold and Low Moist Side T (K)