A Non Isothermal Two-Phase Model for the Air-Side Electrode of PEM Fuel Cells

Transcription

1 A Non Isothermal Two-Phase Model for the Air-Side Electrode of PEM Fuel Cells M. Khakbaz Baboli 1 and M. J. Kermani 2 Department of Mechanical Enineerin Amirkabir University of technoloy (Tehran Polytechnic) Tehran, Iran, P.Code mobinkhakbaz@mail.com and 2 mkermani@aut.ac.ir ABSTRACT A two-dimensional, non isothermal, and two-phase flow in the porous GDL and the open channel of the air-side of a PEM fuel cell is numerically studied in this paper. The mixture is composed of oxyen, nitroen, water vapor and liquid water. The overnin PDE s are the conservation of the species, and a sinle momentum equation for the mobile fluid mixture and an enery equation for the mobil mixture and the solid phase. The slip velocity between the phases are taken into account usin the capillary pressure. The stronly coupled equations are solved usin the finite volume SIMPLER scheme of Patankar (1984). The enery equation was derived by the sum of enery equations of all three phases (solid matrix, liquid and as phases), with the enthalpy as a fitted function of temperature from JANAF tables of thermodynamic properties. The whole computational domain consistin two sub-domains of the porous GDL and open channel is treated in a unified manner, in which a sinle set of overnin equations are solved for both of the sub-domains with the correspondin properties for each of them. Comparisons with the experimental data of Ticianelli et al. [2] (1988) for the polarization curve shows a ood areement. 1 INTRODUCTION Two phase modelin of PEM fuel cells is necessary due to the stron effect of liquid water existence in porous cathode GDL. This consequently affects the performance of the whole cell. The prediction of the phenomena happenin as liquid water forms, is a contribution to optimization of water manaement and fuel cell desin. As it was shown by previous works liquid water appears at hih current densities. This liquid water is an obstacle to the reactant as delivery to the catalyst layer because it partially fills pores actin like solid matrix. Liquid water formation is a function of local temperature and water vapor partial pressure. Temperature does have distribu- Fiure 1: Schematic of the computational domain includin the dimensions. The GDL porosity and permeability are taken: ε = and K = cm 2. tion in cathode electrode due to non uniform heat eneration. There have been some studies on water eneration, its transport and its effect on the performance of the PEM fuel cells. Bernardi and Verbrue [4] performed one-dimensional models that were the beinnin of PEM fuel cell modelin. One-dimensional models are unable to simulate the species and phase distribution alon the channel. The two-dimensional models by Fuller and Newman and Nuyen and White [5] assumed that diffusion was the only mechanism for oxyen transport and did not consider the interaction of the flow with the species field in the channel and as diffusion layer. Gurau and Liu [6] modeled the coupled flow and transport equations in the flow channel and as diffuser in a sinle phase and also incompressible form. Wan et al. [7] presented a model includin two-phase flow and transport of all components but their model did not account for the effect of temperature on flow field and liquid water eneration. Yun Wan and Chao-Yan Wan [1] used the software packae of Fluent and presented a 3D two-phase model of fuel cell includin enery equation for a sinle point on polarization curve, with a very limited information about what is happenin alon the channel. In this paper, A two-dimensional, non isothermal, and two-phase flow in the porous GDL and the open channel of the air-side of a PEM fuel cell is numerically studied. The eometry is shown in Fi. 1 The mixture is composed of oxyen, nitroen, water vapor and

2 liquid water in which the slip velocity between the phases are included. 2 GOVERNING EQUATIONS The overnin equations of the fluid motion used in the present multi-component and two-phase flow are iven as follows: The continuity of the two-phase mixture (as + liquid) [11]: (ερ) + (ερ V ) = 0, (1) where ε is porosity of the GDL taken as constant, ρ is the density of the two-phase mixture, and V is the mixture mass averaed velocity. The continuity equation for the liquid phase (water) [11]: (εsρ l ) + (εsρ l Vl ) = Ṡ l, (2) where ρ l is the liquid density taken as a constant ρ l = 998 k/m 3, s is the saturation, Ṡ l is the condensation rate (the source term), and V l is the liquid water velocity to be determined from Eqn. 12. The source term is obtained as follows [11]: [ Ṡ l = ε ( ) (1 s) t+ t ( ) ] (1 s) t P sat P H2O, R H2 O t T T (3) where P H2 O and P sat are both the local partial pressure of water vapor at next time step, the only difference is that P H2 O is before condensation and thermodynamically unstable but P sat is after condensation and is thermodynamically stable. P sat is the local saturation pressure obtained at the local mixture temperature from [7]: lo 10 P sat = (T ) (4) (T ) (T ) 3, where P sat is obtained in atmosphere. It is noted that Ṡ l is set equal to zero within the open channel as recommended in [8]. The liquid phase includes just water and the liquid and as phases are assumed to be in equilibrium, so, the sum of water diffusion between both phases is zero. That is why the concentration equations are just written in as phase. The concentration equation for the water vapor [11]: (ε(1 s)ρ C H 2O ) + (ε(1 s) N H2 O) = S, (5) where ρ is as phase density and N H2 O is the mass flux of water vapor as: N H2 O = ρ H2 O V + J H2 O, where J H2 O is the diffusive flux of water vapor obtained from the Maxwell-Stefan equation [3], and C H 2O is the mass fraction of water vapor as: C H 2O = ρ H 2O /ρ. In Eqn. 5, S is the condensation rate of water determined by: S = Ṡ l, and V is the as phase mass averaed velocity to be determined from Eqn. 12. The concentration equation for oxyen [11]: (ε(1 s)ρ C O 2 ) + (ε(1 s) N O2 ) = 0, (6) where N O2 is the mass flux of oxyen component as: N O2 = ρ O2 V + J O2, where J O2 is the diffusion of O 2 obtained from the Maxwell-Stefan equation, and C O 2 is the mass fraction of the oxyen component as: C O 2 = ρ O 2 /ρ. The Maxwell-Stefan equation for molar diffusive flux of any component in a mixture with more than two components is as follows: Y i = 1 nd e (Y i J j Y j Ji ), (7) i j where Y i is the mole fraction of component i, n is the as molar density, Ji is the molar diffusion of component i and D e i j is the effective diffusion coefficient iven by Bruemann equation [10]: D e i j = D i j (ε(1 s)) 1.5 (8) Note that J i (k/sm 2 ) is obtained from J i (mol/sm2 ). Momentum equation for the two-phase mixture in open channel and the porous GDL are presented in a unified form as iven below. The unified form is useful as it needs no boundary condition at the channel/gdl interface [11]. (ερ V ) + (ερ V V ) = ε P (9) + (εµ V )+ερ k ε 2 µ K V, where ρ is the two-phase mixture density obtained from Eqn. 17, µ is its viscosity iven by Eqn. 19, ρ k is the kinetic density obtained from Eqn. 18. The enery equation is resulted by summin the enery equations of all of the phases of as, liquid, and the solid matrix with their correspondin volume fractions, as: (εsρ lh l + ε(1 s)ρ h +(1 ε)ρ s h s ) (10) + (εsρ l h l Vl + ε(1 s)ρ h V ) (k e f f T) = Ṡ t, where h and ρ denote the enthalpy and density, respectively, with the subscripts l indicatin the liquid phase,

3 the as phase, and s the solid matrix. The as species enthalpies and that of the liquid are taken as fitted functions of solely temperature from JANAF table. k e f f is the effective thermal conductivity obtained from: k e f f = ε(1 s)k + εsk l +(1 ε)k s. Ṡ t in Eqn. 10 is the enery eneration rate per unit volume, obtained from: Ṡ t = ε [Ṡ l (h h l )+ i2 σ e f f ], (11) within the GDL. The first term in the bracket is the evaporation latent heat and the second term represents the ohmic heat eneration due to the electronic current density i in the solid matrix within the GDL. The current density i within the GDL (as used in Eqn. 11) is taken to posses identical distribution as that of the catalyst layer. In open channel area, Ṡ l = 0 and there exits no current density. This consequents Ṡ t = 0 within the channel. Liquid and as phase velocity components are obtained from: ερ l Vl = j l + ελ l ρ V, ερ V = j + ελ ρ V (12) where j is the phase diffusion encompassin the difference between liquid or as phase velocity and the mixture velocity ( j = j l ): j l = λ lλ K [ P c +(ρ l ρ ) ], (13) ν where ν is the mixture kinematic viscosity (=µ/ρ), and λ is the relative mobility, with the value for liquid and as phases obtained from: λ l = k rl /ν l k r /ν + k rl /ν l, λ = 1 λ l, (14) where k rl and k r are relative permeabilities of liquid and as obtained from Eqn. 20. ν l and ν are kinematic viscosity of liquid and as respectively. P c in Eqn. 13 is the capillary pressure determined from: P c = C[1.417(1 s) 2.120(1 s) (1 s) 3 ], (15) where C = σ c cosθ c (ε/k) 1/2, in which σ c and θ c are the surface tension and contact anle, respectively, taken as constants: σ c = 25N/m, and θ c = 0 for the hydrophilic surface in the present study. The ideal as law is used for the as density: ρ = P R T, R = R/M, (16) where R and M are the as constant and its molecular weiht, respectively, and R is the universal as constant. Density of the total mixture (includin both phases) is obtained from: ρ = (1 s)ρ + sρ l. (17) The kinetic density ρ k appeared in the momentum equation (Eqn. 9) is included to account for the response of each of the phase on ravitational acceleration [7]. ρ k = ρ λ + ρ l λ l. (18) The parameter ρ k reduces to its limitin value of ρ or ρ l when s = 0 and s = 1, respectively. The two-phase mixture viscosity is obtained from [11]: µ= ρ ls + ρ (1 s) k r /ν + k rl /ν l. (19) It is noted that µ reduces to its limitin value of µ or µ l when s = 0 and s = 1, respectively. The relative permeability for the phases [7]: k rl = s 3 and k r = (1 s) 3. (20) These parameters are dimensionless correction coefficients multiplied by the permeability of the porous GDL (K) to obtain each phase permeability. 3 SOLUTION PROCEDURE AND BOUNDARY CONDITIONS 3.1 Solution Procedure The set of equations explained in Sec. 2 contains six unknowns as the mixture pressure, the saturation, water vapor and oxyen concentrations, the mixture velocity components (see Eqns. 1, 2, 5, 6, 9). Other parameters are evaluated in terms of these six variables. The PDE s are discretized usin the power law method. A code was developed for the unsteady solution of this set of PDE s usin the SIMPLER scheme of Patankar (1984). 3.2 Boundary Conditions The boundary conditions applied to the computational domain shown in Fi. 1 are as follows: At boundary location I (channel inlet), uniform velocity and the mass fraction species H 2 O and O 2 were specified, i.e.: u = u in, v = 0, C H 2O = C H 2O in, CO 2 = C O 2 in, T = T operatin (21) where u in is the x-component velocity at as channel inlet, and C H 2O in and CH 2O in are the species H 2O and O 2 concentrations at inlet, respectively. The temperature at inlet is set equal to the operatin temperature of the cell as reported by experimental study [2]. The inlet velocity u in was not explicitly specified in Ticianelli et al. [2]. However, to estimate a reasonable value

4 for u in (or correspondinly the mass flow at the channel inlet, ṁ in = [k/s]) the hihest current density of the cell (I max ) with a stoichiometry coefficient are assumed. Accordin to Ticianelli et al. [2] I max = 700mA/cm 2, which corresponds to an overpotential of η = V. The procedures of the evaluation of ṁ in and u in are explained as follows. The supplied air flow to the channel (ṁ in ) should include, at least, oxyen mass flow rate that is obtained accordin to the Faraday law (ṁ O2 ) consumed = M O2 I max A cl /(2F), where A cl is the GDL/catalyst layer interface area (note that current density on polarization curve is reported based on A cl, so, its dimension is A/cm 2 ). But if the supplied oxyen at the inlet to the channel is limited by (ṁ O2 ) consumed then the oxyen flow rate at the exit would be zero. This is a practically impossible operatin condition for the cell. Moreover, reardless of its practicality it will ive a zero local current density at the exit of the channel, which is not efficient. To be realistic and avoid the deficiency a stoichiometry coefficient ζ = 3 is used in the present study [9], ṁ O2 = ζ ṁ O2 consumed. Therefore, the mass flux of dry air at the inlet to the channel would be (ṁ in ) dry = ṁ O2 + ṁ N2. Notin that: ṁ N2 ṁ O2 = Y N 2 Y O2 M N2 M O2, (22) where Y N2 /Y O2 = 79/21, M O2 = 32k/kmol, and M N2 = 28 k/kmol. Then, the ratio of mass fluxes of nitroen to the oxyen at the inlet to the channel would be: ṁ N2 /ṁ O2 = Hence, the mass flux for the dry air at the inlet can be obtained in terms of current density as: (ṁ in ) dry = ṁ O2 (1+ ṁn 2 ṁ O2 ) = ζ M O 2 I 2F. (23) For the moisten air enterin the channel, one can write: ṁ in = (ṁ in ) dry + ṁ v, (24) where ṁ v is the vapor mass flux at the channel inlet. ṁ v is determined usin the humidity ratio ω, also known as specific humidity, ω = ṁ v (ṁ in ) dry = M v P v M dry air P dry air = M v P v M dry air (P P v ), (25) where M v = 18k/kmol, and at the inlet to the channel M dry air = Y i M i 29k/kmol. For the saturated air flow enterin the channel (i.e. the relative humidity φ = 100%), P v = P sat at the cell operatin temperature, Eqn. 25 reduces to: ṁ v = P sat, (26) (ṁ in ) dry P P sat where the air-side channel pressure P is taken = 5atm in the present study. Replacin ṁ v from Eqn. 25 into Eqn. 24, one can obtain: ṁ in = (ṁ in ) dry [ P sat P P sat ], (27) With the (ṁ in ) dry known from Eqn. 23, ṁ in can be obtained usin Eqn. 27. Finally the u in is obtained knowin the inlet area and density via: u in = ṁ in A cl /ρ in A in. At inlet, all species mole fractions can be calculated from below relations: Y H2 O = P sat P, Y O 2 Y N2 = 21 79, Y O 2 +Y N2 +Y H2 O = 1 (28) Species mass fractions can be obtained as follows, where C O 2 in = Y O 2 M O2 M, C H 2O in = Y H 2 O M H2 O M, (29) M = Y O2 M O2 +Y N2 M N2 +Y H2 OM H2 O. (30) At boundary condition II, fully developed condition was used: u v = 0, = 0, CH2O = 0, CO 2 T = 0, = 0. (31) At boundary condition III, wall boundary condition was applied, this boundary condition requires zero velocity and no mass transfer throuh it, Also the heat transfer throuh this boundary is inored: u = 0, v = 0, C H 2O y = 0, C O 2 y = 0, T y = 0. (32) At boundary condition IV, there are two other wall boundary conditions in porous GDL where: u = 0, v = 0, C H 2O = 0, CO 2 = 0, s T = 0, = 0. (33) At boundary condition V (the catalyst layer-gdl interface), oxyen is consumed and water is enerated as the reaction product. It is assumed that there is one row of computational cells, just above the GDL-catalyst layer interface, for which the conservation equations were applied. The mass flux of species O 2 and H 2 O throuh the northern faces of these cells can be related to the local current density via: N O2 = i M O 2 4F, N H 2 O = (1+2α) i M H 2 O 2F, (34) where N O2 and N H2 O denote the mass fluxes of species O 2 and H 2 O, respectively, and α is the net water transferred throuh the membrane. The catalyst layer is not modeled in this study like what was done by N. Khajeh Hosseini D. [12], so, It is taken as constant equal to in the

5 present study. For the formation of a water molecule two protons are required, that is why the coefficient 2 is multiplied to α. Moreover, the rate of protons transfer throuh the membrane is proportional to the local current density i. So, the mass flux of water throuh the membrane is proportional to 2αi. Consequently, the mass fluxes of oxyen and water vapor throuh the northern faces of the cells (of the strin) can be obtained from Eqns. refupper boundary species mass fluxes. The tanential velocity component at this boundary is zero due to the no slip condition. Therefore, the advection mass fluxes throuh eastern and western faces of the cells for all of the species at this boundary will be zero. It is also noted that the species acts as an inert component, i.e. not participatin the reaction. Hence the vertical component of its mass flux is set to zero at this boundary. Applyin the mass conservation equation for the whole mixture (Eqn. 1) for the computational cells just above GDL, i.e. (ερ)/ + (ε N ) = 0, and inorin all of the horizontal flux components (due to the no slip condition), the discretized form of the equation at this boundary becomes: (ερ) + ε(n) n f x ε(n) s f x = 0, (35) where the subscripts n f and s f corresponds to the north and south faces, respectively. The mass fluxes (N) n f and (N) s f in Eqn. refupper boundary vertical velocity are as follows: i (N) n f = M H2 O 2F (1+2α) M i O 2 4F, (N) s f = ρv. (36) The horizontal velocity is taken zero, so, mass fluxes throuh eastern and western faces are zero and there is a mass flux at northern face formed by the above mentioned oxyen and water vapor mass fluxes, it is noted that nitroen (inert species) mass flux is zero at this boundary it means its diffusion and advection cancel one an other. Equations 36 and 35 are solved for the vertical velocity, this velocity is used as y-direction momentum boundary condition. The species at this boundary are calculated such that there is no mass flux throuh eastern and western faces of the computational cells just above GDL and there is just mass flux in y-direction, usin Eqn. 34 the species concentration equations Eqns. 5, 6 are as follows: c.v. c.v. (ε(1 s)ρ C H 2O ) dxdy (1+2α) i M H 2 O Fiure 2 presents the comparison between the numerical 2F and experimental polarization curves as verification. It is (ε(1 s)n H2 O) s f = S dxdy, (37) evident that the numerical polarization curve is very close (ε(1 s)ρ C O 2 ) dxdy+ i M O 2 4F (ε(1 s)n O 2 ) s f = 0, The enery equation is discretized such that there is not heat transfer throuh the northern faces of the computational cells just above the GDL. The reason is that velocity at this boundary is zero and the thermal conductivity of the membrane is very small in comparison with that of GDL. The horizontal velocity is zero at this boundary, so, there is just conductive heat transfer throuh the eastern and western faces of the computational cells just above the GDL. The heat eneration in the catalyst layer is introduced to the upper boundary of the computational domain usin a heat source term as [1]: [ ( i 2 Ṡ t = ε σ e f f + i η T U )] 0. (38) T The eneral equation expressin the relation between the reversible and the operatin voltae at each cell of boundary V is as follows: V cell = V rev η σi, (39) where σ is the whole ohmic resistance of the fuel cell taken as and η is the activation loss obtained from the Tafel equation expressin the reaction rate in the cell: η = RT αf ln( O C 2 re f i (1 s)c O 2 i 0 ), (40) where i 0 is the exchane current density and C O 2 re f is the reference oxyen mass fraction i 0 /C O 2 re f = 0.8. α c is the cathodic transfer coefficient taken as 5. If Eqn. 40 is substituted in Eqn. 39, the equation from which the local current density is calculated for a specific voltae loss (V rev V cell ) will be: V cell = V rev RT αf ln( O C 2 re f i 4 RESULTS (1 s)c O 2 i 0 ) σi. (41) The results obtained from the present computations at a sample overpotential η = V (i.e. at the cell voltae = 6 V ) are brouht below. to that of experimental, especially in two phase reions. Fiure 3 shows the contours of the as phase density ρ within the computational domain. The as phase density decreases from the lower left corner where oxyen mass fraction (at the entrance) is hiher than that at the

6 upper riht corner where it has been consumed in the favor of water production. Since the molecular weiht of water vapor M H 2O is less than that of the oxyen M O 2 therefore the resultin as molecular weiht M is hiher at the lower left corner. On the other hand the temperature is hiher in the upper riht corner as compared to that at the lower end corner. This is due to the reaction at the catalyst layer and condensation in GDL as shown Fi. 13. Therefore, with the as density obtained from ρ = P/(R T) = PM /( RT), then ρ is much hiher at the entrance (lower end corner) due to the enhanced M and reduced T at this corner. Fiure 4 shows the contours of the oxyen mass fraction C O 2 within the computational domain. Oxyen mass fraction decreases alon the channel especially adjacent to the catalyst layer where reaction exists and oxyen is consumed. As Fi. 5 shows, the reaction rate has similar trend alon boundary V. Fiure 6 shows the contours of the Water vapor mass fraction C H 2O within the computational domain. Water vapor mass fraction increases alon catalyst layer. It has an inverse mass fraction distribution trend in comparison with that of oxyen because the product of the oxyenconsumin reaction is water. (see Fi. 4 for comparisons). Fiure 7 shows the contours of the saturation s within the computational domain. Saturation is the result of water vapor condensation when water vapor partial pressure exceeds the saturation pressure at the mixture temperature. Water vapor partial pressure increases alon the catalyst layer because of its mass fraction increment alon the catalyst layer (see Fi. 6). That is why the saturation ets larer values from left to riht (Fi. 12). Fiure 8 shows the liquid flow field, as it is expected, the liquid is movin out of the GDL into the channel, there, it may drop down and is thrusted by shear force of as velocity towards the as channel outlet. It is noted that the x-component liquid velocity is very small in comparison with y-component liquid velocity Fis. 10 and 11, so, the velocity vectors are normal to the flow stream. Fiure 9 shows the mass fraction distribution of nitroen. It was expected that nitroen mass fraction decreases alon the channel. This is because two water molecules are produced per one oxyen molecule destruction alon the channel that lessens the nitroen presence with respect to the fact that the sum of all species mole fraction is one. Fiure 13 shows the contours of the mixture temperature within the computational domain. The reaction rate is reater around inlet due to the smaller saturation (in comparison with around outlet), but the maximum temperature occurs around outlet. This shows that maximum temperature location is predominantly dependent on the latent heat of condensed water. REFERENCES [1] Y. Wan and C. Y. Wan "A Nonisothermal, Towphase Model for Polymer Electrolyte Fuel Cells", J. of The Electrochem. Soc., 153 (6) A1193 A1200 (2006). [2] E. A. Ticianelli, C. R. Derouin, A. Redondo, and S. Srinivasan, "Methods to Advance Technoloy of Proton Exchane Membrane Fuel Cells", J. of The Electrochem. Soc., 135, 2209 (1988). [3] R. B. Bird, W. Steward, and E. N. Lihtfoot, "Transport Phenomena", Wiley, New York, [4] D. M. Bernardi, M. W. Verbrue, "A mathematical model for the solid-polymer-electrode fuel cell", J. of The Electrochem. Soc., 139, (1992). [5] T. V. Nuyen, R. E. White, "A water and thermal manaement model for proton-exchane-membrane fuel cells", J. Electrochem. Soc. 140, (1993). [6] V. Gurau, H. T. Liu, S. Kakac, "Two-dimensional model for proton exchane membrane fuel cells", AIChE J. 44, (1998). [7] Z. H. Wan, C. Y. Wan, K. S. Chen, "Two-phase flow and transport in the air cathode of PEM fuel cells", J. Power Source 94, (2001). [8] D. Natarajan, T. V. Nuyen, "A Two-Dimensional, Two-phase, Multicomponent, Transient Model for the Cathode of a Proton Exchane Fuel Cell Usin Conventional Gas Distributors", J. of The Electrochem. Soc. 148 (12) A1324 A1335 (2001). [9] B.E.Sc. Phon Thanh Nuyen, "A Three- Dimensional Computational Model of PEM Fuel Cell with Serpentine Gas Channels", University of Western Ontario (2003). [10] J. H. Nam, M. Kaviany "Effective Diffusivity and Water-Saturation Distribution in Sinle- and Two- Layer PEMFC Diffusion Medium", Int. J. Heat and Mass Transfer. 46, , (2003). [11] M. Khakbaz Baboli "A Two-Dimensional Computational model of chatode electrode of PEM fuel cells", Mechanical department of Amirkabir (Tehran polytechnic) [12] N. Khajeh Hosseini D., H. Shabard "Water Manaement in Cathode Side of a PEM Fuel Cell" 15th Annual (International) conference on Mechanical Enineerin, ISME2007, May 15-17, Amirkabir University of Technoloy, Tehran, Iran,(2007).

7 Fuel cell voltae (V) Experimental study by Ticianeli [2] Numerical in the present study Local current density (A/cm 2 ) Current density (A/cm 2 ) Fiure 2: Polarization curves. Fiure 5: Local current density distribution alon catalyst layer (A/cm 2 ). Gas phase density (k/m 3 ) Water vapor mass fraction Fiure 3: Gas phase density ρ contours (k/cm 3 ). Fiure 6: Water vapor mass fraction contours. Oxyen mass fraction Saturation Fiure 4: Oxyen mass fraction contours. Fiure 7: Saturation contours.

8 5 Liquid phase velocity field (cm) 0.05 x = 0 x = 5L x = 0.75L x = L x = L v l (cm/s) Fiure 8: Liquid phase flow field (cm/s). Fiure 11: Comparative profiles of v l (cm/s) at three stations x = L, 0.75L and L alon the channel. Nitroen mass fraction x = 0 x = 5L x = L x = 0.75L x = L 0 0 s Fiure 9: Nitroen mass fraction contours. Fiure 12: Comparative profiles of saturation (s) at five stations x = 0, 5L, L, 0.75L and L alon the channel. 2 1 Temperature x = 0 x = 5L x = L x = 0.75L x = L E-05 2E-05 3E-05 4E-05 5E-05 6E-05 u l (cm/s) Fiure 10: Comparative profiles of u l (cm/s) at three stations x = L, 0.75L and L alon the channel. Fiure 13: Temperature contours (K).