Computational study of heat and mass transfer issues in solid oxide fuel cells

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Computational study of heat and mass transfer issues in solid oxide

fuel cells

COMPUTATIONAL STUDY OF HEAT AND MASS TRANSFER ISSUES IN SOLID OXIDE FUEL CELLS

D.H. Jeon1, S.B.Beale1, H-W Choi2, J.G. Pharoah2 and H. Roth1

1

National Research Council, Montreal Road, Ottawa, ON K1A 0R6, Canada

2

Queen’s-RMC Fuel Cell Research Centre, Queen’s University, Kingston, ON K7L 3N6, Canada ABSTRACT

Temperature uniformity contributes to minimizing material stress and improving lifetime in SOFCs. This paper investigates the temperature distribution in three flow field configurations;. co-flow, counter-flow and cross-flow. A three-dimensional computational fluid dynamics model coupling transport and electrochemistry is developed within the framework of an existing open source library. Porous electrodes with effective diffusivities and interconnect rib geometries are accounted for. Two different approaches were investigated in obtaining effective diffusion coefficients for the porous transport layers. A lumped internal resistance model is implemented for the calculation of electrochemical reaction and Joule heating. The temperature profiles are influenced by the air flow passage and the interconnect rib geometry. For the three geometries considered, the counter-flow case produces the most uniform temperature distribution. It is shown that the prescription for the diffusion coefficients is a matter of importance, as it significantly affects the entire solution.

INTRODUCTION

Much of the development of intermediate temperature solid oxide fuel cells (SOFCs) is presently focused on anode-supported SOFCs, owing to their relatively low operating temperature and high current density [1]. Planar-type anode-supported SOFCs are attractive as they are capable of providing high power density and may be fabricated in a cost-effective manner. However thermally induced stresses can cause sealing problems, due to different coefficients of thermal expansion in the various SOFC components [1,2]. Moreover, repetitious thermal cycling may result in cracking of components and lead to gas leakage [3]. Optimization of various parameters such as the design operating point, geometrical design, and component material properties are all necessary to minimise thermal-stresses. Among the challenges associated with design optimisation is the maintenance of as uniform a temperature distribution as possible over the positive electrode-electrolyte-negative electrode-assembly (PEN), as it minimizes material stresses on the PEN and hence extends the lifetime of the SOFC.

Recent research has centred on minimizing non-uniform temperature distributions, by both experimental methodology, and also numerical calculations. Huang et al. [4] considered the impact of flow uniformity upon cell performance, by both experimental and numerical approaches. Chiang et al. [5] and Yakabe et al. [2] both built thermal-fluid models based on a finite volume method. From the results of their temperature field calculations, they subsequently performed stress analysis using finite element analysis (FEA). Nakajo et al. [6], Cui et al. [7], Selimovic et al. [8] and Lin et al. [9] performed thermal stress calculations using commercial FEA software. Jiang and Chen [3], and Weil and

Koeppel [10] studied heat transfer and thermally-induced-stresses for bonded compliant seal designs. Yuan [11] studied the performance of a SOFC stack with flow maldistributions, which affect the current density distribution. Achenbach [12] considered the impact of different flow geometries, and also took into account internal fuel reforming. It was shown that the cross-flow geometry resulted in the largest temperature gradients. Recknagle et al. [13] simulated planar SOFCs and predicted temperature and current density distributions for flow and counter-flow. It was found that the co-flow configuration resulted in a more uniform temperature distribution and the smallest temperature gradient, for a given fuel utilization and cell temperature. However the authors did not consider the effects of porous electrodes and interconnect rib geometries upon transport phenomena. A similar study was carried out by Xia et al. [14] who investigated the influence of the non-uniformity of gas flow rates on the thermal and electrical performance. It was reported that the counter-flow geometry rendered superior thermoelectric characteristics and a more uniform current density distribution and temperature distribution than for co-flow. Although there is a great deal of active research on SOFCs, especially numerical models of electrochemical performance and transport phenomena, it seems there is relatively little discussion in the literature about the system-level layout of cells and stacks [15].

In this study, three different flow path configurations for an anode-supported SOFC are investigated, in terms of uniform temperature distribution. Co-flow, counter-flow, and cross-flow configurations are all considered. Detailed CFD and thermofluid transport calculations with electrochemical phenomena are implemented by means of the object-oriented code OpenFOAM® (Open Field Operation and Manipulation) [16,17]. The electrochemical model and numerical algorithms used in this study follow previous work with commercial codes [18,19]. The temperature distributions and profiles, together with statistical variance (standard deviation) are considered for a range of mean current densities to show the temperature uniformity/range. Also the distributions of interdependent quantities such as current density and mass fraction are also examined.

GEOMETRY AND OPERATING CONDITIONS

The anode-supported SOFC considered in this study is shown in Fig. 1 (for the case of co-flow). The unit is composed of interconnects, gas channels, and PEN. The flow channels are embedded within the rectangular cross-section of the metallic interconnects.. The fuel and air channels are separated by the PEN which consists of five distinct layers: (1) anode substrate layer (ASL), (2) anode functional layer (AFL), (3) electrolyte, (4) cathode functional layer (CFL), and (5) cathode current collector layer (CCCL).

2-5 November, 2010, Kaohsiung City, Taiwan

Figure 1 Geometry and mesh for planar-type anode-supported SOFC.

Table 1. Operating conditions

Fuel Air

Mass fraction of H2(or O2) (%) 78.2 23.3

Mass fraction of H2O(or N2) (%) 21.8 76.7

Utilisation (%) 14.0 32.4

Operating temperature (K) 1 000

Operating pressure (bar) 1.01325

Although both fuel and air channels are connected to inlet and exhaust manifolds; manifold effects are not considered in this study, ie. uniform inlet flow is presumed. The geometry, properties and operating conditions are detailed in Table 1. A relatively small-scale SOFC having reaction area 16 cm2, with fuel and air passages of 2mm ×1.5mm, is considered here.. The domain was tessellated with a rectilinear computational grid which was uniform in the plane of the PEN.

GOVERNING EQUATIONS

The physico-chemical transport phenomena are presumed to be governed by the following set of coupled equations:

( )

ρ =

div u 0 (1)

(

ρ

)

= − p+

(

μ

)

+

div uu grad div gradu SP (2)

D k μ = − P u S (3)

(

ρ

)

=

(

Γeff

)

i i i y y

div u div grad ₍₄₎

(

ρc_P T

)

−

(

k T

)

=S

div u div grad (5)

( )

2 i S H T i V F ′′′ _′′′ = Δ − (6)

where the enthalpy of formation, ΔH, is obtained from expressions for the heat capacity [20,21]. The continuity boundary condition for the two electrode walls is obtained from the mass flux divided by the mixture density, the former being computed as;

= ′′=

∑

′′  n _i i m m 1 ₍₇₎ where ′′ = H O H O i m M F 2 ₂ 2 (8) 2 ₂ 2 H H i m M F ′′ = −  (9) at the anode 2 ₄ 2 O O i m M F ′′ = −  (10)

at the cathode. Adiabatic and no-slip boundary conditions are applied at the outer walls. Electrochemical reactions are assumed to take place only at the interface between the electrode and electrolyte. Radiation heat transfer [22] is not considered in the present study.

Within the porous media, gas diffusion is calculated using two distinct approaches. In the first methodology, a simple algebraic formulation is adopted, based on. [23,24], Bosanquet’s relationship [25] is used to combine the effects of binary and Knudsen diffusion in terms of a single [26] effective diffusion coefficient, as follows − ⎛ ⎞ ε ⎜ ⎟ = _⎜ + _⎟ τ ⎝ ⎠ eff i ij i Kn D D D_, 1 1 1 (11)

(

)

= + ij ij i j T D pM V12 13 V13 1.75 2 0.00143 (12) = × i Kn pore i T D d M , 97 2 1000 (13) ε = − ε pore p d 2 d 3 1 (14)

with a constant tortuosity factor, τ฀= 3, which is within the range for the porous electrode of SOFCs [27] [28].

In the second approach, numerical reconstructions as described in Kenney et al. [29] of electrode geometries are used to implement random walk (Monte Carlo) calculations and subsequently directly obtain diffusion coefficients accounting for both microstructural and Knudsen effects.

The cell voltage, V, is just the open circuit voltage

(OCV), VOCV_{, less anode and cathode overpotentials and}

ohmic resistance in the electrolyte, as follows = − η − η −A C = − i

V E iR E ir (15)

g g a x R T R T P E E x F x F P 1 2 2 2 2 H 0 O H O 0 ln ln 2 4 ⎛ ⎞ ⎛ ⎞ = + ⎜_⎜ ⎟_⎟+ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (16)

For simplicity, the activation losses in Eq. (15) are lumped into as an overall area specific resistance, ri,

based on experimental measurements [18,30].

= + α + α + α + α i r 2 3 4 0.3044 0.408 0.8687 2.7861 2.9825 (17) α = − T 1000 1.1463 (18) where T is in deg. Celsius. In general, the resistance is

strongly dependent upon temperature, decreasing strongly with increasing temperature [31].

NUMERICAL METHOD

The object-oriented open source software suite OpenFOAM® version 1.5-dev was selected as the development platform for the physics and multi-scale calculation. The set of governing equations is solved by using a finite volume method written in the object-oriented C++ programming language.

The numerical model was implemented as a steady-state OpenFOAM® solver . A useful feature of OpenFOAM is the provision of a full set of implicit finite volume discretisation operators and associated linear system solver classes, allowing transparent representation of partial differential equations in the code. Another useful feature is that the discretisation schemes (eg Gaussian integration, upwind interpolation) used by the operators can be selected at run time, allowing the user free choice of schemes without any recoding or recompiling. The selection of linear solvers and their parameters are also chosen at run time. For the experiments reported here, symmetric linear systems were solved using conjugate gradient with incomplete Cholesky pre-conditioning (ICCG) [32] and asymmetric systems using bi-conjugate gradient schemes, BiCG [33], Bi-CGSTAB [34].

The model is solved on a set of computational domains, one global domain for the entire SOFC, and a subdomain for each of the air, fuel, electrolyte, and interconnect regions. Each domain can support its own fields. There are no fields specific to the interconnects. Pressure, momentum, and species mass fractions are solved in the fluid domains, and electrochemical heating is calculated in the electrolyte. Temperature is solved on the global domain, using heating source terms from the electrolyte and velocities from the air and fuel regions. The porous anode and cathode zones are implemented within the fuel and air regions, respectively, using Darcy’s law. Cell mappings between global domain grid cells and subdomain grid cells are established during grid generation.

Electrochemical reactions are assumed to occur at the electrode-electrolyte interfaces. The resulting mass fluxes are used to derive Dirichlet velocity BCs and Neumann species mass-fraction BCs on the air and fuel boundaries adjacent to the electrolyte.

The solution algorithm proceeds as follows: 1. Initialize meshes, constants, and other

parameters. Specify initial fields and BCs, physical properties, and average current density.

2. Solve Eqs (1)-(3) for pressure and momentum according to PISO algorithm [35].

3. Calculate mass diffusivity and solve Eq (5) for species mass fractions in fluid subdomains. 4. Map regional subdomain fields to global

domain.

5. Solve Eq (7) for global temperature.

6. Map computed temperature from global domain to fluid region subdomains.

7. Calculate Nernst potential, lumped internal resistance, current density, cell voltage, electrochemical heating.

8. Calculate electrochemical mass fluxes, hence fluid region BCs for velocity and species mass fractions at boundaries adjacent to electrolyte. 9. Repeat from step 2 until converged.

At convergence, the average current density is equal to that specified in step 1. Grid independence studies were performed and based on these a mesh size of 2 457 600 (160×160×96) was used in all calculations. RESULTS AND DISCUSSION

The cell is presumed to be operating under galvanostatic conditions with a mean current density of

i = 0.6 A/cm2_{, at a temperature of 1 000 K and}

pressure of

 

1.01325 bar (1 atm).

Table 2. Effective diffusion coefficients computed by two approaches Algebraic Numerical eff i D [m2/s] D [m2/s] AFL 1.9x10-5 4.6x10-5 ASL 8.4x10-5 1.3x10-4 CFL 2.1x10-6 1.1x10-5 CCCL 2.6x10-5 _3.0x10-5

Table 2 shows a comparison for the two diffusivity formulations. It can be seen that for the analytical approach of Eqs. (11)-(14), the diffusion coefficients are smaller, by up to one order of magnitude, than those obtained from the detailed numerical approach. These differences result in local concentration changes in excess of 100%. This demonstrates the importance of properly accounting for representative microstructures which are present in fuel cells when parameterising the governing equations. Accordingly, the results presented below, are for the numerically determined micro-scale model-based values of effective diffusivity.

2-5 November, 2010, Kaohsiung City, Taiwan

Figure 2 Temperature distribution for (a) co-flow, (b) counter-flow, (c) cross-flow.

Figure 3 Temperature [K] along the (a) spanwise y, and (b) streamwise x directions

Figure 4. Observed frequency distribution for AFL surface temperature.

Table 3. Predicted results, showing mean values and standard deviations. V[V] i [A/cm2] T [K] i σ T σ Co-flow 0.753 0.6 0.0435 1011.7 3.63 Counter-flow 0.750 0.6 0.0377 1008.3 1.32 Cross-flow 0.750 0.6 0.0383 1009.5 2.55

The results presented below illustrate the effect of the flow configuration on temperature distribution and uniformity. Figure 2 shows the temperature distribution on the AFL, at i = 0.6 A/cm2, for the 3 cases of co-flow, counter-flow, and cross-flow. Higher temperatures are observed at the interface of the AFL and electrolyte [36,37] due to Joule heating, and the lowest temperatures are always at the flow inlets, i.e., the working fluids act as coolants, with the temperature rising along the flow paths. A high temperature is observed at the outlet for co-flow, Fig. 2(a), at the middle for counter-flow, , Fig. 2(b), and at the outlet corner for cross-flow, Fig. 2(c). From inspection of Table 3, it is readily apparent that the co-flow configuration results in the highest average cell temperature, and counter-flow the lowest. Moreover counter-flow yields the smallest standard deviation in temperature for the three cases. There is little difference in the cell voltage for the three flow cases, however. The reader will note that the AFL surface has a non-uniform temperature distribution across the flow channels; The temperatures underneath the ribs are lower than those beneath the channels, as the current density under the ribs is lower, due to reduced mass transfer [38].

This is further illustrated in Fig. 3 which shows temperature values plotted along the centerline of the AFL surface. It can be seen that along the spanwise y-direction, local extrema caused by variations in the reaction-rate due to the presence of the interconnect ribs, are readily apparent, with local maxima occurring underneath the flow channels and minima underneath the ribs, Fig. 3(a). The counter-flow case exhibits a lower overall temperature than the co-flow case. Both cases exhibit relatively uniform temperature distributions, whereas the cross-flow profile varies significantly along the cross-wise y-direction. The smoothly varying temperature profile in the stream-wise

x direction is shown in Fig. 3(b). Local extrema observed

for cross-flow (only) are somewhat smaller, as might be expected. The counter-flow case has the most uniform temperature distribution, whereas the co-flow case displays the largest change in temperature distribution along the centerline of the ASL. Plots of results obtained at prescribed current density i = 0.2 A/cm2 _{(not shown)}

have qualitatively similar profiles as those shown (

i = 0.6 A/cm2_{.) with slightly lower temperatures.}

The observed frequency distribution (probability density function) of AFL surface temperature is shown as a histogram in Fig. 4. This further assists in a relative evaluation of the temperature uniformity for the three cases to be made. The co-flow and cross-flow cases display a relatively large variance in temperature distribution, whereas the counter-flow case spans a

narrower range consistent with Figs. 2-3 and Table 3. It can be seen that the modal value for counter-flow is a distinct peak, whereas for cross-flow a much broader distribution with no obvious global maxima is apparent. Counter-flow consistently displays the most uniform temperature distribution of the three configurations.

Figures 5 and 6 show current density distribution across the electrolyte. A high current density is observed at the fuel inlet. Local variations in the current density associated with the presence of the interconnect ribs are apparent and are associated with the coupled effects of charge, heat, and mass transfer. This is because the current density is dependent on both the temperature and mass fraction of the reactants, which in turn are a function of Joule heating (i.e. current density) and geometry, respectively. It can be seen from Fig. 6, that local fluctuations in current density are quite significant. The co-flow and counter-flow cases exhibit near-identical profiles in the cross-wise y-direction. Local maxima and minima are observed in both the x and y directions, for cross-flow (not apparent in the corresponding temperature profile). Figure 7 shows observed frequency distribution for current density as a histogram. For all three cases, local current densities are distributed around the (prescribed) identical mean current density in a ‘bell-shaped’ profile (approximately). As is shown in Table 3, the co-flow case has the largest standard deviation, and the counter-flow case the smallest one. However differences in this measure of non-unformity between the three cases, are very small.

Since the ASL is relatively thick, the hydrogen percolates well through the porous media, and therefore the gradient of hydrogen mass fraction is small. However that of oxygen is much larger. This is due to the relatively thin CCCL and the lower effective diffusivity of oxygen which impedes oxygen transport under the rib [39], and will lead to oxygen depletion at higher current densities.

Figures 8-9 show mass fraction profiles of hydrogen; Figures.10-11 are similar representations of the corresponding oxygen mass fractions10. Local mass fractions decrease along the main flow directions in a near-linear fashion, with some oscillations in the cross-wise directions. It was observed that the differences in hydrogen mass fraction between the fuel channels and the reaction-sites are relatively large, compared to the variations at the AFL-electrolyte interface, Fig.9. This is due to the relatively thick ASL, which enables sufficient hydrogen to permeate under the solid ribs. For oxygen, the converse is true, namely that differences in mass fraction between the air channel and the CFL-electrolyte interface are quite small, in comparison to those on the surface itself, Fig. 11. This is caused by the thin CCCL which prevents sufficiently-concentrated oxygen gas from reaching the reaction sites located under the ribs.

Figure 5. Current density [A/m2] for (a) co-flow, (b) counter-flow, (c) cross-flow.

Figure 6 Current density [A/cm2] along y and x directions

Figure 7 Observed frequency distribution for current density.

2-5 November, 2010, Kaohsiung City, Taiwan

For this reason, it can be seen that: while local fluctuations in hydrogen mass fraction, Fig. 9(a), are relatively small in comparison to the overall reduction from inlet to outlet, Fig. 9(b); the same certainly cannot be said for oxygen mass fractions, Fig. 11, where the fluctuations are large in comparison to the overall differences.

While this state-of-affairs might, seemingly, be obviated by increasing the thickness of the CCCL, (or increasing the effective diffusivity, or running at a lower utiliza

tion).

, this may have the adverse effects not only of decreasing overall performance and increasing weight, but also could result in mechanical failure due to stresses associated with the different thermal expansion coefficients typically found in CCCL and the CFL materials.

Figure 8 Mass fraction of hydrogen on electrode surface

Figure 9. Hydrogen mass fraction along y and x directions

Figure 10 Mass fraction of oxygen on electrode surface

Figure 11 Oxygen mass fraction along y and x directions

CONCLUSIONS

Calculations were performed for planar-type anode-supported SOFCs using the object oriented continuum mechanics system, OpenFOAM®. The calculations took into account all significant electrochemical and transport phenomena. The model was based on a coupled set of conservation equations for mass, momentum, species, and energy. A lumped internal resistance correlation which was derived from experimental data was employed to compute potential losses and Joule heating terms. The fuel cell considered was composed of nine layers; lower interconnect, air channel, CCCL, CFL, electrolyte, AFL, ASL, fuel channel and top interconnect.

Effective diffusion coefficients were computed using two distinct approaches (i) using an algebraic correlation based on commonly-used engineering concepts such as ‘pore-size’ and ‘tortuosity’, and (ii) a more sophisticated numerical scheme based on reconstruction of representative geometry, and numerical solution for volume-averaged diffusion coefficients by means of random walk calculations. It was shown that large differences in effective diffusion coefficients arise, and

that these will change the quantitative solutions for the entire fuel cell significantly. Subsequent calculations were based on the latter formulation.

Three distinct cases were considered, co-flow, counter-flow, and cross-flow, in order to investigate the impact of flow configuration on temperature uniformity of the SOFC.

The lowest temperature was observed at the inlet, and the highest at the outlets for the cases of co-flow and cross-flow, whereas for counter-flow, the maximum temperature was located at the middle of the cell. Temperature profiles were substantially influenced both by the air flow paths and the presence of the interconnect ribs, with local extrema being observed on the air side, in particular. The co-flow case exhibited the highest cell voltage and mean temperature, and also the largest temperature difference and associated standard deviation. Conversely, the counter-flow case displayed the smallest temperature difference (and standard deviation). It is concluded that the counter-flow design has the most uniform temperature distribution, for the class of problem and parameters considered in this study. The design that minimizes the variation in temperature distribution throughout the cell is an important consideration for the long-term mechanical stability of this type of planar fuel cell.

ACKNOWLEDGEMNTS

Financial support for this work was provided through the Solid Oxide Fuel Cell Canada Strategic Research Network from the Natural Science and Engineering Research Council¸ Forschungszentrum Jülich GmbH, the National Research Council of Canada, Defence Research and Development Canada, and the Program of Energy Research and Development of Natural Resources Canada.

NOMENCLATURE

p

c heat capacity, J/kg K

D diffusivity, m2_/s

dp diameter of spherical particle, m

E Open circuit voltage, V

F Faraday constant, 96,485 C/mol

enthalpy, J/kmol

i current density, A/m2

i´´´ _{volumetric current density, A/m}3

k thermal conductivity, W/m K

kD permeability, m2

Mi molecular mass of gas species i, kg/mol

m′′ mass flux, kg/s

Mij mean molecular mass, kg/kmol

P pressure, Pa

ri lumped internal resistance, ohm-m2

R resistance, ohm-m2

Rg universal gas constant, 8.314 J/K mol

T temperature, K or °C

u velocity, m/s

V voltage, V

Vi diffusion volume for species i yi mass fraction of species i

ε porosity η overpotential, V μ dynamic viscosity, Pa s ρ density, kg/m3 τ tortuosity Superscripts or Subscripts A _anode C cathode eff effective 0 reference REFERENCES

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