1. FUEL CELL STACK MODEL BASED ON MULTIPLIPLY SHARED SPACE METHOD Steven B. Beale* and Sergei V. Zhubrin** * National Research Council, Ottawa, Canada ** Flowsolve Ltd, London, United Kingdom

2. 2 Contents Introduction to distributed resistance analogy Introduction to solid oxide fuel cells Governing equations Cell and stack geometries considered Comparison of present model with detailed calculations Discussion of results Conclusions and future work

3. 3 DISTRIBUTED RESISTANCE ANALOGY Problem: Not possible to make grid fine enough to capture flow around all tubes. Solution: Prescribe distributed resistance, F, and heat transfer coefficients,, and solve for superficial flow around baffles. Values of f, and Nu, obtained from experiments, analysis, or detailed numerical simulations.

4. 4 DISTRIBUTED RESISTANCE ANALOGY Original scheme developed by Patankar and Spalding (1972) only shell-side flow solved for. Key concept: replace diffusion terms with prescribed rate terms Problem: In general CFD codes only admit to a single value for pressure, p. How to solve for two phases at a given point? Multiply-shared Space method (MUSES) developed in 1990s to solve this problem.

5. 5 SOLID OXIDE FUEL CELL Converts chemical energy to electricity and heat. Basic components anode, cathode, electrolyte. O 2- ions produced at cathode combine with electrons and H 2, at anode to form H 2 O. Operate at 800-1 000 °C, Thermally induced stresses, and performance of cell important. Cells stacked together to increase voltage; Fuel/air introduced through manifolds. Stainless steel interconnects make electrical connection. Uniform flow/pressure important.

6. 6 SOLID OXIDE FUEL CELL Many CFD companies now developing PEM and SOFC models – None of these codes can be scaled to model stack. Numerous flow channels. Same meshing problem as for heat exchangers. Authors have developed original stack model. System treated as sandwich of four materials; air, fuel, electrolyte (including electrodes) and interconnect. Idealised geometry: All fluid and solid regions simple rectangular-shaped zones; planar ducts - Allowed for comparison with detailed model for SOFC stack. Single-cell and 10 cell stack considered: Fuel/air in cross flow.

7. 7 SOLID OXIDE FUEL CELL

8. 8 THERMODYNAMICS Nernst potential, E, given by When current flows, actual potential, V, is lower, i current density (A/m2), are overpotentials, r e is electrolyte resistance All lumped together as a internal resistance, r. Goal to develop DRA model for SOFC stack design. Detailed numerical model (DNM) used to validate model

9. 9 THEORY General form: (i) Transient, (ii) convection, (iii) inter-phase transfer, (iv) diffusion or within phase transfer, and (v) source. 1. Continuity: Mass sources computed from Faradays law. Coded as volumetric sources.

10. 10 THEORY 2. Momentum: F is distributed resistance. If then. Viscous terms in zero fluids, F is zero in manifolds.

11. 11 THEORY 3. Heat transfer Heat sources due to Joule and Peltier effects. Two terms combined as single volumetric term. Heat transfer coefficient Mass transfer effects accounted for by using

12. 12 THEORY 4. Mass transfer Mass sources/sinks T-state value. Wall (not bulk) values are needed for Nernst equation Mass transfer driving force for 1-D convection/diffusion Blowing parameter g* is conductance zero mass transfer

13. 13 MUSES method Multiply-shared Space method. Separate spaces for each material. Porosities used (except manifolds). Inter-phase values fetched across structured rectilinear mesh.

14. 14 Implementation FORTRAN code inPLANTed in PHOENICS code: PATCH (fu2el,VOLUME,NI1,NI2,NJ1,NJ2,NK1,NK2,1,LSTEP) VAL=TEM1[,,+:FTOE:] COVAL(fu2el,TEM1,GRND,GRND) IF(INDVAR.EQ.INAME('TEM1 ').AND.NPATCH.EQ.'FU2EL ') THEN LFVAL =L0F(VAL) LFTEM1=L0F(INAME('TEM1 ')) DO 13801 IX=IXF,IXL IADD=NY*(IX-1) DO 13801 IY=IYF,IYL I=IY+IADD L0TEM1=LFTEM1+I 13801 F(LFVAL+I)=F(L0TEM1+NFM*42) ENDIF

15. 15 COMPUTE ALGORTHM 1. Solve transport equations. 2. Solve Nernst potential, 3. Compute values for cell resistance, r, and heat and mass sources/sinks. 4. Repeat steps 1-3 until convergence is obtained.

16. 16 VOLTAGE CORRECTION Both galvanostatic and potentiostatic situations occur. For former adjust voltage iteratively until the desired current reached. V* is V-value of at previous iteration and is a voltage correction is desired current density. This step arguably redundant (can compute cell voltage from mean current density direct)

17. 17 DETAILED NUMERICAL MODEL (DNM) Used for comparison. Rate eqns. not assumed in DNM. Fine mesh. No volume averaging.

18. 18 SINGLE CELL RESULTS : Current density Both cell temperature and Nernst potential affect the current density since r is inversely proportional to temperature DRA DNM

19. 19 Nernst Potential Nernst potential more influenced by H 2 and H 2 O mass factions than by O 2, due to the stoichiometry. E decreases as concentrations of O 2 and H 2 decrease and H 2 O increase from inlet to outlet. DRA DNM

20. 20 Temperature Temperature variations are inevitable: If heat production uniform, maximum would be at top-right corner. Since both E and i are maximum at top-left; peak shifts Metallic interconnect serves as a thermal fin DRA DNM

21. 21 Anode wall H 2 mass fraction H 2 contours nominally perpendicular to, and decreasing along fuel streamlines NB: DRA values wall values DRA DNM

22. 22 Anode wall H 2 O mass fraction Good agreement between methodologies. DRA DNM

23. 23 Cathode wall O 2 mass fraction Current density identical at anode/cathode (thin electrolyte), so source terms of H 2, H 2 O and O 2 coupled, and mass fraction contours skewed, more pronounced at high current densities. DRA DNM

24. 24 10-CELL STACK RESULTS: Air side velocity vectors Fine detail of motion lost, Characteristic parabolic-shaped (DNM) velocity profile associated with fluid flow in planar ducts absent from DRA Manifold-stack assembly well designed pressure/velocity fields uniform: Manifolds losses small compared to cell. DRA DNM

25. 25 Air side pressure Pressure fields in close agreement. Problems can arise in large stacks, where mass transfer into results in stack pressure gradient decreasing upwards. results in variations in flow field. Minimised by ensuring cell passages small in comparison manifolds. DRA DNM

26. 26 Air side bulk O 2 mass fraction O 2 mass fractions constant from cell-to-cell. Good agreement between methodologies. NB: DRA values bulk values, Significant variation across micro-channels; maximum for high current density (short circuit, V 0) diffusion limit; mass transfer rate-limiting factor. Important that wall values obtained to avoid over-prediction of Nernst potential from bulk values DRA DNM

27. 27 Temperature Zig-zag temperature field in DNM Secondary thermal distribution due to ordering of fluids Occurs even if heating perfectly uniform and flow constant DRA DNM

28. 28 DISCUSSION Problem: In original DRA secondary thermal effects lost, due to volume-averaging. Solution: Computational cells in z direction coincided with actual fuel cells. Inter-phase heat transfer terms computed as pairs eg. air-electrolyte pair, Similarly treatment for air-interconnect, fuel-interconnect but for fuel-to-electrolyte pair sources computed as. Recovered ordering of streams

29. 29 DISCUSSION Complex interaction between physical chemistry and transport phenomena; subject known as physical-chemical hydrodynamics Voltage correction algorithm converged provided a reasonable initial guess is made. If V-i curve is linear, and r is actual resistance, correct cell value predicted after 1 iteration: In practice V-i curve non- linear and r only an estimate Advantage voltage-correction approach, is r need not be exact, Alternative can compute the average resistance and current density by integration, and hence obtain voltage. Same result is obtained.

30. 30 DISCUSSION PLANT/IMMERSOL good for in-line programming, but need way to nest multi-line commands e.g. using curly braces {} or the like. In some aspects (eg voltage correction) PLANT is a little clumsy. (Compromise)

31. 31 CONCLUSIONS Most CFD vendors developing single cell models which cannot be scaled to stack level We developed first CFD stack-level model Excellent agreement between DRA and DNM results Substitution of appropriate values for and F leads to reasonable results at fraction of cost 1-D mass transfer analysis yields wall mass fractions in agreement with detailed calculations even when significant variations in m w and m b

32. 32 steven.beale@nrc.ca svz@flowsolve.com