While the reduced model agrees well with the full set when every cell is operated at identical inlet conditions, we expect that this may not be the case when there are signi…cant perturbations between cells since the leading-order stack model does not consider streamwise di¤usive transport. In order to study how the reduced model fares, we introduce a leading-order perturbation in the inlet velocities at the cathodes for a 10-cell stack, given by
Uc;jin =Ucin+j U; (9.28)
wherej denotes the number of the cell in the stack (see Fig. 9.2);Uc;jin is the velocity of cellj and U is the increment in inlet velocity from cell to cell. In this particular case, U is chosen such that the inlet velocity increases by a factor two between the …rst and the last cell, which is larger than the typical variation in inlet velocities between 5 and 25% that have been studied for U- and Z-shaped external manifolds of stacks comprising between 25 and 100 cells [13, 15, 134]. The maximum relative error between the reduced model and the full set is around 2% for the perturbed global polarization curve, as can be inferred from Fig. 9.4. The order of magnitude of the error is related to the inherent error in neglecting the streamwise di¤usive terms for conservation of charge (electrons) and energy: namely, O(h2stack=L2); where hstack is the height of the stack;
this ratio loosely represents the order of magnitude di¤erence between the streamwise and normal di¤usive terms in the elliptic governing equations (see. Refs. [96, 102, 103]
for more details), and is for the 10-cell stack 10 1.
Proceeding with the distributions of the local current density in Fig. 9.7 for the
…rst, …fth and tenth cell in the stack at Estack = 6 V (corresponds roughly to around 0.6 V for each cell, which is a typical operating point for the PEMFC), we see that
9.7. Verification with perturbations between cells 157
0 0.02 0.04 0.06 0.08 0.1
5600 5800 6000 6200 6400 6600 6800 7000 7200 7400
x / m
Current density / A m-2
Figure 9.7: Local current density distribution for a 10-cell stack (Estack= 6 V) along thex-axis at the interface between the cathode catalyst layer and membrane (VII in Fig. 9.2) in cell ( ) 1, (N) 5, and (H) 10 for the full set of equations and corresponding
values in cell ( ) 1, ( ) 5, and ( ) 10 for the reduced counterpart.
the local current density varies due to redistribution between cells, whereas the reduced model predicts the same local current density for each x-location in the stack – here, the maximum relative error is around 4%, again on the order ofO(h2stack=L2):Similarly, the relative error for the local temperature distribution is around 4% (see Fig. 9.8) and around 0.1% for the oxygen concentration. While the former is expected, the latter is signi…cantly lower than that of the local current density and temperature; the reason can be found in the nature of the dependent …eld variables in the stack: transport of energy and charge (electrons) occur throughout the stack, whereas transport of momentum, charge (ions), species, and mass are limited to each cell of the stack. Thus, heuristically speaking, we expect the error in neglecting the streamwise second order di¤usive terms for energy and charge of electrons to be on the order ofh2stack=L2;and for the remaining transport mechanisms to be on the order ofh2cell=L2; wherehcell is the height of a unit cell in the stack;provided that the perturbations are not so large as to introduce larger errors through the source terms, Eq. 4.13, which couple the global current density with
158 9. Asymptotically Reduced Model for a PEMFC Stack
the variables that are bound to a cell.
0 0.02 0.04 0.06 0.08 0.1
6.5 7.2 7.9 8.6 9.3 10
c O 2 / mol m-3
0 0.02 0.04 0.06 0.08 0.11.5
1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5
x / m
∆T / K
Figure 9.8: Local distributions for a 10-cell stack (Estack= 6 V) along thex-axis at the interface between the cathode catalyst layer and the membrane (VII in Fig. 9.2) for the full set of equations for temperature in cell ( ) 1, (N) 5, and (H) 10; concentration
of oxygen in cell ( ) 1, (J) 5, and (I) 10; and the corresponding predictions of the reduced set in cell ( ) 1, ( ) 5, ( ) 10.
Decreasing the stack voltage to around 1 V, which roughly amounts to 0.1 V or less for each cell, increases the maximum relative errors for temperature, local current density and oxygen concentration to around 100%, 30%, and 1%, respectively (not shown here); i.e. all errors are – loosely speaking – of the order of O(h2stack=L2); suggesting that the perturbation is now so large that even the …eld variables limited to each cell are a¤ected. While the latter comparison at a stack voltage of 1 V is more of an academic interest since it does not make any practical sense to operate a fuel cell stack in the mass-transport limiting region, these …ndings suggest that one may want to include the streamwise second-order terms if perturbations are signi…cant; however, this should be accomplished without solving for the full set of equations since the computational cost would then again become prohibitive if one seeks to include the essential physics at a
9.8. Computational Cost and Efficiency 159
local level. For this purpose, we therefore suggest two possible extensions: The …rst would be to solve the reduced stack model and add the solution for the next order term in an asymptotic series expansion with the "small parameter" related to h2stack=L2 and possibly an estimate of the perturbation. The second would be to split the dependent
…eld variables into two sets of equations; i.e., to solve for conservation of energy and charge of electrons with the full set of elliptic governing equations and couple these in an iterative manner with the reduced model for conservation of momentum, mass, species and charge of ions. Both approaches should lead to substantial savings in computational cost.