Chap.6 Fuel Cell modeling

6.1 Putting it all together: A basic fuel cell model (6.1) Fig. 6.1 Pictorial summary of major factors that contribute to fuel cell performance.

(6.2)

Effect of leakage current on fuel cell performance Fig 6.2 Pictorial illustration of the effect of a leakage current loss on overall fuel cell performance. (6.3) (6.4)

Typical Value for PEMFC Low-temperature PEMFC vs. High-temperature SOFC Parameter Typical Value for PEMFC Typical Value for SOFC Temperature 350 K 1000 K Ethermo 1.22 V 1.06 V j0(H2) 0.10 A/cm2 10 A/cm2 j0(O2) 10-4 A/cm2 α(H2) 0.50 α(O2) 0.30 ASRohmic 0.01 Ω∙cm2 0.04 Ω∙cm2 jleak 10-2 A/cm2 jL 2 A/cm2 c 0.10 V Fig. 6.3 Comparison of our simple model results for typical PEMFC versus typical SOFC.

6.2 A 1D Fuel Cell model 6.2.1 Flux Balance in Fuel Cells Flux 14 = flux 5 = flux 1-flux 4 = flux 8 – flux 13 (6.5) (6.6) Where j : current density (A/cm2) F : Faraday’s constant (96484 C/mol) J : molar flux (mol/s∙cm2) flux 2-flux 3 = flux 6-flux 7 = flux 12-flux 9-flux 5 (6.7) anode membrane cathode

Figure 6.4. Flux details for (a) 1D PEMFC model and (b) 1D SOFC model.

(6.8) Where and represent the net flux into the anode catalyst layer, across the electrolyte, and out of cathode catalyst layer, And j/2F represent the water generation rate at the cathode (6.9) Using Eq. 6.9 and 6.6, we can write Eq. 6.8 in terms of J and α (6.10) By combining Eq. 6.6, 6.8, 6.9 and 6.10 (6.11) For PEMFC (6.12) For SOFC

6.2.2 Simplifying Assumptions

Assumptions Convective transport is ignored. Diffusive transport in the flow channels is ignored. All the ohmic losses come from the electrolyte membrane The anode reaction kinetics is ignored The catalyst layers are extremely thin or act as interfaces. 6. Water exists only as water vapor. By making the following assumptions, most of items in Table 6.2 can be ignored in the model.!!

SOFC structure affects modeling assumptions In SOFC, there are three structures; anode-supported, cathode-supported, and electrolyte-supported SOFCs. However, in anode supported SOFC several of the modeling assumptions listed above prove problematic. High anodic reaction losses in anode-supported SOFCs. This is because there is hydrogen diffusion limitations in thick anode structures. The assumption described above should be used only for cathode- and electrolyte-supported SOFCs.

6.2.3 Governing Equations Electrode layer (6.13) Electrolyte (6.17) For H2, O2, H2O and N2 (6.13) Electrolyte For SOFC Relation between O2- flux to the current density (6.17) Then, we can determine the ohmic voltage loss from Eq. 4.11 (6.18) (where tM is the thickness of the electrolyte) For PEMFC (6.20)

Diffusion models for fuel cells Binary Diffusion Model (6.14) 2. Maxwell-Stefan Model At low density, multi-component gas diffusion can be approximated by the Maxwell-Stefan equation (6.16)

Catalyst Since the oxygen partial pressure at the cathode is the dominant factor in determining the cathodic overvoltage, the simplified form of the Butler-Volmer equation can be used. The 4 in the denominator represents the electron valence number for an oxygen molecule. (6.21) For an ideal gas (p=cRT) , the above equation becomes Where pC isi the total pressure at the cathode xO2 is the oxygen mole fraction at the cathode layer (6.22)

6.2.4 Examples 1D SOFC Model Example (6.23) (6.24) H2 and H2O transport in anode (6.23) By solving the above equations, the linear profiles for the hydrogen and water concentrations can be calculated. (6.24)

(6.25) (6.26) (6.27) Where tA represents anode thickness Following a similar procedure, we can also obtain the oxygen profile at the cathode (6.26) By combining Eq. 6.26 and 6.22, we can calculate the cathode overpotential (6.27)

(6.28) (6.29) From Eq. 6.18 and 6.19, we can calculate the ohmic loss Finally, we obtain the fuel cell voltage (6.29)

Example to predict the performance of a realistic SOFC The parameter values and conditions shown in Table 6.4

At a current density of 500 mA/cm2, output voltage? From Eq. 6.29

1D PEMFC Model Example (6.33) (6.34) (6.35) H2 and H2O transport in anode (6.33) The above equations have the following solutions (6.34) (6.35) Where α is an unknown

From the above equations, the hydrogen and water concentrations at the anode-membrane interface can be calculated. (6.36) (6.37) In a similar manner the oxygen and water concentrations at the cathode-membrane interface can be obtained. (6.38) (6.39)

The biggest challenge of our PEMFC model is to find the ohmic overpotential. The water profile In the membrane along with the unknown α* can be obtained by solving the membrane water flux Eq. 6.20. Eq. 6.37, 6.39 serve as our boundary conditions. The solution to Eq. 6.20 has been previously worked out. (6.40) Using this equation, the water content λ at the anode-membrane Interface “b” and the cathode-membrane interface “c” can be obtained. (6.41) (6.42)

Two assumption to solve nonlinear equations of Nafion water contents Water content in the Nafion membrane increases linearly with water activity. Thus, the following linearized form of Equation 4.34 can be used. (6.43) (6.44) Water diffusivity in Nafion is constant. This is a fairly reasonable assumption since the water diffusivity does not change much over most water content ranges.

(6.45) (6.46) Since , combining Equations 6.43 and 6.37 gives Similarly, combining Equations 6.39 and 6.44 for the cathode side yields (6.46)

Example to predict the performance of a realistic PEMFC The specific fuel cell properties were listed in Table 6.5

Incorporate these properties into Eq. 6.45 and 6.46. (6.47) (6.48) and Eq. 6.41 and 6.42 become (6.49) (6.50)

Equate Eq. 6. 47 with Eq. 6. 49 and Eq. 6. 48 with Eq. 6. 50 Equate Eq. 6.47 with Eq. 6.49 and Eq. 6.48 with Eq. 6.50. And then value of α and C can be calculated.; α=2.25, C=2.0 From Eq. 4.38 and 6.40, the conductivity profile of the membrane can be determined (6.51) Finally, the resistance of the membrane can be determined by using Eq. 4.40 (6.52)

Thus, the ohmic overvoltage due to the membrane resistance in this PEMFC is The cathodic overvoltage is Finally, the fuel cell voltage is

Gas depletion Effects: Modifying the 1D SOFC Model In realistic case, oxygen can be depleted at these boundaries depending on the relative rates of oxygen supply and consumption. At the cathode outlet (6.56) Replace and with known values. From the SOFC flux balance equation 6.12 (6.57) The oxygen inlet flux is regulated according to the stoichiometric number

(6.58) (6.59) (6.60) From the definition of the stoichiometric number Plugging the above equation into Equation 6.57 solve for in terms of the stoichiometric number (6.59) Since there is no nitrogen consumption (6.60) Where ω represents the molar ratio of nitrogen to oxygen in air (typically ω = 0.79/0.21=3.76)

(6.61) (6.63) From Eq. 6.59, 6.60 and 6.56, we can get Plugging Eq.6.61 into 6.29 gives us the final model equation. (6.63)

Stoichiometric number Stoichiometric number is the ratio between the provided rate of species i and consumed rate. λ=2 means that twice as much reactant as needed is being provided to a fuel cell. A large λ is wasteful. For a less λ than 1, reactant depletion effects become more severe. For this SOFC model, the hydrogen and oxygen stoichiometric number are;

In previous SOFC example with λO2=1.5 and j=500 mA/cm2, this modified model gives; A much higher cathodic overvoltage compared to the first example is obtained. This is because the low λ O2 value causes significant gas depletion effects.

6.2.5 Additional Consideration Thermal effect Convective heat transfer via fuel and air streams Conductive heat transfer through the fuel cell structure Heat absorption/release from phase changes of water Entropy losses from the electrochemical reaction Heating due to the various overvoltages Mechanical effect

6.3 Fuel Cell Modeling based on Computational Fluid Dynamics The 3D governing equations containing single phase without water 1. Mass conservation (E.1) rate of mass change per unit volume net rate of mass change per unit volume by convection

2. Momentum conservation net rate of momentum change per unit volume by pressure viscous pore friction structure rate of momentum change per unit volume convection 3. Species conservation (E.3) net rate of species mass change per unit volume by diffusion electrochemical reaction rate of a species mass change per unit volume convection

(E.4) 4. Charge conservation (E.5) From the overall charge neutrality (E.6) (E.7)

By incorporating Ohm’s law (E.8) At the cathode or anode (E.9)

Fig. 6.5. isometric view of serpentine flow channel fuel cell model (500 μm channel feature size).

Table 6.6 Physical Properties used in CFD fuel cell model Fig. 6.6. Cell j-V curves for serpentine flow channel model. Activation, ohmic, and concentration loss are observed.

Fig. 6.7. Oxygen concentration in cathode at 0.8 V overvoltage. The plan view shows the oxygen concentration profile across the cathode surface. Fig. 6.7. Oxygen concentration in cathode at 0.8 V overvoltage.