Model equations

The model equations are described in four subsections: energy and mass balance equations, an electrical model, and a membrane model.

The equations for energy and mass balance are formulated as classical balance laws based on the equation of continuity, i.e. Eq. (2.15) derived in Chapter 2. The

Figure 5.1: 3D view of the cell model, indicating the cross-sectional cell area As, the cross-sectional area of the gas channels of the anode A_aand of the cathode A_c, the effective channel length Le f f, and the cut for Fig. 5.2.

Figure 5.2: Sideview of the cell model along the cut indicated in Fig. 5.1, illustrating the direction of the molar flow N_a,i of species i in the anode gas channel and the molar flow N_c,jof species j in the cathode gas channel from the channel inlets to the channel outlets.

mass balance is set up for H₂,O₂,N₂,H₂O^v,H₂O^l. Energy balance equations are formulated for the gas channels of anode and cathode and for the solid material.

Their basic form is given by Eq. (2.50) in Chapter 2. The different balance equa-tions are described below in detail.

5.3.1 Energy balance

First, the energy balance of the gases is described. The considered species are i=H₂,H₂O^von the anode side and i=O₂,N₂,H₂O^v on the cathode side. Liquid water is assumed to exist in form of small droplets at the surface of the channels. It is further assumed that the liquid water has stack temperature. The energy balance of the gas reads where c_q,i is the concentration of species i and q=a,c denotes the anode side and the cathode side, respectively. Nq,i is the molar flux of species i along the channel of electrode k. Cidenotes the heat capacity and Tqis the gas temperature.

A_q is the cross-sectional area of the channel as indicated in Fig. 5.1, and A_sg is the heat-exchange area between channel and solid material. Furthermore, U_sg describes the heat transfer between the gas and the solid material. T_sol is the stack temperature and ∆Hvap is the enthalpy of evaporation of water. The term on the left-hand side of the equation describes the change of internal energy in a volume element. The terms on the right-hand side describe from left to right:

the transfer of internal energy by convection, the heat transfer between the gas and the components of the stack, and the heat consumption or production due to evaporation and condensation of water.

The energy balance equation of the solid material links the processes in the gas channels of the anode side and the cathode side. It is given by

ρsC_sA_s∂tT_sol(x,t) =A_sk_s∂²_xT_sol(x,t) where ρs is the density of the solid material, C_s is its heat capacity, and A_s is the cross-sectional area of a single cell in the stack (see Fig. 5.1). ks denotes the

heat conduction coefficient, n_chanis the number of channels in one cell, and A_ss is the heat exchange area between the stack and its surroundings. Furthermore, Uss

is the heat transfer coefficient and T_sur denotes the ambient temperature. d_y is a scaling coefficient, which takes into account the enlargement of the contact area between the gas and the porous catalyst layer due to the cross-diffusion under the flow-field bridges. The width of a gas channel is denoted as w and ∆Sq is the entropy of reaction in electrode q. Finally, F is the Faraday constant, ηc

is the activation overpotential of the oxygen reduction reaction, and I denotes the current density. The change of internal energy in a volume element of the solid stack material, which is described by the term on the left-hand side, is given by the following source and sink terms on the right-hand side of the equation:

(a) Heat conduction driven by a temperature gradient within the stack; (b) heat transferred by convection of liquid water; (c) heat transfer between bulk material and gases of anode and cathode sides; (d) heat transfer between bulk material and surroundings; (e) heat generation and consumption due to condensation or evaporation of water in anode or cathode; (f) heat generation due to the reaction entropy and dissipation. As a cell contains several channels, the terms b, c, e, and f are multiplied by the number of channels n_chan.

5.3.2 Mass balance

For the mass balances the convective transport along the channel, the fuel con-sumption of H₂ and O₂, the production of H₂O^v at the cathode, the phase transi-tion of water, and the transport of water vapor through the membrane are taken into account. Below, the mass balance equations for the species on the cathode side are given. The corresponding equations for the anode side are analogous.

The mass balance equation of oxygen in the cathode is

∂t

£cc,O₂(x,t)A_q¤

=−∂x[Nc,O₂(x,t)]−dywI(x,t)

4F . (5.3)

The concentration of oxygen changes due to the convective transport along the channel and the consumption of oxygen at the cathode.

The mass balance equation for water vapor in the cathode is given by

∂t

The change in the concentration of water vapor is balanced by the convective transport of water vapor and convection of liquid water. Moreover, the production of water in the electrochemical reaction and the net-transport of water through

the membrane are taken into account. The net-migration coefficient αnet is the number of water molecules per proton that is transported through the membrane.

It can be positive or negative, depending on the direction of the net-water flux.

The mass balance equation for liquid water in the cathode is

∂t

where k_c is the condensation rate constant and h is the channel height. R denotes the ideal gas constant, p_H2O^v is the partial pressure of water vapor in the cathode gas channel, and Psat is the saturation pressure. The concentration of liquid water changes due to convective transport of liquid water along the channel and the phase transition of water.

5.3.3 Electrical model

The electrical model is similar to the model developed by Golbert et al. [82]. The cell potential is calculated from the following equation

V_cell=V_oc−RT_sol where V_ocis the open circuit voltage and a_O2 denotes the activity of oxygen in the cathode. The exchange current density is denoted as i0,c. The membrane is de-scribed by its thickness t_mand its conductivityσm. The cell potential V_cellis given by the open circuit voltage, which is reduced by the considered losses. The second term on the right-hand side of Eq. (5.6) describes the activation overpotential of the oxygen reduction reaction. The third term describes the ohmic losses due to the limited membrane conductivity.

5.3.4 Membrane model

The transport of water through the membrane is assumed to be a superposition of the electro-osmotic drag and the diffusion due to a concentration gradient across the membrane. The net-migration coefficient of water through the membraneαis modeled as described in Ref. [82]:

αnet=n_drag−F

where n_dragis the electro-osmotic drag coefficient describing the number of water molecules carried through the membrane for each proton. Dm,H₂O is the diffu-sion coefficient of water in the membrane, k_m,p the water permeability, and µ_l the viscosity of water. Three transport mechanisms are considered in the equa-tion above. The first term on the right-hand side describes the transport of water by electro-osmotic drag. The second and third terms describe the diffusion due to a concentration and a pressure gradient in the gas phase. The dependency of the membrane water contentλm on the water vapor activity aq at the membrane interface is modeled by the following equations [25]

λm= The membrane conductivityσm depends on the membrane water contentλm. To account for this dependency, the membrane water content is calculated in each time-step according to The supporting equations that are needed to complete the DAE system are given in the appendix of the thesis.