As noted, the model presented predicts the current density response of an SOC subject to a voltage under specified gas reactant compositions. The Butler–Volmer equation is a standard method for defining the reaction current density giving rise to the aggregate cell current density, see Eq. (6) [45,53,56,62]. In the present work, concentration dependence of the exchange current density is not included. Instead, representative values of the exchange current density are applied. Equation (6) allows a relationship to be formed between the cell current density, the electrochemical potential of electrons and ions, and reactant gases based on their chemical potential. This connection is accomplished by expressing the activation overpotential as a function of the electrochemical and chemical potentials of the charged and neutral reactant species in the cell. Equations (7) and (8) show the definition of the activation overpotential for the fuel-side electrode in H_{2}–H_{2}O and CO–CO_{2} atmospheres, respectively [57]. The activation overpotential for the air-side (O_{2}–N_{2}) electrode is shown in Eq. (9) [58]. Here, it is important to note that the overpotentials defined in Eqs. (7)–(9) vary locally across the solution domain. The ohmic and concentration overpotentials characteristic of transport losses are accounted for within this local variation. These transport losses are determined by the governing equations of mass and charge transport (Eqs. (1)–(5)), and the physical characteristics of the cell set by the defined microstructural parameters and related effective diffusivities and conductivities. A voltage is applied across the cell, and the model calculates a corresponding current density that accounts for all of the overpotentials
Display Formula

(6)$irxn(x)=i0[exp(\beta F\varphi (x)RT)\u2212exp(\u2212(1\u2212\beta )F\varphi (x)RT)]$

Display Formula(7)$\varphi a,H2(x)=1F{12\mu \u0303O2\u2212(x)\u2212\mu \u0303e\u2212(x)\u221212[\Delta GH2Oo+RT\u2009log(pH2O(x)pH2(x))]}$

Display Formula(8)$\varphi a,CO(x)=1F{12\mu \u0303O2\u2212(x)\u2212\mu \u0303e\u2212(x)\u221212[\Delta GCO2o+RT\u2009log(pCO2(x)pCO(x))]}$

Display Formula(9)$\varphi c(x)=1F{12\mu \u0303O2\u2212(x)\u2212\mu \u0303e\u2212(x)\u221214RT\u2009log(pO2(x))}$