Solid oxide fuel cell (SOFC) is one of the most useful of the fuel cell types, and it works in hightemperature conditions and is characterized by higher performance and fuel utilization due to endothermic internal reforming occurring in the anodic porous layer. Tubular SOFC and planar SOFC are two basic types of SOFC currently being developed. The SOFC cell stack includes porous anode/cathode electrodes, the electrolyte, fuel and air channels, and solid connectors. The porous electrode structure of a SOFC allows the gases from the respective gas channels to penetrate through the porous layers, thus facilitating the reforming and electrochemical reactions near the interface between gas and solid phases. Mass and energy exchanges occur within the cell due to the reactions. While the high temperature improves the cell performance and allows the use of different fuels, it can also be responsible for excessive thermal stresses, catalyst degradation (carbon deposition), and thermal instability problems in materials. The SOFC primarily uses hydrogen as the fuel. The fuel reacts with oxygen to produce electricity. Fundamental to SOFC design is an understanding of reaction kinetics and its impact on power output and cell efficiency. The fundamental SOFC reactions and the corresponding enthalpies of formation ЛЯ298° K, are:
Anode : 2H2 + 2O2— ! 2H2O + 4e—
Cathode : O2 + 4e— ! 2O2—, ЛИ298" K = 0,
Overall : 2H2 + O2 ! 2H2O, ЛИ298“ K = —286 kJ/mol
Methane steam reforming and electrochemical reactions occurring at the anode porous layer are the most important factors to affect the heat – and masstransfer processes within the fuel cell. As internal reforming can occur in SOFC’s anode, a mixture of hydrocarbons may be used as the fuel in the SOFC. Moreover, the reforming rate could be enhanced by a catalyst. However, ideally, the catalyst does not take any part in the overall reactions. И2 and CO which are generated by methane and water gas steam reforming are the primary components of the electrochemical reactions in the anodic active layer. H2O and CO2 are then
produced in the anodic active layer, and a certain proportion of the H2O can be reused for methane steam reforming at the anode which in turn improves the fuel usage and overall efficiency. Oxygen which on the other hand is consumed in the cathode side does not contribute to the heat transfer within the cell. The equations corresponding to the reforming of hydrogen must be appended to the electrode electrochemical reaction equations. Hence, in addition, one has
CH4 partial oxidation : CH4 + H2O! CO + 3H2, AH298° K = 206kJ/mol
CO oxidation : +H2O! CO2 + H2, AH298° K = —41 kJ/mol
CH4 oxidation : CH4 + 2H2O! CO2 + 4H2, AH298o K = 165 kJ/mol.
There are two different approaches to reforming: external reforming, which is performed external to the cell before the fuel reaches it, and internal reforming, which takes place within the fuel cell anode. These are illustrated in Figs. 6.6 and 6.7.
reforming catalyst 
Methane t c£ steam ■ 
anode exhaust*^ (C02& steam HjOJ Anode Cathode 
Air A CO7* 
carbon dioxide 
Fig. 6.6 SOFC with internal reforming
Fuel( methane, CH4 or butane, C4Hl0)
(eg. Yttria stabilized zirconia (YSZ))
Fig. 6.7 SOFC with external reforming
In contrast to the steam reforming reactions, two of which, with the exception of the water gas shift reforming equation, are endothermic, the electrode electrochemical reactions are exothermic because of heat production within the fuel cell caused by the internal resistances. The reforming equations do not contribute to the electron flow but are only responsible for the reforming of hydrogen. Thus, for every two moles of hydrogen consumed, four moles of electrons are passed through the electric load. Electron mole flow is converted into charge units using Faradays constant (F = 96,485 coulombs/mole of electrons). Since the current is the rate of charge flow, the fuel cell delivers power to the load and it follows that
Power = Current • Voltage, (coulomb • volt = joule and joule/s = watt). (6.8.1)
The fuel cell gains this power at the expense of the enthalpy released during the overall reaction, which is the oxidation of hydrogen to form water.
2H2 + O2 ! 2H2O.
Yet, only a portion of this enthalpy can be converted into electric power, the remainder being accounted for by heat released by the reaction. This heat is normally transferred to the air and hydrogen fuel by heat exchangers. The performance of a fuel cell is typically expressed in terms of its total efficiency which is defined as energy delivered to the load divided by the total energy available from reaction.
In this section, a typical SOFC is considered with internal reforming only, although the model can be applied to an externally reformed cell. Assuming that the reactor is of a tubular section and narrow in width, the molar flow equations for the ith molar flow rate component N. at the cathode reaction zone may be expressed as
In the above equations, the first two terms correspond to the corresponding inlet and outlet flows of the component. Furthermore, the relation between reaction rate and fuel cell current is a proportional relationship. If we define rt as the total rate of species i generated in the fuel cell (typically with units of mole/s) and 7cell is the total current leaving the cell, then
The reaction rates of the reforming equations may be defined on the basis of the reaction route network theory developed by Fishtik et al. (2004a, b, c) and applied by Master (2010) where coefficients of the rate equations found from experiments. A reaction route is defined as a linear combination of the elementary reactions, such that a certain number of species, either terminal or intermediate, are cancelled, thus producing a new reaction referred to as an overall reaction. By eliminating all the intermediate species, an overall reaction results and it is termed a full route. If all the species are eliminated in a reaction route, an empty route is formed where all the stoichiometric coefficients are equal to zero. Xu and Froment (1989a, b) determined intrinsic reaction rates for the reforming reactions in the form of the LangmuirHinshelwood kinetics model. The reaction rates in mols/s are defined by,
where wca is the density of the catalyst used, Vri is the ratio of the actual volume over which the ith reaction is taking place to the reactor volume, Vr is the reactor physical volume,
A — 1 + Kcopco + Kh2Ph2 + Kch4 Pch4 + Kh2oPh2o/ph2 (6.8.8)
and pco, pH2, pcH4, pH2O are the partial pressures of CO, H2, CH4, and H2O within the reaction volume. These partial pressures are calculated using the ideal gas equations,
PjVanode — NjRT, j — CO, H2, CH4, H2O. (6.8.9)
The reaction rate constants are defined by Arrheniustype relations given by, ki — Aiexp(Ei/RT), i — 1, 2, 3. (6.8.10)
Table 6.5 Values of model constants

Keq1,Keq2, and Keq3 are the equilibrium constants for the reforming reactions, which are calculated from,
Keq = exp^1 – ; i = 1; 2; 3. (6.8.11)
and
Kj = Bjexp(AHj/RT); j = CO; H2; CH4; H2O. (6.8.12)
The constants Ai, Ei, Ti, TRi, Bj, and AHj may be obtained from Xu and Froment (1989a, b) and are tabulated in Table 6.5.
The components Fi, in the component molar flow rate equations, are the components corresponding to the fuel (hydrogen) and the oxidized fuel (water steam). The components Ei are the components corresponding to the effluents from the reforming reactions, methane, carbon monoxide, and carbon dioxide. The coefficients aFi;k and aEi;k are the stoichiometric coefficients of the components Fi and Ei in the reaction k. The total pressure at the anode, which is also the input pressure at the anode, is given by,
4
panode Y, Pj; j = CO; H2; CH4 ; ^O. (6.8.13)
j=1
At the cathode, the partial pressure of Oxygen (O2) is given by,
PO2 Vcathode = N02 RT. (6.8.14)
At the cathode, the total pressure at the cathode, which is also the input pressure at the cathode, is given by,
where Nair is the total molar flow rate at the cathode. Finally, the mass flow rates at the anode and cathode channels are assumed to be given by,
mout, e = JK(pe — Pout. e), e = anode, cathode, (6.8.16)
where ke is a coefficient mass flow rate to the pressure difference.
The temperature within the fuel cell is given by, the energy balance equation,
dT 4′
mCp "dt Panode H Pcathode ^ Dhj rj PDC Pht_loss (6:8.17)
j=1
where Panode is the total power input to cell at the anode, Pcathode is the total power input to the cell at the cathode, Dhj is the molar specific enthalpy change due to the reaction j, PDC is the electrical DC power loss across the output terminals, Pht_loss represents the radiation and conduction heat loss to the surroundings from the SOFC.
The operating cell voltage is related to the opencircuit theoretical voltage and to the various losses as
Ecell = Erev – ga(i) — gc(i) — gconc(i) – gohm(i) (6.8.18)
where ga(i) and gc(i) are the activation losses at the anode and cathode side, respectively, gconc() is the concentration over potential and gohm() is the Ohmic over potential. For a solid oxide fuel cell, the Nernst equation gives the reversible cell potential which is,
where Eo = — AG°rxn/2 F, where AG°rxn is the Gibbs free energy of the overall reaction (evaluated at standard pressures and the operating temperature of the cell), and the pi’s are the partial pressures at the threephase interface within the reaction region. The activation over potentials are obtained by using the ButlerVolmer equation given for each participating species s, i. e., H2, O2 separately. The functional relation between the activation loses and current density is described by the ButlerVolmer equations. For hydrogen oxidation, the ButlerVolmer equation takes the form,
f (anF ( (1 — a)nF
lH2 = i0,H^ exP I – RT gaj — exP I——————— RT ga
In the above equation, i0,H2 is the exchange current given by,
The constants in the equation for the exchange current are obtained by fitting experimental data in the appropriate temperature range (750850 °C for a SOFC). The activation energy barrier, Eact;H2, is 110.3 kJ/mol. A modified version of the ButlerVolmer equation is used for oxygen reduction which takes the form,
where
Eact,02
‘0,02 = ‘00,02 exP RT
and ’00 02 can be found from Zhu et al. (2005). The equations are then solved for the activation potentials numerically, assuming that ‘H2 = io2 = ‘ceii = ‘. A good approximation is obtained by setting a = 0.5and it follows that
Hence,
Moreover,
dgact _ 2RT 1
di nF p4’2 + ‘2
Alternately, the activation loss may be obtained by fitting a Tafellike equation to experimental data, in which case, the activation over potential may be expressed in the form,
RTi
g“=anfК* • (68.27)
The Ohmic loss may be expressed as,
gohm = Rtot/cell (6828)
where Rtot is the total specific resistance of the cell (per unit area), and jcell is the cell current density. The main contribution to the specific total resistance is from the electrolyte with the anode and cathode making a small contribution to it. Thus, it is expressed as,
Rtot R0 F ^elec = relec (6.8.29)
where R0 is a small constant that accounts for the anode and cathode resistivities, telec is the thickness or width of the electrolyte, and relec is the electrolyte conductivity. Yttriastabilized zirconia (YSZ), which has a high ionic conductivity at high temperatures in the vicinity of 1,000 °C, is most common electrolyte used in SOFCs. For YSZ, the electrical conductivity is 6.67 x 102 S cm1 at 1,000 °C. The relationship between electrical conductivity and temperature can be described by a thermally activated Arrhenius equation and expressed as,
where the actuation potential Eact, YSZ — 93kJ/mol for YSZ in the vicinity of 1,000 °C. Thus, the electrolyte conductivity may be expressed as,
where T is in °K, and r1273 is the electrical conductivity at 1,000 °C. In the Nernst equation, the species concentrations are taken as the reference value to calculate the opencircuit voltage. However, during the operation of the cell, the concentration at the reaction sites is not the one at the input channels, due to the depletion of the reactants and to masstransfer resistances at the electrodes which act as porous media. Hence, Nernst equation must be corrected, both in the anode and in the cathode. Thus, the concentration over potentials in the anode and cathode are, respectively, given by,
where the subscripts refer to the input channels and the reaction sites in the cell.