# The Effect of Biogas Origin on the Electricity Production by Solid Oxide Fuel Cells
## 2 Theory
\right\}=0 \tag{3}\]  
By assuming isobaric conditions, no-phase change and equilibrium for all the reactions considered here, the above generic relations can be mathematically described as follows [13]:  
* Energy balance for heat exchangers (see in Figure 1, devices D1, D2 and D3) [14]: \[T_{\text{in}}[\sum_{\text{i}}M_{\text{i}}(C_{P})_{\text{i}}]_{\text{in}}=T_ {\text{out}}[\sum_{\text{i}}M_{\text{i}}(C_{P})_{\text{i}}]_{\text{out}}\] (4) where \(T_{\text{in}}\) and \(T_{\text{out}}\) are the inlet and outlet temperatures (in K), respectively, at each branch of heat exchangers, \(M_{\text{i}}\) is the mass (in mol) of species i, and \((C_{P})_{\text{i}}\) is the specific heat capacity (in J mol\({}^{-1}\) K\({}^{-1}\)) of species i. * Energy balance for biogas reformer (see in Figure 1, device D4) [15]: \[T_{\text{in}}[\sum_{\text{i}}M_{\text{i}}(C_{P})_{\text{i}}]_{\text{in}}+[ Q_{\text{burn}}]_{\text{ref}}+|\Delta H_{\text{WGS}}|=T_{\text{out}}[\sum_{ \text{i}}M_{\text{i}}(C_{P})_{\text{i}}]_{\text{out}}+\Delta H_{\text{ref}}\] (5) where \(T_{\text{in}}\) and \(T_{\text{out}}\) are the inlet and outlet temperatures (in K) at each branch of the reformer, \([Q_{\text{burn}}]_{\text{ref}}\) is the thermal energy (in J) that is transferred from the burner to the reformer, while \(\Delta H_{\text{WGS}}\) and \(\Delta H_{\text{ref}}\) are the enthalpies (in J mol\({}^{-1}\)) for the water gas shift and the methane reforming reaction, respectively. * Exergy balance for heat exchangers (see in Figure 1, devices D1, D2 and D3) [16]: \[[\sum_{1}[M_{\text{tot}}\Delta T\sum_{\text{i}}x_{\text{i}}(C_{P}^{\epsilon}) _{\text{i}}]]_{\text{in}}=[\sum_{1}[M_{\text{tot}}\Delta T\sum_{\text{i}}x_{ \text{i}}(C_{P}^{\epsilon})_{\text{i}}]]_{\text{out}}+I\] (6) where \(M_{\text{tot}}\) is the total mass (in mol), x\({}_{\text{i}}\) is the molar fraction of species i, \(\Delta T\) is the temperature difference (in K), \(\left(C_{P}^{\epsilon}\right)_{\text{i}}\) is the mean isobaric exergy capacity (in J kmol\({}^{-1}\) K\({}^{-1}\)) of species i, and I is the irreversibility rate (in J). * Exergy balance for biogas reformer (see in Figure 1, device D4) [16]: \[[\sum_{1}(M_{\text{tot}}(\sum_{\text{i}}\kappa(c_{0})_{\text{i}}+RT_{0}\sum _{\text{i}}\ln\kappa_{\text{i}})+M_{\text{tot}}\Delta T\sum_{\text{i}}(C_{P}^ {\epsilon})_{\text{i}})]_{\text{in}}+Q_{\text{burn}}\ \
ot=T_{\text{out}}\* Exergy balance for the burner (see in Figure 1, device D6) [16]: \[I^{Q}\ =\ (Q_{\text{burn ref}}+Q_{\text{burn env}})\frac{T_{\text{burn}}-T_{0}}{T_{ \text{burn}}}\] (8) where \(I^{Q}\) is the irreversibility due to heat losses (in J), \([Q_{\text{burn}}]_{\text{env}}\) is the thermal energy (in J) that is emitted to the environment, and \(T_{\text{burn}}\) is the operational temperature (in K) of the burner. * Exergy balance of the SOFC (see in Figure 1, device D5) [16]: \[W_{\text{el}}=\sum_{\text{r}}[-\left(H_{\text{prod}}-TS_{\text{prod}}\right)+ \left(H_{\text{react}}-TS_{\text{react}}\right)]\] (9) where \(W_{\text{el}}\) is the electric energy (in J) produced in the SOFC, \(H_{\text{prod}}\) and \(H_{\text{react}}\) are the enthalpies (in J) of the products and the reactants, respectively, while \(S_{\text{prod}}\) and \(S_{\text{react}}\) are the entropies (in J) of the products and the reactants, respectively.