# Thermal Hydrolysis of Sewage Sludge: A Case Study of a WWTP in Burgos, Spain
## 2 Materials and Methods
### Consumption of Biogas by the Recovery Boiler and Engines
\[<\text{TP}>_{\text{rb}}=\ \text{$\tau$}_{\text{bunner}}*\ <\text{TP}>_{\text{ biogas-rb}}+\text{$\tau$}_{\text{rb}}*\ <\text{TP}>_{\text{eg-useful}}+\ <\text{TP}>_{\text{fw}} \tag{6}\]  
\[\text{TP}_{\text{biogas}}=\ <\text{TP}>_{\text{biogas-rb}}+\text{TP}_{\text{biogas-eng}} \tag{7}\]  
\[\text{TP}_{\text{biogas-st}}=\ <\text{TP}>_{\text{biogas-rb}} \tag{8}\]  
\(Cp\) values for air were estimated using the NASA polynomial equation that is valid for temperatures between 200-1000 K [42]. Table 5 shows the \(Cp\) values obtained for the individual components of exhaust gases and the whole gas stream. These values were also estimated from this approximation when operating under lean conditions, assuming an air factor of 1.57 [43]. Enthalpy was calculated by subtracting the reference temperature (25 \({}^{\circ}\)C). Equations (9) and (10) are the specific enthalpies of inlet and outlet gases as a function of the gas boiler inlet temperature, which is expressed in Equation (11). The outlet temperature was considered constant [44]. Equation (12) estimates the degree of loading for the engine, and Equation (14) is used to calculate the mass flow of exhaust gases [45]. Finally, by applying Equation (14), usable thermal power, available with exhaust gases and used in the boiler, is calculated for the batch cycle. \[\text{h}_{\text{rb-in}}=\ C_{\text{P-eg-in}}*\ \text{T}_{\text{rb-in}} \tag{9}\]  
\[\text{h}_{\text{rb-out}}=\ C_{\text{P-eg-out}}*\ \text{T}_{\text{rb-out}} \tag{10}\]  
\[\text{T}_{\text{rb-in}}=\ -0.735\ *\ \text{DL}\ +\ 547.2 \tag{11}\]  
\[\text{DL}\ =\frac{(\text{TP}_{\text{biogas-eng}}-233.6)}{12.97*\text{N}_{\text{ eng}}} \tag{12}\]  
\[\text{TP}_{\text{biogas-su-gasom}}*\text{N}_{\text{reactor}}*\text{t}_{\text{inj} }=\text{TP}_{\text{biogas-st-gasom}}*\left(\text{t}_{\text{cycle}}-\text{N}_{ \text{reactor}}*\text{t}_{\text{inj}}\right) \tag{5}\]0  
\[\text{TP}_{\text{biogas-su-gasom}}*\text{N}_{\text{reactor}}*\text{t}_{\text{inj} }=\text{TP}_{\text{biogas-st-gasom}}*\left(\text{t}_{\text{cycle}}-\text{N}_{ \text{reactor}}*\text{t}_{\text{inj}}\right) \tag{5}\]1  
where C\({}_{\text{P-eg}}\) is the specific heat of engine gases (kJ/kg K). R\({}_{\text{g}}\) is the ideal gas constant (0287 kJ/kg K). T is the gas temperature (K). h\({}_{\text{rb-in}}\), h\({}_{\text{rb-out}}\), T\({}_{\text{rb-in}}\) and T\({}_{\text{rb-out}}\) are the specific enthalpies and temperatures of the engine gases at the inlet and outlet ports of the boiler (kJ/kg, \({}^{\circ}\)C). C\({}_{\text{P-eg-in}}\) and C\({}_{\text{P-eg-out}}\) are the specific heat values of inlet and outlet gases for the boiler (kJ/kg, \({}^{\circ}\)C). DL is the degree of loading of the engine (%). TP\({}_{\text{eg-useful}}\) is the useful thermal power of the recovery boiler exhaust gas for the batch cycle (kW). The available energy is the energy transferred by these gases, considering an outlet temperature of 142 \({}^{\circ}\)C. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Temperature** (\({}^{\circ}\)**C) & **Cp (kJ/kg K)** & **h (kJ/kg)** \\ \hline Recovery boiler quiet stream & & \\
142 & 1.016 & 118.1 \\ Recovery boiler inlet stream & & \\ Scenario 1, 482.7 & 1.087 & 478.6 \\ Scenario 2, 483.3 & 1.087 & 479.2 \\ Scenario 3, 504.2 & 1.092 & 502.2 \\ Scenario 4, 498.8 & 1.091 & 496.2 \\ Scenario 5, 486.6 & 1.088 & 482.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Specific enthalpies and \(Cp\) values calculated for inlet and outlet streams of the recovery boiler. \(Cp\) values of exhaust gases were assumed to be equivalent to air. The thermal power of the main circuit of the engines (TP\({}_{\text{main-circuit}}\), kW) is used to supply thermal energy to digesters in Scenarios 1 and 3-5 and also to boiler feed water in Scenarios 2-5. Equation (15) allows the calculation of this total thermal power [45]. Equation (16) represents the average thermal power of the feed water of the boiler's main circuit, (\(<\)TP\(>_{\text{fw}}\), kW).