## 3 Process Optimisation
### Pre-Treatment of High-Fat RMP Wastewater Substrates to Enhance Anaerobic Digestion
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(B_{0}\) & **Inhibition** & \(t_{0}\) & \(t_{\text{Finish}}\) & \(k\) & \(U\) & \(R^{2}\) & \(R^{2}\) \\ \hline \multirow{2}{*}{**Treatment**} & **mLN CH\({}_{4}\cdot\)g** & **(A)** & \multirow{2}{*}{**Days**} & **Days** & & **mLN CH\({}_{4}\cdot\)g** & \multirow{2}{*}{*} & **\multirow{2}{*}{**\(\ast\ast\)**} \\  & **VS\({}^{-1}\)added** & & **Days** & & & **VS\({}^{-1}\)added day\({}^{-1}\)** & & \\ \hline Control & \(759\pm 4\) & \(4\pm 0.3\) & \(6.8\pm 0.1\) & \(12\pm 0.00\) & \(0.74\pm 0.05\) & \(149\pm 16\) & 0.993 & 0.979 \\ Thermal & \(821\pm 148\) & \(0.9\pm 0.2\) & \(4.8\pm 0.1\) & \(13\pm 1.73\) & \(0.54\pm 0.02\) & \(114\pm 4\) & 0.996 & 0.995 \\ Chemical & \(782\pm 6\) & \(3.3\pm 0.3\) & \(6.2\pm 0.1\) & \(12\pm 0.58\) & \(0.71\pm 0.05\) & \(147\pm 16\) & 0.993 & 0.977 \\ Thermochem & \(821\pm 11\) & \(2.4\pm 0.3\) & \(6.2\pm 0.1\) & \(13\pm 0.58\) & \(0.55\pm 0.02\) & \(114\pm 9\) & 0.995 & 0.983 \\ \hline Control & \(999\pm 7\) & \(19.2\pm 0.1\) & \(23.2\pm 0\) & \(33\pm 0.60\) & \(0.53\pm 0.1\) & \(136\pm 2\) & 1 & 0.999 \\ Bile 0.2 gL\({}^{-1}\) & \(1056\pm 22\) & \(19.1\pm 0.1\) & \(23.1\pm 0.1\) & \(33\pm 0.60\) & \(0.51\pm 0.1\) & \(139\pm 3\) & 0.999 & 0.998 \\ Bile 0.4 gL\({}^{-1}\) & \(1056\pm 3\) & \(19.2\pm 0.1\) & \(23.4\pm 0.1\) & \(34\pm 0.00\) & \(0.48\pm 0.1\) & \(135\pm 9\) & 0.999 & 0.997 \\ Bile 0.6 gL\({}^{-1}\) & \(1090\pm 12\) & \(19.1\pm 0.1\) & \(23.3\pm 0.2\) & \(33\pm 0.60\) & \(0.5\pm 0.1\) & \(135\pm 4\) & 1 & 0.997 \\ Bile 0.8 gL\({}^{-1}\) & \(1068\pm 4\) & \(19.1\pm 0.1\) & \(23.3\pm 0.1\) & \(33\pm 0.60\) & \(0.5\pm 0.1\) & \(136\pm 4\) & 0.999 & 0.998 \\ Bile 1 gL\({}^{-1}\) & \(1056\pm 12\) & \(19\pm 0.1\) & \(23.2\pm 0.3\) & \(33\pm 1.70\) & \(0.5\pm 0.3\) & \(129\pm 4\) & 1 & 0.997 \\ \hline Control & \(765\pm 11\) & \(7.1\pm 0.2\) & \(10.1\pm 0.1\) & \(17\pm 0\) & \(0.73\pm 0.01\) & \(121\pm 6\) & 0.999 & 0.999 \\ Bile 1 gL\({}^{-1}\) & \(766\pm 12\) & \(7.3\pm 0.1\) & \(10.3\pm 0.1\) & \(17\pm 1\) & \(0.71\pm 0.02\) & \(121\pm 6\) & 0.999 & 0.995 \\ Bile 2 gL\({}^{-1}\) & \(764\pm 3\) & \(7\pm 0.2\) & \(10.1\pm 0.1\) & \(17.7\pm 0.6\) & \(0.68\pm 0.01\) & \(109\pm 5\) & 0.993 & 0.994 \\ Bile 3 gL\({}^{-1}\) & \(761\pm 8\) & \(7.8\pm 0.2\) & \(11.4\pm 0.2\) & \(19.7\pm 1.2\) & \(0.57\pm 0.02\) & \(96\pm 4\) & 0.997 & 0.994 \\ Bile 4 gL\({}^{-1}\) & \(764\pm 10\) & \(8.1\pm 0.2\) & \(12\pm 0.2\) & \(20.7\pm 1.2\) & \(0.53\pm 0.02\) & \(88\pm 4\) & 0.994 & 0.994 \\ Bile 5 gL\({}^{-1}\) & \(756\pm 3\) & \(9.7\pm 0.3\) & \(14.0\pm 0.2\) & \(23.3\pm 1.2\) & \(0.44\pm 0.01\) & \(75\pm 4\) & 0.989 & 0.987 \\ Bile 6 gL\({}^{-1}\) & \(745\pm 11\) & \(12.7\pm 0.4\) & \(17.6\pm 0.3\) & \(28\pm 1\) & \(0.35\pm 0.02\) & \(65\pm 3\) & 0.986 & 0.986 \\ \hline \hline \end{tabular}  
* \({}^{\star}\)R\({}^{2}\) for Logistic equation; **R\({}^{2}\)** for modified Gompertz equation; mLN represents a normal volume (i.e., standard temperature and pressure \(-0\) \({}^{\circ}\)C, 1 atm, 0% humidity). Values are presented as \(\pm\) one standard deviation from the mean.  
\end{table} 
Table 2: Kinetics analysis of treatment groups adapted from [42; 43]. Figure 6: Specific methane potential from DAF sludge treated with bile in doses ranging from: (a) 0.2–1 g bile-L\({}^{-1}\) and; (b) 1–6 g bile-L\({}^{-1}\)[43]. (days) at which maximum production rate occurs. The function is weighted using standard deviation to achieve better fit. Equation (2): Modified Gompertz equation [44]. \[B=B_{0}e^{-e^{(\frac{U_{E}(\lambda-t)}{B_{0}}(\lambda-t)+1)}} \tag{2}\]  
From Equation (2): \(B\) is the cumulative SMP at time t; B\({}_{0}\) is the maximum SMP achieved by end of digestion; \(U\) is the kinetic constant of methane production rate; \(\lambda\) is the duration of lag phase in days, used here to represent inhibition. The equation is unweighted.