A family keeps 6 cattle. Each gives 10 kg of fresh dung in a day, and all of it is fed to the plant…

Methanogens such as Methanobacterium make methane without oxygen. Biogas is mostly methane. Methane gives $35.8$ MJ for each m³ and LPG gives $46$ MJ for each kg. The digester volume is the slurry fed each day times the days that it stays in the tank. The reaction $\mathrm{C_6H_{12}O_6} \rightarrow 3\,\mathrm{CH_4} + 3\,\mathrm{CO_2}$ gives $3$ mol of methane from $1$ mol of glucose, and $1$ mol of gas fills $22.4$ L at $0\ ^\circ\mathrm{C}$ and $1$ atm.