Methane mass flow rate, $mo{H}_{CH,2}(k\frac{g}{?}$ day$),$ and $C{O}_2$ mass flow rate, day

Previously, waste sludges from the MCUP plant's multiple wastewater treatment process trains were dewatered and trucked to an off$-$site landfill. However, in a major environmentally sustainable expansion at MCUP's wastewater treatment plant, the various sludge streams $($from the MCUP #$4$ process and many others$)$ are fed to anaerobic biochemical reactors$-$commonly called anaerobic digesters$-$at a rate of $55,000k\frac{g}{?}$ day producing three residuals:

i$.$ Digester gas $-$ composed of $68$ percent methane $(C{H}_4)$ by volume and $32$ percent $C{O}_2$ by volume with a pressure of $1$ atm and a temperature of $35C.$

ii$.$ Biologically stable humus$-$like solid material called digested sludge $-37$ percent by mass of the raw sludge.

iii. Liquid supernatant $-59$ percent by mass of the raw sludge.

Digester gas is transferred to an on$-$site cogeneration power plant with a thermal efficiency of $0\mathrm{.}55.$ A process flow schematic is shown in Figure $1.$ The methane has a net heat of combustion of $802\mathrm{.}2k\frac{J}{m}$ole and is completely combusted to $C{O}_2$ and water vapor in the power plant. Neglect the minor quantity of $C{O}_2$ in the intake combustion air.

Apply the Ideal Gas Law, stoichiometric relationships, and fundamental conservation laws including total mass balance, mass balance on $C{O}_2,$ and energy balance analyses around appropriate control surfaces to determine:

Digester gas total mass flow rate, $m_2^{}(k\frac{g}{?}$ day