PROBLEM NO$.\backslash (2[80\backslash )$ POINTS$]$

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 $\backslash$#$4$ process and many others$)$ are fed to anaerobic biochemical reactors$-$commonly called anaerobic digesters$-$at a rate of $\backslash (55,000\backslash$mathrm$\{$~kg$\}/\backslash )$ day producing three residuals:

i$.$ Digester gas $-$ composed of $68$ percent methane $\backslash (\backslash$left$(\backslash$mathrm$\{$CH$\}\_\{4\}\backslash$right$)\backslash )$ by volume and $32$ percent $\backslash (\backslash$mathrm$\{$CO$\}\_\{2\}\backslash )$ by volume with a pressure of $1$ atm and a temperature of $\backslash (35^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$

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 $\backslash (802\mathrm{.}2\backslash$mathrm$\{$~kJ$\}/\backslash )$ mole and is completely combusted to $\backslash (\backslash$mathrm$\{$CO$\}\_\{2\}\backslash )$ and water vapor in the power plant. Neglect the minor quantity of $\backslash (\backslash$mathrm$\{$CO$\}\_\{2\}\backslash )$ in the intake combustion air.

Apply the Ideal Gas Law, stoichiometric relationships, and fundamental conservation laws including total mass balance, mass balance on $\backslash (\backslash$mathrm$\{$CO$\}\_\{2\}\backslash ),$ and energy balance analyses around appropriate control surfaces to determine:

$1.$ Digester gas total mass flow rate, $\backslash (\backslash$dot$\{$m$\}\_\{2\}\backslash )(\backslash (\backslash$mathrm$\{$kg$\}/\backslash )$ day$).$ digester gas. $[$Hint: The percentages of $\backslash (\backslash$mathrm$\{$CH$\}\_\{4\}\backslash )$ and $\backslash (\backslash$mathrm$\{$CO$\}\_\{2\}\backslash )$ in digester gas on a mass basis are much different from their given percentages on a volume basis. Use the Ideal Gas Law to determine the gas densities of $\backslash (\backslash$mathrm$\{$CH$\}\_\{4\}\backslash )$ and $\backslash (\backslash$mathrm$\{$CO$\}\_\{2\}\backslash ),$ and then combine them to determine the density of the overall digester gas mixture in a manner identical to that used in the class example for computing the density of air. Then use the ratio of the density of each component gas to the composite density of digester gas to determine the component gas mass ratios $($i$.$e$.,$ kg $\backslash (\backslash$mathrm$\{$CH$\}\_\{4\}/\backslash$mathrm$\{$kg$\}\backslash )$ digester gas and $\backslash (\backslash$mathrm$\{$kg$\}\backslash$mathrm$\{$CO$\}2/\backslash$mathrm$\{$kg$\}\backslash )$ digester gas.$]$

$3.$ Stack gas $\backslash (\backslash$mathrm$\{$CO$\}\_\{2\}\backslash )$ mass emission rate, mcO$2,8($kg$/$day$),$ and volumetric emission rate at STP$,$ QCO$2,8(\backslash (\backslash$mathrm$\{$m$\}^\{3\}/\backslash )$ day$).$

$4.$ Total net rate of waste heat emission to the ambient environment, ewaSTE $(\backslash (\backslash$mathrm$\{$kJ$\}/\backslash$mathrm$\{$day$\}\backslash )),$ and the rate of power generation, $\backslash (\backslash$dot$\{$e$\}\_\{7\}(\backslash$mathrm$\{$~kW$\})\backslash ).$

In your report, comment on the sustainability benefits of the expansion, e$.$g$.,$ reduction in transportation and disposal costs, generation of a useful byproduct fuel, etc.

Figure $1.$ Anaerobic Digester$-$Cogeneration Power Plant