It is required to build a UASB reactor on a laboratory scale. This UASB reactor will work with industrial wastewater. Anaerobic granular sludge from a full$-$scale wastewater treatment plant will be used as inoculum for the UASB reactor. The methanogenic activity test results of the sludge that will be inoculated into the UASB reactor are shown in table.

Digester for test:

Volume $=500$ml

Sludge $=5$g SV$/$L in the digester

Additional data:

Temperature $=30\backslash$deg C$,$ pressure $=1$ atm

Total sludge solids $=180$ g$/$L$,$ of which $30\%$ are volatile solids $($VS$)$

Determine:

a$)$ The methanogenic activity of the sludge in gCOD$/$gSV$*$ia

b$)$ Volume of the laboratory's UASB reactor. Calculate the volume of sludge that must be introduced as inoculum to the UASB reactor to obtain the desired initial sludge concentration.

c$)$ Hydraulic retention time in the laboratory's UASB reactor.

$1\begin{array}{cc}\text{T}\text{i}\text{e}\text{m}\text{p}\text{o}(hr)& VCH4(ml)\\ 0& 0\\ 1\mathrm{.}25& 5\\ 7\mathrm{.}35& 32\mathrm{.}5\\ 7\mathrm{.}75& 37\mathrm{.}5\\ 8\mathrm{.}025& 41\mathrm{.}5\\ 8\mathrm{.}45& 47\mathrm{.}5\\ 8\mathrm{.}65& 51\\ 8\mathrm{.}9& 54\\ 9\mathrm{.}5& 64\\ 9\mathrm{.}65& 66\mathrm{.}5\\ 9\mathrm{.}95& 72\\ 10\mathrm{.}5& 81\\ 11\mathrm{.}85& 106\\ 12\mathrm{.}05& 109\\ 12\mathrm{.}3& 113\\ 19\mathrm{.}4& 158\mathrm{.}5\\ 20\mathrm{.}3& 159\mathrm{.}5\end{array}$