An activated-sludge waste treatment system is required to reduce the amount of $\mathrm{BOD}_{5}$ from $1000 \mathrm{mg} / 1$ to $20 \mathrm{mg} / \mathrm{l}$ at the exit. The sedimentation unit concentrates biomass by a factor of 3. Kinetic parameters are $\mu_{m}=0.2 \mathrm{~h}^{-1}, K_{s}=80 \mathrm{mg} / \mathrm{l}, k_{d}=0.01 \mathrm{~h}^{-1}$, and $Y_{X S}^{M}=0.5 \mathrm{~g} \mathrm{MLVSS} / \mathrm{g}$ $\mathrm{BOD}_{5}$. The flow of waste water is $10000 \mathrm{l} / \mathrm{h}$ and the size of the treatment basin is 50,0001 .
a. What is the value of the solids residence time (i.e., $\theta_{c}$ )?
b. What value of the recycle ratio must be used?