$($a$)$ Briefly describe the characteristics of biological growth in the batch bioreactor of a wastewater treatment plant, and draw the growth diagram.

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$($b$)$ Biological tanks in wastewater treatment plants are designed to use microorganisms $($such as bacteria$)$ to remove pollutants in wastewater through biochemical reactions. The biomass growth rate is related to the biological substrate utilization rate as expressed by the following relationships $($Monod kinetics$)$:

$\frac{d{x}_B}{dt}={}_{\text{m}\text{a}\text{x}}\frac{S}{K_S+S}{x}_B=-Y_{\text{m}\text{a}\text{x}}\frac{dS}{dt}$

Define the parameters in the above equation.

$($c$)$ In a simple flow system with one fully$-$mixed biological reactor shown in Figure Q$4,$ the flow rate is $100\frac{m^3}{h}$ and the following measurements are made: $x_{b1}=100m\frac{g}{L},C_1=800m\frac{g}{L},x_{b3}=200m\frac{g}{L}$ and $C_3=250m\frac{g}{L}.$ Neglecting effects of decay, estimate the observed yield coefficient.

$($d$)$ For the above question $4($c$),$ estimate the volume of the reactor if the growth and substrate uptake follows Monod kinetics with $K_s=50$ and ${}_{\text{m}\text{a}\text{x}}=0\mathrm{.}1h^{-1}.$ Assume that all substrate is present as soluble matter $($i$.$e$.C=S).$

$\frac{10}{25}$