A circular sedimentation basin is $133\mathrm{.}2$ ft in diameter and $12$ ft in depth with the sludge zone comprising the bottom two feet of the basin. The flow rate is $20$ MGD and the particles in the feed stream entering the basin have a particle settling velocity distribution function $($pdf$)$ as shown in Figure $1-1$ and given by the following equation:

$F^{\text{'}}(v_s)=0\mathrm{.}2-0\mathrm{.}02v_s,(h\frac{r}{f}t)$

where $v_s$ is in units of $f\frac{t}{h}r.$ Assume for this problem that the basin overflow rate is equal to the theoretical overflow rate.

Figure $1-1.$

a$)$ Determine the theoretical overflow rate, $v_{o,\text{t}\text{h}\text{e}\text{o}\text{r}\text{y}}(f\frac{t}{h}r)$ for the sedimentation basin.

b$)$ What is the detention time of the basin $($hr$)?$

c$)$ Develop an analytical expression for the cumulative settling velocity distribution function, $F(v_s)$ as a function of $v_s.$

d$)$ Draw the cumulative particle velocity distribution function, $F(v_s)$ vs$.v_s,$ for the particles entering the basin for $0.$

$e$