An adsorption bed $(40cm$ in diameter$)$ is packed with activated carbon as the adsorbent. A feed stream containing $0\mathrm{.}15wt\%$ of impurity A enters the column at $1000kg/$hr$.$ The adsorption of $A$ on activated carbon follows the Langmuir isotherm:

$q_A=\frac{30\mathrm{.}15c_A}{1+5c_A}$

where $c_A$ is the concentration of $A$ in the wastewater stream $(kg$ of $\frac{A}{k}g$ of wastewater feed$),$ and $q_A$ is the corresponding maximum amount of impurity A that can be adsorbed $(kg$ of $\frac{A}{k}g$ adsorbent$).$ Impurity $A$ is first detected in the exiting stream after $15$ hours of operation, and the bed reaches complete saturation after $2$ more hours.

$($a$)$ Starting from Langmuir isotherm expression, determine the amount $($in $kg)$ of adsorbent material needed for the bed to achieve saturation after $15$ hours of operation.

$($b$)$ Given that the density of activated carbon is $550k\frac{g}{m^3},$ what is the length $($in $m)$ of the adsorbent bed?

$($c$)$ Suppose the breakthrough curve is a standard, S$-$shaped curve, what is the percentage of bed that is not used at breakthrough?