$2(20).$ You are tasked with improving the bed efficiency, $,$ of a fixed$-$bed adsorption process from an unacceptable value of $0\mathrm{.}67$ to the target value of $0\mathrm{.}80.$ The target solute feed is dilute and at fixed concentration, $c_F.$ A linear isotherm applies, $q=Kc,$ and the breakthrough curve is described well by an expression based on the error function, erf $($see below$).$ Although the economical breakthrough concentration has not been established yet, an approximate process time corresponding to the midpoint of the breakthrough curve, $t_0,$ at $=0\mathrm{.}7$ has been set. Also, solution composition and temperature have been set, so $K$ cannot be manipulated.

$($a$)$ Given the conditions described above and prior knowledge that the separation is limited by internal diffusion, what parameter$($s$)$ can be manipulated and in what manner $($up or down$)$ to improve $$ given fixed, $C_F,t_0$ and $K?$ Note the table below as well as the expressions for $,$ the breakthrough curve, and the slope of the breakthrough curve at $t=t_0.$ The expression for the slope is valid for any breakthrough curve described using the expression based on erf.

$($b$)$ Determine quantitatively the change in one or more parameters to improve $$ to $0\mathrm{.}80$ given the constraints described in the problem statement above.

$=\frac{2t_B}{t_E+t_B}=1-\frac{t_E-t_B}{t_E+t_B},\frac{c}{c_F}=\frac{1}{2}(1+$erf$[\frac{t-t_o}{\sqrt[\phantom{2}]{2}{t}_o}]),($ slope @$t_o)=\frac{0\mathrm{.}399c_F}{t_o}$

$\backslash$table$[[$Controlling$,{}^2$ is$,t_o$ is$,$NTU is$,R$ is$],[$Step$,$proportional to$,$proportional to$,$proportional to$,$proportional to$]]$

Internal Diff'n $,\frac{v{d}^2}{L},d\sqrt[\phantom{2}]{\frac{L}{v}},\frac{L}{v{d}^2},\frac{1}{d}\sqrt[\phantom{2}]{\frac{L}{v}}$

Ext. Transport $,\frac{v^{\frac{1}{2}}{d}^{\frac{3}{2}}}{L},\frac{L^{\frac{1}{2}}{d}^{\frac{3}{4}}}{v^{\frac{3}{4}}},\frac{L}{v^{\frac{1}{2}}{d}^{\frac{3}{2}}},\frac{L^{\frac{1}{2}}}{d^{\frac{3}{4}}{v}^{\frac{1}{4}}}$

Dispersion $,\frac{v{d}^2}{L},d\sqrt[\phantom{2}]{\frac{L}{v}},\frac{L}{v{d}^2},\frac{1}{d}\sqrt[\phantom{2}]{\frac{L}{v}}$