The results of pilot filter experiments are summarized in the tables below. For each set of experiments, the independent variable was either the media effective size or the media depth, as given in the second column below. For a given set of experiments $($A$,$ B$,C,$ or $D,$ to be selected by instructor$),$ determine equations for how the specific deposit at breakthrough $({}_B)$ and the head loss rate constant $)$ each depend on the independent variable. Also determine the optimal value of the independent variable and the corresponding filter run duration. For all problems, assume $C_O=2\mathrm{.}0m\frac{g}{L}$ and $C_E=0m\frac{g}{L}.$

c$.$ Design conditions: $v_F=33\mathrm{.}8\frac{m}{h},$ media $=$ anthracite, $ES=1\mathrm{.}55mm,$ max available head $=3m($adapted from pilot results for the LADWP Aqueduct Filtration Plant$).$

$\backslash$table$[[\backslash$table$[[$C$],[$Filter$]],\backslash$table$[[$Media$],[$ES$,],[$m$]],\backslash$table$[[$Time to$],[$Breakthrough$,],[$h$]],\backslash$table$[[$Initial$],[$Head Loss,$],[$m$]],\backslash$table$[[$Head Loss When$],[$Breakthrough$],[$Occurred$,m$