$1.$ A mechanically cleaned coarse bar rack will be used in a WWTP for a flowrate of $\backslash (40,000\backslash$mathrm$\{$~m$\}^\{3\}/\backslash$mathrm$\{$d$\}\backslash ).$ The bar rack will be set at an angle of $75$ degrees from horizontal and will have clear bar spacing of $12$ mm $.$ The bar rack uses $\backslash (12-\backslash$mathrm$\{$mm$\}\backslash )-$wide bars. The upstream channel velocity should be greater than $\backslash (0\mathrm{.}4\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ Assume the water depth is $2\mathrm{.}4$ m $.$

$(1)$ Size the channel for a mechanically cleaned coarse bar rack.

$(2)$ Determine the head loss through the bar rack for the clean screen and for a screen that is $40$ percent clogged.