Q$4)(10$ points$)$ The head loss and effluent during a typical filter run at a drinking water treatment plant are shown in Figure below. The maximum head loss at which the filter can be operated efficiently is $2\mathrm{.}5$ m $,$ and the turbidity criterion is $0\mathrm{.}2$ NTU.

$($a$)$ Approximately how long should the operator run the filter before backwashing it$?$

$($b$)$ Explain briefly why the head loss increases at an approximately steady rate over time.

$($c$)$ Describe the likely effects of the following changes on the amount of water that can be processed in a filter run. Assume in all cases that the changes are fairly small, so that whatever limits the run length in the baseline $($initial$)$ condition still limits it after the change has been implemented.

$($i$)$ Adding a small amount of polymeric coagulant $($and thereby improving the destabilization$)$ as the water enters the filter bed. $($The purpose of coagulant is to remove$/$reduce the surface charge and enable better floc formation $($particle size grows$).$

$($ii$)$ Adding a small amount of sand to the bed to make the column deeper.

$($iii$)$ Replacing the sand in the bed with slightly larger grained material with the same depth. $($Assume the porosity remains the same but recognize that the pore size will increase.$).$

Figure: Headloss and effluent quality in an operating muer.