A chemical engineer is designing a suction system for a water treatment plant. Water must be pumped from a river to a primary filtration system. Before reaching the pump, a horizontal packed filter must be installed to remove suspended solids and protect the pump equipment.

System Specifications:

$1.$ Required flow rate: $\backslash (\backslash$mathrm$\{$Q$\}=100\backslash$mathrm$\{$~L$\}/\backslash$mathrm$\{$min$\}\backslash )$

$2.$ Suction height: The pump is located $2$ m above the river level.

$3.$ Suction pipe diameter: $6$ inches, made of PVC$.$

$4.$ Suction line accessories:

$-$ Two $\backslash (90^\{\backslash$circ$\}\backslash )$ flanged elbows

$5.$ Water properties:

$-$ Temperature: $\backslash (20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$

$6.$ Filter material:

$-$ Packing particle size: $\backslash (2\backslash$mathrm$\{$~mm$\}^\{2\}\backslash )$

$-$ Packing porosity $()$: $0\mathrm{.}40$

Design Constraints:

$1.$ The pressure at the pump suction must not drop below $\backslash (-3\backslash$mathrm$\{$~m$\}\backslash$mathrm$\{$H$\}\_\{2\}\backslash$mathrm$\{$O$\}\backslash )$ to prevent cavitation.

$2.$ The water velocity inside the filter must not exceed $\backslash (0\mathrm{.}2\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ to avoid bed fluidization or particle entrainment.

Objective:

Design the horizontal packed filter by determining:

$1.$ The maximum filter length $($L$)$ for a filter diameter $($D$)$ that satisfies the suction pressure constraint. $(30$ points$)$

$2.$ The pressure drops by:

$-$ Friction losses in the pipe, $(10$ points$)$

$-$ Losses due to accessories, $(10$ points$)$

$-$ Pressure drops across the filter. $(10$ points$)$