HW$7-4$: In lecture, we analyzed flow from a water tower in one of the examples. We assumed that the gate valve was fully open and therefore that the loss coefficient was $K=0\mathrm{.}2.$ Evaluation of Table $8\mathrm{.}4$ shows that the loss coefficient increases dramatically as the valve is partially closed, throttling the flow. Determine what percentage open the valve would need to be before the flow rate was decreased $10\%$ from the $297$ gpm value we obtained in the example.TABLE $8-4$

Representative Loss Coefficients for Fittings and Valves

Source: Data from References $[9]$ and $[21]$

$\backslash$table$[[$Fitting$,$Geometry,K$,$Fitting,Geometry,K$],[90$ elbow,Flanged regular,$0\mathrm{.}3,$Globe valve,Open,$10],[,$Flanged long radius,$0\mathrm{.}2,$Angle valve,Open,$5],[,$Threaded regular,$1\mathrm{.}5,$Gate valve,Open,$0\mathrm{.}20],[,$Threaded long radius,$0\mathrm{.}7,,75\%$ open,$1\mathrm{.}10],[,$Miter,$1\mathrm{.}30,,50\%$ open,$3\mathrm{.}6],[,$Miter with vanes,$0\mathrm{.}20,,25\%$ open,$28\mathrm{.}8],[45$ Elbow,Threaded regular,$0\mathrm{.}4,$Ball valve,Open,$0\mathrm{.}5],[,$Flanged long radius,$0\mathrm{.}2,,1/3$ closed,$5\mathrm{.}5],[$Tee$,$ Straight through flow,Threaded,$0\mathrm{.}9,,2/3$ closed,$200],[,$Flanged,$0\mathrm{.}2,$Water meter,,$7],[$Tee$,$ branching,Threaded,$2\mathrm{.}0,$Coupling,,$0\mathrm{.}08],[$flow$,$Flanged,$1\mathrm{.}0,,,]](2)$

A fire protection system is supplied from a water tower and standpipe $80$ ft tall.

The longest pipe in the system is $600$ ft and is made of cast iron about $20$ years old $-$ corrosion may e$/$D factors by $5-10x.$

The pipe contains one gate valve; other minor losses may be neglected.

The pipe diameter is $4$ inches.

Determine the maximum rate of flow $($gpm$)$ through the pipe.