Question $3$: A water pump is used to pump water from one large reservoir to another large reservoir that is at a higher elevation. The free surfaces of both reservoirs are exposed to atmospheric pressure. The dimensions and minor loss coefficients are provided in the figure. The pump's performance is approximated by the expression $H_{\text{a}\text{v}\text{a}\text{i}\text{l}\text{a}\text{b}\text{l}\text{e}}=H_0-Q^2,$ where shutoff head $H_0=24\mathrm{.}4m$ of water column, coefficient $=0\mathrm{.}0678\frac{m}{L}p{m}^2,$ available pump head $H_{\text{a}\text{v}\text{a}\text{i}\text{l}\text{a}\text{b}\text{l}\text{e}}$ is in units of meters of water column, and capacity $Q$ is in units of liters per minute $($Lpm$).$ Estimate the capacity delivered by the pump.

Assumptions $1.$ The water is incompressible. $2.$ The flow is nearly steady since the reservoirs are large. $3.$ The water is at room temperature.

Properties The kinematic viscosity of water at $T=20C$ is $1m\frac{m^2}{s}={10}^{-6}\frac{m^2}{s}.$

$z_2-z_1=7\mathrm{.}85m(elevation$ difference$)$

$D=2\mathrm{.}03cm(pipe$ diameter$)$

$K_{L\text{e}\text{n}\text{t}\text{r}\text{a}\text{n}\text{c}\text{e}}=0\mathrm{.}50(pipe$ entrance$)$

$K_{L\text{v}\text{a}\text{l}\text{v}\text{e}}=17\mathrm{.}5(valve)$

$K_{L\text{e}\text{l}\text{b}\text{o}\text{w}}=0\mathrm{.}92(each$ elbow$-$there are $5)$

$K_{L\text{e}\text{x}\text{i}\text{t}}=1\mathrm{.}05(pipe$ exit$)$

$L=176\mathrm{.}5m(total$ pipe length$)$

$=0\mathrm{.}25mm(pipe$ roughness$)$