Flow Operating Condition in a Hydraulic Network $(100$ Points$)$

Consider the hydraulic network shown below. Assume the pump having a uniform diameter, d$,$ throughout the piping network. Determine the system operating point $($both pressure and flow rate$)$ when a centrifugal pump with the following pump characteristics parameters is used: $H_0=15m,A=2\mathrm{.}63{10}^3\frac{s}{m^2}($where $H_0$ is the shut$-$off head, and $A$ is the parabola coefficient for the pump head curve, refer to my class notes$).$ Assume the minor loss coefficient associated with the discharge from station $2$ exit to the air, k $,$ to be $1\mathrm{.}5.$

Use the following geometric dimensions:

$L_1=1m$

$L_2=1m$

$L_3=10m$

$L_4=1m$

$H=1m$

$d=0\mathrm{.}1m$

Assume the pipe has no surface roughness. There is a gate valve at the pump inlet side.