Please solve this step by step and with details if you can! Please do $1$ through $6!$

A centrifugal pump has a characteristic represented by: $H=a{Q}^2+b,($ H in meters, Q in $\frac{L}{s},$ a and b are

constants$).$ The hydraulic efficiency can be written in the form of: in $\frac{l}{s},c$ and d are

constants$).$ The nominal operating point where the efficiency is at its maximum is: $N=1500rpm,H=$

$100m,Q=40\frac{l}{s},h=0,8.$

calculate the constants $a,b,c$ and d$.$

In particular operating conditions, this pump must deliver a manometric head of $50$ meters when the

rotational speed is $1000$ rpm $.$

what will then be the flow rate and the hydraulic efficiency?

This pump is used to raise water from a tank $A$ to a tank $B$ as shown in the drawing below: We give :

$N=$ pump rotational speed $=1500rpm=$ cte.

$S_A=$ tank A section $=$ cte $=200m^2.$

$D_{\text{p}\text{i}\text{p}\text{i}\text{n}\text{g}}=$ cte $=0,16m.$

$=$ Head loss coefficient $=0,07$

Piping length $L=200m.$

To simplify the calculations, we neglect the head losses in the intake piping and the singular head losses

caused by the elbows in the exhaust piping. The water level in the tank B stays the same and equal to z$3$

$)=(86m$

find the operating point of the pump when the water level in the tank $A$ is constant $)=(20m$

We will consider that the time $t=0$ is when $zO=20m.$ The tank is no longer supplied;

establish the differential equation connecting the water level in the tank $A$ to time.

calculate the date t $1$ corresponding to the level $z1=10m$ at which the pump is stopped.

calculate the flow rate of the pump immediately before it is shut down.