# Question No.3

#### Topic : Pump-Pipeline System Analysis & Design

Calculate the steady discharge of water between the tanks in the system shown in the following figure and the power consumption. Pipe diameter Ds = Dd = 200 mm; length = 2,000 m; k = 0.03 mm (uPVC). Losses in valves, bends plus the velocity head amount to  6.2 v2 / 2g. Static lift = 10 m. Pump Characteristics :

 Discharge (l/sec) 0 10 20 30 40 50 Total Head (m) 25 23.2 20.8 16.5 12.4 7.3 Efficiency (per cent) - 45 65 71 65 45

The efficiencies given are the overall efficiencies of the pump and motor combined.

Given :

• $hminor=6.2v22g$

Steps of solution :

The solution to such problems is basically to solve simultaneously the head-discharge relationships for the pump and pipeline :

For the pump, head delivered at discharge Q may be expressed by :

$Hm=AQ2+BQ+C$

For the pipeline, the head required to produce a discharge Q is given by :

$Hm=Hs+hf+hminor$

$Hm=Hs+8fLπ2gD5Q2+Km2gA2Q2$

A graphical solution is the simplest method and also gives the engineer a visual interpretation of the matching of the pump & pipeline.

First of all, we shall start with the system curve. Values of H corresponding to a range of Q values will be calculated. To summarize the procedure for completing the previous table :

Flow velocity is calculated through :

$v=QA=Qπ4D2$ $→$ $v=Qπ4∗0.22$

The kinematic viscosity for water is taken as :

$ν=1.13∗10−6m2/sec$

Reynolds number is calculated through :

$Re=vDν=v∗0.21.13∗10−6$

The roughness height for uPVC pipes is :

To calculate the friction coefficient, Barr's equation is used :

$1f=−2logk/D3.7+5.1286Re0.89$

The friction losses (using Darcy-Weisbach equation) occurring in the pipeline is calculated through :

$hf=KQ2,K=8fLπ2gD5$

To calculate the minor/secondary losses through the pipeline :

$hminor=Km2gA2Q2$

Finally, the total manometric head required is :

$Hm=Hs+hf+hminor$

The following graph shows the system curve for the previously tabulated values : The following graph shows the pump characteristic curve (Head vs Discharge curve ) : The computed system curve data and pump characteristic curve data are now plotted on the same graph as shown in the following figure : The intersection point gives the operating conditions; in this case :

The operating efficiency is :

$ηtotal=0.70$ To calculate the power consumption :

$→$