# Question No.6

#### Topic : Solving Pipe Networks using Quantity Balance Method

In the system illustrated in the following figure, a pump is installed in pipe BC to provide a flow of 40 lit/sec to reservoir C. Neglecting minor losses, calculate the total head to be generated by the pump and the power consumption assuming an overall efficiency of 60 per cent. Determine also the flow rates in the other pipes.

Data :

 Pipe Length (m) Diameter (mm) AB 10,000 400 BC 4,000 250 BD 5,000 250

Roughness size of all pipes = 0.06 mm

Given :

• $ηtotal=0.60$

Steps of solution :

Step (1) - Estimate ZB (pressure head elevation at B) = 130 m (Note that the elevation of the pipe junction itself does not affect the solution.)

Step (2) - Since the flow in BC is prescribed, it is simply treated as an external outflow at B.

For automatic computer analysis, Darcy-Colebrook-White combination can be used :

$Q=−2A2gDhfLlogk/D3.7+2.51νD2gDhfL$

For each pipe (friction head loss) is initialized to :

$ZI−ZB$

Step (3) – Calculate the correction value for the estimated pressure head elevation. The calculations proceed in tabular form. Note that Q is written in liter/sec simply for convenience; all computations are based on Q in m3/sec.

To calculate the correction required in ZB value :

$ΔZB=2(ΣQIB−FB)ΣQIBhL,IB$

For this problem, the flow in BC is considered as an external outflow at junction B, therefore :

Therefore :

$ΔZB=2(ΣQIB−0.04)ΣQIBhL,IB$

For the first iteration :

Correction to ZB :

For the second iteration :

Correction to ZB :

For the third iteration :

Correction to ZB :

For the fourth iteration :

Correction to ZB :

Final discharges :

Final discharges (after 4 iterations) :

 Pipe Q (l/sec) Flow Direction AB 118.896 A to B BC 40.000 B to C BD 78.896 B to D

To calculate the total head generated by the pump :

$ZB−ZC+Hpump=KBCQBC2$

To calculate Reynolds number :

$Re=vDν=0.815∗0.251.011∗10−6=201,502$

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

$f=0.0174$

To calculate the head loss coefficient K for pipe BC :

$KBC=8fBCLBCπ2gDBC5=8∗0.0174∗4,000π2∗9.81∗(0.25)5=5,892.25$

$ZB−ZC+Hpump=KBCQBC2$

$132.394−145+Hpump=5,892.25∗0.042$

Therefore, the total head provided by the pump is :

To calculate the power consumption :

$→$ $0.60=8,646.14InputPower$