# Question No.4

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

If in the network shown in the following figure the flow to C is regulated by a valve to 100 lit/sec, calculate the effect on the flows to the other reservoirs; determine the head loss to be provided by the valve. Data :

 Pipe Length (m) Diameter (mm) AJ 10,000 450 BJ 2,000 350 CJ 3,000 300 DJ 3,000 250

Roughness size of all pipes = 0.06 mm

Given :

Steps of solution :

Step (1) - Estimate ZJ (pressure head elevation at J) = 150 m (Note that the elevation of the pipe junction itself does not affect the solution.)

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

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 :

$ZI−ZJ$

Step (3) – Calculate the correction value for the assumed 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 ZJ value :

$ΔZJ=2(ΣQIJ−FJ)ΣQIJhL,IJ$

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

Therefore :

$ΔZJ=2(ΣQIJ−0.1)ΣQIJhL,IJ$

For the first iteration : Correction to ZJ :

For the second iteration : Correction to ZJ :

For the third iteration : Correction to ZJ :

For the fourth iteration : Correction to ZJ :

For the fifth iteration : Correction to ZJ :

Final Discharges : Final discharges (after 5 iterations) :

 Pipe Q (l/sec) Flow Direction AJ 340.565 A to J BJ 125.365 J to B CJ 100.000 J to C DJ 115.201 J to D

To calculate the head loss provided by the valve :

$ZJ−ZC=KCJQCJ2+hvalve$

To calculate Reynolds number :

$Re=vDν=1.415∗0.31.011∗10−6=419,795$

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

$1f=−2logk/D3.7+5.1286Re0.89$

$1f=−2log0.06/3003.7+5.1286419,7950.89$

$f=0.0158$

To calculate the head loss coefficient K for pipe CJ :

$KCJ=8fCJLCJπ2gDCJ5=8∗0.0158∗3,000π2∗9.81∗(0.3)5=1,610.271$

$ZJ−ZC=KCJQCJ2+hvalve$ $→$ $127.576−100=1,610.271∗0.12+hvalve$

Finally, the head loss provided by the valve is :