Q & A – Fluid Mechanics – Solving Pipe Networks using Quantity Balance Method – Q.5

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Q & A – Fluid Mechanics – Solving Pipe Networks using Quantity Balance Method – Q.5

Q & A – Fluid Mechanics – Solving Pipe Networks using Quantity Balance Method – Q.5

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Question No.5

Topic : Solving Pipe Networks using Quantity Balance Method

Determine the flows in the network illustrated in the below figure; minor losses are given by Cm v2/2g.

Data :

Pipe

Length (m)

Diameter (mm)

Roughness (mm)

Minor Loss Coefficient Cm

AB

20,000

500

0.3

20

BC

5,000

350

0.3

10

BD1

6,000

300

0.3

10

BD2

6,000

250

0.06

10

Given :

  • ZA=100  mZC=70  mZD=60  m

Steps of solution :

Step (1) - Estimate ZB (pressure head elevation at B) = 80 m (Note that the elevation of the pipe junction itself does not affect the solution.)

ZB=80  m

Step (2) - Allocate estimated flow velocities in the pipes.

v=2m/sec  for  all  pipes

The friction factor f may be obtained from the Moody diagram, or using Barr's equation, using an initially estimated velocity in each pipe. Subsequently f can be based on the previously calculated discharges. However, unless there is a serious error in the initial velocity estimates, much effort is saved by retaining the initial f values until perhaps the final correction.

Step (3) – Evaluate the head loss coefficient K for each pipe. The friction coefficient f is obtained through Moody chart or using Barr's equation. 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 velocity at each pipe :

v=QA=Qπ4D2

To calculate Reynolds number :

Re=vDν,ν=1.011106m2/sec

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

1f=2logk/D3.7+5.1286Re0.89

Barr's equation is based partly on an approximation to the logarithmic smooth turbulent element in the Colebrook-White equation.

To calculate friction head loss coefficient Kf for each pipe  (using Darcy-Weisbach equation) :

Kf=8fLπ2gD5

To calculate minor head loss coefficient Km for each pipe (representing minor losses) :

hminor=Cmv22g=CmQ22gA2 Km=Cm2gA2

To calculate head loss coefficient K for each pipe (representing both friction & minor losses) :

K=Kf+Km=8fLπ2gD5+Cm2gA2

To calculate the correction required in ZB value :

ΔZB=2(ΣQIBFB)ΣQIBhL,IB

For this problem, no outflow occurs at junction B, therefore :

FB=0

Therefore :

ΔZB=2ΣQIBΣQIBhL,IB

For the first iteration :

Initial estimate for velocity is 2 m/sec for all pipes !

Correction to ZB :

ΔZB=2(0.054)0.0210=5.146  mZB=805.146=74.854  m

For the second iteration :

The initial estimates for velocity is updated according to the previously calculated value !

Correction to ZB :

ΔZB=2(0.004)0.0242=0.328  mZB=74.854+0.328=75.183  m

For the third iteration :

The initial estimates for velocity is updated according to the previously calculated value !

Correction to ZB :

ΔZB=2(0.001)0.0237=0.076  mZB=75.183+0.076=75.258  m

For the fourth iteration :

Correction to ZB :

ZB=75.259  m

Final Discharges in the given pipe network :

Final discharges (after 4 iterations) :

Pipe

Q (l/sec)

Flow Direction

AB

156.513

A to B

BC

56.518

B to C

BD1

58.979

B to D

BD2

41.016

B to D

 

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