1. Local Head Losses
Local head losses are the “loss” of energy at point where the pipe changes dimension (and/or direction).
Pipe Expansion
Pipe Contraction
Entry to a pipe from a reservoir
Exit from a pipe to a reservoir
Valve (may change with time)
Orifice plate
Tight bends
They are “velocity head losses” and are represented by

2. Value of kL
For junctions and bends we need experimental measurements
kL may be calculated analytically for
Expansion
Contraction
By considering continuity and momentum exchange and Bernoulli

3. Losses at an Expansion
As the velocity reduces (continuity)
Then the pressure must increase (Bernoulli)
So turbulence is induced and head losses occur
Turbulence and losses

4. Value of kL for Expansion1 2
Apply the momentum equation from 1 to 2
Using the continuity equation we can eliminate Q
From Bernoulli

5. Value of kL for Expansion
Combine
and
- Borda-Carnot Equation
Using the continuity equation again

6. Losses at Contraction
Flow converges as the pipe contracts
Convergence is narrower than the pipe
Due to vena contractor
Experiments show for common pipes
Can ignore losses between 1 and 1’
As Convergent flow is very stable

7. Losses at Contraction
Apply the general local head loss equation between 1’ and 2
And Continuity
The value of k depends on
In general

8. Losses: Junctions

9. Losses: Sharp bends

10. kL values Bell mouth Entry T-branch kL= 1.5 kL= 0.1 Sharp Entry/Exit
T-inline
kL Values
Bellmouth entry
0.1
Sharp entry
0.5
Sharp exit
90o Bend
0.4
90o Tees
flow in line
to line Branch
1.5
Gate value (open)
0.25

11. Pipeline Analysis
Bernoulli Equation
equal to a constant: Total Head, H
Applied from one point to another (A to B)
With head losses

12. Bernoulli Graphically
Reservoir
Pipe of Constant diameter
No Flow A

13. Bernoulli Graphically
Constant Flow
Constant Velocity
No Friction A

14. Change of Pipe Diameter
Bernoulli Graphically
Constant Flow
Constant Velocity
No Friction
Change of Pipe Diameter A

15. Bernoulli Graphically
Constant Flow
Constant Velocity
With Friction A

16. Reservoir Feeding Pipe Example

17. Reservoir Feeding Pipe Example
Apply Bernoulli with head losses
pA=pC = Atmospheric
uA = negligible

18. Find pressure at B: Apply Bernoulli A-B
pA= Atmospheric
u = uB = 2.41 m/s
Negative
i.e. less than Atmospheric pressure

19. Pipes in series
Consider the situation when the pipes joining two reservoirs are connected in series 1 2
Total loss of head for the system is given as
A1U1=A2U2=AnUn=Q
Q1=Q2=Qn=Q

20. Pipes in Series Example
Two reservoirs, height difference 9 m, joined by a pipe that changes diameter. For, 15m d=0.2 m then for 45 m, d=0.25 m. f = 0.01 for both lengths.
Use kL,entry= 0.5, kL,exit = 1.0. Treat the joining of the pipes as a sudden expansion.
Find the flow between the reservoirs.

21. Apply Bernoulli A-B

23. Pipes in parallel
The head loss across the pipes is equal
Diameter, f, length L, Q, andu may differ
Total flow is sum in each pipe

24. Knowing the loss of head, the discharges in each pipe can be obtained
The values of U1, U2 and U3 may be obtained from the above equations and hence the discharges Q1, Q2 and Q3 obtained.

25. The discharge Q is given, the distribution of discharge in different branches is required
The discharges Q1, Q2, Q3 can be expressed in terms of H
Where

26. Similarly
Therefore:
As the discharge is known and K1,K2 and K3 are constants, the value of H is obtained and the discharge in individual pipes obtained

27. Pipes in Parallel Example
Two pipes connect two reservoirs which have a height difference of 10 m. Pipe 1 has diameter 50 mm and length 100 m. Pipe 2 has diameter 100 mm and length 100 m. Both have entry loss kL,entry = 0.5 and exit loss kL,exit =1.0 and Darcy f of
Find Q in each pipe
Diameter D of a pipe 100 m long and same f that could replace the two pipes and provide the same flow.

28. Apply Bernoulli for each pipe separately

29. Pipe 2:

30. Flow required in new pipe =
Replace u using continuity
Must solve iteratively

31. Must solve iteratively for D
Get approximate answer by leaving the 2nd term
Increase deq a little, say to 0.106
… a little more … to 0.107 So

32. Flow through a by-pass
A by-pass is a small diameter pipe connected in parallel to the main pipe
Meter
By-pass
Main pipe
The ratio of q in the by-pass, to the total discharge is known as the by-pass coefficient.
by-pass coefficient

33. Head loss in main pipe = head loss in by-pass
Minor losses in the by-pass due to bend and the meter etc
Divide by
Implies

34. Using continuity equation
Adding one on both sides

35. By-pass coefficient
Knowing the by-pass coefficient, the total discharge in the main pipe can be determined