1. Chapter 4. Analysis of Flows in Pipes
Learning Outcomes:
Classification of pipe flows based on Reynolds number
Evaluate energy losses due to friction, valves, fittings, bends, diameter expansion and contraction.
Design the size of distribution pipes considering head loss.

2. 4.1 Characterization of flow based on Reynolds number
Osborne Reynolds (a British engineer) conducted a flow experiment, i.e. by injecting dye into pipe flow to classify the types of flow.
Turbulent
Near laminar

3. Three types of flow:
Laminar flow
Re < 2000
Transitional flow
2000 < Re < 4000
Turbulent flow
Re > 4000
Reynolds number Re is the ratio of the inertia force on an element of fluid to the viscous force.
Re is dimensionless.
where V = average velocity, D = diameter of pipe,  = fluid density,  = dynamic viscosity, and  = kinematic viscosity
Reynolds number is one of the important dimensionless number used in the dimensional analysis in fluid mechanics and hydraulics.

4. Example 4.1
Determine the range of average velocity of flow for which the flow would be in the transitional region if an oil of S.G. = 0.89 and dynamic viscosity = 0.1 Ns/m2 is flowing in a 2-in pipe.
For transitional flow,
When Re = 2000,
When Re = 4000,
Therefore, for the flow to be in transitional state, the average velocity V should be between m/s and m/s.

5. 4.2 Head Loss due to Friction in Pipes of Constant Cross Section
Head loss or energy loss in pipe flow maybe caused by friction, valves, fittings, expansion and contraction in pipe diameter, and others.
Frictional head loss is known as the major head loss in pipe flow as the loss occurs along the pipe while other head losses such as due valves, fittings, expansion and contraction are known as minor losses.
Friction head loss can be estimated using the Darcy-Weisbach equation:
where hf = head loss due to friction, f = friction factor (dimensionless), L = length of flow (or pipe), D = diameter of pipe, V = average velocity of flow, and g = gravity acceleration.
For laminar flow, the head loss due to friction can also be estimated using the Hagen-Poiseuille equation:
where  = dynamic viscosity, L = length of flow (or pipe), D = diameter of pipe, V = average velocity of flow, and  = specific weight of liquid.

6. For laminar flow, the friction factor f can be calculated based on Reynolds number.
Darcy-Weisbach equation:
Hagen-Poiseuille equation:
Equating both equations:
Friction factor for laminar flow

7. Example 4.2
In a refinery oil (S.G. = 0.85,  = 1.8  105 m2/s) flows through a 30 m long, 100 mm diameter pipe at 0.50 L/s. Is the flow laminar or turbulent? Find the head loss per meter of pipe length.
Reynolds number
< 2000, therefore laminar flow
For laminar flow, friction factor
Head loss per meter pipe length

8. 4.3 Friction factor and Moody Chart
Friction factor can be estimated by the following equations:
Darcy-Weisbach & Hagen-Poiseuille:
for Re < 2000
Blasius (for smooth pipes):
for 3000  Re 
Colebrook (for all pipes):
for 4000  Re  108
For convenience, these friction factor equations are used to prepare a Moody diagram.
Moody diagram provides the relationship between friction factor f, Reynolds number Re and pipe relative roughness e/D, where e is the absolute roughness.
Pipe surface roughness measurement

10. Example 4.3
A 20-in diameter galvanized iron pipe ft long carries 4 cfs of water at 60F. Find the friction head loss using the Moody chart.
L = ft, Q = 4 ft3/s,  =  105 ft2/s
Reynolds number
> 4000, turbulent flow
Galvanized iron pipe, e = ft
= in
Relative roughness
From Moody diagram,
Using Darcy-Weisbach equation, friction head loss

11. f =
0.0003
2.51  105

12. Example 4.4
Determine the friction factor f if water at 70C is flowing at 9.14 m/s in stainless steel pipe having an inside diameter of 25 mm.
Given velocity of flow = 9.14 m/s and diameter of pipe D = m
For water at 70C, kinematic viscosity  = 9.75  107 m2/s
Reynolds number
> 4000, turbulent flow
Stainless steel pipe, e = mm
Relative roughness
From Moody diagram,

13. f =
2.34  105

14. Example 4.5
Water at 20C flows in a 500-mm diameter welded steel pipe. If the friction loss gradient is 0.006, determine the flow rate.
For welded steel pipe, surface roughness e = mm
Relative roughness
For water at 20C, kinematic viscosity  =  106 m2/s
Friction loss gradient
Friction head loss

15. ftrial2 =
ftrial1 =
1.113  106

16. Assume friction factor f by assuming turbulent flow condition,
Based on
, ftrial1 =
Velocity
Reynolds number
Check the friction factor f with the corresponding e/D and Re,
Friction factor ftrial2 =
Velocity
Reynolds number
Check the friction factor f with the corresponding e/D and Re,
Friction factor f =
= ftrial2  converged!

17. Or, trial-and-error in tabulated form,
(m/s)
Re =
Trial f
Check f
0.0118
2.233
1.113  106
0.0131
Try again
0.0131
2.120
1.057  106
0.0131
Converged!
Flow rate

18. Example 4.6
A galvanized iron pipe ft long must convey ethyl alcohol ( = 2.3  105 ft2/s) at a rate of 135 gpm. If the friction head loss must be 215 ft, determine the pipe size using Moody chart. Note: 1000 gpm = 2.23 cfs.
For galvanized iron pipe, surface roughness e = ft
Relative roughness
Velocity
Reynolds number
Friction head loss

19. Trial-and-error in tabulated form,
(ft)
Trial f
Check f
0.03
0.3561
0.0253
Try again
0.0253
0.3442
0.0253
Converged!
Assume mid-range value of f
Therefore, the size of the pipe

20. ftrial1 = 0.03
ftrial2 =
0.0014
4.84  104
4.68  104

21. Example 4.7
In a chemical processing plant, benzene at 50C (S.G. = 0.86) must be delivered to point B with a pressure of 550 kPa. A pump is located at point A 21 m below point B, and the two points are connected by 240 m of plastic pipe having an inside diameter of 50 mm. If the volume flow rate is 110 L/min, calculate the required pressure at the outlet of the pump.
 = 860 kg/m3
 = 4.2  104 Pa.s
= zB  zA
Energy equation between A and B

22. zB  zA = 21 m
 = 860 kg/m3
For plastic pipe, roughness e = 0
Reynolds number
From Moody chart,
f =
Energy equation between A and B:

23. 4.4 Minor Losses in Pipe Flow
a. Loss of head at entrance
Loss due to the entrance of flow into pipe
where V = mean velocity in the pipe, ke = loss coefficient

24. b. Loss of head at submerged discharge
Loss of head when pipe discharges fluid into filled reservoir or tank

25. c. Loss of head due to contraction
Loss of head when flow contracts
Loss coefficients for sudden contraction kc
0.0
0.50
0.1
0.45
0.2
0.42
0.3
0.39
0.4
0.36
0.5
0.33
0.6
0.28
0.7
0.22
0.8
0.15
0.9
0.06
1.0
0.00

26. Loss of head when flow contracts
For smooth curved transition, kc can be as small as 0.05.
For conical transducer with angle 20 to 40, a minimum kc is about 0.10.

27. d. Loss of head due to expansion
Loss of head when flow expands.
Loss of head due to sudden expansion
Loss of head due to gradual expansion

28. Loss of head due to gradual expansionkx

29. e. Loss of head due to pipe fittings
Loss of head due to pipe fittings such as valve, bend, and elbow
Values of loss factors for pipe fittings
Fitting k
Globe valve, wide open 10
Angle valve, wide open 5
Close-return bend
2.2
T, through side outlet
1.8
Short-radius elbow
0.9
Medium-radius elbow
0.75
Long-radius elbow
0.60
45 elbow
0.42
Gate valve, wide open
0.19
Gate valve, half open
2.06

30. f. Loss of head in bends and elbows
Loss of head due to bends and elbows
Loss coefficient due to 90 bend

31. Example 4.8
Water at 15C is being pumped from a stream to a reservoir whose surface is 64 m above the pump. The pipe from the pump to the reservoir is 8 in steel pipe 762 m long. If m3/s is being pumped, compute the pressure at the outlet of the pump. Consider the friction loss in the discharge line, but neglect other losses.
If the pressure at the pump inlet is 16.27 kPa, compute the power delivered by the pump to the water.
Water at 15C,  =1000 kg/m3,  =  106 m2/s B
pB = 0, VB = 0 
zB  zA = 64 m
D = 8 in = m
L = 762 m
Steel pipe, e = mm  A
Q = m3/s

32. B A
From Moody chart, f =
Energy equation between A and B:

33. Power delivered by pump to water

34. Example 4.9
The tanks, pump and pipelines of figure below have the characteristics noted. The suction line entrance from the pressure tank is flush, and the discharge into the open tank is submerged. If the pump P puts 2.0 hp into the liquid, (a) determine the flow rate, and (b) find the pressure in the pipe on the suction side of the pump.
ke = 0.5
kd = 1.0
V1  0
p3 = 0
V3  0
Energy equation between 1 and 3:

35. Energy equation between 1 and 3:

36. To find the pressure in the pipe on the suction side of the pump, write the energy equation between 1 and 2.

37. Example 4.10
Water is being conveyed from 2 reservoirs through a steel pipe of diameter 20 cm. The difference of water level in the two reservoirs is 25 m. Based on your own assumptions, compute the flow rate through the pipe. The kinematic viscosity for water at 20C is  106 m2/s.
D = 20 cm
Steel pipe, e = mm A
zA  zB = 25 m 
pA = 0
Gate valve, half open, kv = 2.06
VA  0 B 
pB = 0
VB  0
Energy equation between A and B,

38. Energy equation between A and B,
Assuming the length of the pipe is 100 m long and it is fully turbulent pipe flow,
From Moody chart, f = 0.014

39. Assignment No. 4 (due March 18, 2011)
1. The tanks, pump and pipelines of figure below have the characteristics noted. The suction line entrance from the pressure tank is flush, and the discharge into the open tank is submerged. If the pump P puts 2.0 kW into the liquid, (a) determine the flow rate, and (b) find the pressure in the pipe on the suction side of the pump.
ke = 0.5
Water m cm
kPa
kd = 1.0

40. Available that V1  0; V3  0; p3 = 0 and
Energy equation between 1 and 3:
Therefore,
and
To find the pressure in the pipe on the suction side of the pump, write the energy equation between 1 and 2.

41. 2. A wrought iron pipe ft long must convey water at a rate of 35 m3/s at 20C. If the friction head loss must be 225 m, determine the pipe size using Moody chart.
For wrought iron pipe, surface roughness e = mm
Relative roughness
Velocity
At 20C,  =  106 m2/s
Reynolds number
Friction head loss

42. Trial-and-error in tabulated form,
Trial f
Check f
(m)
0.03
122.38
3.759  107
3.63  105
0.014
Try again
0.014
105.08
4.378  107
4.23  105
0.0135
Try again
0.0135
104.32
4.410  107
4.26  105
0.0135
Converged!
Assume mid-range value of f
Therefore, the size of the pipe