1. Viscous Flow in Pipes

2. Types of Engineering Problems
How big does the pipe have to be to carry a flow of x m3/s?
What will the pressure in the water distribution system be when a fire hydrant is open?

3. Example Pipe Flow Problem
cs1
D=20 cm
L=500 m
Find the discharge, Q.
100 m
valve
cs2
Describe the process in terms of energy!

4. Viscous Flow: Dimensional Analysis
Remember dimensional analysis?
Two important parameters!
R - Laminar or Turbulent
e/D - Rough or Smooth
Flow geometry
internal _______________________________
external _______________________________
Where
and
in a bounded region (pipes, rivers): find Cp
flow around an immersed object : find Cd

5. Laminar and Turbulent Flows
Reynolds apparatus
inertia
damping
Transition at R of 2000

6. Boundary layer growth: Transition length
What does the water near the pipeline wall experience? _________________________
Why does the water in the center of the pipeline speed up? _________________________
Drag or shear
Conservation of mass v v v
Pipe Entrance
Non-Uniform Flow
Need equation for entrance length here

7. Laminar, Incompressible, Steady, Uniform Flow
Between Parallel Plates
Through circular tubes
Hagen-Poiseuille Equation
Approach
Because it is laminar flow the shear forces can be easily quantified
Velocity profiles can be determined from a force balance
Don’t need to use dimensional analysis

8. Laminar Flow through Circular Tubes
Different geometry, same equation development (see Streeter, et al. p 268)
Apply equation of motion to cylindrical sleeve (use cylindrical coordinates)

9. Laminar Flow through Circular Tubes: Equations
a is radius of the tube
Max velocity when r = 0
Velocity distribution is paraboloid of revolution therefore _____________ _____________
average velocity (V) is 1/2 umax
Q = VA =
Vpa2

10. Laminar Flow through Circular Tubes: Diagram
Shear
Velocity
Laminar flow
Shear at the wall
True for Laminar or Turbulent flow

11. The Hagen-Poiseuille Equation
cv pipe flow
Constant cross section
h or z
Laminar pipe flow equations
CV equations!

12. Example: Laminar Flow (Team work)
Calculate the discharge of 20ºC water through a long vertical section of 0.5 mm ID hypodermic tube. The inlet and outlet pressures are both atmospheric. You may neglect minor losses. What is the total shear force?
What assumption did you make? (Check your assumption!)

13. Turbulent Pipe and Channel Flow: Overview
Velocity distributions
Energy Losses
Steady Incompressible Flow through Simple Pipes
Steady Uniform Flow in Open Channels

14. Turbulence A characteristic of the flow.
How can we characterize turbulence?
intensity of the velocity fluctuations
size of the fluctuations (length scale)
instantaneous
velocity
mean
velocity
velocity
fluctuation t

15. Turbulence: Size of the Fluctuations or Eddies
Eddies must be smaller than the physical dimension of the flow
Generally the largest eddies are of similar size to the smallest dimension of the flow
Examples of turbulence length scales
rivers: ________________
pipes: _________________
lakes: ____________________
Actually a spectrum of eddy sizes
depth (R = 500)
diameter (R = 2000)
depth to thermocline

16. Turbulence: Flow Instability
In turbulent flow (high Reynolds number) the force leading to stability (_________) is small relative to the force leading to instability (_______).
Any disturbance in the flow results in large scale motions superimposed on the mean flow.
Some of the kinetic energy of the flow is transferred to these large scale motions (eddies).
Large scale instabilities gradually lose kinetic energy to smaller scale motions.
The kinetic energy of the smallest eddies is dissipated by viscous resistance and turned into heat. (=___________)
viscosity
inertia
head loss

17. Velocity Distributions
Turbulence causes transfer of momentum from center of pipe to fluid closer to the pipe wall.
Mixing of fluid (transfer of momentum) causes the central region of the pipe to have relatively _______velocity (compared to laminar flow)
Close to the pipe wall eddies are smaller (size proportional to distance to the boundary)
constant

18. Turbulent Flow Velocity Profile
Turbulent shear is from momentum transfer
h = eddy viscosity
Length scale and velocity of “large” eddies
Dimensional analysis y

19. Turbulent Flow Velocity Profile
increases
Size of the eddies __________ as we move further from the wall.
k = 0.4 (from experiments)

20. Log Law for Turbulent, Established Flow, Velocity Profiles
Integration and empirical results
Laminar
Turbulent
Shear velocity y x

21. Pipe Flow: The Problem
We have the control volume energy equation for pipe flow
We need to be able to predict the head loss term.
We will use the results we obtained using dimensional analysis

22. Pipe Flow Energy Losses
Horizontal pipe
Dimensional Analysis
Darcy-Weisbach equation

23. Friction Factor : Major losses
Laminar flow
Turbulent (Smooth, Transition, Rough)
Colebrook Formula
Moody diagram
Swamee-Jain

24. Laminar Flow Friction Factor
Hagen-Poiseuille
Darcy-Weisbach
Slope of ___ on log-log plot -1

25. Turbulent Pipe Flow Head Loss
___________ to the length of the pipe
___________ to the square of the velocity (almost)
________ with the diameter (almost)
________ with surface roughness
Is a function of density and viscosity
Is __________ of pressure
Proportional
Proportional
Inversely
Increase
independent

26. Smooth, Transition, Rough Turbulent Flow
Hydraulically smooth pipe law (von Karman, 1930)
Rough pipe law (von Karman, 1930)
Transition function for both smooth and rough pipe laws (Colebrook)
(used to draw the Moody diagram)

27. Moody Diagram 0.10 0.08 0.06 0.05 0.04 friction factor 0.03 0.02 0.01
0.015
0.04
0.01
0.008
friction factor
0.006
0.03
0.004
laminar
0.002
0.02
0.001
0.0008
0.0004
0.0002
0.0001
0.01
smooth
1E+03
1E+04
1E+05
1E+06
1E+07
1E+08 R

28. Swamee-Jain no f Each equation has two terms. Why? 1976 limitations
/D < 2 x 10-2
Re >3 x 103
less than 3% deviation from results obtained with Moody diagram
easy to program for computer or calculator use
no f
Each equation has two terms. Why?

29. Pipe roughness Must be dimensionless! pipe material pipe roughness 
(mm)
glass, drawn brass, copper
0.0015
commercial steel or wrought iron
0.045
Must be dimensionless!
asphalted cast iron
0.12
galvanized iron
0.15
cast iron
0.26
concrete
rivet steel
corrugated metal 45
0.12
PVC

30. Solution Techniques find head loss given (D, type of pipe, Q)
find flow rate given (head, D, L, type of pipe)
find pipe size given (head, type of pipe,L, Q)

31. Minor Losses
We previously obtained losses through an expansion using conservation of energy, momentum, and mass
Most minor losses can not be obtained analytically, so they must be measured
Minor losses are often expressed as a loss coefficient, K, times the velocity head.
High R

32. Head Loss due to Gradual Expansion (Diffusor)
diffusor angle ()
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 20 40 60 80 KE

33. Sudden Contraction V2 V1 flow separation
losses are reduced with a gradual contraction

34. Sudden Contraction Cc A2/A1 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.2
0.2
0.4
A2/A1 Cc

35. Entrance Losses vena contracta
Losses can be reduced by accelerating the flow gradually and eliminating the
vena contracta

36. Head Loss in Bends
High pressure
Head loss is a function of the ratio of the bend radius to the pipe diameter (R/D)
Velocity distribution returns to normal several pipe diameters downstream
Possible separation from wall R D
Low pressure
Kb varies from

37. Head Loss in Valves Function of valve type and valve position
The complex flow path through valves can result in high head loss (of course, one of the purposes of a valve is to create head loss when it is not fully open)

38. Solution Techniques Neglect minor losses Equivalent pipe lengths
Iterative Techniques
Simultaneous Equations
Pipe Network Software

39. Iterative Techniques for D and Q (given total head loss)
Assume all head loss is major head loss.
Calculate D or Q using Swamee-Jain equations
Calculate minor losses
Find new major losses by subtracting minor losses from total head loss

40. Solution Technique: Head Loss
Can be solved directly

41. Solution Technique: Discharge or Pipe Diameter
Iterative technique
Set up simultaneous equations in Excel
Use goal seek or Solver to find discharge that makes the calculated head loss equal the given head loss.

42. Example: Minor and Major Losses
Find the maximum dependable flow between the reservoirs for a water temperature range of 4ºC to 20ºC.
25 m elevation difference in reservoir water levels
Water
Reentrant pipes at reservoirs
Standard elbows
2500 m of 8” PVC pipe
Sudden contraction
Gate valve wide open
1500 m of 6” PVC pipe

43. Directions Assume fully turbulent (rough pipe law)
find f from Moody (or from von Karman)
Find total head loss
Solve for Q using symbols (must include minor losses) (no iteration required)
Obtain values for minor losses from notes or text

44. Example (Continued) What are the Reynolds number in the two pipes?
Where are we on the Moody Diagram?
What value of K would the valve have to produce to reduce the discharge by 50%?
What is the effect of temperature?
Why is the effect of temperature so small?

45. Example (Continued) Were the minor losses negligible?
Accuracy of head loss calculations?
What happens if the roughness increases by a factor of 10?
If you needed to increase the flow by 30% what could you do?
Suppose I changed 6” pipe, what is minimum diameter needed?

46. Pipe Flow Summary (1)
Shear increases _________ with distance from the center of the pipe (for both laminar and turbulent flow)
Laminar flow losses and velocity distributions can be derived based on momentum and energy conservation
Turbulent flow losses and velocity distributions require ___________ results
linearly
experimental

47. Pipe Flow Summary (2)
Energy equation left us with the elusive head loss term
Dimensional analysis gave us the form of the head loss term (pressure coefficient)
Experiments gave us the relationship between the pressure coefficient and the geometric parameters and the Reynolds number (results summarized on Moody diagram)

48. Pipe Flow Summary (3)
Dimensionally correct equations fit to the empirical results can be incorporated into computer or calculator solution techniques
Minor losses are obtained from the pressure coefficient based on the fact that the pressure coefficient is _______ at high Reynolds numbers
Solutions for discharge or pipe diameter often require iterative or computer solutions
constant

49. Columbia Basin Irrigation Project
The Feeder Canal is a concrete lined canal which runs from the outlet of the pumping plant discharge tubes to the north end of Banks Lake (see below). The original canal was completed in 1951 but has since been widened to accommodate the extra water available from the six new pump/generators added to the pumping plant. The canal is 1.8 miles in length, 25 feet deep and 80 feet wide at the base. It has the capacity to carry 16,000 cubic feet of water per second.

50. Pipes are Everywhere!
Owner: City of Hammond, IN Project: Water Main Relocation Pipe Size: 54"

51. Pipes are Everywhere! Drainage Pipes

52. Pipes

53. Pipes are Everywhere! Water Mains

54. Glycerin

55. Example: Hypodermic Tubing Flow