1. CVE 341 – Water Resources Lecture Notes I: (Combined Chs 7 & 8)
Closed Conduit Flow
Prepared by Ercan Kahya

2. A large slurry pipe. (Copyright Terry Vine/CORBIS)
View of a Stunning Example in CE Practices
A large slurry pipe. (Copyright Terry Vine/CORBIS)
Copyright © 2007 by Nelson, a division of Thomson Canada Limited

3. INTRODUCTION
The flow of water in a conduit may be either open channel flow or pipe flow.
Open channel flow must have a free surface whereas pipe flow has none.
A free surface subject to atmospheric pressure.
In pipe flow, there is no direct atmospheric flow but hydraulic pressure only.

4. INTRODUCTION (Cont`d)
Differences btw open channel flow & pipe flow

5. Steady Uniform Flow z: geometric head P/g: pressure head
V2/2g: velocity head
Steady, uniform flow in closed conduits

6. General Energy Consideration
a: the velocity coefficient & set to unity for regular & symmetrical cross-section like pipe.

7. Figure 8.2: Layout and hydraulic heads for Example 8.1.
Discuss how to attack to this problem…
Given: Q, d, hf, and available pressure at the building
Unknown: z1
Figure 8.2: Layout and hydraulic heads for Example 8.1.

8. Forces acting on steady closed conduit flow.
Resistance Application & Friction Losses in Pipes
General Resistance Equation:
from computing the shear stress of a system in dynamic equilibrium
t = g R Sf
t : Boundary shear stress (N/m2)
g : Specific weight of water (N/m3)
R : Hydraulic radius (m)
Sf : Slope of energy grade line
Forces acting on steady closed conduit flow.

9. Consider the length of pipe to be a control volume & realize the dynamic equilibrium
P: wetted perimeter
- Dividing this equation by area (A)
R: hydraulic radius (A / P)

10. Note that Note that → Piezometric head change So: pipe slope
∆z: change in pipe elevation
→ Piezometric head change

11. We refer this loss to as the friction loss (hf) & express as
: the change in the hydraulic grade line & also
equal to the energy loss across the pipe
We refer this loss to as the friction loss (hf) & express as
For circular pipes:
Since
; then
Finally
Sf : energy grade line
The general shear stress relation in all cases of steady uniform flow

12. Resistance Equations for Steady Uniform Flow
Now a method needed for “ shear stress ↔ velocity ”
From dimensional considerations;
a : constant related to boundary roughness
V : average cross-sectional velocity
Inserting this to the general shear relation,
Solving for V,
Chezy Equation

13. A general flow function relating flow parameters to
the change in piezometric head in pipes:
- In case of steady uniform flow; the left side equals to friction loss (hf)
Darcy-Weisbach Equation
This equation can be considered a special case of Chezy formula.

14. Resistance Application & Friction Losses in Pipes
From dimensional analysis: Resistance Equation
in terms of the average velocity
Chezy Equation
In most cases of closed conduit flow, it is customary to compute energy losses due to resistance by use of the Darcy-Weishbach equation.

15. Development of an analytical relation btw shear stress & velocity
- We earlier had the following relations for the friction loss:
&
For a general case, using d = 4R;
After simplifying;
Note that friction factor is directly
proportional to the boundary shear stress

16. - Let`s define “shear velocity” as=
Then the above equation becomes
→ important in developing the resistance formula

17. Velocity Distributions in Steady, Uniform Flow
Laminar Flow Rn ≤ Rn: Reynolds number
Turbulence Flow Rn ≥ 4000
Typical laminar & turbulent velocity distribution for pipes:
To start for velocity profile, let`s recall Newton`s law of viscosity
→ governs flow in the laminar region

18. Laminar Flow → Poiseuille Equation
Velocity in terms of radial position:
→ paraboloid distribution
Head loss in a pipe element in terms of average velocity:
Energy loss gradient or friction slope
represents the rate of energy dissipation due to boundary shear stress or friction
- Laminar flow case is governed by the Newtonian viscosity principle.
→ Poiseuille Equation
- Darcy friction factor: f = 64 / Rn in laminar flow

19. Turbulent Flow
More complex relations btw wall shear stress & velocity distribution
In all flow cases, Prandtl showed “laminar sub-layer” near the boundary
The thickness of laminar boundary layer decreases as the Re # increases
Flow is turbulent outside of the boundary layer

20. Turbulent Flow General form for turbulent velocity distributions
After statistical and dimensional considerations, Von Karman gives logarithmic dimensionless velocity distribution as
General form for turbulent velocity distributions
: instantaneous velocity
* : shear velocity
k : von Karman constant (= 0.4 for water)
y : distance from boundary
yo : hydraulic depth
C : constant

21. Turbulent Flow velocity distribution
For smooth-walled conduits
For rough-walled conduits
Function of the laminar sub-layer properties (wall Reynolds number)
Function of the wall roughness element

22. Friction Factor in Turbulent flow
Using velocity distribution given previously & shear stress/velocity relation (Ch7), it is possible to solve for friction factor:
In smooth pipes
In rough pipes
Von Karman & Prandtl equation
Nikuradse rough pipe equation
Nikuradse rough pipe equation
in terms of pipe diameter

23. Friction Factor in Turbulent flow
Transition region: Characterized by a flow regime in a particular case follow neither the smooth nor rough pipe formulations
- Colebrook & White proposed the following semi-empirical function:
Note that all analytical expressions are nonlinear; so it is cumbersome to solve !
Lewis Moody developed graphical plots of f as given in preceding expressions ☺

24. Moody Diagram
Figure 8.4: Friction factors for flow in pipes, the Moody diagram (From L.F. Moody, “Friction factors for pipe flow,” Trans. ASME, vol.66,1944.)

25. How to Read the Moody Diagram
♦ The abscissa has the Reynolds number (Re) as the ordinate has the resistance coefficient f values.
♦ Each curve corresponds to a constant relative roughness ks/D (the values of ks/D are given on the right to find correct relative roughness curve).
♦ Find the given value of Re, then with that value move up vertically until the given ks/D curve is reached. Finally, from this point one moves horizontally to the left scale to read off the value of f.

26. Empirical Resistance Equations
Blasius Equation:
In case of smooth-walled pipes; very accurate for Re <100,000
Manning’s Equation: (special case of Chezy)
n : coefficient which is a function of the boundary roughness and hydraulic radius
Velocity: (used in open-channels)
For the SI unit system

27. Empirical Resistance Equations
Manning’s Equation:
For the BG system
Manning’s n is not be a function of turbulence characteristic or Re number but varies slightly with the flow depth (through the hydraulic radius)
In view of (1), therefore one can say that the Manning equation would be strictly applicable to rough pipes only, although it has frequently been employed as a general resistance formula for pipes.
It is, however, employed far more frequently in open-channel situations.

28. Empirical Resistance Equations
Hazen-Williams Equation:
R: hydraulic radius of pipe
Sf : friction slope
CHW : resistance coefficient
(pipe material & roughness conditions)
Important Notes:
Widely used in water supply and irrigation works
Only valid for water flow under turbulent conditions.
It is generally considered to be valid for larger pipe (R>1)

29. Empirical Resistance Equations
Three well-known 1-D resistance laws are the Manning, Chezy & Darcy-Weisbach resistance equations
The interrelationship between these equations is as follows:

30. Minor Losses in Pipes
Energy losses which occur in pipes are due to boundary friction, changes in pipe diameter or geometry or due to control devices such as valves and fittings.
Minor losses also occur at the entrance and exits of pipe sections.
Minor losses are normally expresses in units of velocity head
kl : Loss coefficient associated with a particular type of minor loss and a function of Re, R/D, bend angle, type of valve etc.

31. Pipe connections, bends and reducers

32. Sleeve valve. (Courtesy TVA.)

33. ke: expansion coefficient
Minor Losses in Pipes
In the case of expansions of cross-sectional area, the loss function is sometimes written in terms of the difference between the velocity heads in the original and expanded section due to momentum considerations;
For Gradual Expansion
ke: expansion coefficient

34. Minor Losses in sudden expansions
For Sudden Expansion
Head loss is caused by a rapid increase in the pressure head

35. Minor Losses in sudden contraction
Head loss is caused by a rapid decrease in the pressure head

36. Expansion and contraction coefficients
for threaded fittings
The magnitude of energy loss is a function of the degree and abruptness of the transition as measured by ratio of diameters & angle θ in Table 8.3.

37. Coefficients of entrance loss for pipes (after Wu et al., 1979).
Entrance loss coefficient is strongly affected by the nature of the entrance

38. Summary for minor losses
Please see Table 4 & 5 of Fluid Mechanics & Hydraulics by Ranald V Giles et al. (Schaum’s outlines series)
Copyright © Fluid Mechanics & Hydraulics by Ranald V Giles et. al. (Schaum’s outlines series)

39. Bend Loss Coefficients
r : radius of bend
d: diameter of pipe

40. Loss coefficients for
Some typical valves

41. CLASS EXERCISES

42. CLASS EXERCISES

43. CLASS EXERCISES

44. Three-Reservoir Problem
Determine the discharge
Determine the direction of flow

45. Three-Reservoir Problem
If all flows are considered positive towards the junction then
QA + QB + QC = 0
This implies that one or two of the flows must be outgoing from junction.
The pressure must change through each pipe to provide the same piezometric head at the junction. In other words, let the HGL at the junction have the elevation
pD: gage pressure

46. Three-Reservoir Problem
Head lost through each pipe, assuming PA=PB=PC=0 (gage) at each reservoir surface, must be such that
Guess the value of hD (position of the intersection node)
Assume f for each pipe
Solve the equations for VA, VB & VC and hence for QA, QB & QC
Iterate until flow rate balance at the junction QA+QB+QC=0
If hD too high the QA+QB+QC <0 & reduce hj and vice versa.

47. Example: (Class Exercise)

48. Pipes in Parallel
Head loss in each pipe must be equal to obtain the same pressure difference btw A & B (hf1 = hf2)
Procedure: “trial & error”
Assume f for each pipe  compute V & Q
Check whether the continuity is maintained

49. CLASS EXERCISES

50. Pipe Networks The need  to design the original network
to add additional nodes to an existing network
Two guiding principles (each loop):
1- Continuity must be maintained
2- Head loss btw 2 nodes must be
independent of the route.
The problem is to determine
flow & pressure at each node

51. Pipe Networks Consider 2-loop network:
Procedure: (inflow + outflow + pipe characteristics are known)
1- Taking ABCD loop first, Assume Q in each line
2- Compute head losses in each pipe
& express it in terms of Q
Consider 2-loop network:
For the loop:
The difference is known:

52. Hardy Cross Method
3- If the first Q assumptions were incorrect, Compute a correction to
the assumed flows that will be added to one side of the loop & subtracted
from the other.
► Suppose we need to subtract a ∆Q from the clockwise side and add it to the other side for balancing head losses. Then
► After applying Taylor`s series expansion & math manipulations for the above relation:
4- After balancing of flows in the first loop, Move on to the next one

53. Hardy Cross Method --- Example 8.11
Given info: