CVE 341 – Water Resources

Lecture Notes I:

(Combined Chs 7 & 8)

Closed Conduit Flow

Prepared by Ercan Kahya

Copyright © 2007 by Nelson, a division of Thomson Canada Limited

A large slurry pipe. (Copyright Terry Vine/CORBIS)

View of a Stunning Example in CE Practices

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.

INTRODUCTION (Cont`d)

Differences btw open channel flow & pipe flow

Steady Uniform Flow

Steady, uniform flow in closed conduits

z: geometric head

P/pressure head

V2/2g: velocity head

zP

g

VET

2

2

General Energy Consideration

fT hzP

g

Vz

P

g

VE 2

2

2

221

1

2

11 22

the velocity coefficient & set to unity for regular & symmetrical cross-section like pipe.

Figure 8.2: Layout and hydraulic heads for Example 8.1.

Example 8.1:

Discuss how to attack to this problem…Given: Q, d, hf, and available pressure at the buildingUnknown: z1

Forces acting on steady closed conduit flow.

= R Sf

Boundary shear stress (N/m2)

Specific weight of water (N/m3)

R : Hydraulic radius (m)

Sf : Slope of energy grade line

Resistance Application & Friction Losses in Pipes

General Resistance Equation:from computing the shear stress of a system in dynamic equilibrium

ConsiderConsider the length of pipe to be a control the length of pipe to be a control volume & volume & realize realize the dynamic equilibrium the dynamic equilibrium

P: wetted perimeter

- Dividing this equation by area (A)

R: hydraulic radius (A / P)

Note that So: pipe slope

Note that ∆z: change in pipe elevation

→ Piezometric head change

7-11

: 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

The general shear stress relation in all cases of steady uniform flowThe general shear stress relation in all cases of steady uniform flow

Resistance Equations for Steady Uniform Resistance Equations for Steady Uniform Flow 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

- 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.

Resistance Application & Friction Losses in Pipes

fRSCV Chezy Equation

From dimensional analysis: Resistance Equation in terms of the average velocity

In most cases of closed conduit flow, it is customary to compute energy losses due to resistance by use of the Darcy-Weishbach equation.

g

V

d

Lfhf

2

2

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

- Let`s define “shear velocity” as

=

Then the above equation becomes

→ important in developing the resistance formula

Velocity Distributions in Steady, Uniform Velocity Distributions in Steady, Uniform Flow Flow

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

Laminar Flow Rn ≤ 2000 Rn: Reynolds number Turbulence Flow Rn ≥ 4000

Laminar Flow Laminar Flow Velocity in terms of radial position:

→ paraboloid distribution

Head loss in a pipe element in terms of average velocity:

Energy loss gradient or friction slopeEnergy 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.

2

32

d

VLhf

→ Poiseuille Equation

- Darcy friction factor: f = 64 / Rn in laminar flow

Turbulent Flow 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

Turbulent Flow

Cy

y

k o

ln1

*

: instantaneous velocity* : shear velocityk : von Karman constant (= 0.4 for water)y : distance from boundaryyo : hydraulic depthC : constant

General form for turbulent velocity distributions

After statistical and dimensional considerations, Von Karman gives logarithmic dimensionless velocity distribution as

Turbulent Flow velocity distribution

1) For smooth-walled conduits

2) For rough-walled conduits

5.5ln1 *

*

v

yv

k

48.8ln1

*

y

k

Function of the laminar sub-layer properties (wall Reynolds number)

Function of the wall roughness element

Friction Factor in Turbulent flow

In smooth pipes

In rough pipes

74.1)/(21

orLogf

8.0)(Re21

fLogf

14.1)/(21

dLogf

Nikuradse rough pipe equation

Von Karman & Prandtl equation

Nikuradse rough pipe equation in terms of pipe diameter

Using velocity distribution given previously & shear stress/velocity relation (Ch7), it is possible to solve for friction factor:

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:

)/

7.181log(274.1)/log(21

fR

rr

f n

Friction Factor in Turbulent flow

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

☺

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.)

Moody Diagram

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.

Empirical Resistance Equations

Blasius Equation:

25.0

316.0

nRf In case of smooth-walled pipes;

very accurate for Re <100,000

Manning’s Equation: (special case of Chezy)

6/11Rn

C n : coefficient which is a function of the boundary roughness and hydraulic radius

5.03/21SR

nV For the SI unit system

Velocity: (used in open-channels)

Empirical Resistance Equations

5.03/2485.1SR

nV

Manning’s Equation:

For the BG system

1. 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)

2. 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.

3. It is, however, employed far more frequently in open-channel situations.

Empirical Resistance Equations

54.063.0fHW SRCV

Hazen-Williams Equation:

Important Notes:

1. Widely used in water supply and irrigation works

2. Only valid for water flow under turbulent conditions.

3. It is generally considered to be valid for larger pipe (R>1)

R: hydraulic radius of pipe

Sf : friction slope

CHW : resistance coefficient

(pipe material & roughness conditions)

Empirical Resistance Equations

V

RS

CR

n

g

f f1

8 6/1

•Three well-known 1-D resistance laws are the Manning, Chezy & Darcy-Weisbach resistance equations

• The interrelationship between these equations is as follows:

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

g

Vkh ll 2

2

kl : Loss coefficient associated with a

particular type of minor loss and a function of Re, R/D, bend angle, type of valve etc.

Pipe connections, bends and reducers

Sleeve valve. (Courtesy TVA.)

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;

g

VVkh el 2

221 For Gradual Expansion

ke: expansion coefficient

Minor Losses in sudden expansions

g

VVhl 2

2

21

For Sudden Expansion

Head loss is caused by a rapid increase in the pressure head

Minor Losses in sudden contraction

g

Vkh ll 2

2

2

Head loss is caused by a rapid decrease in the pressure head

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.

Coefficients of entrance loss for pipes (after Wu et al., 1979).

Coefficients of entrance loss for pipes

Entrance loss coefficient is strongly affected by the nature of the entrance

Summary for minor losses

Copyright © Fluid Mechanics & Hydraulics by Ranald V Giles et. al. (Schaum’s outlines series)

Please see Table 4 & 5 of Fluid Mechanics & Hydraulics by Ranald V Giles et al. (Schaum’s outlines series)

Bend Loss Coefficients

r : radius of bend d: diameter of pipe

Loss coefficients for Some typical valves

CLASS EXERCISES

CLASS EXERCISES

CLASS EXERCISES

Three-Reservoir Problem

• Determine the discharge

• Determine the direction of flow

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

D

DD z

ph

pD: gage pressure

Three-Reservoir Problem

Head lost through each pipe, assuming PA=PB=PC=0 (gage) at each reservoir surface, must be such that

DBB

BB hzg

V

d

Lfh

2

2

DCC

CC hzg

V

d

Lfh

2

2

DAA

AA hzg

V

d

Lfh

2

2

1. Guess the value of hD (position of the intersection node)

2. Assume f for each pipe

3. Solve the equations for VA, VB & VC

and hence for QA, QB & QC

4. 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.

Example: (Class Exercise)

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

CLASS EXERCISES

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.

0 iQ

fCCfC hh

The problem is to determineflow & pressure at each node

Pipe Networks

Consider 2-loop network: Procedure: (inflow + outflow + pipe characteristics are known)

1- Taking ABCD loop first, Assume Q in each line2- Compute head losses in each pipe & express it in terms of Q

For the loop:

The difference is known:

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

Hardy Cross Method --- Example 8.11

Given info: