Monroe L. Weber-Shirk School of Civil and

Environmental Engineering

Viscous Flow in PipesViscous Flow in Pipes

CEE 331 Fluid Mechanics

April 18, 2023

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?

Can we increase the flow in this old pipe by adding a smooth liner?

Viscous Flow in Pipes: Overview

Boundary Layer DevelopmentTurbulenceVelocity DistributionsEnergy LossesMajorMinor

Solution Techniques

Transition at Re of 2000

Laminar and Turbulent Flows

Reynolds apparatus

ReVDrm

= =dampinginertia

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? _________________________

v v

Drag or shear

Conservation of mass

Non-Uniform Flow

v

Entrance Region Length

1

10

100

Re

l e /D

1/ 64.4 Reel

D0.06Reel

D

laminar turbulent

Reel fD

Distance for velocity profile to develop

Shear in the entrance region vs shear in long pipes?

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)

uniform

Shear velocity

Dimensional analysis and measurements

Velocity of large eddies

Log Law for Turbulent, Established Flow, Velocity Profiles

*

*

2.5ln 5.0u yuu n= +

0* u

u/umax

rough smooth

y

f0 4

h d

l

f* 4

gh du

l

Force balance

Turbulence produced by shear!

* fu V=

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

,Rep

DC f

l Deæ ö=è ø 2

2C

Vp

p Re

VDrm

=

Viscous Flow: Dimensional Analysis

Where and

in a bounded region (pipes, rivers): find Cp

flow around an immersed object : find Cd

Remember dimensional analysis?

Two important parameters!Re - Laminar or Turbulent/D - Rough or Smooth

Flow geometryinternal _______________________________external _______________________________

Dimensional Analysis

Darcy-Weisbach equation

Pipe Flow Energy Losses

R,f

Df

L

DC p

2

2C

Vp

p

2

2C

V

ghlp

f2

2f

gh D

V L

2

f f2

L Vh

D g

lgh pr =- D

Always true (laminar or turbulent)

Assume horizontal flow

More general

lgh p g zr r=- D - D

Friction Factor : Major losses

Laminar flowTurbulent (Smooth, Transition, Rough) Colebrook FormulaMoody diagramSwamee-Jain

Hagen-Poiseuille

Darcy-Weisbach

Laminar Flow Friction Factor

2

32lhgD

VL

rm

=

f 2

32 LVh

gD

2

f f2

L Vh

D g

gV

DL

gDLV

2f

32 2

2

RVD6464

f

Slope of ___ on log-log plot-1

fh V

f independent of roughness!

Turbulent Flow:Smooth, Rough, Transition

Hydraulically smooth pipe law (von Karman, 1930)

Rough pipe law (von Karman, 1930)

Transition function for both smooth and rough pipe laws (Colebrook)

1 Re f2log

2.51f

æ ö= ç ÷è ø

1 3.72log

f

De

æ ö=è ø

2

f f2

L Vh

D g

(used to draw the Moody diagram)

1 2.512log

3.7f Re f

Deæ ö=- +è ø

*fu

V

Moody Diagram

0.01

0.10

1E+03 1E+04 1E+05 1E+06 1E+07 1E+08Re

fric

tion

fact

or

laminar

0.050.04

0.03

0.020.015

0.010.0080.006

0.004

0.002

0.0010.0008

0.0004

0.0002

0.0001

0.00005

smooth

lD

C pf

D

0.02

0.03

0.04

0.050.06

0.08

Swamee-Jain

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

0.044.75 5.221.25 9.4

f f

0.66LQ L

D Qgh gh

2

0.9

0.25f

5.74log

3.7 ReD

no f

Each equation has two terms. Why?

5/ 2 f3

f

log 2.513.7 22

gh LQ D

L D gh D

Pipe roughness

pipe material pipe roughness (mm)

glass, drawn brass, copper 0.0015

commercial steel or wrought iron 0.045

asphalted cast iron 0.12

galvanized iron 0.15

cast iron 0.26

concrete 0.18-0.6

rivet steel 0.9-9.0

corrugated metal 45PVC 0.12

d Must be

dimensionless!

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)

Solution Techniques

2 f

f

1.7840.965 ln

3.7gDh

Q DL D gDh

DL

e næ öç ÷

=- +ç ÷ç ÷è ø

0.044.75 5.221.25 9.4

f f

0.66LQ L

D Qgh gh

e né ùæ ö æ ö

= +ê úç ÷ ç ÷è ø è øê úë û

2

f 2 5

8f

LQh

g Dp=2

0.9

0.25f

5.74log

3.7 ReDe

=é ùæ ö+ê úè øë û

Re 4QD

Example: Find a pipe diameter

The marine pipeline for the Lake Source Cooling project will be 3.1 km in length, carry a maximum flow of 2 m3/s, and can withstand a maximum pressure differential between the inside and outside of the pipe of 28 kPa. The pipe roughness is 2 mm. What diameter pipe should be used?

Minor Losses: Expansions!

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.

2

ex 2

Vh K

g

( )f geometry,RepC = 2

2C

Vp

p

2

2C ex

p

gh

V

2

C2ex p

Vh

g

Venturi

High Re

Sudden Contraction

V1 V2

EGL

HGL

vena contracta Losses are reduced with a gradual contraction Equation has same form as expansion equation!

g

V

Ch

c

c

21

1 22

2

2A

AC c

c

0.60.650.7

0.750.8

0.850.9

0.951

0 0.2 0.4 0.6 0.8 1

A2/A1

Cc

0.60.650.7

0.750.8

0.850.9

0.951

0 0.2 0.4 0.6 0.8 1

A2/A1

Cc

0.60.650.7

0.750.8

0.850.9

0.951

0 0.2 0.4 0.6 0.8 1

A2/A1

Cc

c 2

22

ex 12

in in

out

V Ah

g A

Prove this!

g

VKh ee

2

2

0.1eK

5.0eK

04.0eK

Entrance Losses

Losses can be reduced by accelerating the flow gradually and eliminating thevena contracta

Estimate based on contraction equations!

Head Loss in Bends

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

High pressure

Low pressure

Possible separation from wall

D

R

g

VKh bb

2

2

Kb varies from 0.6 - 0.9

2Vp dn z Cr g+ + =ó

ôõ R

n

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)

see table 8.2 (page 489 in text)

g

VKh vv

2

2

What is the maximum value of Kv? ______¥

2

2 4

8v v

Qh K

g D

What is V?

Solution Techniques

Neglect minor lossesEquivalent pipe lengthsIterative TechniquesUsing Swamee-Jain equations for D and QUsing Swamee-Jain equations for head lossAssume a friction factor

Pipe Network Software

Solution Technique 1: Find D or Q

Assume all head loss is major head lossCalculate D or Q using Swamee-Jain

equationsCalculate minor lossesFind new major losses by subtracting minor

losses from total head loss 0.044.75 5.22

1.25 9.4

f f

0.66LQ L

D Qgh gh

e né ùæ ö æ ö

= +ê úç ÷ ç ÷è ø è øê úë û

2

2 4

8ex

Qh K

g D

f l exh h h

5/ 2 f3

f

log 2.513.7 22

gh LQ D

L D gh D

Solution Technique: Head Loss

Can be solved explicitly

fl minorh h h= +å å

2

2minor

Vh K

g=å

2

f 2 5

8f

LQh

g Dp=

2

9.0Re

74.5

7.3log

25.0

D

f

2

2 4

8minor

Q Kh

g Dp= å

D

Q4Re

Solution Technique 2:Find D or Q using Solver

Iterative techniqueSolve these equations

fl minorh h h= +å å

42

28

Dg

QKhminor

2

f 2 5

8f

LQh

g Dp=2

0.9

0.25f

5.74log

3.7 ReDe

=é ùæ ö+ê úè øë ûD

Q4Re

Use goal seek or Solver to find discharge that makes the calculated head loss equal the given head loss.

Spreadsheet

Solution Technique 3:Find D or Q by assuming f

The friction factor doesn’t vary greatlyIf Q is known assume f is 0.02, if D is

known assume rough pipe lawUse Darcy Weisbach and minor loss

equationsSolve for Q or DCalculate Re and /DFind new f on Moody diagramIterate

1 3.72log

f

De

æ ö=è ø

Example: Minor and Major Losses

Find the maximum dependable flow between the reservoirs for a water temperature range of 4ºC to 20ºC.

Water

2500 m of 8” PVC pipe

1500 m of 6” PVC pipeGate valve wide open

Standard elbows

Reentrant pipes at reservoirs

25 m elevation difference in reservoir water levels

Sudden contraction

DirectionsSpreadsheet

Example (Continued)

What are the Reynolds numbers in the two pipes?

Where are we on the Moody Diagram?What is the effect of temperature?Why is the effect of temperature so small?What value of K would the valve have to

produce to reduce the discharge by 50%?

0.01

0.1

1E+03 1E+04 1E+05 1E+06 1E+07 1E+08Re

fric

tion

fact

or

laminar

0.050.04

0.03

0.020.015

0.010.0080.006

0.004

0.002

0.0010.0008

0.0004

0.0002

0.0001

0.00005

smooth

90,000 & 125,000

140

e/D= 0.0006, 0.0008

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?

0.01

0.1

1E+03 1E+04 1E+05 1E+06 1E+07 1E+08Re

fric

tion

fact

or

laminar

0.050.04

0.03

0.020.015

0.010.0080.006

0.004

0.002

0.0010.0008

0.0004

0.0002

0.0001

0.00005

smooth

Yes

5%

f goes from 0.02 to 0.035

Increase small pipe diameter

Pipe Flow Summary (1)

linearly

experimental

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 (Navier Stokes) and energy conservation

Turbulent flow losses and velocity distributions require ___________ results

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)

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

Pipes are Everywhere!

Owner: City of Hammond, INProject: Water Main RelocationPipe Size: 54"

Pipes are Everywhere!Drainage Pipes

Pipes

Pipes are Everywhere!Water Mains

Pressure Coefficient for a Venturi Meter

1

10

1E+00 1E+01 1E+02 1E+03 1E+04 1E+05 1E+06

Re

Cp

ReVlrm

=

2

2C

Vp

p

0.01

0.1

1E+03 1E+04 1E+05 1E+06 1E+07 1E+08Re

fric

tion

fact

or

laminar

0.050.04

0.03

0.020.015

0.010.0080.006

0.004

0.002

0.0010.0008

0.0004

0.0002

0.0001

0.00005

smooth

Moody Diagram

0.01

0.1

1E+03 1E+04 1E+05 1E+06 1E+07 1E+08Re

fric

tion

fact

or

laminar

0.050.04

0.03

0.020.015

0.010.0080.006

0.004

0.002

0.0010.0008

0.0004

0.0002

0.0001

0.00005

smooth

lD

C pf

D

Minor Losses

LSC Pipeline

cs1

cs2

2 21 1 2 2

1 1 2 22 2 l e

p V p Vz z h h

g g

z = 0

Ignore entrance losses

≈0

28 kPa is equivalent to 2.85 m of water-2.85 m

D m 154.

DLQgh

QL

ghf f

FHGIKJ

FHGIKJ

LNMM

OQPP0 66 1 25

24 75

9 4

5 2 0 04

. .

.

.

. .

Q m s

m s

L m

m

h mf

2

10

3100

0 002

2 85

3

6 2

/

/

.

.

1.07 /V m s

Directions

Assume fully turbulent (rough pipe law)find f from Moody (or from von Karman)

Find total head loss (draw control volume)Solve for Q using symbols (must include

minor losses) (no iteration required)

0.01

0.1

1E+03 1E+04 1E+05 1E+06 1E+07 1E+08Re

fric

tion

fact

or

laminar

0.050.04

0.03

0.020.015

0.010.0080.006

0.004

0.002

0.0010.0008

0.0004

0.0002

0.0001

0.00005

smooth

Pipe roughness

fl minorh h h= +å å Solution

WaterWater

Find Q given pipe system

fl minorh h h= +å å

42

28

Dg

QKhminor

2

2 5

8ff

LQh

g Dp=

2

2 5 4

8fl

Q L Kh

g D Dpé ùæ ö æ ö= +ê úè ø è øë ûå å

5 48 f

lghQ

L KD D

p=é ùæ ö æ ö+ê úè ø è øë ûå å

WaterWater