Monroe L. Weber-Shirk

School of Civil and

Environmental Engineering

Closed Conduit FlowCEE 332

Closed Conduit Flow

Energy equationEGL and HGLHead loss

major lossesminor losses

Non circular conduits

Conservation of Energy

Kinetic, potential, and thermal energy

hL =

Ltp hhzg

Vphzg

Vp 2

22

22

1

21

11

22

hp =ht =

head supplied by a pumphead given to a turbinemechanical energy converted to thermal

Cross section 2 is ____________ from cross section 1!downstream

Point to point or control volume?Why ? _____________________________________

irreversibleV is average velocity, kinetic energy 2V

Energy Equation Assumptions

hp

p1

1

V12

2g z1 hp

p2

2

V22

2g z2 ht hL

hydrostatic

densitySteady

kinetic

Pressure is _________ in both cross sectionspressure changes are due to elevation only

section is drawn perpendicular to the streamlines (otherwise the _______ energy term is incorrect)

Constant ________at the cross section_______ flow

EGL (or TEL) and HGL

velocityhead

elevationhead (w.r.t.

datum)

pressurehead (w.r.t. reference pressure)

zg

VpEGL

2

2

zγpHGL

downward

lower than reference pressure

The energy grade line must always slope ___________ (in direction of flow) unless energy is added (pump)

The decrease in total energy represents the head loss or energy dissipation per unit weight

EGL and HGL are coincident and lie at the free surface for water at rest (reservoir)

If the HGL falls below the point in the system for which it is plotted, the local pressures are _____ ____ __________ ______

Energy equation

z = 0pump

Energy Grade LineHydraulic G

Lvelocity head

pressure head

elevation

datum

z

2gV2

p

Ltp hhzg

Vphzg

Vp 2

22

22

1

21

11

22

static headWhy is static

head important?

Bernoulli Equation Assumption

constp

g

Vz

2

2

densitySteady

streamline

Frictionless_________ (viscosity can’t be a significant parameter!)

Along a ________________ flowConstant ________No pumps, turbines, or head loss

Why no Does direction matter? ____Useful when head loss is small

point velocityno

Pipe Flow: Review

2 21 1 2 2

1 1 2 22 2p t Lp V p Vz h z h h

g g

dimensional analysis

We have the control volume energy equation for pipe flow.

We need to be able to predict the relationship between head loss and flow.

How do we get this relationship? __________ _______.

Flow Profile for Delaware Aqueduct

Rondout Reservoir(EL. 256 m)

West Branch Reservoir(EL. 153.4 m)

70.5 km

Sea Level

(Designed for 39 m3/s)

2 21 1 2 2

1 1 2 22 2p t lp V p Vz H z H h

g g

Need a relationship between flow rate and head loss

1 2lh z z

Ratio of Forces

Create ratios of the various forcesThe magnitude of the ratio will tell us

which forces are most important and which forces could be ignored

Which force shall we use to create the ratios?

Inertia as our Reference Force

F=maFluids problems (except for statics) include

a velocity (V), a dimension of flow (l), and a density ()

Substitute V, l, for the dimensions MLT

Substitute for the dimensions of specific force

F a F a

f f M

L T2 2

L l T M

fi

lV

l3

Vl

2

Dimensionless Parameters

Reynolds Number

Froude Number

Weber Number

Mach Number

Pressure/Drag Coefficients (dependent parameters that we measure experimentally)

Re Vlrm

=

Fr Vgl

=

2

2C p

pV

lVW

2

cVM

AVd

2

Drag2C

2fuVl

fg g

2fl

2

fvE

cl

r=

2

fiVl

( )p g zrD + D

Problem solving approach

1. Identify relevant forces and any other relevant parameters2. If inertia is a relevant force, than the non dimensional Re,

Fr, W, M, Cp numbers can be used3. If inertia isn’t relevant than create new non dimensional

force numbers using the relevant forces4. Create additional non dimensional terms based on

geometry, velocity, or density if there are repeating parameters

5. If the problem uses different repeating variables then substitute (for example d instead of V)

6. Write the functional relationship

Pipe Flow: Dimensional Analysis

What are the important forces?______, ______,________. Therefore ________number and _______________ .

What are the important geometric parameters? _________________________Create dimensionless geometric groups

______, ______Write the functional relationship

C p f

Re, ,l

D D

Inertial

diameter, length, roughness height

Reynolds

l/D

viscous

/D

22CV

pp

Other repeating parameters?

pressurePressure coefficient

Dimensional Analysis

How will the results of dimensional analysis guide our experiments to determine the relationships that govern pipe flow?

If we hold the other two dimensionless parameters constant and increase the length to diameter ratio, how will Cp change?

,RepDC fl D

f ,RepDC fl D

22CV

pp

Cp proportional to l

f is friction factor

, ,ReplC fD D

Dimensional Analysis

Darcy-Weisbach equation

Pressure Coefficient and Head Loss

2

2C pp

V

2

2C lp

ghV

f2

2f gh DV L

2

f f2

L VhD g

lh p

Always true (laminar or turbulent)

Assume horizontal flow

More general

lh p h

f pDCl

Definition of f!

Darcy-Weisbach

Friction Factor : Major losses

Laminar flowHagen-Poiseuille

Turbulent (Smooth, Transition, Rough) Colebrook FormulaMoody diagramSwamee-Jain

Hagen-Poiseuille

Darcy-Weisbach

Laminar Flow Friction Factor

2

32lhDV

L

f 2

32 LVhgD

2

f f2

L VhD g

2

2

32 f2

LV L VgD D g

64 64fReVD

Slope of ___ on log-log plot

f 4

128 LQhgD

-1

Turbulent Pipe Flow Head Loss

___________ to the length of the pipeProportional to the _______ of the velocity

(almost)________ with surface roughnessIs a function of density and viscosityIs __________ of pressure

Proportional

Increases

independent

2

f f2

L VhD g

square

(used to draw the Moody diagram)

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)

1 Re f2log2.51f

1 2.512log3.7f Re f

D

1 3.72logf

D

2

f2f

L VhD g

Moody Diagram

0.01

0.1

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

fric

tion

fact

or

laminar

0.050.040.03

0.020.015

0.010.0080.0060.004

0.002

0.0010.0008

0.0004

0.00020.00010.00005smooth

f pDCl

D

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

5/ 2 f

3/ 2 f

1.782.22 log3.7

ghQ DL D ghD

L

0.044.75 5.221.25 9.4

f f

0.66 LQ LD Qgh gh

2

0.9

0.25f5.74log

3.7 ReD

no f

Each equation has two terms. Why?

fghL

L hf

Swamee-Jain gets an f

The challenge that S-J solved was deriving explicit equations that are independent of the unknown parameter.

3 potential unknowns (flow, head loss, or diameter): 3 equations for f

that can then be combined with the Darcy Weisbach equation

2

f 2 5

8f LQhg D

2

f f2

L VhD g

2

0.9

0.25f5.74log

3.7 ReD

Colebrook Solution for Q

1 2.512log3.7f Re f

D

2

f 2 5

8f LQhg D

2

2 5f

1 1 8f

LQh g D

Re 4QD

2 5

f 2

4Re f8

Q g DhD LQ

3f21Re f gh D

L

21 2.514 logf 3.7 Re f

D

f2 5 2

8f h gD LQ

Colebrook Solution for Q2

2

2 5 3f f

1 8 2.514 log3.7 21

LQh g D D gh D

L

5/ 2 3f f

2 2.51log3.7 21

L Qgh D D gh D

L

5/ 2 f3

f

log 2.513.7 22

gh LQ DL D gh D

Swamee D?0.045 1/ 4 5 1/52 2 2 2

1.250.66 Q Q Q QDg g Q g g

1/ 251/5 1/ 4 1/52 2 25/ 40.66 Q Q QD

g g Q g

2

f 2 5

8f LQhg D

25

2

8f QDg

25

2

64f8QDg

1/51/ 4 1/52 25/ 4

2

64f Q Qg Q g

1/52

2

64f8QDg

1/51/51/ 4 1/52 2 25/ 4

8Q Q QDg g Q g

1/51/ 4 1/52 2 25/ 41f

4 4Q Qg Q g

Pipe Roughness

pipe material pipe roughness (mm)glass, drawn brass, copper 0.0015commercial steel or wrought iron 0.045asphalted cast iron 0.12galvanized iron 0.15cast iron 0.26concrete 0.18-0.6rivet steel 0.9-9.0corrugated metal 45PVC 0.12

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)0.044.75 5.22

1.25 9.4

f f

0.66 LQ LD Qgh gh

2

2 5

8ffLQh

g D2

0.9

0.25f5.74log

3.7 ReD

Re 4QD

5/ 2 f3

f

log 2.513.7 22

gh LQ DL D gh D

Exponential Friction Formulas

f

n

m

RLQhD

=

units SI 675.10

units USC727.4

n

n

C

CR

1.852

f 4.8704

10.675 SI unitsL QhD C

æ ö= è ø

C = Hazen-Williams coefficient

range of data

Commonly used in commercial and industrial settings

Only applicable over _____ __ ____ collected

Hazen-Williams exponential friction formula

Head loss:Hazen-Williams Coefficient

C Condition150 PVC140 Extremely smooth, straight pipes; asbestos cement130 Very smooth pipes; concrete; new cast iron120 Wood stave; new welded steel110 Vitrified clay; new riveted steel100 Cast iron after years of use95 Riveted steel after years of use60-80 Old pipes in bad condition

Hazen-Williams vs

Darcy-Weisbach

1.852

f 4.8704

10.675 SI unitsL QhD C

2

f 2 5

8f LQhg D

preferred

Both equations are empiricalDarcy-Weisbach is dimensionally correct,

and ________.Hazen-Williams can be considered valid

only over the range of gathered data.Hazen-Williams can’t be extended to other

fluids without further experimentation.

Head Loss: Minor Losses

potential thermalVehicle drag Hydraulic jump

Vena contracta Minor losses!

Head loss due to outlet, inlet, bends, elbows, valves, pipe size changes

Flow expansions have high losses Kinetic energy decreases across expansion Kinetic energy ________ and _________ energy Examples – ________________________________

__________________________________________ Losses can be minimized by gradual transitions

Minor Losses

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

2lVh K

g=

( )geometry,RepC f=

2

2CV

ghlp

g

Vh pl2

C2

High Re

Head Loss due to Sudden Expansion:Conservation of Energy

1 2

ltp hHg

VzpHg

Vzp 22

22

222

22

111

1

1

lhgVVpp

2

21

2221

gVVpphl 2

22

2121

z1 = z2

What is p1 - p2?

Apply in direction of flowNeglect surface shear

Divide by (A2 )

Head Loss due to Sudden Expansion:Conservation of Momentum

Pressure is applied over all of section 1.Momentum is transferred over area corresponding to upstream pipe diameter.V1 is velocity upstream.

sspp FFFWMM 2121

1 2

xx ppxx FFMM2121

12

11 AVM x 22

22 AVM x

222122

212

1 ApApAVAV

gAAVV

pp 2

121

22

21

AA11

AA22

xx

Energy

Head Loss due to Sudden Expansion

g

VVpphl2

22

2121

gAAVV

pp 2

121

22

21

1

2

2

1

VV

AA

gVV

gVVVV

hl 2

22

211

221

22

g

VVVVhl 22 2

1212

2

gVVhl 2

221

2

2

12

1 12

AA

gVhl

2

2

11

AAK

Momentum

Mass

Contraction

V1 V2

EGL

HGL

vena contracta

gVKh cc 2

22

losses are reduced with a gradual contraction

Expansion!!!

Entrance Losses

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

Ke 0.5

Ke 1.0

Ke 0.04

he Ke

V2

2g

reentrant

Head Loss in Valves

Function of valve type and valve position

The complex flow path through valves often results in high head loss

What is the maximum value that Kv can have? _____

hv Kv

V2

2g

How can K be greater than 1?

Questions

What is the head loss when a pipe enters a reservoir?

Draw the EGL and HGL

V

g

V

2

2

EGLHGL

2

2

11

AAK

Questions

Can the Darcy-Weisbach equation and Moody Diagram be used for fluids other than water? _____Yes

No

Yes

Yes

What about the Hazen-Williams equation? ___ Does a perfectly smooth pipe have head loss?

_____ Is it possible to decrease the head loss in a

pipe by installing a smooth liner? ______

Example

D=40 cmL=1000 m

D=20 cmL=500 m

valve100 m

Find the discharge, Q.What additional information do you need?Apply energy equationHow could you get a quick estimate? _________________Or spreadsheet solution: find head loss as function of Q.

Use S-J on small pipe

ltp hHg

VzpHg

Vzp 22

22

222

22

111

1

1

cs1

cs2

22100

2 lVm h

g= +

Non-Circular Conduits:Hydraulic Radius Concept

A is cross sectional areaP is wetted perimeterRh is the “Hydraulic Radius” (Area/Perimeter)Don’t confuse with radius!

2

f2f

L VhD g

=

2

f f4 2h

L VhR g

=2

44h

DA DRP D

p

p= = = 4 hD R=

For a pipe

We can use Moody diagram or Swamee-Jain with D = 4Rh!

Quiz

In the rough pipe law region if the flow rate is doubled (be as specific as possible) What happens to the major head loss? What happens to the minor head loss?

Why do contractions have energy loss? If you wanted to compare the importance of minor

vs. major losses for a specific pipeline, what dimensionless terms could you compare?