1 MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 9: FLOWS IN PIPE Instructor: Professor C. T. HSU

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MECH 221 FLUID MECHANICS(Fall 06/07)

Chapter 9: FLOWS IN PIPE

Instructor: Professor C. T. HSU

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MECH 221 – Chapter 9

9.1 General Concept of Flows in Pipe

As a uniform flow enters a pipe, the velocity at the pipe walls must decrease to zero (no-slip boundary condition). Continuity indicates that the velocity at the center must increase.

Thus, the velocity profile is changing continuously from the pipe entrance until it reaches a fully developed condition. This distance, L, is called the entrance length.

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For fully developed flows (x>>L), flows become parallel, , the mean pressure remains constant over the pipe cross-section

),(u(y), 00v

4



Flows in a long pipe (far away from pipe entrance and exit region, x>>L) are the limit results of boundary layer flows. There are two types of pipe flows: laminar and turbulent

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Whether the flow is laminar or turbulent depends on the Reynolds number, where Um is the cross-sectional mean velocity defined by

Transition from laminar to turbulent for flows in circular pipe of diameter D occur at Re=2300

A

m udAA

U1

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When pipe flow is turbulent. The velocity is unsteadily random (changing randomly with time), the flow is characterized by the mean (time-averaged) velocity defined as:

Due to turbulent mixing, the velocity profile of turbulent pipe flow is more uniform then that of laminar flow.

T

TT

dttyvT

yv ),(2

1lim)(

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Hence, the mean velocity gradient at the wall for turbulent flow is larger than laminar flow.

The wall shear stress, ,is a function of the velocity gradient. The greater the change in with respect to y at the wall, the higher is the wall shear stress. Therefore, the wall shear stress and the frictional losses are higher in turbulent flow.

wu

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9.2 Poiseuille Flow Consider the steady, fully developed laminar flow in

a straight pipe of circular cross section with constant diameter, D.

The coordinate is chosen such that x is along the pipe and y is in the radius direction with the origin at the center of the pipe.

D b

y

x1p 2p

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9.2 Poiseuille Flow

For a control volume of a cylinder near the pipe center, the balance of momentum in integral form in x-direction requires that the pressure force,

acting on the faces of the cylinder be equal to the shear stress acting on the circumferential area, hence

In accordance with the law of friction (Newtonian fluid), have:

221 )( ypp

yb2

b

ypp

2

)( 21

dy

duy )( since u decreases with increasing y

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9.2 Poiseuille Flow Therefore:

when is constant (negative)

Upon integration:

The constant of integration, C, is obtained from the condition of no-slip at the wall. So, u=0 at y=R=D/2, there fore C=R2/4 and finally:

4

1

4)(

22221 y

Cdx

dpyR

b

ppyu

22

)( 12 y

dx

dp

b

y

b

pp

dy

du

22

4

1)( yR

dx

dpyu

b

pp

dy

du )( 12

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9.2 Poiseuille Flow The velocity distribution is parabolic over the radius, and the

maximum velocity on the pipe axis becomes:

Therefore,

The volume flow rate is:

2

4

1R

dx

dpum

2

2

1R

y

u

u

m

24

22

2

1

8

24

12)(

RuRdx

dpQ

ydyyRdx

dpydyyuQ

m

12


9.2 Poiseuille Flow The flow rate is proportional to the first power of the

pressure gradient and to the fourth power of the radius of the pipe.

Define mean velocity as

Therefore,

This solution occurs in practice as long as,

2R

QUm

2

8

1

2

1 R

dx

dpuU mm

2300v

DURe m

mURdx

dp2

8Hence,

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9.2 Poiseuille Flow The relation between the negative pressure gradient and the

mean velocity of the flow is represented in engineering application by introducing a resistance coefficient of pipe flow, f.

This coefficient is a non-dimensional negative pressure gradient using the dynamic head as pressure scale and the pipe diameter as length scale, i.e.,

Introducing the above expression for (-dp/dx),

so,RUR

U

U

Df

m

m

m222

3282

Ref

64

2

2

1mU

D

f

dx

dp

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9.2 Poiseuille Flow

At the wall,

So,

As a result, the wall friction coefficient is:

dx

dpRy

dx

dp

dy

du

RyRy

w 24

2

Re

16

421 2

f

UC

m

wf

2

2

12m

w UD

f

dx

dp

R

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9.3 Head Loss in Pipe For flows in pipes, the total energy per unit of mass is

given by where the correction factor is

defined as,

with being the mass flow rate and A is the cross sectional area.

gy

Up m

2

2

2

2

m

A

Um

udAu

Qm

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9.3 Head Loss in Pipe So the total head loss between section 1 and 2 of

pipes is:

hl=head loss due to frictional effects in fully developed flow in constant area conduits

hlm=minor losses due to entrances, fittings, area changes, etcs.

lmllt

mmlt

hhh

gyUp

gyUp

h

22 2

22

22

1

21

11

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9.3 Head Loss in Pipe So, for a fully developed flow through a constant-area

pipe,

And if y1=y2,

)( 2121 yygpp

hl

ppp

hl

21

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9.3 Head Loss in Pipe

For laminar flow,

Hence

2

32

D

U

L

p

dx

dp m

2

64

2

6432 22

2mm

m

ml

U

D

L

Re

U

D

L

DUD

ULph

2

2m

l

U

D

Lfh

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9.4 Turbulent Pipe Flow For turbulent flows’ we cannot evaluate the pressure

drop analytically. We must use experimental data and dimensional analysis.

In fully developed turbulent pipe flow, the pressure drop, , due to friction in a horizontal constant-area pipe is know to depend on:

Pipe diameter, D Pipe length, L Pipe roughness, e Average flow velocity, Um Fluid density, Fluid viscosity,

p

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9.4 Turbulent Pipe Flow Therefore,

Dimensional analysis,

Experiments show that the non-dimensional head loss is directly proportional to L/D, hence

D

e

D

LRe

U

hph

D

e

D

L

DUU

p

m

ll

mm

,,

,,

12

12

Re,2/ 22

D

e

D

L

U

h

m

l

,,,,, mUeLDpp

21


9.4 Turbulent Pipe Flow Defining the friction factor as, , hence

where f is determined experimentally.

The experimental result are usually plotted in a chart called Moody Diagram.

D

eRef ,2

2

2m

l

U

D

Lfh

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9.4 Turbulent Pipe Flow In order to solve the pipe flow problems numerically,

a mathematical formulation is required for the friction factor, f, in terms of the Reynolds number and the relative roughness.

The most widely used formula for the friction factor is that due to Colebrook,

This an implicit equation, so iteration procedure is needed to determine.

fRe3.7

e/D

f

51.2log2

1

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9.4 Turbulent Pipe Flow

Miller suggested to use for the initial estimate,

That produces results within 1% in a single iteration

251.2

log25.0

0.9o Re3.7

e/Df

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9.5 Minor Loss The minor head loss may be expressed as,

where the loss coefficient, K, must be determined experimentally for each case.

Minor head loss may be expressed as

where Le is an equivalent length of straight pipe

2

U

D

Lfh

2me

lm

2

KUh

2m

lm

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Source of minor loss:

1. Inlets & Outlets2. Enlargements & Contractions3. Valves & Fittings4. Pipe Bends

9.5 Minor Loss

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Pipe flow results sometimes can be used for non-circular ducts or open channel flows to estimate the head loss

Use Hydraulic Diameter,

A - Cross section area; P - Wetted perimeter

For a circular duct,

For rectangular duct,

where Ar =b/a is the geometric aspect ratio

9.6 Non-Circular Ducts

P

ADh

4

DD

DDh

4/4 2

)1(2

4

)(2

4

Ar

aAr

ba

abDh

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9.6 Non-Circular Ducts

Effect of Aspect Ratio (b/a): For square ducts:

For wide rectangular ducts with b>>a:

Thus, flows behave like channel flows

However, pipe flow results can be used with good accuracy only when:

a=b Ar=1 Dh=a

Ar Dh2a

1/3<Ar<3

b

a

ba

b

ab

a

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1 MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 9: FLOWS IN PIPE Instructor: Professor C. T. HSU