unit f: pipe flows - mercury.pr.erau.edumercury.pr.erau.edu/~hayasd87/ae301/es206_master_f.pdf ·...

ES206 Fluid Mechanics

UNIT F: Pipe Flows

ROAD MAP . . .

F-1: Characteristics of Pipe Flow

F-2: Major and Minor Losses of Pipe System


Unit F-1: List of Subjects

Components of Pipe Flows

Laminar and Turbulent Flow

Pipe Entrance Flow

Fully Developed Laminar Flow

Horizontal Straight Pipe

Transition to Turbulence

Major and Minor Losses

Unit F-1Page 1 of 10

Components of Pipe Flows

UNIT GUNIT G--11SLIDE SLIDE 11

Pipe System ComponentsPipe System Components

➢ The basic components of a typical pipe system include the pipes, various fittings used to connect the individual pipes to form a desired system, the flowrate control devices (valves), and pumps or turbines that

add or remove energy from

the fluid

Textbook (Munson, Young, and Okiishi), page 402


General Characteristics General Characteristics

of Pipe Flowof Pipe Flow

➢ For pipe flow, it is assumed that the pipe is completely filled with the fluid being transported

➢ The flows, such as concrete pipe through which rainwater flows without completely filling the pipe, are open-channel flow (not pipe flow)


Reynolds Experiment

• Reynolds conducted a famous “Reynolds experiments.”

• Where, he discovered the Reynolds number being the main factor of

determining the different types of pipe flows.


Laminar and Turbulent Flow (1)

VDD Re


Laminar and Turbulent FlowLaminar and Turbulent Flow (1)(1)

➢ The flow of a fluid in a pipe may be laminar or turbulent (or transitional)


Types of Flow

• Depending on the Reynolds number, three distinctively different types of flow

can be observed.

o Laminar: smooth (predictable and mostly steady) flow

o Turbulent: chaotic (random and mostly unsteady) flow

o Transitional: mixing of laminar and turbulent flows co-existing, changing

from one to another


Laminar and Turbulent Flow (2)

Laminar

Transitional

Turbulent

Re 2,100 4,000D

000,4Re D

Re 2,100D


Laminar

Transitional

Turbulent

Re 2,100 4,000D

Laminar and Turbulent FlowLaminar and Turbulent Flow (3)(3)

000,4Re D

Re 2,100D

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Based on experiment of Reynolds, the flow is laminar if Re 2,100

9 3

22 3

10 m /s

0.08 10 m4 4

Q QV

AD

= 0.199 m/s

Therefore,

3

5 2

0.199 m/s 0.08 10 mRe

1.46 10 m /s

VD VD

= 1.09 << 2,100 (laminar flow)


Class Example Problem

Related Subjects . . . “Laminar and Turbulent Flow”

Pipe Entrance Region


Pipe Entrance Flow


Pipe Entrance FlowPipe Entrance Flow (1)(1)

➢ The fluid typically enters the pipe with a nearly uniform velocity profile: as the fluid moves through the pipe, viscous effects cause it to stick to the pipe flow



Pipe Entrance FlowPipe Entrance Flow (2)(2)

➢ The shape of the velocity profile in the pipe depends on whether the flow is laminar or turbulent

➢ The dimensionless entrance lengthcorrelates with the Reynolds number as:

➢ For very low Reynolds number flows, the entrance length can be quite short, whereas for high Reynolds number flows it may take a length equal to many pipe diameters before the end of entrance region is reached

➢ For many practical engineering problems:

Analytical Solution of Laminar Pipe Flow

If the flow is laminar:

(Non-Horizontal, or “General” Pipe Flow equations)

p : Pressure drop for the pipe length

: Specific weight of the flowing fluid

: Length of the pipe

: Angle of the pipe, measured from horizontal direction (0 for horizontal pipe, and 90

for vertical up-flow pipe, and 90 for down-flow pipe) D: Internal diameter of the pipe

V: Average velocity of the flowing fluid at the given cross section

: Viscosity of the moving fluid

: Shear stress developed at the internal surface of the pipe

r: Radius of the pipe


Fully Developed Laminar Flow


➢ The adjustment to account for nonhorizontal pipes:

➢ All of the results for the horizontal pipe are valid provided the pressure gradient is adjusted for the elevation term ( is replaced by ):

Fully Developed Laminar FlowFully Developed Laminar Flow (7)(7)


p p

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For a laminar flow of a pipe, the average velocity is equal to the half of the centerline

velocity: 1 1

1.0 m/s 0.5 m/s2 2

cV V

First, let us check that the flow is really laminar flow:

3 3

2

1,260 kg/m 0.5 m/s 75 10 mRe

1.50 N s/m

VD

= 31.5 << 2,100 (laminar flow)

For non-horizontal pipe with 90 (vertical):

2 2sin

32 32

p D p DV

Therefore, the pressure drop can be calculated as:

2

2

3 3

23

32

32 1.50 N s/m 10 m 0.5 m/s12.4 10 N/m 10 m

75 10 m

Vp

D

= 166.67103 N/m3 (166.67 kPa)

Applying the energy equation: 2 2

out out in inout in

2g 2gL

p V p Vz z h

(for pipe: 0sh )

where, in outV V , out in 10 mz z

Hence, the head loss is:

3 2in out

out in 3 3

166.67 10 N/m10 m

12.4 10 N/mL

p ph z z

= 3.44 m



Related Subjects . . . “Fully Developed Laminar Flow”

Consider a horizontal straight pipe flows for simplification:

( 0 with constant cross-section)

Head loss of a pipe can be found by applying the energy equation: 2 2

out out in inout in

2g 2gL

p V p Vz z h

(note: for pipe, 0sh )

For a horizontal pipe: in outz z , also for a straight pipe: in outV V (the

pipe cross-section is constant), thus:

2 2 2in out

2 2 2L

p p p V V Vh f f f

D D g D g

(Head Loss due to Pressure Drop for a Horizontal Straight Pipe)


Horizontal Straight Pipe

64 (if flow is )

Ref LAMINAR


Dimensional AnalysisDimensional Analysis

➢ Applying the dimensional analysis for a fully developed laminar flow:

➢ There are five variables that can be described in terms of three reference dimensions: two pi terms

➢ This will lead to the pressure drop:

( f : Darcy Friction Factor)(Laminar Flow)


Dimensional AnalysisDimensional Analysis

➢ Applying the dimensional analysis for a fully developed laminar flow:

➢ There are five variables that can be described in terms of three reference dimensions: two pi terms

➢ This will lead to the pressure drop:

( f : Darcy Friction Factor)(Laminar Flow)

Laminar, Turbulent, and Transitional Flow of a Pipe

Recall, for a horizontal straight pipe:

2

2

Vp f

D

(Pressure Drop)

2

2L

Vh f

D g (Head Loss)

• For laminar flow: 64 Ref

• For turbulent flow: ?f


Transition to Turbulence


Major and Minor LossMajor and Minor Loss

➢ Turbulent flow can be a very complex (difficult) to analyze

➢ Most turbulent pipe flow analyses are based on experimental data and semi-empirical formulas

➢ These data are expressed conveniently in dimensionless form

➢ The overall head loss for the pipe system consists of the head loss due to viscous effects in the straight pipes (major loss) and head loss in the various pipe components (minor loss)

minor major LLL hhh

Major Head Loss

• majorLh : head loss of the “major” component of the pipe (horizontal straight

segment of the pipe)

• The loss equation: 2

major2

L

Vh f

D g

o If the flow is laminar: 64

Ref

o If the flow is turbulent: ?f (the value can only be determined by

experimental results: the MOODY CHART)

Minor Head Loss

• minorLh : head loss of the “minor” components of the pipe (any component that is

not the pipe itself): including, valves, fitting, couplers, etc. etc.,

• The loss equation: 2

minor2

L L

Vh K

g

o LK : Loss coefficient (depends on the minor component)


Major and Minor Losses

minor major LLL hhh


Major and Minor LossMajor and Minor Loss

➢ Turbulent flow can be a very complex (difficult) to analyze

➢ Most turbulent pipe flow analyses are based on experimental data and semi-empirical formulas

➢ These data are expressed conveniently in dimensionless form

➢ The overall head loss for the pipe system consists of the head loss due to viscous effects in the straight pipes (major loss) and head loss in the various pipe components (minor loss)

minor major LLL hhh


UNIT F: Pipe Flows

ROAD MAP . . .

F-1: Characteristics of Pipe Flow

F-2: Major and Minor Losses of Pipe System


Unit F-2: List of Subjects

Major Loss of Pipe System

The Moody Chart

Minor Loss of Pipe System

Entrance/Exit Flows

Pipe Components

Unit F-2Page 1 of 9

Major Loss of Pipe System

?Re

64f


Major LossesMajor Losses (1)(1)

➢ Major losses of pipe flow can be analyzed by dimensional analysis

➢ The pressure drop for steady incompressible turbulent flow in a horizontal round pipe of diameter D can be written in functional form:

➢ Thin viscous “sublayer” is formed near the pipe wall, and the pressure drop is a function of the wall roughness () for turbulent flow


Recall:for laminar flow, p

is independent of



➢ There are seven variables which can be written in terms of the three reference dimensions – hence, the problem can be written with four pi terms:

➢ As was done in laminar flow, the functional representation can be simplified by imposing the reasonable assumption that pressure drop should be proportional to the pipe length:

➢ Note that Darcy friction factor:

In terms of Darcy

friction factor

(if Laminar Flow)Re

64fIf turbulent, obtained

only by experiment

NOTE: “1” is inflow and “2” is outflow

Unit F-2Page 2 of 9

The Moody Chart (1)

/D


➢ Now, consider energy equation for incompressible steady flow:

➢ The head loss between two points (1) and (2) with assumption of a constant diameter, horizontal pipe with fully developed pipe flow is:

➢ If include non-horizontal pipe flows:


(Darcy-Weisbach Equation)

Unit F-2Page 3 of 9

The Moody Chart (2)


Moody ChartMoody Chart (4)(4)

➢ For laminar flow: f = 64/Re (independent of roughness)

➢ For very large Reynolds numbers: f = (/D), which

is independent of the Reynolds number (this is called“wholly turbulent” flow)

➢ Note that even for smooth pipes ( = 0), the friction factor is not zero: there still is the head loss in any pipe due to no-slip surface condition (no matter how smooth it may be)

➢ The minimum friction factor applies for a pipe flow even though the surface roughness is considerably less than the viscous sublayer: such pipes are called “hydraulically smooth”

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Head loss of a horizontal straight pipe can be found by (for laminar or turbulent flow): 2

2L

Vh f

D g

If two pipe flows (one is laminar and the other is turbulent) have the same velocity (V ),

pipe length ( ), pipe diameter ( D ): 2

laminar laminar2

L

Vh f

D g and

2

turbulent turbulent2

L

Vh f

D g

Therefore, laminar laminar

turbulent turbulent

L

L

h f

h f

(a) If the flow is maintained to be laminar at Re 6,000 :

laminar

64 64

Re 6,000f = 0.01067

(b) If the flow is turbulent at Re 6,000 :

Using Moody chart with smooth pipe ( 0D ):turbulent 0.035f

Thus,

laminar laminar

turbulent turbulent

0.01067

0.035

L

L

h f

h f = 0.305

If the flow could be maintained as laminar flow rather than the expected turbulent flow,

the head loss can be reduced by:

turbulent laminar laminar

turbulent turbulent

100 1 100 1 0.305 100L L L

L L

h h h

h h

= 69.52%

Unit F-2Page 4 of 9


Related Subjects . . . “The Moody Chart”

Unit F-2Page 5 of 9

Minor Loss of Pipe System


Minor LossesMinor Losses (1)(1)

➢ In addition to the major losses, additional components of the pipe system (valves, bends, tees, etc.) add losses to the overall head loss: such losses are called “minor losses”

➢ The most common method used to determine these head losses (or pressure drops) is to specify the loss coefficient, KL, which is:

➢ Often, minor losses are given in terms of equivalent length:

In most cases of practical interest, the loss coefficient is a function of geometry only:

Loss at the Pipe Entrance

• Flow separation at the pipe entrance will reduce the effective pipe cross section.

• Then the flow is accelerated (pressure drop) at the reduced cross section.

• The accelerated flow (kinetic energy) is usually not fully recovered back into the

static pressure, due to the presence of viscosity.

Unit F-2Page 6 of 9

Entrance/Exit Flows (1)

KL = 0.8 KL = 0.5

KL = 0.2 KL = 0.04

(Reentrant) (Sharp-Edged)

(Slightly Rounded) (Well-Rounded)


Minor LossesMinor Losses (3)(3)

➢ Entrance Flow Losses (2)


Loss at the Sudden Change of Pipe Diameter

Unit F-2Page 7 of 9

Entrance/Exit Flows (2)


Flow in Sudden ExpansionFlow in Sudden Expansion

➢ The loss coefficient for a flow in sudden expansion can be analytically obtained:

➢ Continuity:

➢ Momentum:

➢ Energy:

➢ These equations can be combined to give:


Loss of Common Pipe (Minor) Components

Flanged v.s. threaded (elbows, return bends, tees, and union)

Valves (examples):

Unit F-2Page 8 of 9

Pipe Components


Internal Structure of ValvesInternal Structure of Valves


Globe Valve

Swing Check Valve

Gate Valve

Stop Check Valve

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Major head loss and minor head loss needs to be determined and compared against.

The head loss (major):

2

2L

Vh f

D g , where 6 6 4 1 17 in and 0.75 inD

Also, 3

2

0.020 ft /s6.519 ft/s

1 ft0.75 in

4 12 in

QV

A

, thus:

5 2

1 ft6.519 ft/s 0.75 in

12 inRe

1.21 10 ft /s

VD

= 3.37104

30.0005 ft8 10

1 ft0.75 in

12 in

D

From the Moody chart, 0.038f

Therefore, the head loss (major) is: 2 2 217 in

0.038 0.8612 0.75 in 2 2

L

V V Vh f

D g g g

The head loss (minor): 2

2L L

Vh K

g where, 90 threaded elbows (2): 2 1.5LK

Tee (branch flow: threaded): 2.0LK

Reducer (from 0.75 to 0.6 in.-diameter): 2 2

2 1 2 1 0.6 0.75 0.64A A d d => 0.15LK

Hence, the head loss (minor) is: 2 2 2

2 1.5 2.0 0.15 5.152 2 2

L L

V V Vh K

g g g

So, Major Loss ( , major

0.861100 14.3%

0.861 5.15Lh

and minor Loss ( , major

5.15100 85.7%

0.861 5.15Lh

Unit F-2Page 9 of 9


Related Subjects . . . “Minor Loss of Pipe System”

unit f: pipe flows - mercury.pr.erau.edumercury.pr.erau.edu/~hayasd87/ae301/es206_master_f.pdf ·...

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