ME2342_Sec6_PipeFlow

7/29/2019 ME2342_Sec6_PipeFlow

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P. S. Krueger ME/CEE 2342 6 - 1

ME/CEE 2342:

Paul S. Krueger

Associate Professor

Department of Mechanical Engineering

Southern Methodist UniversityDallas, TX 75275

[email protected]

(214) 768-1296Office: 301G Embrey

Fluid Mechanics

Section 6 Pipe Flow

[Chapter 8 in the text book]
mailto:[email protected]:[email protected]


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Pipe Flow Terminology:


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Entrance Region:

Fully Developed Region:

Velocity profile changes from uniform to more rounded or blunt.

Pressure gradient is not linear

ReDLe 05.0=

41359.1 ReDLe =

Velocity profile (u(r)) does notchange with distance.

Pressure decreases linearlywith distance.

Mathematically:


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Horizontal pipe:

Assume steady, incompressible, axisymmetric (no

dependence on ). Fully-developed requires:

Shear Stress in the Fully-Developed Region



Step 1: CV & FBD (see previous diagram)

Step 2: Conservation of Mass

Step 3: Forces



Step 4: N2 (x-component only)

But,

Or( )r

r

l

p =

2

Note: The LHS does not depend on r. Therefore, the RHS

doesnt either!

Or ( )

= R

rr w



Laminar vs. Turbulent Flow

ForRe sufficiently low, the flow is smooth and steady. Fluid

at adjacent radii slide over one another:

ForRe too large, the flow becomes unstable and velocity

fluctuations appear


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Laminar Pipe Flow

Recall

For laminar flow

Boundary Condition (BC):

So, ( )

=

22

14 R

rR

l

pru


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Volume Flow Rate:

In terms ofD = 2R:

= l

pD

Q 128

4


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Energy Considerations:

For a stream tube we have

In this case


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For a horizontal pipe

From Poiseuilles Law

Using Q = UA gives

or

g

U

D

lfhL

2

2

=

f= Darcy Friction Factor


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Using laminar flow analysis we have obtained a formula

forhL in pipe flow.

f= 64/Re only for laminar flow.

Since the losses are due only to viscosity, this result also

works for inclined pipes. In that case

Notes:


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As we have already seen, in turbulent flow the velocity

components show random fluctuations due to complex

swirls and eddies:

Here uA(t) shows random fluctuations even though the

average flow rate through the pipe (Q) is constant.

Fully-Developed Turbulent Flow


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For constant Q, however, the time average ofuA is constant.

That is,

The fluctuation is then defined as

which is still a function of time. Similarly, the other velocity

components show fluctuations and may be decomposed as


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The magnitude of the fluctuations is specified by

which is the standard deviation of the fluctuations, squared.

Using this quantity, we define turbulence intensity as


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The mean/average quantities , , have many of the

same properties as the full velocity field for laminar flow:

u v w

1) For axisymmetric pipe flow 0== wv

2) If the flow is fully-developed, then does not changealong the pipe length ( ). In other words, there is a

distinct velocity profile for the mean velocity.

( )ru0= xu

We cannot solve an equation directly to find the velocity

profile in turbulent flow as we did with laminar flow, but

experiments show a good fit for is( )ru


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Graphically:

[Source: Munson, Young Okiishi (4th Ed.)]

Notes:

The profile is nearly uniform for turbulent flow. Thus,

= 1, = 1 is a reasonable assumption for turbulent flow.

The power law fit does not have the correct gradients(slopes) at r= 0 andR.


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Near the wall, the profile is better described by a Linear-Law

region (next to the wall) and a Log-Law region (overlap

between the wall and outer regions). This is discussed inmore detail in the text book.

Shear Stress

Recall that shear stress varies linearly with r, namely,

In turbulent flow, we dont have a simple way of determining

w. Mathematically, however, we can break into two terms:


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Experiments show that turb dominates except near the wall.

The physical meaning of the two terms can be illustrated as

follows:

[Source:

Munson, Young

Okiishi (4th

Ed.)]


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Dimensional Analysis of Turbulent Pipe Flow

We already know that

We would like to use this to find a simple formula relating p

to Q (like we did for laminar flow). The problem is finding aformula forw that is valid for both laminar and turbulentflow. Instead, we must rely on dimensional analysis,

physics, and experiments.

Functional dependence ofp:Dimensional Analysis:

i (h i ht) f h th i id f th i


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= size (height) of roughness on the inside of the pipe.

= fluid density inertia effects

Dimensions

Repeating Variables


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Repeating Variables:

Physics:

Experiment: Measure f for different R and /D => Moody


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Experiment: Measureffor differentRe and /D => MoodyChart

N t


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hL has the same form for laminar and turbulent flow, only

fis more complex for turbulent flow. For laminar flow f= 64/Re. So,fis not a function of/D

for laminar flow.

Values of for common materials can be found in Table8-2 and Fig. A-12 in the textbook.

From the Moody Chart,fis a only function of/D as

Re . Due to uncertainties in actual parameters (l,D, etc.) pipe

flow analysis is likely accurate only to within 10%.

Notes:

Example: Type I (Flow rate and diameter are known)


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Example: Type I (Flow rate and diameter are known)

WhatHis required to achieve flow rates ofa) Q = 6.60 10-5 m3/s

b) Q = 4.40 10-4 m3/s

Ignore entrance effects (assume pipe friction only)

Governing equation: Energy equation


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Governing equation: Energy equation

a) First determineRe:

From the Moody chart or results for laminar flow


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From the Moody chart or results for laminar flow

b)

From the Moody chart (see next slide):

So,


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0.035

Example: Type II (D known h known Q unknown)


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Example: Type II (D known, hL known, Q unknown)

Ignore entrance and exit effects and find Q.

Governing equation:


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Governing equation:

Only losses in the pipe considered, so hL is determined as

Procedure:

(i) Guess Q (orf)(ii)Findf(orQ)

(iii)Calculate new Q (determine newf)

(iv)Use now Q (f) to fine newf(Q)(v)Repeat until the answer converges

(i) Guess:


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(i) Guess:

(ii)

(iii)

(iv)

(v) Repeat


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Minor Losses


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Minor Losses

What happens if we dont have a nice, straight, constant

diameter pipe? Changes in geometry cause disturbanceswhich further increase hL. These are called minor losses.

For example:

Similarly, in a pipe bend we can have separation and


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y, p p p

secondary flows:

In general we express these losses as


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g p

g

UKh LmL 2

2

, =

where

The loss coefficientsKL are typically determined by

experiment and are tabulated for various features (e.g.,

Table 8-4, Figs. 8-34, 8-36, 8-38). Types of minor losses:

Change in pipe diameter

Entrance effects

Exit flow Valves

Pipe bends

Defects in pipes (e.g., couplings)

Example:


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p

How much power does the fan add to the air?

Governing equation


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For systems with minor losses

+

=+=

g

UK

g

U

D

lfhhh LmLfLL

22

22

,,

Elbows:


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Plot

Flow Coefficient (Cv)


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Minor losses are sometimes expressed in terms of a flow

coefficient (Cv), especially among valve manufacturers.The flow coefficient for liquids is defined as

Open Channel Flow (Ch. 13 in textbook)


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p ( )

Front View Side View

Hydraulic Diameter

P

AD ch

4=

Energy Equation:


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Slope:

For uniform flow, flow velocity and depth remain unchanged

d th f b l ith h i i d


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and therefore, a balance with hL is required:

Manning Coefficient (see Table 13-1):

ME2342_Sec6_PipeFlow

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