new fluids lecture 0506 - Michigan Technological Universitypages.mtu.edu/~fmorriso/cm310/new0506fluids.pdf · fmorriso/cm310/Navier.pdf. ... rdr d v rdr d average v velocity Engineering

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Microscopic Balances

We have been doing microscopic h ll b l th t ifi t

dzshell balances that are specific to whatever problem we are solving.

dx

dy

We seek equations for microscopic mass, momentum (and energy) balances that are general.

© Faith A. Morrison, Michigan Tech U.

•equations must not depend on the choice of the control volume,

•equations must capture the appropriate balance

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Continuity Equation

n̂dSS

microscopic mass balance written on an arbitrarily shaped volume, V, enclosed by a surface, S

V

vvv

vvv zyx

zyxz

vy

vx

vt zyx

vvt

Gibbs notation:

© Faith A. Morrison, Michigan Tech U.2

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Equation of Motion

n̂dSS

microscopic momentum balance written on an arbitrarily shaped volume, V, enclosed by a surface, S

V

gPvvt

v

Gibbs notation: general fluid

gvPvvt

v

2Gibbs notation:

Newtonian fluid

Navier‐Stokes Equation


Problem‐Solving Procedure ‐ fluid‐mechanics problems

1. sketch system

amended: when using the microscopic balances

2. choose coordinate system

3. simplify the continuity equation (mass balance)

4. simplify the 3 components of the equation of motion (momentum balance) (note that for a Newtonian fluid, the equation of motion is the Navier‐Stokes equation)

5. solve the differential equations for velocity and pressure (if applicable)

6. apply boundary conditions

7. calculate any engineering values of interest (flow rate, average velocity, force on wall)


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EXAMPLE I: Flow of a Newtonian fluid down an inclined plane x

z

xz

fluid

xvz

air

singgx

cosggz H

g

cos

0

sin

g

g

g

g

g

g

z

y

x

z


(Bird sign convention)


6

www.chem.mtu.edu/~fmorriso/cm310/Navier.pdf

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7


cross-section A:A

r z

r

z

EXAMPLE II: Pressure‐driven flow of a Newtonian fluid in a tube: Poiseuille flow

z

L vz(r)

•steady state•well developed•long tube

Rfluid

g


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9



10


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Poiseuille flow of a Newtonian fluid:

LP P

0

22

( )

( )2

o L

o Lrz

P PP z z P

LL g P P

r rL

R L g P P r

1

4o L

z

R L g P P rv r

L R


2

2

0 0R

R

z

z

drdr

drdrv

vaverage velocity

Engineering Quantities of Interest(tube flow)

0 0

drdr

volumetric flow rate z

R

z vRdrdrvQ 22

0 0

z‐component of force on the wall dzRdF

L

Rrrzz

0

2

0


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A

zdA

dAvn

v

ˆ

average velocity

Engineering Quantities of Interest(any flow)

A

volumetric flow rate

A

dAvnQ ˆ

z‐component of force on the wall ˆ ˆ

z z

A at surface

F e n pI dA



0( ) o LP P

P z z PL

22

( )2

14

o Lrz

o Lz

LL g P P

r rL

R L g P P rv r

L R

4

4

8o L

L R

R L g P PQ

L


Hagen‐Poiseuille Equation**

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2 R

zav rdrdrvv

14

2

2

0 0

22

0 0

Lo

RLo

PPgLR

rdrdR

r

L

PPgLR

2

8

max,zv

L



-1

-0.5

0

0 0 25 0 5 0 75 1z

Lpp

pp

0

0

1

1.5

2

av

z

v

v

0 0.25 0.5 0.75 1L

z


0

0.5

0 0.25 0.5 0.75 1

R

r

16

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vz/<v>

EXAMPLE II: Pressure‐driven flow of a Newtonian fluid in a tube: Poiseuille flow


Reynolds Number

Dv velocityaveragev

densityFlow rate

Data may be organized in terms of two dimensionless parameters:

Fanning Friction Factor

DvzRe

lengthpipeL

droppressurePP

viscosity

diameterpipeD

velocityaveragev

L

z

0

1


Pressure Drop

2

0

21

41

z

L

vDL

PPf

18

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Data correlation for friction factor (P) versus Re (flow rate) in a pipe

Re

16

(Geankoplis)© Faith A. Morrison, Michigan Tech U.

Moody Chart

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What is the Fanning Friction Factor for Laminar Flow?

Hagen-PoiseuilleE i ( i

PPRvRQ L 0

42

Equation (gravity neglected)L

vRQ z

8

Re

16161

41

0

DL

PPf

L

TRUE


Re21 2

Dv

vDL z

z

20

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Turbulent flow generates much more friction than themuch more friction than the (unrealizable) laminar flow at

the same flow rate.

(Geankoplis)© Faith A. Morrison, Michigan Tech U.

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Long‐chain polymers can “laminarize” the flow and

reduce drag.


Bird, Armstrong, Hassager, Dynamics of Polymeric Liquids, Wiley, New York 1987. 22

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The hose connecting the city water supply to the washing machine in a home burst while the homeowner was away Water spilled out of the ½ in pipe for 48 hours before the problem was noticed by a neighbor and the water was cut off.

A problem from real life: REDUX

How much water sprayed into the house over the 2‐day period?

The water utility reports that the water pressure supplied to the house was approximately 60 psig.


Home flood: the cold‐water feed to a washing machine burst and was unattended for two days

REDUX


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Discussion:

•How do we calculate the total amount of water spilled?

•What determines flow rate through a pipe?

•What information do we need about the system to calculate the amount of water spilled over two days?

REDUX

water spilled over two days?

Solution Strategy:

•Apply the laws of physics to the situation•Calculate the velocity field in the pipe

(will be function of pressure)•Calculate the flow rate from the velocity field

(as a function of pressure)C l l t th t t l t f t ill d


•Calculate the total amount of water spilled

Because flow in a tube is a bit complicated to do as a first problem (because of the curves), let’s consider a somewhat simpler problem first – laminar flow in a tube.

cross-section A:A

r z

r

z

The Situation:

Steady, laminar,

REDUX

z

L vz(r)

pressure‐driven flow of water in a pipe

Poiseuille Flow

Rfluid


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0( ) o LP P

P z z PL

22

( )2

14

o Lrz

o Lz

LL g P P

r rL

R L g P P rv r

L R

4

4

8o L

L R

R L g P PQ

L


Hagen‐Poiseuille Equation**

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Example 7.3: The hose connecting the city water supply to the washing machine in a home burst while the homeowner was away. Water gushed out of the ½” diameter pipe for 48 hours before the problem was noticed by a neighbor and the water was cut off. How much water sprayed intthe house over the two‐day period? The water utility reports that the y p y pwater pressure supplied to the house was approximately 60 psig.


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4 Hagen Poiseuille

Volume Q t Solution:

Our laminar flow solution gives the pressure‐drop/flow rate relationship:

4

8o LR P P

QL

Hagen‐Poiseuille

Equation

(gravity neglected)

Water main from street: Pipe in house:


Solution:Water main from street: Pipe in house:

1 48

8800

p psig

Q gpm

Answer (laminar flow):


Reality check: •volume of swimming pool: 13,000 gal•1 ½” fire hose at 300 psi: 100 gpm•4” fire hose: 1000 gpm

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1 48

8800

p psig

Q gpm

Answer (laminar flow):

This answer is unreasonable.Now we must figure out why.


Returning to flow down an incline, what is the force (a vector) on the wall?

xxz

fluid

xvz

air


H

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0 0

W H

z

z W H

v dx dy

v

dx dy

average velocity

H is the height of the film

Engineering Quantities of Interest

0 0

dx dy


0 0

W H

x zQ v dx dy WH v L Wz‐component

of force on the wall

0 0

L W

z xz x HF dy dz


(The expressions are different in different coordinate systems) 33

A

zdA

dAvn

v

ˆ

average velocity


A


ˆsurface

A

Q n v dA z‐component of force on the wall ˆ ˆ

z z

A at surface

F e n pI dA


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z‐component of force on the wall

ˆ ˆz z

A at surface

F e n pI dA

ˆF I dA


A at surface

F n pI dA force on the wall

Tv v

For other coordsystems use the

2

2

yx x x z

xx xy xzy y yx z

yx yy yz

vv v v v

x y x z x

v v vv v

v v use the handout


2

yx yy yz

zx zy zz xyzyx z z z

xyz

y x y z y

vv v v v

z x z y z

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Summary

•Use microscopic balances to calculate velocity fields•Use velocity field to calculate flow rate•Use velocity field to calculate flow rate•Use velocity field to calculate average velocity•Use velocity field to calculate shear stress•Use velocity field to calculate stress tensor•Use shear stress to calculate shear force on a wall•Use stress tensor to calculate total force on a wall


•Laminar flow gives a ridiculous answer for the flood problem •What about non‐Newtonian fluids?

Need to address these last issues . . .

Documents

new fluids lecture 0506 - Michigan Technological Universitypages.mtu.edu/~fmorriso/cm310/new0506fluids.pdf · fmorriso/cm310/Navier.pdf. ... rdr d v rdr d average v velocity Engineering