Fluids Lecture 14

7/27/2019 Fluids Lecture 14

1/17

1

Faith A. Morrison, Michigan Tech U.

Real Flows (continued)

So far we have talked about internal flows

ideal flows (Poiseuille flow in a tube)

real flows (turbulent flow in a tube)

Strategy for handling real flows: Dimensional analysis and data

correlations

How did we arrive at correlations? non-Dimensionalize ideal flow; use to

guide expts on similar non-ideal

flows; take data; develop empirical

correlations from dataWhat do we do with the correlations? use in MEB; calculate pressure-drop

flow-rate relations

Empirical data correlations

friction factor (P) versus Re (Q) in a pipe


4000Re4.0Relog0.41

turbulent

10Re4000Re079.0turbulent

2100Re

Re

16laminar

10

525.0

=

=


2/17

2

rough pipes - need an additional dimensionless group



k- characteristic size of the surface roughness

D

k- relative roughness (dimensionless roughness)

28.2Re

67.4log0.4

110 +

+=

fD

k

f

Colebrook correlation (Re>4000)

Otherinternal flows:

k


Drawn tubing (brass,lead, glass, etc.) 1.5x10-3

Commercial steel or wrought iron 0.05

Asphalted cast iron 0.12

Galvanized iron 0.15

Cast iron 0.46

Wood stave 0.2-.9

Concrete 0.3-3

Riveted steel 0.9-9

Material k(mm)

from Denn, Process Fluid Mechanics,

Prentice-Hall 1980; p46

Surface Roughness for Various

Materials



3/17

3

( ) HHRD 4

perimeterwetted

)areasectionalcross(4=


Empirically, it is found that f vs. Re correlations for circular

conduits matches the data for noncircular conduits if D is

replaced with equivalent hydraulic diameter DH.

Hydraulic radiusEquivalent hydraulic

diameter


flow through noncircular conduits



Flow Through Noncircular Conduits

from Denn, Process Fluid Mechanics,

Prentice-Hall 1980; p48

f

Re

Flow through

equilateral

triangular

conduit

f and Re

calculated

with DH

solid lines

are forcircular pipes

Note: for some shapes the correlation is

somewhat different than the circular pipe

correlation; see Perrys Handbook



4/17

4


Non-Circular Cross-

sections have application

in the new field of

microfluidics


Chemical & Engineering News, 10 Sept2007, p14


5/17

5



entry flow in pipes

flow through a contraction

flow through an expansion

flow through a Venturi meter

flow through a butterfly valve

etc.


see Perrys Handbook

calculate drag - superficial velocity

relations



Now, we will talk about external flows

ideal flows (flow around a sphere)

real flows (turbulent flow around a sphere, other obstacles)

Strategy for handling real flows: Dimensional analysis and data

correlations

How did we arrive at correlations? non-Dimensionalize ideal flow; use to

guide expts on similar non-ideal

flows; take data; develop empiricalcorrelations from data

What do we do with the correlations?


6/17

6


y

z

(r,,)

flow

g

Steady flow of an

incompressible, Newtonian

fluid around a sphere

Creeping Flow

spherical coordinates

symmetry in the dir

calculate v and drag

force on sphere

neglect inertia

upstream = vvz

(equivalent to

sphere falling

through a liquid)


Steady flow of an

incompressible,Newtonian fluid around

a sphere

Creeping Flow

r

r

v

v

v

=0

r

g

g

g

=0

sin

cos

),( rPP=

gvPvvt

v ++=

+

2

steady

state

neglect

inertia

SOLVE

BC1: no slip at sphere surface

BC2: velocity goes to far from spherev

Eqn of

Motion:

Eqn of

Continuity:

( )0

sin

sin

11 2

2=

+

v

rr

vr

r

r


7/17

7


SOLUTION: Creeping Flowaround a sphere

cos2

3cos

2

0

=

r

R

R

vgrPP

( )T

vv += all the stresses can be

calculated from v

r

r

R

r

Rv

r

R

r

Rv

v

+

=

0

sin4

1

4

31

cos2

1

2

31

3

3

0

Bird, Stewart, Lightfoot, Transport Phenomena,

Wiley, 1960, p57; complete solution in Denn

evaluate at the

surface of the

sphere

( )[ ] ==

2

0 0

2 sin ddRIPrFRr


SOLUTION: Creeping Flow

around a sphere

What is the totalz-direction

force on the sphere?

total stressat a point in

the fluid

vector stress on a

scopic surface ofunit normal r

integrate over

the entire

sphere surface

total vector

force on

sphere

Fk=

totalz-

direction

force on the

sphere


8/17

8


Force on a sphere (creeping flow limit)

++== RvRvgRFFk z 423

4 3

buoyant

force

comes frompressure

friction drag

kinetic termsstationary terms

(=0 when v=0)

Stokes law:

kinetic force = RvFkin 6

comes from shearstresses

form drag

Bird, Stewart,

Lightfoot,

Transport

Phenomena,

Wiley, 1960, p59


Steady flow of an

incompressible,Newtonian fluid around

a sphere

TurbulentFlow

**2******

*

* 1

Re

1g

FrvPvv

t

v++=

+

Nondimensionalize eqns of change:

Nondimensionalize eqn forFkin:

define dimensionless

kinetic force

==

2

2,

2

1

4v

D

FCf kineticzD

concludef=f(Re) or

CD=CD(Re)

drag

coefficient

take data, plot, develop correlations


9/17

9


Steady flow of an

incompressible,

Newtonian fluid arounda sphere

TurbulentFlow( )

Re

24

2

1

4

6

22

=

==

vD

RvCf D

take data, plot, develop correlations

Laminar flow: Stokes law

Turbulent flow: Calculate CD from terminal velocity of a falling sphere(see BSL p182; Denn p56)

2

sphere

3

4

==

v

DgCf D

allmeasurable

quantities


Steady flow of an incompressible, Newtonian


McCabe et al., Unit Ops of Chem

Eng, 5th edition, p147

Re

24

graphical correlation


10/17

10


Steady flow of an incompressible, Newtonian


BSL, p194

correlation equations

000,200Re50044.0turbulent

500Re2Re5.18turbulent

10.0ReRe

24laminar

60.0

=

=


11/17

11

internalflows (flow in a conduit)

externalflow (around obstacles)



Now, weve done twoclasses of real flows:

We can apply the techniques we have learned to

more complex engineering flows.

We will discuss two examples briefly:

1. Flow through packed beds

2. Fluidized beds

ion exchange columns

packed bed reactors

packed distillation

columns

filtrationflow through soil

(environmental issues,

enhanced oil recovery)

fluidized bed reactors


Flow through Packed Beds

voids

voids

solids

solids

solids

=

bedofsection-x

solidareasectional-x1

bedofsection-x

voidsareasectional-x

If the hydraulic diameter DH concept works for this flow, cross-

section then we already knowf(Re) from pipe flow.


12/17

12

What is pressure-drop versus flow rate for flow through anunconsolidated bed of monodisperse spherical particles?



More Complex Applications I: Flow through Packed Beds

flowDp=sphere diameter

or for irregular particles:

v

p

a

D 1

particlesofareasurface

particlesofvolume

6=

We will choose to model the flowresistance as flow through tortuous

conduits with equivalent hydraulic

diameterDH=4RH.


Real Flows (continued) Flow through Packed Beds

Hagen-Poiseuille equation:We will choose to model the flow

resistance as flow through tortuous

conduits with equivalent hydraulic

diameterDH=4RH.( )

L

DPPv L

32

20 =

average

velocity in the

interstitial

regions

bedentireofsection-x

voidsofareasectional-x

0

Qv

Qv

=

vvv =

=

bedofsection-x

voidsareasectional-x0

void

fraction

superficial

velocity

BUT, what areDHand average velocity

in terms of things

we know about the

bed?

0vv =


13/17

13



==surfacewettedtotal

flowforavailablevolume

4H

H RD

BUT, what isDHin terms of things we know about the bed?

)1(6)1(

bedofvolume

surfacewettedbedofvolume

voidsofvolume

=

=

= p

v

D

a

from Denn, Process Fluid

Mechanics, Prentice-Hall

1980; p69

bedofvolume

particlesofvolume

particlesofvolume

surfaceparticle

( )

=

13

2 pH

DD



Now, put it all together . . .



1980; p69

0vv = ( )

=

13

2 pH

DD

( )L

DPPv L

32

20 =

( )

=2

0

0

2

1 4

1

vD

L

PP

f

p

L

analogous toffor

for pipes we write:

( )LPPD vfL

p

= 02

02


14/17

14



Now, put it all together . . .



1980; p692

2200

)1(36

= p

Dvfv

)1(

2

72

1)1(

3

0

=

f

Dv p

Now we follow

convention and

define this as 1/Rep

and this as fp

p

p

f721

Re1 =



p

p

f72

1

Re

1 =

When we check this relationship with experimental data we find that a

better fit can be obtained with,

p

p

f=+ 75.1Re

150

Ergun Equation

A data correlation for pressure-drop/flow rate data

for flow through packed beds.



1980; p69

)1(

2

)1(Re

3

0

ff

Dv

p

p

p


15/17

15

pf

pRe


Flow through Packed Beds

from Denn, Process Fluid Mechanics, Prentice-Hall 1980; p709;

original source Ergun, Chem Eng. Progr., 48, 93 (1952).

p

p

f=+ 75.1Re

150



What did we do?

We assumed the same functional form forPandQ as laminar pipe flow with,

hydraulic diameter substituted for diameter

hydraulic diameter expressed in measureables

resulting functional form was fit to experimental data (new Re and f

defined for this system)

scaling was validated by the fit to the experimental data

we have obtained a correlation that will allow us to do design

calculations on packed beds


16/17

16

Can we use the Ergun equation (for pressure drop versus flow rate ina packed bed) to calculate the minimum superficial velocity at which

a bed becomes fluidized?



More Complex Applications II: Fluidized beds

flow

In a fluidized bed reactor, the flow

rate of the gas is adjusted to

overcome the force of gravity and

fluidize a bed of particles; in this

state heat and mass transfer is good

due to the chaotic motion.

v

p

p

f=+ 75.1Re

150The Pvs Qrelationship cancome from the Ergun

eqn atsmallRep

neglect

Now we perform a force balance on the bed:


Real Flows (continued) More Complex Applications II: Fluidized beds

pressure

(Ergun eqn)

gravity

buoyancy

AP

net effect of

gravity and

buoyancy is:

( )ALgp 1

( )AL1bed volume =

When the forces

balance,

incipient

fluidization


17/17

17


Real Flows (continued) More Complex Applications II: Fluidized beds

( )ALgAP p = 1

When the forces balance, incipient fluidization

p

p

f=Re

150eliminate P;solve forv0

( )

( )

=

1150

32

0pp gD

v velocity at the point of

incipient fluidization


Real Flows SUMMARY

internalflows (pipes, pumping)

externalflow (packed beds, fluidized bed reactors)

REAL

ENGINEERINGUNIT OPERATIONS

internalflows (Poiseuille flow in a pipe)

externalflow (flow around a sphere)

IDEAL

FLOWS

internalflows (f vs Re)

externalflow (CD vs Re)

REAL

FLOWSnondimensionalization

scopicbalances

apply engineering approximations using reasonable

concepts and correlations obtained from experiments.

Fluids Lecture 14

Documents

Transcript of Fluids Lecture 14