Real Flows continuedfmorriso/cm310/new09fluids.pdf1 Real Flows (continued) So far we have talked...

1

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

© Faith A. Morrison, Michigan Tech U.

flows; take data; develop empirical correlations from data

What do we do with the correlations? use in MEB; calculate pressure-drop flow-rate relations

Empirical data correlationsfriction factor (P) versus Re (Q) in a pipe

graphical correlations(flow in a pipe)

2100R16

l i f

correlation equations(flow in a pipe)

(Geankoplis 3rd ed)


4000Re4.0Relog0.41

turbulent

10Re4000Re079.0turbulent

2100ReRe

laminar

10

525.0

ff

f

f

from Denn, Process Fluid Mechanics, Prentice-Hall 1980; p35

2


F. A. Morrison “Data Correlation for Friction Factor in Smooth Pipes,” Michigan Technological University, Houghton, MI (2010) www.chem.mtu.edu/~fmorriso/DataCorrelationForSmoothPipes2010.pdf

•rough pipes - need an additional dimensionless group


k - characteristic size of the surface roughness

Other internal flows:

D

k- relative roughness (dimensionless roughness)

28.2Re

67.4log0.4

110

fD

k

f


Colebrook correlation (Re>4000)k

3

Real Flows – Rough Pipes (continued)k


J. Nikuradse, ``Stromungsgesetze in Rauhen Rohren,” VDI Forschungsh, 361 (1933); English translation, NACA Tech. Mem. 1292.

Surface Roughness for Various Materials


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

Commercial steel or wrought iron 0.05Asphalted cast iron 0.12Galvanized iron 0.15Cast iron 0.46Wood stave 0.2-.9Concrete 0.3-3Riveted steel 0 9 9

Material k (mm)


Riveted steel 0.9-9


4

Empirically, it is found that f vs. Re correlations for circular conduits matches the data for noncircular conduits if D is

l d ith i l t h d li di t D


•flow through noncircular conduitsOther internal flows:

HH RD 4perimeter wetted

)area sectionalcross(4

replaced with equivalent hydraulic diameter DH.

Hydraulic radiusEquivalent hydraulic


Hydraulic radiusdiameter

Flow Through Noncircular Conduits

•Flow through equilateral t i l d it


f

triangular conduit

•f and Re calculated with DH

•solid lines are for circular pipes



Re

Note: for some shapes the correlation is somewhat different than the circular pipe

correlation; see Perry’s Handbook

5

Non-Circular Cross-sections have application in the ne


application in the new field of microfluidics


Chemical & Engineering News, 10 Sept 2007, p14

6


•entry flow in pipes

•flow through a contraction

Other internal flows:

•flow through an expansion

•flow through a Venturi meter

•flow through a butterfly valve

•etc.


see Perry’s Handbook


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

calculate drag - superficial velocity relations


flows; take data; develop empirical correlations from data

What do we do with the correlations?

7


A real flow problem (external). What is the speed of a sky diver?


z(r,,

Steady flow of an incompressible,

Newtonian fluid around a sphere

Creeping Flow

(equivalent to sphere falling through a liquid)

y

g

•spherical coordinates

•symmetry in the dir

•calculate v and drag force on sphere


flow

force on sphere

•neglect inertia

•upstream vvz

8

Steady flow of an incompressible, Newtonian fluid

around a sphere

Creeping Flow

r

rvv

v

0

r

gg

g

0sincos

),( rPP

Eqn of sin11 2

vvr

gvPvvt

v

2

steady neglect SOLVE

Eqn of Motion:

Eqn of Continuity:

0

sin

sin

112

v

rr

vr

rr


steady state

neglect inertia

BC1: no slip at sphere surfaceBC2: velocity goes to far from spherev

SOLUTION: Creeping Flow around a sphere

r

R

r

Rv

r

R

r

Rv

v

sin4

1

4

31

cos2

1

2

31

3

3

cos2

3cos

2

0

r

R

R

vgrPP

r

0

0


Tvv all the stresses can be calculated from v

Bird, Stewart, Lightfoot, Transport Phenomena, Wiley, 1960, p57; complete solution in Denn

9

SOLUTION: Creeping Flow around a sphere

What is the total z-direction force on the sphere?

vector stress on a

integrate over the entire

2

2

0 0

ˆ sinrr R

F e PI R d d

vector stress on a microscopic surface of

unit normal ˆre

the entire sphere surface

total vector force on sphere

evaluate at the surface of the

sphere


total stress at a point in

the fluid

ˆze F

total z-direction

force on the sphere

Force on a sphere (creeping flow limit)

comes from pressure

comes from shear stresses

form drag

34ˆ 2 43z ze F F R g Rv Rv

buoyant force

friction drag

form drag


kinetic termsstationary terms (=0 when v=0)

Stokes law:kinetic force RvFkin 6

Bird, Stewart, Lightfoot, Transport Phenomena, Wiley, 1960, p59

10



Turbulent Flow

**2*******

* 1

Re

1g

FrvPvv

t

v

•Nondimensionalize eqns of change:

•Nondimensionalize eqn for Fkin:

define dimensionless kinetic force

2

2,

21

4v

D

FCf kineticz

D

•conclude f=f(Re) or drag


•conclude f=f(Re) or CD=CD(Re)

drag coefficient

•take data, plot, develop correlations



Turbulent Flow

R

24

1

62

D

RvCf D

•take data, plot, develop correlations

Laminar flow:Fsphere=Stokes law

Re21

42

2

vD

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

34( )sR g

Fsphere=net weight in fluid At terminal speed the net weight is exactly

balanced by the viscous


2

sphere

3

4

v

DgCf D

all measurable quantities

22

( )3

14 2

s

D

gf C

Dv

retarding force.

11

Steady flow of an incompressible, Newtonian fluid around a sphere

Re

24

graphical correlation

© Faith A. Morrison, Michigan Tech U.McCabe et al., Unit Ops of Chem Eng, 5th edition, p147

Steady flow of an incompressible, Newtonian fluid around a sphere

correlation equations

500Re2Re5.18turbulent

10.0ReRe

24laminar

60.0

f

f

BSL, p194000,200Re50044.0turbulent f

•use correlations in engineering practice•particle settling

t i d d l t i di till ti l

(See Denn, BSL, Perry’s)


•entrained droplets in distillation columns

•particle separators

•drop coalescence

Perry s)

12


© Faith A. Morrison, Michigan Tech U.F. A. Morrison An Introduction to Fluid Mechanics, Cambridge 2012

•rough spheres

•objects of other shapes


Other external flows:

•flows past walls

•airplane flight


13

internal flows (flow in a conduit)

external flow (around obstacles)


Now, we’ve done two classes of real flows:

We can apply the techniques we have learned to more complex engineering flows.

We will discuss two examples briefly:

•ion exchange columns•packed bed reactors


1. Flow through packed beds

2. Fluidized beds

•packed distillation columns•filtration•flow through soil (environmental issues, enhanced oil recovery)•fluidized bed reactors

Flow through Packed Beds

voidssolids

voids

solids

solids

solidareasectional-x

bed ofsection -x

voidsarea sectional-x


bed ofsection -x

solidarea sectionalx1

If the hydraulic diameter DH concept works for this flow, cross-section then we already know f(Re) from pipe flow.

14

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


More Complex Applications I: Flow through Packed Beds

flowflowDp=sphere diameter

or for irregular particles:

v

p

a

D 1

particles of area surface

particles of volume

6


We will choose to model the flow resistance as flow through tortuous conduits with equivalent hydraulic diameter DH=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 diameter DH=4RH.

L

DPPv L

32

20

average velocity in the

interstitial regions

bed entire ofsection -x

voidsof area sectional-x

0Q

v

Qv

superficial velocity

BUT, what are DH

and average velocity in terms of things

we know about the b d?


vvv

bed ofsection -x

voidsarea sectional-x0

void fraction

bed?

0v

v

15


surface wettedtotal

flowfor available volume

4 HH R

D

BUT, what is DH in terms of things we know about the bed?

from Denn, Process Fluid Mechanics, Prentice-Hall

1980; p69

)1(6)1(bed of volume

surface wettedbed of volume

voidsof volume

p

v

D

a

bedofvolume

particles of volume

particlesofvolume

surface particle


p

13

2 pH

DD


Now, put it all together . . .


1980; p69

0v

v

13

2 pH

DD

L

DPPv L

32

20

01

PP L

analogous to f for for pipes we write:

2


20

0

21

4

vDL

f

p

L

L

PP

D

vf L

p

0

202

16


Now, put it all together . . .


1980; p692

2200

)1(36

pDvfv

)1(

2

72

1)1(

3

0

f

Dv p

Now we follow convention and

and this as fp


define this as 1/Rep

pp

f72

1

Re

1


pp

f72

1

Re

1

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


1980; p69

When we check this relationship with experimental data we find that a better fit can be obtained with,

pp

f 75.1Re

150

Ergun Equation)1(

2

)1(Re

3

0

ff

Dv

p

pp


A data correlation for pressure-drop/flow rate data for flow through packed beds.

)1(

17

Flow through Packed Beds

pf pp

f 75.1Re

150

pRe


from Denn, Process Fluid Mechanics, Prentice-Hall 1980; p709; original source Ergun, Chem Eng. Progr., 48, 93 (1952).


What did we do?

We assumed the same functional form for P and Q 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

18

Can we use the Ergun equation (for pressure drop versus flow rate in a packed bed) to calculate the minimum superficial velocity at which a bed becomes fluidized?


More Complex Applications II: Fluidized beds

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.


flow v

pp

f 75.1Re

150The P vs Q

relationship can come from the Ergun

eqn at small Rep

neglect

Now we perform a force balance on the bed:

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

gravity net effect of gravity and

buoyancy is:

AL1bed volume =

When the forces balance, i i i t ALgp 1incipient

fluidization


pressure(Ergun eqn)

buoyancy

AP

19

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

ALgAP p 1

When the forces balance, incipient fluidization

eliminate P;

pp

fRe

150eliminate P; solve for v0

32D


1150

32

0pp gD

v velocity at the point of incipient fluidization

Real Flows SUMMARY

internal flows (Poiseuille flow in a pipe)IDEAL scopic

l fl ( i i )REAL

external flow (flow around a sphere)FLOWS

internal flows (f vs Re)

external flow (CD vs Re)

REAL FLOWS

nondimensionalization

balances


internal flows (pipes, pumping)

external flow (packed beds, fluidized bed reactors)

REAL ENGINEERING

UNIT OPERATIONSapply engineering approximations using reasonable

concepts and correlations obtained from experiments.

Real Flows continuedfmorriso/cm310/new09fluids.pdf1 Real Flows (continued) So far we have talked...

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