real flows continued - michigan technological universityfmorriso/cm310/fluids_lecture_14.pdf ·...

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 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


4000Re4.0Relog0.41

turbulent

10Re4000Re079.0turbulent

2100ReRe

16laminar

10

525.0

≥−=

≤≤=

<=−

ff

f

f

correlation equations(flow in a pipe)

(Geankoplis 3rd ed)

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

graphical correlations(flow in a pipe)

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)

Other internal flows:

k


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)


Surface Roughness for Various Materials


3

( ) HH RD 4perimeter wetted

)area sectionalcross(4 =−≡


Empirically, it is found that f vs. Re correlations for circularconduits matches the data for noncircular conduits if D is replaced with equivalent hydraulic diameter DH.

Hydraulic radiusEquivalent hydraulic

diameter


•flow through noncircular conduitsOther internal flows:


Flow Through Noncircular Conduits


f

Re

•Flow through equilateral triangular conduit

•f and Re calculated with DH

•solid lines are for circular pipes

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

correlation; see Perry’s Handbook


4


Non-Circular Cross-sections have application in the new field of microfluidics


Chemical & Engineering News, 10 Sept 2007, p14

5



•entry flow in pipes

•flow through a contraction

•flow through an expansion

•flow through a Venturi meter

•flow through a butterfly valve

•etc.

Other internal flows:

see Perry’s 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 empirical correlations from data

What do we do with the correlations?

6


y

z(r,θ,φ)

θ

flow

g

Steady flow of an incompressible, Newtonian

fluid around a sphere

Creeping Flow

•spherical coordinates

•symmetry in the φ 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

θφ

θ

r

rvv

v

=

0 θφ

θθ

r

gg

g

−=

0sincos

),( θrPP =

gvPvvt

v ρµρ +∇+−∇=

∇⋅+

∂∂ 2

steady state

neglect inertia

SOLVE

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

Eqn of Motion:

Eqn of Continuity:

( )0

sin

sin

11 2

2 =

∂∂+

∂∂

θθ

θθv

rr

vr

rr

7


SOLUTION: Creeping Flow around a sphere

θµθρ cos2

3cos

2

0

−−= ∞

r

R

R

vgrPP

( )[ ]Tvv ∇+∇−= µτ all the stresses can be calculated from v

θφ

θ

θ

r

r

R

r

Rv

r

R

r

Rv

v

−−−

+−

= ∞

∞

0

sin41

43

1

cos21

23

1

3

3

0

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

evaluate at the surface of the

sphere

( )[ ]∫ ∫ =−−⋅=

π π

φθθτ2

0 0

2 sinˆ ddRIPrFRr


SOLUTION: Creeping Flow around a sphere

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

total stress at a point in

the fluid

vector stress on a µscopic surface of

unit normal r̂

integrate over the entire

sphere surface

total vector force on sphere

Fk ⋅= ˆ

total z-direction

force on the sphere

8


Force on a sphere (creeping flow limit)

∞∞ ++==⋅ RvRvgRFFk z πµπµρπ 4234ˆ 3

buoyant force

comes from pressure

friction drag

kinetic termsstationary terms (=0 when v=0)

Stokes law:kinetic force ∞=≡ RvFkin πµ6

comes from shear stresses

form drag

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




Turbulent Flow

**2*******

* 1Re1

gFr

vPvvt

v +∇+−∇=

∇⋅+

∂∂

•Nondimensionalize eqns of change:

•Nondimensionalize eqn for Fkin:

define dimensionless kinetic force

==

∞2

2,

21

4v

D

FCf kineticz

D

ρπ

•conclude f=f(Re) or CD=CD(Re)

drag coefficient

•take data, plot, develop correlations

9




Turbulent Flow( )

Re

24

21

4

6

22 =

==

∞

∞

vD

RvCf D

ρππµ

•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 ρ

ρρ all measurable quantities


Steady flow of an incompressible, Newtonian fluid around a sphere

McCabe et al., Unit Ops of Chem Eng, 5th edition, p147

Re

24

graphical correlation

10


Steady flow of an incompressible, Newtonian fluid around a sphere

BSL, p194

correlation equations

000,200Re50044.0turbulent

500Re2Re5.18turbulent

10.0ReRe

24laminar

60.0

≤≤=

≤≤=

<=−

f

f

f

•use correlations in engineering practice•particle settling

•entrained droplets in distillation columns

•particle separators

•drop coalescence

(See Denn, BSL, Perry’s)

•rough spheres

•objects of other shapes

•flows past walls

•airplane flight



Other external flows:

11

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:

1. Flow through packed beds

2. Fluidized beds

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


Flow through Packed Beds

voids

voids

solids

solids

solids

=−

≡

bed ofsection -xsolid area sectional-x

1

bed ofsection -x voidsarea sectional-x

ε

ε

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

12

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

flowDp=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

µ32

20 −=

average velocity in the

interstitial regions

bed entire ofsection -x

voidsof area sectional-x

0Q

v

Qv

≡

=

εvvv =

=

bed ofsection -x

voidsarea sectional-x0

void fraction

superficial velocity

BUT, what are DH

and average velocity in terms of things

we know about the bed?

ε0v

v =

13



==

surface wettedtotal

flowfor available volume

4 HH R

D

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

)1(6)1(bed of volume

surface wettedbed of volume

voidsof volume

εε

εε

−=

−=

= p

v

D

a

from Denn, Process Fluid Mechanics, Prentice-Hall

1980; p69

bed of volume

particles of volume

particles of volume

surface particle

( )εε

−=

13

2 pH

DD



Now, put it all together . . .


1980; p69

ε0v

v = ( )εε

−=

13

2 pH

DD

( )L

DPPv L

µ32

20 −=

( )

−=

20

0

21

41

vDL

PPf

p

L

ρ

analogous to f for for pipes we write:

⇒ ( )L

PP

D

vf L

p

−= 0202 ρ

14



Now, put it all together . . .


1980; p692

2200

)1(36 εµερ

ε −= pDvfv

)1(

2

72

1)1(

3

0 εεε

ρµ

−=− f

Dv p

Now we follow convention and

define this as 1/Rep

and this as fp

pp

f721

Re1 =⇒



pp

f721

Re1 =

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

pp

f=+ 75.1Re150

Ergun Equation

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


1980; p69

)1(2

)1(Re

3

0

εε

εµρ

−≡

−≡

ff

Dv

p

pp

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).

pp

f=+ 75.1Re150



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

16

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

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

pp

f=+ 75.1Re150

The ∆P vs Qrelationship 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

pressure(Ergun eqn)

gravity

buoyancy

AP∆

net effect of gravity and

buoyancy is:

( )( )ALgp ερρ −− 1

( )ALε−1bed volume =

When the forces balance, incipient

fluidization

17


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

( )( )ALgAP p ερρ −−=∆ 1

When the forces balance, incipient fluidization

pp

f=Re150

eliminate ∆P; solve for v0

( )( )εµ

ερρ−

−=

1150

32

0pp gD

v velocity at the point of incipient fluidization


Real Flows SUMMARY

internal flows (pipes, pumping)

external flow (packed beds, fluidized bed reactors)

REAL ENGINEERING

UNIT OPERATIONS

internal flows (Poiseuille flow in a pipe)

external flow (flow around a sphere)

IDEAL FLOWS

internal flows (f vs Re)

external flow (CD vs Re)

REAL FLOWS

nondimensionalization

µscopic balances

apply engineering approximations using reasonable concepts and correlations obtained from experiments.

real flows continued - michigan technological universityfmorriso/cm310/fluids_lecture_14.pdf ·...

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