fluids lecture 14
TRANSCRIPT
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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
Faith A. Morrison, Michigan Tech U.
4000Re4.0Relog0.41
turbulent
10Re4000Re079.0turbulent
2100Re
Re
16laminar
10
525.0
=
=
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rough pipes - need an additional dimensionless group
Faith A. Morrison, Michigan Tech U.
Real Flows (continued)
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
Faith A. Morrison, Michigan Tech U.
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
Real Flows (continued)
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( ) HHRD 4
perimeterwetted
)areasectionalcross(4=
Faith A. Morrison, Michigan Tech U.
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
Real Flows (continued)
flow through noncircular conduits
Otherinternal flows:
Faith A. Morrison, Michigan Tech U.
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
Real Flows (continued)
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Faith A. Morrison, Michigan Tech U.
Non-Circular Cross-
sections have application
in the new field of
microfluidics
Faith A. Morrison, Michigan Tech U.
Chemical & Engineering News, 10 Sept2007, p14
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Faith A. Morrison, Michigan Tech U.
Real Flows (continued)
entry flow in pipes
flow through a contraction
flow through an expansion
flow through a Venturi meter
flow through a butterfly valve
etc.
Otherinternal flows:
see Perrys Handbook
calculate drag - superficial velocity
relations
Faith A. Morrison, Michigan Tech U.
Real Flows (continued)
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?
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Faith A. Morrison, Michigan Tech U.
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)
Faith A. Morrison, Michigan Tech U.
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
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Faith A. Morrison, Michigan Tech U.
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
Faith A. Morrison, Michigan Tech U.
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
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Faith A. Morrison, Michigan Tech U.
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
Faith A. Morrison, Michigan Tech U.
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
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Faith A. Morrison, Michigan Tech U.
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
Faith A. Morrison, Michigan Tech U.
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
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Faith A. Morrison, Michigan Tech U.
Steady flow of an incompressible, Newtonian
fluid around a sphere
BSL, p194
correlation equations
000,200Re50044.0turbulent
500Re2Re5.18turbulent
10.0ReRe
24laminar
60.0
=
=
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internalflows (flow in a conduit)
externalflow (around obstacles)
Faith A. Morrison, Michigan Tech U.
Real Flows (continued)
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
Faith A. Morrison, Michigan Tech U.
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.
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What is pressure-drop versus flow rate for flow through anunconsolidated bed of monodisperse spherical particles?
Faith A. Morrison, Michigan Tech U.
Real Flows (continued)
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.
Faith A. Morrison, Michigan Tech U.
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 =
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Faith A. Morrison, Michigan Tech U.
Real Flows (continued) Flow through Packed Beds
==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
Faith A. Morrison, Michigan Tech U.
Real Flows (continued) Flow through Packed Beds
Now, put it all together . . .
from Denn, Process Fluid
Mechanics, Prentice-Hall
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
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Faith A. Morrison, Michigan Tech U.
Real Flows (continued) Flow through Packed Beds
Now, put it all together . . .
from Denn, Process Fluid
Mechanics, Prentice-Hall
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 =
Faith A. Morrison, Michigan Tech U.
Real Flows (continued) Flow through Packed Beds
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.
from Denn, Process Fluid
Mechanics, Prentice-Hall
1980; p69
)1(
2
)1(Re
3
0
ff
Dv
p
p
p
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pf
pRe
Faith A. Morrison, Michigan Tech U.
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
Faith A. Morrison, Michigan Tech U.
Real Flows (continued) Flow through Packed Beds
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
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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?
Faith A. Morrison, Michigan Tech U.
Real Flows (continued)
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:
Faith A. Morrison, Michigan Tech U.
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
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Faith A. Morrison, Michigan Tech U.
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
Faith A. Morrison, Michigan Tech U.
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.