Real Flows continued - Michigan Technological Universityfmorriso/cm310/fluids_lecture_14.pdf ·...
Transcript of Real Flows continued - Michigan Technological Universityfmorriso/cm310/fluids_lecture_14.pdf ·...
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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 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
© Faith A. Morrison, Michigan Tech U.
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
© 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)
Other internal flows:
k
© Faith A. Morrison, Michigan Tech U.
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)
from Denn, Process Fluid Mechanics, Prentice-Hall 1980; p46
Surface Roughness for Various Materials
Real Flows (continued)
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( ) HH RD 4perimeter wetted
)area sectionalcross(4 =−≡
© Faith A. Morrison, Michigan Tech U.
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
Real Flows (continued)
•flow through noncircular conduitsOther internal 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 for circular pipes
Note: for some shapes the correlation is somewhat different than the circular pipe
correlation; see Perry’s 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 Sept 2007, 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.
Other internal flows:
see Perry’s 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 empirical correlations from data
What do we do with the correlations?
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© Faith A. Morrison, Michigan Tech U.
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)
© Faith A. Morrison, Michigan Tech U.
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
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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
© Faith A. Morrison, Michigan Tech U.
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
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© Faith A. Morrison, Michigan Tech U.
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
© Faith A. Morrison, Michigan Tech U.
Steady flow of an incompressible,
Newtonian fluid around a sphere
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
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© Faith A. Morrison, Michigan Tech U.
Steady flow of an incompressible,
Newtonian fluid around a sphere
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
© 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
≤≤=
≤≤=
<=−
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
© Faith A. Morrison, Michigan Tech U.
Real Flows (continued)
Other external flows:
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internal flows (flow in a conduit)
external flow (around obstacles)
© Faith A. Morrison, Michigan Tech U.
Real Flows (continued)
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
© Faith A. Morrison, Michigan Tech U.
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.
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What is pressure-drop versus flow rate for flow through an unconsolidated 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
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.
© 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 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
© Faith A. Morrison, Michigan Tech U.
Real Flows (continued) Flow through Packed Beds
==
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
© 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
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
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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 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 =⇒
© Faith A. Morrison, Michigan Tech U.
Real Flows (continued) Flow through Packed Beds
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.
from Denn, Process Fluid Mechanics, Prentice-Hall
1980; p69
)1(2
)1(Re
3
0
εε
εµρ
−≡
−≡
ff
Dv
p
pp
15
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).
pp
f=+ 75.1Re150
© Faith A. Morrison, Michigan Tech U.
Real Flows (continued) Flow through Packed Beds
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
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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?
© 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
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:
© 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 ερρ −− 1
( )ALε−1bed 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 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
© Faith A. Morrison, Michigan Tech U.
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.