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)
© Faith A. Morrison, Michigan Tech U.
4000Re4.0Relog0.41
turbulent
10Re4000Re079.0turbulent
2100ReRe
laminar
10
525.0
ff
f
f
from Denn, Process Fluid Mechanics, Prentice-Hall 1980; p35
2
© Faith A. Morrison, Michigan Tech U.
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
Real Flows (continued)
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
© Faith A. Morrison, Michigan Tech U.
Colebrook correlation (Re>4000)k
3
Real Flows – Rough Pipes (continued)k
© Faith A. Morrison, Michigan Tech U.
J. Nikuradse, ``Stromungsgesetze in Rauhen Rohren,” VDI Forschungsh, 361 (1933); English translation, NACA Tech. Mem. 1292.
Surface Roughness for Various Materials
Real Flows (continued)
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)
© Faith A. Morrison, Michigan Tech U.
Riveted steel 0.9-9
from Denn, Process Fluid Mechanics, Prentice-Hall 1980; p46
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
Real Flows (continued)
•flow through noncircular conduitsOther internal flows:
HH RD 4perimeter wetted
)area sectionalcross(4
replaced with equivalent hydraulic diameter DH.
Hydraulic radiusEquivalent hydraulic
© Faith A. Morrison, Michigan Tech U.
Hydraulic radiusdiameter
Flow Through Noncircular Conduits
•Flow through equilateral t i l d it
Real Flows (continued)
f
triangular conduit
•f and Re calculated with DH
•solid lines are for circular pipes
© Faith A. Morrison, Michigan Tech U.
from Denn, Process Fluid Mechanics, Prentice-Hall 1980; p48
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
© Faith A. Morrison, Michigan Tech U.
application in the new field of microfluidics
© Faith A. Morrison, Michigan Tech U.
Chemical & Engineering News, 10 Sept 2007, p14
6
Real Flows (continued)
•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.
© Faith A. Morrison, Michigan Tech U.
see Perry’s Handbook
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
calculate drag - superficial velocity relations
© Faith A. Morrison, Michigan Tech U.
flows; take data; develop empirical correlations from data
What do we do with the correlations?
7
Real Flows (continued)
A real flow problem (external). What is the speed of a sky diver?
© Faith A. Morrison, Michigan Tech U.
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
© Faith A. Morrison, Michigan Tech U.
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
© Faith A. Morrison, Michigan Tech U.
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
© Faith A. Morrison, Michigan Tech U.
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
© Faith A. Morrison, Michigan Tech U.
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
© Faith A. Morrison, Michigan Tech U.
kinetic termsstationary terms (=0 when v=0)
Stokes law:kinetic force RvFkin 6
Bird, Stewart, Lightfoot, Transport Phenomena, Wiley, 1960, p59
10
Steady flow of an incompressible,
Newtonian fluid around a sphere
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
© Faith A. Morrison, Michigan Tech U.
•conclude f=f(Re) or CD=CD(Re)
drag coefficient
•take data, plot, develop correlations
Steady flow of an incompressible,
Newtonian fluid around a sphere
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
© Faith A. Morrison, Michigan Tech U.
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)
© Faith A. Morrison, Michigan Tech U.
•entrained droplets in distillation columns
•particle separators
•drop coalescence
Perry s)
12
Real Flows (continued)
© Faith A. Morrison, Michigan Tech U.F. A. Morrison An Introduction to Fluid Mechanics, Cambridge 2012
•rough spheres
•objects of other shapes
Real Flows (continued)
Other external flows:
•flows past walls
•airplane flight
© Faith A. Morrison, Michigan Tech U.
13
internal flows (flow in a conduit)
external flow (around obstacles)
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:
•ion exchange columns•packed bed reactors
© Faith A. Morrison, Michigan Tech U.
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
© Faith A. Morrison, Michigan Tech U.
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?
Real Flows (continued)
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
© Faith A. Morrison, Michigan Tech U.
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?
© Faith A. Morrison, Michigan Tech U.
vvv
bed ofsection -x
voidsarea sectional-x0
void fraction
bed?
0v
v
15
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?
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
© Faith A. Morrison, Michigan Tech U.
p
13
2 pH
DD
Real Flows (continued) Flow through Packed Beds
Now, put it all together . . .
from Denn, Process Fluid Mechanics, Prentice-Hall
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
© Faith A. Morrison, Michigan Tech U.
20
0
21
4
vDL
f
p
L
L
PP
D
vf L
p
0
202
16
Real Flows (continued) Flow through Packed Beds
Now, put it all together . . .
from Denn, Process Fluid Mechanics, Prentice-Hall
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
© Faith A. Morrison, Michigan Tech U.
define this as 1/Rep
pp
f72
1
Re
1
Real Flows (continued) Flow through Packed Beds
pp
f72
1
Re
1
When we check this relationship with experimental data we find that a
from Denn, Process Fluid Mechanics, Prentice-Hall
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
© Faith A. Morrison, Michigan Tech U.
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
© Faith A. Morrison, Michigan Tech U.
from Denn, Process Fluid Mechanics, Prentice-Hall 1980; p709; original source Ergun, Chem Eng. Progr., 48, 93 (1952).
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
© Faith A. Morrison, Michigan Tech U.
•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?
Real Flows (continued)
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.
© Faith A. Morrison, Michigan Tech U.
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
© Faith A. Morrison, Michigan Tech U.
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
© Faith A. Morrison, Michigan Tech U.
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
© Faith A. Morrison, Michigan Tech U.
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.