Shell Momentum Balances
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Transcript of Shell Momentum Balances
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Shell Momentum Balances
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Outline
1.Flow Through a Vertical Tube
2.Flow Through an Annulus
3.Exercises
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Flow Through a Vertical Tube
The tube is oriented vertically.
What will be the velocity profile of a fluid whose direction of flow is in the +z-direction (downwards)?
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Flow Through a Vertical Tube
Same system, but this time gravity will also cause momentum flux.
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Flow Through a Vertical Tube
rate of momentum rate of momentum force of gravityin by molecular out by molecular 0acting on systemtransport transport
0
1 2
: :
: + (why positive?)
z z L
rz rzr r r
pressure PA PAnet momentumflux A Agravity gV
0
Adding all terms together:2 2 2 2
(2 ) 0rz rzz z L r r r
P r r P r r rL rLg r rL
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Flow Through a Vertical Tube
0
0
Dividing by 2 :
0
Let 0:
0
rz rzz z L r r r
Lrz
L rr rP P r gr
L r
rP P dr r grL dr
02 2 2 2 (2 ) 0rz rzz z L r r rP r r P r r rL rL g r rL
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Flow Through a Vertical Tube
0 0Lrz
P P dr r grL dr
0 0
Rewriting:(0)L L
rzd P P P P g gLr g r rdr L L L
We let: z zP gz 0 Lrz
d r rdr L
0 (0) Lrz
d P g P gLr rdr L L
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Flow Through a Vertical Tube
0 Lrz
d r rdr L
0 Lrz
d P Pr rdr L
Flow through a circular tube
Flow through a vertical tube
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Flow Through a Vertical Tube
2 204
Lzv R r
L
2032
Lavev D
L
Hagen-Poiseuille Equation
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Outline
1.Flow Through a Vertical Tube
2.Flow Through an Annulus
3.Exercises
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Flow Through an Annulus
Liquid is flowing upward through an annulus (space between two concentric cylinders)
Important quantities:R : radius of outer cylinderκR : radius of inner cylinder
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Flow Through an Annulus
Assumptions:1. Steady-state flow2. Incompressible fluid3. Only Vz component is
significant4. At the solid-liquid interface,
no-slip condition5. Significant gravity effects6. Vmax is attained at a
distance λR from the center of the inner cylinder (not necessarily the center)
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Flow Through an Annulus
rate of momentum rate of momentum force of gravityin by molecular out by molecular 0acting on systemtransport transport
0
1 2
: :
: (why negative?)
z z L
rz rzr r r
pressure PA PAnet momentumflux A Agravity gV
0
Adding all terms together:2 2 2 2
(2 ) 0rz rzz z L r r r
P r r P r r rL rLg r rL
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Flow Through an Annulus
0 0Lrz
P P dr r grL dr
0 0
Rewriting:(0)L L
rzd P P P P g gLr g r rdr L L L
We let: z zP gz 0 Lrz
d r rdr L
0 (0) Lrz
d P g P gLr rdr L L
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Flow Through an Annulus
0 Lrz
d r rdr L
0
201
0 1
Solving:
2
2
Lrz
Lrz
Lrz
d r rdr L
r r CL
CrL r
BOUNDARY CONDITION!At a distance λR from the center of the inner cylinder, Vmax is attained in the annulus, or zero momentum flux.
0 10 2L CR
L R
201 2
LC RL
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Flow Through an Annulus
0 2
Rewriting:
2L
rzR r R
L R r
20 02 2
L Lrz
Rr
L L r
From the definition of flux:z
rzdvdr
0 2
2Lz Rdv r R
dr L R r
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Flow Through an Annulus
0 2
2Lz Rdv r R
dr L R r
2
0 22
Solving:1 ln2 2
Lz
R rv R r CL R
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Flow Through an Annulus
2
0 22
1 ln2 2L
zR rv R r C
L R
220 2
2
Rewriting:
2 ln4L
zR r Rv r C
L R R
Take out R/2 Multiply r in log term by R/R (or 1)
Expand log term
Lump all constants into C2
220 2
22 ln ln( )4L
zR r rv R C
L R R
220 2
22 ln4L
zR r rv C
L R R
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Flow Through an Annulus
220 2
22 ln4L
zR r rv C
L R R
We have two unknown constants: C2 and λWe can use two boundary conditions:
No-slip ConditionsAt r = κR, vz = 0At r = R, vz = 0
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Flow Through an Annulus
220 2
22 ln4L
zR r rv C
L R R
20 2 2
2
2 22
Using B.C. #1:
0 2 ln40 2 ln
L R CL
C
22
2
112 ln
C
20
2
2
Using B.C. #2:
0 140 1
L R CL
C
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Flow Through an Annulus
220 2
22 ln4L
zR r rv C
L R R
22
2
112 ln
C
22 20 1 ln 14 ln
Lz
R r rvL R R
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Shell Balances
1. Identify all the forces that influence the flow (pressure, gravity, momentum flux) and their directions. Set the positive directions of your axes.
2. Create a shell with a differential thickness across the direction of the flux that will represent the flow system.
3. Identify the areas (cross-sectional and surface areas) and volumes for which the flow occurs.
4. Formulate the shell balance equation and the corresponding differential equation for the momentum flux.
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Shell Balances
5. Identify all boundary conditions (solid-liquid, liquid-liquid, liquid-free surface, momentum flux values at boundaries, symmetry for zero flux).
6. Integrate the DE for your momentum flux and determine the values of the constants using the BCs.
7. Insert Newton’s law (momentum flux definition) to get the differential equation for velocity.
8. Integrate the DE for velocity and determine values of constants using the BCs.
9. Characterize the flow using this velocity profile.
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Shell Balances
Important Assumptions*1. The flow is always assumed to be at steady-
state.2. Neglect entrance and exit effects. The flow is
always assumed to be fully-developed.3. The fluid is always assumed to be
incompressible.4. Consider the flow to be unidirectional.
*unless otherwise stated
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Design Equations for Laminar and Turbulent Flow in Pipes
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Outline
1.Velocity Profiles in Pipes
2.Pressure Drop and Friction Loss (Laminar
Flow)
3.Friction Loss (Turbulent Flow)
4.Frictional Losses in Piping Systems
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Velocity Profiles in Pipes
Recall velocity profile in a circular tube:
1. What is the shape of this profile?2. The maximum occurs at which region?3. What is the average velocity of the fluid
flowing through this pipe?
2 204
Lz
P Pv R rL
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Velocity Profiles in Pipes
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Velocity Profiles in Pipes
Velocity Profile in a Pipe:
Average Velocity of a Fluid in a Pipe:
2 204
Lz
P Pv R rL
2032
Lave
P Pv DL
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Maximum vs. Average Velocity
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Outline
1.Velocity Profiles in Pipes
2.Pressure Drop and Friction Loss (Laminar
Flow)
3.Friction Loss (Turbulent Flow)
4.Frictional Losses in Piping Systems
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Recall: Hagen-Poiseuille Equation
2032
Lave
P Pv DL
Describes the pressure drop and flow of fluid (in the laminar regime) across a conduit with length L and diameter D
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Hagen-Poiseuille Equation
0 232 ave
LLvP PD
Pressure drop / Pressure loss (P0 – PL):Pressure lost due to skin friction
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Friction Loss
0 232 ave
LLvP PD
In terms of energy lost per unit mass: 2
32O L avef
P P LvFD
Mechanical energy lost due to friction in pipe (because of what?)
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Friction Factor
Definition: Drag force per wetted surface unit area (or shear stress at the surface) divided by the product of density times velocity head
0
2 22 2L C SS
P P A Af
v v
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Friction Factor
24 2
fF
c c
F L vfg D g
Frictional force/loss head is proportional to the velocity head of the flow and to
the ratio of the length to the diameter of the flow stream
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Friction Factor for Laminar Flow
Consider the Hagen-Poiseuille equation (describes laminar flow) and the definition of the friction factor:
Prove:
2032
Lave
P Pv DL
24 2
f O LF
c c
F P P L vfg g D g
Re
16Ff N Valid only for laminar flow
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Outline
1.Velocity Profiles in Pipes
2.Pressure Drop and Friction Loss (Laminar
Flow)
3.Friction Loss (Turbulent Flow)
4.Frictional Losses in Piping Systems
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Friction Factor for Turbulent Flow
1. Friction factor is dependent on NRe and the relative roughness of the pipe.
2. The value of fF is determined empirically.
24 2
fF
c c
F L vfg D g
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Friction Factor for Turbulent Flow
How to compute/find the value of the friction factor for turbulent flow:
1. Use Moody diagrams.- Friction factor vs. Reynolds number with a series of parametric curves related to the relative roughness
2. Use correlations that involve the friction factor f.- Blasius equation, Colebrook formula, Churchill equation (Perry 8th Edition)
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Moody Diagrams
Important notes:1. Both fF and NRe are plotted in logarithmic scales.
Some Moody diagrams show fD (Darcy friction factor). Make the necessary conversions.
2. No curves are shown for the transition region.3. Lowest possible friction factor for a given NRe in
turbulent flow is shown by the smooth pipe line.
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1. Blasius equation for turbulent flow in smooth tubes:
2. Colebrook formula0.25Re
0.079Ff N 5
Re4000 10N
10Re
1 2.512log 3.7D DDf N f
Friction Factor Correlations
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3. Churchill equation (Colebrook formula explicit in fD)
4. Swamee-Jain correlation
0.9
10Re
1 0.27 72logD D Nf
10 0.9Re
0.255.742log 3.7
Df
D N
Friction Factor Correlations
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Materials of Construction Equivalent Roughness (m)Copper, brass, lead (tubing) 1.5 E-06Commercial or welded steel 4.6 E-05Wrought iron 4.6 E-05Ductile iron – coated 1.2 E-04Ductile iron – uncoated 2.4 E-04Concrete 1.2 E-04Riveted Steel 1.8 E-03
Equivalent Roughness, ε
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Instead of deriving new correlations for f, an approximation is developed for an equivalent diameter, Deq, which may be used to calculate NRe and f.
where RH = hydraulic radiusS = cross-sectional areaPw = wetted perimeter: sum of the length of the boundaries of the cross-section actually in contact with the fluid
4 4eq Hw
SD RP
Frictional Losses for Non-Circular Conduits
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Determine the equivalent diameter of the following conduit types:
1.Annular space with outside diameter Do and inside diameter Di
2.Rectangular duct with sides a and b3.Open channels with liquid depth y and liquid
width b
4 4eq Hw
SD RP
Equivalent Diameter (Deq)