Shell Momentum Balances

Outline

1.Flow Through a Vertical Tube

2.Flow Through an Annulus

3.Exercises

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)?


Same system, but this time gravity will also cause momentum flux.


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


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


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


0 Lrz

d r rdr L

0 Lrz

d P Pr rdr L

Flow through a circular tube

Flow through a vertical tube


2 204

Lzv R r

L

2032

Lavev D

L

Hagen-Poiseuille Equation

Outline

1.Flow Through a Vertical Tube

2.Flow Through an Annulus

3.Exercises

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


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)


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


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


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


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


0 2

2Lz Rdv r R

dr L R r

2

0 22

Solving:1 ln2 2

Lz

R rv R r CL R


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


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


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


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

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.

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.

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

Design Equations for Laminar and Turbulent Flow in Pipes

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

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


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

Maximum vs. Average Velocity

Outline



Flow)



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

Hagen-Poiseuille Equation

0 232 ave

LLvP PD

Pressure drop / Pressure loss (P0 – PL):Pressure lost due to skin friction

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?)

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

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

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

Outline



Flow)



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

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)

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.

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

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

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

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

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)

Shell Momentum Balances

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