Momentum Transfer

Momentum Transfer

Jul-Dec 2006Instructor: Dr. S. RamanathanOffice: CHL 210Email: [email protected] Notes: http://www.che.iitm.ac.in/~srinivar

IIT-Madras, Momentum Transfer: July 2005-Dec 2005

Overview Background & Motivation Course Syllabus

What will be covered and what will not be Examples Goals & Pre-requisites Evaluation Tentative Schedule Text Books / References


Transport Phenomena

Chemical Engineering

Heat

MassMomentumReaction Kinetics


Background :

Most of the momentum transfer equations are similar to heat and mass transfers

Momentum transfer: Focus is on fluids

Heat and Mass Transfer: Also include solid

Heat Transfer: Radiation (no corresponding phenomena in momentum and mass transfer)

Similarities in problems will be discussed as appropriate


Motivation

Momentum Transfer: Fluid Mechanics

Design

Manufacturing (Production/ Maintenance)

Troubleshooting

Understanding Lab Results

To do these things, how much do I have to know


Course Syllabus:

Fundamentals (ideal cases)

Some applications (more realistic, but not very)

Most real-life issues, ==> kinetics & heat/mass/momentum transfer together

Analytical solutions not possible in many cases

What will be covered? And to what extent?

Compressible , supersonic flowsOnly limited exposure to non-newtonian fluidsComputational Fluid Dynamics (CFD)limited exposure to Perturbation methods...and so on

What will not be covered?


Course Syllabus:

Statics:To refresh the basics

Dynamics:

Mass Balance

Momentum Balance (Linear & Angular)

Energy Balance

Frictional losses Boundary layer theoryFlow past/through


Examples

• Pumps, Turbines• Heat Exchangers, Distillation column • Fluidized or Fixed bed reactors• CVD reactors (micro electronics)• Artificial blood vessels (Bio)


Examples•Production of Sulfuric Acid

• used in fertilizers, car batteries etc2 2S O SO

2 2 32 2CatalystSO O SO

2 3 2 4H O SO H SO


Examples• Monsanto Process

• Pump air (N2+O2) and burn Sulfur• Provide large area of catalyst• “Scrub” with water• Store the sulfuric acid

• For a given production (ton per day),• What is the pump capacity needed?• Design and operation of reactor • How to measure the flow rate?• What if something goes wrong? How to detect it and how to respond? (Detection of leak through chemical sensor, pressure sensor etc)


Goals:Understanding and approaching problems which involve Momentum Transfer

==> Pumps, flow through pipes ==> Separation (filtration, adsorption etc)

More emphasize on application and less on proof

Also prepare for future courses Momentum Transfer Lab Transport phenomena

Calculus (PDE), Complex VariablesLittle bit of programming

Final Exam - 40Quizzes - 2 * 20 = 40 and Project/Assignment -20


Tentative Schedule

Quiz-1

Quiz-2

Section Focus AreaNo.

Classes

1 Statics 12 Conservation of Mass 13 Conservation of momentum - linear 34 Conservation of Energy (no friction) 25 Friction (Shear Stress) & models 46 Navier Stokes eqn 77 Dimensional Analysis 38 Stream Lines 29 Inviscid flow 2

10 Viscous flow & BL theory, Drag on particles 711 Turbulence 212 pipe flow (with friction factors) 413 Fixed bed & Fluidized beds 214 Pumps and Turbines 1


Text Books / References

•Class Notes / Slides•Slides will be on the internal server

•Reference: Transport Phenomena by Bird, Stewart and Lightfoot, edition, McGraw Hill•Fluid Mechanics and its applications by Gupta & Gupta•Other sources referenced will be mentioned in the class

•Text: Fundamentals of Momentum, Heat and Mass Transfer by Welty, Wicks , Wilson & Rorrer (4th edition) John Wiley & Sons


Statics


Statics

• Fluid: changes shape continuously when a tangential force is applied• Pressure at any point in a stationary fluid is same in all directions

•Pressure vs Distance•Consider only gravity effects•ie. Ignore electromagnetic, chemical (eg.osmosis) and other forces

gdz

dp


Statics

Constant Density (eg Liquids)

h

Po = atm

P bottom = Po + g h

zgP Application: Manometer


Statics

Variable Density eg Gases

dzRT

g

p

dp

TRnVP

RT

P

V

n

Height(km)

Temp (C)

10

50

80

0-120 -60

Approx air temp vs height

Fig from “Introduction to Fluid Mechanics” by Fox & McDonald, page 53


Example

Water

Hg

25 cm

10 cmA

BPbm. 2.13

PA-PB=?

Mercury = 13,600 kg/m3

PB’-PB= 1 * g * h1

B’

PA-PB’ = 2 * g * h2 - 1 * g * h2

Actually used for flow rate measurement


Example

Pbm 2.22

atmPP

e

0

atm21136

gdz

dp

zP

P

PP

dzgdpeatm

atm

0

0

z

Practical depth for a suited diver is ~ 180 mWhat is the error in assuming density is constant?

30 1027m

kg


Example

Coin on water: Surface Tension

gh

dWeight

4

2

sindF Y

F

MAXAngleContact Indication of force between liquid-metal vs liquid/liquid

MAXFloatTo :


Statics

•Acceleration due to other forces •eg centrifuge, accelerating vehicle

•In centrifuge, usually g is negligible compared to a•Otherwise use vector algebra to add g and a

adz

dp

Fgdz

dp


Example: Centrifuge

rdr

dp 2

•To separate materials based on density difference• in case gravity is insufficient (for reasonable separation)

•Acceleration expressed as N times “g”•Typically acceleration >> g

•Ignore gravity effects

r

a


Example: Slow rotation

dzgdrrdp 2

h1

For lesser acceleration

At z=h1, r=0, P = PatmOn the surface, P = Patm

2

1

2

1

2

1

2z

z

r

r

P

P

dzgdrrdp

122

12

2

2

12 2ZZgrrPP

g

rhZ

2

2

1

Equation of free surface


Conservation of Mass

•In any control volume•Mass flux in - mass flux out = Mass accumulation rate.•If (mass in) is taken as -ve, then

•Accumulation rate + Flux(out -in) =0

dAnVInOutFluxMasss . Vol

S

V-velocityn-normal vector

Vol

VoldtRateonAccumulatiMass )(


Conservation of Mass

0)(.

Vols

Voldt

dAnV

•Reynold’s Theorem (generalization)•For a property B (Mass, for example)•and corresponding b (per unit mass)

Vols

Voldbt

dAnVbDt

DB)(.

Rate of change (system) = Flux+ Accumulation

See Transport Phenomena, by Bird Stewart Lightfoot for an analogy

dt

dz

z

A

dt

dy

y

A

dt

dx

x

A

t

A

dt

dA


Reynold’s Transport Thm

•B = Mass•==> b =1•DB/Dt =0; Eqn of Conservation of Mass

•B = Momentum•==> b = velocity•Momentum Eqn

•B = Angular Momentum•==> b = r x v (Angular Momentum Eqn)

•etc..


Mass conservation

•Simplifications•Steady State : (gas or liquid)

•d/dt =0 •Mass in = Mass out•For liquids (Volume in = Volume out)

•Constant density & fixed control Volume:•d/dt (V) =0•Volume in = Volume out•True even for unsteady state

0)(.

Vols

Voldt

dAnV


Examples

•Pbm. 4.8, 4.5, 4.12, 4.18, 4.11,4.9 4.15, 4.20, 4.22, 4.21, 4.24


ExamplesPbm. 4.18, steady state

V1 V2

V3d1 = d2 = 2 cmQ1 = 0.0013 m3/sV2 = 2.1 m/sA3 = 100 * ( 1e-3*1e-3/4)There are 100 holes of 1 mm dia in the shower

0332211 VAVAVA


Examples

Pbm. 4.8

V1,a1

V2,a2A1

A2

Area =A, Velocity =V, Acclrn = a. Find V2, A2

V1 (t) A1 = V2 (t) A2

a1A1 = a2A2


ExamplesPbm. 4.5, steady state

0.5 m Long, 0.1 m R6 m/s V m/s

V/2 m/s

0)(.

Vols

Voldt

dAnV

0 Side

sideoutin dAVVAVA

L

xVV out

side 2

dxRdA 2

L

out

Side

side xdxL

RVdAV

0


ExamplesPbm. 4.11

0)(.

Vols

Voldt

dAnV

VwV2

V1

12

X Y

AYAXM 12

Consider stationary control volume

dt

dy

dt

dxVw

2211. AVAVdAnVs

ww

Vol

AVAVVoldt 12)(


ExamplesPbm. 4.11

0)(.

Vols

Voldt

dAnV

VwV2

V1

12

X Y

AYAXM 12

Consider control volume moving @ Vw

dt

dy

dt

dxVw

2211. VVAVVAdAnV ww

s


ExamplesPbm. 4.9, one dimension, steady flow

0)(.

Vols

Voldt

dAnV

ConstVA 0s

VAd

0

A

dA

V

dVd

VA

VAd

A

dA

MaV

dVlawgasideal

balanceenergygasleCompressib

21

1

,

0A

dA

V

dVd


ExamplesPbm. 4.12

71

max 1

R

rVV

R

?AverageV Average

R

VAreardrVFlowRate **20

R

rdrR

rV

0

7

1

max 12R

rxn ;7

1

1

0

11

0

1

0

1

1

1

1

11 dx

n

xx

n

xxdxx

nnn

r


ExamplesPbm. 4.15

X

Y h

Steady flowliquid film thickness is “h”width “into the paper” is W

2

2

0

2

h

y

h

yVVx

V0

h

xdAVQ0

dyWdAdy

h

y

h

yWVQ

h

02

2

0

2


ExamplesPbm. 4.14

Constant Velocity VVarying thickness “b”Infinitely long plate (in one direction)Exit velocity is (a) flat or (b) parabolic

V

2L

b

0)(.

Vols

Voldt

dAnV Mass Flux Accmln rate

bLVolControlinMass 2

Consider unit depth for control Vol

VLdt

dbLMassofchangeofRate 22

InOutFluxMass

0Y direction


ExamplesPbm. 4.14

V

2L

b

b

y

side

y

dyV

dAVFluxMass

0

)(

)(

2

Velocity of outgoing fluid = V(y)

Y direction

For a flat profile, V(y) = constant, say Vavg

Vb

LVavg

For a parabolic profile,

2)( yyV y

2max)()0( ,0 bb VVVV

2

2

max)( 4b

y

b

yVV y


ExamplesPbm. 4.21

d1d2

d1= 2cm, d2=0.8 mm

31cm

gscmQ

36

V

How fast should the plunger move (ie find V)(a) if there is no leakage(b) if leakage between tube and plunger is 10% of needle flow

InOutFluxMass 0

Leakages

g

cm

g

s

cm 616

3

3

VARateonAccumulatiMass 1

0)(.

Vols

Voldt

dAnV

Mass Flux Accmln rate


ExamplesPbm. 4.13

V0

Vx =V0

Vx =V0

d Height=6d

V0

V0

0

Qn: What is the flow rate across the Horizontal surface?

0)(.

Vols

Voldt

dAnV

03

263

0

00

Horizontal

d

OutMassdyd

yVVd

FluxMass

ddepthunitperArea 6""

Momentum Transfer

Documents

Transcript of Momentum Transfer