Dimensional Analysis & Similarity

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

Dimensional Analysis & Similarity

Uses:Verify if eqn is always usablePredict nature of relationship between quantities (like friction, diameter etc)

Minimize number of experiments. Concept of DOEBuckingham PI theorem

Scale up / downScale factors


Dimensional Analysis

Basic Dimensions:M,L,T (or F,L,T for convenience)

Temp, Electric Charge... (for other problems)

2MCE

LengthForceEnergyE

2212

TLMLLTM

LengthonAcceleratiMass2212

TLMLLTM

LengthonAcceleratiMass

)log(CpH litrepermolegraminC


Dimensional Analysis

0)ln( CPRTG

Not dimensionally consistentCan be used only after defining a standard state

)ln(s

s P

PRTGG

Ideal Gases

Empirical Correlations: Watch out for unitsWrite in dimensionally consistent form, if possible


Dimensional AnalysisIs there a possibility that the equation exists?Effect of parameters on drag on a cylinder

Choose important parametersviscosity of medium, size of cylinder (dia, length?), densityvelocity of fluid?Choose monitoring parameterdrag (force)

Are these parameters sufficient?How many experiments are needed?

1LD 11 TLV

211 TLMF31 LM

111 TLMsPa


Is a particular variable important?

Need more parameters with tempActivation energy & Boltzmann constant

Does Gravity play a role?Density of the particle or medium?


Design of Experiments (DoE)

How many experiments are needed?DOE:

Full factorial and Half factorialNeglect interaction termsCorner, center modelsLevels of experiments (example 5)

Change density (and keep everything else constant) and measure velocity. (5 different density levels)Change viscosity to another value

Repeat density experiments againchange viscosity once more and so on...

5 levels, 4 parameters

Limited physical insightsexperiment62554

models)( quadraticor

linearwisepiece


Pi TheoremCan we reduce the number of experiments and still get the exact same information?Dimensional analysis / Buckingham Pi TheormSimple & “rough” statement

If there are N number of variables in “J” dimensions, then there are “N-J” dimensionless parameters

Accurate statement:If there are N number of variables in “J” dimensions, then the number of dimensionless parameters is given by (N-rank of dimensional exponents matrix)Normally the rank is = J. Sometimes, it is less

,,,,, ForceDV TLM ,,Min of 6-3 = 3 dimensionless groups


Pi TheoremPremise: We can write the equation relating these parameters in dimensionless form

0)....,,,( 321 nessdimensionlisi

“n” is less than the number of dimensional variables (i.e. Original variables, which have dimensions)==> We can write the drag force relation in a similar way if we know the Pi numbersMethod (Thumb rules) for finding Pi numbers


Method for finding Pi numbers1.Decide which factors are important (eg viscosity, density, etc..).

Done2.Minimum number of dimensions needed for the variables (eg M,L,T)

Done3.Write the dimensional exponent matrix

111 TLM031 TLM

010 TLMD010 TLM211 TLMF110 TLMV

110

211

010

010

031

111

M L T


Method for finding Pi numbers4.Find the rank of the matrix

=3 To find the dimensionless groups

Simple examination of the variables

D

5.Choose J variables (ie 3 variables here) as “common” variablesThey should have all the basic dimensions (M,L,T)They should not (on their own) form a dimensionless number (eg do not choose both D and length)They should not have the dependent variableNormally a length, a velocity and a force variables are included


Method for finding pi numbers

,, VDCombine the remaining variables, one by one with the following constraint

0001)variable( TLMVD cbai

Solve for a,b,c etc (If you have J basic dimensions, you will get J equations with J unknowns)Note: “common” variables form dimensionless groups among themselves ==> inconsistent equationsdependent variable (Drag Force) is in the common variable, ==> an implicit equation


Pi numbers: Example,, VD

Consider viscosity

DV

1

Length

D

2

Drag Force

3 2 2

F

V D

DV

F23

21

What if you chose length instead of density? Or velocity?

213 ,

D

DVDVF

,2

1 2

Similarly, pressure drop in a pipe

D

DVVP

,2

1 2


Physical Meaning

Ratio of similar quantitiesMany dimensionless numbers in Momentum Transfer are force ratios

ForceViscous

ForceInertial

DV

DVDV

22

Re

ForceGravity

ForceInertial

Dg

DV

gL

V

gL

VFr

3

222

Force

ForceInertial

PD

DV

P

VEu

Pressure2

222

ForceTensionurface

ForceInertial

D

DVDVWe

S

222

22

2

222

lasticMa

C

V

ForceilityCompressib

ForceInertial

ForceE

ForceInertial

DE

DV

E

VCa

ss

ForceInertial

ForcelCentrifuga

DV

D

V

D

V

DStrouhal

22

422


N-S equation

Use some characteristic length, velocity and pressure to obtain dimensionless groups

gVPDt

DV 2

g

g

FrVPVV

t

V

1

Re

1. *******

*

*2

2,, UUL L

Utt *

L

xx *

U

VV *

2*

U

PP

L*

Reynolds and Froude numbers in equationBoundary conditions may yield other numbers, like Weber number, depending on the problem


Scaling (Similarity/Similitude)

Scale up/downPractical reasons (cost, lack of availability of tools with high resolution)

Geometric, Kinematic and DynamicGeometric - length scaleKinematic - velocity scale (length, time)Dynamic - force scale (length, time, mass)

Concept of scale factorsKL = L FULL SCALE/ L MODEL

KV = (Velocity) FULL SCALE / (Velocity) MODEL


Examples

Impeller

Turbine

No baffles Baffles

Sketch from Treybal


Examples

From “Sharpe Mixers” website

Documents

Dimensional Analysis & Similarity