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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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Is a particular variable important?
Need more parameters with tempActivation energy & Boltzmann constant
Does Gravity play a role?Density of the particle or medium?
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
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
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Examples
Impeller
Turbine
No baffles Baffles
Sketch from Treybal