Transcript No Slide Title
Dimensional Analysis & Similarity
Uses: Verify if eqn is always usable Predict nature of relationship between quantities (like friction, diameter etc) Minimize number of experiments. Concept of DOE Buckingham PI theorem Scale up / down Scale 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)
E
MC
2
E
Energy
Force
Length
Accelerati on
2
pH
log(
C
)
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
C in gram
mole per litre
Dimensional Analysis
Ideal Gases
G
RT
ln(
P
)
C
0 Not dimensionally consistent Can be used only after defining a standard state
G
G
s
RT
ln(
P
)
P
s
Empirical Correlations: Watch out for units Write in dimensionally consistent form, if possible
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Dimensional Analysis
Is there a possibility that the equation exists?
Effect of parameters on drag on a cylinder Choose important parameters viscosity of medium, size of cylinder (dia, length?), density velocity of fluid?
Choose monitoring parameter drag (force) Are these parameters sufficient?
How many experiments are needed?
Pa
s
M
1
L
3
M
1
L
1
T
1
F
M
1
L
1
T
2
D
L
1
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
V
L
1
T
1
Is a particular variable important?
Need more parameters with temp Activation 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 factorial Neglect interaction terms Corner, center models Levels of experiments (example 5)
piece wise linear
(
or quadratic
) models
Change density (and keep everything else constant) and measure velocity. (5 different density levels) Change viscosity to another value Repeat density experiments again change viscosity once more and so on...
5 levels, 4 parameters
5
4
625 experiment
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
s
Limited physical insight
Pi Theorem
Can we reduce the number of experiments and still get the exact same information?
Dimensional analysis / Buckingham Pi Theorm Simple & “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 ,
V
, ,
D
,
Force
, Min of 6-3 = 3 dimensionless groups
M
,
L
,
T
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Pi Theorem
Premise: We can write the equation relating these parameters in dimensionless form
(
1
,
i
is
2
,
3
...., dimensionl
n
) ess
0
“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 numbers Method (Thumb rules) for finding Pi numbers
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Method for finding Pi numbers
1.Decide which factors are important (eg viscosity, density, etc..).
Done 2.Minimum number of dimensions needed for the variables (eg M,L,T) Done 3.Write the dimensional exponent matrix
M
1
L
1
T M
1
L
3
T
0 1
D
M M
0
L
1
T
0
L
1
T
0 0
F V
M
1
L
1
T
2
M
0
L
1
T
1
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
M L T
1 1 1 3 1 0 0 1 0 0 1 0 1 1 1 0 2 1
Method for finding Pi numbers
4.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” variables They 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 variable Normally a length, a velocity and a force variables are included
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Method for finding pi numbers
D
,
V
, Combine the remaining variables, one by one with the following constraint
i
D a V b
c
( variable ) 1
M
0
L
0
T
0 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 equations dependent variable (Drag Force) is in the common variable, ==> an implicit equation
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Pi numbers: Example
D
,
V
, Consider viscosity Length Drag Force 1
DV
2
D
F
2 3 1 2
V F
2 What if you chose length instead of density? Or velocity?
3 1 , 2
F
1 2
V
2
DV
,
D
Similarly, pressure drop in a pipe
P
1 2
V
2
DV
,
D
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Physical Meaning
Ratio of similar quantities Many dimensionless numbers in Momentum Transfer are force ratios Re
DV
V
2
D
DV
2
Inertial Force Viscous Force Fr
Strouhal
D V
V gL
V
2
gL
V
2
D g
D
3 2
Inertial Force Gravity Force Eu
We
V
2
P
DV
2
V
2
D
2
V
2
D D
2
Inertial
Pressure
Force Force Inertial Force
S
urface Tension Force
D V
2
V
2 2
D
4
D
2
Centrifuga l Force Inertial Force Ca
V
2
E s
V
E s
2
D
2
D
2
Inertial Force E
lastic
Force
Inertial Force Compressib ility Force
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
V C
2
Ma
2
N-S equation
DV Dt
P
2
V
g
Use some characteristic length, velocity and pressure to obtain dimensionless groups
L
,
U
,
U
2
x
*
x L V
*
V U t
*
Ut L
V
*
t
*
V
* .
*
V
* *
P
* 1 Re * 2
V
* 1
Fr
g g P
*
P
U
2 *
L
Reynolds and Froude numbers in equation Boundary 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/down Practical reasons (cost, lack of availability of tools with high resolution) Geometric, Kinematic and Dynamic Geometric - length scale Kinematic - velocity scale (length, time) Dynamic - force scale (length, time, mass) Concept of scale factors K L K V = L FULL SCALE / L MODEL = (Velocity) FULL SCALE / (Velocity) MODEL
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Impeller
Examples
No baffles Baffles Sketch from Treybal Turbine
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Examples
From “Sharpe Mixers” website
IIT-Madras, Momentum Transfer: July 2005-Dec 2005