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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. Dimensional Analysis.

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dimensional analysis similarity
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
dimensional analysis
Dimensional Analysis
  • Basic Dimensions:
    • M,L,T (or F,L,T for convenience)
  • Temp, Electric Charge... (for other problems)
dimensional analysis3
Dimensional Analysis
  • Ideal Gases
  • Not dimensionally consistent
  • Can be used only after defining a standard state
  • Empirical Correlations: Watch out for units
    • Write in dimensionally consistent form, if possible
dimensional analysis4
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?
is a particular variable important
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?

design of experiments doe
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)
  • 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
pi theorem
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

Min of 6-3 = 3 dimensionless groups

pi theorem8
Pi Theorem

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

“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

method for finding pi numbers
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
method for finding pi numbers10
Method for finding Pi numbers
  • 4.Find the rank of the matrix
    • =3
  • To find the dimensionless groups
    • Simple examination of the variables
  • 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
method for finding pi numbers11
Method for finding pi numbers

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

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

pi numbers example
Pi numbers: Example

Length

Drag Force

Consider viscosity

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

Similarly, pressure drop in a pipe

physical meaning
Physical Meaning

Ratio of similar quantities

Many dimensionless numbers in Momentum Transfer are force ratios

n s equation
N-S equation

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

Reynolds and Froude numbers in equation

Boundary conditions may yield other numbers, like Weber number, depending on the problem

scaling similarity similitude
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
    • KL = L FULL SCALE/ L MODEL
    • KV = (Velocity) FULL SCALE / (Velocity) MODEL
examples
Examples

No baffles

Baffles

Sketch from

Treybal

Impeller

Turbine

examples17
Examples

From “Sharpe Mixers” website