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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.

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Dimensional Analysis & Similarity

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  1. 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

  2. Dimensional Analysis • Basic Dimensions: • M,L,T (or F,L,T for convenience) • Temp, Electric Charge... (for other problems)

  3. 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

  4. 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?

  5. 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?

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. Pi numbers: Example Length Drag Force Consider viscosity What if you chose length instead of density? Or velocity? Similarly, pressure drop in a pipe

  13. Physical Meaning Ratio of similar quantities Many dimensionless numbers in Momentum Transfer are force ratios

  14. 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

  15. 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

  16. Examples No baffles Baffles Sketch from Treybal Impeller Turbine

  17. Examples From “Sharpe Mixers” website

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