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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

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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**• Basic Dimensions: • M,L,T (or F,L,T for convenience) • Temp, Electric Charge... (for other problems)**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 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?**• Need more parameters with temp • Activation 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 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**• 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 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**• 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 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 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**Length Drag Force Consider viscosity What if you chose length instead of density? Or velocity? Similarly, pressure drop in a pipe**Physical Meaning**Ratio of similar quantities Many dimensionless numbers in Momentum Transfer are force ratios**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)**• 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**No baffles Baffles Sketch from Treybal Impeller Turbine**Examples**From “Sharpe Mixers” website

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