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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Does Gravity play a role?
Density of the particle or medium?
Min of 6-3 = 3 dimensionless groups
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
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
What if you chose length instead of density? Or velocity?
Similarly, pressure drop in a pipe
Ratio of similar quantities
Many dimensionless numbers in Momentum Transfer are force ratios
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
From “Sharpe Mixers” website