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CHAPTER 4: Dimensionless Analysis - Coggle Diagram
CHAPTER 4: Dimensionless Analysis
Dimensional analysis provides a strategy for choosing relevant data and how it should be presented.
Dimension
properties which can be measured
Unit
the standard elements we use to quantify these dimensions
Length = L , Mass = M , Time = T
Importance of dimensional analysis
To generate nondimensional parameters that help in the design of experiments (physical and/or numerical) and in reporting of results, thus simplify the experiment
To obtain scaling laws so that prototype performance can be predicted from model performance.
To predict trends in the relationship between parameters.
Result will be independent of units, easier to apply to other similar system.
Buckingham's π Theorems
STEPS
List the parameters in the problem and count their total number n.
List the primary dimensions of each of the n parameters
Set the reduction j as the number of primary dimensions. Calculate k, the expected number of Π's. According to Buckingham Π Theorems, the expected number of Π's (k) is equal to n minus j (k = n - j).
Write the final functional relationship and check algebra.
Choose j repeating parameters. (Guideline Repeating parameters)
Construct the k Π's, and manipulate as necessary.
Guidelines for choosing Repeating Parameters
Never pick the dependent variable. Otherwise, it may appear in all the Π's.
Chosen repeating parameters must represent all the primary dimensions.
A combination of the repeating variables must not form a dimensionless group. Otherwise, it would be impossible to generate the rest of the Π's.
Never pick parameters that are already dimensionless.
Never pick two parameters with the same dimensions or with dimensions that differ by only an exponent.
Pick common parameters since they may appear in each of the Π's.
Pick simple parameters over complex parameters.