2.3. DIMENSIONAL ANALYSIS METHODOLOGY
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4. Set up the r independent equations of ^-values either by expressing the pi term as the product of each variable raised
to a power (see Example 2.4) or by direct inspection of the
fundamental dimensions matrix (see Example 2.5).
5. For each pi term, (n—r) k-values must be specified, and the
remaining r values are solved from the set of r independent
equations.
6. Form the required number of dimensionless products. Be
certain that each original variable is included in at least
one pi term, and try to incorporate physical reasoning into
the selection of exponents.
7. Check the resulting pi terms to assure they are dimensionless.
8. Think about the possible relationship between the dimensionless products.
2.3.2 Selection of Exponents
Examples 2.5 and 2.6 illustrate that dimensional analysis can often result in
infinitely many different complete sets of dimensionless products for a given
set of variables (unless you are fortunate enough to have the same number
of equations as there are unknowns). Even though every complete set of
dimensionless products is mathematically correct, dimensional analysis itself does not provide any indication whether some of the products are less
important than others. Because most problems will not provide a determinant system of equations, it is worthwhile to review some rules, thoughts,
and advice from others on how to select ^-values in forming dimensionless
products. Several considerations are listed below.
1. Buckingham pointed out that we gain maximum experimental control when each controllable variable occurs in
only one dimensionless product. Therefore, try to select
exponents to make this happen. If later it is shown that
some variable has a minor impact on the process over the
range being examined, it is possible to discard the variable
if it occurs in only one dimensionless product.
2. The dependent variable should not occur in more than one
dimensionless product.
3. When possible, manipulate the exponents to obtain standard dimensionless products such as the Reynolds number,
39
4. Set up the r independent equations of ^-values either by expressing the pi term as the product of each variable raised
to a power (see Example 2.4) or by direct inspection of the
fundamental dimensions matrix (see Example 2.5).
5. For each pi term, (n—r) k-values must be specified, and the
remaining r values are solved from the set of r independent
equations.
6. Form the required number of dimensionless products. Be
certain that each original variable is included in at least
one pi term, and try to incorporate physical reasoning into
the selection of exponents.
7. Check the resulting pi terms to assure they are dimensionless.
8. Think about the possible relationship between the dimensionless products.
2.3.2 Selection of Exponents
Examples 2.5 and 2.6 illustrate that dimensional analysis can often result in
infinitely many different complete sets of dimensionless products for a given
set of variables (unless you are fortunate enough to have the same number
of equations as there are unknowns). Even though every complete set of
dimensionless products is mathematically correct, dimensional analysis itself does not provide any indication whether some of the products are less
important than others. Because most problems will not provide a determinant system of equations, it is worthwhile to review some rules, thoughts,
and advice from others on how to select ^-values in forming dimensionless
products. Several considerations are listed below.
1. Buckingham pointed out that we gain maximum experimental control when each controllable variable occurs in
only one dimensionless product. Therefore, try to select
exponents to make this happen. If later it is shown that
some variable has a minor impact on the process over the
range being examined, it is possible to discard the variable
if it occurs in only one dimensionless product.
2. The dependent variable should not occur in more than one
dimensionless product.
3. When possible, manipulate the exponents to obtain standard dimensionless products such as the Reynolds number,