Dimensionless numbers - Principles of Chemical Separations with Environmental Applications - page 274

Appendix A: Dimensionless numbers

A.1 Definition

A group of physical quantities with each quantity raised to a power such that all the units
associated with the physical quantities cancel, i.e., dimensionless.

A.2 Buckingham Pi Theorem

This theorem provides a method to obtain the dimensionless groups which affect a process.
First, it is important to obtain an understanding of the variables that can influence the
process. Once you have this set of variables, you can use the Buckingham Pi Theorem.
The theorem states that the number of dimensionless groups (designated as i ) is equal

to the number ( n )of independent variables minus the number ( m ) of dimensions. Once
you obtain each

, you can then write an expression:

1 = f ( 2 , 3 ,... ) .

This result only gives you the fact that s.

Normally, the exact functional form comes from data correlation or rearrangement of
analytical solutions. Correlating data using the dimensionless numbers formed by this
method typically allows one to obtain graphical plots which are simpler to use and/or
equations which fit to the data. If the dimensionless terms are properly grouped, they
represent ratios of various effects and one can ascertain the relative importance of these
effects for a given set of conditions.
Basic dimensions include length (L), time (t), and mass (M). Variables which are inde-
pendent means that you cannot form one variable from a combination of the other variables

1 can be written as a function of the other