2.2. PRINCIPLES OF DIMENSIONAL ANALYSIS
25
2.2 Principles of Dimensional Analysis
One of the first steps in the analysis of a physical phenomenon is to decide which physical variables have an effect on the process being studied.
Once this has been accomplished, theoretical or experimental studies can
be conducted with the goal of establishing the functional relationship between all the important variables. If no theoretical formulation is found,
then meaningful and systematic experiments must be performed in an attempt to obtain information about the relationship between variables. The
set of experiments may involve changing one variable in the experiment
while all other variables are held constant. Obviously, the number of necessary experiments increases rapidly as the number of variables increases.
However, if several process variables can be combined to form a dimensionless variable, then the number of experiments can be significantly reduced.
Dimensional analysis is a rational procedure for combining physical variables into dimensionless products, thereby reducing the number of variables
that need to be considered.
Briefly, dimensional analysis involves these steps:
1. Identify the important independent variables of the process.
2. Decide which variable is to be the dependent variable.
3. Determine how many independent dimensionless products
can be formed from the variables.
4. Reduce the system variables to the proper number of independent dimensionless variables.
Forming the dimensionless products can be accomplished either by inspection, by using the more formal procedure given later in this chapter, or by a
combination of both. After performing the dimensional analysis, theoretical or experimental work is required to establish the functional relationship
between the independent dimensionless variables.
Henry Langhaar, who wrote a very good textbook on dimensional analysis, made the following observation about dimensional analysis.
Dimensional analysis is a method by which we deduce information about a phenomenon from the single premise that the
phenomenon can be described by a dimensionally correct equation among certain variables. The generality of the method is
both its strength and its weakness. With little effort, a partial
solution to nearly any problem is obtained. On the other hand,
a complete solution is not obtained, nor is the inner mechanism
25
2.2 Principles of Dimensional Analysis
One of the first steps in the analysis of a physical phenomenon is to decide which physical variables have an effect on the process being studied.
Once this has been accomplished, theoretical or experimental studies can
be conducted with the goal of establishing the functional relationship between all the important variables. If no theoretical formulation is found,
then meaningful and systematic experiments must be performed in an attempt to obtain information about the relationship between variables. The
set of experiments may involve changing one variable in the experiment
while all other variables are held constant. Obviously, the number of necessary experiments increases rapidly as the number of variables increases.
However, if several process variables can be combined to form a dimensionless variable, then the number of experiments can be significantly reduced.
Dimensional analysis is a rational procedure for combining physical variables into dimensionless products, thereby reducing the number of variables
that need to be considered.
Briefly, dimensional analysis involves these steps:
1. Identify the important independent variables of the process.
2. Decide which variable is to be the dependent variable.
3. Determine how many independent dimensionless products
can be formed from the variables.
4. Reduce the system variables to the proper number of independent dimensionless variables.
Forming the dimensionless products can be accomplished either by inspection, by using the more formal procedure given later in this chapter, or by a
combination of both. After performing the dimensional analysis, theoretical or experimental work is required to establish the functional relationship
between the independent dimensionless variables.
Henry Langhaar, who wrote a very good textbook on dimensional analysis, made the following observation about dimensional analysis.
Dimensional analysis is a method by which we deduce information about a phenomenon from the single premise that the
phenomenon can be described by a dimensionally correct equation among certain variables. The generality of the method is
both its strength and its weakness. With little effort, a partial
solution to nearly any problem is obtained. On the other hand,
a complete solution is not obtained, nor is the inner mechanism