The statement of the theorem is as follows : If a relation among n parameters exists in the form

f(q1, q2, ........ qn) = 0

then the n parameters can be grouped into n – m independent dimensionless ratios or π parameters, expressed in the form

g(π1, π2 ........ πn–m) = 0 (8.3.1) or π1 = g1 (π2, π3 ...... πn–m) Eq. 1

where m is the number of dimensions required to specify the dimensions of all the parameters, q1, q2, .... qn.

It is also possible to form new dimensionless π parameters as a discrete function of the (n – m) parameters.

The limitation of this exercise is that the exact functional relationship in equation 1 cannot be obtained from the analysis. The functional relationship is generally arrived at through the use of experimental results.

Determination of π Groups
Irrespective of the method used the following steps will systematise the procedure.

Step 1. List all the parameters that influence the phenomenon concerned.
This has to be very carefully done. If some parameters are left out, π terms may be formed but experiments then will indicate these as inadequate to describe the phenomenon. If unsure the parameter can be added.

Later experiments will show that the π term with the doubtful parameters as useful or otherwise. Hence a careful choice of the parameters will help in solving the problem with least effort.

Usually three type of parameters may be identified in fluid flow namely fluid properties, geometry and flow parameters like velocity and pressure.

Step 2. Select a set of primary dimensions, (mass, length and time), (force, length and time), (mass, length, time and temperature) are some of the sets used popularly.

Step 3. List the dimensions of all parameters in terms of the chosen set of primary dimensions. Table 8.3.1. Lists the dimensions of various parameters involved.

Step 4. Select from the list of parameters a set of repeating parameters equal to the number of primary dimensions. Some guidelines are necessary for the choice. (i) the chosen set should contain all the dimensions (ii) two parameters with same dimensions should not be chosen. say L, L2, L3, (iii) the dependent parameter to be determined should not be chosen.

Step 5. Set up a dimensional equation with the repeating set and one of the remaining parameters, in turn to obtain n – m such equations, to determine π terms numbering n – m. The form of the equation is,

π1 = qm+1 . q1
a . q2
b . q3
c ..... qm

As the LHS term is dimensionless, an equation for each dimension in terms of a, b, c, d can be obtained. The solution of these set of equations will give the values of a, b, c and d. Thus the π term will be defined.

Step 6. Check whether π terms obtained are dimensionless. This step is essential before proceeding with experiments to determine the functional relationship between the π terms.

Related post

No comments:

Post a Comment