THEOREM PI BUCKINGHAM

When a large number of variables are involved, Raleigh’s method becomes lengthy. In such circumstances, the Buckingham's method is useful. This method expressed the variables related to a dimensional homogenous equation as:
                                             

 

          

where, the dimension at each section is the same.

The Buckingham Phi Theorem can also be expressed in terms of Õ as shown in on the right.

                                     
                                 Π1 = function (Π2, Π3, …, Πn-k)

      
      where, m = the primary dimensions

                          n = dimensional variables such as velocity, discharge and density.

                          k = reduction

                                          

THE STEP-BY-STEP METHOD

QUESTION 1

Find the dimensionless form of the solution for the thrust force, FT of a propeller if it depends upon the fluid density ρ, the diameter D, the rotational speed ω, and the relative fluid velocity, V.

Solution :

STEP 1 :

FT = f (ρ, D, ω, V )

                      

FT = Dependent variable

ρ, D, ω, V = Independent variable

STEP 2 :

STEP 3

                                                                                        

Thus, the value of k is equal to 3. Find n-k to find the number dimensionless π groups needed.

5-3=2 , so we can write f (π1,π2)= 0

STEP 4

                                                                             

We need to choose three repeating variables since m=3. These variables must contain all the m.

               Contain all m

STEP 5

                                                                                                                                                      

                                                                                                                                          

STEP 6                                                                                                               

STEP 7                                                                                                    

                                                                                               

STEP 8

STEP 9 

Example 2 :

Solution : http://youtu.be/zxJ4R9qe3Yc