Dimensional Analysis – Similitude (continued)

 

Example 1 (continued)    

 

 

 

5.

 

Use the repeating variables to form the four dimensionless products by inspection.

 

     Pi1  =  L1/L,       Pi12  =  V2/gL,       Pi3  =  ρgL/μ,      Pi4  =  Q2/gL5

 

 

Thus the prediction equation for velocity, V,  in terms of the dimensionless products is:

 

                        V2/gL  =  f( L1/L,  ρgL/μ, Q2/gL5 )     (result for part a)

 

Now for similitude the dimensionless products must be equal for both the model and the prototype.

 

Start with  ρgL/μ.    ρgL/μ | prototype  =  ρgL/μ | model

 

Now water is used for both the model and the prototype.  So  ρ and μ are the same for both.
Also  g  is the same for both.  The conclusion (for similitude)  is that  Lmodel = Lprototype

But that cannot be achieved since the prescribed length scale was given as

Lmodel / Lprototype = 1/13.  This similarity requirement cannot be kept.

 

Next look at  Q2/gL5.     Q2/gL5 | prototype  =  Q2/gL5 | model      Now Lmodel / Lprototype = 1/13

 

So    Q2/L5 | prototype  =  Q2/L5 | model   and   Q model /Qprototype  =    ( L5model/L5prototype)1/2   

 

So   ( L5model/L5prototype)1/2   =  (1/13)5/2 and  therefore  Qmodel =  Q prototype  [(1/13)5/2 ]

 

Recall  Q prototype  =  250 gpm   so  Q model  =  250 [(1/13)5/2 ] = 0.41 gpm   (result)

 

This result is for Froude Number similarity, (V2/gL).

 

Next look at  L1/L.  For similitude    L1/L | model   =  L1/L | prototype 

 

    L1 | model   =  L | model /L | prototype  =  (1/13) (32)  =  2.46 inches  (result)

 

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