Example

Consider pressure drop in a tube of length , hydraulic diameter d, surface roughness Î, with fluid of density ρ and viscosity μ moving with average velocity ν

This can be expressed as

f (DP, U, d, , Î, ρ, μ) = 0

Now m=7 since the phenomenon involves 7 independent parameters.

We select ρ, U, d as repeating variables (so that all 3 dimensions are represented)

Now 4 π (7 - 3) parameters are determined as

Now basic units

All Π parameters ® M⁰ L⁰ T⁰

The above four equations yield

a₁= -1; b₁ = - 2; c₁= 0 
a₂= -1; b₂ = - 1; c₂= -1 
a₃= 0; b₃ = 0; c₃= -1 
a₄= 0; b₄ = 0; c₄= -1

Thus writing                                                  

implies                                                       

Therefore,                                                 

    is called relative roughness

Similarly,   other sets of Π parameters can be chosen to describe the phenomena. Thus though it does not give the actual relationship, but it puts the data in a compact form