Question

The pressure difference p of air that flows through a fan is a function of the diameter D of the blade, its angular rotation, density of the air, and the flow Q. Use the FLT method of repeating variables to generate a dimensionless relationship between these parameters

Answer 1

In FLT method of dimension we take F(Force), L(Length) and T (Time ) as standard dimension and find the dimension of other quantities in terms of FLT

Here pressure difference is function of diameter of blade(D), Angular rotation($\theta$), density of air ($\rho$ ) and the flow (Q).

$\Delta p=k (D)^a(\theta)^b(\rho)^c(Q)^d$

Now here equate the dimension of the above quantities in terms of FLT

$\Delta p=FL^{-2},D= L,\rho=FL^{-4}T^2,Q=L^3T^{-1}$

now equate the dimension of all the quantities in above equation

$(FL^{-2})=k (L)^a(F^0L^0T^0)^b(FL^{-4}T^2)^c(L^3T^{-1})^d$

On comparing with the dimensions of the above quantities we can say that

$F^1L^{-2}T^0= L^{a-4c+3d} F^{c}T^{2c-d}$

on comparing the power of dimension we can say that,

c=1,

2 c-d=0

2 c=d

d=2,

a-4c+3d=-2

1-4+6=-2

a=-5

so from here we can say that the relation is as follows

$\Delta p=$ k$D^{-5}\rho Q^2$

$\Delta p=k (\frac{\rho Q^2}{D^5})$