# Dimensionless anaylsis

1. Aug 5, 2010

### buddy21

1. The problem statement, all variables and given/known data
The volume flow rate of a fluid V ̇ through a circular orifice in the base of a tank depends upon the orifice diameter d, the tank diameter D, the pressure across the orifice delta ∆p, the fluid density and the fluid viscosity. Show by D.A. that:

∴((∆p*d^4)/(V ̇^2*ρ))= ∅(((ρ*V ̇)/(μ*d)),(D/d) )
NOTE: V ̇ is volume flow rate

A fluid having a relative density of 0.8 and viscosity twice that of water, flows through a circular orifice in the base of a circular tank. In order to predict the time taken to empty the tank, tests are carried out on a ¼ scale model using water. Determine the scale factors for V, ∆p, the time, and the rate of change of head in the tank for dynamic similarity. (Ans. 1:10, 3.2:1, 1:6.4, 1.6:1).

NOTE: Im using SI units

2. Relevant equations

Variables:

|(∆p*d^4)/(V ̇^2*ρ)|=((ML^(-1) T^(-2) )*(L^4 ))/((L^3 T^(-1) )*(ML^(-3) ) ) = 1

|ρ|=ML^(-3)

|V ̇ |=L^3 T^(-1)

|μ|=ML^(-1) T^(-1)

|d|=L

|D|=L

3. The attempt at a solution

Dimsionless equations

pi_1=((∆p*d^4)/(V ̇^2*ρ))
pi_2=(μ*d)/(ρ*V ̇ )
pi_3=D/d

Scale Factors

K_V=V_fluid/V_Water

K_∆p=∆p_fluid/∆p_Water

K_time=t_fluid/t_Water