Dimensionless Analysis: Homework on Determining Scale Factors

[buddy21]

Homework Statement

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: I am using SI units

Homework 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

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

K_(Rate of change of head)=?

 

[Valkarie]

From the given condition, K_V=1/10 K_∆p=(ρ_water/ρ_fluid)*(V_water/V_fluid)^2=(1000/800)*(10)^2=3.2 K_time=t_fluid/t_Water=(V_fluid/V_Water)^2=(1/10)^2=1/100=1/6.4 K_(Rate of change of head)=K_∆p*K_time=3.2*1/6.4=1.6