- #1
buddy21
- 1
- 0
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)=?