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 K_(Rate of change of head)=???