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Dimensionless anaylsis

  1. Aug 5, 2010 #1
    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)=???
     
  2. jcsd
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