Question:
A calibration test of a 10 mm circular sharp-edged orifice in the end of a 75 mm pipe showed a discharge of 1.5 liter per second of gasoline (sp. gr. 0.73) when the pressure on the center line of the pipe just upstream from the orifice was 345.50 KPa. The diameter of the jet at the vena contracta was found to be 8 mm. Compute the three orifice coefficients and the headloss on the orifice. Note: determine each coefficient independent from the other coefficients, solve them as equal to the ratio of the actual quantity to the theoretical quantity. No credit points for direct formula substitution solutions.
Answer:
Given
Diameter of orifice (do) = 10mm
Diameter of pipe (D) = 75mm
Diameter of vena contracta (dc) = 8mm
measured discharge or actual discharge (Qact) = 1.5 L/s
Sp. gravity of oil = 0.73
Pressure head (h) = 345.5/(9.81*0.73) = 48.245m
we need to compute three orifice coefficients , that is coefficient of contraction (Cc) , coefficient of velocity (Cv) and coefficient of discharge (Cd)
- Coefficient of contraction (Cc)
Coefficient of contraction is defined as ratio of area of vena contracta to the area of orifice
Area of vena contracta (Ac) = (π/4) dc2 = (π/4) 0.0082 = 0.000050265 m2
Area of orifice (Ao) = (π/4) do2 = (π/4) 0.012 = 0.00007854 m2
So, Coefficient of contraction (Cc) = Ac / Ao
= 0.000050265 /0.00007854
= 0.64
Therefore, Coefficient of contraction (Cc) = 0.64
- Coefficient of discharge (Cd)
Coefficient of discharge is defined as ratio of actual discharge to the theoretical discharge
Actual discharge (Qact) = 1.5 L/s [ given ]
Theoretical discharge (Qth) :
Qth = Area of orifice (Ao) * Velocity (V)
[ h = pressure head = 48.245m and g = acceleration due to gravity = 9.81 m/s2 ]
=3.14/4*0.012*(2*9.81*48.245)0.5
So, Qth = 0.0024164 m3/s = 0.0024164 * 1000 = 2.4164 L/s
So, Coefficient of discharge (Cd) = Qact / Qth
=1.5 /2.4164 = 0.62
Therefore, Coefficient of discharge (Cd) = 0.62
- Coefficient of velocity (Cv)
Coefficient of velocity is defined as ratio of actual velocity to the theoretical velocity
Also, the coefficients are related as :
Coefficient of discharge (Cd) = Coefficient of velocity (Cv) * Coefficient of contraction (Cc)
Cd = Cv * Cc
So, Cv = Cd / Cc
Putting the value of Cd and Cc above, we get the value of Cv as :
Cv = 0.62 / 0.64 =0.97
Therefore, Coefficient of velocity (Cv) =0.97