Question 6.P.24: Water is flowing through a 100 mm diameter pipe and its flow...
Water is flowing through a 100 mm diameter pipe and its flowrate is metered by means of a 50 mm diameter orifice across which the pressure drop is 13.8 kN/m². A second stream, flowing through a 75 mm diameter pipe, is also metered using a 50 mm diameter orifice across which the pressure differential is 150 mm measured on a mercury-underwater manometer. The two streams join and flow through a 150 mm diameter pipe. What would you expect the reading to be on a mercury-under-water manometer connected across a 75 mm diameter orifice plate inserted in this pipe? The coefficients of discharge for all the orifice meters are equal. Density of mercury = 13600 kg/m³.
As in Problem 6.23:
u_2^2=\frac{2\left(P_1-P_2\right) v}{1-\left(A_1 / A_2\right)^2} and Q=C_D A_2 \frac{\sqrt{2\left(P_1-P_2\right) v}}{1-\left(A_2 / A_1\right)^2}
For pipe 1, A_2=(\pi / 4) \times 0.05^2=0.00196 m ^2, A_1=(\pi / 4) \times 0.10^2=0.00785 m ^2, \left(P_1-P_2\right)=13,800 N / m ^2
v=(1 / 1000)=0.001 m ^3 / kg
∴ Q_1=C_D \times 0.00196 \sqrt{\frac{2 \times 13,800 \times 0.001}{1-(0.00196 / 0.00789)^2}}=0.011 C_D
For pipe 2, \sqrt{2\left(P_1-P_2\right) v}=\sqrt{2 g h} (equation 6.10)
A_2=0.00196 m ^2, A_1=(\pi / 4) \times 0.075^2=0.0044 m ^2
Head loss, h = 150 mm Hg-under-water or (150 / 1000) \times[(13,600-1000) / 1000] = 1.89 m water.
∴ Q_2=C_D \times 0.00196 \sqrt{\frac{2 \times 9.81 \times 1.89}{1-(0.00196 / 0.0044)^2}}=0.0133 C_D
Total flow in pipe 3, Q_3=\left(Q_1+Q_2\right)=\left(0.011 C_D+0.0133 C_D\right)=0.0243 C_D
For pipe 3, A_2=(\pi / 4) \times 0.075^2=0.0044 m ^2, A_1=(\pi / 4) \times 0.15^2=0.0176 m ^2
and: Q_3=C_D \times 0.0044 \sqrt{\frac{2 \times 9.81 \times h}{\left(1-(0.0044 / 0.0176)^2\right)}}=0.020 C_D \sqrt{h}
∴ 0.0243 C_D=0.020 C_D \sqrt{h}
and: h = 1.476 m of water
or (1.476 \times 1000) /((13600-1000) / 1000)= \underline{\underline{117 mm \text { of } Hg \text {-under-water }}}.