Question 6.P.5: A venturi meter with a 50 mm throat is used to measure a flo...
A venturi meter with a 50 mm throat is used to measure a flow of slightly salt water in a pipe of inside diameter 100 mm. The meter is checked by adding 20 cm³/s of normal sodium chloride solution above the meter and analysing a sample of water downstream from the meter. Before addition of the salt, 1000 cm³ of water requires 10 cm³ of 0.1 M silver nitrate solution in a titration. 1000 cm³ of the downstream sample required 23.5 cm³ of 0.1 M silver nitrate. If a mercury-under-water manometer connected to the meter gives a reading of 221 mm, what is the discharge coefficient of the meter? Assume that the density of the liquid is not appreciably affected by the salt.
If the flow of the solution is x m³/s, then a mass balance in terms of sodium chloride gives:
(x \times 0.0585)+\left(20 \times 10^{-6} \times 58.5\right)=0.1375\left(20 \times 10^{-6}+x\right)
and: x=0.0148 m ^3 / s
or, assuming the density of the solution is 1000 kg/m³, the mass flowrate is:
(0.0148 \times 1000)=14.8 kg / s
For the venturi meter, the area of the throat is given by:
A_1=(\pi / 4)(50 / 1000)^2=0.00196 m ^2
and the area of the pipe is:
A_2=(\pi / 4)(100 / 1000)^2=0.00785 m ^2
From equations 6.32 and 6.33:
G=C_D \rho \frac{A_1 A_2}{\sqrt{\left(A_1^2-A_1^2\right)}} \sqrt{\left(2 v\left(P_1-P_2\right)\right)} (equation 6.32)
G=C_D \rho C^{\prime} \sqrt{ }\left(2 g h_v\right) (equation 6.33)
C^{\prime}=A_1 A_2 / \sqrt{\left(A_1^2-A_2^2\right)}=0.00204 m ^2
h = 221 mm Hg _{\text {under-water }} =0.221(13500-1000) / 1000=2.78 m water
and hence: 14.8=\left(C_D \times 1000 \times 0.00204\right) \sqrt{(2 \times 9.81 \times 2.78)}
and: C_D=\underline{\underline{0.982}}