Chapter 6
Q. 6.P.16
The flowrate of air at 298 K in a 0.3 m diameter duct is measured with a pitot tube which is used to traverse the crosssection. Readings of the differential pressure recorded on a water manometer are taken with the pitot tube at ten different positions in the crosssection.
These positions are so chosen as to be the midpoints of ten concentric annuli each of the same crosssectional area. The readings are:
Position  1  2  3  4  5 
Manometer reading (mm water)  18.5  18.0  17.5  16.8  15.7 
Position  6  7  8  9  10 
Manometer reading (mm water)  14.7  13.7  12.7  11.4  10.2 
The flow is also metered using a 150 mm orifice plate across which the pressure differential is 50 mm on a mercuryunderwater manometer. What is the coefficient of discharge of the orifice meter?
StepbyStep
Verified Solution
Crosssectional area of duct =(\pi / 4)(0.3)^2=0.0707 m ^2.
Area of each concentric annulus =0.00707 m ^2.
If the diameters of the annuli are designated d_1, d_2 etc., then:
0.00707=(\pi / 4)\left(0.3^2d_1^2\right)
0.00707=(\pi / 4)\left(d^2d_2^2\right)
0.00707=(\pi / 4)\left(d_2^2d_3^2\right) and so on,
and the midpoints of each annulus may be calculated across the duct.
For a pitot tube, the velocity may be calculated from the head h as u=\sqrt{(2 g h)}
For position 1, h = 18.5 mm of water.
The density of the air =(29 / 22.4)(273 / 298)=1.186 kg / m ^3
h=\left(18.5 \times 10^{3} \times 1000 / 1.186\right)=15.6 m of air
and: u=\sqrt{(2 \times 9.81 \times 15.6)}=17.49 m/s
In the same way, the velocity distribution across the tube may be found as shown in the following table.
Mass flowrate, G=(1.107 \times 1.186)=1.313 kg/s
For the orifice, \left[1\left(A_0 / A_1\right)^2\right]=\left[1(0.15 / 0.3)^2\right]=0.938
h = 50 mm Hgunderwater
=(0.05 \times(13.551) \times 1000 / 1.186)=529 m of air
and: 1.313=C_D(\pi / 4)(0.15)^2 \times 1.186 \sqrt{ }(2 \times 9.81 \times 529 / 0.938) and C_D=\underline{\underline{0.61}}
Position 
Distance from axis of duct (mm) 
Manometer reading 
Air velocity (u m/s) 
Velocity × area of annulus (m³/s) 

Water (mm) 
Air (m) 

1  24  18.5  15.6  17.5  0.124 
2  57  18.0  15.17  17.3  0.122 
3  75  17.5  14.75  17.0  0.120 
4  89  16.8  14.16  16.7  0.118 
5  101  15.7  13.23  16.1  0.114 
6  111  14.7  12.39  15.6  0.110 
7  121  13.7  11.55  15.1  0.107 
8  130  12.7  10.71  14.5  0.103 
9  139  11.4  9.61  13.7  0.097 
10  147  10.2  8.60  13.0  0.092 
Total = 1.107 
The velocity profile across the duct is plotted in Fig. 6d.