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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 cross-section. 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 mid-points of ten concentric annuli each of the same cross-sectional 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 mercury-under-water manometer. What is the coefficient of discharge of the orifice meter?

Step-by-Step

Verified Solution

Cross-sectional 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(0.3^2-d_1^2\right)

0.00707=(\pi / 4)\left(d^2-d_2^2\right)

0.00707=(\pi / 4)\left(d_2^2-d_3^2\right) and so on,

and the mid-points 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 Hg-under-water

=(0.05 \times(13.55-1) \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.

6D