FLOW AND PRESSURE MEASUREMENT 95
If the duct radius isr, the velocityuyat a distanceyfrom the wall (andsfrom the
centreline) is given by the one-seventh power law as:
uyDus
(y
r
) 1 / 7
(equation 3.59)
whereusis the velocity at the centreline.
The flow, dQ, through an annulus of thickness dy 1 distanceyfrom the axis is:
dQD 2 sdyus
(y
r
) 1 / 7
Multiplying and dividing through byr^2 gives:
dQD 2 r^2 us
s
r
(y
r
) 1 / 7
d
(y
r
)
or, sincesDry: D 2 r^2 us
(
1
y
r
)(y
r
) 1 / 7
d
(y
r
)
The total flow is: QD 2 r^2 us
∫ 1
0
[(
y
r
) 1 / 7
(y
r
) 8 / 7 ]
d
(y
r
)
D 2 r^2 us
[
7
8
(y
r
) 8 / 7
7
15
(y
r
) 15 / 7 ]^1
0
D 0. 817 r^2 us
The average velocity,uavDQ/r^2 D 0. 817 us
Thus: uyDuav, 0. 817 usDusy/r^1 /^7
∴ y/rD 0 .243 ands/rD 0. 757
PROBLEM 6.20
A gas of molecular weight 44 kg/kmol, temperature 373 K and pressure 202.6kN/m^2 is
flowing in a duct. A pitot tube is located at the centre of the duct and is connected to
a differential manometer containing water. If the differential reading is 38.1 mm water,
what is the velocity at the centre of the duct?
The volume occupied by 1 kmol at 273 K and 101.3kN/m^2 is 22.4m^3.
Solution
As shown in section 6.2.5, for a pitot tube:
u^21 / 2 CP 1 vDu^22 / 2 CP 2 v
u 2 D0, and hence,u 1 D
p
2 P 2 P 1 v
Difference in headD 38 .1 mm of water
∴P 2 P 1 D 38. 1 / 1000 ð 1000 ð 9. 81 D 373 .8N/m^2