CHEMICAL ENGINEERING

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