CHEMICAL ENGINEERING

FLOW AND PRESSURE MEASUREMENT 93

cross-section pipe in which a fluid is in streamline flow, calculate at what point in the
cross-section it should be situated so as to give a direct reading representative of the mean
velocity of flow of the fluid.

Solution

The principle of operation of a pitot tube is discussed in Section 6.3.1. It should be
emphasised that the pitot tube measures the point velocity of a flowing fluid and not the
average velocity so that in order to find the average velocity, a traverse across the duct is
necessary. Treatment of typical results is illustrated in Problem 6.16. The point velocity is
given byuD

p
 2 ghwherehis the difference of head expressed in terms of the flowing
fluid.
For streamline flow, the velocity distribution is discussed in Section 3.3.4 and:

us/uCLD 1 s^2 /r^2  (equation 3.32)

whereusanduCLare the point velocities at a distancesfrom the wall and at the axis
respectively andris the radius of the pipe. The average velocity is:

uavDumax/ 2 (equation 3.36)

WhenusDuavDumax/2,

us/umaxDumax/ 2 /umaxD 1 s^2 /r^2 

and: 0. 5 Ds^2 /r^2 from whichsD 0. 707 r

PROBLEM 6.18

The flowrate of a fluid in a pipe is measured using a pitot tube, which gives a pressure
differential equivalent to 40 mm of water when situated at the centre line of the pipe and
22.5 mm of water when midway between the axis and the wall. Show that these readings
are consistent with streamline flow in the pipe.

Solution

For streamline flow in a pipe, a force balance gives:

Pr^2 D

du

dr

2 rl

and: uD

P

2 l

rand uD

P

2 l

r^2

2

Cconstant.

WhenrDa(at the wall),uD0, the constantDPa^2 / 4 l

and: uD

P

4 l

a^2 r^2