CHEMICAL ENGINEERING

94 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

The maximum velocity,umaxD

Pa^2

4 l

and:

u

umax

D 1 

(r

a

) 2

WhenrDa/2,u/umaxD 0 .75.

The pitot tube is discussed in Section 6.3.1 and:

uDk

p

h (from equation 6.10)

At the centre-line,uDumaxandhD40 mm.

∴ umaxDK

p

40 D 6 .32 K

At a point midway between the axis and the wall,uDu 1 / 2 andhD 22 .5mm.

∴ u 1 / 2 DK

p

22. 5 D 4 .74 K

u 1 / 2 /umaxD 4 .74 K/632 KD 0. 75

and hence the flow is streamline.

PROBLEM 6.19

Derive a relationship between the pressure difference recorded by a pitot tube and the
velocity of flow of an incompressible fluid. A pitot tube is to be situated in a large circular
duct in which fluid is in turbulent flow so that it gives a direct reading of the mean velocity
in the duct. At what radius in the duct should it be located, if the radius of the duct isr?
The point velocity in the duct can be assumed to be proportional to the one-seventh
power of the distance from the wall.

Solution

An energy balance for an incompressible fluid in turbulent flow is given by:

u^2 / 2 CgzCvPCFD 0 (equation 2.55)

Ignoring functional losses and assuming the pitot tube to be horizontal,

u 22 u^21 / 2 DvP 2 P 1 

If the fluid is brought to rest of plane 2, then:

u^21 / 2 DvP 2 P 1 

and: u 1 D

√

2 vP 2 P 1 D

p

2 gh (equation 6.10)