CHEMICAL ENGINEERING

28 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

PROBLEM 3.11

Explain the phenomenon of hydraulic jump which occurs during theflow of a liquid in
an open channel.
A liquid discharges from a tank into an open channel under a gate so that the liquid
is initially travelling at a velocity of 1.5 m/s and a depth of 75 mm. Calculate, fromfirst
principles, the corresponding velocity and depth after the jump.

Solution

See Volume 1, Example 3.9.

PROBLEM 3.12

What is a non-Newtonian fluid? Describe the principal types of behaviour exhibited
by thesefluids. The viscosity of a non-Newtonianfluid changes with the rate of shear
according to the approximate relationship:

(^) aDk

(



dux

dr

) 0. 5

where (^) ais the viscosity, and du/dris the velocity gradient normal to the direction of
motion.
Show that the volumetric rate of streamlineflow through a horizontal tube of radiusa
is:

5
a^5

(

P

2 kl

) 2

wherePis the pressure drop over a lengthlof the tube.

Solution

For a power-lawfluid, the apparent viscosity is given by equation 3.123. Noting that the
velocity gradient dux/dris negative then:

(^) aDk

(∣

∣

∣

∣

dux

dy

∣

∣

∣

∣

)n 1

Dk

(



dux

dr

)n 1

Thus, in this problem:n 1 D 0 .5 and:

n 1 D 0. 5 , givingnD 0. 5

Thefluid is therefore shear-thinning.
Equation 3.136 gives the mean velocity in a pipe:

uD

(

P

4 kl

)^1

n

(

n

6 nC 2

)

d

nC 1

n (equation 3.136)