CHEMICAL ENGINEERING

FLOW IN PIPES AND CHANNELS 51

PROBLEM 3.36

The rheological properties of a particular suspension can be approximated reasonably well
by either a“power law”or a“Bingham plastic”model over the shear rate range of 10
to 50 s^1. If the consistencykis 10 Nsn/m^2 and theflow behaviour indexnis 0.2 in
the power law model, what will be the approximate values of the yield stress and of the
plastic viscosity in the Bingham plastic model?
What will be the pressure drop when the suspension isflowing under laminar condi-
tions in a pipe 200 m long and 40 mm diameter, when the centre line velocity is 1 m/s,
according to the power law model? Calculate the centre line velocity for this pressure
drop for the Bingham plastic model and comment on the result.

Solution

See Volume 1, Example 3.10.

PROBLEM 3.37

Show how, by suitable selection of the indexn, the power-law may be used to describe
the behaviour of both shear-thinning and shear-thickening non-Newtonianfluids over a
limited range of shear rates. What are the main objections to the use of the power law?
Give some examples of different types of shear-thinningfluids.
A power-lawfluid isflowing under laminar conditions through a pipe of circular cross-
section. At what radial position is thefluid velocity equal to the mean velocity in the
pipe? Where does this occur for afluid with ann-value of 0.2?

Solution

Steady state shear-dependent behaviour is discussed in Volume 1, Section 3.7.1.

For a Newtonianfluid, RD

du

dy

(equation 3.4)

For a non-Newtonian power lawfluid, RDk

(

du

dy

)n

(equation 3.119)

Dk

(

du

dy

)n 1 (

du

dy

)

D (^) a
du
dy
where the apparent viscosity (^) aDk

(

du

dy

)n 1

Whenn< 1 , shear-thinning behaviour is represented

n> 1 , shear-thickening behaviour is represented

nD 1 , the behaviour is Newtonian.