CHEMICAL ENGINEERING

48 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

solid surfaces at that radius isrsin 4 whereωis the angular velocity of rotation. The shear
stressRracting on a small element or area drwide will produce a couple 2 rdrRrr
about the axis of rotation. With a uniform velocity gradient at all points in contact with the
cone surface, the surface stressRwill also be uniform, so that the suffix can be omitted
and the total couple about the axis is:

CD

∫r 0

0

2 r^2 RdrD

2

3

r 03 R

The shear stress within thefluid can therefore be evaluated from this equation.

In this problem, 4 D 1 °D 0 .0175 rad andr 0 D 0 .05 m

When the cone speed is 0.1 Hz,ωD 2 ð 0. 1 D 0. 628

Hence the shear rate,ω/sin 4 D 0. 628 / 0. 0175 D36 s^1

The shear stress is given by:RD

3 c

2 7 03

WhencD 4. 6 ð 10 ^2 Nm,RD 3 ð 4. 6 ð 10 ^2 / 2 ð 0. 053 D176 N/m^2

The remaining data may be treated in the same way to give:

Cone speed Shear rate Torque Shear stress

(Hz) (s^1 ) (Nm) (N/m^2 )

0.1 36 0.46 1760

0.5 180 0.70 2670

1 360 1.0 3820

5 1800 3.4 13000

10 3600 6.4 24500

50 18000 30.0 114600

These data may be plotted on linear axes as shown in Fig. 3.24 or on logarithmic axes
as in Fig. 3.26 given here as Figs 3g and 3h, respectively.
It will be seen from Fig. 3g that linear axes produce an excellent straight line with
an intercept of 1500 N/m^2 and this indicates a Bingham plastic type of material whose
characteristics are described by equation 3.122

jRyjRyD (^) p

∣

∣

∣

∣

dux

dy

∣

∣

∣

∣ (equation 3.122)

From Fig. 3g, the slope is (^) pD 6 .4Ns/m^2 and the graph confirms Bingham plastic
behaviour.
PROBLEM 3.35
Tomato puree of density 1300 kg/m ́^3 is pumped through a 50 mm diameter factory
pipeline at aflowrate of 0.00028 m^3 /s. It is suggested that in order to double production:
(a) a similar line with pump should be put in parallel to the existing one, or