CHEMICAL ENGINEERING

52 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

For shear-thinningfluids, (^) a!1at zero shear stress and (^) a!0atinfinite shear
stress. Paint often exhibits shear thinning behaviour as its apparent viscosity is very high
while in the can and when just applied to a wall but its apparent viscosity is very low as
the brush applies it to the surface when itflows readily to give an evenfilm. Toothpaste
remains in its tube and on the brush when not subjected to shear but when sheared, as it
is when the tube is squeezed, itflows readily through the nozzle to the brush.
For afluidflowing in a pipe of radiusa, lengthlwith a central core of radiusr,a
force balance gives:
Pr^2 Dk

[



dur

dr

]n

2 rl

Pr

2 l

Dk

[



dur

dr

]n

or: 

dur

dr

D

[

P

2 kl

] 1 /n

r^1 /n

Integrating: urD

[

P

2 kl

] 1 /n

n

nC 1

r

nC 1

n CC

WhenrDa,urD0andCD

[

P

2 kl

] 1 /n

n

nC 1

a

nC 1

n

∴ urD

[

P

2 kl

] 1 /n

n

nC 1

[

a

nC 1

n r

nC 1

n

]

The mean velocity isugiven by the volumetricflow/area

or: uD

1

a^2

∫Q

0

dQD

1

a^2

∫a

0

2 rdrux

∴ uD

1

a^2

(

P

2 kl

) 1 /n

n

nC 1

2 

∫a

0

[

a

nC 1

n rr

2 nC 1

n

]

dr

∴ uD

(

P

2 kl

) 1 /n

n

3 nC 1

a

nC 1

n

When the mean velocityDaverage velocity, then:
(
P
2 kl

) 1 /n

n

3 nC 1

a

nC 1

n D

(

P

2 kl

) 1 /n

n

nC 1

[

a

nC 1

n r

nC 1

n

]

1 

(r

a

)nC^1

n D nC^1

3 nC 1

or:

r

a

D

(

2 n

3 nC 1

) n

nC 1

WhennD 0 .2 then:

r

a

D

(

0. 4

1. 6

)^0.^2

1. 2

D 0. 794