CHEMICAL ENGINEERING

FLOW IN PIPES AND CHANNELS 53

PROBLEM 3.38

A liquid whose rheology can be represented by the power-law model isflowing under
streamline conditions through a pipe of 5 mm diameter. If the mean velocity offlow is
1 m/s and the velocity at the pipe axis is 1.2 m/s, what is the value of the power law
indexn?
Water, of viscosity 1 mNs/m^2 flowing through the pipe at the same mean velocity gives
rise to a pressure drop of 10^4 N/m^2 compared with 10^5 N/m^2 for the non-Newtonianfluid.
What is the consistency (“k”value) of the non-Newtonianfluid?

Solution

In problem 3.37, the mean velocity,u,isshowntobe:

uD

(

P

2 kl

) 1 /n

a

nC 1

n

n

3 nC 1

and the velocity at any distanceyfrom the pipe axis is:

urD

(

P

2 kl

) 1 /n

n

nC 1

[

a

nC 1

n r

nC 1

n

]

The maximum velocity,umax, will occur whenyD0 and:

umaxD

(

P

2 kl

) 1 /n

n

nC 1

a

nC 1

n

∴

umax

u

D

1. 2

1. 0

D

3 nC 1

nC 1

andnD 0. 111

As shown previously: uD

(

P

2 kl

) 1 /n

a

nC 1

n

n

3 nC 1

WhennD 0 .111 for the non-Newtonianfluid,PD 105 N/m^2 anduD1m/s

∴ 1 D

(

105

2 kl

) 9

a^10 ð 0. 083

WhennD1forwater,PD 104 N/m^2 anduD1m/sandkD.

∴ 1 D

(

104

2 l

)

a^2 ð 0. 25

or: 1 D

(

104

2 l

) 9

a^18 ð 3. 81 ð 10 ^6

∴

(

105

2 kl

) 9

a^10 ð 0. 083 D

(

104

2 l

) 9

a^18 ð 3. 81 ð 10 ^6