56 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 3.40
Afluid which exhibits non-Newtonian behaviour isflowing in a pipe of diameter 70 mm
and the pressure drop over a 2 m length of pipe is 4ð 104 N/m^2. When theflowrate is
doubled, the pressure drop increases by a factor of 1.5. A pitot tube is used to measure
the velocity profile over the cross-section. Confirm that the information given below is
consistent with the laminarflow of apower-lawfluid.
Any equations used must be derived from the basic relation between shear stressRand
shear rate:
RDk,Pn
radial distance from centre velocity
of pipe (smm) (m/s)
00.80
10 0.77
20 0.62
30 0.27
Solution
For a power-lawfluid:
At the initialflowrate: 4 ð 104 Dkun
With aflow of: 6 ð 104 Dk 2 un
Dividing: 1. 5 D 2 n
and hence: nD 0. 585
For the power-lawfluid:
RDk
∣
∣
∣
∣
dux
dy
∣
∣
∣
∣
n
A force balance on afluid core of radiussin pipe of radiusrgives:
Rs 2 slDPs^2
or: RsDk
(
dux
ds
)n
D
Ps
2 l
dux
ds
D
(
P
2 kl
) 1 /n
s^1 /n
Integrating: uxD
(
P
2 ks
) 1 /n(
n
nC 1
)
s
nC 1
n Cconstant