32 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Solution
The shearing characteristics of non-Newtonianfluids are shown in Fig. 3.24 of Volume 1.
This type offluid remains rigid when the shear stress is less than the yield stressRYand
flows like a Newtonianfluid when the shear stress exceedsRY. Examples of Bingham
plastics are manyfine suspensions and pastes including sewage sludge and toothpaste.
The velocity profile in laminarflow is shown in Fig. 3c.
Velocity distribution
R = o
Rc
Ry
Rw
Pipe wall
yro
r
Plug flow
region
Figure 3c.
A force balance over the pipe assuming no slip at the walls gives:Pr^2 DRw 2 rL,
and
P/LD 2 Rw/rwhereRwDshear stress at the wall.i
A force balance over the annular core wherey>r 0 gives:
Py^2 D 2 yLRy
Hence: RyDyRw/randyDrRy/Rw (ii)
when: RyDRYandr 0 DrRY/Rw (iii)
RyRYDkdux/dy
∴from equation (ii):
dux
dy
D
RyRY
k
D
1
k
(
yRw
r
RY
)
(iv)
Integrating: kuxDy^2 Rw/ 2 rRYyCC
WhenyDr,uxD0,CDrRw/ 2 CRYr.
∴ kuxDRw
(
r
2
y^2
2 r
)
RYry (v)
Substituting foryfrom equation (iii) gives:
ku 0 DRw
[
r
2
(
RY
Rw
)
r
2
]
RY
(
r
RY
Rw
r
)
andku 0 D
r
2 Rw
RwRY^2 (vi)