58 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Solution
For a Bingham-plastic material, the shear stressRsat radiussis given by:
RsRYDC (^) p
(
dux
ds
)
RsRY
dux
ds
D 0 RsRY
The central unsheared core has radiusrcDrRY/R(whererDpipe radius andRD
wall shear stress) since the shear stress is proportional to the radiuss.
In the annular region:
dux
ds
D
1
(^) p
RRYD
1
(^) p
(
P
s
2 l
RY
)
from a force balance
uxD
∫
duxD
1
(^) p
(
P
s^2
4 l
RYs
)
Cconstant
For the no-slip condition:uxD0, whensDr
Thus: 0 D
1
(^) p
(
P
r^2
4 l
RYs
)
Cconstant
and: usD
1
(^) p
{
P
4 l
r^2 s^2 RYrs
}
Substituting: PD
2 R
l/r
usD
1
(^) p
{
R
2 r
r^2 s^2 RYrs
}
(i)
The volumetricflowrate through elemental annulus, dQADus 2 sds
Thus: QAD
∫r
rc
1
(^) p
{
R
2 r
r^2 s^2 RYrs
}
2 sds
D
2
(^) p
R
[
1
2 r
(
r^2 s^2
2
s^4
4
)
RY
R
(
rs^2
2
s^3
3
)]r
rc
Writing
RY
R
DXandrcDr
RY
R
,then :
QAD
2
(^) p
R
{
1
2 r
(
r^4
2
r^4
4
)
X
(
r^3
2
r^3
3
)
1
2 r
(
X^2 r^4
2
X^4 r^4
4
)
CX
(
r^3 X^2
2
r^3 X^3
3
)}
D
2 R
(^) p
r^3