CHEMICAL ENGINEERING

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

{

1

8



1

6

X

1

4

X^2 C

1

8

X^4 C

1

2

X^3 

1

3

X^4

}