CHEMICAL ENGINEERING

42 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

or: 

dux

dr

D

(

P

2 k^00 l

) 1 / 2

r^1 /^2 dr

∴ uxD

2

3

(

P

2 k^00 l

) 1 / 2

r^3 /^2 Cconstant.

WhenrDa(at the wall),uxD0 and the constantD

[

2

3

(

P

2 k^00 l

) 1 / 2

a^3 /^2

]

and: uxD

(

P

2 k^00 l

) 1 / 2

2

3

a^3 /^2 r^3 /^2 r

The volumetricflowrateQis:
∫Q

0

dQD

∫a

0

2 ruxdrD 2 

(

P

2 k^00 l

) 1 / 2

2

3

∫a

0

ra^3 /^2 r^5 /^2 dr

D 2 

(

P

2 k^00 l

) 1 / 2

2

3

(

3

14

a^7 /^12

)

D

2 

7

(

P

2 k^00 l

) 1 / 2

a^7 /^2

PROBLEM 3.30

Determine the yield stress of a Binghamfluid of density 2000 kg/m^3 which will justflow
out of an open-ended vertical tube of diameter 300 mm under the influence of its own
weight.

Solution

The shear stress at the pipe wall,R 0 , in a pipe of diameterd, is found by a force balance
as given Volume 1, Section 3.4.1:

R 0 dlDP/ 4 d^2

or: R 0 DPd/ 4 l (equation 3.6)

If thefluid justflows from the vertical tube, then:

P/lDg

and: R 0 Dgd/ 4 D 2000 ð 981 ð 0. 3 / 4 D1472 N/m^2

PROBLEM 3.31

Afluid of density 1200 kg/m^3 flows down an inclined plane at 15°to the horizontal. If
the viscous behaviour is described by the relationship:

RyxDk

(

dux

dy

)n