CHEMICAL ENGINEERING

FLOW IN PIPES AND CHANNELS 57

WhensDr,uxD0 (the no-slip condition):

and hence the constant D

(

P

2 kl

)^1

n

(

n

nC 1

)

r

nC 1

n

Substituting: uxD

(



P

2 kl

) 1 /n(

n

nC 1

)[

r

nC 1

n s

nC 1

n

]

On the centre line: uCLD

(

P

2 kl

) 1 /n(

n

nC 1

)

r

nC 1

n

and hence:

ux

uCL

D 1 

(s

r

)nC^1

n

at whennD 0 .585 : D 1 

(s

r

) 2. 71

The following data are obtained for a pipe radius ofrD35 mm:

experimental

radiuss(mm)

ux

uCL

D 1 

(s

r

) 2. 71

ux(m/s) ux/uCL

0 1 0.80 1

10 0.966 0.77 0.96

20 0.781 0.62 0.77

30 0.341 0.27 0.34

Thus, the calculated and experimental values ofux/uCLagree within reasonable limits of
experimental accuracy.

PROBLEM 3.41

A Bingham-plasticfluid (yield stress 14.35 N/m^2 and plastic viscosity 0.150 Ns/m^2 )is
flowing through a pipe of diameter 40 mm and length 200 m. Starting with the rheological
equation, show that the relation between pressure gradientP/land volumetricflowrate
Qis:

QD

Pr^4

8 lp

[

1 

4

3

XC

1

3

X^4

]

wherelis the pipe radius, (^) pis the plastic viscosity, andXis the ratio of the yield stress
to the shear stress at the pipe wall.
Calculate theflowrate for this pipeline when the pressure drop is 600 kN/m^2 .Itmay
be assumed that theflow is laminar.