CHEMICAL ENGINEERING

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)