CHEMICAL ENGINEERING

50 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

Solution

The Casson equation is a particular form of equation 3.122 which applies to a number of
foodstuffs as well as tomato puree.

Ry^1 /^2 DRY^1 /^2 C

(

 (^) c
dux
dy

) 1 / 2

or:

(

 (^) c
dux
dy

) 1 / 2

DRy^1 /^2 RY^1 /^2

and:  (^) c
dux
dy
DRYCRy 2 RyRY^1 /^2



dux

dy

D

1

(^) c

[

RyCRY 2 RyRY^1 /^2

]

The Rabinowitch–Mooney equation gives the total volumetricflowrateQthrough the
pipe as:

Q

a^3

D

1

(^) cR^3 w
∫Rw
0
Ry^2 fRydRy (equation 3.149)
whereais the pipe radius andRwis the stress at the wall. Substituting forfRy:
Q
a^3

D

1

(^) cR^3 w
∫Rw
0
R^3 yCR^2 yRYC 2 R^5 y/^2 R
1 / 2
Y dRy

D

1

(^) cR^3 w

[

R^4 y

4

C

R^3 yRY

3

C

4

7

R^7 y/^2 R^1 Y/^2

]Rw

0

D

1

(^) c

[

Rw

4

C

RY

3

C

4

7

R^1 w/^2 R^1 Y/^2

]

In this problem,RYD20 N/m^2 , (^) cD5Ns/m^2 ,QD 2. 8 ð 10 ^4 m^3 /s,aD 0 .025 m
and substituting these values,RwD 030 .24 N/m^2.
From equation 3.138,RwDD/ 4 P/lD 30 .24 N/m^2
and: P/lD 2420 N/m^2 /m
For case (a),the pressure drop will remainunchanged.
For case (b),theflowrateD 2 Qand substituting 2QforQenablesRwto be recalculated
as 98.0N/m^2 and (P/l) to be determined as 7860N/m^2 /m.
For case (c),theflowrateD 2 Qand the pipe diameterDa
p

2. Again recalculation of
   Rwgives a value of 14.52 N/m^2 andP/lD 821 N/m^2 /m.