CHEMICAL ENGINEERING

FLOW IN PIPES AND CHANNELS 25

D

Pba^3

(^2) Al

[

(^4) AC (^4) BC (^3) A (^3) B
6 AC (^) B

]

D

Pba^3

(^12) Al

[

(^7) AC (^) B
(^) AC (^) B

]

PROBLEM 3.8

Glycerol is pumped from storage tanks to rail cars through a single 50 mm diameter main
10 m long, which must be used for all grades of glycerol. After the line has been used
for commercial material, how much pure glycerol must be pumped before the issuing
liquid contains not more than 1% of the commercial material? Theflow in the pipeline
is streamline and the two grades of glycerol have identical densities and viscosities.

Solution

Making a force balance over an element distancerfrom the axis of a pipe whose radius
isa, then:
Pr^2 Ddu/dr 2 rl

whereuis the velocity at distancerandlis the length of the pipe.

Hence: duDP/ 2 lrdr

and: uDP/ 4 lr^2 Cconstant

WhenrDa,uD0, the constantDP/ 4 la^2 and henceuDP/ 4 la^2 r^2 .
At a distancerfrom the axis, the time taken for thefluid toflow through a lengthlis
given by 4l^2 /Pa^2 r^2 .
The volumetric rate offlow fromrD0torDris:

D

∫r

0

P/ 4 la^2 r^2  2 rdr

DP/ 8 l 2 a^2 r^2 r^4 

Volumetricflowrate over the whole pipeDPa^4 / 8 l

and the mean velocityDPa^2 / 8 l.

The required condition at the pipe outlet is:

P/ 8 l 2 a^2 r^2 r^4 

Pa^4 / 8 l

D 0. 99

from whichrD 0. 95 a.
The time forfluid at this radius toflow through lengthlD 4 l^2 /Pa^2  1  0. 952 D
 41 l^2 /Pa^2 )

Hence, the volume to be pumpedD 41 l^2 /Pa^2 Pa^4 / 8 l

D 41 a^2 / 8 D 41  0. 25 ^2 / 8 D 0 .10 m^3