CHEMICAL ENGINEERING

26 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

PROBLEM 3.9

A viscousfluidflows through a pipe with slightly porous walls so that there is a leakage
ofkP,wherePis the local pressure measured above the discharge pressure andkis a
constant. After a lengthl, the liquid is discharged into a tank.
If the internal diameter of the pipe isdand the volumetric rate offlow at the inlet is
Qo, show that the pressure drop in the pipe is given byPDQo/kdatanhal,

where: aD 128 k/d^3 ^0.^5

Assume a fully developedflow withR/u^2 D 8 Re^1.

Solution

Across a small element of the pipe, the change in liquidflow is:

dQDkPddl

and the change in velocity is:

duD 4 kPdl/d

Also: R/u^2 D 8 /ud and:RD 8 u/d i

Making a force balance over the element:

dP/ 4 d^2 DRddlD 8 udl

and: dPD 32 udl/d^2 ii

From equations (i) and (ii):

dP/duD 8 u/kPd

and: PdPD 8 udu/kd

Over the whole pipe:

P^2 o/ 2 P^2 / 2 D 8 /kd[u^2 o/ 2 u^2 / 2 ]

and: u^2 Du^2 oCP^2 P^2 okd/ 8 

Assuming zero outlet pressure as a datum, then substituting foruin equation (ii):

∫ 0

Po

dp

/√

[u^2 oCP^2 P^2 okd/ 8 ]D 32 l/d^2

Thus:

√

 8 /kdsinh^1

[

Po

/√

 8 u^2 o/kdP^2 o

]

D 32 l/d^2

and: Po

/√

[8u^2 o/kdP^2 o]Dsinh

√

 128 k/d^3 l