CHEMICAL ENGINEERING

FLOW IN PIPES AND CHANNELS 23

IfR/u^2 is taken as 0.002, then:
∫
dtD 15 , 740

∫ 6

7

dh

p

 2. 95 h 16. 72 

and tD10590 s

Average volumetric flowrateD 38. 48  7  6 / 10590 D 0 .00364 m^3 /s

Cross-sectional area of pipeD 0 .00442 m^2.

Average velocity in the pipeD 0. 00364 / 0. 00442 D 0 .82 m/s.

Reynolds numberD 1000 ð 0. 82 ð 0. 75 / 10 ^3 D 6. 2 ð 104.

From Fig. 3.7, ifeD 0 .05 mm,e/dD 0 .00067 andR/u^2 D 0 .0025, which is near enough
to the assumed value of 0.002 for a first estimate.
Thus the time for the level to fall is approximately 10590 s(2.94 h).

PROBLEM 3.7

Two immiscible fluidsAandB, of viscosities (^) Aand (^) B, flow under streamline conditions
between two horizontal parallel planes of widthb, situated a distance 2aapart (wherea
is much less thanb), as two distinct parallel layers one above the other, each of deptha.
Show that the volumetric rate of flow ofAis:
(
Pa^3 b
(^12) Al

)(

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

)

where,Pis the pressure drop over a lengthlin the direction of flow.

l

Fluid B

Fluid A

CL

s

RA = Shear stress at centre-plane on A

RB = Shear stress at centre-plane on B

Figure.

Solution

Considering a force balance on thefluid lying within a distance s from the centre plane,
then:

For A: PsbDbl A

(

dus

ds

)

A

CRAl

whereRAis the shear stress at the centre plane,

or: d (^) sD

P

(^) Al
sds

RA

(^) A
ds
Integrating: usAD

P

(^) Al
s^2
2



RA

(^) A
sCk 1
Similarly forB: usBD

P

(^) Al
s^2
2



RB

(^) B
sCk 2
whereRBis the shear stress at the centre plane onB.