CHEMICAL ENGINEERING

220 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

In this problem:sDbtand ebt

(^2) / 2
ft
DconstantDk
∴ ft
Dkebt
(^2) / 2
The total area of surface considered is unity and:

∴

∫ 1

0

ft

dtD 1

∴

∫ 1

0

kebt

(^2) / 2
dtD 1
and by substitution as in equation 10.120:
k/ 2 b
^0.^5 D 1
kD 2 b/
^0.^5 and ft
D 2 b/
^1 /^2 ebt
(^2) / 2
PROBLEM 10.5
By consideration of the appropriate element of a sphere show that the general equation
for molecular diffusion in a stationary medium and in the absence of a chemical reaction
is:
∂CA
∂t

DD

(

∂^2 CA

∂r^2

C

1

r^2

∂^2 CA

∂ˇ^2

C

1

r^2 sin^2 ˇ

∂^2 CA

∂^2

C

2

r

∂CA

∂r

C

cotˇ

r^2

∂CA

∂ˇ

)

whereCAis the concentration of the diffusing substance,Dthe molecular diffusivity,t
the time, andr, ˇ, are spherical polar coordinates,ˇbeing the latitude angle.

Solution

The basic equation for unsteady state mass transfer is:

∂CA

∂t

DD

[(

∂^2 CA

∂x^2

)

yz

C

(

∂^2 CA

∂y^2

)

zx

C

(

∂^2 CA

∂z^2

)

xy

]

(equation 10.67) (i)

This equation may be transformed into other systems of orthogonal coordinates, the most
useful being the spherical polar system. (Carslaw and Jaeger,Conduction of Heat in Solids,
gives details of the transformation.) When the operation is performed:

xDrsinˇcos

yDrsinˇsin

zDrcosˇ

and the equation forCAbecomes:

∂CA

∂t

D

D

r^2

[

∂

∂r

(

r^2

∂CA

∂r

)

C

1

sinˇ

∂

∂ˇ

(

sinˇ

∂CA

∂ˇ

)

C

1

sin^2 ˇ

∂^2 CA

∂^2

]

ii