CHEMICAL ENGINEERING

MASS TRANSFER 223

Concentration

CAi

CAo

L

y

0

Figure 10a.

ReplacingCAibyC^0 iandCAbyC^0 where:CADC^0 CCAoandCAiDC^0 iCCAo,then
using these new variables:

At: tD 0 C^0 D 00 <y<L

t> 0 C^0 DC^0 i yD 0

t> 0 C^0 D 0 yDL

The problem states that the solution of the one dimensional diffusion equation is:

C^0 Dsteady state solution C

∑^1

0

expn^2 ^2 Dt/L^2 

Ansinny/L

where the steady state solutionDC^0 iC^0 iy/L.
(A derivation of the analogous equation for heat transfer may be found inConduction
of Heat in Solidsby H. S. Carslaw and J. C. Jaeger, Oxford, 1960.)

AnD

2

L

∫L

0

initial concentration profile – steady state sinny/L

dy

D

2

L

∫L

0

[0CC^0 iy/L

C^0 i]sinny/L

dy

D 2 C^0 i/n(this proof is given at the end of this problem).

Hence: C^0 DC^0 iC^0 iy/L

2 C^0 i



∑^1

nD 0

1

n

expn^2 ^2 Dt/L^2 sinny/L

∴ CDCoCCiCo

[

1 

y

L



2



∑^1

nD 0

1

n

expn^2 ^2 Dt/L^2 sinny/L

]

AnD

2

L

∫L

0

[C^0 iy/L

C^0 i]sinny/L

dy