CHEMICAL ENGINEERING

MASS TRANSFER 241

In general,CA 06 D0andNAD 2 CAiCA 0

p
D/4for mass transfer without surface
renewal.
Random surface renewal is discussed in Section 10.5.2 where it is shown that the age
distribution function is:

Dconstant estDkest

wheresis the rate of production of fresh surface per unit total area of surface.
If the maximum age of the surface is 4 , then:

K

∫ 4

0

estdtD 1



k

s

[est]^40 D 1

1 es4Ds/kandKD

s

1 es4

∴the age distribution function is:

(

s

1 es4

)

est

The mass transfer in time 4 is:

∫ 4

0

√

D

t

CAiCA 0

s

1 es4

estdtD

√

D

t

CAiCA 0

s

1 es4

∫ 4

0

est

p

t

dt

The integral is conveniently solved by substitutingstDˇ^2 and

p

tDˇ/

p
sorsdtD
2 ˇdˇand dtD 2 ˇdˇ/s

Then:

∫ 4

0

p

s

ˇ

eˇ

2 ˇdˇ

s

D

2

p

s

∫ 4

0

eˇ

2

dˇD

√



s

erf

p

s4DCAiCA 0

p

Ds

erf

p

s4

1 es4

The enhancement factorEis given by:

ED

CAiCA 0

p

Ds

erf

p

s4

1 es4

2

√

D

4

CAiCA 0

D

p

s4erf

p

s4

2  1 es4

PROBLEM 10.22

Solute gas is diffusing into a stationary liquid, virtually free of solvent, and of sufficient
depth for it to be regarded as semi-infinite in extent. In what depth of fluid below the
surface will 90% of the material which has been transferred across the interface have
accumulated in the first minute? Diffusivity of gas in liquidD 10 ^9 m^2 /s.