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.