240 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
constant fractional rate of surface renewal,s, but an upper limit on surface age equal to
the life of the jet, 4 , show that the surface age frequency distribution function,ft
,for
this case is given by:
ft
Dsexpst/[1expst
] for 0 <t<4
ft
D0fort>4.
Hence, show that the enhancement,E, for the increase in value of the liquid-phase mass
transfer coefficient is:
ED[s4
^1 /^2 erfs4
^1 /^2 ]/f2[1exps4
]g
whereEis defined as the ratio of the mass transfer coefficient predicted by conditions
described above to the mass transfer coefficient obtained from the penetration theory for
a jet with an undisturbed surface. Assume that the interfacial concentration of acetone is
practically constant.
Solution
For the penetration theory:
∂CA
∂t
DD
∂^2 CA
∂y^2
(equation 10.66)
As shown in Problem 10.19, this equation can be transformed and solved to give:
CNADAe
p
p/D
yCBe
p
p/D
y
The boundary conditions are:
WhenyD 0 ,CADCAi,BDCAi/p
and whenyD1,CAD0andAD 0
∴ CNAD
CAi
p
e
p
p/D
y
dCNA
dy
DCAi
√
1
D
√
1
p
e
p
p/D
y
From Volume 1, Appendix, Table 12, No 84, the inverse:
dCA
dy
DCAi
√
1
D
√
1
t
ey
(^2) / 4 Dt
At the surface: NA (^) tDD
(
dCA
dy
)
yD 0
DCAi
√
D
t
at timet
The average rate over a time 4 is:
1
4
CAi
√
D
∫ 4
0
dt
p
t
D 2 CAi