252 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
CAxatyD0 is now a variable
yD1,CAD 0
Hence: AD0andCNADCNAxe
p
p/D
y
and:
dCNA
dy
D
√
p
D
CNAxe
p
p/D
y
and:
(
dCN 1
dy
)
yD 0
D
√
p
D
CNAx
To maintain mass balance at the film boundary:
D
L
CAiCAx DD
(
∂CA
∂y
)
yD 0
Taking Laplace transforms:
D
L
(
CAi
p
CNAx
)
DD
(
∂CNA
∂y
)
yD 0
D
√
pDCNAx
So:
D
L
CAi
p
D
(
D
L
C
√
pD
)
CNAx
and: CNAxD
D
L
CAi
p
D
L
C
p
pD
Hence:
dCNAx
dy
D
√
p
D
D
L
CAi
p
1
D
L
C
p
pD
e
p
p/D
y
DCAi
1
p
p
1 /L
p
D
L
C
p
p
e
p
p/D
y
Inverting:
∂CA
∂y
D
CAi
L
ey/LeDt/L
2
erfc
(p
Dt
L
C
y
2
p
Dt
)
NA (^) yD 0 DD
(
∂CA
∂y
)
yD 0
DCAi
D
L
eDt/L
2
erfc
(p
Dt
L
)
CAi
D
L
eDt/L
2
erfc
(p
Dt
L
)
√
D
t
CAi