MASS TRANSFER 221
which may be written as:
∂CA
∂tDD
[
∂^2 CA
∂r^2C
2
r∂CA
∂rC
1
r^2υ
∂ (
1 ^2
∂CA
∂
)
C
1
r^2 1 ^2∂^2 CA
∂^2
]
iiiwhere: Dcosˇ. iv
In this problem∂CA/∂tis given by:∂CA
∂tDD
(
∂^2 CA
∂r^2C
1
r^2∂^2 CA
∂ˇ^2
C
1
r^2 sin^2 ˇ∂^2 CA
∂^2
C
2
r∂CA
∂rC
cotˇ
r^2∂CA
∂ˇ
)
vComparing equations (iii) and (v) is necessary to prove that:1
r^2∂
∂
(
1 ^2
∂CA
∂
)
C
1
r^2 1 ^2∂^2 CA
∂^2
D
1
r^2∂^2 CA
∂ˇ^2
C
1
r^2 sin^2 ˇ∂^2 CA
∂^2
C
cotˇ
r^2∂CA
∂ˇ
Dcosˇ, 1 ^2 D 1 cos^2 ˇDsin^2 ˇ∴
1
r^2 1 ^2∂^2 CA
∂^2
D
1
r^2 sin^2 ˇ∂^2 CA
∂^2
It now becomes necessary to prove that:1
r^2∂^2 CA
∂ˇ^2
C
cotˇ
r^2∂CA
∂ˇ
D
1
r^2∂
∂
(
1 ^2
∂CA
∂
)
viFrom equation (iv): Dcosˇ
∴ ∂ /∂ˇDsinˇvii
and: ∂^2 /∂ˇ^2 Dcosˇviii
1
r^2∂
∂
(
1 ^2
∂CA
∂
)
D
1
r^2∂
∂ˇ
(
1 ^2
∂CA
∂ˇ
∂ˇ
∂
)
∂ˇ
∂
Substituting from equation (iv) for from equation (vii) for∂ˇ/∂ gives:D
1
r^2∂
∂ˇ
(
1 cos^2 ˇ
∂CA
∂ˇ
1
sinˇ)
1
sinˇD
1
r^2∂
∂ˇ
(
sinˇ∂CA
∂ˇ
)
1
sinˇD
1
r^2[
sinˇ∂^2 CA
∂ˇ^2
C
∂CA
∂ˇ
cosˇ
]
1
sinˇD
1
r^2∂^2 CA
∂ˇ^2
C
cotˇ
r^2∂CA
∂ˇ
PROBLEM 10.6
Prove that for equimolecular counter diffusion from a sphere to a surrounding stationary,
infinite medium, the Sherwood number based on the diameter of the sphere is equal to 2.