Miscellaneous notes on mass transfer coefficient models

P.C. Chau (UCSD, 1999)
Pe = Re δ Sc = Uδ
D AB
(17)
• Equation (6.4.9)
This equation is derived from the short contact time approximation at the solid/liquid interface
in a laminar liquid film (Section 5.2). We begin by stating that the local flux in Eq. (5.2.13) is to
be put in terms of the local mass transfer coefficient:
N y (x) = D bC s
Γ(4/3)
α
9D b x
1/3 =kC s (18)
where now the coordinate y is based on the solid surface. The local mass transfer coefficient has to
be defined as
k= D b α 1/3 x
Γ(4/3) 9D –1/3 (19)
b
and the average mass transfer coefficient can be evaluated as
k = 1 L
0
L
k dx
= 3 2
D b
Γ(4/3)
α
9D b 1/3 L –1/3 (20)
We now need to get a handle on α. For film flow, the velocity profile is usually derived with
the coordinate based on the gas/liquid interface, z = δ – y,
u x (z) = 3 2 U 1– z δ
2
(21)
Thus the velocity gradient at the solid surface is
α = du x
=– du x
dy y=0 dz
z=δ
= 3U δ
(22)
We finally are ready to find the Sherwood number as defined in Eq. (15):
Sh δ =
δ 3
D AB 2
= 1.165 Uδ
ν
D b
Γ(4/3)
ν
D b
δ
L
1/3
3U 1
δ 9D b L
(23)
1/3
which is Eq. (6.4.9) with the substitution of the definitions of dimensionless groups.
• Equation (6.4.12)
This equation is derived from the long contact time approximation at the solid/liquid interface
in a laminar liquid film (Section 5.3). The basis is the result in Eq. (5.3.32). If we take the P*
terms to be large, the P* will cancel out and leave us with the factor 1.6.
6