Page 555 - Bird R.B. Transport phenomena

§17.7 Mass and Molar Transport by Convection 535

Table 17.7-2 Notation for Velocities in Multicomponent Systems
Basic definitions:
v ft velocity of species a with respect to fixed coordinates (A)
N
v = 2 CtJ v mass average velocity (B)
a a
N
x v
v * = 2 a a molar average velocity (C)
v - v diffusion velocity of species a with respect to the mass average
a
velocity v (D)
v - v* diffusion velocity of species a with respect to the molar average
a
velocity v* (E)
Additional relations:
v - v* = 2 "afro - v*) (F) v* - v = 2 x (v a - v) (G)
a
a = 1 a = 1

other average velocities are sometimes used, such as the volume average velocity (see
Problem 17СЛ). In Table 17.7-2 we give a summary of the various relations among
these velocities.

Molecular Mass and Molar Fluxes
In §17.1 we defined the molecular mass flux of a as the flow of mass of a through a unit
area per unit time: j — p (v — v). That is, we include only the velocity of species a rela-
a a a
tive to the mass average velocity v. Similarly, we define the molecular molar flux of
species a as the number of moles of a flowing through a unit area per unit time: J* =
c (v A — v*). Here we include only the velocity of species a relative to the molar average
A
velocity v*.
Then in §17.1 we presented Fick's (first) law of diffusion, which describes how the
mass of species A in a binary mixture is transported by means of molecular motions.
This law can also be expressed in molar units. Hence we have the pair of relations for bi-
nary systems:

Mass units: ) = p (v A - v) = -рЯЬ ¥а) А (17.7-3)
АВ
A
A
Molar units: ] A = c (v A - v*) = -c9) Vx A (17.7-4)
A
AB
The differences v A - v and v A - v* are sometimes referred to as diffusion velocities. Equa-
tion 17.7-4 can be derived from Eq. 17.7-3 by using some of the relations in Tables 17.7-1
and 2.
Convective Mass and Molar Fluxes

In addition to transport by molecular motion, mass may also be transported by the bulk
motion of the fluid. In Fig. 9.7-1 we show three mutually perpendicular planes of area dS
at a point P where the fluid mass average velocity is v. The volume rate of flow across the
plane perpendicular to the surface element dS perpendicular to the x-axis is v dS. The
x
rate at which mass of species a is being swept across the same surface element is then
p v dS. We can write similar expressions for the mass flows of species a across the sur-
a x
face elements perpendicular to the y- and z-axes as p v dS and p v dS, respectively. If we
a y
a z