Page 46 - Modelling in Transport Phenomena A Conceptual Approach

2.4. TOTAL FLUX 27

When air is pumped through a pipe, it is considered to be a single phase and
a single component system. In this case, there is no ambiguity in defining the
characteristic velocity. However, if the oxygen in the air were reacting, then the
fact that air is composed predominantly of two species, 02 and Na, had to be taken
into account. Hence, air should be considered a single phase, binary component
system. For a single phase system composed of n components, the general definition
of a characteristic velocity is given by
n
(2.3-2)
a
where pi is the weighting factor and vi is the velocity of a constituent. The three
most common characteristic velocities are listed in Table 2.2. The term vi in the
definition of the volume average velocity represents the partial molar volume of a
constituent. The molar average velocity is equal to the volume average velocity
when the total molar concentration, c, is constant. On the other hand, the mass
average velocity is equal to the voIume average velocity when the total mass density,
p, is constant.

Table 2.2 Common characteristic velocities.
Characteristic Velocity Weighting Factor Formulation
Mass average Mass fraction (wi) 21 = xi w;v;
Molar average Mole fraction (xi) v* = xi xivi

Volume average Volume fraction (ciVi) v' = xi ciVivi

The choice of a characteristic velocity is arbitrary. For a given problem, it is
more convenient to select a characteristic velocity which will make the convective
flux zero and thus yield a simpler problem. In the literature, it is common practice
to use the molar average velocity for dilute gases, i.e., c = constant, and the mass
average velocity for liquids, i.e., p = constant.
It should be noted that the molecular mass flux expression given by Eq. (2.1-
6) represents the molecular mass flux with respect to the mass average velocity.
Therefore, in the equation representing the total mass flux, the characteristic ve
locity in the convective mass flux term is taken as the mass average velocity. On
the other hand, Q. (2.1-8) is the molecular molar flux with respect to the mw
lar average velocity. Therefore, the molar average velocity is considered to be the
characteristic velocity in the convective molar flux term.

2.4 TOTAL FLUX

Since the total flux of any quantity is the sum of its molecular and convective
fluxes, then