Page 608 - Bird R.B. Transport phenomena

588 Chapter 19 Equations of Change for Multicomponent Systems

Table 19.2-2 The Combined, Molecular, and Convective Fluxes for
Multicomponent Mixtures (all with the same sign convention)
Combined = Molecular + Convective
Entity flux flux flux

Mass h + pva) a (A)"
(a = 1,2,...,? (Eq. 17.8-1)
Momentum Ф = IT + pvv (B) b
(Eq. 1.7-1)
2
Energy e = q + [IT • v] + pvdi + \v ) (C) c
(Eq. 9.8-5)
" The velocity v appearing in all these expressions is the mass average velocity, defined in
Eq. 17.7-1.
h The molecular momentum flux consists of two parts: тг = p& + т.
L The molecular energy flux is made up of the heat flux vector q and the work flux vector
[тг • v] = pv + [T • v], the latter occurring only in flow systems.

Table 19.2-3 Equations of Change for Multicomponent Mixtures in Terms of
the Molecular Fluxes

Total mass: •v) (A)
(Eq. (A) of Table 3.5-1)

Species mass: U + r. (B) rt
(a = 1, 2, • • •, (Eq. 19.1-7a)
, g -
Momentum: -v -[V-Tl + pg; (O*
P (Eq. (B) of Table 3.5-1)

y
Energy: P g(Lf + ) = -(V- q) - (V • pv) - (V • [T • v]) + (pv • g) (D) b
(Eq. (E) of Table 11.4-1)

" Only N - 1 of these equations are independent, since the sum of the N equations gives
0 = 0.
*" See note (b) of Table 19.2-1 for the modifications needed when the various species are
acted on by different forces.

We conclude this discussion with a few remarks about special forms of the equa-
tions of motion and energy. In §11.3 it was pointed out that the equation of motion as pre-
sented in Chapter 3 is in suitable form for setting up forced-convection problems, but
that an alternate form (Eq. 11.3-2) is desirable for displaying explicitly the buoyant forces
resulting from temperature inequalities in the system. In binary systems with concentra-
tion inequalities as well as temperature inequalities, we write the equation of motion as
in Eq. (B) of Table 3.5-1 and use an approximate equation of state formed by making a
double Taylor expansion of p(T, o) ) about the state T, w :
A A
dp
p(T, o) )= p + — (T-T) + - co ) A
A
T,w A
p - pp(T -T) - - o) A ) (19.2-2)