Page 609 - Bird R.B. Transport phenomena

§19.2 Summary of the Multicomponent Equations of Change 589
Table 19.2-4 The Equations of Energy for Multicomponent Systems, with Gravity as the
1
Only External Force"'"
P§j(U+ Ф + \v ) = -(V-q) - (V-twv]) (A) c
2
2
p^(U + \v ) = -(V-q)-(V-[wv]) + (vpg) (B)
P§~& ) = -(v[V-w]) + (vpg) (C)
2
t
p ^ = -(V-q)-(«:Vv) (D)
P^f = -(?-q)-(T:Vv)+^ (E)

-
| 2 с Н„ + (V • 2 2 N H j = (V • kVT) - (r.Vv) + ^ (НУ
а
O
O
u)
a For multicomponent mixtures q = -kVT + 2 77")« + 4 / where q is a usually negligible term
W
associated with the diffusion-thermo effect (see Eq. 24.2-6).
b
The equations in this table are valid only if the same external force is acting on all species. If this is not
the case, then 2 () • j must be added to Eq. (A) and Eqs. (D-H), the last term in Eq. (B) has to be
g
a
a
replaced by I (n a • g ), and the last term in Eq. (C) has to be replaced by I (v • p g«).
Q
a
a
Q
c Exact only if дФ/dt = 0.
d L. B. Rothfeld, PhD thesis, University of Wisconsin (1961); see also Problem 19D.1.
e (v)
The contribution of q to the heat flux vector has been omitted in this equation.
Here the coefficient £ = -(1/р)(др/дсо ) evaluated at T and a) relates the density to the
A
А
composition. This coefficient is the mass transfer analog of the coefficient /3 introduced
in Eq. 11.3-1. When this approximate equation of state is substituted into the pg term
(but not into the pDv/Dt term) of the equation of motion, we get the Boussinesq equation
of motion for a binary mixture, with gravity as the only external force:
The last two terms in this equation describe the buoyant force resulting from the temper-
ature and composition variations within the fluid.
Next we turn to the equation of energy. Recall that in Table 11.4-1 the energy equation
for pure fluids was given in a variety of forms. The same can be done for mixtures, and a
representative selection of the many possible forms of this equation is given in Table
19.2-4. Note that it is not necessary to add a term S (as we did in Chapter 10) to describe
c
the thermal energy released by homogeneous chemical reactions. This information is in-
cluded implicitly in the functions Я and U, and appears explicitly as -X H R a and
a
a
-l, U R a in Eqs. (F) and (G). Remember that in calculating H and U, the energies of for-
a
a
mation and mixing of the various species must be included (see Example 23.5-1).