Page 619 - Bird R.B. Transport phenomena

§19.5 Dimensional Analysis of the Equations of Change for Nonreacting Binary Mixtures 599

during the change of temperature, whereas C pmon is the heat capacity for a gas in which tran-
sitions between quantum states are not allowed, so that C pmon = |R. When the numerical
value A = 1.106 is inserted in Eq. 19.4-41, we get finally

= 0.115 + 0.354 ^ р = 0.469 + 0.354 — - р (19.4-43)
К / \ К )
which is the formula recommended by Hirschfelder. 7 Although the predictions of Eq. 19.4-43
are not much better than those of the older Eucken formula, the above development does at
8 9
least give some feel for the role of the internal degrees of freedom in heat conduction. '

§19.5 DIMENSIONAL ANALYSIS OF THE EQUATIONS OF
CHANGE FOR NONREACTING BINARY MIXTURES
In this section we dimensionally analyze the equations of change summarized in §19.2,
using special cases of the flux expressions of §19.3. The discussion parallels that of §11.5
and serves analogous purposes: to identify the controlling dimensionless parameters of
representative mass transfer problems, and to provide an introduction to the mass trans-
fer correlations of Chapter 22.
Once again we restrict the discussion primarily to systems of constant physical
properties. The equation of continuity for the mixture then takes the familiar form
Continuity: (V • v) = 0 (19.5-1)
The equation of motion may be approximated in the manner of Boussinesq (see §11.3) by
putting Eqs. 19.3-2 and 19-5.1 into Eq. 19.2-3, and replacing -Vp + pg by -V9>. For a
constant-viscosity Newtonian fluid this gives

2
-
Motion: p =^ = uV v — V& — pg/3(T — T) — ])g£((o A — CJ A) (19.5-2)
The energy equation, in the absence of chemical reactions, viscous dissipation, and exter-
nal forces other than gravity, is obtained from Eq. (F) of Table 19.2-4, with Eq. 19.3-3. In
using the latter we further neglect the diffusional transport of energy relative to the mass
average velocity. For constant thermal conductivity this leads to
2
Energy: Щ = aV T (19.5-3)
in which a = k/pC is the thermal diffusivity. For nonreacting binary mixtures with con-
p
stant p and %b Eq. 19.1-14 takes the form
AB/
2
Continuity of A: - ^ = <3) V a) (19.5-4)
AB A
For the assumptions that have been made, the analogy between Eqs. 19.5-3 and 4 is clear.

7
J. O. Hirschfelder, /. Chem. Phys., 26, 274-281 (1957); see also D. Secrest and J. O. Hirschfelder,
Physics of Fluids, 4, 61-73 (1961) for further development of the theory, in which equilibrium among the
various quantum states is not assumed.
8
For a comparison of the two formulas with experimental data, see Reid, Prausnitz, and Poling,
op. cit., p. 497. The Hirschfelder formula in Eq. 19.4-42 and the Eucken formula of Eq. 9.3-15 tend to
bracket the observed conductivity values.
J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases, North Holland,
9
Amsterdam (1977), §§11.2 and 3.