Page 620 - Bird R.B. Transport phenomena

600 Chapter 19 Equations of Change for Multicomponent Systems

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We now introduce the reference quantities , v , and 2P, used in §3.7 and §11.5, the
0
0
0
reference temperatures T o and T : of §11.5, and the analogous reference mass fractions
cj and o) . Then the dimensionless quantities we will use are
M
A0
x = f y = f z = f t = f (19.5-5)

- T o „ й> л - ш л о
5
5
V ^ ^ = ^ * ^ (19 - " 7)
Here it is understood that v is the mass average velocity of the mixture. It should be
recognized that for some problems other choices of dimensionless variables may be
preferable.
In terms of the dimensionless variables listed above, the equations of change may be
expressed as
Continuity: (V • v) = 0 (19.5-8)

2
Motion: Щ = ±- V v - V* - ^ | (f - T) - ^ f f (&> A - ш ) (19.5-9)
л
z
Dt K e Re л Re X
2
^4- = ^ V V T (19.5-10)
£)^ RePr
2
Continuity of A: ^ = ^ ^ V u A (19.5-11)
The Reynolds, Prandtl, and thermal Grashof numbers have been given in Table 11.5-1.
The other two numbers are new:
Sc = I-£— I = \^-\ = Schmidt number (19.5-12)

6
, = diffusional Grashof number (19.5-13)

The Schmidt number is the ratio of momentum diffusivity to mass diffusivity and repre-
sents the relative ease of molecular momentum and mass transfer. It is analogous to the
Prandtl number, which represents the ratio of the momentum diffusivity to the thermal
diffusivity. The diffusional Grashof number arises because of the buoyant force caused
by the concentration inhomogeneities. The products RePr and ReSc in Eqs. 19.5-10 and 11
are known as Peclet numbers, Pe and Рё , respectively.
лв
The dimensional analysis of mass transfer problems parallels that for heat transfer
problems. We illustrate the technique by three examples: (i) The strong similarity be-
tween Eqs. 19.5-10 and 11 permits the solution of many mass transfer problems by anal-
ogy with previously solved heat transfer problems; such an analogy is used in Example
19.5-1. (ii) Frequently the transfer of mass requires or releases energy, so that the heat
and mass transfer must be considered simultaneously, as is illustrated in Example 19.5-2.
(iii) Sometimes, as in many industrial mixing operations, diffusion plays a subordinate
role in mass transfer and need not be given detailed consideration; this situation is illus-
trated in Example 19.5-3.
We shall see then that, just as for heat transfer, the use of dimensional analysis for
the solution of practical mass transfer problems is an art. This technique is normally
most useful when the effects of at least some of the many dimensionless ratios can be ne-
glected. Estimation of the relative importance of pertinent dimensionless groups nor-
mally requires considerable experience.