Page 621 - Bird R.B. Transport phenomena

§19.5 Dimensional Analysis of the Equations of Change for Nonreacting Binary Mixtures 601

EXAMPLE 19.5-1 We wish to predict the concentration distribution about a long isothermal cylinder of a
volatile solid Л, immersed in a gaseous stream of a species B, which is insoluble in solid A.
Concentration The system is similar to that pictured in Fig. 11.5-1, except that here we consider the transfer
Distribution about a of mass rather than heat. The vapor pressure of the solid is small compared to the total pres-
Long Cylinder sure in the gas, so that the mass transfer system is virtually isothermal.
Can the results of Example 11.5-1 be used to make the desired prediction?

SOLUTION The results of Example 11.5-1 are applicable if it can be shown that suitably defined dimen-
sionless concentration profiles in the mass transfer system are identical to the temperature
profiles in the heat transfer system:
a> (x, y, z) = f(x, y, z) (19.5-14)
A
This equality will be realized if the differential equations and boundary conditions for the
two systems can be put into identical form.
We therefore begin by choosing the same reference length, velocity, and pressure as in
Example 11.5-1, and an analogous composition function: w = (w - о) )/(со - a) ). Here
A A А0 Ах A0
o) is the mass fraction of A in the gas adjacent to the interface, and o) is the value far from
A0 Ax
the cylinder. We also specify that ш А = со so that OJ = 0. The equations of change needed
А0
A
here are then Eqs. 19.5-8, 9, and 11. Thus the differential equations here and in Problem 11.5-1
are analogous except for the viscous heating term in Eq. 11.5-3.
As for the boundary conditions, we have here:
B.C. 1: as x 2 + у 2 -> °°, v->8 (19.5-15)
x
B.C2: atx- = V = R^1T^^) -Vu. (19.5-16)
+ r 5 /
B.C. 3: at* 2 + f = oo and у = 0, SJ> = 0 (19.5-17)
The boundary condition on v, obtained with the help of Fick's first law, states that there is an
interfacial radial velocity resulting from the sublimation of A.
If we compare the above description with that for heat transfer in Example 11.5-1, we see
that there is no mass transfer counterpart of the viscous dissipation term in the energy equa-
tion and no heat transfer counterpart to the interfacial radial velocity component in the
boundary condition of Eq. 19.5-16. The descriptions are otherwise analogous, however, with
o) , Sc, and Gr taking the places of f, Pr, and Gr.
A w
When the Brinkman number is sufficiently small, viscous dissipation will be unimpor-
tant, and that term in the energy equation can be neglected. Neglecting the Brinkman number
term is appropriate, except for flows of very viscous fluids with large velocity gradients, or in
hypersonic boundary layers (§10.4). Similarly, when (l/ReSc)[(^ - a) )/(l - co )] is very
0
A0
Aoo
small, it may be set equal to zero without introducing appreciable error. If these limiting con-
ditions are met, analogous behavior will be obtained for heat and mass transfer. More pre-
cisely, the dimensionless concentration <b will have the same dependence on x, y, z, t, Re, Pr,
A
and Gr as the dimensionless temperature T will have on x, y, z, t, Re, Pr, and Gr. The concen-
w
tration and temperature profiles will then be identical at a given Re whenever Sc = Pr and
Gr w = Gr.
The thermal Grashof number can, at least in principle, be varied at will be changing T o -
Т . Hence it is likely that the desired Grashof numbers can be obtained. However, it can be
ж
seen from Tables 9.1-1 and 17.1-1 that Schmidt numbers for gases can vary over a considerably
wide range than can the Prandtl numbers. Hence it may be difficult to obtain a satisfactory
thermal model of the mass transfer process, except in a limited range of the Schmidt number.
Another possibly serious obstacle to achieving similar heat and mass transfer behavior is
the possible nonuniformity of the surface temperature. The heat of sublimation must be ob-
tained from the surrounding gas, and this in turn will cause the solid temperature to become
lower than that of the gas. Hence it is necessary to consider both heat and mass transfer si-
multaneously. A very simple analysis of simultaneous heat and mass transfer is discussed in
the next example.