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§17.7 Mass and Molar Transport by Convection 533

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Experimental data agree more or less with this result, but the exponent on the molecular
weight may be as great as 3 for some polymers.
Although a very general theory for diffusion of polymers has been developed, 6 not
very much has been done with it. So far it has been used to show that, in flowing dilute
solutions of flowing polymers, the diffusivity tensor (see Eq. 17.1-10) becomes anisotropic
and dependent on the velocity gradients. It has also been shown how to generalize the
Maxwell-Stefan equations (see §17.9 and §24.1) for multicomponent polymeric liquids.
Further advances in this subject can be expected through use of molecular simulations. 7

§17.7 MASS AND MOLAR TRANSPORT BY CONVECTION

In §17.1, the discussion of Fick's (first) law of diffusion was given in terms of mass units:
mass concentration, mass flux, and the mass average velocity. In this section we extend the
previous discussion to include molar units. Thus most of this section deals with questions
of notation and definitions. One might reasonably wonder whether or not this dual set of
notation is really necessary. Unfortunately, it really is. When chemical reactions are in-
volved, molar units are usually preferred. When the diffusion equations are solved to-
gether with the equation of motion, mass units are usually preferable. Therefore it is
necessary to acquire familiarity with both. In this section we also introduce the concept of
the convective flux of mass or moles.

Mass and Molar Concentrations

Earlier we defined the mass concentration p as the mass of species a per unit volume of
a
solution. Now we define the molar concentration c a = p /M a as the number of moles of a
a
per unit volume of solution.
Similarly, in addition to the mass fraction <D a = p /p, we will use the mole fraction x a =
a
c /c. Here p = S p is the total mass of all species per unit volume of solution, and с =
a a a
l, c is the total number of moles of all species per unit volume of solution. By the word
a a
"solution" we mean a one-phase gaseous, liquid, or solid mixture. In Table 17.7-1 we
summarize these concentration units and their interrelation for multicomponent systems.
It is necessary to emphasize that p a is the mass concentration of species a in a mix-
ture. We use the notation p (a) for the density of pure species a when the need arises.

Mass Average and Molar Average Velocity
In a diffusing mixture, the various chemical species are moving at different velocities. By v ,
a
the "velocity of species a," we do not mean the velocity of an individual molecule of species
a. Rather, we mean the average of all the velocities of molecules of species a within a small
volume. Then, for a mixture of N species, the local mass average velocity v is defined as
N N
2 РсУа 2 РоУа N
v = — = Ъ— = Z Wo (17.7-1)
L Pa

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P. F. Green, in Diffusion in Polymers (P. Neogi, ed.), Dekker, New York (1996), Chapter 6.
According to T. P. Lodge, Phys. Rev. Letters, 86, 3218-3221 (1999), measurements on undiluted polymers
show that the exponent on the molecular weight should be about 2.3.
6 С F. Curtiss and R. B. Bird, Adv. Polym. Sci., 125,1-101 (1996) and /. Chem. Phys., l l , 10362-10370
I
(1999).
7 D. N. Theodorou, in Diffusion in Polymers (P. Neogi, ed.), Dekker, New York (1996), Chapter 2.