Читать книгу The Mathematics of Fluid Flow Through Porous Media - Myron B. Allen III - Страница 32

The balance laws for single continua extend to multiconstituent continua in a manner that allows for exchanges of mass, momentum, and other conserved quantities among the constituents.

For the differential mass balance, the extension has the following form:

(2.28)

To see how this equation allows for exchanges of mass among constituents, rewrite it as follows:

(2.29)

where

(2.30)

Mathematically, this new form amounts to a trivial reformulation. Physically, it captures the exchange of mass into each constituent from other constituents, at a rate given by the mass exchange rate , having dimension . Mass exchange can occur via several mechanisms:

 Phase changes, such as melting, freezing, evaporation, and condensation;

 Interphase mass transfer, such as dissolution or adsorption;

 Chemical reactions, which transform molecular species into different molecular species.

For multiphase continua, Eq. (2.29) has an equivalent form:

again subject to the constraint (2.30).

It is common to write the multiconstituent mass balance in terms of constituent mass fractions, defined as and having dimension (mass of )(total mass). Doing so yields the following equivalent forms for the mass balance equation for each constituent , all subject to the constraint (2.30):

where is the diffusive flux of constituent . In the last form, we refer to the terms labeled (I), (II), (III), and (IV) as the accumulation, advection, diffusion, and reaction terms, respectively.

The following exercise reassuringly shows that the multiconstituent mass balance reduces to the single‐constituent mass balance if we use the definitions of the mixture density and the barycentric velocity and ignore the distinctions among constituents.

Exercise 2.15 Use the definitions of the multiconstituent density and the barycentric velocity to show that Eq. (2.28) is equivalent to