Myron B. Allen, III - The Mathematics of Fluid Flow Through Porous Media

Figure 2.13 Conceptual plot of REV‐averaged volume fraction versus radius of averaging window, showing how averaged values can stabilize for a range of averaging radii.

If we account for all of the volume in the continuum, then the volume fractions obey the constraint

In this case, we define the true densityof constituent as

The bulk density has dimension (mass of ) (volume of continuum), while the true density has dimension (mass of ) (volume of ).

In the second category of multiconstituent continua, segregation of constituents is observable only at molecular length scales, so continuum‐scale interfaces between the constituents do not exist. Saltwater is an example: The particles of , , and O are segregated at length scales of roughly m, far smaller than the continuum scale. For such multispeciesor misciblemulticonstituent continua, the concept of a continuum‐scale volume fraction does not apply.

With this framework in place, we define several functions associated with the continuum. The mixture densityis

which we can write for multiphase continua as follows:

The mass‐weighted or barycentric velocityis

Sometimes it is useful to refer to the barycentric derivative, which for a differentiable function has the form

(2.26)

Finally, the diffusion velocityof constituent is

(2.27)

Exercise 2.14 Show that

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):