alt="alpha"/>) 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
With this framework in place, we define several functions associated with the continuum. The mixture density is
which we can write for multiphase continua as follows:
The mass‐weighted or barycentric velocity is
Sometimes it is useful to refer to the barycentric derivative, which for a differentiable function
(2.26)
Finally, the diffusion velocity of constituent
(2.27)
Exercise 2.14 Show that
2.5.3 Multiconstituent Mass Balance
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:
To see how this equation allows for exchanges of mass among constituents, rewrite it as follows:
where
Mathematically, this new form amounts to a trivial reformulation. Physically, it captures the exchange of mass into each constituent
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
