The multicomponent diffusion coefficients, thermal conductivities, and thermal diffusion coefficients are computed from the solution of a system of equations defined by what we call the L matrix [48]. It is convenient to refer to the L matrix in terms of its nine block sub-matrices, and in this form the system is given by

(5–58)

where the right-hand side vector is composed of the mole fraction vectors . The multicomponent diffusion coefficients are given in terms of the inverse of the block as

(5–59)

where

(5–60)

The thermal conductivities are given in terms of the solution to the system of equations by

(5–61)

(5–62)

(5–63)

and the thermal diffusion coefficients are given by

(5–64)

The components of the L matrix are given by Dixon-Lewis,[30]

(5–65)

(5–66)

(5–67)

(5–68)

(5–69)

(5–70)

(5–71)

(5–72)

(5–73)

(5–74)

(5–75)

In these equations is the temperature, is the pressure, is the mole fraction of species , are the binary diffusion coefficients, and is the molecular mass of species . Three ratios of collision integrals , , and are defined by Equation 5–55 through Equation 5–57 . The universal gas constant is represented by and the pure species viscosities are given as . The rotational and internal parts of the species molecular heat capacities are represented by and . For a linear molecule

(5–76)

and for a nonlinear molecule

(5–77)

The internal component of heat capacity is computed by subtracting the translational part from the full heat capacity as evaluated from the Ansys Chemkin Thermodynamic Database.

(5–78)

Following Dixon-Lewis,[30] we assume that the relaxation collision numbers depend only on the species , that is, all . The rotational relaxation collision number at 298 K is one of the parameters in the Transport database, and its temperature dependence was given in Equation 5–33 and Equation 5–34 .

For non-polar gases the binary diffusion coefficients for internal energy are approximated by the ordinary binary diffusion coefficients. However, in the case of collisions between polar molecules, where the exchange is energetically resonant, a large correction of the following form is necessary,

(5–79)

(5–80)

when the temperature is in Kelvins.

There are some special cases that require modification of the matrix. First, for mixtures containing monatomic gases, the rows that refer to the monatomic components in the lower block row and the corresponding columns in the last block column must be omitted. This is apparent by noting that the internal part of the heat capacity appears in the denominator of terms in these rows and columns (for example, ). An additional problem arises as a pure species situation is approached, because all , except one, approach zero, and this causes the matrix to become singular. Therefore, for the purposes of forming the matrix, a pure species situation is not allowed to occur. We always retain a residual amount of each species by computing the mole fractions from

(5–81)

A value of works well; it is small enough to be numerically insignificant compared to any mole fraction of interest, yet it is large enough to be represented on nearly any computer.