Using the Kinetic Theory Method for Molecular Diffusivity

The Kinetic Theory method calculates the molecular diffusivity of an individual component using kinetic theory.

When this method is used, the following properties show up on the mixture component level:

Dipole momentum—a measure of polarity of a covalent bond in the molecule (Debye)
Lennard-Jones characteristic length—the collision diameter (always given in Angstroms)
Lennard-Jones energy—the potential energy of attraction (given in K)
Polarization—polarizability of a molecule (Cubic Angstroms)

These properties are entered as Constant values only. These values are used for calculating the molecular diffusivity of the individual component for the Kinetic Theory method.

To expose this method, select the Material Properties > Molecular Diffusivity node for a multi-component gas or multi-component liquid, then set Method to Kinetic Theory. The Molecular Diffusivity > Kinetic Theory node appears.

The Molecular Diffusivity property on a mixture level is different than other properties – it actually calculates the molecular diffusivities for each individual component.

The molecular diffusivity $D_{m}$ of component $m$ is calculated as:

Figure 1. EQUATION_DISPLAY

D_{m} = \frac{1 - X_{m}}{Σ_{i = 1, i \neq m}^{N} \frac{X_{i}}{D_{i, m}}}

(172)

where the mole fraction,

X_{i}

, is related to the mass fraction,

Y_{i}

, as:

Figure 2. EQUATION_DISPLAY

X_{i} = \frac{Y_{i} M_{w}}{M_{i}}

(173)

$M_{i}$ is the molecular weight of component $i$
$M_{w}$ is the molecular weight of the mixture
$D_{i, m}$ is the binary diffusivity of components $i$ and $m$

From gas kinetic theory, the expression of $D_{i, m}$ is based on Chapman-Enskog [30]:

Figure 3. EQUATION_DISPLAY

D_{i, m} = \frac{2.66 \times 10^{- 7} T^{3 / 2}}{{p M}_{i, m}^{1 / 2} σ_{i, m}^{2} Ω (T *)}

(174)

where:

Figure 4. EQUATION_DISPLAY

M_{i, m} = \frac{2 M_{i} M_{m}}{{M_{i} + M}_{m}}

(175)

and:

$σ_{i, m}$ is the Lennard-Jones characteristic length, the collision diameter for the pair of component $i$ and component $m$ . $σ_{i, m}$ is a function of the Lennard-Jones collision diameter material property, $σ_{i}$ , that is specified for each component $i$ of the mixture. $σ_{i, m}$ is different for polar and non-polar molecules (gases).
$p$ is the (absolute) static pressure
$Ω (T *)$ is a collision integral that is a function of the reduced temperature, $T^{*}$ . $T^{*}$ is defined similarly to the reduced temperature for the viscosity calculation:

Figure 5. EQUATION_DISPLAY

T^{*} = \frac{k T}{ε_{i, m}}

(176)

where:

$k$ is the Boltzmann constant $= 1.3806503 \times 10^{- 23}$ m² kg s^-2 K^-1
$T$ is temperature
$ε_{i, m}$ is the characteristic Lennard-Jones energy for the pair of component $i$ and component $m$ . $ε_{i, m}$ is a function of the Lennard-Jones potential energy material property, $ε_{i}$ , that is specified for each component $i$ of the mixture. $ε_{i, m}$ is different for polar and non-polar molecules (gases).