Multi-Component Mixture Material Properties

For example, mixture property ${\varphi }_{mix}$ is:

where $Y_i$ and ${\varphi }_i$ are the mass fraction and property values of mixture component $i$ and $N$ is the total number of components in the mixture.

For a mixture containing N components, this method requires the specification of $\left(N\left(N-1\right)\right)/2$ diffusion coefficients.

This specification enforces the fact that self-diffusion is zero and the binary diffusion coefficients are symmetric. An array profile specifies these binary diffusion coefficients with all the standard methods available for specifying the individual profiles. The diffusivity of each component is calculated as:

The Schmidt number is a non-dimensional parameter that is defined as:

where $\nu$ is the kinematic viscosity.

From gas kinetic theory, the expression for the binary diffusion coefficient $D_{i,j}$ is based on Chapman-Enskog [30]:

and:

where:

• $M_i$ , $M_j$ are the molecular weights of component $i$ and $j$ , respectively.
• ${\sigma }_{i,j}$ is the Lennard-Jones characteristic length, the collision diameter for the pair of component $i$ and component $j$ . ${\sigma }_{i,j}$ is a function of the Lennard-Jones collision diameter material property, ${\sigma }_i$ , that is specified for each component $i$ of the mixture. ${\sigma }_{i,j}$ is different for polar and non-polar molecules (gases).
• $p$ is the (absolute) static pressure
• $\Omega \left(T\text{*}\right)$ 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:

where:

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

The Lewis number is expressed as follows:

where $\rho$ , $C_p$ , and $k$ are the density, specific heat, and thermal conductivity of the mixture, respectively.

Given values for $L_e$ , $D_m$ , $\rho$ , and $C_p$ , thermal conductivity $k$ can be calculated as follows:

The mixture method calculates the critical temperature $T_{c,m}$ of a mixture as:

where $W_i$ is the molecular weight, $Y_i$ the mass fraction and $T_{c,i}$ the critical temperature of species i. The summation is over all species in the mixture. The critical pressure $P_{c,m}$ of a mixture is similarly calculated as:

When the mixture method is used for surface tension, the mixture surface tension is:

where ${\sigma }_i$ is the surface tension of component i, $X_i$ is its mole fraction, and r is the exponent property of the mixing law.

When the mixture method is used for molecular weight, the mixing law is:

where $W_i$ is the molecular weight of species i, $Y_i$ is the mass fraction of species i, and the summation is over all species in the mixture.

The Mathur-Saxena averaging methods for dynamic viscosity and thermal conductivity are specifications of the multi-component Mathur-Saxena Averaging Property method < $\Phi$ > where $\Phi$ is a material property. This method uses the following formula to compute the property on the mixture level, using values that are given for the individual mixture components ( $X_i$ is the mole fraction of component $i$ ):