\section{Thermal Conductivity of Mixtures}
The determination of the thermal conductivity of gas mixtures is a central aspect of modeling
transport phenomena, particularly in high-temperature and high-pressure processes. Among the
most established approaches is the empirical equation introduced by Wassiljewa, which was
subsequently refined by Mason and Saxena to improve its applicability to multicomponent systems.
This model offers a reliable means of estimating the thermal conductivity of gas mixtures based
on the properties of the pure components and their molar interactions.

The thermal conductivity of a gas mixture, denoted by \(\lambda_{\text{mix}}\), can expressed as
shown in equation \ref{eq:lambda-mixture}.
\begin{equation}\label{eq:lambda-mixture}\lambda_{   ext{mix}} = \sum_{i=1}^{n} \frac{x_i \lambda_i}{\sum_{j=1}^{n} x_j \Phi_{ij}}\end{equation}
In this equation, \(x_i\) represents the molar fraction of component \(i\) within the mixture,
while \(\lambda_i\) denotes the thermal conductivity of the pure substance \(i\). The denominator
contains the interaction parameter \(\Phi_{ij}\), which describes the influence of component
\(j\) on the transport properties of component \(i\).

The interaction parameter \(\Phi_{ij}\) is given by the relation shown in equation \ref{eq:interaction-parameter}.
\begin{equation}\label{eq:interaction-parameter}\Phi_{ij} = \frac{1}{\sqrt{8}} \left(1 + \frac{M_i}{M_j} \right)^{-1/2} \left[ 1 + \left( \frac{\lambda_i}{\lambda_j} \right)^{1/2} \left( \frac{M_j}{M_i} \right)^{1/4} \right]^2\end{equation}
Here, \(M_i\) and \(M_j\) are the molar masses of the components \(i\) and \(j\), respectively.
Molar masses and thermal conductivity of the pure substances are listed in table \ref{table:gas-probs}.
The structure of this expression illustrates the nonlinear dependence of the interaction term on
both the molar mass ratio and the square root of the conductivity ratio of the involved species.

\begin{table}
\centering
\caption{Properties of some gases}
\label{table:gas-probs}
\begin{tabular}{lSS}
\toprule
\text{Gas} & \text{Molar mass in g/mol} & \text{Thermal conductivity in W/m/K} \\
\midrule
H2 & 2.016 & 0.1805 \\
O2 & 32.00 & 0.0263 \\
N2 & 28.02 & 0.0258 \\
CO2 & 44.01 & 0.0166 \\
CH4 & 16.04 & 0.0341 \\
Ar & 39.95 & 0.0177 \\
He & 4.0026 & 0.1513 \\
\bottomrule
\end{tabular}
\end{table}This formulation acknowledges that the transport properties of a gas mixture are not a simple
linear combination of the individual conductivities. Rather, they are governed by intermolecular
interactions, which affect the energy exchange and diffusion behavior of each component. These
interactions are particularly significant at elevated pressures or in cases where the gas components
exhibit widely differing molecular masses or transport properties.

The equation proposed by Wassiljewa and refined by Mason and Saxena assumes that binary interactions
dominate the behavior of the mixture, while higher-order (three-body or more) interactions are
neglected. It also presumes that the gases approximate ideal behavior, although in practical
applications, moderate deviations from ideality are tolerated without significant loss of accuracy.
In figure \ref{fig:mixture} the resulting thermal conductivity of an H2/CO2-mixture is shown.

\begin{figure}
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\caption{Thermal Conductivity of H2/CO2 mixtures}
\label{fig:mixture}
\end{figure}In engineering practice, the accurate determination of \(\lambda_{\text{mix}}\) is essential
for the prediction of heat transfer in systems such as membrane modules, chemical reactors, and
combustion chambers. In the context of membrane-based gas separation, for instance, the thermal
conductivity of the gas mixture influences the local temperature distribution, which in turn affects
both the permeation behavior and the structural stability of the membrane.

It is important to note that the calculated mixture conductivity reflects only the gas phase
behavior. In porous systems such as carbon membranes, additional effects must be considered.
These include the solid-phase thermal conduction through the membrane matrix, radiative transport
in pore channels at high temperatures, and transport in the Knudsen regime for narrow pores.
To account for these complexities, models based on effective medium theory, such as those of
Maxwell-Eucken or Bruggeman, are frequently employed. These models combine the conductivities of
individual phases (gas and solid) with geometrical factors that reflect the morphology of the
porous structure.



\noindent\rule[0.5ex]{\linewidth}{1pt}

Expanded by more or less sensible AI jabbering; based on: \href{https://doi.org/10.14279/depositonce-7390}{doi:10.14279/depositonce-7390}

