mirror of https://github.com/Nonannet/pyladoc.git
614 lines
26 KiB
TeX
614 lines
26 KiB
TeX
\section{Thermal Conductivity of Mixtures}
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The determination of the thermal conductivity of gas mixtures is a central aspect of modeling
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transport phenomena, particularly in high-temperature and high-pressure processes. Among the
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most established approaches is the empirical equation introduced by Wassiljewa, which was
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subsequently refined by Mason and Saxena to improve its applicability to multicomponent systems.
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This model offers a reliable means of estimating the thermal conductivity of gas mixtures based
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on the properties of the pure components and their molar interactions.
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The thermal conductivity of a gas mixture, denoted by \(\lambda_{\text{mix}}\), can expressed as
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shown in equation \ref{eq:lambda-mixture}.
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\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}
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In this equation, \(x_i\) represents the molar fraction of component \(i\) within the mixture,
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while \(\lambda_i\) denotes the thermal conductivity of the pure substance \(i\). The denominator
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contains the interaction parameter \(\Phi_{ij}\), which describes the influence of component
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\(j\) on the transport properties of component \(i\).
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The interaction parameter \(\Phi_{ij}\) is given by the relation shown in equation \ref{eq:interaction-parameter}.
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\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}
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Here, \(M_i\) and \(M_j\) are the molar masses of the components \(i\) and \(j\), respectively.
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Molar masses and thermal conductivity of the pure substances are listed in table \ref{table:gas-probs}.
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The structure of this expression illustrates the nonlinear dependence of the interaction term on
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both the molar mass ratio and the square root of the conductivity ratio of the involved species.
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\begin{table}
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\centering
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\caption{Properties of some gases}
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\label{table:gas-probs}
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\begin{tabular}{lSS}
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\toprule
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\text{Gas} & \text{Molar mass in g/mol} & \text{Thermal conductivity in W/m/K} \\
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\midrule
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H2 & 2.016 & 0.1805 \\
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O2 & 32.00 & 0.0263 \\
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N2 & 28.02 & 0.0258 \\
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CO2 & 44.01 & 0.0166 \\
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CH4 & 16.04 & 0.0341 \\
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Ar & 39.95 & 0.0177 \\
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He & 4.0026 & 0.1513 \\
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\bottomrule
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\end{tabular}
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\end{table}This formulation acknowledges that the transport properties of a gas mixture are not a simple
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linear combination of the individual conductivities. Rather, they are governed by intermolecular
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interactions, which affect the energy exchange and diffusion behavior of each component. These
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interactions are particularly significant at elevated pressures or in cases where the gas components
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exhibit widely differing molecular masses or transport properties.
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The equation proposed by Wassiljewa and refined by Mason and Saxena assumes that binary interactions
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dominate the behavior of the mixture, while higher-order (three-body or more) interactions are
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neglected. It also presumes that the gases approximate ideal behavior, although in practical
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applications, moderate deviations from ideality are tolerated without significant loss of accuracy.
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In figure \ref{fig:mixture} the resulting thermal conductivity of an H2/CO2-mixture is shown.
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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.
|
|
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\noindent\rule[0.5ex]{\linewidth}{1pt}
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Expanded by more or less sensible AI jabbering; based on: \href{https://doi.org/10.14279/depositonce-7390}{doi:10.14279/depositonce-7390}
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