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\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.
|
||||
|
||||
|
||||
|
||||
\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}
|
||||
|
||||
|
|
@ -43,13 +43,17 @@
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|||
text-align: center;
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}
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}
|
||||
|
||||
.equation-container {
|
||||
display: flex;
|
||||
justify-content: space-between;
|
||||
|
|
@ -192,9 +196,9 @@
|
|||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
<hr>
|
||||
<hr></hr>
|
||||
<h1>Equations</h1>
|
||||
<p>This line represents a reference to the equation 1.</p><div class="equation-container"><div class="equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="102.120397pt" height="34.4322pt" viewBox="0 0 102.120397 34.4322" xmlns="http://www.w3.org/2000/svg" version="1.1">
|
||||
<p>This line represents a reference to the equation 1.</p><span class="inline-equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="102.120397pt" height="34.4322pt" viewBox="0 0 102.120397 34.4322" xmlns="http://www.w3.org/2000/svg" version="1.1">
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<defs>
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<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
|
||||
|
|
@ -470,7 +474,7 @@ z
|
|||
</g>
|
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</g>
|
||||
</svg>
|
||||
</div><div class="equation-number">(1)</div></div><div class="figure"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="460.8pt" height="345.6pt" viewBox="0 0 460.8 345.6" xmlns="http://www.w3.org/2000/svg" version="1.1">
|
||||
</span><div class="figure"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="460.8pt" height="345.6pt" viewBox="0 0 460.8 345.6" xmlns="http://www.w3.org/2000/svg" version="1.1">
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|
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<defs>
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<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
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|
|
@ -499,7 +503,7 @@ L 142.181053 307.584
|
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L 142.181053 206.208
|
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L 73.832727 206.208
|
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z
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" clip-path="url(#p69260dc8d6)" style="fill: #d62728"/>
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" clip-path="url(#p8886bfdb0f)" style="fill: #d62728"/>
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</g>
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<g id="patch_4">
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<path d="M 159.268134 307.584
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|
|
@ -507,7 +511,7 @@ L 227.616459 307.584
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L 227.616459 54.144
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L 159.268134 54.144
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z
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" clip-path="url(#p69260dc8d6)" style="fill: #1f77b4"/>
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" clip-path="url(#p8886bfdb0f)" style="fill: #1f77b4"/>
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</g>
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<g id="patch_5">
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<path d="M 244.703541 307.584
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@ -515,7 +519,7 @@ L 313.051866 307.584
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L 313.051866 231.552
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L 244.703541 231.552
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z
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" clip-path="url(#p69260dc8d6)" style="fill: #d62728"/>
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" clip-path="url(#p8886bfdb0f)" style="fill: #d62728"/>
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</g>
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<g id="patch_6">
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<path d="M 330.138947 307.584
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@ -523,18 +527,18 @@ L 398.487273 307.584
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L 398.487273 168.192
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L 330.138947 168.192
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z
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" clip-path="url(#p69260dc8d6)" style="fill: #ff7f0e"/>
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" clip-path="url(#p8886bfdb0f)" style="fill: #ff7f0e"/>
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L 0 3.5
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<use xlink:href="#m5bc4660a75" x="108.00689" y="307.584" style="stroke: #000000; stroke-width: 0.8"/>
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<use xlink:href="#m0a989fc6c9" x="193.442297" y="307.584" style="stroke: #000000; stroke-width: 0.8"/>
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<use xlink:href="#m5bc4660a75" x="193.442297" y="307.584" style="stroke: #000000; stroke-width: 0.8"/>
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<g id="line2d_4">
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<g id="text_4">
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<g id="ytick_1">
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<g id="line2d_5">
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<path id="mb5327dd54d" d="M 0 0
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L -3.5 0
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<g id="text_9">
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<g id="ytick_6">
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<g id="line2d_10">
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<g>
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<g id="text_10">
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Binary file not shown.
Binary file not shown.
|
|
@ -2,4 +2,4 @@
|
|||
<p>Umlaute: ÖÄÜ öäü</p>
|
||||
<p>Other: ß, €, @, $, %, ~, µ</p>
|
||||
<p>Units: m³, cm²</p>
|
||||
<p>Controll characters: <, >, ", ', &, |, /, \</p>
|
||||
<p>Controll characters: <, >, ", ', &, |, /, \</p>
|
||||
|
|
@ -2,7 +2,7 @@
|
|||
unconditional branch instruction—often simply called the
|
||||
“B” instruction—and how its 26‐bit immediate field (imm26)
|
||||
is laid out and later relocated during linking.</p>
|
||||
<hr>
|
||||
<hr></hr>
|
||||
<h2>Instruction Layout</h2>
|
||||
<p>The unconditional branch in AArch64 is encoded in a 32‑bit
|
||||
instruction. Its layout is as follows:</p>
|
||||
|
|
@ -40,5 +40,5 @@ the full register width (typically 64 bits).</li>
|
|||
</li>
|
||||
<li>With a 26‑bit signed field that’s effectively 28 bits
|
||||
after the shift, the branch can cover a range
|
||||
of approximately ±128 MB from the current instruction.</li>
|
||||
of approximately ±128 MB from the current instruction.</li>
|
||||
</ul>
|
||||
Loading…
Reference in New Issue