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<!doctype html>
<html lang="en">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Test template</title>
<style>
body
{
background-color: white;
font-family: Lucida Grande,Lucida Sans Unicode,Lucida Sans,Geneva,Verdana,sans-serif;
}
div.document
{
max-width: 820px;
top: 20px;
overflow: visible;
margin: 0 auto;
width: 90%;
padding-bottom: 50px;
}
div h1
{
font-size: 32px;
font-weight: normal;
margin-bottom: 10px;
margin-top: 24px;
color: black;
}
div h2
{
font-size: 24px;
font-weight: normal;
margin-bottom: 10px;
margin-top: 24px;
color: black;
}
div.figure {
text-align: center;
}
.equation svg {
margin-left: -5%;
max-width: 110%;
height: auto;
object-fit: contain;
}
.inline-equation svg {
vertical-align: middle;
}
.equation-container {
display: flex;
justify-content: space-between;
align-items: center;
width: 100%;
}
.equation {
text-align: center;
width: 100%;
}
.equation-number {
text-align: right;
}
table {
border-collapse: collapse;
margin: 20px auto;
}
table th,
table td {
padding: 8px;
font-variant-numeric: tabular-nums;
}
table th {
background-color: #f4f4f4;
font-weight: bold;
}
/* Alternating row colors for readability */
table tr:nth-child(even) {
background-color: #f9f9f9;
}
table tr:nth-child(odd) {
background-color: #ffffff;
}
/* Caption styling */
table caption {
caption-side: top;
font-weight: bold;
font-size: 16px;
margin-bottom: 10px;
}
/* Fine-tuning text in table */
table td, table th {
vertical-align: middle;
line-height: 1.5;
}
</style>
</head>
<body>
<div class="document"><h1>Thermal Conductivity of Mixtures</h1>
<p>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.</p>
<p>The thermal conductivity of a gas mixture, denoted by <span class="inline-equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="24.566351pt" height="15.0336pt" viewBox="0 0 24.566351 15.0336" xmlns="http://www.w3.org/2000/svg" version="1.1">
<defs>
<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
</defs>
<g id="figure_1">
<g id="patch_1">
<path d="M 0 15.0336
L 24.566351 15.0336
L 24.566351 0
L 0 0
z
" style="fill: #ffffff"/>
</g>
<g id="axes_1">
<g id="text_1">
<!-- $\lambda_{\text{mix}}$ -->
<g transform="translate(3.040255 10.351343) scale(0.1 -0.1)">
<defs>
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L 5113 2094
Q 5113 2597 4934 2840
Q 4756 3084 4391 3084
Q 3944 3084 3684 2787
Q 3425 2491 3425 1978
L 3425 0
L 2847 0
L 2847 2094
Q 2847 2600 2669 2842
Q 2491 3084 2119 3084
Q 1678 3084 1418 2786
Q 1159 2488 1159 1978
L 1159 0
L 581 0
L 581 3500
L 1159 3500
L 1159 2956
Q 1356 3278 1631 3431
Q 1906 3584 2284 3584
Q 2666 3584 2933 3390
Q 3200 3197 3328 2828
z
" transform="scale(0.015625)"/>
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<use xlink:href="#DejaVuSans-6d" transform="translate(59.179688 -16.390625) scale(0.7)"/>
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</span>, can expressed as
shown in equation 1.
<div class="equation-container"><div class="equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="100.888782pt" height="43.88076pt" viewBox="0 0 100.888782 43.88076" xmlns="http://www.w3.org/2000/svg" version="1.1">
<defs>
<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
</defs>
<g id="figure_1">
<g id="patch_1">
<path d="M 0 43.88076
L 100.888782 43.88076
L 100.888782 0
L 0 0
z
" style="fill: #ffffff"/>
</g>
<g id="axes_1">
<g id="text_1">
<!-- $\lambda_{ ext{mix}} = \sum_{i=1}^{n} \frac{x_i \lambda_i}{\sum_{j=1}^{n} x_j \Phi_{ij}}$ -->
<g transform="translate(9.955501 21.209784) scale(0.1 -0.1)">
<defs>
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" transform="scale(0.015625)"/>
<path id="DejaVuSans-Oblique-65" d="M 3078 2063
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L 3078 2063
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M 3578 1613
L 903 1613
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Q 872 872 1139 634
Q 1406 397 1894 397
Q 2269 397 2603 481
Q 2938 566 3225 728
L 3116 159
Q 2806 34 2476 -28
Q 2147 -91 1806 -91
Q 1078 -91 686 257
Q 294 606 294 1247
Q 294 1794 489 2264
Q 684 2734 1063 3103
Q 1306 3334 1642 3459
Q 1978 3584 2356 3584
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" transform="scale(0.015625)"/>
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M 2206 494
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<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(425.668945 -98.748438) scale(0.7)"/>
<use xlink:href="#DejaVuSans-3d" transform="translate(458.754883 -98.748438) scale(0.7)"/>
<use xlink:href="#DejaVuSans-31" transform="translate(531.044922 -98.748438) scale(0.7)"/>
<use xlink:href="#DejaVuSans-Oblique-78" transform="translate(641.581055 55.628125) scale(0.7)"/>
<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(683.006836 44.14375) scale(0.49)"/>
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<use xlink:href="#DejaVuSans-3d" transform="translate(598.741211 -168.230156) scale(0.49)"/>
<use xlink:href="#DejaVuSans-31" transform="translate(649.344238 -168.230156) scale(0.49)"/>
<use xlink:href="#DejaVuSans-Oblique-78" transform="translate(680.519531 -98.863438) scale(0.7)"/>
<use xlink:href="#DejaVuSans-Oblique-6a" transform="translate(721.945312 -110.347813) scale(0.49)"/>
<use xlink:href="#DejaVuSans-3a6" transform="translate(737.473145 -98.863438) scale(0.7)"/>
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<path d="M 575.581055 19.14375
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z
"/>
</g>
</g>
</g>
</g>
</svg>
</div><div class="equation-number">(1)</div></div>
In this equation, <span class="inline-equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.427961pt" height="12.4416pt" viewBox="0 0 11.427961 12.4416" xmlns="http://www.w3.org/2000/svg" version="1.1">
<defs>
<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
</defs>
<g id="figure_1">
<g id="patch_1">
<path d="M 0 12.4416
L 11.427961 12.4416
L 11.427961 0
L 0 0
z
" style="fill: #ffffff"/>
</g>
<g id="axes_1">
<g id="text_1">
<!-- $x_i$ -->
<g transform="translate(1.75683 9.042383) scale(0.1 -0.1)">
<defs>
<path id="DejaVuSans-Oblique-78" d="M 3841 3500
L 2234 1784
L 3219 0
L 2559 0
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L 641 3500
L 1300 3500
L 1972 2234
L 3144 3500
L 3841 3500
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-Oblique-69" d="M 1172 4863
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M 909 3500
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L 909 3500
z
" transform="scale(0.015625)"/>
</defs>
<use xlink:href="#DejaVuSans-Oblique-78" transform="translate(0 0.3125)"/>
<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(59.179688 -16.09375) scale(0.7)"/>
</g>
</g>
</g>
</g>
</svg>
</span> represents the molar fraction of component <span class="inline-equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.995pt" height="12.96pt" viewBox="0 0 4.995 12.96" xmlns="http://www.w3.org/2000/svg" version="1.1">
<defs>
<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
</defs>
<g id="figure_1">
<g id="patch_1">
<path d="M 0 12.96
L 4.995 12.96
L 4.995 0
L 0 0
z
" style="fill: #ffffff"/>
</g>
<g id="axes_1">
<g id="text_1">
<!-- $i$ -->
<g transform="translate(1.159937 9.304175) scale(0.1 -0.1)">
<defs>
<path id="DejaVuSans-Oblique-69" d="M 1172 4863
L 1747 4863
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L 1172 4863
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M 909 3500
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L 225 0
L 909 3500
z
" transform="scale(0.015625)"/>
</defs>
<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(0 0.015625)"/>
</g>
</g>
</g>
</g>
</svg>
</span> within the mixture,
while <span class="inline-equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.643961pt" height="15.0336pt" viewBox="0 0 11.643961 15.0336" xmlns="http://www.w3.org/2000/svg" version="1.1">
<defs>
<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
</defs>
<g id="figure_1">
<g id="patch_1">
<path d="M 0 15.0336
L 11.643961 15.0336
L 11.643961 0
L 0 0
z
" style="fill: #ffffff"/>
</g>
<g id="axes_1">
<g id="text_1">
<!-- $\lambda_i$ -->
<g transform="translate(1.86753 10.351343) scale(0.1 -0.1)">
<defs>
<path id="DejaVuSans-Oblique-3bb" d="M 2350 4316
L 3125 0
L 2516 0
L 2038 2588
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L 1184 4863
L 1550 4856
Q 2253 4847 2350 4316
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-Oblique-69" d="M 1172 4863
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" transform="scale(0.015625)"/>
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<use xlink:href="#DejaVuSans-Oblique-3bb" transform="translate(0 0.015625)"/>
<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(59.179688 -16.390625) scale(0.7)"/>
</g>
</g>
</g>
</g>
</svg>
</span> denotes the thermal conductivity of the pure substance <span class="inline-equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.995pt" height="12.96pt" viewBox="0 0 4.995 12.96" xmlns="http://www.w3.org/2000/svg" version="1.1">
<defs>
<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
</defs>
<g id="figure_1">
<g id="patch_1">
<path d="M 0 12.96
L 4.995 12.96
L 4.995 0
L 0 0
z
" style="fill: #ffffff"/>
</g>
<g id="axes_1">
<g id="text_1">
<!-- $i$ -->
<g transform="translate(1.159937 9.304175) scale(0.1 -0.1)">
<defs>
<path id="DejaVuSans-Oblique-69" d="M 1172 4863
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z
" transform="scale(0.015625)"/>
</defs>
<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(0 0.015625)"/>
</g>
</g>
</g>
</g>
</svg>
</span>. The denominator
contains the interaction parameter <span class="inline-equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.951782pt" height="15.8112pt" viewBox="0 0 15.951782 15.8112" xmlns="http://www.w3.org/2000/svg" version="1.1">
<defs>
<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
</defs>
<g id="figure_1">
<g id="patch_1">
<path d="M 0 15.8112
L 15.951782 15.8112
L 15.951782 0
L 0 0
z
" style="fill: #ffffff"/>
</g>
<g id="axes_1">
<g id="text_1">
<!-- $\Phi_{ij}$ -->
<g transform="translate(2.075289 10.084656) scale(0.1 -0.1)">
<defs>
<path id="DejaVuSans-3a6" d="M 2206 3644
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M 2206 494
Q 1444 563 950 981
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" transform="scale(0.015625)"/>
<path id="DejaVuSans-Oblique-69" d="M 1172 4863
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M 909 3500
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" transform="scale(0.015625)"/>
<path id="DejaVuSans-Oblique-6a" d="M 928 3500
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Q 403 -1106 132 -1218
Q -138 -1331 -506 -1331
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Q -144 -844 -1 -703
Q 141 -563 238 -63
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M 1197 4863
L 1772 4863
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" transform="scale(0.015625)"/>
</defs>
<use xlink:href="#DejaVuSans-3a6" transform="translate(0 0.09375)"/>
<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(79.667969 -16.3125) scale(0.7)"/>
<use xlink:href="#DejaVuSans-Oblique-6a" transform="translate(99.116211 -16.3125) scale(0.7)"/>
</g>
</g>
</g>
</g>
</svg>
</span>, which describes the influence of component
<span class="inline-equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.399pt" height="15.552pt" viewBox="0 0 6.399 15.552" xmlns="http://www.w3.org/2000/svg" version="1.1">
<defs>
<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
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<g id="figure_1">
<g id="patch_1">
<path d="M 0 15.552
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" style="fill: #ffffff"/>
</g>
<g id="axes_1">
<g id="text_1">
<!-- $j$ -->
<g transform="translate(1.879487 10.60376) scale(0.1 -0.1)">
<defs>
<path id="DejaVuSans-Oblique-6a" d="M 928 3500
L 1503 3500
L 813 -63
L 809 -78
Q 694 -675 544 -897
Q 403 -1106 132 -1218
Q -138 -1331 -506 -1331
L -722 -1331
L -628 -844
L -481 -844
Q -144 -844 -1 -703
Q 141 -563 238 -63
L 928 3500
z
M 1197 4863
L 1772 4863
L 1631 4134
L 1056 4134
L 1197 4863
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" transform="scale(0.015625)"/>
</defs>
<use xlink:href="#DejaVuSans-Oblique-6a" transform="translate(0 0.015625)"/>
</g>
</g>
</g>
</g>
</svg>
</span> on the transport properties of component <span class="inline-equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.995pt" height="12.96pt" viewBox="0 0 4.995 12.96" xmlns="http://www.w3.org/2000/svg" version="1.1">
<defs>
<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
</defs>
<g id="figure_1">
<g id="patch_1">
<path d="M 0 12.96
L 4.995 12.96
L 4.995 0
L 0 0
z
" style="fill: #ffffff"/>
</g>
<g id="axes_1">
<g id="text_1">
<!-- $i$ -->
<g transform="translate(1.159937 9.304175) scale(0.1 -0.1)">
<defs>
<path id="DejaVuSans-Oblique-69" d="M 1172 4863
L 1747 4863
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M 909 3500
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L 909 3500
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" transform="scale(0.015625)"/>
</defs>
<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(0 0.015625)"/>
</g>
</g>
</g>
</g>
</svg>
</span>.</p>
<p>The interaction parameter <span class="inline-equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.951782pt" height="15.8112pt" viewBox="0 0 15.951782 15.8112" xmlns="http://www.w3.org/2000/svg" version="1.1">
<defs>
<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
</defs>
<g id="figure_1">
<g id="patch_1">
<path d="M 0 15.8112
L 15.951782 15.8112
L 15.951782 0
L 0 0
z
" style="fill: #ffffff"/>
</g>
<g id="axes_1">
<g id="text_1">
<!-- $\Phi_{ij}$ -->
<g transform="translate(2.075289 10.084656) scale(0.1 -0.1)">
<defs>
<path id="DejaVuSans-3a6" d="M 2206 3644
Q 1738 3578 1431 3306
Q 1025 2947 1025 2328
Q 1025 1713 1431 1353
Q 1738 1081 2206 1016
L 2206 3644
z
M 2838 1016
Q 3306 1081 3613 1353
Q 4013 1713 4013 2328
Q 4013 2947 3613 3306
Q 3306 3578 2838 3644
L 2838 1016
z
M 2206 494
Q 1444 563 950 981
Q 359 1481 359 2328
Q 359 3175 950 3678
Q 1441 4100 2206 4169
L 2206 4666
L 2838 4666
L 2838 4169
Q 3600 4097 4091 3678
Q 4678 3175 4678 2328
Q 4678 1484 4091 981
Q 3600 563 2838 491
L 2838 0
L 2206 0
L 2206 494
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-Oblique-69" d="M 1172 4863
L 1747 4863
L 1606 4134
L 1031 4134
L 1172 4863
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M 909 3500
L 1484 3500
L 800 0
L 225 0
L 909 3500
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-Oblique-6a" d="M 928 3500
L 1503 3500
L 813 -63
L 809 -78
Q 694 -675 544 -897
Q 403 -1106 132 -1218
Q -138 -1331 -506 -1331
L -722 -1331
L -628 -844
L -481 -844
Q -144 -844 -1 -703
Q 141 -563 238 -63
L 928 3500
z
M 1197 4863
L 1772 4863
L 1631 4134
L 1056 4134
L 1197 4863
z
" transform="scale(0.015625)"/>
</defs>
<use xlink:href="#DejaVuSans-3a6" transform="translate(0 0.09375)"/>
<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(79.667969 -16.3125) scale(0.7)"/>
<use xlink:href="#DejaVuSans-Oblique-6a" transform="translate(99.116211 -16.3125) scale(0.7)"/>
</g>
</g>
</g>
</g>
</svg>
</span> is given by the relation shown in equation 2.
<div class="equation-container"><div class="equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="221.766721pt" height="26.13pt" viewBox="0 0 221.766721 26.13" xmlns="http://www.w3.org/2000/svg" version="1.1">
<defs>
<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
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<g id="figure_1">
<g id="patch_1">
<path d="M 0 26.13
L 221.766721 26.13
L 221.766721 0
L 0 0
z
" style="fill: #ffffff"/>
</g>
<g id="axes_1">
<g id="text_1">
<!-- $\Phi_{ij} = \frac{1}{\sqrt{8}} \left(1 + \frac{M_i}{M_j} \right)^{-1/2} \left[ 1 + \left( \frac{\lambda_i}{\lambda_j} \right)^{1/2} \left( \frac{M_j}{M_i} \right)^{1/4} \right]^2$ -->
<g transform="translate(21.255444 15.69565) scale(0.1 -0.1)">
<defs>
<path id="DejaVuSans-3a6" d="M 2206 3644
Q 1738 3578 1431 3306
Q 1025 2947 1025 2328
Q 1025 1713 1431 1353
Q 1738 1081 2206 1016
L 2206 3644
z
M 2838 1016
Q 3306 1081 3613 1353
Q 4013 1713 4013 2328
Q 4013 2947 3613 3306
Q 3306 3578 2838 3644
L 2838 1016
z
M 2206 494
Q 1444 563 950 981
Q 359 1481 359 2328
Q 359 3175 950 3678
Q 1441 4100 2206 4169
L 2206 4666
L 2838 4666
L 2838 4169
Q 3600 4097 4091 3678
Q 4678 3175 4678 2328
Q 4678 1484 4091 981
Q 3600 563 2838 491
L 2838 0
L 2206 0
L 2206 494
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-Oblique-69" d="M 1172 4863
L 1747 4863
L 1606 4134
L 1031 4134
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M 909 3500
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L 800 0
L 225 0
L 909 3500
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-Oblique-6a" d="M 928 3500
L 1503 3500
L 813 -63
L 809 -78
Q 694 -675 544 -897
Q 403 -1106 132 -1218
Q -138 -1331 -506 -1331
L -722 -1331
L -628 -844
L -481 -844
Q -144 -844 -1 -703
Q 141 -563 238 -63
L 928 3500
z
M 1197 4863
L 1772 4863
L 1631 4134
L 1056 4134
L 1197 4863
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-3d" d="M 678 2906
L 4684 2906
L 4684 2381
L 678 2381
L 678 2906
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M 678 1631
L 4684 1631
L 4684 1100
L 678 1100
L 678 1631
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-31" d="M 794 531
L 1825 531
L 1825 4091
L 703 3866
L 703 4441
L 1819 4666
L 2450 4666
L 2450 531
L 3481 531
L 3481 0
L 794 0
L 794 531
z
" transform="scale(0.015625)"/>
<path id="STIXSizeOneSym-Regular-221a" d="M 6970 9933
L 3373 -1888
L 3104 -1888
L 1626 2918
Q 1555 3149 1465 3251
Q 1376 3354 1229 3354
Q 1011 3354 794 3181
L 717 3309
L 1766 4115
L 1926 4115
L 3379 -602
L 3405 -602
L 6605 9933
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" transform="scale(0.015625)"/>
<path id="DejaVuSans-38" d="M 2034 2216
Q 1584 2216 1326 1975
Q 1069 1734 1069 1313
Q 1069 891 1326 650
Q 1584 409 2034 409
Q 2484 409 2743 651
Q 3003 894 3003 1313
Q 3003 1734 2745 1975
Q 2488 2216 2034 2216
z
M 1403 2484
Q 997 2584 770 2862
Q 544 3141 544 3541
Q 544 4100 942 4425
Q 1341 4750 2034 4750
Q 2731 4750 3128 4425
Q 3525 4100 3525 3541
Q 3525 3141 3298 2862
Q 3072 2584 2669 2484
Q 3125 2378 3379 2068
Q 3634 1759 3634 1313
Q 3634 634 3220 271
Q 2806 -91 2034 -91
Q 1263 -91 848 271
Q 434 634 434 1313
Q 434 1759 690 2068
Q 947 2378 1403 2484
z
M 1172 3481
Q 1172 3119 1398 2916
Q 1625 2713 2034 2713
Q 2441 2713 2670 2916
Q 2900 3119 2900 3481
Q 2900 3844 2670 4047
Q 2441 4250 2034 4250
Q 1625 4250 1398 4047
Q 1172 3844 1172 3481
z
" transform="scale(0.015625)"/>
<path id="STIXSizeTwoSym-Regular-28" d="M 3219 -1555
L 3219 -1786
Q 2144 -774 1517 730
Q 890 2234 890 4115
Q 890 6003 1533 7593
Q 2176 9184 3219 10022
L 3219 9811
Q 2490 9062 1994 7606
Q 1498 6150 1498 4115
Q 1498 563 3219 -1555
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-2b" d="M 2944 4013
L 2944 2272
L 4684 2272
L 4684 1741
L 2944 1741
L 2944 0
L 2419 0
L 2419 1741
L 678 1741
L 678 2272
L 2419 2272
L 2419 4013
L 2944 4013
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-Oblique-4d" d="M 1081 4666
L 2028 4666
L 2572 1522
L 4378 4666
L 5350 4666
L 4441 0
L 3828 0
L 4622 4091
L 2791 897
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L 1581 4103
L 788 0
L 172 0
L 1081 4666
z
" transform="scale(0.015625)"/>
<path id="STIXSizeTwoSym-Regular-29" d="M 730 9792
L 730 10022
Q 1805 9011 2432 7507
Q 3059 6003 3059 4115
Q 3059 2234 2416 643
Q 1773 -947 730 -1786
L 730 -1574
Q 1459 -826 1955 633
Q 2451 2093 2451 4115
Q 2451 7674 730 9792
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-2212" d="M 678 2272
L 4684 2272
L 4684 1741
L 678 1741
L 678 2272
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-2f" d="M 1625 4666
L 2156 4666
L 531 -594
L 0 -594
L 1625 4666
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-32" d="M 1228 531
L 3431 531
L 3431 0
L 469 0
L 469 531
Q 828 903 1448 1529
Q 2069 2156 2228 2338
Q 2531 2678 2651 2914
Q 2772 3150 2772 3378
Q 2772 3750 2511 3984
Q 2250 4219 1831 4219
Q 1534 4219 1204 4116
Q 875 4013 500 3803
L 500 4441
Q 881 4594 1212 4672
Q 1544 4750 1819 4750
Q 2544 4750 2975 4387
Q 3406 4025 3406 3419
Q 3406 3131 3298 2873
Q 3191 2616 2906 2266
Q 2828 2175 2409 1742
Q 1991 1309 1228 531
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" transform="scale(0.015625)"/>
<path id="STIXSizeTwoSym-Regular-5b" d="M 2701 -1786
L 1216 -1786
L 1216 10022
L 2701 10022
L 2701 9766
L 1702 9766
L 1702 -1530
L 2701 -1530
L 2701 -1786
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-Oblique-3bb" d="M 2350 4316
L 3125 0
L 2516 0
L 2038 2588
L 328 0
L -281 0
L 1903 3356
L 1794 3975
Q 1725 4369 1391 4369
L 1091 4369
L 1184 4863
L 1550 4856
Q 2253 4847 2350 4316
z
" transform="scale(0.015625)"/>
<path id="DejaVuSans-34" d="M 2419 4116
L 825 1625
L 2419 1625
L 2419 4116
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M 2253 4666
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L 3047 1625
L 3713 1625
L 3713 1100
L 3047 1100
L 3047 0
L 2419 0
L 2419 1100
L 313 1100
L 313 1709
L 2253 4666
z
" transform="scale(0.015625)"/>
<path id="STIXSizeTwoSym-Regular-5d" d="M 1722 -1786
L 237 -1786
L 237 -1530
L 1235 -1530
L 1235 9766
L 237 9766
L 237 10022
L 1722 10022
L 1722 -1786
z
" transform="scale(0.015625)"/>
</defs>
<use xlink:href="#DejaVuSans-3a6" transform="translate(0 0.348281)"/>
<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(79.667969 -16.057969) scale(0.7)"/>
<use xlink:href="#DejaVuSans-Oblique-6a" transform="translate(99.116211 -16.057969) scale(0.7)"/>
<use xlink:href="#DejaVuSans-3d" transform="translate(140.78125 0.348281)"/>
<use xlink:href="#DejaVuSans-31" transform="translate(274.052734 44.145156) scale(0.7)"/>
<use xlink:href="#STIXSizeOneSym-Regular-221a" transform="translate(244.052734 -54.315781) scale(0.405101)"/>
<use xlink:href="#DejaVuSans-38" transform="translate(295.624446 -65.432969) scale(0.7)"/>
<use xlink:href="#STIXSizeTwoSym-Regular-28" transform="translate(361.410579 -34.704687) scale(0.90232)"/>
<use xlink:href="#DejaVuSans-31" transform="translate(414.567821 0.348281)"/>
<use xlink:href="#DejaVuSans-2b" transform="translate(497.673289 0.348281)"/>
<use xlink:href="#DejaVuSans-Oblique-4d" transform="translate(600.944774 55.629531) scale(0.7)"/>
<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(661.340282 44.145156) scale(0.49)"/>
<use xlink:href="#DejaVuSans-Oblique-4d" transform="translate(600.944774 -38.139219) scale(0.7)"/>
<use xlink:href="#DejaVuSans-Oblique-6a" transform="translate(661.340282 -49.623594) scale(0.49)"/>
<use xlink:href="#STIXSizeTwoSym-Regular-29" transform="translate(689.368114 -34.704687) scale(0.90232)"/>
<use xlink:href="#DejaVuSans-2212" transform="translate(745.197124 38.629531) scale(0.7)"/>
<use xlink:href="#DejaVuSans-31" transform="translate(803.849467 38.629531) scale(0.7)"/>
<use xlink:href="#DejaVuSans-2f" transform="translate(848.3856 38.629531) scale(0.7)"/>
<use xlink:href="#DejaVuSans-32" transform="translate(869.432085 38.629531) scale(0.7)"/>
<use xlink:href="#STIXSizeTwoSym-Regular-5b" transform="translate(916.702592 -34.90625) scale(0.969231)"/>
<use xlink:href="#DejaVuSans-31" transform="translate(961.168214 0.348281)"/>
<use xlink:href="#DejaVuSans-2b" transform="translate(1044.273683 0.348281)"/>
<use xlink:href="#STIXSizeTwoSym-Regular-28" transform="translate(1147.545168 -36.015625) scale(0.925677)"/>
<use xlink:href="#DejaVuSans-Oblique-3bb" transform="translate(1202.101274 55.629531) scale(0.7)"/>
<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(1243.527055 44.145156) scale(0.49)"/>
<use xlink:href="#DejaVuSans-Oblique-3bb" transform="translate(1202.101274 -40.293906) scale(0.7)"/>
<use xlink:href="#DejaVuSans-Oblique-6a" transform="translate(1243.527055 -51.778281) scale(0.49)"/>
<use xlink:href="#STIXSizeTwoSym-Regular-29" transform="translate(1271.554887 -36.015625) scale(0.925677)"/>
<use xlink:href="#DejaVuSans-31" transform="translate(1328.827913 38.629531) scale(0.7)"/>
<use xlink:href="#DejaVuSans-2f" transform="translate(1373.364046 38.629531) scale(0.7)"/>
<use xlink:href="#DejaVuSans-32" transform="translate(1394.41053 38.629531) scale(0.7)"/>
<use xlink:href="#STIXSizeTwoSym-Regular-28" transform="translate(1441.681038 -24.514219) scale(0.90232)"/>
<use xlink:href="#DejaVuSans-Oblique-4d" transform="translate(1494.83828 65.82) scale(0.7)"/>
<use xlink:href="#DejaVuSans-Oblique-6a" transform="translate(1555.233788 54.335625) scale(0.49)"/>
<use xlink:href="#DejaVuSans-Oblique-4d" transform="translate(1494.83828 -38.139219) scale(0.7)"/>
<use xlink:href="#DejaVuSans-Oblique-69" transform="translate(1555.233788 -49.623594) scale(0.49)"/>
<use xlink:href="#STIXSizeTwoSym-Regular-29" transform="translate(1583.26162 -24.514219) scale(0.90232)"/>
<use xlink:href="#DejaVuSans-31" transform="translate(1639.09063 38.629531) scale(0.7)"/>
<use xlink:href="#DejaVuSans-2f" transform="translate(1683.626762 38.629531) scale(0.7)"/>
<use xlink:href="#DejaVuSans-34" transform="translate(1707.210747 38.629531) scale(0.7)"/>
<use xlink:href="#STIXSizeTwoSym-Regular-5d" transform="translate(1754.481255 -34.90625) scale(0.969231)"/>
<use xlink:href="#DejaVuSans-32" transform="translate(1799.903908 38.629531) scale(0.7)"/>
<path d="M 244.052734 19.145156
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L 348.910579 19.145156
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"/>
<path d="M 286.874446 4.145156
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"/>
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"/>
<path d="M 1202.101274 19.145156
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L 1259.054887 25.395156
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"/>
<path d="M 1494.83828 19.145156
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"/>
</g>
</g>
</g>
</g>
</svg>
</div><div class="equation-number">(2)</div></div>
Here, <span class="inline-equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.366162pt" height="14.1696pt" viewBox="0 0 14.366162 14.1696" xmlns="http://www.w3.org/2000/svg" version="1.1">
<defs>
<style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
</defs>
<g id="figure_1">
<g id="patch_1">
<path d="M 0 14.1696
L 14.366162 14.1696
L 14.366162 0
L 0 0
z
" style="fill: #ffffff"/>
</g>
<g id="axes_1">
<g id="text_1">
<!-- $M_i$ -->
<g transform="translate(1.912658 9.915023) scale(0.1 -0.1)">
<defs>
<path id="DejaVuSans-Oblique-4d" d="M 1081 4666
L 2028 4666
L 2572 1522
L 4378 4666
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</span>, respectively.
Molar masses and thermal conductivity of the pure substances are listed in table 1.
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.</p>
<table id="T_gas_probs">
<caption>Table 1: Properties of some gases</caption>
<thead>
<tr>
<th id="T_gas_probs_level0_col0" class="col_heading level0 col0" >Gas</th>
<th id="T_gas_probs_level0_col1" class="col_heading level0 col1" >Molar mass in g/mol</th>
<th id="T_gas_probs_level0_col2" class="col_heading level0 col2" >Thermal conductivity in W/m/K</th>
</tr>
</thead>
<tbody>
<tr>
<td id="T_gas_probs_row0_col0" class="data row0 col0" >H2</td>
<td id="T_gas_probs_row0_col1" class="data row0 col1" >2.016</td>
<td id="T_gas_probs_row0_col2" class="data row0 col2" >0.1805</td>
</tr>
<tr>
<td id="T_gas_probs_row1_col0" class="data row1 col0" >O2</td>
<td id="T_gas_probs_row1_col1" class="data row1 col1" >32.00</td>
<td id="T_gas_probs_row1_col2" class="data row1 col2" >0.0263</td>
</tr>
<tr>
<td id="T_gas_probs_row2_col0" class="data row2 col0" >N2</td>
<td id="T_gas_probs_row2_col1" class="data row2 col1" >28.02</td>
<td id="T_gas_probs_row2_col2" class="data row2 col2" >0.0258</td>
</tr>
<tr>
<td id="T_gas_probs_row3_col0" class="data row3 col0" >CO2</td>
<td id="T_gas_probs_row3_col1" class="data row3 col1" >44.01</td>
<td id="T_gas_probs_row3_col2" class="data row3 col2" >0.0166</td>
</tr>
<tr>
<td id="T_gas_probs_row4_col0" class="data row4 col0" >CH4</td>
<td id="T_gas_probs_row4_col1" class="data row4 col1" >16.04</td>
<td id="T_gas_probs_row4_col2" class="data row4 col2" >0.0341</td>
</tr>
<tr>
<td id="T_gas_probs_row5_col0" class="data row5 col0" >Ar</td>
<td id="T_gas_probs_row5_col1" class="data row5 col1" >39.95</td>
<td id="T_gas_probs_row5_col2" class="data row5 col2" >0.0177</td>
</tr>
<tr>
<td id="T_gas_probs_row6_col0" class="data row6 col0" >He</td>
<td id="T_gas_probs_row6_col1" class="data row6 col1" >4.0026</td>
<td id="T_gas_probs_row6_col2" class="data row6 col2" >0.1513</td>
</tr>
</tbody>
</table>
<p>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.</p>
<p>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 1 the resulting thermal conductivity of an H2/CO2-mixture is shown.</p><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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<br>Figure 1: Thermal Conductivity of H2/CO2 mixtures</div><p>In engineering practice, the accurate determination of <span class="inline-equation"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="24.566351pt" height="15.0336pt" viewBox="0 0 24.566351 15.0336" xmlns="http://www.w3.org/2000/svg" version="1.1">
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</span> 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.</p>
<p>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.</p>
<hr></hr>
<p>Expanded by more or less sensible AI jabbering; based on: <a href="https://doi.org/10.14279/depositonce-7390">doi:10.14279/depositonce-7390</a></p>
</div>
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