<p>Species separation is usually achieved in closed vertical thermogravitational columns (TGC). To obtain continuous separation, the initially homogeneous binary solution, saturating the porous medium is introduced at a constant volumetric flow rate through one of the two vertical slots of the TGC and retrieved through the opposite slot. However, this process requires the horizontal dimension of the two vertical walls of the cell to be sufficiently large for the mass regime at the exit slot to reach the steady state. The analytical resolution obtained using the parallel flow approximation and numerical simulations developed are in very good agreement. The vertical mass fraction gradient,<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">${m}^{},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow> <mi>m</mi> </mrow> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> at steady state was shown not to admit an optimum with respect to the thickness <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$e$</EquationSource> <EquationSource Format="MATHML"><math> <mi>e</mi> </math></EquationSource> </InlineEquation> for a fixed <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> or with respect to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> for a fixed <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$e$</EquationSource> <EquationSource Format="MATHML"><math> <mi>e</mi> </math></EquationSource> </InlineEquation>, unlike the gradient <i>m</i> obtained in a vertical TGC. The ratio of the two simplified analytical expressions <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">${m}_{s}^{}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>m</mi> <mrow> <mi>s</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">${m}_{s}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation>, respectively, obtained for the two columns filled with the same binary fluid in porous or fluid media, led to an expression depending only on the ratio of the thermodiffusion coefficient, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq8.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">${D}_{T},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mi>T</mi> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> to the corresponding coefficient, <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq9.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">${D}_{T}^{*},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>D</mi> <mrow> <mi>T</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> in the porous medium, the column thickness <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq10.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$e,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>e</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and the permeability of the porous medium. To increase the degree of separation of this mixture, we could simply add another column of height <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$h$</EquationSource> <EquationSource Format="MATHML"><math> <mi>h</mi> </math></EquationSource> </InlineEquation>, at the outlet of the first column of height <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq12.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$H,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq13.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$h<H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo><</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation>, and restart the process with a mass fraction higher than the initial fraction, <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11242_2025_2173_Article_IEq14.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">${C}_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. This technique could be repeated as often as necessary.</p>

Continuous Species Separation in an Open-Ended Porous Thermogravitational Column Using Mixed Convection

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2025年7月2日

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A. Mojtabi,
K. Sioud,
M. -C. Charrier Mojtabi

摘要

Species separation is usually achieved in closed vertical thermogravitational columns (TGC). To obtain continuous separation, the initially homogeneous binary solution, saturating the porous medium is introduced at a constant volumetric flow rate through one of the two vertical slots of the TGC and retrieved through the opposite slot. However, this process requires the horizontal dimension of the two vertical walls of the cell to be sufficiently large for the mass regime at the exit slot to reach the steady state. The analytical resolution obtained using the parallel flow approximation and numerical simulations developed are in very good agreement. The vertical mass fraction gradient, ${m}^{*},$ $m^{*},$ at steady state was shown not to admit an optimum with respect to the thickness $e$ $e$ for a fixed $\Delta T$ $Δ T$ or with respect to $\Delta T$ $Δ T$ for a fixed $e$ $e$ , unlike the gradient m obtained in a vertical TGC. The ratio of the two simplified analytical expressions ${m}_{s}^{*}$ $m_{s}^{*}$ and ${m}_{s}$ $m_{s}$ , respectively, obtained for the two columns filled with the same binary fluid in porous or fluid media, led to an expression depending only on the ratio of the thermodiffusion coefficient, ${D}_{T},$ $D_{T},$ to the corresponding coefficient, ${D}_{T}^{*},$ $D_{T}^{*},$ in the porous medium, the column thickness $e,$ $e,$ and the permeability of the porous medium. To increase the degree of separation of this mixture, we could simply add another column of height $h$ $h$ , at the outlet of the first column of height $H,$ $H,$ with $h<H$ $h < H$ , and restart the process with a mass fraction higher than the initial fraction, ${C}_{0}$ $C_{0}$ . This technique could be repeated as often as necessary.