{"id":55,"date":"2021-10-02T23:21:35","date_gmt":"2021-10-02T23:21:35","guid":{"rendered":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/molecular-regime-uniform-pressure\/"},"modified":"2022-01-10T21:41:10","modified_gmt":"2022-01-10T21:41:10","slug":"molecular-regime-uniform-pressure","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/molecular-regime-uniform-pressure\/","title":{"raw":"5.1 Molecular Regime \u2013 Uniform Pressure","rendered":"5.1 Molecular Regime \u2013 Uniform Pressure"},"content":{"raw":"
\r\n

Resistance to diffusion of an individual species in the molecular regime is dominated by inter-molecular collisions, so the ratio [latex]D\/D_{A}^{K}[\/latex] is very small and the second term on the right side of Equation\u00a022<\/a> is negligible. Also, Graham\u2019s law and the relation dC<\/em>A<\/em><\/sub>\u00a0=\u00a0<\/em>Cdx<\/em>A<\/em><\/sub> apply for the isobaric condition. Equation\u00a022 becomes Equation\u00a023.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle -N_{A}^{D}=-\\frac{DC\\ dx_{A}\/dl}{1-\\left ( 1-M_{AB}^{0.5} \\right )x_{A}}[\/latex]<\/td>\r\n(23)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The diffusive flux for species B<\/em> is obtained from Equation\u00a023 by interchanging the subscripts (note: M<\/em>BA<\/em><\/sub> \u2261 M<\/em>B<\/em><\/sub>\/M<\/em>A<\/em><\/sub>). Strictly speaking, these results become Fick\u2019s law only if the molecular weights of the species are the same. However, diffusion is closely approximated by Fick\u2019s law when the molecular weights are nearly equal and\/or when one species is present in dilute concentration (i.e., x<\/em>A<\/em><\/sub>\u00a0<<\u00a01). The latter situation is common in environmental applications where the species of interest often appears in only trace amounts (e.g., vapor from a neat liquid with low vapor pressure or evaporation from the dissolved state in aqueous solution).<\/p>\r\n

Integration of Equation\u00a023 for steady-state diffusion between open boundaries on which the pressure is the same is demonstrated in subsequent examples. Non-equimolar diffusion results in the development of a pressure gradient in any system in which the free flux of gas components is prevented on one (a semi-open system) or both boundaries (a closed system). Even in completely open systems, transient pressure gradients are present during unsteady diffusion (Fen and Abriola, 2004). Thus, the isobaric condition under which Equation\u00a023 applies is expected to occur rarely if ever in natural field settings. The case of simultaneous diffusion and viscous flow is addressed in the following section. Click on these exercise links to view example problems <\/a>Exercise\u00a01<\/a> and <\/a>Exercise\u00a02<\/a>.<\/p>\r\n\r\n<\/div>","rendered":"

\n

Resistance to diffusion of an individual species in the molecular regime is dominated by inter-molecular collisions, so the ratio \"D/D_{A}^{K}\" is very small and the second term on the right side of Equation\u00a022<\/a> is negligible. Also, Graham\u2019s law and the relation dC<\/em>A<\/em><\/sub>\u00a0=\u00a0<\/em>Cdx<\/em>A<\/em><\/sub> apply for the isobaric condition. Equation\u00a022 becomes Equation\u00a023.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle -N_{A}^{D}=-\frac{DC\ dx_{A}/dl}{1-\left ( 1-M_{AB}^{0.5} \right )x_{A}}\"<\/td>\n(23)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The diffusive flux for species B<\/em> is obtained from Equation\u00a023 by interchanging the subscripts (note: M<\/em>BA<\/em><\/sub> \u2261 M<\/em>B<\/em><\/sub>\/M<\/em>A<\/em><\/sub>). Strictly speaking, these results become Fick\u2019s law only if the molecular weights of the species are the same. However, diffusion is closely approximated by Fick\u2019s law when the molecular weights are nearly equal and\/or when one species is present in dilute concentration (i.e., x<\/em>A<\/em><\/sub>\u00a0<<\u00a01). The latter situation is common in environmental applications where the species of interest often appears in only trace amounts (e.g., vapor from a neat liquid with low vapor pressure or evaporation from the dissolved state in aqueous solution).<\/p>\n

Integration of Equation\u00a023 for steady-state diffusion between open boundaries on which the pressure is the same is demonstrated in subsequent examples. Non-equimolar diffusion results in the development of a pressure gradient in any system in which the free flux of gas components is prevented on one (a semi-open system) or both boundaries (a closed system). Even in completely open systems, transient pressure gradients are present during unsteady diffusion (Fen and Abriola, 2004). Thus, the isobaric condition under which Equation\u00a023 applies is expected to occur rarely if ever in natural field settings. The case of simultaneous diffusion and viscous flow is addressed in the following section. Click on these exercise links to view example problems <\/a>Exercise\u00a01<\/a> and <\/a>Exercise\u00a02<\/a>.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":18,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-55","chapter","type-chapter","status-publish","hentry"],"part":104,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/55","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":16,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/55\/revisions"}],"predecessor-version":[{"id":425,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/55\/revisions\/425"}],"part":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/parts\/104"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/55\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/media?parent=55"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapter-type?post=55"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/contributor?post=55"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/license?post=55"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}