{"id":57,"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\/transition-regime-constant-and-non-uniform-pressure\/"},"modified":"2022-01-10T21:46:40","modified_gmt":"2022-01-10T21:46:40","slug":"transition-regime-constant-and-non-uniform-pressure","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/transition-regime-constant-and-non-uniform-pressure\/","title":{"raw":"5.3 Transition Regime \u2013 Constant and Non-uniform Pressure","rendered":"5.3 Transition Regime \u2013 Constant and Non-uniform Pressure"},"content":{"raw":"
\r\n

Diffusion in the transition regime is distinguished by the fact that resistance to diffusion offered by both molecule-molecule and molecule-particle collisions must be considered. That is, the second term on the right of Equation\u00a014<\/a> must be retained. For the isobaric condition, the relation between the component fluxes is given by Graham\u2019s law, Equation\u00a021<\/a>. Upon introduction of Equation\u00a021<\/a> into Equation\u00a022<\/a>, followed by some algebraic manipulation, the equation for the flux of species A<\/em> under isobaric conditions is derived to be Equation\u00a027.<\/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}{\\left ( 1+D\/D_{A}^{K} \\right )-\\left ( 1-M_{AB}^{0.5} \\right )x_{A}}[\/latex]<\/td>\r\n(27)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The corresponding result for the non-isobaric condition is obtained by using Equation\u00a020<\/a> in Equation\u00a022<\/a> and adding advection via viscous flow to obtain Equation\u00a028.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle N_{A}=-\\frac{DC\\ dx_{A}\/dl+\\left ( D+D_{B}^{K} \\right )x_{A}dC\/dl}{\\left ( 1+D\/D_{A}^{K} \\right )-\\left ( 1-M_{AB}^{0.5} \\right )x_{A}}[\/latex] [latex]\\displaystyle -x_{A}\\frac{k_{g}p}{\\mu }\\frac{dC}{dl}[\/latex]<\/td>\r\n(28)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

As usual, the corresponding equation for the flux of species B<\/em> is obtained by interchanging the subscripts.<\/p>\r\n

We appealed to the conditions [latex]D\/D_{i}^{K} < < 1[\/latex], i<\/em> = A<\/em>, B<\/em> and [latex]D_{i}^{K}\\mu \/k_{g}p < < 1[\/latex], i<\/em> = A<\/em>, B<\/em> to justify simplifications leading to Equation\u00a026<\/a>, applicable in the molecular regime. Neither of these conditions generally applies in the transition regime now under consideration. However, important to groundwater scientists and engineers is the circumstance in which the species of interest, say species A<\/em>, is present only in trace concentrations. Equation\u00a028 can then be simplified to Equation\u00a029.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle N_{A}=-\\frac{DC\\ dx_{A}\/dl}{(1+D\/D_{A}^{K})}[\/latex] [latex]\\displaystyle -\\left ( \\frac{D+D_{B}^{K}}{1+D\/D_{A}^{K}}+\\frac{k_{g}p}{\\mu } \\right )\\left ( x_{A} \\frac{dC}{dl}\\right )[\/latex]<\/td>\r\n(29)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

This result is of the same mathematical form as the widely used advection-diffusion model. Webb and Pruess (2003) calculated the transport of trace species using a flux equation that can be derived from Equation\u00a029 written on a mass flux basis. Click on this exercise link to view an example problem <\/a>Exercise\u00a05<\/a>.<\/p>\r\n\r\n<\/div>","rendered":"

\n

Diffusion in the transition regime is distinguished by the fact that resistance to diffusion offered by both molecule-molecule and molecule-particle collisions must be considered. That is, the second term on the right of Equation\u00a014<\/a> must be retained. For the isobaric condition, the relation between the component fluxes is given by Graham\u2019s law, Equation\u00a021<\/a>. Upon introduction of Equation\u00a021<\/a> into Equation\u00a022<\/a>, followed by some algebraic manipulation, the equation for the flux of species A<\/em> under isobaric conditions is derived to be Equation\u00a027.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle N_{A}^D=-\frac{DC\ dx_{A}/dl}{\left ( 1+D/D_{A}^{K} \right )-\left ( 1-M_{AB}^{0.5} \right )x_{A}}\"<\/td>\n(27)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The corresponding result for the non-isobaric condition is obtained by using Equation\u00a020<\/a> in Equation\u00a022<\/a> and adding advection via viscous flow to obtain Equation\u00a028.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle N_{A}=-\frac{DC\ dx_{A}/dl+\left ( D+D_{B}^{K} \right )x_{A}dC/dl}{\left ( 1+D/D_{A}^{K} \right )-\left ( 1-M_{AB}^{0.5} \right )x_{A}}\" \"\displaystyle -x_{A}\frac{k_{g}p}{\mu }\frac{dC}{dl}\"<\/td>\n(28)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

As usual, the corresponding equation for the flux of species B<\/em> is obtained by interchanging the subscripts.<\/p>\n

We appealed to the conditions \"D/D_{i}^{K} < < 1\", i<\/em> = A<\/em>, B<\/em> and \"D_{i}^{K}\mu /k_{g}p < < 1\", i<\/em> = A<\/em>, B<\/em> to justify simplifications leading to Equation\u00a026<\/a>, applicable in the molecular regime. Neither of these conditions generally applies in the transition regime now under consideration. However, important to groundwater scientists and engineers is the circumstance in which the species of interest, say species A<\/em>, is present only in trace concentrations. Equation\u00a028 can then be simplified to Equation\u00a029.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle N_{A}=-\frac{DC\ dx_{A}/dl}{(1+D/D_{A}^{K})}\" \"\displaystyle -\left ( \frac{D+D_{B}^{K}}{1+D/D_{A}^{K}}+\frac{k_{g}p}{\mu } \right )\left ( x_{A} \frac{dC}{dl}\right )\"<\/td>\n(29)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

This result is of the same mathematical form as the widely used advection-diffusion model. Webb and Pruess (2003) calculated the transport of trace species using a flux equation that can be derived from Equation\u00a029 written on a mass flux basis. Click on this exercise link to view an example problem <\/a>Exercise\u00a05<\/a>.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":20,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-57","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\/57","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":22,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/57\/revisions"}],"predecessor-version":[{"id":427,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/57\/revisions\/427"}],"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\/57\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/media?parent=57"}],"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=57"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/contributor?post=57"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/license?post=57"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}