{"id":56,"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-non-uniform-pressure\/"},"modified":"2022-01-10T21:43:54","modified_gmt":"2022-01-10T21:43:54","slug":"molecular-regime-non-uniform-pressure","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/molecular-regime-non-uniform-pressure\/","title":{"raw":"5.2 Molecular Regime \u2013 Non-uniform Pressure","rendered":"5.2 Molecular Regime \u2013 Non-uniform Pressure"},"content":{"raw":"
\r\n

The total flux of a component is the sum of the total diffusive flux and the advective flux resulting from viscous flow (Equation\u00a03<\/a>). The advective flux resulting from viscous flow is the product of the mole fraction and viscous flux, as explained previously. Then our task is reduced to determining the total diffusive flux affected by non-uniform pressure. The spadework pertinent to this task has already been accomplished. Neglect the second term on the right side of Equation\u00a022<\/a> because we are considering the molecular regime and substitute Equation\u00a020<\/a> for [latex]N_{B}^{D}[\/latex] in the remaining term. Upon rearrangement we have Equation\u00a024.<\/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 ( D+D_{B}^{K} \\right )x_{A}dC\/dl}{1-\\left ( 1-M_{AB}^{0.5} \\right )x_{A}}[\/latex]<\/td>\r\n(24)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Equation\u00a024 uses dC<\/em>A<\/em><\/sub>\u00a0=\u00a0<\/em>Cdx<\/em>A<\/em><\/sub>\u00a0<\/em>+\u00a0<\/em>x<\/em>A<\/em><\/sub>dC<\/em>. When the ideal gas law is used to replace the total molar concentration gradient with the gradient of gas pressure, we see that the second term in this result calculates the effect of pressure gradient on diffusion. This effect is sometimes referred to as pressure diffusion. Recall the discussion in Section 3<\/a> in which we identified the pressure gradient in the bulk gas as a driving force for diffusion of individual species, as well as for viscous flow.<\/p>\r\n

The flux equation for species A<\/em> (interchange subscripts for species B<\/em>) affected by both diffusion and advection via viscous flow is obtained by simply adding the viscous advective flux to Equation\u00a024. We then have Equation\u00a025.<\/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}{1-\\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(25)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

This equation is readily reduced to the simpler expression of Equation\u00a026 for the condition [latex]D_{B}^{K}\\mu \/k_{g}p< < 1[\/latex].<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle N_{A}=-\\frac{DC\\ dx_{A}\/dl}{1-\\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(26)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

This simplification is tantamount to assuming that dC<\/em>A<\/em><\/sub> \u2248\u00a0 Cdx<\/em>A<\/em><\/sub> (i.e., pressure diffusion is negligible) and that the total diffusion flux is satisfactorily approximated by Equation\u00a023<\/a>. Note that the product DC<\/em> is independent of pressure, owing to the fact that the effective diffusion coefficient is inversely proportional to pressure. Click on these exercise links to view example problems <\/a>Exercise\u00a03<\/a> and <\/a>Exercise\u00a04<\/a>.<\/p>\r\n\r\n<\/div>","rendered":"

\n

The total flux of a component is the sum of the total diffusive flux and the advective flux resulting from viscous flow (Equation\u00a03<\/a>). The advective flux resulting from viscous flow is the product of the mole fraction and viscous flux, as explained previously. Then our task is reduced to determining the total diffusive flux affected by non-uniform pressure. The spadework pertinent to this task has already been accomplished. Neglect the second term on the right side of Equation\u00a022<\/a> because we are considering the molecular regime and substitute Equation\u00a020<\/a> for \"N_{B}^{D}\" in the remaining term. Upon rearrangement we have Equation\u00a024.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle N_{A}^{D}=-\frac{DC\ dx_{A}/dl+\left ( D+D_{B}^{K} \right )x_{A}dC/dl}{1-\left ( 1-M_{AB}^{0.5} \right )x_{A}}\"<\/td>\n(24)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Equation\u00a024 uses dC<\/em>A<\/em><\/sub>\u00a0=\u00a0<\/em>Cdx<\/em>A<\/em><\/sub>\u00a0<\/em>+\u00a0<\/em>x<\/em>A<\/em><\/sub>dC<\/em>. When the ideal gas law is used to replace the total molar concentration gradient with the gradient of gas pressure, we see that the second term in this result calculates the effect of pressure gradient on diffusion. This effect is sometimes referred to as pressure diffusion. Recall the discussion in Section 3<\/a> in which we identified the pressure gradient in the bulk gas as a driving force for diffusion of individual species, as well as for viscous flow.<\/p>\n

The flux equation for species A<\/em> (interchange subscripts for species B<\/em>) affected by both diffusion and advection via viscous flow is obtained by simply adding the viscous advective flux to Equation\u00a024. We then have Equation\u00a025.<\/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}{1-\left ( 1-M_{AB}^{0.5} \right )x_{A}}\" \"\displaystyle -x_{A}\frac{k_{g}p}{\mu }\frac{dC}{dl}\"<\/td>\n(25)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

This equation is readily reduced to the simpler expression of Equation\u00a026 for the condition \"D_{B}^{K}\mu /k_{g}p< < 1\".<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle N_{A}=-\frac{DC\ dx_{A}/dl}{1-\left ( 1-M_{AB}^{0.5} \right )x_{A}}\" \"\displaystyle -x_{A}\frac{k_{g}p}{\mu }\frac{dC}{dl}\"<\/td>\n(26)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

This simplification is tantamount to assuming that dC<\/em>A<\/em><\/sub> \u2248\u00a0 Cdx<\/em>A<\/em><\/sub> (i.e., pressure diffusion is negligible) and that the total diffusion flux is satisfactorily approximated by Equation\u00a023<\/a>. Note that the product DC<\/em> is independent of pressure, owing to the fact that the effective diffusion coefficient is inversely proportional to pressure. Click on these exercise links to view example problems <\/a>Exercise\u00a03<\/a> and <\/a>Exercise\u00a04<\/a>.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":19,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-56","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\/56","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":10,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/56\/revisions"}],"predecessor-version":[{"id":426,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/56\/revisions\/426"}],"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\/56\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/media?parent=56"}],"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=56"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/contributor?post=56"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/license?post=56"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}