{"id":58,"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-a-pure-gas\/"},"modified":"2022-01-10T20:20:04","modified_gmt":"2022-01-10T20:20:04","slug":"transition-regime-a-pure-gas","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/transition-regime-a-pure-gas\/","title":{"raw":"5.4 Transition Regime - A Pure Gas","rendered":"5.4 Transition Regime – A Pure Gas"},"content":{"raw":"
\r\n

We have seen that the fluxes of individual species in a binary mixture are affected by mole-fraction diffusion, pressure diffusion and viscous flow. Only pressure diffusion and viscous flow occur in a pure gas. The diffusion flux for a single-species gas as shown in Equation\u00a030 follows immediately from Equation\u00a022<\/a>.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle N^{D}=-D^{K}\\frac{dC}{dl}[\/latex]<\/td>\r\n(30)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The viscous flux is added to the diffusion flux as usual to obtain the Equation\u00a031 for the mole flux of a pure gas.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle N=-\\left ( D^{K}+\\frac{k_{g}p}{\\mu } \\right )\\frac{dC}{dl}[\/latex]<\/td>\r\n(31)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Equation\u00a031 shows that the viscous flux calculated by Darcy\u2019s law is a satisfactory approximation of the total flux of a single component gas when conditions in the molecular regime satisfy the criterion expressed in Equation\u00a032.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\frac{\\mu D^{K}}{k_{g}p}< < 1[\/latex]<\/td>\r\n(32)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Klinkenberg (1941), as well as Heid and others (1950), present experimental data for the flux of air in response to pressure gradients in porous media with low permeability where the diffusion contribution is significant. These data and Equation\u00a031 play a key role in the following section wherein we address the estimation of numerical values for the Knudsen diffusion coefficients.<\/p>\r\n\r\n<\/div>","rendered":"

\n

We have seen that the fluxes of individual species in a binary mixture are affected by mole-fraction diffusion, pressure diffusion and viscous flow. Only pressure diffusion and viscous flow occur in a pure gas. The diffusion flux for a single-species gas as shown in Equation\u00a030 follows immediately from Equation\u00a022<\/a>.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle N^{D}=-D^{K}\frac{dC}{dl}\"<\/td>\n(30)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The viscous flux is added to the diffusion flux as usual to obtain the Equation\u00a031 for the mole flux of a pure gas.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle N=-\left ( D^{K}+\frac{k_{g}p}{\mu } \right )\frac{dC}{dl}\"<\/td>\n(31)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Equation\u00a031 shows that the viscous flux calculated by Darcy\u2019s law is a satisfactory approximation of the total flux of a single component gas when conditions in the molecular regime satisfy the criterion expressed in Equation\u00a032.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle \frac{\mu D^{K}}{k_{g}p}< < 1\"<\/td>\n(32)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Klinkenberg (1941), as well as Heid and others (1950), present experimental data for the flux of air in response to pressure gradients in porous media with low permeability where the diffusion contribution is significant. These data and Equation\u00a031 play a key role in the following section wherein we address the estimation of numerical values for the Knudsen diffusion coefficients.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":21,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-58","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\/58","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":5,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/58\/revisions"}],"predecessor-version":[{"id":305,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/58\/revisions\/305"}],"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\/58\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/media?parent=58"}],"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=58"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/contributor?post=58"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/license?post=58"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}