{"id":43,"date":"2021-10-02T23:21:34","date_gmt":"2021-10-02T23:21:34","guid":{"rendered":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/fluxes-that-comprise-total-diffusion-flux\/"},"modified":"2022-01-10T18:18:37","modified_gmt":"2022-01-10T18:18:37","slug":"fluxes-that-comprise-total-diffusion-flux","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/fluxes-that-comprise-total-diffusion-flux\/","title":{"raw":"2.3 Fluxes That Comprise Total Diffusion Flux","rendered":"2.3 Fluxes That Comprise Total Diffusion Flux"},"content":{"raw":"
\r\n

Graham\u2019s (1833) experiments clearly demonstrate that the components of a binary gas at uniform pressure in a porous medium diffuse at different rates in general. Consequently, the sum of the total diffusive fluxes of the individual components, [latex]\\inline N_{A}^{D}+N_{B}^{D}[\/latex], is not zero. Rather, this sum contributes to the motion of the fluid as a whole\u2014a feature of diffusion in porous solids that is not observable in systems free of solid obstructions. We refer to this net diffusive flux as the non-equimolar flux (Cunningham and Williams, 1980) and write Equation\u00a05.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle N^{D}=N_{A}^{D}+N_{B}^{D}[\/latex]<\/td>\r\n(5)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Similar to the viscous flux, the non-equimolar flux imparts motion to the individual species in the mixture by advection (i.e., x<\/em>i<\/em><\/sub> N<\/em>D<\/em><\/sup>, <\/em>i<\/em>\u00a0=\u00a0A,\u00a0B<\/em>), but is distinguished from advection via a viscous flux by the fact it arises solely as a result of diffusion. The sum of advection by the non-equimolar flux and by the viscous flux is the total advection by the phase motion.<\/p>\r\n

The increment of motion for component i<\/em> that is in addition to advection via the phase motion is the equimolar diffusion flux J<\/em>i<\/em><\/sub> defined by Equation\u00a06.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle J_{i}=N_{i}-x_{i}N[\/latex] , \u00a0\u00a0\u00a0\u00a0 [latex]\\displaystyle i=A,B[\/latex]<\/td>\r\n(6)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Because equimolar diffusion makes no net contribution to motion of the phase, we have Equation\u00a07.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle J_{A}+J_{B}=0[\/latex]<\/td>\r\n(7)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Equation 7 expresses a condition that holds under all circumstances treated in this book.<\/p>\r\n

It is common to rearrange Equation 6 so that the mole flux is expressed as the sum of the equimolar and advection fluxes as in Equation 8a for constituent A<\/em>. We may then express the flux of component A<\/em> by any one of Equations\u00a08a through 8d. The subscripts can be interchanged to obtain the equivalent expressions for component B<\/em>.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle N_{A}=J_{A}+x_{A}N[\/latex]<\/td>\r\n(8a)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle N_{A}=J_{A}+x_{A}\\left ( N_{A}+N_{B} \\right )[\/latex]<\/td>\r\n(8b)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle N_{A}=J_{A}+x_{A}\\left ( N^{D}+N^{v} \\right )[\/latex]<\/td>\r\n(8c)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle N_{A}=J_{A}+x_{A}\\left ( N_{A}^{D}+N_{B}^{D}+N^{v} \\right )[\/latex]<\/td>\r\n(8d)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The reader is encouraged to become thoroughly familiar with these definitional equations and the various forms they may take. For example, if there is no viscous flux then [latex]N^{v}=0[\/latex], [latex]N_{A}=N_{A}^{D}[\/latex] and Equation\u00a08d becomes Equation\u00a09.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle N_{A}^{D}=J_{A}+x_{A}\\left ( N_{A}^{D}+N_{B}^{D} \\right )[\/latex]<\/td>\r\n(9)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Rearranging and solving for J<\/em>A<\/em><\/sub> gives Equation\u00a010.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle J_{A}=N_{A}^{D}x_{B}-x_{A}N_{B}^{D}[\/latex]<\/td>\r\n(10)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

This is in the form of a Stefan-Maxwell equation for a binary gas that we soon will have occasion to use in our calculations.<\/p>\r\n\r\n<\/div>","rendered":"

\n

Graham\u2019s (1833) experiments clearly demonstrate that the components of a binary gas at uniform pressure in a porous medium diffuse at different rates in general. Consequently, the sum of the total diffusive fluxes of the individual components, \"\inline N_{A}^{D}+N_{B}^{D}\", is not zero. Rather, this sum contributes to the motion of the fluid as a whole\u2014a feature of diffusion in porous solids that is not observable in systems free of solid obstructions. We refer to this net diffusive flux as the non-equimolar flux (Cunningham and Williams, 1980) and write Equation\u00a05.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle N^{D}=N_{A}^{D}+N_{B}^{D}\"<\/td>\n(5)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Similar to the viscous flux, the non-equimolar flux imparts motion to the individual species in the mixture by advection (i.e., x<\/em>i<\/em><\/sub> N<\/em>D<\/em><\/sup>, <\/em>i<\/em>\u00a0=\u00a0A,\u00a0B<\/em>), but is distinguished from advection via a viscous flux by the fact it arises solely as a result of diffusion. The sum of advection by the non-equimolar flux and by the viscous flux is the total advection by the phase motion.<\/p>\n

The increment of motion for component i<\/em> that is in addition to advection via the phase motion is the equimolar diffusion flux J<\/em>i<\/em><\/sub> defined by Equation\u00a06.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle J_{i}=N_{i}-x_{i}N\" , \u00a0\u00a0\u00a0\u00a0 \"\displaystyle i=A,B\"<\/td>\n(6)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Because equimolar diffusion makes no net contribution to motion of the phase, we have Equation\u00a07.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle J_{A}+J_{B}=0\"<\/td>\n(7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Equation 7 expresses a condition that holds under all circumstances treated in this book.<\/p>\n

It is common to rearrange Equation 6 so that the mole flux is expressed as the sum of the equimolar and advection fluxes as in Equation 8a for constituent A<\/em>. We may then express the flux of component A<\/em> by any one of Equations\u00a08a through 8d. The subscripts can be interchanged to obtain the equivalent expressions for component B<\/em>.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle N_{A}=J_{A}+x_{A}N\"<\/td>\n(8a)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n
<\/td>\n\"\displaystyle N_{A}=J_{A}+x_{A}\left ( N_{A}+N_{B} \right )\"<\/td>\n(8b)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n
<\/td>\n\"\displaystyle N_{A}=J_{A}+x_{A}\left ( N^{D}+N^{v} \right )\"<\/td>\n(8c)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n
<\/td>\n\"\displaystyle N_{A}=J_{A}+x_{A}\left ( N_{A}^{D}+N_{B}^{D}+N^{v} \right )\"<\/td>\n(8d)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The reader is encouraged to become thoroughly familiar with these definitional equations and the various forms they may take. For example, if there is no viscous flux then \"N^{v}=0\", \"N_{A}=N_{A}^{D}\" and Equation\u00a08d becomes Equation\u00a09.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle N_{A}^{D}=J_{A}+x_{A}\left ( N_{A}^{D}+N_{B}^{D} \right )\"<\/td>\n(9)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Rearranging and solving for J<\/em>A<\/em><\/sub> gives Equation\u00a010.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle J_{A}=N_{A}^{D}x_{B}-x_{A}N_{B}^{D}\"<\/td>\n(10)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

This is in the form of a Stefan-Maxwell equation for a binary gas that we soon will have occasion to use in our calculations.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":8,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-43","chapter","type-chapter","status-publish","hentry"],"part":86,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/43","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":14,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/43\/revisions"}],"predecessor-version":[{"id":375,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/43\/revisions\/375"}],"part":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/parts\/86"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/43\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/media?parent=43"}],"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=43"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/contributor?post=43"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/license?post=43"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}