{"id":51,"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\/momentum-balance-for-the-gas-as-a-whole\/"},"modified":"2022-01-10T18:53:07","modified_gmt":"2022-01-10T18:53:07","slug":"momentum-balance-for-the-gas-as-a-whole","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/momentum-balance-for-the-gas-as-a-whole\/","title":{"raw":"4.3 Momentum Balance for the Gas as a Whole","rendered":"4.3 Momentum Balance for the Gas as a Whole"},"content":{"raw":"
\r\n

The momentum balance for component B<\/em> is obtained from Equation\u00a014<\/a> by replacing the subscript A<\/em> with B<\/em> to obtain Equation\u00a015.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle -\\frac{dp_{B}}{dl}=\\frac{RTJ_{B}}{D}+\\frac{RTN_{B}^{D}}{D_{B}^{K}}[\/latex]<\/td>\r\n(15)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The sum of Equations\u00a014<\/a> and 15 for the individual components is the momentum balance for the gas as a whole, and is given by Equation\u00a016.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle -\\frac{dp}{dl}=RT\\left \\{ \\frac{N_{A}^{D}}{D_{A}^{K}}+\\frac{N_{B}^{D}}{D_{B}^{K}} \\right \\}[\/latex]<\/td>\r\n(16)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The first term on the right side of Equation\u00a014<\/a> and Equation\u00a015 sum to zero as required by Equation\u00a07<\/a>, and is an expression of conservation of momentum for intermolecular collisions within the gas, as a whole as noted previously. Thus, Equation\u00a016 expresses the momentum balance for diffusion-initiated, molecule-particle collisions for the gas as a whole. No momentum loss due to viscous shear is accounted for in Equation 16. Under isobaric conditions Equation\u00a016 becomes Equation\u00a017.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\frac{N_{B}^{D}}{N_{A}^{D}}=-\\frac{D_{B}^{K}}{D_{A}^{K}}[\/latex]<\/td>\r\n(17)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>","rendered":"
\n

The momentum balance for component B<\/em> is obtained from Equation\u00a014<\/a> by replacing the subscript A<\/em> with B<\/em> to obtain Equation\u00a015.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle -\frac{dp_{B}}{dl}=\frac{RTJ_{B}}{D}+\frac{RTN_{B}^{D}}{D_{B}^{K}}\"<\/td>\n(15)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The sum of Equations\u00a014<\/a> and 15 for the individual components is the momentum balance for the gas as a whole, and is given by Equation\u00a016.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle -\frac{dp}{dl}=RT\left \{ \frac{N_{A}^{D}}{D_{A}^{K}}+\frac{N_{B}^{D}}{D_{B}^{K}} \right \}\"<\/td>\n(16)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The first term on the right side of Equation\u00a014<\/a> and Equation\u00a015 sum to zero as required by Equation\u00a07<\/a>, and is an expression of conservation of momentum for intermolecular collisions within the gas, as a whole as noted previously. Thus, Equation\u00a016 expresses the momentum balance for diffusion-initiated, molecule-particle collisions for the gas as a whole. No momentum loss due to viscous shear is accounted for in Equation 16. Under isobaric conditions Equation\u00a016 becomes Equation\u00a017.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle \frac{N_{B}^{D}}{N_{A}^{D}}=-\frac{D_{B}^{K}}{D_{A}^{K}}\"<\/td>\n(17)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n","protected":false},"author":1,"menu_order":14,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-51","chapter","type-chapter","status-publish","hentry"],"part":96,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/51","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":9,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/51\/revisions"}],"predecessor-version":[{"id":313,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/51\/revisions\/313"}],"part":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/parts\/96"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/51\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/media?parent=51"}],"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=51"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/contributor?post=51"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/license?post=51"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}