{"id":49,"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-in-a-particle-free-system\/"},"modified":"2022-01-10T18:39:25","modified_gmt":"2022-01-10T18:39:25","slug":"momentum-balance-in-a-particle-free-system","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/momentum-balance-in-a-particle-free-system\/","title":{"raw":"4.1 Momentum Balance in a Particle-Free System","rendered":"4.1 Momentum Balance in a Particle-Free System"},"content":{"raw":"
\r\n

Newton\u2019s law of motion relevant to the present discussion can be simply stated: the sum of the forces acting to drive motion is equal in magnitude and opposite in direction to the sum of the forces acting to resist the motion. It is helpful to examine the familiar Darcy\u2019s law in the context of this principle before undertaking the application to diffusion. Equation\u00a01<\/a> can be rearranged to -<\/em>dp<\/em>\/dl\u00a0=\u00a0v<\/em>\u03bc<\/em>\/k<\/em>g<\/em><\/sub>. In this form the left side is the driving force per unit volume and the right side is the force per unit volume that resists the motion. The quantity on the right is a macroscopic manifestation of the pore-scale rate of momentum loss per unit volume due to viscous shear within the complex and unknowable geometry of the pore space (Hubbert, 1956; Corey, 1994).<\/p>\r\n

In a similar way, Fick\u2019s law represents a macroscopic momentum balance for diffusion in systems free of any solid particles or other obstructions. The interpretation of Fick\u2019s law as a momentum balance (i.e., force balance) is easier to grasp if Equation\u00a011<\/a> is written in terms of the gradient of partial pressure. For species A<\/em>, we obtain Equation 13.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle -\\frac{dp_{A}}{dl}=\\frac{RTJ_{A}}{D_{m}}[\/latex]<\/td>\r\n(13)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The molecular diffusion coefficient is used here because there are no obstructions in the case under discussion. The left side of Equation\u00a013 is the driving force per unit volume and the right side is the resisting force per unit volume or, equivalently, the rate of change of momentum per unit volume for species A<\/em> due to intermolecular collisions with species B<\/em> within a local volume element. The molecular diffusion coefficient, D<\/em>m<\/em><\/sub>, is the phenomenological parameter that accounts for the complicated, unresolved molecular-scale process of intermolecular collisions within the volume element, just as the permeability appearing in Darcy\u2019s law is the macroscopic manifestation of the process of viscous shear that occurs in the complex geometry of internal pore space.<\/p>\r\n

In this case of binary diffusion in space free of solid particles, the only mechanism by which the momentum of component A<\/em> can be lost is through momentum exchange with component B<\/em>. In the absence of solid particles, there can be no momentum loss by molecule-particle collisions or by viscous shear. Thus, p<\/em>A<\/em><\/sub>\u00a0+\u00a0p<\/em>B<\/em><\/sub>\u00a0=\u00a0p\u00a0=\u00a0constant<\/em> and the only driving force for the flux, J<\/em>i<\/em><\/sub>, <\/em>i<\/em>\u00a0=\u00a0A,<\/em>\u00a0<\/em>B<\/em> is proportional to the gradient of the mole fraction as given in Equation\u00a011<\/a>. As mentioned previously, J<\/em>A<\/em><\/sub>\u00a0+\u00a0J<\/em>B<\/em><\/sub>\u00a0=\u00a00<\/em>, as required by Equation\u00a07<\/a>. We now see that this means that intermolecular collisions do not contribute to a change in the momentum of the gas as a whole.<\/p>\r\n\r\n<\/div>","rendered":"

\n

Newton\u2019s law of motion relevant to the present discussion can be simply stated: the sum of the forces acting to drive motion is equal in magnitude and opposite in direction to the sum of the forces acting to resist the motion. It is helpful to examine the familiar Darcy\u2019s law in the context of this principle before undertaking the application to diffusion. Equation\u00a01<\/a> can be rearranged to –<\/em>dp<\/em>\/dl\u00a0=\u00a0v<\/em>\u03bc<\/em>\/k<\/em>g<\/em><\/sub>. In this form the left side is the driving force per unit volume and the right side is the force per unit volume that resists the motion. The quantity on the right is a macroscopic manifestation of the pore-scale rate of momentum loss per unit volume due to viscous shear within the complex and unknowable geometry of the pore space (Hubbert, 1956; Corey, 1994).<\/p>\n

In a similar way, Fick\u2019s law represents a macroscopic momentum balance for diffusion in systems free of any solid particles or other obstructions. The interpretation of Fick\u2019s law as a momentum balance (i.e., force balance) is easier to grasp if Equation\u00a011<\/a> is written in terms of the gradient of partial pressure. For species A<\/em>, we obtain Equation 13.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle -\frac{dp_{A}}{dl}=\frac{RTJ_{A}}{D_{m}}\"<\/td>\n(13)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The molecular diffusion coefficient is used here because there are no obstructions in the case under discussion. The left side of Equation\u00a013 is the driving force per unit volume and the right side is the resisting force per unit volume or, equivalently, the rate of change of momentum per unit volume for species A<\/em> due to intermolecular collisions with species B<\/em> within a local volume element. The molecular diffusion coefficient, D<\/em>m<\/em><\/sub>, is the phenomenological parameter that accounts for the complicated, unresolved molecular-scale process of intermolecular collisions within the volume element, just as the permeability appearing in Darcy\u2019s law is the macroscopic manifestation of the process of viscous shear that occurs in the complex geometry of internal pore space.<\/p>\n

In this case of binary diffusion in space free of solid particles, the only mechanism by which the momentum of component A<\/em> can be lost is through momentum exchange with component B<\/em>. In the absence of solid particles, there can be no momentum loss by molecule-particle collisions or by viscous shear. Thus, p<\/em>A<\/em><\/sub>\u00a0+\u00a0p<\/em>B<\/em><\/sub>\u00a0=\u00a0p\u00a0=\u00a0constant<\/em> and the only driving force for the flux, J<\/em>i<\/em><\/sub>, <\/em>i<\/em>\u00a0=\u00a0A,<\/em>\u00a0<\/em>B<\/em> is proportional to the gradient of the mole fraction as given in Equation\u00a011<\/a>. As mentioned previously, J<\/em>A<\/em><\/sub>\u00a0+\u00a0J<\/em>B<\/em><\/sub>\u00a0=\u00a00<\/em>, as required by Equation\u00a07<\/a>. We now see that this means that intermolecular collisions do not contribute to a change in the momentum of the gas as a whole.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":12,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-49","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\/49","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\/49\/revisions"}],"predecessor-version":[{"id":310,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/49\/revisions\/310"}],"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\/49\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/media?parent=49"}],"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=49"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/contributor?post=49"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/license?post=49"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}