{"id":68,"date":"2021-10-02T23:21:37","date_gmt":"2021-10-02T23:21:37","guid":{"rendered":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/exercise-4\/"},"modified":"2022-01-09T17:58:48","modified_gmt":"2022-01-09T17:58:48","slug":"exercise-4","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/exercise-4\/","title":{"raw":"Exercise\u00a04","rendered":"Exercise\u00a04"},"content":{"raw":"
\r\n

Aerobic biodegradation of petroleum hydrocarbons present in the vadose zone is manifest by a predominately upward flux of carbon dioxide. The magnitude of CO2<\/sub> flux is an indicator of the rate of depletion of the liquid-phase hydrocarbon. Among the various methods that have been used to estimate the CO2<\/sub> flux in the field is the so-called gradient method, by which the upward flux is computed from a measured concentration distribution and the effective diffusion coefficient. More than two gas components are involved in the depletion of a liquid-phase hydrocarbon by aerobic biodegradation. Nevertheless, it is instructive to revise the gradient method using the flux equations for a binary gas in which air and carbon dioxide are assumed to be the only two constituents.<\/p>\r\n

Refer to Figure\u00a0Exercise\u00a04-1 and derive an equation by which the steady CO2<\/sub> flux can be calculated from a measured concentration distribution and a known effective diffusion coefficient. Assume the molecular regime prevails, pressure diffusion is negligible, and air can be considered to be a single species (species A<\/em>).<\/p>\r\n

Compute the mole flux of carbon dioxide, phase (bulk gas) flux, non-equimolar flux, and viscous flux using the following data from Tracy (2015).<\/p>\r\n

p<\/em> = 8.3 \u00d7 104<\/sup> Pa<\/p>\r\n

T<\/em> = 21.6 \u00b0C<\/p>\r\n

D<\/em> = 4.7 \u00d7 10-6<\/sup> m2<\/sup>\/s<\/p>\r\n

z<\/em>2<\/sub> - z<\/em>1<\/sub> = 1.29 m<\/p>\r\n

x<\/em>B<\/em><\/sub>(z<\/em>1<\/sub>) = 0.0583<\/p>\r\n

x<\/em>B<\/em><\/sub>(z<\/em>2<\/sub>) = 0.0013<\/p>\r\n

[latex]\\displaystyle M_{BA}^{0.5}= 1.232[\/latex]<\/p>\r\n \r\n

\"Sketch<\/strong><\/p>\r\n

Figure\u00a0<\/strong>E<\/strong>xercise\u00a0<\/strong>4<\/strong>-<\/strong>1<\/strong>-<\/strong>Sketch of steady diffusion of carbon dioxide in the vadose zone from a source at the water table.<\/p>\r\n

Click here for solution to <\/span>Exercise\u00a0<\/span>4<\/span><\/a><\/p>\r\n

Click here <\/span>to return to where the text links to <\/span>Exercise\u00a0<\/span>4.<\/span><\/a><\/p>\r\n\r\n<\/div>","rendered":"

\n

Aerobic biodegradation of petroleum hydrocarbons present in the vadose zone is manifest by a predominately upward flux of carbon dioxide. The magnitude of CO2<\/sub> flux is an indicator of the rate of depletion of the liquid-phase hydrocarbon. Among the various methods that have been used to estimate the CO2<\/sub> flux in the field is the so-called gradient method, by which the upward flux is computed from a measured concentration distribution and the effective diffusion coefficient. More than two gas components are involved in the depletion of a liquid-phase hydrocarbon by aerobic biodegradation. Nevertheless, it is instructive to revise the gradient method using the flux equations for a binary gas in which air and carbon dioxide are assumed to be the only two constituents.<\/p>\n

Refer to Figure\u00a0Exercise\u00a04-1 and derive an equation by which the steady CO2<\/sub> flux can be calculated from a measured concentration distribution and a known effective diffusion coefficient. Assume the molecular regime prevails, pressure diffusion is negligible, and air can be considered to be a single species (species A<\/em>).<\/p>\n

Compute the mole flux of carbon dioxide, phase (bulk gas) flux, non-equimolar flux, and viscous flux using the following data from Tracy (2015).<\/p>\n

p<\/em> = 8.3 \u00d7 104<\/sup> Pa<\/p>\n

T<\/em> = 21.6 \u00b0C<\/p>\n

D<\/em> = 4.7 \u00d7 10-6<\/sup> m2<\/sup>\/s<\/p>\n

z<\/em>2<\/sub> – z<\/em>1<\/sub> = 1.29 m<\/p>\n

x<\/em>B<\/em><\/sub>(z<\/em>1<\/sub>) = 0.0583<\/p>\n

x<\/em>B<\/em><\/sub>(z<\/em>2<\/sub>) = 0.0013<\/p>\n

\"\displaystyle M_{BA}^{0.5}= 1.232\"<\/p>\n

 <\/p>\n

\"Sketch<\/strong><\/p>\n

Figure\u00a0<\/strong>E<\/strong>xercise\u00a0<\/strong>4<\/strong>–<\/strong>1<\/strong>–<\/strong>Sketch of steady diffusion of carbon dioxide in the vadose zone from a source at the water table.<\/p>\n

Click here for solution to <\/span>Exercise\u00a0<\/span>4<\/span><\/a><\/p>\n

Click here <\/span>to return to where the text links to <\/span>Exercise\u00a0<\/span>4.<\/span><\/a><\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":29,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-68","chapter","type-chapter","status-publish","hentry"],"part":119,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/68","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":6,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/68\/revisions"}],"predecessor-version":[{"id":348,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/68\/revisions\/348"}],"part":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/parts\/119"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/68\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/media?parent=68"}],"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=68"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/contributor?post=68"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/license?post=68"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}