{"id":72,"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-2-solution\/"},"modified":"2022-01-10T21:39:29","modified_gmt":"2022-01-10T21:39:29","slug":"exercise-2-solution","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/exercise-2-solution\/","title":{"raw":"Exercise\u00a02 Solution","rendered":"Exercise\u00a02 Solution"},"content":{"raw":"
\r\n

Start with Equation\u00a05<\/a>, which defines the non-equimolar flux.<\/p>\r\n

[latex]\\displaystyle N^{D}\\equiv N_{A}^{D}+N_{B}^{D}=(1+N_{B}^{D}\/N_{A}^{D})N_{A}^{D}[\/latex]<\/p>\r\n

Use Graham\u2019s Law to obtain the following expression.<\/p>\r\n

[latex]\\displaystyle N^{D}=(1-M_{AB}^{0.5})N_{A}^{D}[\/latex]<\/p>\r\n

Then replace the total diffusion flux of component A<\/em> using Equation\u00a023<\/a> to obtain the desired constitutive equation (Equation\u00a0Exercise\u00a0Solution\u00a02-1).<\/p>\r\n\r\n\r\n\r\n\r\n
[latex]\\displaystyle N^{D}=-\\frac{\\left ( 1-M_{AB}^{0.5}DCdx_{A}\/dl \\right )}{1-\\left ( 1-M_{AB}^{0.5} \\right )x_{A}}[\/latex] [latex]\\displaystyle =-\\frac{\\left ( 1-M_{BA}^{0.5}DCdx_{B}\/dl \\right )}{1-\\left ( 1-M_{BA}^{0.5} \\right )x_{B}}[\/latex]<\/td>\r\n(Exercise Solution 2-1)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Underpinning this development is the prescription of uniform pressure. However, Section\u00a05.2<\/a> illustrates that Equation\u00a0Exercise\u00a0Solution\u00a02-1 is a satisfactory approximation under non-isobaric conditions when diffusion that is driven by variable pressure is negligible relative to that driven by gradients of mole fraction.<\/p>\r\n

Return to E<\/span>xercise\u00a0<\/span>2<\/span><\/a><\/p>\r\n

<\/p>\r\n\r\n<\/div>","rendered":"

\n

Start with Equation\u00a05<\/a>, which defines the non-equimolar flux.<\/p>\n

\"\displaystyle N^{D}\equiv N_{A}^{D}+N_{B}^{D}=(1+N_{B}^{D}/N_{A}^{D})N_{A}^{D}\"<\/p>\n

Use Graham\u2019s Law to obtain the following expression.<\/p>\n

\"\displaystyle N^{D}=(1-M_{AB}^{0.5})N_{A}^{D}\"<\/p>\n

Then replace the total diffusion flux of component A<\/em> using Equation\u00a023<\/a> to obtain the desired constitutive equation (Equation\u00a0Exercise\u00a0Solution\u00a02-1).<\/p>\n\n\n\n
\"\displaystyle N^{D}=-\frac{\left ( 1-M_{AB}^{0.5}DCdx_{A}/dl \right )}{1-\left ( 1-M_{AB}^{0.5} \right )x_{A}}\" \"\displaystyle =-\frac{\left ( 1-M_{BA}^{0.5}DCdx_{B}/dl \right )}{1-\left ( 1-M_{BA}^{0.5} \right )x_{B}}\"<\/td>\n(Exercise Solution 2-1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Underpinning this development is the prescription of uniform pressure. However, Section\u00a05.2<\/a> illustrates that Equation\u00a0Exercise\u00a0Solution\u00a02-1 is a satisfactory approximation under non-isobaric conditions when diffusion that is driven by variable pressure is negligible relative to that driven by gradients of mole fraction.<\/p>\n

Return to E<\/span>xercise\u00a0<\/span>2<\/span><\/a><\/p>\n

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