{"id":75,"date":"2021-10-02T23:21:38","date_gmt":"2021-10-02T23:21:38","guid":{"rendered":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/exercise-5-solution\/"},"modified":"2022-01-10T05:53:59","modified_gmt":"2022-01-10T05:53:59","slug":"exercise-5-solution","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/exercise-5-solution\/","title":{"raw":"Exercise\u00a05 Solution","rendered":"Exercise\u00a05 Solution"},"content":{"raw":"
\r\n

Start with Equation\u00a027 for species B<\/em>.<\/p>\r\n

[latex]\\displaystyle N_{B}^{D}=-\\frac{DC\\frac{dx_{B}}{dl}}{\\left ( 1+\\frac{D}{D_{B}^{k}} \\right )-(1-M_{BA}^{0.5})x_{B}}[\/latex]<\/p>\r\n

Then, because x<\/em>A<\/em><\/sub> + x<\/em>B<\/em><\/sub>\u00a0=\u00a01 and [latex]M_{BA}^{0.5}=M_{AB}^{-0.5}[\/latex], the denominator is written as follows.<\/p>\r\n

[latex]\\displaystyle \\frac{D}{D_{B}^{K}}+1-(1-M_{BA}^{0.5})x_{B}[\/latex] [latex]\\displaystyle =\\frac{D}{D_{B}^{K}}+\\left\\{1-(1-M_{AB}^{0.5})x_{A} \\right\\}M_{AB}^{-0.5}[\/latex]<\/p>\r\n

With [latex]\\frac{D}{D_{B}^{K}}M_{AB}^{0.5}=\\frac{D}{D_{A}^{K}}[\/latex] from Equation\u00a019<\/a>, the flux of species B<\/em> is shown below.<\/p>\r\n

[latex]\\displaystyle N_{B}^{D}=\\frac{M_{AB}^{0.5}DC\\frac{dx_{A}}{dl}}{\\frac{D}{D_{A}^{K}}+1-(1-M_{AB}^{0.5})x_{A}}[\/latex]<\/p>\r\n

This combines with Equation\u00a027 to form Graham\u2019s law.<\/p>\r\n

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

Return to E<\/span>xercise\u00a0<\/span>5<\/span><\/a><\/p>\r\n

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

\n

Start with Equation\u00a027 for species B<\/em>.<\/p>\n

\"\displaystyle N_{B}^{D}=-\frac{DC\frac{dx_{B}}{dl}}{\left ( 1+\frac{D}{D_{B}^{k}} \right )-(1-M_{BA}^{0.5})x_{B}}\"<\/p>\n

Then, because x<\/em>A<\/em><\/sub> + x<\/em>B<\/em><\/sub>\u00a0=\u00a01 and \"M_{BA}^{0.5}=M_{AB}^{-0.5}\", the denominator is written as follows.<\/p>\n

\"\displaystyle \frac{D}{D_{B}^{K}}+1-(1-M_{BA}^{0.5})x_{B}\" \"\displaystyle =\frac{D}{D_{B}^{K}}+\left\{1-(1-M_{AB}^{0.5})x_{A} \right\}M_{AB}^{-0.5}\"<\/p>\n

With \"\frac{D}{D_{B}^{K}}M_{AB}^{0.5}=\frac{D}{D_{A}^{K}}\" from Equation\u00a019<\/a>, the flux of species B<\/em> is shown below.<\/p>\n

\"\displaystyle N_{B}^{D}=\frac{M_{AB}^{0.5}DC\frac{dx_{A}}{dl}}{\frac{D}{D_{A}^{K}}+1-(1-M_{AB}^{0.5})x_{A}}\"<\/p>\n

This combines with Equation\u00a027 to form Graham\u2019s law.<\/p>\n

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

Return to E<\/span>xercise\u00a0<\/span>5<\/span><\/a><\/p>\n

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