{"id":60,"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\/effective-molecular-diffusion-coefficient\/"},"modified":"2022-01-10T22:05:45","modified_gmt":"2022-01-10T22:05:45","slug":"effective-molecular-diffusion-coefficient","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/effective-molecular-diffusion-coefficient\/","title":{"raw":"6.1 Effective Molecular Diffusion Coefficient","rendered":"6.1 Effective Molecular Diffusion Coefficient"},"content":{"raw":"
\r\n

The molecular diffusion coefficient, D<\/em>m<\/em><\/sub> (L2<\/sup>\/T), that appears in Equation\u00a013<\/a> applies when the diffusion space is free of obstruction. This is the parameter that has been widely measured and tabulated in handbooks and reference works. Typical values for common gas pairs often fall in the range 1\u00a0\u00d7\u00a010\u2212<\/sup>5<\/sup> to 1\u00a0\u00d7\u00a010\u2212<\/sup>4<\/sup>\u00a0m2<\/sup>\/s. It can be shown from kinetic theory that diffusion in a particular gas pair is characterized by a single molecular diffusion coefficient (Cunningham and Williams, 1980). Further, for the conditions of interest in this book, the molecular diffusion coefficient is proportional to p<\/em>-1<\/sup> and T<\/em>2\/3<\/sup> while being practically independent of composition.<\/p>\r\n

The existence of obstructions in the diffusion space causes the effective (macroscopic) molecular diffusion coefficient to be less than the molecular diffusion coefficient applicable in particle-free space. The effective diffusion coefficient must be determined by an appropriate direct measurement for the problem at hand or estimated by suitable adjustments to the free space coefficient, D<\/em>m<\/em><\/sub>. The effective molecular diffusion coefficient is often calculated from Equation\u00a033.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle D=\\theta _{g}\\tau D_{m}[\/latex]<\/td>\r\n(33)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
\u03b8<\/em>g<\/em><\/sub><\/td>\r\n=<\/td>\r\nfraction of bulk porous medium that is occupied by gas, i.e., volumetric gas content (dimensionless)<\/td>\r\n<\/tr>\r\n
\u03c4<\/em><\/td>\r\n=<\/td>\r\ntortuosity (dimensionless)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The volumetric gas content accounts for the reduction of area available for gas diffusion because part of the macroscopic cross-sectional area is occupied by solids and other fluids. The tortuosity adjusts for the fact that the observable macroscopic distance between two macroscopic points is less than the distance along the tortuous pore-scale travel path. The values of both parameters are less than unity.<\/p>\r\n

A popular empirical expression for tortuosity is shown in Equation\u00a034 (Millington and Quirk, 1961).<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\tau =\\frac{\\theta {_{g}}^{7\/3}}{n^{2}}[\/latex]<\/td>\r\n(34)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n
n<\/em><\/td>\r\n=<\/td>\r\nporosity (dimensionless)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

For a dry porous medium, the volumetric gas content equals porosity and the corresponding tortuosity is n<\/em>0.33<\/sup>. Dry, loose sand with a porosity of 0.35 can be expected to have a tortuosity of about 0.7, according to Equation\u00a034. Notice, however, the tortuosity falls rapidly as the gas content decreases. Furthermore, the volumetric gas content itself is highly sensitive to the heterogeneity of the porous medium. Consequently, the effective diffusion coefficient can be expected to exhibit large, even extreme, spatial variability in field problems. A single fine-grained layer oriented normal to the diffusion path, even if very thin, may effectively block and re-direct gas-phase diffusion because of small gas content.<\/p>\r\n

Calculation of the effective diffusion coefficient by the procedure outlined above (or similar calculation) is popular because it is simple and inexpensive. Field measurements of the effective diffusion coefficient offer an alternative when they can be made with reasonable effort and reliability. Field measurements are accomplished by fitting a suitable diffusion model to appropriate field measurements (e.g., Johnson et al., 1998; Kreamer et al., 1988; Weeks et al., 1982). Values of the effective diffusion coefficient determined in this way are automatically macroscopic values and reflect the effects of spatial variability over some scale.<\/p>\r\n

The method presented by Johnson et al. (1998) is a particularly innovative and practical example of this procedure. A known mass M<\/em>0<\/em><\/sub> of inert tracer gas that does not significantly partition into water is mixed with air in a small volume, V<\/em>0<\/em><\/sub>. This volume is injected at a \u201cpoint\u201d from which it spreads radially by diffusion. A measured volume V<\/em>S<\/em><\/sub> of the gas mixture, larger than the injected volume, is subsequently withdrawn (at time t<\/em>S<\/em><\/sub>) from the point of injection and the mass M<\/em>(t<\/em>S<\/em><\/sub>) of extracted tracer gas is determined. The effective diffusion coefficient is calculated from Equation\u00a035.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\beta =\\frac{\\theta {_{g}}^{1\/3}}{4t_{S}\\beta }\\left ( \\frac{3V_{S}}{4\\pi } \\right )^{2\/3}[\/latex]<\/td>\r\n(35)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n
\u03b2<\/em><\/td>\r\n=<\/td>\r\na dimensionless parameter<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The dimensionless parameter \u03b2<\/em> only depends upon the ratio of recovered to injected tracer mass and is determined from the implicit Equation\u00a036.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\frac{M(t_{S})}{M_{0}}=\\textup{erf}(\\beta ^{1\/2})-2\\left ( \\frac{\\beta }{\\pi } \\right )^{1\/2}\\exp (-\\beta )[\/latex]<\/td>\r\n(36)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The volumetric gas content appearing in Equation\u00a035, <\/span>\u03b8<\/em>g<\/em><\/sub>,<\/span> is usually not measured. However, the calculation of D<\/em> from Equation\u00a035 is not very sensitive to \u03b8<\/em>g<\/em><\/sub> and a reasonable estimate for \u03b8<\/em>g<\/em><\/sub> is sufficient in most cases.<\/p>\r\n

Equations\u00a035 and 36 follow from the solution of Fick\u2019s law-based partial differential equation for unsteady radial diffusion from a point source. The developments in this book predict that transient non-equimolar and viscous fluxes are engendered by unsteady diffusion from a point source and in many other cases of unsteady diffusion. There exists a paucity of solutions for unsteady diffusion that are based on the flux equations of this book, and it remains unknown whether solutions derived from differential equations based on Fick\u2019s law constitute suitable models in the sense used above. However, any differences between effective diffusion coefficients derived by fitting either Fick\u2019s law-based models or models based on the flux equations of this book will likely be overwhelmed by the range of uncertainty in the determination, irrespective of which flux equations are used.<\/p>\r\n\r\n<\/div>","rendered":"

\n

The molecular diffusion coefficient, D<\/em>m<\/em><\/sub> (L2<\/sup>\/T), that appears in Equation\u00a013<\/a> applies when the diffusion space is free of obstruction. This is the parameter that has been widely measured and tabulated in handbooks and reference works. Typical values for common gas pairs often fall in the range 1\u00a0\u00d7\u00a010\u2212<\/sup>5<\/sup> to 1\u00a0\u00d7\u00a010\u2212<\/sup>4<\/sup>\u00a0m2<\/sup>\/s. It can be shown from kinetic theory that diffusion in a particular gas pair is characterized by a single molecular diffusion coefficient (Cunningham and Williams, 1980). Further, for the conditions of interest in this book, the molecular diffusion coefficient is proportional to p<\/em>-1<\/sup> and T<\/em>2\/3<\/sup> while being practically independent of composition.<\/p>\n

The existence of obstructions in the diffusion space causes the effective (macroscopic) molecular diffusion coefficient to be less than the molecular diffusion coefficient applicable in particle-free space. The effective diffusion coefficient must be determined by an appropriate direct measurement for the problem at hand or estimated by suitable adjustments to the free space coefficient, D<\/em>m<\/em><\/sub>. The effective molecular diffusion coefficient is often calculated from Equation\u00a033.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle D=\theta _{g}\tau D_{m}\"<\/td>\n(33)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n
\u03b8<\/em>g<\/em><\/sub><\/td>\n=<\/td>\nfraction of bulk porous medium that is occupied by gas, i.e., volumetric gas content (dimensionless)<\/td>\n<\/tr>\n
\u03c4<\/em><\/td>\n=<\/td>\ntortuosity (dimensionless)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The volumetric gas content accounts for the reduction of area available for gas diffusion because part of the macroscopic cross-sectional area is occupied by solids and other fluids. The tortuosity adjusts for the fact that the observable macroscopic distance between two macroscopic points is less than the distance along the tortuous pore-scale travel path. The values of both parameters are less than unity.<\/p>\n

A popular empirical expression for tortuosity is shown in Equation\u00a034 (Millington and Quirk, 1961).<\/p>\n\n\n\n
<\/td>\n\"\displaystyle \tau =\frac{\theta {_{g}}^{7/3}}{n^{2}}\"<\/td>\n(34)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n
n<\/em><\/td>\n=<\/td>\nporosity (dimensionless)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

For a dry porous medium, the volumetric gas content equals porosity and the corresponding tortuosity is n<\/em>0.33<\/sup>. Dry, loose sand with a porosity of 0.35 can be expected to have a tortuosity of about 0.7, according to Equation\u00a034. Notice, however, the tortuosity falls rapidly as the gas content decreases. Furthermore, the volumetric gas content itself is highly sensitive to the heterogeneity of the porous medium. Consequently, the effective diffusion coefficient can be expected to exhibit large, even extreme, spatial variability in field problems. A single fine-grained layer oriented normal to the diffusion path, even if very thin, may effectively block and re-direct gas-phase diffusion because of small gas content.<\/p>\n

Calculation of the effective diffusion coefficient by the procedure outlined above (or similar calculation) is popular because it is simple and inexpensive. Field measurements of the effective diffusion coefficient offer an alternative when they can be made with reasonable effort and reliability. Field measurements are accomplished by fitting a suitable diffusion model to appropriate field measurements (e.g., Johnson et al., 1998; Kreamer et al., 1988; Weeks et al., 1982). Values of the effective diffusion coefficient determined in this way are automatically macroscopic values and reflect the effects of spatial variability over some scale.<\/p>\n

The method presented by Johnson et al. (1998) is a particularly innovative and practical example of this procedure. A known mass M<\/em>0<\/em><\/sub> of inert tracer gas that does not significantly partition into water is mixed with air in a small volume, V<\/em>0<\/em><\/sub>. This volume is injected at a \u201cpoint\u201d from which it spreads radially by diffusion. A measured volume V<\/em>S<\/em><\/sub> of the gas mixture, larger than the injected volume, is subsequently withdrawn (at time t<\/em>S<\/em><\/sub>) from the point of injection and the mass M<\/em>(t<\/em>S<\/em><\/sub>) of extracted tracer gas is determined. The effective diffusion coefficient is calculated from Equation\u00a035.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle \beta =\frac{\theta {_{g}}^{1/3}}{4t_{S}\beta }\left ( \frac{3V_{S}}{4\pi } \right )^{2/3}\"<\/td>\n(35)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n
\u03b2<\/em><\/td>\n=<\/td>\na dimensionless parameter<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The dimensionless parameter \u03b2<\/em> only depends upon the ratio of recovered to injected tracer mass and is determined from the implicit Equation\u00a036.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle \frac{M(t_{S})}{M_{0}}=\textup{erf}(\beta ^{1/2})-2\left ( \frac{\beta }{\pi } \right )^{1/2}\exp (-\beta )\"<\/td>\n(36)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The volumetric gas content appearing in Equation\u00a035, <\/span>\u03b8<\/em>g<\/em><\/sub>,<\/span> is usually not measured. However, the calculation of D<\/em> from Equation\u00a035 is not very sensitive to \u03b8<\/em>g<\/em><\/sub> and a reasonable estimate for \u03b8<\/em>g<\/em><\/sub> is sufficient in most cases.<\/p>\n

Equations\u00a035 and 36 follow from the solution of Fick\u2019s law-based partial differential equation for unsteady radial diffusion from a point source. The developments in this book predict that transient non-equimolar and viscous fluxes are engendered by unsteady diffusion from a point source and in many other cases of unsteady diffusion. There exists a paucity of solutions for unsteady diffusion that are based on the flux equations of this book, and it remains unknown whether solutions derived from differential equations based on Fick\u2019s law constitute suitable models in the sense used above. However, any differences between effective diffusion coefficients derived by fitting either Fick\u2019s law-based models or models based on the flux equations of this book will likely be overwhelmed by the range of uncertainty in the determination, irrespective of which flux equations are used.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":23,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-60","chapter","type-chapter","status-publish","hentry"],"part":111,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/60","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":10,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/60\/revisions"}],"predecessor-version":[{"id":428,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/60\/revisions\/428"}],"part":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/parts\/111"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/60\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/media?parent=60"}],"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=60"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/contributor?post=60"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/license?post=60"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}