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csh_lecture6_diffusion

# csh_lecture6_diffusion - CWR 6537 Subsurface Contaminant...

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CWR 6537 Subsurface Contaminant Hydrology Lecture 6 1 SOLUTE DIFFUSION IN POROUS MEDIA A. Lecture Goal To understand solute transport via molecular diffusion in porous media. B. Environmental Importance of Diffusion Diffusion results from the thermally induced agitation of molecules (i.e., Brownian motion). In gases diffusion progresses at a rate of approximately 10 cm/min; in liquids about 0.05 cm/min and in solids about 0.00001 cm/min. Important diffusion processes in porous media include: (i) diffusion of water vapor or organic vapors; (ii) diffusion of gases (O 2 , CO 2 , N 2 , etc.); (iii) diffusion of nutrients away from fertilizer granules and/or bands; (iv) diffusion of nutrients towards plant roots; and (v) diffusion of contaminants in the absence of advective flow. (vi) diffusion is the dominant rate-limiting step for many physico-chemical processes of relevance in contaminant advective transport. Diffusion occurs in the fluid phase (liquid and gaseous phases) of the porous medium. Therefore, the structure of the porous medium (porosity and pore-size distribution) determines the cross- sectional area and the geometry of the pores available for diffusion; this determines the intrinsic restrictions to diffusion as a result of pore geometry. We also have to consider the effects of pore- saturation by the fluid phase of interest, either water or gas, in understanding restrictions offered by the medium for solute diffusion. C. Two Models for Diffusion: Mass Transfer and Fick’s First Law Interestingly, there are two alternate mathematical models that can be used to describe diffusion processes. The first is based on an empirically determined mass transfer coefficient k and is common in engineering applications. The mass transfer model assumes that flux is proportional to concentration difference : (1) The second, Fick’s Law, is the more fundamental and physically-based. Fick's First Law is a linear flux law similar to Darcy's law and assumes that flux is proportional to concentration difference per unit length (i.e., concentration gradient) . Fick’s First Law for one- and multi-dimensional diffusion cases is:

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CWR 6537 Subsurface Contaminant Hydrology Lecture 6 2 (2A) (2B) where, M D Mass Flow - amount of solute diffusing per unit time (M A T -1 ); J D Mass Flux - diffusive flux, (M A L -2 A T -1 ); A cross-section across which diffusion occurs (L 2 ); D by definition is solute molecular diffusion coefficient (L 2 A T -1 ); M C / M x concentration gradient (M A L -3 A L -1 ); x distance along the diffusion path (L); and L del operator A careful definition of the frame of reference used to express M D , D, A, and C is essential for unambiguous statement of Fick's law. Generally, (i) M D should have the same quantity (mass) reference as C (concentration), and (ii) the volume should have the same length references as A and x. For example, the frames of reference in porous media could be: (a) entire bulk medium (subscript, m ) (b) fluid phase only (subscript, f ) @ gaseous phase only (subscript, g ) @ liquid phase only (subscript, w ) For a fluid phase alone (i.e., no porous medium), Fick's law (Fick 1855) is:
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csh_lecture6_diffusion - CWR 6537 Subsurface Contaminant...

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