Introduction to the Thermodynamically Constrained Averaging by William G. Gray

By William G. Gray

Thermodynamically restricted averaging conception offers a constant approach for upscaling conservation and thermodynamic equations for software within the learn of porous medium structures. the strategy presents dynamic equations for levels, interfaces, and customary curves which are heavily in response to insights from the entropy inequality. All better scale variables within the equations are explicitly outlined by way of their microscale precursors, facilitating the choice of vital parameters and macroscale nation equations in response to microscale experimental and computational research. the tactic calls for that each one assumptions that bring about a selected equation shape be explicitly indicated, a limit that's necessary in ascertaining the variety of applicability of a version in addition to power resources of errors and possibilities to enhance the analysis.

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1) = fα →Ωβ ,Ωβ , Eq. 2) = fα →Ωα ,Ωα , intrinsic average, Eq. 3) = fα →Ωβ ,Ωβ ,ρα , general density-weighted average, Eq. 4) = fα →Ωα ,Ωα ,ρα , intrinsic density-weighted average, Eq. 5) = fiα →Ωβ ,Ωβ ,ρα ωiα , general species mass density weighted average, Eq. 6) = fiα →Ωα ,Ωα ,ρα ωiα , intrinsic species mass density weighted average, Eq. 1 Overview Mechanistic computational modeling of environmental systems presupposes the existence of a properly posed set of equations that adequately describes the underlying physical, chemical, and biological processes at time and length scales consistent with the problem being addressed.

Daily or hourly values of infiltration in response to a storm event are modeled when one considers watershed response. In laboratory studies or in consideration of the initiation of infiltration into a dry soil, a time scale on the order of minutes or seconds may be appropriate. , 23]. In such cases, the dynamics within a step occur at a smaller interval than the time required for equilibrium to be achieved. Creating links among the models formulated at different time and space scales is an active area of research.

Thus, in working with specific applications, we will not include explicit quantification of the length scale but will only qualitatively categorize the scale of interest. We emphasize, however, that the concept of “scale invariance” as employed here should not be understood to indicate that an equation derived at a particular scale can be applied at any scale. Scale invariance only means that, within a couple of orders of magnitude of length, the equations, variables, and parameters need not be adjusted.

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