Introduction to the Thermodynamically Constrained Averaging by William G. Gray

By William G. Gray

Thermodynamically restricted averaging concept offers a constant procedure for upscaling conservation and thermodynamic equations for program within the examine of porous medium structures. the strategy presents dynamic equations for stages, interfaces, and customary curves which are heavily in accordance with insights from the entropy inequality. All better scale variables within the equations are explicitly outlined when it comes to their microscale precursors, facilitating the choice of vital parameters and macroscale kingdom equations according to microscale experimental and computational research. the tactic calls for that each one assumptions that result in a specific equation shape be explicitly indicated, a limit that's worthy in ascertaining the variety of applicability of a version in addition to strength assets of blunders and possibilities to enhance the analysis.

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Thus one must exercise caution in using these terms. 2 In scientific inquiry, it is always convenient and even comforting to have a precise set of definitions from which to work. We note that the study of porous media tends to deny such comfort. The distinction between a mixture of solids and fluids vs. a porous medium system is imprecise; immiscibility of fluid phases is not complete; the definition of length scales is fraught with caveats. It is important to understand that the definitions are nevertheless made to provide an opportunity to move forward while also acknowledging that the lack of precise definitions impacts one’s ability to describe porous medium processes.

The macroscale, ma , is a scale at which each entity of the porous medium— phase, interface, or common curve—can be treated as a continuum in the full space of the porous medium. Thus at the macroscale, the porous medium system is modeled as being composed of overlapping continua, simultaneously present at each macroscale point. At this scale, the detailed microscale distribution of the locations of adjacent phases is not resolved but is accounted for in an average sense. Phases are described merely as occupying fractions of available space, and the interfacial areas per volume between phases and common curve lengths per volume are considered.

All these are extensive thermodynamic quantities that are additive. Thus, if two identical systems are combined, the values of all four of the quantities in the combined system will be double their values in each of the uncombined elements. The partial derivatives of internal energy with respect to each of entropy, volume, and mass, while hold- 16 1 Elements of Thermodynamically Constrained Averaging Theory ing the other two variables constant, defines the intensive thermodynamic quantities temperature, pressure, and chemical potential, respectively.

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