1 |
Front Matter |
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Abstract
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2 |
,Elements of Thermodynamically Constrained Averaging Theory, |
William G. Gray,Cass T. Miller |
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Abstract
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.
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3 |
,Microscale Conservation Principles, |
William G. Gray,Cass T. Miller |
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Abstract
TCAT models are developed by formally upscaling conservation and balance equations, thermodynamic relations, and equilibrium conditions from the microscale to a macroscale, a megascale, or some combination of the two scales.
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4 |
,Microscale Thermodynamics, |
William G. Gray,Cass T. Miller |
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Abstract
Models of a system that are mechanistically based make use of the full range of physical principles that influence system behavior. In addition to these expressions, the thermodynamic properties of the entities in the system, as well as of the system as a whole, impact the observed behavior. The thermodynamic relations that are hypothesized to describe a system impact how that system is modeled and the fidelity of the model compared to the actual behavior of the system. Thermodynamics plays a role not only in describing system properties and parameters but also in identifying limits on system behavior that might be observed.
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5 |
,Microscale Equilibrium Conditions, |
William G. Gray,Cass T. Miller |
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Abstract
The role of classical thermodynamics is to represent the equilibrium state of a system and the change in system properties that occurs due to a transition between states. Because the TCAT method describes system dynamics at “near” equilibrium dynamic conditions, it is essential to know what the actual equilibrium conditions are. The second law of equilibrium thermodynamics requires that entropy production associated with a spontaneous, irreversible change of system state be nonnegative.
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6 |
,Microscale Closure for a Fluid Phase, |
William G. Gray,Cass T. Miller |
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Abstract
The TCAT approach outlined in Fig. 1.3 includes many interrelated components, including conservation and balance equations, thermodynamics, equilibrium conditions, entropy inequalities, and closure relations. Each of these components must be established at the scale of interest. If the scale of interest is the microscale, then all the necessary components to build a complete, closed, mechanistic model are available from the preceding chapters. In the present chapter, we will illustrate how these components can be employed to deduce a range of models. The concepts will be employed to analyze relatively simple cases that, nevertheless, have wide applicability to continuum modeling of real systems.
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7 |
,Macroscale Conservation Principles, |
William G. Gray,Cass T. Miller |
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Abstract
In the preceding chapter, we demonstrated how the elements of the TCAT approach, as depicted in Fig. 5.1, are used to obtain a closed set of microscale equations. The procedure for the development of macroscale equations is similar. However, an additional step is required to transform the microscale relations to the macroscale prior to applying the closure procedure.
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8 |
,Macroscale Thermodynamics, |
William G. Gray,Cass T. Miller |
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Abstract
A unique feature of the TCAT approach is the treatment of thermodynamics at the macroscale. Alternative averaging theories either ignore thermodynamics completely or introduce thermodynamics directly at the macroscale using the rational thermodynamics approach.
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9 |
,Evolution Equations, |
William G. Gray,Cass T. Miller |
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Abstract
The goal of this chapter is to formulate macroscale evolution equations for geometric densities that can be applied to complete the formulation of closed, solvable TCAT models. These densities are quantities such as the amount of volume per averaging volume occupied by a phase or the amount of area between phases per volume occupied by an interface between phases. The need for the evolution equations for geometric properties is rooted in the fact that these quantities do not exist at the microscale but arise in the averaging process.
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10 |
,Single-Fluid-Phase Flow, |
William G. Gray,Cass T. Miller |
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Abstract
The goal of this chapter is to formulate a set of closed, solvable, macroscale, singlefluid-phase flow models using the TCAT approach. This is the first macroscale TCAT application to be considered.
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11 |
,Single-Fluid-Phase Species Transport, |
William G. Gray,Cass T. Miller |
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Abstract
The goal of this chapter is to extend the macroscale TCAT analysis of Chap. 9 to include compositional effects in single-fluid-phase porous medium systems. Because the development is similar to that employed in the previous chapter, emphasis will be on the extensions to the analysis that are needed.
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12 |
,Two-Phase Flow, |
William G. Gray,Cass T. Miller |
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Abstract
This chapter is the third that presents an example application of TCAT to a porous medium flow and transport problem. Single-phase flow was modeled in Chap. 9. The analysis considered three entities: a fluid phase, ., a solid phase, ., and the interface between the phases, .. In Chap. 10, the additional feature of chemical species transport was added to the analysis.
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13 |
,Modeling Approach and Extensions, |
William G. Gray,Cass T. Miller |
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Abstract
Previous chapters have detailed the general TCAT approach, the steps needed to build TCAT models, and the development of three specific classes of models. While considerable ground has been covered, much more remains to be accomplished in extending the models presented and in going on to other cases and problem types. The purpose of this concluding chapter is to point out a few of the many opportunities that exist for evaluation and extension of TCAT models.
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14 |
Back Matter |
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Abstract
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