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A Pedagogical Approach to the Thermodynamically Constrained Averaging Theory

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Abstract

The thermodynamically constrained averaging theory (TCAT) is an evolving approach for formulating macroscale models that are consistent with both microscale physics and thermodynamics. This consistency requires some mathematical complexity, which can be an impediment to understanding and efficient application of this model-building approach for the non-specialist. To aid understanding of the TCAT approach, a simplified model formulation approach is developed and used to show a more compact, but less general, formulation compared to the standard TCAT approach. This new simplified model formulation approach is applied to the case of binary species diffusion in a single-fluid-phase porous medium system, clearly showing a TCAT approach that is applicable to many other systems as well. Recent extensions to the TCAT approach that enable a priori parameter estimation, and approaches to leverage available TCAT modeling building results are also discussed.

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Abbreviations

b :

Entropy source density

C :

Species concentration in an entity

\(\hat{\varvec{\mathsf {D}}}\) :

Effective diffusivity tensor

\({{\hat{\varvec{\mathsf {D}}}}}^{ABw}\) :

Second-rank symmetric closure tensor for a binary system

\(\varvec{\mathsf {d}}\) :

Rate of strain tensor

\({\mathcal E}_{**}^{{\overline{\overline{w}}}}\) :

Particular material derivative form of a macroscale entity total energy conservation equation, Eq. (10)

\({\mathcal G}_{**}^{{\overline{\overline{w}}}}\) :

Particular material derivative form of a macroscale body force potential balance equation, Eq. (11)

h :

Energy source density

\({\varvec{\mathsf {I}}}\) :

Identity tensor

\(\mathcal {J}_s\) :

Index set of species

i :

Species qualifier

\(K_{E}\) :

Kinetic energy term due to velocity fluctuations

\({\mathcal M}_{**}^{{\overline{\overline{{i}w}}}}\) :

Particular material derivative form of a macroscale species mass conservation equation, Eq. (9)

\(MW_i\) :

Molecular weight of species i

\(MW_w\) :

Molecular weight for entity w

p :

fluid pressure

\(\mathbf {q}\) :

Non-advective energy flux vector

\(\mathbf {q}_{\mathbf {g}0}\) :

Non-advective energy flux vector due to the product of fluctuations

R :

Ideal gas constant

\({\mathcal S}_{**}^{{\overline{\overline{w}}}}\) :

Particular material derivative form of a macroscale entropy balance, Eq. (12)

\({\mathcal T}_*^{{\overline{\overline{w}}}}\) :

Particular material derivative form of a macroscale differential thermodynamic equation, Eq. (13)

\({\mathcal T}_{{\mathcal G}*}^{{\overline{\overline{w}}}}\) :

Particular material derivative form of the body source potential equation, Eq. (14)

t :

Time

\({{\mathbf {u}}^{\,{}}_{}}\) :

Species deviation velocity vector

\(\mathbf {v}\) :

Velocity vector

x :

Mole fraction of a species in an entity

\({{\hat{{\gamma }}}}\) :

Macroscale activity coefficient

\({{{\epsilon }^{{\overline{\overline{{{\alpha }}}}}}}}\) :

Specific entity measure of the \({{\alpha }}\) entity (volume fraction, specific interfacial area)

\(\eta \) :

Entropy density

\({\theta }\) :

Temperature

\({\Lambda }\) :

Entropy production rate density

\({\lambda }_{\mathcal E}^w\) :

Lagrange multiplier for energy conservation equation

\({\lambda }_{{\mathcal G}}^w\) :

Lagrange multiplier for potential energy balance equation

\({\lambda }_{{\mathcal M}}^{{i}w}\) :

Lagrange multiplier for mass conservation equation

\({\lambda }_{{\mathcal M}_{i}}\) :

Is a constant related to the sum of potentials

\({\lambda }_{{\mathcal T}}^w\) :

Lagrange multiplier for thermodynamic equation

\({\lambda }_{{\mathcal T}{\mathcal G}}^w\) :

Lagrange multiplier for derivative of potential energy equation

\(\mu \) :

Chemical potential

\(\mu _0\) :

Reference chemical potential

\(\rho \) :

Mass density

\(\mathbf {\varphi }\) :

Non-advective entropy density flux vector

\(\psi \) :

Body force potential per unit mass (e.g., gravitational potential)

\({{\Omega }_{}}\) :

Spatial domain

\({{\Omega }_{}}_w\) :

Domain occupied by the wetting phase in the REV

\({{\omega }_{{}{}}}\) :

Mass fraction of a species in an entity

A :

Species qualifier (subscript, superscript)

B :

Species qualifier (subscript, superscript)

\({\mathcal E}\) :

Energy equation qualifier (subscript)

\({\mathcal G}\) :

Potential equation qualifier (subscript)

\({i}\) :

General index denoting a species (subscript, superscript)

\({\mathcal M}\) :

Mass equation qualifier (subscript)

s :

Index that indicates a solid phase (subscript, superscript)

\({\mathcal T}\) :

Thermodynamic equation qualifier (subscript)

\({\mathcal T}{\mathcal G}\) :

Fluid potential energy identity qualifier (subscript)

w :

Entity index corresponding to the wetting phase (subscript, superscript)

\(\overline{\ }\) :

Above a superscript refers to a density weighted macroscale average

\(\overline{\overline{\ }}\) :

Above a superscript refers to a uniquely defined macroscale average

\({{{\left\langle f \right\rangle _{{{\Omega }_{{{\alpha }}}},{{\Omega }_{}},w}}}}\) :

Averaging operator, \(\int _{{\Omega }_{{{\alpha }}}}wf\, \mathrm {d}\mathfrak {r}/\int _{{{\Omega }_{}}} w\, \mathrm {d}\mathfrak {r}\)

\(\mathrm {D}^{}_{{}}/\mathrm {D}t\) :

Material derivative

\(\text{ CEI }\) :

Constrained entropy inequality

\(\text{ EI }\) :

Entropy inequality

\(\text{ MVA }\) :

Method of volume averaging

\(\text{ REV }\) :

Representative elementary volume

\(\text{ SEI }\) :

Simplified entropy inequality

\(\text{ TCAT }\) :

Thermodynamically constrained averaging theory

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Acknowledgements

The work of CTM was supported by Army Research Office grant W911NF-14-1-02877 and National Science Foundation grant 1619767. The contributions of BDW were supported by National Science Foundation, grant EAR-1521441. The efforts of W.G. Gray to review and comment on this work are appreciated.

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Authors and Affiliations

  1. University of North Carolina, Chapel Hill, NC, 27599-7431, USA

    Cass T. Miller

  2. Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, 09340, México, D.F, Mexico

    Francisco J. Valdés-Parada

  3. Oregon State University, Corvallis, OR, 97330, USA

    Brian D. Wood

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  1. Cass T. Miller
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  2. Francisco J. Valdés-Parada
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  3. Brian D. Wood
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Correspondence to Cass T. Miller.

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Miller, C.T., Valdés-Parada, F.J. & Wood, B.D. A Pedagogical Approach to the Thermodynamically Constrained Averaging Theory. Transp Porous Med 119, 585–609 (2017). https://doi.org/10.1007/s11242-017-0900-6

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