Non-Isothermal Constitutive Relations and Heat Transfer Equations of a Two-Phase Medium

Open access

Abstract

In the case of a two-phase medium – such as the soil, which consists of an elastic skeleton and is filled with pore fluids – stress and strain within the medium are dependent on both phases. Similarly, in the case of heat transfer, heat is conducted through the two phases at different rates, with an additional heat transfer between the phases. In the classical approach to modelling a porous medium, it is assumed that the fluid filling the pore space is water, which is incompressible. In the case of gas, the volume of which is strongly dependent on temperature and pressure, one should take this behavior into account in the constitutive relations for the medium. This work defines the physical relations of a two-phase medium and provides heat transfer equations, constructed for a porous, elastic skeleton with fluid-filled pores, which may be: liquid, gas, or mixture of liquid and a gas in non-isothermal conditions. The paper will present constitutive relations derived from the laws of irreversible thermodynamics, assuming that pores are filled with either a liquid or a gas. These relations, in the opinion of the authors, may be used as the basis for the construction of a model of the medium filled partly with a liquid and partly with a gas. It includes the possibility of independent heat transfer through any given two-phase medium phase, with the transfer of heat between the phases.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Auriault J.L. Dynamic Behaviour of a Porous Medium Saturated by a Newtonian Fluid Int. J. Engng. Sc. 1980 Vol. 18.

  • [2] Auriault J.L. Strzelecki T. Bauer J. He S. Porous deformable Media Saturated by a Very Compressible Fluid Eur. J. Mech. A/Solid 1990 Vol. 9 4.

  • [3] Auriault J.L. Heterogeneous Medium. Is an Equivalent Macroscopic Description Possible? Int. J. Engng. Sci. 1991 29 7.

  • [4] Bensoussan A. Lions J.L. Papanicolau G. Asymptotic Analysis for Periodic Structures Amsterdam: Holland Publishing Company 1978.

  • [5] Biot M.A. Le probleme de la Consolidation des Matieres Argileuses sous une charge B. 55 Ann. Soc. Sci. 1935 Bruxelles 110/113.

  • [6] Biot M.A. General Theory of three-dimensional Consolidation J. Appl. Physics 1941 Vol. 12.

  • [7] Biot M.A. Theory of Stress-Strain Relation in Anisotropic Viscoelasticity and relaxation Phenomena J. Appl. Phys. 1954 25.

  • [8] Biot M.A. Willis D.G. The Elastic Coefficients of the Theory of Consolidation J. Appl. Mech. 1957 24.

  • [9] Bartlewska M. Strzelecki T. Equations of Biot’s consolidation with Kelvin–Voight rheological frame Studia Geotechnica et Mechanica 2009 31(2).

  • [10] Coussy O. Revisiting the constitutive equations of unsaturated porous solids using a Lagrangian saturation concept Int. J. Numer. Anal. Meth. Geomech. 2007 31.

  • [11] Coussy O. Mechanics and Physics of Porous Solids John Wiley 2010.

  • [12] Derski W. Outline of continuum mechanics PWN Warszawa 1975 (in Polish).

  • [13] De Groot S.R. Mazur P. Non-equilibrium Thermodynamics Amsterdam: North-Holland Publishing Company 1984.

  • [14] FlexPDE 6 (PDE Solutions 2015); www.pdesolutions.com

  • [15] Kisiel I. An outline of soil rheology effect of static loading on soil Arkady Warszawa 1966 (in Polish).

  • [16] Kisiel I. Derski W. Izbicki R. Mróz Z. Rock and soil mechanics PWN Warszawa 1982 (in Polish).

  • [17] Łydżba D. Constitutive equations of gas-coal medium Studia Geotechnica et Mechanica 1991 13(3–4).

  • [18] Łydżba D. Applications of asymptotic homogenisation method in soil and rock mechanics: Oficyna Wydawnicza Politechniki Wrocławskiej Wrocław 2002 (in Polish).

  • [19] Nowacki W. Theory of elasticity PWN Warszawa 1970 (in Polish).

  • [20] Strzelecki T. Bauer J. Auriault J.L. Constitutive equation of a gas-filled two-phase medium Transport in Porous Media 1993 10.

  • [21] Strzelecki T. Auriault J.L. Bauer J. Kostecki S. Puła W. Mechanics of heterogeneous media. Homogenization theory Lower Silesia Educational Publishers Wrocław 1998 (in Polish).

  • [22] Strzelecki T. Kostecki S. Żak S. Modeling of flows through porous media Lower Silesia Educational Publishers Wrocław 2008 (in Polish).

  • [23] Strzelecki T. Strzelecki M. Relation between filtration and soil consolidation theories Studia Geotechnica et Mechanica 2015 37(1) 105–114.

  • [24] Uciechowska A. Strzelecki T. The influence of the type of pore fluid in two phase media on the form of consolidation equations [in:] M. Kwaśniewski & D. Łydżba (eds.) Rock Mechanics for Resources London: Taylor & Francis Group Energy and Environment 2013 491–495.

Search
Journal information
Impact Factor


CiteScore 2018: 1.03

SCImago Journal Rank (SJR) 2018: 0.213
Source Normalized Impact per Paper (SNIP) 2018: 1.106

Metrics
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 200 42 1
PDF Downloads 104 39 1