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2014 | OriginalPaper | Buchkapitel

13. Heat Transport in Porous Media

verfasst von : Hans-Jörg G. Diersch

Erschienen in: FEFLOW

Verlag: Springer Berlin Heidelberg

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Abstract

In this chapter we discuss the finite-element computation of heat (thermal energy) transport in porous media. Nonisothermal porous-medium processes can be found in many areas of application to natural and engineered systems, for instance exploitation of geothermal reservoirs as a viable and renewable source of energy, underground energy storage and recovery for heating and cooling purposes, waste disposal of heat-generating materials, chemical reactor engineering, insulation of buildings, material technology and many others. Modern industrial developments have expanded significantly the fields, where numerical simulation is required as a powerful tool to aid the design and operation of equipments.

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Vorheriges Kapitel Mass Transport in Porous Media with and Without Chemical Reactions

Nächstes Kapitel Discrete Feature Modeling of Flow, Mass and Heat Transport Processes

The divergence form (13.1) assumes that the specific heat capacities c and c ^s are independent of T (cf. Sect. 3.9.1). Contrarily, the convective form (13.2) does not imply such an assumption.

Optionally, FEFLOW suppresses the time derivative term ∂ T∕∂ t for solving steady-state solutions. A specific option exists, named steady flow – transient transport, in which the advective flow vector $\boldsymbol{q}$ is invariant with time.

A boundary with OBC on $\varGamma _{N_{O}}$ can be separated from the Neumann boundary $\varGamma _{N_{T}}$ so that for the divergence form

$$\displaystyle{\int _{\varGamma _{N_{ T}}}wq_{T}^{\dag }d\varGamma =\int _{\varGamma _{ N_{T}}\setminus \varGamma _{N_{O}}}wq_{T}^{\dag }d\varGamma +\int _{\varGamma _{ N_{O}}}w\bigl ((T - T_{0})\rho c\boldsymbol{q} -\boldsymbol{\varLambda }\cdot \nabla T\bigr ) \cdot \boldsymbol{ n}d\varGamma }$$

and for the convective form

$$\displaystyle{\int _{\varGamma _{N_{ T}}}wq_{T}d\varGamma =\int _{\varGamma _{N_{ T}}\setminus \varGamma _{N_{O}}}wq_{T}d\varGamma -\int _{\varGamma _{N_{O}}}w(\boldsymbol{\varLambda }\cdot \nabla T) \cdot \boldsymbol{ n}d\varGamma }$$

The implicit treatment of OBC requires the incorporation of the $\varGamma _{N_{O}}-$ integrals into the LHS of the resulting matrix system (see below). In contrast, a natural Neumann-type BC with $-(\boldsymbol{\varLambda }\cdot \nabla T) \cdot \boldsymbol{ n} \approx 0$ on $\varGamma _{N_{O}}$ is often the preferred alternative formulation for an OBC. Note, however, that for both cases in the divergence form the boundary flux $\boldsymbol{q} \cdot \boldsymbol{ n}$ must be known a priori. The boundary flux $\boldsymbol{q} \cdot \boldsymbol{ n}$ can be either explicitly given from a Neumann-type BC $q_{h} =\boldsymbol{ q} \cdot \boldsymbol{ n}$ for flow or must be computed by a postprocessing budget evaluation of the flow equation on the corresponding outflowing boundary section imposed by Dirichlet-type or Cauchy-type BC of flow.

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Titel: Heat Transport in Porous Media
verfasst von: Hans-Jörg G. Diersch
Verlag: Springer Berlin Heidelberg
Buch: FEFLOW
Print ISBN: 978-3-642-38738-8

Electronic ISBN: 978-3-642-38739-5

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-642-38739-5_13

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