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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Notes
- 1.
- 2.
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.
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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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Diersch, HJ.G. (2014). Heat Transport in Porous Media. In: FEFLOW. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38739-5_13
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DOI: https://doi.org/10.1007/978-3-642-38739-5_13
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