Description

This model implements a generic energy balance for single and multi-phase transport problems. Currently the non-isothermal model can be used on top of the 1p2c, 2p, 2p2c and 3p3c models. Comparison to simple analytical solutions for pure convective and conductive problems are found in the 1p2c test. Also refer to this test for details on how to activate the non-isothermal model.

For the energy balance, local thermal equilibrium is assumed. This results in one energy conservation equation for the porous solid matrix and the fluids,

\begin{align*} \phi \frac{\partial \sum_\alpha \varrho_\alpha S_\alpha (u_\alpha - \mathbf{g}\cdot\mathbf{x})}{\partial t} & + \frac{\partial \left((\left( 1 - \phi \right) \varrho_s c_s T\right)}{\partial t} - \sum_\alpha \nabla \cdot \left\{ \varrho_\alpha (h_\alpha - \mathbf{g}\cdot\mathbf{x}) \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \nabla p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\} \\ & - \nabla \cdot \left(\lambda_{pm} \nabla T \right) - q^h = 0, \end{align*}

where:

\( \phi \) is the porosity of the porous medium,
\( S_\alpha \) represents the saturation of phase \( \alpha \),
\( \rho_\alpha \) is the mass density of phase \( \alpha \),
\( h_\alpha \) is the specific enthalpy of phase \( \alpha \),
\( u_\alpha \) is the specific internal energy of phase \( \alpha \),
\( \lambda_{pm}\) is the effective heat conductivity in the porous medium,
\( T \) is the temperature,
\( \rho_s \) is the mass density of the solid phase,
\( c_s \) is the heat capacity of the solid,
\( k_{r\alpha} \) is the relative permeability of phase \( \alpha \),
\( \mu_\alpha \) is the dynamic viscosity of phase \( \alpha \),
\( \mathbf{K} \) is the intrinsic permeability tensor,
\( p_\alpha \) is the pressure of phase \( \alpha \),
\( \mathbf{g} \) is the gravitational acceleration vector,
\( q^h \) is a source or sink term.

The gravitational potential is approximated by \(\psi \approx \mathbf{g}\cdot\mathbf{x}\) with \( \mathbf{g} = -\nabla\psi\) and where \(\mathbf{x}\) is the position in space. The approximation is exact in case the gravitation vector is constant in space. The implementation via the gravitational potential in the formulation above also assumes that the gravitional potential is independent of time. This formulation is incorrect otherwise.

Classes
struct	Dumux::PorousMediumFlowNIModelTraits< IsothermalT, TDM >
	Specifies a number properties of non-isothermal porous medium flow models based on the specifics of a given isothermal model. More...

Namespaces
namespace	Dumux

Include dependency graph for porousmediumflow/nonisothermal/model.hh:

Description

Classes

Namespaces