Matrix Thermal

Matrix thermal transfer describes heat storage and conduction within the intact solid material. Temperature gradients drive heat from warmer regions toward cooler regions, while heat sources, boundary conditions, and material properties determine the transient temperature field.

Fourier’s law

The conductive heat-flux density is given by Fourier’s law:

\[\mathbf{q}=-\mathbf{K}\nabla T,\]

where \(\mathbf{q}\) is heat flux per unit area, \(T\) is temperature, and \(\mathbf{K}\) is the thermal-conductivity tensor. The negative sign indicates that heat flows down the temperature gradient. For a two-dimensional domain, \(\nabla T=(\partial T/\partial x,\,\partial T/\partial y)\). For an isotropic material, \(\mathbf{K}=k\mathbf{I}\), where \(k\) is the scalar thermal conductivity.

Energy balance

For a stationary solid with constant density and specific heat capacity, local energy conservation is

\[\rho C_p\frac{\partial T}{\partial t} +\nabla\cdot\mathbf{q}=Q,\]

where \(\rho\) is mass density, \(C_p\) is specific heat capacity, and \(Q\) is the volumetric heat-generation rate. Combining this balance with Fourier’s law gives

\[\rho C_p\frac{\partial T}{\partial t} =\nabla\cdot(\mathbf{K}\nabla T)+Q.\]

For homogeneous isotropic material with constant \(k\), this becomes

\[\frac{\partial T}{\partial t} =\frac{k}{\rho C_p}\nabla^2T+\frac{Q}{\rho C_p}.\]

The thermal diffusivity \(a_T=k/(\rho C_p)\) measures how quickly temperature disturbances spread through the matrix. If properties vary with temperature or position, the divergence form should be retained rather than replacing it with \(k\nabla^2T\).

Initial and boundary conditions

Transient conduction requires an initial temperature field \(T(\mathbf{x},0)=T_0(\mathbf{x})\). Common boundary conditions include a prescribed temperature, a prescribed normal heat flux \(-\mathbf{n}\cdot\mathbf{K}\nabla T\), or an exchange condition with the surrounding environment. These conditions define how the matrix receives or loses heat and should be consistent with the physical heating or cooling process being represented.

The equations above assume a continuum matrix and do not by themselves represent heat exchange across a discontinuity. Heat transfer between separated blocks or across a fracture is described by the contact and fracture thermal-resistance models, respectively.