1. Problem overview

The heat equation has strong nonlinearities under certain conditions. This is the case when the material undergoes phase changes: these are accompanied by a sudden variation in characteristic quantities (heat capacity, enthalpy). This nonlinearity is all the more accentuated when one treats the convection-diffusion problem, where the term transport dependent on the enthalpy function appears. The aim of this modeling is to deal with this last problem in a steady state (stationary case).

In all cases, it is assumed that the speed field is known a priory. The case of a mobile solid is quite frequent in practice. In particular, it relates to welding applications or surface treatment that involve a heat source moving in a given direction and at a given speed. The thermal problem is then studied in a frame of reference linked to the source.

The partial derivative problem results from the global heat balance equation for any domain \(\Omega\) which is written as:

\(\begin{array}{ccc}\frac{d}{\text{dt}}\underset{\Omega }{\mathrm{\int }}\mathit{\rho \beta d}\Omega \mathrm{=}& \underset{\Omega }{\mathrm{\int }}\mathit{Qd}\Omega \mathrm{-}& \underset{\mathrm{\partial }\Omega }{\mathrm{\int }}q\text{.}nd\Gamma \\ \text{accumulation}& \text{création}+& \text{entrée-sortie}\end{array}\) eq 1-1

In this equation, \(O\) represents a related domain, within the system under study, which is followed in its movement, \(\beta\) represents the specific enthalpy of the material and \(\rho\) designates its density. \(Q\) is a volume heat source, \(q\) is the heat flow across border \(\partial \Omega\) (\(n\) being the outside normal), and \(d/\text{dt}\) is the particle derivative.

The first term in [éq 1-1] is written (see for example [bib1]):

\(\frac{d}{\text{dt}}\underset{\Omega }{\mathrm{\int }}\mathit{\rho \beta d}\Omega \mathrm{=}\underset{\Omega }{\mathrm{\int }}(\frac{\mathrm{\partial }(\rho \beta )}{\mathrm{\partial }t}+\text{div}(\rho \beta V))d\Omega\) eq 1-2

or, taking into account mass conservation \((\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\text{div}(\rho V)\mathrm{=}0)\) [bib1]:

\(\frac{d}{\text{dt}}\underset{\Omega }{\mathrm{\int }}\rho \beta d\Omega \mathrm{=}\underset{\Omega }{\mathrm{\int }}(\rho \frac{\mathrm{\partial }\beta }{\mathrm{\partial }t}+\rho V\text{.}\text{grad}\beta )d\Omega\) eq 1-3

where \(V\) is the movement speed vector for the \(\Omega\) domain. \(V\) is entered under the simple CONVECTION keyword from commands AFFE_CHAR_THER and AFFE_CHAR_THER_F.

The second term of the second member of [éq 1-1] is written, taking into account the divergence theorem and Fourier’s law \((q\mathrm{=}\mathrm{-}k(T)\text{grad}T)\):

\(\underset{\mathrm{\partial }\Omega }{\mathrm{\int }}q\text{.}nd\Gamma \mathrm{=}\underset{\Omega }{\mathrm{\int }}\text{div}qd\Omega \mathrm{=}\mathrm{-}\underset{\Omega }{\mathrm{\int }}\text{div}(k(T)\text{grad}T)d\Omega\) eq 1-4

where \(T\) is the temperature and \(k(T)\) is the thermal conductivity of the material, a function of temperature.

The equation [éq 1-1] must be satisfied for any \(\Omega\) domain, so it comes:

\(\rho \frac{\mathrm{\partial }\beta }{\mathrm{\partial }t}+\rho V\text{.}\text{grad}\beta \mathrm{-}\text{div}(k(T)\text{grad}T)\mathrm{=}Q\) in \(\Omega\) eq 1-5

Note:

Note that the classical case with, \(k(T)=k\) * (constant) and \(V=0\) , and where the specific enthalpy is a linear function of temperature, \(\beta (T)\mathrm{=}\text{CT}\) returns the well-known classical equation:

\(\rho C\frac{\mathrm{\partial }T}{\mathrm{\partial }t}\mathrm{-}\mathit{kDT}\mathrm{=}Q\) in \(\Omega\)

where \(\Delta\) is Laplacian and \(C\) (constant) represents specific heat.

The partial derivative problem treated by the command THER_NON_LINE_MO [U4.33.04], consists in solving the equation [éq 1-5] in the stationary case (directly in the permanent state) with boundary conditions on the border \(\partial \Omega\).

This problem is formally written in the following form:

\(\begin{array}{cc}V\text{.}\text{grad}u(T)-\text{div}(k(T)\text{grad}T)=Q,& \text{dans}\Omega ,\\ +\text{conditions aux limites}& \text{sur}\partial \Omega \end{array}\) eq 1-6

where we adopted the notation, valid for the whole sequence, \(u(T)\mathrm{=}\rho \beta (T)\) where \(\rho\) is constant, defining the volume enthalpy.