2. Boundary conditions, loading and initial condition

Refer to [R5.02.01] for thermal boundary conditions and loads leading to linear temperature equations as well as for the initial condition.

2.1. Nonlinear normal flow

These are Neumann-type conditions, defining the flow entering the domain.

\(-q(x,t)\mathrm{.}n=g(x,T)\) on the border \(\mathrm{\Gamma }\) eq 2.1-1

where \(g(x,T)\) is a function of the temperature and possibly of the space variable \(x\) and/or of the time variable \(t\) and/or time and \(n\) designates the external normal at the border \(\Gamma\), \(q\) is the heat flow vector (directed according to decreasing temperatures).

This expression makes it possible to introduce, for example, exchange-type conditions with a convective exchange coefficient dependent on temperature:

\(-q(x,t)\mathrm{.}n=g(x,T)=h(x,T)({T}_{\text{ext}}(x,t)-T)\) eq 2.1-2

2.2. Nonlinear normal flow - infinity radiation-like condition

A particular case of the previous boundary conditions is the infinity radiation of gray bodies which results in a special case of function \(g(x,T)\):

\(-q(x,t)\mathrm{.}n=\sigma \epsilon \left[(T(x)+\text{273}\text{.}\text{15}{)}^{4}-({T}_{\infty }+\text{273}\text{.}\text{15}{)}^{4}\right]\) eq 2.2-1

The characteristics to be defined when defining this load are emissivity, the Stefan-Boltzmann constant \(\sigma =\mathrm{5,73}{.10}^{-8}\mathrm{usi}\) and the temperature to infinity.

\(T(r)\) and \({T}_{\infty }\) are then expressed in degrees Celsius. \(–273.15°C\) is the temperature of absolute zero.