Linearity Property of the Temperature Equation
The Heat Transfer interfaces define an elliptic partial differential equation for the temperature, T, of the form:
with Dirichlet and Neumann boundary conditions at some boundaries:
In its basic form, the density, ρ, heat capacity, Cp, thermal conductivity, k, heat sources, Q, constraint temperatures, T0, and heat fluxes, q0, are all constant, which leads to a linear system. Here, linear solvers described in the next paragraphs are completely suited for the resolution.
However, nonlinearities can appear in the equation in the following cases:
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The material properties, ρ, Cp, and k, have a temperature dependency.
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The heat sources are not linear in temperature.
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The Neumann boundary condition is not linear in temperature, hence
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A convective cooling condition of type −n ⋅ q = h(Text − T) keeps the linearity of the problem when the heat transfer coefficient, h, is constant.
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A radiative condition of type −n ⋅ q = εσ(Tamb4 − T4) is strongly nonlinear.
Different nonlinear solvers are also provided for these kinds of problems.