1. Reference problem

1.1. Geometry

Let’s say a bar moving, at speed \(V\), in line with the temperature conditions imposed in \(X=0\) and \(X=L\) expressed in a fixed frame of reference (with respect to the bar moving).

1.2. Material properties

• thermal conductivity is constant: \(K=150W/m°C\)

• the enthalpy function is such that:

\(\beta (T)=\{\begin{array}{cc}{C}_{S}T& \text{;}T\le {T}_{1}\\ {C}_{S}T+{C}_{\mathrm{Sl}}(T-{T}_{1})& \text{;}{T}_{1}\le T\le {T}_{2}\\ {C}_{S}T+{C}_{\mathrm{Sl}}({T}_{2}-{T}_{1})+{C}_{l}(T-{T}_{2})& \text{;}{T}_{1}\le T\le {T}_{2}\end{array}\)

with the following values:

\(\begin{array}{}{C}_{S}={C}_{l}=1/3\mathrm{.}{10}^{7}J/{m}^{3}°C\\ {C}_{\mathrm{Sl}}=8.333{10}^{7}J/{m}^{3}°C\\ {T}_{1}=585.°C\\ {T}_{2}=615°C\end{array}\)

1.3. Boundary conditions and loads

Temperatures imposed at the extremities

\({T}_{0}=200°C\) for \(x=0\)

\({T}_{L}=1000°C\) for \(x=L=\mathrm{1m}\)

Solid movement speed: \(V={10}^{-4}m/s\)