For a global estimation of the extrusion process, the knowledge of the melting process is of a great importance. To calculate melting length and the solids bed profile, a modified TADMOR model is used. This estimates the location de-pendent molten film thickness on the cylinder wall amongst the inclusion of radial leakage flows. The solution of the melting process in the channel direction results from the energy balance in the molten film above the solid bed and the mass balance between the melt pool and the melt film, see figure.

Figure: Modified TADMOR model with location dependent melt film thickness

The equations for the calculation of the melt profile in the channel direction can be seen in the table.

Table: Equations for the calculation of the melt profile

Dimensionless solid base width from the energy balance in the melt film: $$y = \frac{X}{b} = \frac{\frac{k_1 \rho_s v_{0z} \Delta h}{\lambda_z(T_z - T_{Fl})b}\left(\frac{\delta}{1 + c}\right)^2}{2 + \frac{k_2 K(T_{Fl})v_{rel}^{1+n}}{\lambda_z(T_z - T_{Fl})}\left(\frac{\delta}{1 + c}\right)^{1-n}}\left(1 - \frac{s_R}{\delta}\right)$$

Dimensionless melt film width: $$\psi = \frac{\delta}{\delta_0}\psi^* = \frac{\psi - \psi_S}{1 - \psi_S} = y^c$$

Constant gradient for the melt profile: $$c = \frac{\lg\left(\frac{\psi_1 - \psi_S}{\psi_2 - \psi_S}\right)}{\lg\left(\frac{y_1}{y_2}\right)}$$

Function of the location dependent melt film width: $$\delta_0 = (\delta_1 - s_R)y_1^{-c} + s_R$$

Constants for the melt calculation: $$k_1 = 2\left(\frac{1}{1 - e^A} + \frac{1}{A}\right) k_2 = \frac{2}{A^2}\left(\frac{A}{e^A - 1}\right)^{1+n}(e^A - A - 1)$$ $$A = \frac{\beta}{n}(T_z - T_{Fl})$$

Function of the calculation of the dimensionless solid base width in the channel direction: $$y = \frac{X}{b} = [1 - (1 - c)(1 - \psi_S)\pi_1 \zeta]^{\frac{1}{1-c}}$$ $$\pi_1 = \frac{\rho_s k_1 \delta_0 v_0 D_S}{2\dot{m}}\zeta = \frac{z}{D_S} = \frac{L}{D_S \sin(\varphi_S)}$$

The location where the melting process starts can be simplified to the point of first filling (PFF). These assumptions were validated by various arrays of experiments.