The modified disperse melting model introduced in [Thüm08] is basically build on the theoretical remarks on disperse melting according to Melisch [Meli98, Pote96]. At this point references are made to the preliminary considerations introduced in the melting model for disperse fillers.

In the following the assumptions and the boundary conditions introduced are used to find a detailed solution.

The influence of the finite channel dimension is especially pronounced in channel height direction. Therefore a correction depending on the particle diameter-to-channel height-ratio (dP/h) is made. The bases for this correction are among other things CFD simulations, which show the temperature profile of a particle in different dP/h-ratios. The results show that only with particles showing a low dP/h-ratio an undisturbed temperature profile can be formed.

The heat flow which exists at the interface particle/melt in case of a pure thermal conduction is corrected by a thermal conduction correction factor $f_lh$.

$$f_{lh,sim} = 1,30525 + 1,98091 \cdot \frac{d}{h}$$

The correction factor is valid in the range 0,2 < d/h <0,9 and is based on an approximation of the dimensionless temperature field. It shows a good match with the CFD results, and compares the middle temperature gradient at the particle surface with finite channel height to the middle temperature gradient with infinite expansion.

An analytical solution of the energetic differential equation is in this case hard to derive [Pape06].

The factor $f_{lh}$ does not include a consideration of the expansion in channel width direction. In this case numerics are also used because an analytical derivation does not seem possible. The simulations show that the influence of the channel width only occurs with low pitch-screw diameter-ratios. But these are rarely used in practice. A correction for the heat flow with reference to the finite channel width is therefore neglected.

The influence of the convection on the melt process in twin screw extruders was until here neglected. That is why a factor fk to consider the convective heat transfer is introduced in the modified melting model. This factor is also based on CFD simulations. Based on a single rotating particle surrounded by melt in an unwound twin screw channel different relations of the pellet diameter to the channel height dp/h as well as different Péclet numbers were used. The latter balances the ratio of the convection and the thermal conduction and describes the amount of the convection occurring in the flow.

$Pe = \frac{\rho_m \cdot c_{pm} \cdot v_{0z} \cdot d_p}{\lambda_m}$

The simulation results show that for high convective shares a considerably faster melting is to be expected. Mathematically, the correction factor can be described by:

$f_k = 1 + \frac{(b_1 \cdot \kappa + b_2 \cdot \kappa^2 + b_3 \cdot \kappa^3 + b_4 \cdot \kappa^4) \cdot Pe}{(b_5 \cdot \kappa + b_6 \cdot \kappa^2 + b_7 \cdot \kappa^3 + b_8 \cdot \kappa^4) \cdot \left(\frac{3}{4} \cdot \sqrt{Pe} + b_9\right)}$

with: $\kappa = 1 - e^{-\frac{dp}{h}}$

and it compares the heat flow from the analytical calculation (without considering the convection) with the heat flow from the simulations. Therefore the following is true:

$\dot{q}_r = \left(-\lambda_m \frac{\partial T}{\partial r}\right) \cdot f_k \cdot f_{lh}$

The regression constants b1 to b9 can be taken from the following table.

Table: Regression constants

The melting process can not be considered without taking into account the whole process. Interactions between single particles as well as the particle dimension have an influence on the viscosity of the melt-solid mixture.

According to Potente and Melisch [Meli98, Pote96] the interactions are considered by using an effective channel height and width which is dependent on the amount of solid. As shown in the illustration below the melt-solid mixture is considered independent of each component. In this way the flow can be modelled through the melt above the solid layer.

Figure: Consideration of the solid particle amount in the original disperse melting model [Thüm08]

In the modified melting model the flow in the melt-solid mixture is formed with the help of an adjustment of the viscosity. For this purpose the correction of the power law consistency with the factor $f_Φ$ is in introduced.

$K_{sm} = K \cdot f_\phi$

$f_\phi = \frac{\eta_{sm}}{\eta_0} = 1 + \frac{5}{2} \cdot \phi_{V,s}$

The correction factor $f_Φ$ is dependent on the volume content of the solid $Φ_{V,s}$ and compares the viscosity of the fluid-solid mixture $η_{sm}$ with the viscosity of the pure fluid $η_0$. This correction is based on Einstein but for large solid shares the viscosities calculated deviate too much. A further development according to Guth and Simah yields better results. Therefore the following is true for the correction factor:

$f_\phi = \frac{\eta_{sm}}{\eta_0} = 1 + \frac{5}{2} \cdot \phi_{V,s} + \frac{141}{10} \cdot \phi_{V,s}$

A conversion of the volume content on the mass content yields:

$f_\phi = 1 + \frac{5}{2}\left(\frac{\phi_{M,s}}{\phi_{M,s} + \frac{\rho_s}{\rho_m}(1 - \phi_{M,s})}\right)\phi_{V,s} + \frac{141}{10}\left(\frac{\phi_{M,s}}{\phi_{M,s} + \frac{\rho_s}{\rho_m}(1 - \phi_{M,s})}\right)^2$

If the $d_p/h$-ratio approaches 1 the particle diameter considered corresponds approximately to the channel height, and the influence of the particle size on the flow has to be considered. Following Pape [Pap06] another correction of the power law consistency is introduced.

$K_{sm} = K \cdot f_\phi \cdot f_{dh}$

$$f_{dh} = 1 + \frac{\frac{d_p}{h} \cdot \left(1 + \left(\frac{d_p}{h}\right)^2\right)}{\left(\frac{\rho_s}{\rho_m} - \frac{1 - \Phi_{M,s}}{\Phi_{M,s}}\right) \cdot \left[\left(\frac{d_p}{h}\right)^2 + \frac{d_p}{h} + 4\right] \cdot \left(1 - \frac{d_p}{h}\right)}$$

Based on a solid particle in a Newtonian melt the flow conditions around this particle are considered in order to determine the correction. After this the viscosity for the flow of an equivalent volume flow of pure melt is determined. The ratio of the viscosity of this consideration and the viscosity of the solid-melt mixture leads to the correction with the factor fdh.

The physical-mathematical description of the modified disperse melting behavior is basically based on the assumptions of the prior melting model for disperse fillers. Here the following additional aspects are taken into consideration:

• The influence of the particle dimension on the flow (fdh),
• The influence of the particle interactions on the flow (fΦ),
• The Influence of the finite expansion of the channel in height direction on the heat flow in radial direction (flh)
• The influence of the convection on the heat flow in radial direction (fk).

by the assumptions made.

With the influence of the correction factors flh and fk the heat flow in radial direction is given by the following equation (cf:Calculation of the Solid Bed Reduction Along the Melt Path):

$\dot{q}_r = -\lambda_m \cdot \frac{\partial T}{\partial r} \cdot f_k \cdot f_{lh}$

These factors are used in the whole calculation process which was introduced in the previous chapter. Therefore the following is true for the change of the particle radius in the interval considered:

$r_{i+1} = \sqrt{r_i^2 - \frac{2 \cdot \lambda_m \cdot f_{lh} \cdot f_k}{c_m \cdot \rho_s \cdot \bar{v}} \cdot \ln\left(1 + \frac{c_m \cdot (T_m - T_{flow})}{\Delta h \cdot f_{lh} \cdot f_k}\right) \cdot \Delta z}$

The factors fdh and fΦ, which reflect the influence of the particles on the flow, are anchored in the calculation of the middle flow velocity $\bar{v}$.