It is assumed, that the initial strand temperature is constantly distributed over the cross section and corresponds to the temperature at the screw tip.

$$T_{0,j} = T_M \text{ für alle j} \tag{Equation 15}$$

For all radial segments with index j=1, it is assumed that the two predecessor temperatures $T_{i,j}$ und $T_{i,j-1}$ are the same. The Taylor series development is broken off after the first member, resulting in equation 6:

$$T_{i+1,1} = T_{i,1} + \frac{\Delta x \cdot a}{v_{ab} \cdot R^2} \cdot (T_{i,2} - T_{i,1}) \cdot \frac{3}{2} \cdot N^2 \tag{Equation 16}$$

In the last radial segment j=N is no previous temperature with index j+1. Therefore, an auxiliary temperature is needed, which is linked to the water temperature. For this purpose, an auxiliary layer is defined, which is a step size (Δr) behind the last segment and represents the index j + 1.

The calculated temperatures are always at the core of the segments, see figure 2. This means für j=N, that the last segment is not the surface temperature. The surface is displaced by the length Δr/2.

To determine the temperature in the auxiliary layer, a boundary condition of the third kind (or Newton boundary condition) is considered, which is used for the convective heat transfer. The respective auxiliary temperature can be determined graphically, see figure 3. A Taylor series development of the temperature in the radial direction with truncation after the second element results in the following: [BK97]; [VDI06]

$$T_{i,Hilfs.} = T_{i,N} + \frac{\Delta r}{\frac{\lambda}{\alpha} + \frac{\Delta r}{2}} \cdot (T_U - T_{i,N}) \tag{Equation 17}$$

For the calculation of the heat transfer coefficient and the therefor required Nusselt number, the case of a moving cylinder in a resting fluid with the assumption Pr = 10 is used [VDI06].

Figure 3: Exemplary temperature curves at I and i+1