The molten strand is led out of the screw tip into the process water and in doing so it is cooled convectively (see Figure 1). Within the water, there is always the same defined strand length.

Figure 1: Exemplary strand cooling with additional strand extraction and granulation

In order to determine the temperature distribution over the strand length and the strand cross-section, it is discretized as follows:

Figure 2: Discretization strand

In the length direction, the strand is divided into M segments with the running index i. Since the time is relevant for the unsteady heat conduction, it is described over the cooling length L and the line speed v.

The index i stands for any position x, at which the strand has already been in contact with the water for a specific time t. The length of the individual subintervals in the length direction can be determined with:

$$\Delta x = \frac{L}{M} \tag{Equation 1}$$

Further, the cross section in radial direction is divides into N segments with the index j. The subintervals with the radius of the strand R results to:

$$\Delta r = \frac{R}{N} \tag{Equation 2}$$

Therefore, the relevant temperatures have two indices ($T_{i,j}$) and are located in the middle of the defined intervals, see Figure 2.