To determine the time-related temperature fields within a solid, the Fourier differential equation is considered (here in cylinder coordinates): [VDI06]

$$\rho c_p \frac{\partial T}{\partial t} = -\lambda \left[\frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r} + \frac{1}{r^2} \frac{\partial T}{\partial \varphi^2} + \frac{\partial^2 T}{\partial z^2}\right] \pm \dot{q}_s \tag{Equation 3}$$

For the current problem (viz. no external heat source, no angular temperature gradient), the differential equation is simplified as follows:

$$\frac{\partial T}{\partial t} = \frac{\lambda}{\rho \cdot c_p} \cdot \left[\frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r}\right] \tag{Equation 4}$$

The finite difference method provides an approximate solution, whereby the introduced error can be minimized by increasing the degree of discretization. However, this also increases the calculation effort. To approximate the partial derivatives, a Taylor series development is performed.

A Taylor series expansion is carried out to approximate the partial derivatives: The development of the temperature for a small time step results in: [BK97]

$$T_{t+\Delta t,r} = T_{t,r} + \frac{\partial T}{\partial t} \Delta t + \cdots \tag{Equation 5}$$

The Taylor series development is broken off after the first member, resulting in

$$\frac{\partial T}{\partial t} = \frac{T_{t+\Delta t,r} - T_{t,r}}{\Delta t} = \frac{T_{i+1,j} - T_{i,j}}{\Delta t} \tag{Equation 6}$$

The first of the three partial derivatives is approximated with

$$\frac{\partial T}{\partial t} = \frac{T_{i+1,j} - T_{i,j}}{\Delta t} \tag{Equation 7}$$

A Taylor series development of the temperature in the radial direction with truncation after the second element results in the following: [BK97]

$$T_{i,j+1} = T_{i,j} + \frac{\partial T}{\partial r} \Delta r + \frac{1}{2} \frac{\partial^2 T}{\partial r^2} \Delta r^2 + \cdots \tag{Equation 8}$$

$$T_{i,j-1} = T_{i,j} - \frac{\partial T}{\partial r} \Delta r + \frac{1}{2} \frac{\partial^2 T}{\partial r^2} \Delta r^2 + \cdots \tag{Equation 9}$$

These two equations are added together:

$$\frac{\partial^2 T}{\partial r^2} = \frac{T_{i,j+1} - 2T_{i,j} + T_{i,j-1}}{\Delta r^2} \tag{Equation 10}$$

$$\frac{\partial T}{\partial r} = \frac{T_{i,j+1} - T_{i,j-1}}{2 \cdot \Delta r} \tag{Equation 11}$$

This determines all partial derivatives approximately. Inserting them into equation 4 gives the following result:

$$\frac{T_{i+1,j} - T_{i,j}}{\Delta t} = a \left[\frac{T_{i,j+1} - 2T_{i,j} + T_{i,j-1}}{\Delta r^2} + \frac{1}{r} \frac{T_{i,j+1} - T_{i,j-1}}{2 \cdot \Delta r}\right] \tag{Equation 12}$$

After a few simplifications, the following expression results:

$$T_{i+1,j} = T_{i,j} + \frac{\Delta t \cdot a}{R^2} \left[(T_{i,j+1} - 2T_{i,j} + T_{i,j-1}) \cdot N^2 + \frac{N}{j} \cdot \frac{T_{i,j+1} - T_{i,j-1}}{2}\right] \tag{Equation 13}$$

In addition, the time difference Δt can be described by the ratio of the length to the pull-off speed:

$$T_{i+1,j} = T_{i,j} + \frac{\Delta x \cdot a}{v_{ab} \cdot R^2} \left[(T_{i,j+1} - 2T_{i,j} + T_{i,j-1}) \cdot N^2 + \frac{N^2 T_{i,j+1} - T_{i,j-1}}{j \cdot 2}\right] \tag{Equation 14}$$

$$T_{i+1,j} = T_{i,j} + \frac{\Delta x \cdot a \cdot N^2}{v_{ab} \cdot R^2} \left[(T_{i,j+1} - 2T_{i,j} + T_{i,j-1}) + \frac{1}{j} \frac{T_{i,j+1} - T_{i,j-1}}{2}\right] \tag{Equation 14.1}$$

For each shell j, the material-specific properties are determined in each time step i depending on the temperature.

It can be seen that the temperature can be determined on the basis of three temperatures that are located in the preceding length segment i. If the temperature is to be determined in the last radial segment at j=N, there is no preceding temperature with the index j+1; similarly, there is no preceding temperature with j-1 in the middle of the line. Accordingly, boundary conditions must be defined for these limiting cases in order to calculate auxiliary temperatures.