In the work [7] the simplified method of simulation of heat transfer between calendering roll and paper is presented. By combining thermal conduction and existing thermal conditions with energy balance, the movement of the paper web can be neglected, as it is shown (for 2D model)in Figure 3. As can be seen, two types of cylindrical surfaces (in 3D volumes) have been created: rotating (steel roll) and non-rotating (generally representing heat sources and sinks). There is no thermal resistance between these two types of region. The fixed regions may represent this part of roll surface where power is dissipated (regions under inductors), as well as areas of intensive heat removal (in contact with the wet web of paper). This approach may solve the problem of connecting the rotation motion of the roll with linear motion of the endless paper web. The material parameters of the area which represents the paper web can be treated as a function of the observation time. This makes it possible to analyze the dynamic effects of changing these parameters (e.g. moisture and temperature) on the temperature of the rotating roller smoothing the paper.

Figure 3 The 2D model of heat transfer between the rotating cylinder and the moving paper web

In general, two main problems can be highlighted in the considered issue. Firstly, how to present and simulate the induction heating (power generation), and secondly how to simulate the heat transfer to wet web, its drying and heating.

It was assumed that the large roll diameter in comparison to the circumferential size of inductors, as well as the skin effect and high speed of rotation, means the integral of volumetric power density dissipated in the roll (along the circumference and thickness of the cylinder wall)can be used in thermal calculations, instead of the actual distribution of volumetric power density. This leads (in 3D) to the usage of the linear power density distribution along the length of the cylinder in thermal calculations.

Taking into account that the temperature of calendering rolls changes during the work only about a few degrees, the linear power density along the length of cylinder can be calculated (for roll working temperature) by electromagnetic calculation not coupled with the thermal calculations. This allows a significant reduction in considered area and the omission of motion, as is shown in Figure 4. For example, in Figure 4a the volumetric power distribution on the surface roll under the inductor and calculated linear power density along a generatrix of the cylinder, for two values *x* of position of the side inductor (for the same inductor parameters and power supply as assumed above), is shown in Figure 4b.

Figure 4 The volumetric (a) and linear (b) power density distribution, in the considered part of stationary cylinder

The calculated linear power distribution for 3D analysis or power for 2D can be used as the power source in “Main inductor area” or “Control inductor area”, Figure 3, in considered simplified thermal model of rotating cylinder and moving paper web. It can be [7] used for simulating both the process of induction heating of a calender roll without paper web, and when the roll (on part of its circuit) has contact with wet paper web, as shown in Figure 1.

During the time *Δt*_{c}of contact, the paper web is warming up from temperature *ϑ*_{in} to *ϑ*_{out} close to the temperature of cylinder *ϑ*_{cyl}. The average *ϑ*_{av} temperature of the paper cross section achieves *ϑ*_{av}>0.95·*ϑ*_{cyl} usually in a time *Δt* less than the time *Δt*_{c}. Taking this into account the area of the cylinder-paper contact can be divided into two areas, referred to as “cold paper” and “hot paper”. In a such situations it can be assumed that in a “hot paper” area there is no energy accumulation in the web paper but the area (Figure 3) “cold paper” should consume the power (*ϑ*_{av}>100°C>*ϑ*_{in}) with volumetric density [7]:

$$\begin{array}{}{\displaystyle {p}_{v}=-\frac{[{c}_{pap}\cdot ({\vartheta}_{av}-{\vartheta}_{in})+w\cdot {c}_{vap}]\cdot {\rho}_{pap}}{\mathit{\Delta}t}\cdot \frac{g}{{g}_{m}}}\end{array}$$(1)

where: *ρ*_{pap}– paper density, *c*_{pap} – specific heat of the paper, *c*_{vap} - water vaporization heat, *w* – paper moisture, *g* – real thickness of paper web, *g*_{m}– thickness of paper web in the simulation model.

The value *Δt* of time should be chosen in such a way that for a cylinder rotating at speed *θ*, described by (2) the angular part *Δψ* of the cylinder which adjoins the “cold paper” would be less than angular part *Δψ*_{w} of the cylinder that adjoins the whole web of paper,

$$\begin{array}{}{\displaystyle \mathit{\Delta}\psi =\frac{\theta}{60}\cdot 2\pi \cdot \mathit{\Delta}t}\end{array}$$(2)

where: *θ*- rotating speed in rpm.

Using the above method, the calendering process with induction heated roll was simulated. As an example it is presented in the case of paper web width 2·*w*_{p} = 1 m, with *ϑ*_{in} = 120°C and *θ* = 40 rpm (so in expression (1) *w*·*c*_{vap} = 0). It is assumed that the roll has already been heated to a temperature of approximately 200°C and the longitudinal distribution of linear power density is as in Figure 4b, for *x* = 80 mm. In Figure 5 the temperature distribution which occurs along the cylinder, in the half of its angular part adjoining the paper web, was presented.

Figure 5 The temperature distribution along the length of the rotating cylinder with and without paper web

The example of longitudinal temperature distribution, Figure 5, presents the very important problem of calendering.

Non-uniform temperature distribution leads to differences in roll diameter, which leads to differences in thicknesses of the treaded web of paper. As shown this nonuniformity depends on the web of paper and its parameters (temperature, moisture, etc.)

This requires correction which is possible when the induction heating method is used. This can be realized by the set of additional inductors, (“control” inductors) which can be similar, Figure 6, to the side inductors mentioned above or a mobile inductor [8]. The work of such a mobile inductor should be controlled (position and dissipated power) in order to stabilize the uniform temperature distribution regardless of disturbance by the paper web. The exact shape of such an inductor and the associated special control system can be determined by using the presented method to simulate the calendering system.

Figure 6 The simulation model of “control” inductor-steel cylinder system

The distribution of the linear power density in the roll for an additional “control” inductor can be calculated, as done above for a main inductor, by separately solving the electromagnetic numerical model of the inductor, Figure 6.

For magnetic parameters of a steel cylinder described by saturation *J*_{s} = 1.9 T and initial magnetic relative permeability *μ*_{r} = 1000, the distribution of linear power density *P*_{l, con} dissipated in the cylinder along direction *s* (Figure 6), for two different values of current in a 10 turns inductor (*f* = 20 kHz), is shown in Figure 7a. The average value of the linear power density for each of the eight highlighted sections (10 mm width) of the “control” inductor was calculated.

Figure 7 The distribution of the average linear density *P*_{l, con} of power in each *n* section on cylinder (a), the linear resistance *R*_{l, con} of sections and inductance of the system *L*_{total, con} as function of effective value of inductor current (b)

During simulation of the temperature control process, the distribution of power in the roll (induced by the “control” inductor) is controlled to obtain the required temperature distribution. The distribution of the linear power density *P*_{l, con} generated by the “control” inductor is calculated in another numerical model and only then the results are transmitted to the thermal model, which is used for simulation of the control system. Additionally, we must remember that the steel roll is a magnetic nonlinear material, so there is also a nonlinear relationship between current in the inductor and generated power. Taking this into account, for the “control” inductor not only distribution of linear power density *P*_{l, con} but also the linear resistance *R*_{l, con}(*I*_{ind}, *n*) as a function of inductor current *I*_{ind} (has influence on magnetic field strange) for each *n* section of cylinder, Figure 7b, under the “control” inductor must be calculated:

$$\begin{array}{}{\displaystyle {R}_{l,con,n}=\frac{{P}_{l,con,n}}{{I}_{ind}^{2}}}\end{array}$$(3)

The magnetic non-linearity has direct influence not only on the resistance, but on the inductance *L*_{total, con} of the system too (Figure 7b), which in turn has an impact on the resonant frequency of the power supply inverter and in this way (penetration depth) again on resistance. This should be taken into account when in the thermal calculations the distribution of the linear power density *P*_{l, con} is calculated, based on the above linear resistances (3).

The initial electromagnetic calculations are realized as time harmonic for constant frequency values (in that example for *f* = 20 kHz). Assuming, that the imposed frequency occurs, in real setup, for inductor current close to the zero values *I*_{ind}→ 0, the calculated linear resistance *R*_{l, con}(*I*_{ind}, *n*), Figure 7b, should be corrected (in thermal calculations) in accordance with:

$$\begin{array}{}{\displaystyle {R}_{l,con}^{corr}({I}_{ind},n)={R}_{l,con}({I}_{ind},n)\cdot 4\sqrt{\frac{{L}_{total,con}(0)}{{L}_{total,con}({I}_{ind})}}}\end{array}$$(4)

where: *L*_{total, con}(0)- inductance of the system for *I*_{ind}→ 0.

Equation (4) gives an approximate account of the effect of the current value (by the value of inductance *L*_{total, con}) on the resonant frequency and then (by varying the penetration depth) on the linear resistance of the charge.

The presented methodology was tested in the case of simulation of the temperature distribution control system (see Figure 8) by joining the work of two commercial programs, Flux and Portunus. In the FEM program (Flux) the temperature *ϑ* 3D distributions were calculated. Additionally, for paper web the average (on the thickness) temperature *ϑ*_{ave} along the width 2*w*_{p} of paper web and average temperature *ϑ*_{contr} on the “control” inductor width 2*w*_{contr} (used for control) were calculated. In the presented example the “control” inductor was used to correct temperature distribution around the edge of the paper web. It was placed in such a way that its axis of symmetry was located at a distance of 30 mm from the edge of the paper web (location of the minimum temperature), Figure 5.

Figure 8 A block diagram of the simulation model for the control of temperature distribution for induction heating of rotating cylinder with paper web

For a simple P controller with gain 10, Figure 9a shows the surface temperature distribution around the perimeter of the cylinder on its symmetry surface (no “control” inductor influence) and on symmetry surface of “control” inductor (after two and three rotations, 3 sand 4.5 s). Figure 9b shows the longitudinal temperature distribution under the paper web, for 3 sand 4.5 s after switching on the temperature control system.

Figure 9 The temperature distribution along the circumference (a) and the length (b) of the rotating cylinder with temperature control system

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