1 Introduction

To achieve the ambitious targets for CO\(_2\) emission reductions set for the industry, it is necessary to introduce and develop innovative, more efficient processes. Reducing both the amount of material and the energy required by industrial production processes should be the focus of new process development. In this context, Brosius et al. [1] have proposed a new process named thermomechanical tangential profiled ring rolling (TPRR), aiming at achieving efficient production of steel ring shaped parts.

Tangential profiled ring rolling, a subfamily of the ring rolling processes [2], is typically used for the production of small to medium scale ring shaped parts, such as roller bearing tracks, see Fig. 1. The process may be implemented cold or semi-hot to reach near net shape geometries. It is typically followed by heat treatment to achieve the desired microstructure and hardness. According to Profiroll GmbH, a producer of rolling machines, this process already presents significant advantages compared to cutting in terms of material and energy efficiency [3]. In thermomechanical TPRR, the ring is heated to austenization temperature, then formed and rapidly cooled during one single step, aiming to obtain both the correct geometry and required microstructure or hardness. This new process route presents the advantage of reducing the number of process steps and heating cycles, which represents the majority of energy consumption required in hot forming processes. It also helps to simplify the production chain, by combining different steps. Figure 2 presents the process route compared with the traditional approach.

Fig. 1
figure 1

Example of ring rolled product [4]

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In this study, the need for a controlled cooling system for thermomechanical TPRR is demonstrated. Subsequently, a control system is designed and validated, first using numerical simulation, and then using a simplified physical ring rolling model. Finally, the influence on process performance is demonstrated in real ring rolling experiments.

Fig. 2
figure 2

Process route comparison between semi-hot rolling and thermomechanical TPRR [4]

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2 State of the art

2.1 Thermomechanical TPRR

The TPRR machine parts are presented in Fig. 3. In TPRR, the part is formed through the combined action of the rotation and the translation of the main roll. As the main roll translates towards the mandrels, the ring thickness reduce is reduced and the ring diameter grows. The profile of the tool is also impressed on the ring. Finally, similarly to close die forging, the process ends when the tools make contact. The main roll is actively driven by a motor during the process, which then drives the ring, which drives the free rolling mandrel. Two stabilization rolls help to keep the movement of the ring stable during the process. In thermomechanical TPRR, the ring cooling occurs not only through heat conduction at contact between the ring and the tools but also through convection with supplementary compressed air cooling. It is controlled by a valve and enables both a faster and more precise control of the cooling rate.

Fig. 3
figure 3

TPRR machine schematics and photo [4]

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Thermomechanical ring rolling has two major process parameters: the feed rate of the roll and the airflow. The desired output of the process can be seen as the final outside diameter and the part’s average hardness. The influence of both process parameters has been previously studied [5]. It was shown that a significant cross effect exists. Indeed, the growth rate of the ring is dependent on both the feed rate and the temperature, which is also dependent on the airflow. However, the cooling rate is also dependent on the feed speed through a modification of the plastic energy dissipation rate and conduction contact influence. The cooling rate is the main factor to achieve specific hardness, since, the main hardening mechanism in TPRR is phase transformation. By increasing cooling rates, harder phases are formed, achieving higher overall hardness in the ring.

The study also indicated that the process has a limited repeatability in an open loop setup, caused by variation of the two main process parameters, but also other factors such as the initial ring and tool temperatures. Thus, in the open-loop configuration, a significant variation in the hardness and final diameter of the parts produced was observed. Experimental data showed that a significant proportion of this variance could be reduced by implementing a temperature control loop.

2.2 Controlled cooling in forming processes

Compressed air is widely used for the cooling of cutting operations, such as milling [6] and significant literature is available on the subject. This, however, does not compare well with the ring rolling process as the temperature of the parts is much lower during a cutting operation. Furthermore, in cutting operation only a local cooling is necessary whereas, in a rolling process, the complete part must be cooled.

Compressed air or mixed water-air spray are, sometimes, used to achieve targeted cooling of the parts before or after forming operations, but this thematic is very much less developed in the literature. Mohapatra et al. [7] used air and air-mist spray to achieve varying cooling rate in hot steel plates. Pyshmintsev et al. [8] described the use of similar air-mist spray to harden large forgings and obtain targeted microstructures. Finally, compressed air was used by Golovko et al. [9] to obtain tailored blanks for hot stamping by influencing the local cooling rate. Overall, literature analysis shows that that compressed air and liquid-air spray can successfully be used to control the cooling rate of parts. TPRR seems to be a suitable application for the use of compressed air during a forming process:

  • the part presents a high surface to volume ratio,

  • a large section of the part surface is accessible for cooling,

  • the process can be operated at a large range of temperatures.

No description of a system for the closed loop control of temperature during a forming operation was found in the literature.

3 Controller design and test

The cooling system is composed of a proportional valve, an airflow sensor and two outlets. The compressed air is taken directly from the main circuit of the facilities. In open loop, the opening of the valves is set to a constant value during the process and the sensor is only used to monitor the system after the fact. Potential disturbances can impact the cooling rate of the part and include the following fluctuations [5]:

  • of the supplied air pressure,

  • of the heat transfer coefficient with the air due to change in ring size and consequently the distance between the inlet and the ring,

  • of the plastic strain rate, and the plastic energy dissipation during the process,

  • of the tool surface temperatures and consequently the heat conduction between the tools and the ring.

Controlled cooling of the ring serves two purposes: ensuring that the ring follows a desired temperature curve, and the rejection of disturbances.

The proposed control follows the architecture shown in Fig. 4. Two cascading PID (Proportional Integral Derivative) controllers are used to ensure control of the airflow and the temperature. The use of the cascading PID is necessary, because the time constants in the airflow system are expected to be much smaller than for the heating process. Decoupling both problems simplifies the PID tuning [10].

To test and investigate the performance of this controller, a simplified numerical model of the process was implemented. The controller was then tested on a simplified physical model.

Fig. 4
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Proposed controller architecture

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3.1 System modelling

To determine the controller parameters and the suitability of this controller design, a model of the cooling system needs to be implemented. For that purpose, MATLAB Simulink is used. The following sections describe the different parts of the model.

3.1.1 Airflow control system

For the PID tuning of the airflow control system, a practical approach is used. Indeed, the experiments necessary to use the Ziegler–Nichols method [11], which require finding the ultimate gain and the oscillation period, can be easily realized. The system was tuned so that overshoot does not exceed 5%. The system as a whole is then approximated as a second order LTI (Linear Time-Invariant) system. The parameters of the model were computed using a step response of the airflow control system in closed loop.

3.1.2 Forced convection

The heat transfer between the air and the ring occurs through forced convection. This is also the case when the airflow is zero due to the tangential speed of the ring. The heat flux at the surface can be computed using the Newton law of cooling:

$$\begin{aligned} \dot{Q} = A h(af)(T_\text {surf} - T_\text {air}) \end{aligned}$$
(1)

with \(\dot{Q}\) the rate of heat transfer, h the heat transfer coefficient, A the surface, \(T_\text {surf}\) and \(T_\text {air}\) the temperature of the surface and the air and af the airflow. The value of the heat transfer coefficient cannot be determined theoretically in this case. For experimental determination, a new setup, presented in Fig. 5, is used. The ring is rotated at a constant rate of 380 rot/min, which resembles the ring rolling process. The clamping device was designed to limit surface contact with the ring, thereby reducing conduction.

Fig. 5
figure 5

Setup for the determination of the heat transfer coefficient

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The temperature is continuously measured using a pyrometer during the experiments. The ring has a width of 18 mm, an outside diameter of 65 mm, a thickness of 7.5 mm and a weight of 225 g. The measured temperature curve is then fitted to an exponential curve to determine the time constant of the temperature decay, which gives the heat transfer coefficient. By repeating this computation for different values of the airflow, a curve of the heat transfer coefficient was determined by fitting a power function to the measured data. Figure 6 shows the obtained results. The influences of thermal radiation and conduction are neglected.

Fig. 6
figure 6

Measured heat transfer coefficient as a function of the airflow and the fitted power function

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The Biot number is defined by the following equation:

$$\begin{aligned} \text {Bi} = \frac{h}{\lambda }L \end{aligned}$$
(2)

with h the heat transfer coefficient, \(\lambda\) the thermal conductivity of the material, and L a characteristic length. When this number is not significantly smaller than one, the temperature of the body during cooling cannot be assumed as homogeneous.

Here, the Biot number varies between 5 and 40. That means that the surface temperature is significantly different from the averaged temperature. When the airflow is constant, there is only a timing delay, which has limited effect on the computation of the heat transfer coefficient. However, when the value of the airflow varies dynamically, this effect needs to be considered, as it tends to significantly affect phase, especially for lower pulsation. Therefore, the temperature non-uniformity in the ring was modeled.

3.1.3 Thermal ring model

An accurate modelling of the ring thermal behavior during the process would require a complete thermomechanical finite element simulation. Previous studies have shown that this technique is applicable to ring rolling using a digital twin [12]. However, the computational cost of this technique are particularly high, and the design of the controller only requires a simplified model. The following effects need to be captured by the model presented:

  • primarily, the ring undergoes cooling at the surface through forced convection,

  • the cooling of the ring is not uniform in the ring,

  • only the temperature at the surface can be measured,

  • plastic dissipated energy and heat transfer between the tools can be treated as disturbances.

Assuming that the thermal properties of the material are constant, the heat equation is linear, and it is possible to model the ring as an LTI system. The temperature of the ring is assumed to be homogeneous in the tangential direction, since the ring rotates constantly during the process. Because the compressed air is projected on the outer surface, the heat transfer in the axial direction is assumed to be negligible compared to heat transfer in the radial direction.

With the previously introduced hypothesis, the equation governing the temperature in the ring simplifies to:

$$\begin{aligned} \frac{\partial T(t, r)}{\partial t} - \frac{\lambda }{\rho c_\text {p}} \left( \frac{1}{r}\frac{\partial T(t, r)}{\partial r} + \frac{\partial ^2 T(t, r)}{\partial r^2}\right) = \frac{s_\text {pl}}{\rho c_\text {p}} \end{aligned}$$
(3)

with

  • T(t, r) the temperature as a function of time and coordinates,

  • \(\lambda\), \(\rho\), \(c_\text {p}\), respectively the thermal conductivity, the density, and the specific isobaric heat capacity,

  • \(s_\text {pl}\) the plastic dissipated power, assumed homogeneous within the ring.

The problem comes with a set of initial and Neumann boundary conditions:

$$\begin{aligned} {T(t, r, \theta , z)}\big \vert _{t=0} = T_0 \end{aligned}$$
(4)
$$\begin{aligned} {\frac{\partial T(t,)}{\partial r}}\big \vert _{r=r_\text {o}} = s_\text {o}\end{aligned}$$
(5)
$$\begin{aligned} {\frac{\partial T(t)}{\partial r}}\big \vert _{r=r_\text {i}} = 0 \end{aligned}$$
(6)

with

  • \(r_\text {o}\) and \(r_\text {i}\) the outer and inner radii.

  • \(s_\text {o}\) the surface power density conducted at the outer and inner boundary surfaces.

A complete solution of the differential equation is not necessary to model the system. A set of four function transfers are required, as described in the block diagram Fig. 7. The solution for \(F_\text {pl/avg}\), \(F_\text {pl/surf}\) and \(F_\text {o/avg}\) are trivial, they are all integrators. The solution for \(F_\text {o/surf}\) is more difficult to obtain.

Fig. 7
figure 7

Block diagram of the thermal ring model

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While the analytical solution of the equation can be found in the literature, in the EXACT [13] database for example, the deduction of a transfer function is not immediately possible. Therefore, another approach was used. The equation (1) with \(s_{\text {pl}} = 0\) is solved using the Crank–Nicholson method finite difference method, with \(s_\text {o} = sin(\omega t)\). From these computations, a bode diagram can be computed. The result for the surface temperature is presented in Fig. 8. The transfer function can be adequately approximated with the following form:

$$\begin{aligned} F_\text {o/surf} = \frac{A_o}{\sqrt{1+\tau _o \times s}}\frac{1}{s} \end{aligned}$$
(7)

In order to obtain a rational function, necessary for an implementation in Simulink, the Padé approximant method is used [14], with two poles, and one zero. The parameters \(\tau _0\) and \(A_0\) are then determined by fitting in the harmonic domain.

Fig. 8
figure 8

Bode diagram of \(F_\text {o/surf}\), simulated using the Crank-Nicholson method, approximated using an LTI, and then under the form of a rational function using Padé approximant

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It is also interesting to note, that, assuming initial condition are known, it is possible to use this method to determine the temperature at any coordinates in the ring as a function of the surface temperature.

3.1.4 Temperature measurement

The surface temperature is measured using an Optris OPTCTL3MH1 pyrometer, with a 90% response time of the sensor of 1 ms. Because of the ring rotation, the sensor does not measure a fixed point of the ring. Due to local variation of emission values and temperatures, the measured temperature is perturbed with a sinusoidal harmonic noise. To limit this effect, a first order low pass filter is used, with a cutoff frequency of 0.667 Hz, slower than that of the ring rotation. The response time of the pyrometer is negligible compared to this time constant. The sensor is therefore modelled as a first order system with a unitary static gain and a 1.5 s time constant.

3.2 Closed loop control

3.2.1 Implementation

Using the MATLAB PID tuning app [15], parameters for the PID control are selected. The derivate coefficient of the PID is set to zero for two main reasons. Firstly, the temperature signal, even after filtering, is slightly noisy. The computation of the derivate of such a signal is challenging. Secondly, compared results using the MATLAB model showed little improvement between using the PID rather than the PI controller. Therefore, in this study, the PID controller is simplified to a PI controller.

Because the controller will be implemented on a microcontroller, the sampling frequency also needs to be carefully selected. Numerical tests, performed using Simulink, have indicated that a frequency lower than 10 Hz deteriorates the performance of the controller. Therefore, the update frequency of the controller is set to 20 Hz.

As temperature input, an exponential decay curve with a given time constant is used. When the controller is started, the actual temperature is employed to compute the input as a function of time. The controller turns off when the temperature falls below a predetermined value, below the recrystallization temperature.

Fig. 9
figure 9

Untuned and tuned response of the system simulated using MATLAB Simulink

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Figure 9 presents the tuned and untuned response of the system in closed loop mode. Between 10 s and 20 s a disturbance is applied in the form of a homogeneous power source in the ring: \(s_\textrm{pl} = 3 \times c_\textrm{p} \times m\), with \(c_\textrm{p}\) and m respectively the heat thermal capacity and the ring mass. The control error is the difference between the command and the signal, while the true error is the signal between the surface temperature and the command. Due to the delay in the temperature filter, a static error will always be present, but this is considered acceptable.

The complete control loops are then implemented on an Espressif ESP32 microcontroller.

3.2.2 Validation

Using the same physical model as for the heat transfer coefficient determination, a first series of four tests is realized on the system. It is possible to closely monitor the behavior of the system under selected condition using this setup.

The first experiment is with nominal conditions, with a target time constant for the temperature decay of 60 s (T1). During the second test (T2), an artificial disturbance is applied: from around 10–30 s, a supplementary cooling jet is actuated to provide additionally cooling to the ring, all other conditions remaining nominal. A different ring geometry with a significantly higher wall thickness (+ 35% from nominal) is employed for the third test (T3). For the fourth test (T4), the target time constant is increased to 80 s (+ 20% from nominal).

Figure 10 presents the results of these first validation tests, with the value of the filtered surface temperature, the error and the measured airflow. During the experiments, the controller is able to react to the disturbance and adapt the airflow. The experiments indicate that some overshoot occurs, but it remains acceptable. The ability of the controller to achieve a similar result despite the disturbance is demonstrated.

Fig. 10
figure 10

Cooling test with closed loop control under different conditions

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4 Ring rolling tests

Ring rolling experiments were realized on a retrofitted UPWS 31,5.2 from VEB Werkzeugmaschinenfabrik Bad Düben (1986), now known as Profiroll Technologies GmbH. The tools used, shown Fig. 3 correspond to the outer bearing of a cylindrical roller bearing, whose dimension are presented Fig. 1, and were the same as employed in a previous study [3]. Twelve Experiments were performed, six of which in open-loop mode, six in close-loop mode.

The rings were heated for 30 min at 900 \(^\circ\)C. The experiments were realized at a fixed rate of one ring every 5 min. Prior to the recorded test, a few warm-ups rings were realized to ensure that tools reach their nominal temperature.

In closed-loop mode, the time constant was set to 34.7 s, which corresponds to a cooling time between 800 and 400 \(^\circ\)C (\(T_{8-4}\)) of 25 s. In open-loop mode, the air-flow was set to 260 l/min, which should, according to previous works [5], correspond to a similar \(T_{8-4}\) cooling time.

The surface temperature was measured with the pyrometer, with logging data acquired unfiltered, at the rate of 1000 Hz. This was done to alleviate any potential effect of the filtering and to have the same acquisition chain for both open and closed loop experiments. For each curve, the time constant of the temperature decays was determined. Figure 11 shows the results of 12 tests, grouped by control mode, and the standard deviation over the two groups of six tests.

Fig. 11
figure 11

Influence of closed loop control on repeatability

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The coefficient of variation in cooling rate in open loop mode is estimated at 7.6%. This is consistent with the value obtained in previous works, where it was estimated at 7.4%. When comparing the results between open and closed loop, it is clear that the use of the control helps achieve a significant, 5-fold, reduction of the sampled standard deviation.

When testing for the hypothesis that both samples have the same variance, the Bartlett test [16] for variance equality returns a p-value of 0.0026, which means that this hypothesis should be rejected. Although the number of samples is limited, the data shows a clear, statistically significant, improvement when using closed-loop control.

However, the accuracy of the system seems not to be perfect, as the averaged cooling rate is 0.9 s (2.5%) smaller than the required value. This might be improved by reducing the influence of the filtering, and improving the dynamic of the controller using a derivate coefficient.

5 Conclusion

In this study, the possibility of using a closed-loop control system to improve the repeatability of cooling during thermomechanical TPRR was demonstrated. A model of the cooling process was developed and implemented, with a focus on obtaining LTI-models for describing the thermal behavior of the ring. Then the performance of the control loop was illustrated in a simplified setup before being demonstrated in real ring rolling experiments. This study found that the described simplified model was sufficient to effectively tune the PI controller.

It is also clear that this strategy helps to significantly reduce the variance of the cooling rate during ring rolling operation. In a series of ring rolling experiments, the variance of the cooling decay was reduced by a factor of five compared with open loop control. Overall, the performance of the cooling system was significantly improved by the addition of a controller.

However, some questions remain open, and will be the focus of future work:

  • What is the size limit of the ring that could be hardened with this method?

  • How much tuning and simulation is required when the geometry of the ring is modified?

  • Some studies have shown the possibility of inline measurement of the ring microstructure [17, 18], this could enable the direct control of the ring microstructure. This would require the implementation of more complex model process control method, using finite horizon model predictive control, such as proposed in earlier works [12].