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Feedback control of liquid metal coating

par Pino, Fabio
Président du jury Bersini, Hugues
Promoteur Scheid, Benoît ;Mendez, Miguel Alfonso
Publication Non publié, 2024-03-05

Thèse de doctorat

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Résumé :

This project aims to mitigate one of the significant industrial problems in the coating industry affecting thesurface quality of liquid metal coatings: the undulation instability. In the hot dip galvanizing process, a metalsheet is dipped and then withdrawn from a bath of molten zinc. This produces a thin liquid film of zinc flowingover the surface of the metal, which then solidifies into a corrosion-protective coating. With more than 100millions tons of steel coated worldwide every year, the hot dip galvanizing process is used nowadays in mostproduction lines, thanks to its simple, continuous, and cost-effective configuration. However, its operatingconditions are limited by the occurrence of a hydrodynamic instability called undulation. Taking the shapeof wavy patterns in the liquid film of zinc, this instability is detrimental to the final product quality and poseslimitations to the maximum production rates.In this project, we developed a closed-loop control strategy to attenuate the 3D undulation waves. Basedon a limited number of liquid film observations, the controller interacts with the liquid through pneumatic orelectromagnetic actuators. The design of an optimal control strategy followed a multidisciplinary approach,combining reduced order modelling of liquid films, numerical methods for solving hyperbolic differentialequations, and advanced feedback and model predictive control theory. The explored control methods includestability analysis based on integral modelling of thin film equations and machine learning techniques such asGenetic Programming and Reinforcement Learning.A clear knowledge of the liquid film dynamics was instrumental in the design of optimal control actions andthe correct placement of observation points. To this end, we studied the linear and nonlinear stability of flat filmsolutions. By solving the Orr-Sommerfeld eigenvalue problem, we calculated the neutral curves, investigatedthe instability mechanism and computed the absolute/convective threshold in the parameters space consideringliquid zinc and other three liquids with Kapitza numbers (Ka) between 4 and 11525. To understand the instabilitymechanism, we studied the vorticity distribution at the free surface and in the bulk. Moreover, we calculatedthe perturbation’s energy balance and compared the different terms in various conditions. We found that cornoil (Ka = 4) has a smaller area of stability than liquid zinc (Ka = 11525) for all the base state thicknesses. Thesurface tension has both a stabilizing and a destabilizing effect, especially for large Ka. For long waves, itcurves the vorticity lines near the substrate, reducing the flow under the crests. Short waves foster vorticityproduction at the interface and create a region of intense vorticity near the substrate, outweighing the one at theinterface. In addition, we discovered that the surface tension contributes to both the production and dissipationof perturbation’s energy depending on the Ka number. In terms of absolute/convective threshold, we revealed astabilizing mechanism that shrinks the absolute instability region for h > 1 above a certain Reynolds number,and we showed that below a critical Ka, the thin film solution is always convectively unstable. Furthermore,we showed that the liquid film is convectively unstable in hot dip galvanizing conditions. This implies thatperturbation produced by the actuators will solely propagate upwards in the substrate direction. This allows usto place the observation points solely upstream of the actuators’ position since the liquid film thickness is notaffected by the actuators in this area. Concerning the nonlinear stability analysis, we simulated linearly unstable waves. waves using an integral boundary layer model of the liquid film without considering the stabilizing effect ofsurface tension. This revealed a nonlinear stabilizing mechanism that damps these waves in the domain. Thisfinding gives more room for manoeuvre to the control functions, which can rely on this mechanism to stabilizethe undulation.Based on the linearized reduced order model equations, a linear feedback control law was found usinglinear stability and model predictive control methods. This was tested on the control of 2D finite-amplitudeunstable waves. The control problem was addressed with a progressive relaxation of the hypothesis on thecontrol function shape and the system observability. We found that it was possible to control the waves withindependent shear and pressure distributions at the free surface proportional to the film thickness. Moreover,a set of 22 jets and punctual observation of the liquid film thickness and flow rate managed as well to bringthe perturbation to rest but over a longer period of time. Finally, we found that 22 blowing jets were enoughto stabilize the system using linear model predictive control, with maximum impinging pressure below 2[Pa].Unfortunately, the high computational cost and the complexity of handling uncertainties in the observationslimit the applicability of these methods to the 3D undulation control problem, which was addressed solely withmachine learning techniques.Prior to addressing the control of the undulation instability, we tackled simplified control test cases embeddingsome of the liquid film’s dynamical features. These encompass the wave cancellation problem from the 1D linearadvection flow and the viscous Burgers’ flow. Moreover, these test cases represented a valuable playgroundfor getting familiar with machine learning methods. We compared the machine learning methods used in ouranalysis to illustrate their difference in exploration versus exploitation and their balance between ‘model capacity’in the control law definition versus ‘required complexity’. In addition, we used the advection test case to studydifferent reward functions balancing performance over computational time. We found that the DDPG managesto control the waves in the Burgers ’flow using an open-loop component, constant in time, and a closed-loopcomponent, generating high-frequency waves. In terms of the reward function, the exponential function withnegative standard deviation at the exponent resulted in the one providing the best results with a limited numberof simulations.Moving to the control of the undulation instability in 2D, we modelled the undulation through unsteadyDirichlet boundary conditions on the flow rate and the liquid film thickness, producing a single harmonicwave. Leveraging the results of the test cases, a negative exponential reward function was used, and differentmachine-learning methods were tested. We found that the PPO reinforcement learning algorithm obtained thebest performance.For the control of the 3D undulation, we solely used the PPO algorithm from the Stable Baseline librarysince this allowed the training of the agent over multiple simulations at the same time on different CPUs. Twodifferent test cases were run with water and liquid zinc. In the case of water, the liquid film reduced order modelwas discretized with the finite volume method without considering surface tension, and the undulation wasmodelled via unsteady Dirichlet boundary conditions coming from Large Eddy Simulations. We found that aconfiguration with a 2D gas jet reduces the undulation by about 13 % compared with the uncontrolled case. Theoptimal control law was proportional to the liquid film observation with a small phase shift. For the case of zinc,the reduced order model was implemented in a spectral solver, and the undulation instability was modelled witha set of forcing gas jets. We found that a 2D gas jet with a simple open-loop harmonic function reduces the waveamplitudes of 8% by pushing the crests. Moreover, a 2D electromagnet was also capable of reducing the waveamplitudes by pulling the valleys.n addition to deforming the liquid film interface, the electromagnetic actuators also affect the temperatureof the liquid film via the Joule effect linked to the induced current. To define the boundaries of this detrimentaleffect and the impact of the magnetic field intensity and shape, we solve a multi-objective optimization problemaccounting for (1) maximal liquid film sensitivity, (2) maximal smoothness of the wiping meniscus, and (3)minimal Joule heating. We presented the Pareto front, identifying the best configurations of the magneticactuators in steady-state conditions. The optimization was based on a 1D steady-state integral model, whoseprediction scales according to the Hartmann number (Ha). The optimization used a multi-gradient approach,with gradients computed via finite differences in variational and analytical methods. The results showed thatthe liquid film sensitivity depends solely on Ha and not the magnetic field distribution. Moreover, we showedthat the liquid thickness becomes insensitive to the intensity of the magnetic field above a certain threshold andthat the current distribution, and hence the Joule heating, is mildly affected by the magnetic field’s intensity andshape.