Packaging steels are coated with the polymer PET for preserving content quality, preventing corrosion, and enabling printability on the surface. The industrial lamination process must handle large volumes at high speeds, imposing a critical challenge. At elevated production speeds, air bubbles are entrapped between the polymer coating and the steel substrate, significantly impacting the final product's quality. The physics governing this air entrapment process are poorly understood, and establishing the relationship with overall process parameters is crucial. In this presentation, we will address the air entrapment problem using a two-scale finite element modeling strategy, complemented by experiments. The coarse-scale model is developed to establish the connection between processing parameters (line speed, roll pressure, temperature, etc.) and the actual loading conditions in the lamination nip where the polymer film bonds. The fine-scale model delves into the roughness asperities of the steel sheet, explicitly studying the plastic flow of the film at high temperatures and strain rates into the roughness valleys. The Eindhoven Glassy Polymer (EGP) is employed to model the behavior of PET under extreme rate/temperature conditions, where thermo-mechanical parameters are taken from literature or calibrated through experiments. The results from the two-scale finite element model reveal that air bubble formation is highest when the temperature of the substrate steel is low or the lamination speed is high. The model allows us to establish a quantitative link between macro-scale process parameters and incomplete filling or bonding of PET at the micro-scale. This paves the way for controlling the air entrapment, thereby enhancing the quality of the coated steels.