Multi-scale modelling of expanding polyurethane foams: Coupling macro- and bubble-scales

• A new multi-scale framework for the simulation of polyurethane foam is presented.
• Two modelling scales, bubble- and macro-scales, are linked using MoDeNa framework.
• The bubble-scale model computes the bubble growth rate for individual bubbles.
• The macro-scale model solves a PBE for prediction of bubble size distribution.
• Validations showed the successfulness of the proposed modelling framework.

In this work, we apply a general multi-scale methodology to the simulation of reacting and expanding polyurethane (PU) foams. The approach starts from the baseline macro-scale model of Karimi and Marchisio (2015), which describes the foam as a continuum and applies the population balance equation (PBE) to determine the evolution of the bubble size distribution (BSD) during the foaming process. The main novelty of this work is the replacement of the simplistic bubble growth rate model used previously, with the detailed bubble-scale model, which resolves the mass, momentum and energy boundary layers around the bubble. A second important novelty concerns the way in which the macro- and the bubble-scale models are coupled, namely on-the-fly and by using the MoDeNa interface. The performance of the final multi-scale model is evaluated by analysing the predictions for foam density, temperature and BSD on 11 different test cases. Comparison for these test cases demonstrates a significant improvement in the ability of the model to describe the PU foaming process.