**5. Summary and Conclusions**

In this work, a literature review was presented regarding the different possible approaches and add-ons required to simulate a thermal multiphase multicomponent system with a complexity such as in thermal liquid foam using the lattice Boltzmann method (LBM). The choice of each ingredient in the model recipe was explained and justified. All the model details and parameters were shown and clarified.

The Shan-Chen (SC) LBM model was explained, considering both short and mid-range interactions, which was a necessity to reach a stable liquid foam and inhibit coalescence, implying the existence of a positive disjoining pressure. The selected forcing scheme and

the Two Relaxation Time were presented. The inclusion of the thermal field as a third set of LBM distribution function was shown. The required non-dimensional groups were stated.

Later, the Young–Laplace test was shown for two different cases of density ratios. The Rayleigh–Taylor Instability test was shown as a validation case for the implemented multiphase–multicomponent model and compared with numerical results from the literature. Four chosen samples from the well-known shape regimes diagram for bubble rise were simulated and presented as a further validation for different Reynolds and Bonds numbers.

Finally, a case study for bubble rise in a partially filled flask-like container was simulated, where random bubbles are initiated and left to rise, reaching to the top. Coalescence was significantly inhibited and, hence, liquid foam is generated from these rising bubbles. Foam dynamics and bubbles' rearrangements were shown with a very interesting and realistic behavior. Convective diffusive heat transfer is included, showing the effect of the flow field on the thermal one. All the simulation parameters and non-dimensional groups were fairly presented. A novel technique to implement the local variation in surface tension due to temperature gradients in the SC LBM model was proposed. Promising findings using this approach were briefly shown, which can lead after deeper quantification and validation to a unique and delicate way to include the Marangoni effect in the SC LBM model.

The chosen recipe of approaches has succeeded in simulating thermal liquid foam and tracking its dynamics and rearrangements. Although all the presented non-dimensional groups are used, one missing ring is to quantify the disjoining pressure and tune the interaction parameters to reach the experimental physical values and isotherms, which have not been found in the literature yet. In the latest work by Ataei et al. [16], a qualitative validation of the whole numerical simulation of foam with experimental data was also not reachable. The mentioned reason was the unsimilar initial conditions, which is of course an agreeable justification in addition to the unmatched quantity of disjoining pressure. In general, this is to be expected since the numerical simulation of liquid foam as a whole system is far away from maturity.

In order to simulate the full system, the relationship between the effect of surfactants or protein molecules in the microscales, single or two bubbles scale and a full scale of a foam column must be all considered. The current proposed model can be seen as a big step towards the final aim, which is the modelling and parametrization of full-scale industrial rectification columns, where foaming is a critical and an often-occurring issue.

**Author Contributions:** Validation, B.G. and M.M.; writing—original draft preparation, M.M.; writing—review and editing, A.D., B.G. and M.M; visualization, M.M.; supervision, A.D. and B.G; project administration, A.D. and B.G; funding acquisition, A.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research is funded by the DFG (Deutsche Forschungsgemeinschaft), FEI (Forschungskreis der Ernährungsindustrie), AiF (Arbeitsgemeinschaft industrieller Forschungsvereinigungen) and the Federal Ministry of Economics and Technology. This research is part of the DFG/AiF project (Cluster 6, CV 6): Industrial Joint Research (IGF) "Physikalisch basiertes Management störender Schäume in Produktionsanlagen: Prävention, Inhibierung und Zerstörung". The authors would like to acknowledge the financial support by Friedrich-Alexander-Universität Erlangen-Nürnberg and the Universitätsbibliothek for this publication.

**Acknowledgments:** The authors gratefully acknowledge the scientific support and HPC resources provided by the Erlangen National High Performance Computing Center (NHR@FAU) of the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU). The hardware is funded by the German Research Foundation (DFG).

**Conflicts of Interest:** The authors declare no conflict of interest.
