Bridging the gap across scales: Coupling CFD and MD/GCMC in polyurethane foam simulation
Introduction
Polyurethane (PU) foams are interesting materials employed in furniture, construction and automotive industries, just to cite a few, and represent an important share of the global polymer materials market (Mills, 1993, Princen and Kiss, 1986, Woods, 1990). PU are manufactured by mixing the two main PU components (i.e., a mixture of polyols and isocyanates) with additives. These include catalysts (to tune polymerization rate) and emulsifiers (to improve reactant compatibility). To generate gas bubbles leading to foam, physical and chemical blowing agents are employed. Physical blowing agents (PBAs) are volatile hydrocarbons that evaporate by virtue of the exothermicity of the polymerization reaction. On the other hand, the mechanism of action of chemical blowing agents (CBAs) is based on their reaction with the polymerization mixture, and ultimately results in gas production. One of the most popular CBAs is water, which reacts with isocyanates to produce carbon dioxide leading to the foam expansion.
Modeling and simulation of a PU foam expansion process is particularly interesting because, being a rapidly time-evolving system, it is very difficult to characterize experimentally. A model describing PU foam expansion, especially for mold filling applications, could be profitably used in the design and optimization of such processes. On the other hand, the scientific modeling community faces a complex multiphase-reacting system in which various physical phenomena encompassing a wide range of length scales take place. This deters scientists to face the problem as a unified challenge, while the current, practical approach is tackling the problem at each single scale (e.g., nano-, meso-, and macro-scale models). Additionally, the final properties of the manufactured PU foam highly depend on the adopted chemical recipe (i.e., polyol, isocyanate, and blowing agents structure and concentrations) and the flow history of the foam when it is applied for mold filling applications. This, in turn, requires the knowledge of fundamental thermophysical properties of different components (e.g., the density of polymerizing mixture) prior and during foam expansion for large-scale applications. Accordingly, the problem is inherently multiscale and, as such, a multi-model approach must be devised and applied.
A review of the current literature on the bubble-scale modeling tools for PU shows that one crucial point consists in correctly describing how an individual spherical bubble grows within a shell of the reacting mixture. Mass and momentum balances are routinely solved to assess the evolution of bubble radius while the mass transfer coefficient is considered as a model parameter (Feng and Bertelo, 2004, Harikrishnan et al., 2006, Harikrishnan and Khakhar, 2009, Kim and Youn, 2000). Furthermore, the macro-scale characteristics of PU foams are generally modeled by solving either ordinary differential equations (ODEs) or partial differential equations (PDEs). The former approach describes the foam apparent density, temperature, and polymerization progress (i.e., the gelling and blowing reactions) with respect to reaction kinetics (Baser and Khakhar, 1994a, Baser and Khakhar, 1994b, Gupta and Khakhar, 1999). Along the alternative line, Computational Fluid Dynamics (CFD) is applied to account for spatial and temporal variation of the foam properties. This last method has proven to be more attractive for mold filling applications, as the foam mobile interface can be monitored using the Volume-of-Fluid (VOF) approach (Bikard et al., 2005, Geier et al., 2009, Samkhaniani et al., 2013, Seo et al., 2003, Seo and Youn, 2005).
From a general perspective, to model PU-based systems we recently developed NANOTOOLS, an integrated, multiscale molecular modeling software for the prediction of major structural and thermophysical properties of this class of polymers and their nanocomposites (Ferkl et al., 2017, Laurini et al., 2016). Specifically, a hierarchical approach was implemented, which involves running separate models with a parametric coupling, with the ultimate goal of predicting the system under consideration from first principles, i.e. starting from the quantum scale and passing information to molecular scales and eventually to process scales. According to this sequential (aka message-passing) methodology, information is computed at a smaller (finer) scale and passed to a model at a larger (coarser) scale by leaving out (i.e. coarse graining) degrees of freedom (Cosoli et al., 2008a, Fermeglia and Pricl, 2007, Laurini et al., 2016, Scocchi et al., 2009, Scocchi et al., 2007a, Scocchi et al., 2007b, Toth et al., 2012). On the macro-scale level, we presented a base line model that corroborates the lack of population balance modeling for a reactive-expanding PU foam (Karimi and Marchisio, 2015). This has paved our way to implement a population balance equation (PBE) into a CFD solver and introduce a new VOF-based solver, coupled with PBE, for modeling and simulation of PU foams (Karimi et al., 2016). The results we obtained from the validation tests showed that by solving a PBE, one can extract practical information about the foam apparent density and its morphological structure. This, however, comes with the cost of compromising some physical phenomena occurring during the foam expansion, e.g., empirically driven correlations or constant values represent the characteristics of the system under investigation. For instance, we applied a simplified diffusion controlled model for the bubble growth rates. However, later we addressed this by coupling a detailed bubble-scale model with the macro-scale CFD code and showed the benefits of applying a multiscale approach on the accuracy of the numerical predictions (Ferkl et al., 2016).
The present work also follows the same philosophy outlined above. Yet, for the first time in the investigation of PU foams expansion, a macro-scale CFD model is coupled with nano-scale atomistic models. The macro-scale CFD model requires in fact three pieces of information: the density of the liquid mixture undergoing polymerization (prior to foaming), the solubility of chemical blowing agents (in the liquid mixture undergoing polymerization) and the solubility of PBA varying with temperature and degree of polymerization (or cross-linking). Accordingly, instead of using empirical and unreliable expressions for the estimation of these quantities, here the nano-scale model is employed. In particular, molecular dynamics (MD) simulations are run to calculate the density of the networking polymer (Ferkl et al., 2017, Laurini et al., 2016, Maly et al., 2008) while Grand Canonical Monte Carlo (GCMC) are carried out to predict the different gases solubility as a function of temperature and degree of cross-linking (Cosoli et al., 2008a, Cosoli et al., 2008b, Cosoli et al., 2008c, Pricl and Fermeglia, 2003). The final macro-scale CFD model predictions, calculated in turn by using results from the underpinning nano-scale models, are validated against experimental data for density and temperature time evolutions for different test cases. The comparison shows that multiscale modeling is an extremely interesting technique for the simulation of PU foams, as it allows to describe them without the need of performing costly experiments. In fact, the most important properties affecting the final behavior of the PU foam are here calculated rather than measured. This is particularly important since not only some properties are difficult to measure, but some others are impossible to obtain experimentally in a rapidly evolving reacting system such as this one.
Section snippets
Mathematical models
In what follows the nano- and macro-scale models will be presented. Details concerning the specific chemical systems investigated will also be summarized in this section, as they are required to lay down the nano-scale models. In the next two sections the models employed to describe a generic PU foam will be outlined. This generic PU foam is prepared by mixing polyols and isocyanates with water, producing carbon dioxide (i.e. chemical blowing agent), and a physical blowing agent. In this work,
Scale coupling, test cases and operating conditions
Figs. 1 and 2 together with Tables 1and 2 summarize the MD data, the surrogate models and the relevant parameter values obtained for the first chemical recipe.
As can be seen from Fig. 1, Fig. 2, MD simulations predict that both the density of the polymerizing mixture and the solubility of carbon dioxide in the reacting mixture decrease with increasing temperature and degree of cross-linking, as somewhat expected.
The second recipe includes both CBA and PBA. In this case, water and n-pentane
Results and discussion
The final application of the multiscale simulation suite developed in this work was to monitor the PU foam expansion during mold filling. Thus, the preliminary observation focused on how the model handles foam expansion. Fig. 4 displays the volume fraction of surrounding air at four different time instants. The simulation represents batch d in Table 4, including n-pentane as PBA and water as CBA. As explained in the previous section, the first 10% of beaker is filled with the foam phase at the
Conclusions
A multi-scale modeling approach is introduced in this work for the simulation of the expansion of PU foams. The model is formulated by coupling a nano-scale model, based on MD, with a macro-scale model, based on VOF and CFD. The lower scale model provides the inputs of the CFD code including the density of polymerizing liquid mixture and the solubility of blowing agents (chemical and physical). The functionality of the CFD inputs on temperature and cross linking are also accounted for using the
Acknowledgements
This work was funded by the European Commission under the grant agreement number 604271 (Project acronym: MoDeNa; call identifier: FP7-NMP-2013-SMALL-7). Computational resources were provided by HPC@POLITO, a project of Academic Computing within the Department of Control and Computer Engineering at the Politecnico di Torino (http://hpc.polito.it).
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