Validation of a macro-scale CFD-PBE model for the Polyurethane foaming process
Mohsen Karimi (email@example.com), Daniele Marchisio (firstname.lastname@example.org).
Department of Applied Science and Technology (DISAT), Politecnico di Torino Corso, Duca degli Abruzzi 24, 10129 Torino, Italy.
Polyurethane (PU) foams are widely used in a variety of applications such as soft cushioning foam, thermal insulation material, car dashboard and many other applications. Because of the wide usage and good mechanical and thermal properties of the PU foams, there exists a steady increase in its commercial interest. Thus, the transition of the modeling methods from the empirical-based models to the numerical ones receives a lot of attentions, recently.
This class of material forms from the exothermic reaction of isocyanate groups with active groups (e.g. alcohol, amines, and water). The reaction comprises two steps of development, including the gelling reaction, between polyols and isocyanates, and the blowing reaction, between water and isocyanate. Additives such as blowing agents and surfactants can also be incorporated to adjust the reaction rate and the mechanical properties of the final product. The second reaction (i.e., the blowing reaction) produces carbon dioxide gas which triggers the foaming process, together with the addition of a volatile substance, known as physical blowing agent.
Furthermore, the characteristics of the final product depend on the size of the gas bubbles or cells. The foaming process begins with mixing of the two components and proceeds with the generation of small gas nuclei. The diffusion of gas from the blowing reaction an the evaporation of the physical blowing agent into the liquid phase results in the continuous grow of bubbles leading to the bubble growth and coalescence. The final morphology of the foam is determined by the size distribution of the gas bubbles or cells . A typical foaming process is shown in Figure 1.
As mentioned above the final bubble size distribution within the foam can significantly alter the properties of the PU foams. Therefore, the main objective is to formulate a macro-scale computational model for the foaming process capable of predicting the evolution of the bubble size distribution during the process. This work is funded by the European Commission under the grant agreement number 604271 (project name: Modena; project identifier: FP7-NMP-2013-SMALL).
Computational Fluid Dynamics (CFD) is utilized to simulate the behavior of the polymer mixture. This complex multiphase systems is considered to be constituted by two immiscible fluids. One phase is air, which fills the majority of the computational domain at time zero. As the time progresses the expansion of the PU foam, treated as a pseudo-continuous phase, replaces the air phase and the interface between the two phases is captured by using the Volume of Fluid (VOF) method. It is also worth pointing out that the varying properties of the PU foam/mixture, such as the density and viscosity of the foam, are modeled using sub-models that will be developed within the project via multiscale modeling. Both OpenFOAM and Fluent CFD solvers are compared for solving the governing equations.
In addition, a population balance equation (PBE) is incorporated in the CFD framework to assess how the size distribution of the disperse gas bubbles in the PU foam varies in time and space. Appropriate source terms are included in the PBE to take into account the influence of the continuous and discontinuous phenomena such as bubble growth and coalescence. The Quadrature-Based Method of Moment (QBMM) is used to solve the PBE. In this method the number density function, representing the expected number of bubbles with certain size, is approximated with simple basis functions , and the evolution of the system is done by tracking some moments of the bubble size distribution. The time efficiency and accuracy of this approach is demonstrated by Marchisio et al. [4-5].
The model is currently under the validation stage. Twelve different test cases found in the literature in which different recipes (i.e. different polyols, isocyanates and blowing agent concentrations) are being used to validate our model predictions. These twelve cases correspond to simple mixing-cup experiments (such as the one depicted in Fig. 1) in which the time evolution of the foam temperature and foam density is monitored. Figure 2 reports an example of this validation work and contains the comparison between experimental measurements and model predictions concerning the time evolution of temperature and foam density . As it case be seen satisfactory agreement is found. Future research efforts will focus on the development of better sub-models for bubble growth and coalescence, kinetics and rheological behavior of the foam.
 S. Cohen-Addad, R. Hohler, O. Pitios, 2013. "Flow in foams and flowing foams", The Annual Review of Fluid Mechanics, (45), 241-267.
 R. Leppkes, 2012. "Polyurethanes, a versatile specialty plastic" BASF Polyurethanes, Sellier Druck GmbH, 85354 Freising, Germany.
 Marchisio, D., Fox, R.O., 2013. Computational Models for Polydisperse Particulate and Multiphase Systems. Cambridge Series in Chemical Engineering, Cambridge University Press, Cambridge, UK.
 Marchisio, D.L., Dennis Vigil, R., O. Fox, R., 2003. Implementation of the quadrature method of moments in CFD codes for aggregation--breakage problems. Chemical Engineering Science 58, 3337--3351.
 Marchisio, D.L., Fox, R.O., 2005. Solution of population balance equations using the direct quadrature method of moments. Journal of Aerosol Science 36, 43--73.
 Karimi, M. Marchisio, D.L., 2015, A baseline model for the simulation of polyurethane foams via the population balance equation, Macromolecular theory and modelling, in press.