(203h) Validation of a Macro-Scale CFD-PBE Model for the Polyurethane Foaming Process

Validation
of a macro-scale CFD-PBE model for the Polyurethane foaming process

Mohsen
Karimi (mohsen.karimi@polito.it), Daniele Marchisio
(daniele.marchisio@polito.it).

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 [1]. 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 [3], 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 [6]. 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.

References:

[1]
S. Cohen-Addad, R. Hohler, O. Pitios, 2013. "Flow in foams and
flowing foams", The Annual Review of Fluid Mechanics, (45),
241-267.

[2]
R. Leppkes,
2012. "Polyurethanes, a versatile specialty plastic" BASF
Polyurethanes, Sellier Druck GmbH, 85354 Freising, Germany.

[3]
Marchisio, D., Fox, R.O., 2013. Computational Models for Polydisperse
Particulate and Multiphase Systems. Cambridge Series in Chemical
Engineering, Cambridge University Press, Cambridge, UK.

[4]
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.

[5]
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.

[6]
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.