1. Introduction
The commercial relevance of flexible or rigid polyurethane (PU) foams has stimulated extensive research interest in their production and product-optimization processes. In their production phase, PU foams exhibit complex behavior initiated by premixing relevant isocyanate and polyol groups in the presence of suitable catalyst and blowing agents. Depending on the type of blowing agent, among many factors affecting the foaming process [
1], the resulting PU foam matrix is classified as rigid or flexible foam. However, certain factors, such as mixture rheology, amount of nucleated bubbles in the mixture, rate of depletion of the created/injected gas, or other external control mechanisms, affect their expansion process.
In flexible foams, the reaction between isocyanate and water in a chemically blown system results in the formation of amine and CO
gas. This CO
gas diffuses into nucleated bubbles in the mixture due to a pressure difference [
2], leading to a continuous increase in mixture volume until the reacting water is totally converted. More so, the combined effect of the evolution of mixture viscosity via chain-linking/polymerization, urea formation, as well as bubble rupture also limit the expansion process. This process results in a foam matrix with open cells [
3]. On the contrary, physical blowing agents, such as trichlorofluoromethane, cyclopentane, or liquid CO
, do not react in the mixture [
4]. However, due to their relatively low boiling point, the associated hydrocarbon vaporizes (at its boiling temperature) as a result of the exothermic nature of the foaming system [
5]. The vaporized gas diffuses into the nucleated bubbles in the mixture, leading to the expansion of the reacting mixture and the formation of a foam matrix with closed cells and rigid morphology
The PU foam expansion process may be homogeneous or heterogeneous depending on whether the system is controlled or not. This, in turn, influences pore structure and pore-size distribution [
6], as well as the thermophysical and physicomechanical properties of the final foam matrix [
7,
8,
9,
10]. However, some applications of PU foams require homogeneously distributed bubbles in the final product. Situations arise, for example, in a geotechnical engineering process, where the expanding system cannot be controlled, thereby resulting in a heterogeneous expansion of the foaming mixture [
11]. Such inhomogeneity in bubble-size distribution, attributed to the spatial variation of flow properties, results in a subdivided domain consisting of zones where the bubbles grow freely, and zones where their growth is restricted [
7,
12].
Several successful attempts to predict bubble growth and their size distribution in various liquids from a microscopic (cell model) view have been reported in the literature [
7,
8,
9,
13,
14,
15,
16,
17,
18]. For instance, the theoretical modelling and analysis of the evolution of a vapor bubble expanding in a shell of power-law (non-Newtonian) fluid by Street et al. [
13] illustrated the influence of the shear thinning, melt viscosity, and molecular diffusivity of the blowing agent on the initial growth of the bubble. In addition, the mass and momentum transport process of the fluid was also shown to significantly affect the initial growth rate of the nucleated bubble [
13]. In a related study [
7], the effect of blowing-agent concentration and gelling (cream) time on the expanding bubble in a PU foam mixture, as well as bubble-size distribution, was presented. To understand the influence of fluid viscoelasticity on expanding bubbles, Feng and Bertelo [
8] carried out an extensive study on bubble expansion in a viscoelastic (Oldroyd-B) fluid. Adopting the cell model, they simulated and discussed the effect of gas depletion and the proximity of neighboring bubbles in the physically blown polymer, and predicted the evolution and size distribution of the bubbles in reasonable agreement with available experimental data.
In their work, Amon and Denson [
14] presented a mathematical framework for a system of expanding bubbles, with each bubble enclosed in a shell of liquid containing supersaturated gas. Relevant features that allow for the possible extension of the proposed model to the macroscopic investigation of bubble growth in liquids were analyzed. Based on the cell model [
13,
14] and neglecting surface tension effects, Bruchon and Coupez [
15] numerically tracked the evolution of the radius of a single bubble in a Newtonian fluid and further carried out 2D and 3D simulations of a finite number of bubbles expanding in a pseudoplastic fluid. Although the cell model provides relevant qualitative information on bubble dynamics in liquids [
14], it reduces the foam-structure formation to the resolution of tracking the evolution of the radius of a limited number of bubbles within the liquid [
15]. This becomes more complicated and (numerically) expensive when an increasingly large number of bubbles [
15,
16], as in the case of PU foam formation, expands heterogeneously under nonisothermal conditions.
Motivated by the theoretical framework in References [
13,
14,
15] and the challenge in Reference [
15], we sought to understand and predict the nonuniform expansion observed in a thermally uncontrolled PU foam-formation process. We propose a macroscale (continuum) model that accounts for local variations in the foam-expansion process. We adopted the modelling approach of Reference [
19], which summarizes the specie consumption with the Kamal law [
20] for the degree of cure, with an adequately modified expansion source term and accounting for local contributions of the mixture temperature and viscosity to the expanding system. Although this approach does not quantify the bubbles in the foam, it gives a qualitative description of the distribution of void fractions in the domain. Of particular importance in this study was to investigate and understand the observed volume variation of the expanded foam when an equal mass of the reacting foam was injected into different cylinders in free-rise experiments. In this regard, three foam-expansion experiments in different geometries were studied and simulated. With graphical illustrations, we present our results and validate them against available experimental data. Furthermore, we carried-out
CT scan analysis of the expanded foam matrix and compared our observations with results from our simulation. Our results show qualitative agreement with observations from the experiments.
The rest of this paper is structured as follows. In
Section 2, we briefly describe the experimental setup and present a mathematical framework for the nonuniform expansion source term adopted in our simulations. The results and corresponding discussions are presented in
Section 3, and we conclude the study in
Section 4.
2. Experiment and Mathematical Framework
Following the experiment and discussions in Reference [
19] on the free-rise PU foam-expansion process, equal masses (77 and 37 g) of a reacting mixture containing isocyanate and precursors for rigid foam with some water are injected into different cylindrical tubes in a series of experiments at room temperature. Each pair had a diameter of 56 and 112 mm, and a height of 812 and 203 mm, respectively. In related reaction injection molding (RIM) experiments, the same mixture was injected into a rectangular mold of 500 × 50 × 40 mm (
Figure 1) under a fixed wall temperature of 55
C. The expanding foam in each tube was monitored, and their heights and volumes were recorded in time with the aim of understanding the effect of the mold or geometry conditions on the expanding PU-foam system, particularly since the free-rise experiments in the cylinders resulted in a foam matrix of different volumes. This suggested a possible influence of the geometry conditions on the foam-expansion process.
Assuming that the foaming mixture is a quasihomogeneous continuum with a constant rate law for the degree of cure, the experiments presented above were modelled mathematically, subject to appropriate boundary conditions. The equations governing the transport of mass, energy, and degree of polymerization follow directly from Reference [
19] (see
Appendix A). However, the source term in the equation governing the conservation of mass is adequately restructured to account for possible nonuniformity in the expanding system.
The mathematical formulations of Street et al. [
13], and Bruchon and Coupez [
15] from the conservation of mass and momentum transport describe the dynamics of a bubble of radius
R expanding in a pool of viscous liquid under quasistatic motion. Adopting the continuity of stresses at the surface of the bubble and neglecting the surface-tension effect, the evolution of the bubble radius was summarized by:
where
and
are the pressure in the bubble and liquid, respectively.
is the effective viscosity of the surrounding liquid, and
is the rate of change of the radius of the bubble as it expands in time. Equation (
1) implies that, for a given bubble, expanding in a liquid with viscosity
under isothermal condition
The equation governing the conservation of mass of the expanding PU foam relates flow velocity
to foam volume in time (
) (see Reference [
19]) by:
here is the expansion source term. We adopted a growth model [
19,
21,
22] for the volume of the expanding foam, so that at any time
t, foam volume
is described by
where
is the time when the foam starts to expand after being injected into the mold. Constants
A,
, and
are estimated from available volume-expansion experiment data.
In a thermally controlled (adiabatic) foam-expansion setup, the source term, denoted here as
, is ideally obtained from volume
of the expanding PU foam under spatially uniform temperature conditions. Consequently, this results in the homogeneous expansion of the foaming mixture at constant pressure. Hence, suppose there are
n nucleated spherical bubbles in such an expanding system, each bubble with an
R radius, then one can show that the contribution of
to individual bubbles relate to the radius of each bubble by
where
(see
Appendix B). Therefore, under adiabatic conditions, Equation (
1) can be rewritten as:
Here, is the mixture viscosity under adiabatic temperatures.
Thermal conditions in chemorheological fluids significantly affect the rate of reaction, which has direct consequence on the evolution of the chemoviscosity of the reacting fluid [
19,
23]. More so, in a thermally uncontrolled (nonadiabatic) foaming system, the temperature of the expanding foam spatially varies. Under this condition, temperature measurements at the core is often approximated to be adiabatic since temperatures in this region are less diffusive. However, beyond a progressive thermal diffusion layer in the region close to the boundaries, temperature conditions are nonadiabatic. Hence, expanding bubbles in such regions would lead to restricted growth due to the combined effect of temperature, reaction rate, and mixture effective viscosity (
) [
7,
12].
Dividing both sides of Equation (
6) by
and assuming that the pressure difference between the bubble and the surrounding liquid is locally the same under adiabatic or nonadiabatic temperature conditions, we have:
Therefore, considering local thermal influence on the growth of individual bubbles reflected in the local viscosity of the expanding PU foam, we define a linearly averaged local effective viscosity
by
where
is the viscosity of the foaming mixture under a nonadiabatic condition, and thermal sensitivity parameter
is to be determined (see
Section 3.3.1 for details). Hence, Equation (
7) becomes:
Rearranging Equation (
9), we obtain:
The equation governing mixture viscosity
as in Reference [
19] is given by
where
is constant,
is the foam activation energy,
T is the temperature, and
R is the rate constant.
and
are, respectively, the contributions of the degree of cure/polymerization
and the absorbed gas fraction
to the viscosity of the mixture. We assume for simplicity that these contributions are locally uniform so that by substituting Equations (
11) in Equation (
10), and simplifying the resulting expression, we obtain
where local damping factor
is given by
This implies that nonlocal parameter
controls the influence of temperature, at a constant reaction rate, on the expansion of the foam. Therefore, at any thermal conditions, the right-hand side of Equation (
3) is adequately replaced with Equation (
12).
In the next section, we present and discuss the results obtained from both the experiments and our simulations. Further comparison against the experimental data are carried out.
4. Conclusions
Inspired by the cell model for a unit bubble expanding in a viscous liquid, we proposed a mathematical framework that accounts for spatial inhomogeneity in a thermally uncontrolled (nonadiabatic) system of expanding polyurethane foam. We surmised that, under adiabatic conditions, a given foam material would expand in the same manner, irrespective of mold geometry. Hence, with the understanding that temperature distribution in a nonadiabatic system plays significant role in the local expansion of the foaming material, the source term driving foam expansion is structured to depend on the variation between adiabatic and nonadiabatic temperatures. With one setup for the expansion source term, calibrated for adiabatic expansions, we studied all flow conditions considered in this work. To understand the degree of nonuniformity in the expanding foam, we modelled and simulated free-rise PU foam-expansion experiments in cylindrical tubes of different diameters, and the injection molding of the same foam in a rectangular mold. A fundamental issue in this study was to investigate the observed difference in the volumes of the expanded PU foam when equal amounts of mass were injected in different geometries.
The interplay between temperature, material viscosity, and the gap between the core and the external environment were observed to affect the distribution of solid foam fractions in the expanded PU foam. In a nonadiabatic system, the thickness of the thermal diffusion layer near the walls resulted in the appearance of a viscous layer that restricts the expansion process around such locations. Then, a densely packed foam fraction formed in the regions near the wall. This phenomenon was more pronounced in constricted geometries, where the core of the expanding foam was relatively closer to the external environment. Therefore, the observed disparity in the volumes of the expanded foam in the free-rise experiments in the cylinders resulted from the combined effects of temperature, viscosity, and the proximity of the foam core to the bounding surface. To validate our models, we compared the results from our simulations with the volume data of the expanded foams, obtained from the free-rise experiments in the cylinders. Our results showed good correlation with those from the experiments.
Images from CT scans of the expanded foam from the rectangular mold were reconstructed and studied using GeoDict digital material laboratory software. The reconstructed images were analyzed for the distribution of foam fractions in the mold. Analysis showed an average spatial variation of about 1.1% in the foam fraction from the walls to the core of the foam matrix. This observation favorably compared with the results from our injection molding simulation.
This study serves as a platform for our ongoing studies on foam-expansion processes in porous media, with applications in reinforced structures with complex geometry.