Population balance modeling of polyurethane foam formation with pressure‐dependent growth kernel

Abstract

Polyurethane foams are widely used materials often chosen for their useful characteristics such as low thermal conductivity, ease of application, and high strength-toweight ratios. Computational models are needed to predict the dynamics of the flow and expansion, and the resulting material properties, to improve manufacturing processes. In this paper, a model for PMDI, a water-blown polyurethane foam, is presented. By extending a kinetics-based approach by adding bubble-scale information via a population balance equation (PBE) using the quadrature method of moments, we can track bubble size distributions during foaming. We present results from a three-dimensional computational fluid dynamics model using arbitrary Lagrangian-Eulerian interface tracking implemented in finite element software. The model compares favorably with experimental data, including dynamics, bubble distributions measured by both camera and diffusion wave spectroscopy, and post-test bubble size from scanning electron microscopy and density measurements from x-ray computed tomography.

Bubble‐scale observations of polyurethane foam expansion

“…The data presented here have been used to support the parameterization of a model of bubble dynamics that, when combined with our previously developed PU foam expansion models, 4,5 can more accurately describe the mold filling process of PU 38 . Modeling the change in bubble size with time allows us to more accurately predict the density and pressure fields.…”

“…In addition to the qualitative conclusions drawn from the SEM histograms, statistical analysis was performed on the bubble size distribution of selected data sets. Here, we convert the bubble measurements to bubble volumes for ease in the use of subgrid models being developed 38 . Converting the optically obtained bubble areas at the wall to volumes should be done with caution because of possible distortion at the walls.…”

“…Here, we convert the bubble measurements to bubble volumes for ease in the use of subgrid models being developed. 38 Converting the optically obtained bubble areas at the wall to volumes should be done with caution because of possible distortion at the walls. Nevertheless, here we will simply assume that a bubble area from the camera data is that of an equivalent sphere for semiquantitative comparisons.…”

“…The data presented here have been used to support the parameterization of a model of bubble dynamics that, when combined with our previously developed PU foam expansion models, 4,5 can more accurately describe the mold filling process of PU. 38 Modeling the change in bubble size with time allows us to more accurately predict the density and pressure fields. In the future, it will also allow better prediction of the foam compressibility, potentially impacting design decisions when encapsulating delicate electronics with foam.…”

Online In Situ Real-Time Experimental Investigation of the Crystallization Characteristics and Kinetic Models during the Nucleation and Growth Process of Para-Xylene Suspension Crystallization

“…33−35 The moment transformation method regresses the model's parameters using corresponding moments from system analysis to match the experimental data's moments for the system output variables. 36 The eq 3 is the j th order moment of crystal particle density…”