A Baseline Model for the Simulation of Polyurethane Foams via the Population Balance Equation

Karimi, Mohsen; Marchisio, Daniele

doi:10.1002/mats.201500014

Cited by 23 publications

(31 citation statements)

References 14 publications

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“…In the above equation, A W and E W are the pre‐exponential factor and the activation energy and X W is the conversion of water. The mass fraction of carbon dioxide in the liquid of the foam can be obtained as follows:

\frac{\partial ρ_{f} α_{f} w_{C O 2}}{\partial t} + \nabla \cdot true(ρ_{f} α_{f} w_{C O 2} U_{f} true) = ρ_{f} α_{f} true[C_{W}^{0} \frac{D X_{W}}{D t} \frac{M_{C O 2}}{ρ_{P U}} - {true \bar{G}}_{1}^{C O 2} \frac{p}{R T} \frac{M_{C O 2}}{ρ_{P U}} true]

where p is the system pressure, ρ PU is the density of the polymerizing liquid mixture and

{true \bar{G}}_{1}^{C O 2}

is the source term of growth rate due to the diffusion of carbon dioxide, the details of the latter terms are elaborated somewhere else . Likewise, the mass fraction of physical blowing agent in the liquid of the foam ( w BA ) is solved by using Equation :

\frac{\partial ρ_{f} α_{f} w_{B A}}{\partial t} + \nabla \cdot true(ρ_{f} α_{f} w_{B A} U_{f} true) = - {true \bar{G}}_{1}^{B A} ρ_{f} α_{f} \frac{p}{R T} \frac{M_{B L}}{ρ_{P U}}

where

{true \bar{G}}_{1}^{B A}

is the source term of growth rate for moment of order one due to the diffusion of the evaporated physical blowing agent into the gas bubbles.…”

Section: Mathematical Modelling and Numerical Methodologymentioning

confidence: 99%

“…The equilibrium (maximum) concentration of carbon dioxide in the liquid part of the foam is represented as

w_{C O 2}^{\max}

whereas that of the physical blowing agent as

w_{B A}^{\max}

. They can be easily calculated by knowing the Henry coefficients of the two components, more details are reported elsewhere . The last two equations complete the coupling of kinetics and PBE for the solver.…”

Section: Mathematical Modelling and Numerical Methodologymentioning

confidence: 99%

“…where p is the system pressure, r PU is the density of the polymerizing liquid mixture and G CO2 1 is the source term of growth rate due to the diffusion of carbon dioxide, the details of the latter terms are elaborated somewhere else. [18] Likewise, the mass fraction of physical blowing agent in the liquid of the foam (w BA ) is solved by using Equation 11:…”

Section: Mathematical Modelling and Numerical Methodologymentioning

confidence: 99%

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Multiscale Modeling of Expanding Polyurethane Foams via Computational Fluid Dynamics and Population Balance Equation

Karimi

Droghetti

Marchisio

2016

Macromolecular Symposia

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Summary This study is aimed to formulate a numerical modeling recipe for polyurethane foams. The model is capable of simulating the foam principal characteristics during mold filling. The model is formulated upon coupling of Computational Fluid Dynamics (CFD) and Population Balance Equation (PBE) to predict and simulate the evolution of foam features including apparent density and viscosity, bubble (or cell) size distribution (BSD) during the polymerization, as well as its kinetics. The solution of PBE inside the CFD code is performed with Quadrature Method of Moments (QMOM). The foam, constituted by a liquid polymer and gas bubbles, is simulated as a pseudo‐single‐phase system, while the interface between the foam and the surrounding air is tracked by a Volume‐of‐Fluid (VOF) solver within the open‐source CFD code OpenFOAM. The modeling is applied for a simple foaming experiment and attention is paid to the effect of the rheological model on the predictions.

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\frac{\partial ρ_{f} α_{f} w_{C O 2}}{\partial t} + \nabla \cdot true(ρ_{f} α_{f} w_{C O 2} U_{f} true) = ρ_{f} α_{f} true[C_{W}^{0} \frac{D X_{W}}{D t} \frac{M_{C O 2}}{ρ_{P U}} - {true \bar{G}}_{1}^{C O 2} \frac{p}{R T} \frac{M_{C O 2}}{ρ_{P U}} true]

where p is the system pressure, ρ PU is the density of the polymerizing liquid mixture and

{true \bar{G}}_{1}^{C O 2}

\frac{\partial ρ_{f} α_{f} w_{B A}}{\partial t} + \nabla \cdot true(ρ_{f} α_{f} w_{B A} U_{f} true) = - {true \bar{G}}_{1}^{B A} ρ_{f} α_{f} \frac{p}{R T} \frac{M_{B L}}{ρ_{P U}}

where

{true \bar{G}}_{1}^{B A}

is the source term of growth rate for moment of order one due to the diffusion of the evaporated physical blowing agent into the gas bubbles.…”

Section: Mathematical Modelling and Numerical Methodologymentioning

confidence: 99%

“…The equilibrium (maximum) concentration of carbon dioxide in the liquid part of the foam is represented as

w_{C O 2}^{\max}

whereas that of the physical blowing agent as

w_{B A}^{\max}

Section: Mathematical Modelling and Numerical Methodologymentioning

confidence: 99%

Section: Mathematical Modelling and Numerical Methodologymentioning

confidence: 99%

See 1 more Smart Citation

Multiscale Modeling of Expanding Polyurethane Foams via Computational Fluid Dynamics and Population Balance Equation

Karimi

Droghetti

Marchisio

2016

Macromolecular Symposia

Self Cite

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“…The second more general bubble‐shell model , also includes the effect of diffusion limitation. Recently, Karimi and Marchisio proposed a model, which can predict the evolution of the foam density, temperature, and bubble size distribution using the population balance equation and successfully coupled it with bubble‐shell model of Ferkl et al . .…”

Section: Introductionmentioning

confidence: 99%

“…A similar approach, we assumed, could be successfully applied to the formation of PU foams reinforced with fillers . Previously studies using such statistical method for describing the evolution of the bubble size distribution used Dirac delta functions centered on nodes of a quadrature approximation . All previous studies neglected additional important aspects of the foam morphology that is, that the bubbles are not spherical but have any anisotropy.…”

Section: Introductionmentioning

confidence: 99%

Kinetics of formation of microstructure in polyurethane foams infused with micro and nanosized carbonaceous fillers

Pikhurov

Zuev

2018

Polymer Engineering & Sci

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The kinetics of catalyzed by dibutyltin dilaurate reaction of polyurethane foam synthesis infused with 1 wt% of fillers was studied in dependence of Type 1D (nanosized [multi‐wall carbon nanotubes] or macrosized [carbon fibers]) or 2D (thermoexpanded graphite) carbonaceous fillers by IR spectroscopy and optical microscopy. It is shown that the foaming reaction has two stages: the first stage being a first‐order reaction, and the second stage a second‐order reaction. The mean diameter of the bubbles and the bubble size distribution has been determined during the process of foaming. It is shown that the presence of 2D fillers leads to a decrease in the mean diameter of the bubbles. POLYM. ENG. SCI., 59:941–948, 2019. © 2018 Society of Plastics Engineers

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

et al. 2021

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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.

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A Baseline Model for the Simulation of Polyurethane Foams via the Population Balance Equation

Cited by 23 publications

References 14 publications

Multiscale Modeling of Expanding Polyurethane Foams via Computational Fluid Dynamics and Population Balance Equation

Multiscale Modeling of Expanding Polyurethane Foams via Computational Fluid Dynamics and Population Balance Equation

Kinetics of formation of microstructure in polyurethane foams infused with micro and nanosized carbonaceous fillers

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

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