# Numerical Simulation of Flame Retardant Polymers Using a Combined Eulerian–Lagrangian Finite Element Formulation

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

Transport of | $\mathit{\varphi}$ | ${\mathit{H}}_{\mathit{\varphi}}$ | ${\mathit{S}}_{\mathit{\varphi}}$ | |

Mass | 1 | 0 | ${\u03f5}_{v}$ | (2) |

Momentum | $\mathbf{v}$ | $\mu $ | $-{\nabla}_{x}p+\mu {\nabla}_{x}({\nabla}_{x}^{T}\mathbf{v})+\rho \mathbf{f}$ | (3) |

Energy | T | $\kappa /C$ | $\gamma [{w}_{T}/C+(\nabla .{Q}_{R})/C]+(1-\gamma ){Q}_{v}/C$ | (4) |

Species | ${Y}_{k}$ | $\kappa /C$ | $-{w}_{k}/C\phantom{\rule{0.277778em}{0ex}}for\phantom{\rule{0.277778em}{0ex}}k=F\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}O$ | (5) |

Symbol | Parameter |

${\nabla}_{x}={\left\{{\partial}_{{x}_{i}}\right\}}_{i=1}^{3}$ | vectorial operator of spatial derivatives |

$\rho $ | density |

p | pressure |

$\mu $ | viscosity |

$\mathbf{f}$ | gravity force |

C | capacity |

$\kappa $ | thermal conductivity |

${w}_{T}$ | rate of production of heat ${}^{1}$ |

${Q}_{R}$ | radiative heat flux |

${\u03f5}_{v}$ | mass loss |

${Q}_{v}$ | heat absorbed due to pyrolysis |

A | pre-exponential function |

E | activation energy |

R | universal gas constant |

$\alpha $ | absorption coefficient |

$\sigma $ | Stefan–Boltzmann constant |

^{1} Assuming constant values for the Schmidt (Sc = 1) and Prandtl (Pr = 1) number-simplified composition and temperature-dependent transport properties; thus, ρD = κ/C. |

Source Terms | ${\mathit{S}}_{\mathit{\varphi}}$ | |

$\nabla \xb7QR$ | $\alpha \left(4\sigma {\mathrm{T}}^{4}-\mathrm{G}\right)$ | (6) |

${\u03f5}_{v}$ | $-A{e}^{-E/R\mathrm{T}}$ | (7) |

${Q}_{v}$ | $\rho H{\u03f5}_{v}$ | (8) |

${w}_{F}$ | $-C{B}_{c}{\rho}^{2}{\mathrm{Y}}_{F}{\mathrm{Y}}_{O}{exp}^{\left(-{\mathrm{T}}_{a}/\mathrm{T}\right)}$ | (9) |

${w}_{O}$ | $-s{w}_{{C}_{3}{H}_{8}}$ | (10) |

${w}_{T}$ | ${h}_{{C}_{3}{H}_{6}}{B}_{c}{\rho}^{2}{\mathrm{Y}}_{F}{\mathrm{Y}}_{O}{exp}^{\left(-{\mathrm{T}}_{a}/\mathrm{T}\right)}$ | (11) |

## 3. Numerical Strategy

#### 3.1. Particle Finite Element Method for the Polymer

#### 3.2. Finite Element Formulation for the Air

#### 3.3. Overall Solution Strategy

Algorithm 1: Solution algorithm for the simulation of the UL 94 test. |

## 4. Model and Numerical Computation

#### 4.1. Materials, Experimental Methods, and Input Parameters

#### 4.1.1. Material

#### 4.1.2. Thermogravimetric Analysis

#### 4.1.3. Rheological Behavior

#### 4.1.4. UL 94 Test

#### 4.2. Numerical Setup

Algorithm 2: Definition of the experimental viscosity curve. |

1 if(T > 453.15 and T <= 515.88): |

2 mu = 1.25 × 10${}^{4}$ × e${}^{(-3.4\times {10}^{-3}\times T)}$ |

3 else if(T > 515.88 and T <= 533.15): |

4 mu = 1.10 × 10${}^{-25}$ × e${}^{(1.26\times {10}^{-1}\times T)}$ |

#### 4.3. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A

Algorithm 3: Definition of the curve 1. |

1 If(T <= 373.0): |

2 mu=$5.72\times {10}^{6}\times {e}^{(-6\times {10}^{-3}\times T)}$ |

3 else if(T > 373.0 and T <= 453.15): |

4 mu = $1.95\times {10}^{16}\times {e}^{(-6.48\times {10}^{-2}\times T)}$ |

5 else if(T > 453.15 and T <= 515.88): |

6 mu = $1.25\times {10}^{4}\times {e}^{(-3.4\times {10}^{-3}\times T)}$ |

7 else if(T > 515.88 and T <= 533.15): |

8 mu = $1.10\times {10}^{-25}\times {e}^{(1.26\times {10}^{-1}\times T)}$ |

9 else if(T > 533.15000001 and T <= 724.0): |

10 mu = 20000.0 |

11 else |

12
mu = $1.83\times {10}^{16}\times {e}^{(-4.98\times {10}^{-2}\times T)}$ |

Algorithm 4: Definition of the curve 2. |

1 if(T <= 533.15): |

2 Definition Curve 1 |

3 else if(T > 533.15000001 and T <= 723.0): |

4 mu = 20000.0 |

5 else if(T > 723.0 and T <= 762.49): |

6 mu = $1.38\times {10}^{15}\times {e}^{(-0.035\times T)}$ |

7 else if(T > 762.49 and T <= 769.5): |

8 mu = $1.70\times {10}^{266}\times {e}^{(-7.94\times {10}^{-0.001}\times T)}$ |

9 else if(T >769.5 and T <= 791.5): |

10 mu = $1.49\times {10}^{25}\times {e}^{(-7.30\times {10}^{-2}\times T})$ |

11 else if(T >791.5 and T <= 819.0): |

12
mu = $2.99\times {10}^{22}\times {e}^{(-6.52\times {10}^{-2}\times T)}$ |

13 else if(T >819.0 and T <= 850.0): |

14 mu = $1.75\times {10}^{4}\times {e}^{(-1.39\times {10}^{-2}\times T)}$ |

15 else |

16 mu = 0.13 |

Algorithm 5: Definition of the curve 3. |

1 if(T <= 533.15): |

2 Definition Curve 1 |

3 else if(T > 533.15000001 and T <= 723.0): |

4 mu = 20000.0 |

5 else if(T > 723.0 and T <= 802.0): |

6 mu = 2.0 × 10${}^{15}$ × e${}^{(-0.035\times T)}$ |

7 else if(T > 802.0 and T <= 815.0): |

8 mu = $2.0\times {10}^{146}\times {e}^{(-0.411\times T)}$ |

9 else if(T >815.0 and T <= 900.0): |

10 mu = 9.0 × 10${}^{16}$ × e${}^{(-0.046\times T)}$ |

11 else |

12 mu = 0.13 |

## References

- Hastie, J. Molecular Basis of Flame Inhibition. J. Res. Natl. Bur. Stand. Sect. A Phys. Chem.
**1973**, 77, 733–754. [Google Scholar] [CrossRef] - Wilkie, C.; Morgan, A. Fire Retardancy of Polymeric Materials, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2009. [Google Scholar]
- Camino, G.; Costa, L. Mechanism of Intumescence in Fire Retardant Polymers. Rev. Inorg. Chem.
**1986**, 8, 69. [Google Scholar] [CrossRef] - Matzen, M.; Kandola, B.; Huth, C.; Schartel, B. Influence of Flame Retardants on the Melt Dripping Behaviour of Thermoplastic Polymers. Materials
**2015**, 8, 5621–5646. [Google Scholar] [CrossRef][Green Version] - ASTM D3801-10. Standard Test Method for Measuring the Comparative Burning Characteristics of Solid Plastics in a Vertical Position; ASTM International: West Conshohocken, PA, USA, 2010. [Google Scholar] [CrossRef]
- Kempel, F.; Schartel, B.; Marti, J.; Butler, K.; Rossi, R.; Idelsohn, S.; Oñate, E.; Hofmann, A. Modelling the vertical UL 94 test: Competition and collaboration between melt dripping, gasification and combustion. Fire Mater.
**2015**, 39, 570–584. [Google Scholar] [CrossRef] - Matzen, M.; Marti, J.; Oñate, E.; Schartel, B. Particle Finite Element modelling and advanced experiments on dripping-V-0-classified polypropylene. In Proceedings of the 15th International Conference and Exhibition—Fire and Materials 2017, San Francisco, CA, USA, 6–8 February 2017. [Google Scholar]
- Marti, J.; Idelsohn, S.R.; Oñate, E. A Finite Element Model for the Simulation of the UL 94 Burning Test. Fire Technol.
**2018**, 54, 1783–1805. [Google Scholar] [CrossRef] - Wang, N.; Tu, R.; Ma, X.; Xie, Q.; Jiang, X. Melting behavior of typical thermoplastics materials—An experimental and chemical kinetics study. J. Hazard. Mater.
**2013**, 262, 9–15. [Google Scholar] [CrossRef] - Kidder, R.; Troitzsch, J.; Naumann, E.; Roux, H. From Course Work Materials in New Developments and Future Trends in Europe and the United States for Fire Retardant Polymer Products; Technomic Publishing Co., Inc.: Lancaster, PA, USA, 1989. [Google Scholar]
- Dasari, A.; Yu, Z.; Cai, G.; Mai, Y. Recent developments in the retardancy of polymeric materials. Prog. Polym. Sci.
**2013**, 9, 1357–1387. [Google Scholar] [CrossRef] - Oñate, E.; Rossi, R.; Idelsohn, S.; Butler, K. Melting and spread of polymers in fire with the Particle Finite Element Method. Int. J. Numer. Methods Eng.
**2009**, 81, 1046–1072. [Google Scholar] [CrossRef] - Modest, M. Radiative Heat Transfer; Academic Press: San Diego, CA, USA, 2003. [Google Scholar]
- Oñate, E.; Marti, J.; Ryzhakov, P.; Rossi, R.; Idelsohn, S. Analysis of the melting, burning and flame spread of polymers with the Particle Finite Element Method. Comput. Assist. Methods Eng. Sci.
**2013**, 20, 165–184. [Google Scholar] - Butler, K.; Oñate, E.; Idelsohn, S.; Rossi, R. Modeling polymer melt flow using the Particle Finite Element Method (PFEM). In Proceedings of the Interflam Conference 2007, London, UK, 3–5 September 2007; Volume 2. [Google Scholar]
- Butler, K. A model of melting and dripping thermoplastic objects in fire. In Proceedings of the 11th International Conference and Exhibition—Fire and Materials 2009, San Francisco, CA, USA, 26–28 January 2009. [Google Scholar]
- Butler, K.; Cheng, A.; Mullen, C.; Al-Ostaz, A. Fire characteristics of steel members coated with nano-enhanced polymers. Fire Mater.
**2014**, 38, 227–240. [Google Scholar] [CrossRef] - Oñate, E.; Idelsohn, S.; Del Pin, F.; Aubry, R. The Particle Finite Element Method. An overview. Int. J. Numer. Methods Eng.
**2004**, 1, 267–307. [Google Scholar] [CrossRef] - Marti, J.; Ryzhakov, P.; Idelsohn, S.; Oñate, E. Combined Eulerian-PFEM approach for analysis of polymers infire situations. Int. J. Numer. Meth. Eng.
**2010**, 92, 782–801. [Google Scholar] [CrossRef] - Delaunay, B. Sur la sphere vide. Izv. Akad. Nauk SSSR Otdelenie Matematicheskii i Estestvennyka Nauk
**1934**, 7, 793–800. [Google Scholar] - Kundu, P.; Cohen, I. Fluid Mechanics; Academic Press: San Diego, CA, USA, 2002. [Google Scholar]
- Donea, J.; Huerta, A. Finite Element Method for Flow Problems; Wiley: New York City, NY, USA, 2003. [Google Scholar]
- Lohner, R. Applied CFD Techniques, 2nd ed.; John Wiley and Sons: New York City, NY, USA, 2008. [Google Scholar]
- Guermond, J.; Minev, P.; Shen, J. An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng.
**2006**, 195, 6011–6045. [Google Scholar] [CrossRef][Green Version] - Temam, R. Sur l’approximation de la solution des equations de Navier-Stokes par la methode des pase fractionaires. Arch. Ration. Mech. Anal.
**1969**, 32, 135–153. [Google Scholar] [CrossRef] - Chorin, A. A numerical method for solving incompressible viscous problems. J. Comput. Phys.
**1967**, 2, 12–26. [Google Scholar] [CrossRef] - Codina, R. A stabilized finite element method for generalized stationary incompressible flows. Comput. Methods Appl. Mech. Eng.
**2001**, 190, 2681–2706. [Google Scholar] [CrossRef] - Dadvand, P.; Rossi, R.; Oñate, E. An object-oriented environment for developing finite element codes for multi-disciplinary applications. Arch. Comput. Methods Eng.
**2010**, 17, 253–297. [Google Scholar] [CrossRef] - Slopiecka, K.; Bartocci, P.; Fantozzi, F. Thermogravimetric analysis and Kinetic study of poplar wood pyrolysis. Third Int. Conf. Appl. Energy
**2012**, 97, 491–497. [Google Scholar] [CrossRef] - Chen, X.; Yu, J.; Guo, S.; Luo, Z.; He, M. Flammability and thermal oxidative degradation kinetics of magnesium hydroxide and expandable graphite flame retarded polypropylene composites. J. Macromol. Sci. Part Pure Appl. Chem.
**2008**, 45, 712–720. [Google Scholar] [CrossRef] - Simon, P. Isoconversional methods. J. Therm. Anal. Calorim.
**2004**, 76, 123–132. [Google Scholar] [CrossRef] - Opfermann, J.; Kaisersberger, E.; Flammersheim, H. Model-free analysis of thermoanalytical data-advantages and limitations. Thermochim. Acta
**2002**, 391, 119–127. [Google Scholar] [CrossRef] - Laoutid, F.; Bonnaud, L.; Alexandre, M.; Lopez-Cuesta, J.; Dubois, P. New prospects in flame retardant polymer materials: From fundamentals to nanocomposites. Mater. Sci. Eng. R Rep.
**2009**, 63, 100–125. [Google Scholar] [CrossRef] - Wang, Y.; Jow, J.; Su, K.; Zhang, J. Development of the unsteady upward fire model to simulate polymer burning under {UL94} vertical test conditions. Fire Saf. J.
**2012**, 54, 1–13. [Google Scholar] [CrossRef] - Stanislav, I.; Safronava, N.; Lyon, R.E. The effect of variation in polymer properties on the rate of burning. Fire Mater.
**2009**, 33, 257–271. [Google Scholar] [CrossRef] - Wang, Y.; Zhang, J.; Jow, J.; Su, K. Analysis and modeling of ignitability of polymers in the UL 94 vertical burning test condition. J. Fire Sci.
**2009**, 27, 561–581. [Google Scholar] [CrossRef] - Wang, Y.; Zhang, F.; Jiao, C.; Jin, Y.; Zhang, J. Convective heat transfer of the Bunsen flame in the UL94 vertical burning test for polymers. J. Fire Sci.
**2010**, 28, 337–356. [Google Scholar] [CrossRef]

**Figure 5.**Problem definition. (

**a**) Computational domains with boundary conditions. (

**b**) Meshes of the polymer and the air.

**Figure 8.**Comparison between experimental versus PFEM results for curve 1: blue and red correspond to 298 and 1000 K, respectively.

**Figure 9.**Comparison between experimental versus PFEM results for curve 2: blue and red correspond to 298 and 1000 K, respectively.

**Figure 10.**Comparison between experimental versus PFEM results for curve 3: blue and red correspond to 298 and 1000 K, respectively.

**Figure 12.**Evolution of the temperature and fuel distribution in the air domain at 10, 20, 30, 40, 50, and 60 s.

Parameter | Polymer | Air |
---|---|---|

Density | 905 $[\mathrm{kg}/{\mathrm{m}}^{3}]$ | see Section 2 |

Viscosity | ${}^{-}(\mathrm{T})$ | $1\times {10}^{-5}\phantom{\rule{0.166667em}{0ex}}[{\mathrm{m}}^{2}/\mathrm{s}]$ |

Specific heat capacity | 1900.0 $[\mathrm{J}/\mathrm{KgK}]$ | 1310.0 $[\mathrm{J}/\mathrm{KgK}]$ |

Thermal conductivity | 0.16 $[\mathrm{W}/\mathrm{mK}]$ | 0.0131 $[\mathrm{W}/\mathrm{mK}]$ |

Emissivity | 1.0 | – |

absorption coefficient | – | 1000 $\left[{\mathrm{m}}^{-1}\right]$ |

Stefan–Boltzmann | ||

constant | – | $5.67\times {10}^{-8}\phantom{\rule{0.166667em}{0ex}}[\mathrm{W}/{\mathrm{m}}^{2}{\mathrm{K}}^{4}]$ |

Arrhenius coefficient | $1.63\times {10}^{14}\phantom{\rule{0.166667em}{0ex}}\left[{\mathrm{min}}^{-1}\right]$ | – |

Activation energy | 181.9 $[\mathrm{KJ}/\mathrm{mol}]$ | – |

Enthalpy of vaporization | $8\times {10}^{5}\phantom{\rule{0.166667em}{0ex}}[\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K}]$ [12] | – |

${\mathrm{B}}_{\mathrm{c}}$ | – | $5.96\times {10}^{9}\phantom{\rule{0.166667em}{0ex}}[{\mathrm{m}}^{3}/\mathrm{Kgs}]$ |

${\mathrm{T}}_{\mathrm{a}}$ | – | 10,700 [K] |

$\mathrm{C}$ | – | $2.601\times {10}^{4}\phantom{\rule{0.166667em}{0ex}}[\mathrm{Kj}/\mathrm{Kg}]$ |

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**MDPI and ACS Style**

Marti, J.; de la Vega, J.; Wang, D.-Y.; Oñate, E. Numerical Simulation of Flame Retardant Polymers Using a Combined Eulerian–Lagrangian Finite Element Formulation. *Appl. Sci.* **2021**, *11*, 5952.
https://doi.org/10.3390/app11135952

**AMA Style**

Marti J, de la Vega J, Wang D-Y, Oñate E. Numerical Simulation of Flame Retardant Polymers Using a Combined Eulerian–Lagrangian Finite Element Formulation. *Applied Sciences*. 2021; 11(13):5952.
https://doi.org/10.3390/app11135952

**Chicago/Turabian Style**

Marti, Julio, Jimena de la Vega, De-Yi Wang, and Eugenio Oñate. 2021. "Numerical Simulation of Flame Retardant Polymers Using a Combined Eulerian–Lagrangian Finite Element Formulation" *Applied Sciences* 11, no. 13: 5952.
https://doi.org/10.3390/app11135952