# Effect of Thermal Radiation in Porous Structures and Thermophysical Properties Variations in PVC Pyrolysis modelling

Y Abdelhafiz, Abdenour Amokrane, Gerald Debenest, Serge Bourbigot

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Y Abdelhafiz, Abdenour Amokrane, Gerald Debenest, Serge Bourbigot. Effect of Thermal Radiation in Porous Structures and Thermophysical Properties Variations in PVC Pyrolysis modelling. Journal of Physics: Conference Series, 2024, 2885 (1), pp.012020. 10.1088/1742-6596/2885/1/012020. hal-04980118

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# Effect of Thermal Radiation in Porous Structures and Thermophysical Properties Variations in PVC Pyrolysis modelling: A Study Using Optimization Techniques.

To cite this article: Y Abdelhafiz et al 2024 *J. Phys.: Conf. Ser.* 2885 012020

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# Effect of Thermal Radiation in Porous Structures and Thermophysical Properties Variations in PVC Pyrolysis modelling: A Study Using Optimization Techniques.

Y Abdelhafiz<sup>1,2</sup>, A Amokrane<sup>1*</sup>, G Debenest<sup>2</sup>, S Bourbigot<sup>3,4</sup>

<sup>1</sup> EDF R&D Lab Chatou, 6 quai Watier, 78400 Chatou, France

<sup>2</sup> Institut de Mécanique des Fluides de Toulouse (IMFT) - Université de Toulouse, CNRS-INPT-UPS, Toulouse, France

<sup>3</sup> Univ. Lille, CNRS, INRAE, Centrale Lille, UMR 8207 - UMET - Unité Matériaux et Transformations, F-59000 Lille, France

<sup>4</sup> Institut Universitaire de France (IUF)

* Corresponding author's e-mail: abdenour.amokrane@edf.fr

**Abstract.** In this study, numerical simulations are performed using the Gpyro software to model the thermal decomposition of PVC filled with CaCO3 (FPVC) during a cone calorimeter experiment. For its kinetic behaviour, data are provided through TGA measurement, while the most important thermophysical properties in the virgin state before decomposition are obtained from dedicated measurements. Regarding the properties during the decomposition process, as no data is available, hypotheses are formulated and challenged: do the properties remain constant during decomposition or not? Additionally, a particular focus is placed on the consideration of thermal radiation within the pores. The ability of these two different hypotheses to reproduce the experimental behaviour is explored through inverse simulation. The radiative phenomenon is validated on additional experiments dedicated to the measurement of conductivity with temperature, as well as pores size measurement. The findings emphasize the indispensable role of incorporating radiative contributions within pores. Furthermore, by accounting for changes in thermophysical properties during decomposition, the simulations accuracy is enhanced.

## 1. Introduction

Numerical modeling of fire scenarios faces challenges in accurately representing the thermal decomposition of solid materials [1]. Due to the complexity of the phenomena involved and the lack of knowledge regarding physical properties, as these properties must be characterized across various decomposition stages, yet their measurement often proves difficult. To address these issues, a frequently adopted assumption posits that the material retains constant physical properties throughout degradation, except the density for representing volume change during decomposition [2]. Regarding unmeasured properties, they are frequently sourced from literature, or rather used as a calibration parameter. In this article, the assumption of retaining constant physical properties throughout degradation is challenged.

The material examined in this study is FPVC, containing 31.4 wt% CaCO3. Upon decomposition, this material produces char residue, constituting a highly porous medium. The pores formed during the decomposition influence the thermal transfer. This effect can be estimated through the Loeb's law [3] were the effective thermal conductivity (ETC) noted as $k_{eff}$ (W.m⁻¹.K⁻¹), integrating a term corresponding to radiation effects as expressed in equation (1), with k (W.m⁻¹.K⁻¹) being the thermal conductivity, T (K) the temperature, σ the Stefan-Boltzmann constant (5.67x10⁻⁸ W.m⁻².K⁻⁴),

<figure><img src="image_5.png" /></figure>
and $\gamma$ (m) a parameter dependent on pore characteristics. Additionally, radiation in pores may constitute up to 75% of ETC in some cases [4]. However, this effect is not consistently accounted for in the pyrolysis models in the literature. Despite a few studies examining this effect [2,5,6], it is often inadequately characterized. This article investigates this effect further and highlights its significance in pyrolysis modelling.

$$k_{eff} = k + \gamma \sigma T^3 \quad (1)$$

The first objective of this article is to conduct inverse simulations, by testing different hypotheses, on the ranges of variation for the physical properties. Secondly, it aims to examine the effect of the radiative contribution in the pores. In this study, cone calorimeter experiments validate the simulations, while measurements provide properties of FPVC in its virgin state before decomposition. Properties during decomposition are derived from hypotheses regarding the changes in their values during pyrolysis. TGA experiments provide kinetic properties. Additional experiments validate radiation behavior within the pores.

## 2. Material and method

### 2.1. Experimental method

The density of FPVC is determined by measuring the mass of a plaque and dividing it by its volume. Thermal conductivity and the effect of radiation in pores are measured through the Transient Source Plane (TPS), where the conductivity of the sample is determined between 25°C and 800°C.

A Thermogravimetric Analysis (TGA) is carried out to gain insights into the kinetics of the reactions. The TGA is conducted in an inert atmosphere, where the FPVC is subjected to a heating process from 25°C to 800°C at a constant heating rate of 10°C/min. The mass and mass loss rate are recorded as functions of time and temperature. These experiments are complemented by simultaneous Differential Scanning Calorimeter (DSC) analysis to measure the heat of reaction during the thermal analysis of the materials [7]. Heat capacity is determined using the DSC technique.

Cone calorimeter testing is performed, with three tests conducted. The cone heater provides an external flux of 50 kW/m² to the FPVC sample measuring 0.1×0.1 m², with a thickness of 3 mm. The sample is directly placed on a sample holder consisting of two layers: a thin 2mm stainless steel layer and a thick 4 cm brick layer. The mass loss rate (MLR) is recorded. The data reported in this article represent the average of three replicated experiments.

### 2.2. Model setup

The experiment of Cone calorimeter is simulated using a 1D approach with the Gpyro software [8]. The model consists of three layers: a 3 mm FPVC sample, a 2 mm steel layer, and a 40 mm insulation brick layer. A constant heat flux of 50 kW/m² is applied to the sample's upper surface, with an additional 10 kW/m² upon flame ignition to account for flame heat flux, as done in [2].

Concerning the physical properties of FPVC, a reference simulation is conducted where nominal values are chosen for each property. TGA revealed the involvement of three materials in the process ($FPVC_{Virgin}$, $FPVC_{Int}$ and $char$). For the properties of $FPVC_{Virgin}$, some are directly measured, while the few remaining (absorption and emissivity) are derived from previous work on PVC [2,9]. For the intermediate material $FPVC_{Int}$, properties are assumed to be the same as in the virgin state, following common practice in the literature, except for density. For $char$, nominal physical properties are adopted from literature [2,9].

Additionally, for the inverse modelling, the range of variation of each input parameter needs to be specified. For $FPVC_{Virgin}$, The interval range of measured properties corresponds to experimental uncertainties. For both the $FPVC_{Int}$ and $char$, to overcome the lack of data, two hypotheses ($H_1$ and $H_2$) are proposed about the range of variation of properties during decomposition :

$H_1$ hypothesis: the assumption is made that the properties of intermediate FPVC are the same as those of the virgin material. So, the variation range for intermediate properties matches the experimental uncertainties of $FPVC_{Virgin}$, except for density. The same principle applies to $char$.
$\mathcal{H}_2$ hypothesis: this hypothesis allows wider variation in the properties of intermediate and char materials. This explores the possibility of freeing the model from the assumption that the intermediate material's properties are the same as the virgin material. Due to limited information, a wider range of variation is required (up to ±50%). This value is justified by the challenging nature of measuring these states, with an expected high level of experimental uncertainty.

Table 1 summarizes the chosen nominal values and the range of variation considered for each hypothesis. Note that $\rho$ is the density, $k$ the thermal conductivity, $c$ the heat capacity, $\epsilon$ the emissivity, $\kappa$ the absorption.

For the radiation in pores, the parameter $\gamma$ governs this effect (see equation 1), this parameter $\gamma$ is explored across a wide range for all step of decomposition ($FPVC_{Virgin}$, $FPVC_{Int}$, and $char$). $\gamma$ values ranged from no radiation ($\gamma$=0 mm) to 20 mm. These values are of a comparable order from that used in the literature [2,5,6].

<table><caption>**Table 1.** Nominal values for thermo-physical properties and their range of variation [2,9].</caption><thead><tr><th></th><th></th><th>ρ (Kg.m<sup>-3</sup>)</th><th>k (W.m<sup>-1</sup>.K<sup>-1</sup>)</th><th>c (J.kg<sup>-1</sup>.K<sup>-1</sup>)</th><th>ε (-)</th><th>κ (m<sup>-1</sup>)</th></tr></thead><tbody><tr><td rowspan="2">FPVC<sub>Virgin</sub></td><td>Nominal value</td><td>1413<sup>a</sup></td><td>0.28<sup>a</sup></td><td>1310<sup>a</sup></td><td>0.9<sup>b</sup></td><td>4945<sup>b</sup></td></tr><tr><td>Range of variation</td><td>± 5%</td><td>± 15%</td><td>±16%</td><td>±20%</td><td>±50%</td></tr><tr><td rowspan="3">FPVC<sub>int</sub></td><td>Nominal value</td><td>629<sup>b</sup></td><td>0.28<sup>c</sup></td><td>1310<sup>c</sup></td><td>0.9<sup>c</sup></td><td>4945<sup>c</sup></td></tr><tr><td>Range in <b>H</b><sub>1</sub></td><td>± 5%</td><td>± 15%</td><td>±16%</td><td>±20%</td><td>±50%</td></tr><tr><td>Range in <b>H</b><sub>2</sub></td><td>600-1413</td><td>±50%</td><td>±50%</td><td>±20%</td><td>±50%</td></tr><tr><td rowspan="3">Char</td><td>Nominal value</td><td>296<sup>b</sup></td><td>0.26<sup>b</sup></td><td>1720<sup>b</sup></td><td>0.85<sup>b</sup></td><td>29600<sup>b</sup></td></tr><tr><td>Range in <b>H</b><sub>1</sub></td><td>± 5%</td><td>± 15%</td><td>±16%</td><td>±20%</td><td>±50%</td></tr><tr><td>Range in <b>H</b><sub>2</sub></td><td>200-1413</td><td>±50%</td><td>±50%</td><td>±20%</td><td>±50%</td></tr></tbody></table>

a Measured.

b Derived from literature.

c Assumption that the Intermediate material shares the same properties as the virgin material.

## 2.3. Sensitivity analysis and inverse modelling

Before conducting the inverse simulations, a Sensitivity Analysis (SA) is carried out to determine the most influential input parameters. The goal being to assess if the inverse simulations need the use of all the input parameters or just the most influential of them. The SA identifies key physicochemical parameters, by exploring the effect of chemical properties, virgin material properties, $\gamma$ parameter for radiation effects, and properties of intermediate and char materials. The SA is conducted using the “One at a time method” (OAT) method [10]. In this method, individual parameters are varied while keeping others constant to assess their specific impact on the model's results. The SA investigates the influence of parameters on the mass loss rate (MLR). Specifically, multiple output metrics are compared to the nominal simulation, including the ignition time, peak MLR value, time of peak appearance. The variables are categorized into three groups based on their impact on output metrics. A variable is classified as highly sensitive if any output metric deviates by more than ±15% from the nominal simulation. If the deviation of any output metric falls between ±5% and ±15%, the variable is considered moderately sensitive. Variables that result in deviations lower than ±5% for all output metrics are classified as non-sensitive.

The outcomes of the SA guide an inverse simulation by reducing the number of parameters to be optimized. In this study, the Genetic Algorithm (GA) integrated in Gpyro is used for the inverse modelling [8]. This algorithm operates on a population of potential solutions, treating them as individual "chromosomes" that undergo evolution through processes like selection, crossover, and mutation during several generation.

The inverse simulations are carried out to validate our model under diverse assumptions. Initially, hypotheses $\mathcal{H}_1$ and $\mathcal{H}_2$ concerning the properties of intermediate FPVC, and char materials are explored. For each hypothesis, two scenarios are considered: one incorporating radiation effects in pores and the other without. Additionally, the study is conducted twice, first utilizing the most sensitive
parameters identified through the SA and subsequently incorporating all sensitive parameters. The details of the eight tested cases are summarized in table 2.

For the GA, the population size varies based on parameter count to be calibrated note N: 47 individuals for less than N=5 parameters, 95 for less than N=10, and 147 for more than N=10. Each simulation consists of 150 generations to ensure convergence (typically within 20-120 generations).

<table><caption>**Table 2.** Cases under investigation with inverse simulation.</caption><thead><tr><th>Case</th><th>1</th><th>1-bis</th><th>2</th><th>2-bis</th><th>3</th><th>3-bis</th><th>4</th><th>4-bis</th></tr></thead><tbody><tr><td>Hypothesis</td><td><i>H</i><sub>1</sub></td><td><i>H</i><sub>1</sub></td><td><i>H</i><sub>2</sub></td><td><i>H</i><sub>2</sub></td><td><i>H</i><sub>1</sub></td><td><i>H</i><sub>1</sub></td><td><i>H</i><sub>2</sub></td><td><i>H</i><sub>2</sub></td></tr><tr><td>Consideration of radiation ?</td><td>No</td><td>No</td><td>No</td><td>No</td><td>Yes</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Optimized Parameters</td><td>Highly sensitive</td><td>All sensitive</td><td>Highly sensitive</td><td>All sensitive</td><td>Highly sensitive</td><td>All sensitive</td><td>Highly sensitive</td><td>All sensitive</td></tr></tbody></table>

### 3. Results and analysis

#### 3.1. Analysing Chemical Behaviour

Upon ATG, the results show a two-stage decomposition process for FPVC, that is summarized in table 3. The pyrolysis code Gpyro is employed to determine the kinetic parameters through a trial-and-error process. The numerical model for replicating TGA is a simple 0D model. Table 3 summarizes the kinetic values calibrated, the selected decomposition scheme, the Heat of decomposition ΔH obtained from simultaneous TGA/DSC, and the range of variation of chemical properties considered for the SA and inverse modeling.

<table><caption>**Table 3.** Kinetic model and chemical properties with their uncertainties of FPVC pyrolysis.</caption><thead><tr><th>Reaction</th><th>E (KJ/mol)</th><th>Z (s<sup>-1</sup>)</th><th>Ordre (-)</th><th>θ (Kg/Kg)</th><th>ΔH (J/Kg)</th></tr></thead><tbody><tr><td>FPVC<sub>virgin</sub>→θ<sub>1</sub> FPVC<sub>Int</sub> +</td><td>166.0</td><td>2.8E+13</td><td>1.69</td><td>49%</td><td>1.3E+5</td></tr><tr><td>(1 − θ<sub>1</sub>)Gas<sub>1</sub></td><td>± 2%</td><td>± 57%</td><td>-</td><td>-</td><td>± 100%</td></tr><tr><td>FPVC<sub>Int</sub>→θ<sub>2</sub> Char +</td><td>280.0</td><td>1.4E+18</td><td>2.98</td><td>75%</td><td>8.3E+4</td></tr><tr><td>(1 − θ<sub>2</sub>)Gas<sub>2</sub></td><td>± 2%</td><td>± 60%</td><td>-</td><td>-</td><td>± 75%</td></tr></tbody></table>

#### 3.2. SA Results

The SA results for all variables are summarized in table 4. Note that the parameter γ proves to be highly sensitive across all materials. For understanding the individual impact of each property, readers should refer to the SA analysis in [9] for a similar PVC. A further physical explanation of the SA results for this specific FPVC will be provided in the full paper.

<table><caption>**Table 4.** Classification of the parameters according to their sensitivity.</caption><thead><tr><th></th><th>FPVC<sub>virgin</sub></th><th>Chemical property</th><th>FPVC<sub>Int</sub> in <i>H</i><sub>1</sub></th><th>FPVC<sub>Int</sub> in <i>H</i><sub>2</sub></th><th>Char in <i>H</i><sub>1</sub></th><th>Char in <i>H</i><sub>2</sub></th><th>Radiative contribution</th></tr></thead><tbody><tr><td>Highly sensitive</td><td>-</td><td>Δ<i>H</i><sub>1</sub></td><td>ε</td><td>ρ, ε, c, k</td><td>ε</td><td>ε, c</td><td>γ<sub>virgin</sub>, γ<sub>Int</sub>, γ<sub>Char</sub></td></tr><tr><td>Low sensitivity</td><td>ρ, k, c, ε, κ</td><td><i>E</i><sub>1</sub>, <i>Z</i><sub>1</sub>, <i>E</i><sub>2</sub>, <i>Z</i><sub>2</sub></td><td>c, k</td><td>-</td><td>c</td><td>-</td><td>-</td></tr></tbody></table>

#### 3.3. GA Results

Figure 1 provides a clear illustration of the substantial improvement in simulation output achieved by incorporating radiation in pores (Figure 1a and 1c vs 1b and 1d). Overall, simulations with radiation in pores exhibit behaviors closely aligned with experimental observations. Conversely, simulations without radiation in pores fail to replicate experimental behavior, overestimating the initial peak of MLR by up to twice the observed value. The absence of such a pronounced peak, when including radiation, can be explained by the fact that radiation distributes energy more efficiently towards the interior of the
material, thus avoiding the large peak at the beginning. With radiation in pores, thermal transport and temperature profiles become strongly coupled due to the cubic dependence of radiation on temperature (see equation 1). At 400°C, 30% of thermal transport is attributed to radiation, increasing to 50% at 600°C. This substantial change in thermal transport alters the temperature profile and consequently, the degradation state within the material. These make the radiation in pore dominate the behavior of FPVC pyrolysis. To further comprehend this phenomenon, detailed analysis through TPS experiments and pore analysis will be provided in the full paper.

<figure><img src="image_6.png" /><figcaption>**Figure 1.** MLR Derived from best fit obtain with inverse simulations across all study cases: where (a) and (b) are with $\mathcal{H}_1$ hypothesis, (c) and (d) with $\mathcal{H}_2$ hypothesis, and (a) and (c) without radiation in pores, and (b) and (d) with radiation in pores.</figcaption></figure>

Comparing different investigation interval choices yield the following results: in cases with radiation, the improvement between hypotheses $\mathcal{H}_1$ and $\mathcal{H}_2$ is clear. Indeed, in $\mathcal{H}_1$ hypothesis (Figure 1b), the simulation fails to capture the slow decrease in gas production after 200 seconds, with an overestimation before 300 seconds and an underestimation after. In contrast, with hypothesis $\mathcal{H}_2$, a nearly perfect fit is achieved after 200 seconds (Figure 1d). But note that this improvement is on the second order compared to the contribution obtained with radiation in pores, this highlights the predominancy of the radiation in pores, which once calibrated takes the upper hand on the other parameters.

Comparing simulations where only the most sensitive parameters are considered versus cases where all parameters are included (red curve vs blue curve, respectively) reveals negligeable differences. Including all sensitive parameters does not significantly enhance the fit; it may slightly improve it, as seen in cases 3 and 3-bis (figure 1b). However, it can also decrease the quality of the fit, as in the case of 4 vs 4-bis, where fit score drops from 97.9 to 94.3. This can be explained by the fact that including all sensitive parameters adds complexity to the inverse problem, requiring more generations and computational resources for the genetic algorithm to converge.

Additionally, the analysis of the calibrated parameter values reveals that there is a multitude of parameter combinations that yield the same fit in all cases, proving that there are a lot of optimal solutions. This observation is also made in other studies [11,12]. Consequently, no specific properties about the studied material can be deduced using this technique. At best, it can be inferred that under the given hypotheses and ranges of variation the model can reproduce experimental behaviour. It is noteworthy that the technique employed in this paper, which allows for the exploration of all
combinations of hypotheses (see table 2), enables the individual deduction of the effect of each
hypothesis and assists in discerning compensation effects among different phenomena [11].

Further physical explanation on the different above findings, as well as the comparison of the
thermal radiation to the dedicated experiments will be provided in the full paper.

**4. Conclusion**

In conclusion, the inverse simulation reveals that without considering radiation in pores, it is impossible
to explain the pyrolysis of FPVC. Additionally, the radiation behavior is validated through TPS
experiments up to 800°C, all of this highlights the significant role of radiation in pores. Moreover, it
challenges the common assumption in the literature that the intermediate material possesses nearly
identical properties to the original material, highlighting the potential limitations of such an approach.

Furthermore, insights are provided for inverse modeling in pyrolysis. It is demonstrated that conducting sensitivity analysis prior to inverse modeling refines input parameters and enhances the efficiency of the inverse modeling. Findings suggest that adding low-influence parameters minimally impacts optimization results and may decrease efficiency. It is also recommended to conduct inverse simulations with all possible combinations of hypotheses and parameters to evaluate the individual effects of each phenomenon and identify compensatory effects among them.

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