# Pyrolysis Model Development for a Multilayer Floor Covering

^{*}

## Abstract

**:**

^{−2}, which is within the mean experimental uncertainty.

## 1. Introduction

^{2}installed that year [15]. Modern carpet consists of a series of complicated layers made from several different polymers. Low-pile carpet tile, a modular form of the flooring material that is commonly found in commercial and institutional occupancies, particularly in high traffic areas, features at least three distinct layers of polymer mixtures. To the knowledge of the authors, the complicated structure and composition of the carpet that is the subject of this work make this the most complicated material ever to be characterized for a pyrolysis model.

## 2. Modeling

^{−3}). The statement for the conservation of mass accounts for consumption or production of component j due to reactions, the rate of which is defined in Equation (3), mass transport of gaseous components within the condensed phase which is defined in Equation (4), and mass transport associated with contraction or expansion of the material with respect to a stationary boundary (x = 0). Equation (2) is a statement for the conservation of energy of the sample in terms of the material temperature, T (K). The statement for the conservation of energy accounts for heat flow due to thermal degradation reactions and phase transitions, heat conduction within the condensed phase, which is defined in Equation (5), absorption of radiant heat from external sources defined in Equation (6), radiant heat loss from the material to the environment defined in Equation (7), convective heat transfer due to gas transport, and energy flow associated with contraction or expansion of the material with respect to a stationary boundary (x = 0).

^{−3}) denotes density, c (J kg

^{−1}K

^{−1}) is the heat capacity, k (W m

^{−1}K

^{−1}) is the thermal conductivity, κ (m

^{2}kg

^{−1}) is the absorption coefficient, $\u03f5$ is the emissivity, and λ (m

^{2}s

^{−1}) is the mass transport coefficient. All of these thermophysical properties, with the exception of absorption coefficient and emissivity, may be defined as temperature-dependent. ${\nu}_{i}^{j}$ is the stoichiometric coefficient for component j in reaction i which is positive when the component is produced and negative when the component is consumed. h

_{i}(J kg

^{−1}) is the heat absorbed or released in each reaction or phase transition and may be defined as temperature dependent. A

_{i}((m

^{3}kg

^{−1})

^{n−1}s

^{−1}) (for reaction of order n) is the Arrhenius pre-exponential factor for reaction i, E

_{i}(J mol

^{−1}) is the activation energy for reaction i, and R (J mol

^{−1}K

^{−1}) is the universal gas constant. I

_{ex}(W m

^{−2}) is defined as the radiation flux from external sources traveling within the material, the superscript 0 in ${I}_{ex}^{0}$ denotes net external radiation flux through the material boundary, and ϭ (W m

^{−2}K

^{−4}) is the Stefan-Boltzmann constant.

## 3. Experiments and Analysis

#### 3.1. Materials

^{−2}of woven polyamide-6 (PA6) with auxiliary polymers. The primary backing, which is composed of a mesh through which the face yarn is interwoven, includes approximately 0.11 kg·m

^{−2}of a PA6 and polyethylene terephthalate (PET) bicomponent mixture. The precoat is made from approximately 0.42 kg·m

^{−2}of highly-filled vinyl-acetate ethylene (VAE) with other auxiliaries. The base layer consists of approximately 1.18 kg·m

^{−2}of highly filled very-low-density polyethylene (LDPE) (labeled as “thermoplastic compound” in Figure 1) with auxiliary additives as well as 0.05 kg·m

^{−2}of nonwoven fiberglass mat.

**Figure 1.**Schematic of the EcoWorx carpet tile [19].

^{−2}and the thickness of the layer was measured as 0.0017 ± 0.0001 m. The areal density of the face yarn layer was measured as 0.350 ± 0.050 kg·m

^{−2}and the thickness of the layer was 0.0030 ± 0.0002 m. The areal density of the middle layer was measured as 0.970 ± 0.050 kg·m

^{−2}and the thickness was measured as 0.0016 ± 0.0003 m. These measurements led to the following definitions for the density of each virgin component: The ${\text{FaceYarn}}_{\text{virgin}}$ component density was defined as 125 kg·m

^{−3}, the ${\text{Middle}}_{\text{virgin}}$ component density was defined as 582 kg·m

^{−3}, and the ${\text{Base}}_{\text{virgin}}$ component density was defined as 1060 kg·m

^{−3}in individual layer models constructed as described in Section 3.2.4.

^{−3}, and the ${\text{Base}}_{\text{virgin}}$ component density was defined as 1200 kg·m

^{−3}in the full carpet composite models. Construction of the model that required these geometric and gravimetric definitions is outlined in Section 3.2.4.

#### 3.2. Experimental Methods

#### 3.2.1. Simultaneous Thermal Analysis

^{−1}to approximately 100 K above the highest temperature at which mass loss was observed. This heating rate was chosen to ensure the samples did not experience significant temperature or mass gradients, which created conditions where the thermal degradation kinetics and energetics were decoupled from heat and mass transport within the sample. The test chamber was constantly purged with nitrogen flowing at a rate of 50 mL·min

^{−1}to investigate thermal degradation while eliminating oxidation and other unwanted heterogeneous reactions.

_{peak}), the peak MLR (MLR

_{peak}), the initial reactant mass (m

_{0}) and the condensed-phase product yield (v) were required to determine the Arrhenius reaction parameters (A and E). An initial estimate of the activation energy was calculated from Equation (8), which is an approximate solution to the first order Arrhenius kinetics under linear heating conditions [22].

_{peak}, a maximum absolute error in the prediction of ${T}_{\text{peak}}$ of 3 K, and a mean error of less than 2% in the normalized mass versus temperature curve. The resulting reaction schemes are presented in Section 4.1.1.

#### 3.2.2. Microscale Combustion Calorimetry

_{2}/N

_{2}mixture to ensure accurate HRR measurements. A temperature calibration that relied on the melting point of several pure metals was conducted to ensure accurate sample temperature measurements [24]. The combustor temperature was maintained at 900 K.

^{−1}to the final temperature of approximately 1023 K. The heating rate used in this investigation is outside the range of heating rates recommended in the standard, but was chosen to provide data collected under conditions comparable to those used in the STA tests.

#### 3.2.3. Absorption Coefficient Measurement

^{−2}(980 K) while measuring the radiation flux transmitted through the sample with a Schmidt-Boelter heat flux gauge. A rendering of the measurement setup is provided in Figure 3.

_{x}

_{=}

_{δ}in Equation (10). The values of the reflection loss coefficient utilized in these calculations are discussed in Section 4.1.3.

#### 3.2.4. Gasification Experiments and Analysis

_{back}) data are simultaneously collected. The radiant heat flux onto the front surface of the sample is generated by the cone calorimeter heater. The mass of the sample is measured throughout the tests by the mass balance internal to the cone calorimeter system. The apparatus features an infrared camera focused on a gold mirror to reflect the image of the back surface of the sample. The infrared camera records a spatially-resolved temperature measurement of the back surface. The CAPA was designed to enable simultaneous measurement of the sample T

_{back}and the sample mass throughout the duration of the test.

^{−1}(measured at 1 atm and 298 K). At this flow rate, the mean oxygen concentration approximately 0.001 m from the front surface of the sample was measured as 2.2 ± 0.4 vol %. This oxygen concentration prevented autoignition for all samples and appeared to make any effects of oxidation on MLR and T

_{back}measurements inconsequential.

^{−2}. Each test was repeated three times to accumulate statistics. Samples were prepared in a square geometry with a side of 0.08 m. They were located in the center of a square sheet of 0.00625 m thick Kaowool PM board with an edge dimension of 0.105 m.

_{back}. The T

_{back}data was recorded at a rate of 7.5 Hz. In each frame, the image was divided into three regions. Region 1 consisted of the central 0.04 m × 0.04 m square, region 2 consisted of the central 0.06 m × 0.06 m square less region 1, and region 3 consisted of the entire 0.08 m × 0.08 m sample surface less regions 1 and 2. The mean T

_{back}was calculated in each frame using four randomly selected locations from regions 1 and 2, and two randomly selected locations from region 3. The T

_{back}was generally uniform, with the maximum deviation from the mean value on the order of 5%.

^{−2}·K

^{−1}and the ambient temperature at the front surface was measured as 330 K at 30 kW·m

^{−2}and exhibited a linear dependence on the set heat flux up to 370 K at a heat flux of 70 kW·m

^{−2}. The mass transport at the boundary was defined to provide no impedance to the escape of gaseous pyrolyzate produced during degradation.

^{−2}·K

^{−1}and an ambient temperature of 310 K. A radiant heat flux of 500 W·m

^{−2}was applied to the back surface to simulate radiation from the internal walls of the test apparatus (which were assumed to be at ambient temperature). The absorption coefficient of the back surface was defined such that all incident radiation was absorbed at the surface. The emissivity of the back surface was defined as 0.95 to simulate the presence of high emissivity paint on the back surface of the tested samples or on the aluminum foil on which the samples rested. A default value for the mass transport coefficient was defined for all components (2 × 10

^{−5}m

^{2}s

^{−1}). This value was determined as high enough to allow all gaseous pyrolyzate to escape the condensed phase with no impedance to flow, and low enough that it would maintain the stability of the integration [27].

## 4. Results

#### 4.1. Data Analysis for Property Evaluation

#### 4.1.1. Thermal Degradation Kinetics and Energetics Determination

^{−1}) was not achieved instantaneously. Rather, the heating rate measured in each of the tests had reproducible time-dependency that was approximated in ThermaKin by an exponentially-decaying sinusoid, Equation (11). The parameters of this equation were adjusted until it matched the experimental data. The agreement between the experimentally observed and modeled heating rate profiles is evident in Figure 6.

**Figure 6.**Experimentally observed and modeled heating rate histories typical of the Simultaneous Thermal Analysis (STA). The coefficients for Equation (11) that describe the modeled curve are the following: a = 0.166 K s

^{−1}, b = 0.0024 s

^{−1}, f = 0.004 s

^{−1}, g = −0.0623.

**Table 1.**Effective reaction mechanisms for each layer of the carpet composite and the heats of reactions. Positive heats represent endothermic processes.

# | Reaction Equation | A (s^{−1}) | E (J·mol^{−1}) | h (J·kg^{−1}) |
---|---|---|---|---|

Face Yarn | ||||

1 | ${\text{FaceYarn}}_{\text{virgin}}\to {\text{FaceYarn}}_{\text{melt}}$ | 6.0 × 10^{38} | 3.80 × 10^{5} | 6.1 × 10^{4} |

2 | ${\text{FaceYarn}}_{\text{melt}}\to 0.92{\text{FaceYarn}}_{\text{int}.}+0.08{\text{FaceYarn}}_{\text{volatiles}}$ | 1.0 × 10^{9} | 1.41 × 10^{5} | 5.3 × 10^{4} |

3 | ${\text{FaceYarn}}_{\text{int}.}\to 0.06{\text{FaceYarn}}_{\text{char}}+0.94{\text{FaceYarn}}_{\text{volatiles}}$ | 3.0 × 10^{14} | 2.30 × 10^{5} | 5.3 × 10^{5} |

Middle Layer | ||||

1 | ${\text{Middle}}_{4,\text{virgin}}\to {\text{Middle}}_{4,\text{melt}}$ | 9.0 × 10^{38} | 3.84 × 10^{5} | 8.0 × 10^{4} |

2 | ${\text{Middle}}_{3,\text{virgin}}\to {\text{Middle}}_{3,\text{melt}}$ | 1.0 × 10^{28} | 3.00 × 10^{5} | 6.0 × 10^{4} |

3 | ${\text{Middle}}_{1,\text{virgin}}\to 0.334{\text{Middle}}_{\text{char}}+0.666{\text{Middle}}_{\text{volatiles}}$ | 1.0 × 10^{12} | 1.55 × 10^{5} | 2.7 × 10^{6} |

4 | ${\text{Middle}}_{2,\text{virgin}}\to 0.334{\text{Middle}}_{\text{char}}+0.666{\text{Middle}}_{\text{volatiles}}$ | 1.0 × 10^{20} | 2.62 × 10^{5} | 0 |

5 | ${\text{Middle}}_{3,\text{melt}}\to 0.334{\text{Middle}}_{\text{char}}+0.666{\text{Middle}}_{\text{volatiles}}$ | 5.0 × 10^{8} | 1.42 × 10^{5} | 3.5 × 10^{5} |

6 | ${\text{Middle}}_{4,\text{melt}}\to 0.334{\text{Middle}}_{\text{char}}+0.666{\text{Middle}}_{\text{volatiles}}$ | 1.0 × 10^{10} | 1.70 × 10^{5} | 2.0 × 10^{5} |

Base Layer | ||||

1 | ${\text{Base}}_{\text{virgin}}\to {\text{Base}}_{\text{melt}}$ | 1.0 × 10^{21} | 1.72 × 10^{5} | 6.0 × 10^{3} |

2 | ${\text{Base}}_{\text{melt}}\to 0.92{\text{Base}}_{\text{int}.}+0.08{\text{Base}}_{\text{volatiles}}$ | 5.0 × 10^{6} | 1.15 × 10^{5} | 4.0 × 10^{4} |

3 | ${\text{Base}}_{\text{int}.}\to 0.71{\text{Base}}_{\text{char}}+0.29{\text{Base}}_{\text{volatiles}}$ | 1.0 × 10^{16} | 2.58 × 10^{5} | 1.5 × 10^{5} |

_{0}indicates initial mass of the sample. In general, the reaction mechanisms determined in this work generated curves that agreed well with the experimental MLR and total mass curves. All error bars displayed in this work correspond to two standard deviations of the mean.

_{virgin}and Face Yarn

_{melt}components. The heat capacity of all the Middle

_{virgin}components was adequately described with a single temperature-dependent term. It was impossible to assign a heat capacity value to each individual Middle

_{virgin}component, and all were assigned the same value. The base layer sample melted shortly into the tests, and it proved impossible to determine the heat capacity of the Base

_{virgin}component from the collected heat flow rate data. The heat capacity of the Base

_{melt}component was determined with a linear temperature-dependence and it was assumed that the same expression could adequately describe the heat capacity of the Base

_{virgin}component.

**Figure 7.**Normalized mass loss rate (MLR) and normalized mass data collected in STA experiments and model predicted curves for: (

**a**) and (

**b**) face yarn layer; (

**c**) and (

**d**) the middle layer; and (

**e**) and (

**f**) the base layer. Error bars indicate two standard deviations of the mean experimental data.

_{char}and Middle

_{char}components were characterized by low masses and a porous structure that compromised the thermal contact between the sample and the crucible and yielded unreliable heat flow rate measurements. These char components were assigned a single heat capacity that was measured as the mean value for the chars produced by seven common polymers [21]. The heat capacity of the Base

_{char}component was determined by conducting independent tests on the char produced from thermal degradation of the base layer sample. The heat capacity of the char did not follow a recognizable functional form, so the arithmetic mean of the data over the entire tested temperature range was defined as the heat capacity of the Base

_{char}component.

_{int}. component was defined as the mean between the heat capacities of the Face Yarn

_{melt}component evaluated at 560 K and the Face Yarn

_{char}component. The heat capacity of the Middle

_{3,melt}and Middle

_{4,melt}components was defined as the mean between the heat capacity of the Middle

_{virgin}components evaluated at 500 K and the Middle

_{char}component. The heat capacity of the Base

_{int}component was defined as the mean between the heat capacities of the Base

_{melt}component evaluated at 600 K and the Base

_{char}component. The heat capacity of the reactant was evaluated at a different temperature for each layer based on the temperature at which the intermediate was produced and subsequently reacted.

^{−1}·K

^{−1}. The expressions determined for the heat capacity of each component are provided in Table 2.

Component | c [J kg^{−1} K^{−1}] | Method |
---|---|---|

Face Yarn_{virgin} | 8.2T − 1180 | STA |

Face Yarn_{melt} | 3.6T + 580 | STA |

Face Yarn_{int.} | 2150 | Assumed |

Face Yarn_{char}, Middle_{char} | 1700 | [21] |

Middle_{1,virgin}, Middle_{2,virgin}, Middle_{3,virgin}, Middle_{4,virgin} | 4.2T | STA |

Middle_{3,melt}, Middle_{4,melt} | 1900 | Assumed |

Base_{virgin} | 2.0T + 1000 | Assumed |

Base_{melt} | 2.0T + 1000 | STA |

Base_{int.} | 1525 | Assumed |

Base_{char} | 850 | STA |

Face Yarn_{volatiles}, Middle_{volatiles}, Base_{volatiles} | 1800 | Assumed |

**Figure 8.**Normalized heat flow rate and integral heat flow rate data collected in STA experiments and model predicted curves for: (

**a**) and (

**b**) face yarn layer; (

**c**) and (

**d**) the middle layer; and (

**e**) and (

**f**) the base layer. Error bars indicate two standard deviations of the mean experimental data.

#### 4.1.2. Heat of Combustion Determination

^{−1}were analyzed using the degradation kinetics determined from analysis of STA data. The heating rate profile observed in MCC experiments was different than the profile observed in STA experiments, but was adequately described by the form of Equation (11). The mass loss rate was predicted through a simulation adhering to a heating rate profile that approximated the profile observed in MCC experiments. The agreement between the experimentally observed and modeled heating rate profiles is displayed in Figure 9.

**Figure 9.**Experimentally observed and modeled heating rate histories typical of the Microscale Combustion Calorimetry (MCC) experiments conducted in this work. The coefficients for Equation (14) that describe the modeled curve are the following: a = 0.168 K s

^{−1}, b = 0.0039 s

^{−1}, f = 0.0065 s

^{−1}, g = 0.256.

**Table 3.**Effective heat of combustion values for the volatile species released in each reaction. Positive heats of combustion are exothermic.

Volatile Species | h_{c} (J kg^{−1}) |
---|---|

Face Yarn_{volatiles,reaction 2} | 2.4 × 10^{7} |

Face Yarn_{volatiles,reaction 3} | 2.9 × 10^{7} |

Middle_{volatiles,reaction 3} | 0 |

Middle_{volatiles,reaction 4} | 1.6 × 10^{7} |

Middle_{volatiles,reaction 5} | 2.4 × 10^{7} |

Middle_{volatiles,reaction 6} | 5.0 × 10^{7} |

Base_{volatiles,reaction 2} | 3.4 × 10^{7} |

Base_{volatiles,reaction 3} | 5.9 × 10^{7} |

**Figure 10.**Normalized heat release rate and integral heat release rate data collected in MCC experiments and model predicted curves for: (

**a**) and (

**b**) the face yarn layer; (

**c**) and (

**d**) the middle layer; and (

**e**) and (

**f**) the base layer. Error bars were omitted due to small magnitude scatter.

#### 4.1.3. Absorption Coefficient and Emissivity Determination

**Table 4.**Measurements used to calculate the absorption coefficient for each virgin and melt component.

Layer | $\left(\frac{{I}_{x=0}}{{I}_{x=\delta}}\right)$ | δ (m) | ρ (kg m^{−3}) | κ (m^{2} kg^{−1}) |
---|---|---|---|---|

Face Yarn Melt | 0.025 | 0.0008 ± 0.0001 | 625 | 7.17 |

Middle Layer | 0.026 | 0.0013 ± 0.0001 | 582 | 4.69 |

Middle Layer | 0.020 | 0.0016 ± 0.0001 | 582 | 4.09 |

Base Layer | 0.010 | 0.0010 ± 0.0001 | 1060 | 4.25 |

Base Layer | 0.005 | 0.0010 ± 0.0001 | 1060 | 4.90 |

_{melt}component was also used to calculate the absorption coefficient. The densities of the middle layer and the base layer defined in the individual layer models were used to calculate the absorption coefficient of each of those layers. The mean of the individual measurements of the absorption coefficient for each layer was calculated to define the absorption coefficient in the models. Approximate values were assigned to each component based on the transmitted heat flux measurements (7 m

^{2}·kg

^{−1}for the face yarn, 4.4 m

^{2}·kg

^{−1}for the middle layer, and 4.6 m

^{2}·kg

^{−1}for the base layer). The absorption coefficients of the melt and intermediate components were assigned the same absorption coefficient as the virgin component for all layers.

^{2}·kg

^{−1}). The char formed during thermal degradation of the base layer did not appear to be optically dark. The absorption coefficient for the ${\text{Base}}_{\text{char}}$ component was assumed to be equal to the absorption coefficient of all other base layer components.

#### 4.1.4. Thermal Conductivity Determination

_{melt}component was defined five times larger than the density of the Face Yarn

_{virgin}component, and the density of the Face Yarn

_{int}and Face Yarn

_{char}components were defined proportional to the stoichiometric coefficient for the condensed phase product of each reaction to simulate a constant thickness for the layer after melting.

_{virgin}components were defined with the same density because it was impossible to identify and separate each individual initial component. Two of the middle layer components underwent phase changes that did not affect the geometry of the sample, so the Middle

_{melt}components were defined with the same density as the Middle

_{virgin}components. The Base

_{virgin}component went through a phase change without the geometry of the layer changing considerably and the density of the ${\text{Base}}_{\text{melt}}$ component was defined equivalent to the virgin component density. The overall thickness of the middle layer and the base layer remained approximately constant throughout the CAPA tests. The densities of the Middle

_{char}, Base

_{int}and Base

_{char}components were defined proportional to the associated stoichiometric coefficients in the reaction mechanism to maintain a constant thickness in the simulations. The definitions for the densities of all components are provided in Table 5.

**Table 5.**Full set of thermophysical properties used in the individual upper layer model and base layer model.

Component | ρ (kg m^{−1}) | k (W m^{−1} K^{−1}) | $\u03f5$ | κ (m^{2} κg^{−1}) |
---|---|---|---|---|

Face Yarn | ||||

Face Yarn_{virgin} | 125 | 0.05 | 0.95 | 7 |

Face Yarn_{melt} | 625 | 0.05 | 0.95 | 7 |

Face Yarn_{int.} | 575 | 0.025 + 6.5 × 10^{−10}T^{3} | 0.905 | 7 |

Face Yarn_{char} | 34.5 | 11 × 10^{−10}T^{3} | 0.86 | 100 |

Middle Layer | ||||

Middle_{1,virgin}, Middle_{2,virgin}, Middle_{3,virgin}, Middle_{4,virgin}, Middle_{3,melt}, Middle_{4,melt} | 582 | 0.05 | 0.95 | 4.4 |

Middle_{char} | 194.4 | 11 × 10^{−10}T^{3} | 0.86 | 100 |

Base Layer | ||||

Base_{virgin}, Base_{melt} | 1060 | 0.25 – 2.85 × 10^{−4}T | 0.95 | 4.6 |

Base_{int.} | 975.2 | 0.125 – 1.425 × 10^{−4}T + 3.5 × 10^{−10}T^{3} | 0.905 | 4.6 |

Base_{char} | 692.4 | 7 × 10^{−10}T^{3} | 0.86 | 4.6 |

_{back}data collected in the CAPA tests using the ThermaKin modeling environment. The initial rise of the T

_{back}data curve was chosen as the target for the virgin and melt components because these were the only components that affected the T

_{back}curve early in the tests. The model prediction for the upper layer (face yarn and middle layer) is compared to the experimental data in Figure 11. The temperature prediction was not sensitive to the thermal transport parameters of the char and intermediate components for the time range that corresponded to the initial rise of T

_{back}.

^{−1}·K

^{−1}) of the Face Yarn

_{virgin}, Middle

_{virgin}, Face Yarn

_{melt}and Middle

_{melt}components was adequate to describe the rising edge of the T

_{back}curve. There was no evidence in the data of a change in the thermal conductivity from the virgin components to the melt components. Though this thermal conductivity value is low for a mixture of solid polymers and is more typical of air at elevated temperatures (650 K), the structure of the carpet upper layer supports a thermal conductivity value lower than the typical range for polymers. The face yarn was made of fibrous filaments woven into a yarn and the majority of the volume of the defined face yarn layer was air (or, in the case of the gasification tests, nitrogen). Furthermore, the face yarn melted shortly after exposure to the cone heater and the melted face yarn was characterized by rapidly regenerating bubbles. The middle layer was structured as a mesh interwoven with face yarn and although the density of the middle layer was larger than the face yarn, gases still made a large contribution to the volume of the layer.

**Figure 11.**First 120 s of experimental T

_{back}curve measured in Controlled Atmosphere Pyrolysis Apparatus (CAPA) tests and corresponding model predicted curve for the upper layer exposed to a radiant flux of 30 kW·m

^{−2}. The shaded region corresponds to two standard deviations of the mean experimental data.

_{back}profiles.

_{back}data collected on the upper layer subjected to a heat flux of 30 kW·m

^{−2}to determine the thermal conductivities of the char components. The target data for the inverse analysis was chosen as the slowly rising portion of T

_{back}that was observed after 120 s in the gasification tests. Due to the high porosity of the chars produced in each layer, radiation was assumed to be the dominant mode of heat transfer through the char. The radiation diffusion approximation [30] was invoked to describe the thermal conductivity of all the char components. An attempt was made to describe all the char components with a single thermal conductivity expression of the form βT

^{3}and it was determined that a single value of β adequately described the T

_{back}profile in the final 480 s of the curve. The thermal conductivities of all the intermediate components were defined as the mean of the thermal conductivities of the corresponding virgin component and char component. For the face yarn intermediate, this produced an expression with a constant term and a T

^{3}term. The agreement between the T

_{back}predictions and the experimental data collected on the upper layer at a heat flux of 30 kW·m

^{−2}are shown in Figure 12. The full set of parameters that were determined for the upper layer of the carpet composite to describe thermal transport to and within the solid sample are provided in Table 5.

**Figure 12.**Final 480 s of experimental T

_{back}curve measured in CAPA tests and corresponding model predicted curve for the upper layer exposed to a radiant flux of 30 kW·m

^{−2}. The shaded region corresponds to two standard deviations of the mean experimental data.

^{−2}. The target data was identified as the slope of the initial increase in the T

_{back}. Inadequate agreement between the model prediction and the experimental data was produced with a single, constant value for the thermal conductivity of the virgin and melt components. A linearly decreasing thermal conductivity for the ${\text{Base}}_{\text{virgin}}$ and ${\text{Base}}_{\text{melt}}$ components was able to adequately predict both the fast and slow rising portions of the initial 150 s of the T

_{back}curve. The agreement between the experimental curve and the model prediction are provided in Figure 13.

**Figure 13.**First 150 s of experimental T

_{back}curve measured in CAPA tests and corresponding model predicted curve for the base layer exposed to a radiant flux of 30 kW·m

^{−2}. The shaded region corresponds to two standard deviations of the mean experimental data.

_{back}of the base layer are provided in Figure 14. A single, constant value of β in the βT

^{3}expression adequately described the T

_{back}profile in the final 450 s of the curve. The thermal conductivity of the Base

_{int}component was defined as the mean between the Base

_{virgin}and Base

_{char}components, which resulted in a form with a constant, linear, and T

^{3}term. The full set of parameters that define the base layer thermophysical properties are provided in Table 5.

**Figure 14.**Final 450 s of experimental T

_{back}curve measured in CAPA tests and corresponding model predicted curve for the base layer exposed to a radiant flux of 30 kW·m

^{−2}. The shaded region corresponds to two standard deviations of the mean experimental data.

#### 4.1.5. Parameter Uncertainty

#### 4.2. Individual Layer Model Predictions

#### 4.2.1. Upper Layer (Consisting of Face Yarn and Middle Layer)

_{back}data collected at incident heat fluxes of 50 and 70 kW·m

^{−2}as well as MLR curves collected in CAPA tests conducted at incident heat fluxes of 30, 50, and 70 kW·m

^{−2}. The model predicted curves and the experimental data are provided in Figure 15.

**Figure 15.**Experimental T

_{back}and MLR curve collected in CAPA tests and corresponding model predicted curve for the upper layer exposed to radiant fluxes of (

**a**) and (

**b**) 30 kW·m

^{−2}; (

**c**) and (

**d**) 50 kW·m

^{−2}; and (

**e**) and (

**f**) 70 kW·m

^{−2}. The shaded region and error bars correspond to two standard deviations of the mean experimental data.

_{back}profile as well as the slope of the initial increase for all incident heat fluxes. The model also accurately predicts the final steady temperature at each heat flux. The model overpredicts the T

_{back}from approximately 40 to 180 s at a heat flux of 50 kW·m

^{−2}and overpredicts the T

_{back}from 45 to 90 s at a heat flux of 70 kW·m

^{−2}, although the qualitative shape of each temperature prediction agrees with the experimental data. This overprediction may be attributed to systematic errors in the measurement of T

_{back}due to sample deformation of the upper layer and poor thermal contact between the back surface of the sample and the aluminum foil. It may also be due to uncertainty in the measurement because of degradation of the high emissivity paint on the back surface of the foil above 600 K.

#### 4.2.2. Base Layer

_{back}data collected at incident heat fluxes of 50 and 70 kW·m

^{−2}as well as MLR curves collected at incident heat fluxes of 30, 50, and 70 kW·m

^{−2}. The model predicted curves and the experimental data are provided in Figure 16.

_{back}after about 50 s at heat fluxes of 50 and 70 kW·m

^{−2}, although the temperatures measured at times later than 50 s into the tests for the higher heat fluxes correspond to temperatures significantly above 600 K, so there is some uncertainty about the validity of that data due to degradation of the high emissivity paint on the back surface of the sample. It is also possible that the glass reinforcement, which comprises a large fraction of the residual mass in the base layer and has a relatively low emissivity, compromised the well-defined emissivity at the back surface. A decrease in the emissivity of the measured surface manifests as artificially low T

_{back}measurements.

**Figure 16.**Experimental T

_{back}and MLR curves collected in CAPA tests and corresponding model predicted curves for the base layer exposed to radiant fluxes of (

**a**) and (

**b**) 30 kW·m

^{−2}, (

**c**) and (

**d**) 50 kW·m

^{−2}, and (

**e**) and (

**f**) 70 kW·m

^{−2}. The shaded region and error bars correspond to two standard deviations of the mean experimental data.

#### 4.3. Full Carpet Model Predictions

_{back}and MLR data from CAPA tests were combined in a full carpet model to evaluate their ability to predict the pyrolysis behavior of the full carpet composite. The thicknesses and densities of the middle layer and the base layer were modified based on a discrepancy in thickness measurements discussed in Section 3.1. The density definitions for the full carpet composite model are provided in Table 6.

**Table 6.**Thermal conductivity and density values for Final Full Carpet model. Modifications to property values from individual layer models are shown in bold.

Component | ρ (kg m^{−1}) | k (W m^{−1} K^{−1}) |
---|---|---|

Face Yarn | ||

Face Yarn_{virgin} | 125 | 0.12 |

Face Yarn_{melt} | 625 | 0.12 |

Face Yarn_{int.} | 575 | 0.06 + 3.5 × 10^{−10}T^{3} |

Face Yarn_{char} | 34.5 | 7 × 10^{−10}T^{3} |

Middle Layer | ||

Middle_{1,virgin}, Middle_{2,virgin}, Middle_{3,virgin}, Middle_{4,virgin}, Middle_{3,melt}, Middle_{4,melt} | 750 | 0.12 |

Middle_{char} | 250.5 | 7 × 10^{−10}T^{3} |

Base Layer | ||

Base_{virgin}, Base_{melt} | 1200 | 0.25 – 2.85 × 10^{−4}T |

Base_{int.} | 1104 | 0.125 – 1.425 × 10^{−4}T + 3.5 × 10^{−10}T^{3} |

Base_{char} | 783.8 | 7 × 10^{−10}T^{3} |

_{back}and MLR data predicted by the model of the full carpet constructed from the combination of the individually parameterized upper and base layer representations are labeled in Figure 17 as “Initial Model Prediction”. The model was able to predict the qualitative trends in the experimental T

_{back}curves at all heat fluxes. The approximate steady T

_{back}was well predicted at 30 and 50 kW·m

^{−2}and slightly overpredicted at 70 kW·m

^{−2}. The shape of the MLR curve was well predicted at 30 kW·m

^{−2}, but the agreement between the predicted curve and the experimental curve degraded at the higher heat fluxes. Though the model was able to predict the qualitative trends in the experimental data, the quantitative agreement required improvement.

_{back}profile collected on the full carpet samples in CAPA tests conducted at a heat flux of 30 kW·m

^{−2}. It was hypothesized that the individually parameterized base layer model provided sufficient description of the actual tested base layer and the only independent parameters that were adjusted to improve agreement between the experimental data and the model prediction were the thermal conductivities of the upper layer components. The curves that were predicted when the thermal transport parameters were adequate to describe the target experimental data are plotted as the “Final Model Prediction” in Figure 17. The changes made to the thermal conductivity definitions provided in Table 5 to generate the “Final Model Prediction” are provided in Table 6.

**Figure 17.**Experimental T

_{back}and MLR curve collected in CAPA tests and corresponding model predicted curves for the full carpet composite exposed to radiant fluxes of (

**a**) and (

**b**) 30 kW·m

^{−2}, (

**c**) and (

**d**) 50 kW·m

^{−2}, and (

**e**) and (

**f**) 70 kW·m

^{−2}. The shaded region and error bars correspond to two standard deviations of the mean experimental data.

_{back}and the slope of the initial increase at all heat fluxes. The entire T

_{back}curve was well predicted at heat fluxes of 30 and 50 kW·m

^{−2}, and alternated between underpredicting and overpredicting the T

_{back}above 600 K at a heat flux of 70 kW·m

^{−2}. An interesting observation is that the experimental T

_{back}of the base layer and the full composite never reached temperatures higher than approximately 750 K, which corresponds to the peak MLR in the base layer TGA data. This temperature indicates the point in the tests at which the largest fraction of the volatile mass of the base layer is liberated from the solid, which leaves a matrix of fiberglass as residue. It is possible that a more comprehensive definition of the base layer that includes the properties of the fiberglass reinforcement would improve the agreement between the experimental data and the model predictions for the full carpet composite.

^{−2}, but the time to the peak was over predicted by approximately 50 s. At the higher heat fluxes, the time to the peak MLR was better predicted, but the peak value was underpredicted by larger percentages with each increase in the incident heat flux. The qualitative shape of the predicted curve at 30 and 50 kW·m

^{−2}agree with the experimental curve. The curve predicted at a heat flux of 70 kW·m

^{−2}showed a slowly decaying plateau from approximately 120 to 210 s that did not occur in the experiments. The mean error between the predicted MLR and the mean experimental MLR was calculated as 13% for 30 kW·m

^{−2}(mean MLR value of 0.00171 kg m

^{−2}·s

^{−1}), 18% for 50 kW·m

^{−2}(mean MLR value of 0.00427 kg·m

^{−2}·s

^{−1}), and 28% for 70 kW·m

^{−2}(mean MLR value of 0.00850 kg·m

^{−2}·s

^{−1}). The mean error between the predicted MLR and the mean experimental MLR was within the mean experimental uncertainty for all tested heat fluxes.

## 5. Conclusions

_{back}predictions that agreed well with the experimental data. The combination of these two layers produced predictions that had a fair agreement with experimental data collected on the full carpet composite. It was likely that, by separating the layers of the carpet and effectively compromising the structure of the composite, the thermal transport characteristics of the layers were affected. Qualitatively and quantitatively improved predictions were produced by re-parameterizing the thermal conductivity of the upper layer components in the context of the full carpet composite.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Symbols | Description |

A | Arrhenius pre-exponential factor ((m^{3} kg^{−1})^{n−1}s^{−1}) (for reaction of order n) |

E | activation energy (J mol^{−1}) |

I | radiant flux (W m^{−2}) |

J | mass flux (kg m^{−2} s^{−1}) |

R | universal gas constant (J mol^{−1} K^{−1}) |

T | temperature (K) |

c | heat capacity (J kg^{−1} K^{−1}) |

h | heat evolved in reaction (J kg^{−1}) |

k | thermal conductivity (W m^{−1} K^{−1}) |

m | mass (kg) |

q | heat flux due to thermal conduction (W m^{−2}) |

r | reaction rate (kg m^{−3} s^{−1}) |

t | time (s) |

x | Cartesian coordinate (m) |

Greek | |

β | coefficient of cubic term in effective thermal conductivity (W m^{−1} K^{−4}) |

$\gamma $ | reflection loss coefficient |

δ | thickness of sample (m) |

$\u03f5$ | emissivity |

κ | absorption coefficient (m^{2} kg^{−1}) |

λ | mass transport coefficient (m^{2} s^{−1}) |

ν | stoichiometric coefficient |

ξ | mass concentration (kg m^{−3}) |

ρ | density (kg m^{−3}) |

σ | Stefan-Boltzmann constant (W m^{−2} K^{−4}) |

Acronyms | |

CAPA | Controlled Atmosphere Pyrolysis Apparatus |

DSC | Differential Scanning Calorimetry |

HRR | heat release rate |

MLR | mass loss rate |

MCC | Microscale Combustion Calorimeter |

STA | Simultaneous Thermal Analysis |

TGA | Thermogravimetric Analysis |

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**MDPI and ACS Style**

McKinnon, M.B.; Stoliarov, S.I.
Pyrolysis Model Development for a Multilayer Floor Covering. *Materials* **2015**, *8*, 6117-6153.
https://doi.org/10.3390/ma8095295

**AMA Style**

McKinnon MB, Stoliarov SI.
Pyrolysis Model Development for a Multilayer Floor Covering. *Materials*. 2015; 8(9):6117-6153.
https://doi.org/10.3390/ma8095295

**Chicago/Turabian Style**

McKinnon, Mark B., and Stanislav I. Stoliarov.
2015. "Pyrolysis Model Development for a Multilayer Floor Covering" *Materials* 8, no. 9: 6117-6153.
https://doi.org/10.3390/ma8095295