# Improvement of the modelling of the conduction-radiation flash method by a fully stochastic approach. A numerical validation

Loïc Seyer, Franck Enguehard, Denis Rochais

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Loïc Seyer, Franck Enguehard, Denis Rochais. Improvement of the modelling of the conduction-radiation flash method by a fully stochastic approach. A numerical validation. Journal of Physics: Conference Series, 2024, 2766 (1), pp.012035. 10.1088/1742-6596/2766/1/012035. http://dx.doi.org/10.1088/1742-6596/2766/1/012035

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# Improvement of the modelling of the conduction-radiation flash method by a fully stochastic approach. A numerical validation

To cite this article: L Seyer *et al* 2024 *J. Phys.: Conf. Ser.* **2766** 012035

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# Improvement of the modelling of the conduction-radiation flash method by a fully stochastic approach. A numerical validation

L Seyer¹, F Enguehard² and D Rochais¹

¹CEA, DAM, Le Ripault, F-37260 Monts, France

²Institut Pprime, CNRS, Université de Poitiers, ISAE-ENSMA, F-86962 Futuroscope Chasseneuil, France

E-mail: seyer.loic@gmail.com

**Abstract.** Three different numerical models for simulating the well-known experimental flash method are presented and compared. These models consider conduction-radiation coupling and are applied within a one-dimensional grey semi-transparent slab enclosed between two opaque surfaces. The first model (A) is deterministic, dating back to about a decade, and is based on three assumptions regarding radiation transfer: isotropic intensity within each of the two hemispheres, the scattering phenomenon is supposed to be isotropic, and linearization of terms involving differences of temperatures to the fourth power. The other two models are completely new and do not make these assumptions. While the first one (B) is purely deterministic and serves as a reference for validation, the other one (C) is fully stochastic, and is the focus of this paper. A comparison shows excellent agreement between (B) and (C), validating our new model of interest (C). On the other hand, (A) exhibits significant discrepancies with the other two, highlighting the importance of its limiting assumptions.

## 1. Introduction

The flash method is a widely used experimental technique for thermal characterization. Despite its experimental nature, the method is often coupled with a numerical model to determine the properties of the medium, mainly through inversion techniques [1, 2, 3]. In the quest for high-temperature, high-performance insulators, porous ceramic semi-transparent media have been shown to exhibit excellent properties [4, 5, 6]. Unfortunately, modelling the coupled conduction-radiation heat transfer in such a medium is becoming increasingly difficult. In the case of complex 3D structures (pores, fibers), and when transient states are desired (as in the flash method), most deterministic models reach their limits, especially in terms of memory requirements. To overcome this challenge, one solution is to reduce the size of the numerical structure, but at the expense of the accuracy of the results. An alternative is to use stochastic models, which require less memory, generally at the cost of increased computational requirements.

This paper presents recent progress in the development of a purely stochastic method using a ray-tracing approach to model radiation heat transfer and a Brownian walker method to model conduction heat transfer. This work follows those presented in other articles [7, 8, 9]. In the present paper, we extend these methods to model the flash method applied to a one-dimensional grey semi-transparent slab enclosed between two opaque surfaces.

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Three different methods are compared, all based on an assumption of grey Beerian radiative behavior. The first model (A) is described in detail in [3] and relies on three more assumptions concerning the radiative transfer: isotropic scattering, isotropic intensity within each of the two hemispheres and linearization of terms involving differences of temperatures to the fourth power. This model is used to demonstrate the importance of these assumptions in such a problem. The other two models do not make these three assumptions, one is deterministic and is used as a reference for validation, while the other is stochastic and is the main focus of this paper. These models are briefly introduced in Section 2. The results are compared in the following section.

## 2. Presentation of the models

In all cases, the problem concerns a one-dimensional homogeneous grey Beerian semi-transparent slab enclosed between two opaque black surfaces, whose radiative effects appear as boundary conditions of the radiative transfer problem.

### 2.1. Model (A) – ancient deterministic

This model is described in detail in [3]. It is based on three strong assumptions regarding the radiative transfer: the intensity field is supposed to be isotropic within each of the two hemispheres (the “forward” and the “backward”), the scattering phenomenon is supposed to be isotropic, and terms involving differences of temperatures to the fourth power are linearized. This allows the authors to derive the analytical expression of the temperature field in the Laplace space, which is then Laplace-inversed numerically.

### 2.2. Model (B) – new deterministic

This model uses finite differences and discrete ordinates to solve the transient heat and radiation transfer equations using an implicit scheme. The three assumptions of model (A) are not made here.

### 2.3. Model (C) – fully stochastic

This model uses a Monte Carlo ray tracing method to model the radiation heat transfer [9, 10] and a Brownian walker method to model the transient conduction heat transfer [7, 8, 9]. Again, the three assumptions of model (A) are not needed here. A previous paper [9] shows this model in the case of a one-dimensional grey semi-transparent slab enclosed between two opaque surfaces imposing their intensities and temperatures. We now extend this model to the simulation of a flash experiment.

#### 2.3.1. Reminder of the basics.
The radiative heat transfer is solved using a classical Monte Carlo ray-tracing algorithm. A large number of rays $N_r$ is emitted from each cell of the numerical discretisation, and followed until absorption. This, associated with the temperature field, allows the evaluation of the energy deposition in each cell, and thus of a radiative power field $P^R$ which can be interpreted as an injection of Brownian walkers as described in a previous publication [7]. A Brownian walker is an energy carrier of elementary enthalpy $\delta h$. Consequently, in a cell $i$, the number of Brownian walkers $W_{P,i}$ to be injected during a time step $\delta t$ is:

$$W_{P,i} = \frac{P_i^R \delta t}{\delta h} \qquad (1)$$

The elementary enthalpy is defined by discretizing an enthalpy $H_T$ representative of the thermal problem by a number $W_r$ (chosen arbitrarily) of Brownian walkers. For example, let us suppose that, from the thermal problem under consideration, a reference temperature $T_h$ and a reference temperature difference $\Delta T_c$ can be defined. Then $T_h$ will logically be retained as the temperature origin for enthalpy, whereas $H_T$ will be defined as the enthalpy of a cell whose temperature is $(T_h + \Delta T_c)$, i.e. $H_T = \rho C_p V \Delta T_c$ where $\rho$ and $C_p$ are the density and specific heat of the material and $V$ is the volume of the cell. $\delta h$ will then be:
$$ \delta h = \frac{H_T}{W_r} = \frac{\rho C_p V \Delta T_c}{W_r} \qquad (2) $$

The temperature $T_i$ of the cell $i$ can then be calculated by counting the number of Brownian walkers $W_i$ in that cell:

$$ \rho C_p V (T_i - T_h) = W_i \delta h \Rightarrow T_i = T_h + \Delta T_c \frac{W_i}{W_r} \qquad (3) $$

To solve the transient conductive heat transfer, the Brownian walkers move inside the medium according to an Ito-Taylor scheme, depending on the thermal diffusivity $a$ of the medium and the time step $\delta t$:

$$ \delta z = r_N \sqrt{2a\delta t} \qquad (4) $$

where $\delta z$ represents the algebraic displacement of the walker in the direction of the heat transfer and $r_N$ is a random number following the standard normal distribution.

### 2.3.2. Flash heat flux injection.
Since the opaque surfaces are considered black, all the energy of the flash is absorbed at the front face of the sample, i.e. at $z = 0$. If the flash is characterized by a flux $\Phi_{flash}$ impinging the front surface during a time $t_{flash}$, then the number of Brownian walkers to be injected is:

$$ W_{flash} = \frac{\Phi_{flash} t_{flash}}{\delta h} \qquad (5) $$

The Brownian walkers are injected randomly over the time interval $[0; t_{flash}]$, i.e. at instants $t_{injection} = r_U t_{flash}$ where $r_U$ are random numbers following the uniform distribution over $[0; 1]$. If the instant of injection $t_{injection}$ of a walker is located between two successive instants $t$ and $(t + \delta t)$ of the temporal discretization, then the walker undergoes a random walk during the remaining time $(t + \delta t - t_{injection})$.

### 2.3.3. Boundary conditions.
The two challenging boundary conditions concern the two black surfaces at $z = 0$ and $z = L$. This type of condition is a problem for our Brownian walker method. Indeed, contrary to the previous deterministic method, we do not define temperatures at points, but at the centers of cells (or voxels). This is because, in order to calculate a temperature value, we need to count the number of Brownian walkers inside a volume. Consequently, we cannot compute the temperatures at the black surfaces. This was not a problem in [9], where the temperatures were imposed on both surfaces. To overcome this, we make the following assumptions:

* To compute the temperatures at the surfaces, the first cell V0 (next to the surface S0 at $z = 0$) and the last cell VL (next to the surface SL at $z = L$) give their temperatures to S0 and SL. In other words, $T(S0) = T(V0)$ and $T(SL) = T(VL)$.

* To compute the radiative power field, and taking into account the black surfaces, the cells V0 and VL accumulate the powers emitted and absorbed by themselves and by S0 and SL. In other words, $P_{V0}^R = (\Phi_{V0}^a + \Phi_{S0}^a) - (\Phi_{V0}^e + \Phi_{S0}^e)$ and $P_{VL}^R = (\Phi_{VL}^a + \Phi_{SL}^a) - (\Phi_{VL}^e + \Phi_{SL}^e)$.

Finally, convection losses at boundaries are treated using the explicit “flux method” described in [8].

### 2.3.4. Implicit calculation.
In our previous work, the conduction-radiation coupling was treated in an explicit way, meaning that the radiative power field to be injected at a time $(t + \delta t)$ was computed using the temperature profile at the previous time step $t$. This is strictly wrong because of the instantaneous nature of the radiative transfer (compared with the conductive one). However, this works well as long
as the temperature profiles between two time steps are close enough, as in problems with imposed temperatures. In our new flash method model, this cannot work: the difference between the temperature profiles at the initial time $t = 0$ and at the next time $t = \delta t$ is too important, because of the sudden injection of enthalpy due to the flash. Thus, as with an implicit deterministic method, we have to proceed iteratively until convergence of the temperature field within a time step.

### 3. Results

After rewriting the different equations of the problem in dimensionless forms, several dimensionless numbers appear: (i) the optical thickness $E = \beta L$, where $L$ is the thickness of the wall (m) and $\beta$ is the extinction coefficient (m⁻¹), (ii) the Planck number $Pl = \lambda\beta/n^2\sigma_{SB}T_0^3$, where $\lambda$ is the thermal conductivity (W.m⁻¹.K⁻¹), $n$ is the refraction index, $\sigma_{SB}$ is the Stefan-Boltzmann constant (W.m⁻².K⁻⁴), and $T_0$ is the initial temperature of the slab (K), (iii) the Biot number $Bi = hL/\lambda$, where $h$ is the convective coefficient (W.m⁻².K⁻¹). The anisotropy of the radiation scattering is represented by a Henyey-Greenstein phase function of parameter $g$. Finally, the reference temperature $T_h$ is taken equal to the initial temperature $T_0$, and the reference temperature elevation $\Delta T_c$ is taken equal to $\varphi_{flash}t_{flash}/\rho C_p L$, i.e. the asymptotic uniform temperature elevation within the slab in the absence of heat losses at the boundaries. Figures 1 (a), (b), (c), and (d) show the dimensionless temperature rise $\theta^* = (T - T_0)/\Delta T_c$ at the rear side of the slab as a function of the dimensionless time $t^* = at/L^2$ as obtained with the three modelling approaches.

<table><caption>**Table 1.** Parameters of figures 1 (a), (b), (c) and (d).</caption><thead><tr><th></th><th>1(a)</th><th>1(b)</th><th>1(c)</th><th>1(d)</th><th>Concerned models</th><th>Description</th></tr></thead><tbody><tr><td>W<sub>r</sub></td><td>1e4</td><td>1e4</td><td>1e4</td><td>1e4</td><td>-</td><td>(C) Number of reference Brownian walkers</td></tr><tr><td>N<sub>r</sub></td><td>1e6</td><td>1e6</td><td>1e6</td><td>1e6</td><td>-</td><td>(C) Number of rays launched per cell/surface</td></tr><tr><td>N<sub>t</sub></td><td>100</td><td>100</td><td>100</td><td>100</td><td>(A)</td><td>(B) Number of time steps</td></tr><tr><td>N<sub>z</sub></td><td>100</td><td>100</td><td>100</td><td>100</td><td>-</td><td>(B) Number of space steps</td></tr><tr><td>a.t<sub>f</sub>/L<sup>2</sup></td><td>0.7</td><td>0.35</td><td>0.35</td><td>0.35</td><td>(A)</td><td>(B) Dimensionless final time</td></tr><tr><td>Bi</td><td>0.5</td><td>0.5</td><td>0.5</td><td>0.5</td><td>(A)</td><td>(B) Biot number</td></tr><tr><td>E</td><td>10</td><td>1</td><td>1</td><td>1</td><td>(A)</td><td>(B) Optical thickness</td></tr><tr><td>Pl</td><td>10</td><td>1</td><td>1</td><td>1</td><td>(A)</td><td>(B) Planck number</td></tr><tr><td>ω</td><td>0.5</td><td>0.5</td><td>0.5</td><td>0.5</td><td>(A)</td><td>(B) Scattering albedo</td></tr><tr><td>g</td><td>0.5</td><td>0.5</td><td>0.0</td><td>0 or 0.5</td><td>-</td><td>(B) Asymmetry factor of scattering<br>Ratio between the initial temperature and the</td></tr><tr><td>T<sub>0</sub>/ΔT<sub>c</sub></td><td>30</td><td>30</td><td>3000</td><td>3000</td><td>-</td><td>(B) reference temperature elevation</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>Figure 1(a)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 1(b)</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 1.** Figures (a) and (b): Comparison of various thermograms obtained from the three models (A), (B), and (C). Figure (c): Comparison of the thermograms obtained from model (A) and a degraded version of model (B) where the intensities are artificially made isotropic on both hemispheres. Figure (d): Comparison of the thermograms obtained from model (A) and model (B) used in three configurations (i), (ii), and (iii). The parameters are shown in table 1.</figcaption></figure>

In figures 1(a) and 1(b) our new models (B) and (C) give very similar results despite their radically different modelling approaches, which validates both of them. On the other hand, model (A) shows important deviations. In figure 1(a) the deviation is especially interesting because the three thermograms have shapes generally associated with pure conduction, and yet assumptions made on radiation in model (A) are already visible. In figure 1(b) the deviation is even stronger, particularly at the beginning of the thermogram. Here the radiative transfer contribution is more important than in the previous case, because the Planck number has been divided by 10. Moreover, the first instants of the thermograms are representative of the radiative transfer, quickly arriving from the front surface to the rear surface. Hence, the deviations of model (A) versus models (B) and (C) could be due to the fact that model (A) makes more limiting assumptions about the radiative heat transfer.

In figure 1(c), the parameters of model (A) are identical to those of figure 1(b), but model (B) has been degraded in order to align with the three limiting assumptions of model (A): (i) $g = 0$: isotropic scattering, (ii) $T_0/\Delta T_c = 3000 \gg 1$: the temperature elevations induced by the flash are very low compared to the initial temperature, thus linearization of differences of temperatures to the fourth power is justified, (iii) the intensities are artificially made isotropic on each of the two hemispheres. This last point justifies the name “degraded new deterministic”: model (B) has been degraded to impose “hemispherically isotropic” intensities. In fact, the intensities are first computed as usual, then mean values for each hemisphere are computed and imposed on each hemisphere. The results of model (C) (the fully stochastic model) do not appear in figure 1(c) because it is impossible in this model to impose “hemispherically isotropic” intensities. Results show an excellent agreement between model (A) and the degraded version of model (B), which confirms that the deviations observed previously are the consequences of the assumptions of model (A).

As for the individual importances of these three assumptions, simulations have been made in figure 1(d) with the same thermogram for model (A) and (i) the degraded model (B) (with “hemispherically isotropic” intensities) with a high value for $T_0/\Delta T_c$ (3000) and an anisotropic scattering ($g = 0.5$), (ii) the exact model (B) with the same high value for $T_0/\Delta T_c$ and an isotropic scattering ($g = 0$), (iii) the degraded model (B) with a low value for $T_0/\Delta T_c$ (30) and an isotropic scattering ($g = 0$). The superposition of effects (i), (ii) and (iii) (anisotropic scattering, anisotropic intensities and low value of $T_0/\Delta T_c$) results in the exact thermogram of model (B) in figure 1(b). We can therefore visualize in figure 1(d) the impact of each assumption.
## 4. Conclusion

This paper presents and compares three methods for modelling flash thermograms resulting from conduction-radiation coupling. The first one is an old deterministic model relying on three limiting assumptions about the radiative transfer, the second one is a new deterministic model that serves as a reference for validation, and the third one is a fully stochastic model, which represents a novelty and paves the way for modelling more complex 3D structures. The comparison between the new deterministic model and the fully stochastic one shows excellent agreement, validating these new methods. The old deterministic model shows significant deviations from the others in terms of radiative transfer, and simulations have been made to highlight its weaknesses. In future works, inversion methods will be applied to experimental thermograms with the new models to determine to which extent the choice of the model impacts the identified values of the radiative parameters. Finally, the fully stochastic flash method modelling could be extended to non-Beerian media [9] as well as multilayer samples based on recent progress regarding the Brownian walkers dynamics method.

## References

[1] Sans M, Schick V, Parent G and Farges O 2020 *Int. J. Heat Mass Transf.* **148** 119077

[2] Sans M, Farges O, Schick V and Parent G 2022 *Int. J. Therm. Sci.* **179** 107656

[3] Niezgoda M, Rochais D, Enguehard F, Echegut P and Rousseau B 2011 *Appl. Phys. Lett.* **99** 224101

[4] Ramousse J, Didierjean S, Lottin O and Maillet D 2008 *Int. J. Therm. Sci.* **47** 1–6

[5] Ortona A et al 2014 *Int. J. Appl. Ceram.* **11** 927–937

[6] Ferrari L et al 2016 *Compos. Struct.* **154** 61–68

[7] Gonneau V, Rochais D and Enguehard F 2022 *Int. J. Heat Mass Transf.* **184** 122261

[8] Seyer L, Gonneau V, Enguehard F and Rochais D 2023 *Int. J. Heat Mass Transf.* **211** 124248

[9] Seyer L, Enguehard F and Rochais D 2023 *Proceedings of the 10th International Symposium on Radiative Transfer, RAD-23* pp 195–202

[10] Dauvois Y 2016 *Modélisation du transfert thermique couplé conductif et radiatif au sein de milieux fibreux portés à haute température* (Université Paris-Saclay: France)