# Temperature field measurement of a burning aluminum droplet

Hugo Keck, Christian Chauveau, Guillaume Legros, Stany Gallier, Fabien Halter

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Hugo Keck, Christian Chauveau, Guillaume Legros, Stany Gallier, Fabien Halter. Temperature field measurement of a burning aluminum droplet. Combustion and Flame, 2025, 277, pp.114163. 10.1016/j.combustflame.2025.114163. hal-05044690

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Submitted on 17 May 2025

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# Temperature field measurement of a burning aluminum droplet

Hugo Keck<sup>a, ID</sup>,*, Christian Chauveau<sup>a ID</sup>, Guillaume Legros<sup>a,b</sup>, Stany Gallier<sup>c ID</sup>, Fabien Halter<sup>a,d ID</sup>

## ARTICLE INFO

**Keywords:**

Aluminum

Alumina nanoparticles

Temperature

Absorption

Emission

## ABSTRACT

In a diffusive combustion regime, an aluminum droplet undergoing combustion forms an oxide cloud that surrounds the burning droplet. Thorough characterization of this cloud is crucial to the validation of the subsequent modeling. This paper makes a significant contribution to the field by providing an experimental procedure to resolve the spatial temperature profile within the oxide cloud. An electrodynamic levitator is used to observe the self-sustained combustion of aluminum particles with a radius of 35 µm in atmospheric air, with negligible convective effects. The levitating device is coupled to an optical apparatus that allows for a light extinction method, thereby enabling the determination of size and concentration profiles of the nanometric alumina droplets, as introduced in previous works. The data from the previous study are employed in conjunction with a modulated absorption–emission (MAE) technique to ascertain a temperature profile that does not rely on the grey-body assumption. This technique is further enhanced by an optimization method to account for gaseous phase emissions, which typically hinder conventional temperature evaluation. Consequently, a spatially resolved temperature profile of the oxide cloud surrounding the burning droplet is obtained. Close to the surface of the droplet, a temperature of 2580 K is assessed. Then, a maximum temperature of about 3615 K is measured. As an additional outcome, gaseous emission profiles are obtained for three wavelengths and exhibit a notable correlation with a simulated gaseous suboxide concentration profile. The results presented in this work demonstrate a relatively high degree of consistency with expected temperatures.

**Novelty and Significance Statement**

This work presents a novel experimental method to obtain an unique temperature profile surrounding an isolated aluminum droplet in combustion. In conjunction with previous work, a non-intrusive, complete, and instantaneous characterization of the oxide smoke is now made possible, with the addition of the temperature profile to the known alumina particle size and concentration profiles. This comprehensive data set is presented for a fundamental case of a single levitating particle. The incorporation of the temperature profile provides an incomparable insight into alumina condensation processes and a detailed reference case for simulation purposes. The results presented in this work document the intricate condensation process of nanoparticles and highlight the limitations of current simulation methods.

## 1. Introduction

Solid fuel rockets are widely used in space exploration and military applications due to their simplicity and high specific impulse. Aluminum is often incorporated into these fuels as an energetic additive in particle form, as its high combustion temperature significantly boosts gas temperature and overall rocket performance [1,2]. Furthermore,

aluminum combustion is attracting attention as a carbon-free renewable energy carrier due to its availability and the potential to recycle combustion products (alumina) back into aluminum [3]. For both applications mentioned, it is crucial to fully characterize the combustion process of an aluminum particle. Indeed, while burning in atmospheric air, aluminum burns in a diffusive regime according to the Glassman criterion [4]. Consequently, as aluminum evaporates from the surface

<footnote><sup>a</sup> ICARE-CNRS, 1C Avenue de la Recherche Scientifique, Orléans, 45100, France</footnote>

<footnote><sup>b</sup> Inst. d'Alembert (Sorbonne U./CNRS), 4, place Jussieu, Paris, 75005, France</footnote>

<footnote><sup>c</sup> ArianeGroup, Le Bouchet Research Center, Vert-le-Petit, 91710, France</footnote>

<footnote><sup>d</sup> University of Orléans, Château de la Source, Orléans, 45100, France</footnote>

<footnote>* Correspondence to: Laboratoire ICARE-CNRS, 1C Avenue de la Recherche Scientifique, 45100, Orléans, France.
E-mail address: hugo.keck@cnrs-orleans.fr (H. Keck).</footnote>
of the burning droplet, it reacts with the ambient oxidizer to form
suboxides, which subsequently react to form the final state of oxida-
tion, alumina ($Al_2O_3$), in a flame detached from the droplet's surface.
The condensation of alumina results in nanometric alumina droplets
surrounding the burning aluminum droplet during combustion. Char-
acterizing this oxide smoke is essential, as the oxide formed must be
reduced if aluminum combustion is to be used as an energy carrier.
Furthermore, the presence of oxide leads to two-phase flow related
performance losses and thermo-acoustic instabilities when aluminum is
used in a solid propellant rocket motor [5,6]. Understanding the size,
concentration, and temperature profiles of alumina nanoparticles near
a burning aluminum droplet is crucial for developing accurate models.
Considering the size of the alumina particles, thermal equilibrium is
achieved between the gaseous environment and alumina which makes
the assessed temperature profile a correct tracer of the overall tempera-
ture within the cloud. Our previous research using a multispectral light
extinction method has determined the concentration and size profiles
of oxide smoke for a 35 micron-radius aluminum particle burning
in air [7]. This constitutes, to the best of our knowledge, the first
in situ measurement of such data. However, the temperature profile
remains unknown and is the primary focus of this study. While some
numerical studies have simulated the temperature of the cloud sur-
rounding the burning droplet, discrepancies exist in the literature, and
experimental data are scarce [8,9]. The only experimental temperature
profiles we are aware of are those by Bucher [10] and Nishii [11]
in the case of magnesium in water vapor. In this study, the results
obtained from the previously employed multispectral light extinction
method are combined with a two-dimensional modulated absorption-
emission technique (MAE), which is based on the methodology outlined
in Ref. [12]. This enhanced approach accounts for gaseous phase
emissions and aims to determine in situ profiles of the condensed phase
temperature within the oxide cloud for a 35-micron radius aluminum
droplet burning in atmospheric air. Section 2 provides a brief overview
of the experimental setup and associated diagnostics. In Section 3, the
methods employed in this study are presented, including the light ex-
tinction method, the assessment of local black body emissions, and the
temperature evaluation. The results are then presented and discussed
in Section 4, followed by the conclusion of the paper.

## 2. Experimental setup and diagnostics

This section provides a brief overview of the experimental setup
used in this study, as it has been thoroughly detailed in previous
work [7].

An electrodynamic levitator device is used to isolate a single alu-
minum particle of approximately 35 µm in radius at the center of
a combustion chamber. Once isolated, the particle is impacted from
both sides by a 50 W CO₂ laser until ignition of the particle occurs
and self-sustained combustion is achieved. The signal from one of
the photomultipliers is used to stop the laser heating once the parti-
cle's emission reaches a specified threshold, allowing the combustion
process to continue without additional energy input. A high-speed
color camera PHANTOM T-2410, equipped with a QUESTAR QM100
teleobjective and a triple band filter (450 nm, 530 nm, and 630 nm),
captures the combustion of the particle at a frequency up to 18 kHz and
with a resolution of about 1.3 µm per pixel. The camera's frequency is
adjusted to ensure proper sensor illumination and adequate temporal
resolution of the sequential images. A white LED SOLIS 3-C serves
as a pulsed backlight for the Light Extinction Method described later.
The LED is pulsed at half the camera's frequency, so two consecutive
images captured by the camera show one backlit droplet and one
burning droplet without backlight. The triple band filter allows the
RGB matrices of the color camera to capture light emission at the three
filter wavelengths. Calibration is necessary to determine the spectral
overlap of the color camera, with more details available in [13]. The
figure 8 from [7] illustrates the complete experimental setup used to

gather the data presented in this work. These devices and diagnostics
enable the assessment of the temperature profile of the condensed
phase within the oxide cloud surrounding a burning aluminum droplet
in atmospheric air, using the methodology described in the next section.

## 3. Methodology

This section presents the Modulated Absorption/Emission (MAE)
technique implemented to determine the spatially resolved temper-
ature profile of the condensed phase around the aluminum droplet
during combustion. The content here is adapted from Legros' work, and
readers are encouraged to consult Ref. [12] for a more detailed and
alternative approach. The system that will be considered throughout
the section, illustrated in Fig. 2, consists of a single isolated aluminum
droplet, burning in a diffusion regime, as well as its surrounding cloud,
composed of a gaseous phase and nanometric alumina particles. The
entire methodology to obtain the temperature profile presented in this
work as well as the alumina size and concentration profile from Ref. [7]
is summarized in Fig. 1

The MAE technique involves evaluating the local absorbance of the
species within the cloud surrounding the droplet for different wave-
lengths. Assuming that the gaseous phase within the cloud is optically
thin, the only absorbing phase is the nanometric alumina particles.
Since absorbance is directly related to the emissivity, measuring the
local absorbance of the cloud equates to assessing the local emissivity
of the alumina particles at various wavelengths. With the knowledge
of the local emissivity of the condensed phase, local emission can be
divided by local emissivity to calculate a local black body emission
within the oxide smoke for different wavelengths. A black body calibra-
tion for the sensor and the ratios of experimental black body emissions
enable the determination of the local temperature at all points in the
cloud surrounding the burning droplet. Throughout this work, each
experimental value is assessed for the three wavelengths of the study
$\lambda_1 = 450$ nm, $\lambda_2 = 530$ nm and $\lambda_3 = 630$ nm. Unless otherwise stated,
all processes are conducted using data averaged over 30 experimental
combustion times selected from 5 distinct cases of aluminum particle
combustion. Each of these five cases involves particles with an initial
diameter of $d = 70 \pm 4.5$ µm burning in atmospheric air. For all cases
presented, the combustion times range from 15 ms to 17 ms and the
images selected to determine the temperature profiles are chosen at
instants ranging from 15% to 32% of the total combustion time. At
these instants, the particles burn in a symmetric regime, allowing for
the correct application of the methods introduced in this section, unlike
later stages of combustion where spinning and jetting due to the oxide
cap are commonly observed. These five cases will be the references
cases of this work and will be abundantly referred to. For each of these
cases, 4 to 9 pairs of images were used and averaged to determine
the required data (emissivity, emission...). Combustion images of these
cases are available in supplementary material. Subsection 3.1 briefly
introduces the light extinction method used to determine the local
absorbance and emissivity of the cloud. Section 3.2 explains the method
to determine local black body emission within the cloud and Section 3.3
details the method to determine the local temperature profile within the
oxide cloud. The calibration process of the camera using a black body
is available in the supplementary material.

### 3.1. *Light extinction method*

This subsection provides a brief introduction to the Light Extinction
Method (LEM), with further details available in previous work [7].
Initially, the extinction profile within the oxide smoke is evaluated
using LEM. A white pulsed LED serves as a backlight to measure the
cloud's absorbance. This setup enables a high-speed color camera to
capture successive images of the burning particle both with and without
the backlight. By subtracting the image without the backlight from
the one with it, the contribution of the flame emission is eliminated,
<figure><img src="image_4.png" /><figcaption>Fig. 1. Flow chart highlighting the quantities of interest experimentally accessed through the presented methodology.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. Top: Image of an aluminum droplet of radius $R_0 = 35$ µm burning in atmospheric air captured with a color camera. Non-dimensional scale and deconvolution line are added for clarity and further purpose. The frame is captured 1.4 ms after ignition of the droplet. Bottom: Corresponding emission intensity profiles on the R, G and B matrices.</figcaption></figure>

resulting in an image showing the backlight absorbed by the cloud. The absorbance $κ(λ)$ at any point on the cloud is then calculated as the ratio between this resulting image and the image of the backlight alone (without the burning droplet), using Beer-Lambert's law:

$$κ(λ) = -\ln\left(\frac{I(λ)}{I_0(λ)}\right) \cdot \frac{1}{\ln(10)} \qquad (1)$$

In Beer-Lambert's law, $I_0(λ)$ is the initial intensity of the light ray and $I(λ)$ is the intensity of the light ray after absorption by the medium of absorbance $κ(λ)$. In the context of this work, this law can be applied for each pixel of the image: $I_0(λ)$ is the pixel intensity of the backlight image alone, and $I(λ)$ is the pixel intensity of the image obtained by subtracting the image of the burning droplet from an image of the backlit burning droplet. Here, $I_{px}^{1, on}$ is the intensity of the pixels from the image of the backlit burning droplet, $I_{px}^{1, off}$ is the intensity of the pixels from the image of the burning droplet alone, $I_{px}^{0, on}$ is the intensity of the pixels from the image of the LED alone and $I_{px}^{0, off}$ is the intensity of the pixels from the dark noise image. The integrated absorbance is then computed as follows:

$$κ(λ) = -\ln\left(\frac{I_{px}^{1, on} - I_{px}^{1, off}}{I_{px}^{0, on} - I_{px}^{0, off}}\right) \cdot \frac{1}{\ln(10)} \qquad (2)$$

In this work, $λ$ takes the value of one of the three wavelengths of interest introduced before. As a result of this process, the integrated absorbance is determined as a matrix, which has the same dimensions as a camera image (with $i$ lines and $j$ rows) and contains the integrated absorbance values for each element and is referred to as the integrated absorbance image $κ_{i,j}$. A deconvolution process is thus required to access the local values of absorbance as the previously determined absorbance is integrated along the optical path of the backlight. An onion peeling method is used to extract local absorbance information. As illustrated by Fig. 3, considering the spherical geometry of the system, it is assumed that the system consists of $k = 1, ..., N$ layers, with each layer having a thickness of one pixel. More details about the deconvolution process can be found in Refs. [12,14].

As represented on Fig. 3, each value of the integrated absorbance image contains the contribution of the layers crossed by the line of sight. As an example, from Fig. 3, following the line of sight, the integrated absorbance is expressed as $κ_{i,j} = δκ_{i,k=3}^{loc} + εκ_{i,k=2}^{loc} + φκ_{i,k=1}^{loc}$. The local absorbance values are denoted by the superscript loc and $δ, ε$ and $φ$ are coefficients resulting from the geometry of the system, taking into account the length of the line of sight within each layer $k$ of absorbance $κ_{i,k}^{loc}$. Local values of absorbance are calculated with the introduction
<figure><img src="image_6.png" /><figcaption>Fig. 3. Schematic of the onion peeling representation of the system. The light perceived by the sensor in the figure is emitted by a cross section of the burning aluminum droplet and its surrounding cloud at height i. The green line corresponds to the line of sight. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

of a geometrical deconvolution matrix A that takes into account the previously mentioned layer lengths:

$$A_{j,k} = \begin{cases} 0 & \text{if } j < k \\ \Delta x_{pix} \cdot [k^2 - (j - 0.5)^2]^{0.5} & \text{if } j = k \\ \Delta x_{pix} \cdot [(k^2 - (j - 0.5)^2)^{0.5} - [(k - 1)^2 - (j - 0.5)^2]^{0.5}] & \text{else} \end{cases} \quad (3)$$

With $\Delta x_{pix}$ the pixel length. For each line i of the integrated absorbance image, the equation $2 \cdot A \cdot X = b$ is solved with X the vector of local absorbance at height i as the unknown, defined as $X = (\kappa_{i,k}^{loc})_{k=1,...,N}$ and b the vector of integrated absorbance at height i along the rows j defined as $b = (\kappa_{i,j}^{int})_{j=1,...,N}$

To address the potential ill-conditioning of the problem, each line (i) of the integrated image is solved using Tikhonov regularization. This process results in an image depicting local absorbance, thereby providing the local absorbance at each point within the oxide smoke. An example of local absorbance profiles is presented in Fig. 4. Since absorbance is the sum of the contributions of absorption and scattering and since alumina size is known, the contribution of each can be assessed and the local emissivity within the cloud is thus evaluated, enabling the assessment of local black body emissions using the method described in the next subsection. Note that local absorbance at a location z and wavelength $\lambda$ can be related to alumina particle radius $r(z)$ and volume fraction $f_v(z)$ at this same location as $\kappa_{loc}(z, \lambda) \cdot \ln(10) = \sigma(r(z), \lambda) \frac{f_v(z)}{4/3 \pi r(z)}$ with $\sigma(r(z), \lambda)$ the extinction cross section of an alumina particle of radius $r$ at the wavelength $\lambda$. Previous work [7] introduces a method to evaluate alumina particle size and volume fraction profiles. These profiles are now considered known for each processed image based on the LEM method. The MAE method introduced in this study utilizes these known profiles to determine the temperature profiles.

## 3.2. Local black body emission

Local absorbance, denoted as $\kappa_{i,k}^{loc}$, is determined at each point within the cloud, as discussed in the previous subsection. Utilizing this local absorbance data, the integrated emission from the cloud can be deconvoluted, accounting for self-attenuation, to evaluate the local emission. Local emissivity $e_{i,k}^{loc}$ is obtained by computing the share of absorbance that is due to absorption and $e_{i,k}^{loc} = \kappa_{i,k}^{loc} \cdot \sigma_{abs}(r(z), \lambda) / \sigma(r(z), \lambda)$ with $\sigma_{abs}(r(z))$ the absorption cross section of an alumina particle of radius $r$. Dividing local emission by local emissivity $e_{i,k}^{loc}$ yields the local

<figure><img src="image_7.png" /><figcaption>Fig. 4. Local absorbance profiles as a function of the distance from the surface of the burning aluminum droplet. The profiles are experimentally determined with the Light Extinction Method for the three wavelengths of interest.</figcaption></figure>

black body emission required for temperature assessment through a black body calibration. It is important to note that in the previous subsection, subtracting an image of the droplet without backlight from a backlit image allowed us to disregard the flame emission contribution, focusing solely on the condensed phase absorption, as the gas phase is considered optically thin. In this subsection, the assumption is made that light emission from the cloud originates exclusively from the condensed phase, with the gaseous phase considered non-emitting. This assumption is further discussed in this work.

In this subsection, an image of the burning droplet without backlight called flame image is deconvoluted. Following a similar principle as before, a deconvolution matrix C is introduced. Details about the matrix elements are not introduced in this work but can be found in the work of Legros [12]. However, unlike the previous subsection where matrix A allowed for the deconvolution of any line i of the image, in this subsection, the deconvolution matrix is a function of local absorbance and is thus unique for each height i as local absorbance changes along i. The deconvolution matrix C is computed as follows:

$$C_{j,k} = \begin{cases} 0 & \text{if } k < j \\ A_{j,k} \cdot e^{-\sum_{m=k+1}^{N} A_{j,m} \kappa_{i,m}^{loc}} (1 + e^{-A_{j,k} \kappa_{i,k}^{loc}}) & \text{if } j = k < N \\ A_{j,k} \cdot e^{-\sum_{m=k+1}^{N} A_{j,m} \kappa_{i,m}^{loc}} (1 + e^{-2 \sum_{m=j}^{k-1} A_{j,m} \kappa_{i,m}^{loc} - A_{j,k} \kappa_{i,k}^{loc}}) & \text{if } j < k < N \\ A_{j,k} \cdot (1 + e^{-2 \sum_{m=j}^{k-1} A_{j,m} \kappa_{i,m}^{loc} - A_{j,k} \kappa_{i,k}^{loc}}) & \text{if } j < k = N \\ A_{j,k} \cdot (1 + e^{-A_{j,k} \kappa_{i,k}^{loc}}) & \text{if } j = k = N \end{cases} \quad (4)$$

For each line i of a flame image, we introduce $C_o(\lambda)$ as a spectral optical constant. The equation

$$C \cdot C_o(\lambda) X = d \quad (5)$$

can be solved at the three wavelengths of interest. The vector of integrated emissions, denoted as d, represents the pixel intensities from the flame image at height i. This is written as $d = (I_{px,i,j}^{1,off})_{j=1,...,N}$. The unknown vector of local emission is expressed as $X = (e_{i,k}^{loc} \cdot B_{i,k})_{k=1,...,N}$ where $B_{i,k}$ is the local black body emission. Light emitted from the cloud reaches the camera sensor with a given solid angle, passing through a triple band filter with non-constant spectral response and impinging on pixels with a given spectral response. All these geometrical and spectral optical parameters are taken into account through the deconvolution process using the optical constant $C_o(\lambda)$. This equation is solved using a Tikhonov regularization to improve the stability of the resolution. Upon this resolution, the vector $C_o(\lambda) \cdot X$ is assessed. The previously mentioned equation can be solved for each line of the flame image, but this requires computing a different C matrix for each line, which is time-consuming. To reduce computation time, it is sufficient to deconvolute a line that includes all the different positions within the oxide cloud. The line that lies just above the particle, as illustrated by Fig. 2, is selected. This line contains a range of distances from
the surface of the droplet that is complete and that allows for the reconstruction of a spatially resolved profile. This line is also selected because it reaches the center of the image without passing through the aluminum droplet, which simplifies the deconvolution process. To avoid the deconvolution of a noisy signal, the pixel intensities along the green line of Fig. 2 are averaged. Specifically, for each point on the green line at a distance L from the droplet's center, the average value of all pixels on the circle centered on the particle with radius L is computed, and this average replaces the pixel value on the green profile.

This reconstructed profile serves as the vector d from Eq. (5). The deconvolution matrix for this line is then computed using Eq. (4), with $A_{j,k}$ being the geometrical deconvolution matrix introduced in Eq. (3), and $(\kappa_{i,k}^{loc})_{k=1,...,N}$ representing the vector of local absorbance corresponding to the green line in Fig. 2.

Using the vectors d and the deconvolution matrices C for the three wavelengths in the study, we evaluate three vectors $C_o(\lambda) \cdot X$ by solving Eq. (5). Recalling that local emissivity profiles $\epsilon^{loc}$ are obtained from the previous subsection, the product of the optical constant with the local emission profile $B_{i,k}$ can be obtained:

$$C_o(\lambda) \cdot B_{i,k}(\lambda) = C_o(\lambda) \frac{X(\lambda)}{\epsilon^{loc}_{i,k}(\lambda)} \quad (6)$$

If the optical constants were consistently equal to 1, the temperature could be directly derived using Planck's law. However, since these constants are unknown, a black body calibration for the setup, detailed in the supplementary material, is required. For the sake of clarity, we now write $C_o(\lambda) \cdot B(\lambda) = B_{cam}(\lambda)$ and to further simplify, we now only use the variables R, G and B respectively corresponding to $\lambda_3 = 630$ nm, $\lambda_2 = 530$ nm, $\lambda_1 = 450$ nm defined such as:

$$\left\{ \begin{array}{l} B_{cam}(\lambda_3) = R \\ B_{cam}(\lambda_2) = G \\ B_{cam}(\lambda_1) = B \end{array} \right. \quad (7)$$

The black body calibration in supplementary material establishes a relationship between the three local black body emission ratios ($R/G$, $R/B$ and $G/B$) and the local temperature of the observed body. The next subsection introduces the method to assess the local temperature of alumina at all points within the oxide cloud using the black body calibration.

### 3.3. Temperature determination

#### 3.3.1. Impact of gaseous emissions

In the literature on multicolor pyrometry methods, it is often noted that the gaseous emissions are either neglected when determining the condensed phase temperature or that filter wavelengths are chosen to avoid these emissions [15,16]. This subsection presents quantitative data demonstrating that gaseous emissions cannot be ignored when evaluating a temperature profile within the oxide cloud surrounding a burning aluminum droplet using a RGB camera. Figures S3 and S4 in the supplementary material show that camera calibration with a 3200 K black body resulted in a minimum $R/G$ greater than 4 and a minimum $R/B$ ratio greater than 2.5 with the additional observation that these minimum ratios tend to decrease less and less with an increase in the temperature. In comparison, Figs. 5, 6 display local black body emission values within the oxide cloud for the five experimental cases, showing minimal discrepancies between experiments. The measured $R/G$ ratios present a minimum of about 1 to 2 at about 2 droplet radii and the measured $R/B$ ratios present a minimum of about 0.6 to 1 at the same location. According to the calibration laws, such black body

<figure><img src="image_8.png" /><figcaption>Normalized distance from the surface of the burning droplet $R/R_0$ (-)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Normalized distance from the surface of the burning droplet $R/R_0$ (-)</figcaption></figure>

<caption>Fig. 6. Profiles of local ratios of red black body emissions by blue black body emissions ($R/B$) for the five experimental cases of interest. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</caption>

emission ratios would correspond to temperatures well above 8000 K while the adiabatic flame temperature in atmospheric air is expected to be around 3600 K [17]. Fig. 2 shows a bluish halo between 1 and 2.5 $R/R_0$, indicating that blue emission prevails in this region, which aligns with the measured ratios from Figs. 5 and 6. This bluish halo is primarily due to gaseous emissions in the blue-green region of the visible spectrum, lowering the $R/G$ and $R/B$ ratios below the expected values.

The works from Dabos [18] et al. and Lomba [19] et al. identify potential gaseous emissions within the relevant wavelength range, which align with our observations. Specifically, AIO emission lines span from 450 nm to 550 nm, while other aluminum suboxides exhibit emission peaks outside the filter wavelengths. As an example, $Al_2O_3$ presents a peak at 429 nm. It is thus deduced that gaseous emissions modify the black body emissions (and thus the black body ratios), preventing direct inference of black body temperature from the measurements. Our method allows to measure the temperature of the condensed phase. Due to the very small size of alumina particles, thermal equilibrium is assumed between the gaseous environment and alumina, making the alumina temperature an indicator of the gas phase temperature. However, literature reports low energy accommodation coefficients [20, 21] and the measured oxide temperature could be slightly different from gas temperature; a local temperature distribution might exist but cannot be tracked down with the current experimental approach. The difference between gas temperature and oxide temperature is therefore supposed to be limited. Further theoretical and numerical work is required in the future to evaluate more precisely the possible temperature difference between both phases. The following subsection introduces a numerical method to separate gaseous emissions from black body emissions, enabling accurate temperature evaluation.
### 3.3.2. Numerical method to account for gaseous emissions

As mentioned in the previous subsection, the presence of these gaseous emissions hinders the direct assessment of the condensed phase temperature. This is because the local black body emissions determined in Section 3.2. are in reality a sum of black body emissions and gaseous phase emissions. In this subsection, a numerical method is introduced to subtract the gaseous emissions from the emission signal. Local emissions within the cloud are denoted R, G and B. These emissions are equal to the previously called 'local black body emissions' assessed in Section 3.2. The 'black body' part is now discarded as gaseous emissions are considered. Let $R_{bb}$ and $R_g$ represent the local emission from black body radiation and from gaseous emission at $\lambda_3 = 630$ nm, respectively. Similarly, $G_{bb}$ and $G_g$ denote the local emission from black body radiation and from gaseous emission at $\lambda_2 = 530$ nm, and $B_{bb}$ and $B_g$ represent the local emission from black body radiation and from gaseous emission at $\lambda_1 = 450$ nm. We can write:

$$ \begin{cases} R = R_{bb} + R_g \\ G = G_{bb} + G_g \\ B = B_{bb} + B_g \end{cases} \qquad (8) $$

Three parameters $\alpha, \beta$ and $\gamma$ are defined such that:

$$ \begin{cases} R_{bb} = \alpha R \\ G_{bb} = \beta G \\ B_{bb} = \gamma B \end{cases} \qquad (9) $$

These parameters, ranging from 0 to 1, represent the proportion of the total local emission attributed to black body radiation. When applying the calibration laws at any point in the cloud using the ratios R/G, R/B and G/B, three incorrect temperatures are obtained. However upon using the three calibration laws with the so far unknown ratios $\frac{R_{bb}}{G_{bb}}$, $\frac{R_{bb}}{B_{bb}}$ and $\frac{G_{bb}}{B_{bb}}$, the three computed temperatures are expected to be equal and correct, as all the ratios follow a black body emission law. The numerical method introduced in this work relies on assessing the triplet $(\alpha, \beta, \gamma)$ along the entire profile, based on an optimization procedure relying on physical constraints.

Optimizing $\alpha, \beta$ and $\gamma$ with the constraint that the three temperatures resulting from the three black body radiation ratios (obtained as part of the total radiation) must be equal is insufficient, as there is no unique triplet $(\alpha, \beta, \gamma)$ that yields three identical temperatures. Therefore, an additional constraint is necessary to ensure the uniqueness of the temperature at each point of the profile.

To constraint the problem, the three local emission profiles R, G and B are utilized. Using the system of Eq. (9) as well as Planck's law with the Wien's approximation, we can write for the R channel:

$$ R \propto \frac{2hc^2}{\alpha} e^{\left(-\frac{hc}{\lambda_3 k_b T}\right)} \cdot S_{loc} \qquad (10) $$

With $h$ representing Planck's constant, $k_b$ Boltzmann's constant, $c$ the celerity of light and $S_{loc}$ the local alumina surface, the right-hand side of this equation reconstructs the black body emission profile at all points in the cloud. The $\alpha$ parameter accounts for the gaseous emissions. Based on previous work [7], the volume fraction of alumina as well as alumina nanoparticle size have been determined at all point in the cloud. Denoting $f_v$ as the local volume fraction of alumina and $r_{nano}$ as the local radius of alumina nanoparticles, we can write

$$ S_{loc} = \frac{f_v \cdot V_{loc}}{4/3 \cdot \pi r_{nano}^3} \pi r_{nano}^2 \qquad (11) $$

Since the local volume $V_{loc}$ corresponding to the volume of the local voxels is the same at all point in the cloud, we can simplify this equation by writing that at all points in the cloud

$$ S_{loc} \propto \frac{f_v}{r_{nano}} \qquad (12) $$

And thus Eq. (10) can be rewritten

$$ R \propto \frac{2hc^2}{\alpha} e^{\left(-\frac{hc}{\lambda_3 k_b T}\right)} \cdot \frac{f_v}{r_{nano}} = R_{rec} \qquad (13) $$

Let $R_{rec}$ represent the reconstructed R channel. From Eq. (13), we can establish a constraint for the temperature profile to ensure that the reconstructed R channel ($R_{rec}$) mirrors the evolution of the measured R channel.

Let T represent the function resulting from the black body calibration, where $T(A/B, n)$ yields the black body temperature from the ratio of channel A by the channel B at the point n in the cloud. The triplet $(\alpha, \beta, \gamma)$ is thus optimized along the following three constraints:

1. Black Body Constraint: At any point n within the cloud, the black body temperature should be the same independently of the ratio considered: $T(\frac{\alpha R}{\beta G}, n) = T(\frac{\alpha R}{\gamma B}, n) = T(\frac{\beta G}{\gamma B}, n) = T(n)$

2. Camera Fitting Constraint: The ratios of the measured profiles to the reconstructed profiles remain constant: $R/R_{rec} = constant$; $G/G_{rec} = constant$ and $B/B_{rec} = constant$.

3. Continuity Constraint: To prevent temperature jumps during optimization, the temperature at point n should be close to the temperature at point n + 1: $T(n) \approx T(n+1)$. The associated error is calculated as the square of the difference between the temperatures at points n and n + 1.

The success of the optimization procedure is evaluated by the consistency of the $(X_{rec}/X)_{X=R,G,B}$ ratios and the absence of significant differences between the three temperature profiles $T(\frac{\alpha R}{\beta G}, T(\frac{\alpha R}{\gamma B}))$ and $T(\frac{\beta G}{\gamma B})$ at each point of the profile. The respect of the third constraint is straightforward and observed by the absence of discontinuities in the temperature profile. Thus, errors regarding the continuity constraint will not be assessed. The results of this optimization procedure are introduced in the following section.

## 4. Results and discussion

In this section, we discuss the efficiency of the optimization process and introduce the experimentally assessed temperature profile of nanometric alumina droplets, followed by a discussion about these results.

### 4.1. Optimization efficiency

For each case of the reference cases a mean temperature profile is assessed. Averaging these five temperature profiles yields the resulting temperature profile shown in Fig. 8 introduced in the next subsection.

Three optimization constraints have been introduced in the previous section. Since a quantitative evaluation of the continuity constraint cannot be performed, the efficiency of the optimization procedure is evaluated with respect to the two remaining constraints.

For each point of the five profiles, an absolute error is defined to evaluate adherence to the black body constraint. Denoting $T_{RG}$, $T_{RB}$ and $T_{GB}$ as the temperatures obtained from the respective channel ratios R/G, R/B and G/B and $T_m$ as the average of these three temperature, the black body constraint error $e_{bb}$ is computed as follows:

$$ e_{bb} = \left( \frac{1}{3} \cdot \left[ (T_m - T_{RG})^2 + (T_m - T_{RB})^2 + (T_m - T_{GB})^2 \right] \right)^{0.5} \qquad (14) $$

To assess adherence to the camera fitting constraint, the emission profiles from the camera and the reconstructed profiles are both normalized for comparison. Denoting $R^{norm}$, $G^{norm}$ and $B^{norm}$ as the normalized measured profiles and $R^{norm}_{rec}$, $G^{norm}_{rec}$ and $B^{norm}_{rec}$ as the normalized reconstructed profiles, the camera fitting relative error $e_{cf}$ is computed at each point as follows:

$$ e_{cf} = \frac{100}{3} \cdot \left( \frac{|R^{norm} - R^{norm}_{rec}|}{R^{norm}} + \frac{|G^{norm} - G^{norm}_{rec}|}{G^{norm}} + \frac{|B^{norm} - B^{norm}_{rec}|}{B^{norm}} \right) \qquad (15) $$

Table 1 presents values of the black body constraint error and the camera fitting error for the five experiments of this study. The
<table><caption>**Table 1**
Summary of the optimization constraint errors for separate cases and averaged errors.</caption><thead><tr><th>Experimental case</th><th>e<sub>bb</sub> (K)</th><th>e<sub>cf</sub> (%)</th><th>σ<sub>bb</sub> (K)</th><th>σ<sub>cf</sub> (%)</th><th>e<sup>mean</sup><sub>bb</sub> (K)</th><th>e<sup>mean</sup><sub>cf</sub> (%)</th></tr></thead><tbody><tr><td>1</td><td>20.2</td><td>6.9</td><td>21.4</td><td>6.4</td><td></td><td></td></tr><tr><td>2</td><td>10.6</td><td>4.6</td><td>11.9</td><td>1.9</td><td></td><td></td></tr><tr><td>3</td><td>18.7</td><td>3.5</td><td>15.6</td><td>2.3</td><td>14.8</td><td>4.8</td></tr><tr><td>4</td><td>12.0</td><td>4.6</td><td>12.0</td><td>4.3</td><td></td><td></td></tr><tr><td>5</td><td>12.5</td><td>4.4</td><td>15.8</td><td>4.5</td><td></td><td></td></tr></tbody></table>

presented error is averaged over the entire profile of each case, and the standard deviation for each error is also provided. The last column presents the errors averaged over the five cases. As an example, the three temperature profiles (from R/G, R/B and G/B) for the experimental case 1 present in average an error of 20 K to the mean of the three profiles. The reconstructed normalized emission profile of the experimental case 1 presents a 6.9% relative error to the normalized measured emission profile. The errors presented most likely result from measurement noise as convergence towards a solution is observed upon averaging the results.

Table 1 shows that the absolute black body constraint errors and their standard deviations are quite acceptable given the temperatures involved. Similarly, the camera fitting constraint relative errors are also acceptable, with a maximum error of less than 7% and low standard deviations. Considering the low optimization errors for the main constraints and the close alignment between the mean assessed temperature and its associated 95% confidence interval from Fig. 8, we conclude that the optimization process is successful, allowing for a detailed presentation of the temperature profile.

## 4.2. Temperature profile

In this subsection, the temperature profile of nanometric alumina droplets surrounding a 35 µm radius aluminum droplet burning in atmospheric air introduced in Fig. 8 is discussed. The 95% confidence interval associated to the temperature profile from Fig. 8, represented by the shaded area, is calculated using the standard deviation of each case relative to the averaged profile and indicates relatively limited variation between the different experiments. The increase of the shaded area, and thus uncertainty, as the distance from the surface of the droplet approaches 1 droplet radius is attributed to the deconvolution noise.

At the surface of the droplet, a mean temperature of 2580 K is measured. Although the droplet burns in a diffusive regime and aluminum evaporates from its surface, the surface temperature is often found in the literature to be lower than the evaporation temperature of aluminum due to the aluminum vapor pressure differing from ambient pressure. Values varying from 2570 K in Ref. [22] to 2650 K in Ref. [23] can be found in the literature. The mean temperature assessed at the droplet surface shows a very satisfying correlation to theory and simulations. A temperature lower than the boiling point of aluminum is expected as the partial pressure of aluminum at the surface is not the atmospheric pressure [7]. In such circumstances, other species can diffuse to the surface of the particles and heterogeneous surface reactions are allowed to happen in this region. Note that for low normalized radii, deconvolution noise is significant, reducing data accuracy in this region. However, it is believed that the averaging process tends to smooth out this deconvolution noise.

Starting from the droplet's surface, the temperature increases to reach a global maximum at approximately $3.3R/R_0$. This peak temperature location appears to contradict the black body emission ratios shown in Figs. 5 and 6, where the minimum $R/G$ ratio at around $1.9R/R_0$ should correspond to a maximum in black body temperature. This discrepancy is explained by the presence of an emitting gaseous phase, which is accurately accounted for by the numerical method. Fig. 7 presents normalized and averaged local R, G and B emission

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Profiles of averaged normalized local R, G and B emissions.</figcaption></figure>

profiles from five experimental cases. As observed from the previously introduced ratios in Fig. 5, 6, blue and green emissions are relatively high around $1.9R/R_0$ matching with the blue halo from Fig. 2. However, a peak of R, G and B emission is also observed around $3.3R/R_0$. These local profiles indicate that a peak in emission following Planck's law (implying it is not from gaseous emission) results from either an increase in emitting surface or temperature. Previous work [7] has shown that particle size remains almost constant within the oxide cloud, and volume fraction decreases as normalized radii increase, leading to a decreasing emitting surface. Therefore, the local emission peak observed for the three channels at about $3.3R/R_0$ is attributed to a local temperature increase, consistent with the temperature profile assessed in this work. As widely mentioned in the literature [24], the condensation process of alumina is highly exothermic and is often considered as the main source of heat release, leading to a significant temperature increase. From the temperature profile shown in Fig. 8, and in correlation with alumina particle size from previous work [7], displayed with the alumina volume fraction in Fig. 9, it is assumed that the main condensation region of alumina for a 35 µm radius aluminum droplet burning in atmospheric air is located around 3.3 normalized radii from the droplet's surface.

For normalized radii increasing from 3.3 to 5, the temperature decreases monotonously. Note that temperature is not assessed for normalized radii greater than 5, as cloud emission becomes too low, as observed in Figs. 2 and 7, mainly due to a decrease in alumina concentration coupled to a decrease in alumina temperature.

The temperature profile obtained can be compared with those found in the literature. Fig. 8 presents experimental data from Bucher [10] for a 210-micron diameter droplet in atmospheric air, simulation data from Beckstead calculated for a 230 µm diameter droplet in atmospheric air, and current simulation data calculated for a 70 µm diameter droplet in air [22].

Fig. 8 displays various temperature profile shapes that need to be discussed. Due to the lack of robust experimental data, most simulation procedures have set Bucher's work [10] as their main target regarding the temperature profiles. As observed from Fig. 8, simulations from Beckstead [8] for a particle size similar to Bucher's experimental particle size, show difficulties to predict the experiments. As it is expected, simulations and experiments for a droplet of a 230 µm are not directly
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Comparison of the temperature profile obtained in this work to literature data. Beckstead data are issued from Ref. [8] and Bucher data are issued from Ref. [10]. Current simulation details can be found in Ref. [22]. Each profile corresponds to an aluminum droplet burning in atmospheric air. The droplet diameter is specified for each case.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Profiles of alumina volume fraction and alumina particle radius from [7].</figcaption></figure>

comparable to data obtained for a 70 µm diameter droplet as the combustion regimes, stand off flame ratios and global scales might impact the overall phenomenology of the combustion. Furthermore, Bucher's experimental temperature profile is obtained via a PLIF method with a camera resolution about ten times lower than the resolution used in this work (1.3 µm/pixel). The temperature profile obtained with a PLIF method depends on the concentration of AlO. As it can be observed in Figure 11, AlO concentration is important at a given region but decreases rapidly upon consumption to form alumina. The absence of AlO in certain regions of the oxide cloud is thus a imitation to the assessment of a temperature profile using a PLIF method. The differences in resolution (leading to different accuracies when applying Abel transforms to the measure profiles) and experimental method might also lead to discrepancies in the experimental results. Note however that experimental data from Bucher display correlations to the experimental data from this work that seem to not be captured by simulations such as relatively low temperature gradient close to the droplet or the relative location of the temperature peak. Additionally, current simulations [22] executed for the same conditions in size and environment as current experiments show significant discrepancies. Temperature peak locations and temperature gradients are poorly predicted. Although gaseous diffusion, kinetics or heat production have been widely studied and are now simulated with correct accuracy, we believe that the condensation processes responsible for significant amount of energy production are not correctly accounted for due to their high complexity. The newly introduced temperature profile from Fig. 8 provides an incomparable target for simulation and suggests that improvements have to be made regarding our understanding of alumina condensation. The hypothesis of a local monodisperse alumina particle size has also been challenged. A parametric study with considerations

<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Top: Log-normal probability density functions of alumina particle radius centered at 75 nm for standard deviations of 0.1 (red), 0.2 (blue) and 0.4 (green). Bottom: Associated temperature profiles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

of log-normal particle size distributions at each location of the cloud has been carried out. The top figure of Fig. 10 illustrates the chosen probability functions and the bottom figure displays the resulting temperature profiles. The implementation of a PSD shifts the temperature profile up by about 200 K but no changes are observed for the structure of the profile. This simple test shows the robustness of the method and shows that considering a local PSD at each point of the cloud would be the object of future work.

As an additional outcome of the optimization procedure, gaseous emission profiles are assessed allowing for a qualitative study of the latter. The profiles of (α, β, γ) result from the optimization procedure, and light emission profiles attributed to gaseous phases can be retrieved using equation systems 8 and 9. Fig. 11 displays the resulting normalized gaseous emission profiles. Significant gaseous emissions are observed for the G and B channels, corresponding to 450 nm and 530 nm respectively, within 1 to 2.7 droplet radii, while no emission is detected for the R channel at 630 nm. According to Ref. [19], the gaseous emissions in this location can be attributed to AlO emission lines within the bandwidth of the B and G bands of the triple band filter. A relatively good agreement is found between the blue and green gaseous emissions and the simulated AlO concentration profile, supporting, furthermore, the hypothesis of AlO emission. As mentioned earlier, such agreement is expected from simulation despite the erroneous predictions in terms of temperature profiles as gaseous flame dynamics are well known and accurately modeled by current simulation methods. It is thus believed that concentration of gaseous species are correctly predicted from simulations. Current work distinguishes from the more classical approach relying on pyrometers [25] by its ability to locally assess quantity of interest and separate the contributions of the different species. An integrated (with no possibility to apply an Abel procedure to the data) pyrometric method is limited due to the inevitable capture of aluminum radiation, alumina radiation and gaseous phase emissions as a single signal.
<figure><img src="image_14.png" /><figcaption>Fig. 11. Profiles of averaged normalized local R, G and B gaseous emissions. A simulated and normalized AlO concentration profile surrounding a 35 µm radius aluminum droplet burning in a (80%N₂/20%O₂) (vol.) mixture has been added. Details about current simulation methods are available in Ref. [22].</figcaption></figure>

Beyond 2.7 droplet radii, a significant increase in gaseous emission is observed for all the three wavelengths of interest. In this region, the detected emission follows a continuum emission and cannot be directly attributed to any known gaseous phase. However, as previously mentioned, this region matches with the high temperature region as well as the alumina condensation region. It is noteworthy that recent studies [26,27] have correlated the alumina condensation phenomenon to a process of electronic emission. From Fig. 11, the remaining continuous spectrum between 2.7 and 4 droplet radii can be isolated and compared to a normalized Bremsstrahlung emission spectrum. According to Ref. [28], the spectral intensity of the Bremsstrahlung spectrum at a wavelength λ is related to the electronic temperature as:

$$I_B(\lambda) \propto e^{\frac{-hc}{\lambda k_b T_e}} \cdot g_f \quad (16)$$

with h the Planck's constant, k_b Boltzmann's constant, c the celerity of light, T_e the electronic temperature and g_f the gaunt factor, it is noted from the same reference that the gaunt factor is proportional to the natural logarithm of the wavelength of the radiation. Therefore, we can write $I_B(\lambda) \propto e^{\frac{-hc}{\lambda k_b T_e}} \cdot \ln(\lambda)$. Using an electronic temperature of approximately 21 500 K, a reasonable correlation is observed between the spectrum measured in the condensation zone and the Bremsstrahlung spectrum as shown in Fig. 12. A linear extrapolation is applied to the measured spectrum between the three wavelengths of the triple band filter. Although Bremsstrahlung light emission is not commonly reported in the combustion field, other groups have observed electronic emissions in terms of electronic current [29]. While more common in plasma studies, Bremsstrahlung emission spectra measurements have been performed [30]. It is supposed that alumina condensation results in significant electron emission, with an electronic temperature of about 21 500 K, leading to light emission through the free-free radiation process, following a Bremsstrahlung spectrum. Note that in the literature, a work function for alumina of 2.32 eV is reported corresponding to about 26 900 K [31] leading to a difference of about 0.5 eV with the value used for the current estimate. This discrepancy could be attributed to the combustion process, as opposed to the absence of reactions in the referenced paper, or to the lack of precision of the current experimental approach regarding electronic temperature. We would like to emphasize that this is purely hypothetical and is one possible explanation we propose to account for the continuum spectrum observed in the condensation zone. Bremsstrahlung emissions may prevent light measurement method from accurately assessing the real temperature of the observed bodies [32], so subtracting this contribution from condensed phase radiations is critical. Currently, it is estimated that in the region where Bremsstrahlung is detected, its flux amounts to just over 30% of the oxide radiation. Furthermore, this

<figure><img src="image_15.png" /><figcaption>Fig. 12. Comparison of the normalized measured gaseous spectrum with experimental points to a normalized Bremstrahlung spectrum computed with an electronic temperature of 21 500 K.</figcaption></figure>

spectrum is observed in a specific region of the cloud and detected as a share of the condensed phase emissions, thus electronic emissions do not appear to be the main light emitting sources within the oxide cloud. The question of thermal equilibrium arises when electrons of such temperatures are present within the cloud. A short calculation is available in the supplementary material that highlights the weak energy transfer from electrons to the gaseous species of the cloud. Additionally, electrons might be impacted by the electrodynamic field that allows the aluminum droplet to levitate. Given the density of the medium, electrons are expected to recombine on an extremely short time scale, limiting the impact of the electric field on their behavior.

## 5. Conclusion

The research presented in this article builds upon previous work to introduce a Modulated Absorption/Emission technique combined with a numerical method to account for the gaseous emissions. This approach aims to determine the spatial temperature profile of the oxide cloud surrounding a 35 µm radius aluminum droplet burning in atmospheric air. Previously determined optical depth of the cloud enables the quantification of its local emissivity. Through a deconvolution process, the local black body emissions and temperature of the cloud are assessed using a black body calibration of the camera. The surface temperature of the droplet is assessed at about 2580 K while a maximum temperature of 3615 K is measured in the cloud at approximately 3.3 droplet radii. Upon comparison with simulations from the literature, the same temperature range is observed, although the shape of the simulated profiles shows significant discrepancies to this work. As a supplementary result, gaseous emission profiles are deduced for the three wavelengths of interest in this study. Close to the droplet, the blue and green gaseous emission profiles are similar to simulated AlO concentration profiles. However, further from the droplet, a continuous spectrum is observed, attributed to a Bremsstrahlung emission process facilitated by the presence of electrons emitted during the condensation of alumina nanometric droplets. This work, in conjunction with previous research cited in Ref. [7], provides a comprehensive methodology to thoroughly characterize the oxide cloud surrounding a burning aluminum droplet, yielding alumina size, concentration and temperature profiles. Future work will apply this methodology to aluminum burning in pressurized environments and varying mixtures.

## CRediT authorship contribution statement

**Hugo Keck:** Writing – original draft, Investigation, Conceptualization. **Christian Chauveau:** Writing – review & editing, Validation, Supervision, Investigation, Conceptualization. **Guillaume Legros:** Writing – review & editing, Supervision, Methodology, Investigation,
**Conceptualization.** **Stany Gallier:** Writing – review & editing, Supervision, Investigation, Conceptualization. **Fabien Halter:** Writing – review & editing, Supervision, Investigation, Conceptualization.

**Declaration of competing interest**

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

**Acknowledgments**

The authors express their gratitude to the French Defense Procurement Agency DGA (Direction Générale de l'Armement) for their financial support. The authors extend their gratitude to Viviana LAGO from ICARE, Orléans, for her invaluable insights into the Bremsstrahlung emission process.

**Appendix A. Supplementary data**

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.combustflame.2025.114163.

**References**

[1] V.N. Emelyanov, I.V. Teterina, K.N. Volkov, Dynamics and combustion of single aluminium agglomerate in solid propellant environment, Acta Astronaut. 176 (2020) 682-694.

[2] X. Zou, N. Wang, L. Han, R. Xue, C. Xu, W. Zhuang, B. Shi, Investigation on burning behaviors of aluminum agglomerates in solid rocket motor with detailed combustion model, Acta Astronaut. 206 (2023) 243-256.

[3] K. Trowell, S. Goroshin, D. Frost, J. Bergthorson, Aluminum and its role as a recyclable, sustainable carrier of renewable energy, Appl. Energy 275 (2020) 115112.

[4] I. Glassman, R. Yetter, Combustion, Elsevier Inc., San Diego, USA, 2008.

[5] M. Salita, Survey of recent Al2O3 droplet size data in solid rocket chambers, nozzles, and plumes, in: JANNAF Meeting, 1994.

[6] A. Genot, S. Gallier, T. Schuller, Thermo-acoustic instabilities driven by fuel droplet lifetime oscillations, Proc. Combust. Inst. 37 (2019) 5359-5366.

[7] H. Keck, C. Chauveau, G. Legros, S. Gallier, F. Halter, New experimental method for the simultaneous determination of concentration and size profiles of condensed combustion products around a burning aluminum droplet, Combust. Flame 268 (2024) 113616.

[8] M.W. Beckstead, A summary of aluminum combustion, in: RTO/VKI Special Course on "Internal Aerodynamics in Solid Rocket Propulsion", NATO, 2002.

[9] J. Finke, F. Sewerin, Combining a population balance approach with detailed chemistry to model the condensation of oxide smoke during aluminum combustion in spatially homogeneous reactors, Combust. Flame 248 (2023) 112510.

[10] P. Bucher, L. Ernst, F. Dryer, R.A. Yetter, T.P. Parr, D.M. Hanson-Parr, Detailed studies on the flame structure of aluminium particle combustion, solid propellant chemistry, combustion, and motor interior ballistics, progress in astronautics and aeronautics, AIAA J. 185 (2000) 689-722.

[11] K. Nishii, H. Koizumi, Temperature measurement on condensed combustion products of magnesium with oxygen and water vapor, Combust. Flame 262 (2024) 113362.

[12] G. Legros, Q. Wang, J. Bonnety, M. Kashif, C. Morin, J.-L. Consalvi, F. Liu, Simultaneous soot temperature and volume fraction measurements in axis-symmetric flames by a two-dimensional modulated absorption/emission technique, Combust. Flame 162 (2015) 2705-2719.

[13] K. McNesby, S. Dean, R. Benjamin, J. Grant, J. Anderson, J. Densmore, Imaging pyrometry for most color cameras using a triple pass filter, Rev. Sci. Instrum. 92 (2021) 063102.

[14] Q. Wang, G. Legros, C. Morin, M. Yao, W. Cai, L. Jiang, Optical measurements of temperature fields in sooting flames: Influence of soot self-absorption, Appl. Phys. B 125 (2019) 63.

[15] R. Khatami, Y.A. Levendis, On the deduction of single coal particle combustion temperature from three-color optical pyrometry, Combust. Flame 158 (2011) 1822-1836.

[16] C.-H. Yeh, Y.-H. Liu, Development of two-color pyrometry for flame impingement on oxidized metal surfaces, Exp. Therm. Fluid Sci. 152 (2024) 111108.

[17] D.S. Sundaram, P. Puri, V. Yang, A general theory of ignition and combustion of nano- and micron-sized aluminum particles, Combust. Flame 169 (2016) 94-109.

[18] M. Dabos, K.-H. Tran, N. Lecysyn, G. Baudin, M. Genetier, I. Ranc, B. Serio, A. Osmont, Investigation of Alumina's Emission in the Deflagration of Gas-Aluminum Particles Mixtures, Europyro, 2019.

[19] R. Lomba, Utilisation de la combustion métallique dans les machines thermiques (Ph.D. thesis), 2017.

[20] D. Allen, H. Krier, N. Glumac, Heat transfer effects in nano-aluminum combustion at high temperatures, Combust. Flame 161 (2014) 295-302.

[21] I. Altman, High-temperature estimation of energy accommodation coefficient of gas molecules on the surface, J. Phys. Stud. 3 (1999) 456-457.

[22] S. Gallier, A. Braconnier, F. Godfroy, F. Halter, C. Chauveau, The role of thermophoresis on aluminum oxide lobe formation, Combust. Flame 228 (2021) 142-153.

[23] M. Muller, D. Davidenko, V. Giovangigli, Computational study of aluminum droplet combustion in different atmospheres, in: 7th European Conference for Aeronautics and Space Sciences, EUCASS, 2017.

[24] J. Finke, F. Sewerin, A combined PBE-cfd approach for modelling the formation and dispersion of oxide smoke in homogeneous aluminum combustors and around a burning aluminum particle, in: Proceedings of the European Combustion Meeting 2023, 2023.

[25] V. Glasziou, C. Chauveau, S. Courtiaud, F. Halter, Measurement of the temperature of burning aluminum particles using multi-spectral pyrometry, Combust. Sci. Technol. 196 (2024) 2138-2151.

[26] I. Altman, E. Fomenko, I.E. Agranovski, Fundamental insight into critical phenomena in condensation growth of nanoparticles in a flame, Sci. Rep. 12 (2022) 15699.

[27] Q. Tran, M.L. Pantoya, I. Altman, Condense-luminescence and global characterization of metal particle suspension combustion, Appl. Energy Combust. Sci. 11 (2022) 100080.

[28] G.B. Rybicki, A.P. Lightman, Radiative Processes in Astrophysics, first ed., Wiley, 1985.

[29] A.N. Zolotko, N.I. Poletaev, Y.I. Vovchuk, Gas-disperse synthesis of metal oxide particles, Combust. Explos. Shock. Waves 51 (2015) 252-268.

[30] A. Pavone, U. Hergenhahn, M. Krychowiak, U. Hoefel, S. Kwak, J. Svensson, P. Kornejew, V. Winters, R. Koenig, M. Hirsch, K.-J. Brunner, E. Pasch, J. Knauer, G. Fuchert, E. Scott, M. Beurskens, F. Effenberg, D. Zhang, O. Ford, L. Vanó, R. Wolf, Measurements of visible bremsstrahlung and automatic Bayesian inference of the effective plasma charge $Z_{eff}$ at W7-X, J. Inst. 14 (2019) C10003.

[31] Y.I. Semov, Work function of oxidized metal surfaces and estimation of $Al_2O_3$ film band structure parameters, Phys. Status Solidi b 32 (1969).

[32] S. Talebi Moghaddam, K.J. Daun, Plasma emission during time-resolved laser-induced incandescence measurements of aerosolized metal nanoparticles, Appl. Phys. B 124 (2018) 159.