# Predicting supercooling of water for large-scale volumes using differential scanning calorimetry data

Jawad Rabbi, José Lara Cruz, Jean-Pierre Bédécarrats

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Jawad Rabbi, José Lara Cruz, Jean-Pierre Bédécarrats. Predicting supercooling of water for large-scale volumes using differential scanning calorimetry data. Journal of Energy Storage, 2025, 140, pp.119122. 10.1016/j.est.2025.119122. hal-05367158

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Research Papers

# Predicting supercooling of water for large-scale volumes using differential scanning calorimetry data

Jawad Rabbi *, José Lara Cruz, Jean-Pierre Bédécarrats

Université de Pau et des Pays de l'Adour, E2S UPPA, LaTEP, Laboratoire de Thermique, Énergétique et Procédés, 64000, Pau, France

## ARTICLE INFO

### Keywords:

Thermal energy storage

PCMs

Supercooling

Statistical model

Prediction

## ABSTRACT

Thermal energy storages (TES) have become essential to the renewable energy ecosystem, tackling the intermittent nature of renewable energy sources and enhancing energy efficiency. This makes TES a cornerstone for decarbonization and energy optimization. Phase change materials (PCMs) are a fundamental element of a latent heat TES system, which provides a higher energy storage density than sensible heat storages. However, the supercooling (delay in the crystallization) of PCMs is a crucial factor to consider when developing a latent heat TES. Due to its considerable volume dependence, supercooling that is characterized at the laboratory scale cannot be applied to large-scale TES systems. Designing a reliable and effective TES system becomes highly challenging if the degree of supercooling corresponding to the system's volume is not precisely predicted. In this work, a statistical model is developed using Differential scanning calorimetry (DSC) data to predict the degree of supercooling for larger volumes. DSC is used to obtain supercooling experimental data for two distinct volumes and cooling rates for type 2 pure water. From these data, an extrapolation is made using a statistical model, and the model's predictions are validated by comparing them with experimental supercooling results for type 2 pure water at two larger volumes (3 mL and 500 mL). The model demonstrated high accuracy, with errors of 0.82 % and 5.36 % for 3 mL and 500 mL volumes at a cooling rate of 1 °C/min, respectively. Finally, data analysis was conducted on the DSC results to establish an optimal protocol by determining the minimum number of DSC experiments to accurately predict the degree of supercooling. In this study, the minimum number of total combined samples is found to be 40, and the minimum number of cycles (repetitions) is found to be 6 for each sample.

## 1. Introduction

Nearly half of the world's energy is consumed as heat in buildings (for space heating, water heating, etc.) and industrial processes (such as manufacturing, chemical production, and steelmaking) [1]. The main source for this heat generation is fossil fuels, which contribute to global carbon emissions [2]. Thermal energy storage (TES) systems are causing an increase in the utilization of renewable energy sources by effectively storing and transferring thermal energy, thereby reducing the gap between the supply and demand for energy. It is especially crucial when considering intermittent renewable energy sources like solar and wind [3]. In addition, TES systems also increase the energy efficiency of industrial systems, significantly lowering greenhouse gas emissions and dependency on fossil fuels, making them an essential part of global efforts to mitigate climate change and move towards a sustainable energy future [4].

TES are of different types: sensible heat storages (SHS), latent heat thermal energy storages (LHTES), and thermochemical storages. Sensible heat storages store heat in the form of sensible heat by changing its temperature while the physical state of the storing materials remains the same. Sensible heat storing materials can be further classified into solid (rocks, concrete, etc.) and liquid (thermal oils, molten salts, etc) [5]. LHTES, stores heat or cold during the phase change of the material. This is why these materials are called phase change materials (PCMs). In industrial applications, the phase change used are solid-liquid and, to a lesser extent, solid-solid where heat is stored during a change of crystalline structure while the physical state of the material remains the same [6]. On the scope of this study, the focus is on solid-liquid transition. Solid-liquid PCMs can be classified into organic, inorganic, and eutectic. Eutectic PCMs are composed of two or more different PCMs. Eutectic mixtures can offer desirable properties, which may not be offered by pure organic or inorganic PCMs [6,7]. PCMs offer more

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: jawad.rabbi@univ-pau.fr (J. Rabbi).</footnote>
storage capacity in comparison to sensible heat storages because they benefit from the latent heat, unlike sensible heat storage. On the other hand, these materials face several challenges as well. Depending on the type of PCM, the different challenges faced can be: phase separation, chemical stability, low thermal conductivity, volume change, corrosion, flammability, and supercooling [8].

Supercooling is one of the major problems faced by some PCMs. It is expected that PCMs will solidify at the solid-liquid equilibrium temperature during a phase transition from liquid to solid. Nevertheless, under some conditions, the PCM's temperature falls below this point without crystallizing. The supercooling degree is the temperature difference between the solid-liquid equilibrium temperature and the crystallization temperature. [9]. This behavior is shown in Fig. 1. In the PCM, crystallization begins when solid nuclei are nucleated. Heterogeneous and homogeneous nucleations are the two categories of nucleation processes [10]. Homogeneous nucleation occurs when a liquid PCM is uniformly cooled below its freezing point without the presence of any nucleation sites, which could lead to solidification. Unlike heterogeneous nucleation, in this situation, solid nuclei form independently of an already-existing solid object. On the other hand, the process by which a cooling liquid encounters foreign materials such as fins, heat exchangers, and other structures results in heterogeneous nucleation [11].

## 1.1. Disadvantages of supercooling

Two significant downsides are associated with the supercooling.

* The nucleation temperature can be out of the system's operating temperature range.

* Second, the system will consume extra energy [12,13].

The fact that a system's operating temperature range may not include the freezing point of the supercooled PCM is the most undesired of these two restrictions. The second issue that supercooling has brought about in energy systems is the rise in energy consumption since the liquid PCM needs additional energy to lower the temperature below its solid-liquid equilibrium point, which in certain situations can be a significant energy expense [12].

## 1.2. Factors affecting supercooling

In cases when supercooling is undesirable, it is essential to minimize

<figure><img src="image_3.png" /><figcaption>Fig. 1. Temperature evolution within the phase change material (PCM) during a cooling process.</figcaption></figure>

it. Investigating the factors that affect the supercooling is essential. Supercooling is most affected by the following factors: (i) Volume; (ii) Container surface roughness; (iii) Thermal history; (iv) PCM purity; (v) Rate of cooling; (vi) Pressure [11,14,15]. Researchers have studied the impact of the cooling rate on the supercooling of PCMs. Table 1 summarizes the literature review regarding the effect of the cooling rate on the degree of supercooling.

The literature provides three different trends regarding the impact of the cooling rate: increasing supercooling with cooling rates (Taylor et al. [17]; Solomon et al. [18]; Song et al. [20]), decreasing supercooling with increasing cooling rates (Chen & Lee [16]; Yoon et al. [19]) and no impact of varying cooling rates (Mossop [21]; Song et al. [20], Rabbi et al. [22]).

Volume is another important factor that impacts supercooling. Table 2 presents the literature review about the dependence of supercooling on sample volume.

From the literature review, it can be seen that the degree of supercooling decreases with increasing volume. Fig. 2 summarizes part of the literature results obtained for water with different purities. The general trend remains the same, i.e., increasing supercooling with decreasing logarithm of volume, but the slopes of these plots differ from each other. This variation is due to factors such as different purities of water, experimental (e.g., surface roughness of the container, forced or natural convection, etc.), and environmental conditions.

## 1.3. Research gap and objectives

Supercooling can be characterized using differential scanning calorimetry.

<caption>Table 1</caption>

A comparison of literature review presenting the impact of cooling rate on the degree of supercooling.

<table><thead><tr><th>Ref</th><th>Cooling rate (°C/min)</th><th>PCM</th><th>Number of samples analysed</th><th>Results</th></tr></thead><tbody><tr><td>Chen &amp; Lee [16]</td><td>0.5-3.2</td><td>Distilled water</td><td>24</td><td>At the lower coolant temperatures (high cooling rates), the degree of supercooling decreases.</td></tr><tr><td>Taylor et al. [17]</td><td>20-80</td><td>Hydrated CaCl<sub>2</sub> based (PC25)</td><td>4</td><td>The degree of supercooling increased from 9 to 22 °C with increasing cooling rate.</td></tr><tr><td>Solomon et al. [18]</td><td>0.1-0.25</td><td>RT21 (Paraffin-based)</td><td>-</td><td>The supercooling increases when the cooling rate increases.</td></tr><tr><td>Yoon et al. [19]</td><td>1.2-18</td><td>Distilled water</td><td>-</td><td>The high cooling rates (10-18 °C/min) had a lower degree of supercooling (5 °C), while for the lower cooling rates, a higher degree of supercooling (7 °C) was observed.</td></tr><tr><td>Song et al. [20]</td><td>1-10</td><td>Decanoic acid</td><td>120</td><td>There was no impact of cooling rate on the degree of supercooling</td></tr><tr><td>Song et al. [20]</td><td>1-10</td><td>MgCl<sub>2</sub> 6H<sub>2</sub>O</td><td>150</td><td>Higher values of the degree of supercooling were observed for higher cooling rates.</td></tr><tr><td>Mossop [21]</td><td>0.05-0.5</td><td>Distilled water</td><td>36</td><td>The difference between the degree of supercooling was less than 0.4 °C, which is very insignificant.</td></tr><tr><td>Rabbi et al. [22]</td><td>0.083-1</td><td>Distilled water</td><td>300</td><td>The degree of supercooling was not changing with a change in cooling rate.</td></tr></tbody></table>
<table><caption>**Table 2**

Impact of volume on the degree of supercooling of various PCMs.</caption><thead><tr><th>Ref.</th><th>PCM</th><th>Volume of PCM (mL)</th><th>Number of samples analysed</th><th>Results</th></tr></thead><tbody><tr><td>Song et al. [20]</td><td>Decanoic acid</td><td>8E-4-1.2E-2</td><td>120</td><td>The degree of supercooling decreased with increasing volume.</td></tr><tr><td>Song et al. [20]</td><td>MgCl<sub>2</sub> 6H<sub>2</sub>O</td><td>1.1E-3 – 1.3E-2</td><td>150</td><td>The degree of supercooling decreased with increasing volume.</td></tr><tr><td>Mossop [21]</td><td>Distilled Water</td><td>5E-5 – 12.5E-3</td><td>168</td><td>The average supercooling was 33 °C. The results do not show any clear relationship between the freezing point of the specimen and volume.</td></tr><tr><td>Adachi et al. [23]</td><td>Erythritol</td><td>0.004–16</td><td>46</td><td>By increasing the volume, supercooling decreases from 80 to 60 °C.</td></tr><tr><td>Chen & Chen [24]</td><td>Distilled water</td><td>19–1000</td><td>24</td><td>By increasing the volume, supercooling decreases, and nucleation probability increases.</td></tr><tr><td>Bigg [25]</td><td>Distilled water</td><td>6E-8 – 8</td><td>100</td><td>By increasing the volume, supercooling decreases. The logarithm of the diameter of the drop has a linear relationship with the degree of supercooling.</td></tr><tr><td>Dumas [26]</td><td>Benzene</td><td>≤1</td><td>–</td><td>Increase of the supercooling degree from 19.5 °C to 71.5 °C when the volume decreased from 1cm<sup>3</sup> to a few cubic micrometres.</td></tr><tr><td>Kuhns & Mason [27]</td><td>Distilled water</td><td>2.6E-7 – 4.1E-9</td><td>100 (all samples)</td><td>A decrease in supercooling degree was observed with increasing volumes.</td></tr><tr><td>Carte [28]</td><td>Distilled Water</td><td>3.8E-10 – 1.8E-8</td><td>65/100/191</td><td>The same trend was found as in other studies. But the decrease in supercooling was only 1 °C following a volume increase of 50 times.</td></tr><tr><td>Kubota et al. [29]</td><td>KNO<sub>3</sub></td><td>0.2–1.2</td><td>1000</td><td>The degree of supercooling decreased with increasing volume. The logarithm of the volume of the drop has a linear relationship with the degree of supercooling.</td></tr><tr><td>Langham & Mason [30]</td><td>Triple-distilled water</td><td>2E-3 – 1.5E-2</td><td>–</td><td>The degree of supercooling decreased with increasing volume.</td></tr></tbody></table>

rimetry (DSC) in the laboratory. However, this supercooling cannot be applied to the PCMs in the real systems because of their larger volumes. Researchers have tried to calculate supercooling using physics-based models [31]. On the other hand, supercooling can be predicted statistically. The erratic nature of the supercooling makes statistical models more reliable and suitable for the prediction of supercooling in larger

volumes from DSC experimental data. Therefore, statistical approaches have potential for improving the prediction of supercooling, but further investigation is required. Statistical models have been initially used to predict the nucleation of metals [32,33]. Later, the degree of supercooling in PCMs was predicted statistically by Lilley et al. [34]. They characterized the supercooling (nucleation time) of Neopentyl Glycol (NPG), a solid-solid type PCM, at laboratory scale using DSC. The supercooling was experimentally characterized for a 10 mg (0.01032 mL) sample at a cooling rate of 10 °C/min. Using a statistical model involving a non-homogeneous Poisson process, they predicted the nucleation time for larger volumes of Neopentyl Glycol. The predictions of the statistical model were validated against the experimentally obtained nucleation time for NPG with a volume of 2.6 × 10⁻⁶m³ (2.6 mL), at a cooling rate of 0.1 °C/min. Later, the same laboratory worked on solid-liquid phase change. Song et al. [20] predicted the nucleation time for Magnesium chloride hexahydrate (MgCl₂·6H₂O) using the same statistical framework developed by Lilley et al. [34]. DSC experiments were performed for MgCl₂·6H₂O with 1.4 mg (8.9 × 10⁻⁷m³ or 0.00089 mL) and 19.8 mg (1.2 × 10⁻⁵m³ or 0.0126 mL) at two different cooling rates, 1 °C/min and 10 °C/min. The authors then statistically predicted the nucleation time for the mass of 31 g (19.7 mL) of MgCl₂·6H₂O and validated their prediction with the experimental results.

In this work, the statistical model previously developed on the work of Lilley et al. [34] is employed to predict the degree of supercooling of another type of PCM (type 2 pure water) for larger volumes. The following are the goals of this study, along with the motivations behind them.

* The primary goal of this study is to predict the degree of supercooling of water for large volumes using a statistical model. The prediction of supercooling of PCMs using a statistical model has not been extensively studied, making the application of this framework to PCMs very novel. The earliest studies were performed by Lilley et al. [34] and Song et al. [20]. They applied and validated this model to the solid-solid PCMs (Neopentyl Glycol), and an inorganic solid-liquid PCM (MgCl₂·6H₂O). In order to enhance the validity of this model and to identify the constraints, it is important to extend its application to a wider range of materials. Water, being one of the TES materials experiencing supercooling, it is important to predict the degree of supercooling of water.

* The second goal of this research is to validate the employed statistical model for significantly larger volumes (up to 500 mL) than used in the previous studies. Song et al. [20] validated the model with only one larger volume of MgCl₂·6H₂O, having a mass of 31 g (19.7 mL). A thorough validation of this model is needed across a wider range of volumes. Therefore, it is important to extend the validation to include additional larger volumes. In addition to validation with further larger volumes, the degree of supercooling was validated along with the nucleation time, whereas previous studies validated only the nucleation time for samples with non-uniform temperature distribution [20].

* The third objective is to establish a protocol for DSC experiments used for supercooling analysis. To minimize error in the predicted results, it is important to overcome the erratic nature of supercooling. This erratic behavior can be mitigated by increasing the number of DSC results. However, this should not increase unreasonably the duration of any experimental campaign. Hence it is very important to suggest the minimum number of DSC experiments necessary to produce reliable results from the model. The minimum number of DSC experiments to minimize the error in predicting the degree of supercooling has not been suggested in any study.

* The final objective is to utilize the statistical model developed by Lilley et al. [34] for uniform temperature distribution samples to predict the nucleation time and the degree of supercooling for large volume samples with non-uniform temperature distributions. The aim of this approach is to better understand the limitations of the
<figure><img src="image_4.png" /><figcaption>Fig. 2. Comparison of results from different researchers on the volume dependence of supercooling in water; a common trend can be seen, but with slightly varying results.</figcaption></figure>

statistical model adopted in this work when it is applied on its simplified version to our PCM of interest. Previously, in the open literature, those potential limitations were only highlighted by Lilley et al. for the solid-solid phase change of Neopentyl Glycol and it is relevant to extend this analysis to other PCMs.

## 2. Statistical modelling

### 2.1. Statistical modelling of supercooling

Nucleation is a key step in the crystallization process. It is a stochastic process. In order to establish that nucleation is a stochastic process, let us consider a sample of a PCM, experiencing several numbers of heating and cooling cycles. In each cooling cycle, nucleation may occur at different times and temperatures, while all experimental conditions remain the same in each cycle. This type of data can be modelled with a non-homogeneous Poisson process. In probability theory, a Poisson process is a stochastic process which counts the number of events and records the time at which these events occur within a given time interval. The time between each pair of consecutive events has an exponential distribution with the rate parameter (λ), and each of these inter-arrival times is assumed to be independent of other inter-arrival times. Poisson processes can be classified into homogeneous and non-homogeneous types, depending on whether the event rate λ remains constant over time or varies as a function of time. Homogeneous poisons process is a type of Poisson process where the rate parameter “λ” remains constant and the distribution of event counts depends only on the length of the interval. Non-homogeneous Poisson processes, on the other

hand, are a type of Poisson process where the rate parameter “λ” varies as a function of time, and the distribution of events depends on both the time and the interval length [35]. Broadly speaking, the following characteristics of the nucleation process are consistent with those of the non-homogeneous Poisson process (NHPP):

* Nucleation, which is the event occurring in this statistical process, is discrete.

* The time or temperature with which these events occur is continuous.

* The nucleation process has a rate parameter, which is the nucleation rate, J.

* The average rate parameter (nucleation rate) is not constant throughout time or temperature, otherwise, it would be modelled by the homogeneous Poisson process [35].

One of the most important functions of the NHPP is the cumulative distribution function, given in Eq. (1).

$$ CDF(t) = 1 - e^{-\int_{0}^{t} \lambda(s) ds} \qquad (1) $$

Here λ, is the rate parameter. The rate parameter is a quantity that characterizes the speed or intensity of a process. In the nucleation context, the rate parameter λ(t) is defined as: the expected number of nucleation events occurring per unit of time in the whole system in time, t. This rate parameter, λ(t), can be related to the nucleation rate, J(t). The nucleation rate is the rate at which stable nuclei (tiny clusters of a new phase) form within a parent phase per unit volume (or unit area) and per unit time. The relationship between the rate parameter and
nucleation rate is given by Eq. (2).

$$ \lambda(t) = VJ_V(t) + AJ_A(t) \qquad (2) $$

where, $J_V$ is the volume specific nucleation rate that includes nucleation due to both homogeneous nucleation and heterogeneous nucleation, while $J_A$ is the area specific nucleation rate that considers nucleation only in the case of heterogeneous nucleation, V is the volume of the system. A is the surface area of the system. In this study, we assume that the system is large enough with chemically stable surface so that $J_V$ is much greater than $J_A$, which is the same assumption previously done in the work of Lilley et al. [20]. This makes the term $VJ_V(t)$ much greater than $AJ_A(t)$ so that we can neglect the latter term. After this assumption, Eq. (2) becomes.

$$ \lambda(t) = VJ_V(t) \qquad (3) $$

Combining Eqs. (1) and (3) gives,

$$ CDF(t) = 1 - e^{-\int_{0}^{t} VJ_{V}(t)dt} \qquad (4) $$

Nucleation can be recorded as a function of either time or temperature. Among these two parameters, temperature is the point of interest for the current study, i.e., to predict the degree of supercooling. Now we can replace time (t) in the non-homogeneous Poisson process with the temperature or a variable $\theta_s$, which is defined as $\theta_s = T_m - T$. $T_m$ is the solid-liquid equilibrium temperature, and if the sample crystallizes at any other temperature (T), we can refer to the variable $\theta_s$ as the degree of supercooling. Cooling rate $\beta$ can be related to the derivative of $\theta_s$. This is given in Eq. (5).

$$ \frac{d\theta_s}{dt} = \dot{\theta}_s = \beta \qquad (5) $$

The initial condition at time $t = 0$ becomes $T = T_m$ and consequently $\theta_s(T = T_m) = 0$. Replacing $t$ with $\theta_s$ in Eq. (4) gives.

$$ CDF = 1 - e^{-\frac{V}{\beta} \int_{0}^{\theta_s} J_v(\theta_s) d\theta_s} \qquad (6) $$

Another important parameter to model the nucleation is the survival function because it can also be calculated experimentally. Survival function $\chi(\theta_s)$ describes the probability that a nucleation site will survive (non-nucleated) over a certain period of time. Physically, it represents the likelihood that the system remains in its metastable supercooled state without the formation of a stable nucleus. It is given in Eq. (7) below.

$$ \chi(\theta_s) = 1 - CDF \qquad (7) $$

$$ \chi(\theta_s) = e^{\frac{V}{\beta} \int_{0}^{\theta_s} J_v(\theta_s) d\theta_s} \qquad (8) $$

Rearranging Eq. (8),

$$ J_V(\theta_s) = - \frac{1}{V} \frac{\beta}{\chi(\theta_s)} \frac{d\chi(\theta_s)}{d\theta_s} \qquad (9) $$

In Eq. (9), Volume (V) and cooling rate ($\beta$) are the known parameters while $\chi(\theta_s)$ can be determined experimentally with DSC experiments (see Section 4.1.2). Thus, the nucleation rate of the DSC-scale PCM samples for the applied experimental conditions (volume and cooling rate) can be calculated. This calculated nucleation rate for the DSC samples is normalized by using a power law fitting function. This data fitting results in two important parameters $\gamma$ and $n$, given in Eq. (10). These parameters are particularly significant because they are later used in the prediction of the degree of supercooling for different volumes as the statistical model relies on the assumption that the nucleation rate observed at DSC-scale (Eqs. (9) and (10)) is also valid at higher volumes (see Sections 2.2 and 2.3).

$$ J_V(\theta_s) = \gamma \theta_s^n \qquad (10) $$

There are two different cases to obtain a solution to this statistical

model based on the temperature distributions of the samples. The first case is the solution for the samples with uniform temperature distribution, and the second case is the solution for the samples with non-uniform temperature distribution.

## 2.2. Solution of the model for samples with uniform temperature distribution

Nucleation is independent of the space when the samples are small enough to exhibit uniform temperature distribution. In this case analytical solution to the statistical model can predict the degree of supercooling. This analytical solution is presented in this section. Combining Eqs. (6) and (10) gives:

$$ CDF(t) = 1 - e^{-\frac{\gamma V}{\beta(n+1)} \theta_s^{n+1}} \qquad (11) $$

The probability distribution function, PDF($\theta_s$), is obtained by differentiating Eq. (11) with respect to $\theta_s$.

$$ PDF(\theta_s) = \frac{\gamma V}{\beta} \theta_s^n e^{-\frac{\gamma V}{\beta(n+1)} \theta_s^{n+1}} \qquad (12) $$

The average temperature when the first nucleation event occurs can be calculated as the first moment of the PDF, which is given in Eq. (13).

$$ \theta_{s,\text{avg}}(V, \beta) = \int_{0}^{\infty} \theta_s PDF(\theta_s) d\theta_s \qquad (13) $$

Solving Eq. (13) gives,

$$ \theta_{s,\text{avg}}(V, \beta) = \beta^{\frac{1}{n+1}} \left( \frac{n+1}{\gamma V} \right)^{\frac{1}{n+1}} \Gamma\left( \frac{n+2}{n+1} \right) \qquad (14) $$

Eq. (14) is used to predict the degree of supercooling of the system with uniform temperature distribution, having the known parameters $\beta$, $V$, $n$ and $\gamma$.

Eq. (15) presents the equation for uncertainty calculation for the predicted degree of supercooling of the system with a uniform temperature distribution. It was obtained by taking the second moment of the PDF.

$$ \sigma_{\theta_s} = \beta^{\frac{1}{n+1}} \left( \frac{n+1}{\gamma V} \right)^{\frac{1}{n+1}} \left\{ \Gamma\left( \frac{n+3}{n+1} \right) - \left[ \Gamma\left( \frac{n+2}{n+1} \right) \right]^2 \right\}^{\frac{1}{2}} \qquad (15) $$

## 2.3. Solution of the model for non-uniform temperature samples

Systems with larger volumes will have non-uniform temperature distribution, where temperature will change along with time and space. Consequently, nucleation will depend on both time and space. This system can be modelled by dividing the system into infinitesimally small elements. Each element with the volume $dV_i$ will behave as a sample where temperature distribution is uniform, and the probability of the nucleation will be given by a Poisson process with its own rate parameter $\lambda_i$. In order to find the rate parameter of the complete system, the rate parameter of each element must be added into one global rate parameter. For a non-homogeneous Poisson process, the convolution is simply the sum of the individual rate parameters. Consider the system with N number of small elements having volume $dV_i$ and each element having its own rate parameter $\lambda_i$. For this system, we can then define a global, or an effective rate parameter $\lambda_{\text{eff}}(t)$, given in Eq. (16).

$$ \lambda_{\text{eff}}(t) = \sum_{i=1}^{N} \lambda_i(x, y, z, t) \qquad (16) $$

From Eq. (3), we know that $\lambda(t) = VJ_V(t)$, and combining it with Eq. (16).

$$ \lambda_{\text{eff}}(t) = \sum_{i=1}^{N} J_{V_i}(x, y, z, t) V_i(x, y, z) \qquad (17) $$
$$ \lambda_{\text{eff}}(t) = \iiint J_V(x, y, z, t) dxdydz \quad (18) $$

$$ \lambda_{\text{eff}}(t) = \iiint \gamma(T_m - T(x, y, z, t))^{n} dxdydz \quad (19) $$

where $\gamma$ and $n$ are the same parameters of the power law determined by the normalization of the nucleation rate (see Section 2.1). The average time of the nucleation can be found by the first moment of the PDF, which is given in Eq. (22).

$$ CDF(t) = 1 - e^{-\int_{0}^{t} \lambda_{\text{eff}}(t)dt} \quad (20) $$

$$ PDF(t) = \frac{dCDF(t)}{dt} \quad (21) $$

$$ t_{\text{avg}} = \int_{0}^{\infty} tPDF(t)dt \quad (22) $$

The error bounds of the predicted nucleation time are calculated by the standard deviation, which is calculated by the second moment of the PDF, as in Eq. (23).

$$ \sigma_{t_p} = \pm \sqrt{\int_{0}^{\infty} t^2 PDF(t)dt - t_{\text{avg}}^2} \quad (23) $$

where, $\sigma_{t_p}$ is the error bound of the predicted nucleation time.

### 3. General methodology

The presented statistical model is developed using data from DSC experiments. The application of the model involves calculating the survival function using experimental data, and then the nucleation rate for the samples using Eq. (9). The calculated nucleation rate is normalized using a power law function, as defined by Eq. (10). This normalization of the nucleation rates results in two parameters of the power law, $\gamma$ and $n$.

The determination of the power law parameters ($\gamma$ and $n$) allows the statistical model to predict the degree of supercooling for large volumes samples with uniform temperature distribution using Eq. (14).

On the other hand, the degree of supercooling of samples with non-uniform temperature distribution can be predicted with the help of a CFD model implemented through COMSOL Multiphysics (see Section 4.2.2. Numerical Modelling). The model is used to characterize the thermal behavior of the samples, and its results are combined to the power law parameters ($\gamma$ and $n$). In this case, the sample is divided into N number of infinitesimally small elements having volume $dV_i$ using CFD software. The rate parameter $\lambda_i(x, y, z, t)$ for each small volume element is obtained, and then an effective rate parameter $\lambda_{\text{eff}}(t)$ is calculated by using Eq. (19). These calculations are possible thanks to the temperature field of the sample under the thermal conditions of interest, which can be predicted with the CFD model. This effective rate parameter $\lambda_{\text{eff}}(t, \gamma, n)$ is necessary for calculating the probability density function (PDF) of the nucleation time and its first moment, representing the average time to first nucleation, as defined in Eqs. (21) and (22). Effective rate parameter in Eq. (19) was calculated by performing integration using COMSOL Multiphysics. While PDF, CDF and nucleation time in Eqs. (20), (21), and (22) were calculated using MATLAB differential and integration functions. Following the prediction of nucleation time, CFD modelling of these samples is used again to calculate the degree of supercooling from the predicted nucleation time. The CFD model records the evolution of temperature over time in the sample, and the minimum temperature in the geometry at the predicted nucleation time provides the average degree of supercooling.

After the prediction of supercooling of large-volume samples, experiments are conducted to compare the results of the statistical model

to the real supercooling. These steps for non-homogeneous temperature distribution samples are summarized in Fig. 3.

### 4. Application of the statistical model

The material used as the phase change material (PCM) throughout the experimental campaign in this work was Type 2 pure water. It was produced by the Elix® Essential 10 Water Purification System using an electrodeionization (EDI) technology with resistivity of 15 MΩ·cm at 25 °C. It meets the standard requirements for laboratory-grade water (Type 2) suitable for thermal and physical characterization studies.

#### 4.1. Characterization using DSC

##### 4.1.1. DSC experimental setup

The first step to predict the degree of supercooling is to perform DSC experiments. These DSC experiments, together with the statistical model, will provide two power law parameters, “n” and “$\gamma$”. Two separate devices (Pyris Diamond from Perkin-Elmer and MICRO-DSC7-EVO from Setaram) were used to study two different volumes. The DSC analysis provides detailed insights into the phase change equilibrium temperature and crystallization temperature. The uncertainty of the DSC measurement was determined by comparing the experimentally obtained solid-liquid equilibrium temperature of a 23.3 wt% NaCl-Water solution with the reference temperature obtained from literature data [36]. This eutectic mixture melts close to the temperature of crystallization of supercooled water at the studied volume. The uncertainty for Pyris Diamond and Micro-DSC7-EVO is found to be less than $\pm 0.2$ °C.

The samples analysed using the Pyris Diamond had a mass of 1.5 mg, while those analysed on the MICRO-DSC7-EVO had a mass of 30 mg. Each sample was analysed at two different cooling rates, 1 °C/min and 0.166 °C/min. The degree of supercooling was calculated for each case. An example is done in Fig. 4 representing the heat flux versus furnace temperature for a sample of water of 30 mg. This obtained thermogram shows endothermic and exothermic peaks, with the onset of the exothermic peak representing the crystallization temperature and the onset of the endothermic peak representing the melting temperature. The degree of supercooling is calculated by taking the difference between the solid-liquid equilibrium temperature of type 2 pure water and the crystallization temperature. The total number of samples for each case is given in Table 3. In a previous research by our team, it was found that the degree of supercooling of type 2 pure water is not dependent on the cooling rate when this parameter varies between 0.166 °C/min and 1 °C/min [22]. Therefore, fewer samples were characterized at the lower cooling rate of 0.166 °C/min.

##### 4.1.2. Calculating $\gamma$ and $n$ using DSC data and statistical model

After calculating the degree of supercooling, the survival function for each dataset was determined. It is calculated by dividing the number of non-nucleated (surviving) samples at a given temperature by the total number of samples. The survival function is then plotted against the degree of supercooling (Fig. 5a). The derivative of the survival function with respect to temperature is also calculated. i.e. $dX/dT$. After determining the survival function, the derivative of the survival function, and along with remaining known parameters (volume and cooling rate), the nucleation rate for each sample is calculated using Eq. (9). These nucleation rates are fitted with the help of the power law as shown in Eq. (10), giving the parameters $n$ and $\gamma$. Fig. 5b portrays the fitting of the nucleation rates by using a power law. The coefficients of the power law are found to be $\gamma = 5.803E-19$ and $n = 20.035$.

#### 4.2. Large volume experiments and modelling

##### 4.2.1. Experimental setup and supercooling degree determination

To validate the prediction of the statistical model, an experimental
<figure><img src="image_5.png" /><figcaption>Fig. 3. Application of the developed statistical model to predict the degree of supercooling for large-volume samples with non-homogeneous temperature distribution.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. DSC thermogram – representing peaks, properties and nucleation temperature.</figcaption></figure>
<table><caption>**Table 3**
Number of samples for different combinations of volume and cooling rate.</caption><thead><tr><th></th><th>1.5 mg</th><th>30 mg</th></tr></thead><tbody><tr><td>1 °C/min</td><td>162</td><td>216</td></tr><tr><td>0.166 °C/min</td><td>50</td><td>65</td></tr></tbody></table>

campaign with three different cases was developed:

* The first case (Case I) corresponds to a volume of 3 mL of water in tubes, with a cooling rate set at 1 °C/min. The tubes are made up of polypropylene, with an inner diameter of 10 mm and a thickness of 1 mm. The setup includes T-type thermocouple positioned outside the sample. The thermocouple probe is wire-type and is positioned in contact with the tube wall using a zip-tie (Fig. 6a). The thermocouples are installed outside the sample rather than being placed in it because a thermocouple in direct contact with water can change the degree of supercooling. Inserting a thermocouple probe into the PCM sample can trigger nucleation and consequently reduces the degree of supercooling, resulting in inaccurate measurement of the degree of supercooling [37].

* The second case (Case II) corresponds to the same volume of 3 mL of water in tubes but with a slower cooling rate of 0.166 °C/min. The tubes and position of thermocouples are the same as case I. The sample tubes for cases I and II are shown in Fig. 6a.

* The third case (Case III) corresponds to a larger volume of 500 mL of water in a borosilicate beaker. In this case, cooling cycles were applied at a cooling rate of 1 °C/min. The beaker wall has a thickness of 2 mm and an outer diameter of 90 mm. The setup includes a T-type thermocouple positioned outside the sample. The case-III setup is shown in Fig. 6b.

In all three cases no external factors (e.g. mechanical agitation, seeding, or impurities) were applied to trigger nucleation.

Fig. 7 shows the complete experimental setup. The heat transfer fluid's (HTF) temperature is controlled by a Julabo FP 50, a refrigerated heating circulator with a bath, which provides precise cooling and heating control over the entire operating range. This device ensures a stable and uniform thermal environment in the bath, allowing the water samples to be subjected to well-controlled cooling rates during the experiments. The water samples are inserted into a holder that is fully immersed in the bath. The samples' temperature is measured and

recorded over time using the KEYSIGHT DAQ970A data acquisition system. The thermocouple's accuracy is ±0.1 °C, which was determined by measuring the solid-liquid equilibrium temperature of a mixture of ice and liquid water (0 °C). Any deviation from this temperature will indicate the uncertainty. An aqueous solution of 50 % water and 50 %

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** (a) Schematic of the temperature measurement for a 3 mL sample. (b) Schematic of the temperature measurement for a 500 mL sample.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Experimental setup to study the supercooling.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Statistical analysis of data obtained from DSC (a) survival function as a function of the degree of supercooling for different volumes and cooling rates. (b) Nucleation rates of all the datasets collected using DSC, along with the data fitting curve based on the power law.</figcaption></figure>

<figure><img src="image_10.png" /></figure>
monoethylene glycol (Neutragel®) serves as the HTF.

The degree of supercooling was calculated by using the lowest temperature in the supercooled sample (using a CFD numerical model: see Section 4.2.2. Numerical modelling for more information) at the nucleation time, which was determined experimentally by a thermocouple installed outside the container.

When the PCM is cooled below the solid-liquid equilibrium temperature, its temperature suddenly increases when it starts to crystallize, due to the release of the latent heat. The moment at which latent heat is released corresponds to crystallization. It can be detected using a thermocouple placed on the outer surface of the PCM container. The time and temperature when the nucleation is triggered are determined from the temperature evolution as a function of time. This nucleation point is represented in Fig. 8. Even if the crystallization does not begin exactly at the position where the thermocouple is placed, it spreads rapidly, making it possible to determine the moment of crystallization with a good accuracy. However, since this thermocouple is not placed into the PCM, the temperature inside the PCM must be determined with the CFD model in order to know the degree of supercooling. In large volume samples 3 mL and 500 mL, significant spatial temperature variations can occur within the sample, which makes it necessary to use the CFD modelling to predict the transient internal temperature distribution in both sample sizes. Finally, since nucleation is erratic, we perform multiple tests and take the average nucleation time to estimate the nucleation temperature, which improves the accuracy in the determination of the supercooling degree. The details of three experimental cases are summarized in Table 4.

### 4.2.2. Numerical modelling

Besides the calculation of the effective rate parameter $\lambda_{eff}(t)$ of the statistical model (see Section 3), the objective of the numerical simulation is to estimate the internal temperature of the samples (3 mL and 500 mL) immediately prior to the onset of crystallization. This approach ensures that the degree of supercooling determined in this study reflects the actual internal temperature of the sample and is not affected by the temperature gradient across the container wall. Since the focus was on the pre-crystallization temperature distribution in the geometry, modelling the liquid-solid phase transition was not necessary.

Each large volume case was modelled using COMSOL Multiphysics. The sample's geometry was designed to be 2D axisymmetric.

Fig. 9 displays the geometry and the boundary conditions. The modelled geometry has identical dimensions to the materials (tubes and beaker) utilized in the experiments. Natural convection is considered

<figure><img src="image_11.png" /><figcaption>Fig. 8. Temperature evolution of the PCM versus time measured by the thermocouple installed outside a sample container and representation of nucleation temperature.</figcaption></figure>

<table><caption>**Table 4**
Specifications of the large volume samples used to validate the model.</caption><thead><tr><th></th><th>Case - I</th><th>Case-II</th><th>Case-III</th></tr></thead><tbody><tr><td>Material</td><td>Type 2 pure water</td><td>Type 2 pure water</td><td>Type 2 pure water</td></tr><tr><td>Volume (mL)</td><td>3</td><td>3</td><td>500</td></tr><tr><td>Cooling rate (°C/min)</td><td>1</td><td>0.166</td><td>1</td></tr><tr><td>Number of samples</td><td>304</td><td>304</td><td>30</td></tr></tbody></table>

inside the water, the air and the HTF:

$$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 \qquad (24) $$

$$ \rho \frac{\partial \vec{u}}{\partial t} + \rho(\vec{u} \cdot \nabla)\vec{u} = \nabla \cdot [-p\mathbf{I} + \mathbf{K}] + \rho \vec{g} \qquad (25) $$

$$ \mathbf{K} = \mu (\nabla \vec{u} + (\nabla \vec{u})^T) \qquad (26) $$

$$ \rho C_p \frac{\partial T}{\partial t} + \rho C_p \vec{u} \cdot \nabla T = \nabla \cdot (k \nabla T) \qquad (27) $$

Conduction is considered in the wall that separates water from the HTF:

$$ \rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) \qquad (28) $$

The specific heat capacity, thermal conductivity, dynamic viscosity, and density of the supercooled water were obtained from the data in the published literature [38–40]. While for the glycol-water (50 % composition), properties were taken from the Monoethylene Glycol MEG Technical Product Brochure [41]. A mesh refinement study was conducted, and increasing the number of domain elements from 700 to 2400 did not yield any substantial impact on the outcomes. A time-dependent study was adopted. For time-dependent convergence, error tolerance was set at 0.001. The output time step in the time-dependent study was set to 0.1 s.

To validate the numerical model, a series of experiments were conducted under controlled conditions. Two sample volumes, 3 mL and 500 mL, were tested across a range of cooling rates from 0.166 to 1 °C/min. To validate CFD model for 3 mL volume samples, probes positioned on the tube's outside wall and at the center of the sample were used to collect temperature data during the experiments. On the other hand, to validate 500 mL volume CFD model, probes were positioned inside the beaker at the bottom near the wall, and at the top of the sample to collect temperature data during the experiments. In the case of the 500 mL volume, validation was performed using measurements taken at two distinct locations inside the tube rather than outside, due to the significant temperature gradients, which was confirmed by the CFD model. This approach was essential to assess whether the CFD model accurately captures the non-uniform temperature distribution within the 500 mL volume. These probe readings were then compared with the numerical results at the corresponding location. Because the numerical model is unable to forecast the nucleation point, the comparison between the model and the experiment is only made prior to crystallization.

Fig. 10 displays the experimental and numerical change in temperature over time for a specific cooling rate. The difference between the CFD-predicted and experimentally measured temperatures is less than 3 % for all the examined cooling rates (0.166 and 1 °C/min) and volumes (3 mL and 500 mL). As a result, the model is validated.

In Fig. 10b, there is a sudden change in temperature around 4 °C which corresponds to the temperature of maximum density for water. During the cooling process down to 4 °C, the density of water increases steadily with decreasing temperature. Below 4 °C, the density of water decreases as temperature decreases. This fact causes a change in natural convection currents at 4 °C. Even if this phenomenon is present in all the
<figure><img src="image_12.png" /><figcaption>Fig. 9. Geometry and Boundary conditions for CFD simulations of (a) 3 mL volume (b) 500 mL volume.</figcaption></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 10. Comparison of the numerical model with experimental results at 1 °C/min for (a) 3 mL volume (b) 500 mL volume.</figcaption></figure>

<figure><img src="image_15.png" /></figure>

cases, it is more prominent in the 500 mL (see Fig. 10b) due to the high temperature gradient inside the sample, along the tube axis, as compared to the other cases. Therefore, the lowest temperature in the PCM below the maximum density point is observed at the top of the 500 mL sample inside the inner wall of the container. On the other hand, in the 3 mL sample, depending on the cooling rate, either the thermal gradient inside the sample is low, or the lowest temperature remains always in the bottom.

Fig. 11a and b illustrates the temperature distribution at the nucleation time in 3 mL tubes and in the 500 mL beaker at a rate of 1 °C/min. The Minimum/Maximum function in COMSOL was used to extract the exact coordinates of the lowest temperature for each case. This was also the approach used to determine the minimum temperature at the nucleation time for the different experiments of this study in order to calculate the degree of supercooling. These results show that the minimum temperature is not always at the same position. It depends on the

heat transfer which can vary considerably depending on the volume and cooling rate, which is a consequence of how each sample reacts to the change in the evolution of water density with temperature at 4 °C.

### 4.2.3. Uncertainty calculation for experimental nucleation time and temperature

The error bounds in the nucleation time of the large volume experimental samples are obtained by calculating the standard error and considering 95 % confidence intervals (in statistics 95 % corresponds to a ± 1.96 standard deviation (σ) range from the mean in a normal (Gaussian) distribution), which are given by Eq. (29) [42].

$$ \sigma_{t_{exp}} = \pm 1.96 \times \frac{\sigma_t}{\sqrt{n}} \qquad (29) $$

where, $\sigma_{t_{exp}}$ represents the error bounds in the nucleation time of larger
<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 11. Temperature distribution in the geometry at the time interval when nucleation is triggered for (a) 3 mL (b) 500 mL.</figcaption></figure>

volume samples, $\sigma_t$ is the standard deviation of the nucleation time data, and $n$ is the total number of samples (see Table 4 for the number of samples in each experiment). Standard deviation is a measure of the dispersion or spread of data points around the mean (average) of a dataset. The term $\sigma_t/\sqrt{n}$ represents the standard error of the data. Although this analysis pertains to the uncertainty in nucleation time, the evaluation of the supercooling degree error is carried out separately. Combining the accuracy of the thermocouple with the other error sources allows for the calculation of the overall error bounds for the experimental degree of supercooling. Because the thermocouple is outside the PCM, the temperature within the sample tube was determined using a numerical model (see Section 4.2.1. Experimental Setup and Supercooling Degree Determination). The maximum error associated to this calculation is also considered in the determination of the overall error bounds. One should also include an uncertainty source associated to the standard error of the nucleation temperature ($\sigma_t/\sqrt{n}$) of each experimental case, observed at the nucleation point as previously presented in Fig. 8. Although this temperature is measured outside the PCM, it allows one to account for the uncertainty associated to the erratic nature of supercooling. A 95 % confidence interval is considered to have the uncertainty in the experimental degree of supercooling presented in Eq. (30):

$$ \sigma_{T_{\text{exp}}} = \pm 1.96 \times \sqrt{\left(\frac{\sigma_T}{\sqrt{n}}\right)^2 + (\delta_{T_C})^2 + (\delta_s)^2} \quad (30) $$

where, $\sigma_{T_{\text{exp}}}$ is the error in the degree of supercooling of larger volume samples, $\delta_{T_C}$ is the error associated with the thermocouple (±0.1 °C), and $\delta_s$ is the maximum error associated with the validation of the numerical model. The calculated uncertainties for experimental nucleation time ($\sigma_{T_{\text{exp}}}$), and experimental nucleation temperature ($\sigma_{T_{\text{exp}}}$) are presented in Table 5.

<table><caption>**Table 5**
Uncertainties calculated in different cases.</caption><thead><tr><th></th><th>σ<sub>T<sub>exp</sub></sub><br>(sec)</th><th>σ<sub>T<sub>exp</sub></sub><br>(°C)</th></tr></thead><tbody><tr><td>Case I</td><td>14</td><td>0.47</td></tr><tr><td>Case II</td><td>68</td><td>0.48</td></tr><tr><td>Case III</td><td>40</td><td>0.86</td></tr></tbody></table>

### 4.3. Predicting supercooling using statistical model

#### 4.3.1. Prediction of nucleation time for non-uniform temperature case

The methods used to obtain the average nucleation time for three different cases have been described in the previous section, “4.2. Large Volume Experiments”. Systems with larger volumes will exhibit non-uniform temperature distribution, where the temperature varies with both time and space. This system can be modelled by dividing the system into infinitesimally small elements using computational fluid dynamics (CFD). Each element, with volume $dV_i$, behaves as a sample where temperature distribution is uniform, with its own rate parameter $\lambda_i$, as previously explained in the section “2.3 Solution of the model for the non-uniform temperature samples”. Thanks to the CFD model, for each element i of the system, the term $\gamma(T_m - T(x, y, z, t))^n$ is determined. This will give the nucleation rate $J_i(x, y, z, t)$ of each infinitesimally small volume element. Subsequently, the integration of the term $\gamma(T_m - T(x, y, z, t))^n$ over the volume is performed to obtain the effective rate parameters $\lambda_{eff}$ and the global nucleation rate $J(t)$. Now, the CDF and PDF of nucleation time are calculated using Eqs. (20) and (21). The PDF of nucleation time for different cases is plotted in Fig. 12a, b and c. Finally, by taking the first moment of the PDF as in Eq. (22), the average nucleation time is calculated for each case. The uncertainties for predicted nucleation time ($\sigma_{T_p}$) was calculated using Eq. (23). They were found to be 44, 243, and 42 s for case - I, case-II, and case-III, respectively.

The experimentally obtained nucleation time is compared with the predicted time in Fig. 13a. The difference found between experimental and modelling results in cases I, II and III is 0.6, 4.8 and 4 % respectively.

#### 4.3.2. Calculating degree of supercooling from predicted nucleation time for non-uniform temperature case

From Fig. 13a, one may notice that the nucleation times of Case I and Case III are quite similar in spite of the difference in volume between those two cases. However, the different volumes imply that the temperature field is not the same in each sample for a same period of time since the same thermal treatment (temperature program) is applied in both cases. Therefore, it is important to calculate the degree of supercooling for non-uniform temperature cases. The supercooling degree is defined using the lowest temperature one may observe in the supercooled liquid before it crystallizes as explained in Section 4.2.1. Therefore, to determine the degree of supercooling, it is necessary to find the lowest temperature in the sample at the measured nucleation time using numerical simulation. For the 3 mL samples (Cases I and II),
<figure><img src="image_18.png" /><figcaption>Fig. 12. Probability density function of nucleation time; (a) PDF of the nucleation time for 3 mL – 1 °C/min, (b) 3 mL – 0.166 °C/min and (c) 500 mL – 1 °C/min.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 13. Predicted time and temperature for larger volumes (a) average time at which nucleation is triggered compared to the average experimental results (b) estimated degree of supercooling from nucleation time.</figcaption></figure>

the lowest temperature was observed at the bottom of the container, while for the 500 mL samples, it was found at the top near the inner wall. This behavior is explained in detail in Section 4.2.2. Fig. 11 illustrates these results. For all three experimental cases, the degree of supercooling is shown in Fig. 13b. This predicted degree of supercooling from the statistical model is compared with the degree of supercooling obtained from the experiments. The error between predicted and experimental degree of supercooling for cases I, II and III is 0.82, 7.7, and 5.36 %, respectively. This confirms that the model is adequately validated for sample volumes up to 500 mL. The predicted degree of supercooling in Fig. 13b has no uncertainty bounds. The uncertainty bounds were originally calculated for predicted nucleation time. Since CFD was used to determine the supercooling degree from the analysis of each geometry at the predicted nucleation time, it was not possible to obtain a relevant estimate of the uncertainty bounds for the predicted degree of supercooling.

### 4.3.3. Predicting large volume supercooling with a statistical model developed for uniform temperature distribution samples

In a previous section “2.2. Solution of the Model for Samples with Uniform Temperature Distribution”, a solution to the statistical model is developed for the samples with uniform temperature distribution. In this section, this solution is applied to the large-volume samples with non-uniform temperature distribution, using Eq. (14). The results are compared with the experimental results. The experimentally obtained

<figure><img src="image_20.png" /><figcaption>Fig. 14. Comparison of the degree of supercooling obtained experimentally and predicted by uniform and non-uniform temperature distribution statistical models.</figcaption></figure>

nucleation temperature is compared with the predicted temperature in Fig. 14. The difference found between experimental and modelling (uniform temperature model) results in cases I (3 mL, 1 °C/min), II (3 mL, 0.166 °C/min) and III (500 mL, 1 °C/min) is 14.2 %, 10.68 % and
<table><caption>**Table 6**
Seven different cases to analyze DSC data based on the number of samples and cycles.</caption><thead><tr><th rowspan="2"></th><th colspan="2">30 mg/1 °C/min</th><th colspan="2">1.5 mg/1 °C/min</th><th colspan="2">30 mg/0.166 °C/min</th><th colspan="2">1.5 mg/0.166 °C/min</th><th rowspan="2">Total number of experiments</th></tr><tr><th>Samples</th><th>Cycles</th><th>Samples</th><th>Cycles</th><th>Samples</th><th>Cycles</th><th>Samples</th><th>Cycles</th></tr></thead><tbody><tr><td>Case 1</td><td>5</td><td>1</td><td>5</td><td>1</td><td>5</td><td>1</td><td>5</td><td>1</td><td>20</td></tr><tr><td>Case 2</td><td>10</td><td>1</td><td>10</td><td>1</td><td>5</td><td>1</td><td>5</td><td>1</td><td>30</td></tr><tr><td>Case 3</td><td>15</td><td>1</td><td>15</td><td>1</td><td>5</td><td>1</td><td>5</td><td>1</td><td>40</td></tr><tr><td>Case 4</td><td>25</td><td>1</td><td>15</td><td>1</td><td>5</td><td>1</td><td>5</td><td>1</td><td>50</td></tr><tr><td>Case 5</td><td>25</td><td>3</td><td>15</td><td>3</td><td>5</td><td>3</td><td>5</td><td>3</td><td>150</td></tr><tr><td>Case 6</td><td>25</td><td>6</td><td>15</td><td>6</td><td>5</td><td>6</td><td>5</td><td>6</td><td>300</td></tr><tr><td>Case 7</td><td>1</td><td>12</td><td>1</td><td>12</td><td>1</td><td>12</td><td>1</td><td>12</td><td>48</td></tr><tr><td>Case 8</td><td>All data</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>490</td></tr></tbody></table>

<figure><img src="image_21.png" /><figcaption>**Fig. 15.** Error in the predicted nucleation time and experimental results for different cases – The highest error was found in case-7 when one sample, with several cycles, were considered; the error was decreased in case-6 when the results up to 6 cycles were considered on the analysed samples.</figcaption></figure>

10 %, respectively. The results indicate that, in the analysed range of volume, the error will vary between 10 % and 14 %, making it more relevant to use the statistical model for non-uniform temperature distribution to predict the supercooling degree of those samples. As one may observe, the large volume samples behavior, analysed with statistical model for uniform temperature distribution samples, was not predicted with the same degree of accuracy as it was when the statistical model for non-uniform temperature distribution was used (see Section 4.3.2). This is due to the fact that the statistical model for uniform temperature distribution does not take into account temperature variations within the sample and it highlights the limits imposed by the hypothesis of this model on the quality of the predictions obtained.

**5. Data analysis to establish protocol for DSC experiments**

The first and most important step to predict the degree of supercooling is to obtain data by using DSC devices. One may discuss what can be the minimum number of DSC samples, and the minimum number of cycles to correctly predict the degree of supercooling for large volume samples. In order to establish this protocol, the statistical model was applied to different combinations of DSC data, ranging from fewer samples and fewer cycles to larger numbers of samples with more cycles. Here, the cycles refer to the number of repetitions of experiments on the same sample inside the DSC.

**Table 6** summarizes different combinations of the datasets, ranging from fewer samples to all samples. These cases are designed in such a manner that initially only a few samples are taken from the data, and

only 1 cycle is analysed (case 1). In the second step, the number of samples increases with again only 1 cycle considered (cases 2, 3, 4). Later, in case 5 and 6, all samples are used along with three and six number of cycles, respectively. Finally in case 8, all the datasets are included to analyze the error in predictions. In case 7, the dataset includes only one sample repeated several times.

The data collection methods can be analysed through predicted results for different data collection cases and compared to the experimental results (Figs. 15 and 16). These results demonstrate the inherent difficulty in predicting the true degree of supercooling due to the erratic nature of nucleation.

In case 7, the error (difference between model prediction and experimental data) was found very high. This clearly indicates that the degree of supercooling measured from a single sample, even when repeated over several cycles, may fail to predict the actual supercooling behavior. Therefore, it is crucial to measure the degree of supercooling of several samples rather than one sample.

Case 1 causes an error quite high. This result shows that using 5 samples per experimental condition but with one repetition is not sufficient. From case 2 to case 3, error is decreasing while the total number of samples increases, considering all the experimental conditions in terms of volume and cooling rate. Therefore, when we go from a total of 20 (case 1) to 40 (case 3) samples, with 1 cycle each, the error of the model reduces sharply.

Then, from case 3 to 5, a limited improvement in the model prediction is observed by keeping approximately the same number of samples and increasing the number of cycles.
<figure><img src="image_22.png" /><figcaption>Fig. 16. Error in the estimated degree of supercooling at the predicted nucleation time and experimental results for different cases – the highest error was found in case-7, when one sample, with several cycles was considered, the error was decreased in case-6 when results up to 6 cycles were considered on the analysed samples.</figcaption></figure>

Finally, repeating 6 cycles while also using multiple samples leads to a significant reduction in error, as observed in the results of case 6 which presented an error similar to that of case 8. The results suggest that the most reliable methodology to characterize supercooling is to consider a sufficiently large number of samples (in this study, combined for all cases, more than 40 samples, as in case 3) combined with several repetitions (in this study, at least 6 repetitions as in case 6). Besides, case 6 required 39 % fewer experiments than case 8, yet still provided accurate results. This highlights that those insights on the optimization of the number of DSC analysis required for investigating the supercooling degree can considerably reduce the number of experiments. However, it is

not simple to generalize the minimum number of samples and cycles from this study, since the required values may vary depending on the material under investigation.

This dependence arises because the required dataset size is strongly influenced by the variability of the nucleation rate. When the nucleation rate shows less variation across different volumes and cooling rates, reliable power-law fitting can be achieved with fewer samples and cycles dataset (according to the protocol presented in Section "4.1. Characterization using DSC"). In contrast, when the nucleation rate is more scattered or erratic (as is often the case in supercooling), a larger dataset is required to minimize the influence of outliers and obtain accurate

<figure><img src="image_23.png" /><figcaption>Fig. 17. Standard deviation and standard error of the DSC data arranged in different cases.</figcaption></figure>
model parameters. Fig. 17 shows the standard deviation and standard error in each case. Standard deviation indicates the spread of the data as explained in Section 4.2.3. Therefore, we can see that by increasing the dataset, the standard deviation decreases. Besides, the standard error decreases with the increasing number of samples.

## 6. Conclusion

Designing a robust TES system requires the knowledge of the correct degree of supercooling of the PCM used. However, the supercooling characterized in the laboratory using DSC cannot be used directly for real TES systems, in particular due to the volume dependence of the supercooling. Hence, it is important to predict the degree of super-cooling of large volumes of the PCMs. In this work, the degree of supercooling of type 2 pure water is characterized by DSC, and predictions are made for different orders of volumes by using a statistical model based on a non-homogeneous Poisson process.

* The degree of supercooling for 1.5 mg and 30 mg DSC samples at two different cooling rates (1 and 0.166 °C/min) was characterized using DSC.

* This data is used to predict the degree of supercooling of two larger volumes (3 mL and 500 mL) in samples where a thermal gradient is observed.

* First, the average nucleation time is obtained using a statistical model, and then the lowest temperature inside the sample (super-cooling) is estimated in order to predict the supercooling degree of the sample.

* The maximum error obtained in the prediction of nucleation time and degree of supercooling was 4.8 % and 7.7 %, respectively, for 3 mL volume at 0.166 °C/min.

Data analysis was conducted to suggest a protocol for DSC experiments. It was found that in order to make predictions with the least error it is important to characterize several samples and also to characterize each sample several times to overcome the erratic nature of super-cooling. The number of samples and cycles may vary depending on the nature of the PCMs, but the protocol of having several samples and cycles remains the same. In this study, the minimum number of total combined samples for all four cases is found to be 40, and the minimum number of cycles (repetitions) is found to be 6.

It is important to note that to increase the validity of this model and to test its limits, it is recommended to apply this framework to a wider range of materials and further higher volumes. To increase its robustness of the model, it is recommended to develop a statistical model incorporating other important parameters, such as surface roughness of the container and purity of the PCM.

## Acronyms

CDF Cumulative distribution function

CFD Computational fluid dynamics

DSC Differential scanning calorimetry

LHTES Latent heat thermal energy storage

NHPP Non-homogeneous Poisson process (NHPP)

PCM Phase change material

PDF Probability distribution function

SHS Sensible heat storage

TES Thermal energy storage

Cp Specific heat capacity (J.kg⁻¹.K⁻¹)

D Diameter (m)

g Gravitational acceleration vector (m s⁻²)

h Heat transfer coefficient (W.m⁻².K⁻¹)

I Identity tensor

J Nucleation rate (Events.m⁻³.min⁻¹)

k Thermal conductivity (W.m⁻¹.K⁻¹)

Pr Prandtl number

P Pressure (N.m⁻²)

Re Reynolds number

t Time

T Temperature (K)

Tm Solid-liquid equilibrium temperature (K)

V Volume (m³)

u Velocity vector (m.s⁻¹)

Qwall Heat flux (W.m⁻²)

ρ Density (kg.m⁻³)

μ Dynamic viscosity (N.s.m⁻²)

λ Rate parameter

λeff Effective rate parameter

θs Degree of supercooling (K)

β Cooling rate (°C.min⁻¹)

χ Survival function

σtexp Error in the nucleation time for validation experiments

σ Standard deviation

n Total number of the samples (validation experiments)

σTexp Error in the degree of supercooling of validation experiments

δTC Error associated with the thermocouple

δs Error in validation of the numerical model

σtp Error bounds of the predicted nucleation time with a non-uniform temperature distribution model

σθs Error bounds of the predicted degree of supercooling with a uniform temperature distribution model

## CRediT authorship contribution statement

**Jawad Rabbi:** Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization. **José Lara Cruz:** Writing – review & editing, Supervision, Project administration, Formal analysis. **Jean-Pierre Bédécarrats:** Writing – review & editing, Supervision, Project administration, Funding acquisition, Formal analysis.

## Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work the author(s) used ChatGPT in order to improve writing, to make it more understandable. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the published article.

## 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.

## Acknowledgement

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 945416 and was carried out through the research project KSP PCM-STORE (308847) supported by the Research Council of Norway and industry partners. PCM-STORE aims at building knowledge on novel PCM technologies for low- and medium temperature thermal energy storage systems. The authors would like to express their gratitude to Jorge Salgado-Beceiro, Magnus Rotan, Ragnhild Sæterli, Fride Vullum-Bruer, and Alexis Sevault at SINTEF Energy Research for their valuable insights and discussions that contributed to this work.
# Data availability

Data will be made available on request.

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