# Experimental study of thermal convection and melting dynamics in a Phase Change Material (PCM) embedded in a solid foam

Nicolò R Sgreva, Christel Métivier, Anthony Teixeira, Sébastien Leclerc

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Nicolò R Sgreva, Christel Métivier, Anthony Teixeira, Sébastien Leclerc. Experimental study of thermal convection and melting dynamics in a Phase Change Material (PCM) embedded in a solid foam. International Journal of Heat and Mass Transfer, 2025, 251, pp.127283. 10.1016/j.ijheatmasstransfer.2025.127283. hal-05154992

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# Experimental study of thermal convection and melting dynamics in a Phase Change Material (PCM) embedded in a solid foam

<figure><img src="image_3.png" /></figure>

Nicolò R. Sgreva<sup>a, *</sup>, Christel Métivier<sup>a</sup>, Anthony Teixeira<sup>a,b, ∗</sup>, Sébastien Leclerc<sup>a, †</sup>

## ARTICLE INFO

Keywords:

Thermal convection

Phase change materials

Composite systems

Melting interface

## ABSTRACT

We investigate the role of thermal convection in the melting of a Phase Change Material (PCM) embedded in a highly porous plastic foam. Using Magnetic Resonance Imaging (MRI), we examine the melting of the composite PCM system heated from below and cooled from above. Our results show that, under convective conditions, the melting interface develops a wavy surface shaped by thermo-convective flows. Convection significantly enhances the melting rate, and can lead to final liquid heights more than twice those observed in the conductive regime, depending on the imposed temperature boundary conditions at the top and bottom surfaces. Flow field analysis reveals different convection patterns, from cross-rolls to hexagonal and square convective cells, with local velocity fields strongly influenced by the porous medium geometry. Finally, heat transfer analysis shows a not-clear relation between Nusselt number (Nu) and porous medium Rayleigh number (Ra*) for Ra* > Ra*_c, where Ra*_c is the critical Rayleigh number for convection. However, when Nu is normalized by the temperature difference ratio (Θ) and plotted as a function of Ra* and the phase change ratio A (the average height of the solid PCM relative to that of the liquid PCM), a good data collapse is achieved, leading to the scaling Nu/Θ ~ (Ra*/A)0.5. These findings provide a basis for further experimental and numerical studies on heat and mass transfer in more complex composite systems.

## 1. Introduction

The solid-to-liquid (and vice versa) phase change is a fundamental process that influences the dynamics of numerous systems across various fields. In engineering, Phase Change Materials (PCMs) are particularly valued for their ability to store and release large amounts of energy through latent heat, making them key components in thermal energy storage and thermal management systems [1]. Due to these properties, PCMs find widespread application in diverse sectors, including battery thermal management [2], cold chain logistics [3], and healthcare [4]. They also play a significant role in improving the energy efficiency of buildings [5], where they are integrated into floors, roofs, wallboards, and other structural components to help maintain indoor thermal comfort by mitigating temperature variations. In this context, when the external temperature rises above the PCM's melting point, the material undergoes phase transition, storing a significant amount of latent heat. With appropriate system design and insulation to minimize heat losses, the stored energy is then gradually released when the external temperature drops below the phase transition temperature.

Despite their advantages, practical drawbacks such as liquid phase leakage and not-ideal thermal conductivity often limit PCMs performance in applications. To address these issues, PCMs are frequently

combined with solid supports [6], including macroscopic open-cell solid foams, regular porous grids, or packed beds of spheres. These composite systems help minimize liquid PCM leakage [7] and improve the solidification process by enhancing guest-host interactions [8]. Moreover, depending on the application and the type of composite, they can enhance the system's thermal performance by improving heat transfer [9] or provide a mechanically stable medium with low thermal conductivity for thermal insulation systems [10]. The overall thermal behavior of these systems depends on multiple factors, including the geometry and thermal properties of the porous matrix [11], the rheological behavior of the liquid PCM [12], variations in the thermophysical properties of the PCM during the phase transition process [13], and the capacity of the porous support to accommodate volume changes associated with phase transitions [12].

In systems with highly conductive porous media such as metal foams, enhancing the rate of heat transfer is a primary objective. Beyond increasing overall thermal conductivity [14], the geometry of the matrix, including pore size, surface area, and pore density, plays a crucial role in thermal performance. For a given porosity, smaller

<footnote><sup>a</sup> Université de Lorraine, CNRS, LEMTA, F-54000 Nancy, France</footnote>

<footnote><sup>b</sup> Institut de Mécanique des Fluides de Toulouse, IMFT, Université de Toulouse, CNRS, Toulouse, France</footnote>

<footnote>* Corresponding author.
E-mail address: nicolo.sgreva@univ-lorraine.fr (N.R. Sgreva).</footnote>
pores provide a larger surface area for heat exchange, improving con-
ductive heat transfer during phase change [7], while larger pores
enhance convective effects, allowing the system to reach steady-state
temperatures more rapidly [15]. Porosity gradients further affect both
the melting rate and the development of thermal convection [16],
though the enhanced heat conduction in metallic matrices generally
hampers convective flows. Advanced porous structures based on triply
periodic minimal surfaces structures offer additional improvements
in heat transfer performance compared to conventional porous me-
dia [17]. On the other hand, in low-conductivity porous media, such
as thermally insulating foams, the role of the porous matrix shifts from
heat transfer enhancement to providing structural support, improving
mechanical properties, and minimizing heat losses to ensure long-term
thermal energy storage [18]. In these systems, highly porous structures
are beneficial for increasing latent heat storage capacity. However, a
decrease in pore size can reduce permeability, which may limit the flow
of liquid PCM and suppress natural convection, thereby affecting the
efficiency of the phase change process.

When a PCM-based system is subjected to a temperature gradient, heat transfer occurs through conduction, convection, or a combination of both mechanisms. In systems without porous media, the interplay between these processes becomes particularly significant during the melting process, where convection can substantially enhance phase change by improving the melting rate and increasing the melted fraction [16]. For example, different studies have showed that in a horizontal PCM layer heated from below and cooled from above, convective flows significantly enhance the transient growth of the melting interface, leading to liquid layer heights up to four times greater than those observed under purely conductive conditions [19,20]. Additionally, since the system is cooled from above, a temperature gradient is induced in the upper solid PCM layer, which behaves as an imperfect thermal conductor. As showed by Davis et al. [21], this configuration can lead to different convective patterns within the liquid layer, depending on the Rayleigh number in the liquid layer (Ra_l), and the phase change ratio (A), which is the ratio of the solid PCM layer height to the liquid PCM height. When the solid layer is thin (small A), convection organizes into roll-like patterns, while thicker solid layers (large A) lead to hexagonal patterns. This occurs because the increasing thickness of the solid layer induces a degree of vertical asymmetry sufficient to establish hexagonal convection patterns. The interplay between thermo-convective flows and the morphology of the melting interface has also been extensively explored by numerical simulations. Favier et al. [22], for example, showed that not-flat melting interface topography can lead to baroclinic flow, destabilizing the conductive state even for subcritical conditions. Similarly, Madruga and Curbelo [23] and Satbhai et al. [24] showed that when convection develops, buoyancy flows lead to wavy melting interfaces, with higher liquid heights over upwelling regions and lower heights over downwelling regions.

In a composite system where the phase change takes place within
a porous medium, investigating the melting rate, melting interface
behavior, and convective flow of the liquid PCM is complicated by
the presence of the porous matrix. Experimentally, the opacity of
the porous structure limits the use of conventional optical techniques
for flow visualization, making it more difficult to develop detailed
analyses. As a result, most experimental studies on heat transfer in
such systems rely on invasive local measurements, such as thermo-
couples embedded in the system [25], or non-invasive techniques that
gather information at the boundary surfaces, such as infrared ther-
mography [26]. However, both approaches have significant drawbacks.
Invasive methods can have a strong impact on the flow of melted PCM
and may act as nucleation sites, thereby promoting heterogeneous crys-
tallization during solidification. Measurements at boundary surfaces, on
the other hand, are often influenced by heat losses at the walls [27] and
defects in the porous medium that influence flow conditions in these
regions [28]. Moreover, obtaining precise local velocity measurements
of the melted PCM remains challenging, with most studies limited to

macroscopic flow rate measurements [7]. Numerical simulations of
PCMs in porous media are also subject to difficulties, including the
accurate modeling of the phase transition [22] and phenomena at
the solid–liquid interfaces [29]. Simulations also face complications
arising from the multi-scale nature of the system, where the presence of
different pore scales can significantly impact local flow dynamics [30].
Additionally, coupling the evolution of crystalline structures with the
mesoscale flow of liquid PCM near the phase transition temperature
remains a major challenge [31].

Despite the extensive literature on composite systems, detailed ex-
perimental investigations of the melting process under conditions dom-
inated by convection, particularly in systems heated from below and
cooled from above, remain limited. Key aspects, including accurate
temperature and flow velocity measurements during phase transitions,
as well as the characterization of local flow dynamics and their interac-
tions with the porous matrix, are still poorly investigated. Additionally,
to our knowledge, experimental studies focusing on the time-evolution
and final shape of the phase change interface within the bulk of the
composite system, away from boundary walls, are very rare. This
lack of analysis limits the ability to fully characterize the melting
process in porous media under convective conditions, despite its critical
importance in numerous applications and fields.

In this study, we present an experimental investigation that simul-
taneously addresses thermal convection, phase change, and flow in
porous media by measuring local flow properties during melting. The
objective is to improve the understanding of the physical mechanisms
at play when these phenomena are coupled in a composite system.
To this end, we employ a robust experimental approach based on
Magnetic Resonance Imaging (MRI). The investigated system consists of
a paraffin embedded within a plastic solid foam with high porosity and
a thermal conductivity comparable to that of the PCM. These character-
istics facilitate the development of thermal convection within the liquid
layer. Our experimental setup imposes continuous cooling at one of the
system's surfaces, which can lead to the formation of a stationary melt
front within the composite system. MRI is used to track the transient
evolution of the melting interface in the bulk of the system and to
analyze flow velocities in the melted layer under convective conditions.
The final shape of the interface and the overall heat transfer are also
investigated under different conditions. Beyond providing fundamen-
tal insights into melting behavior under such conditions, the studied
configuration is also representative of applications in which PCMs are
integrated with additional heat removal mechanisms [e.g., 32].

The paper is structured as follows. In Section 2, we describe the porous medium and PCM used in our experiments, along with the experimental methodologies and protocols. In particular, MRI was employed to analyze both the velocity fields in the melted PCM and the shape and position of the melting interface. Section 3 presents the results of the melting process, focusing on the melting front under conductive or convective conditions. The analysis of flow fields during steady convection is detailed in Section 4. A discussion of the results is provided in Section 5, followed by final remarks in Section 6.

## 2. Materials and methods

### 2.1. Porous medium and PCM

All relevant thermophysical properties for this work are listed in Table 1.

We used Hexadecane as PCM (supplied by Sigma-Aldrich, 99%
purity). It was used as received, without any further purification or
removal of impurities. A complete characterization of the thermophysi-
cal properties of hexadecane, both in liquid and solid phase and during
the solid–liquid phase change, is available in [13,33]. A key property
of PCMs is their melting temperature, which is crucial for determining
both Rayleigh number in the liquid layer (Ra_l) and Nusselt number
(Nu) when describing the convective regime and heat transfer. For
<table><caption>**Table 1**

Thermophysical properties of hexadecane [13,33] and solid foam [37].</caption><thead><tr><th>System</th><th>Property</th><th>Value</th></tr></thead><tbody><tr><td rowspan="8">Hexadecane</td><td>Density, ρ</td><td>770 (kg m<sup>-3</sup>)</td></tr><tr><td>Viscosity, η</td><td>0.003 (Pa s)</td></tr><tr><td>Thermal conductivity (liquid phase), λ<sub>l</sub></td><td>0.150 (W m<sup>-1</sup> K<sup>-1</sup>)</td></tr><tr><td>Thermal conductivity (solid phase), λ<sub>s</sub></td><td>0.341 (W m<sup>-1</sup> K<sup>-1</sup>)</td></tr><tr><td>Thermal expansion, β</td><td>8.86 × 10<sup>-4</sup> (°C<sup>-1</sup>)</td></tr><tr><td>Thermal diffusivity (liquid phase), κ</td><td>8.70 × 10<sup>-8</sup> (m<sup>2</sup> s<sup>-1</sup>)</td></tr><tr><td>Heat capacity (liquid phase), c<sub>p</sub></td><td>2.24 (kJ kg<sup>-1</sup> K<sup>-1</sup>)</td></tr><tr><td>Melting temperature, T<sub>m</sub></td><td>17.9 (°C)</td></tr><tr><td rowspan="2">Foam</td><td>Permeability, K</td><td>2.836 × 10<sup>-7</sup> (m<sup>2</sup>)</td></tr><tr><td>Porosity, ε</td><td>0.83 (–)</td></tr><tr><td rowspan="5">Foam + hexadecane</td><td>Thermal conductivity, λ<sub>PM</sub></td><td>0.192 (W m<sup>-1</sup> K<sup>-1</sup>)</td></tr><tr><td>Effective thermal conductivity (liquid PCM), λ<sub>eff</sub></td><td>0.157 (W m<sup>-1</sup> K<sup>-1</sup>)</td></tr><tr><td>Effective thermal conductivity (solid PCM), λ<sub>eff</sub></td><td>0.310 (W m<sup>-1</sup> K<sup>-1</sup>)</td></tr><tr><td>Effective volumetric heat capacity (liquid PCM), ρc<sub>p,eff</sub></td><td>1.75 × 10<sup>6</sup> (J m<sup>-3</sup> K<sup>-1</sup>)</td></tr><tr><td>Effective thermal diffusivity (liquid PCM), κ<sub>eff</sub></td><td>8.98 × 10<sup>-8</sup> (m<sup>2</sup> s<sup>-1</sup>)</td></tr></tbody></table>

hexadecane, we use a melting temperature ($T_m$) of 17.9 °C, which is the average value derived from different sources [13,34–36]. Small deviation in the chosen value of $T_m$ (up to a maximum of 1.5%) result in negligible variations in $Ra_l$ and $Nu$.

The porous medium consists of a solid structured plastic foam with Kelvin-like unit cells (Fig. 1), fabricated by 3D printing using a light-sensitive photopolymer plastic. The final sample consists of a cylindrical disk of diameter $D = 0.114$ m and thickness $H = 0.026$ m. Each individual Kelvin cell has a diameter of 6.5 mm (the distance between two opposite square faces) and cylindrical struts with a diameter of 1.2 mm. This results in a pore-length scale ($L_p$) of 7.7 mm, defined as the sum of one unit cell’s effective diameter and the strut size. The foam sample has a porosity of 0.83, a tortuosity of 1.3, a permeability $K = 2.84 \times 10^{-7}$ m<sup>2</sup>, and a thermal conductivity of $\lambda_{PM} = 0.192$ W m<sup>-1</sup> K<sup>-1</sup> [37].

The effective thermal conductivity of the hexadecane-saturated foam is estimated using the Hashin–Shtrikman model [38], an equivalent model based on the volume fraction and conductivity of the components, suitable for homogeneous, isotropic, two-phase composite materials with relatively close thermal conductivities. The effective volumetric heat capacity is obtained as the weighted volumetric average of the heat capacity of the PCM and the solid plastic.

At $T = 25$ °C, the hexadecane-saturated porous medium has a moderate Prandtl number, $Pr = \eta/(\rho\kappa_{eff}) = 43.5$, and a small Stefan number, defined as the ratio of sensible heat to latent heat, $St = (c_{p,eff}\Delta T)/\Delta H_m = 8.9 \times 10^{-3}$, for $\Delta T = 1$ °C and a latent heat of melting of $\Delta H_m = 232.55$ kJ kg<sup>-1</sup> [13]. Since $St \ll 1$, heat transfer is dominated by latent heat rather than sensible heat.

## 2.2. Experimental setup and melting protocol

As shown in Fig. 1, the PCM-saturated foam sample was enclosed in a PMMA cylinder with the same dimensions (inner diameter and height) as the foam cylinder. Good contacts between all surfaces were ensured. Two small holes (approximately 5 mm in diameter) were drilled on opposite sides of the PMMA cylinder, to which two plastic tubes were attached. These tubes were used to fill the setup with liquid PCM, minimizing the introduction of air bubbles during preparation. Moreover, an additional amount of PCM inside both tubes accommodated the PCM volume variations due to phase change and temperature changes within the experimental cavity.

All melting experiments were conducted from a totally frozen system, maintained at an initial temperature $T_{init} \approx 6.0$ °C $< T_m$ for at least 12 h. This value of $T_{init}$ was chosen to ensure complete solidification of the PCM throughout the volume. We have not observed differences in the melting process for a $T_{init}$ up to $\approx 10.0$ °C. A temperature difference $\Delta T = T_{dw} - T_{up}$ was then applied to the system by imposing controlled temperatures on the lower surface ($T_{dw}$) and upper surface ($T_{up}$) by water flows from temperature-controlled water baths. The value $T_{dw}$

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Experimental setup. The cavity dimensions match those of the foam disk, with a diameter $D = 0.114$ m and a height $H = 0.026$ m.</figcaption></figure>

represents the average of two thermocouple measurements (with 1% uncertainty) at the inlet and outlet of the lower water flow, while $T_{up}$ is obtained in the same way for the upper water flow. Throughout all experiments, the imposed temperatures remained above $T_{init}$. To ensure good thermal contact between the water flows and the PCM-saturated foam, we used two 3.1-mm-thick sapphire plates with relatively high thermal conductivity (35–40 W m<sup>-1</sup> K<sup>-1</sup>). Finally, the PMMA cylinder was insulated with a thick foam layer to minimize heat loss to the external environment. Further details on heat losses and the influence of environmental temperature on the experiments can be found in the supplementary materials.

## 2.3. Magnetic resonance imaging

Magnetic Resonance Imaging (MRI) was employed to resolve the structure of the porous medium and to identify the phase of the PCM, allowing for precise localization and characterization of the melting interface. Additionally, MRI was used to perform magnetic resonance velocimetry experiments [39] to obtain local flow velocity measurements within the liquid PCM during convection.

MRI experiments were performed with a Bruker Avance Biospec 24/40 spectrometer equipped with a 200-mm wide gradient coil delivering a 200 mT m<sup>-1</sup> maximum gradient strength and a 160-mm wide quadrature resonator. The horizontal magnetic field was 2.34 T and the proton resonance frequency 100.3 MHz. The spectrometer was located in a temperature-controlled room at 24 °C. All MRI images were acquired with a slice thickness of 2 mm and a field of view
of 120 × 120 mm. We acquired images at different cross-sections. Horizontal images were acquired in planes parallel to the top and bottom surfaces (xy-plane) at different heights (i.e., at H/2, H/4, and H/6). Vertical images were acquired along the plane that crosses the cavity perpendicularly through its center (zx-plane).

For phase identification, we used Rapid Acquisition with Relaxation Enhancement (RARE) experiments [40]. This type of acquisition takes advantage of the fact that solid phases (both porous medium and solid PCM) have a very fast transverse relaxation time, resulting in the absence of signal in the acquired image. For this reason, RARE images provide a clear contrast between the liquid and solid phases. We used an echo time of 14 ms, repetition time of 1000 ms, RARE factor of 8, and we acquired images at resolutions of 128 × 128 and 256 × 256 pixels.

For the flow velocity field in the melted PCM, we used spin-echo imaging sequences together with flow encoding gradient pulses. We followed the same protocol described in our previous work [41]. We used a gradient pulse duration of 4 ms, a gradient separation time of 12 ms, an echo time of 28 ms, a repetition time of 1000 ms, a minimum flow encoding gradient pulse intensity of 0 mTm⁻¹, and a maximum gradient intensity of 175 mTm⁻¹. Images were acquired with a 128 × 128 pixel resolution, corresponding to a voxel of 0.94 × 0.94 × 2 mm³. Velocity images were then obtained from the difference between the images acquired at the maximum gradient pulse intensity and the images acquired at the minimum gradient pulse intensity. The final images were normalized to the RARE images acquired at the same resolution to account for partial volume effects. As indicated in [41], with this protocol, the estimated uncertainty for the velocity is 7 × 10⁻⁵ m s⁻¹.

### 3. The melting process

This section focuses on the influence of different heat transfer mechanisms on the melting process in a fluid-saturated porous foam. We present the results for the melting process in the conductive regime, followed by those for the convective regime. In both cases, the analysis focuses on the transient behavior of the melting interface and the final shape of the solid-liquid interface at steady-state. Notably, the interface is evaluated within the bulk of the system, limiting effects at the boundaries, such as heat losses or preferential flow along the PMMA sidewall, on the proper characterization of the interface.

#### 3.1. Conductive regime

Melting experiments in the purely conductive regime were conducted starting from a system initially frozen at $T_{init} \approx 6.0$ °C $< T_m$. The system was then heated from above and cooled from below to prevent the onset of convection. Data on the transient evolution of the melting interface until steady state was reached were collected by acquiring 40 vertical RARE images at intervals of 571 s from the onset of heating, over a total duration of approximately 6 h.

An example of the development and time evolution of the melting interface is shown in Fig. 2a for $T_{up} = 25.4$ °C and $T_{dw} = 10.4$ °C. To describe the progressive increase in the melt fraction over time, we use the following dimensionless quantities: the normalized liquid height $h^*$, the phase change ratio $A$, and the Fourier number $Fo$, which represents the dimensionless time. They are defined as:

$$h^* = \frac{h_l}{H}, \quad A = \frac{H - h_l}{h_l}, \quad Fo = \frac{t \kappa_{eff}}{H^2}, \qquad (1)$$

where $h_l$ is the average liquid height relative to the heating surface, $H$ is the height of the cavity, $t$ is the time in seconds from the start of heating, and $\kappa_{eff}$ is the effective thermal diffusivity (provided in Table 1).

For the purely conductive regime, we recover the conditions of the Stefan problem. Indeed, Fig. 2a shows that the transient evolution of

<figure><img src="image_5.png" /><figcaption>Fig. 2. (a) Transient evolution of the melting interface in the conductive regime. $h^*$ is the mean normalized height of the liquid PCM relative to the heating surface (in this case, the top surface). The black solid line corresponds to $h^* \sim Fo^{0.5}$. The inset shows the final melting stage, where the yellow line highlights the melting interface. (b) Reconstructed 3D melting interface. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

the melting interface scales as expected, following $h^* \sim Fo^{0.5}$ [19]. The shape of the melting interface at steady-state is shown in the vertical RARE image acquired at the center of the cavity (see inset of Fig. 2a). From this 2D image, the melting interface exhibits a well-defined flat profile, with the liquid height resulting in $h_l = 8.2 \times 10^{-3}$ m with $u_a(h_l) = 0.3 \times 10^{-3}$ m. The standard uncertainty ($u_a$) for $h_l$ is estimated from its standard deviation along this flat profile.

To better evaluate the global shape of the melting surface at the steady-state, we reconstructed its 3D shape from 20 parallel vertical RARE images, acquired along the cavity's diameter at the same time. The images were spaced 4 mm apart and covered the central area of the cavity up to about 17 mm from the sidewall along the y direction. Missing data were interpolated using a moving average method over a 9-pixel window (corresponding to about 4 mm). The resulting melting surface (Fig. 2b) is found to be flat across both the bulk of the sample and near the lateral walls. The mean liquid height evaluated from the 3D figure is $h_l = 7.5 \times 10^{-3}$ m with $u_a(h_l) = 0.2 \times 10^{-3}$ m, in good agreement with the value estimated directly from the 2D vertical image at the center of the cavity. The flatness of the entire melting interface suggests that, under our conditions and within MRI resolution, the porous medium has a negligible effect on shaping the interface. This can be due to the similar thermal properties of the plastic foam and the PCM.

#### 3.2. Convective regime

Melting experiments in the convective regime were conducted following a protocol similar to that used for the conductive case. The system was initially frozen at $T_{init} \approx 6.0$ °C and then, in this case, heated from below and cooled from above. The imposed bottom temperature ranged from 19.9 °C to 23.1 °C, while the upper temperature ranged from 6.3 °C to 17.5 °C, so that $T_{up}$ remained below $T_m$ throughout the experiments. In the case of convection, the flow regime is characterized by the porous medium Rayleigh number ($Ra^*$), defined as

$$Ra^* = Ra_l \times Da = \frac{\rho g \Delta T_l K h_l}{\eta \kappa_{eff}}, \qquad (2)$$
<figure><img src="image_6.png" /><figcaption>Fig. 3. Transient melting interface evolution for a system heated from below and in presence of convection.</figcaption></figure>

where Ra_l is the Rayleigh number for a non-porous liquid layer, Da = K/h_l^2 is the Darcy number, and ΔT_l = T_dw - T_m is the temperature difference across the melted layer. In Eq. (2), K is the permeability of the porous layer, h_l is the height of the melted layer, ρ is the density of the liquid PCM, η is its viscosity, and g is the acceleration due to gravity. For a fully liquid layer, above the critical value Ra_c^* = 4π^2, the conductive regime becomes unstable, and convection begins [42].

An example of the development and time evolution of the melting interface in the presence of convection is shown in Fig. 3. Here the applied temperatures were T_dw = 22.4 °C and T_up = 10.0 °C. Initially, the melting interface grows relatively slowly due to heat conduction, maintaining a flat profile across the cavity. The onset of convection is captured by a sharp change in the h^* vs. Fo curve: once convection begins, the melting interface grows more rapidly, scaling as h^* ~ Fo. This relationship characterizes the transient convective regime until a steady-state is reached, where a constant final mean height is achieved. Notably, the same scaling has been observed in a system with convection and phase change but without the porous matrix [22].

As in the previous case, we present the reconstructed 3D shape of the melting interface for a convective experiment in Fig. 4. This experiment was carried out under the following conditions: ΔT = 18.4 °C, T_dw = 23.1 °C, T_up = 10.7 °C, and Ra^* = 633.9. Unlike the conductive regime, in the presence of convection, the steady-state melting interface exhibits a distinct wavy surface caused by the thermo-convective flows within the liquid PCM layer. In this specific case, the mean liquid height is estimated to be h_l = 1.56×10⁻² m with a standard uncertainty of u_q(h_l) = 0.3 × 10⁻³ m. For the case in Fig. 4, the melting interface exhibits a periodic waviness. This pattern is characterized by a waviness spacing of 21.8 × 10⁻³ m, which corresponds to the average distance between adjacent interface uplifts, and an average amplitude of 6×10⁻³ m, defined as the mean vertical distance between adjacent interface uplifts and downlifts.

Finally, we report a scenario where convection was triggered in an already partially melted system. In this case, a system at steady-state with a liquid layer melted in the conductive regime was subjected to an increased ΔT in order to initiate convection. This scenario is particularly relevant for applications in thermal management systems using PCMs, where sudden temperature variations can lead to a rapid change in the heat transfer regime [7]. The evolution of the melting interface in this case is shown in Fig. 5. Here, the initially frozen system was melted under purely conductive conditions until reaching steady-state by imposing ΔT = 10.2 °C (T_dw = 20.0 °C, T_up = 9.8 °C). The maximum liquid height under these conditions is h^* = 0.12, and the melting interface has a flat profile. At time t = 4.96 × 10^4 s from the start of the experiment, the bottom temperature was increased to T_dw = 22.4 °C, while T_up remained constant. In response, the melting interface began

to grow again due to heat conduction and, upon reaching the critical condition of motion, also due to convection. As noted earlier, the onset of convection triggers a sharp increase in the interface growth rate, with a rapid transient growth (h^* ~ Fo) persisting until steady-state convection is achieved, at which point the mean liquid height stabilizes. The final liquid height reached in steady-state convection is compared in Fig. 5 with the theoretical liquid height for pure conduction under the same temperature gradient (h^*^cond). The normalized liquid height in stationary conduction is given by:

$$ \frac{h_l^{\text{cond}}}{H} = \frac{1 - \Theta}{1 - \Theta + \frac{\lambda_{\text{eff}_s}}{\lambda_{\text{eff}_l}} \Theta} \quad \text{with} \quad \Theta = \frac{\Delta T - \Delta T_l}{\Delta T} \qquad (3) $$

where λ_{eff} and λ_{eff_l} are the effective thermal conductivities of the solid and liquid layer, respectively, and Θ is the temperature difference ratio. Eq. (3) indicates that, under steady-state conditions, the heat flux φ at the melting interface is continuous and there is no latent heat contribution. That is, φ = φ_s = φ_l, where φ_s and φ_l are the heat fluxes from the solid and liquid layers, respectively.

As shown in Fig. 5, this theoretical liquid height was achieved just before convection began, highlighting the significant role of convection in our system. Convection facilitates melting more PCM in a shorter time than in the conductive-only case. In fact, at the end of the experiment shown in Fig. 5, the h^* achieved in steady convection exceeds h^*^cond/H by more than a factor of two.

## 4. Steady-state convection in the liquid layer

This section presents the results of steady-state convection within the melted liquid layer. MRI was used to obtain velocity fields in both horizontal and vertical planes across the fluid-saturated foam. MRI images of the horizontal plane were acquired at different heights within the cavity, depending on the liquid layer's thickness. To ensure steady-state conditions, images were acquired at least 12 h after the temperature change and repeated to confirm time-independence.

### 4.1. Convective patterns

An overview of the different convective patterns developed within the melted layer is given in Fig. 6. The figure presents the vertical component of the velocity field in both a horizontal and vertical plane within the sample.

Fig. 6a was obtained for ΔT = 16.2 °C (T_dw = 22.5 °C, T_up = 6.3 °C), resulting in A = 1.7 and Ra^* = 311.6. The vertical cross-section shows a wavy melting interface, shaped by upwelling (positive vertical velocity) and downwelling (negative vertical velocity) flows of the convective liquid PCM. In this case, convection is well developed, which is consistent with the relatively high Ra^*, and forms a narrow, cross-rolls “worm-like” pattern aligned with the porous medium's geometry. To characterize the convective rolls, we estimated the associated wavelength (λ_w) and the dimensionless wavenumber, a = (2πλ_w)/h_l. For Fig. 6a, we found λ_w = (1.9 ± 0.2) × 10⁻² m, corresponding to a = 3.2 ± 0.3. Although in this case Ra^* > Ra_c^*, the wavenumber value is close to the critical wavenumber for the onset of convection in a porous medium without phase change, i.e. a_c = π [42]. When considering phase change, however, the critical wavenumber obtained from linear stability analysis [43] is affected by the phase-change ratio A. For values of A and Da similar to those in our experiment, the critical wavenumber becomes a_c ≈ 2.7 [43]. The wavenumber obtained for Fig. 6a thus exceeds this critical value, consistently with what is expected for supercritical conditions without phase change, where convective rolls have a wavenumber larger than the critical one [42].

Furthermore, by relating λ_w to the pore-length scale (L_p), defined as the sum of the foam's unit cell diameter and strut size, we establish a link between the convection characteristic length scale and the length scale of the porous medium. For the convective pattern shown in Fig.
<figure><img src="image_7.png" /><figcaption>Fig. 4. Reconstructed 3D melting interface for an experiment in convective regime. $\Delta T = 18.4^{\circ}C$, $T_{dw} = 23.1^{\circ}C$, $T_{up} = 10.7^{\circ}C$, $Ra^* = 633.9$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Different stages of melting for an experiment initially in the conductive regime and subsequently in the convective regime. The upper temperature was kept constant through the entire experiment at $T_{up} \approx 9.9^{\circ}C$. The bottom temperature in conductive regime was $T_{dw} = 20.0^{\circ}C$, while it was increased afterwards to $T_{dw} = 22.4^{\circ}C$. The red dashed line indicate the theoretical normalized liquid height for pure conduction obtained from Eq. (3).</figcaption></figure>

6a, we find $\lambda_w/L_p = 2.5$, indicating that the pattern consists of one stable roll every 2.5 pore lengths.

Fig. 6b was obtained for $\Delta T = 12.3^{\circ}C$ ($T_{dw} = 22.5^{\circ}C$, $T_{up} = 10.2^{\circ}C$), with $A = 0.9$ and $Ra^* = 436.6$. Here, $T_{dw}$ was the same as in the previous case (Fig. 6a), while a higher $T_{up}$ was applied to the top surface, leading to a higher liquid height. The convective pattern now shows merging branches, forming a transitional pattern with polygonal structures. The wavelength of this convective pattern has increased compared to the previous case, leading to $\lambda_w = (2.8 \pm 0.4) \times 10^{-2}$ m and a wavenumber of $a = 13.0 \pm 1.9$. The larger convective cell size is also reflected in an increased ratio of $\lambda_w/L_p$ compared to the previous case, which results now in $\lambda_w/L_p = 3.6$.

Finally, Fig. 6c shows the case for $\Delta T = 4.3^{\circ}C$ ($T_{dw} = 20.9^{\circ}C$, $T_{up} = 16.6^{\circ}C$), $A = 0.3$ and $Ra^* = 427.5$. Due to the higher $T_{up}$, the melting interface rises significantly, resulting in a relatively thin upper solid layer. Here, convection organizes into large, square-shaped polygonal cells, with downwelling at the cell centers and upwelling along the boundaries. For this case, the wavelength is estimated in $\lambda_w = (4.0 \pm 0.3) \times 10^{-2}$ m, leading to $a = 12.4 \pm 0.9$ and $\lambda_w/L_p = 5.2$. While the convective cells here have a distinct square shape, similar conditions of $A$ and $Ra^*$ also led to large hexagonal cells. Given the limited number of convective cells in the system, the melting interface in the vertical cross section appears less wavy than in previous cases. Notably, for the same range of $Ra^*$ and for the same system, the convective pattern in

the limit case of $A = 0$ consists of deformed rolls (Fig. 6d), and only at larger $Ra^*$ ($\approx 10^3$) cross-rolls convection was observed in a fully liquid system [41].

## 4.2. Flow velocity

Fig. 7 shows the flow velocity field for square convection ($Ra^* = 488.5$, $A = 0.1$), divided into its vertical ($v_z$) and horizontal components ($v_x$ and $v_y$). The figure shows the relative extent of upward and downward flow in the vertical $z$ direction, alongside the corresponding horizontal flow in both the $x$ and $y$ directions. Velocity fields were measured at three different heights within the liquid layer, corresponding to $H/6$ (Fig. 7a), $H/4$ (Fig. 7b), and $H/2$ (Fig. 7c).

Near the bottom surface (Fig. 7a), convection is already well organized into a distinct square pattern, with downwelling flow at the centers of the squares and upflow occurring along narrow, straight boundaries that intersect perpendicularly. Here, the horizontal flow is relatively intense, with velocities in the $x$ and $y$ directions comparable to the vertical flow velocity. The upwelling flow remains relatively narrow at $H/6$ (Fig. 7a) and $H/4$ (Fig. 7b), but it begins to spread laterally moving up into the melted layer (Fig. 7c). In fact, at $H/6$ the distinct square convective pattern is less prominent, and only regular patches of upward and downward flow are recognizable in the 2D plane.

Similarly, the vertical flow velocity ($v_z$) intensity gradually increases from the bottom regions of the melted layer ($H/6$) to higher regions of the convective cell ($H/2$). This is associated with a progressive decrease in the horizontal flow velocities $v_x$ and $v_y$. This is shown in Fig. 8 where we plot the velocity component ratio ($\Gamma$) for all experiments with $A < 0.5$ (i.e., cases where $h_l > H/2$). This velocity ratio is defined as the squared magnitude of the vertical velocity divided by the squared magnitude of the horizontal velocity and is calculated at the three different heights within the melted layer:

$$ \Gamma = \frac{\sum |v_z|^2}{\sum (|v_x|^2 + |v_y|^2)}. \qquad (4) $$

Each squared velocity magnitude is obtained by summing the squares of the velocity component for each voxel in the corresponding MRI image. For measurements at $H/6$, the velocity ratio is less than 1, indicating that the horizontal flow dominates at the base of the convective cells. At $H/4$, the ratio rises above 1, highlighting a stronger vertical flow component. The ratio reaches its maximum for $H/2$, where it also increases with $Ra^*$ due to the related growth of $h_l$. Indeed, as $Ra^*$ increases, so does $h_l$, resulting in larger convective cells. Consequently, the $H/2$ image captures regions closer to the mid-height of these cells, where vertical flow dominates while horizontal flow is relatively less intense.
<figure><img src="image_9.png" /><figcaption>Fig. 6. Steady-state convective patterns in the melted layer. Vertical (top) and horizontal (bottom) cross-sections for the vertical velocity fields obtained from MRI. (a) Cross-rolls for Ra* = 311.6, Da = 3.1 × 10⁻³ and A = 1.7. The horizontal image is taken at a height of H/6. (b) Transition pattern for Ra* = 436.6, Da = 1.6 × 10⁻³ and A = 0.9. The horizontal image is taken at a height of H/6. (c) Square cells for Ra* = 427.5, Da = 6.9 × 10⁻⁴ and A = 0.3. The horizontal image is taken at a height of H/4. (d) Deformed rolls for a fully melted layer (Ra* = 302, Da = 4.2 × 10⁻⁴ and A = 0).</figcaption></figure>

Source: Modified from [41].

<figure><img src="image_10.png" /><figcaption>Fig. 7. Velocity components for square convection. Ra* = 488.5, Da = 5.4 × 10⁻⁴, A = 0.1. Images are taken at (a) H/6, (b) H/4, and (c) H/2.</figcaption></figure>

Finally, velocity measurements allow us to define a large-scale flow velocity, $U^2$, as

$$U^2 = \frac{1}{S} \int_S (v_x^2 + v_y^2 + v_z^2) dS \qquad (5)$$

where S is the area of the sample's cavity in the horizontal MRI image. Since the large-scale flow velocity scales proportionally to the kinetic energy ($U^2 \sim E_K$), it can be used as a parameter to investigate the intensity of convection within the melted layer. The dependence of the large-scale flow velocity on Ra* is shown in Fig. 9. Here, the velocity is normalized by $\kappa_{eff}/h_l$, representing the characteristic thermal diffusion velocity within the melted layer. Experiments were carried out with A varying from A = 0.1 (relatively thick liquid layer) to A = 3.6 (relatively thin liquid layer). We find that the intensity of convection increases with the Rayleigh number, following $U^2 \sim Ra^*$ (as shown by the fit in Fig. 9). This scaling is independent of the depth at which the velocity is measured and the height of the liquid

layer. This result corresponds to the scaling observed for convection in porous media within a fully melted system [41], and with results from weakly-nonlinear theory for convection at low Ra in the classical Rayleigh-Bénard configuration [44].

## 5. Discussion

This work investigates thermal convection in a material undergoing solid/liquid phase change within a low-conductivity porous foam, using MRI to resolve flow dynamics and visualize the melting interface within the system's bulk. Experiments were conducted at low-to-moderate Ra* for various phase change ratios A. Due to the low viscosity of hexadecane, the critical conditions for convection are reached easily in our case, leading to relatively high liquid heights even for small temperature differences. As a consequence, most of our experiments resulted in A < 4.
<figure><img src="image_11.png" /><figcaption>Fig. 8. Velocity's component ratio evaluated at three different heights as a function of Ra*, for experiments where A < 0.5.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. Intensity of convection, defined as the normalized large-scale flow velocity ($U h_l)/\kappa_{eff}$, as a function of Ra*. The solid line represents the data fit with the equation $y(x) = a x^{0.5} + b$, where $a = 245.5$ and $b = -2077$ ($R^2 = 0.92$).</figcaption></figure>

## 5.1. Flow patterns and phase-change interface

Fig. 10 summarizes the experiments and highlights the different convective patterns observed. Notably, we identified two patterns not previously reported for convection in porous media with phase change: cross-rolls convection and square convection (Fig. 6). During PCM melting without porous media, convection typically forms hexagonal cells or rolls, depending on A [21]. Hexagonal patterns arise from the top-down thermal boundary asymmetry caused by the solid PCM layer on top, which acts as an imperfect thermal conductor. Roll patterns, on the other hand, were observed only for relatively small A ($A < 0.05$), corresponding to a very thin solid PCM layer. A transition between these two patterns in the form of a mixed polygonal pattern has been described by Davis et al. [21] within the range $0.13 < A < 0.19$. In our system, we observed similar patterns but at much higher A values (Fig. 10). Cross-roll patterns occurred for $1.4 \lesssim A < 3.6$, while transitional patterns for $0.9 < A \lesssim 1.4$. Interestingly, we observed an opposite

trend compared to systems without porous media [21]. In our case, as the temperature difference within the liquid layer increases, Ra* rises and A decreases, and the pattern evolves from stable cross-rolls to transitional forms and, finally, to polygons (squares or hexagons). This trend, however, could be attributed to the relatively large Ra*. Since we do not have access to convective patterns for small A and Ra* ($\approx Ra_c^*$), we cannot extend the analysis to conditions near the onset of motion.

Under our conditions, the melting interface separating the liquid PCM from the solid PCM exhibits a wavy surface in the convective regime, reflecting the organization of convective cells in the melted layer, whereas a flat interface is observed under conductive conditions. Thermo-convective flows within the melted PCM are also affected by the structure of the porous medium. The impact of the porous medium on convection can be evaluated from vertical velocity field images. An example of this pores-induced effect on the velocity field is shown in Fig. 11 for a roll-like pattern in the transitional region where local maxima of positive and negative velocities consistently appear at the center of vertically open porous unit cells. These peaks are separated from one another due to the lateral boundaries of the unit cells, which increase flow resistance in their vicinity.

Another factor influencing the melting interface behavior in our configuration is the low thermal conductivity of the plastic foam. The similar thermal conductivities of the PCM and foam minimizes the local temperature gradients between the solid matrix and the PCM. In contrast, in high-conductivity foams, more conductive pathways would promote local melting near the solid matrix, reducing the influence of convective flows on shaping the interface. This leads to a locally bent melting interface near the porous support [45] and the formation of wider mushy regions with strong liquid fraction variations [46]. At the pore scale, melting progresses from the pore edges toward the core, where localized solid drops may form and settle due to density differences between solid and liquid PCM phases [47]. This scenario differs significantly from our case, where the melting process and interface dynamics are found to be primarily governed by large-scale convection within the liquid PCM layer.

## 5.2. Impact of convection on melting height

A key aspect in the analysis of heat transfer in PCM-saturated porous media is the evaluation of how convection can enhance the
<figure><img src="image_13.png" /><figcaption>Fig. 10. Patterns of convection for Ra vs. A.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 11. Vertical velocity field for convection in the melted layer. Ra\* = 197.2, Da = 1.7 × 10⁻³, A = 1.0. The horizontal image was acquired at H/4.</figcaption></figure>

phase change process. To this end, we compared the height of the melted layer during convection with the height predicted for purely conductive heat transfer. The results for the steady-state are shown in Fig. 12. The figure shows the final normalized average liquid height measured in our experiments, $h^* = h_l/H$, and $h_l^{\text{cond}}/H$, the theoretical normalized liquid height for pure conduction with the same applied temperature gradient, as defined by Eq. (3). From Fig. 12, we can see that the theoretical steady melting height is recovered in our system for experiments in the conductive regime and for cases where $Ra^* < Ra_c^*$. Under our conditions, once stable convection develops, the melting height increases significantly compared to the purely conductive case. For $h^* < 0.6$, the melting height under convection is more than twice that obtained from conduction alone. However, this enhancement

decreases as the liquid layer grows thicker ($h^* \to 1$). In this case, $h^*$ converges toward $h_l^{\text{cond}}/H$ due to the imposed temperature boundary conditions (i.e., $\Theta$). For example, if $T_{up} < T_m$, a thin layer of solid PCM remains near the top wall, as in our experiments, thus limiting the upper bound of $h^*$ to a value below 1. Reciprocally, if $T_{up} \ge T_m$, the entire PCM can melt, and $h^* = 1$ is reached. Obviously, the lower bound for $h^*$ corresponds to the conductive solution. The initial part of the curve ($h^* < 0.6$) displays a higher slope than for thicker liquid layers, and we expect even larger variations in $h^*$ by increasing the convection intensity through $Ra^*$. This aspect could be further tested in our setup by varying the type of PCM to explore a broader range of $Ra^*$ (via changes in liquid properties) while maintaining a comparable range of liquid heights. Lastly, Fig. 12 also includes data from [19] for a similar
<figure><img src="image_15.png" /><figcaption>Fig. 12. Experimental normalized melting height ($h_l/H$) vs. the theoretical normalized melting height for the conductive regime ($h_l^{cond}/H$). The black line represents a slope of 1. Red symbols indicate the results from the experiments presented in this work, while empty symbols represent the experiments from [19], which were carried out without a porous medium. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

system without a porous medium. For the range of $h^*$ where their convection results are available (i.e., $h^* < 0.7$), the impact of convection on enhancing the melting height observed in our experiments appears comparable to that in systems without a porous medium.

## 5.3. Heat transfer

The heat transfer within the melted layer is evaluated in terms of Nusselt number ($Nu$), which represents the ratio of total heat transfer ($\phi_l^{tot}$) to the conductive heat transfer ($\phi_l^{cond}$). In the liquid layer, the conductive heat flux is given by:

$$ \phi_l^{cond} = \lambda_{eff} \frac{T_{dw} - T_m}{h_l}. \qquad (6) $$

At the melting interface, the total heat flux from the liquid layer is balanced by the total heat flux from the solid layer, i.e. $\phi_l^{tot} = \phi_s^{tot}$. The latter accounts for both the conductive heat flux through the solid layer and the volumetric enthalpy due to the melting process:

$$ \phi_s^{tot} = \lambda_{eff} \frac{T_m - T_{up}}{H - h_l} - \epsilon \rho_l \Delta H_m \frac{\partial h_l}{\partial t}. \qquad (7) $$

In the liquid layer, the Nusselt number writes as

$$ Nu = \frac{\phi_l^{tot}}{\phi_l^{cond}}, \qquad (8) $$

by substituting Eqs. (6)–(7) into Eq. (8) we obtain

$$ Nu = \frac{\lambda_{eff} T_m - T_{up}}{\lambda_{eff} T_{dw} - T_m} \frac{h_l}{H - h_l} - \frac{1}{\lambda_{eff} T_{dw} - T_m} \frac{1}{\epsilon \rho_l \Delta H_m h_l} \frac{\partial h_l}{\partial t}, \qquad (9) $$

that in dimensionless form writes

$$ Nu = \frac{\lambda_{eff}}{\lambda_{eff}} \frac{\Theta h^*}{(1 - h^*)(1 - \Theta)} + \frac{\epsilon St^{-1} \frac{\partial(h^*)^2}{\partial Fo}}{2}, \qquad (10) $$

with $St$ being the Stefan number, $\Delta H_m$ the latent heat of melting, $\epsilon$ the porosity of the medium, $\Theta$ the temperature difference ratio defined in Eq. (3), and $h^* = h_l/H$. The second term on the right-hand side of the equation represents the time-dependent term that accounts for the phase change contributions. Fig. 13 shows the variation of $Nu$ with dimensionless time for one of our convection experiments. Initially, $Nu$ remains close to one, reflecting a conduction-dominated heat transfer

<figure><img src="image_16.png" /><figcaption>Fig. 13. Nusselt number $Nu$ as a function of $Fo$ for the convective experiment of Fig. 3.</figcaption></figure>

mechanism. At the onset of convection, a sharp increase in heat transfer occurs. In the transient part, $Nu$ increases proportionally with the growth of the liquid layer, following $Nu \sim h^* \sim Fo$. When the steady-state is reached, and the melting height reaches a final height, $Nu$ becomes constant in time and can be determined directly from the first term on the right-hand side of Eq. (10), that is

$$ Nu = \frac{\lambda_{eff} \Theta h^*}{\lambda_{eff} (1 - h^*)(1 - \Theta)}. \qquad (11) $$

In Fig. 14 we report the variation of $Nu$ obtained from Eq. (11) with $Ra^*$ for all our experiments at steady-state conditions. As expected, for $Ra^* < Ra_c^*$, we recover $Nu \to 1$ since convection has not yet begun. For $Ra^* > Ra_c^*$, the onset of convection leads to an enhancement in heat transfer, as highlighted by the sharp increase in $Nu$. In classical Rayleigh-Bénard convection, in the laminar regime, $Nu$ scales with the Rayleigh number for a fully liquid layer ($Ra_l$) as $Nu \sim Ra_l^{1/4}$. This same relation was verified for convection in phase change material by Noel et al. [19]. In porous media, the slope of the convective portion of the $Nu-Ra^*$ curve has been widely studied, with theoretical predictions often differing from experimental results [48]. For low and moderated $Ra^*$, Joseph [49] predicted for an unbounded porous layer a linear dependence with a slope $dNu/dRa^* = 1/(2\pi^2)$. However, experimental studies have reported lower slopes, with differences arising due to the cavity’s aspect ratio [50], the homogeneity of the porous medium [51], and the structure and thermal properties of the porous material [52]. In our case, a direct relationship between $Nu$ and $Ra^*$ is difficult to establish (Fig. 14). This is because the two main parameters controlling $Ra^*$, i.e. the temperature difference in the liquid layer ($\Delta T_l$) and the liquid height ($h_l$), can vary independently. As a result, the same $Ra^*$ can correspond to melted layers of different heights, each having the same temperature gradient within the liquid layer but subjected to different total temperature gradients $\Delta T$ (because of a colder top boundary condition).

To account for this aspect, we normalize $Nu$ by the temperature gradient ratio ($\Theta$), which is the ratio between the temperature difference within the solid PCM layer and total temperature difference. In Fig. 15 we show the trend of $Nu/\Theta$ as a function of $A$ (inset) and $Ra^*/A$. In both cases, we observed a good collapse of all data, highlighting the relationship between the normalized Nusselt number, the phase change ratio, and the porous Rayleigh number. Our experiments follow the scaling

$$ Nu/\Theta \sim (Ra^*/A)^{0.5}. \qquad (12) $$
<figure><img src="image_17.png" /><figcaption>Fig. 14. Nusselt number Nu as a function of Ra*.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 15. Variation of the ratio Nu/Θ with Ra*/A. The inset shows Nu/Θ as a function of the phase change ratio A.</figcaption></figure>

This indicates that heat transfer is enhanced with increasing convection intensity due to the higher values of h* and ΔT, both represented by A and Θ. Consequently, Eq. (12) suggests that a complete characterization of heat transfer in such systems requires not only parameters controlling the melted layer but also global parameters such as the total temperature difference.

## 6. Conclusions

In this study, we carried out an experimental investigation of thermal convection during the melting of a PCM embedded in a highly porous structured plastic foam. This work was motivated by the limited availability of detailed analyses of flow fields and melting interface behaviors under convective conditions in PCM-saturated porous media. This is mainly due to the difficulties in performing experiments in opaque media without perturbing the system or obtaining only partial information.

Using MRI, we provide a characterization of the melting process under both conductive and convective regimes. In the conductive regime, the melting interface grows as h* ~ Fo^0.5, consistent with expectations for non-porous systems, and its final shape is flat across the whole sample. When convection is present, the melting interface grows faster

(h* ~ Fo) and exhibits a wavy surface influenced by thermo-convective flows in the liquid PCM layer. The convective regime leads to liquid heights up to more than twice those observed in conduction for h* < 0.6. However, this enhancement decreases for thicker liquid layers due to the imposed temperature boundary conditions (i.e., Θ).

We also resolved the flow velocity field in the melted layer under steady convection. The porous medium strongly influences local flow dynamics, resulting in velocity peaks at the centers of foam unit cells. Under steady-state, different convection patterns were observed, from cross-rolls and transitional patterns to polygonal patterns in the forms of hexagons or squares, depending on Ra* and the phase change ratio A. Within convective cells, the vertical velocity component increases from the bottom of the cavity toward the central regions of the cell, associated with a progressive decrease in horizontal velocity. The overall intensity of convection is found to scale with the Rayleigh number as U² ~ Ra.

Finally, we investigated the heat transfer in steady convection in terms of Nusselt number. A sharp transition is observed in the Nu-Ra relationship, with Nu = 1 under conductive conditions, followed by a significant increase in the convective regime. Although a direct correlation between Nu and Ra was not found, a good data collapse was achieved by taking into account global parameters such as the total applied temperature gradient and the phase change ratio, that is Nu/Θ ~ (Ra*/A)⁰.⁵.

In conclusion, this work provides a detailed characterization of the melting interface, flow field, and heat transfer in a composite PCM system. It establishes a basis for further experimental and numerical studies to investigate the physical mechanisms at play when combining thermal convection, phase change and flow in porous media in more complex systems. These can include systems with PCMs characterized by mushy layers or porous media with different geometries and thermal-physical properties.

## CRediT authorship contribution statement

Nicolò R. Sgreva: Writing – original draft, Supervision, Investigation, Formal analysis, Data curation, Conceptualization. Christel

Métivier: Writing – original draft, Supervision, Funding acquisition,

Formal analysis, Conceptualization. Anthony Teixeira: Validation, In-

vestigation, Formal analysis. Sébastien Leclerc: Software, Methodol-

ogy.

## 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

This work was founded by the operation STOCK'NRJ, co-financed by the European Regional Development Fund within the framework of the Program FEDER-FSE Lorraine and Massif des Vosges 2019–2023. This work is also part of the project CONVINCES, financially supported by the French National Research Agency (Grant No. ANR-21-CE50-0029).

## Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.ijheatmasstransfer.2025.127283.

## Data availability

Data will be made available on request.
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