# Effective thermal diffusivity and conductivity of a fluid-saturated solid foam

Nicolò R. Sgreva, Yves Jannot, Christel Metivier

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Nicolò R. Sgreva, Yves Jannot, Christel Metivier. Effective thermal diffusivity and conductivity of a fluid-saturated solid foam. Review of Scientific Instruments, 2023, 94 (7), 10.1063/5.0154737. hal-04310649

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# Effective thermal diffusivity and conductivity of a fluid-saturated solid foam

Nicolò R. Sgreva 🐶 ; Yves Jannot 🐶 ; Christel Métivier 🐶

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Rev. Sci. Instrum. 94, 074901 (2023)

https://doi.org/10.1063/5.0154737

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# Effective thermal diffusivity and conductivity of a fluid-saturated solid foam

Cite as: Rev. Sci. Instrum. 94, 074901 (2023); doi: 10.1063/5.0154737

Submitted: 15 April 2023 • Accepted: 11 June 2023 •

Published Online: 5 July 2023

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Nicolò R. Sgreva,ª ID Yves Jannot, ID and Christel Métivier ID

## AFFILIATIONS

Université de Lorraine, CNRS, LEMTA, 54000 Nancy, France

## ABSTRACT

The thermal properties of composite materials, such as fluid-saturated solid foams, can be estimated in two ways: using equivalent models based on both the thermal properties of the liquid and solid phases or by direct measurements, which, however, are not always straightforward. In this paper, we provide a new experimental device based on the four-layer (4L) method to measure the effective thermal diffusivity of a solid foam filled with different fluids (glycerol and water). The specific heat of the solid part is measured using differential scanning calorimetry, and the volumetric heat capacity of the composite system is estimated using an additive law. The effective thermal conductivity deduced from experiments is then compared with the maximum and minimum values obtained from the parallel and series equivalent models. The proposed 4L method is first validated by measuring the thermal diffusivity of pure water and then used afterward to measure the effective thermal diffusivity of the fluid-saturated foam. Experimental results converge with those obtained from equivalent models in the case where the different components of the system have similar thermal conductivities (e.g., glycerol-saturated foam). On the other hand, when the liquid and solid phases have very different thermal properties (e.g., water-saturated foam), the experimental results differ from what equivalent models predict. This highlights the need to make appropriate experimental measurements to estimate the overall thermal properties of these latter multicomponent systems or to consider more realistic equivalent models.

Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0154737

## I. INTRODUCTION

In the field of thermal energy storage, a way to make latent energy storage systems more efficient is to combine phase change materials with macroscopic solid structures. The latter are typically porous materials, such as packed beds of beads or, in most cases, open-cell solid foams with high porosity (Zhang et al., 2021; Zhao et al., 2022). In these composite systems, latent heat is stored during the melting of the saturating phase change material, and released during its solidification. The use of porous solid structures improves some of the typical drawbacks of phase change materials such as the low thermal conductivity (Xiao et al., 2014) and the high degree of supercooling (Beaupere et al., 2018). To develop these energy storage systems and make them work properly and efficiently, it is essential to know the properties of each of their components. This concerns the thermophysical properties of the saturating phase change material, especially during or near the phase transition (Sgreva et al., 2022; Klochko et al., 2022), as well as the properties of the solid porous medium, such as, for example,

its geometry (Righetti et al., 2020) or permeability if fluid flow is involved (Lafdi et al., 2007).

In addition to the properties of each individual component, another crucial point in developing these systems involves the estimation of the overall effective properties of the composite material. In the latter case, the estimation of the effective thermal properties is not an easy task and is generally carried out using two different approaches.

The first involves the use of equivalent or empirical models based on ideal multiphase systems (Kumar and Topin, 2017a) or derived from experiments with specific composite materials (Rouhani and Bahrami, 2018). In this case, the thermal property that is most often investigated is the thermal conductivity $k$ (W m$^{-1}$ K$^{-1}$), defined as

$$k = \alpha(\rho c)^{-1}, \quad (1)$$

where $\alpha$ is the thermal diffusivity (m$^2$ s$^{-1}$) and $\rho c$ is the volumetric heat capacity (J m$^{-3}$ K$^{-1}$). The simplest models for determining the

<footnote>ª) Author to whom correspondence should be addressed: nicolo.sgreva@univ-lorraine.fr</footnote>
effective thermal conductivity of composite materials are the series model and the parallel model (Nield and Bejan, 2006). These assume that the composite material consists of several layers with different thermal conductivities arranged in series or parallel, respectively. Apart from these simple models, when the solid structure is a high-porous open-cell foam, other more complex models can be used. For instance, Smith et al. (2013) report that the most suitable model for determining the effective conductivity in this type of fluid-saturated solid foam is given by the Hashin-Shtrikman upper bound (Hashin and Shtrikman, 1962). This model is based on the volume fraction and conductivity of the components and allows the evaluation of the effective thermal conductivity limits for homogeneous, isotropic, and two-phase composite materials.

Another way to estimate the overall thermal properties is to directly measure them. From an experimental point of view, the thermal conductivity of a material can be measured by various techniques and methods. The most common techniques are the heat flow meter (HFM), the hot wire, the flash method, and the hot disk method.

The HFM method is a steady-state method that can be used to measure the thermal conductivity of a solid material. It consists of a sample holder with a heat source on one side and a heat sink on the other, with a temperature sensor in between. The sample holder is designed to minimize heat loss to the surroundings so that the measured heat flux goes mainly through the sample. For example, this method has been employed for the determination of the effective thermal conductivity of a porous skeleton made by packed beds of beads (Rouhani et al., 2018) or by autoclaved aerated concrete (Capanale and Moro, 2016). Another method often used is the hot wire. This is a transient method in which a wire is heated by a constant electric current and its temperature is measured at different times. The rate of temperature rise is proportional to the thermal conductivity of the material, and the thermal conductivity can be calculated from the slope of the temperature curve (Kim and Oh, 2020; Bicer and Kar, 2019).

The flash method involves heating a spot on the surface of a sample material by using a high-intensity laser pulse and measuring the resulting temperature rise (Smith et al., 2013). The latter is used to calculate the effective thermal diffusivity. The thermal conductivity is retrieved afterward from Eq. (1). Finally, in the hot disk method, a small disk-shaped heating element is placed in contact with the surface of the sample material. The temperature rise is measured by the electrical resistance variation of the heating element with a known temperature coefficient. The rate of temperature variation is then used to determine the thermal conductivity of the material (Gustafsson, 1991).

In this paper, we aim to estimate the effective thermal conductivity of a composite system consisting of a high-porous solid foam saturated with a liquid. The main issues encountered when measuring the effective thermal conductivity of a multiphase composite system are related to: (i) the occurrence of thermal convection in the liquid phase; (ii) the possibility of liquid phase leakage from the system; and (iii) the characteristic length scale of the solid foam that can induce a local and not averaged (or not effective) measurement of the sample. For these reasons, the classical experimental techniques described above reveal some limitations that strongly limit their use. In detail, the classical HFM apparatus is designed to work mainly with solid samples rather than a composite system with a fluid phase

in it. In the hot wire method, as pointed out by Jannot and Degiovanni (2013), the number of contact points between the wire and the solid foam will be small due to the small diameter of the wire (less than 0.5 mm) compared to the mean pore diameter (several millimeters). Fluid convection may also develop around the wire. Finally, the sample must have small pores (<500 µm) and not be transparent to use the flash method properly (Smith et al., 2013; Jannot and Degiovanni, 2018). In addition, the hot disk method can lead to problems related to fluid convection on the side walls of the sample, where a temperature gradient develops.

This paper aims to propose a new method dedicated to measuring the effective thermal diffusivity of multiphase composite systems. This method is based on the four layer (4L) device proposed by Jannot et al. (2010) to measure the thermal diffusivity of a solid sample. The thermal conductivity is then deduced from Eq. (1) after measuring the specific heat by Differential Scanning Calorimetry (DSC). The results are then compared with those obtained for the same system from equivalent models for thermal conductivity.

The article is organized as follows: In Sec. II, we describe the main equivalent models for estimating the effective conductivity of a saturated porous medium. Then we describe the 4L method used to experimentally measure the effective $k$ of our samples. Section II concludes with a description of the estimation method used during the experiments and validation of the assumptions on which the technique is based. Section III details the materials and devices used during the experiments. In Sec. IV, we report first the results of the determination of effective thermal conductivity with the equivalent models described above, then those from the modified 4L device. The article concludes with a summary of the main findings in Sec. V.

## II. PRINCIPLES AND MODELS

### A. Effective thermal conductivity models

The effective thermal conductivity of a multi-component system depends strongly on the thermal conductivity of each component and the geometry of the system (Wiener, 1912). The lower and upper limits of this effective thermal conductivity can be defined by two equivalent models based on the internal arrangement of the structure. The lowest possible value is given by the series model, while the highest one is given by the parallel model (Nield and Bejan, 2006). In both models, the components of the material are assumed to be arranged in parallel layers. In the parallel model, the heat flow is parallel to these layers [Fig. 1(a)], and the effective thermal conductivity $k_\parallel$ is given by

$$k_\parallel = \sum_{i=1}^{N} \varepsilon_i k_i, \qquad (2)$$

where $\varepsilon_i$ and $k_i$ are the volume fraction and the thermal conductivity of the component $i$, respectively.

In the series model, the heat flow is perpendicular to component layers [Fig. 1(b)], and the effective thermal conductivity $k_\perp$ is given by

$$k_\perp = \frac{1}{\sum_{i=1}^{N} \varepsilon_i k_i}. \qquad (3)$$
<figure><img src="image_14.png" /><figcaption>FIG. 1. Diagram of the (a) parallel and (b) series models.</figcaption></figure>

From these definitions, in a multi-component material such as a fluid-saturated porous medium, we always have $k_\parallel \ge k_\perp$. The two models return the same overall thermal conductivity only when the thermal conductivity of each component is the same.

In addition to these two models, other empirical mixed-models based on different internal arrangements of system components have been proposed (e.g., Xu et al., 2019; Kumar and Topin, 2017a). However, they are not universal models and are applicable only under specific conditions. For the specific case of a highly porous solid foam, Smith et al. (2013) show that the most suitable formulation for evaluating the effective thermal conductivity is the Hashin-Shtrikman upper bound (Hashin and Shtrikman, 1962), that is

$$k_{\text{HS}} = k_s \frac{k_l + 2k_s + 2\varepsilon_l(k_l - k_s)}{k_l + 2k_s - \varepsilon_l(k_l - k_s)}, \quad (4)$$

where the subscripts “s” and “l” refer to the solid and fluid phases, respectively. This latter relation is based on the volume fraction and conductivity of the components of the system and allows the effective conductivity to be evaluated for a two-phase homogeneous composite material.

## B. Modified 4L method

The 4L method is a method developed to measure the thermal diffusivity of a solid material (Jannot et al., 2010). The thermal conductivity of the material is then obtained by knowing its volumetric heat capacity. A schematic representation of the 4L method for solid samples is shown in Fig. 2. This method is based on the following principle: a modulated heat flux produced by a heating element is applied to a system initially at the steady state, and the temporal variations of the three temperatures $T_1$, $T_2$, and $T_3$ are recorded simultaneously. The temperature $T_2$ can be expressed as $T_2(t) = f[T_1(t), T_3(t), \alpha]$, where $\alpha$ is the thermal diffusivity of the sample. Here, the thermal diffusivity is considered an unknown parameter. An inverse procedure is carried out to estimate the value of $\alpha$, which minimizes the squared differences between the experimental thermogram and a simulated thermogram obtained with a direct model.

When the sample consists of a fluid that is free to move instead of a solid, the original configuration in Fig. 2 must be modified to avoid a number of problems. To address the main issues already

<figure><img src="image_15.png" /><figcaption>FIG. 2. Sketch diagram of the 4L method. Modified from Jannot et al. (2010). Reproduced with permission from Meas. Sci. Technol. 21 035106 (2010). Copyright 2010, IOP Publishing Ltd.</figcaption></figure>

mentioned in the introduction, we propose the following modifications. A first modification is made to avoid fluid convection, which is done by moving the heating element above the sample. A second issue arises from the difficulties in placing the thermocouple measuring $T_2$ right in the middle of a fluid layer or accurately measuring its position if it is displaced from the center. A solution is to replace the lower sample (the one in between $T_1$ and $T_2$) with a sample of insulating material with known thermal properties. The device arranged in this way must then be enclosed in a cylindrical tank made of poly(methyl methacrylate) (PMMA) to prevent fluid leakages. Finally, two small pieces of adhesive aluminum foil are added on top of the thermocouple tips to enable the measurement of average temperature values $T_0$ and $T_1$ on 1-cm² surfaces centered on the axis. The presence of these foils does not affect the proposed model because their thickness is negligible and their thermal conductivity is high. The final representation of the modified 4L device that we propose is shown in Fig. 3.

In this configuration, the temperature on the sample's unheated face $T_1(t)$ is a function of the thermal properties of the insulating
<figure><img src="image_16.png" /><figcaption>FIG. 3. Sketch of the asymmetrical modified 4L device used to estimate the thermal diffusivity.</figcaption></figure>

material, the temperature on the sample's heated face $T_0(t)$, and the effective thermal diffusivity $\alpha$ of the sample.

In what follows, we develop the model used to determine $T_1(t)$ within the 4L configuration. The direct model used to evaluate the temperature $T_1(t)$ is based on the following assumptions: (1) The heat transfer is unidirectional (1D) at the center of the device, and (2) the bottom insulating material sample is considered semi-infinite in the axial-direction. Hence, the governing equation to be solved is the heat equation along the center-line of the sample,

$$ \frac{\partial^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial t}. \qquad (5) $$

The whole system is at the initial temperature $T_i$, and the following boundary conditions are considered:

$$ \lim_{x \to \infty} T(x, t) = T_i, \qquad (6) $$

$$ T(0, t) = T_0(t), \qquad (7) $$

$$ T(x, 0) = T_i, \qquad (8) $$

where $e$ is the thickness of the sample and $e_i$ is the thickness of the lower insulating layer (Fig. 3). By setting

$$ \theta(x, p) = \mathcal{L}[\tilde{T}(x, t)], \qquad (9) $$

where $\tilde{T}(x, t) = T(x, t) - T_i$ and $\mathcal{L}$ is the Laplace transform operator, Eq. (6) becomes

$$ \lim_{x \to \infty} \theta(x, p) = 0. \qquad (10) $$

The system given by Eqs. (5)–(8) is solved in the Laplacian's space by using the quadrupolar formalism for a multilayer system (Maillet et al., 2000). In this way, we can write

$$ \begin{bmatrix} \theta_0 \\ \Phi_0 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} \theta_1 \\ E_i \sqrt{p} \theta_1 \end{bmatrix}, \qquad (11) $$

where $E_i$ is the thermal effusivity of the lower insulating layer in Fig. 3. In Eq. (11), $\Phi(x, p)$ is the Laplace transform of the heat flux $\phi(x, t)$. According to Maillet et al. (2000), the coefficients of the matrices are

$$ \left\{ \begin{array}{l} A = D = \cosh (q e) \\ B = (k q)^{-1} \sinh (q e) \\ C = (k q) \sinh (q e) \\ q = \sqrt{(p/\alpha)} \end{array} \right. \qquad (12) $$

From Eq. (11) we deduce

$$ \theta_1 = \theta_0 H(p) \qquad (13) $$

with

$$ H(p) = \frac{1}{A + B E_i \sqrt{p}}, \qquad (14) $$

and, hence,

$$ \tilde{T}_1(t) = \tilde{T}_0(t) \otimes \mathcal{F}^{-1}[H(p)] = \tilde{T}_0(t) \otimes \mathcal{F}(t). \qquad (15) $$

In our case, $\mathcal{F}(t)$ is calculated by an inverse Laplace transform applied to $H(p)$ using the De Hoog algorithm (De Hoog et al., 1982).

### C. Estimation method

The transfer function $H(p)$ defined in Eq. (14) depends on the parameters $\alpha$, $k$, and $E_i$. The thermal effusivity $E_i$ of the insulating material is assumed to be known. According to the definitions of the sample's thermal conductivity ($k$), the two remaining parameters to be estimated are $\alpha$ and $\rho c$. The estimation procedure uses a Levenberg-Marquardt algorithm (Marquardt, 1963) to find the values of the unknown parameters that minimize the sum $S$ of the squared errors between the measured experimental temperature $T_{exp}$ and the modeled temperature $T_{mod}$. Hence, the residuals between the two temperatures are calculated as

$$ S = \sum_{n=1}^{N} [T_{\text{exp}}(t_n) - T_{\text{mod}}(t_n)]^2, \qquad (16) $$

where $T_{mod}$ is calculated using Eq. (15) and $N$ is the number of experimental points.

### D. Validation of the 1D hypothesis and estimation of the uncertainty

In order to validate the 1D heat transfer hypothesis, we carried out a three dimensional (3D) direct numerical simulation of the
<figure><img src="image_17.png" /><figcaption>FIG. 4. Numerical simulation of heat transfer in the 4L system with COMSOL. (a) Mesh and applied boundary conditions. The convective heat transfer coefficient is $h = 10 \text{ W m}^{-2} \text{ K}^{-1}$. (b) Resulting temperature field after 1000 s.</figcaption></figure>

whole 4L system using COMSOL Multiphysics. The implemented geometry is shown in Fig. 4. It consists of an axisymmetric configuration in which the sample lies on an insulating layer, which in turn lies on an aluminum block. All layers have a diameter of 0.1 m. The thermal properties and thicknesses used for the simulation are listed in Table I. Figure 4(a) shows the mesh and the boundary conditions used during the simulation. We used an insulated boundary condition for the bottom surface, a no-flux axial symmetry along the center-line [dotted axis in Fig. 4(a)], and a convective heat transfer on the side wall (i.e., the interface in between the external air and the PMMA tank) characterized by a convective heat coefficient of $h = 10 \text{ W m}^{-2} \text{ K}^{-1}$. The system at time $t = 0$ s is set at the initial temperature $T_i = 20^{\circ}\text{C}$. At $t > 0$ s, a temperature of $T_0 = 24^{\circ}\text{C}$ is imposed on the sample's top surface. The latter temperature is chosen to be close to that used later during experiments. The temperature field obtained after 1000 s is shown in Fig. 4(b).

The temperature values $\tilde{T}_0(t)$ and $\tilde{T}_1(t)$ obtained from the 3D simulation are used to estimate the thermal properties of the sample according to the 1D model-based estimation process described in the previous Secs. II B and II C.

A first estimation is performed by simultaneously evaluating both the thermal diffusivity $\alpha$ and the volume heat capacity $\rho c$ over

the time interval $[0, 600 \text{ s}]$. Then the reduced sensitivities of $\tilde{T}_1(t)$ to $\alpha$ and $\rho c$ are calculated. The reduced sensitivity of $\tilde{T}_1(t)$ to a parameter $x$ is calculated as $x(\partial\tilde{T}_1/\partial x)$. This gives the variation (in K) of $\tilde{T}_1$ for a 1% variation in the parameter $x$ (Beck and Arnold, 1977). Figure 5(a) shows the temporal evolution of $\tilde{T}_1$ obtained from the 3D simulation and that from the 1D model with the estimated values of $\alpha$ and $\rho c$. The differences between the two are highlighted by the estimated residuals multiplied by ten times. The latter is not perfectly flat on zero. Moreover, from the plot of the reduced sensitivities [Fig. 5(b)], we find an almost null sensitivity of $\tilde{T}_1$ to $\rho c$. Using the proposed estimation protocol, we get $\alpha = 8.10 \times 10^{-8} \text{ m}^2\text{s}^{-1}$ and $\rho c = -8 \times 10^{-28} \text{ J m}^{-3} \text{ K}^{-1}$. The estimated value of $\alpha$ deviates by 2.8% from the nominal value (Table I). It is worth noting that the estimated value of $\rho c$ is clearly meaningless. The proposed model fails to determine $\rho c$ due to an almost zero sensitivity of $T_1(t)$ to $\rho c$. For this reason, it is, therefore, more convenient to fix the value of $\rho c$ (even to an approximate value) and estimate only the value of $\alpha$ (Jannot and Degiovanni, 2018).

Another estimation is carried out by setting a priori the value of $\rho c$ in the modeling equations. For a fixed value of $\rho c = 3 \times 10^6 \text{ J m}^{-3} \text{ K}^{-1}$, the estimated value of thermal diffusivity is now $\alpha = 8.32 \times 10^{-8} \text{ m}^2 \text{ s}^{-1}$ (a deviation of 0.1% from the nominal value), and the residuals obtained during the estimation process are perfectly flat within the estimation time interval $[0, 600 \text{ s}]$ (Fig. 6). This validates the assumption of 1D heat transfer at the center of the system for at least 600 s, which is a long enough time to allow an accurate estimation of the thermal diffusivity.

The estimation error, produced by the error on the assumed known parameter on which Eq. (15) depends, was evaluated by varying the parameters values in the estimation process. The deviation between the estimated thermal diffusivity and its nominal value was subsequently calculated. The results are given in Table II. They show that an error in the sample's volumetric heat capacity ($\rho c$) and the thermal effusivity ($E_i$) of the insulating material has a negligible effect. An uncertainty of 0.1 mm (i.e., 0.5%) on the sample thick-

<table><caption>TABLE I. Material properties used for simulations and thickness $e$ of the different layers.</caption><thead><tr><th>Material</th><th>k<br>(W m<sup>-1</sup> K<sup>-1</sup>)</th><th>ρc<br>(J m<sup>-3</sup> K<sup>-1</sup>)</th><th>α<br>(m s<sup>-1</sup>)</th><th>e (m)</th></tr></thead><tbody><tr><td>Aluminum block</td><td>200.0</td><td>1.5 × 10<sup>6</sup></td><td>1.33 × 10<sup>-4</sup></td><td>0.020</td></tr><tr><td>Insulating layer</td><td>0.0304</td><td>7.6 × 10<sup>4</sup></td><td>4.00 × 10<sup>-7</sup></td><td>0.020</td></tr><tr><td>Plastic tank wall</td><td>0.200</td><td>1.5 × 10<sup>6</sup></td><td>1.33 × 10<sup>-7</sup></td><td>0.002</td></tr><tr><td>Sample</td><td>0.250</td><td>3.0 × 10<sup>6</sup></td><td>8.33 × 10<sup>-8</sup></td><td>0.018</td></tr></tbody></table>
<figure><img src="image_18.png" /><figcaption>FIG. 5. (a) Temperature on the unheated face over time. $\bar{T}_{1_{\text{mod}}}$ is the temperature obtained from the 1D model, and $\bar{T}_{1_{\text{num}}}$ the one from the numerical simulation. The difference between the two is shown by the residuals multiplied by ten times. (b) Reduced sensitivity of $\bar{T}_{1_{\text{num}}}$ to thermal diffusivity $\alpha$ and volume heat capacity $\rho c$.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>FIG. 6. Temperature on the unheated face over time for an estimation process with fixed $\rho c = 3 \times 10^6 \text{ J m}^{-3} \text{ K}^{-1}$. $\bar{T}_{1_{\text{mod}}}$ is the temperature obtained from the 1D model, and $\bar{T}_{1_{\text{num}}}$ the one from the numerical simulation. The difference between the two is shown by the residuals multiplied by ten times.</figcaption></figure>

ness $e$ leads to an error of 1.1% on the estimated $\alpha$. This is consistent with the results of the flash method, in which the relative error in the estimation of $\alpha$ is twice as large as the relative error in the thickness of $e$ (Jannot and Degiovanni, 2018).

<table><caption>TABLE II. Estimated value of the sample's thermal diffusivity $\alpha_{\text{est}}$ for a given error in the known parameters and its percentage difference with the nominal value of $\alpha$ defined in Table I.</caption><thead><tr><th>Parameter</th><th>Error in the parameter</th><th>α<sub>est</sub> (m<sup>2</sup> s<sup>-1</sup>)</th><th>α<sub>est</sub> - α / α × 100</th></tr></thead><tbody><tr><td>e</td><td>+0.5%</td><td>8.420 × 10<sup>-8</sup></td><td>1.10%</td></tr><tr><td>ρc</td><td>+10%</td><td>8.338 × 10<sup>-8</sup></td><td>0.10%</td></tr><tr><td>E<sub>i</sub></td><td>+10%</td><td>8.343 × 10<sup>-8</sup></td><td>0.15%</td></tr></tbody></table>

As a partial conclusion, we have shown that the assumption of 1D heat transfer at the center of our system is valid. Moreover, we have shown that it is better to fix, in the estimation process, a value of $\rho c$, even an approximated one, since an uncertainty of up to 10% of its value has a minimal influence on the estimation of $\alpha$ (see Table II).

### III. MATERIALS AND EXPERIMENTAL DEVICES

#### A. Solid foam and saturating fluids

A structured solid foam sample with open Kelvin-like (Tetrakaidecahedron-shape) unit cells was 3D printed using a light-sensitive photopolymer plastic (Monocure 3D Rapid, Monocure Pty Ltd., Australia). The print was based on a 3D model where Kelvin cells were assembled together to form a cylindrical disk of diameter $d = 98.18$ mm and thickness $e = 18.10$ mm [Fig. 7(a)]. A single Kelvin unit cell has a constant diameter of 6.5 mm (namely the distance between two opposite square faces) and cylindrical struts with a diameter of around 1.2 mm.

As a saturating fluid, we used three different fluid samples: demineralized water, glycerol, and an 85 wt. % glycerol aqueous solution. Glycerol was used as received from the supplier (Fisher Chemical, with a stated purity of 99%), and no further purification was performed. The 85% glycerol solution was prepared by adding demineralized water to the glycerol.

The reference values of fluid properties at a temperature of 20 °C, along with those measured for solid plastic (see Sec. IV B), are listed in Table III.

#### B. Experimental devices

To estimate the effective thermal conductivity of the fluid-saturated foam of thickness $e$, we first measured the effective thermal diffusivity using the modified version of the 4L method shown in Fig. 3. In this case, an aluminum disk of 5 mm thickness was placed in between the sample's top surface and the heating element in order to guarantee a homogeneous temperature. The heating element was a rounded Omega polyimide heater (model KHRA-4/10-P). As an insulating material, we used two rigid disks of polyurethane foam
<figure><img src="image_20.png" /><figcaption>FIG. 7. (a) Foam sample used for the experiments. The diameter is $d = 98.18$ mm, and the thickness is $e = 18.10$ mm. (b) Plastic spacer used for the experiments where the foam was not used. The diameter is the same as the foam sample, whereas the height is 15.10 mm.</figcaption></figure>

with a thickness of $e_i = 20.0$ mm and a known effusivity of $E_i$. In the case of a measurement with a pure liquid (i.e., without the foam), a 3D printed plastic spacer [see Fig. 7(b)] was inserted between the aluminum disk and the insulating disk to control the thickness of the liquid layer. All the different layers in Fig. 3 were compressed together by two thick aluminum blocks. Constant pressure was applied to the top one by means of a press. This was done to ensure good thermal contact between each surface.

We used two K-type thermocouples to record $T_0$ and $T_1$. The tips of the thermocouples were covered by a square piece of aluminum foil (blue plates in Fig. 3) in order to avoid single contact points between the thermocouple and the saturated foam (i.e., to avoid the thermocouple being in contact only with the liquid or the solid phase). The added square aluminum foils were 10 mm long and 0.06 mm thick. In this way, the mean temperature on the aluminum foil was recorded as a function of time $t$ by a Picolog TC-08 apparatus. The mass of the aluminum folds (around 0.016 g) is not taken into account in the 1D model described in Sec. II B. The entire device

was kept in a climatic enclosure, Binder KB 53-UL, at a controlled temperature of 20 °C. The electrical tension was measured with a Tektronix DMM 4040, while the electrical current was provided by a Tektronix PWS2185 power supplier. In order to avoid convection, the applied intensity of current (and, thus, the heat flow rate) was continuously adjusted during the measurements so that the temperature on the heated sample's face $\tilde{T}_0 = T_0 - T_i$ remained close to 4 °C, with $T_i$ being the initial temperature of the system.

To obtain the unknown thermal properties of single components, we proceeded as follows: the thermal conductivity of the solid plastic alone was measured with the stationary hot-plate method detailed in Jannot et al. (2016). For the measurement, we used a 3D printed parallelepiped sample with the same plastic material and printing technique as the foam sample. The global measuring device error is estimated to be 2.5% (Jannot et al., 2016). To measure the specific heat capacity $c$ of the solid plastic, we used a Setaram µdSc3 differential calorimeter. The measuring device error is estimated to be within 2%. A recent calibration and validation of this instrument is provided by Noel et al. (2022).

The transient hot-plate method (Jannot and Degiovanni, 2018) was used to measure the thermal effusivity $E_i$ of the insulating disks used in the modified 4L device. In this case, we used a rigid parallelepiped sample of polyurethane foam with a thickness of 20.0 mm.

## IV. RESULTS

### A. Thermal effusivity of the insulating material

The estimation process described in Sec. II C requires that the thermal effusivity ($E_i$) of the insulating material be known. We measured the $E_i$ of the polyurethane foam sample with the transient hot-plate method (Jannot and Degiovanni, 2018) at an initial temperature of 20 °C. Figure 8(a) shows the temperature $\tilde{T}$ recorded on the heated face as a function of the square root of time $t$ after applying a current $I$. As described by Jannot and Degiovanni (2018), the thermal effusivity of the sample can be estimated by

$$E = \frac{\phi}{\beta\pi^{0.5}}, \qquad (17)$$

where $\phi$ is the heat flux and $\beta$ is the slope of the linear part of the recorded $\tilde{T}(t^{0.5})$. In the case of Fig. 8(a), the slope of the curve is estimated in the time interval 50–400 s [Fig. 8(b)]. The resulting thermal

<table><caption>TABLE III. Materials properties at temperatures of 20 °C (the fluids) and 25 °C (the solid plastic). The thermal effusivity is defined as $E = \sqrt{k\rho c}$ and the thermal diffusivity as $\alpha = k(\rho c)^{-1}$. Data sources: (*) this study, (a) Ginnings and Furukawa (1953), (b) Ramires et al. (1995), (c) Association et al. (1963), and (d) Linstrom and Mallard (2001).</caption><thead><tr><th>Material</th><th>ρ (kg m<sup>-3</sup>)</th><th>c (J kg<sup>-1</sup> K<sup>-1</sup>)</th><th>k (W m<sup>-1</sup> K<sup>-1</sup>)</th><th>E (W m<sup>-2</sup> K<sup>-1</sup> s<sup>-0.5</sup>)</th><th>α (m<sup>2</sup> s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>Water</td><td>998</td><td>4184 <sup>(a)</sup></td><td>0.598 <sup>(b)</sup></td><td>1580</td><td>1.43 × 10<sup>-7</sup></td></tr><tr><td>Pure glycerol</td><td>1261 <sup>(c)</sup></td><td>2377 <sup>(d)</sup></td><td>0.285 <sup>(c)</sup></td><td>924</td><td>9.51 × 10<sup>-8</sup></td></tr><tr><td>85 wt. % glycerol</td><td>1222 <sup>(c)</sup></td><td>2648 <sup>(d)</sup></td><td>0.310 <sup>(c)</sup></td><td>1002</td><td>9.58 × 10<sup>-8</sup></td></tr><tr><td>Solid plastic</td><td>1192 <sup>(*)</sup></td><td>1560 <sup>(*)</sup></td><td>0.192 <sup>(*)</sup></td><td>598</td><td>1.03 × 10<sup>-7</sup></td></tr></tbody></table>
<figure><img src="image_21.png" /><figcaption>FIG. 8. Estimation of the thermal effusivity of the insulating material with the transient hot-plate method. (a) Temperature $\bar{T}$ recorded on the heated face as a function of the square root of time $t$. (b) Linear fit (black line) in the time interval 50–400 s. The data are linearly fitted by the equation $y(x) = 0.3998x - 0.8964$.</figcaption></figure>

effusivity of the insulating material is $E_i = 48.1 \text{ W m}^{-2} \text{ K}^{-1} \text{ s}^{-0.5}$, with a standard uncertainty of $u(E_i) = 1.9 \text{ W m}^{-2} \text{ K}^{-1} \text{ s}^{-0.5}$.

## B. Properties of the solid plastic used for 3D printing the foam

The thermal properties of the solid plastic that makes up the foam were not provided by the producer. For this reason, we used standard procedures to estimate the thermal conductivity ($k_s$), the density ($\rho_s$), and the specific heat capacity ($c_s$) of the solid plastic at a temperature of 25 °C.

The thermal conductivity was measured with the steady-state hot-plate method on a parallelepiped sample of thickness $e_s = 15.0$ mm and length $d_s = 99.2$ mm. The resulting thermal conductivity is $k_s = 0.192 \text{ W m}^{-1} \text{ K}^{-1}$ with an estimated standard uncertainty of $u(k_s) = 0.009 \text{ W m}^{-1} \text{ K}^{-1}$.

A density equal to $\rho_s = 1192 \text{ kg m}^{-3}$ is obtained from the weight (175.88 g) and the volume of the parallelepiped sample, assuming a whole sample without any defect in it.

The heat capacity $c_s$ of the solid plastic was determined by Differential Scanning Calorimetry (DSC). The heat capacity is

calculated from the heat transferred to a sample of 188.9 mg after temperature variations within the range of 15–96 °C [Fig. 9(a)]. The resulting $c_s$ as a function of $T$ is shown in Fig. 9(b). At a temperature of 25 °C, we have $c_s = 1.56 \text{ kJ kg}^{-1} \text{ K}^{-1}$, with a standard uncertainty of $u(c_s) = 0.05 \text{ kJ kg}^{-1} \text{ K}^{-1}$. The volumetric heat capacity of the solid plastic is then $1.86 \times 10^6 \text{ J m}^{-3} \text{ K}^{-1}$.

Finally, we estimated the porosity $\epsilon$ of the foam disk from the density $\rho_s$ and the foam's weight of 27.76 g. In this way, we obtain a porosity for the foam sample of $\epsilon = 0.83$. The same porosity was also obtained by water absorption. This value of $\epsilon$ falls within the expected porosity range for this type of structured foam (Kumar and Topin, 2017b).

## C. Equivalent thermal conductivity models

In Table IV, we report the overall thermal properties obtained for our fluid-saturated solid foam considering the two simple equivalent thermal conductivity models described in Sec. II A. In this case, the parallel and series models lead to the upper and lower limits of the overall thermal conductivity of the given multi-component system, respectively. The effective thermal effusivity and diffusivity

<figure><img src="image_22.png" /><figcaption>FIG. 9. DSC measurements of the solid plastic. (a) Temporal evolution of the heat flow rate (blue) and imposed temperature (red). (b) Specific heat capacity of the solid plastic $c_s$ as a function of temperature $T$. The data are fitted with the second order polynomial fit $y(x) = -0.08035x^2 + 15.44x + 1222$ ($R^2 = 0.997$).</figcaption></figure>
<table><caption>TABLE IV. Estimated effective thermal properties for the fluid-saturated solid foam obtained from the parallel and series models with $\varepsilon = 0.83$.</caption><thead><tr><th>System</th><th>Model</th><th>k<br>(W m<sup>-1</sup> K<sup>-1</sup>)</th><th>ρc<br>(J m<sup>-3</sup> K<sup>-1</sup>)</th><th>E<br>(W m<sup>-2</sup> K<sup>-1</sup> s<sup>-0.5</sup>)</th><th>α (m<sup>2</sup> s<sup>-1</sup>)</th></tr></thead><tbody><tr><td rowspan="2">Foam + water</td><td>Parallel</td><td>0.529</td><td>3.78 × 10<sup>6</sup></td><td>1414</td><td>1.40 × 10<sup>-7</sup></td></tr><tr><td>Series</td><td>0.440</td><td>3.78 × 10<sup>6</sup></td><td>1290</td><td>1.16 × 10<sup>-7</sup></td></tr><tr><td rowspan="2">Foam + pure glycerol</td><td>Parallel</td><td>0.269</td><td>2.80 × 10<sup>6</sup></td><td>868</td><td>9.61 × 10<sup>-8</sup></td></tr><tr><td>Series</td><td>0.263</td><td>2.80 × 10<sup>6</sup></td><td>858</td><td>9.39 × 10<sup>-8</sup></td></tr><tr><td rowspan="2">Foam + 85% glycerol</td><td>Parallel</td><td>0.290</td><td>2.95 × 10<sup>6</sup></td><td>925</td><td>9.83 × 10<sup>-8</sup></td></tr><tr><td>Series</td><td>0.281</td><td>2.95 × 10<sup>6</sup></td><td>910</td><td>9.53 × 10<sup>-8</sup></td></tr></tbody></table>

are then estimated from the weighted volumetric average of the heat capacity, i.e.,

$$ \rho c = (1 - \varepsilon)(\rho c)_s + \varepsilon(\rho c)_f, \quad (18) $$

where $(\rho c)_f$ indicates the volume heat capacity of the fluid saturating phase, $(\rho c)_s$ is the one of the solid plastic, and $\rho c$ is the one of the entire system.

When the saturating fluid is pure glycerol, the difference between the maximum and minimum values of effective thermal conductivity is only 2.3%. This is due to the very similar thermal conductivity of the solid and liquid phases. Since the difference between the results obtained with the two models is less than the uncertainty of classical measurements, we can conclude that the measurement of effective thermal conductivity is not necessary in this case.

On the other hand, for a water-saturated system, this difference reaches 18%. In this case, a measurement must be carried out since the mere use of one of these two models clearly leads to a non-negligible error in estimating the effective thermal conductivity of the system. The difference between the two values increases with the difference between the thermal conductivities of the two materials, so it would be even greater in the case of a metal or ceramic

composite material. In these cases, and in the absence of a more complex conductivity model suited to the specific geometry of the porous medium under consideration, appropriate measurements of the effective thermal properties must be made.

## D. Experimental results from the modified 4L device

We first validated the modified 4L device (Fig. 3) with a water-only sample. The water layer in this case had a thickness of $e = 15.1$ mm. The results for this validation are shown in Fig. 10 and summarized along with all the others in Table V. The intensity of heating was adjusted during the experiment so that the temperature on the top of the fluid layer $T_0(t) = T_0(t) - T_i$ was kept around 4 °C. The figure shows the experimental temperature $\tilde{T}_{1\text{exp}}(t)$ measured at the bottom of the water layer compared with $\tilde{T}_{1\text{mod}}(t)$ obtained from Eq. (15) for the value of thermal diffusivity $\alpha$ that minimizes the squared errors between $\tilde{T}_{1\text{exp}}$ and $\tilde{T}_{1\text{mod}}$. For the case of water, the estimation procedure is carried out in a time interval of 10–300 s since convection occurs after this time (residues are no longer flat after 300 s). In the specific case displayed in Fig. 10, we obtain $\alpha = 1.41 \times 10^{-7} \text{ m}^2 \text{ s}^{-1}$. By repeating the measure three times, we obtained a mean thermal diffusivity for water of $\alpha = 1.42 \times 10^{-7} \text{ m}^2 \text{ s}^{-1}$.

<figure><img src="image_23.png" /><figcaption>FIG. 10. Validation of the modified 4L device by using water. The temperature $\tilde{T}$ is the difference with the initial temperature, $\tilde{T} = T - T_i$. The thickness of the water layer is $e = 15.1$ mm. (a) Experimental temperature of the heated ($\tilde{T}_{0\text{exp}}$) and unheated ($\tilde{T}_{1\text{exp}}$) sample's face. (b) Comparison between experimental ($\tilde{T}_{1\text{exp}}$) and modeled ($\tilde{T}_{1\text{mod}}$) temperature. The estimation of $\alpha$ is carried out in the time interval of 10–300 s, where the residuals between $\tilde{T}_{1\text{exp}}$ and $\tilde{T}_{1\text{mod}}$ are flat and centered on zero. For this experiment, the latter is minimized for $\alpha = 1.41 \times 10^{-7} \text{ m}^2 \text{ s}^{-1}$.</figcaption></figure>
<table><caption>**TABLE V.** Experimental mean effective thermal diffusivity $\alpha_{\text{exp}}$ with the relative standard deviation (std). $\alpha_{\text{min}}$ and $\alpha_{\text{max}}$ are the values obtained from the series model and the parallel model, respectively.</caption><thead><tr><th>Material</th><th>Mean α<sub>exp</sub> (m<sup>2</sup> s<sup>-1</sup>)</th><th>Relative std (%)</th><th>α<sub>min</sub> (m<sup>2</sup> s<sup>-1</sup>)</th><th>α<sub>max</sub> (m<sup>2</sup> s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>Water</td><td>1.42 × 10<sup>-7</sup></td><td>3.3</td><td>1.43 × 10<sup>-7</sup></td><td>1.43 × 10<sup>-7</sup></td></tr><tr><td>Foam + pure glycerol</td><td>9.36 × 10<sup>-8</sup></td><td>0.5</td><td>9.39 × 10<sup>-8</sup></td><td>9.61 × 10<sup>-8</sup></td></tr><tr><td>Foam + 85% glycerol</td><td>9.80 × 10<sup>-8</sup></td><td>0.6</td><td>9.53 × 10<sup>-8</sup></td><td>9.83 × 10<sup>-8</sup></td></tr><tr><td>Foam + water</td><td>1.26 × 10<sup>-7</sup></td><td>1.8</td><td>1.16 × 10<sup>-7</sup></td><td>1.40 × 10<sup>-7</sup></td></tr></tbody></table>

with a standard uncertainty of $u(\alpha) = 7 \times 10^{-9} \text{ m}^2 \text{ s}^{-1}$ (Table V). This result deviates by less than 1% from the reference value reported in Table III.

After the device was validated, we measured the effective thermal diffusivity for the case of the fluid-saturated solid foam (Table V). In this case, the thickness of the sample was $e = 18.1$ mm. Measurements were carried out at a controlled external temperature of 20 °C, and they were repeated a minimum of three times to ensure reproducibility. The measurement procedure followed is the same as that used during device validation.

Figure 11 shows the estimation process for the case where the saturating fluid is 85% glycerol [Fig. 11(a)] and pure glycerol [Fig. 11(b)]. For a foam filled with 85% glycerol, the time interval in which Eq. (15) accurately models the experimental data is found to be [10–400] s. After this time, residuals depart from zero due to the onset of thermal convection. For this case, we found a mean thermal diffusivity of $\alpha_{\text{exp}} = 9.80 \times 10^{-8} \text{ m}^2 \text{ s}^{-1}$ with a standard uncertainty of $u(\alpha_{\text{exp}}) = 3.5 \times 10^{-9} \text{ m}^2 \text{ s}^{-1}$. When the solid foam is filled with pure glycerol, the residuals between $\bar{T}_{1\text{exp}}(t)$ and $\bar{T}_{1\text{mod}}(t)$ are found flat on zero within the interval of 10–500 s [Fig. 11(b)]. Here, we found a mean $\alpha_{\text{exp}} = 9.36 \times 10^{-8} \text{ m}^2 \text{ s}^{-1}$ with a standard uncertainty of $u(\alpha_{\text{exp}}) = 3.3 \times 10^{-9} \text{ m}^2 \text{ s}^{-1}$.

In both of these two cases, the mean experimental thermal diffusivity $\alpha_{\text{exp}}$ falls in the range defined by the thermal conductivities

$k_{\perp}$ and $k_{\parallel}$. With pure glycerol, $\alpha_{\text{exp}}$ differs from $\alpha_{\text{min}} = k_{\perp}(\rho c)^{-1}$ by only 0.3%, while for the aqueous glycerol solution, $\alpha_{\text{exp}}$ differs from $\alpha_{\text{min}}$ by 3.0% and from $\alpha_{\text{max}} = k_{\parallel}(\rho c)^{-1}$ by 0.3%. Since the thermal properties of the liquid and solid phases are very similar in this case, a small deviation between the values obtained with the equivalent models and the measured values is indeed expected. It also shows that the modified 4L device can accurately measure the thermal diffusivity of a composite system such as a fluid-saturated foam.

Figure 12 shows the thermal diffusivity estimation process in the case of a water-saturated foam. We found a good agreement between the measured and modeled temperature of the sample’s unheated face in the time interval between 10 and 500 s. In this case, the resulting mean effective thermal diffusivity is $\alpha_{\text{exp}} = 1.26 \times 10^{-7} \text{ m}^2 \text{ s}^{-1}$ with a standard uncertainty of $u(\alpha_{\text{exp}}) = 5 \times 10^{-9} \text{ m}^2 \text{ s}^{-1}$. This marks a decrease in $\alpha_{\text{exp}}$ of about 12% compared with the case where the sample was only water. In terms of effective thermal conductivity, the value of $\alpha_{\text{exp}}$ that we have found leads to $k_{\text{exp}} = 0.476 \text{ W m}^{-1} \text{ K}^{-1}$. The latter is well within the range imposed by the series and parallel models (see Table IV for a water-saturated foam). The percentage difference between $k_{\text{exp}}$ and $k_{\perp}$ is about 11%, while with $k_{\parallel}$ it is about 8%. The Hashin–Shtrikman upper bound formulation, Eq. (4), provides a better estimation of $k$. In this case, the difference between the experimental value

<figure><img src="image_24.png" /><figcaption>**FIG. 11.** Estimation of thermal diffusivity for the solid foam filled with (a) 85 wt. % glycerol and (b) pure glycerol. The temperature $\bar{T}$ is the difference with the initial temperature, i.e., $\bar{T} = T - T_i$. The thickness of the saturated foam is $e = 18.1$ mm. In (a), the time interval for estimating $\alpha$ is from 10 to 400 s. In this interval, residuals between $\bar{T}_{1\text{exp}}$ and $\bar{T}_{1\text{mod}}$ are flat and centered on zero. For the case shown here, they are minimized for $\alpha = 9.73 \times 10^{-8} \text{ m}^2 \text{ s}^{-1}$. In (b), the time interval where residuals are flat is from 10 to 500 s, and $\alpha = 9.41 \times 10^{-8} \text{ m}^2 \text{ s}^{-1}$.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>FIG. 12. Estimation of thermal diffusivity for a water-saturated structured solid foam. The temperature $\bar{T}$ is the difference with the initial temperature, i.e., $\bar{T} = T - T_i$. The thickness of the saturated foam is $e = 18.1$ mm. The time interval where residuals are flat is from 10 to 500 s. Here, residuals are minimized for $\alpha = 1.23 \times 10^{-7} \text{ m}^2 \text{ s}^{-1}$.</figcaption></figure>

and the one obtained from Eq. (4) is about 4%. This difference can be due to the formulation of the Hashin–Shtrikman model, which assumes a continuum porous medium. In our case, the solid foam used in the experiments has only about three unit cells in the vertical direction, a number that may not be sufficient to homogenize the sample.

To conclude, when the thermal properties between the saturating fluid and the solid that forms the structure of the composite system begin to differ substantially, as in this last case, the effective thermal properties of the system need to be measured appropriately. In light of the results obtained, this can be done through accurate measurements of the effective thermal diffusivity using the modified 4L device discussed in this study.

## V. CONCLUSION

In this paper, we estimate the overall thermal properties of a composite system consisting of a fluid-saturated solid foam. We used both a saturating fluid with a thermal conductivity similar to that of the solid plastic structure (i.e., glycerol or an aqueous solution of glycerol) and one with a higher thermal conductivity (i.e., water). The effective thermal conductivity $k$ of the composite system was first estimated using two equivalent models: the parallel model to obtain its maximum possible value and the series model to obtain its minimum value. Second, the value of the overall thermal diffusivity $\alpha$ was experimentally measured for each different composite system through a modified 4L device.

Considering the difficulty of measuring the effective thermal conductivity of composite materials by standard methods, the modified 4L device proposed here provides a reliable and accurate approach. Our experimental results converge with those obtained from existing simple equivalent models in the case where the different components of the system have similar thermal conductivities (e.g., glycerol-saturated foam). On the other hand, when the liquid and solid phases have very different thermal properties (e.g., in the case of a water-saturated foam), the values obtained by the various models are widely dispersed. This observation highlights the need to make appropriate experimental measurements to estimate the actual

thermal properties of these multicomponent systems or to consider more realistic equivalent models.

## 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 Mas-sif des Vosges 2019–2023. This work was also part of the project “CONVINCES,” financially supported by the French National Research Agency (Grant No. ANR-21-CE50-0029).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Nicolò R. Sgreva:** Conceptualization (equal); Data curation (equal); Investigation (equal); Validation (equal); Writing – original draft (equal). **Yves Jannot:** Formal analysis (equal); Methodology (equal); Validation (equal); Writing – original draft (equal).

**Christel Métivier:** Conceptualization (equal); Investigation (equal); Writing – original draft (equal).

## DATA AVAILABILITY

The data that support the finding of this study are available from the corresponding author upon reasonable request.

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