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<figure><img src="image_1.png" /></figure>

# Experimental and numerical investigation of porous heat exchangers with Kelvin cell structured foam at high temperatures: Coupled conduction-convection and radiation heat transfer

Raj Narayan Konduru<sup>a,b</sup>, Olivier Farges<sup>a,b</sup>, Vincent Schick<sup>a,b</sup>, Patrick Hairy<sup>c</sup>, Yves Gaillard<sup>c</sup>, Gilles Parent<sup>a,b,*</sup>

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

**Keywords:**

Porous heat exchanger

High temperatures

Conduction

Convection

Radiation

Coupled phenomenon

Complex geometries

Numerical methods

Finite volume method

## ABSTRACT

Heat recovery is crucial in the energy sector, especially for industries operating at high temperatures. One way to achieve high-efficiency heat recovery is to use porous materials in heat exchangers. In this work, the performance of a high temperature heat exchanger was investigated. For this purpose, a tube containing a porous cast iron structure was externally heated by radiation inside a tubular furnace and air at ambient temperature was blown into the tube. Experiments were conducted at different temperatures of up to 800 °C and different airflow velocities in a specialized bench, considering the combined effects of conduction, convection, and radiation. Temperature profiles in three sections of the porous structure were measured by using sheathed thermocouples brazed to the porous structure. Power and exergy flow rate extracted from the porous structure to the air flow were calculated and analyzed to quantify recovered heat. In addition, a numerical modeling performed at the pore level using Fluent showed an average relative error of 15% compared to experimental results, indicating acceptable agreement with observed trends. This numerical model will serve as a valuable reference tool for future parametric analyses of porous structures in heat exchanger configurations.

## 1. Introduction

Greenhouse gas emissions profoundly affect Earth's climate by trapping heat in the atmosphere, leading to global climate change primarily driven by human activities like fossil fuel combustion and deforestation [1]. These emissions, especially from industries, are often treated as waste, exacerbating the issue. The energy sector, responsible for a significant share of emissions, has seen a continuous rise in gases like nitrous oxide, CO₂, and methane, according to data from 2020 and 2021. The IPCC Sixth Assessment Report (AR6) in 2023 confirms human-induced global warming, with temperatures surpassing 1.1 °C above pre-industrial levels from 2011 to 2020.

To combat this, measures must be taken: enhancing energy efficiency, managing consumption, adopting renewable energy sources, and implementing emission-reducing strategies. Industries, particularly high-temperature ones like metallurgy and cement, need to prioritize energy efficiency and heat recovery to reduce emissions significantly.

This article focuses on porous structures as a possible way to harness waste heat, particularly in the form of radiation.

Utilizing porous structures within heat exchanger tubes is a promising approach to enhance heat recovery efficiency in a compact design. Though it increases pressure drop, the overall system efficiency remains favorable due to the inherent characteristics of porous materials, such as their substantial surface area and tortuous structure. This allows heat exchangers to perform better while occupying less industrial space. In these foam structures, there exists an interconnected network comprising repeating units with specific dimensional arrangements referred to as unit cells. These unit cells can take various forms, such as cubes, Kelvin cells, as well as stochastic shapes, among others.

Numerous researchers have explored the utilization of porous media in heat exchangers. Hunt and Tien (1988) [2] presented a study of heat transfer in fibrous media. Numerically, the volume-averaged continuity and momentum equations in a porous medium were used. Experimentally, water flowed through a channel filled with a fibrous medium, with

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

<footnote><sup>b</sup> Laboratoire Commun Canopée, CNRS, Université de Lorraine, Saint-Gobain, France</footnote>

<footnote><sup>c</sup> CETIM Pôle FMC, 44 avenue de la Division Leclerc, F-92318 Sevres cedex, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: gilles parent@univ-lorraine.fr (G. Parent).</footnote>
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## Autres
Nomenclature

Acronyms

DM Deterministic Method
FVM Finite Volume Method

Greek Symbols

<table><tr><td>α</td><td>Absorbance</td><td>-</td></tr><tr><td>δT<sub>exp</sub></td><td>Experimental Temperature difference</td><td>°C</td></tr><tr><td>δT<sub>num</sub></td><td>Numerical Temperature difference</td><td>°C</td></tr><tr><td>ε</td><td>Emissivity (-)</td><td></td></tr><tr><td>η</td><td>Thermal diffusivity</td><td>m<sup>2</sup>·s<sup>-1</sup></td></tr><tr><td>λ</td><td>Wavelength</td><td>m</td></tr><tr><td>φ</td><td>Porosity</td><td>%</td></tr><tr><td>ρ</td><td>Density</td><td>kg·m<sup>-3</sup></td></tr><tr><td>k</td><td>Thermal Conductivity</td><td>W·m<sup>-1</sup>·K<sup>-1</sup></td></tr></table>

Latin Symbols

B Exergy flow rate kW

<table><tr><td>C<sub>p</sub></td><td>Heat Capacity</td><td>J·kg<sup>-1</sup>·K<sup>-1</sup></td></tr><tr><td>d</td><td>Diameter</td><td>m</td></tr><tr><td>d<sub>c</sub></td><td>Cell Diameter</td><td>m</td></tr><tr><td>d<sub>i</sub></td><td>Inner Diameter</td><td>m</td></tr><tr><td>d<sub>o</sub></td><td>Outer Diameter</td><td>m</td></tr><tr><td>d<sub>strut</sub></td><td>Strut Diameter</td><td>m</td></tr><tr><td>h<sub>specific</sub></td><td>Specific enthalpy of air</td><td>kJ·kg<sup>-1</sup></td></tr><tr><td>I<sub>0</sub></td><td>Blackbody intensity</td><td>W·m<sup>-2</sup>·sr<sup>-1</sup>·µm<sup>-1</sup></td></tr><tr><td>l</td><td>Length</td><td>m</td></tr><tr><td>m</td><td>Mass of the fluid</td><td>kg</td></tr><tr><td>P</td><td>Power extracted by air</td><td>kW</td></tr><tr><td>t</td><td>time</td><td>s</td></tr><tr><td>T<sub>ambient</sub></td><td>Ambient Temperature</td><td>°C</td></tr><tr><td>T<sub>avg</sub></td><td>Average Temperature</td><td>°C</td></tr><tr><td>T<sub>furnace</sub></td><td>Furnace Temperature</td><td>°C</td></tr><tr><td>T<sub>in</sub></td><td>Inlet Temperature</td><td>°C</td></tr><tr><td>T<sub>out</sub></td><td>Outlet Temperature</td><td>°C</td></tr><tr><td>T<sub>radial</sub></td><td>Radial Temperature</td><td>°C</td></tr><tr><td>T<sub>section</sub></td><td>Temperature at sections</td><td>°C</td></tr><tr><td>T<sub>surface</sub></td><td>Temperature of the surface</td><td>°C</td></tr><tr><td>V<sub>in</sub></td><td>Inlet Velocity</td><td>m·s<sup>-1</sup></td></tr></table>

one side of the channel heated. An averaged heat transfer coefficient was determined from the experiment. Both experiment and calculations showed the heat transfer enhancement provided by the porous medium. In Lu et al. (1998) [3], an analytical approach was developed to model foams with simple cubic unit cells consisting of heated slender cylinders, based on existing heat transfer data on convective crossflow through cylinder banks. In another investigation by Bhattacharya et al. (2002) [4], an analytical and experimental approach was employed to determine the effective thermal conductivity, permeability, and inertial coefficient of high-porosity metal foams. As effective conductivity was studied in this work, no fluid flow was considered

Boomsma et al. (2003) [5] investigated the use of open-cell metal foams as compact heat exchangers and found that they exhibit thermal resistances that are approximately one-third of those in the best available conventional heat exchangers. In this study, water was used as fluid. In the work conducted by N. Delalic et al. (2004) [6], the development of devices with porous burners to increase energy efficiency and reduce emissions was explored. The research concluded that the implementation of low-cost, highly efficient, compact heat exchangers based on porous media as supplemental installations in existing boilers can significantly improve energy efficiency and mitigate environmental pollution. Shadi Mahjoob and Kambiz Vafai [7] provided valuable insights into fluid and thermal transport models for metal foam exchangers, along with correlations established by numerous researchers.

In the work of Ranut et al. [8], high-resolution X-ray microtomography-based CFD simulations were performed on three open-cell aluminum foam samples with different pore porosities (10-20-30 PPI). The computed values of permeability and effective thermal conductivity are reported and compared with the corresponding experimental values available in the literature. As in the Bhattacharya et al.'s paper [4], as far as effective conductivity is concerned, it implies that there is no fluid flow. Kuruneru et al. [9] provided a review on various applications of porous metal foam heat exchangers, focusing primarily on fouling mechanisms and their impact on efficiency.

In the realm of metallic foam heat exchangers specifically focusing on the combination of conduction and convection while excluding radiation effects, Emad Sepehri and Majid Siavashi (2022) [10] have studied the effect of heat transfer at different levels of porosity using a pore-scale OPEN-FOAM direct numerical simulation of flow and

convection-conduction heat transfer in open-cell metal foams with a complex structure. In this work, permeability, Forchheimer coefficient, pressure drop, volumetric heat transfer coefficient, temperature and pressure distributions were investigated. Fluid was water and a uniform heat generation inside the solid matrix was considered. Mingrui Sun et al.'s study [11] involved experimental comparisons between kelvin-cell and open-cell stochastic metal foams, examining the effects of pore per inch (PPI) and porosity on the pressure drop and the overall heat transfer coefficient (HTC). Bogdan I. Pavel et al. (2004) [12], conducted experimental and numerical investigations on the impact of metallic porous materials inserted in a pipe on heat transfer rates. The results indicated that the use of porous inserts can enhance heat transfer, albeit with a reasonable pressure drop. Mechanisms such as flow redistribution, thermal conductivity modification, and improved radiative heat transfer were identified as the main factors contributing to the heat transfer enhancement. It should be noted that while the numerical simulation did not account for conduction in the pipe walls, experimental observations showed its influence. Lu et al. (2006) [13] presented an analytical study of the forced convection heat transfer characteristics in high porosity open-cell metal-foam filled pipes. The Brinkman-extended Darcy momentum model and two-equation heat transfer model for porous media were employed. Based on the analytical solutions, the velocity and temperature distributions in metal-foam filled pipes were obtained. The effects of the microstructure of metal foams on overall heat transfer were examined and the results showed that the pore size and porosity of metal-foams play important roles on overall heat transfer performance.

In high-temperature scenarios, thermal radiation takes precedence as a significant heat transfer mechanism. Morgan Sans et al. [14] undertook a parametric investigation, focusing on the interplay of conduction and radiation within ceramic foams under vacuum conditions. The work established a correlation between the structural attributes and the corresponding equivalent thermal properties of these foams. No fluid flow is involved in this work since equivalent thermal conductivity was concerned. Significant investigations have been carried out concerning the combined effects of conduction, convection and radiation on thermal performance. Researchers have extensively analyzed the influence of porous medium structure parameters, temperature variations, and radiation parameters. Li et al. (2018) [15] performed simulations to explore the interplay of thermal radiation and convection in a metal foam (13
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<figure><img src="image_3.png" /><figcaption>Fig. 1. General setup of the porous heat exchanger.</figcaption></figure>

and 35 PPI) inside a rectangular channel heated to 1000 K on two opposite sides of the channel. Both Continuous Scale Approach (the medium is homogenized) and Discrete Scale Approach (full simulation of the actual structure of the material at the pore level) have been performed and compared. Fluent was used for both approaches. For DSA, surface-to-surface radiation transfer was considered, whereas for CSA the Monte Carlo method was used to solve the radiation. The comparison between the two methods showed very good agreement. An analysis conducted in 2005 by X. Chen and W. H. Sutton [16] revealed that the combined convection and radiation process involving porous inserts enhanced convective and radiative transfer by 35% and 105%, respectively. Hatami et al. (2013) investigated heat transfer and temperature distribution equations for circular convective-radiative porous fins with various section shapes and materials [17].

Further, researchers explored on coupled conduction-convection-radiation heat transfer in different applications. L. Ibarrart et al. (2018) [18] introduced a numerical model that allows the coupling of all three heat transfer modes using the Monte Carlo method, emphasizing memory efficiency and fast computation. In addition, the Monte Carlo method has shown a remarkable insensitivity to complexity, making it particularly interesting for the study of porous media without having to resort to homogenization. M. Pelanconi et al. (2019) [19] studied ceramic lattices (C111 structured foam based on cubic cells) inserted into a tube and heated radially at temperatures up to 700 °C. They showed a significant increase in heat transfer (in the range +160% - +280%) when the tube is filled with a foam compared to a plain tube. It is worth noting that in this work the heat transfer was quantified by measuring the outlet air temperature only. Numerically, both 2D axisymmetric homogeneous approach and full 3D calculations were performed. Radiative transfer was modeled using the Discrete Ordinate Method. In another paper, M. Pelanconi et al. (2021) [20] numerically studied structures consisting of one to three layers of structured Kelvin cells sandwiched between two parallel plates. Hot (from 600 °C to 1200 °C) fluid flow was applied, while a temperature 33 °C lower than the fluid temperature was imposed on the plates. OpenFOAM was used for the 3D calculations without radiation, Fluent was used for the calculations with radiation and the Surface-to-Surface model was used to handle the radiation. A similar study is presented in Zavattoni, S.A. et al. (2022) [21]. Jing Ma et al. (2017) [22] introduced a spectral element method (SEM) for analyzing heat transfer in irregular porous fins, demonstrating accuracy and computational efficiency. Zhiyong Wu et al. (2017) [23] investigated temperature distribution in ceramic foam-based solar receivers, emphasizing the effect of mean cell size and thermal conductivity on temperature fields while accounting for pressure drop and radiative heat transfer. In this work, a macroscopic model using the Local Thermal Non-Equilibrium (LTNE) approach was employed. Fluid and solid phases exchange heat via a volumetric convection coefficient. Radiative transfer was solved using the P1 method and the radiative properties of the foam were obtained from geometrical optics as described in K. Vafai (2005)'s Handbook of porous media [24]. The solid

and fluid (air) conductivities were described by correlations, as well as the pressure drop, and the coupled equations were solved in 2D using Fluent. T. Fischedick et al. (2017) [25] experimentally determined the radial two-phase thermal conductivity of ceramic sponges (10 to 30 PPI) under forced convection up to 800 °C. Radial two-phase thermal conductivity is a central parameter in homogeneous (assuming Local Thermal Equilibrium (LTE)) macroscopic approaches.

Based on the current literature, it has been seen that several researchers have explored the integration of metal foam structures into heat exchangers. This exploration has encompassed various combinations of heat transfer mechanisms, including coupled conduction-convection and convection-radiation in metal foam heat exchangers, as well as conduction-radiation and coupled conduction-convection-radiation, primarily utilizing ceramic foam heat exchangers.

However, high-temperature studies are still not very numerous. Most experimental work includes very few temperature measurements, often limited to the outlet temperature, due to the difficulty of instrumenting a foam at high temperature. Numerically, studies at the pore level are quite recent and remain not numerous. To the best of our knowledge, there is no study of high temperature heat transfer in a porous structure combining both experimental and numerical work, allowing a direct comparison between the two. The objectives of this research study are as follows

* To conduct experimental investigations within a setup employing a tubular furnace as the heat source, where a porous metal foam is included inside a tube within the furnace. These experiments will encompass the exploration of heat transfer involving conduction, convection, and radiation,

* to provide temperature measurements at numerous locations within the foam structure, using thermocouples brazed directly to the foam,

* to perform numerical simulations at the pore level, including all three modes of heat transfer, including accurate radiation transfer between foam struts using the surface-to-surface model. Geometry and boundary conditions will align as closely as possible to the experiment, including the geometry and location of the furnace heating resistors,

* To compare experimental and numerical results,

* To assess the power transferred to the airflow and analyze the potential for using the energy contained in the hot air stream.

The subsequent sections of this paper provide a comprehensive account of our experimental setup, the procedures employed during experimentation, the findings from our experimental endeavors, details of our modeling approach, and a comparative analysis of numerical results versus experimental observations.
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<figure><img src="image_4.png" /><figcaption>Fig. 2. Kelvin-cell structure: geometrical representation. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)</figcaption></figure>

## 2. Experimental method

In this study, the porous material, namely Kelvin-cell structured foam, is inserted into a plain tube. The overall concept of the experimental setup is illustrated in Fig. 1. The principles, construction, and operational procedures will be discussed in the subsequent sections.

### 2.1. Principle of porous heat exchanger

The primary principle behind incorporating porous materials in a heat exchanger is to significantly enhance the heat transfer rate compared to traditional methods that do not utilize porous materials. The rate of heat transfer is directly influenced by the surface area, porosity and the geometrical properties of the porous material. By increasing the surface area of the material or wall, the heat transfer can be greatly improved.

After subjecting the porous material to high temperatures, the airflow begins at the inlet and traverses through the tube, ultimately reaching the outlet. During this process, the air molecules come into contact with the hot solid, absorbing thermal energy as they carry it towards the outlet. The effectiveness of heat transfer is directly linked to the number of air molecules exposed to the heat source.

The foam employed in this study features a structured lattice design known as the Kelvin-cell pattern, as clearly illustrated in Fig. 2. The geometrical model of the Kelvin-cell foam was provided by CETIM pole FCM (formerly CTIF, Centre Technique des Industries de la Fonderie), and samples made of this foam were manufactured in cast iron by CETIM too. The sample was manufactured using a molding process, ensuring a seamless integration of the tube and the foam, eliminating any thermal resistance between the two. The foam structure consists of cells with a diameter $d_c$ equal to 14 mm and struts with a diameter $d_{strut}$ equal to 2.5 mm. With a length ($l$) of 200 mm, an inner diameter ($d_i$) of 100 mm, and a porosity ($\phi$) of approximately 85%, the foam part exhibits desirable characteristics for efficient heat transfer. Foam parameters were chosen based on the constraints and availability in the manufacturing process.

### 2.2. Measurement of foam sample properties

The thermo-physical and radiative properties of the sample are important factors that play a significant role in heat transfer. In order to conduct accurate experiments and simulations, it is important to determine these properties with a sufficient degree of precision. By knowing the thermo-physical properties, we can gain insights into how the sample will behave under different conditions, such as changes in temperature. Similarly, the radiative properties of the sample provide information about its ability to absorb or emit radiation, which is important for understanding its overall heat transfer characteristics. Therefore, the determination of these properties was an important step in our research.

**Thermo-physical properties** Accurate measurement of the thermo-physical properties of the foam sample was achieved using a laser flash measurement system (LFA1000) provided by Linseis company. This instrument allows the determination of the sample's thermal diffusivity.

The properties were measured using a cast iron small sample with a diameter of 25.58 mm and a thickness of 2.29 mm. The sample had a weight of 8.39 g, and was subjected to an annealing process prior to the measurements. The density of the sample, $\rho$, was determined from the sample dimensions and weight and found to be 6970(10) kg·m⁻³.

The laser flash system was utilized to measure the thermal diffusivity ($\eta = \frac{k}{\rho C_p}$) of the sample and its heat capacity ($C_p$) was determined using Differential Scanning Calorimetry (Setaram Multi HTC96). The thermal conductivity ($k$) of the sample was calculated from the thermal diffusivity formula.

**Radiative properties** It is worth noting that accurate measurement of radiative properties is important in the study of heat transfer in porous materials, since it directly affects the amount of radiative heat transfer occurring between the surfaces of the porous medium. To further elaborate on the radiative properties measurement, an infrared spectrometer equipped with an integrating sphere was used to measure the reflectance ($R(\lambda)$) of the sample as a function of the wavelength $\lambda$. This measurement was made after oxidation of the sample (exposed to 800 °C for several hours). Using Kirchhoff's law which expresses that the spectral emissivity ($\epsilon(\lambda)$) is equal to the spectral absorbance ($\alpha(\lambda)$),
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<figure><img src="image_5.png" /><figcaption>Fig. 3. Position of sections A, B and C.</figcaption></figure>

the emissivity was obtained from the measured reflectance (the sample being opaque).

$$<\epsilon> = \frac{\epsilon(\lambda) \times I_0(\lambda, T_{\text{surface}}) \, \text{d}\lambda}{I_0(\lambda, T_{\text{surface}}) \, \text{d}\lambda} = \frac{(1 - R(\lambda)) \times I_0(\lambda, T_{\text{surface}}) \, \text{d}\lambda}{I_0(\lambda, T_{\text{surface}}) \, \text{d}\lambda} \quad (1)$$

$T_{\text{surface}}$ is the temperature of the porous surface and $I_0(\lambda, T_{\text{surface}})$ is the blackbody spectral intensity.

## 2.3. Construction

The experiment involves the use of a Kelvin cell foam structure integrated into a cylindrical tube. As mentioned earlier, the foam and tube were made as a single piece using a casting process, so there is no thermal resistance between the tube and the foam. 100 mm long pieces were fabricated and two of them were welded together to form a 200 mm long tube-and-foam piece. Prior to this operation, 1 mm diameter sheathed K-type thermocouples were brazed to the foam struts at the ends of the pieces. This allowed temperature measurements to be taken in three sections A, B and C, as shown in Fig. 3, including in the mid-plane (i.e. section B) of the 200 mm long structure. The thermocouples were placed in the center of the struts along the vertical and the horizontal centerlines of each section, as shown in Fig. 4a, thus providing temperatures at 10 positions along the foam radius. In sections A and B, one of the thermocouples placed closest to the tube inner surface was not functioning, so there were only 9 temperatures available in those two sections. Additional 0.5 mm diameter sheathed thermocouples with naked junction were also placed near the outlet section with their tips in air to measure the fluid temperature. These thermocouples were twisted around the sheath of 1 mm thermocouples which are brazed to the foam struts. Images of the foam equipped with the thermocouples are shown in Fig. 4.

Fig. 5 shows a detailed schematic of the experimental setup, which consists of a tubular furnace (Carbolite GHA 12/1050) and the cylindrical tube and porous structure described above. Only 2 heating zones of the furnace were powered and the remaining part of the tube (located downstream the foam part) was insulated using Saint Gobain Silpower® material. The experimental setup is designed to allow air to flow through the heat exchanger in a controlled manner, and the goal of the experiment was to investigate the efficiency of the porous structure to extract heat and transfer it to the air.

* A 12 mm diameter pipe was connected to a pressure regulator from the air station to provide and control the airflow.

* A hot wire anemometer was placed before entering the tube to measure the air flow rate, which could measure up to 0.35 m·s⁻¹ in our case.

* The tube was constructed as shown in the schematic diagram Fig. 5

* A wheel anemometer was placed at part 'c' of the tube to measure air flow rates up to 1 m·s⁻¹ in our case.

* The Kelvin-cell foam made of cast iron material, along with thermocouples welded as shown in Fig. 4, was placed at part 'd' of the tube, followed by a cylindrical tube made of stainless steel.

<table><caption>**Table 1**
Maximum uncertainty of experiment in the study.</caption><thead><tr><td>Parameter</td><td>Uncertainty</td></tr></thead><tbody><tr><td>Temperature (K-Type Thermocouple)</td><td>± 2 °C</td></tr><tr><td>Flow rate (Hot wire anemometer)</td><td>± 3%</td></tr><tr><td>Flow rate (Wheel anemometer)</td><td>± 0.05 m·s⁻¹</td></tr></tbody></table>

<table><caption>**Table 2**
Operating conditions of the experiment in the study.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Furnace Temperature ($T_{\text{furnace}}$)</td><td>200 °C - 800 °C</td></tr><tr><td>Inlet velocity ($V_{\text{in}}$)</td><td>0.1 m·s⁻¹ - 0.5 m·s⁻¹</td></tr></tbody></table>

* 30 sheathed thermocouples of type K (two of them being not functioning) were used to measure the temperature of the foam at different locations i.e. entry, mid and exit sections (sections A, B and C) as above described.

* 6 thermocouples were electrically welded (separated contact) on the surface of the tube at three sections uniformly on the surface of porous sample at distance of 100 mm to measure the temperature of the external surface of the tube.

* All thermocouples, anemometers, pressure regulators, and the furnace's power measuring device were connected to a data acquisition system (National Instruments) to record the data.

* The real view of the experimental setup is shown in Fig. 6.

Additionally, the maximum uncertainties and operating conditions of the above parameters, including temperature, flow rate are shown in Table 1 and Table 2 respectively.

## 2.4. Working protocol

i. The furnace was heated up to the desired temperature set point without any airflow.

ii. Internal and external temperatures were measured during this phase to establish a baseline

iii. The system was allowed to reach thermal equilibrium.

iv. The airflow was started at the desired speed with a stepwise increase in velocity to observe the transient effects. Thermal equilibrium was established after each velocity step.

v. The airflow was stopped and the system was allowed to return to thermal equilibrium.

vi. The procedure was repeated from step ii with a new airflow velocity set point.

This process was carried out for all the experiments by varying the furnace operating temperature $T_{\text{furnace}}$ and inlet velocity $V_{\text{in}}$. The data collected from each experiment was recorded and analyzed. The experimental results are discussed in section 4.1 followed by numerical comparison in section 4.2.

# 3. Numerical simulation

Because of the complexity involved in setting up experiments for testing various configurations of metallic foams in a porous heat exchanger, conducting such experiments is not feasible for a large number of cases. Therefore, numerical modeling serves as a valuable tool to assess the heat exchanger's performance under various foam configurations. It is essential for this numerical model to be validated against experimental results, enabling its use for further parametric analysis of the heat exchanger. The numerical model enables the execution of 3D Computational Fluid Dynamics (CFD) simulations, providing a thorough comprehension of thermo-fluid dynamics behavior. Experimental results were obtained at various combinations of $T_{\text{furnace}}$ and $V_{\text{in}}$ conditions. For each of these experimental results, numerical simulations were conducted and compared, yielding a comprehensive understanding of the system's behavior.
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<figure><img src="image_6.png" /><figcaption>All Dimensions are in mm</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(b) Foam - Inlet section (section A)</figcaption></figure>

<caption>(c) Foam - Outlet section (section C)</caption>

<caption>Fig. 4. Positions and photos of the thermocouples brazed to the porous structure (Kelvin-cell).</caption>
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<figure><img src="image_8.png" /><figcaption>Fig. 5. Experimental setup - schematic view.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Experimental setup - real view built at Laboratoire Énergies & Mécanique Théorique et Appliquée (LEMTA).</figcaption></figure>

## 3.1. Computational domain

The entire domain consisted of three components: the solid medium (porous structure), the fluid medium (enclosure), and the furnace. The Kelvin-cell structure, represented by the pink color in Fig. 2, was obtained from the CETIM company, France.

It served as the solid medium within the system. To allow the interaction between the solid and fluid media, an enclosure was developed around the Kelvin-cell structure. This enclosure, depicted in gray in Fig. 2 (color: gray), acted as the fluid medium. The two media were perfectly interconnected, forming
<figure><img src="image_10.png" /><figcaption>Fig. 7. Computational domain of the simulation.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Polyhedral meshing for plain surface (Furnace) and complex structure (Kelvin-cell foam).</figcaption></figure>

a seamless interface between the solid and fluid regions. This geometric arrangement allowed for efficient heat transfer between the solid Kelvin-cell structure and the surrounding fluid medium.

In the experiment, only two heating zones of the furnace were used as shown in Fig. 7a. The temperature of the resistors was measured by a thermocouple, which was then imposed on the surface of the geometry as shown in Fig. 7c for simulation. The insulation (color: white) as shown in Fig. 7a was designed according to the geometry of the furnace as shown in Fig. 7b. The geometry was thoroughly checked for errors and, once confirmed to be error-free, proceeded to the next operation.

## 3.2. Computational grid generation

The mesh grid independence test was conducted to determine the appropriate element size for the simulation. Element sizes of 1.5 mm, 1 mm, and 0.7 mm were considered. After analyzing the stability, con-

vergence, and computational time, an element size of 1 mm was selected for the simulation.

In addition to the element size, the mesh element type was also tested using tetrahedral and polyhedral element types. Polyhedral elements were used as a mesh type for better stability when dealing with complex geometries [26], [27]. Tetrahedral mesh type resulted in a higher skewness and poorer meshing of sharp corners compared to polyhedral mesh type. Illustrations of both meshes are shown in appendix (Fig. B.22a and Fig. B.22b). Moreover, the polyhedral mesh type showed good meshing at sharp corners with a non-uniform distribution of elements, which made the computational time faster with good accuracy. Hence, polyhedral element type with 1 mm element size was considered for all subsequent cases. The sample meshing of the furnace is as shown in Fig. 8a and porous structures in Fig. 8b.
<figure><img src="image_12.png" /><figcaption>Fig. 9. Pressure-based coupled solver.</figcaption></figure>

### 3.3. Numerical modeling

The steady-state numerical model employs the finite volume method [28] and is implemented using ANSYS Fluent® v17.2 (ANSYS, Pittsburgh, PA, USA) [29] to solve equations governing mass, momentum, and energy conservation. The numerical analysis was subjected to the following assumptions:

* All analyses were conducted under steady-state conditions, which were deemed appropriate for the application, as heat exchangers typically operate for extended periods.

* The Knudsen number was calculated based on the maximum operating conditions and was determined to be less than 0.01. This validates the use of a continuum model for fluid analysis.

* The Mach number was computed to be 0.003, indicating incompressible flow, and therefore, incompressible flow equations were employed.

* The ideal gas equation of state was utilized to describe air expansion due to the air heating.

* To simulate fluid motion, the realizable k-epsilon turbulence model [30,31], was employed.

* For radiation modeling, the solid medium was assumed to be opaque, while the fluid medium (air) was considered transparent. As a result, a surface-to-surface model was used to address radiation.

* The thermal conductivity of air was assumed to be constant and its viscosity was calculated using Sutherland's law.

* The solid properties of cast iron were adopted from the properties listed in section 2.2. User-defined functions (UDFs) were used to define temperature-dependent properties for the simulations.

* It was assumed that both phases were in a state of thermal equilibrium due to a well-established interface. Therefore, the Local Thermodynamic Equilibrium (LTE) model was applied.

The conservation of mass can be written with index notation as

$$ \frac{\partial u_i}{\partial x_i} = 0 \qquad (2) $$

where $u_i$ is the velocity vector. To compute the viscous shear drags induced pressure drop across the domain, the Reynolds' conservation of momentum equation can be considered as

$$ \rho u_j \frac{\partial u_i}{\partial x_j} = \rho g_i - \frac{\partial p}{\partial x_i} + \mu \frac{\partial^2 u_i}{\partial x_j \partial x_j} + \frac{\partial}{\partial x_j} \left( -\rho \overline{u'_i u'_j} \right) \qquad (3) $$

where $\rho$ is the fluid density, $g_i$ is the vector of the conservative gravity field, $p$ is the hydrodynamic pressure, $\mu$ is the dynamic viscosity of the fluid, and $u'_i$ is the velocity fluctuation vector. The elements of

the Reynolds' stress tensor $R_{ij} = -\overline{\rho u'_i u'_j}$ in eq. (3) can be modeled by applying engineering turbulence models for the closure problem. The energy equation is solved to take into account the thermal energy exchange from the heated surfaces to the cooling air, including the viscous dissipation as

$$ \rho c_p u_j \frac{\partial T}{\partial x_j} = u_j \frac{\partial p}{\partial x_j} + \frac{\partial}{\partial x_j} \left( k \frac{\partial T}{\partial x_j} \right) - \rho c_p \frac{\partial}{\partial x_j} \left( \overline{u'_j T'} \right) + \phi_D + \rho \epsilon + \phi_S \qquad (4) $$

where $c_p$ is the specific heat capacity, $k$ is the thermal conductivity, $T'$ is the local temperature fluctuation, $\phi_D$ is the viscous dissipation function, $\epsilon$ is the turbulent energy dissipation, and $\phi_S$ represents heat sources in the computational domain i.e. radiation heat flux ($q_{rad}$). Each turbulence model predicts the Reynolds' stress distribution with different mathematical assumptions through modeling velocity and temperature fluctuations. The system of equations is solved using the pressure-based coupled solver, as referenced in Fig. 9, which is a feature provided within ANSYS Fluent®. Spatial discretization of the transport equations employed a second-order upwind scheme, and gradient calculations were performed using the Least Squares Cell-Based method. In addition to tracking the progress of key properties like the air outlet temperature, convergence of the iterative process was deemed successful when residuals reached values below $10^{-5}$ for momentum and continuity equations, $10^{-8}$ for energy.

### 3.4. Boundary conditions

The boundary conditions applied to the model, as detailed in Table 3, closely align with the experimental setup and test conditions described in section 2. Specifically, the inner surfaces of the furnace corresponding to the heating resistors (as shown in Fig. 7c) were subjected to a constant temperature ($T_{furnace}$) boundary condition. All other external surfaces were defined to be adiabatic. At the fluid medium's inlet, a velocity-inlet boundary condition was imposed, while at the outlet section, a constant relative pressure boundary condition was applied. For all simulations, it was assumed that the inlet air flow was fully developed. The system was subjected to testing at four different furnace wall temperatures, namely 200 °C, 400 °C, 600 °C and 800 °C, each combined with four distinct inlet air velocities, namely 0.1 m·s⁻¹, 0.2 m·s⁻¹, 0.3 m·s⁻¹ and 0.5 m·s⁻¹. This approach allowed for a more comprehensive exploration of the influence of thermal fluid behavior and facilitated a more robust comparison with the experimental results.

A well defined interface was established between solid kelvin-cell structure and fluid medium. This interface allowed for the efficient transfer of heat and other properties between the solid and fluid regions, ensuring accurate representation of the physical phenomena occurring within the system. Additionally, all external walls of the system were assumed to be adiabatic, meaning that no heat transfer occurred through
<figure><img src="image_13.png" /><figcaption>Fig. 10. Temperature contours at steady state: $T_{\text{furnace}} = 1073$ K and $V_{\text{in}} = 0.1 \text{ m} \cdot \text{s}^{-1}$. Positions of sections A, B and C are shown in Fig. 3.</figcaption></figure>

<table><caption>**Table 3**

Boundary conditions for numerical simulation.</caption><thead><tr><th>Boundary</th><th>Condition</th></tr></thead><tbody><tr><td>Inlet</td><td>Velocity-inlet<br>Ambient temperature ($T_{in}$ = 300 K)</td></tr><tr><td>Outlet</td><td>Pressure-outlet</td></tr><tr><td>Walls</td><td>Adiabatic</td></tr><tr><td>Interface</td><td>Coupled and Mapped</td></tr><tr><td>Furnace resistors</td><td>Furnace Temperature value</td></tr></tbody></table>

the walls. This assumption simplified the analysis by focusing on the heat transfer processes occurring within the Kelvin-cell structure and the fluid medium, without considering heat losses through the system walls. The carefully defined boundary conditions, interface connectivity, and adiabatic walls provided the necessary framework for simulating the heat transfer behavior and capturing the essential interactions within the system.

### 3.5. Sample test

A sample test simulation was carried out using the numerical configuration parameters outlined previously. In this particular test, the sample was exposed to a furnace temperature ($T_{\text{furnace}}$) of 800 °C and an inlet velocity ($V_{\text{in}}$) of 0.1 ms. The temperature contours in steady-state, as illustrated in Fig. 10, offer a visual depiction of the temperature distribution within the system.

The temperature distribution on the surface of the furnace, where the imposed temperature was present, can be observed in Fig. 10a. Notably, the heat was transferred from the surface of the furnace to the surface of the cast iron or solid media primarily through radiation. This temperature distribution on the cast iron surface is depicted in Fig. 10b.

To gain further insights into the heat transfer behavior, the overall temperature distribution in the system was examined in the axial plane view (2D) as shown in Fig. 10c. This visualization allows to ob-
serve how the flow through the porous structure affects the heat transfer from the inlet to the outlet. Additionally, radial views at sections A, B, and C (refer to Fig. 3 for the location of these sections) were analyzed to provide localized temperature representations. These views, displayed in Fig. 10d, Fig. 10e and Fig. 10f respectively, shed light on the temperature distribution at specific locations within the system. Heating resistors are clearly visible in Fig. 10a, Fig. 10d, and Fig. 10e (there is no heating resistor in section C since this section is between two heating zones). We can also see that the air is significantly colder than the foam, especially at the foam inlet. Therefore, it would not be possible to use a model that homogenizes the medium and considers a single temperature (LTE), since the air and the solid have a significantly different temperature. At the foam outlet, the temperature difference is not so high

# 4. Results and discussion

## 4.1. Experimental results

The experiments were conducted at various furnace temperatures ($T_{\text{furnace}}$) ranging from 200 °C to 800 °C and different inlet velocities ($V_{\text{in}}$) ranging from 0.1 m·s⁻¹ to 0.5 m·s⁻¹. The experimental protocol described in section 2.4 was followed to ensure accurate measurements. The temperatures were recorded at different sections, as illustrated in Fig. 3 from exact thermocouple location as in Fig. 4. As mentioned, welded thermocouples provide temperature readings of the solid, while additional thermocouples which were placed in close proximity to the thermocouples located in the solid medium served to measure the temperature of the fluid medium. During the experiments, it was observed that the temperature obtained from both sets of thermocouples were close to each other indicating that the temperature of the fluid and the solid at the outlet section were close to equilibrium.

The power extracted between the air entering at room temperature and leaving immediately after passing through the porous structure (i.e., section C) was calculated using eq. (5), which assumes revolution symmetry

$$P = \int 2\pi\nu(r)\rho(r) (h_{\text{specific}}(T_{\text{out}}(r)) - h_{\text{specific}}(T_{\text{in}})) r dr \quad (5)$$

where $h_{\text{specific}}(T_{\text{out}}(r))$ is the air specific enthalpy at temperature $T(r)$, $\nu(r)$ and $\rho(r)$ is the air velocity and density, respectively, at the radial position $r$. It is to be noted that the calculation of the power from the experimental results and from the numerical results is different. Indeed, in the experiment, we have the information of the temperature at the outlet section at some specific positions but no information of velocity and density at these same positions. So it was considered a constant value for velocity and density in eq. (5) obtaining eq. (6). In this equation, $\dot{m}$ is equal to the mass flow rate at the inlet of the system, since mass flow rate should remain constant.

$$P = \dot{m} \left( \frac{2}{R^2} \int h_{\text{specific}}(T_{\text{out}}(r)) r dr - h_{\text{specific}}(T_{\text{in}}) \right) \quad (6)$$

R being the tube radius. Considering that $h_{\text{specific}}(T_{\text{out}}(r))$ varies almost linearly over the narrow temperature range in the exit section, eq. (6) can be simplified even further:

$$\begin{aligned} P &= \dot{m} \left( h_{\text{specific}} \left( \frac{2}{R^2} \int (T_{\text{out}}(r)) r dr \right) - h_{\text{specific}}(T_{\text{in}}) \right) \\ &= \dot{m} (h_{\text{specific}}(T_{\text{out}}) - h_{\text{specific}}(T_{\text{in}})) \end{aligned} \quad (7)$$

$<T_{\text{out}}>$ being the average temperature in the outlet section. In the numerical simulations, we have all the information at local distribution at every location, eq. (5) can be used directly without any assumptions.

### 4.1.1. Porous heat exchanger

The experiments were conducted with a furnace temperature set-point varying from 200 °C to 800 °C. Hot wire anemometer or wheel

<figure><img src="image_14.png" /><figcaption>Fig. 11. Measured air $V_{\text{in}}$. Error bands are derived from uncertainties given in Table 1.</figcaption></figure>

anemometer was used to measure the inlet air velocities, which were from 0.1 m·s⁻¹ to 0.3 m·s⁻¹ and 0.5 m·s⁻¹ respectively, depending on which anemometer was used.

For the paper clarity, detailed graphs are presented only for the 200 °C setpoint temperature, graphs for other temperatures are provided in appendix since they are similar to those obtained at 200 °C. Variation of velocity is presented in Fig. 11. As can be seen in this figure, the inlet velocity was kept constant throughout the experiment duration, with only small variations around the chosen velocity.

Furnace temperature is presented in Fig. 12a to Fig. 12d. It is observed that $T_{\text{furnace}}$ was maintained constant through the experiment even when $V_{\text{in}}$ increased, which implies that the furnace has sufficient power to maintain the set point temperature throughout the time in all experiments whatever the inlet velocity in the range 0.1 m·s⁻¹ – 0.5 m·s⁻¹, except for the 800 °C temperature, as will be seen later.

The temperature of the outer surface of the tube is shown in Fig. 12e. As can be seen in this figure, this temperature decreased as the air flow was turned on. It took about 1000 s to reach a new thermal equilibrium. This delay is due to the thermal inertia of the foam-and-tube system. Logically, the higher the air velocity, the colder the surface temperature of the tube. Indeed, when the air velocity increases, the power extracted by the air increases, as will be shown later, so a greater temperature difference between the furnace surface and the tube surface is required to provide that power. The average temperature in each section over time has been plotted in Fig. 12f to Fig. 12h. The same observation applies as for the tube surface temperature, i.e. the higher the air velocity, the lower the temperatures. It is also observed that, as expected, the temperature increases from section A to section C. The steady state temperature profiles are shown in Fig. 13. At steady state, the temperature fluctuates around a mean value with a given standard deviation. In Fig. 13 both the mean and the standard deviation are shown using error bands. Due to the finite thermal conductivity, the center of the foam is colder than the tube, but the temperature gradient is not as high, indicating that the conduction is able to efficiently transfer the heat within the foam. Thus the limiting factor for the global heat transfer in the present experiment is the amount of heat that can be transferred from the furnace to the tube, mainly by radiation.

From the radial temperature profile at the exit of the foam, the power $P$ transferred to the air was calculated using eq. (7). The variation of this power as a function of time is shown in Fig. 12i. As can be seen in this figure, the power is very high at the beginning of the air blowing, because at this moment the whole foam is very hot and stores a lot of energy. After about 1000 s, the steady state is reached, as previously observed in the thermograms. This steady state power is shown in Table 4. As expected, the higher the velocity, the higher the power transferred to the air. However, this increase in power is accompanied by a decrease in foam outlet temperature, so this heat is of lower “quality”. One way to quantify this is to use exergy instead of energy. Exergy can be seen as the amount of work that can be extracted from a heat
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Experiment with Porous structure: Furnace power, temperatures at different positions with operating temperature of 200 °C at different velocities and power extracted by the air. Temperature uncertainty (2K) is too small to be represented, power uncertainty comes mainly from the velocity uncertainty.</figcaption></figure>
<caption>**Table 4**</caption>

Calculated power extracted by air, Exergy flow rate in kW and Carnot efficiency based on experimental results at steady state.

<table><thead><tr><th>T<sub>furnace</sub><br>(°C)</th><th>V<sub>in</sub><br>(m·s<sup>-1</sup>)</th><th>Power extracted by air<br>(Experimental results) (kW)</th><th>Exergy flow rate<br>(kW)</th><th>Carnot efficiency<br>(%)</th></tr></thead><tbody><tr><td rowspan="4">200</td><td>0.1</td><td>0.1083</td><td>0.0323</td><td>27</td></tr><tr><td>0.2</td><td>0.1343</td><td>0.0236</td><td>18</td></tr><tr><td>0.3</td><td>0.1399</td><td>0.0183</td><td>12</td></tr><tr><td>0.5</td><td>0.1763</td><td>0.0178</td><td>10</td></tr><tr><td rowspan="4">400</td><td>0.1</td><td>0.2975</td><td>0.1532</td><td>51</td></tr><tr><td>0.2</td><td>0.4330</td><td>0.1725</td><td>44</td></tr><tr><td>0.3</td><td>0.5051</td><td>0.1821</td><td>37</td></tr><tr><td>0.5</td><td>0.7051</td><td>0.2086</td><td>32</td></tr><tr><td rowspan="4">600</td><td>0.1</td><td>0.5456</td><td>0.335</td><td>65</td></tr><tr><td>0.2</td><td>0.9482</td><td>0.4746</td><td>60</td></tr><tr><td>0.3</td><td>1.0176</td><td>0.5256</td><td>53</td></tr><tr><td>0.5</td><td>1.3784</td><td>0.6329</td><td>47</td></tr><tr><td>800</td><td>0.1</td><td>0.8163</td><td>0.5763</td><td>72</td></tr></tbody></table>

<figure><img src="image_16.png" /><figcaption>(i) (cont.) Power extracted at $V_{in} = 0.1 \text{ m} \cdot \text{s}^{-1}$, $0.2 \text{ m} \cdot \text{s}^{-1}$, $0.3 \text{ m} \cdot \text{s}^{-1}$ and $0.5 \text{ m} \cdot \text{s}^{-1}$</figcaption></figure>

<caption>**Fig. 12. (continued)**</caption>

source using a perfect thermodynamic cycle, i.e. a Carnot machine. The formula used to estimate the exergy flow rate, $B$, is

$$B = P \left( 1 - \frac{T_{\text{ambient}}}{\langle T_{\text{out}} \rangle} \right) \qquad (8)$$

considering the ambient temperature as the cold temperature source (temperatures being considered in Kelvin in eq. (8)). This quantity is presented together with the power in Table 4.

Experiments were carried out at temperatures 400 °C, 600 °C and 800 °C. Results at these temperatures are presented in Figs. A.19–A.21 in Appendix A. Similar results were obtained as for 200 °C. Of course, as expected, the hotter the furnace, the higher the extracted power. Power and exergy flow rate for all the temperatures are given in Table 4. For 800 °C temperature the furnace was not powerful enough to maintain the setpoint temperature for air velocity higher than $0.1 \text{ m} \cdot \text{s}^{-1}$, as can be seen in Fig. A.21c.

Based on Table 4, the power extracted by the air was calculated at steady state at different furnace temperatures and inlet velocities. The key observations are

* Increasing the furnace temperature: As the furnace temperature increases from 200 °C to 800 °C, there is a noticeable increase in the power extracted by the air. The power does not increase linearly with the furnace temperature, mainly because the heat transfer between the furnace and the tube is primarily by radiation, and radiative heat transfer is essentially non-linear.

* Influence of Inlet Velocity: Increasing the inlet velocity of the air also results in an increase in extracted power. This increase is also non-linear because increasing the air velocity causes the foam temperature to decrease. The maximum achievable power is determined by the amount of heat that can be transferred from the furnace to the tube, or the maximum power of the furnace. As mentioned above, an increase in air velocity is accompanied by a decrease in outlet temperature, making the recovered heat less and less useful. If it is intended to convert this heat into work (for example, to produce electricity), the relevant quantity is exergy and not energy (or power). If the recovered heat is to be used for another process, a certain temperature level is required for that process. Or the number of possible processes that could use that heat is reduced.

* For a low furnace temperature, the exergy flow rate decreases as the air velocity increases. For a higher furnace temperature, the exergy continues to increase with air velocity in the range studied here, but it would reach a maximum and then decrease as the air velocity increases further.

## 4.2. Numerical comparison

In this section, numerical simulations were compared to the experimental results. The focus will be on steady-state conditions for validation. Although transient simulations and validations would be interesting to be done, they were not conducted due to two main reasons. Firstly, in practical applications, heat exchangers generally operate for extended periods and eventually reach an equilibrium state. Secondly, transient simulations require significantly more computational resources compared to steady-state simulations. However, it is worth mentioning that transient simulations would be relevant for heat storage applications which is also another potential solution for waste heat recovery.

### 4.2.1. Temperature cross sections

The simulation was performed at the same furnace temperature as used for the experiment i.e. 200 °C, 400 °C, 600 °C and 800 °C and the same for inlet velocities $V_{in}$ equal to $0.1 \text{ m} \cdot \text{s}^{-1}$, $0.2 \text{ m} \cdot \text{s}^{-1}$, $0.3 \text{ m} \cdot \text{s}^{-1}$ and $0.5 \text{ m} \cdot \text{s}^{-1}$.

**Results** Average temperature across all probe locations at each section in axial direction and temperature profiles in the different sections for furnace temperature equal to 200 °C are shown in Figs. 14, 15, 16 and 17. Experimental results are also shown in these graphs, which correspond to time-averaged temperatures in steady state. Graphs for the others furnace temperatures are provided in appendix (see
<figure><img src="image_17.png" /><figcaption>Fig. 13. Average steady-state temperature radial profiles at set-point temperature equal to 200 °C. Error band corresponds to temperature uncertainty and temperature standard deviation.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 14. Validation at $T_{\text{furnace}} = 200$ °C and $V_{\text{in}} = 0.1 \text{ m} \cdot \text{s}^{-1}$.</figcaption></figure>

Figs. C.23–C.31), since they look very similar to those obtained at 200 °C.

Table 5 gathers the relative differences between the numerical and the experimental results, these differences being calculated as

$$ Error(\%) = \left( \frac{\delta T_{\text{exp}} - \delta T_{\text{num}}}{\delta T_{\text{exp}}} \right) * 100 \quad (9) $$

where $\delta T$ is the temperature rise in °C given by

$$ \delta T = T_{\text{section}} - T_{\text{in}} $$

**Discussion of results** It should be noted that in contrast to previous research works cited in the literature review, such as those by researchers mentioned in [20,23], which typically employed a single thermocouple for temperature measurement in the area of interest, this study utilized ten thermocouples in each section providing temperature profiles.

As depicted in section 4.2.1, the observation reveals that the temperature profiles exhibit a relatively flat, characterized by a low concavity. This observation aligns with the fact that temperature is lower in the center of the foam, that is logical since the foam is heated from the external surface of the tube. However, it is worth noting that heat transfer
<figure><img src="image_19.png" /><figcaption>Fig. 15. Validation at $T_{\text{furnace}} = 200^{\circ}\text{C}$ and $V_{\text{in}} = 0.2 \, \text{m} \cdot \text{s}^{-1}$.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 16. Validation at $T_{\text{furnace}} = 200^{\circ}\text{C}$ and $V_{\text{in}} = 0.3 \, \text{m} \cdot \text{s}^{-1}$.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 17. Validation at $T_{\text{furnace}} = 200^{\circ}\text{C}$ and $V_{\text{in}} = 0.5 \, \text{m} \cdot \text{s}^{-1}$.</figcaption></figure>
<caption>**Table 5**</caption>

Numerical model was compared and validated with the experimental results for different furnace temperatures. Error % is used to represent the discrepancies. (+) and (-) signs indicate overestimation and underestimation, respectively, of the numerical results compared to the experimental results.

<table><thead><tr><th rowspan="2">T<sub>furnace</sub><br>(°C)</th><th rowspan="2">V<sub>in</sub><br>(m·s<sup>-1</sup>)</th><th colspan="3">Model validation with Experimental results<br>δT (in Error %)</th></tr><tr><th>Section A</th><th>Section B</th><th>Section C</th></tr></thead><tbody><tr><td rowspan="4">200</td><td>0.1</td><td>(+) 4.5%</td><td>(-) 0.25%</td><td>(-) 20%</td></tr><tr><td>0.2</td><td>(+) 1.25%</td><td>(+) 2.5%</td><td>(-) 21%</td></tr><tr><td>0.3</td><td>(+) 14%</td><td>(+) 21%</td><td>(-) 14%</td></tr><tr><td>0.5</td><td>(+) 20%</td><td>(+) 0.5%</td><td>(-) 33.5%</td></tr><tr><td rowspan="4">400</td><td>0.1</td><td>(+) 2%</td><td>(+) 2.5%</td><td>(-) 15%</td></tr><tr><td>0.2</td><td>(+) 4.5%</td><td>(+) 1%</td><td>(-) 21%</td></tr><tr><td>0.3</td><td>(+) 2%</td><td>(+) 0.5%</td><td>(-) 25%</td></tr><tr><td>0.5</td><td>(+) 4%</td><td>(+) 17.5%</td><td>(-) 43%</td></tr><tr><td rowspan="4">600</td><td>0.1</td><td>(+) 13%</td><td>(+) 9%</td><td>(-) 9%</td></tr><tr><td>0.2</td><td>(+) 23%</td><td>(+) 13%</td><td>(-) 13.5%</td></tr><tr><td>0.3</td><td>(+) 25%</td><td>(+) 18%</td><td>(-) 12%</td></tr><tr><td>0.5</td><td>(+) 1.5%</td><td>(+) 3.5%</td><td>(-) 27.5%</td></tr><tr><td>800</td><td>0.1</td><td>(+) 16%</td><td>(+) 12%</td><td>(-) 9%</td></tr></tbody></table>

mechanisms prove effective in achieving temperature homogenization throughout the foam.

Furthermore, it is noteworthy that the numerical results demonstrate a good agreement with the experimental findings, albeit with a reasonable overestimation evident in sections A and B. In contrast, there exists a consistent and notably significant underestimation in section C, among other observations.

**Section A:** Section A corresponds to the entrance portion of the foam. At a furnace temperature of 200 °C, the model exhibits a tendency to overestimate the temperature in Section A. This overestimation is more pronounced as the fluid inlet velocity increases, with errors of 4.5% and 1.25% at 0.1 m·s⁻¹ and 0.2 m·s⁻¹, respectively. At higher fluid velocities of 0.3 m·s⁻¹ and 0.5 m·s⁻¹, the overestimation becomes even more significant, reaching 14% and 20%.

This trend of overestimation continues at a furnace temperature of 400 °C. At lower fluid velocities (0.1 m·s⁻¹ and 0.2 m·s⁻¹), the model overestimates the temperature by 2% and 4.5%, respectively. Higher fluid velocities (0.3 m·s⁻¹ and 0.5 m·s⁻¹) result in similar overestimations of 2% and 4%. Moving to a furnace temperature of 600 °C, model consistently overestimates temperatures in Section A, with errors ranging from 1.5% to 25% across different fluid velocities.

However, at a furnace temperature of 800 °C, the model's performance in Section A continues to demonstrate a tendency to overestimate temperatures. At a fluid velocity of 0.1 m·s⁻¹ the error remains at 16%. The model consistently shows a propensity to overestimate temperatures in Section A, particularly at higher fluid velocities and furnace temperatures.

**Section B:** Within Section B, which corresponds to the middle portion of the foam-filled tube, the numerical model's performance is notably influenced by variations in both furnace temperature ($T_{furnace}$) and inlet fluid velocity ($V_{in}$). At a furnace temperature of 200 °C, the model demonstrates a nuanced response. It closely matches experimental data at lower fluid velocities, with a negligible underestimation of 0.25% at 0.1 m·s⁻¹ and a very reasonable overestimation of 2.5% at 0.2 m·s⁻¹. However, at a higher velocity of 0.3 m·s⁻¹, the model exhibits a substantial overestimation of 21%, suggesting an overemphasis on convective heat transfer, possibly at the expense of other influential factors. At the highest fluid velocity tested, 0.5 m·s⁻¹, the model's accuracy improves since the error is only 0.5%. As the furnace temperature increases to 400 °C, the model's performance remains relatively consistent at lower velocities, maintaining accuracy within a range of 0.5% to 2.5% overestimation. However, at 0.5 m·s⁻¹, the model shows

a notable overestimation of 17.5% under these higher-velocity conditions.

At furnace temperatures of 600 °C and 800 °C, the model consistently overestimates temperature differences across all fluid velocities in Section B. This observation suggests a general trend of overprediction as the furnace temperature increases. These findings underscore the significance of achieving a well-balanced representation of conduction, convection, and radiation heat transfer mechanisms within the mode.

**Section C:** It is located at the end of the foam-filled tube. When the furnace temperature is set at 200 °C, the model exhibits a consistent trend of substantial underestimation in Section C across various fluid velocities. At $V_{in} = 0.1 m·s⁻¹$, the model underestimates the temperature by a significant 20%. This underestimation continues at $V_{in} = 0.2 m·s⁻¹$ and $V_{in} = 0.3 m·s⁻¹$, with percentage errors of 21% and 14%, respectively. The highest fluid velocity of 0.5 m·s⁻¹ results in the most significant underestimation, with the model predicting a temperature 33.5% lower than the experimental data.

As the furnace temperature increases to 400 °C and beyond, the trend of underestimation in Section C persists. Regardless of the furnace temperature, the model consistently predicts lower temperatures in Section C compared to the experimental results. At higher temperatures of 600 °C and 800 °C, the underestimation remains substantial, with percentage errors ranging from 9% to 27.5%.

It is worthy to note that the numerical simulations were conducted under ideal conditions with a smooth and clean geometry. However, in the actual experimental setup, there is the presence of sheathed thermocouples traversing the foam, as illustrated in Fig. 4. In Section A (Fig. 3), the thermocouple wires are directed upstream, while in Sections B and C, they are oriented downstream. This arrangement of thermocouples had the potential to hinder both fluid flow and temperature distribution. Consequently, when the fluid flows from upstream to downstream, noteworthy differences were revealed in the experimental and numerical comparison. Between Sections A and B, where there are no thermocouples, the geometry resembled that used in simulations, but disturbances in fluid flow can occur between Sections B and C due to the presence of thermocouples. This can explain difference between the temperatures recorded experimentally in Section C and temperatures predicted numerically.

### 4.2.2. Recovered power and exergy

While temperature profiles are important to analyze temperature gradients within the porous structure and to validate that the model is capable of predicting sufficiently accurate temperature fields, the global performance of the system is characterized by how much power/exergy flow rate can be extracted and transferred to the air. These two quantities depend mainly on the average outlet temperature $\langle T_{out} \rangle$ as shown in eq. (7) and eq. (8).

Starting with the outlet temperature, $\langle T_{out} \rangle$, as shown in Fig. 18b, it was observed that, for different furnace temperatures and inlet velocities, the simulated values closely aligned the experimental measurements. This alignment demonstrates the simulation model's capability to accurately predict the global thermal behavior of the heat exchanger, successfully capturing the influence of changing operational parameters. The percentage differences between experimental and simulated values for temperature differentials ($\Delta T = \langle T_{out} \rangle - T_{in}$) remain within an acceptable range, affirming the model's accuracy in reproducing the actual thermal gradients within the foam.

The power extracted by the air as shown in Fig. 18c found that the simulated power values exhibit good agreement with the experimental values. The percentage differences in power extraction requirements are consistent, further underscoring the model's ability to predict the system's energy demands accurately. This parameter not only validates the model but also provides valuable insights into the energy dynamics of the system.
<table><caption>**Table 6**
Comprehensive data of all experiments and 3D simulations ($\Delta T = \langle T_{out} \rangle - T_{in}$).</caption><thead><tr><th colspan="2">Variables</th><th colspan="4">Experiment</th><th colspan="5">3D simulations</th></tr><tr><th>T<sub>furnace</sub><br>°C</th><th>V<sub>in</sub><br>m·s<sup>-1</sup></th><th>⟨T<sub>out</sub>⟩<br>°C</th><th>ΔT<br>°C</th><th>Power<br>kW</th><th>B<br>kW</th><th>⟨T<sub>out</sub>⟩<br>°C</th><th>ΔT<br>°C</th><th>δP<br>Pa</th><th>Power<br>kW</th><th>B<br>kW</th></tr></thead><tbody><tr><td rowspan="4">200</td><td>0.1</td><td>137</td><td>110</td><td>0.1083</td><td>0.0323</td><td>119</td><td>92</td><td>1.46</td><td>0.0893</td><td>0.0201</td></tr><tr><td>0.2</td><td>94</td><td>67</td><td>0.1343</td><td>0.0236</td><td>94</td><td>67</td><td>4.94</td><td>0.1298</td><td>0.0237</td></tr><tr><td>0.3</td><td>69</td><td>42</td><td>0.1399</td><td>0.0183</td><td>77</td><td>50</td><td>10.71</td><td>0.1444</td><td>0.0205</td></tr><tr><td>0.5</td><td>61</td><td>34</td><td>0.1763</td><td>0.0178</td><td>60</td><td>33</td><td>28.91</td><td>0.1575</td><td>0.0155</td></tr><tr><td rowspan="4">400</td><td>0.1</td><td>340</td><td>313</td><td>0.2975</td><td>0.1532</td><td>310</td><td>283</td><td>2.12</td><td>0.2832</td><td>0.1374</td></tr><tr><td>0.2</td><td>255</td><td>228</td><td>0.4330</td><td>0.1725</td><td>238</td><td>211</td><td>6.36</td><td>0.4162</td><td>0.1717</td></tr><tr><td>0.3</td><td>199</td><td>172</td><td>0.5051</td><td>0.1821</td><td>190</td><td>163</td><td>12.94</td><td>0.4788</td><td>0.1684</td></tr><tr><td>0.5</td><td>169</td><td>142</td><td>0.7051</td><td>0.2086</td><td>138</td><td>111</td><td>32.93</td><td>0.5398</td><td>0.1457</td></tr><tr><td rowspan="4">600</td><td>0.1</td><td>563</td><td>536</td><td>0.5456</td><td>0.3335</td><td>524</td><td>497</td><td>3.00</td><td>0.5217</td><td>0.3252</td></tr><tr><td>0.2</td><td>465</td><td>438</td><td>0.9482</td><td>0.4746</td><td>443</td><td>416</td><td>8.57</td><td>0.8577</td><td>0.4982</td></tr><tr><td>0.3</td><td>359</td><td>332</td><td>1.0176</td><td>0.5256</td><td>372</td><td>345</td><td>16.63</td><td>1.049</td><td>0.5608</td></tr><tr><td>0.5</td><td>292</td><td>265</td><td>1.3784</td><td>0.6329</td><td>276</td><td>249</td><td>39.84</td><td>1.2394</td><td>0.5624</td></tr><tr><td rowspan="4">800</td><td>0.1</td><td>800</td><td>773</td><td>0.8163</td><td>0.5763</td><td>744</td><td>717</td><td>4.00</td><td>0.7858</td><td>0.5539</td></tr><tr><td>0.2</td><td>-</td><td>-</td><td>-</td><td>-</td><td>672</td><td>645</td><td>11.28</td><td>1.3982</td><td>0.9545</td></tr><tr><td>0.3</td><td>-</td><td>-</td><td>-</td><td>-</td><td>597</td><td>570</td><td>21.60</td><td>1.8258</td><td>1.1964</td></tr><tr><td>0.5</td><td>-</td><td>-</td><td>-</td><td>-</td><td>482</td><td>446</td><td>50.07</td><td>2.3132</td><td>1.3825</td></tr></tbody></table>

<figure><img src="image_22.png" /><figcaption>**Fig. 18.** Numerical analysis using the validated model at all $T_{furnace}$ and $V_{in}$.</figcaption></figure>

Additionally, the exergy flow rate values (B) in Fig. 18d exhibit good agreement between experimental and simulated data, capturing the overall trends in exergy loss as furnace temperature and inlet velocity vary. This consistency between the two datasets emphasizes the model's competence in assessing the energy efficiency of the system under different conditions.

As can be seen in Table 6 and Fig. 18a, the pressure drop increases non-linearly with the air velocity, as predicted by the Darcy-

Forchheimer model. The pressure drop increases also with the temperature because, due to the thermal expansion, the velocity of the air increases along the tube and thus the pressure drop. In practical heat recovery systems, the pressure drop should be kept low enough because the higher the pressure drop, the more energy is required to blow air through the system.

This consistency between experimental and simulated data, considering outlet temperature, temperature rise, power, exergy flow rate and
pressure drop, reflects the simulation model's reliability. It accounts for the coupled heat transfer mechanisms within the system and provides valuable insights into varying furnace temperature and inlet velocity affect its thermal and energy performance. As a result, the model can be used with good confidence to predict or optimize the performance of a system by playing with its geometry and foam parameters (porosity, cell size and shape, conductivity of the solid material, etc), which is the next step of our research.

## 5. Conclusion

This paper has presented an experimental and numerical study on porous heat exchangers, focusing on the use of Kelvin-cell foam as a metallic porous structure to efficiently harness heat at high temperatures by considering the interplay between conduction, convection, and radiation. For this purpose, a tube filled with such a foam was placed inside a tubular furnace and air was blown through the tube. The foam was equipped with thermocouples brazed to the foam struts to measure the temperature in three sections i.e. inlet, midsection and outlet. Measurements were carried out from 200 °C up to 800 °C with air velocity varying from 0.1 m·s⁻¹ to 0.5 m·s⁻¹.

The experiment was modeled using ANSYS Fluent, with great care taken to be as close to the experimental configuration as possible, including heating by localized resistors. The simulation considered all three coupled heat transfer modes, with radiation calculated using the Fluent surface-to-surface approach, which is the most appropriate way to describe radiation transfer between the struts that make up the foam.

Key points from this investigation to conclude:

* Porous heat exchanger experiment

- Increasing extracted power was obtained when increasing furnace temperature or air velocity. As expected, increasing air velocity results in a lower outlet temperature, making this heat source less useful, so a compromise between power (which increases with velocity) and temperature (which decreases with velocity) should be found.

- Considering the conversion of the hot air into work (and possibly in a further step into electricity), it has been shown that this potential work decreased with air velocity (in the studied range) for the 200 °C temperature, whereas it continued to increase at higher temperature, the optimum to be found at a velocity higher than those studied.

* Numerical comparison

- When comparing the numerical simulations with the experimental data at specific probe locations, a satisfactory agreement was observed in Sections A and B. However, in Section C, the simulation results exhibited an underestimation of up to 43% and with a average discrepancy of 20%. This discrepancy may be attributed to the presence of thermocouples wires, which can disturb flow dynamics, an aspect not considered in the simulations.

- In our sectional analysis, the focus was primarily on temperature comparisons at specific probe locations. However, in engineering applications, it becomes crucial to study the overall system's performance, particularly in terms of power and exergy flow rates, rather than relying solely on temperature data.

- In analyzing the overall system behavior, we found a satisfactory agreement between the numerical model and experimental results, with a maximum error of 15%. This demonstrates the model's reliability in predicting the heat exchanger's overall performance.

The overall trend is good, so the simulation can be used to study the performance of the system under different conditions (furnace temperature and air velocity) than those used experimentally, and also with foams with different characteristics: porosity, cell size and shape, material conductivity, etc. Using these engineered metallic foams, it becomes feasible to halve the tube length in order to achieve more compact heat exchangers compared to using empty tubular solutions.

Additionally, it's worth emphasizing that performing coupled heat transfer simulations with complex geometries for characteristic studies demands substantial computational resources. As part of our future work, we plan to explore the use of the numerical approach proposed by Loris Ibarratt et al. [18] for simulating the exchanger configuration.

The experimental and numerical findings of this study can serve as references for future research on porous heat exchangers with similar constructions. By leveraging these insights, researchers can further advance the application and understanding of porous heat exchangers, contributing to the development of more efficient and sustainable heat transfer technologies.

## CRediT authorship contribution statement

Raj Narayan Konduru: Data curation, Investigation, Software, Writing – original draft. Olivier Farges: Methodology, Writing – review & editing, Supervision. Vincent Schick: Methodology, Writing – review & editing. Patrick Hairy: Writing – review & editing. Yves Gaillard: Resources. Gilles Parent: Conceptualization, Funding acquisition, Methodology, Supervision, Writing – original draft.

## Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Raj Narayan Konduru reports financial support was provided by Région Grand Est.

## Data availability

Data will be made available on request.

## Acknowledgements

Région Grand Est, France is acknowledged for funding Raj Konduru Phd Thesis (grant number LS 218239). This work is part of the THERMOUSSE project funded by ICEEL Carnot institute, France (grant number 20 CARN 0013 01). High Performance Computing resources were partially provided by the EXPLOR centre hosted by the University de Lorraine (Project: 2017EMPPX0317). We are grateful to the Halle des matériaux, EEIGM, Nancy, France for the execution of the thermo-couple brazing. Saint Gobain company is acknowledged for providing the Silpower® insulation parts.
## Appendix A. Experimental graphs at other temperatures

<figure><img src="image_23.png" /><figcaption>**Fig. A.19.** Experiment with Porous structure: furnace power, temperatures at different positions with operating temperature of 400 °C at different velocities and Power extracted by the air.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Fig. A.20.** Experiment with Porous structure: furnace power, temperatures at different positions with operating temperature of 600 °C at different velocities and Power extracted by the air.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Fig. A.21. Experiment with Porous structure: furnace power, temperatures at different positions with operating temperature of 800 °C at different velocities and Power extracted by the air.</figcaption></figure>
## Appendix B. Tetrahedral and polyhedral meshes

<figure><img src="image_26.png" /><figcaption>Fig. B.22. Representation of different element types for 1 mm element size.</figcaption></figure>
## Appendix C. Graphs for simulations at 400 °C, 600 °C and 800 °C

<figure><img src="image_27.png" /><figcaption>Fig. C.23. Validation at $T_{\text{furnace}}$ = 400 °C and $V_{\text{in}}$ = 0.1 m·s⁻¹.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>Fig. C.24. Validation at $T_{\text{furnace}}$ = 400 °C and $V_{\text{in}}$ = 0.2 m·s⁻¹.</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>Fig. C.25. Validation at $T_{\text{furnace}}$ = 400 °C and $V_{\text{in}}$ = 0.3 m·s⁻¹.</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>Fig. C.26. Validation at $T_{\text{furnace}}$ = 400 °C and $V_{\text{in}}$ = 0.5 m·s⁻¹.</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Fig. C.27. Validation at $T_{\text{furnace}}$ = 600 °C and $V_{\text{in}}$ = 0.1 m·s⁻¹.</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>Fig. C.28. Validation at $T_{\text{furnace}}$ = 600 °C and $V_{\text{in}}$ = 0.2 m·s⁻¹.</figcaption></figure>
<figure><img src="image_33.png" /><figcaption>Fig. C.29. Validation at $T_{\text{furnace}} = 600^{\circ}\text{C}$ and $V_{\text{in}} = 0.3 \text{ m} \cdot \text{s}^{-1}$.</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>Fig. C.30. Validation at $T_{\text{furnace}} = 600^{\circ}\text{C}$ and $V_{\text{in}} = 0.5 \text{ m} \cdot \text{s}^{-1}$.</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>Fig. C.31. Validation at $T_{\text{furnace}} = 800^{\circ}\text{C}$ and $V_{\text{in}} = 0.1 \text{ m} \cdot \text{s}^{-1}$.</figcaption></figure>
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