Experimental Tests of Conduction/Convection Heat Transfer in Very High Porosity Foams with Lattice Structures, Immersed in Different Fluids

: This experimental work presents the results of measurements of thermal conductivity λ and convection heat transfer coefﬁcient h on regular structure PLA and aluminium foams with low density ratio (~0.15), carried out with a TCP (thermal conductivity probe), built by the authors’ laboratory. Measurements were performed with two ﬂuids, water and air: pure ﬂuids, and samples with the PLA and aluminium foams immersed in both ﬂuids have been tested. Four temperatures (10, 20, 30, 40 ◦ C) and various temperature differences during the tests ∆ T (between 0.35 and 9 ◦ C) were applied. Also, tests in water mixed with 0.5% of a gel (agar agar) have been run in order to increase the water viscosity and to avoid convection starting. For these tests, at the end of the heating, the temperature of the probe reaches steady-state values, when all the thermal power supplied by the probe is transferred to the cooled cell wall; thermal conductivity was also evaluated through the guarded hot ring (GHR) method. A difference was found between the results of λ in steady-state and transient regimes, likely due to the difference of the sample volume interested by heating during the tests. Also, the effect of the temperature difference ∆ T on the behaviour of the pure ﬂuid and foams was outlined. The mutual effect of thermal conductivity and free convection heat transfer results in being extremely important to describe the behaviour of such kinds of composites when they are used to increase or to reduce the heat transfer, as heat conductors or insulators. Very few works are present in the literature about this subject, above all, ones regarding low-density regular structures.


Introduction
Metallic, polymeric and ceramic foams have recently received the attention of the scientific community due to their peculiar mechanical, thermal and fluid dynamic properties [1,2]. These properties are mainly due to the reduced presence of the base material (the matrix) and the large amount of interstitial fluid. Thus, these characteristics result in a very low density of the composite and the consequent lightness of the manufactured goods. Furthermore, when the solid structure is made of high-cost material, the sparing of said material can contribute to reducing the cost of components.
Even if thermal and fluid dynamic properties could be lower than the analogous ones of the full materials, a comparison at equal density and weight necessarily privileges the porous composites. Thus, these structures are very well suited to build heat exchangers and to substitute finned surfaces.
Considering that composites can be used either as thermal insulators or conductors, it becomes extremely important to evaluate the thermal properties and the temperature -Assessing the possibility of measuring the thermal conductivity with the probe method [3] of the above-described structures; -Identifying which heat transfer mechanism was present in the structures during the tests, if pure conduction, pure convection or mixed conduction and convection, and also identifying the difference in steady and transient state (for thermal conductivity); -Finding empirical relations to interpolate the experimental results to foresee the thermophysical properties in the examined temperature ranges and for the tested structures.
Calmidi et Mahajan [10] measured the thermal conductivity of aluminium-based composites with a method similar to the guarded hot plate, both in water and air. Thus, only steady-state measurements were carried out. Results were compared with a series/parallel empirical model and arranged in empirical correlations. The same authors [11] measured the forced convection heat transfer coefficient in aluminium foams crossed by air and found the empirical parameters of a series/parallel model describing the phenomenon. Potenza et al. [4] numerically evaluated the effect of porosity in thermal diffusivity measurements with the flash method, when transient or steady-state procedure is applied. Shih et al. [12] studied how heat transfer in aluminium foams is affected by pore density, sample length and air velocity, and flow cross-section. The latter resulted in being the most important factor. Kuang et al. [13] measured the heat transfer convection coefficient in different flow regimes and concluded that a characteristic length is the major factor influencing it. Hong et al. [5] analytically evaluated the border effects in the free convection of a heated vertical wall covered by metallic foam: these effects are meaningful only for high porosity foams. Zhao et al. [15]  performance of aluminium and copper metal foams with a change in geometrical (pores per inch, porosity, sample height) and process (mass flow rate of air, thermal flux) parameters. Duarte et al. [21] evaluated the fire resistance of polymeric foams (polyurethane and PET) used as building materials. Dukhan et al. [14] carried out experimental tests on open-cell aluminium foam in the shape of a compact block. A connect analytical model was developed, based on the hypothesis of thermal equilibrium between fluid and solid. In order to study the metallic foam contribution to condensers, Abdul-Sahib et al. measured mass flow rates, temperatures, and pressure drops of the condensing fluid, in order to correlate condensation pressure, thermophysical properties, pore density and number to the heat transfer. Pulvirenti et al. [2] carried out the evaluation of the fluid dynamic behaviour, in order to establish the non-validity of the Darcy-Forchheimer law. Singh et al. [16] both calculated and experimentally verified the best configuration of an air jet impinging on a thin metal foam, in terms of ratio between jet diameter and foam thickness. Andreozzi et al. [6] and Piller at al. [8] numerically studied the effect of the inclination of a foamcovered cylinder during free convection heat transfer. Ranut et al. [7] CFD calculated the performance of aluminium foam, detecting the foam structure through an X-ray tomography. Buonomo et al. [19] evaluated both experimentally and numerically the heat transfer in horizontal cylinders covered or filled with aluminium foam subjected to free convection. Corasaniti et al. [9] theoretically evaluated the effective thermal conductivity of an open foam with cubic cell structure, in steady-state regime, verified through the comparison with literature reference data. Among the reviews, ref. [2] give support to theoretical modelling when the Darcy-Forchheimer law is not respected; ref. [22] gives an outlook on aluminium foams in water subjected to forced convection; ref. [23] deals with non-equilibrium thermal transport associated with phase change in metal foams filled with phase change materials (PCM); ref. [24] presents a comparison between the traditional and metallic foam covered heat exchangers; in [25], a wide review is undertaken of theoretical calculations of convective heat transfer coefficient and effective thermal conductivity in metal foams containing PCM; ref. [26] presents the experimental and analytical methods to evaluate the effective thermal conductivity, and the resulting empirical correlations.
Summarizing, the general tendency of the study of low-density composites is to numerically simulate, experimentally measure, or both, the thermal properties of these materials and devices made with them. To carry on this goal, sample shapes are changed, as well as porosity, fluid, fluid dynamic parameters and the heat transfer mechanism.
The present work is a part of the international tendency to develop new materials especially suited to the following applications: energy saving through the reduction or increase of heat transfer in devices devoted to it; ambient impact reduction through the sparing of valuable materials; an increase in the efficiency of devices, as heat exchangers or thermal insulators, whose performances are strictly connected with energy efficiency. Even if the measurement of thermophysical properties can be regarded as a basic subject, its importance is fundamental for the cited applications.

Materials and Sample Preparations
Regular structure foams with very high porosity and low solid material content have been produced through the method described in [27,28]. Based on fused deposition modelling (FDM) 3D printing from an Autocad ® 2023 model (Autodesk, San Francisco, CA, USA), shown in Figure 1, first the sample in PLA (polylactic acid) was realized, and from it, through a metal replacement procedure, the aluminium sample. The complete step sequence of the procedure is described in Table 1. The relative density, that is the ratio of the density of the foam to the density of the solid, resulted in 0.95. Both samples are shown in Figure 2. from it, through a metal replacement procedure, the aluminium sample. The complete step sequence of the procedure is described in Table 1. The relative density, that is the ratio of the density of the foam to the density of the solid, resulted in 0.95. Both samples are shown in Figure 2.     from it, through a metal replacement procedure, the aluminium sample. The complete step sequence of the procedure is described in Table 1. The relative density, that is the ratio of the density of the foam to the density of the solid, resulted in 0.95. Both samples are shown in Figure 2.   3D printing of the model using PLA  The sizes of the samples were chosen so that they fit within the internal volume of the measuring cell of the apparatus for thermal conductivity tests ( Figure 3); that is, they are cylinders of 60 mm height and with a diameter of 50 mm (reported in Figure 2).
Energies 2023, 16, x FOR PEER REVIEW 5 8 plaster removal to obtain a foam The sizes of the samples were chosen so that they fit within the internal volum the measuring cell of the apparatus for thermal conductivity tests ( Figure 3); that is, are cylinders of 60 mm height and with a diameter of 50 mm (reported in Figure 2).

Experimental Section
This section may be divided by subheadings. It should provide a concise and pr description of the experimental results, their interpretation, as well as the experim conclusions that can be drawn.

Thermal Conductivity Probe (TCP)
Thermal conductivity of the two samples of Figures 1 and 2 were measured wit probe method [29][30][31], with a special probe realized by the authors' laboratory [3 ported in Figure 4. Due to its very fine sizes (60 mm in length and 0.6 mm in diame the probe presents a very high length-to-diameter ratio (~100), so particularly suit apply the hot wire theory [30,31]: in fact, the linear portion of the ΔT vs. ln t curve sta few seconds after the beginning of the sample heating (e.g., see Section 3.3). The prob be easily inserted into the large voids of the foam structure, without touching it.

Experimental Section
This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.

Thermal Conductivity Probe (TCP)
Thermal conductivity of the two samples of Figures 1 and 2 were measured with the probe method [29][30][31], with a special probe realized by the authors' laboratory [3] reported in Figure 4. Due to its very fine sizes (60 mm in length and 0.6 mm in diameter), the probe presents a very high length-to-diameter ratio (~100), so particularly suited to apply the hot wire theory [30,31]: in fact, the linear portion of the ∆T vs. ln t curve starts a few seconds after the beginning of the sample heating (e.g., see Section 3.3). The probe can be easily inserted into the large voids of the foam structure, without touching it.
Energies 2023, 16, x FOR PEER REVIEW 5 8 plaster removal to obtain a foam The sizes of the samples were chosen so that they fit within the internal volum the measuring cell of the apparatus for thermal conductivity tests ( Figure 3); that is, are cylinders of 60 mm height and with a diameter of 50 mm (reported in Figure 2).

Experimental Section
This section may be divided by subheadings. It should provide a concise and pr description of the experimental results, their interpretation, as well as the experim conclusions that can be drawn.

Thermal Conductivity Probe (TCP)
Thermal conductivity of the two samples of Figures 1 and 2 were measured wit probe method [29][30][31], with a special probe realized by the authors' laboratory [3 ported in Figure 4. Due to its very fine sizes (60 mm in length and 0.6 mm in diam the probe presents a very high length-to-diameter ratio (~100), so particularly suit apply the hot wire theory [30,31]: in fact, the linear portion of the ΔT vs. ln t curve st few seconds after the beginning of the sample heating (e.g., see Section 3.3). The prob be easily inserted into the large voids of the foam structure, without touching it.

Experimental Apparatus
A sketch of the experimental apparatus with a short description of its components is reported in Figure 5. Both the apparatus and the test procedure have already been described in [3].

Test Procedure
After mounting one of the two samples and the TCP in the measuring cell, the thermostat is switched on and the thermosetting fluid is flowed through the cell and cover walls. After temperatures of the TCP and the wall are stabilized at the desired values, the PS is switched on to supply the desired electric current value, depending on the fluid (water requires a higher value than air to meaningfully increase the probe temperature) and the power to be given to the TCP.
During all the measurement phases, the four signals (probe TC, wall TC, Shunt voltage drop, and probe platinum heater voltage drop) are continuously recorded by the DAS. Data are then processed according to the TCP theory: first the trend of the probe temperature rise is plotted versus time, then the linear portion of this trend is identified; afterwards, the slope of this trend is evaluated and connected to the thermal conductivity of the medium (a typical trend of the temperature increase vs. logarithm of time is reported in Fig.6). Unfortunately, with the medium being totally or mainly liquid, free convection starts easily. In this case, the curve deviates from the linear trend till it becomes flat (constant temperature). From this time, heat produced by the current flowing in the probe platinum Pt wire is transferred to the wall (cooled by the thermostatic fluid flow) and only free convection from the probe (vertical small diameter cylinder, i.e., a wire) and the cell wall are present. Convection heat transfer coefficient is evaluated from the temperature difference between the probe and the wall, in the cylindrical reference system. Figure 6 presents a typical trend of the temperature increase vs. the logarithm of time for the Al foam in air, where the linear portion is clearly recognizable, and the constant ΔT zone

Experimental Apparatus
A sketch of the experimental apparatus with a short description of its components is reported in Figure 5. Both the apparatus and the test procedure have already been described in [3].

Test Procedure
After mounting one of the two samples and the TCP in the measuring cell, the thermostat is switched on and the thermosetting fluid is flowed through the cell and cover walls. After temperatures of the TCP and the wall are stabilized at the desired values, the PS is switched on to supply the desired electric current value, depending on the fluid (water requires a higher value than air to meaningfully increase the probe temperature) and the power to be given to the TCP.
During all the measurement phases, the four signals (probe TC, wall TC, Shunt voltage drop, and probe platinum heater voltage drop) are continuously recorded by the DAS. Data are then processed according to the TCP theory: first the trend of the probe temperature rise is plotted versus time, then the linear portion of this trend is identified; afterwards, the slope of this trend is evaluated and connected to the thermal conductivity of the medium (a typical trend of the temperature increase vs. logarithm of time is reported in Figure 6). Unfortunately, with the medium being totally or mainly liquid, free convection starts easily. In this case, the curve deviates from the linear trend till it becomes flat (constant temperature). From this time, heat produced by the current flowing in the probe platinum Pt wire is transferred to the wall (cooled by the thermostatic fluid flow) and only free convection from the probe (vertical small diameter cylinder, i.e., a wire) and the cell wall are present. Convection heat transfer coefficient is evaluated from the temperature difference between the probe and the wall, in the cylindrical reference system. Figure 6 presents a typical trend of the temperature increase vs. the logarithm of time for the Al foam in air, where the linear portion is clearly recognizable, and the constant ∆T zone where only free convection is present. The free convection heat transfer coefficient h is calculated from Newton convection law: is the thermal power supplied by the wire inside t by the steady-state thermal balance with the power dissipated throug ΔT is the temperature difference between the probe (assumed unifor small diameter) and the average temperature inside the measuremen undisturbed.
Unfortunately, during pure water and pure air tests (without the starts almost at the beginning of heating, so it is very difficult to identi of the trend; so λ values result in being affected by high uncertainties. A is possible to overcome this problem by adding a small percentage (~0. gel (agar agar) which highly increases the water viscosity, leaving the n practically the same. In this case, the linear zone results in being much w ure 7, see the results of a test of the Al foam and water + agar agar). In linear zone, used to calculate thermal conductivity, spans from ln t = 1.5 6 (t = 600 s). The free convection heat transfer coefficient h is calculated from the well-known Newton convection law: where . Q = R·I 2 is the thermal power supplied by the wire inside the probe, derived by the steady-state thermal balance with the power dissipated through free convection. ∆T is the temperature difference between the probe (assumed uniform due to its very small diameter) and the average temperature inside the measurement cell, assumed as undisturbed.
Unfortunately, during pure water and pure air tests (without the foam), convection starts almost at the beginning of heating, so it is very difficult to identify a linear portion of the trend; so λ values result in being affected by high uncertainties. At least for water it is possible to overcome this problem by adding a small percentage (~0.5% in weight) of a gel (agar agar) which highly increases the water viscosity, leaving the nature of the liquid practically the same. In this case, the linear zone results in being much wider (e.g., in Figure 7, see the results of a test of the Al foam and water + agar agar). In this last case, the linear zone, used to calculate thermal conductivity, spans from ln t = 1.5 (t = 4.5 s) to ln t = 6 (t = 600 s).
Also, at the end of these tests the temperature difference reaches constant values. In this case, the reason is not the start of convection, but reaching the steady state, where the whole heat supplied by the probe heater is transferred to the cell wall. This corresponds to measuring thermal conductivity with a steady-state method: the guarded hot ring (GHR) method, where the conditioning fluid circulating in the interlayer of the cell behaves as a control ring. A comparison can be caried out between the transient (TCP) and steady-state (GHR) results (see the Results paragraph).

Tests
The following tests have been carried out: practically the same. In this case, the linear zone results in being much ure 7, see the results of a test of the Al foam and water + agar agar). I linear zone, used to calculate thermal conductivity, spans from ln t = 1 6 (t = 600 s). Also, at the end of these tests the temperature difference reaches this case, the reason is not the start of convection, but reaching the stea whole heat supplied by the probe heater is transferred to the cell wal to measuring thermal conductivity with a steady-state method: the

Figures 8-11
report all the results of tests in air and water, for thermal conductivity λ and convection heat transfer coefficient h, respectively. The figures show all of the repetitions of the tests carried out. Averages and standard deviations can be easily calculated from the data. Blue interpolating lines are the trends of experimental data (constant, linear, or with some specific behaviour). Orange data or lines are values derived from the literature (LeFevre and Ede [32] and Haar et al. [33], Incropera et al. [34]), and are clearly present only for pure water and air. In some cases, the other colours represent a comparison between different data, as indicated.

Tests
The following tests have been carried out: -Pure air tests; -Pure water tests; -Tests on pure water mixed with agar agar; -Tests on Al structures with water and agar agar; -PLA test in water; -PLA test in water +agar agar; -Al tests in air; -PLA tests in air.  [32] and Haar et al. [33], Incropera et al. [34]), and are clearly present only for pure water and air. In some cases, the other colours represent a comparison between different data, as indicated.       Empirical correlations of h as a function of test temperature T in the case of foams inserted in air and water, obtained from the trendlines, are reported in the graphs.

Discussion of Results
Looking at the reported graphs ( Figure 8-11), the following conclusions can be drawn. Empirical correlations of h as a function of test temperature T in the case of foams inserted in air and water, obtained from the trendlines, are reported in the graphs.

Discussion of Results
Looking at the reported graphs (Figures 8-11), the following conclusions can be drawn.

Thermal Conductivity
(1) ∆T vs. ln t trends (see, for instance, Figure 6) often present different slopes: only the first is generally connected to the true λ value, and the others are due to mixed convection/conduction, or steady-state convection when this trend is horizontal. (2) Only water and only air (above all, this latter) tests give values affected by high uncertainties, due to the difficulty in identifying the linear zone in the ∆T vs. ln t trends: see, for instance, Figures 8a-c and 9a,b, which show a relevant data spread (till 30%) and difference with respect to references. This is the reason why, at least for water, adding agar agar highly reduces the uncertainty (see in Figure 9a this reduction for water, from 9% to 2.3‰). Moreover, the obtained values of water fall within 1.7% from tables of ref. [33]. (3) Comparing λ values of the PLA foam in water + agar agar, obtained from the TCP theory (from the slope of the ∆T vs. ln t) and in steady state (GHP), different trends appear, as seen in Figure 9c. This is likely due to the different volume interested by the two procedures: the whole composite volume for the GHP, and only a layer around the probe in the TCP, which increases during the measurement. Furthermore, the presence of a pure liquid layer around the probe needle results in an effect similar to the so-called wall effect. Water + agar agar viscosity lowers with temperature, producing an apparent thermal conductivity which increases with temperature. When the same comparison is made on the Al foam in water, this shift appears also at lower temperatures (Figure 9d), probably due to the high conductivity of aluminium, which produces this effect (apparent thermal conductivity) at lower temperatures as well. (4) PLA λ values (0.16 W/m K, [35]) lower than pure water (0.6 W/m K) explain why the PLA foam + water composite decreases its λ from 0.6 to 0.41 ÷ 0.46 W/m K (Figure 9c). (5) In Figure 9d, while the aluminium foam in water+agar agar presents the same λ steady-state value of pure water, in the TCP tests, λ results in being increased about 3.6 times, from 0.6 to 2 ÷ 2.25 W/m K: this demonstrates the effect of the solid material λ, at least in the neighbourhood of the heat source.

Convection Heat Transfer Coefficient
(1) Figure 10a shows the comparison of h values measured with the TCP immersed in air and computed with the empirical correlation of LeFevre & Ede [32]: both trends present the same slope but different absolute values. However, the difference lay within 20%, as usually occurs when dealing with a comparison between empirical correlations and experimental data. In conclusion, the test temperature from 10 to 40 • C increases h by 12% in air in PLA foam, and 7% in Al foam, while the same foams give a fairly constant h in water. No or little effect of ∆T in air was found, while ∆T increasing from 0.4 to 3 • C makes h of Al foam in water 20% higher.
Thus, using Al foams as heat exchangers at higher temperatures in air increases their performances, while the same cannot be said for water. When PLA foams are used, the heat exchanger performances also increase, but care must be taken to avoid temperatures that are too high, which can damage the material.

Uncertainty Analysis
Following the ISO GUM rule [36], both of the two components of uncertainty (type A and type B) were evaluated.
6.1. Thermal Conductivity 6.1.1. Type A Uncertainty This is the uncertainty source evaluated with statistical methods. Repetitions of tests (see multiple data in the graphs of  belong to this uncertainty, as well as the prevision uncertainty obtained from least square regression procedure, e.g., in evaluating the slope of the linear portion of the ∆T vs. ln t trends. This last uncertainty is obtained from the diagonal elements of the covariance matrix of the unknown [37]. In any case, it results in being much lower than dispersion due to the repetitions of tests and the other type B sources, so it has been neglected in the sum in quadrature of the components [36].

Type B Uncertainty
The following uncertainty causes belong to this category: -Calibration uncertainty of thermocouple and TCP: when comparing the results obtained on a reference material (glycerol), an uncertainty of 4% was obtained [3]; -Uncertainty due to uncalibrated thermocouples: generally, a 0.3 • C is attributed to this source, but considering the high number of TC measurements carried out during each test, this cause is negligible, and furthermore, calibration uncertainty already takes it into account; the same can be said about the uncertainty available in manuals and books, and the one due to the experience of the experimenter and previous tests.
Finally, a ±5% (one standard deviation) can be attributed to λ carried out measurements.

Convection Heat Transfer Coefficient
For what concerns the free convection heat transfer coefficient, its uncertainty is calculated by the defining equation: (2) Its relative standard deviation σ h /h results in: (the bar over different quantities mean the average value over the whole considered period, e.g., in Figure 12 from 1000 to 2500 s). Among the terms in Equation (3), the greatest is the last, σ ∆T /∆T. For instance, Figure 12 reports the temperature difference ∆T trend of a typical test (Al foam in water at 10 • C mean temperature and with a specific thermal power supplied of 139 W/m 2 ). For this test, σ ∆T /∆T results in 0.04%. Adding in quadrature to the other terms of Equation (3) does not meaningfully increase this value. This procedure, however, highly underestimates the uncertainty, because it does not take into account the intrinsic variability of the free convection phenomenon. In fact, an uncertainty of at least 15 ÷ 20% is generally attributed to this quantity.
In conclusion, the random causes that influence the uncertainty associated with the phenomenon are: - The convective cell generation, and the secondary and third order cells; - The influence of the ambient conditions: temperature, humidity, pressure, etc.; - The statistical nature of the phenomenon: even in the same experimental conditions, the exact repetition of the results is very difficult to obtain; - The presence of obstacles in the fluid movement, as the foam can be considered, which leads to the declared uncertainty (20%).
riod, e.g., in Figure 12 from 1000 to 2500 s). Among the terms in Equation (3), the greatest is the last, / T T σ Δ Δ . For instance, Figure 12 reports the temperature difference ΔT trend of a typical test (Al foam in water at 10 °C mean temperature and with a specific thermal power supplied of 139 W/m 2 ). For this test, / T T σ Δ Δ results in 0.04%. Adding in quadrature to the other terms of Equation (3) does not meaningfully increase this value. This procedure, however, highly underestimates the uncertainty, because it does not take into account the intrinsic variability of the free convection phenomenon. In fact, an uncertainty of at least 15÷20% is generally attributed to this quantity.
In conclusion, the random causes that influence the uncertainty associated with the phenomenon are: - The convective cell generation, and the secondary and third order cells; - The influence of the ambient conditions: temperature, humidity, pressure, etc.; - The statistical nature of the phenomenon: even in the same experimental conditions, the exact repetition of the results is very difficult to obtain; - The presence of obstacles in the fluid movement, as the foam can be considered, which leads to the declared uncertainty (20%).

Conclusions
A difference was found between the thermal conductivity measured with the probe method in transient state and with the guarded hot ring in steady state, and this difference is more evident in aluminium (Figure 9d) than in PLA foams ( Figure 9c): this is due to the high λ of the metal which moves the heat source far from the probe at the beginning of tests. Also, the wall effect (measuring λ of a layer of pure fluid surrounding the TCP at the beginning of the test) can influence the difference between the TCP and the GHR tests.
The presence of the solid foam can have a double effect on h: generally, the very low percentage of solid in the foam makes the h values in free fluid comparable with that of the fluid-solid composite. But the presence of the solid structure can increase the conduction heat transfer (at least for aluminium), decreasing the h value. This could be the reason

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.