## Abstract

Metal foams have been often used for thermal management due to their favorable characteristics including high specific surface area (SSA), high thermal conductivity, and low relative density. However, they are accompanied by shortcomings including the significant contact resistances due to attachment method, as well as the need for characterization of foam parameters such as pore diameter and SSA. Additive manufactured (AM) metal foams would eliminate the substrate/foam thermal resistance, decrease the need for pre-usage characterization, and allow for tailoring structures, while also taking advantage of the characteristics of traditionally manufactured foams. A commercial, aluminum foam (nominally 5 pores per inch (PPI), 86.5% porosity) was analyzed using X-ray microcomputed tomography, and a custom-designed metal foam based on the cell diameter and porosity of the commercial sample was subsequently manufactured. Reduced domain computational fluid dynamics/heat transfer (CFD-HT) models were compared against experimental data. Postvalidation, the flow behavior, effect of varying attachment thermal conductivities, and thermal performance were numerically investigated, demonstrating the usefulness of validated pore-scale models, as well as the potential for improved performance using AM metal foams over traditionally manufactured foams.

## Introduction

Metal foams are characterized by the dispersion of voids throughout a solid matrix. They can be characterized as periodic/stochastic and open/closed structures. Open-cell foams are typically characterized as a network of interconnecting nodes and ligaments with high porosity, tortuous flow paths, and high specific surface area (SSA). Examples of open, stochastic, aluminum foams can be seen in Fig. 1. Figure 2 illustrates common terminology used in describing metal foams. These foams are attractive for a wide variety of applications, ranging from catalysts for chemical reactions to impact absorption [1,2]. There is also extensive literature examining their potential to increase heat transfer (HT) performance due to the aforementioned qualities [3].

Heat transfer in metal foams is most commonly studied using stochastic, open-cell, metal foams. A variety of cooling fluids, solid phases, and heat transfer mechanisms have been investigated. Boomsma and Poulikakos [4] examined the effect of compression and nominal pore size on the pressure drop of water across aluminum foams at liquid velocities up to 1.042 m/s. They found that the flowrate range influenced the permeability and form coefficient, particularly in the transitional regime, as the flow exits the linear Darcy regime and exhibits a quadratic pressure drop response. Zhao et al. [5] investigated natural convection in open-cell FeCrAlY foams both experimentally and numerically and showed that it contributes more than 50% of the effective thermal conductivity. They also studied radiation in these FeCrAlY foams [6]. The spectral transmittance and reflectance were measured and used to calculate emissivity and the extinction coefficient. They found that fixing porosity causes radiative conductivity to increase with cell size and temperature.

Kim et al. investigated single-phase forced convection of water, and single-phase flow, and flow boiling of FC-72 in a copper foam-filled channel [7]. Single-phase heat transfer coefficients of 10 kW/m2 K and 2.85 kW/m2 K were achieved with water and FC-72, respectively, in foams with 95% porosity. Two-phase performance with FC-72 showed a near four-times improvement, with a maximum heat transfer coefficient of 10 kW/m2 K at a heat flux of 50 kW/m2. The porosity was also seen to have a moderate effect on the heat transfer coefficient at higher heat fluxes. Li and Leong [8] performed experiments and numerical simulations involving single-phase and flow boiling with water and FC-72 in aluminum foams. They observed the onset of nucleate boiling, the hysteresis effect, and critical heat flux (CHF). However, while the CHF for FC-72 was 8.7 W/cm2, the heaters were not able to provide sufficient heat to achieve CHF for water. Lafdi et al. [9] analyzed the effects of porosity and pore size on solid–liquid phase change material melting on phase-change material-filled metal foams, and found that higher porosity and larger pore sizes allowed for quicker achievement of steady-state temperatures.

Additive manufactured (AM) has recently been investigated to create foam-type structures. There has been research regarding metal foams made using AM, including their microstructure, mechanical properties, and their ability for building functionally graded next-generation biomedical implants [1012]. There has also been some literature regarding additive manufactured heat sinks for electronics cooling [13], as well as limited research regarding the utilizing AM metal foams for thermal management [14]. Combining metal foams and AM allows for the realization of a thermal management solution that leverages both areas' positive attributes. In addition to the high SSA, low relative density, and tortuous flow path, AM provided numerous advantages that support it as a viable cooling solution. De-Schampheleire et al. [15] discussed the extensive list of foam properties necessary for adequate characterization to properly study metal foams and their thermal applications. The AM process is less random than a gas blowing process, which mitigates the need for significant structural characterization. Zhao [3] reviewed previous thermal applications of metal foams and observed that the thermal contact resistance between the tube and foam provided a limiting effect on enhancement. De-Jaeger et al. [16] and Sadeghi et al. [17] found that the thermal contact resistance between the substrate and the metal foam represented a significant portion of the total thermal resistance. Eliminating the resistance between the two sections can be easily and naturally done with AM and would therefore markedly improve thermal performance. This can be done by directly printing on the substrate, with substrate materials including metals (such as the aluminum alloy used in this work) or silicon for direct cooling of integrated circuits [18]. It could also allow for more control of thermal transport characteristics, whether it is global control of properties such as porosity and pore orientation, or local control for hotspot mitigation.

This paper investigated the thermo-hydraulic performance of both traditionally manufactured, stochastic metal foams and AM foams as a method of heat transfer enhancement. The stochastic sample was purchased from ERG Aerospace, Inc., (Oakland, CA) and characterized using X-ray microcomputed tomography (X-ray μCT) and imagej/bonej. The AM sample was based on a unit cell seen in literature that imitates stochastic foams, and the unit cell selection was justified. The two structures were tested with a uniform heat flux boundary condition and then thermo-hydraulically compared with respect to pressure drop and effective wall heat transfer coefficient in an experimental test bed with de-ionized (DI) water as the working fluid. Computational fluid dynamics (CFD)-HT pore-scale models of both structures were developed, and the experimental results were used to verify their validity. The validated models were then used for further investigations regarding flow behavior, attachment method, and thermal performance.

## Characterization and Experimental Sample Descriptions

Obtaining sufficient relevant geometrical characteristics is required for studying thermal applications of metal foams [15]. Existing studies often do not give comprehensive characterization of all relevant geometric parameters. In addition to measuring an adequate number of parameters, the measurement method should also be clearly stated, which allows for repeatable experiments and comparison between studies. De-Schampheleire et al. mentioned the lack of characterization by many authors, such as presenting only information on sample porosity and pores per inch (PPI) [15]. This is particularly problematic as PPI is three-dimensional, varies depending on orientation, and more a nominal value given by manufacturers rather than a physical quantity of the sample. Table 1 shows characterization information from several past studies. Only literature that reported values for nominally 5 PPI samples were listed in this table, as the samples studied for this study also used the same PPI. The commercial metal foam sample utilized can be seen in Fig. 3(a) and the AM sample Fig. 3(b). The test bed for the samples can be seen in Fig. 3(c). Each of the properties in Table 1 is characterized for the sample used. The characterization is done in imagej, an open-source scientific image analysis tool, as well as bonej, a plugin developed for imagej that was developed for both trabecular and whole bone research that proved useful because of the porous nature of bone geometries [25].

The metal foam sample (referred to as commercial/stochastic/ERG foam) was purchased from ERG Aerospace, Inc, Oakland, CA. The Duocel® aluminum foam was nominally labeled as 5 PPI, processed from 6101 alloy, and underwent T6 heat treatment. A Metrotom 800 (Zeiss, Oberkochen, Germany) was used to perform X-ray μCT, a nondestructive imaging technique. A small section of the aluminum foam was scanned at 0.25 deg rotations with a voxel size of 18.43 μm, and the resulting two-dimensional image files were subsequently imported to imagej. Ideally, the voxel size should be minimized, as smaller voxel sizes would yield increased accuracy albeit with decreasing returns, but the conflicting desires for minimizing voxel size while maximizing numerical domain size limited the voxel size for these simulations. De-Schampheleire et al. [15] concluded that it was not necessary to have a voxel size of less than 5 μm. This change in accuracy due to smaller voxel size mostly affects SSA, which saw a 19% change going from 37.5 μm to 8.5 μm, as opposed to interfacial strut (also known as fiber or ligament) area, which only saw a 4% change for the same change in voxel size. The images near both the top and bottom of the scan were deleted to remove distorted sections, and the files were cropped to only include the foam, without the dead space. Optimize Threshold, a bonej command, was used for binary segmentation into black and white. The analysis of the properties (porosity, pore diameter, ligament diameter, ligament length, cell diameter, and specific surface area) began with the modified image files, and a flowchart illustrating the methodology can be seen in Fig. 4.

Porosity (ε) was obtained by using the volume fraction command in bonej and subtracting the result from one, yielding a value of 86.5%. The pore diameter was found by locating unbroken windows, finding the best-fit ellipse, and averaging the major and minor axes of 30 different windows for the effective pore diameter. As the struts taper so that the thinnest region is near its middle, the ligament diameter was calculated at the smallest cross section, as defined in Eq. (1). The Heywood circularity factor H, which compares the strut's perimeter to the perimeter of a circle with the same cross-sectional area, was calculated to be 1.086 from Eq. (2) [24]. A cross section with H =1.086 is shown as a graph next to “Ligament diameter” in Fig. 4. The ligament length was found by running Skeletonize 3D in bonej to create a skeleton image of the image sequence in order to more easily measure ligament length. Thirty ligament lengths from node to node were found and averaged. The cell diameter was calculated by finding the equivalent spherical diameter, when compared to an ellipsoid of ten unbroken cells. The three ellipsoid axes were obtained by finding the two axes at the middle of the cell and the distance between the two pores that make up the cell. bonej's Isosurface was utilized to find the specific surface area. The calculated values can be seen in Table 2
$dh=4Ax/P$
(1)
$H=Pobject/Pcircle=Pobject/4πAeq$
(2)

The first step to creating the additive AM foam structures was to choose an appropriate unit cell. A unit cell can be utilized to imitate foams with good thermo-hydraulic agreement as seen in Refs. [26] and [27]. Unit cells that were seen in literature include cubic unit cells [28], rhombic dodecahedron unit cells [29], pentagonal dodecahedron unit cells [29], the Lord Kelvin model [30], and the Weaire–Phelan model [30], illustrated in Fig. 5. Although certain models such as the Lord Kelvin and Weaire–Phelan model were more accurate as their curved edges minimize surface energy (which is what occurs during the foaming process), whether it could be manufactured and well packed had to be considered. The first unit cell was deemed too far from a physically accurate model and was not selected as a result. The last two models were not chosen because of manufacturer warnings that angles of too small a rise would necessitate irremovable support structures. The middle model was not selected as it does not pack well and cannot form a three-dimensional (3D) honeycomb structure—it was more appropriate for correlations or modeling of a single unit cell as opposed to a larger macroscale structure. Because of these concerns, a rhombic dodecahedron was chosen for the AM unit cell. Two views of this unit cell can be seen in Fig. 6.

Table 2 compares the foam geometric properties obtained from the analysis discussed earlier in this section to both an example in the literature that used similar analyses and the AM foam. The rhombic dodecahedron-based foam was constructed by setting the cell size and porosity equal to that found from the X-ray μCT analysis. The results matched closely for all parameters except the pore diameter and specific surface area. The reason for the former was illustrated in Fig. 5 with the two ellipses. The nature of the rhombic windows and its pronounced elliptical shape as opposed to the Lord Kelvin model's more circular pore windows was such that the effective pore diameter of the AM foam was noticeably smaller. The latter could be explained by the imperfections of the stochastic foam since only structurally ideal cells were chosen for analysis of cell size. As a result, the true cell size may have been larger, and this would have cause the surface area to be an underestimate.

The ERG foam was cut to the appropriate nominal size of 100 mm × 40 mm × 9.3 mm using electrical discharge machining. The foam was then attached to an aluminum plate (sanded with 1000 grit sandpaper) using Omegabond 200 Epoxy Adhesive (k =1.38 W/m K, thickness 0.3 mm). The assembled foam and substrate were placed in a Thermo Scientific Lindberg Blue M oven for 3 h at 150 °C to cure per supplier recommendation. Figure 3(a) shows the attached foam with the black layer of cured epoxy between the substrate and the foam. The AM foam was assembled using materialisemagics, a software for STL file editing for 3D printing. The lattice unit cell was created in Solidworks then imported into materialise. STL file corruptions and errors were handled using the Fix Wizard in-software. The AM manufacturing was outsourced to Forecast 3D who used direct metal laser sintering of AlSi10 Mg particles to manufacture the designed foam structure. A vertical band saw was used to cut the structure out of the baseplate. Then, an end mill and electric discharge machining were used to bring the plate to the correct dimensions, and the plate was heat treated at 600 °F for 2 h to relieve stresses created during machining so as to not break or warp the structures. The finished product postmachining and heat treatment can be seen in Fig. 3(b).

## Computational Model

The numerical modeling examined and compared the thermo-hydraulic performance of both the ERG foam and the AM foam. Although a macroscopic, volume-averaged approach can be used for decreased computational time, it comes at the expense of lost details and potentially decreased accuracy. Unit cell models have also been used but they also fail to capture some characteristics, particularly resulting from the random nature of foam morphology [26,31]. The pore-scale models seen in this paper will also allow us to understand and visualize microscopic flow phenomena without difficult and costly flow visualization techniques. The numerical model was experimentally validated to ensure model fidelity.

### Model Description and Assumptions.

The numerically simulated AM geometry measured 9.30 mm × 2.33 mm × 46.5 mm and can be seen in Fig. 7(a). The commercial foam geometry used in CFD-HT simulations was obtained from the image stack from the X-ray μCT analysis. A surface was created using the Isosurface command in bonej and then exported as a binary STL file. The final foam geometry used in the modeling measured 9.30 mm × 4.65 mm × 25.0 mm, with the directions oriented in the same way as the experimental sample to ensure any directional bias in the foam is replicated in the simulations. Floating mesh points and triangles were removed using meshlab, an open-source software used for mesh processing and repair [32], which was then subsequently used to smooth the geometry to decrease the mesh resolution. The geometry was assembled in Solidworks and subsequently exported to Fluent for CFD-HT analysis. The assembly can be seen in Fig. 7(b), with an inlet and outlet section added on either side. The 0.3 mm thick thermal interface material (TIM) layer between the substrate plate and the metal foam matrix is shown in the magnified view as the grayed out layer. A qualitative comparison between the two geometries is shown in Fig. 8, which illustrates the similar dimensions of the two foams. The AM foam's simulation boundary conditions were the same as the ERG foam model. Both samples had Al 5083 substrates, and the thermophysical properties for the materials in the numerical analysis can be seen in Table 3. Despite the difference in materials used for the foams, their values were nearly identical so that the thermal performance differences are not attributable to the material selection. The ERG foam was Al 6101, the AM foam was AlSi10 Mg, and the working fluid was DI water.

The stochastic foam domain was reduced lengthwise, as a full length simulation would not import properly into the 3D modeling software utilized unless the mesh was simplified to the point of losing too much of the geometrical details. The domain reduction was necessitated by hardware and software, and it was assumed that the reduction in total domain length would have a small enough impact on the thermo-hydraulic parameters studied that it would be possible for us to still maintain model accuracy. Additionally, it was assumed that the scanned section was large enough that it would encompass a sufficient quantity of foam cells that it could be considered a representative elementary volume. De-Schlampheleire et al. [15] found that eight foam cells would give an appropriately sized section for a representative elementary volume, and with a conservative estimate of each unit cell having a cubic shape with edges of length dc, the current geometry would be able to hold over ten cells total. It was also assumed that the inlet effects would be small enough so as to still give reasonable agreement between the CFD-HT simulation and experimental results. Dukhan and Suleiman [36] investigated the entrance effects in open-cell foams, and they demonstrated that the entrance length increases with velocity. As the inlet speeds seen in this study were low (≤12.5 cm/s), this assumption was reasonable.

The simulations assumed steady-state, laminar, incompressible flow with negligible viscous dissipation. Buoyancy and radiative heat transfer effects were ignored, and material properties were considered constant. Prior studies defined the transition for open-cell, high-porosity foams in terms of pore diameter-based Reynolds number Rep and permeability-based Reynolds ReK, where the length scale is the square of permeability [37,38]. However, Dukhan et al. [38] recommend a permeability-based Reynolds number to be most appropriate for porous media. They found that the Darcy–Forchheimer region, where a second-order polynomial describes hydraulic performance, extends until ReK = 37.5, and turbulence is seen at values greater than 50. The pressure drop, ΔP, can be described as
$ΔPΔL=μKu+ρCfKu2$
(3)

where ΔL is the test section length, μ is the dynamic viscosity, K is the permeability, u is the superficial inlet velocity, ρ is the density, and Cf is the inertial coefficient. The turbulent Reynolds numbers corresponded to 10.2 cm/s for the ERG foam and 15.5 cm/s for the AM foam, so the assumption of laminar flow was reasonable. Figure 8 illustrates the qualitatively similar nature of the two structures. It must be noted that the experimental overall heat transfer coefficient differed from the computed values due to the additional resistances from the thermal paste applied onto the thermocouples. The thermal conductivity of various TIMs was quantified by Narumanchi et al., and the product utilized for this work (Wacker Silicone P12 thermal grease) measured at 0.54 W/m K [39]. The simulation results were adjusted accordingly to account for the additional thermal resistance, by considering a thermal paste layer of estimated thickness t =0.14 mm.

### Governing Equations and Boundary Conditions.

The governing equations in Cartesian coordinates used for the fluid mass conservation and force–momentum balance were as follows:
$∂u∂x+∂v∂y+∂w∂z=0$
(4)
$ρ(u∂u∂x+v∂u∂y+w∂u∂z)=−dPdx+μ∇2u$
(5)
$ρ(u∂v∂x+v∂v∂y+w∂v∂z)=−dPdy+μ∇2v$
(6)
$ρ(u∂w∂x+v∂w∂y+w∂w∂z)=−dPdz+μ∇2w$
(7)
The fluid energy conservation equation and the heat conduction equation for the solid region in Cartesian coordinates were given as
$ρcP(u∂T∂x+v∂T∂y+z∂T∂z)=k∇2T$
(8)
$∂2T∂x2+∂2T∂y2+∂2T∂z2=0$
(9)

A uniform inlet velocity condition was applied on one end of the computational domain normal to the boundary. The inlet speeds (u) were limited by the equipment discussed in the section detailing experimental setup and consequentially varied from 2.5 cm/s to 12.5 cm/s. The inlet temperature (Ti) was set as 300 K. Both side walls were set as symmetry conditions, as this computational domain represents a relatively thin slice of the overall width. The solid–fluid interfaces were set as no-slip walls with temperature and heat flux continuity. The top section directly above the porous medium was a no-slip wall with a prescribed a heat flux of q″ = 10 W/cm2, and the outlet was prescribed a pressure outlet boundary condition.

### Numerical Procedure.

ansysfluent 19.2 was utilized for the CFD-HT simulations. A second-order upwind scheme was used for the momentum and energy equations discretization. The pressure was interpolated using a second-order scheme, and the gradients were discretized using the least squares cell based method. The SIMPLE algorithm [40] was used for pressure–velocity coupling. Convergence was reached after the average topside temperature and pressure drop qualitatively leveled off and the residuals dropped below 10−3 for mass and momentum and 10−6 for energy. Both pore-scale simulation geometries were examined for mesh independence. This was achieved by refining three separate meshes with u =10 cm/s and examining both the pressure drop and average surface temperature for convergence. The ERG foam model elements numbered from 2.7 × 106 to 14.8 × 106, with the coarsest mesh being used for analysis. The AM model ranged from 5.2 × 106 to 14.8 × 106 and the mesh of 5.2 × 106 elements was selected. Both the solid and fluid domains for the two geometries were meshed using tetrahedral elements.

## Experimental Testing Setup and Procedures

The samples were tested in the closed flow loop shown in Fig. 9. Prior to charging with degassed, DI water, the flow loop was evacuated using a commercial vacuum pump. A gear pump (Micropump® GJ-N27) mounted on a variable-speed gear pump system (Cole-Parmer® EW-75211-30) continuously circulated the working fluid around the loop. The coolant then passed through a liquid-to-liquid heat exchanger (Lytron® LL520FG12) connected to a constant temperature bath (Thermo Scientific™ A25 refrigerated circulator) for finer temperature control, and a flowmeter (McMillan® S-114-8-D-S6) measured the volumetric flowrate. Three pressure sensors (Omega® PX219 series) and in-line type-T thermocouples (Omega® MQSS series) were placed in various locations around the loop for continuous monitoring of both values. For the test section itself, a differential pressure sensor (Omega® PX2300 series) measured the pressure drop across the samples. Five additional type-T thermocouples were attached to the test section along the midline for calculating the heat transfer coefficient using the average wall temperature, and the data acquisition was handled by an Agilent® 37940 A DAQ unit. A liquid-to-air heat exchanger (Lytron® M14-120) cooled the fluid back to near room temperature. A DC power supply (Agilent® E3620A) provided the electrical power necessary for the instrumentation.

The uniform heat flux heating condition was achieved by placing five cartridge heaters into an aluminum block. A DC power supply (Keysight N8742A, 600 V, 5.5 A) provided electrical power to the heaters. The test samples were nominally 4 cm × 10 cm × 0.93 cm, and the heated section had the same area of 4 cm × 10 cm. A thermal pad was used to decrease the contact resistance between the heater block and the test section. The system reached steady-state before data collection, and this was ensured by ensuring the thermocouples measured a < 1 °C change over 5 min, which was typically reached in 25 min between each collection point. The measurement uncertainty can be seen in Table 4. The flowrate was measured by filling a graduated cylinder and timing it while simultaneously recording flowmeter output voltages. The pressure drop was calibrated using an Omega DPI 610 portable pressure calibrator, and the thermocouples were calibrated with an Omega CL122-4 block calibrator.

## Results

Before using the computational models for further analysis, they were first validated using experimental data. The total pressure drop across the test section and the Nusselt number calculated from effective heat transfer coefficient were compared for both the ERG foam and AM samples. The heat transfer coefficient was defined by the following:
$h=q″/(Tw−Ti)$
(10)
The pressure drop normalized per unit length as well as the Nusselt number, which was defined as
$Nu=hdc/kf$
(11)

was given as a function of the Rec in Fig. 10. The pressure drop results were curve-fit to a second-order polynomial to conform to the Darcy-Forchheimer Law, which applies for the range studied. A power law fit was selected for the Nusselt number, as is commonly done for thermal behavior with varying Reynolds numbers. Figure 10 showed good agreement between experimental and numerical results, which ensured model validity. Regarding the pressure drops for experimental and CFD datasets, there was a slight behavioral difference for the ERG sample. This was attributed to using a smaller representative section, which does not perfectly replicate the entirety of the foam sample and therefore introduces small discrepancies (within 0.36 kPa/m). Modeling more of, or the entirety of the sample, would yield better agreement but at a severe computational cost relative to the small improvements in accuracy. The figure also demonstrated that although the pressure drop is higher (66%) for the AM sample (as to be expected due to the rhombic shape as opposed to more circular shape of the ERG foams), the penalty came with approximately 60% increase in effective heat transfer coefficient.

As a pressure drop penalty typically accompanies heat transfer enhancement, it was not surprising that the designed sample will have a higher pressure drop. But it was necessary to ensure that the increase in thermal performance was not outweighed by pressure drop penalties. Figure 11 compared the nondimensional thermal performance of the two thermal management solutions with respect to pressure drop per unit length. The increase in pressure drop by using the AM structure was contributed primarily to the pore window shape, which was rhombic, instead of a more circular shape of traditional foams. The pressure drop was approximately 60% higher for the designed structure when compared to the ERG structure, and the Nusselt number was also around 60% higher for the same inlet speeds as seen in Fig. 10. However, when comparing the thermal performance as a function of pumping power, the AM sample was clearly superior at the studied pressure drops.

In these structures, the variations in local fluid velocity about the mean flow velocity caused mixing and increased heat transfer. Flow mixing can be seen in Fig. 12, which shows the streamlines in the ERG foam at two different inlet flow speeds and compared it to the AM sample's streamlines. Unlike pore-level simulations, volume-averaging can decrease computational time at the expense of intrafoam detail. The contribution to heat transfer from mixing was accounted for by augmenting the effective fluid conductivity kf with the dispersion conductivity, kd. Generally, the dispersion conductivity increases with flow speed and streamline tortuosity. Calmidi and Mahajan [20] gave an equation for kd which simplifies to the following:
$kd=CDρuKcP$
(12)

Figures 12(a) and 12(b) display the streamlines for u =2.5 cm/s and 12.5 cm/s with both the streamlines with the solid structure and just the streamlines in black without the solid to emphasize the path of the streamlines. It was clearly visualized that the fluid path was more erratic with increasing inlet velocity, which would increase flow mixing and supported the assumption of increasing dispersion conductivity with respect to velocity. Examination of the streamlines also illustrated that the majority of the particle paths are not heavily disturbed by the foam, as they traveled in a mostly linear path. This was also confirmed by the later tortuosity calculations, which demonstrated a total average increase in streamline length of 9.3% when compared to the length of the ERG foam.

The streamlines were also compared for the traditional and designed foams. The position coordinates for several hundred particles were exported to matlab, where the particle paths were analyzed for their total distance traveled over the foam structure. The code ensured that the tortuosity calculation only used paths that traversed the entirety of the structure (as some streamlines were truncated before exiting the foam). 500 and 161 particles were used after the elimination of the aforementioned incomplete pathlines, respectively. The average distances traveled were 2.732 cm and 4.828 cm for the ERG and AM foams. The tortuosity τ was defined as
$τ=sL$
(13)
where s is the total distance traveled and L is the shortest travel length possible (i.e., the length of the foam). The tortuosity values were calculated to be 1.093 and 1.038. Increasing τ corresponds to increasing thermal dispersion conductivities. For example, Du et al. [41] gave the following as a way to calculate kd while considering the effect of tortuosity
$kdkf=c(τ−1)(1−ετ)RepPr$
(14)

where c is a constant, ReP is pore-based Reynolds number, and Pr is the fluid Prandtl number. As the extra distance traveled grew by 2.45× when comparing the ERG and AM samples, it implied that the dispersion conductivity should be greater for the stochastic foam. It was also reasonable to assume that the randomness of the ERG foam promotes greater mixing, and although the effect of the dispersion conductivity is small at these velocities, it may warrant additional investigation at higher flow speeds and Reynolds numbers. Comparing the streamlines for u =10 cm/s in Fig. 12(b) with those in Fig. 12(c) visually demonstrated the differences in mixing, with some streamlines in the ERG geometry spanning more than half the breadth of the sample height, as opposed to the AM's streamlines, which did not mix significantly with the fluid in adjacent foam cells.

Figure 13 shows the pressure and velocity contours for the commercial and AM geometries along a plane parallel to the flow direction. The flow patterns for the AM foam were repeating due to the ordered structure, whereas for the stochastic sample a consistent flow pattern was not established. Stagnation regions can be seen in both geometries when the working fluid encountered a ligament. Downstream of the ligament, there were recirculation zones, where the velocity was significantly lower than the bulk fluid velocity. The pressure values decreased to below 0 Pa relative pressure due to the outlet being constrained to 0 Pa, and the fluid velocities were higher due to the presence of the metal foam. The pressure contours implied significantly more mixing in the stochastic structure as well, due to the uneven contour lines. Another difference between the two structures was the existence of large stagnant zones in the ERG sample, contrasting with the relative lack of such large low velocity zones in the AM geometry.

While Fig. 10 demonstrated the AM sample's superior thermal performance, the causes of improved performance, namely, structural differences in the foam versus elimination of the thermal interface material between the foam and substrate, were further investigated. Using a one-dimensional resistive network (Fig. 14(a)), the effective Nusselt number was recalculated with an updated overall heat transfer coefficient Unew for three kTIM values (kTIM = 4.0, 40, and ∞ W/m K), in addition to the original kTIM (1.38 W/m K). These values were chosen to be representative of a high conductivity epoxy, a solder, and no TIM layer. This was done by using the following equation that substitutes the highlighted resistance in Fig. 14(a) as:
$Unew=[1Uk=1.38−tTIMkoriginal+tTIMknew]−1$
(15)

The results of the calculations can be seen in Fig. 14(b), where (N) designated the data points obtained by using a resistance network. To ensure the accuracy of the one-dimensional assumption, additional simulations with kTIM = 4.0 W/m K were performed, and the numerical results were compared graphically with the resistance network results in Fig. 14(b). As the numerical and semi-analytical results demonstrated good agreement, the one-dimensional assumption was considered reasonable. The effect of incrementally improving TIM thermal conductivity had a significant impact at lower values. However, the returns diminished at higher kTIM values, and increasing kTIM from 40 W/m K to effectively infinite had much smaller performance benefits. Figure 14(b) also shows the effect of eliminating the TIM versus changing the structure to the AM's rhombic dodecahedron unit cell. At lower Reynolds numbers, the difference in Nusselt number was approximately half due to the TIM layer, and the other half was attributed to the structural differences. However, at higher Reynolds numbers, the performance reduction due to the ERG foam structure was much smaller than the TIM's thermal resistance effect. Additionally, the performance of the ERG foam approached the AM foam with increasing flow speeds, which may have been caused by the differences in kd.

Generally, the wall heat transfer coefficient is presented for metal foams, as it is more easily applicable for use in electronics cooling via resistance networks and simple numerical simulations. However, the interfacial heat transfer coefficient (hsf), which represents the heat transfer between the solid phase and the working fluid, has also been reported and can be used in volume-averaging simulations. Calmidi and Mahajan [20] modified a correlation developed for cylinders in crossflow
$Nusf=hsfdlkf=0.52Rel0.5Pr0.37$
(16)
where hsf is the interfacial heat transfer coefficient, dl is the ligament diameter, kf is fluid conductivity, Rel is the Reynolds number based on ligament diameter, and Pr is the fluid Prandtl number. The constant 0.52 was determined by comparing numerical results to their experimental air/aluminum results, as well as the water/aluminum data from Hunt and Tien [42]. Mancin et al. Produced a similar correlation based on their extensive experimental work with heat transfer using air flowing through copper foams [19]. Their correlation was given as
$Nusf=hsfdlkf=0.418Rel0.53Pr1/3$
(17)
Mancin et al. based their correlation on an extensive experimental dataset. The similarities between the correlations presented are evident with regard to similar numerical values as well as basing the Reynolds number's length-scale on ligament diameter. The numerical simulations were adjusted by setting the lattice structure to a uniform wall temperature Tw = 320 K and using the following to obtain the foams' interfacial heat transfer coefficient:
$hsf=m˙cpAsfln(Tw−ToTw−Ti)$
(18)

where $m˙$, Asf, Tw, To, and Ti are the mass flowrate, interfacial area, wall temperature, outlet temperature, and inlet temperature. Figure 15 compares the numerically calculated hsf with the values found using the correlations from Refs. [19] and [20]. For the ERG foams, Mancin et al.'s correlation matched more closely with the other, resulting in a noteworthy overestimate of hsf, but the opposite held true for the AM foams, where the same correlation predicted lower values than the numerical predictions, likely due to the structural differences between the two samples. Although the AM foam was designed to be an imitation of the commercial foam, the structural differences between the two may necessitate either an adjustment of the constants in the interfacial heat transfer coefficient correlations or a modification to make them more widely applicable.

A fin efficiency (η) analysis was conducted for both models where η is defined as the ratio of convected heat to the heat that would be convected with an ideal fin. For the ideal fin, all solid geometries were set to a uniform temperature of 320 K. The real fin simulations were done by setting the lower substrate surface or the fin base to Tb = 320 K. Additionally, to provide a more one-to-one comparison of the two foam structures, the heat dissipated by the fluid was measured at 2.5 cm into the AM structure as opposed to at the exit, which was at an additional 2.15 cm downstream. Temperature contours of the solid phase can be seen in Fig. 16 for an inlet velocity of u =10 cm/s. Increasing hsf and kf or decreasing ks were accompanied by a resulting decrease in η, as the heat was convected before it can travel down the finning surface. Figure 16 demonstrates how the temperature decreases more quickly for the ERG foam, which implied that the fin efficiency would be lower. This is shown quantitatively in Fig. 17, which illustrated the fin efficiencies, as well as the nondimensional outlet temperatures. The reason the heat was able to penetrate further down the foam was primarily due to the thicker nodes, which allowed for more heat transfer in the fins. Figure 17(a) shows that the ERG foam had better fin efficiency than the AM foam, except at lower inlet velocities. Figure 17(b) illustrates the outlet temperatures at a location 2.5 cm downstream of the foam inlet, which were used for calculating η. Although upon initial examination it seemed that the commercial foam performs as a better finning surface than the AM foam, the isothermal simulation results for both geometries showed that the AM geometry had a higher heat transfer rate at a given Reynolds number. The AM geometry also transported more heat than the commercial foam in the conjugate heat transfer model, which was due to the TIM and to the AM cell's more favorable geometry, particularly the rhombic windows.

## Conclusion

This study demonstrated the advantages of traditionally manufactured and AM metal foams for thermal management. The commercial foam was extensively characterized and exported to CAD modeling software using X-ray μCT and imagej/bonej, and the AM unit cell was based on the porosity and cell diameter found from the analysis. Both reduced geometries were analyzed using commercial CFD-HT software, and extensive numerical studies were undertaken after validating the CFD-HT commercial and AM models' thermo-hydraulic performance with experimental data. Both structures' tortuosity, which could impact the thermal dispersion conductivity in volume-averaged simulations, was numerically compared. The average increase in streamline length in the ERG foam was substantially higher than the AM sample, suggesting that the thermal dispersion conductivity would have to be adjusted for the AM structure. The numerical data also indicated that the commercial foams could be viable if the interface material's thermal conductivity was markedly increased, and that the elimination of an attachment layer would bring the performance of the ERG foam much closer to the AM foam. The interfacial heat transfer coefficients were quantified and compared with correlations found in the literature, and the numerical results were found to be in reasonable agreement with the correlations. The fin efficiencies were also numerically examined and compared for either structure. AM metal foams demonstrated improved performance over traditional metal foams largely due to the elimination of the TIM, and the models gave valuable insight into the pore-scale phenomena of both foams.

## Acknowledgment

This material is based upon work supported by the U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy (EERE) under the Vehicle Technologies Program Office Award Number DE-EE0008708. The authors acknowledge support from Haipeng Qiao and Professor Christopher Saldana in the tomographic imaging.

## Disclaimer

This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

## Nomenclature

• A =

area, m2

•
• cp =

specific heat, J kg−1 K−1

•
• Cf =

inertial coefficient

•
• d =

diameter, m

•
• h =

heat transfer coefficient, W m−2 K−1

•
• H =

Heywood circularity factor

•
• k =

thermal conductivity, W m−1 K−1

•
• K =

permeability, m2

•
• L =

length, m

•
• ΔL =

test section length, m

•
• $m˙$ =

mass flowrate, kg s−1

•
• Nu =

Nusselt number

•
• P =

perimeter, m

•
• ΔP =

pressure drop, Pa

•
• PPI =

pores per inch

•
• q″ =

heat flux, W m−2

•
• Re =

Reynolds number

•
• s =

distance traveled, m

•
• SSA =

specific surface area, m−1

•
• t =

thickness, m

•
• T =

temperature, K

•
• u =

superficial inlet velocity, m s−1

•
• U =

overall heat transfer coefficient, W m−2 K−1

### Greek Symbols

Greek Symbols

• ε =

porosity

•
• η =

fin efficiency

•
• μ =

dynamic viscosity, kg m−1 s−1

•
• ρ =

density, kg m−3

•
• τ =

tortuosity

### Subscripts

Subscripts

• b =

base

•
• c =

cell

•
• d =

dispersion

•
• eff =

effective

•
• f =

fluid

•
• i =

inlet

•
• K =

permeability

•
• l =

ligament

•
• o =

outlet

•
• p =

pore

•
• sf =

interfacial

•
• TIM =

thermal interface material

•
• w =

wall

•
• x =

cross-sectional

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