Abstract
This paper presents the convective heat transfer coefficient of cubic lattices under both buoyancy-induced and forced convection. Additionally, it examines the effective thermal conductivity, permeability, and inertial coefficient of a cubic unit cell of porosity ∼0.87. The test specimens were additively manufactured using stainless steel 420 (with 40% bronze infiltration) using the binder jetting technique. In the buoyancy-driven convection experiments, three different aspect ratios (width/height) varying from 0.5 to 2 were tested across three different heating orientations, viz., bottom wall (0 deg), side wall (90 deg), and top wall (180 deg). The lattice with the lowest aspect ratio had the highest convective heat transfer coefficient in all three heating orientations. The forced convection heat transfer coefficient was determined for an additively manufactured part comprising 10 × 10 cubic unit cell array in the plane perpendicular to the flow and 20 unit cells in the streamwise direction. Additionally, the flow characteristics of the cubic lattice were characterized through permeability (K) and inertial coefficient (Cf), determined by conducting separate pressure drop experiments over a wide range of flow velocities. The thermal hydraulic performance (THP) of the cubic lattice was assessed by combining the periodic regime convective heat transfer coefficient with the pressure drop data obtained from the experimentally determined values of K and Cf. The comprehensive characterization of flow and thermal transport, including K and Cf, along with hsf, keff, presented in this paper, provides a robust foundation for their application in volume-averaged computations for detailed parametric study.
1 Introduction
High porosity metal foams have superior capabilities to dissipate large heat flux levels under both buoyancy- and forced-convection scenarios. Typically, the unit cell of metal foams produced by foaming process can be approximated as a tetrakaidecahedron (TKD) shape. These metallic foams offer high specific surface area (m2/m3), highly tortuous flow paths, and complex interconnections of struts, resulting in enhanced flow mixing and thermal dispersion. Metal foams significantly enhance the effective thermal conductivity of the medium as well as the interstitial heat transfer coefficient at the strut level, leading to an overall high rate of heat dissipation. The metal foams obtained with this manufacturing process have porosities greater than 0.9, with pore densities ranging from 5 to 40 pores per inch. These metallic foams are typically manufactured in aluminum, copper, and nickel. The flow and thermal transport properties of such metal foams (aluminum) have been experimentally characterized by Mahajan and coworkers under different conditions [1–6]. A comprehensive review on the thermal transport of such metal foams can be found in Refs. [7,8].
Despite the superior thermal transport characteristics of metal foams, their widespread adoption has been limited due to several challenges, including geometrical constraints (pore density, porosity), limited material options, limitations on unit cell topology and metal foam block's form factor, machining difficulties, and the need for additional bonding with the substrate/plate subjected to heat loads. These challenges, however, present an opportunity to innovate and retain the benefits of metal foams while eliminating these limitations.
To this end, metal additive manufacturing presents an exciting opportunity by enabling the production of desired unit cell topology at different length-scales, porosities, materials, and form factors, which significantly expands the application domain of metal foams. Further, additively manufactured (AM) parts can be integrally manufactured where the plate subjected to the heat load and the lattice structures are built as a single solid body, thereby eliminating the thermal contact resistance issue, which is prominent in the foaming process enabled metal foam and its assembly with heated plates. Additionally, as pointed out earlier, metal foams produced via the traditional foaming processes are available only in high porosities, and uniform porosity distribution throughout the volume, limiting their functional versatility. In contrast, metal additive manufacturing enables the creation of porous media with moderately low porosities, along with functionalized grading, to achieve a desired local porosity distribution within the metal foam volume. Given these advantages, there has been a growing number of investigations to develop advanced concepts pertaining to lattice topology concepts for a variety of applications including, forced and buoyancy-induced convection with different working fluids, phase change material integration with lattices, flow and pool boiling, heat pipes etc. In this study, our focus is on forced, and buoyancy-induced convection. A brief review of the literature in these two convective heat transfer categories is presented below.
1.1 Lattice Structures for Forced Convection.
In forced convection research, the primary objective is to enhance the heat dissipation rate while balancing this against the associated cost of increase in pressure drop or pumping power. Enhanced heat transfer performance is typically quantified by comparing it to a similar geometry without enhancements (baseline configuration). Thermal-hydraulic performance, on the other hand, considers both, the enhanced heat transfer and the increased pressure drop in reference to the baseline configuration.
As discussed above, incorporating metal foams enhances effective thermal conductivity and local interstitial heat transfer coefficient. In addition, metal foams provide a high specific surface area, leading to increased heat dissipation rates. However, the complex strut interconnections and high flow blockage also cause a significant increase in pressure drop. Hence, the motivation behind the development of advanced porous media concepts is to retain and further improve on the benefits offered by metal foams while addressing the above-mentioned challenges and striking a balance between the enhanced heat transfer and increased pressure drop. One promising approach to achieving these objectives is the use of lattice structures with tunable unit cells. With recent advancements in metal additive manufacturing, complex and ordered unit cell topologies can be realized with high precision. Some of the recent studies related to forced convection on additively manufactured lattice structures are discussed below.
Liang et al. [9] carried out an experimental study to characterize different lattice topologies including, body centered cubic (BCC), face centered cubic (FCC), Kagome, and X-type in a lattice frame configuration with a single unit cell in the thickness of the porous volume. Selective laser sintering (SLS) method was employed to additively manufacture the test articles. The authors concluded that the X-type lattice provided the highest heat transfer enhancement for air forced convection. Takarazawa et al. [10] conducted an experimental study on lattice blocks based on BCC, FCC, and their variants with a higher number of struts, where the samples were additively manufactured using selective laser melting technique. The results showed that FCC lattice and its variants had higher effective thermal conductivity compared to their BCC counterparts, whereas BCC and its variants were more effective in forced convection scenarios.
As discussed earlier, the unit cell of the metal foams produced via foaming process can be approximated to a TKD shape. Typically, the struts of the TKD unit cell are of tricuspid shape [11], which is a result of the minimal free surface energy principle on which this manufacturing process is based on. With metal AM, the strut shape can also be controlled to achieve the desired flow and thermal transport characteristics. Lorenzon et al. [12] recently evaluated BCC type unit cells in a lattice frame configuration, where the strut shapes were varied. The authors investigated cam-like, drop-like, and elliptical cross section struts for the BCC unit cell configuration where laser powder bed fusion technique was used to print the samples in AlSi10 Mg. They found that the strut shape had a significant effect on the pressure drop, and elliptical shape struts had heat transfer performance similar to that of circular struts while exhibiting low pressure drop penalty. Chaudhari et al. [13] carried out an experimental study on lattice frame configuration based on Octet-shaped unit cell, where the parts were printed in AlSi10 Mg for porosities between 0.7 and 0.9. The authors characterized the permeability, inertial coefficient, effective thermal conductivity, and interstitial heat transfer for the three samples within the above porosity range.
Recently, a series of experimental studies investigated four different unit cell topologies—cubic, face diagonal cube (FD-Cube), TKD, and octet—in different configurations using air as the working fluid. In the lattice frame configuration [14], it was experimentally found that at a porosity of ∼0.87, the FD-Cube exhibited the highest heat transfer coefficient for a given flow velocity, while the simple cubic lattice showed the highest heat transfer coefficient at a given pumping power for the same porosity. However, as porosity decreased, thermal hydraulic performance improved as the gain in heat transfer compensated for the increase in pressure drop. It is important to note that in lattice frame configuration, where only one unit cell is present in the thickness of the porous media volume, the endwall also plays a significant role in flow and thermal transport. The endwall characteristics for three different unit cell topologies—octet, octa, and V-octet—were determined experimentally by Kaur and Singh [15], using liquid crystal thermography to determine the local convective heat transfer coefficient. For the three unit cell topologies tested, the configuration featured a single column of five interconnected unit cells in the streamwise direction. The presence of X-shaped half struts at the endwall in the octet and V-octet topologies had a distinct effect on local heat transfer, as the flow separated and reattached in the immediate downstream vicinity of the half struts. Additionally, the lattice topology affected the flow mixing.
Aider et al. [16] carried out an experimental study to characterize the convective heat transfer coefficients of large lattice structures composed of unit cells similar to those described in [14]. The foam volume consisted of 5 × 5 unit cells in the plane perpendicular to bulk flow and 10 unit cells in the streamwise direction, in contrast to the lattice frame configuration of 5 × 1 × 5 unit cells in [14]. The selection of the foam volume with several repeated unit cells by Aider and coauthors [16] was deliberate, aiming to minimize endwall effects and focus solely on the effects of individual unit cell topology. The authors reported heat transfer coefficient values for four unit cell topologies in the periodic heat transfer regime, where convective heat transfer coefficients at the unit cell level remained nearly constant, implying that the temperature rise of the working fluid across a unit cell (along the flow direction) in the periodic regime was similar. It was found that heat transfer coefficients in the periodic regime for the octet, FD-Cube, and TKD unit cells were comparable, while the cubic unit cell exhibited distinctly lower heat transfer coefficients compared to the other topologies. However, when analyzing thermal hydraulic performance (THP) versus channel hydraulic diameter, it was observed that all structures had nearly similar THP values across the investigated range of Reynolds number. This observation suggests that while complex topologies can potentially yield higher convective heat dissipation, they may also incur a fairly large concomitant pressure drop. Further research is needed to explore the thermal-hydraulic performance of lattice topologies with varying degrees of complexity. The unique benefits offered by additively manufactured lattice structures are comprehensively discussed in Kaur and Singh [17,18].
1.2 Lattice Structures for Buoyancy-Induced Convection.
Buoyancy-driven convection is a passive cooling technology known for its reliability, ease in maintenance, and no requirement of pumping power for coolant flow. The convective heat transfer rates for a typical heated plate can be enhanced by installing high porosity metal foams, which provide high specific area per unit volume and high local interstitial heat transfer as the fluid moves through the pores and interacts with the struts. Phanikumar and Mahajan [2] conducted a numerical study on aluminum foams under buoyancy-driven convection, where the pore density was varied between 5 and 40 PPI. The authors demonstrated that metal foams significantly enhanced heat transfer, and low pore density foam was superior to the higher pore density foam, as it allowed easier escape of the spent fluid to the ambient. This observation was confirmed experimentally by Bhattacharya and Mahajan [4] where the 5 PPI foam significantly outperformed the 40 PPI foams for Rayleigh numbers varying between 105 and 106. Note that despite the high surface area offered by 40 PPI foams relative to 5 PPI, the higher permeability of 5 PPI foams dominated the resultant buoyancy-driven convection heat transfer. The authors extended this concept to longitudinal finned heat sink where the passage between the adjacent fins (solid) was packed with high porosity metal foams. The permeability of metal foams was found to be an important parameter which affected the convective heat transfer under buoyancy-driven convection involving metal foams. Zhao et al. [19] studied the natural convection in FeCrAlY metal foams and demonstrated that the local thermal nonequilibrium effects must be considered for accurate prediction of the thermal performance, a finding similar to the one reported in Ref. [2]. In contrast with Ref. [4], Feng et al. [20] studied the thermal performance of metal foam strips made from copper, where the space occupied by solid longitudinal fins as in Ref. [4] was kept empty to allow the movement of air. The number of metal foam strips varied along with the individual strip height. It was found that an intermediate configuration with 7 foam strips performed better than 4 or 10 foam strip configurations for the different heights studied. The hypothesis behind such a configuration was to increase the effective permeability of the volume occupied by the heat sink by allowing free motion of air between adjacent strips. An experimental study was conducted by Qu et al. [21] where copper metal foams' orientation with respect to the gravity vector was studied under buoyancy-driven convection. It was shown that 60-75 deg orientation resulted in optimum thermal performance for different metal foam configurations, where the inclination was varied between 0 deg (horizontal) and 90 deg (vertical).
High porosity metal foams investigated in the aforementioned buoyancy-driven convection studies were manufactured via foaming process, resulting in a randomized arrangement of struts and pores, where a typical pore can be closely represented by a TKD shaped unit cell topology. From the above studies, it can be concluded that an easy escape passage of spent fluid may result in enhanced thermal performance, a characteristic feature of low pore density foams. This advantage compensates for the reduced surface area available for heat dissipation in buoyancy-driven convection transport. A more structured arrangement of unit cell topologies has the potential to yield higher heat transfer rates compared to a randomized arrangement.
To this end, recently Aider et al. [22] carried out an experimental study on buoyancy-driven convection transport in configurations involving two parallel plates with structured lattice unit cells packed between them. They investigated five unit cell topologies—cube, FD-cube (two patterns of adjacent unit cell attachment), TKD, and Octet—under two different heating configurations (one and both side walls heated). The samples were manufactured via binder jetting in SS420 (40% bronze infiltration). The results indicated that the cubic lattice (the simplest topology) exhibited the highest heat transfer among the five unit cell topologies, which was ∼six times that of the baseline configuration of two vertically heated parallel plates. This was attributed to its remarkably high permeability, which allowed easy departure of heated air, while the Octet topology, despite its high surface area to volume ratio, resulted in the weakest thermal transport due to its low permeability.
From the literature review presented above, it can be concluded that relatively simpler unit cell topologies may outperform in terms of thermal hydraulic performance in forced convection and overall heat dissipation in buoyancy-driven convection due to larger permeability. In this study, a comprehensive experimental investigation of flow and thermal transport of lattice configurations based on simple cubic unit cells has been carried out. The novelty of this research lies in its employment of simple unit cell topology for both, forced and buoyancy-driven convection scenarios, to demonstrate their effectiveness in flow and thermal transport, and also provide motivation for their fabrication via conventional manufacturing processes such as subtracting manufacturing, wire-woven techniques, brazing.
The experiments were conducted to characterize both forced and buoyancy-induced convection transport coefficients, along with the effective thermal conductivity, permeability, and inertial coefficient, using different test sections and test specimens that were printed through Binder jet technology with Stainless Steel 420 (with 40% bronze infiltration) as the material. Section. 2 provides the details of experimental test setups used for the flow and heat transfer experiments, details of test articles, experimental procedures, data reduction methodologies, and results and discussion.
2 Experimental Setups
This section provides details of the test setups used for effective thermal conductivity, buoyancy-induced convection, and forced convection experiments.
2.1 Effective Thermal Conductivity Experimental Setup.
The effective thermal conductivity of the cubic unit cell was determined through steady-state experiments, as illustrated in Fig. 1. This measurement is critical for understanding the thermal transport properties of the cubic lattice sample by virtue of its topology, while the air occupying the void space is mostly stagnant. The sample was positioned on an aluminum slab partially submerged in an ice-water slurry, which resulted in the slab temperature of ∼0 °C in steady-state conditions. A patch heater attached to the top plate of the lattice sample ensured a uniform heat flux input to the lattice frame. The patch heater was powered by a direct current (DC) source, which also displayed the current in the circuit. By knowing the patch heater's resistance and the current in the circuit, the total heat input was determined. The stray heat losses were minimized by attaching a rubber foam insulation (k ∼ 0.037 W/mK) of 3.2 mm thickness.
The heat flow through the insulator was also determined by measuring the temperature drop across it and applying Fourier's law of heat conduction. The bottom plate of the lattice sample was glued onto the aluminum slab, maintained at a steady temperature of ∼0 °C. The large thermal mass of the aluminum slab effectively served as a cold reservoir, and the heat input to the top plate did not result in noticeable temperature rise in the cold reservoir. This experimental design ensured unidirectional heat flow through the thickness of the lattice frame, which was essentially one unit cell thick. With the known value of heat conducted through the thickness of the lattice sample, the effective thermal conductivity was calculated by measuring the temperature drop across the thickness of the lattice. The heat flux used in the effective thermal conductivity experiments was chosen to keep the temperature difference between the top and bottom plate temperature small. This ensured low Rayleigh numbers for the air trapped in the void spaces, thereby minimizing the buoyancy-driven convection. Note that since the solid-phase thermal conductivity was much larger than that of the fluid phase, the effective thermal conductivity determined via the above procedure depends on the lattice topology, its porosity, and the solid material used.
In addition to determining of the lattice frame configuration cubic unit cell sample, the thermal conductivity of the solid sample made in SS420 (with 40% bronze infiltration) via the same AM process used to manufacture the lattice samples, was also measured. This solid block was 25.4 × 25.4 mm in cross section and 100 mm in height. Four slots for thermocouple-based measurements were incorporated in the AM process itself. These samples were also tested on the same test bench as the lattice frame materials, as shown in Fig. 1. The thermal conductivity of the solid (ks) was found to be 20.8 W/mK.
2.2 Buoyancy-Driven Convection Experimental Setup.
The experimental setup for conducting buoyancy-driven convection experiments is shown in Fig. 2. The setup comprised four wooden supports and two plastic wires, which suspended the additively manufactured cubic lattices in various heating orientations. This setup ensured minimal heat loss from the lattice to the supporting accessories, allowing free air flow through the lattice for all three heating orientations. It was crucial to maintain constant laboratory ambient conditions throughout the course of steady-state experiments to prevent their influence on the convective heat transport within the lattice. The additively manufactured samples used in the buoyancy-driven convection experiments are shown in Fig. 3. Three different aspect ratios (width/height) of lattices were investigated for the three different heating orientations, viz., bottom wall (BW), top wall (TW), and side wall (SW) (see Fig. 2). As mentioned earlier, these samples were manufactured through binder jetting process in stainless steel 420 with 40% bronze infiltration, featuring unit cells of cube shape with cylindrical struts and unit cell edge length of 5.08 mm.

Schematic representation of test setup for buoyancy-driven convection experiments, sample cubic lattice, and three different heating orientations

Three different aspect ratios (width-to-height ratio) for buoyancy-driven convection experiments (sample images are not representative of their true size, the scale shown at the bottom of each sample's image should be used for the intercomparisons of the size of samples)
Steady-state experiments were conducted for the three samples for different heating orientations. A constant heat flux boundary condition was applied on the substrate via a firmly glued flexible polyamide patch heater on its back. The backside of the patch heater was insulted by polystyrene. Two thin thermocouples were placed inside and outside of the insulation to calculate backside heat loss. Experiments were performed over a wide range of Rayleigh numbers such that the difference between the substrate temperature and ambient fluid temperature varied between 10 and 60 . The substrate, 3 mm thick, featured a hole (also included in the AM original computer-aided design (CAD) model) to accommodate a thermocouple. This hole terminated at the center of the 50 mm × 50 mm substrate and was situated 1 mm beneath the solid–fluid interface. Temperature readings obtained at this particular location were considered as representative wall temperature (Tw), owing to the material's high thermal diffusivity and the small size of the substrate.
The test sample was a single, integrated piece, comprising both the cubic lattices and the substrate, which were additively manufactured together, thereby eliminating contact resistance between the substrate and lattice. This is one of the many advantages offered by AM. Figure 3 displays the CAD models of the three lattice configurations with different aspect ratios. The cubic unit cell configuration is also shown. The lattices were designed with a porosity (ε) of 0.9 where the cylindrical struts were ∼1 mm in diameter (df). These dimensions were selected based on recommendations for accurate manufacturability using the Binder jetting process. The measured porosity of the final printed parts used in the buoyancy-driven convection experiments was ∼0.87.
2.3 Forced Convection Experimental Setup.
The experimental setup for the forced convection experiments is shown in Fig. 4. These experiments were conducted using air drawn from a compressed air tank maintained at 100 psi pressure. The supply pressure was regulated down to a lower value based on the required air flowrate corresponding to the desired Reynolds number at the test section. Following pressure regulation, the air flowrate was metered using an orifice plate. Pressure, temperature, and differential pressure across the orifice plate were measured using Dwyer DPG-002, T-type thermocouple and Dwyer 477 AV-2 devices. These measurements were used in the in-house computer program to iteratively determine the mass flowrate through the determination of flow coefficient and discharge coefficient of the orifice plate. The air flow was then routed to a diffuser section connected to the test section. The diffuser was designed such that a uniform supply pressure could be obtained in the plane perpendicular to the bulk flow. The test section comprised a straight channel ( in length) leading to the test article, allowing hydrodynamic flow development. The pressure drop at the test section was measured slightly upstream of the test section via a static pressure transducer. An exit length was also provided to avoid any back pressure effects resulting from sudden discharge of spent air into the laboratory ambient.

Schematic of the forced convection experimental setup, test sample and its placement in the duct as seen by the flow entering the test section
Figure 4 also shows the test article used in the forced convection experiments. The heat transfer test article was manufactured in the same way as the test articles used in the buoyancy-driven convection experiments described above. In forced convection, the aim was to determine the convective heat transfer coefficient in the regime where net convective heat transport across one unit cell in streamwise direction does not change with streamwise location. The test article contained 10 × 10 cubic unit cells in the plane perpendicular to the bulk flow direction, while 20 unit cells were present in the streamwise direction. This arrangement resulted in unit aspect ratio of the channel. The unit aspect ratio is common in different heat exchanger applications, such as a typical circular duct in a shell-and-tube heat exchanger, square cross section channels in crossflow heat exchanger with alternate hot and cold streams, midchord region of gas turbine airfoil internal which typically features angled rib turbulator on its two opposite walls. In several heat exchanger configurations, heat exchange from one channel typically occurs through the two opposite walls, while the other two confining walls are attached with adjacent channels carrying the same working fluid. Hence, the heating orientation in this study featured constant heat flux applied to the opposite walls, while the other two confining walls were adiabatic.
Forced convection experiments were conducted under steady-state conditions, with the two opposite walls (endwalls) subjected to constant heat flux via patch heaters. Stray heat losses from the thermal system were minimized by installing rubber foam insulation (k ∼ 0.037 W/mK) of 3.2 mm thickness on the backsides of the two opposite base plates subjected to heat flux. The remainder of the test section was made from clear acrylic which has low thermal conductivity, to further minimize stray heat loss. Although the heat loss was minimal due to the above-mentioned measures, the loss was still determined by measuring temperature drop across the insulator on both ends. The wall temperatures were measured at pre-existing slots for thermocouples distributed along the length of the test article. The patch heaters were connected to a DC power supply. The power supplied to individual patch heaters was determined by measuring the patch heaters' resistance and the current in the circuit as displayed in the DC power source. The forced convection experiments were conducted for Reynolds numbers between 5000 and 20,000, which lies in the turbulent flow regime and finds its application in wide range of heat exchanger and thermal energy dissipation applications.
3 Data Reduction Procedures
3.1 Effective Thermal Conductivity.
The total heat input () was determined by using the relationship , where is the current and is the resistance of the patch heater. The heat loss () was determined by measuring temperature drop across the thermal insulator (used to minimize the heat loss). It was found that the heat loss was less than 3% of the total heat input via the patch heater. In Eq. (1), is the area through which the unidirectional heat was being diffused, is the spatially averaged wall temperature, measured at both, the top and bottom walls, and is the height of the cubic unit cell (∼5.08 mm).
3.2 Buoyancy-Driven Convection Experiment.
where is the convective heat transfer determined by subtracting the heat loss () from the backside of the patch heater from the total heat supplied (), and is the base area. Heat transfer coefficient is also presented in normalized form of Nusselt number (Nu), where , Lc is the characteristic length scale and is fluid thermal conductivity evaluated at the film temperature . For calculation of Rayleigh numbers (Ra), the required thermophysical properties were evaluated at the film temperature ().
3.3 Forced Convection Experiment.
In Eq. (4), is the fluid temperature at the inlet, is the mass flowrate of air, is the specific heat capacity of air, is the width of the channel, and “” is the local position in the streamwise direction at which the local bulk fluid temperature was determined.
4 Results and Discussion
In this section, the flow and thermal characteristics of lattice structures constructed from a cubic unit cell are presented. The section is divided in three parts, (i) effective thermal conductivity, (ii) buoyancy-induced convection, and (iii) forced convection. The forced convection section also includes results for permeability and inertial coefficient results, followed by thermal hydraulic performance analysis.
4.1 Effective Thermal Conductivity.
Thermal transport in cellular materials is governed by the effective thermal conductivity of the medium, as well as the interstitial heat transfer coefficient at the outer surfaces of the struts and the endwalls to which the porous block is attached for heat dissipation. Accurate determination of is, therefore, imperative for designing high heat dissipation concepts and facilitating volume-averaged computations that rely on the empirical correlations for .
Figure 5 shows the results of the cubic unit cell, determined from the lattice frame configuration featuring a single unit cell in the thickness. Such a configuration with cubic unit cell is essentially an inline array of cylindrical pin fins joining the two endwalls between which the cubic unit cell is sandwiched. To ensure that the reported is independent of the thermal boundary conditions, the experiments were conducted by subjecting the top plate to three different heat flux levels. The normalized was found to be independent of the applied heat flux level. The results were normalized with (solid intrinsic thermal conductivity), which was determined through separate experiments on a solid block printed using the exact same procedure as for the lattices. Detailed results are presented in Ref. [23], with the average value of found to be 20.8 W/mK. As mentioned earlier, the porosity of the samples used in the buoyancy-driven and forced convection experiments were measured, and found to be ∼0.87. Thus, samples for experiments were designed to have a porosity of 0.87 to ensure that flow and heat transfer results could be presented for the cubic unit cell at a fixed porosity (0.87). However, the actual porosity of the three-dimensional printed part turned out to be 0.85. Although the difference between the designed and actual porosity was not significant, the obtained values were still modified to reflect the intended 0.87 porosity. This calculation was carried out using the generalized correlations prescribed by Kaur et al. [24] and the cubic unit cell specific correlation prescribed by Wang et al. [23]. Both these approaches yielded similar values of the cubic unit cell for lattice porosity of 0.87, as shown in Fig. 5.
4.2 Buoyancy-Induced Convection.
Buoyancy-driven convection over a small plate with a cubic lattice structure integrally manufactured on top of it was investigated for three different cubic lattice configurations, each for three different heating orientations. For all three configurations, the heated area was 50 × 50 mm2, essentially a small plate where edge effects are expected to influence local heat transfer. For further insight, see, for example, references [25] and [26]. The authors presented a qualitative sketch (Fig. 6) of typical flow over a small hot plate subjected to buoyancy-driven convection, with the hot side facing upwards [26]. The sketch reveals that in this particular configuration, boundary layer flows interact with each other, resulting in a complex flow with plume rising upwards from the center of the plate, which significantly impacts the overall heat transfer.
![A qualitative description of convection over a flat plate (small) with hot surface facing upwards, based on [26]](https://asmedc.silverchair-cdn.com/asmedc/content_public/journal/heattransfer/147/2/10.1115_1.4066775/2/m_ht_147_02_022701_f006.png?Expires=1742371208&Signature=UcLnaLEtfQKC0LMU-Wz0dD7Zrcxva5jwrxnOx6tlDaiC8o0YrIvp~mDPIOSW29dzwvVAExgTVQJZGLQ6qUQFkn13FJbtvNc3eU~2mDH7fXFXHRSOfm-5gDYp5eSwT6sHF2OMnySruSYffEg3iRO0xQNs7EqIhFKK6ylcGYQFyhyO4iOALGTULYRJNaWpEMzIC9IDRP5UO9a8TP5yL~NlgHYY-eMrvb3q4BW29c-wQ5CY2rgWbU5P8ZhcFcU0jkCCL3Rz7C4rEhIsdcMiUz5m5BGaZItWJPH3PaOkq1E8ZrW-adu2gYZ4ClmGLsBoIFgVtn7soIdYwV6lcnLvysoKdw__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
A qualitative description of convection over a flat plate (small) with hot surface facing upwards, based on [26]
![A qualitative description of convection over a flat plate (small) with hot surface facing upwards, based on [26]](https://asmedc.silverchair-cdn.com/asmedc/content_public/journal/heattransfer/147/2/10.1115_1.4066775/2/m_ht_147_02_022701_f006.png?Expires=1742371208&Signature=UcLnaLEtfQKC0LMU-Wz0dD7Zrcxva5jwrxnOx6tlDaiC8o0YrIvp~mDPIOSW29dzwvVAExgTVQJZGLQ6qUQFkn13FJbtvNc3eU~2mDH7fXFXHRSOfm-5gDYp5eSwT6sHF2OMnySruSYffEg3iRO0xQNs7EqIhFKK6ylcGYQFyhyO4iOALGTULYRJNaWpEMzIC9IDRP5UO9a8TP5yL~NlgHYY-eMrvb3q4BW29c-wQ5CY2rgWbU5P8ZhcFcU0jkCCL3Rz7C4rEhIsdcMiUz5m5BGaZItWJPH3PaOkq1E8ZrW-adu2gYZ4ClmGLsBoIFgVtn7soIdYwV6lcnLvysoKdw__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
A qualitative description of convection over a flat plate (small) with hot surface facing upwards, based on [26]
Figure 7 illustrates the variation in the convective heat transfer coefficient for the three configurations shown in Fig. 3 tested under three different heating orientations. Across all heating orientations, it was observed that the lower aspect ratio lattice configuration (taller samples) exhibited higher heat transfer compared to the shorter, higher aspect ratio lattice (Fig. 3). The lower aspect ratio lattice provided a higher wetted surface area but imposed greater blockage to the rising flow. While the larger wetted surface area aids in heat dissipation, the increased blockage diverts the flow to the path of least resistance, often sideways.
For taller samples, the smaller plate length and width (in physical dimensions), made it easier for spent air to escape along the four open faces. Considering the two competing factors—high wetted surface area and increased flow resistance —the presence of more surface area appears to be dominant for low aspect ratio samples, yielding the highest heat transfer. Among the three heating orientations, sideways heating (90 deg) yielded the highest heat transfer, followed by bottom wall (0 deg) and top wall heating (180 deg), where the bottom wall and top wall heating correspond to the classic buoyancy-induced convection adjacent a heated plate facing upwards and a heated surface facing downwards, respectively. Since the purge through the side planes is common in all configurations, the overall trends were driven by the plate heating orientation [25]. Huang et al. [27] investigated three heating orientations for a flat plate featuring an array of pin fins and found that the heat transfer rates for sideways and bottom wall heating were comparable to each other and higher than the values for the top wall heating configuration. The authors also carried out experiments on a flat plate (100 × 100 mm2) and found that sideways heating produced slightly higher heat transfer rates than the bottom wall heating configuration, and that the rates for both heating orientations were significantly higher than those corresponding to the top heating orientations. The authors also validated their results with those of Al-Arabi and El-Riedy [26].
Figure 8 shows the normalized form of Fig. 7, where the Nusselt number is presented as a function of the Rayleigh number, and the characteristic length scale was based on the side length (=W) of the heated plate. The overall trends discussed for Fig. 7 are the same for Fig. 8 as well. For the BW heating orientation, a comparison has been presented with respect to a baseline flat plate correlation [26] to show the enhancement in heat transfer due to the presence of the cubic lattice structures.
![Nusselt number versus Rayleigh number based on characteristic length scale of length or width of heated plate, also shown Al-Arabi and El-Riedy [26] correlation for flat heated plate facing upwards (NuL=0.7RaL1/4)](https://asmedc.silverchair-cdn.com/asmedc/content_public/journal/heattransfer/147/2/10.1115_1.4066775/2/m_ht_147_02_022701_f008.png?Expires=1742371208&Signature=NnlvV0MqrNhGARubK3Bcq2SiVMjcbDyntc4MdHWEQ~rJxQYj99Kj9gBj2OqmpG~uNjpk1EI7cdUBYsrU-mJaBV7mb0Bv6KyngAEOFp9Qb6Nvh574mE51VU6PmefzR6MOnu7sWR9o4I8P5C-kWimvf7VVpq6W6BCRcLTZrbXEmaaoXP3tbKGkBBoC8UKR4vOvt9JUi7zAPs0dEiAh4JsdeTK90f8WM~9Dq0VTE9AJWyMNwV9l3g4P6kKvCTC-SmU8euHzm7U151lpNU-dPlrCDXbVJouuqlV0IWFQj6ymCKp5oOpsgZNiMZVu-7kLY9aqfQ51KDUPZphiCvfaH4FT3Q__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
Nusselt number versus Rayleigh number based on characteristic length scale of length or width of heated plate, also shown Al-Arabi and El-Riedy [26] correlation for flat heated plate facing upwards ()
![Nusselt number versus Rayleigh number based on characteristic length scale of length or width of heated plate, also shown Al-Arabi and El-Riedy [26] correlation for flat heated plate facing upwards (NuL=0.7RaL1/4)](https://asmedc.silverchair-cdn.com/asmedc/content_public/journal/heattransfer/147/2/10.1115_1.4066775/2/m_ht_147_02_022701_f008.png?Expires=1742371208&Signature=NnlvV0MqrNhGARubK3Bcq2SiVMjcbDyntc4MdHWEQ~rJxQYj99Kj9gBj2OqmpG~uNjpk1EI7cdUBYsrU-mJaBV7mb0Bv6KyngAEOFp9Qb6Nvh574mE51VU6PmefzR6MOnu7sWR9o4I8P5C-kWimvf7VVpq6W6BCRcLTZrbXEmaaoXP3tbKGkBBoC8UKR4vOvt9JUi7zAPs0dEiAh4JsdeTK90f8WM~9Dq0VTE9AJWyMNwV9l3g4P6kKvCTC-SmU8euHzm7U151lpNU-dPlrCDXbVJouuqlV0IWFQj6ymCKp5oOpsgZNiMZVu-7kLY9aqfQ51KDUPZphiCvfaH4FT3Q__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
Nusselt number versus Rayleigh number based on characteristic length scale of length or width of heated plate, also shown Al-Arabi and El-Riedy [26] correlation for flat heated plate facing upwards ()
The buoyancy-induced convection mode does not require additional pumping power to facilitate air flow through these lattice structures, making them highly reliable for cooling. This concept can be effectively applied in data center cooling, where the volume between adjacent racks can be packed with cube-shaped unit cells to enhance heat dissipation. Such an arrangement would result in a sideways heating configuration, as reported above. Additionally, while the cubic lattice samples studied here were additively manufactured, such geometries can also be manufactured via other techniques such as wire-based methods, which are considerably more cost-effective and are scalable. Such a manufacturing approach would require bonding of the lattice structure with the base plate from where the heat must be dissipated. Both the cubic lattice structure studied in the present work and those examined in Ref. [22] have the potential for a wide range of applications requiring reliable and efficient cooling driven by buoyancy-induced convection.
4.3 Forced Convection.
Forced convection experiments were conducted using the cubic unit cell tested for effective thermal conductivity and buoyancy-driven convection. The sample featured 10 × 10 cubic unit cells in the plane perpendicular to bulk flow, with 20 unit cells arranged in the streamwise direction. This configuration was chosen to achieve periodic heat transfer at the unit cell level. Figure 9 shows the unit cell level convective heat transfer coefficient variation with streamwise direction for four different flow conditions. Wall temperatures were measured at both top and bottom plates, as depicted in the sample image in Fig. 4. These measurements at the four discrete locations were used to extrapolate the measured temperature to the entrance and exit edges of the top and bottom plates and interpolate to obtain the wall temperatures at the unit cell level. The wall temperatures corresponding to each unit cell location in the streamwise direction were then used in Eq. (3) to derive the local convective heat transfer coefficient.

Streamwise variation of local convective heat transfer coefficient determined at the top and bottom walls, also shown is the averaged heat transfer coefficient () by taking the local mean of the top and bottom plate h
The data in Fig. 9 can be divided into three domains. First, the entry region (i) covering approximately the first 40% of the streamwise length exhibited a hydrodynamically and thermally developing flow as the fluid entered the porous medium. The unheated entry length also contributed to the high heat transfer observed in this region. The next region (ii), accounting for ∼35% of the streamwise length, represents a domain of nearly constant heat transfer coefficient. The convective heat transfer data in this regime was ∼20% lower than that in the entrance region for the lowest flow velocity of 2.05 m/s. Similar levels of reduction were observed for other flow velocities, as shown in Fig. 9. The convective heat transfer values corresponding to region (ii) represent a “periodic” nature, where the values are nearly similar at the unit cell level. Here, the periodicity in heat transfer refers to the case when the temperature rise of the working fluid when it passes through one unit cell in the streamwise direction is nearly the same. The convective heat transfer coefficient in the periodic regime can be used in the volume-averaged computations, as it is free from the entrance and exit effects and is representative of the sole effect of the unit cell topology on heat dissipation. Finally, in the last ∼25% of the streamwise length (region iii), exit effects came into play.
Figure 9 also shows the convective heat transfer coefficient averaged between the top and bottom walls at the respective unit cell locations. The heat transfer () reported in Fig. 10 was taken to be the average of top and bottom wall heat transfer coefficient in the periodic regime as highlighted in Fig. 9. As expected, a power law dependence between and was observed. Note that is the average flow velocity at the entrance of the lattice sample and is determined by the total mass flowrate and total flow area in the supply duct leading to the test section (Fig. 4). The average flow velocity inside the lattice sample will be higher than , and is dependent on the lattice porosity ( ∼), where is the porosity of the lattice sample (in this case, = 0.87). For volume-averaged simulations, users can determine the flow velocity based on the respective formulations and appropriately apply the correlation, .
In this study, the permeability and inertial coefficient of the lattice sample used in the forced convection study were determined by conducting pressure drop experiments for a wide range of flow velocities, as shown in Fig. 11. The relationship between the pressure gradient and the flow velocity was then used to perform regression analysis to determine the values of K and . The flow experiments were conducted separately to obtain the pressure drop data for multiple flow velocities over a wide range.

Variation of pressure gradient with flow velocity, dependence of determined values of on the choice of for regression analysis
The K and values shown in Fig. 11 were determined by carrying out regression analysis for different values, where represents the velocity up to which the pressure gradient dependence on flow velocity was considered.
Although theoretically, the and values should be independent of the choice of , our study showed otherwise. It was found that the knowledge of pressure drop for a broad range of flow velocities is essential to accurately capture and of the lattice structure. Such a behavior was also observed by Boomsma and Poulikakos [28] in an experimental study on compressed metal foams, where and values were found to be sensitive toward the choice of .
Figure 12 shows the variation of Nusselt number and enhancement in Nusselt number with Reynolds number, where a power law dependence was observed for the Nu variation with Re. The ratio showed a monotonically decreasing trend with increasing Reynolds number, which is also an expected trend in enhanced heat transfer studies in turbulent flow regime [14,16,30,31]. The heat transfer enhancement ranged between 12 and 18 times that of the smooth channel configuration across the investigated range of Reynolds number. Compared to smooth channel, lattice structures integrated with the smooth walls offer superior heat transfer due to increase in wetted surface area augmenting the conjugate heat transfer, and endwall- and strut-based convective heat transfer coefficient enhancement due to strut-endwall interfaces and strut interconnections.
Figure 12 also shows the variation of the THP and versus Reynolds number. Although the pressure drop penalty was large, the significant enhancement in heat transfer compensates for this loss, resulting in an overall higher thermal hydraulic performance was observed for the cubic lattice structure. Across the range of Reynolds number, the THP was found to vary between 1.5 and 2.5.
5 Conclusions
A comprehensive experimental study was conducted to characterize the flow and thermal transport in lattice structures made from cubic unit cells in buoyancy-induced and forced convection. These structures were made using binder jet additive manufacturing technique in stainless steel 420 (with 40% bronze infiltration). The experiments revealed crucial insights into the effective thermal conductivity, permeability, inertial coefficient, and convective heat transfer coefficients of these structures, as follows:
The effective thermal conductivity closely followed the correlation proposed by Kaur et al. [23], with ∼ 0.06 for porosity between 0.85-0.87, where ∼ 20.8 W/mK. The results were minimally affected by roughness since the air trapped in the void spaces was mostly stagnant, and local roughness was accounted for in the reported measured porosity values.
Permeability and inertial coefficient for cubic lattice structure with 0.87 porosity were ∼ 3.7 × 10−7 m2 and 0.047, respectively.
The buoyancy-driven convection data taken for three cubic lattice configurations with aspect ratios of 0.5, 1.0 and 2.0, using three different heating orientations—sideways heating (90 deg), bottom wall heating (0 deg) and top wall heating (180 deg)—revealed that sideways heating and bottom wall heating exhibit similar heat transfer performances, with both being superior to top wall heating. Additionally, lattice structures with low aspect ratios exhibited higher heat transfer performance under all heating orientations.
The steady-state forced convection data with cubic unit cells placed in a channel with top and bottom walls subjected to constant heat flux indicated that the presence of cubic lattice structures enhanced heat transfer by 12–18 times compared to a smooth channel, across the investigated range of Reynolds number. The THP varied between 1.5 and 2.5. Furthermore, the unit cell heat transfer coefficient in the periodic heat transfer regime correlated with the average flow velocity at the inlet, by .
In conclusion, these findings suggest that cubic lattice structures offer promising cost-effective and reliable solutions for cooling problems. While the structures in this study were additively manufactured, similar configurations can be achieved using conventional wire-based techniques or subtractive manufacturing, potentially with lower roughness length scales.
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.
Nomenclature
- =
plate surface area
- =
inertial coefficient
- =
channel hydraulic diameter
- f =
friction factor
- h =
convective heat transfer coefficient
- =
height of the vertical plates
- =
effective thermal conductivity
- =
air thermal conductivity
- K =
permeability
- L =
length of sample used in forced convection
- =
characteristic length scale
- =
Nusselt number
- p =
pressure
- =
total heat supplied
- =
Rayleigh number
- Re =
Reynolds number
- =
heater resistance
- =
ambient temperature
- =
film temperature
- =
wall temperature
- THP =
thermal hydraulic performance
- =
inlet velocity in forced convection experiments