## Abstract

Additive manufacturing is now a promising option to obtain porous customized structures at relatively low scales. The capability to design structures with tunable heat transfer performance compared to conventional porous materials, such as open-cell foams, is very interesting to the user. In this study, we investigated heat conduction in a drilled-hollow-sphere architected foam, inspired by Triply-Periodic Minimal Surfaces (TPMS) and foam structures, generated using perforated spherical hollow shells connected with cylindrical binders. Temperature fields and heat fluxes in the foam were predicted numerically, and the effective thermal conductivity of the foam was calculated for different sets of the binder angle, the shell thickness, and the perforation radius. The dependence of the foam porosity on the binder angle and perforation radius was also pointed out. Predictions were validated by comparing them with data available from the literature. Results showed that varying the characteristics of the investigated drilled-hollow-sphere architected foam, its predicted effective thermal conductivity can be adjusted by more than one order of magnitude larger or smaller than that of conventional foams, making architected foams promising enhancers of their heat transfer performance. Finally, new dimensionless correlations among the effective thermal conductivity and some significant morphological parameters of the foam were derived and presented for practical use.

## 1 Introduction

Many materials in nature, such as cork, wood, sponges, and even bones, have a cellular structure. In engineering, there is interest in such structures since they provide unique properties that have wide applicability due to their lightweight. Cellular solids can be divided into two categories: (1) foams, which have random cell architecture, and (2) lattices which are periodic assemblies of a unit cell [1]. Closed-cell foams generally have low thermal conductivity, which makes them excellent insulators [2], while open-cell foams have high thermal conductivity, which may be exploited in heat exchangers for heating, ventilation, air conditioning and refrigeration systems, heat sinks, electronic power dissipation fire barriers, and thermal energy storage because of their interconnected porosity. Cellular solids also have promising mechanical properties, such as high ultimate tensile and yield strength [3,4], that may be used for energy absorption in structural applications, sound absorption, electromagnetic wave shields, and porous scaffolds in tissue engineering.

Even though open-cell foams are currently reported to be promising for heat transfer enhancement, due to their high thermal conductivity, their microstructure is quite limited by the manufacturing routes, like casting method from a preform; thus, cellular microstructures with a high degree of customization would be preferable. Recent interest in the field is in architected materials, due to new developments in manufacturing techniques, namely additive manufacturing and laser cutting. The term *architected materials* was coined by Ashby and Bréchet [5] to link how architecture and topological optimization can be used to produce light and reliable structures. Architected materials are porous solids that have well-defined microstructures, that may be additively manufactured or 3D printed [6,7], and can be tailored to the desired application. One class of novel architected materials is Triply-Periodic Minimal Surfaces (TPMS) cellular material having low density and promising acoustic, electric, thermal, and mechanical properties [8,9]. TPMS are created from the local area minimization principles which generate structures with a local zero mean curvature [10–12]. Isotropic TPMS generally have cubic symmetry and can be constructed through periodic repetition of a unit cell [10]. Unlike lattices, they do not contain joints, lumps, or struts. Researchers have investigated how to employ the TPMS-based material to design and create biomaterials [6,13] since minimal surface geometries often appear in the biological tissue of living organisms [14].

### 1.1 Heat Transfer in TPMS Structures: State-of-the-Art.

A certain amount of research has been devoted to investigate conduction heat transfer in TPMS structures, in order to improve the thermal performance of heat transfer devices. Abueidda et al. [15,16] used a finite element analysis to study the effective mechanical and thermo-electrical characteristics of several TPMS-based foams, such as Primitive, I-graph and wrapped package-graph (IWP), Neovius, Gyroids, Fischer-Koch S, and crossed layer of parallel (CLP)-sheet foams. CLP-sheet foams have a tetragonal symmetry [13], while the other foams all have cubic symmetry [2]. The authors demonstrated that such foams have higher effective thermal conductivities and superior mechanical performance than the gyroid lattice, which depends on the interconnectivity of its struts. Li et al. [17] proposed TPMS structures like Gyroid and Schwarz-D surfaces to increase heat transfer in a supercritical carbon dioxide heat exchanger for a Brayton cycle, showing a 15–100% increase in thermal performance. Fluid flow and heat transfer in defined IWP, Primitive, Diamond, and Gyroid surfaces were analyzed by Cheng et al. [18]. The authors compared heat transfer coefficients and pressure drop in the investigated structures, concluding that IWP and Gyroid structures exhibited higher volumetric heat transfer coefficients, while the best ratio of the Colburn j-factor to the friction factor was found for the Primitive structures. TPMS foams have also been proposed to increase the conduction heat transfer of phase change materials. Qureshi et al. [19] compared Gyroid with IWP, Primitive, and Kelvin’s foams in a cell stack heated from below, reporting that the melting time in TPMS foams was 31–40% lower than in Kelvin’s foams when an isothermal heating condition was employed. The same authors [20] also proposed a finned metal foam-phase change material heat sink, where TPMS and Kelvin’s foam employed in Ref. [19] were compared. The authors concluded that TPMS increased the heat transfer performance since the melting time reduced and the overall heat transfer coefficient increased.

Mirabolghasemi et al. [6] computationally studied a sample of Representative Volume Elements of 2D architected cellular foams, which were compared to TPMS-based foams. A wide range of topological parameters was investigated, and the manufacturability of the structures through 3D printing was demonstrated. The authors provided theoretical bases for the foam morphology and the prediction of its effective thermal conductivity (ETC). Thermal conduction in TPMS based on Schoen’s F-RD surface and in tomography-based 40 Pores Per Inch (PPI) foams was compared by Wang et al. [21]. The authors found a 103% increased thermal performance in the TPMS under investigation.

### 1.2 Aim and Novelty of the Present Work.

The literature survey shows that TPMS structures are very promising for heat transfer enhancement. However, the thermal characteristics of the different kinds of TPMS and other cellular materials have not been fully investigated yet as there are very few studies of heat transfer, which deserves to be further investigated. Furthermore, starting from well-established concepts about TPMS, one can generate very similar cellular materials with characteristics enhancing their thermal performance. It is also worth remarking that, to the authors’ knowledge, no correlations for TPMS’ applications have been proposed in the open literature thus far.

Conduction heat transfer in a three-dimensional (3D) drilled-hollow-sphere architected foam was analyzed in this paper. The generated foam was inspired by hollow-sphere foam models reported by Jiang et al. [22], derived from well-established TPMS structures, aiming at overcoming the limiting low level of microstructure customization of casted foams. Polymer foams are printed using a layer-by-layer manufacturing process, whose high design flexibility allows for the tailoring of their mechanical performance. We detailed the drilled-hollow-sphere architected foam morphology and studied the heat conduction characteristics of Jiang et al.’s [22] hollow-sphere foam models. Namely, we numerically simulated conduction heat transfer and estimated the effective thermal conductivity of the generated drilled-hollow-sphere architected foam, in order to study the effects of the hollow-sphere foam’s structural parameters. We analyzed the nature of heat conduction in the foam and identified where the largest temperature gradients occur. The effects of the foam’s structure on its effective thermal conductivity were also pointed out. Finally, we derived two sets of parametrized dimensionless correlations, where two different independent structural parameters were varied, useful for engineering applications.

## 2 Foam Generation

*a*of its sides; thus, we have

*R*is the outer radius of the spherical shell and

*l*is the length of the binder, which is the distance between the two spherical shells connected by the binder along the cube diagonal. As described in Fig. 1(c), there is one shell of outer diameter 2

*R*along any diagonal line across a cube bounding the foam, which connects two binders, each of length

*l*and attached to a quarter of a shell.

One-eighth of the unit cell size of the custom drilled-hollow-sphere architected foam and its relevant geometrical parameters, constructed along the diagonal of an *a/*2 cube, are shown in Fig. 1(b). A unit cell of the foam consists of a spherical shell with eight binders attached to it. The shell thickness is *t = R–R _{i}*, where

*R*is the inner radius of the shell. The circular perforation holes in the spherical shell have a radius

_{i}*r.*

*b*, and the length,

*l*, of the binder, which is similar to a catenoid surface, the binder angle,

*θ*, as well as the outer radius of the spherical shell,

*R*, are correlated as follows:

*h*, can be expressed as

The foam was generated using Grasshopper, a visual programming language that runs with the computer-aided design (CAD) software Rhinoceros 3D [23]. The generation of the binder and the spherical shell is sketched in Figs. 2 and 3. Assuming the center of the spherical shell as the origin of the axes (Fig. 2(a)), the binder was generated by creating a circle of radius *b* with its center located at the point (*x* = *l*/2 + *R*, *y* = *h*/2 + *b*) on the *xy* plane. The circle was intersected by a *R* + *b* long line starting from the origin at an angle *θ* from the *x*-axis. The same line, drawn at an angle *θ*, was reflected by 180 deg about the line drawn in the *y* direction through the point (*x* = *l*/2 + *R*, *y* = *h*/2 + *b*) which is the other point of intersection, as shown in Fig. 2(a). The intersected area revolved around the *x*-axis to generate the binder element shown in Fig. 2(b); the surface was converted into a solid. Then, a cube of side length *a/*2 was generated, with a vertex placed at the origin, and rotated the center of the binder surface along the diagonal of the cube (Fig. 3(a)).

To create the spherical shell, we generated two concentric spheres with a common center. The other set of two concentric spheres has a center located at the point directly across the cube diagonal. The difference between the radii of the spheres is the shell thickness, *t*. After subtracting the spheres from the cube, we created the perforation holes using three cylinders of radius *r*, whose axes are mutually orthogonal, and intersect the shell on each side of the binder, as shown in Fig. 3(b). The endpoints of the shell are cut out to align with the axis of each of the cylinders to ensure that the perforation holes have radius *r*.

## 3 Mathematical Model

### 3.1 Governing Equations and Boundary Conditions.

In this paper, we use a metal-based drilled-hollow-sphere architected foam. Aluminum T-6210 and air were assumed to be isotropic, with thermal conductivities at room temperature *k _{s}* = 218 W/(m K) and

*k*= 0.027 W/(m K), respectively. The ratio of the solid to fluid phase thermal conductivity is of order 10

_{f}^{3}, meaning that the solid phase of the foam contributes the most to the effective thermal conductivity,

*k*, which represents the conductivity of the volume-averaged equivalent single-phase porous medium, defined in Eq. (5). Therefore, the “under vacuum condition” was assumed in the following analysis. It is worth noticing that the same assumption, which reduces the size of the computational domain, had been made in many previous works in the evaluation of the effective thermal conductivity of the foam, provided

_{eff}*k*be large enough (e.g.,

_{s}/k_{f}*k*>> 10

_{s}/k_{f}^{2}). Boomsma et al. [24] observed that, because of their low thermal conductivities, liquids like water or ethylene glycol poorly contribute to the effective thermal conductivity of the foam and concluded that it can be dramatically improved only by increasing the thermal conductivity of the solid phase. Mendes et al. [25] also showed that for

*k*>> 10

_{s}/k_{f}^{2},

*k*can be accurately predicted by accounting for the solid phase of the porous media only, even for high porosities. Similarly, Iasiello et al. [26] showed that for large solid-to-fluid thermal conductivity ratios, such as metal with air, the solid phase contribution to the effective thermal conductivity of the foam is predominant.

_{eff}The unit cell of the drilled-hollow-sphere architected foam was assumed as the Representative Elementary Volume (REV) of the foam, that is the smallest volume in which the sizes of morphological characteristics are the same as those of the entire foam; each variable is averaged in the REV. The cubic unit cell, bounded by a fictitious cubic box, with *a* the side length and *A* the heat transfer area, is sketched in Fig. 4(a). The cross-sectional area of the solid phase, *A _{s}*, is highlighted in Fig. 4(b); since the ratio

*A*varies throughout the foam herein, we assumed

_{s}/A*A*to be proportional to

_{s}*A*(1–

*ɛ*),

*A*∝

_{s}*A*(1

*–ɛ*), with the porosity,

*ɛ*=

*V*/

_{f}*a*

^{3}; the ratio of the fluid phase volume to the total volume of the foam.

In order to predict the effective thermal conductivity of the foam along the *x* direction the well-known 3D steady-state heat conduction equation with no heat generation was applied to the solid phase domain depicted in Fig. 1(c) and shaded in Fig. 4(b). The equation was solved by assuming the inlet (*x* = 0) and outlet (*x* = *a*) cross sections of the cell (Fig. 4(b)) to be isothermal at *T _{h}* and

*T*temperatures, respectively; all other internal and external boundaries were adiabatic as well as convection and radiation effects were assumed to be negligible. We prescribed

_{c}*T*= 313 K and

_{h}*T*= 303 K. Due to the small temperature difference, the thermal conductivities of the solid and fluid phases of the foam were assumed to be independent of the temperature.

_{c}*x*= 0)

*T*–

_{h}*T*, flows through the same cubic material, having an

_{c}*a*side length and an

*A*cross-section area, an effective thermal conductivity of the foam,

*k*, defined by the following correlation

_{eff}### 3.2 Numerical Modeling.

The commercial finite-element software COMSOL Multiphysics was used to investigate various ranges of the binder angle, thickness, and perforation radius. The foams were exported from Rhinoceros 3D as .iges files and imported into COMSOL. The grid convergence for the calculated effective thermal conductivity of the foam was checked for five different mesh settings and for 5 deg, 35 deg, 70 deg, and 89 deg binder angles, to ensure we were choosing an appropriate mesh for the investigated ranges. PARDISO solver with quadratic Lagrange discretization was used, with a relative tolerance set to 10^{−5}. The grid independence check for the calculation of the effective thermal conductivity of the foam is reported in Fig. 5.

For binder angles in the 5–70 deg range, 197,000 free tetrahedral elements were chosen and 150,00 free tetrahedral elements were chosen for binder angles greater than 70 deg. For each simulation, energy conservation was checked and differences less than 10^{−5} in the heat rate were observed.

## 4 Results and Discussion

Simulations were carried out for different ranges of foam characteristics. A length of the cell side *a* = 25.0 mm was assumed. The effective thermal conductivity was verified to scale linearly with the size of the domain, as indicated by Eq. (4). Consequently, according to the correlation $R=63a/25$ proposed by Jiang et al. [22], an outer radius of the spherical shell *R* = 10.39 mm was selected. Then, using $R=63a/25$ in Eq. (1), we calculate length of the binder $l=3a/50=0.87mm$. When the binder angle increases past 40 deg, the diameter of the maximum cross section of the binder becomes greater than the bounding box side, as it is represented in Fig. 6(a).

We note that the binder length, *l*, is still a fixed quantity, and the outer radius of the sphere, which is subtracted from the cube, was still assumed to have the fixed value $R=63a/25$. However, since the box cannot expand, due to the constraint of a fixed cell, size, *a*, the shell intersects the binder and the box, causing the thickness of the shells, *t*, to vary throughout the foam, as it is sketched in Fig. 6(b). As a result, we record the minimum thickness of the shell, which in this case is the difference between its minimum outer radius, *R*_{min}, which occurs along the planes where any two one-eighth-size foams are connected to form a full unit cell (Fig. 6(b)), and its inner radius. Thus, it is the solid intersection between the cube, the concentric spheres, and the binder, which results in a varying shell thickness. In the following, we assume *R* = *R*_{min} for *θ* ≤ 40 deg and a variable outer radius for *θ* > 40 deg, in order to quantify the spherical shell thickness. However, *b* and *h* are still determined by Eqs. (2) and (3), with $R=63a/25$.

The geometrical parameters of the investigated foam are reported in Tables 1 and 2. Table 1 refers to a fixed radius of the perforation hole *r* = 3.12 mm and a *θ* = 5–89 deg range of the binder angle; Table 2 refers to a fixed inner radius of the shell, *R _{i}* = 9.35 mm, and a

*θ*= 5–89 deg range of the binder angle. The thickness of the spherical shell,

*t*, was investigated in the 0.28–5.98 mm range for

*θ*= 5–40 deg and in the 0.29–8.50 mm range for the

*θ*= 45–89 deg. The lower limit of the thickness was set up based on the minimum feature sizes of current 3D printing technology, while the upper limit was set up as one-half of the outer radius of the shell, so that the integrity of the design could be kept (the shell remained hollow).

θ (deg) | R (mm) | R (mm)_{i} | t = R –R (mm)_{i} |
---|---|---|---|

5–40 | $63a/25=10.39$ | 4.41–10.10 | 0.28–5.98 |

45–89 | 10.4–12.88 | 4.41–10.10 | 0.29–8.47 |

θ (deg) | R (mm) | R (mm)_{i} | t = R –R (mm)_{i} |
---|---|---|---|

5–40 | $63a/25=10.39$ | 4.41–10.10 | 0.28–5.98 |

45–89 | 10.4–12.88 | 4.41–10.10 | 0.29–8.47 |

θ (deg) | R (mm) | R (mm) | t = R–R (mm)_{i} |
---|---|---|---|

5–40 | 0.52–5.20 | $63a/25=10.39$ | 1.04 |

45–89 | 0.52–5.20 | 10.39–13.54 | 1.04–4.19 |

θ (deg) | R (mm) | R (mm) | t = R–R (mm)_{i} |
---|---|---|---|

5–40 | 0.52–5.20 | $63a/25=10.39$ | 1.04 |

45–89 | 0.52–5.20 | 10.39–13.54 | 1.04–4.19 |

It is worth noticing that for binder angles in the 45–89 deg range, the binder submerges within the spherical shell, which results in varying shell thickness. In this region, we kept track of the minimum thickness of the foam, which was measured in Rhino 3D or COMSOL.

### 4.1 Temperature Fields and Gradients.

Surface temperature fields in the foam, for an inner radius of the shell *R _{i}* = 9.35 mm, outer radii of the shell

*R*= 10.39 mm for

*θ*≤ 40 deg,

*R*= 10.85 mm for

*θ*= 55 deg, and

*R*= 12.15 mm for

*θ*= 70 deg, a radius of the perforation hole

*r*= 3.12 mm, and different binder angles, are shown in Fig. 7. The figure shows that increasing the binder angle, thus increasing the cross section of the binder and decreasing its conductive thermal resistance, makes the distribution of temperatures in the binder and in the shell more uniform. The foam with the smallest binder angle (Fig. 7(a)) exhibits the highest temperatures at the binder, because of its large conduction resistance. At increasing binder angles, the heat flows more uniformly throughout the foam, implying that increasing the binder angle enhances the heat transfer efficiency.

*x*direction at different binder angles are reported in Fig. 8, for a 9.35-mm inner radius of the spherical shell, and 3.12-mm radius of the perforation. The dimensionless temperature gradient component in the

*x*direction, $(\u2202T/\u2202x)*$, is given by

The knowledge of temperature gradients is helpful in finding the regions of the foam where the thermal resistance is the highest. As far as the temperature gradients in Figs. 8(a)–8(d) are concerned, darker gray and lighter gray on the scale denote their maximum and minimum values, respectively; as to the ratio of the temperature gradient in the *x* direction to the overall temperature gradient magnitude in Figs. 8(i)–8(l), lighter and darker gray on the scale denote the highest heat flux in the *x* direction (parallel flow) and no heat flux in the same direction (series flow), respectively.

Figure 8(a) shows that the maximum temperature gradient is attained at the binders, which strongly contribute to the thermal resistance, because of their small cross-sectional area. In fact, the thermal resistance of a binder is proportional to *l/h*^{2} (Fig. 1(b)), which is inversely proportional to the binder angle. At the same time, directional effects also arise, as will be discussed in detail later. At increasing binder angles, the maximum temperature gradient gradually decreases and moves away from the binder into the shell, as exhibited in Figs. 8(b)–8(d). Since the unit cell of the investigated foam consists of eight binders and a perforated spherical shell constructed along the diagonal of a cubic box, whereas heat is applied in the *x* direction, directional effects in the way heat spreads throughout the foam are non-negligible.

Figures 8(e)–8(h) display the heat flux lines throughout the cell. One can distinguish four regions with different line patterns. The first region, for *θ* = 5 deg (Fig. 8(e)), shows that the heat flux lines “tangle,” due to the small binder diameter. Therefore, heat no longer flows parallel in the *x* direction, along which it is applied at the boundary; on the contrary, the components of heat flow in the *x* direction are small, particularly in the upstream regions adjacent to the entrance section of the shell. Figures 8(f)–8(h) show that, as the binder diameter increases by increasing the binder angle, a larger fraction of heat flows in the *x* direction. In other words, in the entrance region of the cell, heat is unobstructed and flows parallel in the *x* direction; then, it follows the curvature of the shell until it reaches the binder.

Since the curvature changes from positive along the spherical shell to negative along the catenoid-like binder, heat flowing in the *x* direction varies at the binder, as denoted by the dark blue regions in Figs. 8(i)–8(l), where the temperature gradient component is oriented along the *y* and *z* directions instead. As the diameter of the binder connecting the shells increases, the change in the curvature is less marked and a greater fraction of the heat flows parallel. Even small increases in the binder angle increase parallel heat flow, thereby increasing the effective thermal conductivity of the foam. This explains the rapid change to a more uniform temperature field for small binder angles in Fig. 7(b), despite the negligible change in the porosity. In the *θ →* 90 deg limiting case, the binder approaches the shape of a cylinder, and the effect of curvature is only apparent along the edges of the shell since a cylinder has zero curvature.

The shift of the maximum temperature gradients along the *x* direction, noticed in Figs. 8(a)–8(d), is detailed in Fig. 9, where the dimensionless maximum temperature gradient component in the *x* direction d*T**/d*x**, with *T* =* (*T*–*T _{c}*)/(

*T*–

_{h}*T*) and

_{c}*x**

*=*

*x/a*, as a function of the dimensionless coordinate

*x**is presented, for an inner radius of the shell

*R*= 9.35 mm, a radius of the perforation hole

_{i}*r*= 3.12 mm, and

*R*= 10.39 mm, for

*θ*≤ 40 deg,

*R*= 10.85 mm for

*θ*= 55 deg,

*R*= 12.15 mm for

*θ*= 70 deg. It is worth reminding that the coordinates are located as reported in Fig. 4(b) and that the shift depends on the diameter of the binder compared to the thickness of the shell. For the 5 deg binder angle, we have the minimum diameter of the binder,

*h*= 0.94 mm, while the shell thickness is

*t*= 1.04 mm. Since the binder is the thinnest region in the foam, the maximum thermal gradient occurs at the binders. As the binder angle increases, there roles between the shell thickness and the diameter of the binder are reversed. Once, the binder submerges within the shell, for

*θ*> 45 deg, the thickness throughout the shell varies, so that the minimum cross-sectional area in the foam occurs at the center plane of the spherical shell, at

*x*/

*a*= 0.5. As the binder angle is increased to 70 deg and beyond, the largest value of the minimum diameter of the binder, 2

*R*sin

*θ*(Fig. 1(b)), approaches 2

*R*, submerging almost completely within the shell, except for its corner, which remains the thinnest region in the foam; this is the region where the maximum temperature gradient occurs.

### 4.2 Effective Thermal Conductivity.

The predicted dimensionless effective thermal conductivity and porosity of the foam as a function of the binder angle, with an inner radius of the shell *R _{i}* = 9.35 mm, a thickness of the shell

*t*= 1.04 mm for

*θ ≤*40 deg and

*t*= 1.05 mm–4.19 mm for

*θ*= 45–89 deg and perforation radius

*r*= 0.52–5.20 mm, are reported in Fig. 10. Figure 10(a) exhibits a non-linear decrease in the porosity at increasing binder angles (Eq. (3)). The figure also shows slightly decreasing porosity up to

*θ*= 20 deg and a marked decrease in the

*θ*= 20–60 deg range. The porosity is nearly independent of the binder angle for

*θ*> 70 deg. In the 40–70 deg region, the rate of decrease in the porosity is larger, due to the increasing shell thickness in the range

*t*= 1.05 mm–2.80 mm at increasing binder angles. Notice that in the limit of

*θ*tending to 90 deg, according to Eq. (3), the binder approaches a cylinder of diameter 2

*R*, causing the spheres to fully overlap (Fig. 3(a)). As to the dependence of the porosity on the perforation radius, one can notice that at increasing binder angles it increases at a faster rate.

Figure 10(b) exhibits a monotonic decrease in the dimensionless effective thermal conductivity of the foam at increasing perforation radius. For all perforation radii, one can remark that for small binder angles, up to *θ* = 7.5 deg, the effective thermal conductivity is poorly affected by the perforation radius, whereas the dependence is more marked at increasing binder angles. For *θ* > 7.5 deg, at each binder angle, *k _{eff}/k_{s}* approximately decreases according to a factor of two with increasing perforation radii. For

*θ*≤ 10 deg, the rate of increase in the

*k*is large compared to that for

_{eff}/k_{s}*θ*= 10–30 deg. In the range

*θ*= 30–70 deg, the rate of increase in

*k*is further enhanced, until it attains a plateau for

_{eff}/k_{s}*θ*> 70 deg, similar to what is exhibited by the dependence of porosity on binder angle in Fig. 10(a).

To summarize, Figs. 10(a) and 10(b) highlight the rates of variation in the porosity and in the effective thermal conductivity with the binder angle, respectively, from which we can identify four regions (I–IV). Region, I, for *θ* ≤ 10 deg, exhibits mild variations in the porosity and large increases in the effective thermal conductivity of the foam, which can be attributed to directional effects remarked in Fig. 4(b). Region II, for *θ* = 10–30 deg, has a faster rate of decrease in the porosity than in region I, due to the increasing cross section of the binder. On the contrary, the increase in the effective thermal conductivity is lower, because directional effects are less dominant than in region I. Region III, for *θ* = 30–70 deg, is where the shell thickness varies, since the binder submerges within the shell. Again, one can remark that the rate of increase in *k _{eff}*/

*k*is higher than that in region II, because the foam thickness increases as the binder angle increases. Region IV, for

_{s}*θ*> 70 deg, is where the dimensionless effective thermal conductivity attains a plateau, similar to the plateau in the porosity.

The effects of the shell thickness and the binder angle on the effective thermal conductivity of the foam are illustrated in Fig. 11, where the dimensionless effective thermal conductivity as a function of the ratio of the thickness of the shell to its outer radius, *t*/*R*, for values of the inner radius of the shell in the *R _{i}* = 4.41–10.10 mm range, a perforation radius

*r*= 3.12 mm and binder angle values in the

*θ =*5–89 deg range, is reported. The figure shows the logarithmic dependence of

*k*on

_{eff}/k_{s}*t/R*. We can similarly identify four different regions here: (I)

*k*monotonically increases with

_{eff}/k_{s}*θ*for small binder angles, up to 10 deg; (II) in the 10–45 deg range,

*k*increases at a faster rate, due to the combined effects of increasing binder diameter and the shell thickness; (III) for binder angles past 45 deg, the binder submerges within the spherical shell and causes the thickness throughout the shell cross section to vary (Sec. 2), which further impacts the effective thermal conductivity; (IV) for

_{eff}/k_{s}*θ*= 70–89 deg porosities are the same as those in Fig. 10(a); therefore, both binder angles provide similar values of

*k*, as reported in Fig. 10(b).

_{eff}/k_{s}Figure 11 also shows that up to *θ* = 60 deg the effective thermal conductivity at equal *t/R* is higher for increasing binder angles, with reduced differences between the curves for higher constrained values of *t*/*R*, due to the varying cross section of the shell. Recall that *t*/*R* is referred to the minimum value of the shell thickness, which is easier to compute than its averaged value over the shell. For *t*/*R* values larger than 0.5, the differences discussed above become negligible since the shell thickness variation caused by the binder submersion is dominated by the increase in the minimum shell thickness. One can finally remark that at binder angles 60 deg and above, curves intersect each other due to the interplay between the shell thickness and the submerging binder. The effects of the decrease in the effective thermal conductivity due to small shell thicknesses are overcome by the extra material gained from binder submersion. This offset is more pronounced at the largest binder angles because of the greater thickness of the foam due to the binder submersion.

### 4.3 Comparison and Validation with Available Data.

As to the authors’ knowledge, very few experimental data for conduction heat transfer in drilled-hollow-sphere architected foams investigated in the present paper are available in the open literature. We validated the numerical results in this paper by comparing to experimental data and numerical predictions of similar TPMS structures at lower porosities, as well as experimental data for real casted foams similar to the investigated drilled-hollow-sphere architected foams at higher porosities.

The predicted dimensionless effective thermal conductivity of the foam as a function of the porosity, for a thickness of the shell *t* = 0.28 mm–5.98 mm in the *θ* = 5–40 deg range and *t* = 0.29 mm–8.47 mm in the *θ* = 45–89 deg range, and for different binder angles, together with data taken from the literature, is presented in Fig. 12.

Abueidda et al. [16] predicted the effective thermal conductivity of architected cellular foams based on TPMS. Their predicted values of the effective thermal conductivity lie in the light gray shaded region in the figure, which indicates the ETC range typical of TPMS foams. The figure conveys that the dimensionless ETC as a function of the porosity both in the investigated drilled—hollow—sphere architected foam and in the TPMS foams have similar profiles; for binder angles higher than 15 deg, despite differences in the mathematical basis of the construction of the foams. It is also worth remarking that, depending on the choice of the foam geometric parameters, the investigated drilled-hollow-sphere architected foams can be adjusted to have either high or low effective thermal conductivities; higher ETCs are obtained with porosities up to 0.90 and binder angles higher than 15 deg.

The values of the apparent thermal conductivity of Hastelloy-X and Ti6Al4V gyroid, diamond, and Schwarz-P architected structures evaluated experimentally by Catchpole-Smith et al. [27] are reported in Fig. 12. One can remark that for lower porosities there is good agreement with the dimensionless effective thermal conductivity predicted in this paper for higher binder angles. In the 0.70–0.80 porosity range, the numerical predictions underpredict experimental data. The underestimation can be attributed to the high temperature, up to 170 °C, attained in experiments by Catchpole-Smith et al. [27], which is higher than the 100 °C average temperature used in their experiments and used in this study as the reference temperature to compute *k _{s}* for comparison. In fact, at high temperatures, radiation, and natural convection could affect the ETC, as it was reported in Refs. [28,29]. Zhao et al. [29] found up to 50% contribution of natural convection to the effective thermal conductivity in high porosity cell foams.

The dimensionless effective thermal conductivity predicted by the empirical model for polyhedral foams proposed by Lemlich [30], together with the distribution of the dimensionless ETC under the parallel heat flow assumption, is also reported in Fig. 12.

Finally, experimental data for commonly used high porosity metal foams from various sources [31–35] are reported in Fig. 12, in order to test the validity of the approach and the mathematical model employed in the present work at larger porosities. Further, numerical predictions for casted foams obtained from Computer Tomography (CT) scans of real samples [36] are also reported. There is good agreement between numerical and experimental data for real casted foam samples obtained via tomographic scans. The realistic representation of the foam geometry is presented in Refs. [31–35] validates the approach used here to predict effective thermal conductivity.

As to the comparison between the ETC values of investigated drilled-hollow-sphere architected foams predicted in the present work and those of available casted foams, the figure shows that the thermal performance of the investigated drilled-hollow-sphere architected foam with binder angles larger than 10 deg is better than that of conventional high porosity metal foams [31–35] and foams built up according to the polyhedral model by Lemlich [30] and not so worse than that of parallel conduction heat flow model. We can notice that the dimensionless ETC of the drilled-hollow-sphere architected foam with an 89 deg binder angle exhibits a 15% lower and a 150% higher deviation from the ETC of both casted and parallel flow foams. On the other hand, the differences discussed above are 35% smaller and 95% larger for a 0.70 porosity and a 60 deg binder angle as well as 45% smaller and 110% larger for a 0.80 porosity and 30 deg binder angle.

It is worth remarking that the investigated foams with binder angles lower than 10 deg can be usefully employed as thermal insulators, also for very high-temperature applications, such as fire barriers [37,38]. One can remark that the drilled-hollow-sphere architected foam with a 5 deg binder angle exhibited the lowest dimensionless effective thermal conductivity, which is 90% and 70% lower than that for the parallel heat flow model and casted foams, respectively. Finally, the differences in the dimensionless ETC were lower than about 85% and 55% for the parallel flow model and the casted foams, respectively, with a 0.75 porosity as well as they were lower than nearly 80% and 40%, respectively, for a 0.85 porosity.

## 5 Correlations

It is worth reminding that in Sec. 4.2 we reported that for binder angles *θ**≤* 10 deg the binder drives heat transfer because it is the thinnest region in the foam and the effective thermal conductivity of the foam is sensitive to changes in the binder angle and not porosity. As the binder angle is increased past 40 deg, the outer radius of the shell, *R*, and therefore, the shell thickness, *t*, varies throughout the foam because the binder merges with the spherical shell. As a result, the binder angle plays a minor role because it is no longer an independent characteristic of the foam.

*k*and

_{eff, FIT}*k*are the fitted and the simulated effective thermal conductivity, respectively, and

_{eff, SIM}*N*is the number of data points.

*θ =*5–10 deg a logarithmic dependence of

*k*on

_{eff}/k_{s}*t/R*and on monotonically increasing binder angles was found, here we assumed a generic power law fit, where the effects of the binder angle and the shell thickness on the effective thermal conductivity are not coupled. In the

*θ =*10–45 deg range, the dimensionless effective thermal conductivity increases at a faster rate due to the combined effects of increasing the binder diameter and shell thickness. The coupled effects of the binder angle and the thickness of the shell on the effective thermal conductivity of the foam are captured by the following parameter:

For binder angles past 45 deg, the binder submerges within the spherical shell and makes the thickness of the shell vary in the 0.29–8.50 mm range. For *θ* ≥ 70 deg the porosity of the foam plateaus; thus, for some set *t/R* ratios, within this binder angle range the porosity of the foam and, therefore, *k _{eff}/k_{s}* will be the same as those of foams with lower binder angles. In the

*θ =*45–70 deg range, the minimum thickness of the shell varies in the 0.29–8.50 mm range, and as shown in Fig. 11, the shell, rather than the binder, is driving heat transfer; thus, the effective thermal conductivity depends less on the binder angle than on the shell thickness. Lastly, for

*θ =*70–89 deg, where the minimum thickness of the shell varies in the 0.34–8.50 mm range and the perforation radius is

*r*= 3.12 mm, the porosity and the effective thermal conductivity plateaus with the binder angle.

*θ*ranges, the following correlations are proposed:

*R*= 4.41–10.10 mm,

_{i}*r*= 3.12 mm, and

*θ =*5–89 deg, referring to data presented in Figs. 11 and 12 and to the foam characteristics detailed in Table 1.

*N*, RMSE, RMSPE, MAPE, r

^{2}, and

*χ*

^{2}for Eq. (11) are reported in Table 3.

θ (deg) | N | RMSE | RMSPE (%) | MAPE (%) | r^{2} | χ^{2} |
---|---|---|---|---|---|---|

5–10 | 27 | 0.006 | 2.51 | 9.93 | 0.976 | 0.016 |

10–45 | 54 | 0.018 | 4.15 | 9.28 | 0.971 | 0.091 |

45–70 | 48 | 0.042 | 6.73 | 8.67 | 0.913 | 0.190 |

70–89 | 36 | 0.027 | 4.75 | 6.31 | 0.999 | 0.055 |

θ (deg) | N | RMSE | RMSPE (%) | MAPE (%) | r^{2} | χ^{2} |
---|---|---|---|---|---|---|

5–10 | 27 | 0.006 | 2.51 | 9.93 | 0.976 | 0.016 |

10–45 | 54 | 0.018 | 4.15 | 9.28 | 0.971 | 0.091 |

45–70 | 48 | 0.042 | 6.73 | 8.67 | 0.913 | 0.190 |

70–89 | 36 | 0.027 | 4.75 | 6.31 | 0.999 | 0.055 |

The second and third correlations show that the effective thermal conductivity is larger in the $45deg\u2264\theta <70deg$ range than for $10deg\u2264\theta <45deg$. We can also remark that the porosity does not affect explicitly the effective thermal conductivity except for $70deg\u2264\theta \u226489deg$, where *k _{eff}/k_{s}* is correlated only to

*ɛ*, that does not vary for a fixed inner radius of the shell (Fig. 10).

The fitted dimensionless effective thermal conductivity of the foam versus its simulated value, for *R _{i}* = 4.41–10.10 mm,

*r*= 3.12 mm, and

*θ =*5–89 deg, is presented in Fig. 13. The figure shows that the proposed correlations capture below 20% deviations most of the scatter in each of the data sets. Table 3 exhibits RMSE, RMSPE, and MAPE values below 10% for each range of the binder angle and

*χ*

^{2}denotes that each correlation captures the data with about 0.99 significance. This indicates that 99% of the variation in the data sets for each region can be accounted for with the correlations in Eq. (11).

*θ*range:

*r =*0.52–5.20 mm range,

*R =*10.39–13.54 mm and

*θ*= 5–89 deg, referring to data presented in Fig. 12(b) and to the foam characteristics detailed in Table 2. The

*N*, RMSE, RMSPE, MAPE, r

^{2}, and

*χ*

^{2}for Eq. (12) are reported in Table 4. We remark that in the

*θ*= 5 –45 deg range

*k*is correlated to

_{eff}/k_{s}*R*/

*r*and

*θ*by the same equation, since we found that for binder angles up to 40 deg, with

*R*= 10.39 mm and

*t*= 1.04 mm, the outer radius of the foam does not vary. For larger binder angles, up to 70 deg,

*R*varies between 10.39 mm and 13.55 mm, with a variation of the shell thickness between 1.04 mm and 4.20 mm. One can also notice that the binder angle plays a major role in the first correlation, because of the smaller exponent of the sin

*θ*term, which is smaller than the unity. The second correlation poins out that, as it was already remarked, above

*θ*= 40 deg, the thickness of the sphere increases, therefore increasing the conduction heat rate through the foam.

θ (deg) | N | RMSE | RMSPE (%) | MAPE (%) | r^{2} | χ^{2} |
---|---|---|---|---|---|---|

5–45 | 42 | 0.002 | 2.38 | 7.46 | 0.977 | 0.023 |

45–70 | 18 | 0.006 | 3.93 | 6.57 | 0.953 | 0.028 |

70–89 | 18 | 0.030 | 3.60 | 5.21 | 0.992 | 0.003 |

θ (deg) | N | RMSE | RMSPE (%) | MAPE (%) | r^{2} | χ^{2} |
---|---|---|---|---|---|---|

5–45 | 42 | 0.002 | 2.38 | 7.46 | 0.977 | 0.023 |

45–70 | 18 | 0.006 | 3.93 | 6.57 | 0.953 | 0.028 |

70–89 | 18 | 0.030 | 3.60 | 5.21 | 0.992 | 0.003 |

Comparing the terms containing the perforation radius, we see that the ratio *R/r* increases as the perforation radius decreases and its overall contribution in the first and second correlations is comparable. We can reasonably think that the coefficient 0.42 in the second correlation is twice the 0.21 in the first correlation because of the lower porosities achieved with higher binder angles.

The third correlation, for binder angles greater than 70 deg, accounts for the plateau attained by the effective thermal conductivity, due to the plateau attained by the porosity too, reported in Fig. 10(a). The dimensionless effective thermal conductivity of the foam was found to be 78% that of the parallel model, which is the upper bound of the effective thermal conductivity of a porous material.

The ratio of the fitted effective thermal conductivity to the thermal conductivity of the solid phase versus its simulated value, for *r =* 0.52–5.20 mm; *R =* 10.39–13.54 mm; *R _{i}* = 9.35 mm;

*θ =*5–89 deg, is presented in Fig. 14. The figure shows that the proposed correlations capture below 10% deviations most of the scatter in each of the data sets. Table 4 exhibits RMSE, RMSPE, MAPE values below 8% for each region, and

*χ*

^{2}denotes that each correlation captures the data with an about 0.99 significance; that is, 99% of variation in the data sets for each region can be accounted for with the correlations in Eq. (12).

Boomsma et al.’s analytical model [24] for a tetrakaidecahedron foam with cylindrical ligaments and cubic nodes was corrected and extended by Dai et al. [39] to include also the effects of ligament orientation and the authors reported a relative 12.2% RMS deviation for the improved model. Bhattacharya et al’s empirical model [31] exhibited a 10.1% deviation [40] and r* ^{2}* = 0.97 in the porosity range 0.89–0.98. Singh and Kasana [40], using a resistor method, proposed an empirical correlation for Aluminum/Air and Aluminum/Water foams with a 6% maximum deviation with a 3% mean deviation. Yang et al’s. analytical model [41] accounted for the cross-section area of the ligaments and an RMS = 9.8–11.1% for the fit of experimental data of aluminum/air as well as aluminum/water foams was reported. Though the models cited above referred to foams different from the structures investigated in this paper, the basis for deriving the empirical correlations was similar to the one used in this paper, thus making the comparison among correlations meaningful. We can conclude that the errors in the correlations derived here have the same order of magnitude in the models cited above.

## 6 Conclusions

A drilled-hollow-sphere architected foam was generated starting from a unit made up of perforated spherical shells connected with a cylindrical binder, then translated, and rotated eight times along the diagonal direction of a cube to obtain a full REV. The structure was inspired by TPMS and conventional casted foams. Temperature fields due to heat conduction in the foam were simulated numerically, and the effective thermal conductivity of the foam was calculated for different sets of the binder angle, the shell thickness, and the perforation radius of the shell. The dependence of the foam porosity on the binder angle and perforation radius were also pointed out. Numerical predictions were compared and validated with data taken from the open literature. Finally, dimensionless correlations were derived, which take into account the dependence of the effective thermal conductivity of the drilled-hollow-sphere architected foam on its geometrical characteristics.

The main results are summarized as follows:

Increasing the binder angle, which increases the binder diameter and its cross section, decreases the conductive thermal resistance of the drilled-hollow-sphere architected foam and makes the temperature distribution in both the binder and the shell more uniform; the temperature gradients in the investigated foam also decrease and the fraction of parallel heat flow increases.

The dimensionless effective thermal conductivity of the foam is nearly independent of the ratio of the shell thickness to the outer radius of the shell for angles smaller than 7.5 deg since the binder was found to dominate the conduction heat transfer. For larger binder angles, the dimensionless effective thermal conductivity was found to increase.

For binder angles smaller than 20 deg, the porosity, in the 0.55–0.90 range typical of conventional foams, is poorly affected by the angle; on the contrary, it is found to markedly decrease at increasing angles.

The comparison of the predicted dimensionless effective thermal conductivity of the investigated drilled-hollow-sphere architected foams with conventional high porosity open-cell foams [24] and parallel heat flow model exhibits higher dimensionless effective thermal conductivities in drilled-hollow-sphere architected foams with binder angles greater than 10 deg than those typically exhibited by conventional foams, with differences of 95% and 110% for 0.70 and 0.80 porosity, respectively. On the contrary, lower effective thermal conductivities of the investigated foam (down to 70%) than in conventional foams were found with differences around 55% and 40% for 0.75 and 0.85 porosities, respectively.

The tunable effective thermal conductivity of the investigated drilled-hollow-sphere architected foam can be useful both in applications where one needs to enhance heat transfer rates and when, on the contrary, lower effective thermal conductivities are required in thermal insulation applications.

Finally, new dimensionless correlations for the effective thermal conductivity as functions the binder angle, the shell thickness, and perforation radius, were derived.

## Acknowledgment

This work was carried out under a Learning Agreement between the University of Connecticut (USA) and the Università degli Studi di Napoli Federico II (Italy) and was supported by the Italian Government MIUR Grant No. PRIN-2017F7KZWS. The authors acknowledge Dr. Yanyu Chen and Huan Jiang for the discussion on hollow-sphere foam models, and their assistance in generating the foam structures.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*a*=side length of the cell, m

*b*=radius of curvature, m

*h*=minimum diameter of the binder, m

*k*=thermal conductivity, W/m K

*l*=length of the binder, m

*r*=radius of the perforated shell, m

*t*=thickness of the shell, m

*A*=heat transfer area, m

^{2}*C*=coefficient in Eq. (10)

*N*=number of data points

*R*=outer radius of the shell, m

*T*=temperature, K

*V*=volume, m

^{3}- $Q\u02d9$ =
heat transfer rate, W

*R*=_{i}inner radius of the shell, m

*x, y, z*=Cartesian coordinates, m

### Greek Symbols

*ɛ*=porosity

*χ*=^{2}statistics chi-squared test

### Subscripts

- c =
cold

*eff*=effective

*f*=fluid

*FIT*=fitted

*h*=hot

*s*=solid

*SIM*=simulated

### Superscript

- * =
dimensionless

### Abbreviations

- MAPE =
mean absolute percent error

- RMSE =
root mean square error

- RMSPE =
root mean square percent error