Numerical analysis of the binder angle effect on convective heat transfer and pressure drop in drilled-hollow sphere architected foams

Due to their capability of generating customized microstructures, additive manufactured cellular materials are promising to being employed in heat transfer devices. To this aim, among printable cellular materials Drilled-Hollow Spheres Architected (DHSA) foams are investigated. However, at the present status, limited data on pressure drop and heat transfer in DHSAs are available. Starting from hollow spheres, a metal DHSA foam is generated with CAD software in this study. Forced air convective heat transfer in the foam is investigated numerically, under the assumptions of air incompressible laminar flow and uniform wall heat flux from the solid to the fluid phase. Mass, momentum and energy equations in the fluid region are written and solved numerically, for various values of the foam binder angle and the velocity of the inlet air. The convective heat transfer coefficients, the pressure drop and the friction factor are predicted. The effects of the binder angle and the air inlet velocity on heat transfer and pressure drop are highlighted.


Introduction
Cellular materials are a promising class of materials in applications where heat transfer plays a primary role, such as heat exchangers, thermal energy storage systems, and heat sinks for electronics [1].Their large ratio of the heat transfer surface area to the volume, relatively high thermal conductivity, and tortuosity, that promotes flow mixing and heat transport, make them suitable to enhance heat transfer.Some examples of cellular materials are body-cubic centered structures, wiremeshes, honeycombs, open-cell foams [2].Attention was focused on the prediction of convective heat transfer, which plays an important role in foams applications [3].Younis and Viskanta [4] employed a single-blow transient technique, coupled with inverse heat transfer, to predict the volumetric heat transfer coefficients for different foams and volumetric flow rates.Numerical techniques have been used through the years to predict convective heat transfer.By using foam reconstruction from computed tomography, Bodla et al. [5] predicted heat transfer coefficients simulating the flow of air through a foam.The same technique was used by Iasiello et al. [6] on Kelvin's foam-based geometrical models, to evaluate the possibility of reconstructing the foam with the help of ideal foam models.Wu et al. [7] analyzed Kelvin's foams under a uniform surface temperature boundary condition and analyzed also the thermally-developing effects.Iasiello et al. [8] carried out a similar analysis by tuning the strut shape of Kelvin's foams as well.
Recent advances in additive manufacturing allowed for the design of cellular materials aimed to enhance heat transfer [9,10].3D-printed structures based on well-established structures, such as bodycubic centered crystals [11], cubic lattices [12], Kelvin's foams [13], also triply-periodic minimal surfaces [14], were developed.3D printing looks really promising in designing innovative heat transfer devices, even if there are still some limits because of printing costs and time [15] as well as surface roughness [16]; however, depending on the applications and on the operating conditions, surface roughness of 3D-printed parts can even be an advantage for heat transfer devices [17].Drilledhollow spheres architected foams were proposed for heat transfer applications [18] and looked promising compared to other porous materials.
The literature survey shows that the employment of additive-manufactured structures in heat transfer devices deserves further research.In this study, air forced convective heat transfer and pressure drop in drilled-hollow sphere architected foams are analyzed numerically.Different binder angles of the foams and inlet velocities of the entering air are taken into account.Their effects on heat transfer and pressure drop are pointed out.

Mathematical model 2.1. Foams design and governing equations
The structure and the generation of the herein investigated Drilled-Hollow Sphere Architected (DHSA) foams were described in [18].The elementary cell, which, repeating itself periodically, generates the structure, and one eighth of it are reported in figure 1a, with a the side of the cubic cell,  Ri and R the inner and outer radii of the shell, respectively, r the perforation radius of the shell, h the minimum radius of the cylindrical binder, l its length,  the binder angle, b the radius of curvature.
The above geometrical parameters of the foam are correlated by the following equations: ( ) Once R is chosen, the other parameters can be varied according to equations ( 1) and ( 2).The dependence of the binder shape on the binder angle is shown in figure 1b.Also the performance of a Kelvin's foam, whose generation was described in [6], will be reported in this study.

Numerical modeling
Air forced convection through the foam was simulated with the elementary cell surrounded by a bounding box.After carrying out Boolean operations, the air domain, which includes half-cell side long regions upstream and downstream of the cell, was simulated by the mathematical model.The investigated domain, for a  = 10° binder angle, is sketched in figure 2.
The porosity,  = Vf/(Vf + Vs), and the ratio of the heat transfer area to the foam volume, S/(Vf + Vs), computed via a numerical solver as a function of the binder angle, for all the investigated DHSA foams, are reported in figure 3. It shows that, in the investigated range, increasing the binder angle decreases both the porosity and the ratio of the heat transfer area to the volume because of the increasing binder volume; one can also remark that the porosity of the foams is fairly affected by the binder angle.
The governing equations for continuity, momentum and energy conservation, under the assumption of steady-state laminar incompressible flow are:

Boundary conditions
Boundary conditions, presented in detail in [6,8], are herein assumed.Air at uniform temperature and uniform velocity enters the foam; a zero gauge pressure is assumed at the domain exit section.The side walls of the domain are assumed to be adiabatic and symmetric, i. e. a slip condition that allows to analyze an elementary cell extracted from a larger set of cells.At the solid/fluid interface, a no-slip condition and a uniform solid-to-fluid heat flux are imposed; the heat relased to the air is such as to neglect temperature-dependent property effects.Equations were solved by means of a commercial finite-element code.For each case, about 800.000 of both tethraedral and boundary layer elements were employed.Deviations always lower than 3% on both interfacial heat transfer coefficient and pressure drop were found with 1,500,000 and 2,000,000 elements.The linear system was solved with the PARDISO algorithm, using crosswind and streamline stabilization tools.

Results and discussion
Velocity fields, convective heat transfer coefficients, friction factors were predicted for a cubic cell with a = 5.0 mm length of the cell side,  = 5° -20° binder angles, uin = 0.2 m/s -1.0 m/s inlet air velocities, Tin = 293.15K inlet temperature of the air, qin = 1,000 W/m 2 heat flux released to the air.
The air velocity fields in the cell at y = 2.5 mm, for uin = 1.0 m/s,  = 10° and  = 20°, are reported in figures 4a and 4b, respectively.Both figures exhibit a significant increase in the velocity in the core of the cell because of the decreasing area of the cross-section available to the air; namely, the maximum velocity of the fluid is more than about 5 times higher than the air inlet velocity.The increase in the fluid velocity promotes both pressure drop and heat transfer.The comparison of figures 4a and 4b shows that the velocity increase in the foam with  = 20° is larger than that occurring in the foam with  = 10°, because the greater volume of the binder reduces the cross section available to the air.
The velocity fields in the cell, for uin = 1.0 m/s,  = 10° and  = 20°, are presented in figures 5a and 5b, respectively.Both figures point out that in the regions close to the bounding box, where the area of the cross section available to the fluid is the smallest, the largest values of the air velocity, up to about 6 times higher than that in the inlet section, are attained.
In the following, reference will be made to the interfacial convection heat transfer coefficient, hc: where Ts,av and Tf,av are the temperatures of the volumetric averaged solid/fluid interface and of the fluid phase, respectively.Reference will be made also to the volumetric interfacial convective heat transfer coefficient, hcv, obtained by scaling hc with S/(Vf + Vs): that allows to account for the actual convection heat transfer at the fluid/solid boundaries, as well as to the Reynolds number: the friction factor: the shear stress at solid-fluid interface walls: where n is the outgoing direction normal to the surface.The predicted interfacial heat transfer coefficient and friction factor in the investigated DHSAs are presented in the following.Simulations for a Kelvin's foam are also presented; consistently with the values of the porosity and the ratio of the heat transfer area to the volume in figure 3, reference was made to a 0.85 porosity Kelvin's foam.It is worth remarking that the S/(Vf + Vs) = 843 m -1 of the chosen Kelvin's foam is far lower than that of the investigated DHSAs, implying that the DHSAs investigated should perform better; on the other hand, one might also expect higher pressure drop.
The interfacial and volumetric interfacial heat transfer coefficients as a function of the inlet air velocity with various binder angles, for the DHSA and the Kelvin's foam, are reported in figures 6 and 7.The figures exhibit both interfacial convection heat transfer coefficients decreasing at increasing binder angles and at any air inlet velocity, as it was also reported in [19] for Kelvin's foams.This occurs because a decrease in the binder volume will reduce the crossflow section of the solid phase  and the wake region (see figure 1) and, therefore, less flow recirculation is available to enhance convective heat transfer.Moreover, decreasing the binder angle will reduce the free local cross-section to the air between two shells (see figure 1), thus increasing the velocity of the air and the heat transfer coefficient.It is also worth noting that, as it was shown in figures 4 and 5, increasing the angle and the volume of the binder reduces the free flow cross section, thus increasing the velocity and the heat transfer coefficient.One can notice that in the investigated DHSA foams, both heat transfer coefficients are fairly affected by the binder angle, with the volumetric heat transfer coefficient appearing to be more sensitive to binder angle.Finally, the comparison between figures 6 and 7 shows that the interfacial convection heat transfer coefficients in the Kelvin's foam are higher than in DHSA foams, with larger differences observed at increasing binder angles.The opposite is observed to occur to the volumetric convection heat transfer coefficient.This is due to the ratio of the heat transfer area to the foam volume that is larger in the Kelvin's foam than in the DHSA foams, as it was remarked in sub-section 2.2.
Temperature contours on solid-fluid interface walls of the shell, for uin = 1.0 m/s: and  = 10° and 20°, are reported in figures 8a and 8b, respectively.The figures exhibit far closer contours for  = 10° than for  = 20°, suggesting that higher temperature gradients arise at smaller binder angles.Since temperature gradients are proportional to the convection heat transfer coefficients via the solid/fluid temperature difference, we can conclude that heat transfer is higher for  = 10°, as depicted in figure 6.
The pressure drop per unit length as a function of the inlet air velocity and the friction factor as a function of the Reynolds number in DHSA and Kelvin's foams, for various binder angles, are reported in figures 9 and 10, respectively.Figure 9 shows that at the range of inlet air velocities considered, the binder angle has a weak effect on the pressure drop in DHSA foams, with a larger angle resulting in higher pressure drop.This occurs because, even if an increase of binder angle results in a higher cross section area available for flow (see figure 1) and lower velocities, the wake region becomes smaller, thus increasing the dissipation and the momentum transfer between the adjacent shells.It is worth remarking that the rate of variation of the pressure drop in DHSA foams is practically the same as that in the Kelvin's foam as well as one can notice that the pressure drop in DHSA foams is slightly lower than in Kelvin's foam.
Figure 10 highlights that the friction factor in DHSA foams is essentially independent of the binder angle, apart from a fair increase at the largest angle.The figure points also out that the friction factor in the Kelvin's foam is less dependent on the Reynolds number than it is in DHSA, foams due to the Kelvin's foam's more uniform structure.
Shear stresses at solid-fluid interface walls, for uin = 1.0 m/s and  = 10° and 20°, are reported in figures 11a and 11b, respectively.Both figures show that the binders are more stressed than the shell, since the velocity of the air flow normal to the binders axis attains its largest values, thus causing highest dissipations.One can also notice that shear stresses are larger at the walls of the binder with the smaller angle ( = 10°), their maximum values being attained at the mid-height external circumference of the binder.This occurs because a smaller angle implies a smaller length of the binder and a shorter distance between the shells, thus increasing the air velocity and the shear stresses.However, pressure drop are generally higher for higher binders because of the larger wake region downstream.

Conclusions
The effect of the binder angle on forced air convective heat transfer and pressure drop in select drilledhollow sphere architected (DHSA) foams has been analyzed numerically.The simulations showed that at any velocity of the air entering the DHSA foams the interfacial and volumetric convective heat transfer coefficients slightly decrease at increasing binder angles; they are fairly affected by the binder angle, the effects of the angle being larger on the volumetric heat transfer coefficient.Interfacial convection heat transfer coefficients in a Kelvin's foam are higher than in DHSA foams, with increasing differences at increasing binder angles; the opposite occurs to the volumetric heat transfer coefficient, due to the larger heat transfer area in DHSA foams.
In the range of investigated inlet air velocities, the binder angle has a weak effect on the pressure drop in DHSA foams; these pressure drop are higher with larger binder angles.The rate of variation of the pressure drop in DHSA foams is similar as that in the Kelvin's foam while pressure drop in DHSA is slightly lower than in Kelvin's foams.The friction factor in DHSA foams is independent of the binder angle, apart from a fair increase in the foam with the largest investigated angle.The friction factor in the Kelvin's foam is less dependent on the Reynolds number than it is in DHSA foams.Temperature gradients at the solid/fluid interface of DHSA foams are far higher at lower binders; the binders are more stressed than the shell, with shear stresses larger in the binder with a smaller angle.This can be attributed to the higher local velocities faced when the shells are closer.
Nomenclature a cubic cell side (m) T temperature (K) b radius of curvature (m) u velocity vector (m/s) cp heat capacity (J/kg K) V volume (m 3 ) h minimum radius of the binder (m) x, y, z Cartesian coordinates (m) hc interfacial heat transfer coefficient (W/m 2 K) Greek symbols hv volumetric heat transfer coefficient (W/m 3 K) area of the solid surface (m 2 ) s Solid

Figure 1 .
Figure 1.The typical elementary DHSA cell: a) the geometrical parameters; b) the binder shape for different binder angles.

Figure 3 .
Figure 3. Porosity and surface-to-volume ratio vs. the binder angle.

Figure 7 .
Figure 7. Volumetric interfacial heat transfer coefficient vs. inlet air velocity, for various binder angles.

Figure 10 .
Figure 10.Friction factor vs. Reynolds number for different binder angles.