An Introduction to Triply Periodic Minimal Surfaces in Thermal Applications
1. Introduction and Motivation
Advancements in additive manufacturing technologies have reduced cost and increased access to a manufacturing technique with significantly reduced design constraints. This has created many opportunities for researchers to investigate structures with complex geometries that are otherwise impossible to create with conventional manufacturing techniques, such as structures consisting of Triply Periodic Minimal Surfaces (TPMS). TPMS are porous cellular-like structures (Fig. 1) that can be univocally defined by a set of trigonometric functions which by definition share the properties of a zero-mean curvature with a significantly increased surface area/volume ratio (Schoen, 1970) compared to conventional foam material. These unique properties lead to promising results in several fields of application, such as structural (Qiu et al. 2023), tissue engineering (Giannitelli et al. 2014), acoustics (Deshmukh et al. 2019), chemical engineering (Baena-Moreno et al. 2021), and heat transfer enhancement (Dutkowski et al. 2022).
(a) | (b) | (c) |
(d) | (e) | (f) |
Figure 1. Different TPMS structures: (a) diamond, (b) Fisher–Koch, (c) gyroid, (d) I-WP, (e) neovius, (f) primitive
2. Heat Transfer in TPMS
2.1 Conduction Heat Transfer Analysis of TPMS Structures
The thermal performance of TPMS structures can be finely tuned by controlling the TPMS unit type, relative density, and structural parameters. Thermal conductivity may be minimized in applications where a thermal insulator is desired, as demonstrated by Tang et al. (2023). On the other hand, many researchers utilize the advantageous surface area/volume ratio and tunable thermal conductivity of TPMS by embedding the structures into a Phase Change Material (PCM) (Qureshi et al. 2021). Given that PCMs used in low-to-mid temperature applications suffer from a low thermal conductivity, the coupling PCMs with TPMS structure's “tunable” thermal conductivity is a promising option. The advantageous properties of TPMS complement many of the shortcomings of PCMs. Results show reduced PCM melting times and an overall higher heat transfer coefficient when compared to stochastic foam-based PCM systems.
Analysis techniques have been developed to quantify the effects of TPMS geometry and porosity on thermal conductivity (Gurra et al., 2023). In particular, tortuosity has been adapted and applied from flow and diffusion through porous media modeled off of Epstein (1989) for use in TPMS. Tortuosity quantifies the curvature of a taken pathlength between two points. A higher tortuosity has a less direct path due to higher curvature. In the context of TPMS it can describe how heat energy must weave through the complex inner structures of TPMS. Figure 2 shows heat transfer streamlines in a Gyroid TPMS generated by COMSOL Multiphysics, acting as a visual representation of the tortuous heat transfer paths. It has been discovered that for a given porosity, a TPMS structure with a lower tortuosity will have a higher thermal conductivity (Bartlett et al., 2023).
(a) | (b) | (c) |
Figure 2. COMSOL Multiphysics generated heat flow streamlines in Gyroid TPMS. (a) Geometry with boundary conditions shown in (b) isometric view, and (c) side view.
2.2 Convection Heat Transfer Analysis of TPMS Structures
Several works have also aimed to identify the effectiveness of TPMS architectures in convection-based heat transfer applications (Fig. 3) and to optimize the thermal performance of these structures by acting on both structural and boundary parameters. Cheng et al. (2017) studied the effect of morphology of porous structures on fluid flow and heat/mass transport by comparing the performance of four TPMS architectures with a structure that approximates metal-sintered porous media. Results show that TPMS can achieve both higher volumetric heat transfer coefficients and interstitial heat transfer coefficients (up to 25% higher). Tang et al. (2023) conducted an empirical comparison of the heat transfer efficiency between TPMS and straight-finned structures. Three TPMS variants (Gyroid, Diamond, and IWP) have demonstrated superior performance over the finned model in several flow conditions. Moreover, advantages in terms of the convective heat transfer coefficient grew while increasing the inlet flow rate. Functional grading of a TPMS structure was also found to control heat transfer (Anacreonte et al., 2024). Al-Ketan et al. (2020) evaluated the heat transfer characteristics exhibited by graded TPMS structures. The experimental findings delineated that modifying the topology of the heat sink in a graded manner, whereby the porosity extends longitudinally along the flow path, resulted in a noteworthy reduction in pressure drop of about 27.6%. Nevertheless, this decrease was accompanied by a consequential compromise in the convective heat transfer coefficient, diminishing by an estimated 16%.
Figure 3. Qualitative example of a numerical conjugated heat transfer study showing the temperature distribution of water flow in a square channel containing Gyroid TPMS material. A heat flux of 1 kW/m^{2} is imposed at the bottom of the duct. The lateral walls are set to adiabatic with a fluid dynamic no-slip condition. Water flow in the channel inlet is fully developed with an average inlet velocity of 0.02 m/s.
TPMS structures are versatile and readily modifiable. The unique inner topology of these structures can be divided to create two non-intersecting channels, sparking investigations into the use of TPMS as crossflow heat exchangers (Yan et al., 2023). Several researchers have focused on further varying TPMS functions' trigonometrical parameters to obtain a multitude of slightly different structures, thus comparing the performance of heat exchangers with different TPMS derived geometries (Reynolds et al., 2023; Xu et al., 2023; Yan et al., 2023).
2.3 Radiation Heat Transfer Analysis of TPMS Structures
The TPMS defining characteristics which prove useful in convection applications also prove useful in radiative applications. TPMS have high surface area to volume ratios, allowing for proportionally more absorbing or emitting surface area when compared to stochastically generated porous materials. Researchers such as Vignoles et al. (2021) have investigated using TPMS as volumetric solar receivers, soaking up radiative heat energy from the sun. Comparatively few papers analyzing radiative heat transfer in TPMS structures have been published, especially when compared to the existing body of literature evaluating the convective performance of TPMS.
3. Multi-Objective Optimization
Using a variety of optimization approaches, TPMS performance can be further controlled by tuning some of the parameters that define a TPMS structure, such as TPMS type, relative density, and structural parameters. A number of studies have focused on this subject, with results showing the possibility of highly increasing both volumetric and interstitial heat transfer coefficients (Cheng et al., 2021), while specifying that these increments come at the cost of higher pressure drops (Tang et al., 2023). Since TPMS have been demonstrated to be controllable, they can be optimized to the needs of the user. Depending on the application, these structures can be tailored to either prioritise higher heat transfer coefficients or lower pressure drop as desired. Nevertheless, ongoing research is envisioned to eventually reach an advanced stage suitable for addressing complex applications. However, a comprehensive exploration of the thermal characteristics pertaining to graded, heterogeneous, and multi-scale TPMS remains lacking.
Our research group focuses on the topology optimization of TPMS structures. A multi-objective genetic algorithm is used to generate structures with pre-determined characteristics and analyze their thermo-fluid-dynamic performance. This algorithm uses a Pareto front to identify solutions with optimal trade-off between pressure drop and heat transfer rate. Preliminary results may be observed in Fig. 4. The optimization study shown is carried out by imposing a heat flux at the solid-fluid interface, while tuning various parameters such as porosity, structural anisotropy, and functional porosity gradients. The performance is evaluated in terms of both pressure drop per unit length and volumetric heat transfer coefficient. While being preliminary, results show a promising outline of TPMS capability in outperforming more conventional porous structures (such as Kelvin foams) in some scenarios.
Figure 4. Convective heat transfer optimization study carried out on four TPMS structures, where multi-objective optimization is performed by comparing the volumetric convection heat transfer coefficient (h_{v}) to the pressure drop per unit length (ΔP/L). Results are compared with a Pareto front obtained for a Kelvin foam.
4. Summary and Outlook
TPMS architectures represent a promising choice for heat transfer applications owing to their increased surface-to-volume ratios and “tunable” structure and architectures. Moreover, their capability to be printed out though additive manufacturing (AM) using many materials (including metals and ceramics) while maintaining a reasonable surface roughness allows these structures to be extremely interesting in a multitude of applications. Notably, the thermal efficiency exhibited by TPMS may surpass that of conventional porous frameworks, such as foams or lattices, in specific applications. This performance attribute can be effectively controlled by manipulating the TPMS unit types, relative densities, and structural parameters. TPMS show promising results in reducing pressure drop and increasing heat transfer coefficients in heat exchangers, increasing the performance of thermal batteries, and generally giving researchers fine control over the thermal performance of an architect porous material. Nevertheless, the depth of existing research is not sufficient to cater to the many intricate applications and comprehensive investigations into the thermal behaviors of graded, heterogeneous, and multi-scale TPMS architectures.
REFERENCES
Al-Ketan, O., Hassan, M.I., Khalil, M., Al-Rub, R.A., Rowshan, R., and Khan, K.A. (2020) Forced convection CFD analysis of architected and 3D printable heat sinks based on triply periodic minimal surfaces, J. Therm. Sci. Eng. Appl., vol. 13, no. 2, pp. 1–33.
Anacreonte, V.A., Iasiello, M., Mauro, G.M., Andreozzi, A., Bianco, N., and Chiu, W.K.S. (2024) A thermal analysis of a functionally-graded gyroid as a heat sink, 9th Int. Symp. on Advances in Computational Heat Transfer (CHT-24), Istanbul, Turkey, May 26–30, 2024.
Baena-Moreno, F.M., Gonzales-Castano, M., Navarro de Miguel, J.C., Miah, K.U.M., Ossenbrik, R., Odriozola, J.A., and Arellano-García, H. (2021) Stepping toward efficient microreactors for CO_{2} methanation: 3D-printed gyroid geometry, Sustain. Chem. Eng., vol. 9, no. 24, pp. 8198–8206.
Bartlett, M.T., Iasiello, M., Anacreonte, A., Peracchio, A.A., Mauro, G.M., Bianco, N., and Chiu, W.K.S. (2023) Analysis of effective thermal conductivity in triply periodic minimal surface foam structures, 40th Unione Italiana Termofluidodinamica (UIT) Int. Heat Transfer Conf., Assisi, Italy, June 26–28, 2023.
Cheng, Z., Xu, R., and Jiang, P. (2021) Morphology, flow and heat transfer in triply periodic minimal surface based porous structures, Int. J. Heat Mass Trans., vol. 170, p. 120902.
Deshmukh, S., Ronge, H., and Ramamoorthy, S. (2019) Design of periodic foam structures for acoustic applications: Concept, parametric study and experimental validation, Mater. Des., vol. 175, p. 107830.
Dutkowski, K., Kruzel, M., and Rokosz, K. (2022) Review of the state-of-the-art uses of minimal surfaces in heat transfer, Energies, vol. 15, no. 21, p. 7994.
N. Epstein (1989) On tortuosity and the tortuosity factor in flow and diffusion through porous media, Chem. Eng. Sci., vol. 44, no. 3, pp. 777–779.
Giannitelli, S., Accoto, D., Trombetta, M., and Rainer, A. (2014) Current trends in the design of scaffolds for computer-aided tissue engineering, Acta Biomater., vol. 10, pp. 580–594.
Gurra, E., Iasiello, M., Naso, V., and Chiu, W.K.S. (2023) Numerical prediction and correlations of effective thermal conductivity in a drilled-hollow-sphere architected foam, J. Therm. Sci. Eng. Appl., vol. 15, no. 4, p. 041002.
Qui, N., Wan, Y., Shen, Y., and Fang, J. (2024) Experimental and numerical studies on mechanical properties of TPMS structures, Int. J. Mech. Sci., vol. 261, p. 108657.
Qureshi, Z., Al-Omari, S., Elnajjar, E., Al-Ketan, O., and Al-Rub, R. (2021) On the effect of porosity and functional grading of 3D printable triply periodic minimal surface (TPMS) based architected lattices embedded with a phase change material, Int. J. Heat Mass Trans., vol. 183B, p. 122111.
Reynolds, B.W., Conan, J.F., Morison, K.R., and Holland, D.J. (2023) Characterisation of heat transfer within 3D printed TPMS heat, Int. J. Heat Mass Trans., vol. 212, p. 124264.
Schoen, A.H. (1970) Infinite periodic minimal surfaces without self-intersections, NASA Technical Note, Report No. NASA-TN-D-5541.
Tang, D., Xu, S., Yang, K., Gao, T., and Tang, H. (2023) Effects of porosity on effective thermal conductivities of thermal insulation SiC sandwich panels with Schoen-gyroid structure, Ceram. Int. (in press).
Tang, W., Zhou, H., Zeng, Y., Yan, M., Jiang, C., and Yang, P. (2023) Analysis on the convective heat transfer process and performance evaluation of triply periodic minimal surface (TPMS) based on diamond, gyroid and Iwp, Int. J. Heat Mass Trans., vol. 201, p. 123642.
Vignoles, G.L., Rochais, D., and Chupin, S. (2021) Computation of the conducto-radiative effective heat conductivity of porous media defined by triply periodic minimal surfaces, Int. J. Therm. Sci., vol. 159, p. 106598.
Xu, Y., Pan, H., Wang, R., Du, Q., and Lu, L. (2023) New families of triply periodic minimal surface-like shell lattices, Addit. Manuf., vol. 773, p. 103779.
Yan, G., Sun, M., Zhang, Z., Liang, Y., Jiang, N., Pang, X., Song, Y., Liu, Y., and Zhao, J. (2023) Experimental study on flow and heat transfer performance of triply periodic minimal surface structures and their hybrid form as disturbance structure, ICHMT, vol. 147, p. 106942.
Yan, K., Wang, J., Li, L., and Deng, H. (2023) Numerical investigation into thermo-hydraulic characteristics and mixing performance of triply periodic minimal surface-structured heat exchangers, Appl. Therm. Eng., vol. 230, p. 120748.
References
- Al-Ketan, O., Hassan, M.I., Khalil, M., Al-Rub, R.A., Rowshan, R., and Khan, K.A. (2020) Forced convection CFD analysis of architected and 3D printable heat sinks based on triply periodic minimal surfaces, J. Therm. Sci. Eng. Appl., vol. 13, no. 2, pp. 1–33.
- Anacreonte, V.A., Iasiello, M., Mauro, G.M., Andreozzi, A., Bianco, N., and Chiu, W.K.S. (2024) A thermal analysis of a functionally-graded gyroid as a heat sink, 9th Int. Symp. on Advances in Computational Heat Transfer (CHT-24), Istanbul, Turkey, May 26–30, 2024.
- Baena-Moreno, F.M., Gonzales-Castano, M., Navarro de Miguel, J.C., Miah, K.U.M., Ossenbrik, R., Odriozola, J.A., and Arellano-García, H. (2021) Stepping toward efficient microreactors for CO_{2} methanation: 3D-printed gyroid geometry, Sustain. Chem. Eng., vol. 9, no. 24, pp. 8198–8206.
- Bartlett, M.T., Iasiello, M., Anacreonte, A., Peracchio, A.A., Mauro, G.M., Bianco, N., and Chiu, W.K.S. (2023) Analysis of effective thermal conductivity in triply periodic minimal surface foam structures, 40th Unione Italiana Termofluidodinamica (UIT) Int. Heat Transfer Conf., Assisi, Italy, June 26–28, 2023.
- Cheng, Z., Xu, R., and Jiang, P. (2021) Morphology, flow and heat transfer in triply periodic minimal surface based porous structures, Int. J. Heat Mass Trans., vol. 170, p. 120902.
- Deshmukh, S., Ronge, H., and Ramamoorthy, S. (2019) Design of periodic foam structures for acoustic applications: Concept, parametric study and experimental validation, Mater. Des., vol. 175, p. 107830.
- Dutkowski, K., Kruzel, M., and Rokosz, K. (2022) Review of the state-of-the-art uses of minimal surfaces in heat transfer, Energies, vol. 15, no. 21, p. 7994.
- N. Epstein (1989) On tortuosity and the tortuosity factor in flow and diffusion through porous media, Chem. Eng. Sci., vol. 44, no. 3, pp. 777–779.
- Giannitelli, S., Accoto, D., Trombetta, M., and Rainer, A. (2014) Current trends in the design of scaffolds for computer-aided tissue engineering, Acta Biomater., vol. 10, pp. 580–594.
- Gurra, E., Iasiello, M., Naso, V., and Chiu, W.K.S. (2023) Numerical prediction and correlations of effective thermal conductivity in a drilled-hollow-sphere architected foam, J. Therm. Sci. Eng. Appl., vol. 15, no. 4, p. 041002.
- Qui, N., Wan, Y., Shen, Y., and Fang, J. (2024) Experimental and numerical studies on mechanical properties of TPMS structures, Int. J. Mech. Sci., vol. 261, p. 108657.
- Qureshi, Z., Al-Omari, S., Elnajjar, E., Al-Ketan, O., and Al-Rub, R. (2021) On the effect of porosity and functional grading of 3D printable triply periodic minimal surface (TPMS) based architected lattices embedded with a phase change material, Int. J. Heat Mass Trans., vol. 183B, p. 122111.
- Reynolds, B.W., Conan, J.F., Morison, K.R., and Holland, D.J. (2023) Characterisation of heat transfer within 3D printed TPMS heat, Int. J. Heat Mass Trans., vol. 212, p. 124264.
- Schoen, A.H. (1970) Infinite periodic minimal surfaces without self-intersections, NASA Technical Note, Report No. NASA-TN-D-5541.
- Tang, D., Xu, S., Yang, K., Gao, T., and Tang, H. (2023) Effects of porosity on effective thermal conductivities of thermal insulation SiC sandwich panels with Schoen-gyroid structure, Ceram. Int. (in press).
- Tang, W., Zhou, H., Zeng, Y., Yan, M., Jiang, C., and Yang, P. (2023) Analysis on the convective heat transfer process and performance evaluation of triply periodic minimal surface (TPMS) based on diamond, gyroid and Iwp, Int. J. Heat Mass Trans., vol. 201, p. 123642.
- Vignoles, G.L., Rochais, D., and Chupin, S. (2021) Computation of the conducto-radiative effective heat conductivity of porous media defined by triply periodic minimal surfaces, Int. J. Therm. Sci., vol. 159, p. 106598.
- Xu, Y., Pan, H., Wang, R., Du, Q., and Lu, L. (2023) New families of triply periodic minimal surface-like shell lattices, Addit. Manuf., vol. 773, p. 103779.
- Yan, G., Sun, M., Zhang, Z., Liang, Y., Jiang, N., Pang, X., Song, Y., Liu, Y., and Zhao, J. (2023) Experimental study on flow and heat transfer performance of triply periodic minimal surface structures and their hybrid form as disturbance structure, ICHMT, vol. 147, p. 106942.
- Yan, K., Wang, J., Li, L., and Deng, H. (2023) Numerical investigation into thermo-hydraulic characteristics and mixing performance of triply periodic minimal surface-structured heat exchangers, Appl. Therm. Eng., vol. 230, p. 120748.