Review of Triply Periodic Minimal Surface (TPMS) Structures for Cooling Heat Sinks

Abstract

This review paper deals with Triply Periodic Minimal Surfaces (TPMS) and lattice structures as a new generation of heat exchangers. Especially, their manufacturing is becoming feasible with technological progress. While some intricate structures are fabricated, challenges persist concerning manufacturing limitations, cost-effectiveness, and performance under transient operating conditions. Studies reported that these complex geometries, such as diamond, gyroid, and hexagonal lattices, outperform traditional finned and porous materials in thermal management, particularly under forced and turbulent convection regimes. However, TPMS necessitates the optimization of geometric parameters such as cell size, porosity, and topology stretching. The complex geometries enhance uniform heat exchange and reduce thermal boundary layers. Moreover, the integration of high thermal conductivity materials (e.g., aluminum and silver) and advanced coolants (including nanofluids and ethylene glycol mixtures) further improves performance. However, the drawback of complex geometries, confirmed by both numerical and experimental investigations, is the critical trade-off between heat transfer performance and pressure drop. The potential of TPMS-based heatsinks transpires as a trend for next-generation thermal management systems, besides identifying key directions for future research, including design optimization, Multiphysics modeling, and practical implementation.

1. Introduction

Thermal management remains a critical challenge across multiple engineering domains, including power electronics, aerospace, thermal energy storage systems, and compact heat exchangers. Conventional cooling solutions such as plate fins and metal foams often fall short in meeting the increasingly stringent requirements for heat transfer efficiency, compactness, and manufacturability. In this context, TPMS has emerged as promising alternatives due to their unique geometric and functional properties. These periodic metamaterials, derived from minimal surface theory, offer a high specific surface area, isotropic thermal distribution, and superior fluid connectivity, making them highly suitable for high-performance heat transfer applications. Recent studies have focused on optimizing TPMS architectures to maximize thermal performance while maintaining manageable hydraulic resistance. Notably, the works of Shen et al. [1], Chen et al. [2], and Wang et al. [3] have demonstrated the superior performance of specific TPMS topologies such as gyroid, diamond, and Schwarz-P over conventional heat exchangers, particularly in terms of elevated Nusselt numbers, reduced surface temperatures, and enhanced thermal uniformity. Furthermore, the integration of TPMS structures with phase change materials (PCMs) and advanced working fluids, such as nanofluids, has been shown to significantly boost energy efficiency through latent heat storage and improved thermal conductivity. Beyond material integration, advanced geometrical design strategies including porosity tuning, functional grading, anisotropic stretching, and multiphysics-driven topology optimization have been employed to tailor TPMS structures to specific cooling scenarios. Studies by Ibhadode [4] and Kerme et al. [5] highlight that local modifications to cell size, porosity distribution, and orientation can lead to substantial improvements in thermal-hydraulic performance. Importantly, the practical implementation of these structures depends on their manufacturability, particularly via additive manufacturing (AM). Incorporating AM constraints during the design phase, such as minimum feature size and overhang angles, are essential for producing functional TPMS heat exchangers in metals like aluminum or silver. Contributions from Kilic [6] and Saghir et al. [7] underscore the need to reconcile thermal performance with manufacturing feasibility. In this context, the present review aims to critically synthesize recent advances (from 2020 to the present) in the application of TPMS structures for heat exchangers and heat sinks. The discussion centers on thermal and hydraulic performance metrics, geometry-driven design enhancements, material integration strategies, and AM compatibility. By highlighting the most impactful developments, this work seeks to provide a roadmap for future research aimed at creating next-generation thermal management solutions that are both efficient and scalable.

2. Fundamentals of TPMS-Based Structures

2.1. Definition of TPMS

TPMS constitutes a distinct class of continuous, boundaryless mathematical surfaces that repeat periodically in all three spatial dimensions. Characterized by zero mean curvature at every point, TPMS locally minimizes surface area while forming a robust and interconnected three-dimensional network. This property is mathematically defined by the condition H= ${\displaystyle \frac{1}{2}}$ (${\mathsf{\kappa }}_1$ +${\mathsf{\kappa }}_2$), new where ${\mathsf{\kappa }}_1$ and ${\mathsf{\kappa }}_2$ are the principal curvature. TPMS divides space into two interpenetrating yet non-overlapping domains, resulting in a highly uniform distribution of fluid pathways and enhanced thermal dissipation. Topologically, these surfaces are typically described by implicit periodic equations involving trigonometric functions. Among the most widely investigated geometries are the Gyroid, Diamond, Primitive, Schwarz-P, and I-WP structures (Figure 1), each exhibiting distinct characteristics in terms of porosity, specific surface area, and tortuosity. Their inherently isotropic connectivity enables homogeneous temperature distribution and effective control of both thermal and hydraulic transport, making them highly suitable for heat transfer applications. Several studies (e.g., Chen et al. [8], Saghir et al. [7,9], Kilic [6], Kerme et al. [5]) have demonstrated that TPMS provide an optimal balance between heat transfer performance and flow resistance. This balance can be fine-tuned by adjusting geometric parameters such as unit cell size, porosity, and deformation factors (e.g., stretching or warping). Moreover, their smooth, continuous surfaces are well-suited for fabrication via additive manufacturing techniques, allowing the production of intricate porous architectures for advanced thermal management systems. In summary, TPMS represents a promising and high-performance geometric approach for the development of next-generation heat exchangers and heat sinks, where efficient, uniform, and adaptable thermal regulation is essential.

2.2. TPMS Unit Design

The design of TPMS units is based on implicit mathematical equations that define surfaces with zero mean curvature and periodicity in three-dimensional space. These surfaces are typically generated using trigonometric or elliptic functions (Figure 2), which control the geometric morphology and topological connectivity of the structure. Among the most extensively studied TPMS geometries are the Gyroid, Schwarz-P, Diamond, Primitive, and I-WP surfaces. Researchers such as Wang et al. [3] and Baobaid et al. [11] have used these equations to parameterize the architecture of 3D-printed heat exchangers, allowing precise control over critical structural parameters such as relative density, mesh size, and deformation factors (e.g., torsion or stretching). Barakat et al. [12] explored geometric variants modulated by a control parameter α, which adjusts the surface’s compactness, curvature distribution, and overall complexity. Additionally, Tang et al. [13] proposed gradual morphological transitions between different TPMS topologies to optimize thermal and hydraulic performance under varying operating conditions. These geometric tunings and topological transitions establish a versatile design framework for advanced heat exchangers and heat sinks, particularly in applications demanding a trade-off between high heat transfer efficiency and low flow resistance.

The C parameter governs the porosity of the TPMS structure. By adjusting this parameter, researchers can modulate the wall thickness and the connectivity of the pore network, thereby influencing the balance between thermal performance and pressure drop. This parametric design approach offers significant flexibility in modeling and optimizing TPMS-based heat exchangers and has become a foundational strategy in the development of advanced thermal management materials.

2.3. Geometric Properties and Thermal Advantages

TPMS are characterized by unique geometric features that offer significant thermal advantages. These surfaces exhibit zero mean curvature and form a highly interconnected and porous architecture, resulting in a high surface-to-volume ratio and isotropic flow distribution. Such features promote uniform heat transfer in all spatial directions, which is particularly advantageous for thermal management in high heat-flux electronic systems. Among the most studied TPMS geometries are the Gyroid, Diamond, and Primitive structures, which are known for maintaining homogeneous fluid distribution and minimizing the development of thermal boundary layers. The intrinsic tortuosity of TPMS channels enhances fluid mixing and mitigates stagnant flow regions, leading to an increase in the Nusselt number (Nu) and overall thermal efficiency. Experimental and numerical studies, such as those conducted by Kerme et al. [5] and Kilic [6], have demonstrated that TPMS structures fabricated from thermally conductive metals like aluminum or silver, when combined with advanced working fluids (e.g., hybrid nanofluids), can achieve a reduction in peak temperature of up to 15% and an enhancement in the performance evaluation criterion (PEC) by approximately 25–30%, compared to conventional finned heat exchangers. In conclusion, the geometrical and physical properties of TPMS enable enhanced heat transfer, uniform temperature distribution, and lower pressure losses, making them highly promising for the development of next-generation heat exchangers and thermal management systems.

2.4. Additive Manufacturing Methods Applicable to TPMS Structures

In

Table 1. Additive manufacturing methods have been applied to TPMS structures in various studies.

Reference	Additive Manufacturing Method	Material Used	TPMS Type	Comments/Benefits
Saghir et al. [7]	FDM (Fused Deposition Modeling)	Polymer	Gyroid	Suitable for visualization and cold testing; resolution limit for microstructures
Kilic [6]	SLM (Selective Laser Melting)	Aluminum, Silver	Gyroid, Diamond	High precision, high thermal conductivity, high cost
Al-Omari et al. [15]	SLM/DMLS	Aluminum	Schwarz-P, Primitive	Excellent thermal performance; good PCM integration
Beer et al. [16]	SLA (Sterelithogrphy)	Photopolymer resin	Gyroid	High resolution for complex geometries; proper for rapid prototyping
Barakat et al. [18]	SLA/SLS (Selective Laser Sintering)	PhotopolymerCeramic	Diamond, Gyroid	Suitable for small scales; allow complex structures with controlled porosity
Liu et al. [17]	DMLS (Direct Metal Laser Sintering)	Stainless steel, Alloys	Gyroid, Diamond	Good mechanical and thermal resistance; ideal for functional exchangers

highlights various additive manufacturing methods used to fabricate Triply Periodic Minimal Surface (TPMS) structures, comparing the materials used, the types of geometries fabricated, and their associated performance. Polymer-based techniques such as Fused Deposition Modeling (FDM), used by Saghir et al. [7], and Stereolithography (SLA), employed by Beer et al. [15], allow the creation of Gyroid structures with good geometric fidelity. These approaches are particularly well-suited for visualization and rapid prototyping, though they suffer from limitations in resolution (especially for FDM) and low thermal conductivity, making them suitable mainly for cold testing. In contrast, metal-based laser sintering techniques, such as Selective Laser Melting (SLM) and Direct Metal Laser Sintering (DMLS), used respectively by Kilic [6], Al-Omari et al. [16], and Liu et al. [17], enable the fabrication of TPMS structures using aluminum, stainless steel, and high thermal conductivity alloys, thereby offering excellent thermal and mechanical performance. These structures are particularly appropriate for functional applications such as heat exchangers with phase change material (PCM) integration, as demonstrated by Al-Omari et al. [16] with Schwarz-P and Primitive geometries. However, these methods come with high production costs and require advanced equipment. Finally, the combined use of SLA and Selective Laser Sintering (SLS) for photopolymers and ceramics (Barakat et al. [18]) enables the fabrication of complex structures with controlled porosity, although their thermal applications are limited due to the low conductivity of non-metallic materials. In summary, the choice of manufacturing method and material strongly depends on the intended application whether it be low-cost visual prototyping or the production of high-performance thermal components.

3. Thermal and Hydraulic Performance of TPMS Structures

3.1. Thermal Performance: Temperature Distribution, Nusselt

TPMS structures have demonstrated excellent thermal management capabilities owing to their highly interconnected, isotropic geometries and high surface-to-volume ratios. These features (

Table 2. Summary of authors’ contributions on the thermal performance of TPMS structures, including temperature distribution, Nusselt number (Nu) and performance evaluation criterion (PEC).

Author	Year	Contribution
Saghir et al. [7]	2024	Experimental study of cooling by water-glycol mixture; good thermal homogeneity in TPMS.
Kilic et al. [6]	2023	Use of nanofluids in Al and Ag TPMS; 8–12% gain in Nu; thermal uniformity.
Al-Omari et al. [16]	2023	Comparison of TPMS geometries (Schwarz-P, Diamond); good Nu and low thermal gradient.
Beer et al. [15]	2024	Thermal simulation on TPMS for electronics; reduction of hot spots, stable distribution.
Barakat et al. [18]	2024	Study of the morphological parameter α; spatial control of temperature via evolving Gyroid geometry.
Liu et al. [17]	2023	Analysis of hybrid TPMS structures; Nu optimization and efficient thermal distribution.

) promote uniform heat distribution and enhanced convective heat transfer, making TPMS ideal for demanding thermal applications. Saghir et al. [7] investigated the thermal behavior of a water–glycol mixture circulating through aluminum and silver TPMS structures. Their results showed a homogeneous temperature distribution across the structure, even at low ambient temperatures, a crucial feature for cold climate operations. Notably, the addition of 5% glycol provided an effective compromise between antifreeze protection and thermal performance. Kilic [6] conducted both experimental and numerical studies demonstrating that the use of hybrid nanofluids in metallic TPMS structures (aluminum and silver) enhanced thermal performance by 8–12%, as evidenced by increased Nusselt numbers. However, this enhancement came at the cost of a moderate increase in pressure drop due to the higher viscosity of the nanofluids. Al-Omari et al. [16] compared various TPMS geometries under forced convection. They found that structures such as Schwarz-P and Diamond achieved superior thermal performance and higher Nusselt numbers compared to conventional designs while maintaining low temperature gradients. Complementarily, Beer et al. [15] employed numerical simulations to assess TPMS performance for electronics cooling. Their findings indicated a significant reduction in thermal hot spots and improved temperature uniformity, attributed to the regular arrangement of fluid pathways within the TPMS geometry. Barakat et al. [18] explored the morphological variations in gyroid structures by introducing a geometric control parameter α. Their study revealed that local modulation of α enabled spatial tuning of thermal performance, allowing for targeted temperature control within heat exchangers. Similarly, Liu et al. [17] reported that hybrid TPMS designs, tailored through structural optimization, significantly improved heat dissipation in embedded thermal systems, as evidenced by enhanced Nusselt numbers. These studies collectively confirm the strong potential of TPMS architectures for advanced thermal management, particularly in compact heat exchangers, electronic cooling systems, and cryogenic applications, where high thermal efficiency and spatial temperature uniformity are critical.

3.2. Hydraulic Performance: Pressure Losses and Tortuosity

TPMS structures, due to their continuous, periodic, and interpenetrating geometries, exhibit distinctive hydraulic characteristics that critically influence the overall performance of heat exchangers. One of the key parameters in this context is the pressure drop, which quantifies the resistance to fluid flow through the structure. In TPMS-based systems, pressure drop is strongly influenced by the tortuosity of the flow paths that is, the complexity and length of the routes the fluid must navigate. While increased tortuosity enhances fluid mixing and promotes convective heat transfer, it also leads to higher pressure losses, which can compromise the overall energy efficiency of the thermal system. Some studies have investigated this fundamental trade-off between thermal performance and hydraulic efficiency. For instance, Ibhadode [4] showed that stretching TPMS unit cells along the flow direction reduces path tortuosity, thereby lowering the pressure drop. However, this geometric alteration may slightly degrade heat transfer performance due to reduced mixing intensity. Similarly, Saghir et al. [7] and Kilic [6] demonstrated that the use of more viscous working fluids, such as nanofluids or water-glycol mixtures, results in increased pressure losses; although it can yield moderate improvements in heat transfer coefficients and temperature uniformity. Kerme et al. [5] conducted a comparative study on Diamond and Gyroid structures with varying unit cell sizes and porosities. They observed that configurations with larger unit cells and lower porosity significantly reduced the pressure drop, while still delivering satisfactory thermal performance. In another study, Saghir et al. [9] reported that Gyroid structures exhibit, on average, an 18% higher pressure drop compared to conventional metal foams. However, this drawback is offset by enhanced temperature uniformity and superior heat transfer capability, due to the optimized surface area and flow organization intrinsic to TPMS. These findings underscore the importance of geometric and topological optimization in TPMS design to strike an optimal balance between thermal efficiency and hydraulic cost. Advanced strategies such as conformal lattice modeling, anisotropic stretching, or the development of hybrid TPMS configurations (combining stretched and unstretched regions) offer promising avenues to achieve this balance in high-performance thermal systems. According to later studies, it is evident that the pressure drop presents a drawback; However, it can be reduced by adding the nanoparticles, selecting the nature of the fluid, and increasing the TPMS unit cells. For a satisfactory solution, more research could be done.

3.3. Comparison with Conventional Structures (Fins, Metal Foams)

TPMS structures are increasingly recognized as efficient alternatives to conventional heat exchangers (

Table 3. Comparative Analysis of TPMS Structures vs. Conventional Cooling Media (Fins and Metal Foams) in Terms of Thermal and Hydraulic Performance.

References	TPMS (Gyroid, Diamond, Etc.)	Conventional Fins	Metal Foams	Criteria
Saghir et al. [7], Kilic [6], Kerme et al. [5]	Very homogeneous, no Distinct boundary layer formation	Inhomogeneous, frequent hot spots	Non-uniform heat accumulation near the outlet	Temperature distribution
Saghir et al. [9], Kilic [6], Ibha-dode [4]	High, especially with nanofluids or optimized structures	Moderate to low, Depending on fin geometry	Moderate, strongly dependent on porosity	Nusselt number (Nu)
Saghir et al. [9], Ibhadode [4]	Medium to high (but can be reduced via cell stretching or topological optimization)	Low to moderate	Low (due to high porosity), but sensitive to clogging	Pressure drop (ΔP)
Ibhadode [4], Kerme et al. [5]	Varies with geometry; stretching cells reduces tortu- osity	Low tortuosity	Very high, which may hinder laminar flow	Fluid tortuosity
Saghir et al. [9], Kilic [6], Beer et al. [15]	Isotropic (especially Gyroid), allows uniform cooling	Anisotropic	Random, depends on the fabrication process	Structural homogeneity
Al-Omari [16], Barakat [18], Liu [17]	Yes, via SLM, FDM, EBM, DMLS	Limited to certain designs	Difficult to fabricate with high precision	Additive manufacturing possible
Kilic [6], Kerme et al. [5], Ibhadode [4]	Very high, especially with topological optimization or nanofluid used	Moderate	Moderate to good if porosity is well-designed	Overall thermal efficiency

), such as metal fins and foams, due to their highly interconnected geometry, isotropic pore distribution, and large specific surface area. Unlike traditional straight-fin configurations, TPMS structures exhibit continuous surfaces without dead zones, promoting uniform temperature distribution and minimizing thermal boundary layer development. For instance, Saghir et al. [9] conducted both experimental and numerical analyses using the Darcy–Brinkman model to compare a Gyroid TPMS structure with a metal foam. Their results demonstrated that the Gyroid provided more uniform cooling and achieved a higher Nusselt number. Although the metal foam exhibited a slightly lower pressure drop, it showed signs of thermal stratification along the flow path, indicating diminished cooling effectiveness. Furthermore, Ibhadode et al. [4] emphasized that certain stretched TPMS configurations, particularly the Diamond structure, can surpass flat-fin heat exchangers under low Reynolds number conditions, achieving enhanced thermal performance with comparable or even reduced pressure losses. This finding is supported by Kilic et al. [6] and Kerme et al. [5], who demonstrated that TPMS structures fabricated from aluminum or silver exhibited superior heat dissipation performance compared to conventional finned heat sinks, for equivalent volumetric footprints. Additionally, the inherent geometric modularity of TPMS structures enables advanced topological optimization, which remains difficult to implement with traditional geometries. These cumulative advantages underline the growing potential of TPMS-based designs for next-generation thermal management applications, especially in sectors such as electronics, automotive, and aerospace, where performance, compactness, and structural customizability are essential.

The comparative table highlights the superior performance of TPMS, such as Gyroid and Diamond, over conventional fins and metal foams across several thermal and structural criteria. In terms of temperature distribution, TPMS structures offer highly uniform thermal behavior, thanks to the absence of a defined boundary layer, which ensures homogeneous cooling. In contrast, conventional fins often experience hot spots due to limited conduction, while metal foams exhibit uneven heat accumulation, especially near the outlet. Regarding the Nusselt number (Nu), which indicates convective heat transfer performance, TPMS again outperforms the alternatives, particularly when enhanced with nanofluids or topological optimization. Conventional fins show moderate to low performance depending on their geometry, while metal foams are strongly influenced by porosity and generally offer moderate efficiency. However, TPMS tend to produce medium to high pressure drops (ΔP), which can be mitigated through geometry adjustments such as cell stretching. Conventional fins have low to moderate pressure losses, while metal foams suffer from significant pressure drops, due to their high porosity, and risk of clogging. In terms of fluid tortuosity, TPMSs are versatile; this parameter can be finely tuned through structural modifications. Conventional fins exhibit low tortuosity, while metal foams are characterized by high tortuosity, which can hinder smooth laminar flow. From a structural homogeneity standpoint, TPMS, especially Gyroid, are isotropic, promoting uniform cooling in all directions. Conventional fins are inherently anisotropic, and metal foams display random, process-dependent architectures. Concerning additive manufacturing, TPMSs are particularly well-suited for advanced techniques such as SLM, EBM, FDM, and DMLS. This gives them a clear advantage over conventional fins, which are limited in shape diversity, and metal foams, which are difficult to fabricate with high precision. By looking at overall thermal efficiency, TPMS structures consistently deliver superior results, especially when paired with optimized design and enhanced working fluids. Conventional fins and metal foams can be moderately efficient, but their performance is highly dependent on design and porosity. In summary, TPMS structures outperform traditional fins and metal foams in most categories, making them highly promising for next-generation thermal management systems. The main challenge remains managing the higher pressure drops, which must be addressed through careful design optimization.

4. Effects of TPMS Design Variables on Flow and Heat Transfer

The design of TPMS structures plays a pivotal role in governing fluid flow and heat transfer characteristics, making them highly suitable for advanced thermal management systems. Numerous studies have explored the interdependence of TPMS topology, porosity, pore density, tortuosity, and geometric modifications (e.g., cell stretching) to evaluate their thermal and hydraulic behavior under varying operating conditions. Cheng et al. [19] compared four canonical TPMS geometries, W, P, D, and G, and demonstrated that the W-type structure delivers the highest thermal performance per unit of flow resistance. Conversely, the D-type structure exhibits the highest pressure drop. However, both D and G types offer enhanced mechanical strength due to more uniform stress distribution and larger load-bearing surfaces. These results highlighted the critical need to balance thermal efficiency, pressure drop, and structural integrity in TPMS design. To address this multi-objective challenge, Lv et al. [20] implemented a field-driven design approach that locally adapts porosity based on the temperature distribution. Compared to structures with uniform porosity, these optimized TPMS designs (P, G, D) yielded a 94.8% reduction in pressure drop and a 19.2% increase in Nusselt number for the Gyroid, underscoring the benefits of spatial geometric control in improving thermal-hydraulic performance. Tang et al. [21] further examined the topology effects by comparing five TPMS geometries to conventional fin structures through both numerical simulations and experiments. Fischer-Koch S and Diamond geometries exhibited the highest heat transfer enhancements is a 177% increase in Nusselt number, but at the expense of higher-pressure losses. Interestingly, the IWP structure surpassed the Gyroid in thermal performance at higher Reynolds numbers (Re > 1587), while maintaining a lower pressure drop. These findings suggest that TPMS performance rankings are sensitive to flow conditions. Rathore et al. [22] investigated the impact of tortuosity and domain modeling sketched (solid, fluid, microporous) on local temperature fields and the applicability of the local thermal equilibrium (LTE) assumption. Highly tortuous geometries, such as Diamond and Gyroid, showed substantial variations in Nusselt number and temperature gradients depending on the secondary domain treatment, emphasizing the importance of accurate modeling for predictive reliability. The influence of geometric stretching was explored by Ibhadode et al. [4], who reported that elongating TPMS cells along the main flow direction reduces flow resistance but slightly compromises heat transfer due to a reduced surface-area-to-volume ratio. Nonetheless, certain stretched Diamond configurations outperformed conventional plate-fin heat exchangers under laminar flow regimes, indicating their suitability for low-Reynolds-number applications. Kerme et al. [5] emphasized the importance of tuning both cell size and porosity. While Diamond structures generally offered better thermal performance than Gyroids, larger-cell Gyroid configurations (e.g., G1P7) provided the best trade-off between heat transfer and pressure drop, demonstrating that fine morphological tuning enables performance optimization tailored to specific operating regimes. From a material and fluid perspective, Kilic [6] showed that employing hybrid nanofluids (HNA) in silver TPMS structures enhanced thermal performance by 8–12% compared to water, though at the cost of increased pressure losses. Similarly, Saghir et al. [7,9] found that Gyroid TPMS outperformed metallic foams in terms of heat dissipation and temperature uniformity under forced convection, despite inducing approximately 18% higher pressure drop. These findings reinforce the superior thermal capabilities of TPMS structures, particularly where compactness and uniform temperature distribution are critical. Overall, the literature highlights that TPMS topology, geometric tuning (e.g., stretching, porosity grading), and material–fluid pairing are levers for optimizing flow and heat transfer performance. Future designs should aim to integrate these factors through multi-objective topology optimization and manufacturability-aware modeling to fully harness the thermal advantages of TPMS structures in real-world applications.

4.1. Parameter Definitions

The sheet gyroid, skeletal gyroid, and combined gyroid TPMS structures proposed by the authors were compared based on numerical simulations. The comparison involved two-dimensional visualizations of key parameters such as flow velocity and temperature distribution (

Table 4. Utilization of Reynolds Number in TPMS-Based Heat Sink Studies.

Author(s)	Use of Reynolds Number (Re)
Rathore et al. [22]	Studied a wide range of Reynolds numbers (0.01 to 100) to analyze thermal behavior in TPMS mini-channels.
Ibhadode [4]	Focused on laminar flow regime; assessed the impact of cell stretching under low Reynolds number conditions.
Saghir et al. [7]	Evaluated the impact of coolant type under various flow rates, implicitly involving Reynolds number variation.
Kilic [6]	Investigated turbulent flow regimes; tested performance under varying flow rates and Reynolds numbers.
Saghir et al. [9]	Compared metallic foam and TPMS (Gyroid) structures under different flow conditions influenced by Re values.
Kerme et al. [5]	Examined TPMS structures under various flow rates; Nusselt number and pressure drop correlated with Re.

) across different regions of the model, as well as three-dimensional visualizations of flow trajectories. The assessment of Hydraulic performance has been done based on the values of the Reynolds number, the friction factor, and the pressure drop.

The Nusselt number (Nu) is a dimensionless quantity that characterizes the ratio of convective to conductive heat transfer at a fluid–solid interface. In the context of TPMS structures, it serves as a key indicator for evaluating the efficiency of thermal energy exchange within their complex geometries. Key characteristics from the literature include:

The Nusselt number (Nu) in triply periodic minimal surface (TPMS) heat exchangers is highly dependent on several design and operating parameters.

• Geometry: plays a decisive role, as different morphologies present distinct surface-to-volume ratios; for instance, Kerme et al. [5] showed that Diamond-type structures generally achieve higher Nu values than Gyroid designs under identical conditions.
• Flow regime also influences performance, with Nu increasing significantly as the Reynolds number rises within the range 0.01–100, particularly when local thermal equilibrium (LTE) is maintained, as confirmed by Rathore et al. [22].
• Material properties are another contributing factor—high thermal conductivity materials, such as silver, enhance Nu and provide more uniform temperature distribution compared to aluminum, according to Kilic et al. [6].
• Fluid type impacts heat transfer, with hybrid nanofluids (HNA) outperforming pure water due to their superior thermal conductivity.
• Porosity and cell size directly affect convective behavior; Kerme et al. [5] observed that structures with larger pores (e.g., G1P7) deliver higher Nu values and improved thermal performance, albeit at the cost of increased pressure drop. These factors underscore the importance of Nu as a key indicator for evaluating and optimizing TPMS heat sinks in a wide range of thermal management applications.

4.2. Porosity

Porosity (ε), also known as the volume fraction, is defined as the ratio of the fluid volume to the total volume of the system under consideration. To enable fair comparisons of flow and heat transfer performance across different TPMS topologies, researchers often standardize the porosity by maintaining it at equivalent values. This strategy helps isolate the influence of geometric configuration from that of the void fraction. In both uniform and graded TPMS structures, porosity can be precisely controlled to remain consistent across different designs. Furthermore, several studies have investigated the effect of varying porosity within a single TPMS topology to assess its impact on fluid flow behavior and thermal transport efficiency.

4.2.1. Equivalent Porosity

Table 5. Summary of Studies Addressing the Effect of Porosity on TPMS-Based Heat Transfer.

Author(s)	Year	TPMS Type(s)	Porosity Studied	Key Findings
Kerme et al. [5]	2023	Gyroid, Diamond	Multiple porosities (e.g., G1P7)	Higher porosity = lower pressure drops, but reduced heat transfer. G1P7 shows the best overall balance.
Rathore et al. [22]	2023	Diamond, Gyroid, Primitive, IWP	Implicit via secondary domain	Impact of solid vs microporous zones on local thermal equilibrium and heat transfer rate.
Orakwe et al. [23]	2022	TPMS (Gyroid-like)	Porosity optimized topologies	Porosity distribution controlled via conformal and field-driven designs for enhanced performance.
Tang et al. [21]	2023	Multiple TPMS	Uniform vs non-uniform porosity	Non-uniform porosity enhances local cooling without incurring a significant penalty in pressure drop.
Cheng et al. [19]	2023	Multiscale TPMS	Varying hierarchical porosity	Hierarchical porosity facilitates improved heat transfer in the natural convection regime.
Chen et al. [8]	2022	Gyroid	Variable porosity (α parameter)	Geometric tuning of porosity enhances thermal-hydraulic balance.

provides a comparative synthesis of recent studies on the effect of porosity in TPMS structures on thermal and hydraulic performance. In the study by Kerme et al. [5], Gyroid and Diamond structures with varying porosities are evaluated. It is shown that higher porosity reduces pressure drop but also lowers heat transfer efficiency; the G1P7 configuration offers the best overall balance. Rathore et al. [22] investigate the impact of solid and microporous zones within Diamond, Gyroid, Primitive, and IWP structures, revealing that the spatial distribution of domains significantly influences local thermal equilibrium. Orakwe et al. [23] highlight the importance of optimized porosity distribution through conformal and field-driven designs, which enhance both thermal performance and pressure management. Similarly, Tang et al. [21] show that non-uniform porosity can enhance local cooling without significantly increasing pressure drop. Cheng et al. [19] went one step further by studying multiscale TPMS, demonstrating that hierarchical porosity is particularly effective under natural convection regimes. Finally, Chen et al. [8] introduce geometric tuning via an α parameter to locally adjust porosity in Gyroid structures, achieving an optimal balance between heat transfer and pressure drop. Overall, these studies confirm that precise control of porosity, whether uniform, locally tuned, or hierarchical, is a key factor in improving the performance of TPMS-based heat exchangers.

The flow characteristics and heat transfer performance of cooling channels in heat sinks. Similarly, Cheng et al. [19] observed that, for TPMS structures, adjusting porosity affects both flow resistance and thermal performance, with certain TPMS types offering superior heat transfer (Figure 3) while maintaining manageable pressure drops. Tang et al. [21] highlighted that increasing the internal contact area within TPMS-based cooling channels enhances conduction, which in turn positively impacts convective heat transfer. Furthermore, Kerme et al. [5] demonstrated that optimizing porosity alongside cell size in TPMS structures significantly improves thermal dissipation and flow distribution, resulting in better overall heat sink performance. Collectively, these studies suggest that porosity must be carefully tailored in the design of cooling channels to achieve an optimal balance between heat transfer enhancement and fluid flow resistance, thereby ensuring efficient thermal management in heat sink applications.

Al-Ketan et al. [25] attributed the variation in pressure loss within the channel to differences in pore size. As illustrated in Figure 4, pore size was quantified by the diameter of the largest sphere (shown in red) that can pass through the narrowest constriction of the TPMS topologies. It was observed that the sheet-Gyroid structure, which exhibited the highest-pressure loss, also had the smallest pore size, followed by the solid-Diamond and solid-Gyroid structures.

Al-Ketan et al. [10] have indeed studied the effects of wall thickness on fluid velocity and pressure drop in TPMS structures. Their results indicate that thicker walls reduce pore size, leading to higher flow resistance. For example, in the Split P lattice TPMS network, reducing wall thickness from 1.2 mm to 0.8 mm and then to 0.4 mm increased the total surface area and porosity, thereby promoting higher fluid velocities and a more homogeneous flow distribution. Conversely, thicker walls generate smaller pores, which cause higher localized velocities and greater velocity gradients, potentially leading to significant variations in flow speed throughout the structure. In summary, research highlights the complex relationship between wall thickness, pore size, and fluid velocity in TPMS structures. While thicker walls may improve the surface area and enhance thermal performance, they also increase flow resistance, requiring a carefully considered trade-off to optimize both thermal and hydraulic performance.

4.2.2. Varying the Porosity

Porosity is one of the most influential parameters in the design of TPMS-based heat sinks and cooling channels, as it directly affects both thermal performance (Figure 5). Numerous studies have investigated its impact across various geometries and flow conditions. For instance, Alteneiji et al. [26] and Samson et al. [27] had shown that decreasing porosity enhances convective heat transfer but leads to a higher pressure drop, particularly under forced convection regimes. To address this trade-off, Lv et al. [20] and Tang et al. [13] have implemented gradient or field-driven porosity control strategies to improve local cooling effectiveness while limiting hydraulic losses. In a comparative study, Cheng et al. [19] demonstrated that the influence of porosity variation on flow and thermal behavior is highly dependent on the TPMS topology, with W-type structures offering better thermal performance at moderate porosities. Kerme et al. [5] and Rathore et al. [22] further investigated this relationship, indicating that specific combinations of porosity and cell size can enhance Nusselt numbers while keeping pressure drops within acceptable limits. Finally, Saghir et al. [7,9] emphasized that applying porosity gradients, particularly in combination with optimized coolants (e.g., ethylene glycol–water mixtures), promotes uniform temperature distributions even at low Reynolds numbers. Overall, both uniform and graded porosity designs are essential tools for tailoring TPMS structures to achieve optimal thermal and hydraulic performance.

4.3. Wall Thickness

Wall thickness is a crucial design parameter influencing both the thermal and hydraulic performance of TPMS structures used in cooling devices. Numerous studies have shown that increasing wall thickness generally enhances the overall thermal conductivity of the structure by increasing the solid surface area available for heat conduction. For instance, Attarzadeh et al. [28] demonstrated that, in Schwarz D structures, greater wall thickness improves thermal dissipation under laminar flow conditions. However, this increase also reduces porosity, which can significantly raise the pressure drop and thus the hydraulic resistance of the system, negatively affecting coolant flow. Further emphasizes, by Yeranee et al. [29], that wall thickness affects laminar and turbulent flow regimes differently and must be optimized for specific operating conditions.

In summary, wall thickness must be carefully balanced to maximize heat transfer while minimizing pressure losses, especially in applications where energy efficiency and reduced pumping power are critical. Recent advances in additive manufacturing now enable the fabrication of TPMS structures with spatially variable wall thicknesses, paving the way for optimized, functionally graded designs. Tang et al. [21] highlighted that increasing wall thickness in specific TPMS topologies (Figure 6), such as Gyroid and Diamond, augment the heat exchange surface and enhances conduction. However, these thermal improvements are typically accompanied by a rise in flow resistance, resulting in greater pressure drops and higher pumping energy requirements. Kerme et al. [5] investigated several TPMS configurations with varying cell sizes and wall thicknesses. Their findings revealed that structures with thicker walls, particularly Diamond, exhibited superior thermal dissipation. Nonetheless, this thermal benefit must be weighed against the associated hydraulic load, which can become prohibitive in certain cases. Similarly, Saghir et al. [9] reported that thicker walls in Gyroid structures promote a more uniform temperature distribution, effectively reducing local thermal gradients and mitigating hot spots. They have also noted that the improved thermal has been accompanied by a significant increase in pressure drop yet. Expanding the analysis to phase change applications, Qureshi et al. [30] explored the influence of wall thickness in TPMS structures (

Table 6. The heat transfer surface area, porosity, and hydraulic diameter of the models by Tang et al. [21].

TPMS & Fins	Heat Exchange $\mathbf{Area}({\mathbf{m}}^2$)	Porosity (%)	Hydraulic Diameter (m)	The Perforation Area Ratio, P (%)
Gyroid	1.11 × ${10}^{-2}$	82.0	9.44 × ${10}^{-3}$	10.78
Diamond	1.33 × ${10}^{-2}$	77.9	7.48 × ${10}^{-3}$	/
IWP	1.25 × ${10}^{-2}$	80.0	8.20 × ${10}^{-3}$	15.15
Primitive	8.91 × ${10}^{-3}$	86.2	1.24 × ${10}^{-3}$	22.28
Fischer-Koch-S	1.81 × ${10}^{-2}$	70.6	5.00 × ${10}^{-3}$	/
Fins	1.09 × ${10}^{-2}$	82.0	9.62 × ${10}^{-3}$	82.00

), incorporating phase change materials (PCMs). Their results confirmed that wall thickness directly affects the heat transfer rate, the melting duration of the PCM, and consequently, the overall efficiency of latent heat storage systems. Collectively, these studies indicate that wall thickness cannot be optimized in isolation. It must be considered in conjunction with other parameters such as porosity, unit cell size, and geometric topology to achieve an optimal trade-off between thermal performance and hydraulic efficiency.

In Table 6, several TPMS geometries (Gyroid, Diamond, IWP, Primitive, Fischer-Koch-S) as well as a conventional fin structure based on key heat exchange parameters: exchange area, porosity, hydraulic diameter, and perforation area ratio (P), are compared. We can conclude that the exchange area geometries vary significantly; Fischer-Koch-S showing is the largest area (1.81 × 10⁻² m²), which could enhance heat dissipation, while Primitive has the smallest area (8.91 × 10⁻³ m²). Porosity follows a somewhat inverse trend: Primitive has the highest porosity (86.2%), which favors fluid flow but may reduce solid contact surface, whereas Fischer-Koch-S has the lowest porosity (70.6%), indicating a denser structure.

The hydraulic diameter, an indicator of the effective fluid passage size, also varies. Primitive has the smallest hydraulic diameter (1.24 × 10⁻³ m), which could cause a significant pressure drop, while Gyroid and the fins have approximately the same diameter, about 9.4 × 10⁻³ m. This suggests that some TPMS structures, as Primitive ones, might generate higher hydraulic resistance than conventional fins. Finally, the perforation area ratio (P) is much lower for the TPMS, generally below 23%, whereas it is extremely high for conventional fins (82%). This parameter is crucial as it affects flow resistance: a high ratio may increase convection with a pressure drop. In summary, this table highlights a typical trade-off between exchange surface area, porosity, and hydraulic characteristics. TPMS offer varied surfaces with high porosities that favor fluid circulation but generally have a much lower perforation area ratio compared to conventional fins, which can influence their thermal and hydraulic performance differently. Some structures, as Primitive, despite having high porosity, may have a smaller hydraulic diameter, potentially limiting their use in forced convection due to increased flow resistance. The table would benefit from including thermal performance indicators and pressure drop data to better assess these trade-offs.

For the heat exchanger configuration consisting of two domains separated by infinitesimally thin walls, Iyer et al. [31] found that the sheet-Primitive structure exhibited the lowest Nusselt number (Figure 7), indicating the least effective heat transfer. In contrast, the sheet-Diamond structure demonstrated the best heat transfer performance, followed by the C (I2-Y**) topology, as illustrated in Figure 7a. In terms of flow resistance, the sheet-Neovius structure showed the highest friction factor, followed by the sheet-FRD design. The other TPMS geometries exhibited relatively similar pressure drops across a Reynolds number range of 0 to 300, as presented in Figure 7b. In a complementary study, Kaur and et al. [32] reported lower frictional pressure losses in their simulations, attributing this to the semi-infinite curvature profiles of TPMS structures, which promote smoother flow paths and reduced hydrodynamic resistance.

Tang et al. [21] highlighted that increasing the wall thickness in certain TPMS topologies, such as Gyroid and Diamond, enhances the heat exchange surface area, thereby improving solid-phase conduction. However, this thermal enhancement is often accompanied by a significant increase in pressure drop, which negatively affects the energy consumption required for fluid pumping. Saghir et al. [9] confirmed that thicker walls in Gyroid structures contribute to a more uniform temperature distribution, reducing local thermal gradients and preventing hot spots; however, the benefit also results in a substantial rise in pressure drop. Al-Ketan et al. [25] investigated the effect of pore size on pressure losses and demonstrated that TPMS structures with smaller. Pores, particularly a sheet-Gyroid configuration, induce higher hydraulic resistance. Meanwhile, Kaur et al. [32] emphasized that certain TPMS topologies can achieve lower pressure losses due to their semi-infinite curvatures, which promote smoother fluid flow and contribute to better overall energy efficiency. All studies agree that balancing wall thickness, pore size, and topology is necessary to improve heat transmission and hydraulic performance in cooling applications. Wall thickness plays a crucial role in determining the effective thermal conductivity of TPMS-based heat exchangers. As wall thickness increases, the solid volume fraction also rises, thereby enhancing conductive heat transfer through the solid matrix. This is particularly important for TPMS structures, where heat conduction predominantly occurs through the solid phase. Increasing wall thickness enhances the number of conduction pathways, thereby raising the effective thermal conductivity (Figure 8), especially in metallic TPMS fabricated from high-conductivity materials such as aluminum or silver. However, this improvement comes at a trade-off: thicker walls reduce the overall porosity, which decreases the fluid–solid interfacial area available for convection. This reduction can lower the convective heat transfer coefficient, particularly in low-Reynolds-number flows where surface contact plays a critical role. As highlighted by Kilic [6] and Saghir et al. [7], an optimal wall thickness exists that balances solid-phase conduction with convective heat transfer performance. Furthermore, the effect of wall thickness is topology-specific; Gyroid and Diamond geometries respond differently to thickness variation due to their distinct curvature profiles and surface characteristics. Recent work also indicates that wall thickness can serve as a tunable parameter in functionally graded TPMS, allowing localized enhancement of thermal conductivity in targeted regions.

4.4. Unit Cell Size

The unit cell size in TPMS structures (

Table 7. Geometric characteristics of the investigated gyroid and diamond samples by Kerme et al. [5].

Parameters			SAMPLES
		Gyroid			Diamond
	G3P6	G3P7	G3P8	G1P7	D1P7	D3P7
Porosity	0.60	0.70	0.80	0.70	0.70	0.70
Unit Cell Size (mm)	12.5	12.5	12.5	15	15	12.5
Sample dimensions
Length	37.5	37.5	37.5	37.5	37.5	37.5
Width	37.5	37.5	37.5	37.5	37.5	37.5
Height	12.7	12.7	12.7	12.7	12.7	12.7
Surface Area (mm2)	6228.4	5584.3	4622.4	4727.6	5611	6203.1
Specific SurfaceArea (${\mathrm{m}}^{-1}$)	3487.4	3126.8	2588.2	2645.5	3141.8	3473.3

) plays a critical role in determining hydraulic pressure drop. Smaller cells, while increasing the heat exchange surface area, also create greater resistance to fluid flow due to a higher density of internal walls and more complex flow paths. Conversely, larger cells facilitate a more direct and less tortuous flow, thereby reducing pressure drop. Ibhadode [4] demonstrated that stretching the cells along the flow direction reduces tortuosity and facilitates fluid circulation. Similarly, Kerme et al. [5] showed that increasing the cell size in Gyroid and Diamond structures lowers reduces hydraulic resistance, although it may also compromise thermal efficiency. Therefore, optimizing cell size is essential for achieving a balance between thermal and hydraulic performance in TPMS-based heat exchangers. Several studies in the literature have investigated the impact of TPMS unit cell size on both thermal and hydraulic performance, highlighting important design trade-offs. Ibhadode [4] emphasized that increasing the unit cell size reduces the tortuosity of fluid paths, resulting in more direct flow and lower pressure drop. However, this geometric adjustment can also shorten thermal conduction paths and reduce the available surface area for convective heat transfer, thereby decreasing overall thermal efficiency. Likewise, Kerme et al. [5] conducted a comparative analysis of Gyroid and Diamond TPMS structures and found that while larger unit cells reduce flow resistance, they also offer a smaller internal surface area for heat exchange. Their study identified the G1P7 configuration as an effective compromise between low hydraulic resistance and high Nusselt numbers. Saghir et al. [7], through experimental evaluation of TPMS heat sinks using water–glycol mixtures, observed that although cell geometry significantly affects the temperature distribution, larger cell structures can support more uniform heat extraction under specific flow conditions. Kilic [6], using both numerical and experimental methods, further confirmed that TPMS structures with optimized cell configurations, particularly when combined with high-conductivity materials or nanofluids, can significantly enhance cooling performance while maintaining pressure drops within acceptable limits. Collectively, these studies demonstrate that careful tuning of unit cell size is essential to maximize the thermofluidic efficiency of TPMS-based heat exchangers.

5. Comparison Between Conventional Cooling and TPMS-Based Structures

Conventional cooling structures, such as plate-fin heat sinks, pin-fin arrays, and metallic foams, have been widely adopted due to their relatively simple design, ease of manufacturing, and effective heat dissipation capabilities. However, these traditional designs often face limitations in achieving optimal thermal performance while maintaining low pressure drops, particularly in compact systems or under high heat flux conditions. In contrast, TPMS-based structures have emerged as promising alternatives, owing to their smooth, continuous surfaces and highly interconnected porous networks. The literature demonstrates that TPMS geometries such as Gyroid, Diamond, and Schwarz surfaces provide enhanced surface-to-volume ratios and more uniform fluid flow distribution, resulting in superior heat transfer coefficients and more homogeneous temperature fields compared to conventional designs. Studies by Saghir et al. [9] and Kerme et al. [5], for instance, have shown that TPMS structures outperform metallic foams by achieving higher Nusselt numbers and better thermal uniformity, albeit with a moderate increase in pressure drop. Furthermore, the tunability of key TPMS parameters such as porosity, unit cell size, and wall thickness enables targeted optimization to balance thermal and hydraulic performance according to specific application requirements. Additive manufacturing techniques further facilitate the fabrication of these complex geometries, which would be difficult or impossible to produce using conventional methods. Nonetheless, challenges remain, including increased manufacturing complexity and cost, as well as pressure drop penalties in certain configurations, which can hinder widespread industrial adoption. Despite these limitations, TPMS-based cooling structures represent a substantial advancement over conventional solutions, particularly for high-performance and miniaturized thermal management systems.

5.1. Two-Fluid Heat Exchangers

In

Table 8. Reviews of the investigation of the TPMS structures for two fluid heat exchangers.

Reference	TPMS Geometry	Method	Flow Conditions	Significant Findings
Saghir et al. [9]	Gyroid vs Metal Foam	Numerical (Navie—Stokes & Energy), Experimental	Laminar, various heat fluxes and flow rate.	Gyroid showed more uniform temperature and better cooling, despite an 18% higher pressure drops
Kerme et al. [5]	Diamond, Gyroid	Numerical & Experimental	Varying flow rates, constant heat flux	Diamond outperformed Gyroid; G1P7 Gyroid balanced heat transfer and pressure drop well
Attarzadeh et al. [28]	Schwarz D	3D Steady-State CHT Simulations (Numerical)	Laminar, various gas velocities	One geometry with a specific wall thickness yielded optimal heat recovery performance
Qureshi et al. [30]	Primitive, Gyroid, IWP	Numerical (Porosity & Grading Study)	Latent Heat Storage, varied porosities	Lower porosity and positive grading reduced melting time; topology significantly impacts heat transfer
Feng et al. [14] (review)	Multiple (General TPMS)	Literature Review + Design Framework	General (laminar/turbulent applications)	TPMS is functional across fields; AM and design methods remain key challenge

, we provide a concise overview of recent key studies comparing different TPMS geometries and their thermal-hydraulic performance under various flow and thermal conditions. Saghir et al. [9] highlight the superior cooling uniformity and temperature distribution of the Gyroid structure compared to traditional metal foam, despite incurring an 18% higher pressure drop, which emphasizes the trade-off between thermal performance and hydraulic resistance. Kerme et al. [5] compare Diamond and Gyroid geometries, finding that the Diamond structure outperforms Gyroid overall, though a specific Gyroid variant (G1P7) achieves a good balance between heat transfer and pressure drop, showcasing the importance of geometry tuning within the same TPMS family. Attarzadeh et al. [28] focus on Schwarz D geometry, demonstrating that wall thickness is a critical parameter for optimizing heat recovery in laminar gas flows, which points to the sensitivity of TPMS performance to geometric detail beyond just shape. Qureshi et al. [30] investigate porosity and grading effects on Primitive, Gyroid, and IWP structures for latent heat storage applications, confirming that lower porosity and positive porosity grading significantly improve melting time, thus reinforcing the impact of internal structural gradation on phase change heat transfer efficiency. Finally, Feng et al. [14] provide a broad literature review and design framework for TPMS, emphasizing their versatility across laminar and turbulent applications, while noting that additive manufacturing capabilities and advanced design methodologies remain critical challenges for practical deployment.

Overall, the table effectively summarizes the diversity of TPMS studies, underlining the key role of geometry, porosity, and manufacturing constraints on thermal-hydraulic performance, but it could benefit from a more explicit comparison of performance metrics and operational limits to better guide design choices.

5.2. Forced Convective Heat Sinks

Several studies emphasize the importance of forced convection in cooling systems such as heat sinks, particularly in advanced structures like TPMS. Saghir et al. [33] highlight that forced convection enables more uniform and efficient heat transfer by minimizing the formation of thermal boundary layers (Figure 9), which can impede heat dissipation. Cheng et al. [19] demonstrate that a controlled increase in fluid velocity, characteristic of forced convection, enhances heat transfer by improving the convection coefficient and overall thermal performance. Tang et al. [21], through comparisons between TPMS structures and conventional fin-based designs, show that forced convection is essential for maximizing the Nusselt number and thereby boosting cooling capacity in high heat flux applications such as electronics and aerospace. Kilic et al. [6] adds that although turbulent flow induced by forced convection can lead to increased pressure drop, the use of nanofluids helps sustain efficient heat transfer. Finally, Yeranee et al. [29] note that forced convection, when combined with optimized geometric parameters such as porosity and unit cell size in TPMS structures, offers a critical balance between enhanced heat transfer and manageable pressure losses. Therefore, forced convection is widely adopted because it accelerates fluid movement, improves thermal dissipation, and ensures effective temperature control in electronic and industrial devices.

Several authors have extensively studied the specific applications of TPMS structures in heat sinks, demonstrating their significant potential in advanced thermal management systems. Saghir et al. [7,9,33] highlighted the superior thermal performance of TPMS heat sinks, showing a uniform temperature distribution and enhanced heat transfer, particularly when combined with high-conductivity materials and nanofluids. Cheng et al. [19] developed rapid customization methods for TPMS topologies, enabling opti mization of porosity, pore size, and mechanical strength to design efficient and robust heat sinks. Tang et al. [21] experimentally and numerically compared various TPMS ge- ometries with conventional fins, concluding that TPMS structures outperform traditional designs in terms of convective heat transfer, despite incurring higher pressure drops. Lv et al. [20] introduced a field-driven design approach to locally adjust porosity within TPMS liquid-cooled heat sinks, significantly reducing pressure drop while maintaining or improving thermal performance. Kilic et al. [6] investigated the use of nanofluids in TPMS heat sinks under turbulent flow, finding enhanced thermal efficiency at the cost of increased pressure loss, with silver-based structures showing particularly promising results. Yeranee et al. [29] provided a comprehensive review of TPMS cooling channels, emphasizing that complex geometries such as Gyroid and Diamond outperform convenconfirmtional structures thermally, albeit with higher hydraulic resistance. Finally, Orakwe et al. [23] proposed an innovative thermo-fluidic topology optimization combined with con formal latticing techniques to design TPMS heat sinks compatible with additive manufacturing, validating their designs experimentally with laser powder bed fusion proto types. Collectively, these studies underline the crucial role of design parameters such as porosity, cell size, and material choice in tailoring TPMS heat sinks for specific applications, balancing enhanced heat transfer with manageable pressure drops. The studies by Feng et al. [14] and Kerme et al. [5] demonstrated that forced convection plays a crucial role in the thermal management of TPMS structures by significantly reducing hot spots within these complex configurations. They also showed that heat flux removal increases with fluid velocity up to an optimal point, beyond which further increases cause pressure losses that offset the thermal gain. Therefore, the ratio between thermal performance and pressure drop emerges as a key criterion for evaluating the overall efficiency of cooling devices. Specifically, Kerme et al. [5] found that certain TPMS structures, such as Gyroid and Diamond, when combined with forced convection, achieve an optimal balance between cooling capacity and energy consumption related to pumping power. Conduction alone is insufficient to effectively cool electronic components that generate large amounts of heat, such as CPUs or insulated-gate bipolar transistors (IGBTs). Conduction relies solely on heat transfer through a solid material, which limits its ability to dissipate heat quickly to the environment. On the other hand, natural convection, although it allows heat transfer to the surrounding fluid without mechanical intervention, is generally too slow, especially in confined spaces, where air circulation is limited. This is why forced convection is preferred: it uses an imposed fluid flow, often generated by a fan or pump, to accelerate heat transfer. According to Lv et al. [20], this technique can improve thermal efficiency by 2 to 5 times compared to natural convection, making heat dissipation significantly more effective in high-power electronic systems. Forced convection in TPMS heat sinks aims to achieve several key objectives to optimize thermal management. Firstly, it enables rapid heat removal by significantly reducing surface temperatures, which is critical for the reliability of electronic components. Secondly, it provides precise control of the flow by optimizing the speed and direction of the cooling fluid, thereby enhancing heat transfer efficiency. Another important goal is to prevent stagnant zones within the structure, improving thermal uniformity and avoiding hot spots. Additionally, forced convection seeks to establish an optimal balance between thermal and hydraulic performance by maximizing the Nusselt number (Nu) while minimizing pressure drop (ΔP). Finally, it must be easily integrated into real systems, being compatible with devices such as fans and pumps for effective and practical operation. The use of TPMS (Triply Periodic Minimal Surfaces) structures under forced convection offers several significant advantages due to their unique geometry. Their high specific surface area and smooth, continuous walls promote intense heat exchange between the fluid and solid surfaces. Unlike traditional finned designs, which can generate dead zones or unidirectional flow, TPMS structures enable multidirectional flow, improving thermal uniformity throughout the volume. Compared to metallic foams, TPMS also offer better geometric control, reducing pressure drops while maintaining excellent thermal performance. These benefits have been highlighted in studies by Tang et al. [21], Saghir et al. [33] and Yeranee et al. [29], which demonstrate that integrating TPMS into forced convection heat exchangers significantly enhances heat transfer while requiring less pumping power than conventional solutions. Several authors have employed the globally averaged Nusselt number as a key metric to compare the heat transfer performance of TPMS-based structures. Kerme et al. [5] used this parameter to evaluate six TPMS designs, including Gyroid and Diamond structures. They found that Diamond geometries consistently exhibited higher Nusselt numbers, indicating better thermal efficiency under forced convection. Similarly, Saghir et al. [9] conducted a comparative analysis between the metallic foam and the Gyroid TPMS structure. Their findings showed that the Gyroid not only delivered a more uniform temperature distribution but also achieved a higher Nusselt number, confirming its superior heat transfer capabilities despite a slightly higher pressure drop. Lv et al. [20] also utilized the Nusselt number to evaluate the effect of varying flow velocities on TPMS heat sinks, observing that thermal performance increased with velocity up to an optimal threshold. Yeranee et al. [29] reinforced the importance of the globally averaged Nusselt number as a standard benchmark for evaluating and comparing the thermal efficiency of different TPMS topologies across a wide range of porosities, wall thicknesses, and flow regimes.

5.3. Other Applications Related to Flow, Heat, and Mass Transfer

Triply Periodic Minimal Surface (TPMS) structures have demonstrated considerable potential in various applications beyond conventional heat sinks, particularly in systems involving combined heat, fluid, and mass transfer. Yeranee et al. [29] highlighted that internal cooling channels based on TPMS geometries such as Gyroid and Diamond significantly enhance thermal performance due to their high surface area and smooth curvature, which promote optimized fluid flow. However, this geometric complexity also leads to increased pressure drop, requiring a trade-off between thermal efficiency and hydraulic resistance. Moreover, Lv et al. [20] proposed a locally variable porosity design tailored to the temperature field distribution, which enables a reduction in pressure drop by up to 94.8% while improving heat transfer in TPMS-based heat exchangers. These developments confirm the relevance of TPMS for efficient liquid cooling systems, particularly in power electronics applications. Furthermore, Feng et al. [14] emphasized the role of additive manufacturing in enabling the fabrication of these complex structures, paving the way for multifunctional materials that simultaneously optimize heat transfer, fluid permeability, and mass transport, especially in biomedical and catalytic applications. Tang et al. [21] demonstrated that various TPMS topologies can be tailored to maximize convective heat transfer in diverse systems such as batteries and aerospace components, while also considering hydraulic resistance. Finally, Qureshi et al. [30] investigated the integration of phase change materials (PCM) into functionally graded TPMS structures, showing that adjusting porosity can accelerate PCM melting and thereby optimize thermal energy storage and management. These studies demonstrate the broad versatility of TPMS in systems combining heat transfer, fluid flow, and mass transport, offering promising perspectives for the development of advanced thermal-fluidic devices. TPMS structures are increasingly recognized for their potential to enhance the performance of high-efficiency heat exchangers, particularly in air/water and gas/liquid configurations, as well as in low- to medium-temperature heat recovery systems. Their complex geometry and large specific surface area promote efficient heat transfer while maintaining low pressure drops. Attarzadeh et al. [28] conducted a numerical study on heat exchangers based on the Schwarz D structure, demonstrating that this topology optimizes heat transfer in laminar flow gas heat recovery systems. Their analysis, covering various wall thicknesses and flow velocities, identified an optimal configuration that effectively balances thermal exchange and hydraulic resistance. This research highlights the promising role of TPMS in designing compact and high-performance thermal devices suitable for emerging industrial applications. The integration of phase change materials (PCMs) into TPMS architectures represents a promising advancement for thermal energy storage (TES) systems. Due to their controllable porous geometry, TPMS structures enable improved distribution and heat transfer of PCMs, enhancing the management of both melting and solidification processes. Qureshi et al. [30] demonstrated that the porosity and topology of TPMS significantly influence the thermal behavior of PCMs, reducing melting time and improving the overall energy efficiency of the system. These findings highlight the potential of functionalized TPMS structures to develop more efficient and compact TES systems. Triply Periodic Minimal Surfaces (TPMS) have gained attention not only for their thermal management capabilities but also for their applications in fluid filtration, separation, and catalysis. Their unique, continuous, and highly interconnected porous networks provide large surface areas and optimized flow pathways, making them ideal candidates for high-performance filters and catalyst supports. For instance, TPMS-based structures have been employed as dry catalysts and advanced filtration media, demonstrating enhanced mass transfer and reduced pressure drop compared to conventional porous materials (Feng et al. [14]; Yeranee et al. [29]). These geometries enable improved fluid distribution and catalytic activity due to their smooth curvature and uniform pore size, which promote efficient interaction between fluids and solid surfaces. Moreover, additive manufacturing techniques have facilitated the fabrication of complex TPMS filters tailored for specific fluid treatment applications, paving the way for multifunctional porous devices that combine filtration, catalytic conversion, and heat management (Feng et al. [14]). Additional research papers focused on heat transfer and fluid flow using TPMS [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68]. Both experimental and numerical approaches were conducted. The findings are like the one discussed earlier.

6. Conclusions and Perspectives

Recent studies confirm the strong potential of TPMS structures, particularly the Gyroid, Diamond, and Primitive variants, for enhancing the performance of heat exchangers under both natural and forced convection. Mastery of geometric parameters (e.g., cell size, porosity, and stretching), materials (such as aluminum and silver), and working fluids (including water and nanofluids) is crucial for optimizing the trade-off between heat transfer efficiency and pressure drop. Advances in multiscale topology optimization and additive manufacturing offer unique opportunities to design custom heat exchangers tailored to specific industrial constraints. However, several challenges remain, including the integration of variable porosity and topology, the consideration of real dynamic operating conditions, comprehensive experimental validation, and cost-effectiveness analysis. These issues represent key directions for future research aimed at promoting the industrial adoption of TPMS-based heat exchangers and enabling their deployment in high-energy-efficiency applications.