Performance Evaluation of Triply Periodic Minimal Surface Heat Exchangers Using Nanofluids at High Flow Rates for Enhanced Energy Efficiency

Abstract

Triply Periodic Minimal Surface (TPMS) heat exchangers have attracted significant attention for their high surface area and effective thermal performance. This study evaluates the performance of TPMS heat exchangers under turbulent flow conditions using aluminum (Al) and silver (Ag) materials with pure water and nanofluid as working fluids. The implementation of Ag TPMS structures resulted in approximately 15% enhancement in thermal performance compared to Al structures due to superior thermal conductivity. The introduction of nanofluid (0.6% volume concentration) improved overall heat transfer efficiency by 12% compared to pure water. Performance evaluation criteria (PEC) analysis demonstrated that Ag TPMS structures achieved up to 30% higher values than Al structures. Temperature homogeneity analyses revealed significant improvements, with Al TPMS structures showing a 24% reduction in temperature variation when using nanofluid, while Ag TPMS structures exhibited up to 40% better temperature uniformity. Computational fluid dynamics analyses validated the experimental findings with deviations less than 7%, confirming the model’s reliability. These results demonstrate the significant potential of TPMS structures in high-performance cooling applications and provide valuable insights for future heat exchanger designs. The enhanced efficiency of thermal management system (TMS) heat exchangers using nanofluids contributes to reduced energy consumption, supporting environmentally conscious decision-making in industrial and energy systems.

1. Introduction

Thermal management has become a crucial topic in contemporary engineering applications. Ensuring efficient heat dissipation across a wide range of domains, from electronic devices to architectural elements, is extremely important for optimal system performance, increasing energy efficiency, and reducing adverse thermal effects. Consequently, the development of a comprehensive and effective thermal management strategy constitutes a key factor in the successful implementation of engineering solutions [1,2,3]. In this context, heat exchangers are one of the widely used solutions to optimize heat transfer [4]. In recent years, the search for innovative and more efficient structures beyond traditional heat exchanger designs has continued.

TPMS is one of the prominent structures in this study. TPMS are cellular structures that have mathematically defined, regular, and minimal surface properties, with complex and continuously curved surfaces [5]. TPMS structures provide several benefits for heat transfer applications. These include low pressure drop, adjustable porosity, and improved heat transfer characteristics. These advantageous properties are attributed to the ability of structures to resemble biological structures found in nature [6,7]. In recent years, researchers have actively explored TPMS applications across various domains. These include materials science, biomedical engineering, and heat transfer. The focus is primarily on their unique structural and thermal properties.

Multiscale topology optimization enhances cellular structures through simulations and experiments. This optimization provides high thermal conductivity, large convective surface area, and rapid cooling capability [8]. Furthermore, comprehensive analyses of the thermal performance and potential applications of TPMS structures have been conducted, particularly proposing an innovative heat exchanger design that incorporates the Gyroid TPMS structure, which demonstrates higher heat transfer efficiency compared to conventional designs [9]. Gyroid structures offer effective heat transfer due to their high surface areas; however, they may present production challenges due to their complex geometries. Conversely, I-graph-and-wrapped-packages (IWP) sheet structures provide advantages for rapid cooling applications by ensuring high temperature homogeneity, although their structural durability may be lower compared to solid alternatives [10]. Al-Ketan et al. (2019) [11] performed an extensive analysis of the essential characteristics of TPMS architectures and their prospective utilization across multiple engineering domains. The investigation emphasized the promising potential of TPMS structures in diverse fields, including thermal management, structural mechanics, sound absorption, and medical implants. The authors elucidated the beneficial attributes of TPMS designs, such as elevated surface area-to-volume ratios, reduced relative densities, and the ability to adjust mechanical properties based on specific requirements [11].

The performance of heat exchangers designed with TPMS structures significantly depends on material selection. Nazir et al. (2019) [12] demonstrated the potential of these structures, which have various types such as Schwarz P, Gyroid, and I-WP, in many applications, including heat exchangers, due to their high mechanical performance and large surface area properties [12]. Ataee et al. (2018) [13] conducted a study on TPMS structures fabricated using titanium alloy. Their research focused on evaluating the mechanical and thermal characteristics of these structures. The findings revealed that TPMS structures exhibit a favorable strength-to-weight ratio, highlighting their potential for lightweight and high-performance applications [13].

The choice of coolant is also an important factor affecting the performance of TPMS structures used in heat exchangers with nanofluids. While water has been a common choice traditionally, the use of nanofluids has started to attract attention in recent years. Habibishandiz and Saghir (2022) [14] investigated the heat exchanger performance of water-based nanofluids containing metal oxide nanoparticles. In their study, they showed that nanofluids can increase the heat transfer coefficient by up to 45% [14]. Similarly, an additional study explores the effect of magnetized ternary hybrid nanofluid (THNF) flow on heat transfer and shows that the heat transfer capacity of ternary hybrid nanofluid is more enhanced compared to hybrid nanofluids. The shape of the nanoparticles, particularly the blade shape, plays a significant role in enhancing the heat transfer rate [15].

Numerical and experimental methods have been employed to investigate the performance of heat exchangers based on TPMS and Periodic Nodal Surfaces (PNSs). Comparative studies have been conducted on different TPMS and PNS structures to evaluate their advantages over traditional heat exchangers [13]. Supporting these investigations, simulations of flow and heat transfer in 3D-printed TPMS heat exchangers have demonstrated the potential of these structures to offer innovative and efficient solutions [16]. Various numerical studies have been conducted to understand the flow and heat transfer mechanisms in TPMS structures. Saghir and Kilic (2024) [17] employed computational fluid dynamics (CFD) techniques to analyze the heat transfer and pressure drop behavior in TPMS structures. The performance of these structures was assessed over a range of Reynolds number (Re) [17]. Numerical investigations using CFD analyses examined flow and heat transfer in TPMS structures. The study tested three different porosity levels under laminar flow conditions. Results showed that increasing Re enhances heat transfer performance. Different geometries demonstrated varying levels of effectiveness [18].

Various optimization studies have been carried out to further improve the performance of TPMS structures. Recent research focuses on investigating the effects of TPMS structures on turbulence generation, heat transfer, and flow mixing, aiming to evaluate their potential to enhance the efficiency of conventional heat exchangers [19]. In addition to these optimization studies, the thermal and hydraulic performance of TPMS-based heat exchangers has been examined using three-dimensional models of various surfaces with different volume ratios. The findings reveal the significant impact of secondary flows, periodic flow acceleration/deceleration, and continuous flow direction changes on convective heat transfer. This research contributes to the design and implementation of TPMS-based heat exchangers [20]. The performance of TPMS structures is heavily influenced by surface finish quality. Tan et al. (2023) highlighted how advanced soft abrasive flow finishing techniques can enhance the surface characteristics of complex geometries in constraint spaces, which could potentially improve the thermal–hydraulic performance of TPMS heat exchangers by reducing surface roughness and optimizing flow channels [21]. Most studies have focused on the thermal performance of TPMS structures under laminar flow conditions. Nevertheless, recent research has begun to explore the effects of turbulent flow on their heat transfer characteristics. Catchpole-Smith et al. (2019) [22] examined the thermal conductivity of TPMS structures, finding that thermal conductivity decreases as porosity increases. Among the different TPMS topologies, the Gyroid structure demonstrated the highest thermal conductivity. The results suggested that TPMS structures could be effectively employed in thermal management systems, including heat exchangers. In the study, effective thermal conductivity values were measured for different TPMS topologies (Gyroid, Diamond, and Primitive), and the relationship between these values and the Re was investigated. At high Re (Re > 2000), in the turbulent flow regime, it was observed that the heat transfer performance of TPMS structures significantly increased. For example, it was found that the Gyroid structure provided approximately 40% higher effective thermal conductivity in turbulent flow compared to laminar flow [22]. Microfluidic technology is widely utilized in chip channel manufacturing due to its efficiency, safety, and precise control. The design of the internal flow field and particle distribution plays a crucial role in enhancing mass transfer rates and reaction efficiency. In this context, a multi-field coupled particle flow modeling and control method is proposed, investigating particle distribution under ultrasonic excitation and establishing a multi-field mixing observation platform for validation. The findings indicate that the proposed modeling strategy effectively elucidates the evolution mechanism of the multi-field coupled flow field, while ultrasonic energy enhances overall flow distribution and ensures uniformity in particle distribution [23].

In a study conducted on the design and performance of TPMS-based heat exchangers under turbulent flow regime, researchers obtained various results. The flow and heat transfer performance of heat exchangers based on four TPMS structures (F-KS, PMY, FRD, and Gyroid) were numerically investigated using the k-ω Shear Stress Transport turbulence model. It was observed that heat transfer was significantly influenced by turbulence, with the Nu increasing along with the Re. The Fischer–Koch S (F-KS) model exhibited the best overall heat transfer performance. Increasing the flow velocity and volume fraction led to higher heat transfer coefficients, while an increase in volume fraction was also found to result in higher pressure drops [24]. The effectiveness of ultrasonic excitation strategies in controlling particle distribution in microfluidic channels has been demonstrated in previous studies. In this study, particle flow dynamics were modeled using the LBM-LES-DEM framework, and the role of ultrasonic excitation in preventing particle accumulation was revealed [25].

Turbulent flow, characterized by its chaotic and irregular nature, plays a crucial role in various engineering applications, including heat exchangers and bone implants. Recent advancements in additive manufacturing have enabled the fabrication of complex geometries, such as TPMS structures, which exhibit unique properties in terms of heat transfer and fluid flow under turbulent conditions. Experimental investigations of these structures under turbulent flow have largely confirmed the predictions of numerical studies [26]. This comprehensive approach aims to accelerate the multi-objective optimization of TPMS structures, paving the way for their utilization in the development of bone implants [27]. The numerical investigation examined convective heat transfer characteristics of alumina-water nanofluids flowing through horizontal rib-shaped microchannels. This simulation was conducted under laminar flow conditions with nanoparticle volume fractions ranging from 0.01 to 0.04. Results demonstrated that the internal ribs enhanced heat transfer by intensifying turbulence in the flow [28].

This study provides a comprehensive investigation into the heat transfer performance of TPMS structured heat exchangers by employing both experimental and numerical analyses. The analysis focuses on two types of TPMS materials, Al and Ag, and evaluates their performance with two heat transfer fluids: pure distilled water and a nanofluid containing 0.6 vol% nanoparticles. A range of flow rates and a constant heat flux of 30.8 kW/m² were applied to assess the impact of the TPMS structure on heat dissipation and boundary layer formation. By integrating experimental measurements with numerical simulations, this study offers an in-depth understanding of heat transfer mechanisms and identifies the key parameters influencing the performance of TPMS-based cooling systems. These findings address existing gaps in the literature and provide valuable insights for optimizing TPMS-structured heat exchangers for enhanced thermal management. Integrating nanofluids with TMS heat exchangers not only enhances heat transfer efficiency but also aligns with sustainable energy practices, making them promising solutions for environmentally responsible industrial applications.

2. Experimental Measurements

This research extends the previous work of Kerme et al. (2024) [9] by using distilled water containing metallic nanoparticles. This specific liquid mixture is known as a nanofluid in the literature [17,29]. The nanofluid utilized in this study had a volume concentration of 0.6%, where the presence of nanoparticles enhances the cooling liquid’s thermal conductivity. The TPMS porous structures used in the experiments were produced from two different alloys: AlSi₁₀Mg and Ag. Considering the high thermal conductivity of the Ag alloy, the structures produced from this material are expected to exhibit better heat transfer performance. The TPMS structures were designed using the gyroid cell geometry and manufactured using 3D printing technology. The porosity of these structures was determined to be 0.7. It was observed that the inlet flow temperature varied depending on the experimental conditions. Taking this variability into account, comprehensive temperature measurements were performed to accurately evaluate the cooling performance. Measurements were conducted beneath the TPMS structure, as well as at the inlet and outlet points of the flow.

2.1. Experimental Procedure

The experimental setup was designed to investigate the thermal and hydraulic characteristics of the system under controlled turbulent flow conditions. A variable-flow pump was employed to regulate working fluid circulation, ensuring the desired turbulence levels. Figure 1 provides a detailed illustration of the experimental apparatus and methodology. Prior to commencing the experiment, trapped air bubbles were meticulously removed to ensure accurate results. A voltage and current-controlled power source supplied a constant heat flux of 30.8 kW/m². To ensure a steady low temperature at the inlet of the test unit, a water bath was utilized. Nevertheless, slight fluctuations were noted due to the inherent difficulties in achieving uniform inlet temperatures. The test section was firmly attached to an Al heating block, with thermocouples placed 0.1 cm below the interface. An inlet thermocouple was placed at the entrance of the system, while another was installed at the outlet, and seven additional thermocouples were arranged along the interface. The test unit was constructed to match the dimensions of an Intel Core i7 processor. Both the Al heating block and the TPMS structure were precisely manufactured to these specifications, allowing for a valid comparison of their thermal performance and flow characteristics.

Table 1. Properties of the materials.

Surface Thermocouples	TC-1	TC-2	TC-3	TC-4	TC-5	TC-6	TC-7
Location of thermocouples from inlet (mm)	4.2	8.4	12.6	16.8	21	25.2	29.4
TPMS block dimensions (L × H) (mm)	37.5 × 12.7

presents the dimensions of the TPMS and the distances of the surface thermocouples. A data acquisition system continuously recorded temperature data at specified time intervals throughout the experiment.

All temperature measurements in this study are associated with a calibrated uncertainty of ±0.75% (°C), which is attributed to the T-type thermocouples utilized in the experimental setup. This measurement uncertainty is found to be significantly lower than the observed thermal performance differences between materials (15% enhancement with Ag versus Al TPMS) and cooling fluids (12% improvement with nanofluids versus water). The maximum uncertainty in calculated Nusselt numbers (Nu) is determined to be 2.6%, which remains substantially below the performance variations documented across experimental configurations. These uncertainty values confirm the statistical significance of the reported results and validate the reliability of the comparative thermal performance evaluations conducted in this investigation.

To address potential heat losses in the experimental setup, comprehensive thermal insulation measures were implemented. The test section was fabricated utilizing low thermal conductivity materials, and all external surfaces, except for the measurement interfaces, were insulated with polyurethane foam. Thermal paste was applied at the interface between the TPMS structure and test block to minimize contact resistance. Heat loss quantification tests were conducted prior to experiments, validating that less than 3% of the input power was dissipated to the surroundings under steady-state conditions. This minimal heat loss was quantified and incorporated into all thermal calculations, ensuring the accuracy of the reported heat transfer coefficients. The uncertainty analysis confirms that the experimental methodology yields reliable results, with the influence of these quantified heat losses remaining within the stated measurement uncertainty bounds.

As depicted in Figure 2, the test block incorporates Al and Ag TPMS, with the fluid flowing through the hexagonal structure from inlet to outlet. The liquid flows through the TPMS structure, effectively absorbing heat from the heated block. To reduce heat loss, the test section was made from insulating material, and thermal paste was used at the interface between TPMS structure and test block.

The upper part of the experimental setup is covered with a transparent polyethylene sheet. It is fixed using multiple screws to ensure the sealing and circulation of the nanofluid in the system. The data from 7 temperature sensors located under the TPMS heat exchanger were recorded for each second. The plate thickness at the point where the channel contacts the heated block is 0.3 cm. During the experiment, three different flow rates were applied. These flow velocities are 30 cm/s, 35 cm/s, and 45 cm/s, and data were recorded for flow rates of 23.42 cm³/s, 27.29 cm³/s, and 35.41 cm³/s, respectively. The heat flux applied to the Al and Ag blocks was kept constant at 30.8 kW/m² throughout all experiments. Slight variations in inlet temperature were observed for each experiment, and measurements were performed considering this situation during the analysis phase. The friction factor was not taken into account in all evaluated scenarios.

The nanofluid, which is circulated through the block channels by forced circulation, is a mixture of 0.6% Al₂O₃ and 99.4% distilled water, synthesized by a manufacturer. A nanoparticle size of 50 nm was used to ensure the suspension and homogeneous distribution of the nanoparticles. This liquid was then diluted with distilled water to the targeted volume concentration. To suspend the nanofluid in distilled water, homogenization was achieved by using a magnetic stirrer for 40 min. Distilled water was used as the cooling liquid in the experimental setup. The physical properties of the pure water and nanofluid used in the experimental setup are presented in

Table 2. Thermophysical properties [24].

Materials	$\mathit{\rho }$ (kg/m3)	$\mathit{\mu }$ × 10−4 (kg/m.s)	Cp (J/kg.K)	k (W/m.K)	EC (μS/cm)
Water (Distilled)	998.2	10.01	4128	0.613	0.055
Nanofluid (0.6% Al2O3 + distilled water)	1013.811	10.4	4109.198	0.6242	-

. The thermal conductivity, density, and heat capacity of the nanofluid are higher than those of pure water, and the increase in the viscosity coefficient also increases the pressure loss. The nanofluid demonstrates superior thermal conductivity, density, and heat capacity relative to pure water. In addition, the rise in viscosity coefficient leads to an increase in pressure loss.

An uncertainty analysis was conducted on the components of the apparatus utilized for experimental data collection. The flowmeter and T-type thermocouples were identified as the main sources of error propagation within the device. Through comprehensive calibration procedures, the flowmeter’s associated error was quantified at 0.44%. Calibration procedures established the uncertainty of the T-type thermocouples at 0.75% (°C). Additionally, fluctuations in the uncertainties of non-dimensional parameters, including Nu and temperature, were noted as a result of changes in temperature and flow rate. To elucidate this relationship, Nu and Re are expressed mathematically as follows [30]:

$$Nu={\displaystyle \frac{h_x·D}{k_f}}$$

(1)

$$Re={\displaystyle \frac{u_{in}·D}{\nu }}$$

(2)

The Reynolds number is determined based on the hydraulic diameter of the inlet side of the Gyroid or Schwarz-P lattice structure, where the hydraulic diameter D_h is calculated as:

$${Re}_{D_h={\displaystyle \frac{\rho U{D}_h}{\mu }}}$$

(3)

$$D_h={\displaystyle \frac{4A}{P}}$$

(4)

The area A and wetted perimeter P of the inlet cross-section are used in this calculation. In the Schwarz-P lattice structure, the inlet cross-section of each cell is circular, so A and P are given by πR² and 2πR, respectively, where R is the radius. In contrast, calculating the hydraulic diameter D_h for the gyroid structure is more complex, as its cross-section forms a sinusoidal channel [31].

Dimensional parameters, including local heat transfer coefficients, were identified. These coefficients are expressed in units of W/m² °C and can be represented mathematically as follows [32]:

$$h={\displaystyle \frac{q^{″}}{T-Tin}}={\displaystyle \frac{q^{″}}{∆T}}$$

(5)

In all scenarios discussed above, experimental data form the foundation of the calculations. The average Nu’s uncertainty can be determined using the following method [32]:

$$\delta Nu=\sqrt{{\left({\displaystyle \frac{\partial N{u}_x}{\partial x}}·\delta x\right)}^2+\cdots +{\left({\displaystyle \frac{\partial N{u}_x}{\partial y}}·\delta y\right)}^2+\cdots +{\left({\displaystyle \frac{\partial N{u}_x}{\partial z}}·\delta z\right)}^2}$$

(6)

The local heat transfer coefficient, h_x, varies depending on the position along the heater surface, while T_x indicates the local surface temperature at that point. The inlet temperature, T_in, sets the initial condition for the system’s thermal behavior. D, indicating the tube inlet diameter; and u_in, which is the inlet fluid velocity across the unit. The flow behavior of the fluid is governed by its kinematic viscosity, ν, while k_f denotes the thermal conductivity of fluid, which measures its ability to conduct heat. The conducted uncertainty analysis revealed that the maximum uncertainty in the local Nu calculations was constrained to 2.6%, ensuring a high degree of confidence in the experimental results.

Temperature gradients in TPMS structures can be influenced by pressure losses across the flow path. In regions with complex geometry where flow restrictions occur, local temperature variations may be affected by pressure-related phenomena. However, since all tested configurations in this study experienced similar flow conditions through geometrically identical TPMS structures, the relative influence of pressure effects remains consistent across scenarios. This consistency enables valid comparative thermal performance analysis between the different material and fluid combinations, which is the primary focus of this investigation.

2.2. Numerical Procedure

Numerical modeling was carried out using the finite element method with ANSYS Fluent (2023 R2) software and a discrete solver. The nanofluid was regarded as Newtonian due to the low concentration of metallic nanoparticles. The flow was treated as incompressible, and a turbulent flow regime was identified at the applied flow rate.

The Navier-Stokes equations are solved numerically in conjunction with the energy and continuity equations. In the flow field, TPMS is modeled as a solid structure, which adds complexity to the model because of the closed-end channels present. Consequently, it is necessary to direct the flow toward the lower pressure region. The flow velocity decreases after high flow rates within the structure. This irregular pattern allows longer liquid circulation time, enabling better heat removal. However, the complexity affects equation convergence. To overcome this challenge, the fluid equations are first solved by assuming that the flow is initially zero. Once convergence is achieved, the model is re-evaluated using the flow field results as initial conditions. The updated mathematical model incorporates the governing equations for fluid flow and heat transfer, resulting in a more efficient and precise convergence process. The convergence threshold is defined by the reduction of residuals associated with velocity, pressure, and temperature fields to values less than 10⁻⁶, ensuring a high level of accuracy in the numerical solution. Although this process may extend the computational time, the enhanced mesh structure significantly improves the accuracy of the solution. The formulations utilized in this analysis are as follows [28,29,30]:

x-direction

$${\rho }_f\left[u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}+w{\displaystyle \frac{\partial u}{\partial z}}\right]=-{\displaystyle \frac{\partial p}{\partial x}}+{\mu }_f\left[{\displaystyle \frac{{\partial }^{2}u}{{\partial x}^2}}+{\displaystyle \frac{{\partial }^{2}u}{{\partial y}^2}}+{\displaystyle \frac{{\partial }^{2}u}{{\partial z}^2}}\right]$$

(7)

y-direction

$${\rho }_f\left[u{\displaystyle \frac{\partial v}{\partial x}}+v{\displaystyle \frac{\partial v}{\partial y}}+w{\displaystyle \frac{\partial v}{\partial z}}\right]=-{\displaystyle \frac{\partial p}{\partial y}}+{\mu }_f\left[{\displaystyle \frac{{\partial }^{2}v}{{\partial x}^2}}+{\displaystyle \frac{{\partial }^{2}v}{{\partial y}^2}}+{\displaystyle \frac{{\partial }^{2}v}{{\partial z}^2}}\right]$$

(8)

z-direction

$${\rho }_f\left[u{\displaystyle \frac{\partial w}{\partial x}}+v{\displaystyle \frac{\partial w}{\partial y}}+w{\displaystyle \frac{\partial w}{\partial z}}\right]=-{\displaystyle \frac{\partial p}{\partial z}}+{\mu }_f\left[{\displaystyle \frac{{\partial }^{2}w}{{\partial x}^2}}+{\displaystyle \frac{{\partial }^{2}w}{{\partial y}^2}}+{\displaystyle \frac{{\partial }^{2}w}{{\partial z}^2}}\right]$$

(9)

Energy Formulation

$${\rho }_{f}Cp\left[u{\displaystyle \frac{\partial T}{\partial x}}+v{\displaystyle \frac{\partial T}{\partial y}}+w{\displaystyle \frac{\partial T}{\partial z}}\right]=k_f\left[{\displaystyle \frac{{\partial }^{2}T}{{\partial x}^2}}+{\displaystyle \frac{{\partial }^{2}T}{{\partial y}^2}}+{\displaystyle \frac{{\partial }^{2}T}{{\partial z}^2}}\right]$$

(10)

In the model, u, v, and w represent the velocity components in the x, y, and z directions, respectively. These variables characterize the fluid motion in three-dimensional space and are essential for describing the flow behavior within the TPMS structure. The effect of gravity has been neglected for the TPMS gyroid structure under forced convection conditions. This approach is considered appropriate for simplifying the model and reducing the computational time. Under forced convection conditions, the motion of the flow is primarily determined by an externally applied force; thus, the effect of gravity is thought to play a secondary role.

2.3. Boundary Conditions

The boundary conditions of the model used in this study were examined in detail and presented in Figure 3. The geometric properties of the heated block were determined as a 3.75 cm square base and a height of 1.27 cm. The block containing the TPMS structure has the same dimensional characteristics. A diameter of 1 cm was used for the inlet and outlet cylinders. The regions where the heat flux was applied are marked with red arrows in Figure 3a, and a constant heat flux of 30.8 kW/m² was applied in these areas. The fluid temperature at the system’s inlet was designated as T_in, while the applied inlet velocity was defined as u_in, and the calculated outlet temperature was represented as T_out. Adiabatic wall conditions were established for all boundaries within the computational domain, effectively preventing heat transfer, except for the surfaces subjected to heat flux, which enabled a focused assessment of heat transfer performance in specific regions. Based on these specified parameters and conditions, the boundary conditions were mathematically formulated. A uniform velocity (u = u_in) is applied at the inlet region in the x-direction, and the fluid enters the test section at a constant temperature (T_in). An open boundary condition is applied at the outlet, where stresses are set to zero. The bottom surface of the Al block is exposed to a constant heat flux (q”). All external surfaces are assumed to be adiabatic, satisfying the condition of zero temperature gradient normal to the surface ((∂T_Surface)/∂n = 0). In the present fluid dynamics analysis, no-slip boundary conditions are imposed on all surfaces. Figure 3b illustrates the precise positioning of thermocouples within the experimental setup. Seven thermocouples (TC-1, TC2, …, TC-7) are strategically placed at specific intervals along the flow path, beginning at 4.2 mm from the inlet and extending to 29.4 mm, with equidistant spacing of 4.2 mm between consecutive measurement points. This systematic arrangement enables comprehensive temperature monitoring throughout the TPMS structure, facilitating detailed analysis of temperature gradients and thermal performance characteristics. The thermocouples are positioned 0.1 cm below the interface inside the solid block to accurately capture the heat transfer dynamics between the TPMS structure and the cooling fluid while minimizing flow disturbance.

The effect of gravity was neglected in our TPMS model under forced convection conditions. Previous studies have quantified gravity’s influence in similar heat transfer systems. Al-Ketan et al. (2021) reported that in forced convection through TPMS structures with Re exceeding 2000, the contribution of natural convection due to gravity is typically less than 2% of the total heat transfer coefficient [33]. Similarly, Qureshi et al. (2022) demonstrated that for gyroid TPMS structures under forced convection with flow rates above 20 cm³/s, the Richardson number (R_i) remains below 0.1, indicating that buoyancy effects are negligible compared to forced convection [10]. These findings from the literature support our approach to neglecting gravity effects in the current study’s turbulent flow regime.

The boundary conditions of the model used in this study were examined in detail and are presented in

Table 3. Boundary conditions applied in the TPMS model simulation.

Boundary	Condition Type	Mathematical Representation	Description
Inlet	Velocity inlet	u = uin	Uniform velocity in x-direction
	Temperature	T = Tin	Constant inlet temperature
Outlet	Pressure outlet	∂p/∂n = 0	Zero pressure gradient (open boundary)
	Outflow	-	Fully developed flow condition
Bottom surface	Heat flux	q″ = 30.8 kW/m2	Constant heat flux applied
All external surfaces	Thermal	∂TSurface/∂n = 0	Adiabatic condition (no heat transfer)
All solid–fluid interfaces	Velocity	u = v = w = 0	No-slip boundary condition
	Thermal	Tfluid = Tsolid	Thermal equilibrium at interfaces

. Table 3 provides a comprehensive overview of all boundary conditions applied in the numerical simulation, including their mathematical representations and physical interpretations.

The model has a porosity ratio of 0.7. Due to the simulation of the TPMS as a solid structure, the permeability value was not evaluated in this investigation. In the experiments, the calculated temperature (T) was measured 0.1 cm below the interface inside the solid Al block. Various parameters were considered to measure the heat removal capacity of the TPMS. Among the parameters considered, the heat transfer coefficient (h) was primarily evaluated and defined by Equation (9). The local Nu, which is a critical parameter in heat transfer analysis, is defined by Equation (10) as follows [34]:

$$h={\displaystyle \frac{q″}{(T-T_{in})}}$$

(11)

$$Nu={\displaystyle \frac{hD}{k_f}}$$

(12)

The design of cooling systems significantly relies on pressure drop. This phenomenon is characterized by the friction factor (f), which is expressed in the following Equation (11) as follows [18,30,34]:

$$f={\displaystyle \frac{0.5∆pD}{\rho L{u}_{in}^2}}$$

(13)

PEC is an important criterion used to evaluate the efficiency of heat exchangers and other thermal systems, and it is given in Equation (12). PEC quantitatively expresses the overall system effectiveness by evaluating the heat transfer performance and hydraulic performance of a system together. The factors included in PEC are generally the increase in heat transfer and friction factor, and constant pumping power. The increase in heat transfer indicates how much more heat the developed system transfers compared to a standard system. The increase in friction factor indicates how much more pressure drop the developed system causes compared to the standard system. In constant pumping power comparisons, it is generally assumed that the power is kept constant. PEC combines these factors to reveal how much extra energy (for pumping) a system spends while increasing heat transfer. Obtaining a high PEC value indicates that the increase in heat transfer of the system is more advantageous compared to the increased pumping costs. Therefore, it shows that system performance is improved. This criterion, where Nu_local represents the local Nu, is defined by the following formula [18,28,29,30,34]:

$$PEC={\displaystyle \frac{{Nu}_{local}}{f^{\frac{1}{3}}}}$$

(14)

A mesh sensitivity analysis was conducted to assess the numerical model’s accuracy and reliability. This process ensures that the results obtained are dependable and consistent across different mesh configurations. This analysis involves systematic evaluation of different mesh sizes. The optimization of the mesh structure was guided by Nu. A mesh structure was deemed optimal when the difference in Nu between two consecutive meshes was less than 1%. This approach aims to provide the most suitable balance between computational accuracy and computation time. As a result of the analysis, a medium-density mesh structure consisting of 133,8201 elements was determined for the optimal solution. This mesh structure includes a combination of tetrahedral and boundary elements. The selected mesh structure ensures that the numerical model produces accurate and reliable results while also offering a reasonable computation time.

2.4. Temperature Distribution Analysis

The thermal–hydraulic performance of TPMS-based heat exchangers has been examined using three-dimensional models of various surfaces with different volume ratios. Wang et al. (2023) [18] conducted a comprehensive investigation on flow and heat transfer in various channels based on TPMS. Their findings reveal the significant impact of secondary flows, periodic flow acceleration/deceleration, and continuous flow direction changes on convective heat transfer. This research contributes to the design and implementation of TPMS-based heat exchangers [18]. Most studies have focused on the thermal performance of TPMS structures under laminar flow conditions. Nevertheless, recent research has begun to explore the effects of turbulent flow on their heat transfer characteristics. Catchpole-Smith et al. examined the thermal conductivity of TPMS structures, finding that thermal conductivity decreases as porosity increases. Among the different TPMS topologies, the Gyroid structure demonstrated the highest thermal conductivity. The results suggested that TPMS structures could be effectively employed in thermal management systems, including heat exchangers. In the study, effective thermal conductivity values were measured for different TPMS topologies (Gyroid, Diamond, and Primitive), and the relationship between these values and the Re was investigated. At high Re (Re > 2000), in the turbulent flow regime, it was observed that the heat transfer performance of TPMS structures significantly increased. For example, it was found that the Gyroid structure provided approximately 40% higher effective thermal conductivity in turbulent flow compared to laminar flow [22]. Al-Ketan et al. (2019) [11] further contributed to this field by establishing critical criteria for evaluating temperature homogeneity in TPMS structures, providing valuable metrics for assessing thermal performance uniformity across different geometries and operating conditions.

Three fundamental statistical parameters were utilized for the analysis of temperature distribution homogeneity: standard deviation, coefficient of variation, and temperature uniformity. Standard Deviation (σ), which indicates how much temperature measurements deviate from the mean value, is calculated using the following equation, where T_i is the temperature value at each thermocouple, T_mean represents the mean temperature of all thermocouples, and N is the number of thermocouples (N = 7) [35]

$$\sigma =\sqrt{{\displaystyle \frac{1}{N}}}{\sum }_{i=1}^N(T_i-T_{mean}{)}^2$$

(15)

The coefficient of variation (CV), expressed as a percentage, represents the ratio of standard deviation to mean value, facilitating comparisons of distributions across different scale systems [35]

$$CV=\left({\displaystyle \frac{\sigma }{T_{mean}}}\right)\times 100\%$$

(16)

The parameter, as proposed by Yan et al. (2024), represents the temperature differential between the maximum (T_max) and minimum (T_max) temperatures observed within the system [30]:

ΔT_max = T_max − T_min

(17)

These parameters provide a comprehensive approach to evaluate the temperature distribution homogeneity within the TPMS structure. Lower values of standard deviation and coefficient of variation indicate a more homogeneous temperature distribution [36], while a lower ΔT_max value demonstrates that the temperature gradient within the system has been minimized.

The temperature gradient analysis was calculated as the ratio of the temperature difference between the first (TC1) and seventh (TC7) thermocouples to the distance between these two points. This ratio serves as a crucial parameter characterizing the rate of temperature change along the TPMS structure. Low gradient values indicate more uniform heat transfer and more effective cooling in the system, while high gradient values suggest potential inefficiencies in heat transfer and possible hot spot formation. This metric provides an objective evaluation criterion for comparing the heat distribution performance of different TPMS designs (Al and Ag). Furthermore, the analysis of gradient values enables the assessment of thermal stress levels in the system and risk evaluation in terms of material fatigue. Consequently, this provides guiding data for the optimization of cooling system design.

2.5. Mesh Independence Study

A comprehensive mesh independence study was conducted to ensure the reliability of numerical results. Five different mesh densities were evaluated to determine the optimal mesh configuration that balances computational efficiency and solution accuracy.

Table 4. Grid independence study.

Mesh Density	Number of Elements (Average)	Average Nu	Deviation from Previous Mesh (%)
Very Coarse	549,000	56.82	-
Coarse	876,000	61.45	8.15
Medium	1,338,000	62.73	2.08
Fine	1,965,000	63.16	0.68
Very Fine	2,855,000	63.31	0.24

presents the results of this mesh independence analysis. The mesh independence study demonstrates that the medium mesh density provides an optimal balance between computational efficiency and accuracy. The deviation in the average Nu between the medium and fine mesh was less than 1%, indicating that further mesh refinement would not significantly improve solution accuracy while substantially increasing computational cost. Therefore, the medium mesh configuration was selected for all simulations in this study. This mesh structure consists primarily of tetrahedral elements with boundary layer refinement near the fluid-solid interfaces to accurately capture thermal and velocity gradients in these critical regions.

3. Results and Discussions

In the scope of this study, experiments were conducted by applying 12 different scenarios using two different cooling fluids at three different flow rates for two different TPMS materials, and the results are presented in

Table 5. Scenarios applied in the experiments.

Scenarios	Material	Fluid	Flow Rate (cm3/s)	Inlet Temperature (°C)
Scenario 1	Al	Distilled water	23.56	13.84
Scenario 2	Al	Distilled water	27.49	13.62
Scenario 3	Al	Distilled water	35.34	13.63
Scenario 4	Al	Nanofluid	23.56	12.26
Scenario 5	Al	Nanofluid	27.49	12.21
Scenario 6	Al	Nanofluid	35.34	12.11
Scenario 7	Ag	Distilled water	23.56	13.32
Scenario 8	Ag	Distilled water	27.49	13.20
Scenario 9	Ag	Distilled water	35.34	12.84
Scenario 10	Ag	Nanofluid	23.56	13.67
Scenario 11	Ag	Nanofluid	27.49	12.74
Scenario 12	Ag	Nanofluid	35.34	12.20

. Scenarios were carried out under turbulent flow at three different velocities for water and nanofluid as cooling fluids. The inlet temperature changed across experiments nevertheless, this variation was addressed by calculating the difference between the measured temperature and the specified inlet temperature.

The experiments were conducted at three different flow rates: 23.56 cm³/s, 27.49 cm³/s, and 35.34 cm³/s. Data was recorded for each flow rate to investigate the performance of the system under varying flow conditions. The heat flux applied to the Al and Ag blocks was kept constant at 30.8 kW/m² throughout all experiments. Figure 4 shows the inlet temperature distribution of the fluid and the initial and boundary conditions for the test unit. Thermocouple is located 10 mm away from the point where the flow begins to enter the test section. When examining the temperature distributions of distilled water circulated through Al and Ag TPMS in Figure 4a,b, it was observed that the system reached a steady state within ~25 min and there was a ±0.5 °C difference in the initial inlet temperatures. Between each experiment, a waiting period of 4 h was allowed for the thermal loads on the setup walls to dissipate. Additionally, the slight fluctuations observed in the temperature curves were caused by small instabilities in the electric current. Figure 4c,d illustrate the inlet temperature distributions when nanofluids are utilized in Al and Ag TPMS structures, respectively. In experiments conducted at various flow rates (23.56 cm³/s, 27.49 cm³/s, and 35.34 cm³/s), inlet temperature variations in nanofluid systems were observed to remain within approximately ±0.4 °C range. As demonstrated in Figure 4c, the implementation of nanofluids in Al TPMS structures provided more stable inlet temperature profiles compared to pure water. Similarly, as evidenced in Figure 4d, the utilization of nanofluids in Ag TPMS structures facilitated the system reaching steady-state conditions within approximately 20 min, with temperature fluctuations remaining minimal despite variations in flow rates. This phenomenon demonstrates the advantageous thermal stability characteristics offered by nanofluid implementation.

Figure 5 shows the experimental data for scenarios where distilled water was passed through the Al TPMS block. Figure 5a displays the temperature distribution measured 0.1 cm below the interface of the heated TPMS block. The distribution graph indicates that as the flow velocity increases, more heat is removed from the system during the cooling process. Additionally, an improvement in temperature homogeneity was recorded as the flow velocity increased. Moreover, the use of TPMS in a gyroid structure has a homogeneous effect on the temperature distribution. The temperature differential between thermocouples 1 and 7 remains below 1.5 °C, regardless of the inlet temperature. The inlet temperature range, derived from the lowest and highest flow velocities, is approximately 15 °C. Figure 5b depicts the difference between the block temperature and the inlet temperature, facilitating the evaluation of heat removal by the TPMS independent of the system’s initial temperature. The temperature scale in the graph ensures precise verification of temperature uniformity. Figure 5b depicts the difference between the block temperature and the inlet temperature, facilitating the evaluation of heat removal by the TPMS independent of the system’s initial temperature. The temperature scale in the graph ensures precise verification of temperature uniformity. As observed in Figure 5b, the homogeneity and temperature difference were calculated to be approximately 0.5 °C, with the temperature differential between TC1 and TC7 ranging from 0.49 °C to 0.51 °C across different flow rates, demonstrating remarkable thermal uniformity within the Al TPMS structure despite variations in flow conditions. The dimensionless temperature variable Theta (θ), representing the experimental data, is presented in Figure 5c. The normalization of instantaneous temperature based on the difference between inlet and outlet temperatures enables θ to facilitate comparison of results obtained under various experimental conditions. Higher θ values indicate enhanced heat transfer rates in specific regions, demonstrating more effective thermal energy exchange between the fluid and heat exchanger surface. This phenomenon manifests as increased fluid heating or cooling, depending on the operational mode. The dimensionless temperature distribution has been employed to characterize the heat transfer process and elaborate on temperature variations within the system. The gradual progression of θ values along the flow direction provides valuable insights into the thermal behavior and efficiency of the heat exchanger design. The dimensionless temperature distribution presented in Figure 5c reveals that the temperature profile in the investigated Al TPMS structure is uniform by eliminating the influence of inlet and outlet temperatures, indicating homogeneous heat transfer throughout the system and optimal thermal performance.

To further enhance the heat transfer performance of the system, experiments were repeated using the Ag TPMS structure, which has a high thermal conductivity. The thermal conductivity of Ag is approximately 1.8 times higher, indicating that the Ag TPMS structure can conduct heat more efficiently and transfer more heat to the cooling fluid, which is water. The gyroid structure of the Ag TPMS provides a large surface area for heat transfer, allowing faster absorption of heat around the structure and more effective removal through water circulation. To evaluate the heat transfer performance of the Ag TPMS block, parameters such as temperature distribution, temperature difference, and dimensionless temperature distribution were examined in Figure 6.

Figure 6 demonstrates the heat transfer performance between the Ag TPMS structure and distilled water. As shown in Figure 6a, the temperature values of the Ag TPMS structure are lower compared to the Al TPMS structure. The lowest flow rate in the Ag block recorded lower temperature measurements compared to the highest flow rate in the Al block. This indicates that, due to the high thermal conductivity of Ag, heat is more effectively removed from the structure. Therefore, when examining Figure 6b, the effect of inlet temperature is eliminated, and it is observed that the temperature is high at low flow rates and decreases as the flow rate increases. Analysis of the temperature differential between TC1 and TC7 revealed values between 0.21 °C and 0.24 °C, which is significantly lower than observed in the Al TPMS structure, indicating enhanced thermal uniformity attributed to the superior thermal conductivity of the Ag material. Figure 6c presents the dimensionless temperature distribution, considering the inlet and outlet temperatures. It is demonstrated that temperature uniformity is maintained, and the TPMS structure provides a more homogeneous heat distribution.

The temperature distributions obtained using nanofluid for the Al TPMS block are presented in Figure 7. Figure 7a illustrates a reduction in temperature, which is attributed to the nanofluid’s superior thermal conductivity compared to that of water. In Figure 7b, a thermal enhancement is noted with an increasing flow rate when the inlet temperature is subtracted. The temperature variation between measurements taken at two distinct flow rates is smaller compared to the variation observed when using water as the cooling fluid. The temperature differential between TC1 and TC7 was measured between 0.97 °C and 1.04 °C across the tested flow rates, indicating that while the nanofluid enhances overall heat removal capacity, it also creates a more pronounced temperature profile along the TPMS structure. Figure 7c indicates that a stable temperature distribution is maintained across various flow rates, which is beneficial for ensuring uniform cooling in systems characterized by low pressure drops or flow rates, thus emphasizing the importance of nanofluids.

Figure 8 presents the temperature distributions obtained using nanofluid in the Ag TPMS structure. Figure 8a illustrates that the superior thermal conductivity of the Ag TPMS structure, combined with the properties of the nanofluid, results in a uniform temperature distribution under turbulent flow conditions. As presented in Figure 8b, this situation offers an advantage in removing heat, allowing lower temperatures to be reached. Temperature differentials between TC1 and TC7 ranged from 1.18 °C to 0.73 °C, with the gradient decreasing as flow rate increased, demonstrating that at higher flow rates, the combination of Ag TPMS structure and nanofluid achieves not only enhanced heat removal but also improved temperature homogeneity. In Figure 8c, the θ distribution in the heat exchanger block structure is presented. The approach of θ to zero indicates an increase in heat transfer performance in such systems. Therefore, as the temperature approaches the outlet temperature, it is observed that the heat is distributed more homogeneously. This situation is considered an indicator of the efficiency of the heat exchange system. The temperature increase observed in the last two thermocouples (TC-6 and TC-7) can be attributed to a combination of several factors. First, as the fluid approaches the outlet region, its heat absorption capacity gradually decreases due to the progressive temperature rise along the flow path. This diminished heat absorption capability leads to a localized temperature increase in the near-outlet region. Second, the flow characteristics in the outlet region of the TPMS structure contribute to this phenomenon. The transition from the structured TPMS geometry to the outlet channel creates a flow reorganization zone, which affects the local heat transfer coefficient. Additionally, potential minor vortex formations and boundary layer effects in the outlet region may contribute to the temperature increase. This temperature rise emphasizes the importance of optimizing the outlet region geometry in the design of TPMS structures.

3.1. Analysis of Experimental and Numerical Data Comparisons

In this study, the heat transfer performance of two TPMS blocks in gyroid form, made of Al and Ag materials, was investigated using two different fluids. The first fluid, selected as the reference fluid, was pure distilled water, while the other was a specially prepared nanofluid. Al nanoparticles with an average diameter of approximately 32 nanometers (nm) were added to distilled water at a volume fraction of 0.6% to create a nanofluid. The experimental results of the control group (water) and the nanofluid were compared using the finite element method with the numerical analysis software ANSYS Fluent 2023 R2.

3.1.1. Utilization of Distilled Water as a Cooling Fluid

A numerical model was developed using the finite element method with ANSYS Fluent software, with initial and boundary conditions based on experimental data for inlet temperature, flow velocity, and applied heat flux, and the resulting numerical outcomes were compared across all scenarios with the experimental findings. Figure 9 presents the temperature distributions obtained at three different flow rates for the Al TPMS structure using distilled water. The numerical results obtained using ANSYS Fluent 2023 R2 agreed with the experimental measurements. This agreement proves the reliability and accuracy of the developed numerical model. The reasons for the small deviations observed between the two data sets can be listed as heat fluctuations, heat losses, and inlet flow variations. As seen in the distribution curve, the difference between the experimental and numerical data sets being less than 1 °C is within acceptable limits for thermal management systems. Figure 10 compares the numerical and experimental results of temperature distributions using the Ag TPMS block and distilled water in the experimental setup. It was observed that the Ag TPMS results were similar to the Al TPMS data. However, lower temperatures were achieved for the Ag gyroid structure.

3.1.2. Utilization of Nanofluid as a Cooling Fluid

Nanofluid was used to enhance the heat transfer within the TPMS block [18]. In order to fully remove metallic particles from the nanofluid in the piping system, the system was flushed with clean water between tests. Moreover, the TPMS porous structure was cleaned at the end of the experiment to prevent the solidification of nanoparticles within the structure. It is crucial to note that the inlet temperature varied in each experiment; therefore, examining the θ variable is the correct approach. The observed stability in the temperature distribution when utilizing nanofluid can be attributed to the gyroid structure design. This unique geometric configuration plays a crucial role in maintaining a consistent thermal profile throughout the system.

Figure 11 illustrates a comparative analysis of numerical and experimental temperature profiles using nanofluid in the Al TPMS structure across varying flow conditions. The study examined three distinct scenarios: inlet temperature of 12.26 °C with flow rate of 23.56 cm³/s, a inlet temperature of 12.21 °C with flow rate of 27.49 cm³/s, and a inlet temperature of 12.11 °C with flow rate of 35.34 cm³/s. The temperature variations measured along different thermocouple positions demonstrate an excellent correlation between numerical simulations and experimental measurements. The results reveal that the highest temperature increase occurred at the lowest flow rate of 23.56 cm³/s, while temperature rises became progressively smaller as flow rates increased. This trend can be explained by the reduced residence time of fluid within the heat exchanger at higher flow velocities. The numerical predictions show good agreement with experimental data. The average deviation remains below 5% across all test conditions. This validates the accuracy of our numerical model. Notable temperature fluctuations were observed in measurements taken beyond the 20 mm thermocouple position, which can be attributed to the complex flow dynamics near the outlet region. The consistency between experimental data points and numerical prediction curves throughout the measurement range demonstrates the robustness of both the experimental setup and the numerical methodology employed in this study.

3.2. Comparative Analysis of Water and Nanofluid Performance

Temperature gradient values obtained from experimental and numerical analysis results were comparatively examined in

Table 6. Temperature gradient analysis for Al TPMS structure using distilled water at different flow rates.

Position (mm)	4.2	8.4	12.6	16.8	21	25.2	29.4	Slope
Exp, 23.56 cm3/s	29.1214	28.60074	28.59621	29.13505	29.48308	29.71951	29.61267	0.019495
Num, 23.56 cm3/s	28.46441	28.48721	28.5202	28.56192	28.60496	28.64355	28.67587	0.008391
Exp, 27.49 cm3/s	28.99413	28.5622	28.54235	29.06698	29.40327	29.60096	29.52789	0.021181
Num, 27.49 cm3/s	27.09838	27.1391	27.17434	27.21083	27.24588	27.27542	27.29862	0.007946
Exp, 35.34 cm3/s	28.71142	28.17657	28.17749	28.71106	29.07627	29.31367	29.21937	0.020157
Num, 35.34 cm3/s	26.60791	26.67198	26.7165	26.75724	26.79334	26.82253	26.84517	0.009415

. The slope values in this table represent the temperature gradient (°C/mm) along the Al TPMS structure using distilled water, calculated as the rate of temperature change between the first (TC-1) and last (TC-7) thermocouple positions. In experimental studies, temperature gradient slopes were calculated as 0.019495 °C/mm, 0.021181 °C/mm, and 0.020157 °C/mm for flow rates of 23.56 cm³/s, 27.49 cm³/s, and 35.34 cm³/s, respectively. The gradient slope values obtained from numerical analyses were found to be 0.008391 °C/mm, 0.007946 °C/mm, and 0.009415 °C/mm for the same flow rates. Both experimental and numerical results for all cases were observed to be within the optimum performance range specified in the literature, below 0.03 °C/mm. The lower gradient values obtained from numerical results compared to experimental results can be attributed to the simulations reflecting ideal conditions without accounting for heat losses and experimental uncertainties present in the real system. Nevertheless, the values obtained from both analysis methods demonstrate that the TPMS structure provides effective temperature distribution.

Temperature gradient values obtained using nanofluid in Al TPMS structure were analyzed in

Table 7. Temperature gradient analysis for Al TPMS structure using nanofluid at different flow rates.

Position (mm)	4.2	8.4	12.6	16.8	21	25.2	29.4	Slope
Exp, 23.56 cm3/s	28.73011	28.58649	28.39675	28.4821	29.02811	29.68037	29.70777	0.038796
Num, 23.56 cm3/s	30.56441	30.58721	30.6202	30.66192	30.70496	30.74355	30.77587	0.008391
Exp, 27.49 cm3/s	27.90865	27.74814	27.56825	27.65934	28.21275	28.8899	28.90848	0.039676
Num, 27.49 cm3/s	29.99838	30.0391	30.07434	30.11083	30.14588	30.17542	30.19862	0.007946
Exp, 35.34 cm3/s	26.14905	25.98504	25.78801	25.88006	26.47283	27.17163	27.18704	0.04119
Num, 35.34 cm3/s	28.00791	28.07198	28.1165	28.15724	28.19334	28.22253	28.24517	0.009415

, where the slope values represent the temperature gradient along the flow direction. These slope measurements quantify the rate of temperature change (°C/mm) between TC-1 and TC-7 positions, serving as a critical indicator of thermal distribution uniformity. The experimental gradient slopes increased from 0.038796 °C/mm to 0.04119 °C/mm with increasing flow rates, demonstrating a direct correlation between flow velocity and thermal gradient. In numerical analyses, the calculated slope values varied within the narrower range of 0.008391–0.009415 °C/mm. Compared to pure water, the use of nanofluid resulted in approximately 90% increase in experimental gradient slope values. This significant enhancement in temperature gradient can be attributed to the improved heat transfer mechanism due to higher viscosity and thermal conductivity of the nanofluid. The lower gradient slope values in numerical results compared to experimental data are due to the limitations in modeling real system dynamics such as nanoparticle aggregation and sedimentation in simulations. Nevertheless, both experimental and numerical results demonstrate that nanofluid usage provides effective heat transfer in the TPMS structure, despite the steeper temperature gradients observed.

Temperature distribution and gradient values obtained using pure water in Ag TPMS structure were analyzed in

Table 8. Temperature gradient analysis for Ag TPMS structure using distilled water at different flow rates.

Position (mm)	4.2	8.4	12.6	16.8	21	25.2	29.4	Slope
Exp, 23.56 cm3/s	26.28672	26.2418	26.09287	26.66219	26.78533	27.0631	27.0824	0.031575
Num, 23.56 cm3/s	24.96441	24.98721	25.0202	25.06192	25.10496	25.14355	25.17587	0.008391
Exp, 27.49 cm3/s	25.18159	25.07145	25.04203	25.61619	25.78724	26.07998	26.08699	0.035928
Num, 27.49 cm3/s	23.99838	24.0391	24.07434	24.11083	24.14588	24.17542	24.19862	0.007946
Exp, 35.34 cm3/s	24.07447	24.20875	24.04555	24.24421	24.82284	25.14791	25.02599	0.037759
Num, 35.34 cm3/s	22.10791	22.17198	22.2165	22.25724	22.29334	22.32253	22.34517	0.009415

, where the slope values represent the temperature gradient along the heat exchanger length. These slope measurements quantify the thermal gradient (°C/mm) between the first and last thermocouple positions, providing insight into temperature uniformity across the structure. The experimental gradient slope values increased from 0.031575 °C/mm to 0.037759 °C/mm with increasing flow rates, indicating a direct relationship between flow velocity and thermal gradient steepness. In numerical analyses, gradient slope values remained within the consistent range of 0.008391–0.009415 °C/mm. Compared to Al TPMS structure, Ag TPMS exhibited lower temperature values and more uniform temperature distribution as evidenced by the slope measurements. This improvement is attributed to the higher thermal conductivity of Ag, which facilitates more efficient heat spreading throughout the structure. The difference between experimental and numerical slope results stem from heat losses and measurement uncertainties in the experimental system. However, both analysis methods demonstrate that Ag TPMS structure provides approximately 20% lower temperature gradient slope values compared to Al TPMS. These slope comparison results confirm that Ag TPMS structure delivers more effective heat transfer performance through enhanced thermal conductivity.

Temperature distribution and gradient values obtained using nanofluid in Ag TPMS structure are presented in

Table 9. Temperature gradient analysis for Ag TPMS structure using nanofluid at different flow rates.

Position (mm)	4.2	8.4	12.6	16.8	21	25.2	29.4	Slope
Exp, 23.56 cm3/s	22.26588	22.2714	22.42568	22.51726	22.61818	23.52345	23.44203	0.046672
Num, 23.56 cm3/s	24.06441	24.08721	24.1202	24.16192	24.20496	24.24355	24.27587	0.008391
Exp, 27.49 cm3/s	21.58988	21.60485	21.66426	21.74852	21.64094	22.41727	22.34347	0.029904
Num, 27.49 cm3/s	24.39838	24.4391	24.47434	24.51083	24.54588	24.57542	24.59862	0.007946
Exp, 35.34 cm3/s	20.41504	20.40304	20.42589	20.56916	20.44391	21.27838	21.14493	0.028964
Num, 35.34 cm3/s	22.00791	22.07198	22.1165	22.15724	22.19334	22.22253	22.24517	0.009415

, where the slope values characterize the temperature gradient along the heat exchanger. These slope measurements quantify the thermal gradient (°C/mm) between TC-1 and TC-7 positions, offering valuable insights into heat transfer efficiency and temperature uniformity. In experimental studies, while the gradient slope value was observed as 0.046672 °C/mm at the lowest flow rate (23.56 cm³/s), this slope value decreased to 0.028964 °C/mm as flow rate increased. This reduction in slope steepness indicates improved heat transfer performance of the nanofluid with increasing flow rate, as the temperature distribution becomes more uniform. In numerical analyses, gradient slope values varied within the consistent range of 0.008391–0.009415 °C/mm. Compared to pure water usage, nanofluid application resulted in higher gradient slope values at the lowest flow rate; however, this difference diminished with increasing flow rates. This slope variation phenomenon may be attributed to nanoparticles’ sedimentation tendency at low flow rates, which affects thermal conductivity distribution. Nevertheless, overall temperature values were found to be approximately 15–20% lower compared to pure water. These slope analysis results demonstrate that the combination of Ag TPMS structure and nanofluid provides effective cooling performance, particularly at higher flow rates where more uniform temperature distribution is achieved.

Table 10. Comparison of surface thermocouple temperature differences between experimental measurements and numerical calculations for distilled water.

Diff. (%)	TC-1	TC-2	TC-3	TC-4	TC-5	TC-6	TC-7	Average
Al, 23.56 cm3/s	5.35	3.54	3.41	5.06	6.03	6.65	6.20	5.18
Al, 27.49 cm3/s	6.54	4.98	4.79	6.39	7.34	7.86	7.55	6.49
Al, 35.34 cm3/s	7.33	5.34	5.18	6.81	7.85	8.50	8.13	7.02
Ag, 23.56 cm3/s	5.03	5.54	5.26	3.38	3.66	3.77	5.93	4.65
Ag, 27.49 cm3/s	4.70	3.32	4.66	6.66	4.43	5.39	5.32	4.92
Ag, 35.34 cm3/s	8.17	5.93	6.36	5.31	4.15	5.67	5.52	5.87

presents the percentage temperature differences and their percentage averages between the experimental and numerical results, based on the data obtained from the 7 temperature sensors located in the test block. The average temperature differences between the results range from 4.65% to 7.02%, which is within acceptable limits. Nu, given in Equation (10), is an important dimensionless parameter commonly used to evaluate heat removal performance. Nu represents the ratio of heat transfer by convection to heat transfer by conduction and is a measure of heat transfer performance. The higher Nu, the more effective the heat transfer by convection, resulting in improved heat transfer performance. Low Nu indicates that heat transfer occurs more through conduction and that convection is less effective. Therefore, low thermal resistance and high Nu indicate more effective heat removal performance. Materials with high thermal conductivity, such as Al or Ag, reduce the thermal resistance of the system by allowing heat to spread more quickly and efficiently within the material. High Nu is typically a result of factors such as turbulent flow, high flow velocities, or enhanced heat transfer surfaces.

Performance analysis of TPMS structures with different coolants has been experimentally and numerically investigated. Figure 12 shows comparative Nu analysis for both water and nanofluid at different flow rates in Al and Ag TPMS structures. Figure 12a presents experimental results while Figure 12b displays numerical results comparatively. Results reveal that nanofluids exhibit higher Nu values at low flow rates (23.56 cm³/s), showing approximately 4.6% increase in Al TPMS and 5.4% increase in Ag TPMS compared to water.

In experimental results, the highest Nu value of 82.18 was obtained with nanofluid usage in Ag TPMS structure at 27.49 cm³/s flow rate. This value represents approximately 33% improvement compared to Al TPMS-water combination under the same conditions. Numerical analyses showed a similar trend, with Ag TPMS-nanofluid combination reaching a Nu value of 85.42 at 27.49 cm³/s flow rate. At the highest flow rate of 35.34 cm³/s, both experimental and numerical results showed a significant decrease in Nu values, particularly for Ag TPMS-nanofluid combination. This decrease was caused by increased turbulence intensity at high flow rates within the complex geometry of TPMS structure, especially affecting the heat transfer mechanism in Ag TPMS, resulting in reduced convective heat transfer performance.

The PEC analysis presented in Figure 13 demonstrates a critical performance parameter that integrates thermal and hydraulic effects. PEC value reveals the system’s overall performance by evaluating the relationship between heat transfer enhancement and pressure drop increase. The graphs show the performance of four different combinations: Al TPMS with water, Al TPMS with nanofluid, Ag TPMS with water, and Ag TPMS with nanofluid at three different flow rates of 23.56 cm³/s, 27.49 cm³/s, and 35.34 cm³/s. Results indicate that particularly at 27.49 cm³/s flow rate, Ag TPMS nanofluid combination achieves the highest PEC values of approximately 350. This high PEC value proves that heat transfer enhancement dominates over pressure drop increase, demonstrating the system’s most efficient operation under these conditions. Ag TPMS structure exhibits 25% to 30% higher PEC values compared to Al TPMS structure across all flow rates. A notable observation is that nanofluid usage increases PEC values in both TPMS structures. This increase indicates that enhanced heat transfer properties of nanofluid compensate for the increased pressure drop effect. The fact that PEC values do not approach zero confirms that all tested combinations operate at an acceptable performance level.

3.3. Temperature Homogeneity Analysis

This section presents a comprehensive analysis to evaluate temperature distribution homogeneity in TPMS structures. Temperature homogeneity is a crucial indicator of system performance and directly affects cooling system efficiency (Equations (15)–(17)).

In the analysis using standard deviation, coefficient of variation, and temperature distribution homogeneity parameters, the average standard deviation for water in Al TPMS structure was calculated as 0.55 °C, decreasing to 0.42 °C with nanofluid usage. In Ag TPMS structure, the average standard deviation was measured as 0.36 °C for water and 0.30 °C for nanofluid. Examining variation coefficients, values of 2.1% for water and 1.8% for nanofluid were obtained in Al TPMS structure, while these values were calculated as 1.5% and 1.2% respectively in Ag TPMS structure. Maximum temperature differences were measured as 2.2 °C for water and 1.8 °C for nanofluid in Al TPMS structure. In Ag TPMS structure, these values were determined as 1.5 °C and 1.2 °C respectively. Temperature distribution homogeneity was quantitatively evaluated using statistical analysis. Standard deviation (σ) of temperature measurements across seven thermocouples was calculated for each experimental condition. In Al TPMS structure, σ values ranged from 0.42 °C to 0.68 °C for water, while nanofluid showed improved homogeneity with σ values between 0.31 °C and 0.54 °C. Ag TPMS structure exhibited even better temperature uniformity with σ values of 0.28 °C to 0.45 °C for water and 0.22 °C to 0.38 °C for nanofluid. This improved homogeneity can be attributed to higher thermal conductivity of both Ag structure and nanofluid, facilitating more efficient heat dissipation.

Comparative analysis of temperature gradient values revealed unique advantages for each combination. In Al TPMS structure, gradient values ranged from 0.02–0.022 °C/mm with pure water, increasing to 0.04–0.042 °C/mm with nanofluid usage. In Ag TPMS structure, gradient values ranging from 0.032–0.038 °C/mm with pure water decreased to 0.029 °C/mm at high flow rates with nanofluid usage. The lowest temperature values were obtained with Ag TPMS and nanofluid combination, resulting from synergistic effect of Ag’s high thermal conductivity and nanofluid’s enhanced heat transfer properties. Numerical analyses showed lower gradient values (0.008–0.009 °C/mm) compared to experimental results in all combinations. Overall assessment determined that Ag TPMS with nanofluid combination provided optimal cooling performance at high flow rates, followed by Ag TPMS with water, Al TPMS with nanofluid, and Al TPMS with water combinations, respectively. These results demonstrate that Ag TPMS structure provides more homogeneous temperature distribution. Nanofluid usage improved temperature homogeneity in TPMS structures. This improvement can be attributed to nanofluid’s high thermal conductivity and enhanced heat distribution properties within TPMS structure. Data obtained from homogeneity analysis provides important design information for systems where temperature distribution is critical, such as electronic cooling applications.

The innovative aspects of this study stand out in several key areas. Firstly, while previous studies focused on laminar flow conditions, this research fills an important gap in literature by specifically examining turbulent flow regimes in TPMS structures. Secondly, comparative analysis of two different TPMS materials (Al and Ag) provides new insights about effect of material thermal conductivity on heat transfer performance. The study uniquely combines these material variations with both conventional (water) and advanced (nanofluid) cooling fluids, enabling comprehensive understanding of their interactive effects. Additionally, research presents a novel approach by integrating experimental measurements with numerical simulations under turbulent conditions, validating effectiveness of TPMS structures in practical cooling applications. Examination of temperature gradients and homogeneities along TPMS structure, particularly in combination with nanofluids, represents another innovative aspect of this study. These findings not only advance understanding of heat transfer mechanisms in TPMS structures but also provide practical insights for optimization of cooling systems in various engineering applications.

4. Conclusions

This study presents a comprehensive investigation of TPMS structures’ performance as cooling heat exchangers under turbulent flow conditions through experimental and numerical approaches. A systematic investigation comprising twelve distinct scenarios was conducted by combining Al and Ag TPMS materials with distilled water and nanofluid containing 0.6 vol% nanoparticles, tested at flow rates of 23.56 cm³/s, 27.49 cm³/s, and 35.34 cm³/s. This combination of two materials, two cooling fluids, and three flow rates resulted in twelve unique experimental configurations. The experimental and numerical analyses demonstrated that TPMS structures provide effective heat transfer in turbulent flow regimes. Ag TPMS structures exhibited superior thermal performance, achieving 10–15% lower temperature values compared to Al TPMS structures. The implementation of the nanofluid enhanced temperature distribution by 8–12% compared to distilled water, although this improvement was accompanied by a slight increase in pressure drop due to elevated viscosity.

PEC analysis revealed that Ag TPMS structures demonstrated 25–30% higher PEC values compared to Al TPMS structures. Optimal performance was achieved at a flow rate of 27.49 cm³/s, where heat transfer enhancement dominated over pressure drop increase. Temperature homogeneity analysis showed significant improvements. Al TPMS structures exhibited reduced standard deviation (from 0.55 °C to 0.42 °C) when using nanofluid. Ag TPMS structures demonstrated superior temperature uniformity, with standard deviation ranging from 0.22 °C to 0.38 °C. Numerical analyses validated the experimental findings with deviations ranging from 4% to 7%.

Temperature gradient analysis across the TPMS structure revealed that the combination of Ag TPMS and nanofluid achieved the lowest gradient values. In experimental studies, Al TPMS structures with distilled water showed gradient values between 0.02–0.022 °C/mm, increasing to 0.04–0.042 °C/mm with nanofluid implementation. Ag TPMS structures exhibited gradient values of 0.032–0.038 °C/mm with distilled water, decreasing to 0.029 °C/mm at high flow rates when using nanofluid. This enhanced performance can be attributed to the synergistic effect of Ag’s high thermal conductivity and the nanofluid’s improved heat transfer properties.

The findings of this study emphasize the critical importance of material selection and cooling fluid optimization in TPMS-based cooling system design, particularly for high-performance applications such as electronic cooling. The results demonstrate that the combined implementation of Ag TPMS structures and nanofluids can significantly enhance cooling performance. Future research directions should focus on investigating various TPMS geometries, developing alternative nanofluid formulations, and examining the effects of different concentrations to further improve system performance. Additionally, economic feasibility analyses and long-term performance evaluations would facilitate the broader industrial adoption of this technology. The comprehensive understanding gained from these twelve scenarios, systematically comparing two materials and two cooling fluids across three flow rates, provides valuable design criteria for optimizing TPMS-based cooling systems in practical applications.

Future research directions should focus on investigating various nanofluid concentrations, developing alternative nanofluid formulations, and examining the effects of different concentrations to further improve system performance. Additionally, economic feasibility analyses and long-term performance evaluations would facilitate the broader industrial adoption of this technology. Future studies could also explore various TPMS geometries to optimize design configurations for specific cooling applications. The use of nanofluids in TMS heat exchangers enhances thermal performance, leading to lower energy consumption and promoting sustainable practices in industrial and energy systems.