Optimization of the TPMS Heat Exchanger Toward Cooling the Heat Sink

Abstract

The subject of the current paper is cooling heat sinks using the TPMS structure. An experiment was conducted using water and a mixture of 10% vol. ethylene glycol in water, which was used to cool heat sinks in the presence of the TPMS structure. The gyroid was developed using 3D printing with three different porosities: 0.7, 0.8, and 0.9, respectively. The shell network is a single domain, and fluid is circulated at various flow rates. A comparison with the numerical model, as simulated using COMSOL software (version 6.2), showed good agreement. A uniform temperature distribution is a clear indication of uniform cooling. Then, the TPMS structure is changed from one domain to two unconnected domains, and a different flow rate is applied to each domain entry. This approach is unique in that it investigates the cooling of the heat sink with a two-domain structure, which has not been previously studied. The novelty of this paper lies in utilizing two TPMS structure domains to cool the heat sink. Thus, dual-domain TPMS heat sinks are implemented and optimized with separate inlets. Statistical testing of the model for the Nusselt number and the performance evaluation criterion is performed using Fisher’s statistical test to analyze variance (ANOVA). It was found that the cooling heat sink is more accurate with two-domain systems. The average Nusselt number polynomial is found to vary linearly with the two-inlet velocity, the porosity and the fluid Prandtl number. Similar linearity is found for the performance evaluation criterion. The optimum Nusselt number equals 77, the PEC equals 49 for a porosity of 0.85, and the Prandtl number is 36.9.

1. Introduction

Heat exchangers are a recurrent element found in many mechanical engineering systems. The design of these heat exchangers has remained static due to manufacturing limitations. However, additive manufacturing has recently provided an opportunity to produce new and previously impossible heat exchanger geometries by fabricating systems layer by layer. Additive manufacturing is receiving considerable attention due to its application in the triply periodic minimal surface (TPMS) structure. Many advantages of using TPMS have been identified, among them the lightweight structure. With the availability of 3D printing of metal in various sizes and high accuracy, other materials for heat exchangers are becoming accessible.

In 2015, Thompson et al. [1] discussed the possibility of designing a heat exchanger using 3D printing with a Ti-6Al-4V oscillating heat pipe. They employed layer-by-layer fabrication of very complex structures that are typically unobtainable with conventional manufacturing methods. They utilized a selective laser melting technique to fabricate a compact, flat plate oscillating heat pipe with innovative design features. Their experimental testing demonstrated the manufacture and functionality of a novel flat plate oscillating heat pipe. Such a design was difficult to achieve using conventional fabrication approaches. They stated that selective laser melting provides the opportunity to fabricate complex heat exchangers and any other structure for a specific application. This was the beginning of using TPMS in thermos-fluid.

Later, Bacellar et al. [2] introduced a design optimization for high-performance heat exchangers using additive manufacturing. Initially, they discussed the introduction of fins on a standard air-fluid heat exchanger. Then, a proof of concept for a finless tube is introduced, and finally, a comprehensive analysis with shape optimization is introduced. Different cross-sections of tubes are studied, with some having an elliptical shape. Their argument is that the use of tubes and the presence of fins should be eliminated. This concept has been demonstrated both numerically and experimentally by the authors. Their work continues as a traditional heat exchanger without mentioning the TPMS structure.

Klein et al. [3] reviewed recent advances in additively manufactured heat exchangers. They presented a list of researchers who have dedicated their time to developing new heat exchangers using additive manufacturing (AM) techniques. Metal, polymer, and ceramic heat exchangers are being fabricated and experimentally tested. However, they stated numerous challenges still need to be addressed, such as material characteristics, cost-effectiveness, and surface quality. However, their reviews helped determine the type of heat exchanger produced and the quality of its structure. Cardone and Guargiulo [4] designed a compact heat exchanger to operate in fluid, air, and water. It consists of an aluminum mini-channel heat exchanger with a crossflow configuration. Additive manufacturing has been utilized to produce the designed heat exchanger using the powder bed fusion technique. The rig is tested both numerically and experimentally under various conditions. A discrepancy is detected between the calculated data and the measured data. The reason for these differences is in the heat exchanger geometry, where the material roughness exists, resulting in a high friction factor. What is noticeable is that the use of AM is taking shape for different applications.

At the same time, Under et al. [5] developed an air-side thermal flow heat exchanger constructed using additive manufacturing with fins. The innovative approach involves fins of various design shapes surrounding the oval tube to assess its performance. The selective laser melting technique is used for this purpose. They identified the best design by examining the Nusselt number and evaluating its performance. They also investigated the fin design and spacing as part of the heat exchanger’s performance. An optimization approach should have been employed to support the conclusion presented by the author. Kim and Yoo [6] proposed a new compact heat exchanger based on volumetric distance fields (VDF) calculations. This VDF-based geometric component description allowed for the computational efficiency design of complex-shaped heat exchangers. Sa Silva et al. [7] compared the thermal and hydrodynamic analytical model with experimental tests in a crossflow heat exchanger produced by the SLM technique. The hydrodynamics theoretical model predicts the influence of electropolishing and re-melting processes on the total pressure loss. They demonstrated that SLM technology could manufacture compact heat exchangers with high accuracy.

Moderk et al. [8] conducted an optimization study to design an additively manufacturable porous heat sink based on the TPMS structure. At the same time, Dixit et al. [9] proposed a high-performance micromachined compact heat exchanger using 3D printing. They demonstrated a 55% increase in heat exchanger effectiveness for the additively manufactured gyroid lattice compared to the equivalent counterflow heat exchanger. This suggests that TPMS is the most suitable approach for designing high-efficiency, compact heat exchangers. To explore the concept of using TPMS in heat exchangers, Iyer et al. [10] investigated the heat transfer and pressure drop characteristics of heat exchangers based on TPMS. They showed that TPMS can be used to design heat exchangers with superior performance. Reynolds et al. [11], continuing the investigation of the use of TPMS in heat exchangers, proposed a correlation for designing a micro-channel heat exchanger that operates at Reynolds numbers ranging between 100 and 2500. Since heat exchangers with TPMS demonstrated high performance and are lightweight, interest in them started to rise in the aerospace industry.

Careri et al. [12] conducted a comprehensive literature review of additive heat exchanger manufacturing for the aerospace industry. They proposed a new criterion for down-selecting compact heat exchanger materials for additive manufacturing, with a particular focus on aluminum alloys. Kaur followed this finding and Sing [13] by highlighting the state-of-the-art heat exchanger additive manufacturing. This was followed by Sabau et al. [14] and Niknam et al. [15], who investigated an additive manufacturing approach for heat exchangers. Finally, Brambati et al. [16] and Min et al. [17] investigated the use of TPMS in heat exchangers. Their findings supported the idea of higher performance and lightweight. Izadi et al. and Azzouz and Hamida [18,19] have recently proposed additional work related to heat exchangers.

The literature review demonstrates that the TPMS heat exchanger has proven its usefulness in industry. The present paper proposes that the TPMS heat exchanger design be used for cooling heat sinks. Such an approach is not found in the literature. The advantage of such an approach is to use two different liquids at two different temperatures to study the heat enhancement toward the heat sink. Then, an optimization approach defines the optimal scenario for this finding. Section 2 describes the problem under study, followed by Section 3, which outlines the experimental setup for a single-domain flow. The numerical modelling is shown in Section 4. The investigation begins by comparing the numerical model with experimental data for a single domain, demonstrating the accuracy of the numerical model as presented in Section 5. Section 6 will show the numerical results obtained for cooling heat sinks with two domains of the heat exchanger. The optimization will be discussed in Section 7, and the concluding remarks are presented in Section 8. The novelty of this paper lies in utilizing two TPMS structure domains to cool the heat sink. Thus, dual-domain TPMS heat sinks are implemented and optimized with separate inlets.

2. Problem Description

The triply periodic minimal surface (TPMS) structure has attracted the interest of researchers in the thermofluid area of research [20]. The uniqueness of this structure lies in its ability to extract heat from a hot surface or to serve as a heat exchanger for two circulating fluids. The structural wall of the TPMS could be thin enough to behave like fins. Additionally, it is lightweight, which attracts aerospace researchers to incorporate it into new designs of heat exchangers.

The present paper utilizes the gyroid to create the TPMS structure, which is manufactured using 3D printing. The material is composed of aluminum with some minor additional metallic components (AlSi₁₀Mg), utilizing laser power technology. The structure under investigation is a block with a square base measuring 37.5 mm and a height of 12.7 mm. The porosity of the structure is 0.7, 0.8, and 0.9, respectively.

Two types of models are used in the present study. In the first model, the TPMS is modelled as a single domain and used experimentally to examine the performance of such a structure in cooling a heat sink. The measured experimental results are compared against numerical modelling to calibrate the model. The experiment used water as a circulating fluid [20] and a mixture of ethylene glycol with water [20]. The TPMS structure used has a porosity of 0.7.

In the second model, using identical dimensions and conditions, numerical modelling is employed to study the performance of heat enhancement by treating the block as two domains. In these two domains, water is circulating in one domain, and a mixture of ethylene glycol is in the second. Different flow rate ratios are investigated by maintaining heating conditions identical to the first model. Thus, the data generated for 17 runs was issued to study the optimization and determine the most suitable cases to be conducted experimentally. A detailed description of the experiment and the model will follow, leading to the optimization process.

3. Description of the Experiment

The experimental setup and data generated are taken from Reference [20] by Saghir et al. As shown in Figure 1a, a data acquisition system, a fluid pump, a test section, and a flow metre are used to conduct accurate experiments. In this experiment, a single flow inlet and outlet are used, and the entire fluid circulates in the TPMS in a single domain, as shown in Figure 2. The thermocouples, as shown in Figure 1b, are located 1 mm below the interface between the heated block and the TPMS in the heated block. The test section is mounted on top of an aluminum block with dimensions of 37.5 mm × 37.5 mm × 12.7 mm and heated from the bottom side with a constant heat flux of 38,400 W/m².

The TPMS structure has a porosity of 0.7. The liquid, identified as a mixture of water and ethylene glycol (10% vol) and water (90% vol), enters the test section at different flow rates. As shown in Figure 2, the test section comprises the TPMS structure mounted on top of an aluminum plate, which is connected to a heated aluminum block. Flow enters at a constant T_in temperature and velocity, circulates through the heated TPMS, and exits from the opposite side. The outlet temperature is also measured. The test section is made of Teflon to suppress any heat losses to the environment. Additional details about the experiment can be found in Saghir et al.’s manuscript [20].

4. Numerical Modelling

Numerical modelling is becoming a potent tool for solving complex engineering problems. In our present case, numerical modelling is used to study heat and fluid distribution on a structure under thermal conditions. The finite element technique is adopted by using the commercial software COMSOL (version 6.2) [21]. Because the TPMS structure is used and heated from below, the complete Navier–Stokes equation combined with the energy equation is solved numerically. The heat conduction equation is also used since the TPMS is modelled as a solid structure. As the experimental setup describes, the numerical model aims to model the test section. In that contest, the formulation used in the modelling is as follows.

4.1. Fluid Flow Formulation

The flow is assumed to be Newtonian and in a steady state condition. Water and water-ethylene glycol mixtures, which are incompressible fluids, are used in the current simulation. The flow rate applied at the inlet shows that the flow is in a laminar regime.

Continuity equation

$${\displaystyle \frac{𝜕\mathrm{u}}{𝜕\mathrm{x}}}+{\displaystyle \frac{𝜕\mathrm{v}}{𝜕\mathrm{y}}}+{\displaystyle \frac{𝜕\mathrm{w}}{𝜕\mathrm{z}}}=0$$

(1)

x-direction

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

(2)

y-direction

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

(3)

z-direction

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

(4)

In Equations (1)–(4), the velocities u, v, and w are in the directions of x, y, and z, respectively, and p is the pressure in N/m². Because the flow is in forced convection, the gravity vector is negligible. The fluid density is $\rho$ in kg/m³, and the dynamic viscosity is $\mu$ in kg/m.s.

Table 1. Fluid mixture properties.

Fluid Mixture	$\mathsf{\rho }$ (kg/m3)	$\mathsf{\mu }$ (kg/m.s)	k (w/m.K)	Cp (J/kg.C)
Water 100%	998.2	0.001001	0.6	4128
Water 90%, EG 10%	1010	0.003	0.5766	4002
Water 75%, EG 25%	1068	0.00369	0.391	3617
Water 50%, EG 50%	1094	0.00761	0.316	3213

shows the physical properties of the fluids used in the numerical simulation.

4.2. Heat Transfer Formulation

The energy equation for the fluid portion is as follows.

$$\rho \mathrm{C}\mathrm{p}\left[\mathrm{u}{\displaystyle \frac{𝜕\mathrm{T}}{𝜕\mathrm{x}}}+\mathrm{v}{\displaystyle \frac{𝜕\mathrm{T}}{𝜕\mathrm{y}}}+\mathrm{w}{\displaystyle \frac{𝜕\mathrm{T}}{𝜕\mathrm{z}}}\right]=\mathrm{k}\left[{\displaystyle \frac{{𝜕}^2\mathrm{T}}{{𝜕\mathrm{x}}^2}}+{\displaystyle \frac{{𝜕}^2\mathrm{T}}{{𝜕\mathrm{y}}^2}}+{\displaystyle \frac{{𝜕}^2\mathrm{T}}{{𝜕\mathrm{z}}^2}}\right]$$

(5)

The specific heat capacity, Cp, is expressed in J/kg·K, and T is the temperature in degrees K for the entire model. Here, the conductivity of the flow is k, and the unit is W/m·K. The temperature in the solid part of the model is determined by solving the energy formulation shown in Equation (5) without the convective term. Two critical parameters will be determined numerically: the Nusselt number (Nu) and the performance evaluation criterion (PEC). In the current analysis, the Nusselt number is defined as

$$\mathrm{N}\mathrm{u}={\displaystyle \frac{\mathrm{h}{\mathrm{D}}_{\mathrm{h}}}{\mathrm{k}}}$$

(6)

where h is the heat convection coefficient defined as $\mathrm{h}=\frac{\mathrm{q}\mathrm{″}}{(\mathrm{T}-\mathrm{T}\mathrm{i}\mathrm{n})}$ and k is the fluid conductivity and ${\mathrm{D}}_{\mathrm{h}}$ is the hydraulic diameter. The measured temperature is T, and Tin is the inlet temperature of the fluid. The performance evaluation criterion is the ratio of the average Nusselt number to the friction coefficient, raised to the power of 1/3. Thus, PEC is

$$\mathrm{P}\mathrm{E}\mathrm{C}={\displaystyle \frac{{\mathrm{N}\mathrm{u}}_{\mathrm{average}}}{{\mathrm{f}}^{1/3}}}$$

(7)

The friction coefficient factor f is known as

$$\mathrm{f}={\displaystyle \frac{4\times (∆\mathrm{p})}{\frac{\mathrm{L}}{{\mathrm{D}}_{\mathrm{h}}}\times \rho \times {{(\mathrm{u}}_{\mathrm{i}\mathrm{n}})}_{\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}}}}$$

(8)

where $∆p$ is the average pressure difference between the inlet and the outlet, u_in is the average inlet velocity, and L is the TPMS structure length along the flow direction.

Different approaches are available in COMSOL to address the convergence criteria. In this model, the default solver used was the segregated method. Details about this approach could be found in any finite element textbook. The convergence criteria are clearly explained in the COMSOL manual. The convergence criteria were set at every iteration, and the average relative error of u, v, w, p and $\mathrm{T}$ were computed. These were obtained using the following relation:

$${\mathrm{R}}_{\mathrm{c}}={\displaystyle \frac{1}{\mathrm{n}\times \mathrm{m}}}{\sum }_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{m}}{\sum }_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{n}}\left|{\displaystyle \frac{({\mathrm{F}}_{\mathrm{i},\mathrm{j}}^{\mathrm{s}+1}-{\mathrm{F}}_{\mathrm{i},\mathrm{j}}^{\mathrm{s}})}{{\mathrm{F}}_{\mathrm{i},\mathrm{j}}^{\mathrm{s}+1}}}\right|$$

(9)

where F represents one of the unknowns, viz., u, v, w, p, or $\mathrm{T}$, s is the iteration number, and (i, j) represents the coordinates on the grid. Convergence is reached if R_c for all the unknowns is below 1 × 10⁻⁶ in two successive iterations. For further information on detailed solution methods, the reader is referred to the COMSOL software manual [21].

4.3. Boundary Conditions

Figure 3 shows the boundary condition used in the model. It is worth mentioning that the heated block has a square base of 37.5 mm and a height of 12.7 mm. Similarly, the block containing the TPMS structure has the same dimensions.

The four cylinders’ inlets and outlets have a diameter of 10 mm. A constant heat flux of 38,400 W/m² is applied from the bottom side to the test sample, as shown by the red arrows. At the inlet, the fluid temperature, denoted as Tin, is the inlet temperature for domain 1, which is u₁. From the domain, it is u₂, as shown in the Figure. Additionally, the outlet temperatures from domain 1 and domain 2 are T_1out and T_2out, respectively. All model boundaries, except where heat flux is applied, are insulated and assume adiabatic walls. Thus, the boundary condition is expressed in equation form.

(i)

The velocity u = u₁ in the x direction and u₂ in the y direction is applied at the inlet.

(ii)

At the inlet, the temperature of the fluid enters the test section at T = Tin for both domains.

(iii)

At the outlet, an open boundary is applied where the stresses are equal to zero.

(iv)

The bottom surface of the aluminum block is heated with a heat flux q″, as shown in red.

(v)

All external surfaces are assumed adiabatic, $\frac{𝜕{\mathrm{T}}_{\mathrm{S}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}}{𝜕\mathrm{n}}=0,$ and for the flow, no-slip boundary conditions are applied.

4.4. Mesh Sensitivity Analysis

It is necessary to conduct a mesh sensitivity analysis to ensure that the selected mesh is the most suitable for predicting accurate results.

Table 2. Mesh sensitivity analysis.

Mesh Size	Domain Elements	Boundary Elements	Edge Elements	Average	Nuaverage
Extra coarse	55,245	11,637	1290	25.98	35.46
Coarser	100,955	17,088	1620	27.14	31.55
Coarse	218,213	28,495	2149	28.88	27.08
Normal	517,272	52,683	3033	30.69	23.62
Fine	1,361,827	105,901	4368	31.35	22.6

presents the study conducted for different mesh elements. As shown, three different mesh elements are needed to mesh the model. The domain mesh consists of tetrahedral elements in 3D and triangular elements. A fine mesh with 1,472,096 elements showed the best mesh for accurate results. The indicators are two variables: the average temperature calculated 1 mm below the interface between the aluminum block and the TPMS in the aluminum block, and the corresponding Nusselt number. However, a standard mesh size is ideal for the simulation, reducing computation time. Figure 4 presents the meshed model.

5. Comparison Between Experimental and Numerical Data

A previous study by Saghir et al. [20,22,23] focused on using a single-domain TPMS to investigate cooling heat sinks. The amount of heat generated by the heat sink is 38,400 W/m². Different flow rate ranges were applied, and experimental data on the temperature variation were obtained. The temperatures were measured in the heated block, 1 mm below the heated block, and at the TPMS interface. Seven thermocouples were used, each equidistant from the other by 4.2 mm. The experiment used two different fluids: purified water and a mixture of 10% vol ethylene glycol in 90% vol water. Figure 5 presents the single-domain model, which is the subject of both experimental and numerical investigation.

Figure 6a compares the temperature between the experimental method and the numerical approach for two different flow rates using water as the circulating liquid. A good agreement is observed between the two approaches, with the maximum difference between the experiment and numerical measurement being around 8%. Later, the same experiment was repeated with two sets of higher flow rates: 11.8 cm³/s (corresponding to a Reynolds number Re = 498) and 15.73 cm³/s (corresponding to a Reynolds number Re = 664), with a mixture of 10% vol of ethylene glycol in water. An uncertainty analysis was conducted in detail in previous work by Saghir et al. [20,22,23] and is omitted here to reduce duplication. Generally, the maximum uncertainty is approximately 5%. In this mixture, the fluid is more viscous and has lower thermal conductivity than distilled water. Figure 6b,c presents the comparison.

Notably, the measured experimental data are lower than the numerical ones. The main reason is that heating losses occur experimentally, or the heater performs poorly. However, the maximum difference between the experimental data and the numerical one is 4% at a flow rate of 11.8 cm³/s and 6% at a flow rate of 15.73 cm³/s. It is evident from Figure 5 that the numerical model is sufficiently accurate for other predictions. Figure 7 shows the three-dimensional temperature and velocity profile across the test section.

6. Heat Exchanger Design in the Presence of a Heat Sink

In the previous section, the numerical model for a single domain structure has been shown to be accurate when the data collected aligns with the experimental data. The intention is to repeat the modelling using two domains within the gyroid structure. The materials used in the numerical model are identical to those in the single-domain model, AlSi10Mg. The conductivity of AlSi10Mg is set equal to 150 W/m·K, the specific heat is 920 J/kg·K, and its density is 2670 kg/m³. The shell structure has porosities of 0.7, 0.8, and 0.9, respectively. Similarly, the gyroid structure has a similar dimension, with a base of 37.5 mm and a height of 12.7 mm. The unit cell dimension used is 18 mm. The boundary conditions applied here are identical to those in the previous case, with a heat flux applied to the aluminum block of 38,400 W/m². The only difference is that fluid enters from two different inlets with different flow rates.

As shown in Figure 8, two rectangular chambers with dimensions of 10 mm in width, 37.5 mm in depth, and 12.7 mm in height are located at opposite sides in the x-direction. Two similar chambers are located in the y-direction. Cylinders having a diameter of 10 mm and a length of 10 mm are connected to the four fluid chambers. Fluid enters from cylinder 1 and exits from cylinder 2. Similarly, different fluid enters from cylinder 3 and exit from cylinder 4. A heat flux q″ is applied at the bottom of the aluminum solid block. The fluid enters at a specific inlet temperature, Tin.

Justification of This Model

Previous experimental and numerical papers by Saghir et al. [20,22,23] focused on a single flow domain for cooling small surfaces. On one hand, the gyroid provided a uniform cooling along the flow direction. However, as flow moves inside the gyroid, the fluid temperature rises, providing better cooling near the inlet than at the outlet. Around 1.5 °C, a temperature difference is measured. By splitting the flow into two domains to mimic a heat exchanger model, it is believed that the plate will undergo a more efficient cooling process, with a temperature difference of less than one degree Celsius from the inlet to the outlet. Additionally, this configuration has not been utilized yet for cooling small surfaces.

The two domains under Investigation are shown In Figure 9. The first one in yellow is for inlet 1, where velocity u₁ is applied, and the inlet temperature is Tin, as shown in Figure 9a.

For the second domain shown in Figure 9b, the inlet temperature velocity is u₂, and the inlet temperature is again Tin. To investigate further, Figure 9 displays the gyroid structure with domain 1 (Figure 10a), domain 2 (Figure 10c), and both combined (Figure 10b).

To investigate the numerical performance of this new model, different flow rates are applied at inlets 1 and 2 using three different fluid compositions: 25% vol ethylene glycol water mixture and 50% vol ethylene glycol water mixture, for separate cases.

The Reynolds number for all cases varies between Re = 40 and Re = 600.

This corresponds to Prandtl numbers of 7, 34, and 77, respectively. It is essential to indicate that the fluid between the two domains does not mix; thus, the heat exchanger approach is unique for cooling the heat sink.

Table 3. Different runs for identical heating conditions.

Run	u1 (m/s)	u2 (m/s)	$\mathit{\phi }$	Run#	u1 (m/s)	u2 (m/s)	$\mathit{\phi }$	Run#	u1 (m/s)	u2 (m/s)	$\mathit{\phi }$
1	0.042	0.0165	0.8	10	0.0165	0.025	0.7	19	0.00785	0.0165	0.8
2	0.025	0.0165	0.8	11	0.033	0.00785	0.9	20	0.03303	0.00785	0.9
3	0.025	0.0165	0.8	12	0.0165	0.00785	0.9	21	0.025	0.0165	0.8
4	0.0330	0.025	0.7	13	0.033	0.025	0.9	22	0.0165	0.025	0.9
5	0.0165	0.025	0.7	14	0.0165	0.00785	0.7	23	0.03303	0.00785	0.7
6	0.033	0.025	0.9	15	0.033	0.00785	0.7	24	0.03303	0.025	0.7
7	0.0165	0.00785	0.9	16	0.025	0.033	0.8	25	0.0165	0.00785	0.7
8	0.025	0.0165	0.8	17	0.025	0.0165	0.8
9	0.0165	0.025	0.9	18	0.025	0.0165	0.8

presents the cases under investigation. In total, 25 runs were conducted to optimize the results to achieve the best scenario.

As indicated in Table 3, multiple cases have been investigated. Two critical parameters will be discussed: the Nusselt number, as defined in Equation (5), and the performance evaluation criterion, as outlined in Equation (6). However, the average pressure drop between inlet and outlet one and inlet and outlet two will be calculated to calculate the friction factor. Figure 11 presents the temperature variation, calculated 1 mm below the solid/TPMS interface, at seven locations for all porosities and different inlet velocities. Water is the circulating liquid with the highest thermal conductivity among the three fluids under investigation. For simplicity, the graph shows the velocity ratio between the two inlets.

The velocity ratio influences the temperature variation. Every three sets are regrouped depending on this ratio for porosity levels of 0.7, 0.8, and 0.9. The almost flat temperature distribution indicates that the TPMS provides uniform cooling. The lowest cooling occurred when the velocity ratio was equal to 1.32, and the porosity was 0.7. Figure 12 presents a similar temperature variation for the remaining 50% vol EG/50% vol water mixture.

A similar temperature distribution is observed, but the magnitude of the temperature is more significant. That indicates that it is due to the lower conductivity of the mixture. Similar behaviour is detected when the fluid mixture is 25% vol EG and 75% vol water.

Figure 13 presents the Nusselt number and the performance evaluation criterion for different inlet velocity ratios when water is the circulating liquid, further investigating the importance of two-domain cooling. It is evident from these results that the velocity ratio of 1.32, in the presence of water, exhibits the highest Nusselt number and performance criteria. For the case where 25% vol EG/75% vol water, the highest Nusselt number and the performance evaluation criterion is when the velocity ratio is equal to 1.27 and finally, for the case when the fluid is 50% vol EG and 50% vol water, the highest indicator of Nu and PEC is when the velocity ratio is equal to 1.32. Among the three liquid conditions, a porosity of 0.8 is the most effective for heat removal.

We ask whether the two flow domains would exhibit better cooling than a single domain for the same flow condition. The answer can be determined by comparing Figure 6a and Figure 10. Using a gyroid with similar porosity and flow rate, the temperature distribution is lower when two flow domains are used. This investigation is further studied by examining the flow behaviour inside the structure. When two domains are adopted, the fluid is forced to circulate throughout the entire structure, unlike when applied to a single domain. In this case, the flow attempts to reach a close distance to the exit, thereby omitting some areas that need to be cooled. Two domain structures enable the flow to flood the TPMS structure, resulting in improved cooling completely.

7. Optimization

Based on the data generated, it is interesting to determine the relationship between the average Nusselt number and all variables used in the modelling: the two inlet velocities (u₁ and u₂), porosity, and the Prandtl number. Similarly, it is essential to determine the relationship between this variable and the performance evaluation criterion, as the friction factor varies for each case studied. Response surface methodology has been employed to determine the optimal scenario for heat enhancement in a model with two domains. As stated earlier, the variables used are the inlet velocity (u₁) for domain 1, the inlet velocity (u₂) for domain 2, the porosity of the TPMS structure, the type of fluid used in the model which was water and a mixture of water with 50% vol ethylene glycol (EG) (Pr = 77.4), and a mixture of 25% vol EG in water (Pr = 34.14).

Table 4. Range of variables used in the model.

Variables	Values
Inlet velocity of domain 1 u1	$0.0785\le {\mathrm{u}}_2\le 0.042$
Inlet velocity for domain 2 u2	$0.00785\le {\mathrm{u}}_2\le 0.0423$
Porosity of the structure $\phi$	$0.7\le \mathrm{\phi }\le 0.9$
Fluid (Pr)	$6.88\le \mathrm{P}\mathrm{r}\le 77$.4

presents the range of the variables used in the model and to be used in the RSM technique.

The design expert software determines the relationship between the variables and their effect on Nusselt and PEC numbers. To achieve the objectives, the software required 25 runs for different values of variables based on the central composite design [24,25,26].

Table 5. Composite Central Design (CCD) matrix of four factors and two numerical data points.

		Factor 1	Factor 2	Factor 3	Factor 4	Response 1	Response 2
Std.	Run	u1 (m/s)	$\mathit{\phi }$	Pr	u2 (m/s)	Nuaverage	PEC
18	1	0.042	0.8	34.14	0.016425	123.25	89.5
20	2	0.025	0.8	34.14	0.016425	100.26	72.8
21	3	0.025	0.8	34.14	0.016425	100.26	72.8
14	4	0.03303	0.7	77.4	0.025	131.71	81.9
13	5	0.0165	0.7	77.4	0.025	109.77	61.1
12	6	0.03303	0.9	6.88	0.025	113.85	97.3
3	7	0.0165	0.9	6.88	0.00785	58.39	39.6
25	8	0.025	0.8	34.14	0.016425	100.26	72.8
15	9	0.0165	0.9	77.4	0.025	112.06	74.7
9	10	0.0165	0.7	6.88	0.025	101.84	81.1
4	11	0.03303	0.9	6.88	0.00785	79.35	41
7	12	0.0165	0.9	77.4	0.00785	62.65	33
16	13	0.03303	0.9	77.4	0.025	130.57	97.1
1	14	0.0165	0.7	6.88	0.00785	58.92	40
6	15	0.03303	0.7	77.4	0.00785	82.23	30.7
19	16	0.025	0.8	34.14	0.03303	145.89	116
22	17	0.025	0.8	34.14	0.016425	100.26	72.8
24	18	0.025	0.8	34.14	0.016425	100.26	72.8
17	19	0.00785	0.8	34.14	0.016425	76.72	44.8
8	20	0.03303	0.9	77.4	0.00785	82.36	36.5
23	21	0.025	0.8	34.14	0.016425	100.26	72.8
11	22	0.0165	0.9	6.88	0.025	94.1	75
2	23	0.03303	0.7	6.88	0.00785	83.97	43.4
10	24	0.03303	0.7	6.88	0.025	126.5	108
5	25	0.0165	0.7	77.4	0.00785	59.46	26.5

presents the four variables (i.e., factors) and the responses of the Nusselt and PEC numbers for all runs.

Statistical testing of the model for each response is performed using Fisher’s statistical test for analysis of variance (ANOVA). The fitted polynomial equation is expressed as a function of all variables. The accuracy of the model’s fit is investigated by calculating the coefficient R² and using the adjusted R² and the predicted R².

Table 6. Presents the ANOVA for the response surface linear model.

Source	Sum of Squares	Degree of Freedom	Mean Square	F-Value	p-Value
Nu
Linear vs. Mean	13,326.42	4	3331.60	115.84	<0.0001
PEC
Linear vs. Mean	14,079.07	4	3519.77	52.39	<0.0001

and

Table 7. Composite Central Design (CCD) matrix and numerical data using RSM for the average Nusselt number.

Nuaverage					RSM
Run	u1 (m/s)	$\mathit{\phi }$	Pr	u2 (m/s)	Numerical Value	Model Value	Relative Error, %
1	0.042	0.8	34.14	0.016425	123.25	118.13	4.14
2	0.025	0.8	34.14	0.016425	100.26	95.44	4.79
3	0.025	0.8	34.14	0.016425	100.26	95.44	4.79
4	0.03303	0.7	77.4	0.025	131.71	133.66	1.48
5	0.0165	0.7	77.4	0.025	109.77	111.60	1.67
6	0.03303	0.9	6.88	0.025	113.85	125.55	10.28
7	0.0165	0.9	6.88	0.00785	58.39	57.84	0.93
8	0.025	0.8	34.14	0.016425	100.26	95.44	4.79
9	0.0165	0.9	77.4	0.025	112.06	108.97	2.75
10	0.0165	0.7	6.88	0.025	101.84	106.12	4.2
11	0.03303	0.9	6.88	0.00785	79.35	79.90	0.7
12	0.0165	0.9	77.4	0.00785	62.65	63.32	1.07
13	0.03303	0.9	77.4	0.025	130.57	131.03	0.35
14	0.0165	0.7	6.88	0.00785	58.92	60.48	2.64
15	0.03303	0.7	77.4	0.00785	82.23	88.02	7.04
16	0.025	0.8	34.14	0.03303	145.89	139.64	4.28
17	0.025	0.8	34.14	0.016425	100.26	95.44	4.79
18	0.025	0.8	34.14	0.016425	100.26	95.44	4.79
19	0.00785	0.8	34.14	0.016425	76.72	72.55	5.42
20	0.03303	0.9	77.4	0.00785	82.36	85.38	3.67
21	0.025	0.8	34.14	0.016425	100.26	95.44	4.79
22	0.0165	0.9	6.88	0.025	94.1	103.49	9.98
23	0.03303	0.7	6.88	0.00785	83.97	82.54	1.69
24	0.03303	0.7	6.88	0.025	126.5	128.18	1.33
25	0.0165	0.7	77.4	0.00785	59.46	65.95	10.92

provide the test output from the ANOVA. Equations (10) and (11) present the fitted formulation for the variation in the average Nusselt number and PEC with the four-input parameter.

$${\mathrm{N}\mathrm{u}}_{\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}}=26.25+1334.6491{\mathrm{u}}_1-13.16875\phi +0.077674\mathrm{P}\mathrm{r}+2661.54887{\mathrm{u}}_2$$

(10)

And

$$\mathrm{P}\mathrm{E}\mathrm{C}=-11.7816+979.812931{\mathrm{u}}_1+13.43750\phi -0.169731\mathrm{P}\mathrm{r}+2856.17412{\mathrm{u}}_2$$

(11)

ANOVA analysis predicts a linear variation in the four variables for the average Nusselt number and the PEC. The study of variance for the designed experiments showed that the adjusted and predicted R² values for Nu_average were 0.9503 and 0.9345, respectively, and for PEC, 0.8953 and 0.8593, respectively.

As shown in Table 4, the relative error between the numerical and model values is 4%. The analysis of the 3D surface and contour plots of the maximum average Nusselt number and PEC, based on the response surface linear model and Equations (1) and (2), is shown in Figure 14 and Figure 15. By carefully examining Figure 13, the contours were linear, meaning that the interaction between the two velocities and between velocity and porosity on the maximum average Nusselt number was significant. Additionally, Figure 14 presents a similar case for the PEC. Based on the optimization, the optimum average Nusselt number and Performance Evaluation Criteria number (PEC) are approximately 77 and 49, respectively, for a porosity of 0.85 and a Prandtl number of 36.9, with velocities u₁ = m/s and u₂ = m/s.

8. Conclusions

The triply periodic minimal surface structure has demonstrated its capability to cool hot surfaces uniformly. The temperature distribution revealed a superior cooling performance of the TPMS structure compared to the foam structure. The availability of metal 3D printing is becoming a valuable tool for constructing heat exchangers. The experimental setup, which utilized water and a mixture of ethylene glycol in water, demonstrated the model’s accuracy in comparison to a numerical model. In this experiment, the TPMS is a single-domain block-shaped structure. Later, a single domain structure is divided into two independent domains that are not interconnected, allowing two liquids to flow without mixing. An identical fluid is circulating, but at different inlet flow rates. This structure is located on top of the heat sink. The results revealed the following:

• A two-domain structure exhibits a better cooling mechanism than a single-domain structure.
• The flow in the domain structure circulates in almost all the structures compared to a single domain.
• A relation between the average Nusselt number and the four variables—two inlet velocities, porosity, and Prandtl number—is obtained, exhibiting a linear variation.
• Performance evaluation criteria were obtained, and a relationship was obtained between the four variables. Again, linear variation is detected.
• Optimization of the system using the response surface methodology provided the optimum average Nusselt number and performance evaluation criteria when the porosity is 0.85 and with a 10% vol EG/water solution. The inlet velocity for domain 1 is 0.0182 m/s, and for domain 2 is 0.013 m/s.