Optimization of Triply Periodic Minimal Surface Heat Exchanger to Achieve Compactness, High Efficiency, and Low-Pressure Drop

Abstract

With advancements in additive manufacturing (AM) techniques, high-quality triply periodic minimal surface (TPMS) structures can now be produced. TPMS walled heat exchangers (HX) hold significant potential for industrial applications and are receiving increasing attention. This paper explores the impact of various TPMS design variables on flow and thermal performance to optimize TPMS heat exchangers for compactness, high efficiency, and low pressure drop. The design variables examined include the type of TPMS lattice, unit cell size, wall thickness, aspect ratio, TPMS orientation, and equivalent thickness. The study reveals that the flow and heat transfer performance of TPMS structures are significantly affected by these design variables. For the Gyroid, Diamond, and SplitP lattices, performance is nearly identical when the surface-to-volume ratio is kept constant. The average velocity of the fluid in the TPMS HX should be 0.3 m/s. The corresponding Re is between 300~800. Thin wall thickness, small equivalent thickness, and flat lattice configurations can significantly reduce pressure drop while maintaining the overall heat transfer coefficient. Additionally, the angle between the flow direction and TPMS orientation can increase pressure drop. Three aluminum heat exchangers were successfully printed using an AM machine, and testing results are comparable with theoretical prediction.

1. Introduction

Compact heat exchangers (HXs) have economic and engineering benefits in the aviation and automotive industry, with reduced volume, weight, and pressure drop while having increased heat transfer efficiency [1,2].

A triply periodic minimal surface (TPMS) structure is very promising for the application in the compact heat exchanger design due to the large turbulent kinetic energy (TKE) in the fluid field and high surface-to-volume ratio [3,4]. A walled TPMS structure divides a three-dimensional space into two separated continuous volumes intertwined with each other [5]. The flow in the walled TPMS structure is very complex and can generate a large amount of TKE. Unlike the conventional heat exchanger, the TPMS heat exchanger may not need some passive heat transfer enhancement techniques like a vortex generator or surface roughness to improve its performance [6,7]. At the same time, the surface-to-volume ratio of the TPMS can be increased to a high level by reducing the TPMS lattice size. Due to the large amount of TKE and high surface-to-volume ratio, the TPMS heat exchanger has a variety of applications in space aircrafts, defense, high power electronics, nuclear energy, and waste energy recovery which have stringent requirements regarding size, efficiency, weight, and cost [8].

Figure 1 shows some typical examples of walled lattices of TPMS structures that can be used as a heat exchanger (HX), such as Gyroid, Diamond, and SplitP.

In order to evaluate the complex flow field in the TPMS HX, parameters such as the Reynolds number, Nusselt number, and friction factor and pumping power are used to evaluate the flow performance in TPMS heat exchangers. The parameters are defined as follows.

The turbulence of the flow within the TPMS structure can be assessed by the Reynolds number (Re), which is the ratio of inertial forces to viscous forces [9].

$$Re={\displaystyle \frac{\rho V{d}_h}{\mu }}$$

(1)

where $\rho$ is the fluid density (kg/m³), $V$ is the inlet velocity (m/s), $\mu$ is the fluid viscosity (Pa.s), and $d_h$ is the hydraulic diameter. The definition for the hydraulic diameter employed in the TPMS structures is ${\displaystyle \frac{4\epsilon }{S_v}}$, where $S_v$ is the specific area surface (m²/m³). The porosity $\epsilon$ is defined as the fraction of the volume of voids over the total volume in heat exchanger.

The thermal and hydraulic performance of TPMS heat exchangers can be assessed using the Nusselt number (Nu). It is determined by the following expressions [10].

$$Nu={\displaystyle \frac{Q_{tot}}{A_w∆T}}{\displaystyle \frac{d_h}{{\lambda }_f}}$$

(2)

where $Q_{tot}$ is the total heat transfer power (W), $A_w$ is the surface area between the interface of fluid and solid or total wetted surface area (m²), ${\lambda }_f$ is the fluid thermal conductivity (W/(m*K)), and $∆T$ is the temperature difference between wall and fluid temperatures (K).

The friction factor ($f$) is commonly utilized to compare pressure loss in cooling channels and can be calculated using the following formula [11,12].

$$f={\displaystyle \frac{2∆p{d}_h}{\rho V^2{L}_c}}$$

(3)

where $∆p$ is the pressure loss in the system (Pa), and $L_c$ is the channel length (m).

The pumping power ($P_e$) is the product of the volume flow rates ($\dot{V}$) and pressure loss ($∆p$):

$$P_e=∆p\dot{V}$$

(4)

where $\dot{V}$ is the volume flow rate (m³/s).

The Re, Nu, f, and $P_e$ can be used together to evaluate the performance of the heat exchanger and to determine the working point of the heat exchanger.

The heat transfer performance of the TPMS structures was strongly influenced by the flow field within the structure, which was determined by the structure design variables. The design variables include the lattice type, Reynold number, lattice wall thickness, unit lattice size, aspect ratio, and surface area at the inlet and outlet. Passos, A.G.P. et al. investigated the laminar steady inertial regime and obtained the transition Reynolds numbers. Schwarz-P and Schoen-Gyroid presented a transition at Re = 75 and Schwarz-D presented a transition at Re = 150 [5]. Reynolds, B.W. et al. simulated the flow and heat transfer in 3D TPMS heat exchangers based on lattice Boltzmann method at a low Reynold number and found that Schwarz primitive TPMS design has a much lower pressure drop when compared to the Gyroid due to a lesser tortuosity of the channel [13]. Zimmer, A. et al. studied the effect of manufacturing techniques in a pressure drop on triple periodical minimal surface packings [14]. These articles investigate the flow field in the TPMS structure used in the heat exchanger at a low Reynold number smaller than 200, which means the flow field is a laminar flow. The compact heat exchanger that works at the low Reynold number may not be economical. The flow field with a higher Re was studied in this paper to choose the proper working point for the TPMS HX.

Reza Attarzadeh et al. developed a multi-objective optimization workflow to study the underlying relationships between design variables. They found that TPMS periodic lattice size and the wall thickness are the most sensitive. In their study, only several lattices were simulated [15]. Li, W. et al. found large turbulent kinetic energy production in a Gyroid structure at various Reynolds numbers and TPMS structures show a much higher heat transfer rate than the printed circuit heat exchanger (PCHE) [4]. The models containing one or several lattices in the paper are relatively small. In order to design the TPMS HX, it is better to build the whole model of the heat exchanger. Based on the whole TPMS HX model in this paper, more design variables such as the TPMS orientation, surface area of fluid at the inlet and outlet of the TPMS structure, and the equivalent thickness can be proposed and investigated.

Reza Attarzadeh et al. built a conjugate heat transfer (CHT) simulations for laminar incompressible flow to quantify the performance of a TPMS-based heat exchanger and found that the smaller the wall thickness, the higher the thermal performance of the heat exchanger [16]. The unit cell size and the wall thickness, which have a big influence on the performance of the heat exchanger, were thoroughly investigated in this paper considering the actual manufacturability of the additive manufacturing (AM) machine.

In this paper, a computational fluid dynamics (CFD) model of the flow field in the TPMS HX was developed to analyze flow and heat transfer within a heat exchanger based on TPMS structures. Section 2 presents a detailed investigation of various TPMS design variables—such as the Reynold number, lattice unit type, unit cell size, wall thickness, lattice aspect ratio, TPMS orientation, sheet Gyroid structure, and equivalent thickness of the lattice structure—to optimize compactness, efficiency, and pressure drop. In Section 3, three TPMS-HX samples were fabricated using an additive manufacturing machine and subsequently tested. The testing results are comparable with theoretical predictions. Section 4 provides a conclusion.

2. ANSYS Fluent Model of HX40 Infilled with TPMS Structure

As illustrated in Figure 2, the HX40 heat exchanger was designed to investigate the efficiency and pressure drop characteristics of TPMS heat exchangers. In Figure 2, the red volume represents the hot fluid, the blue volume represents the cold fluid, and the grey volume represents the solids. The dimensions of the HX40 are detailed in Figure 3. The core of the heat exchanger was filled with various types of TPMS lattice structures.

Due to the symmetry, Fluid A and Fluid B are identical and have the same performance. Both Fluid A and B contain liquid water. To minimize computation time, Fluid A in Figure 2 was modeled in ANSYS Fluent 2023 R1 to determine the performance for the HX40 with various TPMS lattices.

The K-Omega (k-ω) turbulence models are used in computational fluid dynamics (CFD) model to describe the behavior of turbulent flows. There are two transport equations in the model: one for the turbulent kinetic energy (k) and another for the specific dissipation rate (ω).

Key Equations:

Turbulent Kinetic Energy Equation:

$${\displaystyle \frac{\partial k}{\partial t}}+{\displaystyle \frac{\partial \left(k{u}_j\right)}{\partial x_j}}=P_k-\beta k\omega +{\displaystyle \frac{\partial }{\partial x_j}}(\left(\mu +{\mu }_t\right){\displaystyle \frac{\partial k}{\partial x_j}})$$

(5)

$P_k$: production of turbulent kinetic energy

$\beta$: Empirical constant

${\mu }_t$: Turbulent viscosity

Specific Dissipation Rate ($\omega$) Equation:

$${\displaystyle \frac{\partial \omega }{\partial t}}+{\displaystyle \frac{\partial \left(\omega u_j\right)}{\partial x_j}}={{\displaystyle \frac{\alpha }{k}}P}_k-\gamma {\omega }^2+{\displaystyle \frac{\partial }{\partial x_j}}(\left(\mu +{\mu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_j}})$$

(6)

$P_k$: same as above

$\alpha$ and $\gamma$: Empirical constant

The boundary conditions are shown in Figure 4. The inlet flow rate varies from 0.05 to 0.4 kg/s, and the inlet temperature is set at 288 K. The wall temperature of the lattice structure within Fluid A is kept constant at 343 K. Heat will be transferred from the wall to the fluid.

Figure 5 presents the 3D and 2D qualitative results of the flow field in volume A, providing valuable insights into the flow characteristics.

2.1. TPMS Type Comparison

In Figure 6, three representative TPMS lattices were used in the simulation to examine how different lattice types affect the efficiency and pressure drop of the HX40.

The Gyroid, Diamond, and SplitP lattices have different surface-to-volume ratios at the same lattice size. By varying the lattice size, the three TPMS structures achieve nearly the same surface area for heat transfer within the same volume which are shown in

Table 1. Parameters for different lattice in HX40.

Lattice Type	Lattice Unit Size	Wall Thickness	Surface Area in HX40
	mm	mm	m2
Gyroid	5 × 5 × 5	0.7	0.0364
SplitP	8 × 8 × 8	0.7	0.0374
Diamond	6 × 6 × 6	0.7	0.0376

. The inlet, outlet, and thermal boundary conditions are consistent across all cases. The overall heat transfer coefficient and pressure drop are compared.

The relationship of the flow rate, pressure drop and overall heat transfer coefficient (Uc × A) in different lattice are shown in Figure 7. Uc was average heat transfer coefficient and A is the surface area of the interface in the HX for heat transfer. The pressure drop has quadratic relationship with the flow rate. And the overall heat transfer coefficient (Uc × A) increases with the flow rate for all the lattices. In Figure 7, the overall heat transfer coefficient with the SplitP is about 6~8% higher than the Gyroid lattice. The pressure drop of the SplitP is 3~7% higher than the Gyroid lattice. Considering both the overall heat transfer coefficient and pressure drop, the SplitP and Gyroid show similar performances.

As shown in Figure 8, the heat transfer power of HX40 increase rapidly with the pump power of the flow under flow rate 0.1 kg/s. To save energy, it is efficient to make the HX40 works near 0.1 kg/s. The Reynolds number near this flow rate is within 300~800. The average velocity in the lattice is about 0.3 m/s.

In Figure 9, SplitP lattice with 8 × 8 × 8 mm has higher Nusselt number (Nu) at the same Reynolds number and Gyroid lattice has lower friction factor.

In

Table 2. Performance comparison of three lattice at 0.1 kg/s.

Lattice Type	Flow Rate	Inlet P	Uc × A	Re	Nu	f
	kg/s	Pa	W/K	-	-	-
Gyroid	0.1	1579	326	674	31	1.491
SplitP	0.1	1635	354	766	36	1.564
Diamond	0.1	1522	327	583	29	1.732

, at the same flow rate of 0.1 kg/s, the three types of TPMS lattices have nearly identical heat transfer efficiency.

2.2. Lattice Size Impact

To study the efficiency of the Gyroid lattice with different sizes, the lattice size was changed between 5 mm, 7 mm, and 10 mm.

As the lattice size decreased from 10 mm to 5 mm in Figure 10, the overall heat transfer coefficient Uc × A was doubled. The pressure drop increased by four times as the lattice size decreased from 10 mm to 5 mm.

As shown in Figure 11, when the lattice size was decreased from 10 mm to 5 mm, Nu and f will be reduced. For all the lattice, the friction factor will change slowly at higher Re.

As illustrated in Figure 12, at the same pump power, the heat exchanger with a smaller lattice size is capable of transferring more heat power.

2.3. Wall Thickness Impact

Wall thickness has important effects on the heat transfer efficiency of HX40 with different lattice size.

Table 3. Effect of the wall thickness on the HX40.

Lattice Type	Unit Size	Wall Thickness	Surface Area	Inlet P	Uc × A	Re	Nu
	mm	mm	m2	Pa	W/K	-	-
Gyroid	7 × 7 × 7	0	0.031	236	208	696	44
Gyroid	7 × 7 × 7	0.35	0.027	401	215	850	51
Gyroid	7 × 7 × 7	0.7	0.027	660	224	895	47
Gyroid	7 × 7 × 7	1.05	0.026	1137	227	978	41
Gyroid	10 × 10 × 10	0	0.023	175	146	986	59
Gyroid	10 × 10 × 10	0.35	0.019	249	139	1200	71
Gyroid	10 × 10 × 10	0.7	0.019	339	148	1239	68
Gyroid	10 × 10 × 10	1.05	0.019	481	152	1291	62
Gyroid	5 × 5 × 5	0	0.043	331	294	496	34
Gyroid	5 × 5 × 5	0.35	0.038	695	318	604	36
Gyroid	5 × 5 × 5	0.7	0.036	1579	326	674	31
Gyroid	5 × 5 × 5	1.05	0.033	4735	341	757	26

shows the simulation results for different wall thickness in the Gyroid lattice with different size at flow rate 0.1 kg/s.

As shown in Figure 13, when the lattice size is 5 × 5 × 5 mm, the Nusselt number (Nu) increases significantly at the same Reynolds number (Re) as the wall thickness decreases. When the lattice size 10 × 10 × 10 mm, the Nu is not very sensitive for the wall thickness. As shown in Figure 13c, the Nu of the Gyroid (10 × 10 × 10-1.05 mm) is significantly low. This is mainly caused by the reduction of the void volume which was reduced by 21.5%. And the hydraulic diameter is also reduced by 20%. The 1.05 mm thick wall occupied much volume and make the volume for the fluid much smaller. The hydraulic diameter is much smaller than that with 0.35 mm wall thickness.

As illustrated in Figure 14, the pressure drop of HX40 at 0.1 kg/s is sensitive to wall thickness of the lattice with small size such as 5 × 5 × 5 mm. As for 10 × 10 × 10 mm lattice, the pressure drop doesn’t change much with the wall thickness.

As shown in Figure 15, the overall heat transfer coefficient doesn’t change much with the lattice wall thickness. The wall thickness should be kept minimal to help reduce the pressure drop while maintaining the same overall heat transfer coefficient.

2.4. Aspect Ratio

The surface-to-volume ratio of the Gyroid lattice structure is determined by the unit lattice volume and wall thickness. When the wall thickness is 0.7 mm, the surface-to-volume ratio decreases with larger Gyroid unit lattice volume, as illustrated in Figure 16.

In

Table 4. Gyroid unit lattices with different dimensions.

Case	Lattice Type	Lattice Size	Volume of Unit Lattice	Wall Thickness	Surface-to-Volume Ratio	Aspect Ratio	Description
		mm	mm3	mm	mm−1	-
1	Gyroid	6.3 × 6.3 × 6.3	250	0.7	0.52	1	Cubic
2	Gyroid	10 × 5 × 5	250	0.7	0.56	2	Thin
3	Gyroid	8 × 8 × 4	256	0.7	0.56	2	Flat

, the volumes of the three types of Gyroid lattice unit is about 255 mm³. For the Gyroid lattice with the same unit lattice volume, the aspect ratio of the unit lattice is defined as the ratio of the longest edge and shortest edge. In case 1, the edge lengths in three directions are the same which is 6.3 mm and this lattice can be called cubic lattice. In case 2, the longest edge length is 10 mm and the rest two edge length is 5 mm. This lattice can be called thin lattice. In case 3, the shortest edge length is 4 mm and the other two edge length is 8 mm. The unit lattice can be called flat lattice. The three Gyroid unit lattices have identical volumes and nearly identical surface-to-volume ratios.

Using the lattices listed in Table 4, the volume and surface area of the HX40 shown in Figure 17 are kept consistent. The boundary conditions for the simulation are identical across all cases. The pressure drop and the overall heat transfer coefficients (Uc × A) are compared under the same flow rate 0.1 kg/s.

As shown in

Table 5. Simulation results for different lattice of varying sizes.

Lattice Type	Unit Length	Fluid Volume	Area for Heat Transfer	Inlet P	Uc × A
	mm	-	m2	Pa	W/K
Gyroid	6.3 × 6.3 × 6.3	A	0.0297	837	249
Gyroid	8 × 8 × 4	A	0.0322	543	334
Gyroid	10 × 5 × 5	A	0.0317	311	230
Gyroid	10 × 5 × 5	B	0.0319	1806	257

, the 10 × 5 × 5 mm lattice has nearly the same overall heat transfer coefficient with the 6.3 × 6.3 × 6.3 mm lattice. The pressure drop of 10 × 5 × 5 mm lattice is only 37% of the 6.3 × 6.3 × 6.3 mm lattice. However, for the second volume B of 10 × 5 × 5 mm lattice, the pressure drop is pretty high which is 2.16 times of the pressure drop of 6.3 × 6.3 × 6.3 mm lattice.

Compared with the 6.3 × 6.3 × 6.3 mm lattice, the flat lattice with 8 × 8 × 4 mm edge length was more efficient. The pressure drop was reduced by 35% and the overall heat transfer coefficient (Uc × A) was increased by 34%. Because of the symmetry of the flat lattice, the other fluid volume has similar performance compared to the first fluid volume.

As shown in

Table 6. Re, Nu, and f of different lattice size at 0.1 kg/s.

Unit Cell Size (mm)	Re	Nu	f
6.3 × 6.3 × 6.3	782	40	1.57
10 × 5 × 5	639	35	0.868
8 × 8 × 4	625	46	1.266

, the flat lattice with 8 × 8 × 4 mm size has a higher Nu and a lower f value, which means the 8 × 8 × 4 mm lattice has better performance than the other two lattices with the same surface-to-volume ratio.

2.5. TPMS Orientation Impact

The flow angle shown in Figure 18 is the angle between the flow direction and the lattice’s longest edge direction in the 10 × 5 × 5 mm gyroid lattice. The pressure drop can be influenced by the flow angle.

As shown in

Table 7. Simulation results for different flow angles.

Lattice Type	Angle	Flow Rate	Inlet P	Uc × A	Re	Nu	f
	degree	kg/s	Pa	W/K	-	-	-
Gyroid	0	0.1	311	230	635	34	0.862
Gyroid	20	0.1	466	208	618	31	1.364
Gyroid	45	0.1	1130	168	819	25	1.879
Gyroid	90	0.1	1806	257	988	38	2.072

and Figure 19, the angle between the flow direction and the lattice’s longer edge has an influence on the pressure drop. The big flow angle can increase the pressure drop significantly.

2.6. Surface Area of Fluid at the Inlet and Outlet

The HX model, which was infilled with a Gyroid lattice in Figure 20, has the same volume for heat transfer as HX40. As shown in Figure 21, the surface area at the inlet and outlet of the two fluid volumes are different.

The surface area at the inlet and outlet in fluid A is four times that in fluid B and is about two times of that in HX40. The flow rate in

Table 8. Simulation results for different equivalent thicknesses at 0.05 kg/s.

	Surface Area Inlet	Volume	Equivalent Thickness	Surface Area	Velocity	Re	Nu	f	Inlet P	Uc × A
	mm2	cm3	mm	m2	m/s	Re	Nu	f	Pa	W/K
Fluid B	146	18.9	129.7	0.034	0.34	756	29.2	2.31	4952	274
HX40	405	18.9	54.7	0.036	0.18	375	25.4	1.38	452	266
Fluid A	857	18.9	22.1	0.038	0.08	164	17.2	2.19	76	198

is 0.05 kg/s.

The equivalent thickness of the Gyroid lattice structure in Table 8 is defined as the ratio of the volume to the average surface area at the inlet and outlet. When the equivalent thickness increased from 22 mm to 129.7 mm, the pressure drop increased by 65 times at the same flow rate while the overall heat transfer coefficient did not change much. The equivalent thickness can have a substantial impact on the pressure drop. Reducing the equivalent thickness of the lattice structure in the HX design can be an effective method for lowering the pressure drop in one fluid volume.

2.7. Sheet Gyroid Lattice Channel

The sheet Gyroid lattice can be used as the fluid channel in the HX design. The generation of the sheet Gyroid channel is shown in Figure 22. In Figure 23, the fluid B is a sheet Gyroid channel with an average width of 1.1 mm and fluid A is the Gyroid lattice channel. The lattice size is 10 × 10 × 10 mm.

As shown in

Table 9. Design parameters for HX with a sheet Gyroid channel.

Lattice Type	Lattice Size		Surface Area—In Lattice Structure	Volume—In Lattice Structure
	mm		m2	cm3
Gyroid	10 × 10 × 10	Fluid-A	0.0254	12.25
		Fluid-B: Sheet Gyroid	0.0389	25.36
		Solid-S	0.0639	24.38

, the surface area of fluid B with sheet Gyroid channel is 50% bigger than that in fluid A.

In

Table 10. Simulation results for HX with sheet Gyroid channel.

Flow Rate	Average Velocity—Fluid A	Pressure Drop—Fluid A	(Uc × A)—Fluid A	Average Velocity—Fluid B	Pressure Drop—Fluid B	(Uc × A)—Fluid B
kg/s	m/s	Pa	W/K	m/s	Pa	W/K
0.05	0.345	1201	204	0.103	85	161
0.1	0.705	3933	288	0.214	294	267
0.2	1.432	13,566	400	0.445	1076	400
0.3	2.142	29,223	425	0.665	2245	474

, the sheet Gyroid channel exhibits a much lower pressure drop at the same flow rate compared to the other fluid volume. However, the Uc × A remains nearly identical for both volumes at the same flow rate. The volume of the sheet Gyroid channel can be reduced to increase the volume of the other fluid. This adjustment can help balance the pressure drops between the two fluid volumes.

3. Test on Various HXs

3.1. Design of Three HXs

As shown in Figure 24 and Figure 25, three HX samples that have the same dimensions were printed by the AM machine. The HX cores that are infilled with TPMS lattice in the Sample #1 and Sample #2 are the same.

The design parameters of the three samples are listed in

Table 11. Design parameters of three HX samples.

	Lattice Type	Lattice Size	Design Mass	Volume of Core		Surface Area-In Lattice Structure	Volume—In Lattice Structure	Whole Volume
		mm	g	cm3		m2	cm3	cm3
Sample #1	Gyroid	5 × 5 × 5	513	166.27	Fluid-A	0.079	25.4	56.8
					Fluid-B	0.079	25.4	56.6
					Solid-S	0.168	115.47	190.2
Sample #2	SplitP	10 × 10 × 10	428	166.27	Fluid-A	0.084	44.74	75.68
					Fluid-B	0.078	42.93	73.78
					Solid-S	0.160	78.67	158.55
Sample #3	Sheet Gyroid	10 × 10 × 10	369	153.12	Fluid-A	0.099	62.74	101.95
					Fluid-B	0.061	28.79	71.33
					Solid-S	0.160	60.10	136.74

. The material of the HX is Aluminum. The lattice type and lattice dimension for Sample #1 and Sample #2 are different. The wall thickness of the lattice structure is adjusted to ensure that the fluid volume matches the test volume. For Sample #1 and Sample #2, the equivalent wall thicknesses are 1.22 mm and 0.9 mm, respectively. Sample #1 has a smaller fluid volume compared to Sample #2 and both samples have similar surface areas in their lattice structures. Additionally, the average channel width in Sample #1 is narrower than that in Sample #2.

3.2. Test Results of Three HXs

Schematic of testing setup of heat exchanger was shown in Figure 26. There are two water loops in the test rig. The flow meter, pressure transducer, and temperature sensors are used to test the flow rate, pressure drop, and the temperature at the inlet and outlet. The experimental tests of the three samples are in accordance with ANSI/AHRI Standard 400 (I-P) [17]. The data are valid only if the heat balance is within ±5%. The error in the measured heat capacity due to sensor inaccuracies is within ±1.2%, which is relatively low.

As shown in

Table 12. Test results of three samples.

	Test ID	Hot Side Inlet Temp	Hot Side Outlet Temp	Cold Side Inlet Temp	Cold Side Outlet Temp	Hot Side Flow	Cold Side Flow	Hot Side Heat Transfer	Cold Side Heat Transfer	Hot Side Pressure Drop	Cold Side Pressure Drop	Uc × A
		°C	°C	°C	°C	kg/s	kg/s	KW	KW	Pa	Pa	W/K
Sample #1	1	76.64	46.77	18.37	48	0.10	0.10	12.9	12.8	9446	9308	451
	2	76.5	48.64	18.31	45.98	0.20	0.20	23.3	23.3	35,784	35,232	766
	3	74.14	48.47	18.33	42.79	0.29	0.30	30.8	30.5	73,016	77,290	998
Sample #2	1	76.76	43.99	18.10	48.50	0.05	0.05	6.8	6.5	1774	1878	246
	2	76.63	46.69	18.34	46.47	0.10	0.10	12.8	12.2	5376	5323	428
	3	76.50	49.40	18.21	44.53	0.21	0.20	23.6	22.5	21,562	21,712	731
	4	73.38	49.90	18.62	42.01	0.31	0.30	30.6	29.2	46,557	46,484	954
Sample #3	1	76.90	48.75	18.12	46.60	0.05	0.05	6.3	6.2	2243	339	205
	2	76.54	50.30	18.26	44.16	0.11	0.10	11.7	11.2	5485	1181	354
	3	76.54	52.65	18.23	41.95	0.21	0.20	20.8	19.9	19,002	4878	591
	4	76.66	54.09	18.36	40.64	0.31	0.30	29.0	28.0	42,491	10,890	794

, for Sample #1 and Sample #2, the hot fluid volume and cold fluid volume are identical due to the symmetry and have the same pressure drop. For Sample #3, the cold fluid volume and hot fluid volume have different pressure drops.

3.3. Simulation Results of Three HXs

As shown in Figure 27, the heat transfer rate $q$ in the heat exchanger is:

$$q=U_{c}A*\Delta T_{lm}={\displaystyle \frac{\Delta T_{lm}}{\frac{1}{U_1{A}_1}+R_w+\frac{1}{U_2{A}_2}}}$$

(7)

where $U_1$ and $U_2$ are the heat transfer coefficients for hot and cold fluids, respectively, and $A_1$ and $A_2$ are the heat transfer surface areas for the hot and cold fluids, respectively, and $R_w$ is the wall thermal resistance. $\mathsf{\Delta }{T}_{lm}$ is the log mean temperature difference [18].

The wall thermal resistance is

$$R_w={\displaystyle \frac{{\delta }_w}{k_w{A}_w}}$$

(8)

where ${\delta }_w$ is the thickness of the wall, $k_w$ is the thermal conductivity of the wall and $A_w$ is the heat transfer area of the wall, which is the same as $A_1$ or $A_2$ in this case.

The heat exchanger overall heat transfer coefficient

$$U_{c}A={\displaystyle \frac{1}{\frac{1}{U_1{A}_1}+R_w+\frac{1}{U_2{A}_2}}}$$

(9)

As shown in

Table 13. Simulation results of three samples.

	Flow Rate	Average Velocity—Hot	Average Velocity—Cold	Pressure Drop—Hot	Pressure Drop—Cold	Re—Hot	Re—Cold	${\mathit{U}}_1{\mathit{A}}_1$	${\mathit{U}}_2{\mathit{A}}_2$	${\mathit{U}}_{\mathit{c}}\times \mathbf{A}$
	kg/s	m/s	m/s	Pa	Pa	-	-	W/K	W/K	W/K
Sample #1	0.05	0.25	0.25	2428	2428	446	446	392	392	193
	0.1	0.52	0.52	7950	7950	938	938	657	657	322
	0.2	1.06	1.06	26,332	26,332	1919	1919	992	992	480
	0.3	1.59	1.59	54,001	54,001	2870	2870	1261	1261	605
Sample #2	0.05	0.12	0.12	761	761	255	255	351	351	174
	0.1	0.24	0.24	2618	2618	515	515	576	576	283
	0.2	0.48	0.48	8961	8961	1042	1042	852	852	415
	0.3	0.72	0.72	18,699	18,699	1564	1564	1056	1056	511
Sample #3	0.05	0.22	0.06	789	78	405	177	311	267	142
	0.1	0.45	0.13	2683	264	842	369	485	447	230
	0.2	0.93	0.28	8989	939	1731	758	705	701	345
	0.3	1.40	0.42	18,277	1982	2602	1142	882	857	426

, for Sample #1 and Sample #2, one fluid volume was calculated as the two volumes are symmetrical. The flow rate is equal to the test flow rate. The inlet temperature of the water is 291 K and the wall temperature of the lattice structure is constant at 349 K. The pressure drop and the overall heat transfer coefficient of the single volume ($U_1{A}_1$) are calculated and the other volume has the same results. The overall heat transfer coefficient $U_c\mathrm{A}$ of the whole heat exchanger are calculated according to the Equation (9). For Sample #3, the two fluid volumes are different and both are modeled to calculate the pressure drop and overall heat transfer coefficient.

3.4. Comparison of Simulation and Test Results

The pressure drop is proportional to the square of the flow rate in the Sample #1 and Sample #2. And the Uc × A is proportional to the square root of the flow rate.

The sheet Gyroid channel has very low pressure drop and the other fluid volume has high pressure drop.

According to Figure 28, Figure 29, Figure 30 and Figure 31, the numerical simulation can predict the performance trends of the heat exchanger at different flow rate. However, both the pressure drop and Uc × A calculated by CFD model are about 40% lower than the test results. The high surface roughness of the lattice structure could be the possible reason for this difference.

As shown in Figure 32, the Gyroid lattice surfaces printed by AM machines often have many small particles adhering to them, resulting in higher surface roughness. These particles can increase the total surface area of the lattice and may enhance surface turbulence, potentially improving heat transfer efficiency. However, the presence of the small particles also reduce the fluid volume within the lattice and heighten turbulence, which could significantly raise the pressure drop across the heat exchanger. The simulation model needs to be optimized to account for the impact of the small particles on the surface.

In Figure 33, Samples #2 and #3 exhibit similar pressure drops when Uc × A equals 600 W/K, which is significantly lower than that of Sample #1. Sample #1 has a bigger equivalent wall thickness which causes a higher pressure drop.

4. Conclusions

The paper presents a detailed investigation on the TPMS design variables to optimize the TPMS-HX. Some conclusions can be made as follows:

• The Gyroid, Diamond, and SplitP lattices show similar performances when the surface-to-volume ratio is kept the same.
• It is efficient to make the HX infilled with Gyroid lattice works at the point where the Re is about 300~800 and the average velocity in the lattice structure is about 0.3 m/s.
• The thin wall thickness of the lattice structure can reduce the pressure drop of the HX and keep the overall heat transfer coefficient the same.
• The flat lattice can reduce the pressure drop of HX compared with the cubic lattice with the same lattice unit volume.
• The bigger angle between the flow direction and the long lattice edge can increase the pressure drop.
• Reducing the equivalent thickness of the lattice structure is an effective method for decreasing the pressure drop.
• The sheet Gyroid channel has a much lower pressure drop compared with the other fluid volumes.
• The simulation predicted a lower pressure drop and overall heat transfer coefficient by about 40% compared to the test results. The high surface roughness of the lattice structure may be the possible reason for the difference; however, more investigation is needed to confirm.