A Comprehensive Review of Printed Circuit Heat Exchangers: Fabrication, Thermal–Hydraulic Performance, and Reliability

Abstract

Printed circuit heat exchangers (PCHEs) represent a critical technology for application in advanced energy systems due to their compact configuration, exceptional thermal efficiency, and robustness under extreme temperatures and pressures. This review systematically examines PCHE technology, covering key fabrication processes—including material selection, channel etching, and diffusion bonding—alongside the thermal–hydraulic performances of various channel geometries and optimization strategies. Although considerable progress has been made in geometric optimization—with reported heat transfer enhancements of up to 250% and flow resistance reductions of up to 84.7%—most of the available optimized designs remain confined to numerical analyses. A significant gap persists between these idealized models and real manufacturing constraints, where etching and inducing geometric deviations inherently affect both performance and mechanical integrity. Moreover, current Nusselt number and friction factor correlations lack universality, and mechanical integrity assessments often neglect long-term degradation mechanisms such as fouling. This review identifies these critical gaps and proposes that future research should prioritize integrating geometric optimization with fabrication feasibility and mechanical integrity. Also, there is a need to develop generalized correlations that incorporate both fluid property variations and geometric effects, and to systematically evaluate long-term performance via component-scale testing.

1. Introduction

The continuous increase in atmospheric carbon dioxide emissions has exacerbated global warming, posing a severe threat to both natural ecosystems and human society [1]. Consequently, curbing CO₂ emissions to mitigate climate change and enable sustainable development has become an urgent global priority [2]. Advanced high-efficiency energy systems, such as those concentrating solar power [3] and Generation IV nuclear reactors [4], are pivotal in the transition toward carbon neutrality. These advanced energy systems, however, often operate under severe conditions of high temperature and pressure, imposing stringent demands on the performance and reliability of their key components, particularly heat exchangers [5].

Printed circuit heat exchangers (PCHEs), characterized by their high compactness, excellent heat-transfer performance, and robustness in extreme environments [6], demonstrate significant potential for application in these advanced energy systems. As a novel type of compact heat exchangers, PCHEs are manufactured through photochemical etching and diffusion bonding techniques [7]. Compared to shell-and-tube heat exchangers, PCHEs offer a volume reduction of up to five or six times while maintaining equivalent thermal load [8], demonstrating a significant advancement. Recent progress in manufacturing has enabled the growing adoption of PCHEs across various industries, including advanced nuclear power systems [9], solar power systems [10], floating liquefied natural gas systems [11], and the automotive [12] and aerospace sectors [13].

Extensive research has been devoted to improving the thermal–hydraulic performance of PCHEs through channel–geometry optimization, flow analysis, and performance enhancement strategies [14,15]. In parallel, data-driven modeling and correlation development have been conducted to establish predictive relationships for heat transfer and pressure drop under various operating conditions [16,17,18,19]. Moreover, studies on material selection, fabrication reliability, and flow distribution uniformity have been carried out to enhance mechanical safety and long-term operational stability in extreme environments [20,21,22,23,24].

However, these studies remain fragmented across different research aspects, and a unified synthesis is still lacking. Therefore, a systematic review integrating fabrication processes, thermal–hydraulic behavior, correlation development, mechanical integrity, and flow-distribution issues is essential to provide a comprehensive understanding and guide future design and application of PCHEs in advanced energy systems.

In recent years, several review articles have been published focusing on the thermal–hydraulic performance of PCHEs under various working fluids and application conditions. Liu et al. [7] and Kwon et al. [25] reviewed the thermal–hydraulic characteristics of PCHEs with different channel configurations for sCO₂ power cycle application. Ma et al. [6] and Huang et al. [26] further extended the discussion to PCHEs employed with various supercritical gases. In addition, some reviews have provided detailed discussions on fabrication processes. Chai and Tassou [27] comprehensively analyzed material selection for PCHEs applied in helium and sCO₂ environments, whereas Ma et al. [8] conducted an extensive review of diffusion bonding manufacturing processes. Moreover, Chen et al. [28] summarized the improved structures and dynamic responses of existing PCHEs.

Overall, despite extensive research and reviews on PCHEs, several limitations remain. Regarding fabrication, the process parameters of photochemical etching and diffusion bonding exert a significant influence on dimensional accuracy and the resulting thermal–hydraulic performance. Although diffusion-bonding models have been developed to elucidate the bonding mechanisms, few reviews have systematically addressed the influence of process parameters or the theoretical modeling of diffusion bonding. Similarly, only limited reviews have examined how the precision of photochemical etching induces deviations from the designed channel geometries, which in turn modify flow and heat-transfer characteristics. Comprehensive assessment of the effects of such manufacturing variances across different PCHE channel types is still required.

In terms of thermal–hydraulic performance, numerous geometric parameters and optimization strategies have been proposed and partially reviewed; however, a systematic synthesis and comparative assessment of these improvement approaches are still lacking. Furthermore, although most existing reviews focus on sCO₂ and other supercritical gases, PCHEs are also applicable to various extreme working fluids and environments, such as molten salts and petroleum-based liquids with high viscosity. The thermal–hydraulic behavior of these fluids in different channel configurations has rarely been reviewed. In addition, the influence of inlet and header configurations on flow distribution and overall exchanger uniformity has not yet been systematically summarized.

With respect to flow and heat-transfer correlations, a large number of empirical models have been developed, and several reviews have provided partial evaluations. However, most of the existing reviews provide incomplete coverage and lack systematic discussion on the evolution and mathematical development of correlations for PCHEs. Finally, reviews addressing the mechanical integrity and fouling behavior of PCHEs remain scarce, despite the critical importance of these aspects for ensuring long-term mechanical reliability under high-temperature and high-pressure conditions.

To address the aforementioned issues, the present review provides a comprehensive examination of PCHE fabrication, encompassing material selection, photochemical etching, and diffusion bonding, with emphasis on the influence of process parameters, advances in diffusion-bonding models, and the impact of manufacturing on channel geometry and thermal–hydraulic performance. The thermal–hydraulic performance of three representative channel types—straight, zigzag/wavy, and airfoil fin channels—is subsequently analyzed, focusing on flow mechanisms, optimization strategies, and potential applications. Moreover, the review includes an evaluation of header configuration and flow distribution and assesses existing heat-transfer and friction correlations with attention to applicability, mathematical formulation, and limitations. Finally, research concerning mechanical integrity and long-term fouling behavior is analyzed to clarify the mechanical safety and operational reliability of PCHEs under high-temperature and high-pressure conditions.

2. Fabrication Processes

The fabrication of PCHEs involves several essential steps, including material selection, channel etching of plates, diffusion bonding of stacked assemblies, and secondary finishing for header and port integration. This section reviews the key aspects of these processes and their impact on heat exchanger performance.

2.1. Material Selection

PCHEs are widely employed in high-temperature and high-pressure conditions, such as supercritical carbon dioxide (sCO₂) Brayton cycles, helium and natural gas systems, as well as molten salt applications, along with significant corrosion risks. These environments demand materials with high thermal conductivity and mechanical strengths, as well as excellent resistance to creep, fatigue, corrosion, and oxidation to ensure long-term reliability.

For medium–low-temperature conditions, the austenitic stainless steels from the 300 series are commonly used. For example, Chu et al. [29] fabricated PCHEs with SUS304 Stainless Steel (SS) for operating with water at room temperature. Park et al. [30] and Wang et al. [31] selected SS316L for use with sCO₂ at approximately 100 °C and molten salt around 250 °C, respectively.

As operating temperatures rise, however, nickel-based alloys demonstrate markedly superior service performance [5,27]. According to the study of Pint et al. [32], SS316/SS316L/SS347 are generally limited to 600–650 °C owing to creep and corrosion constraints. As shown in Figure 1, Maziasz et al. [33] observed rapid creep deformation and failure in SS347 under 750 °C and 100 MPa. Several studies support the use of advanced alloys for high-temperature service. Cao et al. [34] compared SS316, SS310, and Alloy 800H in sCO₂ at 650 °C and 20 MPa, and identified Alloy 800H as the most corrosion-resistant. Kim et al. [35] further investigated the creep response of diffusion-bonded Alloy 800H tubing at 700 °C under 70–140 MPa, developing a modified Garofalo correlation to accurately predict creep strain evolution. Mylavarapu et al. [36] validated the thermal–hydraulic and mechanical performance of Alloy 617 PCHEs in a high-temperature helium loop at 650 °C and 2.7 MPa, confirming its excellent strength and corrosion resistance.

For even more aggressive high-temperature and high-pressure environments, metallic materials pose considerable risks of oxidation and carburization corrosion [37], which may compromise long-term performance. In contrast, ceramic matrix composites (CMCs) have demonstrated great potential due to their superior chemical stability and mechanical robustness. Schmidt et al. [38] produced C/SiSiC composite plates via metal infiltration and tested them in molten fluoride salt at 850 °C. The results showed that uncoated samples experienced severe corrosion after 500 h in molten FLiNaK at 850 °C, whereas samples protected with a CVD-deposited C + SiC bilayer coating remained structurally intact, demonstrating the coating’s potential for application in molten salt reactors. Similarly, Zhu et al. [39] developed a SiC-based PCHE capable of withstanding a maximum pressure of 30 MPa by incorporating SiC nanoparticles for toughening, employing high-pressure densification molding, optimizing the sintering process, and achieving integral diffusion bonding.

In summary, 300-series stainless steels are suitable for PCHEs at low–medium temperatures (below 600 °C), whereas nickel-based alloys are preferred for high-temperature applications. Moreover, ceramic composites exhibit remarkable corrosion resistance in high-temperature molten salts, warranting further research for advanced energy systems.

2.2. Manufacturing Technologies

This section examines the two most critical manufacturing processes for PCHEs—photochemical etching and diffusion bonding—by reviewing different channel etching techniques and the influence of bonding parameters, establishing a foundation for understanding their impact on mechanical characteristics and operational performance.

The core fabrication of a PCHE, outlined in Figure 2, consists of four main stages: (i) photochemical etching of flow channels into metal sheets, (ii) precise stacking of these etched plates in a clean-room environment, (iii) diffusion bonding to form a monolithic block, and (iv) secondary machining to integrate headers and ports.

Channel Etching: Channel formation is accomplished through photochemical etching. This precision subtractive process involves applying a photoresist layer, transferring a channel pattern by a mask through exposure and development, and then selectively dissolving the unprotected metal in a chemical etchant. The etching rate and final dimensional accuracy are controlled by process parameters such as temperature, immersion time, and etchant composition [41]. According to Yadav et al. [42], increasing the etchant concentration accelerates the etching rate and deepens material removal, but excessive concentration leads to greater surface roughness and undercutting. Higher temperatures similarly enhance reaction rates and etching efficiency, yet excessive heat can cause uncontrolled lateral etching and precision loss. The study emphasizes that optimal machining performance requires a balanced control of etchant concentration, temperature, and etching time.

Conventional immersion etching where the plate is fully submerged in the etchant bath, often leads to prolonged liquid retention within the channel cavity. This cause localized over-etching and enhanced lateral corrosion, reducing dimensional accuracy [21]. To address this limitation, Xin et al. [43] proposed a spray-type etching technique, shown in Figure 3, where pressurized etchant is sprayed upward onto the plate, promoting rapid drainage from channels, mitigating stagnant-liquid effects, and yielding superior geometric uniformity.

In addition, photochemical etching is a subtractive process that can generate significant material waste. Electrically Assisted Manufacturing (EAM) offers a promising alternative for channel formation [44]. Pan et al. [45] introduced the EAM approach to enhance material formability, enabling more efficient fabrication of microchannels. This technique was used to produce a high-performance PCHE prototype with 23 MPa pressure capacity and a heat-transfer coefficient of 22 kW/(m²·K), offering an efficient and environmentally friendly alternative for microchannel fabrication.

Figure 3. Experimental setups for photochemical etching [46].

Figure 3. Experimental setups for photochemical etching [46].

Diffusion Bonding: Following channel formation, plates are joined by diffusion bonding, a solid-state process performed under elevated temperature and pressure that promotes atomic diffusion across mating surfaces without melting the base material. The process occurs in three sequential stages [47].

Stage A: Plastic deformation and interfacial boundary formation—applied heat and pressure cause surface asperities to deform, increasing the true contact area and creating a continuous interface.

Stage B: Grain-boundary migration with pore elimination—boundaries migrate across the interface, aided by short-range diffusion, thereby closing residual voids and reducing interfacial defects.

Stage C: Volume diffusion and joint consolidation—long-range diffusion homogenizes the microstructure and eliminates any remaining porosity, resulting in a fully consolidated joint with mechanical and thermal properties comparable to those of the parent material.

The quality of diffusion bonding is primarily governed by key process parameters such as welding temperature, applied pressure, and holding time. Kim et al. [48] reported that raising the bonding temperature from 800 °C to 900 °C significantly reduced interfacial defects, with the non-integrated grain-boundary fraction dropping from about 75% to 18%. Extending the holding time from 60 to 90 min further improved joint consolidation, while increasing pressure from 4 to 8 MPa showed an unstable effect, slightly varying with bonding conditions. Similarly, Hu et al. [49] found that bonding temperature and holding time strongly affect the bonding ratio and grain growth of diffusion-bonded 316 L stainless steel. As the temperature increased from 900 °C to 1100 °C, the bonding ratio rose from 55.99% to 99.22%, while the grain size expanded from 23.04 μm to 197.51 μm. When the holding time was extended from 30 to 120 min, the bonding ratio increased from 74.15% to 98.61%, and the grain size from 57.31 μm to 132.59 μm. Nevertheless, the optimal bonding conditions can vary significantly with operating environments and material combinations and should therefore be determined on a case-by-case basis [50]. Furthermore, the long-term mechanical integrity requirements of PCHEs in service must be carefully considered when selecting diffusion-bonding parameters [51].

In addition, the process parameters can significantly influence manufacturing deformation and cost. Excessive bonding temperature or pressure may lead to slight plate distortion or dimensional inaccuracy, while longer holding times increase energy consumption and production duration. These factors collectively affect fabrication efficiency, cost, and yield. The selection of diffusion-bonding parameters should therefore also account for the long-term mechanical reliability of PCHEs in service.

Recent advances in diffusion-bonding modeling have evolved from empirical descriptions to physics-based frameworks, enhancing understanding of interfacial evolution and bonding mechanisms. Zhao et al. [52] established an experimental mapping model linking surface roughness with interfacial bonding performance for 316H stainless steel joints, which serves as a semi-empirical framework for analyzing surface morphology effects and optimizing process parameters. Peng et al. [53,54] developed a coupled analytical model integrating deformation and diffusion mechanisms to describe void-closure kinetics and predict bonding time. Chang et al. [55] proposed a comprehensive kinetic model incorporating surface diffusion and multicomponent effects to capture void evolution in dissimilar joints. Building upon these developments, Wang et al. [56] introduced a multi-phase, multicomponent phase-field model that simultaneously captures void and grain evolution, offering a robust numerical platform for optimizing high-temperature diffusion bonding and improving the reliability of compact heat exchangers.

In summary, photochemical etching of PCHE channels can be achieved through immersion, spray-assisted, and electrically assisted approaches. Immersion etching is mature and simple. Spray-assisted etching enhances dimensional accuracy and uniformity by reducing stagnant-liquid effects, while electrically assisted approaches offer greater efficiency and material utilization. Subsequently, diffusion bonding plays a decisive role in consolidating the stacked plates. Key processing parameters—including temperature, pressure, and holding time—directly influence the extent of atomic diffusion, interfacial defect elimination, and microstructural consolidation, thereby determining the overall mechanical integrity and service reliability of the heat exchanger. However, although the effects of process parameters and the underlying mechanisms of photochemical etching and diffusion bonding have been extensively investigated, studies focusing on their applicability to PCHE structures remain limited. Most existing models and conclusions are derived from experiments on flat-plate substrates, and their feasibility in predicting the fabrication accuracy and bonding reliability of complex PCHE geometries still requires further evaluation. Moreover, the overall cost, yield losses, and scalability of these manufacturing processes have not yet been systematically assessed, which are essential factors for large-scale engineering application.

2.3. Influence of Etching on the Thermal–Hydraulic Performance

The photochemical etching process used to fabricate PCHE channels is inherently isotropic. Material is removed not only vertically into the metal but also laterally beneath the photoresist mask, as shown in Figure 4a [57]. This results in a widened channel profile and a transition from the originally designed semicircular shape to a slightly flattened or oval-like cross-section.

The dimensional deviations caused by lateral etching are generally minimal. Tang et al. [22,57] experimentally measured the geometrical dimensions of stainless-steel channels after photochemical etching. Ten test plates with a design diameter of 2 mm were fabricated, yielding 4238 measurements of channel depth and 479 of channel width. CFD analysis based on these measured geometries indicated that, for the most probable etched shape, the width and depth decreased by 0.15% and 1.3% relative to the design, leading to increases of +0.55% and +4.08% in Nu and ΔP, respectively. Under the 95% confidence interval extremes, where the width and depth variations reached +3.75% and +4.5%, the deviations in Nu and ΔP could exceed 3% and 17%.

These results suggest that geometric deviations induced by photochemical etching generally exert only a minor influence on the thermal–hydraulic performance, though the effect may become non-negligible for channels exhibiting larger etching-induced deviations. For PCHEs with relatively small heat loads and a limited number of channels, the smaller statistical sample of etched passages increases the likelihood of extreme geometrical variations, which may introduce noticeable discrepancies in the overall performance evaluation. Therefore, it is necessary to examine the geometry of individual channels to ensure measurement consistency and reduce uncertainty in performance assessment.

The influence of lateral etching is more pronounced and varies significantly across different channel geometries.

Zigzag Channels: Lateral etching rounds the originally sharp bend corners. Figure 4b [58] presents the velocity contour plots for zigzag channels with varying rounding radii. Wang et al. [58] demonstrated that these smoother arcs reduce flow separation and reattachment behavior, thereby lowering the pressure drop. However, the attenuation of flow disturbances also leads to a slight reduction in heat-transfer performance. Notably, appropriately rounded corners can further streamline the flow at the bends, thereby improving velocity distribution and enhancing the overall thermal–hydraulic performance [59].

Airfoil fin Channels: Lateral etching forms distinct filets at the end walls, as shown in Figure 4c [60]. Ma et al. [61] reported that such rounded structures generate small local vortices at the leading and trailing edges, enhancing near-wall turbulence and making the velocity and temperature distributions more uniform, thereby reducing upstream–downstream differences and improving heat-transfer performance.

Lateral etching introduced during photochemical processing introduces geometry-dependent alterations that distinctly influence thermal–hydraulic performance. While its effects are minimal in semicircular channels, it reduces pressure drop in zigzag channels (at the cost of slight heat transfer reduction) and enhances heat transfer in airfoil fin channels via vortex-induced mixing. These impacts must be carefully considered in PCHE design and optimization.

Figure 4. Schematic and geometrical representations of channel deformation and characteristic flow fields during the bonding process: (a) channel deformation during diffusion bonding [57]; (b) velocity contours of zigzag channels with different rounding radii [58]; (c) 3D modeling of airfoil fin channel and unit fluid volume: (i) NACA0020 airfoil fin based on real etched plate, (ii) real unit volume of airfoil fin channel based on the etched shape, and (iii) ideal unit volume of airfoil fin channel [60].

Figure 4. Schematic and geometrical representations of channel deformation and characteristic flow fields during the bonding process: (a) channel deformation during diffusion bonding [57]; (b) velocity contours of zigzag channels with different rounding radii [58]; (c) 3D modeling of airfoil fin channel and unit fluid volume: (i) NACA0020 airfoil fin based on real etched plate, (ii) real unit volume of airfoil fin channel based on the etched shape, and (iii) ideal unit volume of airfoil fin channel [60].

3. Thermal–Hydraulic Characteristics of PCHEs

The thermal–hydraulic characteristics of PCHEs with different channel configurations are reviewed in this section, with emphasis on the effects of channel geometry on flow and heat transfer mechanisms as well as directions for further optimization. In addition, correlations developed for various channel configurations are systematically organized and discussed, with attention given to their applicable ranges and accuracy.

3.1. Channel Geometries

The thermal–hydraulic performance of PCHEs is strongly dependent on the geometric configuration of the flow channels. The classical channel types, as illustrated in Figure 5 [62], include straight, zigzag, wavy, S-shaped fin, and airfoil fin channels. This evolution reflects a continuous effort to optimize the balance between enhancing heat transfer and minimizing flow resistance.

Straight Channels: The earliest and simplest design, straight channels are characterized by structural simplicity and ease of fabrication. However, they offer only limited heat transfer enhancement due to the development of laminar flows.

Zigzag Channels: Developed to overcome the limitation of straight channels, the zigzag geometry promotes flow reattachment and turbulence generation, significantly improving heat transfer. This enhancement comes at the cost of a substantially increased pressure drop. For instance, Chen et al. [63] reported that the heat-transfer performance of zigzag channels was 2–3 times higher than that of straight channels under laminar flow and 1.5–3 times higher under transitional flow, but with a friction factor increase of 2–3 times.

Wavy Channels: To reduce the excessive pressure drop of zigzag designs, wavy channels were proposed. Their smooth curvature weakens flow separation, maintaining favorable heat-transfer performance while reducing resistance. Jin et al. [64] reported that while wavy channels exhibited about 33% lower heat transfer than zigzag channels, they reduced the pressure drop by approximately 80%.

S-Shaped Fin Channels: Based on this development, Ngo et al. [65] proposed a discontinuous configuration, the S-shaped fin channel, to further balance performance metrics. This design reduces the pressure drop to only one-fifth of that of a zigzag channel at an equivalent heat transfer level. However, its high geometry complexity requires advanced manufacturing techniques, and consequently, it has been the subject of limited research.

Airfoil Fin Channels: Following the discontinuous channel concept, Kim et al. [66] proposed the airfoil fin channel. This design features a more uniform and streamlined structure that simplifies fabrication. The geometry effectively reduces form drag, resulting in a pressure drop that is only one-fifth that of wavy channels for the same unit volume heat-transfer rate. Consequently, the airfoil fin channel has become an important development direction for PCHEs.

Collectively, the innovation in PCHE channel geometries demonstrates a clear trajectory toward achieving an optimal balance between thermal efficiency and hydraulic performance.

3.2. Thermal–Hydraulic Characteristics of Straight-Channel PCHEs

Straight-channel PCHEs, as the earliest and most mature configuration, have been extensively investigated owing to their simple geometry and well-established manufacturing processes [7]. Previous review studies have summarized the effects of different experimental conditions, inlet parameters, and preliminary optimization of channel arrangements and structures. However, systematic reviews focusing on how the fundamental geometric parameters of straight channels influence flow and heat-transfer performance, as well as the combined effects of geometry and channel arrangement, remain limited. This subsection reviews their thermal–hydraulic characteristics emphasizing on the effects of cross-sectional shape, flow arrangement, and fluid properties on heat transfer and pressure drop. It also covers various modified designs aiming to enhance turbulence or reduce flow resistance and examines their applications across diverse energy systems.

3.2.1. Effect of the Cross-Sectional Geometry

The semicircular cross-section is the most common geometry for straight-channel PCHEs. Experimental studies have shown that, while the flow friction and heat transfer trends are similar to those in circular pipes (see Figure 6), a key difference exists: the transition from laminar to transitional flow occurs at a lower Reynolds number (~1700) compared to circular pipes (~2300) [67]. This is primarily attributed to the semicircular shape, as well as the relatively rough inlet profile resulting from the practical fabrication process of PCHEs, which together accelerate the onset of flow instability.

The geometric parameters of straight channels exert a significant influence on the thermal–hydraulic performance of PCHEs. Reducing the channel diameter enhances turbulence intensity, while reducing the wall thickness decreases thermal resistance. Reducing the channel diameter enhances turbulence intensity, while reducing the wall thickness decreases thermal resistance. Quantitatively, when the channel diameter increased from 1.8 mm to 2.4 mm, the overall heat-transfer coefficient decreased by up to 37%, whereas increasing the wall thickness from 0.4 mm to 0.7 mm reduced it by about 5% [58], improving overall heat-transfer efficiency [68]. Increasing the ridge width and channel length can also enhance heat transfer, but this comes at the cost of a substantial increase in the PCHE volume [69].

Other cross-sectional shapes also significantly influence thermal–hydraulic performance. Rectangular channels demonstrate superior heat transfer to semicircular channels when using sCO₂ [70]. The rectangular structure increases the heat-transfer rate per unit volume by approximately 20–47% relative to the baseline semicircular channel, and the performance further improves with increasing channel width. The temperature distributions for different cross-sectional shapes are shown in Figure 7a. This enhancement is attributed to the rectangular geometry, which promotes uniform fluid distribution and avoids the asymmetric heat transfer inherent to semicircular channels, thereby further improving heat-transfer performance.

Studies with liquid metals [71] show that trapezoidal, rectangular, and semicircular straight channels have comparable heat-transfer performance. However, in terms of overall thermal–hydraulic performance, the trapezoidal channel ranks first, followed by the rectangular and semicircular channels, though the differences remain relatively small. This suggests that the optimal channel cross-section depends on the thermophysical properties of the working fluid, and conclusions drawn for one fluid may not be universally applicable to others.

Figure 7. (a) Temperature distribution at different cross-sections, (i) cold side (ii) hot side [70]. (b) Physical models of horizontal semicircular channels: (i) uniform cross-section semicircular channel, (ii) diverging channel, and (iii) converging channel [72].

Figure 7. (a) Temperature distribution at different cross-sections, (i) cold side (ii) hot side [70]. (b) Physical models of horizontal semicircular channels: (i) uniform cross-section semicircular channel, (ii) diverging channel, and (iii) converging channel [72].

3.2.2. Geometric Optimization Strategies for Straight Channels

An existing review [28] showed that, although straight-channel PCHEs feature simple geometry, the thermal efficiency remains relatively low. In addition to the basic straight-channel design, several modified geometries have been proposed to intentionally disturb the flow and strengthen heat transfer, as summarized in Table 1.

Variable Cross-Sections: Converging channels, as shown in Figure 7b, improve heat transfer by enhancing field synergy, where the heat-transfer coefficient increases by 42.26% compared with uniform cross-sectional channels [72]. Meanwhile, diverging channels improve hydraulic performance by reducing the pressure drop by approximately 33.3% compared with uniform channels [73]. Further investigations revealed that a hybrid setup with diverging hot channels and uniform cold channels yields the optimal overall performance index, with the overall heat-transfer coefficient improved by more than 20% [73].

Surface Modifications: Introducing dimples or ribs on channel walls significantly enhances heat transfer despite a higher pressure drop. Chen et al. [74] proposed a square straight-channel structure incorporating an array of dimples. These dimples generated vortices and increased turbulent kinetic energy, thereby improving the heat-transfer performance; the heat-transfer coefficient improved by approximately 50% while the friction factor increased only by about 15%, as shown in Figure 8. Aneesh et al. [75] proposed hemispherical dimples in semicircular straight channels. Although this configuration increases the pressure drop, it significantly enhances the heat-transfer efficiency. Han et al. [76] arranged rib structures along the flat wall at the top of semicircular channels, as shown in Figure 9. The ribs significantly increased the turbulent kinetic energy and improved the coordination between the velocity and temperature gradient fields, thus enhancing overall heat-transfer performance, with the heat-transfer coefficient improved by 0.4–9.5% and the comprehensive performance enhanced by 19.3–19.8%.

Figure 8. The distributions of turbulent kinetic energy, velocity vector, and streamlined dimple structure [74].

Figure 8. The distributions of turbulent kinetic energy, velocity vector, and streamlined dimple structure [74].

Figure 9. Distributions of turbulent kinetic energy on cross-sections of channels with ribs [76].

Figure 9. Distributions of turbulent kinetic energy on cross-sections of channels with ribs [76].

Interconnected Channels: Designs featuring interconnecting pipes between adjacent channels augment mixing and strengthen heat transfer. Fu et al. [77] recently proposed semicircular channels with interconnecting pipes, highlighting this design as a newly emerging concept. The induced vortical structures augmented the turbulent kinetic energy and further strengthened heat transfer; the heat-transfer performance increased by 110–250% compared with conventional semicircular channels.

Flow Arrangement Optimization: The flow directions of hot and cold streams, as well as the channel arrangement, also play crucial roles in the thermal–hydraulic performance. An experimental study by Seo et al. [78] demonstrated that, under parallel-flow conditions, both the average heat-transfer rate and overall heat-transfer performance were inferior to those under counter-flow conditions, with reductions of 6.8% and 10–15%, respectively. Hou et al. [79] reported that a double-layer arrangement of micron-scale PCHEs provides superior heat transfer and flow performance compared to a single-layer configuration. In contrast, Aneesh et al. [75] observed that parallel and staggered configurations yielded nearly identical performances, while single-layer arrangements exhibited better thermal–hydraulic characteristics than double-layer ones; these findings were derived through investigations into different channel arrangements (see Figure 10). These apparently inconsistent findings are primarily due to differences in operating conditions (e.g., pressure affecting fluid properties) and channel cross-sectional geometry between the studies.

Table 1. Geometry optimization of straight channels in PCHEs.

Reference	Geometry Modification	Geometrical Parameter	Optimization Target	Mechanism	Performance Improvement
Wei et al. [72]	Converging/Diverging semicircular channels	D = 0.75 mm (uniform); D = 0.5 → 1.0 mm (diverging); D = 1.0 mm → D = 0.5 mm (converging)	Heat transfer enhancement	Converging channels improved field synergy, thereby enhancing heat transfer, while diverging channels exhibited the opposite effect.	Converging channel: heat-transfer coefficient improved by 42.26%. Diverging channel: heat-transfer performance reduced.
Morteza et al. [73]	Converging channels (hot side and cold side)	D = 1.2 mm (uniform); D = 0.9 → 1.5 mm (diverging); D = 0.9 mm → D = 1.5 mm (converging)	Heat transfer enhancement (converging); flow performance improvement (diverging)	Same as above	Converging channel: heat-transfer coefficient improved by 42.26%. Diverging channel: heat-transfer performance reduced. Converging channel (both sides): overall heat-transfer coefficient improved by >20%. Diverging channel (both sides): pressure drop reduced by ≈33.3%.
Chen et al. [74]	Straight square channel with dimples	Square cross-section, Df = 1 mm; dimple diameter D = 0.3 mm, depth H = 0.075 mm, spacing Sx = 0.5 mm, Sy = 0.25 mm	Heat transfer enhancement	Dimples induced vortices and increased turbulent kinetic energy, thus improving heat transfer.	Heat-transfer coefficient improved by ~50%, while friction factor increased only by ~15%.
Fu et al. [77]	Semicircular channels with interconnecting pipes	Semicircular channel D = 1.51 mm; Connecting pipe: radius r2 = 0.1–0.2 mm; deviation angle θ = 0–15°	Heat transfer enhancement	Interconnecting pipes induced vortical structures, increasing turbulent kinetic energy and strengthening heat transfer.	Heat-transfer performance increased by 110–250%.
Han et al. [76]	Ribbed semicircular channels (ribs on flat wall)	Semicircular channel D = 0.9 mm; Rib size 0.2 mm × 0.1 mm; Rib pitch 1.25 mm.	Heat transfer enhancement	Ribs enhanced turbulent kinetic energy and improved velocity–temperature gradient coordination.	Heat-transfer coefficient improved by 0.4–9.5%; comprehensive performance enhanced by 19.3–19.8%.
Aneesh et al. [75]	Semicircular straight channel with hemispherical dimples	Semicircular channel D = 2.0 mm Hemispherical dimple diameter D = 1.0 mm	Heat transfer enhancement	Dimples increased flow disturbance, resulting in higher pressure loss but significantly improving heat-transfer efficiency.	Volumetric heat-transfer rate improved by 27.3–38.3%.

Table 1. Geometry optimization of straight channels in PCHEs.

Reference	Geometry Modification	Geometrical Parameter	Optimization Target	Mechanism	Performance Improvement
Wei et al. [72]	Converging/Diverging semicircular channels	D = 0.75 mm (uniform); D = 0.5 → 1.0 mm (diverging); D = 1.0 mm → D = 0.5 mm (converging)	Heat transfer enhancement	Converging channels improved field synergy, thereby enhancing heat transfer, while diverging channels exhibited the opposite effect.	Converging channel: heat-transfer coefficient improved by 42.26%. Diverging channel: heat-transfer performance reduced.
Morteza et al. [73]	Converging channels (hot side and cold side)	D = 1.2 mm (uniform); D = 0.9 → 1.5 mm (diverging); D = 0.9 mm → D = 1.5 mm (converging)	Heat transfer enhancement (converging); flow performance improvement (diverging)	Same as above	Converging channel: heat-transfer coefficient improved by 42.26%. Diverging channel: heat-transfer performance reduced. Converging channel (both sides): overall heat-transfer coefficient improved by >20%. Diverging channel (both sides): pressure drop reduced by ≈33.3%.
Chen et al. [74]	Straight square channel with dimples	Square cross-section, Df = 1 mm; dimple diameter D = 0.3 mm, depth H = 0.075 mm, spacing Sx = 0.5 mm, Sy = 0.25 mm	Heat transfer enhancement	Dimples induced vortices and increased turbulent kinetic energy, thus improving heat transfer.	Heat-transfer coefficient improved by ~50%, while friction factor increased only by ~15%.
Fu et al. [77]	Semicircular channels with interconnecting pipes	Semicircular channel D = 1.51 mm; Connecting pipe: radius r2 = 0.1–0.2 mm; deviation angle θ = 0–15°	Heat transfer enhancement	Interconnecting pipes induced vortical structures, increasing turbulent kinetic energy and strengthening heat transfer.	Heat-transfer performance increased by 110–250%.
Han et al. [76]	Ribbed semicircular channels (ribs on flat wall)	Semicircular channel D = 0.9 mm; Rib size 0.2 mm × 0.1 mm; Rib pitch 1.25 mm.	Heat transfer enhancement	Ribs enhanced turbulent kinetic energy and improved velocity–temperature gradient coordination.	Heat-transfer coefficient improved by 0.4–9.5%; comprehensive performance enhanced by 19.3–19.8%.
Aneesh et al. [75]	Semicircular straight channel with hemispherical dimples	Semicircular channel D = 2.0 mm Hemispherical dimple diameter D = 1.0 mm	Heat transfer enhancement	Dimples increased flow disturbance, resulting in higher pressure loss but significantly improving heat-transfer efficiency.	Volumetric heat-transfer rate improved by 27.3–38.3%.

3.2.3. Performance of Different Working Fluids

Extensive research has been conducted on the thermal–hydraulic performance of straight-channel PCHEs under different working fluids and boundary constraints. Among these, studies with sCO₂ are the most prominent, primarily because the strong pressure resistance of PCHEs aligns well with the high-pressure operating conditions required in sCO₂ Brayton cycles [80]. Heat transfer is highly variable in the pseudo-critical region, necessitating CFD models for accurate prediction [81]. Local heat-transfer coefficients vary significantly along the channel length [82], and specialized methods, such as a discretization method introduced by Park et al. [30], are required to reduce prediction errors near the critical point.

Beyond sCO₂, investigations on other supercritical gases and mixed working fluids have also indicated that fluid properties exert a significant influence on heat transfer and flow resistance. Nitrogen showed strong nonlinear property changes near its pseudo-critical point [64], and rolling motion enhances local heat transfer but simultaneously increases pressure drop [83]. The sCO₂/propane mixtures can reduce pressure drop and enhance the Nusselt number, with increased propane fractions enhancing suitability for high-temperature power cycles [84]. Thermal performance was boosted (by 43%) by nanofluids with thicker interfacial nanolayers [85], but pulsating flow must be carefully controlled to avoid instability [86].

3.2.4. Design and Optimization

On this basis, the design and optimization of straight-channel PCHEs for different energy systems have become an important research direction. Designs have been developed for lead–bismuth fast reactor [87], and molten chloride salt–sCO₂ systems operating above 700 °C, with materials like ZrC/W outperforming traditional nickel-based alloys and stainless steels in both performance and cost [88].

Analysis codes have been created for thermal design and cost evaluation of advanced SMRs, emphasizing the considerable impact of heat transfer correlations on temperature distribution and identifying cost variations under different coolant–material combinations [89]. Dynamic models have been validated for predicting PCHE responses in very-high-temperature gas-cooled reactor conditions [90]. Improved design methods, such as those based on the Generalized Mean Temperature Difference (GMTD), have achieved higher accuracy than the conventional LMTD method [91].

3.3. Thermal–Hydraulic Characteristics of Zigzag (Wavy)-Channel PCHEs

This subsection critically reviews the current state of research on PCHEs with zigzag and wavy channels. It examines the fundamental flow and heat transfer mechanisms, with particular attention to the influence of key geometric parameters on thermal–hydraulic performance, and various optimization strategies aimed at balancing performance. Finally, it synthesizes their applications with different working fluids.

3.3.1. Flow and Heat Transfer Mechanisms of Zigzag Channels

The flow and heat transfer mechanisms of zigzag channels are primarily governed by their periodic, sharp bends. As shown in Figure 11a [92], these bends force the core flow to accelerate toward the inner wall, causing flow contraction. This action thins or disrupts the thermal boundary layer, thereby enhancing local convective heat transfer. However, these same bends also induce flow separation and generate vortices in the downstream region [93]. While these vortices enhance turbulence and promote fluid mixing, they are the primary source of the significant flow resistance and pressure drop characteristics of zigzag channels.

The wavy channel was developed as a direct improvement on the zigzag design. By replacing sharp angles with smooth, sinusoidal curves, wavy channels mitigate flow separation and the associated pressure loss. This modification retains the heat transfer enhancement while markedly alleviating the excessive flow resistance caused by sharp bends, as shown in Figure 11b [64]. According to the comparative results reported by Jin et al. [64], the wavy channel exhibited approximately 33% lower heat-transfer performance but about 80% lower pressure drop compared with the zigzag channel, demonstrating a more balanced trade-off between heat-transfer efficiency and hydraulic performance.

3.3.2. Effect of Geometric Parameters

The characteristic geometric parameters of zigzag channels, such as bending angle and pitch, exert significant influence on their flow and heat-transfer performance [94].

Bending Angle: As shown in Figure 12, an increase in the bending angle strengthens the secondary vortices and accelerates the core flow toward the inner wall, leading to flow contraction and thinning of the thermal boundary layer. The intensified vortical motion enhances convective heat transfer but simultaneously promotes flow separation and increases pressure losses. Yu et al. [95] reported that, when the bending angle increases from 0° to 45°, the Nusselt number rises by approximately 90%, while the friction factor increases by more than 200%, indicating a distinct trade-off between heat-transfer enhancement and flow resistance. The same study further revealed that, when the bending angle becomes excessively large, vortex structures lose stability, resulting in a decline in heat-transfer performance.

Channel Pitch: For a fixed total channel length, a smaller pitch increases the number of bends, which intensifies flow redirection and disturbance, thereby enhancing heat transfer at the expense of a higher pressure drop. According to the results reported by Yin et al. [96], when the pitch decreases from 24 to 8 mm, the Nusselt number increases by approximately 40–50%, while the friction factor rises by about 70–90% over the investigated Reynolds number range.

Cross-Sectional Shape and Size: While the semi-elliptical cross-section is common, its corners occupy volume with low heat flux density, reducing efficiency. By reducing the channel’s cross-sectional size, the volume fraction occupied by these corners is significantly decreased, while the wall heat flux density increases [97]. Therefore, the unit volume heat-transfer rate of the PCHE is enhanced, thereby contributing to a more compact configuration and improved cost-effectiveness from a design perspective. Increasing the channel width can effectively enhance heat-transfer efficiency by up to 41.5% and reduce the friction factor when the bending angle is α ≤ 20°, with the friction factor of arcuate sections reduced by up to 7.25%. However, this improvement is limited to a critical width; beyond this point, the friction factor increases again. When the bending angle is α > 20°, increasing the channel width generally leads to higher friction factors, with a sharp rise observed particularly at s = 2.4 mm [98].

In addition, some studies have explored alternative cross-sectional geometries, such as rectangular [99], elliptical [100], and circular shapes [101]. The numerical investigation conducted by Lee and Kim [102] on PCHEs with semicircular, rectangular, trapezoidal, and circular cross-sections revealed that the heat transfer effectiveness of the channels is closely correlated with their heat transfer surface area. Among the tested geometries, the rectangular cross-section exhibited the highest heat transfer effectiveness, whereas the circular cross-section demonstrated the most favorable hydraulic performance.

3.3.3. Geometric Optimization Strategies for Zigzag Channel

Extensive research has focused on the structural optimization of zigzag channels to mitigate the excessive pressure drop or to further enhance heat-transfer performance. A summary of geometry optimization studies of zigzag channels is presented in Table 2.

(a) Strategies for Reducing Flow Resistance

Inserting Straight Sections: From the perspective of reducing flow resistance, several studies have attempted to improve the overall thermal–hydraulic performance of zigzag channels by introducing straight segments at the bending regions [18,58,103]. This inserted straight section contributes to more uniform flow, yielding a substantial reduction in pressure loss (36–44%) while incurring only a relatively small decrease in heat transfer (3.6–30.3%) [18].

Lee et al. [104] numerically studied zigzag channels with straight inserts (0.5–2 mm) at the bends (Figure 13a). For 0.5 and 1.0 mm inserts, the pressure drop fell below that of the conventional zigzag and was comparable to the wavy channel. Heat-transfer performance for these two cases remained similar to the zigzag but better than the wavy channel. Notably, the 1.0 mm insert increased the volume goodness factor by ~26–28% relative to the original zigzag design.

Nature-Inspired Fins: On the basis of inserting straight segments into zigzag channels, Bi et al. [105] proposed a novel heat exchanger by further adding fins within these straight sections, inspired by the formation of river islands at natural river bends. The design of the nature-inspired channel is shown in Figure 13b. Experimental investigations demonstrated that, compared with the conventional zigzag channel, this new channel structure increased the average heat-transfer rate by 1.5% while significantly reducing the pressure drop by 34.85%. These findings indicate that the nature-inspired channel can substantially reduce flow resistance while maintaining heat-transfer performance.

Based on the conventional zigzag channel, a novel configuration with staggered sinusoidal fins was proposed to reduce pressure drop by eliminating recirculation and dead zones [16]. This design was optimized using response surface methodology and a genetic algorithm, and numerical simulations showed that, under identical heat transfer, its hydraulic performance improved by up to 2.5 times compared with the conventional zigzag channel. Moreover, Liu et al. [106] introduced a cellular arrangement with hole–obstacle structures in zigzag channels to induce periodic cross-layer switching between plates. This design enhances mixing, improves fin utilization, and enlarges the effective heat transfer surface. Compared with conventional zigzag channels, the cellular arrangement only alters the hot–cold channel sequence; therefore, pressure drop does not significantly increase, while heat transfer capability is enhanced.

(b) Strategies for Enhancing Heat Transfer

Trapezoidal Wavy Channels: From the perspective of enhancing heat transfer, Aneesh et al. [107] further proposed a trapezoidal wavy channel, and numerical results indicated that its heat-transfer rate was improved by approximately 6% compared with that of the zigzag channel. The enhanced heat transfer in trapezoidal channels arises from multiple sharp bends that intensify vortex formation and fluid mixing, thereby improving boundary layer disruption at the cost of higher pressure drop.

Sandwiched Trapezoidal Channels: In addition, Ji et al. [108,109] introduced a sandwiched trapezoidal channel structure in which two hot channels are coupled with one cold channel to match the heat transfer capacity and flow velocity of sCO₂ on the hot and cold sides, and carried out both numerical simulations and experimental studies. The results showed that this design reduces the pressure drop by 75% and improves the regenerative efficiency by 5%, and the overall heat-transfer coefficient of the heat exchanger exceeds 1.10 kW/(m²·K).

Tangled and Helical Configurations: The tangled configuration of hot and cold channels provides an alternative to enhance heat transfer. Li et al. [110] proposed a helical twine channel and numerically studied its performance with water. Compared with the zigzag channel, the 3D helical design induced stronger transverse disturbances and boundary layer disruption, improving thermal–hydraulic performance by 37.6%, though fabrication remains difficult. As a simpler option, Sung and Lee [111] developed a tangled channel using three-layer plates. Experimental and numerical results showed that improved temperature uniformity in mixing zones enhances heat transfer, allowing the tangled channel to outperform straight channels even at low Reynolds numbers.

However, despite the remarkable progress achieved through these structural optimization strategies, most of the aforementioned studies remain limited to numerical simulations, lacking sufficient experimental validation. Furthermore, for some complex configurations—such as nature-inspired or helical designs—their fabrication feasibility and mechanical reliability still require further evaluation before practical implementation.

Table 2. Geometry optimization of zigzag channels in PCHEs.

Reference	Geometry Modification	Geometrical Parameter	Optimization Target	Mechanism	Performance Improvement
Lee et al. [104]	Semicircular channels with inserted straight sections	Semicircular hot channel: D = 1.9 mm, Ph = 9.0 mm, α = 32.5°; Cold channel: D = 1.8 mm, Ph = 7.24 mm, α = 40°; Straight section length = 0.5–5 mm	Reduce pressure drop with limited heat transfer penalty	Straight segments reduce recirculation at bends, resulting in more uniform flow and lower pressure loss	Friction factor reduced by up to 50%; Colburn j factor decreased by ≈8–10% compared with zigzag channel.
Wang et al. [58]		Semicircular channel: D = 2 mm, Ph = 18 mm, α = 30°; Straight section length = 0.5–4 mm			Friction factor decreased by 44–48%, while the Nusselt number decreased by 11–14%
Zhao et al. [103]		Semicircular channel: D = 1.5 mm, Ph = 20 mm, α = 45°; Straight section length = 1–5 mm			Friction factor decreased by 36–44%, while the Nusselt number decreased slightly by only 4–9%
Wang et al. [18]		Semicircular channel: D = 2 mm; Ph = 24.46 mm, α = 0–45° Straight section length = 2.46–9.84 mm			Friction factor decreased by 33.1–84.7%, while the Nusselt number decreased by 3.6–30.3%
Bi et al. [105]	Rectangular channels with nature-inspired fins in straight sections	Rectangular channel: Dx = 1.5 mm, Dy = 1.0 mm Ph = 15 mm α = 30° Nature-inspired fin, chord length = 2.5 mm	Reduce pressure drop and maintain heat transfer	Fins in straight sections induce stable flow similar to “river islands,” improving flow uniformity	Pressure drop reduced by 34.85%, heat transfer rate increased by 1.5%
Saeed and Kim [112]	Semicircular channels with staggered sinusoidal fins	Semi-elliptic channel: Hot side: Dh = 1.106 mm, Ph = 9.0 mm, α = 40°; Cold side: Dh = 1.10 mm, Ph = 7.24 mm, α = 40°; Sinusoidal fins: Hot side: h = 3.03 mm, pt = 2.35 mm; Cold side: h = 2.87 mm, pt = 2.25 mm	Reduce pressure drop	Sinusoidal fins eliminate recirculation zones, reducing dead flow and pressure loss	Pressure drop reduced by 58.3%
Liu et al. [106]	Semicircular channels channel with cellular arrangement (hole–obstacle structures)	Semicircular channel D = 2.0 mm, Ph = 10 mm, α = 30°	Heat transfer enhancement	Induces cross-layer fluid switching, which improves mixing and fin utilization	The overall heat transfer coefficient increased by 8.6%
Aneesh et al. [107]	Trapezoidal wavy channel	Semicircular channel D = 2 mm; Ph = 24.72 mm, α = 45°; Trapezoidal wavy channel: Amplitude A = 1.65 mm	Heat transfer enhancement	Multiple sharp bends intensify vortex formation and boundary layer disruption	Heat transfer density improved by up to 41%
Ji et al. [108,109]		Semicircular channel D = 2 mm; Ph = 10 mm; Trapezoidal wavy channel: Amplitude A = 1 mm			Nusselt number enhanced by 10.67%, while Friction factor increased by 20%
Li et al. [110]	Helical twine channel	Semicircular channel D = 1.5 mm; Ph = 20 mm, α = 10°	Heat transfer enhancement	Stronger transverse disturbance and boundary layer disruption	Colburn j factor increased up to 100%; friction factor increased up to 80%; overall performance index improved by 37.6%

Table 2. Geometry optimization of zigzag channels in PCHEs.

Reference	Geometry Modification	Geometrical Parameter	Optimization Target	Mechanism	Performance Improvement
Lee et al. [104]	Semicircular channels with inserted straight sections	Semicircular hot channel: D = 1.9 mm, Ph = 9.0 mm, α = 32.5°; Cold channel: D = 1.8 mm, Ph = 7.24 mm, α = 40°; Straight section length = 0.5–5 mm	Reduce pressure drop with limited heat transfer penalty	Straight segments reduce recirculation at bends, resulting in more uniform flow and lower pressure loss	Friction factor reduced by up to 50%; Colburn j factor decreased by ≈8–10% compared with zigzag channel.
Wang et al. [58]		Semicircular channel: D = 2 mm, Ph = 18 mm, α = 30°; Straight section length = 0.5–4 mm			Friction factor decreased by 44–48%, while the Nusselt number decreased by 11–14%
Zhao et al. [103]		Semicircular channel: D = 1.5 mm, Ph = 20 mm, α = 45°; Straight section length = 1–5 mm			Friction factor decreased by 36–44%, while the Nusselt number decreased slightly by only 4–9%
Wang et al. [18]		Semicircular channel: D = 2 mm; Ph = 24.46 mm, α = 0–45° Straight section length = 2.46–9.84 mm			Friction factor decreased by 33.1–84.7%, while the Nusselt number decreased by 3.6–30.3%
Bi et al. [105]	Rectangular channels with nature-inspired fins in straight sections	Rectangular channel: Dx = 1.5 mm, Dy = 1.0 mm Ph = 15 mm α = 30° Nature-inspired fin, chord length = 2.5 mm	Reduce pressure drop and maintain heat transfer	Fins in straight sections induce stable flow similar to “river islands,” improving flow uniformity	Pressure drop reduced by 34.85%, heat transfer rate increased by 1.5%
Saeed and Kim [112]	Semicircular channels with staggered sinusoidal fins	Semi-elliptic channel: Hot side: Dh = 1.106 mm, Ph = 9.0 mm, α = 40°; Cold side: Dh = 1.10 mm, Ph = 7.24 mm, α = 40°; Sinusoidal fins: Hot side: h = 3.03 mm, pt = 2.35 mm; Cold side: h = 2.87 mm, pt = 2.25 mm	Reduce pressure drop	Sinusoidal fins eliminate recirculation zones, reducing dead flow and pressure loss	Pressure drop reduced by 58.3%
Liu et al. [106]	Semicircular channels channel with cellular arrangement (hole–obstacle structures)	Semicircular channel D = 2.0 mm, Ph = 10 mm, α = 30°	Heat transfer enhancement	Induces cross-layer fluid switching, which improves mixing and fin utilization	The overall heat transfer coefficient increased by 8.6%
Aneesh et al. [107]	Trapezoidal wavy channel	Semicircular channel D = 2 mm; Ph = 24.72 mm, α = 45°; Trapezoidal wavy channel: Amplitude A = 1.65 mm	Heat transfer enhancement	Multiple sharp bends intensify vortex formation and boundary layer disruption	Heat transfer density improved by up to 41%
Ji et al. [108,109]		Semicircular channel D = 2 mm; Ph = 10 mm; Trapezoidal wavy channel: Amplitude A = 1 mm			Nusselt number enhanced by 10.67%, while Friction factor increased by 20%
Li et al. [110]	Helical twine channel	Semicircular channel D = 1.5 mm; Ph = 20 mm, α = 10°	Heat transfer enhancement	Stronger transverse disturbance and boundary layer disruption	Colburn j factor increased up to 100%; friction factor increased up to 80%; overall performance index improved by 37.6%

3.3.4. Applications of Zigzag Channels with Different Working Fluids

Zigzag channel PCHEs have been widely studied with different gaseous working fluids. Wang et al. [113] reported that, in H₂/He systems, the hot and cold sides exhibit distinct behaviors, with the cold side being more sensitive to gravity and plate thickness. Yin et al. [114] showed that, in He/Xe Brayton cycles, higher mass flow rates and inlet temperatures enhance heat transfer but increase pressure drop, while elevated pressure improves heat transfer with limited pumping penalties. Beyond steady-state operation, Chen et al. [90,115] investigated PCHEs with helium under transient inlet disturbances, confirming model applicability and stability. Together, these studies highlight the complexity of thermal–hydraulic behavior in light, noble gas working fluids.

In liquid and supercritical fluid applications, Zhou et al. [116] demonstrated that, in hydrocarbon fuel/sCO₂ systems, zigzag channels help suppress local heat transfer deterioration under trans-critical and buoyancy effects, and resistance optimization can reduce pressure drop by over 70%. Similarly, Aakre and Anderson [100] found that, in molten salt/sCO₂ systems, zigzag channels outperform smooth tubes in heat transfer but face challenges such as fouling and blockage. These results underscore the potential of zigzag PCHEs for liquid and supercritical media, while pointing to practical issues including nonlinear properties, buoyancy, corrosion, and long-term reliability that remain to be addressed.

In summary, extensive studies on zigzag and wavy channels in PCHEs have clarified flow and heat transfer mechanisms, geometric effects, and optimization strategies. Zigzag channels enhance heat transfer but incur high pressure drops, motivating alternatives such as wavy channels, straight inserts, sinusoidal fins, cellular or trapezoidal geometries, and tangled/helical designs. Applications span nuclear reactors, sCO₂ cycles, molten salt systems, and aerospace propulsion, with research also addressing nonlinear fluid properties, buoyancy, fouling, blockage, and transient responses. Overall, while significant progress has been made, further efforts are needed to balance heat transfer and pressure drop while ensuring reliable operation under advanced energy system conditions.

3.4. Thermal–Hydraulic Characteristics of Airfoil Channel PCHEs

The airfoil fin PCHE is designed to achieve high thermal efficiency with low flow resistance through streamlined fin geometry.

Quantitative comparisons among different channel geometries have demonstrated the advantages of the airfoil fin configuration. Chen et al. [63] experimentally showed that the heat-transfer performance of zigzag PCHEs is 1.5–3 times higher than that of straight channels. Zhao [117] numerically reported that the local heat-transfer coefficient of the zigzag channel is approximately 2.5 times that of the airfoil fin channel, whereas its pressure drop is 4.0–8.3 times higher. Park [60] further demonstrated that, for an equivalent unit-volume heat-transfer rate, the pressure drop of the airfoil fin PCHE is only one-fifth that of the wavy channel. Overall, the airfoil fin channel achieves a well-balanced compromise between heat-transfer enhancement and flow resistance, making it suitable for high-efficiency, compact heat-exchanger applications.

This subsection further discusses the flow and heat-transfer mechanisms of airfoil fin PCHEs, the effects of key geometric parameters, optimization approaches, and their representative applications in advanced energy systems.

3.4.1. Flow and Heat Transfer Mechanisms of Airfoil-Fin Channels

Airfoil fins are generally designed based on the National Advisory Committee for Aeronautics (NACA) airfoil database, with the geometric parameters illustrated in Figure 14. As shown in Figure 15 [118], the incoming stream impinges on the fin leading edge and divides along both sides, forming a stagnation region. Flow accelerates along the surface, reaching a velocity peak near the maximum thickness, where the boundary layer becomes thinner and heat transfer is enhanced. Downstream, periodic deceleration and mixing occur near the trailing edge, sustaining turbulence between adjacent fins. Compared with smooth channels, airfoil fins generate periodic acceleration and mixing zones that suppress boundary layer growth, strengthen convection, and significantly influence overall thermal–hydraulic performance [119].

3.4.2. Effect of the Fin Geometry

The geometry of individual fins is a critical determinant of performance:

Fin Thickness: Increasing fin thickness enhances heat transfer but also raises pressure drop [120,121]. Li et al. [122] reported that, when the maximum thickness increases from 0.5 mm to 1.5 mm, the Nu rises by about 90%, while the f increases by approximately 100%, indicating stronger turbulence and wake mixing that improve convection but increase flow resistance (as shown in Figure 16).

Position of Maximum Thickness: The rearward shift in the maximum-thickness position causes the turbulent kinetic energy region formed at the wake convergence zone to shrink, reducing overall turbulence intensity within the channel. Li et al. [122] reported that, when the maximum-thickness position moves backward from 0.6 mm to 3.6 mm, the Nu decreases by about 17%, while the f increases by approximately 50%.

Camber Magnitude and Position: Greater camber magnitudes lead to significant pressure drop penalties with limited heat transfer gains. Deng et al. [123] attributed this to increased flow separation, vortex generation, and wall shear stresses caused by enhanced asymmetry. Wang et al. [14] further showed that shifting camber position mainly influences the rear airfoil region, with only minor changes in heat transfer and pressure drop.

Surface Roughness: The latest research indicates that surface roughness alters boundary layer development and flow resistance. Deng et al. [123] found that moderate roughness, with height close to viscous sublayer thickness, balances heat transfer enhancement with acceptable resistance. However, excessive roughness markedly increases frictional losses and degrades overall thermal–hydraulic performance (as shown in Figure 17).

However, it must be noted that existing studies on airfoil fins have generally been conducted with fixed fin length. The influence of varying fin length on the flow and heat transfer characteristics of airfoil channels remains unclear.

3.4.3. Effect of the Fin Arrangement

The arrangement of fins within a channel mainly includes parallel and staggered configurations, defined by longitudinal, transverse, and staggered pitches (Figure 18). Staggered layouts generally yield superior overall performance, reducing flow resistance while maintaining heat-transfer rates close to parallel designs [66,124,125,126].

Among the geometric parameters, the longitudinal pitch has the most significant influence on the thermal–hydraulic performance. Reducing the longitudinal pitch increases flow velocity and suppresses boundary layer growth, thereby enhancing heat transfer but also causing a higher pressure drop [21,117]. Liu et al. [21] reported that decreasing the longitudinal pitch from 4 mm to 2 mm increased the heat-transfer coefficient by approximately 12.6% while raising the pressure drop by about 46.2%.

In contrast, variations in transverse pitch and staggered spacing induced only minor performance changes [127]. Zhu et al. [128] found that transverse pitch had a negligible effect on overall performance, and Wang et al. [12] similarly showed that staggered spacing exerted only a slight influence on the Nusselt number. Under molten salt conditions, range analysis by Liu et al. [127] further confirmed that longitudinal pitch has the greatest impact, followed by transverse pitch and staggered spacing.

3.4.4. Geometric Optimization Strategies for Airfoil-Fin Channels

A considerable number of studies have focused on geometric optimization. Similarly to the aforementioned investigations, such optimization can be categorized into two main directions: improvements to the fin geometry itself and modifications to the channel arrangement. Both of these are implemented with the aim of enhancing heat-transfer performance or improving flow characteristics. The geometry optimization studies of airfoil-fin channels in PCHEs are summarized in Table 3.

(a) Strategies for Reducing Flow Resistance.

Leading-Edge Modification: Several studies have emphasized that the sharp curvature at the leading edge of conventional airfoil fins intensifies adverse pressure gradients and increases flow resistance [129]. To mitigate this, Xu et al. [124] introduced a swordfish-type fin, as shown in Figure 19a [124], which reduced heat-transfer performance by about 10.4% but lowered flow resistance by 22.8%, thereby improving overall performance [130,131]. Further investigations under supercritical CO₂ and molten salt conditions confirmed simultaneous reductions in both the Colburn j factor and f, verifying the suitability of this design for low-resistance applications [132,133].

Slotted Designs:

In addition to leading-edge modifications, Ma et al. [134] proposed slotted airfoil geometries that maintain comparable heat-transfer performance while reducing the pressure drop by approximately 15%, demonstrating an effective means of lowering flow resistance without sacrificing thermal efficiency.

(b) Strategies for Enhancing Heat Transfer.

Conventional Geometry Alternatives: Some studies have investigated the use of regular-shaped fins as alternatives to conventional airfoil fins [129]. Xu et al. [126] numerically examined rectangular, rounded rectangular, elliptical, and airfoil fins with sCO₂, and found that elliptical fins achieved the highest Nu at high Re, whereas airfoil fins exhibited superior overall thermal–hydraulic performance.

Modified Airfoil Geometries: To suppress boundary layer thickening at the trailing edge and enhance local convective heat transfer, Cui et al. [118] proposed modified airfoil structures, Fin-II and Fin-III, as illustrated in Figure 19b. Wu et al. [122] further demonstrated, through analyses based on entropy generation and field synergy principles, that these designs improved the uniformity of the heat-transfer coefficient distribution and effectively reduced local irreversibility.

Vortex-Generating and 3D Designs: To enhance turbulence and promote convective mixing, Wu et al. [135] proposed a crossed airfoil fin configuration that generates longitudinal vortices, as shown in Figure 19c. Numerical results indicated that the Nu increased by about 25–30% compared with conventional airfoils. Building on this concept, Wang et al. [136] developed a three-dimensional airfoil fin structure using CFD-based optimization, which reduced local flow resistance caused by fluid impingement and improved overall heat-transfer capacity by approximately 25–29% relative to cylindrical and conventional airfoil fins.

Bionic Designs: Yang et al. [137] introduced a leaf-vein bionic channel structure, as illustrated in Figure 19d. Experimental and numerical studies demonstrated that this design improved the overall thermal–hydraulic performance by approximately 17%, providing a novel direction for the development of efficient and compact heat exchangers.

Figure 19. Geometry optimization of airfoil fin channels: (a) schematic diagram of a swordfish-type fin [124]; (b) sketch of modified fin geometries [118]; (c) 3D flow diagram of crossed airfoil fin channels [135]; (d) model diagram of leaf-vein-based bionic channel experimental plate [137].

Figure 19. Geometry optimization of airfoil fin channels: (a) schematic diagram of a swordfish-type fin [124]; (b) sketch of modified fin geometries [118]; (c) 3D flow diagram of crossed airfoil fin channels [135]; (d) model diagram of leaf-vein-based bionic channel experimental plate [137].

(c) Optimizing Fin Layout.

Non-Uniform Distribution: Several studies have highlighted that non-uniform distributions can improve thermal–hydraulic performance compared with uniform layouts. Han [138] examined two types of non-uniform arrangements: fins densely packed in the front and sparsely packed in the rear, and the opposite configuration. The dense front– sparse back design exhibited the best overall performance. Building on this concept, Xi et al. [139] proposed an interleaved distribution of NACA 0025 airfoil fins along the channel height, which maintained comparable heat-transfer performance to traditional layouts while reducing fin volume by 50%, thereby enabling a more lightweight PCHE design.

Additional Disturbance Structures: Other researchers have incorporated small geometric features into fin channels to induce local disturbances. Tang et al. [140], Yang et al. [141], and Zhang et al. [142] introduced miniature vortex generators between adjacent fins to enhance fluid mixing and strengthen heat transfer. Shi et al. [143] placed dimples downstream of airfoils, generating recirculation zones and lateral vortices that improved heat transfer. Importantly, these modifications had little effect on the mainstream flow, ensuring enhanced heat transfer without significant additional resistance.

Table 3. Geometry optimization of airfoil fin channels in PCHEs.

Reference	Geometry Modification	Geometrical Parameter	Optimization Target	Mechanism	Performance Improvement
Xu et al. [126]	Rounded rectangular fin	Fin arrangement: La = 6 mm, Lb = 3.5 mm, Ls = 4 mm; Fin geometry: Lf = 4 mm, Wf = 1 mm	Heat transfer enhancement	High-velocity and negative-pressure regions form near fin edges or maximum thickness, inducing strong flow disturbances	Nusselt number improved by 24%, pressure drop increased by 173%
	Elliptical fin		Heat transfer enhancement		Nusselt number improved by 12%, pressure drop increased by 83%
Wu et al. [130]	Outer convex edges replaced by inner concave edges	Fin arrangement: La = 12 mm, Lb = 4 mm, Ls = 15 mm; Fin geometry: Lf = 6 mm, Wf = 1.2 mm, Hf = 0.8 mm	Heat transfer enhancement and reduce pressure drop	The concave trailing-edge geometry improves fluid attachment and reduces flow separation, which lowers friction and enhances thermal–hydraulic performance	Colburn j factor improved by up to 8%, friction factor reduced by up to 10%
Cui et al. [118]		Fin arrangement: La = 12 mm, Lb = 4.2 mm, Ls = 6 mm, Fin geometry: Lf = 6 mm, Wf = 1.2 mm
	Maximal circle moved to midline; leading and trailing edges modified	Fin arrangement: La = 12 mm, Lb = 4.2 mm, Ls = 6 mm, Fin geometry: Lf = 6 mm, Wf = 1.2 mm, Hf = 0.95 mm	Reduce pressure drop	The smooth trailing-edge geometry reduces flow separation and resistance	Friction factor reduced by up to 30%, while the Colburn j factor decreased by only 8%
Wu et al. [135]	Crossed airfoil fins	Fin arrangement: La = 12 mm, Lb = 3.6 mm, Ls = 6 mm, cross angle: θ = 30–150° Fin geometry: Lf = 6 mm, Wf = 1 mm, Hf = 0.5 mm	Heat transfer enhancement	Generated longitudinal vortices, disrupted boundary layers	Thermal performance coefficient improved by 10.6–17.1%
Wang et al. [136]	Three-dimensional optimized fins	Fin arrangement: La = 6 mm, Lb = 3.5 mm, Ls = 0 mm; Fin geometry: Baseline cylindrical fin diameter D = 1 mm, height H = 1 mm;	Reduce pressure drop	Reduced fluid–solid contact area, effectively lowering local flow resistance caused by high velocity gradients	Friction factor reduced by 23–41%
Yang et al. [137]	Leaf-vein bionic airfoil fin channel	Fin arrangement: La = 8 mm, Lb = 3 mm, Ls = 4 mm; Fin geometry: Lf = 3 mm, Wf = 1 mm, Hf = 1.4 mm Primary vein geometry: width WM = 2 mm; height HV = 1.4 mm; angle between main and secondary veins A = 45°; starting position 58 mm from inlet; length = 128 mm; spacing between secondary veins = 40 mm	Heat transfer enhancement	Leaf veins promoted the formation of vortex structures, which in turn enhanced the heat transfer process between fluids	Heat transfer capacity increased by 16–31%
Xu et al. [124]	Swordfish-type fin	Fin arrangement: La = 10 mm, Lb = 3.5 mm, Ls = 4 mm; Fin geometry: Lf = 4 mm, Wf = 1 mm, Hf = 1 mm	Reduce pressure drop	The blunted leading edge of swordfish-type fins reduced adverse pressure gradients and flow separation, thereby lowering flow resistance	Flow resistance was reduced by ~22.8%, while heat-transfer performance decreased by ~10.4%
Wu et al. [130]		Fin arrangement: La = 16 mm, Lb = 3.6 mm, Ls = 8 mm; Fin geometry: Lf = 6 mm, Wf = 1.8 mm, Hf = 0.8 mm			Heat-transfer performance reduced by ~10–20%, while flow resistance reduced by ~15–25%
Chu et al. [131]		Fin arrangement: La = 6 mm, Lb = 3.15 mm, Ls = 3 mm; Fin geometry: Lf = 4 mm, Wf = 1 mm, Hf = 0.88 mm			Nusselt number reduced by ~5–10%, while pressure drop reduced by ~10–20%
Yang et al. [132,133]		Fin arrangement: La = 8 mm, Lb = 3 mm, Ls = 4 mm; Fin geometry: Lf = 4 mm, Wf = 0.5 mm, Hf = 1.5 mm			Nusselt number reduced by ~9–12% and friction factor reduced by ~25–29%
Ma et al. [134]	A central groove was introduced into the airfoil fins	Fin arrangement: La = 12 mm, Lb = 2.4 mm, Ls = 6 mm; Fin geometry: Lf = 6 mm, Wf = 1.2 mm, Hf = 0.95 mm Groove thickness Ds = 0–0.3 mm	Heat transfer enhancement or reduce pressure drop	The groove structure increases the heat transfer area and enhances thermal performance at moderate thickness; meanwhile, it weakens trailing-edge flow separation and reduces local resistance, causing the hydraulic performance to first deteriorate and then improve with increasing groove thickness	At Ds = 0.05 mm: Nusselt number increased by ~14–15%, while friction factor increased by ~9–12%
					At Ds = 0.20 mm: Nusselt number reduced by ~1–2%, while friction factor reduced by ~15–20%
Han [138]	Non-uniform fin distribution (dense front–sparse back; sparse front–dense back)	Fin arrangement: La = 12 mm, Lb = 4 mm, Ls = 6 mm; Fin geometry: Lf = 6 mm, Wf = 1.2 mm, Hf = 0.8 mm Dense in front and sparse in back: front section pitch Lh3 = 4 mm, middle sparse section Lh2 = 20 mm, rear sparse section Lh4 = 15 mm Sparse in front and dense in back: front sparse section Lh4 = 15 mm, rear dense section Lh3 = 4 mm, middle sparse section Lh2 = 20 mm	Heat transfer enhancement	The performance difference is closely related to the thermal–physical properties of supercritical CO2, and the distribution of fins further regulates the balance between heat transfer and flow resistance	Nusselt number increased by ~5–7%
Xi et al. [139]	Introduced staggered arrangement in the height direction	Fin arrangement: La = 8 mm, Lb = 4 mm, Ls = 6 mm; Fin geometry: Lf = 4 mm, Wf = 1.25 mm, Hf = 0.6 mm staggered in the vertical direction: Lhs = 6 mm;	Lightweight structure design	Staggering fins vertically and varying fin height changed the channel cross-sectional area, which in turn affected flow velocity, vortex formation, and heat transfer	Reduced fin volume by up to 50% while maintaining comparable thermal–hydraulic performance
Tang et al. [140]	Vortex generators between fins	Fin arrangement: La = 8 mm, Lb = 1 mm, Ls = 4 mm; Fin geometry: Lf = 4 mm, Wf = 0.96 mm, Hf = 1.0 mm Delta vortex generator attack angle: 45° length: 0.5 mm, thickness: 0.05 mm, height: 0.8 mm	Heat transfer enhancement	Vortex generators produce longitudinal vortices, inducing up-wash and down-wash flows that disrupt the boundary layer and enhance mixing	Nusselt number increased by ~25–72%, while the friction factor increased by ~120–130%
Yang et al. [141]		Fin arrangement: La = 10 mm, Lb = 3 mm, Ls = 5 mm; Fin geometry: Lf = 5 mm, Wf = 1 mm, Hf = 1.5 mm Shuttle fins vortex generator: Length: 1.0 mm Width: 0.3 mm, Height: 1.5 mm Oval fins vortex generator: Long axis: 0.5 mm Short axis: 0.25 mm, Height: 1.5 mm			Shuttle fins vortex generator: Nusselt number increased by 6.7–26%, friction factor increased by 8.3–18.6%
					Oval fins vortex generator, Nusselt number increased by 9–27.3% while friction factor increased by 26.6–43.4%
Zhang et al. [142]		Fin arrangement: La = 8 mm, Lb = 3 mm, Ls = 4 mm; Fin geometry: Lf = 4 mm, Wf = 0.96 mm, Hf = 0.5 mm Barchan dune vortex generators: Width: 1.83 mm, Height: 0.5 mm, Angle: 45°			Nusselt number increased by 23.9–26.1%
Shi et al. [143]	Introduced dimples on airfoil fin PCHE channels	Fin arrangement: La = 6 mm, Lb = 3.5 mm, Ls = 0 mm; Fin geometry: Lf = 4 mm, Wf = 1 mm, Hf = 1 mm Dimple structure: Dimple diameter = 1.0 mm, depth = 0.2 mm	Heat transfer enhancement	Dimples generate strong recirculation and lateral vortices; the recirculating flow enhances fluid mixing near the wall, while the lateral vortices create up-wash and down-wash motions that intensify local heat transfer	Overall performance improved by up to 8.7%

Table 3. Geometry optimization of airfoil fin channels in PCHEs.

Reference	Geometry Modification	Geometrical Parameter	Optimization Target	Mechanism	Performance Improvement
Xu et al. [126]	Rounded rectangular fin	Fin arrangement: La = 6 mm, Lb = 3.5 mm, Ls = 4 mm; Fin geometry: Lf = 4 mm, Wf = 1 mm	Heat transfer enhancement	High-velocity and negative-pressure regions form near fin edges or maximum thickness, inducing strong flow disturbances	Nusselt number improved by 24%, pressure drop increased by 173%
	Elliptical fin		Heat transfer enhancement		Nusselt number improved by 12%, pressure drop increased by 83%
Wu et al. [130]	Outer convex edges replaced by inner concave edges	Fin arrangement: La = 12 mm, Lb = 4 mm, Ls = 15 mm; Fin geometry: Lf = 6 mm, Wf = 1.2 mm, Hf = 0.8 mm	Heat transfer enhancement and reduce pressure drop	The concave trailing-edge geometry improves fluid attachment and reduces flow separation, which lowers friction and enhances thermal–hydraulic performance	Colburn j factor improved by up to 8%, friction factor reduced by up to 10%
Cui et al. [118]		Fin arrangement: La = 12 mm, Lb = 4.2 mm, Ls = 6 mm, Fin geometry: Lf = 6 mm, Wf = 1.2 mm
	Maximal circle moved to midline; leading and trailing edges modified	Fin arrangement: La = 12 mm, Lb = 4.2 mm, Ls = 6 mm, Fin geometry: Lf = 6 mm, Wf = 1.2 mm, Hf = 0.95 mm	Reduce pressure drop	The smooth trailing-edge geometry reduces flow separation and resistance	Friction factor reduced by up to 30%, while the Colburn j factor decreased by only 8%
Wu et al. [135]	Crossed airfoil fins	Fin arrangement: La = 12 mm, Lb = 3.6 mm, Ls = 6 mm, cross angle: θ = 30–150° Fin geometry: Lf = 6 mm, Wf = 1 mm, Hf = 0.5 mm	Heat transfer enhancement	Generated longitudinal vortices, disrupted boundary layers	Thermal performance coefficient improved by 10.6–17.1%
Wang et al. [136]	Three-dimensional optimized fins	Fin arrangement: La = 6 mm, Lb = 3.5 mm, Ls = 0 mm; Fin geometry: Baseline cylindrical fin diameter D = 1 mm, height H = 1 mm;	Reduce pressure drop	Reduced fluid–solid contact area, effectively lowering local flow resistance caused by high velocity gradients	Friction factor reduced by 23–41%
Yang et al. [137]	Leaf-vein bionic airfoil fin channel	Fin arrangement: La = 8 mm, Lb = 3 mm, Ls = 4 mm; Fin geometry: Lf = 3 mm, Wf = 1 mm, Hf = 1.4 mm Primary vein geometry: width WM = 2 mm; height HV = 1.4 mm; angle between main and secondary veins A = 45°; starting position 58 mm from inlet; length = 128 mm; spacing between secondary veins = 40 mm	Heat transfer enhancement	Leaf veins promoted the formation of vortex structures, which in turn enhanced the heat transfer process between fluids	Heat transfer capacity increased by 16–31%
Xu et al. [124]	Swordfish-type fin	Fin arrangement: La = 10 mm, Lb = 3.5 mm, Ls = 4 mm; Fin geometry: Lf = 4 mm, Wf = 1 mm, Hf = 1 mm	Reduce pressure drop	The blunted leading edge of swordfish-type fins reduced adverse pressure gradients and flow separation, thereby lowering flow resistance	Flow resistance was reduced by ~22.8%, while heat-transfer performance decreased by ~10.4%
Wu et al. [130]		Fin arrangement: La = 16 mm, Lb = 3.6 mm, Ls = 8 mm; Fin geometry: Lf = 6 mm, Wf = 1.8 mm, Hf = 0.8 mm			Heat-transfer performance reduced by ~10–20%, while flow resistance reduced by ~15–25%
Chu et al. [131]		Fin arrangement: La = 6 mm, Lb = 3.15 mm, Ls = 3 mm; Fin geometry: Lf = 4 mm, Wf = 1 mm, Hf = 0.88 mm			Nusselt number reduced by ~5–10%, while pressure drop reduced by ~10–20%
Yang et al. [132,133]		Fin arrangement: La = 8 mm, Lb = 3 mm, Ls = 4 mm; Fin geometry: Lf = 4 mm, Wf = 0.5 mm, Hf = 1.5 mm			Nusselt number reduced by ~9–12% and friction factor reduced by ~25–29%
Ma et al. [134]	A central groove was introduced into the airfoil fins	Fin arrangement: La = 12 mm, Lb = 2.4 mm, Ls = 6 mm; Fin geometry: Lf = 6 mm, Wf = 1.2 mm, Hf = 0.95 mm Groove thickness Ds = 0–0.3 mm	Heat transfer enhancement or reduce pressure drop	The groove structure increases the heat transfer area and enhances thermal performance at moderate thickness; meanwhile, it weakens trailing-edge flow separation and reduces local resistance, causing the hydraulic performance to first deteriorate and then improve with increasing groove thickness	At Ds = 0.05 mm: Nusselt number increased by ~14–15%, while friction factor increased by ~9–12%
					At Ds = 0.20 mm: Nusselt number reduced by ~1–2%, while friction factor reduced by ~15–20%
Han [138]	Non-uniform fin distribution (dense front–sparse back; sparse front–dense back)	Fin arrangement: La = 12 mm, Lb = 4 mm, Ls = 6 mm; Fin geometry: Lf = 6 mm, Wf = 1.2 mm, Hf = 0.8 mm Dense in front and sparse in back: front section pitch Lh3 = 4 mm, middle sparse section Lh2 = 20 mm, rear sparse section Lh4 = 15 mm Sparse in front and dense in back: front sparse section Lh4 = 15 mm, rear dense section Lh3 = 4 mm, middle sparse section Lh2 = 20 mm	Heat transfer enhancement	The performance difference is closely related to the thermal–physical properties of supercritical CO2, and the distribution of fins further regulates the balance between heat transfer and flow resistance	Nusselt number increased by ~5–7%
Xi et al. [139]	Introduced staggered arrangement in the height direction	Fin arrangement: La = 8 mm, Lb = 4 mm, Ls = 6 mm; Fin geometry: Lf = 4 mm, Wf = 1.25 mm, Hf = 0.6 mm staggered in the vertical direction: Lhs = 6 mm;	Lightweight structure design	Staggering fins vertically and varying fin height changed the channel cross-sectional area, which in turn affected flow velocity, vortex formation, and heat transfer	Reduced fin volume by up to 50% while maintaining comparable thermal–hydraulic performance
Tang et al. [140]	Vortex generators between fins	Fin arrangement: La = 8 mm, Lb = 1 mm, Ls = 4 mm; Fin geometry: Lf = 4 mm, Wf = 0.96 mm, Hf = 1.0 mm Delta vortex generator attack angle: 45° length: 0.5 mm, thickness: 0.05 mm, height: 0.8 mm	Heat transfer enhancement	Vortex generators produce longitudinal vortices, inducing up-wash and down-wash flows that disrupt the boundary layer and enhance mixing	Nusselt number increased by ~25–72%, while the friction factor increased by ~120–130%
Yang et al. [141]		Fin arrangement: La = 10 mm, Lb = 3 mm, Ls = 5 mm; Fin geometry: Lf = 5 mm, Wf = 1 mm, Hf = 1.5 mm Shuttle fins vortex generator: Length: 1.0 mm Width: 0.3 mm, Height: 1.5 mm Oval fins vortex generator: Long axis: 0.5 mm Short axis: 0.25 mm, Height: 1.5 mm			Shuttle fins vortex generator: Nusselt number increased by 6.7–26%, friction factor increased by 8.3–18.6%
					Oval fins vortex generator, Nusselt number increased by 9–27.3% while friction factor increased by 26.6–43.4%
Zhang et al. [142]		Fin arrangement: La = 8 mm, Lb = 3 mm, Ls = 4 mm; Fin geometry: Lf = 4 mm, Wf = 0.96 mm, Hf = 0.5 mm Barchan dune vortex generators: Width: 1.83 mm, Height: 0.5 mm, Angle: 45°			Nusselt number increased by 23.9–26.1%
Shi et al. [143]	Introduced dimples on airfoil fin PCHE channels	Fin arrangement: La = 6 mm, Lb = 3.5 mm, Ls = 0 mm; Fin geometry: Lf = 4 mm, Wf = 1 mm, Hf = 1 mm Dimple structure: Dimple diameter = 1.0 mm, depth = 0.2 mm	Heat transfer enhancement	Dimples generate strong recirculation and lateral vortices; the recirculating flow enhances fluid mixing near the wall, while the lateral vortices create up-wash and down-wash motions that intensify local heat transfer	Overall performance improved by up to 8.7%

3.4.5. Applications of Airfoil-Fin Channels with Different Working Fluids

Airfoil fin PCHEs are less prone to blockage due to their discontinuous fin channels, making them particularly suitable for high-viscosity fluids. This characteristic suggests promising applications in petroleum processing [144] and molten salt heat transfer [31], where clogging and fouling are critical concerns. By maintaining stable operation under such demanding conditions, airfoil fin PCHEs provide advantages over conventional channel configurations.

The streamlined geometry of airfoil fins contributes to stable flow fields and reduced pressure drops, which is especially beneficial for supercritical fluid applications. Tang et al. [11] studied LNG under rolling motion and demonstrated that dynamic operation enhances heat transfer but also increases hydraulic resistance. Chang et al. [145] evaluated supercritical CO₂ and introduced the dimensionless parameter T and the average thermal-resistance ratio, finding optimal thermal performance near T ≈ 1.03. Li et al. [146] further proposed a hybrid airfoil fin design (Figure 20) that accommodates variations in specific heat, density, and other sCO₂ properties along the flow direction, achieving a 2.8–13.2% improvement in overall performance compared with conventional channels.

In summary, the thermal–hydraulic performance of airfoil fin PCHEs is primarily governed by fin geometry and arrangement, with fin thickness and longitudinal pitch identified as dominant factors. Optimization has focused on novel fin shapes, bio-inspired structures, and innovative layouts to either enhance heat transfer or reduce flow resistance. Benefiting from discontinuous channels that resist blockage and streamlined geometries that lower pressure drop, airfoil fin PCHEs show strong potential for high-viscosity fluids and supercritical applications, highlighting their broad prospects in advanced energy systems.

However, research on surface roughness effects remains limited, with insufficient numerical and experimental validation to quantify its influence on boundary layer behavior and overall thermal–hydraulic performance. In addition, since the surrounding metal must be completely removed to form open flow passages during the photochemical etching process, discontinuous airfoil fin channels exhibit lower material utilization than continuous zigzag and straight channels, resulting in significant material loss and higher fabrication cost. These limitations remain major obstacles to large-scale industrial application and call for further optimization of the manufacturing process.

3.5. Influence of Header Configuration on the Performance of PCHEs

Beyond the heat transfer core, composed of microchannels, the headers of PCHEs play a decisive role in overall performance due to their direct influence on flow distribution. This section reviews recent studies on header and manifold design, focusing on the impact on flow distribution uniformity.

The schematic diagram of a PCHE is shown in Figure 21 [147], where the headers, composed of end caps and manifolds, are considered critical for determining the uniformity of flow distribution among the channels. Baek et al. [148] demonstrated that flow maldistribution can emerge at the very onset of operation and, if not addressed through optimized design, will persist throughout service, thereby continuously impairing heat-transfer performance.

To improve flow distribution uniformity, various studies have proposed optimizations to header configurations, as follows.

Streamlined Header Geometries: Chu et al. [149] developed four header configurations (Figure 22). Among these, the streamlined MHIH design markedly improved flow uniformity and thermal–hydraulic performance in straight-channel PCHEs, offering an effective optimization strategy under high-pressure conditions. Lee et al. [150] further investigated the effects of inlet plenum inclination angle, curvature radius, and inlet pipe diameter, showing that curvature radius was the dominant factor. Optimal flow uniformity was achieved at an inclination angle of 30°, while a larger curvature radius enhanced uniformity but reduced frictional performance.

Perforated Baffles: Wang et al. [151] introduced perforated baffles with varying layouts and aperture sizes into inlet manifolds. This approach reduced the maldistribution coefficient to 0.7, while simultaneously enhancing heat transfer and significantly lowering pressure drop, thereby providing a practical solution for improving inlet flow distribution.

Integrated Manifold Designs: Pasquier et al. [152] proposed multiple manifold configurations for zigzag PCHEs, and found that a single-layer merged-channel-integrated manifold reduced the maldistribution coefficient by 91% and enhanced heat transfer, though this comes at the cost of a considerable pressure drop. Jin et al. [153] numerically compared straight, wavy, and airfoil fin PCHEs under free and ribbed manifold conditions and proposed improved designs incorporating bent and airfoil fins. These reduced maldistribution by 39.4% and 61.8% compared with free and ribbed manifolds, respectively, while raising overall performance by 5% and 8.5%. Son et al. [154] further developed a PCHE analysis code accounting for crossflow and parallel-flow paths in the manifold region, demonstrating that accurate modeling of inlet flow paths is critical for predicting heat transfer and pressure drop, and for improving the reliability of manifold design.

In summary, flow maldistribution in PCHE headers can severely impair heat transfer, making header optimization a critical focus. Proposed strategies include streamlined plenum geometries, perforated baffles, inclination and curvature adjustments, integrated merged-channel manifolds, and improved designs with bent or airfoil fins. These approaches have achieved notable reductions in maldistribution coefficients and, in many cases, simultaneous improvements in heat transfer and pressure drop. Furthermore, numerical tools that model realistic manifold flow paths are essential for accurate performance prediction and reliable design.

3.6. Flow and Heat Transfer Correlations for PCHEs

During the past two decades, a large number of correlations have been proposed for PCHEs with different channel configurations, including straight, zigzag, and airfoil fin channels. In terms of development, the correlations can generally be divided into three stages.

Initial Stage: Correlations were largely derived from classical expressions for smooth tubes or conventional heat exchangers (e.g., Dittus–Boelter, Colburn j factor), with the correlations further developed by modifying the constant terms to account for the distinctive features of PCHEs.

Correction Stage:

(1)

Property correction: For working conditions where thermal–physical properties vary significantly with temperature, additional correction factors such as density and viscosity ratios were incorporated alongside Re and Pr, enabling a more accurate description of non-isothermal flow and near-critical effects.

(2)

Geometric correction: With an improved understanding of the influence of geometric configuration, dimensionless parameters representing these geometrical features were gradually incorporated into the correlations to enhance applicability.

Despite the progress, the currently available correlations still exhibit several limitations. The majority are restricted to specific geometries or narrow Reynolds number ranges, which constrains extrapolation capability. The diversity of mathematical forms and the absence of a systematic framework also hinder direct application in engineering practice. Moreover, emphasis has often been placed on fitting accuracy, whereas the underlying flow and heat transfer mechanisms have received less attention. Consequently, a systematic review and comparison of existing correlations is required to clarify developmental trajectories and applicable ranges, while identifying current shortcomings and potential directions for improvement. Such efforts are expected to provide both a clearer understanding of previous achievements and valuable guidance for the future development of new correlations.

3.6.1. Flow and Heat Transfer Correlations for PCHEs with Straight Channels

The straight channel represents the most fundamental configuration of PCHEs. Compared with zigzag and airfoil fin channels, the straight channel has a simpler structure, and the number of geometric variables influencing flow and heat-transfer performance is relatively limited. As a result, improvements in the formulation of flow and heat transfer correlations have primarily been achieved through the introduction of property-based correction factors for specific working fluid. In this section, the development of flow and heat transfer correlations for straight-channel PCHEs is systematically reviewed and the limitations of existing correlations are highlighted. The flow and heat transfer correlations for straight channels are summarized in

Table 4. Summary of heat transfer and friction factor correlations for PCHEs with straight channels.

Reference	Method	Working Fluid	Geometry	Correlations		Application Range
Xu et al. [155]	EXP.	sCO2	Semicircular D = 2 mm	$Nu=3.564\times {10}^{-3}R{e}^{1.035}P{r}^{0.0341}$	(1)	$9.29\times {10}^2\le R\mathrm{e}\le 3.286\times {10}^3$ $1.9\le Pr\le 8.6$
				$f=2.403\times {10}^{3}R{e}^{-1.399}$	(2)	$1.354\times {10}^3\le R\mathrm{e}\le 3.224\times {10}^3$
Chu et al. [29]	EXP.	sCO2	Semicircular D = 1.4 mm	$\begin{array}{c}Nu=\\ 0.0183R{e}^{0.82}P{r}^{0.5}{\left({\displaystyle \frac{{\rho }_b}{{\rho }_w}}\right)}^{-0.3}\left[0.58-53{\left({\displaystyle \frac{Gr}{R{e}^{2.7}}}\right)}^{0.36}\right]\end{array}$	(3)	$3\times {10}^4<Re<6\times {10}^4$ $0.95<$$\mathrm{Pr}*<$4.29 Gr: Grashof number ${\displaystyle \frac{{\rho }_b}{{\rho }_w}}$: bulk-to-wall density ratio
				$\begin{array}{c}Nu=\\ 0.0183R{e}^{0.82}P{r}^{0.5}{\left({\displaystyle \frac{{\rho }_b}{{\rho }_w}}\right)}^{-0.3}\left[0.36-22{\left({\displaystyle \frac{Gr}{R{e}^{2.7}}}\right)}^{0.42}\right]\end{array}$	(4)	$3\times {10}^4<Re<7\times {10}^4$
Liu et al. [156]	EXP.	sCO2	Semicircular D = 1.87 mm	$Nu=0.1229R{e}^{0.6021}P{r}^{0.3}{\left({\displaystyle \frac{c_{pw}}{c_{pb}}}\right)}^{0.1310}$	(5)	$3.6\times {10}^3<Re<3.65\times {10}^4$ $0.88<$$\mathrm{Pr}*<$8.31 ${\displaystyle \frac{c_{pw}}{c_{pb}}}$: wall-to-bulk specific heat ratio
				$f=0.05776R{e}^{-0.2192}$	(6)
Li et al. [81]	EXP.+CFD	sCO2	Semicircular D = 1.17 mm	$\mathrm{H}\mathrm{o}\mathrm{t}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}:$ $N{u}_b=0.0409R{e}_b^{0.681}P{r}_b^{0.650}{\left({\displaystyle \frac{{\rho }_w}{{\rho }_b}}\right)}^{0.223}{\left({\displaystyle \frac{\overline{c_p}}{c_{p,b}}}\right)}^{0.397}$	(7)	/ The corresponding application range is not provided in the referenced study.
				$\mathrm{C}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}:$ $N{u}_b=0.0781R{e}_{\mathrm{b}}^{0.662}P{r}_{\mathrm{b}}^{0.455}{\left({\displaystyle \frac{{\rho }_{\mathrm{w}}}{{\rho }_{\mathrm{b}}}}\right)}^{0.392}{\left({\displaystyle \frac{\overline{c_p}}{c_{p,\mathrm{b}}}}\right)}^{0.334}$	(8)
Liu et al. [82]	EXP.+CFD	sCO2	Semicircular D = 2 mm	$Nu=0.0112R{e}^{0.9285}P{r}^{0.6}{\left({\displaystyle \frac{c_{pb}}{c_{pw}}}\right)}^{-0.1}{\left({\displaystyle \frac{{\rho }_b}{{\rho }_w}}\right)}^{0.713}$	(9)	$3\times {10}^3<Re<3\times {10}^4$ $1.0<$$\mathrm{Pr}*<$7.2
				$f=(4.2\mathrm{l}\mathrm{g} Re-5.64{)}^{-2}$	(10)
Chen et al. [90]	EXP.+CFD	He	Semicircular D = 2 mm	$Nu=\left\{\begin{array}{c}(0.01352\pm 0.0094)R{e}^{(0.80058\pm 0.0921)}\\ (3.6361\times {10}^{-4}\pm 7.855\times {10}^{-5})R{e}^{(1.2804\pm 0.0273)}\end{array}\right.$	(11)	$1.2\times {10}^3\le \mathrm{R}\mathrm{e}\le 1.85\times {10}^3$ $1.85\times {10}^3<\mathrm{R}\mathrm{e}\le 2.9\times {10}^3$ $0.64<$$\mathrm{Pr}*<$0.66
				$\mathrm{N}\mathrm{u}=$ $\left\{\begin{array}{c}(0.047516\pm 0.015662){\mathrm{R}\mathrm{e}}^{(0.633151\pm 0.044606)}\\ (3.68\times {10}^{-4}\pm 1.184389\times {10}^{-4}){\mathrm{R}\mathrm{e}}^{(1.282182\pm 0.0421)}\end{array}\right.$	(12)	$1.2\times {10}^3\le \mathrm{R}\mathrm{e}\le 1.85\times {10}^3$ $0.64<$$\mathrm{Pr}*<$0.66 $1.85\times {10}^3<\mathrm{R}\mathrm{e}\le 2.9\times {10}^3$
Meshram et al. [157]	CFD	sCO2	Semicircular D = 2 mm	$$\begin{gathered} \mathrm{Hot} \mathrm{side} (500 \mathrm{K}<\mathrm{T}<630 \mathrm{K}): \\ Nu=0.0493R{e}^{0.77}P{r}^{0.55} \end{gathered}$$	(13)	$4.4\times {10}^3\le \mathrm{R}\mathrm{e}\le 2\times {10}^4$ Pr = 0.76
				$f=0.8386R{e}^{-0.5985}+0.00295$	(14)	$4.94\times {10}^3\le \mathrm{R}\mathrm{e}\le 2\times {10}^4$
				Hot side (600 K < T < 730 K): $Nu=0.0514R{e}^{0.76}P{r}^{0.55}$	(15)	$5.1\times {10}^3\le \mathrm{R}\mathrm{e}\le 2.48\times {10}^4$ Pr = 0.74
				$f=0.8385R{e}^{-0.5978}+0.00331$	(16)	$5.04\times {10}^3\le \mathrm{R}\mathrm{e}\le 2.51\times {10}^4$
				Cold side (400 K < T < 500 K): $Nu=0.0718R{e}^{0.71}P{r}^{0.55}$	(17)	$4.94\times {10}^3\le \mathrm{R}\mathrm{e}\le 2.47\times {10}^4$ Pr = 1.08
				$f=0.8657R{e}^{-0.5755}+0.00405$	(18)	$4.43\times {10}^3\le \mathrm{R}\mathrm{e}\le 2.48\times {10}^4$
				Cold side (500 K < T < 600 K): $Nu=0.0661R{e}^{0.743}P{r}^{0.55}$	(19)	$5.02\times {10}^3\le \mathrm{R}\mathrm{e}\le 2.53\times {10}^4$ Pr = 0.83
				$f=0.8796R{e}^{-0.5705}+0.00353$	(20)	$5.19\times {10}^3\le \mathrm{R}\mathrm{e}\le 2.58\times {10}^4$
			Semicircular D = 1.2∼2.6 mm	Hot side (500 K < T < 630 K): $Nu=0.0685R{e}^{0.705}{\left({\displaystyle \frac{D}{2}}\right)}^{-0.122}$	(21)	$8.5\times {10}^3\le \mathrm{R}\mathrm{e}\le 2.05\times {10}^4$ $0.76<$$\mathrm{Pr}*<$1.08
				$f=0.0648R{e}^{-0.254}{\left({\displaystyle \frac{D}{2}}\right)}^{-0.0411}$	(22)
				Cold side (400 K < T < 500 K): $Nu=0.0117R{e}^{0.843}{\left({\displaystyle \frac{D}{2}}\right)}^{0.0405}$	(23)
				$f=0.0759R{e}^{-0.241}{\left({\displaystyle \frac{D}{2}}\right)}^{0.089}$	(24)
Liu et al. [158]	CFD	sCO2	Semicircular D = 1.87 mm	$f=\left\{\begin{array}{c}16/Re\\ 4.40\times {10}^{-5}R{e}^{0.6542}\\ 0.0791R{e}^{-0.25}\end{array}\right.$	(25)	$Re<2300$ $2300\le R\mathrm{e}<3980$ $3980\le R\mathrm{e}<R{e}_{\mathrm{t}}$ $R{e}_{\mathrm{t}}$: threshold value of Reynolds number
Ren et al. [91]	CFD	sCO2	Semicircular D = 2.8 mm	$Nu=\left[1+9.5{\left({\displaystyle \frac{Gr}{R{e}^2}}\right)}^{0.958}\right]\cdot 0.01882R{e}_b^{0.82}P{r}^{0.5}{\left({\displaystyle \frac{{\rho }_b}{{\rho }_w}}\right)}^{-0.3}{\left({\displaystyle \frac{{\mu }_b}{{\mu }_w}}\right)}^{0.2887}$	(26)	$4.7\times {10}^3\le \mathrm{R}\mathrm{e}\le 7\times {10}^4$ $0.95<Pr<49,$ $4\times {10}^{-5}<{\displaystyle \frac{Gr}{R{e}^2}}<0.07$ ${\displaystyle \frac{{\mu }_b}{{\mu }_w}}$: bulk-to-wall viscosity radio
Chu et al. [149]	CFD	sCO2	Semicircular D = 2.12 mm	$Nu=0.0084{\sigma }^{0.31}R{e}_{in}^{0.85}$ $f=15.125{\sigma }^{0.16}R{e}_{in}^{-0.56}$	(27) (28)	/
Zhou et al. [159]	CFD	Molten salt	Semicircular D = 2 mm	$Nu=0.76148R{e}^{0.32091}P{r}^{0.19981}{\left({\displaystyle \frac{C{p}_w}{C{p}_b}}\right)}^{4.20945}{\left({\displaystyle \frac{{\lambda }_w}{{\lambda }_b}}\right)}^{0.14014}$	(29)	$3.09\times {10}^2\le \mathrm{R}\mathrm{e}\le 5.15\times {10}^2$ $4.27<$$\mathrm{Pr}*<$4.47
				$f=10.79585R{e}^{-0.92183}{\left({\displaystyle \frac{{\rho }_w}{{\rho }_b}}\right)}^{-1.52735}{\left({\displaystyle \frac{{\mu }_w}{{\mu }_b}}\right)}^{0.1991}$	(30)
		sCO2-based mixtures	Semicircular D = 2 mm	$\begin{array}{c}N{u}_c=0.02986R{e}^{0.77734}P{r}^{0.30678}{\left({\displaystyle \frac{{\rho }_w}{{\rho }_b}}\right)}^{0.34962}\\ {{\left({\displaystyle \frac{{\mu }_w}{{\mu }_b}}\right)}^{-0.5639}\left({\displaystyle \frac{C{p}_w}{C{p}_b}}\right)}^{0.347}{\left({\displaystyle \frac{{\lambda }_w}{{\lambda }_b}}\right)}^{-0.02617}\end{array}$	(31)	$1.38\times {10}^4\le \mathrm{R}\mathrm{e}\le 3.15\times {10}^4$ $0.95<$$\mathrm{Pr}*<$1.45 ${\displaystyle \frac{{\lambda }_w}{{\lambda }_b}}$: wall-to-bulk thermal conductivity
				$\begin{array}{c}{f}_c=0.09579R{e}^{-0.26093}{\left({\displaystyle \frac{{\rho }_w}{{\rho }_b}}\right)}^{-0.19607}{\left({\displaystyle \frac{{\mu }_w}{{\mu }_b}}\right)}^{0.22885}\\ {\left({\displaystyle \frac{C{p}_w}{C{p}_b}}\right)}^{-0.03687}{\left({\displaystyle \frac{{\lambda }_w}{{\lambda }_b}}\right)}^{0.09512}\end{array}$	(32)

Pr*: Prandtl number estimated under working conditions (based on inlet and outlet temperatures and pressures).

.

Most correlations have been developed for semicircular channel cross-sections, with supercritical CO₂ being the dominant working fluid, while only a limited number involve molten salts, water, or gas mixtures. The applicable Reynolds number range extends from laminar to turbulent regimes. However, it must be noted that the majority of the available correlations do not explicitly provide the applicability range with respect to the Prandtl number. To clearly present the Prandtl number range covered by heat transfer correlations for straight-channel PCHEs, Pr values were estimated from the working conditions of each correlation, including hot- and cold-side inlet temperatures, operating pressures, and reported thermophysical properties of the fluids. The existing research ranges for straight-channel PCHEs are shown in Figure 23, indicating that PCHEs essentially cover different flow regimes and a relatively wide range of Prandtl numbers. Nevertheless, it should be noted that only a limited number of correlations are available for Reynolds numbers below 1000, and future efforts should be directed toward developing correlations applicable to this regime.

From the perspective of formulation, a portion of the correlations were developed on the basis of the classical Dittus–Boelter and Blasius correlations, in which the constant terms were modified to account for the characteristics of PCHEs; this is also the case in the correlations proposed by Xu et al. [155] and Meshram et al. [157] derived from experimental and numerical data for sCO₂ (corresponding to Equations (1) and (2), and Equations (13)–(21)). Figure 24 presents a comparison of straight-channel heat transfer correlations derived from the Dittus–Boelter equation. It can be observed that the Reynolds number has the most significant influence on the Nusselt number—at a fixed channel diameter, larger Re values correspond to higher Nu. Moreover, at a given Re, Nu increases with Pr, indicating the important role of thermophysical properties in heat-transfer performance. In cases where the thermal–physical property variation in the working fluid was relatively small, the Prandtl number term was omitted during correlation development, such as in the correlations established by Chen et al. [90] based on experimental and numerical studies with water and helium (corresponding to Equations (11) and (12)).

For working fluids with significant variations in thermal–physical properties inside the channel, such as sCO₂, property-correction terms have been introduced in order to improve the predictive accuracy of flow and heat transfer correlations. The corrections are generally expressed as ratios of dimensionless quantities evaluated at the bulk temperature to those at the wall temperature. Liu et al. [156] introduced a dimensionless correction term for specific heat in the correlation presented in Equation (5). Building upon the inclusion of specific heat, Li et al. [81] and Liu et al. [82] further incorporated density correction terms in the correlations reported as Equations (7)–(9), respectively. A viscosity correction term was subsequently adopted in the correlation developed by Ren et al. [91], reported as Equation (26).

In conditions close to the critical point, the thermal–physical properties of supercritical fluids vary drastically with temperature, which induces large density gradients and generates strong buoyancy effects. To represent buoyancy influences, the non-dimensional parameter Gr/Reⁿ was incorporated into the correlations of Chu et al. [29], given in Equations (3) and (4), and Ren et al. [91], reported as Equation (26). In addition, Zhou et al. [159] developed correlations from numerical studies with molten salt on the hot side and sCO₂-based gas mixtures on the cold side, presented as Equations (29)–(32), where dimensionless correction terms for density, specific heat, thermal conductivity, and viscosity were simultaneously included to capture property variations with temperature in different fluids.

Furthermore, Meshram et al. [157] proposed correlations including the diameter-based correction terms in Equations (21)–(24). The correction terms introduced in this work were dimensional, which requires attention when assessing their applicability. Since Nusselt number and friction factor are defined in a non-dimensional form, the introduction of future correction terms should adopt non-dimensional geometric variables in order to ensure both mathematical and physical consistency and to enhance the general applicability of the correlations.

3.6.2. Flow and Heat Transfer Correlations for PCHEs with Zigzag Channels

Zigzag channels have been extensively investigated due to their significant enhancement of heat-transfer performance and relatively mature fabrication techniques, leading to the development of a large number of flow and heat transfer correlations. In this subsection, the existing correlations for zigzag channels are critically reviewed, the covered ranges are examined, and the influencing factors on prediction accuracy are discussed, with particular attention to correlations derived from the classical Dittus–Boelter equation. Furthermore, correlations incorporating property-based and geometry-based correction terms are surveyed, and the limitations of current research are identified. The summary of heat transfer and friction factor correlations for PCHEs with zigzag channels are shown in

Table 5. Summary of heat transfer and friction factor correlations for PCHEs with zigzag channels.

Reference	Method	Working Fluid	Geometry	Correlations		Application Range
Nikitin et al. [160]	EXP.	sCO2	Semicircular αhot = 32.5°, Dhot = 1.8 mm, Ph,hot = 9 mm αcold = 40°, Dcold = 1.9 mm, Ph,cold = 9 mm	${\mathit{h}}_{\mathrm{hot}}=2.52\mathit{R}{\mathit{e}}^{0.681}$	(33)	$2.8\times {10}^3\le {\mathit{Re}}_{\mathrm{hot}}\le 5.8\times {10}^3$
				$f_{\mathrm{P},\mathit{h}\mathit{o}\mathit{t}}=$ $\left(-1.402\times {10}^{-6}\pm 0.087\times {10}^{-6}\right)Re$ $+(0.04495\pm 0.00038)$	(34)
				${\mathit{h}}_{\mathrm{cold}}=5.49\mathit{R}{\mathit{e}}^{0.625}$	(35)	$6.2\times {10}^3\le {\mathit{Re}}_{\mathrm{cold}}\le 1.21\times {10}^4$
				$f_{\mathrm{P},\mathit{c}\mathit{o}\mathit{l}\mathit{d}}$ $=\left(-1.545\times {10}^{-6}\pm 0.099\times {10}^{-6}\right)Re$ $+(0.09318\pm 0.00090)$	(36)
Ngo et al. [65]	CFD	sCO2	Rectangular α = 32.5°, Dy = 1.31 mm, Dz = 0.94 mm, Ph = 7.565 mm	$Nu$ $=(0.1696\pm 0.0144)R{e}^{0.629\pm 0.009}P{r}^{0.317\pm 0.014}$	(37)	$3.5\times {10}^3<\mathit{Re}<2.2\times {10}^4,$ $0.75<\mathit{Pr}<2.2$
				$f=(0.1924\pm 0.0299)R{e}^{-0.091\pm 0.016}$	(38)
Kim, et al. [161]	EXP.	He	Semicircular α = 15°, D = 1.51 mm, Ph = 24.6 mm	$\mathrm{Nu}=3.255+0.00729·(\mathit{Re}-350)$	(39)	$3.5\times {10}^2<$$\mathrm{Re}<$$1.2\times {10}^3$ Pr = 0.66
				$\mathit{f}·\mathit{Re}=16.51+0.01627·\mathit{Re}$	(40)
Kim and No [162]	EXP	He	Semicircular α = 15°, D = 1.51 mm, Ph = 24.6 mm	$\mathrm{Nu}=4.089+0.00365·\mathit{Re}·{\mathit{Pr}}^{0.58}$	(41)	$\mathit{C}.\mathit{F}=$ $10.939–11.014{\nu }_{\mathit{s}}/{\nu }_{\mathit{b}}$ ${\nu }_{\mathit{s}}/{\nu }_{\mathit{b}}$: surface-to-bulk dynamic viscosity
				$\mathit{f}·\mathit{Re}=15.78+4.868\times {10}^{-3}·{\mathit{Re}}^{0.8416}-\mathit{C}.\mathit{F}$	(42)
Chen et al. [63]	EXP	He	Semicircular α = 15°, D = 2 mm, Ph = 24.6 mm	$Nu=$ $\left\{\begin{array}{c}(0.05516\pm 0.00160)R{e}^{(0.69195\pm 0.00559)}\\ (0.09221\pm 0.01397)R{e}^{(0.62507\pm 0.01949)}\end{array}\right.$	(43)	$1.4\times {10}^3\le \mathit{Re}\le 2.2\times {10}^3$ $2.2\times {10}^3<\mathit{Re}\le 3.558\times {10}^3$
				$f=\left\{\begin{array}{c}{\displaystyle \frac{17.639}{R{e}^{0.8861\pm 0.0017}}}\\ 0.019044\pm 0.001692\end{array}\right.$	(44)
Meshram et al. [157]	CFD	sCO2	Semicircular α = 36°, D = 2 mm, Ph = 12 mm	${\mathit{Nu}}_{\mathrm{hot},1}=0.0174\mathit{R}{\mathit{e}}^{0.893}\mathit{P}{\mathit{r}}^{0.7}$	(45)	$5.0\times {10}^3<\mathit{Re}<3.2\times {10}^4$ $\mathit{Pr}=0.74-0.76$
				${\mathit{f}}_{\mathrm{hot},1}=0.867\mathit{R}{\mathit{e}}^{-0.522}+0.040$	(46)
				${\mathit{Nu}}_{\mathrm{hot},2}=0.0205\mathit{R}{\mathit{e}}^{0.869}\mathit{P}{\mathit{r}}^{0.7}$	(47)
				${\mathit{f}}_{\mathrm{hot},2}=0.819\mathit{R}{\mathit{e}}^{-0.671}+0.044$	(48)
				${\mathrm{Nu}}_{\mathrm{cold},1}=0.0177\mathit{R}{\mathit{e}}^{0.871}\mathit{P}{\mathit{r}}^{0.7}$	(49)	$5.0\times {10}^3<\mathit{Re}<3.2\times {10}^4$ $\mathit{Pr}=0.83-1.08$
				${\mathit{f}}_{\mathrm{cold},1}=0.869\mathit{R}{\mathit{e}}^{-0.512}+0.041$	(50)
				${\mathit{Nu}}_{\mathrm{cold},1}=0.0213\mathit{R}{\mathit{e}}^{0.876}\mathit{P}{\mathit{r}}^{0.7}$	(51)
				${\mathit{f}}_{\mathrm{cold},2}=0.804\mathit{R}{\mathit{e}}^{-0.711}+0.045$	(52)
Kim et al. [163]	CFD	sCO2	Semicircular α32.5° = 32.5°, D32.5° = 1.8 mm, Ph32.5° = 9 mm A40° = 40°, D40° = 1.9 mm, Ph, 40° = 9 mm	${Nu}_{32.5°}=(0.0292\pm 0.0015){\mathit{Re}}^{0.8138\pm 0.0050}$	(53)	$2.0\times {10}^3<\mathit{Re}<5.8\times {10}^4$ $0.7<\mathit{Pr}<1.0$
				$f_{32.5°}=(0.2515\pm 0.0097){\mathit{Re}}^{-0.2031\pm 0.0041}$	(54)
				${Nu}_{40°}=(0.0188\pm 0.0032){\mathit{Re}}^{0.8742\pm 0.0162}$	(55)	$2.0\times {10}^3<\mathit{Re}<5.5\times {10}^4$ $0.7<\mathit{Pr}<1.0$
				$f_{40°}=(0.2881\pm 0.0212){\mathit{Re}}^{-0.1322\pm 0.0079}$	(56)
Baik et al. [101]	EXP	sCO2	Semicircular α = 32.5°, D = 1.8 mm, Ph = 9 mm	$\mathit{Nu}=0.8405\mathit{R}{\mathit{e}}^{0.5704}\mathit{P}{\mathit{r}}^{1.08}$	(57)	$1.5\times {10}^4<\mathit{Re}<8.5\times {10}^4$
				$\mathit{f}=0.0748\mathit{R}{\mathit{e}}^{-0.19}$	(58)
				$\mathit{Nu}=0.2829\mathit{R}{\mathit{e}}^{0.6686}$	(59)	$5.0\times {10}^1<\mathit{Re}<2.0\times {10}^2$
				$\mathit{f}=6.9982\mathit{R}{\mathit{e}}^{-0.766}$	(60)
Yoon et al. [164]	CFD	sCO2	Semicircular α = 90–150°, D = 2 mm, Ph = 5–40 mm	$\mathit{Nu}=5.05+(0.02·\alpha +0.003){\mathit{RePr}}^{0.6}$	(61)	$5.0\times {10}^1<\mathit{Re}<5.5\times {10}^2$
				$\begin{array}{c}\mathit{f}={\displaystyle \frac{15.78}{\mathit{Re}}}+\\ {\displaystyle \frac{2.9311}{100}}\mathrm{e}\mathrm{x}\mathrm{p}(1.9216\alpha ){\left({\displaystyle \frac{{\mathrm{l}}_{\mathrm{R}}}{{\mathrm{D}}_{\mathrm{h}}}}\right)}^{-0.8261\cdot \alpha +3.1254\times {10}^{-2}}\\ +{\displaystyle \frac{(4.7659\cdot \alpha -2.8674)}{100}}\end{array}$	(62)	$5.0\times {10}^1<\mathit{Re}<2.0\times {10}^3$ $l_R={\displaystyle \frac{P_h}{2\cos \alpha }}$
				$\begin{array}{c}\mathit{N}{\mathit{u}}_{\mathrm{hot}}=\\ (0.71\alpha +0.289){\left({\displaystyle \frac{{\mathit{l}}_{\mathit{R}}}{{\mathit{D}}_{\mathit{h}}}}\right)}^{-0.087}\\ \mathit{R}{\mathit{e}}^{(-0.11(\alpha -0.55{)}^2-0.004({\mathit{l}}_{\mathit{R}}/{\mathit{D}}_{\mathit{h}})\alpha +0.54)}\mathit{P}{\mathit{r}}^{0.56}\end{array}$	(63)	$5.5\times {10}^2<\mathit{Re}<2.0\times {10}^3$
				$\begin{array}{c}\mathit{N}{\mathit{u}}_{\mathrm{cold}}=\\ (0.18\alpha +0.457){\left({\displaystyle \frac{{\mathit{l}}_{\mathit{R}}}{{\mathit{D}}_{\mathit{h}}}}\right)}^{-0.038}\\ \mathit{R}{\mathit{e}}^{(-0.23(\alpha -0.74{)}^2-0.004({\mathit{l}}_{\mathit{R}}/{\mathit{D}}_{\mathit{h}})\alpha +0.56)}\mathit{P}{\mathit{r}}^{0.58}\end{array}$	(64)
Jiang et al. [165]	/	sCO2		$\mathit{Nu}=0.0176\mathit{R}{\mathit{e}}^{0.809}\mathit{P}{\mathit{r}}^{1/3}$	(65)	$3.0\times {10}^3<\mathit{Re}<2.06\times {10}^4$
				$\mathit{f}=0.3905\mathit{R}{\mathit{e}}^{-0.0355}$	(66)
				$\mathit{Nu}=0.0845\mathit{R}{\mathit{e}}^{0.721}\mathit{P}{\mathit{r}}^{1/3}$	(67)
				$\mathit{f}=1.336\mathit{R}{\mathit{e}}^{-0.1268}$	(68)
Saeed and Kim [112]	CFD	sCO2	Semicircular α = 40°, D = 1.81 mm, Ph = 9.0 mm	$\mathit{Nu}=0.041\mathit{R}{\mathit{e}}^{0.83}\mathit{P}{\mathit{r}}^{0.95}$	(69)	$3.0\times {10}^3<\mathit{Re}<6.0\times {10}^4$ $0.7<\mathit{Pr}<1.2$
				$f_{\mathrm{hot}}=0.115\mathit{R}{\mathit{e}}^{-0.13}$	(70)
				${\mathit{f}}_{\mathrm{cold}}=0.19\mathit{R}{\mathit{e}}^{-0.089}$	(71)
Chu et al. [166]	EXP	sCO2 Water	Semicircular α = 0–25°, D = 2.8 mm, Ph = 20 mm	$\mathit{Nu}=0.0183{\mathit{Re}}^{0.82}{\mathit{Pr}}^{0.5}{\left({\displaystyle \frac{{\rho }_{\mathit{b}}}{{\rho }_{\mathit{w}}}}\right)}^{-0.3}$	(72)	$2.5\times {10}^4<\mathit{Re}<6.8\times {10}^4$ $2.08<\mathit{Pr}<4.72$ ${\displaystyle \frac{{\rho }_{\mathit{b}}}{{\rho }_{\mathit{w}}}}$: bulk-to-wall density radio
Saeed et al. [167]	CFD	SCO2	Semicircular α = 40°, D = 1.81 mm, Ph = 7.24 mm	${\mathit{Nu}}_{\mathrm{water}}=0.08\mathit{R}{\mathit{e}}^{0.77}$	(73)	$5.0\times {10}^1<\mathit{Re}<1.5\times {10}^4$
				${\mathit{f}}_{\mathrm{water}}=3.04\mathit{R}{\mathit{e}}^{-0.37}$	(74)
				${\mathit{Nu}}_{{\mathrm{sco}}_2}=0.475{\mathit{Re}}^{0.61}\mathit{P}{\mathit{r}}^{0.17}$	(75)	$3.0\times {10}^3<\mathit{Re}<6.0\times {10}^4$ $2.0<\mathit{Pr}<13.0$
				${\mathit{f}}_{{\mathrm{sco}}_2}=0.13\mathit{R}{\mathit{e}}^{-0.044}$	(76)
Alvarez et al. [99]	EXP	Water	Square α = 45°, Dx = 3 mm, Ph = 13.51 mm	$\mathit{Nu}=0.5656\mathit{R}{\mathit{e}}^{0.5424}\mathit{P}{\mathit{r}}^{0.01140}$	(77)	$1.299\times {10}^3<\mathit{Re}<8.313\times {10}^3$ $3.8<\mathit{Pr}<6.2$
Cheng et al. [168]	EXP	sCO2 Water	Semicircular α = 32.5°, D = 1.5 mm, Ph = 9 mm	${\mathit{Nu}}_{{\mathrm{sco}}_2}=(0.02475\pm 0.002657){\mathit{Re}}^{(0.76214\pm 0.03899)}$	(78)	$4.987\times {10}^3<\mathit{Re}<2.389\times {10}^4$ $0.765<\mathit{Pr}<0.784$
				${\mathit{f}}_{{\mathrm{sco}}_2}=(0.7510\pm 0.09037){\mathit{Re}}^{-0.2834\pm 0.08859}$	(79)
				${\mathit{Nu}}_{\mathrm{water}}=(0.02063\pm 0.002562)\mathit{R}{\mathit{e}}^{0.7678\pm 0.04928}$	(80)	$3.231\times {10}^3<\mathit{Re}<1.563\times {10}^4$ $1.01<\mathit{Pr}<1.10$
				${\mathit{f}}_{\mathrm{water}}=(12.74\pm 3.815){\mathit{Re}}^{-0.4806\pm 0.07792}$	(81)
Zhang et al. [169]	EXP, CFD	sCO2	Semicircular αhot = 32.5°, Dhot = 1.5 mm, Ph,hot = 9 mm αcold = 15°, Dcold = 1.6 mm, Ph,cold = 24.6 mm	$\mathit{Nu}=0.007604\mathit{R}{\mathit{e}}^{0.9407}\mathit{P}{\mathit{r}}^{0.3884}$	(82)	$6.9\times {10}^3<\mathit{Re}<5.85\times {10}^4$ $1<\mathit{Pr}<9$
				$\mathit{f}=0.03528\mathit{R}{\mathit{e}}^{-0.09027}$	(83)
Katz et al. [170]	EXP	sCO2 He	Semi-elliptical α = 37°, D = 1.4 mm, Ph = 7.4 mm	$\mathit{N}{\mathit{u}}_{\mathrm{S}\mathrm{C}{\mathrm{O}}_2}=0.02609{\mathit{Re}}^{0.8765}{\mathit{Pr}}^{1/3}$	(84)	$5.0\times {10}^2<\mathit{Re}<1.8\times {10}^4$ $0.66<\mathit{Pr}<1.25$
				${\mathit{f}}_{{\mathrm{SCO}}_2}=0.587\mathit{R}{\mathit{e}}^{-0.303}$	(85)
				${\mathit{f}}_{{\mathit{SCO}}_2}=0.09\mathit{R}{\mathit{e}}^{-0.103}+{\displaystyle \frac{28.593}{\mathit{Re}}}$	(86)
				$\mathit{N}{\mathit{u}}_{\mathrm{He}}=0.8072\mathit{R}{\mathit{e}}^{0.4359}{\mathit{Pr}}^{1/3}$	(87)	$4.0\times {10}^2<\mathit{Re}<3.0\times {10}^3$ $0.66<\mathit{Pr}<1.25$
				${\mathit{f}}_{\mathrm{He}}=0.508\mathit{R}{\mathit{e}}^{-0.276}$	(88)
				${\mathit{f}}_{\mathrm{He}}=0.34\mathit{R}{\mathit{e}}^{-0.229}+{\displaystyle \frac{5.532}{\mathit{Re}}}$	(89)
Cai, et al. [171]	CFD	sCO2	Semicircular α = 15°, D = 1.5 mm, Ph = 24.6 mm	$\mathit{Nu}=0.0901\mathit{R}{\mathit{e}}^{0.694}\mathit{P}{\mathit{r}}^{0.621}$	(90)	$1.0\times {10}^4<\mathit{Re}<5.4\times {10}^4$ $0.75<\mathit{Pr}<1.0$
				$\mathit{f}=0.332\mathit{R}{\mathit{e}}^{-0.437}+0.00521$	(91)
Liu, et al. [172]	EXP	sCO2	Semicircular α = 15°, D = 1.87 mm, Ph = 7.24 mm	$\mathit{Nu}=0.01358\mathit{R}{\mathit{e}}^{0.8087}\mathit{P}{\mathit{r}}^{0.5185}$	(92)	$2.64\times {10}^3\le \mathit{Re}\le 4.56\times {10}^4$
				$\mathit{f}={\displaystyle \frac{18.52}{{\mathit{Re}}^{0.8561}}}+0.0035$	(93)
Aakre and Anderson [100]	EXP	Nitrate salt sCO2 Water	Hot side: elliptical αhot = 30°, Dhot = 1.75 mm, Ph,hot = 9.5 mm Cold side: semi-elliptical αcold = 37°, Dcold = 1.43 mm, Ph,cold = 7.2 mm	${\mathit{Nu}}_{\mathrm{salt}}=0.412\mathit{R}{\mathit{e}}^{0.51}{\mathit{Pr}}^{1/3}$	(94)	$1.0\times {10}^2<\mathit{Re}<7.5\times {10}^2$ $3.4<\mathit{Pr}<7.2$
				${\mathit{f}}_{\mathrm{salt}}=5.419\mathit{R}{\mathit{e}}^{-0.664}+0.042$	(95)	$6.0\times {10}^1<\mathit{Re}<1.1\times {10}^3$
				${\mathit{Nu}}_{{\mathrm{sco}}_2}=0.0163\mathit{R}{\mathit{e}}^{0.922}{\mathit{Pr}}^{1/3}$	(96)	$5.0\times {10}^3<\mathit{Re}<4.0\times {10}^4$ $0.733<\mathit{Pr}<0.777$
				${\mathit{f}}_{{\mathrm{sco}}_2}=1.022\mathit{R}{\mathit{e}}^{-0.537}+0.028$	(97)	$5.0\times {10}^2<\mathit{Re}<3.0\times {10}^4$
				${\mathit{f}}_{\mathrm{water},37°}=0.774\mathit{R}{\mathit{e}}^{-0.425}$	(98)	$2.0\times {10}^2<\mathit{Re}<1.95\times {10}^3$
				${\mathit{f}}_{\mathrm{water},30°}=4.477\mathit{R}{\mathit{e}}^{-0.639}+0.041$	(99)	$1.0\times {10}^2<\mathit{Re}<1.65\times {10}^3$
Aye et al. [173]	CFD	He	Semicircular α = 0–30°, D = 2 mm, Ph = 24.6 mm	$\mathit{N}{\mathit{u}}_{\mathrm{hot}}=4.089+1.83\times {10}^{-5}\mathit{R}{\mathit{e}}^{1.8108}{\alpha }^{0.35952}$	(100)	$4.78\times {10}^2<\mathit{Re}<1.364\times {10}^3$ $\mathit{Pr}=0.66$
				${\mathit{f}}_{\mathrm{hot}}\mathit{Re}=15.78+1.35\times {10}^{-7}\mathit{R}{\mathit{e}}^{2.6162}{\alpha }^{0.52221}$	(101)
				$\mathit{N}{\mathit{u}}_{\mathrm{cold}}=4.089+8.5\times {10}^{-5}\mathit{R}{\mathit{e}}^{1.5732}{\alpha }^{0.47157}$	(102)	$7.0\times {10}^2<\mathit{Re}<2.0\times {10}^3$ $\mathit{Pr}=0.66$
				${\mathit{f}}_{\mathrm{cold}}\mathit{Re}=15.78+4.52\times {10}^{-2}\mathit{R}{\mathit{e}}^{0.90287}{\alpha }^{0.32941}$	(103)
Liu et al. [174]	EXP	sCO2	Semicircular α = 30°, D = 1.95 mm, Ph = 7.24 mm	$\mathit{Nu}=0.0335\mathit{R}{\mathit{e}}^{0.829}\mathit{P}{\mathit{r}}^{0.291}$	(104)	$3.201\times {10}^3<\mathit{Re}<4.326\times {10}^4$ $1.1<\mathit{Pr}<6.0$
				$\mathit{f}=1.051\mathit{R}{\mathit{e}}^{-0.303}$	(105)
Jin et al. [175]	EXP	sCO2	Semicircular α = 40°, D = 2 mm, Ph = 7.24 mm	$Nu=2.3198{Re}^{0.6456}{Pr}^{0.2369}{\left({\displaystyle \frac{\overline{C_p}}{C_p}}\right)}^{0.3167}(q^{+}{)}^{0.4575}$  $q^{+}={\displaystyle \frac{4q\beta }{G{c}_p}}$ ${\overline{C}}_p={\displaystyle \frac{h_b-h_w}{T_b-T_w}}$	(106)	$5.0\times {10}^3<\mathit{Re}<3.46\times {10}^4$ q represents the wall heat flux (taken as absolute value), G represents the mass flux, and β represents the thermal expansion coefficient of sCO2 hb and hw, respectively, represent the enthalpy values corresponding to Tb and Tw
Yin, et al. [96]	CFD	He-Xe	Semicircular α = 5–35°, D = 1.0–3.0 mm, Ph = 8.0–32.0 mm	${\mathit{Nu}}_{\mathrm{hot}}=$ ${\displaystyle \frac{-1.8\alpha +2.5\times {10}^{-2}({\mathit{P}}_{\mathit{h}}/{\mathit{w}}_{\mathit{f}})+3.3\times {10}^{-1}(\mathit{D}/{\mathit{w}}_{\mathit{f}})+2.5\times {10}^{-3}{\mathit{Re}}_{\mathrm{hot}}}{1-1.13\alpha +1.1\times {10}^{-2}({\mathit{P}}_{\mathit{h}}/{\mathit{w}}_{\mathit{f}})}}$	(107)	$2.2\times {10}^2<\mathit{Re}<5.0\times {10}^3$ wf: Distance between adjacent channels
				${\mathit{Nu}}_{\mathrm{cold}}=$ ${\displaystyle \frac{-2.1\alpha +2.5\times {10}^{-2}({\mathit{P}}_{\mathit{h}}/{\mathit{w}}_{\mathit{f}})+3.6\times {10}^{-1}(\mathit{D}/{\mathit{w}}_{\mathit{f}})+2.6\times {10}^{-3}{\mathit{Re}}_{\mathrm{cold}}}{1-1.14\alpha +1.1\times {10}^{-2}({\mathit{P}}_{\mathit{h}}/{\mathit{w}}_{\mathit{f}})}}$	(108)
				$\mathit{f}·{\mathit{Re}}_{\mathrm{hot}}=$ ${\displaystyle \frac{-10.8\alpha +1.0\times {10}^{-1}({\mathit{P}}_{\mathit{h}}/{\mathit{w}}_{\mathit{f}})+8.1\times {10}^{-1}(\mathit{D}/{\mathit{w}}_{\mathit{f}})+1.0\times {10}^{-2}{\mathit{Re}}_{\mathrm{hot}}}{1-1.9\alpha +1.3\times {10}^{-2}({\mathit{P}}_{\mathit{h}}/{\mathit{w}}_{\mathit{f}})}}$	(109)
				$\mathit{f}·{\mathit{Re}}_{\mathrm{cold}}=$ ${\displaystyle \frac{-15.6\alpha +1.3\times {10}^{-1}({\mathit{P}}_{\mathit{h}}/{\mathit{w}}_{\mathit{f}})+1.2(\mathit{D}/{\mathit{w}}_{\mathit{f}})+1.3\times {10}^{-2}{\mathit{Re}}_{\mathrm{cold}}}{1-1.8\alpha +1.1\times {10}^{-2}({\mathit{P}}_{\mathit{h}}/{\mathit{w}}_{\mathit{f}})}}$	(110)
Yu et al. [95]	CFD	Nature Gas Air	Semicircular α = 15°, D = 1.51 mm, Ph = 25 mm	$\mathit{N}{\mathit{u}}_{\mathrm{air}}=0.16\mathit{R}{\mathit{e}}^{0.63}\mathit{P}{\mathit{r}}^{0.41}$	(111)	$2.2\times {10}^2<\mathit{Re}<5.0\times {10}^3$
				${\mathit{f}}_{\mathrm{air}}={\displaystyle \frac{15.767}{\mathit{Re}}}+{\displaystyle \frac{0.085}{\mathit{R}{\mathit{e}}^{0.078}}}$	(112)
				$\mathit{N}{\mathit{u}}_{\mathrm{NG}}=0.29\mathit{R}{\mathit{e}}^{0.54}\mathit{P}{\mathit{r}}^{0.25}$	(113)	$4.47\times {10}^2<\mathit{Re}<3.416\times {10}^3$ $0.8<\mathit{Pr}<1.4$
				${\mathit{f}}_{\mathrm{NG}}={\displaystyle \frac{15.767}{\mathit{Re}}}+0.037\mathit{R}{\mathit{e}}^{0.047}$	(114)
Torre et al. [19]	CFD	He	Semicircular α = 5–45°, D = 1.5 mm, Ph = 9.225−18.45 mm	Hot side: $\begin{array}{c}Nu=5.54+\\ 2.2655\times {10}^{-6}{\alpha }^{0.2263}\\ \times \left(50.954-l_Z/D_h\right)(47.234-R/D_h)\\ \times Re-{\displaystyle \frac{0.3119}{\alpha +0.06419}}\end{array}$	(115)	$5.5\times {10}^2\le Re\le 2.2\times {10}^3,$ $Pr=0.66$ R: Bend radius $l_z={\displaystyle \frac{P_h}{2}}$
				$\begin{array}{c}fRe=0.01731{\displaystyle \frac{\left(\alpha +0.3185\right)}{{1.0321}^{{\left(0.6343\frac{l_z}{D_h}-8.414\right)}^2}}}\\ \times \left(0.5908+{\left({\displaystyle \frac{R}{D_h}}+1.2849\right)}^{-1}\right)Re\\ +0.72425\times \left(29.525-{\alpha }^{-0.9007}\right)\end{array}$	(116)
				Cold side: $\begin{array}{c}Nu=6.63\times {10}^{-6}(\alpha +2.9223)\\ \times \left(170.627-{\displaystyle \frac{lz}{D_h}}\right)Re\\ +1.3046\times {10}^{-1}{\alpha }^{0.1893}\\ \times (49.097-R/D_h)\end{array}$	(117)	$2.2\times {10}^3\le Re\le 2.2\times {10}^3,$ $Pr=0.66$
				$\begin{array}{c}f=2.4169\times {10}^{-8}\\ \times \left({\alpha }^2-1.36\alpha -0.5939\right)\left(92.513-{\displaystyle \frac{l_Z}{D_h}}\right)Re\\ +1.5267\times {10}^{-5}\left({\alpha }^2-2.314\alpha -1.406\right)\\ \times \left({\left({\displaystyle \frac{l_z}{D_h}}\right)}^2-26.902{\displaystyle \frac{l_z}{D_h}}-42.563\right)\\ \times \left(3.043+{\displaystyle \frac{1}{\frac{R}{D_h}+0.9821}}\right)\end{array}$	(118)
Bi et al. [105]	EXP	Air	Rectangular α = 30°, Dy = 1.5 mm, Dz = 1 mm, Ph = 15 mm	$\mathit{Nu}=3.792+0.00361\mathit{R}{\mathit{e}}^{1.087}$	(119)	$4.35\times {10}^2<\mathit{Re}<2.3\times {10}^3$ $\mathit{Pr}=0.741$
				$\mathit{f}·\mathit{Re}=14.773+0.318\mathit{R}{\mathit{e}}^{0.785}$	(120)
Yin et al. [114]	EXP	He-Xe	Semicircular α = 15°, D = 1.5 mm, Ph = 25.6 mm	$\mathit{Nu}=0.025\mathit{R}{\mathit{e}}^{0.907}\mathit{P}{\mathit{r}}^{0.118}$	(121)	$1.5\times {10}^2<\mathit{Re}<1.2\times {10}^3$ $0.26<\mathit{Pr}<0.32$
				$\mathit{f}=(4.289\pm 0.951)\mathit{R}{\mathit{e}}^{-0.762\pm 0.0353}$	(122)
				$\mathit{f}=(3.423\pm 0.719)\mathit{R}{\mathit{e}}^{-0.747\pm 0.0348}$	(123)
Wang et al. [176]	CFD	He	Semicircular α = 10–30°, D = 2 mm, Ph = 10–20 mm	Hot side: $Nu=\left[\begin{array}{c}0.103+0.358{\alpha }^{0.335}{l}_r^{-0.808}\end{array}\right]$ $R{e}^{-8.36+8.977{\alpha }^{0.00234}{l}_r^{0.00122}}P{r}^{0.2}$	(124)	$2\times {10}^3<\mathit{Re}<1\times {10}^4$ $l_r={\displaystyle \frac{P_h}{D_h}}$
				Cold side: $Nu=\left[0.0629+0.462{\alpha }^{0.125}{l}_r^{-0.454}\right]$ $R{e}^{-8.851+9.407{\alpha }^{0.00209}{l}_r^{0.00228}}P{r}^{0.483}$	(125)	$1.3\times {10}^3<\mathit{Re}<1.08\times {10}^4$
				$\begin{array}{c}f={\displaystyle \frac{0.504{\alpha }^{1.193}{l}_r^{-0.320}}{R{e}^{0.0822}}}+\\ \left[\begin{array}{c}458.973{\alpha }^{1.485}{l}_r^{-1.361}+11.535\end{array}\right]/\left[\begin{array}{c}R{e}^{0.702}\cos \left(\alpha \right)\end{array}\right]\end{array}$	(126)

.

Figure 25 presents the applicable ranges of current zigzag channel correlations, which collectively encompass the majority of PCHEs operating conditions. Nevertheless, several critical gaps remain. In the low-Reynolds-number region (Re < 400), correlations are lacking for Prandtl numbers between 0.4 and 3, which represent typical conditions for supercritical fluids and water. At higher Reynolds numbers, correlations applicable to Pr < 0.5 are scarce, limiting the predictive capability for certain supercritical gas mixtures. For high-viscosity fluids with Pr > 10, such as molten salts and oils, no reliable correlations have been established. These gaps indicate that further development of predictive models is essential.

Correlations derived from the classical Dittus–Boelter equation are compared in Figure 25, showing the dependence of Nusselt number on Reynolds and Prandtl numbers. As seen in Figure 26a, an increase in the considered Prandtl number leads to a significant rise in the predicted Nusselt number surface, from Meshram et al. [157]’s Equation (45) at Pr ≈ 0.74–0.76 to Saeed et al. [167]’s Equation (75) at Pr ≈ 2–13, with a maximum increase of approximately 4.59 times. Figure 26b,c further reveal that at constant Pr, variations in Nusselt number under the same Reynolds number are governed primarily by geometric parameters. In zigzag channels, larger bending angles and channel diameters, combined with shorter periodic pitches, yield higher Nusselt numbers. These observations confirm that, in addition to Reynolds number, both fluid properties and geometric configuration are critical variables in correlation development.

For supercritical CO₂ near the critical point, several studies introduced property-based correction terms. Kim and No [162]’s Equation (42) accounted for flow acceleration caused by inlet–outlet temperature differences by incorporating a viscosity-related dimensionless variable in the friction factor expression. Chu et al. [166]’s Equation (72) introduced a density-based correction term into the Nusselt number correlation, while Jin et al. [175]’s Equation (106) considered dimensionless corrections for velocity variation induced by rapid property changes.

Numerous studies have advanced zigzag correlations by incorporating geometry-related correction terms. Typically, non-dimensionalized forms such as bending angle in radians or the ratio of periodic pitch to hydraulic diameter are used. In some cases, such as Aye et al. [173] (Equations (100)–(103)), angular and constant corrections were directly added to the classical formulations. More complex approaches, including those of Yoon et al. [164] (Equations (62)–(64)) and Wang et al. [176] (Equations (124)–(126)), expanded the constant terms of the Dittus–Boelter or Blasius equations into functions of geometry ratios (angle and pitch-to-diameter). These functions were then reintegrated into Nusselt number and friction factor correlations, with functional forms determined as shown in Figure 27. Torre et al. [19] (Equations (115)–(118)) further introduced curvature radii at the turning regions as additional correction parameters. Although such formulations yield high accuracy within specific ranges, their complexity restricts wider applicability and obscures the direct influence of geometry on heat transfer and flow performance.

The review of the existing correlations for zigzag channels demonstrates that most PCHE operating conditions are covered, yet important gaps persist at low Reynolds numbers, low Prandtl numbers, and for high-viscosity fluids. Analyses of extensions based on the Dittus–Boelter and Blasius equations confirm that fluid properties and geometric configuration exert equally significant influence on predictive accuracy. Property- and geometry-based correction terms improve fidelity but often result in correlations of excessive complexity. Therefore, future work should focus on the development of simplified correlation forms that retain accuracy while ensuring interpretability and broader applicability.

3.6.3. Flow and Heat Transfer Correlations for PCHEs with Airfoil Fin Channels

Airfoil fin channels, which can reduce pressure drop while maintaining considerable heat-transfer performance, have been recognized as promising configurations and have become a recent focus in the study of PCHEs. A number of correlations have been proposed, and in this subsection, correlations for airfoil fin channels are critically reviewed. The summary of heat transfer and friction factor correlations for PCHEs with airfoil fin channels is shown in

Table 6. Summary of heat transfer and friction factor correlations for PCHEs with airfoil fin channels.

Publications	Method	Working Fluid	Channel Geometry	Correlations		Application Range
Chung et al. [177]	Exp.	sCO2	NACA 0020 La = 10.8 mm, Lb = 2.4 mm, Ls = 5.4 mm Lf = 6 mm, Wf = 1.2 mm, H = 0.7 mm	$\begin{array}{c}Nu=0.04548R{e}^{0.7398}P{r}^{1/3}\end{array}$	(127)	$4\times {10}^3<\mathit{Re}<1.6\times {10}^4,$ $0.6<\mathit{Pr}<0.8$
				$f=0.1622\mathit{R}{\mathit{e}}^{-0.2832}$	(128)
Wang et al. [31]	Exp.	KNO3 -NaNO2 -NaNO3	NACA0025 La = 8 mm, Lb = 3 mm, Ls = 4 mm Lf = 4 mm, Wf = 1 mm, H = 1.5 mm	$\begin{array}{c}\mathit{N}{\mathit{u}}_{\mathit{s}}=0.0129{{\mathit{Re}}_{\mathit{s}}}^{1.0537}{{\mathit{Pr}}_{\mathit{s}}}^{1/3}{\left({\mu }_{\mathit{s}}/{\mu }_{\mathit{w}}\right)}^{0.14}\end{array}$	(129)	$5\times {10}^2<{\mathit{Re}}_{\mathit{s}}<1.548\times {10}^3,$ $19.4<{\mathit{Pr}}_{\mathit{s}}<23.8,$ $0.73<{\mu }_{\mathit{s}}/{\mu }_{\mathit{w}}<0.85$
				$\begin{array}{c}\mathit{N}{\mathit{u}}_{\mathit{s}}=0.0090{{\mathit{Re}}_{\mathit{s}}}^{1.0731}{{\mathit{Pr}}_{\mathit{s}}}^{0.4}\end{array}$	(130)
Pidaparti et al. [129]	Exp.	sCO2	NACA0020 La = 7.5 mm, Lb = 1.8 mm Lf = 4 mm, Wf = 0.95 mm	$\begin{array}{c}Nu=0.0601{Re}^{0.7326}{Pr}^{0.3453}\end{array}$	(131)	/
Chang et al. [145]	Exp.	sCO2	NACA0050 La = 8 mm, Lb = 4 mm, Ls = 2 mm Lf = 4 mm, Wf = 2 mm, H = 1.2 mm	$\begin{array}{c}Nu=0.023R{e}^{0.87}P{r}^{0.63}{T}^{*}\end{array}$	(132)	$7.578\times {10}^3\le \mathit{Re}\le 2.3257\times {10}^4,$ $1.48\le Pr\le 10.35,$ $0.76\le {\mathit{T}}^{*}\le 1.42$ ${\mathit{T}}^{*}$: Ratio of fluid temperature to pseudo-critical temperature
Chang et al. [178]	Exp.	SCO2	NACA6350 La = 8 mm, Lb = 4 mm, Ls = 1.72 mm Lf = 4 mm, Wf = 2 mm, H = 1.2 mm	$\begin{array}{c}Nu=CR{e}^{0.93}P{r}^{0.1}\end{array}$	(133)	$1.2086\times {10}^4\le \mathit{Re}\le 2.4780\times {10}^4,$ $2.63\le \mathit{Pr}\le 6.36$ For the forward flow, C = 0.026, For the reverse flow, C = 0.02
Park and Kim [60]	Exp.	sCO2	NACA 0020 La = 10.8 mm, Lb = 2.4 mm, Ls = 5.4 mm Lf = 1.2 mm, Wf = 6 mm, H = 0.6 mm	$Nu=0.000241R{e}^{1.2878}P{r}^{0.3}$	(134)	$1.2\times {10}^4<\mathit{Re}<3.0\times {10}^4,$ $1.29<\mathit{Pr}<1.59$
				$f=0.1354R{e}^{-0.2832}$	(135)
Han et al. [179]	Exp. +CFD	sCO2	La = 12 mm, Lb = 4 mm, Ls = 6 mm Lf = 6 mm, Wf = 1.2 mm, H = 0.8 mm	$\begin{array}{c}Nu=0.09192R{e}^{0.73552}P{r}^{0.20285}{\left({\mathit{T}}^{*}\right)}^{-2.72471}\end{array}$	(136)	$\begin{array}{c}1.0\times {10}^4<\mathit{R}\mathit{e}<5.0\times {10}^4,\\ 1<\mathit{P}\mathit{r}<11,\\ 0.95<{\mathit{T}}^{*}<1.2\end{array}$
				$f=0.13247R{e}^{-0.26552}{\left({\mathit{T}}^{*}\right)}^{-0.61892}$	(137)
Zhao et al. [180]	CFD	N2	La = 2.4 mm, Lb = 1.25 mm, Ls = 1.2 mm Lf = 0.6 mm, Wf = 1.2 mm, H = 0.75 mm	$Nu=0.1238R{e}^{0.7202}P{r}^{-0.146}$	(138)	$1.2\times {10}^4\le \mathit{Re}\le 1.45\times {10}^4$
				$f=0.3218R{e}^{-0.4419}+0.003674$	(139)
Yoon et al. [181]	CFD	He, sCO2	NACA 0020 La = 5 mm, Lb = 1.67 mm, Ls = 1 mm Lf = 0.8 mm, Wf = 4 mm, H = 0.5 mm	$\begin{array}{c}Nu=3.7+0.0013{R{e}_{min}}^{1.12}P{r}^{0.38}\end{array}$ $\begin{array}{c}Nu=0.027{R{e}_{min}}^{0.78}P{r}^{0.4}\end{array}$	(140)	$0<\mathit{R}{\mathit{e}}_{\mathit{min}}<2.5\times {10}^3,$ $0.6<\mathit{Pr}<0.8$
				$fR{e}_{min}=9.31+0.028R{e}_{min}^{0.86}$	(141)	$0<\mathit{R}{\mathit{e}}_{\mathit{min}}<1.5\times {10}^4$
Chu et al. [182]	CFD	sCO2	NACA0025 La = 8.0~16.0 mm, Lb = 2.0~4.0 mm, Ls = 4.0~8.0 mm Lf = 4 mm, Wf = 1 mm, H = 1 mm	$\begin{array}{c}j=0.026\times {\left({\displaystyle \frac{L_s}{L_f}}\right)}^{-0.170}{\left({\displaystyle \frac{W_f}{L_y}}\right)}^{-0.248}R{e}_{in}^{-0.19\times (W_f/L_y{)}^{-0.187}}\end{array}$	(142)	$8.0\times {10}^3<\mathit{Re}<1.0\times {10}^5$
				$\begin{array}{c}f=0.357\times {\left({\displaystyle \frac{L_s}{L_f}}\right)}^{-0.252}{\left({\displaystyle \frac{W_f}{L_y}}\right)}^{-0.255}\\ \times R{e}_{in}^{-0.173\times (W_f/L_y{)}^{-0.274}}\end{array}$	(143)
Shi et al. [183]	CFD	MgCl2-KClL Salt – sCO2	NACA0025 La = 8 mm, Lb = 3 mm, Ls = 4 mm Lf = 4 mm, Wf = 1 mm, H = 1.5 mm	$\begin{array}{c}{Nu}_{salt}=0.063{{Re}_{salt}}^{0.755}{Pr}^{1/3}({\displaystyle \frac{{\mu }_{salt}}{{\mu }_w}}{)}^{0.14}\end{array}$	(144)	$5.09\times {10}^2<{\mathit{Re}}_{\mathit{salt}}<6.773\times {10}^3,$ $7.5<\mathit{Pr}<9.5,$ $0.84<{\displaystyle \frac{{\mu }_{salt}}{{\mu }_w}}<0.93$
				$f_{salt}=3.07{{Re}_{salt}}^{-0.462}$	(145)
				${Nu}_{{sCO}_2}=0.0986{{Re}_{{sCO}_2}}^{0.687}{Pr}^{0.4}{\left({\displaystyle \frac{{\mu }_{{sCO}_2}}{{\mu }_w}}\right)}^{0.14}$	(146)	$1.1671\times {10}^4<{\mathit{Re}}_{{\mathrm{sCO}}_2}<1.23483\times {10}^5,$ $0.73\times Pr\times 0.75,$ $1.03<{\displaystyle \frac{{\mu }_{{\mathrm{sCO}}_2}}{{\mu }_{\mathit{w}}}}<1.09$
				$f_{{sCO}_2}=0.513{{Re}_{{sCO}_2}}^{-0.667}$	(147)
Kwon et al. [184]	CFD	sCO2	NACA 0020 La = 12~21 mm, Lb = 2.4~4.2 mm, Ls = 6~10.5 mm Lf = 6 mm, Wf = 1.2 mm, H = 0.8 mm	$\begin{array}{c}N{u}_{hot}=0.02671{\xi }_a^{-0.09177}{\xi }_b^{-0.01118}R{e}^{0.8}P{r}^{0.3}\end{array}$	(148)	$1.2\times {10}^4<\mathit{Re}<2.0\times {10}^4$ $1.5<\xi \mathit{a}<4.0, 2.0<\xi \mathit{b}<3.5$
				$\begin{array}{c}N{u}_{cold}=0.02745{\xi }_a^{-0.09177}{\xi }_b^{-0.01118}R{e}^{0.8}P{r}^{0.4}\end{array}$	(149)
				$\begin{array}{c}f=0.05754{\xi }_a^{-0.2264}{\xi }_b^{-0.03108}R{e}^{-0.1923}\end{array}$	(150)
Zhu et al. [128]	CFD	He	NACA8415 La = 3.6~7.2 mm, Lb = 0~6 mm, Ls = 1.8~3.6 mm Lf = 6 mm, Wf = 0.9 mm, H = 0.95 mm	$\begin{array}{c}Nu=0.5306{\xi }_a^{-0.9243}{\xi }_b^{-0.0027}{\xi }_{\alpha }^{0.1167}R{e}^{0.6674}P{r}^{0.4623}\end{array}$	(151)	$2.0\times {10}^3<\mathit{R}\mathit{e}<1.0\times {10}^4$ $4<\xi \mathit{a}<8,0<\xi \mathit{b}<1,$ $1.010<\xi \alpha <1.269$
				$\begin{array}{c}f=1.6341{\xi }_a^{-1.1108}{\xi }_b^{-0.0064}{\xi }_{\alpha }^{0.0693}R{e}^{-0.3107}\end{array}$	(152)
Cal et al. [185]	CFD	Air	NACA 0021 La = 4~8 mm, Lb = 1.5~1.9 mm Lf = 4 mm, Wf = 0.84 mm, H = 1 mm	$\begin{array}{c}Nu=0.20756N^{-0.10518}{\xi }_a^{0.155274}{\xi }_b^{0.268556}{Re}^{0.615001}\end{array}$	(153)	$1.2\times {10}^4<\mathit{Re}<2.0\times {10}^4$ $1.79<\xi \mathit{a}<2.26, 1.0<\xi \mathit{b}<2.0$
				$\begin{array}{c}f=47.00654N^{-0.35035}{\xi }_a^{0.726493}{{\xi }_b^{1.033682}Re}^{-0.49475}\end{array}$	(154)
Liu et al. [127]	CFD	FNaBe	NACA0025 La = 9 mm, Lb = 4 mm, Ls = 4 mm Lf = 3.6 mm, Wf = 1 mm, H = 1.5 mm	$\begin{array}{c}Nu=\left\{\begin{array}{c}0.681\cdot R{e}^{0.512}\cdot P{r}^{1/3}\cdot (\mu /{\mu }_w{)}^{0.14}(Re<500)\\ 0.274\cdot R{e}^{0.648}\cdot P{r}^{1/3}\cdot (\mu /{\mu }_w{)}^{0.14}(Re>500)\end{array}\right.\end{array}$	(155)	$1.1\times {10}^2<\mathit{Re}<4.3\times {10}^3, 23.3<\mathit{Pr}<47.1,$ $0.71<\mu /{\mu }_{\mathit{w}}<0.98$
				$\begin{array}{c}f=\left\{\begin{array}{c}7.549\cdot R{e}^{-0.822}(Re<500)\\ 0.998\cdot R{e}^{-0.499}(Re>500)\end{array}\right.\end{array}$	(156)
Liu et al. [186]	CFD	sCO2	NACA 0020 La = 12 mm, Lb = 3.6 mm, Ls = 6 mm Lf = 6 mm, H = 0.7 mm	$\begin{array}{c}Nu=0.02282R{e}^{0.8094}P{r}^{0.47562}\end{array}$	(157)	${1.0\times 10}^4\le \mathit{Re}\le {1.0\times 10}^5$
				$\begin{array}{c}f=0.13954R{e}^{-0.28405}\end{array}$	(158)
Liu et al. [187]	CFD	Flue gGas Water	NACA 0020 La = 4.02~15 mm, Lb = 1.2~6.0 mm, Ls = 2.01~7.5 mm Lf = 6 mm, Wf = 1.2 mm, H = 1.2 mm	$\begin{array}{c}{Nu}_{gas}=0.1804{\xi }_a^{0.1352}{\xi }_b^{-0.0549}R{e}^{0.5912}P{r}^{1/3}\end{array}$ $\begin{array}{c}{f}_{gas}=0.2446{\xi }_a^{-0.0535}{\xi }_b^{-0.2125}R{e}^{-0.3225}\end{array}$	(159)	$9.0\times {10}^2\le \mathit{Re}\le 2.1\times {10}^3,$ $0.6\le \mathit{Pr}\le 0.8,$ $0.67\le {\xi }_{\mathit{a}}\le 2.5,$ $1\le {\xi }_{\mathit{b}}\le 5$
				$\begin{array}{c}\begin{array}{c}{Nu}_{water}=0.0568{\xi }_a^{0.0575}{\xi }_b^{0.2218}R{e}^{0.7606}P{r}^{1/3}\end{array}\\ f_{water}=0.9038{\xi }_a^{-0.0273}{\xi }_b^{-0.2289}R{e}^{-0.4907}\end{array}$	(160)	$9.0\times {10}^2\le \mathit{Re}\le 2.1\times {10}^3,\mathit{Pr}=6.1,$ $0.67\le \xi \mathit{a}\le 2.5, 1\le {\xi }_{\mathit{b}}\le 5$
Ding et al. [188]	CFD	FNaBe and sCO2	NACA 0015, 20, 25 La = 6~10 mm, Lb = 3 mm, Ls = 3~5 mm Lf = 3~5 mm, Wf = 1.5~2.5 mm, H = 1 mm	$\begin{array}{c}Nu=0.5294R{e}^{0.6032}P{r}^{0.4}\end{array}$	(161)	$2.0\times {10}^2<\mathit{Re}<1.8\times {10}^3,$ $17.94<\mathit{Pr}<22.33$
				$\begin{array}{c}f=15.3059R{e}^{-0.70601}\end{array}$	(162)
Jiang et al. [189]	CFD	sCO2	NACA 0010, 20, 25 La = 6~10 mm, Lb = 2~4 mm, Ls = 3~5 mm Lf = 4 mm, Wf = 0.4~1.0 mm, H = 1 mm	$\begin{array}{c}N{u}_c=0.318{\xi }_t^{*0.080}{\xi }_s^{*-0.101}{\xi }_a^{*0.265}{\delta }^{*-0.003}R{e}^{0.604}P{r}^{0.4}\end{array}\begin{array}{c}\end{array}$	(163)	$2.36\times {10}^4<\mathit{Re}<9.75\times {10}^4,$ $0.895<\mathit{Pr}<0.922$
				$f_c=1.012{\xi }_t^{*0.137}{\xi }_s^{*-0.300}{\xi }_a^{*0.366}{\delta }^{*-0.024}R{e}^{-0.247}$	(164)
				$N{u}_h=0.095{\xi }_t^{*0.028}{\xi }_s^{*-0.026}{\xi }_a^{*0.506}{\delta }^{*0.048}R{e}^{0.773}P{r}^{0.3}$	(165)	$2.35\times {10}^4<\mathit{Re}<9.65\times {10}^4,$ $0.734<\mathit{Pr}<0.735$ $0.5<{\xi }_{\mathit{t}}^{*}<1.0,0.75<{\xi }_{\mathit{s}}^{*}<1.25,$ $0.15<{\xi }_{\mathit{a}}^{*}<0.25,$ $0.2<{\delta }^{*}<0.4$ ${\xi }_{\mathit{t}}^{*}:$ non-dimensional transverse pitch ${\xi }_{\mathit{s}}^{*}$: non-dimensional staggered pitch ${\xi }_{\mathit{a}}^{*}$: non-dimensional thickness ${\delta }^{*}$: maximum thickness location
				$f_h=1.688{\xi }_t^{*0.239}{\xi }_s^{*-0.227}{\xi }_a^{*0.326}{\delta }^{*0.034}R{e}^{-0.310}$	(166)
Li et al. [119]	/	/	/	$Nu=0.04448{{\xi }_a}^{-0.0991}{{\xi }_b}^{-0.11333}{\left({\xi }_s+1\right)}^{-0.00959}{{\xi }_c}^{-0.06871}(R{\left.e+271.9358\right)}^{0.777566}P{r}^{0.540671}$	(167)	1.13 × 102 ≤ Re ≤ 1.23 × 105, 0.64 ≤ Pr ≤ 23.87, 1.25 ≤ ξa ≤ 4.00, 1.83 ≤ ξb ≤ 10, 0 ≤ ξs ≤ 2.67, 1.33 ≤ ξc ≤ 5.14
				$f=14.029/Re+0.1942{{\xi }_a}^{-0.5528}{{\xi }_b}^{-0.0473}{\left({\xi }_s+1\right)}^{-0.6740}{{\xi }_c}^{-0.5255}R{e}^{-0.1722}$	(168)	1.1 × 102 ≤ Re ≤ 1.23 × 105, 1.25 ≤ ξa ≤ 4.00, 2.00 ≤ ξb ≤ 10.00, 0 ≤ ξs ≤ 2.67, 1.33 ≤ ξc ≤ 5.14

.

The applicable ranges of existing correlations are outlined in Figure 28. Compared with correlations for zigzag channels, correlations for airfoil fin channels are fewer in number but still cover the principal operating conditions: supercritical gases across the full flow regime with Prandtl numbers between 0.6 and 0.8, near-critical gases under turbulent conditions with Prandtl numbers in the range 1–10, and high-density molten salts with Prandtl numbers between 15 and 40. In addition, the work of Li et al. [119] integrated previous developments and proposed a series of correlations spanning the full flow regime and Prandtl numbers from 0.6 to 24, thereby providing predictive tools with broad applicability for the design and analysis of airfoil fin PCHEs.

Correlations derived from the Dittus–Boelter equation are compared in Figure 28. Similarly to zigzag channel correlations, correlations with wider Prandtl number applicability show a more rapid increase in the Nusselt number with Reynolds and Prandtl numbers, highlighting the decisive role of fluid thermal–physical properties in determining heat-transfer performance under fixed-flow conditions. At Re = 1 × 10⁴, the correlation of Chang et al. [145] Chang et al. [63] predicts Nusselt numbers 380% higher than those given by the correlation of Chung et al. [177]. Figure 29 further indicate that, under constant Prandtl number and identical flow conditions, differences in channel geometry strongly affect turbulence intensity and heat transfer capability. At Pr = 4, the correlation of Han et al. [179] produces Nusselt numbers 18.7% lower than those predicted by the correlation of Chang et al. [145], whose more compact fin configuration enhances turbulence and improves heat transfer. These comparisons confirm that fluid flow regime, fluid thermal–physical properties, and channel geometry are the primary factors governing the flow and heat-transfer performance of airfoil fin channels.

Property-based modifications of correlations for airfoil fin channels have been concentrated mainly on sCO₂ and molten salts. For sCO₂, temperature-based dimensionless variables are widely employed, reflecting the fact that variations in thermal–physical properties near the critical point are predominantly induced by temperature changes. This approach is represented by correlations such as those proposed by Chang et al. [145] and Han et al. [179]. For molten salts, where high viscosity constitutes the most distinctive property, corrections typically involve viscosity-related terms, as adopted in the correlations of Shi et al. [183] and Wang et al. [31].

Geometric corrections of airfoil fin channel correlations have been developed through the introduction of dimensionless descriptors representing fin arrangement in different orientations. Three parameters are commonly used in previous studies: the staggered number ξs, which characterizes staggered spacing relative to streamwise pitch; the streamwise number ξa, defined as the ratio of streamwise pitch to fin height; and the spanwise number ξb, representing the ratio of spanwise pitch to fin width. By incorporating such descriptors, Kwon et al. [184] and Liu et al. [187] refined predictive models using streamwise and spanwise parameters. Subsequent studies expanded this framework: Zhu et al. [128] introduced an attack-angle correction to account for the orientation of fins relative to the flow direction; Jiang et al. [189] incorporated dimensionless variables representing the maximum thickness position and fin stagger spacing; Li et al. [119] proposed an aspect-ratio correction to consider the influence of channel height. Comparative analyses further demonstrated that the inclusion of geometric correction terms reduced the prediction errors of Nusselt number and friction factor by 11.10% and 41.52%, respectively, confirming the necessity of geometric correction factors in predictive models for airfoil fin channels.

In summary, existing correlations for airfoil fin channels cover major operating conditions of PCHEs and incorporate both property-based and geometry-based modifications to improve accuracy. The development of correlations for conventional airfoil fin channels is already relatively mature, and the formulations are generally concise and can intuitively reflect the influence of different physical quantities on flow and heat-transfer performance. Attention should therefore also be directed to novel fin geometries and arrangement strategies that have been derived from the airfoil fin concept. For these emerging channel configurations, the establishment of reliable and widely applicable correlations represents an important direction for future research.

3.7. Machine-Learning-Based Prediction and Optimization of PCHEs

In addition to empirical correlations, data-driven approaches based on machine learning have been employed to predict the heat-transfer and frictional characteristics of complex heat-exchanger geometries. These models are capable of capturing the complex nonlinear interactions between dimensionless parameters and flow characteristics, thereby achieving high prediction accuracy even with limited datasets.

Li et al. [119] compared the prediction accuracy of neural network models and conventional empirical correlations for airfoil fin PCHEs. In their study, the artificial neural network (ANN) was trained using both flow and geometric parameters, including Re, Pr, Nu, and f, together with four non-dimensional geometric ratios: streamwise pitch ratio, spanwise pitch ratio, fin height ratio, and staggered pitch ratio. By capturing the coupled influences among these factors, the ANN achieved a mean relative error of 4.95%, which was significantly lower than the 10.24% obtained from empirical correlations. This finding highlights the superior capability of neural networks in accounting for the complex coupled effects of geometric and flow parameters on the thermal–hydraulic performance of PCHEs.

Additionally, Li et al. [190] developed a machine learning framework based on CFD simulation data of PCHEs, employing ANN, extreme gradient boosting, light gradient boosting machine, and random forest algorithms to predict the local heat-transfer coefficient and unit pressure drop. After hyper-parameter optimization, the ANN model exhibited the best performance.

Furthermore, several investigations were conducted in which machine learning was combined with multi-objective optimization algorithms to enhance PCHE design. In a series of studies, Saeed et al. [16,191] utilized machine-learning-assisted frameworks for optimizing PCHE channel geometries, where deep neural networks (DNN) and ANN were trained using CFD data and coupled with a multi-objective genetic algorithm to improve thermal–hydraulic performance. In these works, the optimized C-shaped channel exhibited an overall performance improvement of approximately 1.24 times compared with the conventional zigzag channel, while the optimized straight channel reduced pumping power by nearly 50% despite an increase in exchanger length. In another contribution, Jiang et al. [189] employed an ANN to train and predict the thermal–hydraulic parameters of an airfoil fin PCHE, establishing correlations between geometric and layout parameters. Subsequently, sequential quadratic programming and the non-dominated sorting genetic algorithm II (NSGA-II) were applied for optimization. The results indicated that increasing the location of maximum thickness enhanced the overall performance of the PCHE, with the performance factor improving by approximately 6.2% at Re = 45,000.

Although such machine learning approaches have demonstrated superior predictive capability and optimization efficiency, their physical interpretability remains limited. The independent influence of geometric and thermophysical parameters on flow and heat transfer cannot be explicitly identified through these models. Therefore, despite the promising potential of machine-learning-based frameworks, the development of physics-based correlations continues to play a fundamental role in mechanism understanding and engineering design.

3.8. Summary and Perspectives

This section systematically reviews the heat transfer and flow characteristics of PCHEs with different channel configurations, focusing on the influence of channel geometry on thermal–hydraulic performance and on the development of predictive correlations. The content is organized in a logical sequence from straight channels to zigzag/wavy channels, and finally to airfoil channels, reflecting both the research trajectory of PCHEs and the gradual increase in geometric complexity.

For straight channels, performance is mainly affected by cross-sectional geometry. Compared with semicircular sections, rectangular channels enhance heat transfer, while trapezoidal channels exhibit the best overall performance. Although straight channels feature stable flow fields, their heat transfer capability is limited. Current studies have improved heat transfer by introducing flow disturbances through converging/diverging passages, dimples, and interconnecting layouts, with maximum enhancement up to 250%.

For zigzag channels, sharp bends strengthen turbulence and increase heat transfer by 1.5–3 times compared with straight channels, but the friction factor also rises by 2–3 times. Wavy channels alleviate this drawback, reducing pressure drops by about 80%, though heat transfer decreases by about 33%. To mitigate the high-pressure drop of bent channels, measures such as inserting straight sections at bends and introducing bionic river-type fins and sinusoidal fins have been adopted, which improve flow performance while limiting heat transfer reduction, with pressure drops reduced by up to 84.7%.

For airfoil channels, streamlined geometries maintain high heat transfer while reducing pressure drops to one-fifth of those in wavy channels. In airfoil fin channels, the fin geometry and arrangement parameters with the greatest influence are fin thickness and longitudinal pitch. By optimizing fin geometry and varying fin arrangements, flow and heat-transfer performance can be improved. For example, swordfish-type leading-edge fins reduce pressure drops by 22.8% with only 10.4% heat transfer loss, while non-uniform arrangements can reduce fin volume by 50% without performance degradation.

For PCHE headers, flow maldistribution is a key issue limiting performance. Existing studies show that streamlined plenums achieve optimal uniformity at an inclination angle of 30°, while perforated baffles can reduce the maldistribution coefficient to 0.7, simultaneously improving heat transfer and lowering pressure drops.

In terms of correlation development, two stages can be distinguished: (i) initial correlations modified from smooth-tube expressions (e.g., Dittus–Boelter, Blasius) by adjusting constants; and (ii) the introduction of corrected dimensionless factors, including property corrections (density, viscosity, specific heat ratios) to reflect non-isothermal and near-critical effects, and geometric corrections (dimensionless geometric parameters) to improve generality. Despite significant progress, correlations for straight channels at low Reynolds numbers (Re < 1000) remain scarce, zigzag correlations are complex and lack universality, and many works emphasize fitting accuracy while neglecting mechanistic interpretation.

Overall, the existing optimization methods can enhance heat-transfer performance by up to 250% and reduce pressure drops by up to 84.7%, but they are mostly based on numerical studies, with insufficient analysis of the feasibility of fabrication and practical application. Current correlations remain fragmented, with limited applicability ranges, and research on long-term fouling, corrosion, and transient operating conditions is still lacking. Future work should strengthen feasibility assessments of new optimized structures, develop simplified, accurate, widely applicable, and physically interpretable correlations, and reinforce experimental validation under fouling, corrosion, and transient conditions. Through these efforts, the safe and reliable application of PCHEs in advanced energy systems can be ensured.

4. Mechanical Integrity Analysis of PCHEs

The mechanical integrity of Printed Circuit Heat Exchangers (PCHEs) is a critical issue for long-term and safe operation in advanced energy systems. PCHEs are well recognized for their compactness and excellent thermal–hydraulic performance, but the extreme operating conditions of high temperature and high pressure inevitably introduce significant stresses. The stresses mainly include the following.

(1)

Thermal Stresses: Generated from the mismatch between thermal expansion and contraction of metallic materials under temperature gradients when constrained.

(2)

Mechanical Stresses: Generated by the pressure load exerted by the working fluid inside the channels [192].

The evaluation of stresses is commonly conducted according to the allowable stress limits defined in the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code (BPVC), Section III and Section VIII, which provide standardized guidelines for strength assessment.

In this section, a comprehensive review of the existing mechanical integrity analyses of PCHEs is presented, focusing on the issue of tip stress concentration in semicircular straight channels, the corresponding optimization strategies, and the mechanical integrity of alternative channels.

4.1. Stress Analysis

At present, research on the mechanical integrity of PCHEs has mainly focused on straight-channel configurations under supercritical fluids, as high temperature and pressure impose the most severe structural challenges. For semicircular channels, stress concentration at the tips has been identified as the dominant concern [23]. Wang et al. [193] analyzed PCHEs used as sCO₂ Brayton cycle recuperators and showed that pressure-induced stresses were higher than thermal stresses, with distinct stress concentrations forming at the semicircular tips on the high-pressure side, making the cold-side outlet the weakest region.

To address this, Xu et al. [194] developed a homogenization-based simplified stress method, transforming semicircular channels into mechanically equivalent rectangular ones. The maximum stresses were 151.6 MPa for semicircular channels, 253.0 MPa for simplified rectangular channels, and 184.3 MPa for optimized rectangular channels. The optimized rectangular design achieved stress levels close to semicircular channels while improving compactness and reducing weight, enhancing safety and applicability. However, the persistent issue of tip stress concentration remained unresolved.

4.2. Optimization Strategies for Stress Mitigation

To mitigate this issue, Torre et al. [195] introduced rounded tip designs at channel ends, which effectively alleviated local stress concentration and reduced both thermal and mechanical stresses, the thermal stress variation across cold path of designs with different rounded tips is shown in Figure 30 [195]. Simanjuntak and Lee [196] further proposed adding an elliptical upper channel, which decreased stress intensity by up to 24% and reduced the stress concentration factor as the elliptical radius increased.

4.3. Advanced Modeling Techniques

Most of the aforementioned studies focused on local channel sections, while direct modeling and simulation of the entire PCHE still presents substantial computational difficulty. To address this limitation, Xu et al. [197] developed an asymptotic homogenization method that introduced channel periodicity as a small parameter, derived the equivalent elastic matrix in the form of unit-cell strain energy, and fitted temperature-dependent correlations for Young’s modulus, Poisson’s ratio, and shear modulus. Validation against actual models under tensile, bending, and large temperature difference conditions confirmed that the relative error was below 2.4%, while computation time was reduced by a factor of 45–190, maintaining high accuracy with dramatically improved efficiency.

4.4. Mechanical Integrity

Several studies have extended the investigation of PCHE mechanical integrity to alternative channel designs. Bennett and Chen [198] numerically analyzed zigzag channel PCHEs under sCO₂ Brayton cycle conditions, showing that designs with Alloy N06617 satisfied ASME BPVC stress limits, while those with 316 stainless steel exceeded allowable stress intensities and were therefore unsuitable for nuclear use. Similarly, Zhang et al. [10] examined S-shaped fin channels and found that stress concentration and partial yielding occurred at fin tips under high-pressure differentials, indicating the need for higher-strength alloys or optimized geometries.

Jentz et al. [13] developed finite element models for airfoil fin PCHEs to assess local stress and yielding behavior. Results revealed that tip regions yielded under pressurization below 20 MPa. However, when airfoil coverage exceeded 13% of the diffusion-bonded surface, the total yielding was limited to less than 20%, preventing plastic failure. This established airfoil coverage of at least 13% as a critical safety criterion.

Building on this, Tao et al. [199] proposed a hybrid design combining straight and airfoil channels. Their optimized configuration achieved flow and heat-transfer performance comparable to reference designs, with differences below 1.5%, while reducing maximum stress by nearly 70%. This demonstrated a significant improvement in mechanical integrity without compromising thermal–hydraulic efficiency.

In addition, the headers and their welded attachment regions have been identified as the primary failure mechanisms of PCHEs. Aakre et al. [200] conducted destructive experiments under extreme pressurization using specially designed test units with reinforced headers, applying strain gauges and digital image correlation to monitor strain evolution. Results showed that the diffusion-bonded cores maintained a level of strength comparable to the base material and could still contain fluids even after partial internal failure. However, failures consistently initiated in the headers and welded joints, confirming that these regions govern the mechanical limits of PCHEs. These findings not only clarified dominant failure modes but also suggested that code qualification could be simplified by focusing on header design and reliability.

Current research on the mechanical integrity of PCHEs has mainly addressed stress concentration at the tips of semicircular straight channels. Optimization strategies such as mechanically equivalent rectangular channels, rounded tips, and elliptical upper channels have been proposed, though with varying effectiveness. Studies on alternative geometries—including zigzag, S-shaped, and airfoil fin channels—have further highlighted the critical roles of material selection and fin design, with alloy choice, localized yielding, and hybrid configurations strongly influencing reliability.

Although advanced modeling approaches, such as the asymptotic homogenization method, have improved computational efficiency, most investigations remain restricted to local channel sections rather than full-scale PCHEs. Comprehensive assessments at the component level and under long-term operating conditions, including thermal fatigue, are still lacking. Future research should integrate high-fidelity simulations with experimental validation to build systematic evaluation frameworks and ensure safe deployment of PCHEs in high-temperature and high-pressure energy systems.

5. Fouling Behavior Analysis of PCHEs

Fouling is recognized as one of the dominant degradation mechanisms that can significantly influence the long-term performance of heat exchangers. It is generally classified into corrosion, biological, chemical reaction, freezing, particulate and crystallization [201].

Owing to the extremely small hydraulic diameter of PCHE channels, particle–wall interactions become more pronounced. Under practical operating conditions, when the working fluid contains suspended solids or dissolved inorganic salts, particulate deposition and crystallization fouling are likely to occur. Particulate fouling can be analyzed using particle transport and deposition models, such as those based on the Euler–Lagrange framework [202]. Crystallization fouling can be investigated using nucleation–growth kinetics or solubility-based models; the deposition process is often interpreted through the Kern–Seaton model and its diffusion- or reaction-controlled extensions [203], which describe the interplay between mass transfer and surface growth as functions of concentration gradients or supersaturation. However, it should be noted that existing studies on particulate and crystallization fouling have mainly focused on conventional tubular and plate heat exchangers, while systematic investigations on different geometrical types of PCHE channels remain limited. The spatial distribution and morphological evolution of deposition along the channel, from the inlet to the core region, also lack reliable numerical and experimental support. Therefore, the development of predictive models that can comprehensively account for the unique thermal–hydraulic and surface characteristics of PCHEs remains an important direction for future research.

Chemical and corrosion fouling may also be observed, particularly in molten salt or other chemically aggressive environments. These mechanisms are generally analyzed using reaction–diffusion or electrochemical models [204]. It should be emphasized that most existing electrochemical corrosion models have been formulated for aqueous systems, whereas in high-temperature molten salt environments, corrosion behavior is primarily governed by reaction–diffusion coupled chemical processes, which require further investigation. Mitigation can be achieved through the use of chemically compatible materials or protective surface coatings.

Although several empirical and semi-empirical fouling models have been successfully applied to conventional tubular heat exchangers [205], their applicability to geometrically complex and enhanced heat-transfer structures such as PCHEs is limited. Consequently, new fouling models tailored to PCHEs should be developed, with an emphasis on integrating experimental observations and high-fidelity CFD simulations to systematically capture the coupling among microchannel geometry, surface conditions, and thermal–hydraulic characteristics.

6. Conclusions

PCHEs have emerged as leading candidates for advanced energy systems due to their exceptional compactness, structural robustness, and capability to operate under high temperature and pressure conditions. This comprehensive review systematically examines recent advances in PCHE technologies, covering fabrication processes, thermal–hydraulic characteristics of various channel geometries, and the evaluation of Nusselt number and friction factor correlations. Data-driven prediction and multi-objective optimization methods, as well as research on mechanical integrity and fouling behavior are further introduced and analyzed.

The key remarks from this review are summarized here, as follows.

(1)

Manufacturing and Materials: Material selection and fabrication processes fundamentally determine mechanical reliability.

• Material suitability: Austenitic stainless steels prove suitable below 600 °C, while nickel-based alloys and ceramic–matrix composites exhibit superior performance in higher-temperature and higher-pressure environments.
• Fabrication processes: Immersion, spray, and electrically assisted etching each provide distinct advantages in accuracy, simplicity, and material efficiency, with diffusion bonding serving as the critical consolidation step. However, quantitative studies specific to PCHE structures remain limited, with most findings derived from simplified flat-plate specimens. The scalability, cost-effectiveness, and manufacturing yield of these manufacturing processes require thorough evaluation to support industrial implementation.
• Etching-induced geometric deviations exhibit channel-dependent effects. While these etching-induced effects on Nu and ∆p are generally minimal in semicircular channels, more significant variations may occur when larger geometric deviations are present. In contrast, corner rounding in zigzag and airfoil fin channels can noticeably modify local turbulence intensity and velocity distribution, thereby altering flow patterns and overall thermal performance.

(2)

Thermal–Hydraulic Performance: Channel geometry profoundly influences thermal–hydraulic behavior.

• Baseline geometries: Straight channels offer structural simplicity but limited performance enhancement; zigzag channels intensify heat transfer at the cost of elevated pressure drop; wavy and S-shaped fins provide improved thermal-flow balance; and airfoil fin channels deliver superior thermal performance with minimal flow resistance.
• Improved structures: Various optimized configurations have demonstrated remarkable improvements-up to 250% enhancement in heat transfer and 84.7% reduction in flow resistance. However, these promising results primarily originate from numerical simulations under idealized conditions, with experimental validation of manufacturability and mechanical reliability remaining insufficient. The long-term stability and scalability of improved structures require further validation through component-scale tests.

(3)

Correlations and Modeling: A large number of empirical flow and heat transfer correlations, as well as data-driven modeling are analyzed.

• Correlations: The majority of existing Nusselt number and friction factor correlations lack universality, often failing to adequately capture geometric parameter and fluid property effects across broad Reynolds and Prandtl number ranges. Further efforts are needed to establish generalized models capable of accurately representing both geometric and fluid property effects.
• Machine learning approaches: ANN, DNN, and NSGA-II have shown high predictive accuracy and optimization efficiency. However, their physical interpretability requires enhancement through hybrid, physics-informed machine learning frameworks.

(4)

Mechanical Reliability: Mechanical integrity analyses show that design features, such as rounded tips, elliptical channels, and hybrid straight–airfoil geometries can reduce stress concentrations by up to 70%. However, most studies focus on localized regions without considering cyclic loading or thermal fatigue effects. Fouling represents another critical degradation mechanism, with particulate, crystallization, and corrosion fouling significantly compromising performance in molten salt and particulate-laden environments. Existing fouling models developed for conventional heat exchangers show limited applicability to PCHEs’ complex geometries, underscoring the need to develop PCHE-specific fouling models.