2.1. Research process

(Fig 1) illustrates the research workflow of this study. Based on the operational and thermal management requirements of the laser chip, the fixed dimensions of the model were established. The initial dimensions of the micro-channel liquid cooling plate were defined, and its structural configuration was selected. Subsequently, a three-dimensional model of the designed cooling plate was constructed. The model’s boundary conditions were then input into CFD software for numerical simulation. Under turbulent flow conditions, the effects of Lr, Wr, Hr, and Re, Nu, f, and PEC were analyzed to identify the optimal structural design. Finally, neural networks and genetic algorithms were employed to determine the optimal structural parameter combinations.

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Fig 1. Flow chart of the study.

https://doi.org/10.1371/journal.pone.0346804.g001

2.2. Physical model and boundary conditions

2.2.1. Physical model.

Micro-channels featuring capsule-shaped ribs are illustrated in (Fig 2a), where two rows of these ribs are arranged diagonally across the cross-section. The computational model is divided into solid and fluid domain, where the solid domain comprises the capsule-shaped rib structures. As shown in (Fig 2), The dimensions of the solid computational domain are length L = 50 mm, width W = 45 mm, and height H = 12 mm. In contrast, the fluid computational domain has a height H = 7 mm and a width W = 3 mm. The rib spacing is defined as L_s = 6.5 mm. The rib design parameters investigated are L_r, W_r, and H_r, with values listed in Table 1.

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Table 1. Investigated parameter values of the ribs.

https://doi.org/10.1371/journal.pone.0346804.t001

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Fig 2. Schematic diagrams and dimensions of the designed micro-channel.

a Physical model, b Cross-sectional view, c Top view.

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2.2.2. Boundary conditions.

The material properties and boundary conditions for this study are shown in Tables 2 and Table 3:

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Table 2. Material Properties.

https://doi.org/10.1371/journal.pone.0346804.t002

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Table 3. Parameter values of the boundary conditions.

https://doi.org/10.1371/journal.pone.0346804.t003

2.3. Mathematical model

In this study, several justified assumptions were established prior to the simulation to facilitate accurate numerical computations of heat transfer and fluid flow phenomena:

1. (1) The fluid flow is single-phase;
2. (2) Gravity, viscous dissipation, and thermal radiation are neglected;
3. (3) The flow is three-dimensional, steady, and incompressible;
4. (4) Based on the operating conditions and rib-induced flow disturbance, the fluid flow in the micro-channel is assumed to be fully turbulent;
5. (5) The solid material is copper, with a thermal conductivity of approximately 400 W/(m·K). The cooling fluid is deionized water, whose density (ρ_f), specific heat (C_p), thermal conductivity (k), and dynamic viscosity coefficient (μ_f) are temperature-dependent functions, defined as follows:

(1)

(2)

(3)

(4)

In Equation (1), the unit of T is °C. In equations (2), (3), and (4), the unit of T is K. The applied boundary conditions are as follows:

1. (1) The inlet velocity V_in of the micro-channel is 1.14 m/s, and the inlet temperature T_in is 341 K.
2. (2) Since the outlet of the micro-channel is in contact with the atmosphere, a constant relative pressure P_out = 101.325 kPa is selected.
3. (3) On the heat sink surface at z = 0, the bottom boundary is heated in the form of uniform heat flow:

(5)

CFD is an engineering discipline that integrates fluid mechanics and numerical methods to simulate fluid movement by solving conservation equations related to mass, momentum, and energy within a continuous medium. These fundamental flow equations are typically represented by partial differential equations, and the principles governing fluid motion are based on three essential conservation laws: mass, momentum, and energy.

(1) Conservation of Mass:

(6)

In the formula: ρ refers to density; t refers to time; u, υ, w, x, y, z refers to velocity vectors in the respective directions.

(2) Conservation of Momentum:

(7)

(8)

(9)

In the formula: P refers to the pressure on the fluid microelement. τ_xx、τ_xy、τ_yx、τ_yy、τ_xz、τ_zx、τ_zy、τ_yz、τ_zz: The viscous stress component acting on the surface of the fluid element due to molecular viscosity; F_x、F_y、F_z: Body force acting on the fluid element(N).

(3) Conservation of Energy:

(10)

In this formula, C_p denotes the specific heat capacity at constant pressure, while T represents temperature. The variable k signifies the thermal conductivity of the fluid. Additionally, ST represents the volumetric heat source within the fluid and the viscous dissipation term.

The influence of thermal radiation on temperature distribution is neglected in this simulation. This oversight is primarily due to the system’s structure and the environmental conditions surrounding the liquid cooling plate, such as surface properties, channel configuration, and insulation from the external environment—which significantly limit the effective transmission of thermal radiation. Consequently, convection and conduction dominate the heat transfer. Furthermore, the turbulent and dynamic coolant flow emphasizes the dominance of convection. Under these conditions, radiative heat transfer is negligible compared to convection. Additionally, incorporating radiation would significantly increase model complexity, requiring 3D view factor calculations and emissivity specification. Therefore, neglecting radiation simplifies the model while maintaining sufficient accuracy. This approach ensures the model’s applicability and reliability across a broad spectrum of engineering applications.

2.3.1. Heat conduction.

The chip, comprising various materials, is modeled as a solid entity. Heat transfer within the chip occurs via conduction, governed by Fourier law. The formula for this calculation is presented below:

(11)

In the formula:

Q₁: Total heat transferred between media(w);

k: The thermal conductivity coefficient of the calculated heat transfer surface(W/(m·K));

A₁: Cross-sectional area perpendicular to the direction of heat transfer(m²);

σt/σx: Temperature gradient(K/m)；

2.3.2. Thermal convection.

Convective heat transfer describes the heat exchange mechanism between a fluid and a solid surface in direct contact. The equation representing convective heat transfer can be formulated based on Newton’s cooling principle, as illustrated below:

(12)

In the formula:

Q₂: Convective heat transfer(W);

h: Convective heat transfer coefficient(W/(m²·K));

A₂: Effective convective heat transfer area of the wall(m²);

t_w: Temperature of solid surfaces(K);

t_f: Coolant temperature(K);

2.3.3. Verification of the continuum hypothesis and basis for turbulence model selection.

Prior to conducting numerical simulations, this study first evaluated the validity of the continuous medium assumption for fluid flow within microchannels. The rationale for this assumption was primarily demonstrated through the following aspects:

1. In this study, Reynolds numbers (Re) far exceed 4000. Even within smooth channels, flow has fully developed into turbulence. Inertial forces dominate over viscous forces, resulting in highly random and chaotic motion of fluid particles.
2. Within fin channels, the fins themselves constitute significant geometric discontinuities. As fluid flows past the fins, flow separation occurs on the leeward side, forming a shear layer. A stagnation zone with high pressure exists at the leading edge of the fin, an acceleration zone on the side, and a low-pressure recirculation zone on the back. This intense pressure gradient and velocity shear ensure the continuous generation and transport of turbulent pulsations throughout the flow field, precluding any possibility of laminar flow.
3. The Knudsen number serves as the gold standard for determining whether a fluid satisfies the continuous medium assumption:

K_n < 0.001: Continuous medium region;

0.001 < K_n < 0.1: Slip flow region;

0.1 < K_n < 10: Transition region;

K_n > 10: Free molecular flow.

As shown in Equation (13), the Knudsen number is defined as the ratio of the mean free path of molecules to a characteristic physical length. In this study, the working fluid is deionized water (liquid). Liquid molecules are densely packed, lacking a “free path” analogous to gases. For liquids, lattice spacing is typically used to approximate the mean free path. The effective diameter or lattice spacing of water molecules is approximately 0.3 nm. Substituting this into Equation (13) yields K_n far below 0.001. This indicates that the fluid can be treated as a continuous medium at the microscopic level. Collisions between fluid molecules and wall molecules at the interface occur far more frequently than interactions between fluid molecules themselves. Consequently, the macroscopic velocity at the wall surface is strictly zero. Micro-scale effects (such as rarefied gas effects) can be neglected in this study.

(13)

In the formula: K_n denotes the Knudsen number, γ represents the average free path of fluid molecules(m), while D₁ denotes the feature length(m).

2.4. Evaluation criteria

The thermal efficiency of liquid cooling plates is primarily determined by the thermal conductivity of the solid material that constitutes the plate, as well as the convective heat transfer associated with fluid flow through its channels. The analysis of heat and mass transfer within these plates employs established fluid mechanics equations. Typically, these calculations utilize dimensionless fluid parameters, notably the Re and the Nu. To evaluate thermal performance, key metrics include thermal resistance, temperature standard deviation, friction coefficient, and the heat transfer enhancement factor.

The flow channel dimensions of cold plates are typically characterized by the hydraulic diameter:

(14)

In the formula, A_c is the channel cross-sectional area, P_w is the channel wet perimeter, D is the hydraulic diameter, m; c is the channel width, m; b is the channel depth, m.

The Re is used as a variable to compare the Nu. The Re is defined as follows:

(15)

In the formula, ρ represents the fluid density (kg/m³), μ denotes the dynamic viscosity of the fluid (Pa·s), D₁ signifies the feature length (m), and V_in indicates the velocity of the fluid at the inlet of the liquid cooling plate (m/s). The heat transfer characteristics of the fluid within the liquid cooling plate are assessed using the average Nu, which is defined as follows:

(16)

In the formula, h is the convective heat transfer coefficient of the liquid cooling plate, (W/(m²·K)); D is the hydraulic diameter, m; λ is the thermal conductivity of the fluid, which varies according to the temperature of the fluid, (W/(m·K)).

The Prandtl number represents the relative magnitude of a fluid’s ability to transfer momentum and heat energy, and is defined as follows:

(17)

In the formula, μ represents the dynamic viscosity of the fluid measured in Pascal-seconds (Pa·s). The symbol λ denotes the thermal conductivity of the fluid, which varies with temperature and is expressed in watts per meter per Kelvin (W/(m·K)). Additionally, C_p signifies the specific heat capacity of the fluid, measured in joules per kilogram per Kelvin (J/(kg·K)).

The heat transfer coefficient is a critical parameter that quantifies the efficiency of heat transfer between a surface and the adjacent fluid. It represents the amount of heat transferred per unit area per unit time between the surface and the fluid, whether by convection or conduction. The average heat transfer coefficient is defined as follows:

(18)

In the formula, T_w1 denotes the average temperature of the solid bottom surface of the flow channel (K); T_f1 denotes the average temperature of the fluid (K); A_m denotes the total area of the wetted surface in the flow channel (m²).

(19)

In the formula, T_w2 represents the average temperature of the wall surface surrounding the flow channel (K), while T_f2 denotes the average temperature of the fluid (K). A_m denotes the total area of the wetted surface in the flow channel (m²).

Thermal resistance (R_TH) quantifies the resistance to heat flow through a material or system. It is defined as:

(20)

The total thermal resistance of the liquid cooling plate is shown in (Fig 3):

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Fig 3. Thermal resistance model layer diagram.

https://doi.org/10.1371/journal.pone.0346804.g003

In this formula, T_max represents the maximum temperature of the heat flow surface (°C), while T_in denotes the inlet temperature of the fluid (°C). Q indicates the rate of heat transfer through the heat flow surface per unit time (W). The uniformity of the cold plate is characterized by the average absolute temperature difference, denoted as δ_T (°C), defined as:

(21)

In the formula, T_max and T_min represent the maximum and minimum temperatures (°C) of the heat flux surface, respectively. A larger δ_T indicates poorer temperature uniformity of the cold plate and a greater temperature gradient on the surface of electronic components. While evaluating the temperature uniformity of liquid cooling plates, the selected grid nodes are representative; however, the sample size is too small, leading to significant errors. Consequently, this paper utilizes the temperature standard deviation to characterize the temperature uniformity of liquid cooling plates. The temperature standard deviation is defined as follows:

(22)

In the formula: X is the temperature value on each grid surface (°C), and X₀ is the average value of X (°C). The error is reduced by increasing the number of samples, which enhances the reference value of the average temperature evaluation. As illustrated in (Fig 4), nine points are captured on the bottom surface of the flow channel in the model, with distances between the points measuring 12 mm and 14 mm, respectively. The wall temperatures at these nine points are used to calculate the temperature standard deviation.

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Fig 4. Schematic diagram of grid nine-point selection.

https://doi.org/10.1371/journal.pone.0346804.g004

The f is defined as the ratio of the wall shear stress to the fluid kinetic energy per unit volume, thereby reflecting the frictional resistance within the MCHS. It is defined as:

(23)

In the formula, v represents the fluid velocity(m/s), l represents the perimeter of the flow channel cross-section (m)，ΔP represents the pressure drop between the inlet and outlet (kPa).

The thermal enhancement efficiency is a crucial metric that quantifies the improvement in heat transfer performance by comparing the heat transfer coefficients of various cavity micro-channels to those of a smooth micro-channel. This ratio provides insight into the overall effectiveness of different micro-channel designs in enhancing heat transfer, while also reflecting the characteristics of fluid flow within these channels. It is calculated as:

(24)

In this study, Nu denotes the Nu of the investigated micro-channel, while Nu₀ represents that of the original smooth micro-channel. Similarly, f signifies the Fanning friction coefficient of the micro-channel under analysis, and f₀ denotes the coefficient for the original smooth channel.

2.5. Grid independence verification

The modeling work in this study was conducted using the Creo 6.0.0.0 platform, while numerical simulations of the thermal characteristics of the microchannel were performed with the FLOEFD 2020.2 platform. Before commencing numerical simulation experiments, it is crucial to select an appropriate grid size. In simulation contexts, an increase in grid count enhances computational accuracy but concurrently elevates computational costs. Conversely, a reduction in grid count decreases computational accuracy while also reducing computational expenses. Therefore, it is imperative to conduct grid independence tests to determine the optimal grid size. A formula for relative error is formulated as an average metric to assess the adequacy of the grid, defined as follows:

(25)

Here, M1 can represent numbers such as the Nu and the f. M1 denotes the optimal grid calculation value, while M2 represents other grid calculation values. By comparing the different grid counts M1 and M2, the value e is obtained. The smaller the value of e, the higher the accuracy of the grid. In this study, a micro-channel model with rib length L_r = 1.5 mm, rib width W_r = 1 mm, rib height H_r = 1.5 mm, and Re = 11353 was selected to investigate the Nu and f values of the cold plate corresponding to different grid numbers. Using the same grid generation method, five grid sets were created, with mesh counts of 106,603; 153,292; 319,836; 507,868; and 1,067,300. As shown in (Fig 5a), the capsule-shaped fin microchannel was meshed using a tetrahedral grid. In (Fig 5a), 1, 2, and 3 represent the solid mesh of the entire structure, the internal fluid mesh, and the heat source surface mesh. The global mesh for the entire structure is denoted as 1. To ensure computational efficiency, the element size is set to 1 mm. 2 denotes the fluid domain. Considering the minimum fin dimension L_r of 0.5 mm and the fact that this region is critical for flow separation and vortex generation, a fine mesh is necessary to capture flow details. Accordingly, the mesh size in this region ranges from 0.2 mm to 0.5 mm. Adopting a mesh strategy of “refining near boundaries and coarsening in the main flow area” achieves a balance between computational accuracy and cost. Local mesh refinement was applied near the wall surface and on the fins to conform to the wall surface and accurately capture velocity and temperature in the near-wall region, the mesh size was set to 0.2 mm. 3 represents the heat source surface, which serves as the core heat transfer boundary. A local mesh is applied to the solid domain containing the heat source, with the element size refined to 0.2 mm. This ensures node matching at the fluid–solid interface, thereby minimizing numerical interpolation errors across boundaries. (Fig 5b) presents the Nu and f values obtained from each mesh system. As observed in Table 4, compared to the mesh system with 1,067,300 meshes, when the number of grids reaches 507,868, the relative errors for Nu and f are 1.8% and 0.7%, respectively, indicating sufficient accuracy. This indicates that the impact of further increasing the number of grids on the simulation results can be considered negligible. Considering both computational accuracy and time cost, the grid system with 507,868 grids is adopted for subsequent simulations.

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Table 4. Grid independence test (Re = 11353).

https://doi.org/10.1371/journal.pone.0346804.t004

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Fig 5. Grid Independence Verification.

(a) Geometric mesh, (b) Verification of grid independence.

https://doi.org/10.1371/journal.pone.0346804.g005

2.6. Numerical model validation

To validate the accuracy of the model, this study constructed an experimental platform for physical testing. As illustrated in (Fig 6), a physical prototype of the smooth microchannel was fabricated to assess the effects of varying Re on the maximum temperature and pressure drop of the heat sink. Furthermore, as depicted in (Fig 7), the maximum deviation in temperature observed in the current numerical simulation occurred at Re = 7353, with an error of 4.3%. In contrast, the maximum deviation in pressure drop was noted at Re = 11353, with an error of 4.45%. Both errors are below 5%, which indicates that the current method is reliable and that the numerical simulation demonstrates reasonable accuracy. Consequently, it can be effectively utilized for performance prediction and analysis in this study.

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Fig 6. Experimental platform.

https://doi.org/10.1371/journal.pone.0346804.g006

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Fig 7. Experimental and numerical simulation error validation.

https://doi.org/10.1371/journal.pone.0346804.g007

3.1. Preliminary simulation results and analysis of their causes

To establish an optimization baseline for this study and identify the bottlenecks of traditional structures in thermal management, numerical simulations were first conducted on a smooth micro-channel. (Fig 8a) illustrates the temperature distribution at the bottom of the flow channel and the velocity streamline distribution within the channel under turbulent conditions at a Reynolds number of 11353. As shown in (Fig 8a) and (Fig 8b), the fluid flow within the smooth channel remains relatively stable, without forming significant disturbances. Consequently, heat cannot be effectively dissipated and primarily accumulates in the lower region of the chip, resulting in a notable heat accumulation phenomenon. This finding indicates that convective heat transfer in a smooth channel alone is insufficient to meet the cooling requirements of high-power density devices. To address the aforementioned heat accumulation issue, targeted measures for enhancing heat transfer must be implemented.

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Fig 8. Flow channel temperature and velocity distribution diagram.

(a) Temperature distribution and flow velocity cloud charts of the smooth channel, (b) Thermal boundary.

https://doi.org/10.1371/journal.pone.0346804.g008

To clearly elucidate the logical progression from problem identification to strategy formulation, (Fig 9) presents a flowchart of the preliminary simulation analysis. As depicted in (Fig 8b), the primary physical causes of heat concentration are an excessive boundary layer thickness and insufficient vortex disturbance. Based on Newton’s law of cooling, strategies to enhance cooling performance primarily involve reducing boundary layer thickness, increasing fluid velocity, expanding the wall heat transfer area, and selecting thermal interface materials with high thermal conductivity. This study focuses on modifying the internal geometry of the channel by incorporating fin structures to increase the effective heat transfer area. This design induces fluid turbulence, serving as a key solution to mitigate heat accumulation issues. The preliminary simulation results and the analysis of the underlying causes establish a theoretical foundation for the detailed investigation of various rib parameters in the subsequent sections.

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Fig 9. Preliminary simulation analysis diagram.

https://doi.org/10.1371/journal.pone.0346804.g009

3.2. The influence of different fin shapes

A preliminary investigation into fin selection was conducted. (Fig 10) illustrates the velocity contours, streamlines, and temperature distributions along the longitudinal section at z = 9.5 mm. When fluid flows over a solid surface, a narrow region called the boundary layer develops. Viscous effects within this layer reduce the fluid velocity to zero at the wall, creating a significant velocity gradient; this is known as the flow boundary layer. Simultaneously, the temperature difference between the surface and the fluid leads to the formation of an adjacent thin layer exhibiting a steep temperature gradient. This layer is referred to as the thermal boundary layer and constitutes the primary thermal resistance to heat transfer. As depicted in (Fig 10), geometric protrusions of the fin structure disrupt the development of the flow boundary layer [61]. In smooth channels, the thermal boundary layer progressively thickens in the flow direction, forming a substantial thermal resistance layer that leads to heat accumulation at the channel bottom. Fins increase wall roughness, thereby elevating shear stress [61]. Moreover, momentum exchange resulting from flow separation and secondary flow [2] intensifies mechanical energy dissipation, significantly enhancing the mixing between the main flow and the near-wall fluid regions. Triangular fins exhibit the highest friction resistance due to their sharp vertex angles, which cause abrupt bending of the streamline at the leading edge, thereby increasing the likelihood of flow separation [2]. Hexagonal fins generate organized vortex pairs at the trailing edge, enhancing near-wall radial fluid mixing [62]. This effectively thins the thermal boundary layer and elevates the local Nu. Quadrilateral fins, characterized by near-right-angle leading edges, are prone to flow separation [2], resulting in multiple end vortices and increased friction resistance. The droplet-shaped fin combines a streamlined leading edge with a tapered trailing edge. Conversely, as shown in (Fig 11)，the capsule-shaped fin achieves the highest PEC, signifying an optimal balance between heat transfer enhancement and friction cost [63]. Despite having identical projected areas, the capsule fin offers a larger actual surface area. Crucially, its streamlined leading edge minimizes the initial separation region, reduces flow dead zones, disrupts the thermal boundary layer, and enhances heat transfer, leading to an overall improvement in thermal-hydraulic performance [63]. In summary, since capsule-shaped fins demonstrate the most optimal comprehensive thermohydraulic performance, this study will focus on capsule-shaped fin microchannels for further in-depth investigation.

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Fig 10. Graphs of different performance parameters.

(a) Contours of velocity, streamlines, and temperature at the segmented longitudinal sections of z = 9.5 mm for different fin shapes at Re = 11353, L_r = 2 mm, W_r = 1.2 mm, H_r = 1.5 mm, L_s = 3.5 mm, (b) Localized enlarged view.

https://doi.org/10.1371/journal.pone.0346804.g010

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Fig 11. Graphs of different performance parameters.

(a) Nu with fins shape, (b) f with fins shape, (c) PEC with fins shape Re = 11353, L_r = 2 mm, W_r = 1.2 mm, H_r = 1.5 mm, L_s = 3.5 mm.

https://doi.org/10.1371/journal.pone.0346804.g011

3.3. Comparison of flow field characteristics

The impact of capsule-shaped ribs on the micro-channel flow field was examined. (Fig 12) illustrates a comparison of velocity and streamline profiles across different cross-sections for both the smooth micro-channel and the proposed micro-channel at Re = 11,353. The dimensions for the capsule-shaped ribs in the designed micro-channel are as follows, L_r = 1.5 mm, W_r = 1 mm, and H_r = 1.5 mm. As depicted in (Fig 12), smooth microchannels exhibit smoother flow due to the absence of fins, making boundary layers prone to form within the flow channels, particularly in the channel corners where a substantial boundary layer develops, leading to heat accumulation in the lower region of the chip. In contrast, in the micro-channel with capsule-shaped ribs, these ribs periodically disturb the fluid flow, generating longitudinal vortices [62]. These vortices vary in size within the channel and effectively inhibit the development of the boundary layer, thereby enhancing heat transfer efficiency. As the fluid approaches the ribs, vortices gradually emerge and expand from the region adjacent to the rib wall toward the center of the channel. Upon passing through the fins, the fluid flow experiences a sudden contraction, which increases the local velocity and enhances the local jet effect. Compared to smooth micro-channels, the incorporation of semi-capsule-shaped fins significantly reduces the thickness of the boundary layer. On one hand, the increase in local velocity facilitates the disruption of the initial boundary layer; on the other hand, the alternating fins periodically disturb the evolution of the boundary layer, ultimately leading to a reduction in its thickness. The observed phenomena reveal that the geometric properties of the ribs have a direct impact on disturbances in flow field intensity and vortex dynamics. Specifically, the longitudinal rib dimension (L_r) determines the available space for vortex generation, while the transverse dimension (W_r) affects the extent of jet impact. Additionally, the vertical height (H_r) is intricately related to the strength of secondary flow [63]. Furthermore, the L_s between staggered fins can modify the periodic behavior of flow separation and reattachment by altering the spatial arrangement of vortex units. Optimizing the shape of the fins could further help balance flow resistance with heat transfer benefits. Given these insights, the present study aims to comprehensively investigate how variations in fin length, width, height, spacing, and geometric shape influence flow and heat transfer characteristics. This research seeks to establish a quantitative relationship between these geometric parameters and thermal-hydraulic performance, thereby providing a foundation for optimizing fin design in heat transfer applications.

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Fig 12. Comparison of the velocity and streamlines contours at different cross sections between the smooth and designed micro-channel at Re = 11353.

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3.4. The effect of rib length

(Fig 13) is the physical model structure diagram. (Fig 14a) illustrates the contour plots of velocity, streamline, and temperature distribution across two longitudinal cross-sections at Re = 11353, a rib width of 1 mm, and a rib height of 1.5 mm. These cross-sections are positioned at z = 7.5 mm and z = 9.5 mm, respectively, and both traverse the center of a rib arrangement within the channel unit along the x-axis. It is evident that for smaller values of L_r, the ribs exhibit greater curvature. After fluid separation occurs at the leading edge of the fin, the insufficient lateral length of the fin prevents the separated shear layer from reattaching to the fin wall before reaching the trailing edge. This results in the formation of a broad and unstable wake region on the leeward side of the fin. Although this wake exhibits high turbulence, its extremely low flow velocity leads to poor convective heat transfer capability. Consequently, a distinct low-velocity zone forms behind the fin, which hinders vortex formation. Furthermore, low-speed zones are also present at the front end of the ribs, resulting from flow originating from the preceding rib unit within the channel, which has not completely dissipated. As L_r increases, the strength of the vortex effect intensifies, causing a gradual contraction of the low-speed zones. Changes in the fluid separation point and the wake flow behavior behind the ribs contribute to reducing the size of the low-speed regions. The temperature distribution contour plot depicted in (Fig 14b) correlates with the velocity distribution shown in part (a). With smaller L_r values, a clear stratification of temperature is observed, accompanied by a significant temperature rise at the rear end of the rib, aligning with the low-velocity region noted in part (a). As L_r increases, the enhanced vortex effect promotes fluid mixing and weakens the thermal boundary layer, which improves temperature uniformity and decreases stratification. Notably, optimal heat transfer performance is achieved when L_r is set to 2 mm.

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Fig 13. Diagram of flow channel dimensions with different L_r at Re = 11353, W_r = 1 mm, H_r = 1.5 mm.

https://doi.org/10.1371/journal.pone.0346804.g013

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Fig 14. Flow channel temperature and velocity streamline distribution diagram.

(a) Contours of velocity, streamlines, and temperature at the segmented longitudinal sections of z = 7.5 mm and z = 9.5 mm for different L_r at Re = 11353, (b) localized enlarged view.

https://doi.org/10.1371/journal.pone.0346804.g014

The variation curves for Nu, f, and PEC concerning L_r across different Re are illustrated in (Fig 15). As the fin length increases, the Nu value show an upward trend. Primarily because longer fins not only expand the effective heat transfer area but, more importantly, enhance the ‘flow-guiding effect.’ Longer fins compel the fluid to deflect over a greater distance within the flow channel, thereby intensifying the fluid’s impact on the side walls.This phenomenon leads to a reduction in the thickness of the thermal boundary layer, resulting in greater temperature uniformity and improved thermal performance of the liquid-cooled plate [32,33]. Additionally, as Re increases, the thermal performance of micro-channels improves, facilitating faster flow and a thinner boundary layer. The friction factor f initially rises with longer L_r but subsequently declines as L_r continues to increase. Initially, longer fin lengths exacerbate fluid flow separation, elevate local turbulence intensity, wall shear stress, and increase friction resistance. However, as the fin length is further extended, the flow tends to stabilize into an attached flow state, reducing turbulence in the wake region and minimizing friction energy loss. The decreased curvature of the fins may enhance the pressure distribution within the flow channel, alleviating the adverse effects caused by secondary flow. Additionally, higher values of Re correlate with lower f, a phenomenon linked to the diminishing influence of viscous forces and the increased turbulent mixing primarily governed by inertial forces. Both factors collectively reduce energy loss per unit of flow length. The PEC increases progressively, reflecting a balanced interplay between improvements in heat transfer and resistance losses. While the f peaks at a length ratio (L_r) of 2 mm, the continuous increase in the Nu predominantly drives the overall enhancement in performance. Lengthening the fins increases the disruption of the thermal boundary layer, thereby improving the coupling efficiency of heat flux density with the velocity field, which leads to a higher heat transfer efficiency per unit of pressure drop.

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Fig 15. Graphs of different performance parameters.

(a) Nu with L_r at different Re, (b) f with L_r at different Re, (c) PEC with L_r at different Re.

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3.5. The effect of rib width

(Fig 16) is a schematic diagram of the physical model structure. (Fig 17) illustrates the distribution maps of velocity, streamlines, and temperature at the x = 18.5 mm cross-section under identical conditions of Re = 11353, L_r = 1.5 mm, and H_r = 1.5 mm for varying W_r. In (Fig 17), it is observed that when W_r is small, the ribs exert a negligible influence on the flow field, resulting in minimal vortex formation. As W_r increases, the regions of vortex activity expand, and their influence becomes more pronounced. The alternating arrangement of the ribs leads to a diagonal distribution of the vortices. A larger W_r enhances the jet effect, according to the mass conservation equation, maintaining a constant mass flow rate while increasing W_r reduces the flow area, which in turn elevates the local flow velocity. This increase in local velocity is directly proportional to the cross-sectional contraction ratio. Consequently, as W_r increases, the fluid within the fin gaps accelerates, forming high-velocity jets. This high-momentum fluid significantly reduces the thickness of the thermal boundary layer. The velocity distribution depicted in (Fig 17) aligns with the temperature distribution presented in (Fig 17), where regions of higher velocity correspond to lower temperature zones. The ribs positioned on the bottom and top walls of the micro-channel contribute to increased turbulence intensity, thereby facilitating- thorough mixing of hot and cold flows.

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Fig 16. Diagram of flow channel dimensions with different W_r at Re = 11353, L_r = 1.5 mm, H_r = 1.5 mm, L_s = 6.5 mm.

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Fig 17. Contours of velocity, streamlines, and temperature at the micro-channel cross section of x = 18.5 mm for different W_r at Re = 11353.

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To examine the influence of W_r on the thermal-hydraulic performance of micro-channels, curves representing the Nu, f, and PEC in relation to W_r were plotted at various Re, as illustrated in (Fig 18a). The analyzed Re range was between 11,353 and 17,353, with Nu exhibiting an upward trend as W_r increases. This observation can be attributed to the enhancement of local jet effects and vortex activity caused by a larger W_r, leading to a reduced thermal boundary layer within the channel, improved temperature uniformity, and enhanced thermal efficiency. The relatively modest increase in Nu is attributed to the presence of low-velocity zones located at the rear of the ribs, which somewhat diminishes the overall heat transfer capability. Furthermore, as Re increases, the thermal performance of the micro-channel tends to improve, resulting in a thinner thermal boundary layer and elevated velocities. According to (Fig 18b), with an increase in W_r, there is a gradual rise in f, accompanied by an increased growth rate. This phenomenon is attributed to the increased local flow velocity. According to Bernoulli’s principle, the ΔP is proportional to the square of the flow velocity. Thus, while the increase in velocity induced by W_r enhances heat transfer linearly, it simultaneously raises flow resistance. As depicted in Figs 18a and b, the growth rate of f is significantly higher than that of Nu, suggesting that the pump power loss surpasses the gain in heat transfer. (Fig 18c) illustrates the relationship curves of PEC concerning W_r across different Re. As W_r escalates, both Nu and f experience an increase, with the rise in heat transfer outperforming the flow resistance, resulting in enhanced overall performance and an increase in PEC.

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Fig 18. Graphs of different performance parameters.

(a) Nu with W_r at different Re, (b) f with W_r at different Re, (c) PEC with W_r at different Re.

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3.6. The effect of rib height-

(Fig 19) is the physical model structure diagram. Under conditions of L_r = 1.5 mm, W_r = 1 mm, and Re = 11,353, the impact of H_r on the thermal-hydraulic performance of micro-channels was investigated. Fig 18 presents contour plots that illustrate the velocity, streamline, and temperature distributions at the cross-section located at x = 18.5 mm. As depicted in (Fig 20), H_r induces vortices within the flow channel. With an increase in H_r, these vortices tend to concentrate in two specific regions: the base of the upper rib and the top of the lower rib. This observation suggests that an increase in H_r elevates the channel blockage ratio, compelling the fluid to accelerate through the narrowed cross-section. According to the principle of conservation of mass, the local core flow velocity increases significantly, converting static pressure into dynamic pressure. This enhancement in kinetic energy facilitates the formation of larger-scale horseshoe vortices at the rib base and shear layer separation at the rib top. The expanded vortex area is not merely a geometric scaling; rather, it results from the increased dissipation of turbulent kinetic energy (TKE), which promotes vigorous mixing between the near-wall hot fluid and the core cold fluid. Furthermore, (Fig 20) also demonstrates that the thermal boundary layer becomes progressively thinner, and as H_r increases, the efficiency of heat transfer improves. This enhancement is primarily due to the increase in flow velocity and the amplification of disturbance effects as H_r rises. (Fig 21) illustrates the variations of Nu, f, and PEC concerning H_r across different Re values. As shown in (Fig 21a), the growth rate of Nu consistently increases with an increase in H_r. This phenomenon can be attributed to the reduction of the low-velocity zone caused by a larger H_r, which enhances fluid mixing and intensifies vortex formation. Similarly, the growth rate of f exhibits a gradual increase with higher H_r, which is associated with increased viscous stress and an overall rise in mechanical energy loss. A notable drop is observed at H_r = 2.5 mm, which can be largely attributed to the significant increase in f at this specific H_r, as depicted in (Fig 21b). As shown in (Fig 21c), the PEC initially increases but then decreases with increasing H_r. This trend primarily occurs because, within the H_r range of 1.5 mm to 2 mm, the enhancements in heat transfer efficiency outweigh the increases in flow resistance, making heat transfer the principal contributor to overall performance. Conversely, within the H_r interval of 2 mm to 3.5 mm, the advantages gained from heat transfer diminish relative to the increasing flow resistance, leading to a reduction in PEC.

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Fig 19. Diagram of flow channel dimensions with different H_r at Re = 11353, L_r = 1.5 mm, W_r = 1 mm, L_s = 6.5 mm.

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Fig 20. Contours of velocity, streamlines, and temperature at the micro-channel cross section of- x = 18.5 mm for different H_r at Re = 11353.

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Fig 21. Graphs of different performance parameters.

(a) Nu with H_r at different Re, (b) f with H_r at different Re, (c) PEC with H_r at different Re.

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3.7. The effect of rib spacing

Based on the analysis presented in Sections 3.4, 3.5, and 3.6, the optimal dimensions for the micro-channel model have been determined to be 2.0 mm, 1.2 mm, and 1.5 mm. To further enhance performance, the impact of L_s on thermal-hydraulic performance is examined for ribs with L_r = 2.0 mm, W_r = 1.2 mm, H_r = 1.5 mm, and Re = 11,353. The investigated L_s values are 0, 1.5, 2.5, 3.5, 4.5, 5.5, and 6.5 mm.

(Fig 22) shows contours of velocity, streamlines, and temperature at the longitudinal section z = 9.5 mm for different L_s. The figure clearly shows that when the fin spacing is 0 mm, a continuous solid structure forms between the fins. As the coolant flows through the fins, the flow boundary layers of adjacent fins rapidly merge, resulting in a skimming flow regime. When L_s is significantly smaller than the reattachment length of the separated shear layer, the fluid glides over the rib crests without penetrating the gap. The stable vortices trapped within these gaps effectively insulate the heat transfer surface from the main coolant flow, creating thermal stagnation zones. Consequently, the effective convective heat transfer area is reduced to merely the top surface of the ribs. This phenomenon significantly reduces the convective heat transfer coefficient and diminishes the efficiency of heat transfer from the fin surface to the fluid. As seen in (Fig 22a) (L_s = 0 mm), stagnant zones form at rib junctions, reducing the effective heat transfer area. At the optimal spacing (Ls = 3.5 mm), fluid distributes uniformly between ribs, maximizing surface cont act area while minimizing flow separation [2]. On the right side of (Fig 22a), it can be observed that when the spacing L_s is 0 mm, the heat near the fins that are distant from the inlet fluid cannot be effectively dissipated, leading to increased local temperature gradients and the formation of heat accumulation, which further deteriorates heat dissipation performance. (Fig 23) demonstrates that the Nu initially increases and then decreases with increasing spacing, while the f also follows a similar trend, gradually smoothing out. At L_s = 3.5 mm, the PEC reaches its maximum value, indicating optimal thermohydraulic performance.

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Fig 22. Flow channel temperature and velocity streamline distribution diagram.

(a) Contours of velocity, streamlines, and temperature at the segmented longitudinal sections of z = 9.5 mm for different L_s at Re = 11353, L_r = 2 mm, W_r = 1.2 mm, H_r = 1.5 mm, (b) Localized enlarged view.

https://doi.org/10.1371/journal.pone.0346804.g022

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Fig 23. Graphs of different performance parameters.

(a) Nu with L_s at different Re, (b) f with L_s at different Re, (c) PEC with L_s at different Re.

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4.1. Description of multi-objective optimization problems

As discussed in the previous section, the dimensions of the rib, L_r, W_r, H_r, and L_s, play a crucial role in influencing the hydraulic and thermal efficiency of the channel. However, these individual studies do not identify the global optimum across the entire design space. Therefore, multi-objective optimization is employed to achieve the optimal overall performance. Under the condition of Re = 11353, two objective functions, J1(Nu/Nu₀) and J2(f/f₀), have been selected, with the goal of enhancing J1 while minimizing J2. These two optimization functions illustrate the improvement in hydraulic and thermal efficiency of the designed micro-channel compared to the original smooth channel. Based on the simulation results, the parameters L_r, W_r, H_r, and L_s are established at (0.5–2.5 mm), (0.6–1.4 mm), (1–3.5 mm), and (0–6.5 mm), respectively. Consequently, the multi-objective optimization problem is thus defined as finding the optimal combination (L_r, W_r, H_r, L_s) within these ranges to maximize overall micro-channel performance.

4.2. Combination of neural network algorithms and genetic algorithms

Microchannel heat sinks exhibit complex flow separation, vortex evolution, and boundary layer reconstruction within their structures, revealing highly nonlinear and nonconvex mapping relationships between geometric parameters and thermofluid performance. Traditional response surface methods typically rely on second-order polynomial assumptions, which introduce inherent structural rigidity that can lead to underfitting when addressing high-dimensional complex surfaces. This rigidity often smooths out local optimal peaks in the design space. Conversely, while deep neural networks possess universal function approximation capabilities, the use of sparse small-sample data, coupled with high CFD computational costs, can easily result in model overfitting. To effectively balance model generalization and fitting accuracy within limited computational resources, this study does not rely solely on traditional algorithms. Instead, it constructs a deep learning optimization framework that utilizes Gaussian Process Regression (GPR)-enhanced data augmentation. (Fig 24) illustrates the comparison of the Pareto front between the traditional NSGA-II and the ANN/GPR-enhanced method proposed in this paper. The ANN/GPR method achieves broader coverage of the Pareto frontier and attains higher heat transfer performance at the same flow resistance level, approaching the ideal point more closely. This indicates that the traditional method, which relies on a limited number of raw samples and simple interpolation approximations, results in a relatively concentrated solution set that fails to fully explore the performance boundaries of the design space. Consequently, it is prone to local optimization, whereas the deep learning model successfully uncovers a design space with superior performance. In summary, the algorithmic framework is illustrated in (Fig 25), with the innovative aspects highlighted in red. Additionally, as shown in Table 5, we compared the traditional NSGA-II algorithm with the algorithm developed in this study and summarized the advantages of the latter.

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Table 5. Technical comparison between hybrid GPR-DNN enhanced NSGA-II and the traditional NSGA-II.

https://doi.org/10.1371/journal.pone.0346804.t005

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Fig 24. Comparison of Pareto Fronts between traditional NSGA-II and hybrid GPR-DNN enhanced NSGA-II.

https://doi.org/10.1371/journal.pone.0346804.g024

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Fig 25. Schematic diagram of a multi-objective optimization Framework for microchannel heat sink based on GPR data augmentation and DNN ensemble learning.

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(Fig 26) illustrates the NSGA-II implementation workflow. Neural network algorithms and genetic algorithms are biologically inspired computational techniques [64–66]. Initially, the neural network algorithm is employed to train the provided samples, followed by a series of learning and training phases until the target accuracy is achieved. This trained ANN model is then used to predict the system outputs. The predicted data samples are subsequently reintroduced into the trained neural network model to validate the model’s predictions. The resulting function of the accurate prediction model is integrated into the genetic algorithm model as the objective function, while the combinations of sample input parameters serve as the targets for optimization. Through iterative calculations performed by the genetic algorithm, the most suitable parameter combinations are ultimately identified, thereby concluding the multi-objective parameter optimization process. Artificial Neural Networks (ANN) have been successfully employed to predict and optimize the thermal performance of heat exchangers with high accuracy [67–69]. In this study, an ANN serves as the surrogate model for training and utilization. Regarding the dataset, 23 sets of base samples were generated via numerical simulation, and an additional 100 new samples were produced through Latin hypercube sampling. The 23 base samples were subsequently used to develop two regression models, J1 and J2, employing Gaussian Process Regression. These trained regression models were then applied to predict the 100 new sample points. The evaluation of the model’s fitting performance during the training process involves the use of the Mean Squared Error (MSE) and the coefficient of determination (R²). The equations for MSE and R² are presented below:

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Fig 26. Flowchart of the NSGA-II implementation.

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(26)

(27)

A smaller MSE and an R² value close to 1 indicate superior performance of the prediction model. Z-score standardization is employed to enhance both the performance and stability of the algorithm. Additionally, the Levenberg-Marquardt [70] algorithm is utilized for training. In this study, simulation experiments were conducted using MATLAB 2024b, and the results of the model training are illustrated in ():

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Fig 27. Neural network training error MSE result.

https://doi.org/10.1371/journal.pone.0346804.g027

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Fig 28. Neural network prediction results.

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Fig 29. Neural network model fitting results.

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To verify the accuracy of the training results, seven sets of parameter combinations were randomly designed within the specified parameter range. The corresponding test results, denoted as PEC, were calculated using CFD software. Subsequently, these test results were compared with the training results, referred to as PEC_Te, as illustrated in Table 6:

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Table 6. Randomized data training results and test results.

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Based on the random training results and the seven test results presented in Table 6, (Fig 30) illustrates the relationship curves between J1, J2, and the corresponding errors. The figure shows some discrepancy between the training outcomes and the test calculation results, with a maximum error of 5.7%. This level of error is considered acceptable for the surrogate model in this optimization context. This finding suggests that the model adequately represents the nonlinear problem to a certain extent.

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Fig 30. Training results (J1_Te, J2_Te) and test results (J1, J2), together with corresponding error curves (e₁, e₂).

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In this study, the NSGA-II [71]algorithm was employed for multi-objective optimization. This algorithm iteratively searches for the optimal solution, with the parameter settings detailed in Table 7. By integrating the trained ANN proxy model, the resulting Pareto front is illustrated in (Fig 31a). As one progresses to the right along the Pareto front, a gradual decline in heat transfer performance is observed, while hydraulic performance exhibits a steady improvement. To achieve an optimal balance between hydraulic and thermal performance, the TOPSIS decision-making technique is utilized to identify a compromise solution. The findings presented in (Fig 31b) indicate that the parameters Lr, Wr, Hr, and Ls are measured at 2.4928 mm, 1.3944 mm, 1.3490 mm, and 2.9742 mm, respectively, corresponding to performance metrics J1 (1.2179), J2 (1.1090), and PEC (1.18). To validate the optimized results, numerical simulations were conducted using the micro-channel featuring the identified optimal parameter set at Re = 11353. The numerical analysis reveals that the three key performance indicators of the optimized micro-channel, J1, J2, and PEC, are 1.202, 1.1535, and 1.156, respectively. The relative discrepancies for these metrics are calculated at 1.32%, 4.01%, and 2.08%, all of which fall below the 5% threshold, thus confirming the reliability of the optimization outcomes.

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Table 7. Parameter settings for NSGA-II.

https://doi.org/10.1371/journal.pone.0346804.t007

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Fig 31. Pareto Frontier Comparison Chart.

(a) The obtained Pareto front, (b) Optimal compromise solution using TOPSIS.

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4.3. Comparison between the optimal micro-channel design and the initial micro-channel

(Fig 32) presents the temperature contour map of the micro-channel bottom surface derived from simulation data. (Fig 33) provides a comparative analysis of thermal resistance and temperature uniformity between smooth and optimal micro-channels. As illustrated in (Fig 32), the temperature gradient in the central region of the smooth channel is notably higher than that of the optimal structure channel. This gradient serves as an indicator of temperature uniformity within the system; it is evident that the temperature uniformity on the right is superior to that on the left. This improvement is attributed to the increased heat transfer area, which facilitates heat conduction through the internal structure of the fins to a larger surface area for dissipation. Furthermore, in the smooth channel, the flow boundary layer and thermal boundary layer continue to grow and stabilize within the region, necessitating that heat traverse this relatively thick and stable thermal boundary layer before being transferred to the main fluid flow, resulting in elevated thermal resistance. Conversely, the fins compel the main fluid flow to change direction, accelerate, and decelerate, periodically disrupting the flow and the stable development of the thermal boundary layer. This action reduces the boundary layer thickness and, consequently, thermal resistance. The boundary layer continuously redevelops at the leading edge or top of the fins, enhancing its integration with the fluid for efficient heat dissipation.

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Fig 32. Temperature distribution map of the bottom surface of the flow channel.

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Fig 33. A comparison between the smooth and optimal micro-channels based on standard deviation and thermal.

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(Figs 34-36) illustrates the temperature, velocity, and streamline distribution contour plots for both the smooth channel and the optimized channel at the cross-section located at x = 21 mm and z = 7.7 of Re = 11353. As shown in (Figs 34 and 35), the smooth channel displays stable, spatially symmetric vortex pairs, with hot fluid near the wall and cold fluid at the center of the channel contained within their respective vortices. The efficiency of radial mixing is low due to the large and stable vortices, which hinder effective wall scraping and disrupt the thermal boundary layer adjacent to the wall. As shown in (Fig 36), heat transfer predominantly occurs through slow molecular diffusion across the thick thermal boundary layer. The presence of this thick boundary layer in the smooth channel results in reduced heat transfer efficiency. However, the introduction of capsule-shaped fins disrupts the original symmetry of the flow field. In the regions adjacent to the fins, numerous small-scale vortices are generated, which are significantly smaller than the dimensions of the channel. These vortices penetrate and disturb the boundary layer directly, thereby decreasing the primary thermal resistance to heat transfer and enhancing heat transfer efficiency. As shown in Table 8, compared to the smooth micro-channel, the optimized micro-channel exhibits a 3.1% improvement in temperature uniformity (σ), an 8.3% reduction in thermal resistance (R_TH), and a 20.2% increase in the Nu. Although the f increased by 15.3%, the PEC improved by 15.6%. These results indicate that the optimized micro-channel exhibits higher heat transfer efficiency and achieves superior overall performance.

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Table 8. Comparison of σ, R_TH, Nu, f, e between smooth and optimal micro-channel.

https://doi.org/10.1371/journal.pone.0346804.t008

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Fig 34. Comparison of the temperature and streamlines contours at the cross sections of x = 21 mm at Re = 11353.

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Fig 35. Comparison of the velocity and streamlines contours at the cross sections of x = 21 mm at Re = 11353.

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Fig 36. Comparison of the velocity and streamlines contours at the segmented longitudinal sections of z = 7.7 mm at Re = 11353.

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Nomenclature

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https://doi.org/10.1371/journal.pone.0346804.t009