A Parametric Design Study of Natural-Convection-Cooled Heat Sinks

: Effective natural-convection-cooled heat sinks are vital to the future of electronics cooling due to their low energy demand in the absence of an external pumping agency in comparison to other cooling methods. The present numerical study was carried out with ANSYS Fluent and aimed at identifying a more-effective ﬁn design for enhancing heat transfer in natural convection applications for a ﬁxed base-plate size of 100 mm × 100 mm under an applied heat ﬂux of 4000 W/m 2 . The Rayleigh number used in the present study lied within the range of 2.6 × 10 6 to 4.5 × 10 6 . Initially, a baseline case with rectangular ﬁns was considered in the present study, and it was optimized with respect to ﬁn spacing. This optimized baseline case was then validated against the semi-empirical correlation from the scientiﬁc literature. Upon good agreement, the validated model was used for comparative analysis of different heat sink conﬁgurations with rectangular, trapezoidal, curved, and angled ﬁns by constraining the surface area of the heat transfer. The optimized ﬁn spacing obtained for the baseline case was also used for the other heat sink conﬁgurations, and then, the ﬁn designs were further optimized for better performance. However, for the angled ﬁn case, the optimized conﬁguration found in the scientiﬁc literature was adopted in the present study. The proposed novel curved ﬁn design with a shroud showed a 4.1% decrease in the system’s thermal resistance with an increase in the heat transfer coefﬁcient of 4.4% when compared to the optimized baseline ﬁn case. The obtained results were further non-dimensionalized with the proposed scaling in terms of the baseline case for the two novel heat sink cases (trapezoidal, curved).


Introduction
As the world is becoming more reliant on digital devices along with the improving level of the systems' compactness, hence, the cooling of electronics is gaining increased importance.Thermal energy produced by various electronic components can devastate the integrity of devices if it is not properly extracted.Passive cooling systems are very advantageous due to their low energy demand in the absence of external pumping sources such as blowers, fans etc. [1], and the fluid's motion in such systems will be induced by its non-uniform density [2,3].The industry standard for a natural-convection-cooled heat sink usually involves a vertically oriented base-plate with an evenly spaced rectangular plate fin array covered with or without a shroud [4].The sizing, arrangement, and shape of the heat sink fins will affect its thermal performance.A significant amount of research has been carried out with the aim of improving the system's overall heat transfer performance by varying the fin parameters, as well as by proposing novel fin designs.Numerous designs have been proposed in the scientific literature over the past many years, and many of these claimed to achieve significant performance enhancement when compared to the rectangular fin baseline case.While some of these designs hold merit, many often compare their results to an unoptimized baseline case, which puts into question the validity of their findings.design.Several other studies on angled fin configurations mainly with chevron fins were also carried out [13][14][15], which showed a significant impact of chevron angle on the thermal resistance and heat transfer performance.The observations found in [14] showed that, by decreasing the chevron angle, an increase in the total thermal resistance and heat transfer was achieved.
Since there is a lack of agreement as to which heat sink design achieves the best heat transfer performance, the present study aimed at a comparative analysis of different optimized heat sink configurations with rectangular, trapezoidal, curved, and angled fins.As seen in [12], the W-type fin-array-based heat sink containing angled fins arranged in columns of alternating directions performed better than other designs.As a result of this, the present study focused on proposing an optimized novel fin design for a fixed dimension base-plate heat sink on a comparative basis.These novel designs took inspiration from the geometries proposed in the recent scientific literature, which exhibited a promising performance.The final results are presented in comparison to an optimized rectangular baseline heat sink case to ensure any actual improvement is genuine.

Materials and Methods
The present study aimed to identify, by means of computational fluid dynamics (CFD) and numerical shape optimization, the most-effective fin design for enhancing heat transfer in natural air convection.The target end applications are predominantly in electronics cooling, where the Rayleigh number is generally low enough for the flow to remain laminar.Initially, a baseline case with rectangular fins was considered, and it was optimized with respect to fin spacing.This optimized baseline case was used for comparative analysis of different heat sink configurations with rectangular, trapezoidal, curved, and angled fins.The optimized fin spacing obtained for the baseline case was also used for the other heat sink configurations, and then, the fin designs were further optimized for better performance.However, for the angled fin case, the optimized configuration proposed by Zhang et al. [12] was adopted.
To ensure continuity in the testing, a consistent meshing approach, domain size, and CFD solver setup were used across each of these design cases.Furthermore, for reliable relative comparison between tests, several parameters were fixed across all the cases.Table 1 shows the list of parameters fixed for all the design cases.Each case was optimized by carrying out a parametric design study on their geometry to find the fin shape that minimized the thermal resistance for that specific heat sink configuration.

Computational Setup and Boundary Conditions
To decrease the computational time and complexity, only half of the channel width was modelled (i.e., from midway between one fin to midway through the channel), which also relates to half of the fin spacing (S/2).This was achieved using symmetry boundary conditions on the front and back faces of the domain [16], as shown in Figure 1 (right).A pressure inlet boundary condition was used on the bottom and left side faces of the domain, while a pressure outlet boundary condition was used on the top face [17].A constant heat flux boundary condition was used at the right side (back face of the heat sink) region.Adiabatic and no slip boundary conditions were used for the rest of the back wall.Similar to the findings of [12], a domain height of 3-to 5-times the fin height above the heat sink and 1-to 2-times the fin height below and a depth of 2-to 5-times the fin length was sufficient to ensure the results were independent of the domain size.Using these values as a reference point, a domain independence study was conducted, which resulted in a domain height of 2 H below and 3 H above, with a domain depth of 2 L to the left of the top of the heat sink fins.This equals an overall height and depth of 600 mm × 300 mm for a fin height H = 100 mm and fin length L = 100 mm.Adiabatic and no slip boundary conditions were used for the rest of the back wall.Similar to the findings of [12], a domain height of 3-to 5-times the fin height above the heat sink and 1-to 2-times the fin height below and a depth of 2-to 5-times the fin length was sufficient to ensure the results were independent of the domain size.Using these values as a reference point, a domain independence study was conducted, which resulted in a domain height of 2 H below and 3 H above, with a domain depth of 2 L to the left of the top of the heat sink fins.This equals an overall height and depth of 600 mm × 300 mm for a fin height H = 100 mm and fin length L = 100 mm.Since this was a natural convection study, a pressure-based solver was chosen given its ability to better handle such conditions.By estimating the Rayleigh number present in this scenario, the flow regime that will occur can be estimated.For a Rayleigh number of less than 1 × 10 9 (based on the vertical length of the heated surface), it can be assumed that the resulting flow will be laminar [18].Using an estimated heat sink temperature difference of 35-60 K and an ambient air temperature of 300 K, the Rayleigh number was calculated to vary from 2.6 × 10 to 4.5 × 10 .Since this is significantly smaller than 10 , the laminar model was selected and should be sufficient to capture the fluid flow.To verify other scenarios of interest, a model capable of resolving the transition from laminar to turbulent flow was also used.The transition (γ − Re ) shear stress transport (SST) model was chosen given its capability of modelling both laminar and turbulent flows [18].The governing equations adopted in the present study for both the laminar and transition-SST models are given in [18].A coupled pressure-velocity scheme was employed to solve the Navier-Stokes equations and energy equation, which govern the fluid flow and heat transfer in the system, respectively.This method was chosen over the SIMPLE scheme since it deals with natural convection scenarios better because it simultaneously solves the governing equations, rather than separately, producing a more-accurate answer [18].
The list of material properties used in the present study is shown in Table 2.The circulating fluid was chosen to be air, whose temperature was initially set to the ambient temperature (300 K).The density variation was approximated by using the Boussinesq approach as given in [19,20] with a reference value of 1.225 kg/m 3 and a thermal expansion coefficient β = 0.0033 K −1 .The heat sink material was chosen to be aluminum alloy 2024, and a heat flux of 4000 W/m 2 was applied uniformly to the bottom of the heat sink base.Heat radiation was not modelled.Since this was a natural convection study, a pressure-based solver was chosen given its ability to better handle such conditions.By estimating the Rayleigh number present in this scenario, the flow regime that will occur can be estimated.For a Rayleigh number of less than 1 × 10 9 (based on the vertical length of the heated surface), it can be assumed that the resulting flow will be laminar [18].Using an estimated heat sink temperature difference of 35-60 K and an ambient air temperature of 300 K, the Rayleigh number was calculated to vary from 2.6 × 10 6 to 4.5 × 10 6 .Since this is significantly smaller than 10 9 , the laminar model was selected and should be sufficient to capture the fluid flow.To verify other scenarios of interest, a model capable of resolving the transition from laminar to turbulent flow was also used.The transition (γ − Re θ ) shear stress transport (SST) model was chosen given its capability of modelling both laminar and turbulent flows [18].The governing equations adopted in the present study for both the laminar and transition-SST models are given in [18].A coupled pressure-velocity scheme was employed to solve the Navier-Stokes equations and energy equation, which govern the fluid flow and heat transfer in the system, respectively.This method was chosen over the SIMPLE scheme since it deals with natural convection scenarios better because it simultaneously solves the governing equations, rather than separately, producing a more-accurate answer [18].
The list of material properties used in the present study is shown in Table 2.The circulating fluid was chosen to be air, whose temperature was initially set to the ambient temperature (300 K).The density variation was approximated by using the Boussinesq approach as given in [19,20] with a reference value of 1.225 kg/m 3 and a thermal expansion coefficient β = 0.0033 K −1 .The heat sink material was chosen to be aluminum alloy 2024, and a heat flux of 4000 W/m 2 was applied uniformly to the bottom of the heat sink base.Heat radiation was not modelled.The geometry used for the simulations contained two domains: the heat sink as solid Al-2024 and air as the fluid domain.The fluid domain was meshed using tetrahedrons, while the solid domain was meshed with hexahedrons.An inflation layer was added in the fluid domain with the contact surfaces between the two bodies set as the boundary.This inflation layer was initially set to a first layer height of 0.1 mm and a growth rate of 1.1 for 15 layers.A mesh convergence study was carried out to (i) verify solution independence and (ii) obtain a sufficiently fine mesh that would provide accurate results without excessively high computational cost, thus facilitating numerical optimization.The number of mesh elements was varied mainly using the global element size as an input parameter, while the inflation layer was kept constant at the aforementioned values.The mesh convergence was assessed by measuring the average heat source temperature across varying mesh density, the results of which can be seen in Figure 2. The variation of the inflation layers was also carried out in terms of the growth rate, first layer height, and maximum number of layers to ensure the solution independence.The present study reported a heat source temperature of 334.48 K for a growth rate of 1.1, first layer height of 0.1 mm, and maximum number of layers of 15.This value was found to be 334.54K when the number of layers was increased to 20.It was observed that no significant change in this temperature was recorded when the number of layers was increased and also when the first layer height was decreased.For a growth rate of 1.1 and for a first layer height of 0.05 mm with the maximum layers as 15, the value of the heat source temperature was found to be 334.498K.The geometry used for the simulations contained two domains: the heat sink as solid Al-2024 and air as the fluid domain.The fluid domain was meshed using tetrahedrons, while the solid domain was meshed with hexahedrons.An inflation layer was added in the fluid domain with the contact surfaces between the two bodies set as the boundary.This inflation layer was initially set to a first layer height of 0.1 mm and a growth rate of 1.1 for 15 layers.A mesh convergence study was carried out to (i) verify solution independence and (ii) obtain a sufficiently fine mesh that would provide accurate results without excessively high computational cost, thus facilitating numerical optimization.The number of mesh elements was varied mainly using the global element size as an input parameter, while the inflation layer was kept constant at the aforementioned values.The mesh convergence was assessed by measuring the average heat source temperature across varying mesh density, the results of which can be seen in Figure 2. The variation of the inflation layers was also carried out in terms of the growth rate, first layer height, and maximum number of layers to ensure the solution independence.The present study reported a heat source temperature of 334.48 K for a growth rate of 1.1, first layer height of 0.1 mm, and maximum number of layers of 15.This value was found to be 334.54K when the number of layers was increased to 20.It was observed that no significant change in this temperature was recorded when the number of layers was increased and also when the first layer height was decreased.For a growth rate of 1.1 and for a first layer height of 0.05 mm with the maximum layers as 15, the value of the heat source temperature was found to be 334.498K.   3, Figure 3).(b) Detail of the mesh at the boundary between the solid and fluid domain, where the inflation layer is used.3, Figure 3).(b) Detail of the mesh at the boundary between the solid and fluid domain, where the inflation layer is used.
The temperature converged at approximately 334.48 K after the mesh was refined.A mesh element size of 0.0018 m (approximately 500,000 elements) was chosen as sufficiently fine to produce accurate results.This mesh sizing was kept consistent throughout the testing of the baseline case.However, when the subsequent cases needed to be analyzed by using a turbulence model, then the mesh had to undergo further refinements.For the transition-SST turbulence model, the value of y+ in the wall-adjacent cell layers needs to be less than or equal to 1 to provide accurate results [18].To do so, the inflation layers were adjusted for each case in which the turbulence model was used, and the maximum y+ value for such cases was ensured to be below 1.The temperature converged at approximately 334.48 K after the mesh was refined.A mesh element size of 0.0018 m (approximately 500,000 elements) was chosen as sufficiently fine to produce accurate results.This mesh sizing was kept consistent throughout the testing of the baseline case.However, when the subsequent cases needed to be analyzed by using a turbulence model, then the mesh had to undergo further refinements.For the transition-SST turbulence model, the value of y+ in the wall-adjacent cell layers needs to be less than or equal to 1 to provide accurate results [18].To do so, the inflation layers were adjusted for each case in which the turbulence model was used, and the maximum y+ value for such cases was ensured to be below 1.

Design Cases
Table 3 shows the summary of the heat sink cases adopted in the present study.Each of the design cases can be identified with a specific design code starting from A1 to D2.Four different heat sink configurations with (A) rectangular, (B) trapezoidal, (C) curved, and (D) angled fins were examined.For the novel cases of trapezoidal and curved heat sink configurations, a shroud located at the end of the fins was considered.However, for

Design Cases
Table 3 shows the summary of the heat sink cases adopted in the present study.Each of the design cases can be identified with a specific design code starting from A1 to D2.Four different heat sink configurations with (A) rectangular, (B) trapezoidal, (C) curved, and (D) angled fins were examined.For the novel cases of trapezoidal and curved heat sink configurations, a shroud located at the end of the fins was considered.However, for the angled fin case, the optimized configuration without the shroud as proposed by Zhang et al. [12] was tested and compared with the others.A schematic representation of all the design cases used in the present study is shown in Figure 3. Certain repeating cases (A2, C5, D2, D3, and D4) on the basis of the variation in their fin length and spacing are not included in Figure 3.

Rectangular Fin (Case A)
Figure 3 shows the schematic of the rectangular baseline fin design with Code A1.Before any novel design ideas could be implemented, a set of baseline results had to be established with a rectangular profiled fin.As already established by Bar-Cohen and Rohsenow [5], there exists an optimal spacing for an array of vertical plate fins that can minimize the thermal resistance of a heat sink with a fixed base area.As the spacing increases between each fin, the number of fins that can fit on the fixed base-plate decreases.To ensure a more-consistent comparison between spacings, only fin spacings that correlated to a whole number of fins fitting on the fixed base-plate were considered.To find this spacing, the geometry was initially set to have a channel width of 4 mm, which corresponded to 25 fins fit onto the base-plate and increased until the fin spacing of 50 mm was achieved, corresponding to 2 fins on the base-plate.With the spacing set to the newly found optimal value, the fin length was incrementally varied from 30 mm to 200 mm.The size of the domain was configured to adjust to the varying fin length to provide a sufficiently accurate domain size.

Trapezoidal Fin (Case B)
Figure 3 shows the schematic of trapezoidal fin designs with Codes B1 and B2.Once a suitably optimized baseline has been established, testing could begin on novel designs.This innovative design featured the inclusion of a planar shroud at the end of the fin length.This shroud served to add more surface area to the array and acted as a boundary to force the air to flow upwards.These new fin profiles had the same base depth, width, and fin thickness, but differing fin upper and lower length.The fin's midpoint length was fixed at 100 mm, while the bottom length of the fin (fin lower length), as shown in Figure 4a, was controlled by a design parameter, where the length of the top profile (fin upper length) would change accordingly to create the trapezoidal shape.By configuring the geometry in this way, many permutations could be generated by only varying one design parameter, thus making it far simpler to find an optimal value.When the lower length was set to 100 mm, a rectangular fin profile was generated.This fin was identical to the baseline, but included the 1 mm shroud at its end.For a lower length less than 100 mm, a diverging fin was created, while a bottom length greater than 100 mm resulted in a converging fin shape.Using an identical simulation setup, domain size, and mesh as used for optimizing the baseline heat sink, a natural convection study was conducted for this novel design.The bottom parameter varied from 150 mm to 50 mm in 10 mm increments.The trapezoidal design was also tested for a midpoint of 50 mm in an equivalent manner to assess whether the behavior present was consistent for a different fin length (Case B2).

Curved Fin (Case C)
Figure 3 shows the schematic of the curved fin designs with Codes C1 to C4.Similar to the trapezoidal fin design case, a new model was created here featuring a shroud attached to the ends of each fin, following a curved spline.This spline was equation-driven and connected with two vertex points at its ends.While each end was fixed, the spline's

Curved Fin (Case C)
Figure 3 shows the schematic of the curved fin designs with Codes C1 to C4.Similar to the trapezoidal fin design case, a new model was created here featuring a shroud attached to the ends of each fin, following a curved spline.This spline was equation-driven and connected with two vertex points at its ends.While each end was fixed, the spline's midpoint was controlled in the x-y position, as shown in Figure 4b.Using this new curved fin model, several different parameters were individually varied across various design cases.First, the horizontal position of the spline's midpoint was varied from 50 mm to 150 mm, while its vertical position was fixed at 50 mm (Case C1).Second, its vertical position was varied from 20 mm to 80 mm, while the horizontal position of the spline was fixed at 50 mm from the base-plate (Case C2).Third, the bottom fin length was also kept constant at 100 mm, and the upper profile's length was varied, creating a variety of hyperbola-like curves, while the splines x and y positions were both fixed at 50 mm (Case C3).Since these fin shapes would suffer from an increased thermal resistance as the area of the fin decreased, the heat transfer coefficient was used as the parameter to be maximized.With a local optimal fin shape from Case C3, a variation of the overall fin length was conducted to investigate the presence of an optimal fin length that maximized the benefits of the novel curved shape (Case C4).To accurately compare the curved fins to rectangular fins, the lengths of the curved fins were chosen so that their surface area would be equal to a corresponding-length rectangular fin.This enabled a fair comparison between both fin types in terms of their equal surface areas for heat transfer.With the optimal spacing of the rectangular fin, testing was carried out on this innovative design (Case C5) to see whether any better heat transfer performance could be achieved with a different fin spacing.Using the same methodology as for Design Case A1, the fin spacing was varied for an increasing fin spacing that corresponded to a whole number of fins per the size of the base-plate.The curved fin was set to the dimensions that corresponded to a fin area equal to the 100 mm rectangular baseline, so a comparison of the spacings would be as accurate as possible.

Angled Fin (Case D)
Figure 3 shows the schematic of the angled fin design with Code D1.Angled fins can result in a W-type configuration, as shown in Figure 5 from Zhang et al. [12].The Design Code D2 configuration was similar to D1, but had a different boundary condition, as shown in Figure 6.For a valid comparison, all the design cases in the present study were tested for the same boundary conditions.The fin design in [12] covered a 200 mm-wide base-plate with four vertical columns of angled fins, which worked out to be one column per 50 mm base-plate width.To conform to the fixed 100 mm width in the present study, only a 2-column section was used.Figure 5 shows the domain adopted in the present study for the angled fin cases.
Zhang et al. [12] examined several parameters and their effect on the heat transfer performance of the overall system.Rather than conducting our own variation of these parameters, the optimal values proposed in [12] were chosen and used for the present design cases of angled fins.The base-plate was applied with a heat flux of 4000 W/m 2 .Zhang et al. [12] used a turbulence model including radiation.However, in the present study, all other heat sink configurations were tested with a laminar flow model without radiation.As such, to ensure a reliable comparison, the angled fin Design Cases D1 and D2, as well as the baseline rectangular fin case were tested for the following three simulation setups, each of which used their optimal geometry configuration: (i) laminar model with radiation disabled, (ii) laminar model with radiation enabled, and (iii) transition-SST turbulence model with radiation disabled.Zhang et al. [12] used a surface-to-surface (S2S) radiation model in their study.While the S2S model can produce accurate results, it can be extremely computationally expensive, especially with a high number of surfaces present in the model, as in the case for the angled fins.To overcome this, a discrete ordinance (DO) model was chosen in the present study instead, as it can handle complex geometries such as those in the case for angled fins [18].The design parameters used for the angled fin cases in the present study are shown in Table 4 compared to those used in Zhang et al. [12] for their full W-type heat sink.
Fluids 2023, 8, x FOR PEER REVIEW 9 of 24 shown in Figure 6.For a valid comparison, all the design cases in the present study were tested for the same boundary conditions.The fin design in [12] covered a 200 mm-wide base-plate with four vertical columns of angled fins, which worked out to be one column per 50 mm base-plate width.To conform to the fixed 100 mm width in the present study, only a 2-column section was used.Figure 5 shows the domain adopted in the present study for the angled fin cases.Zhang et al. [12] examined several parameters and their effect on the heat transfer performance of the overall system.Rather than conducting our own variation of these parameters, the optimal values proposed in [12] were chosen and used for the present design cases of angled fins.The base-plate was applied with a heat flux of 4000 W/m 2 .Zhang et al. [12] used a turbulence model including radiation.However, in the present study, all other heat sink configurations were tested with a laminar flow model without radiation.As such, to ensure a reliable comparison, the angled fin Design Cases D1 and D2, as well as the baseline rectangular fin case were tested for the following three simulation setups, each of which used their optimal geometry configuration: (i) laminar model with radiation disabled, (ii) laminar model with radiation enabled, and (iii) transition-SST turbulence model with radiation disabled.Zhang et al. [12] used a surface-to-surface (S2S) radiation model in their study.While the S2S model can produce accurate results, it can be extremely computationally expensive, especially with a high number of surfaces present in the model, as in the case for the angled fins.To overcome this, a discrete ordinance (DO) model was chosen in the present study instead, as it can handle complex geometries such as those in the case for angled fins [18].The design parameters used for the angled fin cases in the present study are shown in Table 4 compared to those used in Zhang et al. [12] for their full W-type heat sink.The computational domains adopted in the present study for the cases with angled fins are shown in Figure 6.Cases D1 and D2 differed only in terms of the boundary conditions applied on either side of the angled fin heat sink.Case D1 represents a confined geometry with symmetry boundary conditions applied on both sides, whereas Case D2 only used a single symmetry condition on the right-hand side.Case D2 provided additional exhaust space along the left side of the heat sink for the natural convection buoyant plumes to exit the heat sink volume.Case D2 also featured a pressure outlet boundary condition instead of a symmetry boundary condition on the left side, as shown in Figure 6.

Results
The performance of the fin designs was assessed by measuring the thermal resistance of the entire fin array with a fixed 100 mm × 100 mm base area, for a uniform application of heat flux (4000 W/m 2 ).This can be calculated as the ratio between the temperature difference and input heat power.An adapted version of this equation as shown in Equation  The computational domains adopted in the present study for the cases with angled fins are shown in Figure 6.Cases D1 and D2 differed only in terms of the boundary conditions applied on either side of the angled fin heat sink.Case D1 represents a confined geometry with symmetry boundary conditions applied on both sides, whereas Case D2 only used a single symmetry condition on the right-hand side.Case D2 provided additional exhaust space along the left side of the heat sink for the natural convection buoyant plumes to exit the heat sink volume.Case D2 also featured a pressure outlet boundary condition instead of a symmetry boundary condition on the left side, as shown in Figure 6.

Results
The performance of the fin designs was assessed by measuring the thermal resistance of the entire fin array with a fixed 100 mm × 100 mm base area, for a uniform application of heat flux (4000 W/m 2 ).This can be calculated as the ratio between the temperature difference and input heat power.An adapted version of this equation as shown in Equation ( 1) was used to calculate the thermal resistance based on the temperatures obtained from the CFD study.Since only half a channel width was modelled and not the entire array, Equation (1) features 2N fins Q in,hc in the denominator.Here, T b,m and Q in,hc are the mean base temperature and applied power for half of the channel width, respectively, and N fins is the whole number of fins resulting from that channel spacing.This T b,m corresponds to the area-weighted average temperature of the heat source.The heat transfer coefficient of the system was also used as a secondary metric to assess thermal performance and calculated as shown in Equation (2) taken from [21] in which A hc represents the surface area of half of the channel including both the surface areas of the fin and base-plate exposed to the air.This equation also includes the conductive thermal resistance arising from the base-plate, as well as the fins.
3.1.Rectangular Fin 3.1.1.Optimal Baseline Fin Spacing An optimal fin spacing was found to exist for the 100 mm fin length for the baseline case of rectangular fins.At a spacing of 8.33 mm, the thermal resistance for the entire fin array was at its minimum.For a smaller fin spacing, the additional surface area effect was counteracted by the confinement effect on the boundary layers.This caused a dramatic increase in the thermal resistance due to a decrease in the heat transfer coefficient.For spacing larger than the optimal, the thermal resistance steadily increased while the heat transfer coefficient plateaued at a value of approximately 6.2 W/m 2 K from a spacing of 14 mm onward.For an array of fins, the configuration that minimizes the overall thermal resistance is the most desirable for increased performance.Note that the optimal fin spacing was dictated purely by the configuration that produced the lowest thermal resistance.This choice excluded looking at the heat transfer coefficient, which is also an important metric when assessing thermal performance.When considering the heat transfer coefficient, it appeared that a spacing of 9.091 mm may be a better choice.As can be seen from Table 5, while increasing the fin spacing to 9.091 mm resulted in a very minor increase in the thermal resistance (only 0.4%), the heat transfer coefficient of the array increased dramatically (an 8.3% increase) as this new configuration used one less fin.The latter spacing was chosen to be used in the further design cases as it was determined to provide the best performance.Model validation is of the utmost importance to ensure that CFD-derived results are representative of reality.The parameters used to validate the obtained results were the heat transfer coefficient (h) and thermal resistance (R th ) of the heat sink array, calculated using Equation (5).Equation ( 3) is a semi-empirical correlation for two symmetrically heated isothermal plates for a specific fin spacing S proposed by Elenbaas [2], wherein the average Nusselt number (Nu S ) and Rayleigh number ( Ra S ) for a specific fin spacing are given by Equation ( 4).The thermal resistance and heat transfer coefficient are, thus, calculated using Equation ( 5), in which the specific values of parameters used are given in Tables 1 and 2. The equations as in (4) represent the Nusselt number, as well as the Rayleigh number.The Nusselt number is the ratio of convective heat transfer to the conductive heat transfer across a fluid boundary layer adjacent to a solid surface, and it is used to quantify the improvement of heat transfer due to convective effects compared to pure conduction.The convective heat transfer coefficient is represented in terms of Fourier's law of heat transfer, wherein the rate of heat transfer through a body is directly proportional to the negative temperature gradient in the direction of heat flow.The Rayleigh number is the ratio of buoyancy to viscous forces present in the fluid, and it is used to determine the type of heat transfer present within the fluid layer.A relatively small Rayleigh number indicates that the heat transfer is dominated by conduction, and a relatively higher number represents natural convection as bulk motion becomes significant, leading to higher heat transfer rates.
As observed in Figure 7, the results obtained in the present study were in good agreement with the Elenbaas correlation [2].For the heat transfer coefficient, the present values diverged from the experimental correlation as the fin spacing increased beyond 15 mm.The thermal resistance maintained a similar value to the experimental values as the fin spacing increased; it only deviated when the fin spacing was less than the optimal and, even then, only in a minor way.This deviation was because the obtained results slowly turned to move closer to the maximum possible resistance that would arise when the air flow path for natural convection is blocked.The R th curve as seen in Figure 7 tends to rise up sharply for a fin spacing below 8 mm.Eventually, this shows that, even if the fin spacing was reduced further to much smaller values below 4 mm, the R th would remain almost the same as the maximum value observed.As a result, this particular 4 mm to 8 mm region of fin spacing was very sensitive, and one may reasonably expect some degree of deviation in the results.Thus, the computational results of the rectangular fin variation were considered validated.Furthermore, to ensure that the present modelling of only half of the channel width was in line with the full heat sink analysis, a larger model encompassing half of the entire heat sink with symmetry boundary condition on one side and exhaust into the fluid domain on the other side was tested for Case A1.The obtained results for the case with the half heat sink exhibited a minor deviation from that of the half channel analysis.It was observed that only a 2% increase in the thermal resistance, as well as a corresponding decrease in the heat transfer coefficient were observed in the half channel analysis when compared with the case of half heat sink.
Fluids 2023, 8, 234 where T fin represents the area-weighted average temperature of the surface area of the fin that is in contact with the fluid domain and T b,m represents the area-weighted average temperature of the heat source, i.e., the mean base temperature.The ambient temperature of the fluid was considered to be T ∞ = 300 K.
sidered validated.Furthermore, to ensure that the present modelling of only half of the channel width was in line with the full heat sink analysis, a larger model encompassing half of the entire heat sink with symmetry boundary condition on one side and exhaust into the fluid domain on the other side was tested for Case A1.The obtained results for the case with the half heat sink exhibited a minor deviation from that of the half channel analysis.It was observed that only a 2% increase in the thermal resistance, as well as a corresponding decrease in the heat transfer coefficient were observed in the half channel analysis when compared with the case of half heat sink.
where Tfin represents the area-weighted average temperature of the surface area of the fin that is in contact with the fluid domain and Tb,m represents the area-weighted average temperature of the heat source, i.e., the mean base temperature.The ambient temperature of the fluid was considered to be T∞ = 300 K.

Rectangular Fin Length Variation
Figure 8 shows the variation of the rectangular fin length with optimal spacing.The fin length was varied with the fin spacing set to the optimal value 9.09 mm, which was found as a result of optimization.As the fin length increased, the fin array thermal resistance decreased, resulting in a higher rate of heat transfer.This was due to the increased surface area for a longer fin, which allowed the provision for more heat to transfer (see Figure 8a for details).As can be seen in Figure 8b, the fin efficiency dropped significantly when the length of the fin increased.The fin efficiency was directly proportional to its heat transfer coefficient.For the shorter fin case with reduced L, the fin efficiency was higher, and thus, the heat transfer coefficient would be high as well.This shorter fin case added more compactness with a capability of delivering a higher heat transfer rate over a small area.This was also the case for the mass specific heat transfer coefficient.

Rectangular Fin Length Variation
Figure 8 shows the variation of the rectangular fin length with optimal spacing.The fin length was varied with the fin spacing set to the optimal value 9.09 mm, which was found as a result of optimization.As the fin length increased, the fin array thermal resistance decreased, resulting in a higher rate of heat transfer.This was due to the increased surface area for a longer fin, which allowed the provision for more heat to transfer (see Figure 8a for details).As can be seen in Figure 8b, the fin efficiency dropped significantly when the length of the fin increased.The fin efficiency was directly proportional to its heat transfer coefficient.For the shorter fin case with reduced L, the fin efficiency was higher, and thus, the heat transfer coefficient would be high as well.This shorter fin case added more compactness with a capability of delivering a higher heat transfer rate over a small area.This was also the case for the mass specific heat transfer coefficient.Figure 9 shows the variation of the trapezoidal fin lower length with optimal spacing for a midpoint length of 100 mm.As observed in Figure 9a, the variation of the lower length of the trapezoidal fin heat sink showed that, when its value was set to 98 mm (corresponding to an upper length of 102 mm), the results showed a decrement in the thermal resistance (R th ) and an increment in the heat transfer coefficient (HTC).

Design Case B1
Figure 9 shows the variation of the trapezoidal fin lower length with optimal spacing for a midpoint length of 100 mm.As observed in Figure 9a, the variation of the lower length of the trapezoidal fin heat sink showed that, when its value was set to 98 mm (corresponding to an upper length of 102 mm), the results showed a decrement in the thermal resistance (Rth) and an increment in the heat transfer coefficient (HTC).While the change in R th and HTC was minor at a lower fin length of 98 mm compared to 100 mm (−0.07%for R th and 0.07% for HTC), the difference in performance compared to the baseline was significant (−4.7% for R th and 0.2% for HTC), showing the clear benefits of adding a planar shroud to the baseline design.The volumetric flow rate was calculated by using the area-weighted average velocity passing through a plane in the middle of the fin and plotted as shown in Figure 9b.Since the midpoint of the fin was fixed, the area of the plane remained constant throughout the bottom length variation.The volumetric flow rate peaked at a fin lower length of 90 mm.These thermal improvements at the local optimum may be due to this increase in flow rate for the configuration combined with the higher fin efficiency, as can be seen in Figure 9b.

Design Case B2
Figure 10 shows the variation of the trapezoidal fin lower length with optimal spacing for a midpoint length of 50 mm.The testing was similar to Case B1, but for a midpoint length of 50 mm.The trends exhibited in the heat transfer coefficient and thermal resistance of the array were nearly identical to those observed for the lower length variation of the trapezoidal fin when the midpoint was set to 100 mm in Case B1.This implies that the change in performance was highly dependent on the angle of the shroud irrespective of overall length.It is worth noting that only two overall fin lengths (i.e., two different fin midlengths) were tested.A local optimal was found with a bottom fin length of 48 mm (corresponding to an upper length of 52 mm).At this length, there was a decrease in the thermal resistance and an increase in the heat transfer coefficient, as seen in Figure 10a.Both Cases B1 and B2 displayed optimal heat transfer when the trapezoidal fin was angled at 87.7 • (i.e., 2.3 • degrees from vertical), while only two fin midpoint lengths were tested.This seemed to indicate that this may be the best angle for a shroud for a varying fin midpoint length.In this configuration, the volumetric flow rate was close to its maximum for the length variation.Similarly, the fin efficiency also increased as the lower length of the fin decreased.The improved performance to the rectangular shape was likely due to a combination of these factors, i.e., increased fluid flow past the heated surface and a more-effective fin. at 87.7° (i.e., 2.3° degrees from vertical), while only two fin midpoint lengths were tested.This seemed to indicate that this may be the best angle for a shroud for a varying fin midpoint length.In this configuration, the volumetric flow rate was close to its maximum for the length variation.Similarly, the fin efficiency also increased as the lower length of the fin decreased.The improved performance to the rectangular shape was likely due to a combination of these factors, i.e., increased fluid flow past the heated surface and a moreeffective fin.  Figure 11 shows the variation of the curved fin midpoint length with optimal spacing for the fin upper and lower lengths of 100 mm.As can be seen in Figure 11a, as the fin's midpoint length increased, the heat sink experienced a reduced thermal resistance, thus increasing the overall heat transfer of the system.While this improvement was genuine, it is worth noting that, as the fin midpoint length increased, so did the fin's surface area, which was directly proportional to the heat transferred.Interestingly, the improvement in thermal resistance began to converge after a midpoint length of 120 mm even with an increasing area.This indicates the existence of a local optimum geometry for a convex curved fin.This local optimum may be due to the high flow rate present under these geometry conditions.The volumetric flow rate passing through the inside of the fin channel can be seen in Figure 11b.The peak flow rate occurred at approximately a 110 mm midpoint length, with only a minor decrease at 120 mm.Looking at the heat transfer coefficient, as shown in Figure 11a, it began converging on a maximum as the fin midpoint length approached 50 mm.The heat transfer coefficient is essentially a measure of how effectively something dissipates heat, i.e., the higher the heat transfer coefficient, the greater amount of heat that can be transferred for a fixed area.In the context of heat sinks, the size of them can often be a factor of importance.Thus, designs that can transfer the same amount of heat, but across a smaller area, and thus take up a smaller volume, are more desirable.Since that configuration of the curved fin showed potential promise as a more-effective design, the further geometry variations were focused on designs with a midpoint length of 50 mm.Figure 11 shows the variation of the curved fin midpoint length with optimal spacing for the fin upper and lower lengths of 100 mm.As can be seen in Figure 11a, as the fin's midpoint length increased, the heat sink experienced a reduced thermal resistance, thus increasing the overall heat transfer of the system.While this improvement was genuine, it is worth noting that, as the fin midpoint length increased, so did the fin's surface area, which was directly proportional to the heat transferred.Interestingly, the improvement in thermal resistance began to converge after a midpoint length of 120 mm even with an increasing area.This indicates the existence of a local optimum geometry for a convex curved fin.This local optimum may be due to the high flow rate present under these geometry conditions.The volumetric flow rate passing through the inside of the fin channel can be seen in Figure 11b.The peak flow rate occurred at approximately a 110 mm midpoint length, with only a minor decrease at 120 mm.Looking at the heat transfer coefficient, as shown in Figure 11a, it began converging on a maximum as the fin midpoint length approached 50 mm.The heat transfer coefficient is essentially a measure of how effectively something dissipates heat, i.e., the higher the heat transfer coefficient, the greater amount of heat that can be transferred for a fixed area.In the context of heat sinks, the size of them can often be a factor of importance.Thus, designs that can transfer the same amount of heat, but across a smaller area, and thus take up a smaller volume, are more desirable.Since that configuration of the curved fin showed potential promise as a more-effective design, the further geometry variations were focused on designs with a midpoint length of 50 mm.

Design Case C2
The midpoint height (i.e., the vertical distance from the bottom of the heat sink to the curved spline's midpoint; see Figure 4) was varied from 20 mm to 80 mm.Any closer midpoint to the top or bottom of the profiles resulted in too extreme of a curve to be reli-

Design Case C2
The midpoint height (i.e., the vertical distance from the bottom of the heat sink to the curved spline's midpoint; see Figure 4) was varied from 20 mm to 80 mm.Any closer midpoint to the top or bottom of the profiles resulted in too extreme of a curve to be reliably meshed.The different permutations resulted in a change in the type of curve on the fin's shroud.A midpoint height of 30 mm will produce a curved fin of equal area to a height at 70 mm due to the symmetry of the design and linearity of the height variation.The thermal resistance and heat transfer coefficient would be expected to be symmetrical as the midpoint height increases.
Interestingly, Figure 12 shows only a minor increase in the thermal resistance as the midpoint height increased.The opposite was true for the heat transfer coefficient, which decreased as the thermal resistance increased.While it may seem like a curve with a lower midpoint height provided the lowest thermal resistance and highest heat transfer coefficient, it should be noted that the numerical stability of the solution decreased significantly as the midpoint height varied in either direction from 50 mm, which may be indicative of unsteady flow separation or transition.

Design Case C3
The fin's bottom profile was fixed at a length of 100 mm, while its midpoint length was fixed at 50 mm in the horizontal and vertical direction.Only the top profile length was allowed to vary.As the upper profile decreased from 100 mm, the thermal resistance of the system increased as observed in Figure 13a.This was expected as the decrease in the upper length reduced the overall fin area.Interestingly, the heat transfer coefficient peaked at a value of 75 mm.This indicated the presence of a local optimum.Why this was the case is unclear, as the maximum flow rate through the fin channel occurred at a length of 100 mm and a maximum fin efficiency at 40 mm as seen in Figure 13b.The answer may be found in the resemblance of this geometry to the hyperbolic curve found in natural draft cooling towers.The shape of these cooling towers forces the rising hot air through a narrow throat section, resulting in an acceleration of the fluid.To assess whether flow unsteadiness or transition would be observed in such configurations, this same case was run using the transition-SST turbulence model and compared to the laminar flow model results.When run with the turbulent model, the fluid velocities inside the fin channels were significantly lower with the maximum velocity through the mid-plane of the fin being 0.158 m/s and 0.076 m/s for the laminar and turbulent model, respectively.While the fluid flow was much slower inside the fin channel for the turbulent model, outside, the speed of the fluid significantly increased.The transition model resulted in a higher base temperature of 359.1 K compared to 338.4 K for the laminar model.

Design Case C3
The fin's bottom profile was fixed at a length of 100 mm, while its midpoint length was fixed at 50 mm in the horizontal and vertical direction.Only the top profile length was allowed to vary.As the upper profile decreased from 100 mm, the thermal resistance of the system increased as observed in Figure 13a.This was expected as the decrease in the upper length reduced the overall fin area.Interestingly, the heat transfer coefficient peaked at a value of 75 mm.This indicated the presence of a local optimum.Why this was the case is unclear, as the maximum flow rate through the fin channel occurred at a length of 100 mm and a maximum fin efficiency at 40 mm as seen in Figure 13b.The answer may be found in the resemblance of this geometry to the hyperbolic curve found in natural draft cooling towers.The shape of these cooling towers forces the rising hot air through a narrow throat section, resulting in an acceleration of the fluid.To assess whether flow unsteadiness or transition would be observed in such configurations, this same case was run using the transition-SST turbulence model and compared to the laminar flow model results.When run with the turbulent model, the fluid velocities inside the fin channels were significantly lower with the maximum velocity through the mid-plane of the fin being 0.158 m/s and 0.076 m/s for the laminar and turbulent model, respectively.While the fluid flow was much slower inside the fin channel for the turbulent model, outside, the speed of the fluid significantly increased.The transition model resulted in a higher base temperature of 359.1 K compared to 338.4 K for the laminar model.

Design Case C4
In Figure 14a, the heat transfer coefficient and thermal resistance of rectangular and curved fins for varying lengths are plotted.The x-axis indicates the length of the rectangular fins, with the curved fin at that point plotted, which represents an equal area to the corresponding length of the rectangular fin.By comparing these two fin designs based on an equal area, the thermal improvements of one design over the other are easier to see.As was expected, the thermal resistance of both arrays was at its lowest when the fin length was at 200 mm.This is obviously the configuration with the highest fin surface area, so it would be expected to provide the largest temperature drop for the heat source.This has been shown to be the case in numerous other studies and is a logical result here.The heat transfer coefficient was at its maximum when the fin length was 50 mm.This is where the fins had their highest fin efficiency and could dissipate the most heat per unit area.For nearly all rectangular fin lengths tested, the curved fin with an equivalent area provided a decreased thermal resistance and increased heat transfer coefficient, demonstrating its design superiority over the baseline design.The difference in performance was not constant with the maximum improvements in the design when the rectangular fin had a length of 80 mm.The curved fin for this length had a decreased thermal resistance of 6.6% and an increased heat transfer coefficient of 5.6%.These results demonstrate the benefit of this curved fin design over the traditional rectangular baseline regardless of fin length not just in its ability to produce a lower heat source temperature, but also a more-effective design.

Design Case C4
In Figure 14a, the heat transfer coefficient and thermal resistance of rectangular and curved fins for varying lengths are plotted.The x-axis indicates the length of the rectangular fins, with the curved fin at that point plotted, which represents an equal area to the corresponding length of the rectangular fin.By comparing these two fin designs based on an equal area, the thermal improvements of one design over the other are easier to see.As was expected, the thermal resistance of both arrays was at its lowest when the fin length was at 200 mm.This is obviously the configuration with the highest fin surface area, so it would be expected to provide the largest temperature drop for the heat source.This has been shown to be the case in numerous other studies and is a logical result here.The heat transfer coefficient was at its maximum when the fin length was 50 mm.This is where the fins had their highest fin efficiency and could dissipate the most heat per unit area.For nearly all rectangular fin lengths tested, the curved fin with an equivalent area provided a decreased thermal resistance and increased heat transfer coefficient, demonstrating its design superiority over the baseline design.The difference in performance was not constant with the maximum improvements in the design when the rectangular fin had a length of 80 mm.The curved fin for this length had a decreased thermal resistance of 6.6% and an increased heat transfer coefficient of 5.6%.These results demonstrate the benefit of this curved fin design over the traditional rectangular baseline regardless of fin length not just in its ability to produce a lower heat source temperature, but also a more-effective design.

Design Case C4
In Figure 14a, the heat transfer coefficient and thermal resistance of rectangular and curved fins for varying lengths are plotted.The x-axis indicates the length of the rectangular fins, with the curved fin at that point plotted, which represents an equal area to the corresponding length of the rectangular fin.By comparing these two fin designs based on an equal area, the thermal improvements of one design over the other are easier to see.As was expected, the thermal resistance of both arrays was at its lowest when the fin length was at 200 mm.This is obviously the configuration with the highest fin surface area, so it would be expected to provide the largest temperature drop for the heat source.This has been shown to be the case in numerous other studies and is a logical result here.The heat transfer coefficient was at its maximum when the fin length was 50 mm.This is where the fins had their highest fin efficiency and could dissipate the most heat per unit area.For nearly all rectangular fin lengths tested, the curved fin with an equivalent area provided a decreased thermal resistance and increased heat transfer coefficient, demonstrating its design superiority over the baseline design.The difference in performance was not constant with the maximum improvements in the design when the rectangular fin had a length of 80 mm.The curved fin for this length had a decreased thermal resistance of 6.6% and an increased heat transfer coefficient of 5.6%.These results demonstrate the benefit of this curved fin design over the traditional rectangular baseline regardless of fin length not just in its ability to produce a lower heat source temperature, but also a more-effective design.

Design Case C5
The behavior of the heat transfer coefficient and thermal resistance for a curved fin array with varying fin spacing was similar to that of the rectangular fins for the same spacing variation.For the curved fin, there existed a fin spacing that minimized the array's thermal resistances.As the spacings decreased from that optimal spacing, the thermal resistance increased dramatically, while the heat transfer coefficient decreased in the same manner.This was nearly identical to the behavior expected of a rectangular fin and can be seen in the closeness in values in Figure 14b.As the spacing increased beyond the optimum, the heat transfer coefficient plateaued after a spacing of approximately 20 mm, while the thermal resistance steadily increased.This behavior was similar to that displayed by the rectangular fin case, except for the rectangular fins providing a more-pronounced change in the thermal resistance and heat transfer coefficient for an increasing fin spacing than the curved fin case.Due to the array being a fixed width, the spacings available were limited to those values that corresponded to a whole number of fins on the base-plate.Because of this, the optimal spacing of two fins can not only be considered, but also the whole number of fins resulting from that spacing.As a result, there may exist an optimal spacing between two fins, but it fell between the increments available for the specific base-plate size chosen for this study (i.e., 100 mm × 100 mm).In Figure 14b, the optimal fin spacing for both the rectangular and novel curved fin design was dictated by the spacing that minimized the thermal resistance for the array.While this is a reasonable definition, it does not fully consider the efficiency of the design.The difference between the thermal resistance at 8.33 mm and 9.09 mm was essentially negligible, while the increase in the heat transfer coefficient was significant.A more-informed decision would suggest that the optimal fin spacing for both the baseline and curved fin would be at 9.09 mm, where the thermal resistance is sufficiently low, but the heat transfer coefficient is also at a high level.

Angled Fin (Design Cases D1 and D2)
The optimized rectangular baseline fin case was compared against the optimized angled fin case adopted from Zhang et al. [12], for identical heat inputs and boundary conditions.Their geometrical parameters were chosen to be identical apart from their respective fin arrangements and fin spacings.The fins were tested against one another for three different simulation setups as follows: (i, ii) laminar flow model with discrete ordinance (DO) radiation (i) disabled and (ii) enabled and the (iii) transition-SST model with radiation disabled.This led to a total of six simulations, the results of which are summarized in Tables 6-8.The percentage difference for the thermal resistance and heat transfer coefficient between each tested case with the baseline rectangular fin (for laminar with no radiation) are also shown in Table 6.A better design is one that produces a decrease in the thermal resistance compared to the baseline (i.e., a negative percentage difference).An increase in the heat transfer coefficient is also desirable (i.e., a positive percentage difference).heat transfer by radiation was included with the overall transfer of heat from the heat sink to the surroundings.This additional mode of heat transfer increased the overall amount of energy that the heat sink could dissipate, thus decreasing the base temperature.Like in the first simulation case, the angled fins as in Case D2 here as well performed significantly better with a base temperature of 340.5 K compared to 345.7 K for the rectangular fins.
When tested with the transition-SST turbulence model, the highest individual improvements were achieved for each of the tested cases in comparison to its corresponding R th , HTC, and temperatures mentioned in Tables 6 and 7.However, as shown in Table 8, the difference in this improvement when compared to the baseline case was observed to be smallest.In comparison with the laminar flow results of Table 6, the angled fins here in Case D1 with a 55 mm fin length showed a temperature drop of almost 19 K compared to a 10 K drop observed for the rectangular fin Case A1.This resulted in an increased heat transfer coefficient, which was mainly due to a turbulence model being used over the laminar model.Even in globally laminar flow (with a Rayleigh number below 10 9 ), the transition-SST model could predict locally transitional or turbulent flow in certain regions.The model could, thus, predict more-chaotic motion and effective fluid mixing, resulting in higher heat transfer rates between the heat sink and the fluid [22].
The findings of Zhang et al. [12] were not entirely confirmed by these results.The closest comparison would be Case D2 (angled fins with symmetry on one side and pressure outlet on the other side) compared to Case A1.Based on the present results, Case D2 only outperformed the rectangular fins for a shorter fin length of 55 mm.However, for longer fin lengths (100 mm), this proposition failed as the D2 heat sink's base temperature almost matched that of the rectangular array (A1).Moreover, Case D1 with symmetry on both sides performed worse altogether, which seems to indicate that the angled fin approach would not be beneficial for a general heat sink base area, but rather that specific combinations of base dimensions (and possibly Rayleigh number) could lead to enhancements, as observed in Zhang et al. [12].

Optimal Design and the Choice of Objective Function
The best-case scenario for each of the design cases was taken and compared against the optimally spaced rectangular fin baseline in terms of thermal resistance and heat transfer coefficient.The percentage difference in the thermal performance of the heat sink's array for each of the subsequent design cases in comparison to the rectangular fin baseline was also calculated.Table 9 details the varying thermal performance of the different design cases when configured to their respective local optima.Several trends can be observed from the variation of thermal performance based on the differing geometries:

•
Firstly, the thermal resistance is inherently linked to the total heat transfer area.Case B1 compared to B2 had a significantly lower thermal resistance and, thus, lower temperature.Similarly, Case C1, which provided the lowest thermal resistance out of the 12 cases listed in Table 9, also had the highest surface area.The converse of this was true for the heat transfer coefficient, with more-compact designs producing higher heat transfer coefficients given their higher fin efficiencies due to their closer proximity to the heat source.

•
The most-marked improvement was arguably seen in Case C4.This design had the same exact area as the baseline, yet provided a noteworthy decrease in the thermal resistance while simultaneously improving the heat transfer coefficient.This proved that the design itself was more effective and was not merely reflecting a performance enhancement through an increase in the heat transfer area.

•
The "optimal" design is highly dependent on the relevant objective function and constrictions placed on a specific scenario.For example, in light-weight applications, where the mass of the heat sink is to be limited, then a design with a high heat transfer coefficient per unit mass takes priority, whereas if the overall thermal performance of the system is the priority and there are no limitations on size, then a larger-area fin such as Case C1 would prove to be optimal.There may exist a global optimal in some configuration of Case C, but from the various parametric design studies conducted, there existed several local optima that provided enhancement over the baseline.

Non-Dimensional Analysis and Scaling of Design Cases
In Figure 15, the horizontal axis shows the average fin length L of the trapezoidal (Cases B1 and B2) and curved (Cases C1 to C5) fins, normalized by the fixed fin height of the baseline Case A1 (H A1 = 100 mm).For clarity, "height" refers to the direction parallel to gravity and "length" refers to the direction perpendicular to the base area, i.e., perpendicular to gravity.Similarly, the vertical axes in Figure 15 show the thermal resistance, heat transfer coefficient, heat transfer coefficient by mass, and fin efficiency, all normalized by the values of the baseline case (Case A1).Recall that Design Cases B1 and C4 were the only two cases that showed simultaneous improvement of all the performance parameters, with C4 outperforming B1.The results of Design Case C4 showed better performance than the baseline rectangular fin design for the thermal resistance, heat transfer coefficient, heat transfer coefficient by mass, and fin efficiency for all the simulated L/H A1 ratios.The optimal performance was achieved at an L/H A1 ratio of 0.8 (approximately), with local maxima and minima for the performance parameters.However, a significant decline in the performance was observed when there was a decrease or increase in the L/H A1 ratio.Hence, a design recommendation of L/H A1 = 0.8 emerged for Design Case C4.
The obtained scaling factors could be used for designing these two novel heat sinks for any given dimensions without running further simulations, as long as the Rayleigh number, ambient conditions, and fluid properties remain comparable.It is important to note that this is a general strategy and can be adopted for the design of any new natural convection heat sink designs.For instance, at a given heat load condition, the effective heat transfer coefficient will be obtained by multiplying it with the scaling factor in Equation ( 4) and back calculating the optimized spacing using Equation (3).However, the validity of this hypothesis must be further tested and is the subject of future work.The obtained scaling factors could be used for designing these two novel heat sinks for any given dimensions without running further simulations, as long as the Rayleigh number, ambient conditions, and fluid properties remain comparable.It is important to note that this is a general strategy and can be adopted for the design of any new natural convection heat sink designs.For instance, at a given heat load condition, the effective heat transfer coefficient will be obtained by multiplying it with the scaling factor in Equation (4) and back calculating the optimized spacing using Equation (3).However, the validity of this hypothesis must be further tested and is the subject of future work.

Conclusions
Based on the scientific literature review on natural convection heat sink design, there is currently no established strategy for creating a new heat sink that outperforms a classical rectangular plate fin array design with optimized fin spacing.The optimal fin spacing for a baseline rectangular fin case studied here was between 8.33 mm and 9.09 mm.This value was obtained by minimizing the thermal resistance for a given base area (100 mm × 100 mm) and the thermal boundary conditions (air at atmospheric pressure and 300 K, uniform heat flux of 4000 W/m 2 at the base) and simultaneously maintaining a high average heat transfer coefficient.These values were in close agreement with the semi-empirical model of Elenbaas [2].This optimized baseline rectangular fin generally performed better than the proposed novel designs in the scientific literature.In specific conditions, some enhancement has been observed [12], which was partly confirmed in this study.However, it is difficult to achieve a consistent marked improvement over the optimized rectangular fin baseline case.That said, there is still scope for designing novel heat sinks that can outperform the optimized baseline rectangular fins.Trapezoidal and curved fin shapes are

Conclusions
Based on the scientific literature review on natural convection heat sink design, there is currently no established strategy for creating a new heat sink that outperforms a classical rectangular plate fin array design with optimized fin spacing.The optimal fin spacing for a baseline rectangular fin case studied here was between 8.33 mm and 9.09 mm.This value was obtained by minimizing the thermal resistance for a given base area (100 mm × 100 mm) and the thermal boundary conditions (air at atmospheric pressure and 300 K, uniform heat flux of 4000 W/m 2 at the base) and simultaneously maintaining a high average heat transfer coefficient.These values were in close agreement with the semi-empirical model of Elenbaas [2].This optimized baseline rectangular fin generally performed better than the proposed novel designs in the scientific literature.In specific conditions, some enhancement has been observed [12], which was partly confirmed in this study.However, it is difficult to achieve a consistent marked improvement over the optimized rectangular fin baseline case.That said, there is still scope for designing novel heat sinks that can outperform the optimized baseline rectangular fins.Trapezoidal and curved fin shapes are two such proposed fin designs that outperform baseline rectangular fins, yet still represent very basic and easily manufactured geometries.Case C1 with curved fins had the most-pronounced reduction in thermal resistance at −5.5%, albeit with a 4.5% lower heat transfer coefficient.It is worth noting that this design had a larger surface area than the rectangular baseline.On the other hand, for a fixed surface area, the hyperboliclike curved fins of Case C4 featured the most-significant reduction in thermal resistance compared to the baseline at −4.1% and an increase in average heat transfer coefficient by +4.4%.The study also examined if the performance enhancements remained constant at varying fin lengths.The maximum improvement was achieved at a rectangular fin length of 80 mm.The curved fin Case C4 with an 80 mm fin length produced a −6.6% reduction in the thermal resistance and a +5.6% increase in the heat transfer coefficient.

Figure 1 .
Figure 1.Schematic representation of the full heat sink (left) and the computational geometry (right) adopted for CFD simulations.Not shown to scale.Gravity acts in the downward or negative ydirection.

Figure 1 .
Figure 1.Schematic representation of the full heat sink (left) and the computational geometry (right) adopted for CFD simulations.Not shown to scale.Gravity acts in the downward or negative y-direction.

Figure 2 .
Figure 2. (a) Mesh convergence study to identify sufficiently small element size for Case A1 (see Table3, Figure3).(b) Detail of the mesh at the boundary between the solid and fluid domain, where the inflation layer is used.

Figure 2 .
Figure 2. (a) Mesh convergence study to identify sufficiently small element size for Case A1 (see Table3, Figure3).(b) Detail of the mesh at the boundary between the solid and fluid domain, where the inflation layer is used.

Fluids 2023, 8 , 24 Figure 3 .
Figure 3. Schematic representation of all the design cases used with their respective design codes.

Figure 3 .
Figure 3. Schematic representation of all the design cases used with their respective design codes.

Figure 4 .
Figure 4. Boundary sketches used to generate fin shapes for trapezoidal and curved fin design cases.(a) The trapezoidal fin boundary sketch representing a possible variation of fin midpoint length and fin lower length corresponding to B1 and B2.(b) The curved fin boundary sketch representing a possible variation of the upper and lower length of the fin and the x-y position of the curve's midpoint corresponding C1 to C4.The curve is a Bezier curve defined by its start point, end point, and a vertex between which the resulting curve is formed.The x-y positions of all three points are defined in the figure.

Figure 4 .
Figure 4. Boundary sketches used to generate fin shapes for trapezoidal and curved fin design cases.(a) The trapezoidal fin boundary sketch representing a possible variation of fin midpoint length and fin lower length corresponding to B1 and B2.(b) The curved fin boundary sketch representing a possible variation of the upper and lower length of the fin and the x-y position of the curve's midpoint corresponding C1 to C4.The curve is a Bezier curve defined by its start point, end point, and a vertex between which the resulting curve is formed.The x-y positions of all three points are defined in the figure.

Figure 5 .
Figure 5. Angled fin adjusted domain used for the study.

Figure 5 .
Figure 5. Angled fin adjusted domain used for the study.

FluidsFigure 6 .
Figure 6.Angled fin design cases with Codes D1 (left) and D2 (right) having different computational domains.Not shown to scale.

Figure 6 .
Figure 6.Angled fin design cases with Codes D1 (left) and D2 (right) having different computational domains.Not shown to scale.

Figure 7 .
Figure 7. Validation of the variation of the fin spacing for a rectangular fin with the Elenbaas [2] semi-empirical correlation.

FluidsFigure 7 .
Figure 7. Validation of the variation of the fin spacing for a rectangular fin with the Elenbaas [2] semi-empirical correlation.

Figure 8 .Figure 9
Figure 8. Variation of heat transfer performance with rectangular fin length at optimal spacing.(a) Heat transfer coefficient (HTC) and thermal resistance (Rth).(b) Percentage fin efficiency and mass specific heat transfer coefficient.3.2.Trapezoidal Fin 3.2.1.Design Case B1 Figure 9 shows the variation of the trapezoidal fin lower length with optimal spacing

Figure 8 .
Figure 8. Variation of heat transfer performance with rectangular fin length at optimal spacing.(a) Heat transfer coefficient (HTC) and thermal resistance (R th ).(b) Percentage fin efficiency and mass specific heat transfer coefficient.

Figure 8 .
Figure 8. Variation of heat transfer performance with rectangular fin length at optimal spacing.(a) Heat transfer coefficient (HTC) and thermal resistance (Rth).(b) Percentage fin efficiency and mass specific heat transfer coefficient.

Figure 9 .
Figure 9. Variation of trapezoidal fin lower length with optimal spacing for a midpoint length of 100 mm.(a) Heat transfer coefficient (HTC) and thermal resistance (R th ).(b) Percentage fin efficiency and volumetric flow rate.

Figure 11 .
Figure 11.Variation of curved fin midpoint length with optimal spacing for the fin upper and lower lengths of 100 mm.(a) Heat transfer coefficient (HTC) and thermal resistance (Rth).(b) Percentage fin efficiency and volumetric flow rate.

Figure 11 .
Figure 11.Variation of curved fin midpoint length with optimal spacing for the fin upper and lower lengths of 100 mm.(a) Heat transfer coefficient (HTC) and thermal resistance (R th ).(b) Percentage fin efficiency and volumetric flow rate.

Fluids 2023, 8 ,
x FOR PEER REVIEW 16 of 24 as the midpoint height varied in either direction from 50 mm, which may be indicative of unsteady flow separation or transition.

Figure 12 .
Figure 12.Heat transfer coefficient (HTC) and thermal resistance (Rth) for variation of curved fin midpoint height.

Figure 12 .
Figure 12.Heat transfer coefficient (HTC) and thermal resistance (R th ) for variation of curved fin midpoint height.

Figure 13 .
Figure 13.Variation of curved fin upper length with optimal spacing for midpoint length fixed at 50 mm and lower length of 100 mm.(a) Heat transfer coefficient (HTC) and thermal resistance (Rth).(b) Percentage fin efficiency and volumetric flow rate.

Figure 14 .Figure 13 .
Figure 14.Variation of curved fin midpoint length with optimal spacing with upper and lower length of 100 mm.(a) The heat transfer coefficient and thermal resistance are plotted for variation of

Fluids 2023, 8 ,Figure 13 .
Figure 13.Variation of curved fin upper length with optimal spacing for midpoint length fixed at 50 mm and lower length of 100 mm.(a) Heat transfer coefficient (HTC) and thermal resistance (Rth).(b) Percentage fin efficiency and volumetric flow rate.

Figure 14 .
Figure 14.Variation of curved fin midpoint length with optimal spacing with upper and lower length of 100 mm.(a) The heat transfer coefficient and thermal resistance are plotted for variation of Figure 14.Variation of curved fin midpoint length with optimal spacing with upper and lower length of 100 mm.(a) The heat transfer coefficient and thermal resistance are plotted for variation of the mid fin length.(b) The fin efficiency and volumetric flow rate through the fin channel for variation of the mid fin length.

Figure 15 .
Figure 15.Non-dimensional representation of performance parameters for all the design cases: (a) thermal resistance, (b) heat transfer coefficient, (c) heat transfer coefficient by mass, and (d) fin efficiency.

Figure 15 .
Figure 15.Non-dimensional representation of performance parameters for all the design cases: (a) thermal resistance, (b) heat transfer coefficient, (c) heat transfer coefficient by mass, and (d) fin efficiency.

Table 1 .
Fixed parameter values used in this CFD-based heat sink optimization study.

Table 2 .
Material properties for this CFD-based heat sink optimization study.

Table 2 .
Material properties for this CFD-based heat sink optimization study.

Table 3 .
Summary of the heat sink configurations studied.

Design Code Fin Shape Parameter of Investigation Range of Testing (mm)
* Equivalent length of rectangular fin of equal area.

Table 3 .
Summary of the heat sink configurations studied.

Design Code Fin Shape Parameter of Investigation Range of Testing (mm)
* Equivalent length of rectangular fin of equal area.

Table 5 .
Optimal fin spacing for a rectangular fin heat sink of length 100 mm.

Table 6 .
Summary of results using the laminar flow model without radiation.Relative differences ∆ are shown in % with respect to Case A1 (55 mm or 100 mm fin length, respectively).

Table 9 .
Thermal performance of all the design cases with optimized fin spacing in terms of R th and HTC.Relative differences ∆ are shown in % with respect to Case A1 (55 mm or 100 mm fin length, respectively).