## 1. Introduction

The progressive size reduction of electronic devices is inevitably leading to a crucial heat dissipation issue in designing novel components [

1]. In fact, overheating conditions could significantly affect the performances and durability of electronic devices, eventually reducing their expected life [

2]. Even though a plethora of methods have been investigated to enhance heat dissipation in the last twenty years, air cooling by means of a fan coupled with properly designed heat sinks installed on the electronic device is currently the most robust, cost-effective and commonly adopted solution for electronics cooling [

3].

However, heat sinks are also finding increasing applications outside the traditional field of high-powered electronic devices. For example, heat sinks are currently applied for cooling purposes in industrial (e.g., solar applications) as well as consumer (e.g., LED) products, which are rapidly causing a change in their design and manufacture processes [

4,

5,

6,

7,

8,

9,

10]. Indeed, such newer application fields are adding further requirements to the traditional structural integrity and thermal performance ones: low cost and mass production needs are leading to a substantial rethinking of the materials and processes adopted for manufacturing heat sinks.

In particular, heat sinks with plate fins are amongst the most common and studied thermal management solutions for electronic and automotive devices [

11,

12]. Plate fin heat sinks (PFHS) differ in manufacturing methods (stamping, extrusion, bonding, folding, additive manufacturing), type of coolant (air, liquid), material (aluminum, copper, alloys, polymeric or composite materials) and flow regime (natural, forced) [

13]. Nowadays, air-cooled heat sinks manufactured by extrusion processes are the most widespread solution, being a satisfactory compromise between costs and thermal performances [

13,

14], both in

active (purposely designed fan) and

semi-active (fan already existing in the system) configurations. However, the choice of the heat sink characteristics maximizing thermal performances and minimizing production and operating costs for a particular application may be extremely complex, due to the multiple design options and restraints to be considered at the same time.

Several theoretical, numerical and experimental thermo-fluid dynamics studies on air-cooled PFHS have been carried out [

15,

16,

17,

18,

19,

20,

21], in order to achieve a more fundamental understanding of the involved phenomena. More recently, the inverse method coupled with numerical simulations and experimental temperature data have been adopted to determine the average convective heat transfer coefficient in PFHS [

22]; whereas a practical model for predicting the hydraulic and thermal performance of PFHS has proved accuracy within −14% to 12% for a wide range of Reynolds numbers [

23]. In summary, the main phenomena to be taken into account for a comprehensive thermal model are the (1) baseplate-to-fins spreading resistance; (2) thermal conduction through fins; (3) boundary layer development along fins; and (4) flow bypass (i.e., unshrouded heat sinks) [

13].

Starting from an adequate thermo-fluid model, the optimal configuration for the air-cooled PFHS can be then investigated. To this purpose, conventional thermal analysis tools are usually inadequate because both geometry and boundary conditions are not known

a priori. Therefore, numerous optimization procedures have been proposed so far. Saini and Webb [

24] adopted an analytical approach to identify the best configurations of plate-fin and plate-pin-fin heat sinks. Entropy generation minimization has been also used to optimize heat sink geometries [

25], as for example reported in the studies by Culham et al. [

26], Shih et al. [

27] and Betchen et al. [

28]. Iyengar and Bar-Cohen, instead, discussed a coefficient of performance (COP) analysis for PFHS in natural or forced convection regimes, aiming to minimize—thanks to a least-energy optimization by entropy minimization methodology—both fan pumping power and the thermodynamic work needed to manufacture and assemble the heat sink [

29,

30,

31,

32]. Furthermore, multi-objective genetic algorithms [

33,

34], artificial neural networks (ANNs) [

35], the Kriging method associated with computational fluid dynamics [

36] and integrated approaches [

37] have been investigated to maximize the thermal performance of heat sinks while considering multi-constraints (e.g., pressure drop, mass, space limitations).

One of the most valid studies on this topic has been conducted by Culham et al. [

38], where a thermal model for shrouded (i.e., no flow bypass) heat sinks and an optimization procedure based on entropy generation minimization have been coupled. However, optimization strategies based on entropy generation minimization only focus on operating costs, while neglecting production ones. This may lead to oversizing the heat sink, which implies an increased amount of material and thus production costs [

34].

In this work, we report both a novel comprehensive thermal model and a methodological approach to designing air-cooled PFHS, which allows for optimizing the production costs at given thermal performances. First, a comprehensive thermal model suited for unshrouded heat sinks is developed, considering the effect of flow bypass, developing boundary layer along fins, heat conduction through fins and baseplate-to-fins spreading resistance. It is worth stressing that the novel thermal model has rather general validity, being easily applicable also to shrouded heat sink (no flow bypass). Second, this novel thermal model is validated by the experimental characterization of a commercial heat sink for the thermal management of an Heating, Ventilating and Air Conditioning (HVAC) control unit in the automotive field. Finally, the thermal model is coupled to an optimization procedure based on genetic algorithms, in order to find the optimal geometric parameters of PFHS in semi-active configurations. This procedure is rather general, taking into account both the full range of possible working conditions and the technological constraints due to the adopted manufacturing process (e.g., extrusion and additive manufacturing). In particular, this optimization methodology is adopted to find the most cost-effective configuration for the considered commercial heat sink in a semi-active configuration.

The paper is structured as follows: in

Section 2, the comprehensive thermal model for unshrouded heat sinks is presented; in

Section 3.1, the proposed model is experimentally validated; in

Section 3.2, the cost optimization methodology for heat sinks is proposed and tested; in

Section 4, conclusions and perspectives are drawn.

## 2. Theoretical Analysis

A comprehensive model for predicting the thermo-fluid dynamics behavior of air-cooled, unshrouded PFHS under forced convection regime is introduced here.

As schematically depicted in

Figure 1, the heat sink geometry considered in this work is characterized by

L,

H,

t, and

N parameters, which are the fins length, height, thickness and number, respectively. Moreover,

W and

${t}_{b}$ are the baseplate width and thickness, while

p is the spacing between neighboring fins.

The effectiveness of heat dissipation by heat sinks depends on their effective thermal resistance, which is given by the overall effect of thermal resistances along the heat flux path, from the source to the ambient. Here, the overall junction-to-ambient thermal resistance (

${R}_{ja}$) of the electronic package is estimated relying on the classical equivalent thermal resistance network depicted in

Figure 2 [

13].

In fact, despite the fact that the film resistance at the fluid-solid boundary usually dominates, it has been demonstrated that the effect of other resistance elements on the heat path cannot be safely neglected in the optimization of the design characteristics of heat sinks [

38].

Therefore,

${R}_{ja}$ is decomposed here in four discrete components, namely:

where

${R}_{jc}$,

${R}_{cs}$,

${R}_{sa}$, and

${R}_{spr}$ are the junction-to-case, case-to-sink, sink-to-ambient, and spreading resistances, respectively.

First, in most applications, ${R}_{jc}$ is given by the manufacturer, who provides the electronic components already embedded into the cases. Hence, ${R}_{jc}$ is not directly modeled in the present work, being specific to the considered application and usually experimentally known.

Second, case-to-sink resistance

${R}_{cs}$ is calculated as:

where

k is the thermal conductivity of the heat sink, and

${t}_{b}$ and

${A}_{b}$ are the thickness and cross section surface area of baseplate.

Sink-to-ambient resistance

${R}_{sa}$ is then computed by analytically modeling the air flow and pressure across the unshrouded heat sink. Note that the following considerations are safely applicable to any type of flow bypass (top, side or both) or to fully shrouded configurations.

Figure 3 reports the average air flow velocities in the different sections of the duct where the heat sink is placed, namely: the approach velocity (

${v}_{d}$) at the entrance of the duct; the channel velocity (

${v}_{ch}$) through the channels made by neighboring fins (fin channels); the side bypass velocity (

${v}_{bs}$) and the top bypass velocity (

${v}_{bt}$) in the sections of the duct not occupied by the heat sink. Note that, by considering a flush-mounted installation of the heat sink on the lower wall of the duct (see

Figure 3, inset), it is possible to safely approximate a 2D, laminar and fully-developed air flow through the heat sink, without longitudinal vortices.

The air flowing through the fin channels experiences pressure drops due to contraction and expansion of the flow field at channel inlet and outlet, respectively. Let us introduce the cross section area of the duct

${A}_{d}=CB\times CH$, the overall cross section area of the fin channels

${A}_{ch}=(N-1)pH$, the cross section area of the side

${A}_{bs}$ and the top

${A}_{bt}$ bypass. Furthermore, let us assume the following conditions [

39]: channel aspect ratio (

${\alpha}_{ch}=p/H$) less than 0.75; 2D flow through the channel of the heat sink; laminar developing flow; no air leakage from the top heat sink; uniform approach velocity; constant

μ and

ρ of the fluid; negligible expansion and contraction losses in the bypass sections. The air flow velocities in the different sections of the duct can be therefore obtained by applying the conservation of mass and momentum in the duct [

39], namely:

ρ being the air density, whereas

$\Delta {p}_{hs}$,

$\Delta {p}_{bt}$,

$\Delta {p}_{bs}$ are the pressure drops experienced by air flowing through fin channels, top bypass and side bypass, respectively. The complete procedure to compute

$\Delta {p}_{hs}$,

$\Delta {p}_{bt}$ and

$\Delta {p}_{bs}$ is reported in

Appendix A. By the simultaneous solution of Equations (

3), the values of air velocity through the fin channels (

${v}_{ch}$), the side bypass (

${v}_{bs}$) and the top bypass (

${v}_{bt}$) can be then obtained.

Once the fluid dynamics quantities within the duct have been fully determined, it is possible to compute the modified spacing channel Reynolds

$R{e}_{p}^{*}=R{e}_{ch}p{L}^{-1}$, where

$R{e}_{ch}=\rho {v}_{ch}{D}_{{h}_{ch}}{\mu}^{-1}$ is the channel Reynolds number and

${D}_{{h}_{ch}}=2p$ is the hydraulic diameter of the fin channel. The average convective heat transfer coefficient can be then calculated as:

where

${k}_{a}$ is the thermal conductivity of air and

$N{u}_{p}$ is estimated as

in the range

$0.1<R{e}_{p}^{*}<100$ [

40],

$Pr=\mu {c}_{p,a}/{k}_{a}$ being the Prandtl number and

${c}_{p,a}$ the specific heat capacity of air at constant pressure. The fin efficiency (

η) can be also determined as [

40]:

The sink-to-ambient thermal resistance can be finally computed as:

${A}_{hs}$ being the total heat sink surface involved in the convective heat transfer.

The fourth thermal resistance considered in the model depicted in

Figure 2 takes into account the spreading resistance between the heat source and the baseplate of heat sink. As sketched in

Figure 4, spreading resistance occurs in configurations where heat flows from a small heat source to the base of a larger heat sink. In this way, the heat cannot uniformly distribute through the baseplate, therefore limiting the convective cooling effect by the fins.

Spreading resistance is progressively becoming an important issue in modern microelectronics, and it can be mitigated by either increasing the thickness of the baseplate or by adopting materials with higher thermal conductivity (e.g., novel micro- and nano-structured materials [

41]). Here, a perfect thermal contact between case and heat sink baseplate (i.e., negligible thermal contact resistances) is assumed, and

${R}_{spr}$ is estimated following the work by Lee et al. [

42], where further details are available. Let us define the baseplate and heat source (i.e., case) equivalent radii as

${r}_{b}=\sqrt{{A}_{b}/\pi}$ and

${r}_{s}=\sqrt{{A}_{s}/\pi}$, respectively, where

${A}_{s}$ is the cross section area of the heat source.

Moreover, let us introduce the dimensionless contact radius (

ϵ) and plate thickness (

τ):

Considering the dimensionless Biot number of the baseplate as

$B{i}_{b}=h{r}_{b}/k$, it is possible to calculate the dimensionless spreading resistance

$\mathsf{\Psi}=\mathsf{\Psi}(\u03f5,\tau ,B{i}_{b})$ as:

where

$\lambda =\lambda \left(\u03f5\right)$ is an empirical parameter found as

$\lambda =\pi +{\left(\sqrt{\pi}\u03f5\right)}^{-1}$ [

42]. Finally, the spreading resistance

${R}_{spr}$ is calculated as:

Even though the overall thermal resistance model for ${R}_{ja}$ has been developed for unshrouded heat sinks, it is worth stressing that it has general validity. In fact, it can be easily applied to shrouded heat sinks by neglecting bypass phenomenon.

## 4. Conclusions

Given the large amount of geometric and thermo-fluid dynamics parameters strongly coupled to each other, the cost-effective selection and design of heat sinks for electronics cooling can be often a complex procedure. In this work, a novel, comprehensive thermal model of unshrouded plate fin heat sinks including (1) flow bypass; (2) developing boundary layer along fins; (3) conduction through fins (fin efficiency); and (4) baseplate to fins spreading resistance, has been developed and experimentally validated. In particular, the original combination of thermal and fluid dynamics models already reported and validated in the literature allows for basing the heat sink optimization methodology on a robust modeling framework. The experimental campaign has been carried out to characterize a commercially available plate fins heat sink, which is currently adopted for automotive applications (millions of units manufactured per year). The limited deviations between experimental results and modeling predictions allow for considering the developed thermal model as an accurate reference for optimizing the plate fins configuration.

Then, a novel approach for the cost optimization of unshrouded PFHS operating in semi-active configurations has been proposed. Such an approach is based on computationally efficient genetic algorithms and, as a test case, it has been adopted to optimize the geometry parameters of the commercial heat sink. The optimized heat sink configuration shows a remarkable 53% volume (i.e., production costs) decrease with respect to the commercial one, while guaranteeing the same thermal performances. For the sake of simplicity, the optimization approach has been experimentally validated only for heat sinks with plate fins and approaching velocities in the range ${v}_{d}$ = 5.6–13.9 m/s (i.e., channel Reynolds number $R{e}_{ch}$ = 528–2481); however, the procedure could be further extended to other heat sink geometries or air flow regimes.

The suggested optimization methodology has been designed for heat sinks operating in semi-active configurations, where the relevant cost factor is the amount of material used to manufacture the heat sink (i.e., production cost). However, we highlight that the reported procedure is rather general, being easily transferable to several thermal management solutions for electronic and automotive components. In fact, this methodology could be scaled-up for a broad variety of: (i) heat sink geometries, e.g., with pin or flared fins; (ii) air flow regimes, e.g., with natural convection; (iii) heat sink installations, e.g., fully shrouded or top bypass ones; (iv) optimization targets, e.g., minimization of pressure drops in case of active configurations. Therefore, the industrial relevance of the methodology introduced in this article is the possible a priori identification of a range of optimal geometrical parameters for PFHS, according to the specific working conditions and manufacturing/performance/cost boundaries to be complied. This has the potential to drastically reduce the amount of experiments (and thus time-to-market) needed to design and characterize PFHS tailored for specific applications (i.e., ”Long Tail” products), especially if coupled with additive manufacturing techniques.