Optimal Thermal Management Using the Taguchi Method for LED Lighting Squared Heat Sink, Including Statistical Approaches

Abstract

The global development of LED lighting in all applications for both public and indoor lighting establishes a very important lever for reducing the carbon impact by significantly reducing energy consumption. Smart lighting will therefore constitute an essential lever in the smart city of tomorrow. The latter is more sustainable and less energy-intensive than other light sources, contributing towards the Sustainable Development Goals set forth by the United Nations (SDGs 7 and 11). With its ease of integration, ergonomics, lightness, and high quality of light, this innovative light source has two major qualities: its energy efficiency and its long service life. However, poor thermal management has catastrophic effects on these two essential criteria. This necessarily requires optimizing thermal management and heat sinks. In some cases, thermal models and simulations can lead to considerable calculation times because they implement many parameters. This work therefore aims to reduce the number of these simulations by applying the method of experimental designs (Taguchi) and analysis of variance (ANOVA) to simulations intended to optimize the dissipation of LED luminaires. We applied the method to a simple finned heat sink model for a unit input power and then to a concrete case of a high-power LED. The control parameters and their respective contributions in the proposed model were studied. The ANOVA results corroborated the conclusions of the Taguchi method, demonstrating a strong agreement between these analytical methods, in which the temperature was adjusted by identifying optimal combinations of parameters. The fields of application relevant to this study include both indoor lighting (false ceiling) and confined spaces with severe sealing constraints such as car headlights or the optical blocks of urban luminaires.

1. Introduction

In recent decades, the global focus on sustainable development has gained unprecedented momentum, as industries and policymakers seek solutions to minimize environmental degradation while maintaining economic growth [1,2,3,4]. A central element in this push for sustainability is energy efficiency, which plays a pivotal role in reducing resource consumption and mitigating climate change [5]. Within this context, lighting technology has undergone a revolutionary transformation, with light-emitting diodes (LEDs) emerging as a key component of energy-efficient solutions [6,7]. LEDs have proven to be one of the most significant advancements in modern lighting, offering an ideal combination of energy savings, durability, and environmental benefits [8,9,10]. One of the key benefits of LEDs is their extended lifespan, which can be up to 25 times longer than that of incandescent bulbs. This longevity, combined with their energy efficiency, leads to significant reductions in both operational costs and environmental impacts. According to the U.S. Department of Energy, widespread adoption of LED technology could result in substantial reductions in global energy consumption, which would translate into lower greenhouse gas emissions and reduced dependency on fossil fuels [8,9]. Beyond environmental impacts, LEDs also play a crucial role in achieving various Sustainable Development Goals (SDGs), particularly those related to affordable and clean energy, sustainable industrial practices, but also climate action (through reduction of carbon emission), due to their high efficiency [10]. These advantages underscore the vital contribution of LEDs to a more sustainable future. In the lighting market, high-powered LEDs, recognized as the fourth generation of light sources, dominate due to their versatility and superior performance [11,12,13,14,15,16]. LEDs provide numerous benefits over traditional lighting sources, including extended lifespans, high reliability, low energy consumption, high efficiency, rapid response times, and adjustable color options [14,15,16,17]. To remain competitive and aligned with market trends, achieving low-cost, lightweight designs, and compact sizes has become a crucial development objective [18].

All these qualities have made LED lighting ubiquitous. This has radically changed our lighting usages for our comfort and well-being as well as for environmental protection (avoiding light pollution) and has led to reductions in our energy consumption and carbon footprint [19,20,21]. Due to its deep integration into the urban landscape, smart lighting also constitutes a neural network, allowing for the transfer of information necessary for smart cities [22]. Its impact on our society is such that it meets many sustainable development goals, such as the clean energy goal and sustainable cities goal (respectively, SDG 7 and SDG 11) [1,4,23]. By continuously enhancing LED technology, we contribute to global efforts to promote sustainable energy solutions and minimize environmental impact. This ensures that lighting systems are both economically viable and environmentally friendly.

However, LEDs are sensitive to temperature. This seriously impacts their energy efficiency but especially LED lifespans, and, therefore, all the positive assets associated with the SDGs. Without thermal management, all the advantages of LED lighting are impacted.

LEDs operate as distinct heat sources, releasing heat via electronic packages into printed circuit boards (PCBs) and heat sinks, similar to other power semiconductor devices. The primary factors responsible for temperature elevation in these devices are the thermal resistance barriers that arise during this process [24,25]. In optimal current densities for LEDs, approximately 60 to 70% of the electrical power input is converted into heat, which means that it is important to keep the thermal management of LEDs in mind when designing new lighting solutions [26]. Therefore, it is crucial to prioritize heat dissipation in technological advancements related to LEDs. The failure to manage heat effectively can have catastrophic effects on LED chips, resulting in decreased luminous efficiency and a significantly reduced lifespan [27,28]. General guidelines suggest that by reducing the junction temperature of electronic components by 10 °C, their failure rate can be cut in half. Consequently, devices operating at lower temperatures tend to exhibit higher reliability [29]. Efficient thermal management is crucial for ensuring the optimal performance and longevity of LED devices. Several studies have recently focused on the design, orientation, and convection of heat sinks for high-power LEDs [30,31,32]. Seung-Hwan Yu et al. [33] found that inclining cylindrical heat sinks at an angle between 25 and 30 degrees reduces thermal resistance and increases LED bulb energy conversion efficiency. In another comprehensive study, Paul Morgan et al. [15] conducted a study to investigate the impact of different geometric factors on the thermal resistance and effectiveness of a heat sink setup. The research showed that adjusting the lengths of the long and middle fins, and varying the number of fin arrays, had a significant impact on the overall performance of the system. Based on their findings, they recommended a new radial heat sink design that aimed to reduce overall mass while maintaining cooling efficiency comparable to traditional plate-fin heat sinks. Shen et al. [34] investigated natural convection from vertical cylindrical heat sinks. These heat sinks contain longitudinal fins that are meant to dissipate heat in LED bulbs and other electronic devices. The researchers formulated a sinusoidal function to define the boundary layer’s edge. Their study found a direct correlation between the mean Nusselt number along the length of the fin and an exponential function of the distance from the cylinder’s base. Based on their observations, they introduced a new correlation to explain this relationship. In another study, Huang et al. [35] attempted to enhance the efficiency of LED heat transfer by using heat sinks that had grooved heat pipes on them. Seok-Hwan Moon et al. [36], developed a flat heat pipe featuring a U-shaped design intended for a 100 W LED lamp. Their research demonstrated that this design effectively maintained a junction temperature below 85 °C. In parallel, Mousavi et al. [37] investigated 10 configurations of discontinuous, staggered, and covered fin heat sinks to determine an optimal design. Complementing these findings, Yiwei Wang et al. [38] developed a flat plate heat pipe featuring micro-fins on the condensation surface. This heat pipe was employed for dissipating heat from LEDs, leading to a 10% decrease in thermal resistance. Wang et al. [39] conducted a numerical simulation to compare the effectiveness of two types of aluminum heat sinks: one with straight radial fins; and the other with aluminum heat pipes. Complementing this research, R. Singh et al. [40] designed an LED headlamp that used a heat pipe for cooling. The heat sinks used in this design were 40% to 50% lighter than traditional die-cast heat sinks. Heat sinks that use natural convection are often used for cooling LEDs because they are energy efficient and do not have any acoustic or overload problems. Canale et al. [41] proposed silent, active cooling using ionic wind solutions, showing a reduced temperature of more than 65% compared to the same device without ionic wind. These heat sinks help to dissipate heat into the surrounding environment while keeping the temperature at the LED junction lower. Several recent studies have investigated the use of heat sinks to manage heat dissipation, with both numerical and experimental investigations on natural convection heat transfer [40,42,43,44,45]. Xu [46] proposed a design featuring a rectangular heat sink coupled with a thermosyphon of varying heights to efficiently cool high-power LED bulbs. Similarly, Seung-Jae Park et al. [47] presented a cooling system for LED downlights that utilized a hollow cylinder and radial heat sink. The hollow cylinder size and material were analyzed, showcasing an up to 43% thermal improvement. The airflow pattern, which resembles a chimney, enhanced thermal performance by increasing the air mass flow rate to the heat sink. The performance of LED thermal management is significantly influenced by heat sink configurations and heat dissipation methods [48]. To provide a comprehensive overview of the literature and the context of the current study, Table 1 summarizes classifications based on heat dissipation methods. Despite the extensive use of LEDs in various applications due to their energy efficiency and long lifespan, effective thermal management remains a critical challenge. Numerous studies have investigated various cooling techniques and materials to improve heat dissipation in LEDs [49,50]. However, these studies often lack systematic optimization methodologies to identify the most effective parameters influencing thermal performance. Most existing research relies primarily on empirical or simulation-based approaches without employing robust statistical optimization methods. This has left a significant gap in systematically understanding and optimizing the key factors affecting LED thermal management. Addressing this gap is crucial because inadequate thermal management leads to reduced efficiency, color shift, and the shorter lifespan of LEDs. Implementing a statistical approach can significantly enhance the performance and reliability of LEDs, which are increasingly used in critical applications such as automotive lighting, display technologies, and general illumination. Here, the Taguchi and ANOVA are highly regarded statistical techniques that complement each other in experimental methodologies, particularly in quality engineering and process optimization [51]. By using the Taguchi method, engineers can precisely determine the optimal combination of factors, leading to improvements in both product reliability and production cost reductions [52].

Previous studies in the literature have widely embraced the Taguchi method to optimize processes and enhance product quality. For instance, W. Yip et al. [53] developed a novel method that combined Taguchi and principal component analysis to optimize optical lenses for LEDs. The objective of the method was to achieve uniformity and high efficiency. The approach overcame traditional limitations and achieved over 92% light efficiency and improved uniformity in LED lighting lenses. C. Wang et al. [54] improved the thermal performance of LEDs by using an optimized heat sink design and Cu₂O coating. The study utilized Taguchi’s method to identify crucial factors that contribute to 91.06% of the variations in LED junction temperatures. The optimized heat sink and coating increased junction temperature efficiency by 23.88%, which reduced variance and improved reproducibility. Further advancing thermal management in LEDs, Liu et al. [55] investigated thermal issues in high-power blue LEDs based on InGaN, which are caused by high thermal resistance due to lattice dislocations. The authors proposed an optimization of heat transfer modules with an AlN film-coated LED substrate, copper fins, and parameters determined using the Taguchi method. Through experiments and simulations, including Finite Element Method (FEM) analysis, they demonstrated a significant reduction in temperature from 102 °C to 65 °C for a 12 W LED when optimal parameters were identified. Addressing these thermal challenges will enable the development of more efficient LED systems, contributing to the overall reduction in energy demand across various sectors.

Table 1. Classification approach review according to heat dissipation type based on recent studies in the literature.

Ref.	Basic Approach and Methods	Heat Dissipation Type
[56]	Using the Levenberg–Marquardt method and CFD-ACE+, this study optimized LED heat dissipation fins, achieving a 29.1% reduction in thermal resistance by incorporating perforations.	Natural convection
[57]	Numerical study optimized the geometric parameters of a branching radial heat sink for efficient LED cooling and improved heat transfer.
[58]	Nanofluids were used to improve LED cooling in thermoelectric cooler and microchannel heat sink systems by enhancing heat dissipation.	Forced convection
[59]	The thermal management of LED luminaires by scaling heat sink dimensions for efficiency.
[60]	Cubic boron nitride films enhanced LED thermal management, reducing temperatures by up to 16.8 °C.	Conduction
[61]	Aluminum with carbon nanotube coatings enhanced LED heat dissipation while thinner coatings provided better performance.
[62]	Improved convective fluid flow and heat transfer enhanced LED heat sink design for better heat dissipation.	Natural convection and conduction
[63]	LED thermal management was improved with an integrated heat sink using vapor chamber technology, which outperformed conventional heat sinks in temperature distribution and luminous efficacy.
[64]	A new micro heat sink with a porous structure efficiently dissipated heat from LEDs, maintaining low temperatures even at high fluxes.	Conduction and forced convection
[65]	Oriented cut copper fiber heat sink design enhanced LED cooling and achieved higher heat transfer efficiency compared to other types of heat sinks.
[66]	A spray cooling system dissipated heat from LED lights and tested various nozzle configurations.	Conduction, natural and forced convection
[67]	A composite of CuO and silicon resin was applied to the aluminum heat sink, improving LED heat dissipation.	Radiation

Table 1. Classification approach review according to heat dissipation type based on recent studies in the literature.

Ref.	Basic Approach and Methods	Heat Dissipation Type
[56]	Using the Levenberg–Marquardt method and CFD-ACE+, this study optimized LED heat dissipation fins, achieving a 29.1% reduction in thermal resistance by incorporating perforations.	Natural convection
[57]	Numerical study optimized the geometric parameters of a branching radial heat sink for efficient LED cooling and improved heat transfer.
[58]	Nanofluids were used to improve LED cooling in thermoelectric cooler and microchannel heat sink systems by enhancing heat dissipation.	Forced convection
[59]	The thermal management of LED luminaires by scaling heat sink dimensions for efficiency.
[60]	Cubic boron nitride films enhanced LED thermal management, reducing temperatures by up to 16.8 °C.	Conduction
[61]	Aluminum with carbon nanotube coatings enhanced LED heat dissipation while thinner coatings provided better performance.
[62]	Improved convective fluid flow and heat transfer enhanced LED heat sink design for better heat dissipation.	Natural convection and conduction
[63]	LED thermal management was improved with an integrated heat sink using vapor chamber technology, which outperformed conventional heat sinks in temperature distribution and luminous efficacy.
[64]	A new micro heat sink with a porous structure efficiently dissipated heat from LEDs, maintaining low temperatures even at high fluxes.	Conduction and forced convection
[65]	Oriented cut copper fiber heat sink design enhanced LED cooling and achieved higher heat transfer efficiency compared to other types of heat sinks.
[66]	A spray cooling system dissipated heat from LED lights and tested various nozzle configurations.	Conduction, natural and forced convection
[67]	A composite of CuO and silicon resin was applied to the aluminum heat sink, improving LED heat dissipation.	Radiation

This article offers a comprehensive overview of how the Taguchi method can be applied to optimize thermal management in LED systems. It discusses key considerations for effective LED thermal management, explains the principles and steps involved in the Taguchi method, and presents relevant case studies that demonstrate its effectiveness. We begin our analysis with a 1 W CoB LED to illustrate the principles of the Taguchi method. Following this, we conduct a confirmation test using the XLAMP^® CXA1304, [68] which is a 13 W CoB LED, using an optimized heat sink design through the Taguchi method.

2. Numerical Model

2.1. Modelling Object

The numerical simulations are performed in a Cartesian coordinate system based on the reference configuration represented in Figure 1. The configuration consists of a rectangular cavity L × W × W’ (where W = b + H + H’ expresses the total height of the cavity) that is filled with air. Positioned along its lower side is a heat source, specifically, a finned heat sink designed for cooling LED lamps. The heat sink comprises a base size of L × W_b, a number of fins N, with each fin having a length H and spaced at a distance s. The distance H’ represents the gap between the upper walls of the cavity and the fins. The base of the heat sink has a thickness b, while the fins have a thickness t. The lateral walls of the cavity, corresponding to the cold boundaries, are uniformly isothermal at ambient temperatures, while the lower and upper sides are considered adiabatic. A constant heat flux Q is directly imposed on the radiator base to simulate the heat generated by the active zone of the LED lamp.

There are much more elaborate and optimized heat sinks with much more complex geometries (circular, multi-branched, integrating heat pipes, etc.). The choice of a simple heat sink in the case of our study (as well as in the following paragraph implementing a real industrial LED power) has two advantages: (1) it aims to present a demonstration of the method in a simple way and the chosen model allows a clearer understanding; and (2) the straight fin geometry is the most commonly used geometry in the industry.

The materials and properties used in this study are shown in

Table 2. Thermo-physical properties of air and aluminum [69] (inside the cavity at t₀, T = 298.15 K).

Material	Thermal Conductivity $\left(\mathbf{W}/\mathbf{m}\mathbf{K}\right)$	$\mathbf{Density} \left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3\right)$	$\mathbf{Specific} \mathbf{Heat} \mathbf{Capacity} \left(\mathbf{J}/\mathbf{k}\mathbf{g} \mathbf{K}\right)$	Dynamic Viscosity (kg/(m s))
Aluminum	$237.5$	$2689$	$951$	$----$
Air (Tcav.init)	$0.026$	$1184$	$1005$	$1.849\times {10}^{-5}$

at t₀, initial time, for T_cav.init = 298.15 K (initial cavity temperature); the dimensions of the model are shown in

Table 3. Dimensions of heat sink and cavity used in the model.

Parameters	Values
N: Number of fins	7
$W_b:$ Heat sink base width (mm)	57
$t:$ Heat sink fin thickness (mm)	2
$S:$ Distance between fins (mm)	23
$W:$ Cavity length (m m)	157
$W’:$ Cavity width (mm)	$2\times W_b$

.

2.2. Assumptions for Numerical Model

These assumptions are adopted in order to simplify the formulation of the numerical model while ensuring the accuracy of our results under the specified conditions. These approximations are valid for low-velocity flows and moderate temperature gradients, where density changes due to compressibility are negligible, as follows:

• Flow is considered to be Newtonian (with constant viscosity), laminar, and incompressible with constant thermo-physical properties;
• Steady-state flow is assumed;
• The work of pressure forces and thermal radiation exchange are assumed to be negligible;
• Air properties are deemed temperature-independent, with the exception of density;
• It is assumed that the density of air depends on temperature in the buoyancy term and is determined based on the ideal gas law, as follows:

$$\rho ={\displaystyle \frac{P_{atm}}{\left(\frac{R}{M_a}\right)T}}$$

(1)

where P_atm is the atmospheric pressure, the ideal gas constant is denoted by R, and Ma represents the molar mass of air.

2.3. Mathematical Modelling

In these conditions, the equations governing the problem are written in the following form.

The equations governing the air side in three-dimensional cartesian coordinates are then expressed as follows:

Continuity equation:

$${\displaystyle \frac{\partial \left(\rho u\right)}{\partial x}}+{\displaystyle \frac{\partial \left(\rho v\right)}{\partial y}}+{\displaystyle \frac{\partial \left(\rho w\right)}{\partial z}}=0$$

(2)

Equation of conservation of momentum in the x, y and z directions:

$${\displaystyle \frac{\partial \left(\rho u^2\right)}{\partial x}}+{\displaystyle \frac{\partial \left(\rho vu\right)}{\partial y}}+{\displaystyle \frac{\partial \left(\rho wu\right)}{\partial z}}=-{\displaystyle \frac{\partial P}{\partial x}}+\mu \left({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial z^2}}\right)$$

(3)

$${\displaystyle \frac{\partial \left(\rho uv\right)}{\partial x}}+{\displaystyle \frac{\partial \left(\rho v^2\right)}{\partial y}}+{\displaystyle \frac{\partial \left(\rho wv\right)}{\partial z}}=-{\displaystyle \frac{\partial P}{\partial y}}+\mu \left({\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}v}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}v}{\partial z^2}}\right)-\rho g$$

(4)

$${\displaystyle \frac{\partial \left(\rho uw\right)}{\partial x}}+{\displaystyle \frac{\partial \left(\rho vw\right)}{\partial y}}+{\displaystyle \frac{\partial \left(\rho w^2\right)}{\partial z}}=-{\displaystyle \frac{\partial P}{\partial z}}+\mu \left({\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}w}{\partial z^2}}\right)$$

(5)

Energy equation:

$${\displaystyle \frac{\partial \left(\rho uT\right)}{\partial x}}+{\displaystyle \frac{\partial \left(\rho vT\right)}{\partial y}}+{\displaystyle \frac{\partial \left(\rho wT\right)}{\partial z}}={\displaystyle \frac{k}{c_p}}\left({\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}\right)$$

(6)

The temperature distribution on the solid side is determined by Fourier’s law, as follows:

$${\displaystyle \frac{{\partial }^{2}T}{\partial x}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}=0$$

(7)

2.4. Boundary and Initial Conditions for the Heat Sink and the Cavity

The computational domain’s boundary conditions are illustrated in Figure 2, and their corresponding explanations are provided below:

A uniform heat flux boundary condition is applied at the bottom of the heat sink base, as follows:

$$u=v=w=0;-k_s{\displaystyle \frac{\partial T}{\partial n}}=Q$$

(8)

The vertical sidewalls of the cavity are kept at a constant low temperature, as follows:

$$u=v=\mathrm{w}=0; T=T_c$$

(9)

The sides running horizontally in the cavity are adiabatic, as follows:

$$u=v=w=0; {\displaystyle \frac{\partial T}{\partial n}}=0$$

(10)

where $u, v$ and $w$ are the velocity components in the $x, y$ and $z$ directions, respectively.

In Figure 2, $u=v=w=0$ along the boundaries due to the no-slip boundary condition, which states that a fluid’s velocity matches that of a stationary solid surface. As the walls are stationary, the fluid in contact with them has zero velocity.

Conjugated heat transfer is considered at the solid/fluid interface, as follows:

$$T_{fluid,interface}=T_{solid,interface}$$

(11)

and

$$k_{fluid}{{\displaystyle \frac{\partial T_{fluid}}{\partial n}}|}_{interface}=k_{solid}{{\displaystyle \frac{\partial T_{solid}}{\partial n}}|}_{interface}$$

(12)

• The heat flux that is applied directly to the base of the heat sink: $Q=1 \mathrm{W}$;
• The temperature of the lateral cold walls of the cavity: $Tc=300 \mathrm{K}$;
• A uniform convection heat transfer coefficient of $h=10 \mathrm{W}/{\mathrm{m}}^2·\mathrm{K}$ was assumed for the heat sink because we have a vertical wall exposed to natural airflow at temperatures near room temperature (300 K), which is consistent with that in the literature and practical experiences with similar systems [30,66,70];
• The ambient temperature was taken as $T_{amb}=298.15 \mathrm{K}$.

3. Numerical Validation and Methods

3.1. Model Validation of the Thermal Model

The computational framework employed for solving the physical model utilizes COMSOL Multiphysics 6.0, a simulation software program that employs finite technical elements. The grid dependence was managed by adjusting the element count from 12,621 to 1,291,404. Consequently, a fine mesh with 226,048 elements was chosen, as the additional elements did not significantly alter the temperature values, as shown in

Table 4. Mesh independence test.

Predefined Mesh Size	Mesh Elements	Junction Temperature (°C)
Extremely coarse	3971	59.104
Extra coarse	6370	59.125
Coarser	11,658	59.163
Coarse	21,896	59.179
Normal	52,343	59.188
Fine	266,048	59.190
Finer	1,014,715	59.194
Extra Fine	2,336,717	59.197
Extremely Fine	5,730,930	59.197

.

In Figure 3, we present a comparative analysis of how ambient temperature impacts the junction temperature of the LED package module with a Chip-on-Board (CoB) configuration. The thermal resistance between the junction and the heat sink (Rth) in our model is based on data provided by the manufacturer for the specific LED package. For this package, Rth is approximately 5.6 K/W, which takes into account the thermal pathway through the solder layer, substrate, and thermal interface material (TIM).

While the simulation accounts for the heat conducted to other regions of the heat sink, our analysis focused primarily on the junction-to-ambient heat dissipation path, as it is the dominant thermal pathway under the operating conditions of interest.

This investigation was conducted under geometrically identical conditions as outlined in a study by M. Ha and S. Graham [71], with a power input of 1 W, an ambient temperature set at $25 °\mathrm{C}$, and a heat transfer coefficient of $h=10 \mathrm{W}/{\mathrm{m}}^2\mathrm{K}$. To enhance accuracy, the simulation data were intentionally aligned with experimental measurements by applying an offset, accounting for specific factors such as calibration discrepancies and variations in ambient conditions.

In our computations, we determined that the maximum deviation between our numerical findings and those of M. Ha and S. Graham was approximately 3.045%.

Figure 4 provides a visual representation of the vertical temperature profile along the center line, demonstrating a noteworthy level of concurrence between our results and the referenced study. It is evident that the results obtained align well with existing studies in the literature [71,72].

In addition to these references, another reference [66] showed similar trends; however, the values differed because our study included additional parameters such as varying ambient conditions and thermal interface assumptions.

3.2. Taguchi Method

The Taguchi method, pioneered by Genichi Taguchi [73], is an extensively employed statistical approach used across diverse engineering fields for experimental designs. This technique aims to identify optimal parameters and rank influential factors. It enhances experimental design by pinpointing critical factors affecting the process. Additionally, it facilitates a substantial reduction in the necessary number of experiments without compromising result accuracy [74].

The Taguchi method is designed to discern the key parameters influencing the system’s response. It assesses the relative impact of each parameter on the response and determines the optimal values for all the parameters under consideration [75].

In this investigation, a method was employed to assess the impact of different process parameters on the temperature of the heat sink base. The goal was to determine their respective contribution ratios, establish their order of importance, and optimize the thermal management of the LED package system. Three parameters, each with three levels, were chosen as control factors, as detailed in

Table 5. Control factors and their levels.

Factors	Level 1	Level 2	Level 3
$H’:$ Distance between the upper walls of the cavity and the fins (mm)	30	90	180
$H:$ Fin length (mm)	15	45	90
$b:$ Base of the heat sink thickness (mm)	4	12	24

.

After defining the parameters and levels, a Taguchi orthogonal array (OA) was created for 27 simulation runs, resulting in an L27 OA.

In each run, the objective function was attained. The details of the L27 orthogonal array and each response, denoted by Y_n, are shown in

Table 6. Experimental plan used in the Taguchi analysis.

Tests	Factors			Response
	H’	H	b	Yn
1	1	1	1	Y1
2	1	1	2	Y2
3	1	1	3	Y3
4	1	2	1	Y4
5	1	2	2	Y5
6	1	2	3	Y6
7	1	3	1	Y7
8	1	3	2	Y8
9	1	3	3	Y9
10	2	1	1	Y10
11	2	1	2	Y11
12	2	1	3	Y12
13	2	2	1	Y13
14	2	2	2	Y14
15	2	2	3	Y15
16	2	3	1	Y16
17	2	3	2	Y17
18	2	3	3	Y18
19	3	1	1	Y19
20	3	1	2	Y20
21	3	1	3	Y21
22	3	2	1	Y22
23	3	2	2	Y23
24	3	2	3	Y24
25	3	3	1	Y25
26	3	3	2	Y26
27	3	3	3	Y27

.

After determining the outcome of each simulation, the subsequent stage in implementing the Taguchi method involved calculating the signal-to-noise S/N ratio for each response, commonly referred to as the S/N ratio [75].

The signal-to-noise (S/N) ratio is a valuable tool for assessing the importance of various parameters that affect a process by analyzing their respective variances.

While a statistical analysis of a simulation provides information on the accuracy of the simulation itself, in the case of our study, the S/N ratio serves as a robust measure to assess both the performance and variability of the system under different conditions, particularly on thermal performance. The SNR approach thus enables us to quantify the robustness of different parameter configurations under varying conditions.

The S/N ratio serves as a robust metric for evaluating both system performance and variability under different conditions.

This approach enables us to effectively explore design parameters and their interactions, providing a comprehensive optimization framework that is often overlooked in traditional methods as a quality function.

When analyzing the S/N ratio, the three categories of quality characteristics are as follows: smaller is better; larger is better; and nominal is best.

Table 7 provides the S/N ratio characteristics based on the quality category being considered. In this table, n refers to the number of experiments conducted, y represents the simulation result value, $\overline{y}$ denotes the average of the simulation result value, and ${\sigma }_y$ signifies the standard deviation.

Since the objective is to minimize the heat sink base temperature, the “smaller the better” S/N ratio was chosen, as outlined in Equation (14).

Table 7. The S/N ratio characteristics as a function of the considered category of quality.

Category of Quality	S/N Ratio Characteristics
Nominal-the better	$$S/N=10lo{g}_{10}{\displaystyle \frac{1}{n}}\left({\displaystyle \sum }{\displaystyle \frac{\overline{y}}{{\sigma }_y}}\right)$$ (13)
Smaller-the-better	$$S/N=-10lo{g}_{10}{\displaystyle \frac{1}{n}}\left({\displaystyle \sum }{y}^2\right)$$ (14)
Higher-the-better	$$S/N=-10lo{g}_{10}{\displaystyle \frac{1}{n}}\left({\displaystyle \sum }{\displaystyle \frac{1}{y^2}}\right)$$ (15)

Table 7. The S/N ratio characteristics as a function of the considered category of quality.

Category of Quality	S/N Ratio Characteristics
Nominal-the better	$$S/N=10lo{g}_{10}{\displaystyle \frac{1}{n}}\left({\displaystyle \sum }{\displaystyle \frac{\overline{y}}{{\sigma }_y}}\right)$$ (13)
Smaller-the-better	$$S/N=-10lo{g}_{10}{\displaystyle \frac{1}{n}}\left({\displaystyle \sum }{y}^2\right)$$ (14)
Higher-the-better	$$S/N=-10lo{g}_{10}{\displaystyle \frac{1}{n}}\left({\displaystyle \sum }{\displaystyle \frac{1}{y^2}}\right)$$ (15)

3.3. Analysis of Variance (ANOVA)

The analysis of variance (ANOVA) serves as a statistical technique employed in decision-making processes to ascertain the proportional contribution of individual parameters to the overall performance system. Its primary purpose is to establish the significance of each parameter’s impact on the response level. In the realm of optimization, the primary objective of ANOVA lies in systematically categorizing a series of experimental outcomes based on a shared variable or parameter, coupled with an objective function or response. Subsequently, the ANOVA analysis illuminates the extent of significance within the variations of these outcomes and elucidates the intricate interplay between system parameters and the corresponding system response.

It also affirms the dependability of the outcomes acquired through the Taguchi method [76]. The primary indicator of the ANOVA method was determined using the following Equations (16)–(20). These equations encompass the calculation of degrees of freedom (DOF), sum of squares (SS), mean of squares (MS), F-values, and contribution ratio (CR) corresponding to each factor [76,77], as follows:

$$DO{F}_T={\displaystyle \sum }_{i=1}^{n_f}\left(L_i-1\right)+DO{F}_{error}$$

(16)

where $L_i$ represents the number of levels, $n_f$ represents the number of factors, and $DO{F}_{error}$ represents the degrees of freedom for the error, as follows:

$$S{S}_i={\displaystyle \sum }_{i=1}^n{\left(\widehat{y_i}-\overline{y}\right)}^2$$

(17)

where $\overline{y}$ is the mean response value and $\widehat{y_i}$ is the i-th response value.

The mean squares corresponding to Equation (18) are as follows:

$$M{S}_i={\displaystyle \frac{S{S}_i}{DO{F}_i}}$$

(18)

The F-factor is a measure of statistical reliability and is calculated as a ratio of the parameter’s mean square to the mean square error (Equation (19)), as follows:

$$F_i={\displaystyle \frac{M{S}_i}{S{S}_T}}$$

(19)

The contribution ratio (CR) is the sum of squared deviations divided by the total sum of squared deviations (Equation (20)), as follows:

$$C{R}_i={\displaystyle \frac{S{S}_i}{S{S}_T}}$$

(20)

4. Results and Discussion

4.1. Thermal Analysis for an Input of 1 W

Furthermore, let us thoroughly investigate the thermal study of the issue, focusing on the examination of heat transfer and the velocity magnitude within the cavity. In any natural convection scenario, the transfer of heat is intimately tied to the movement of the fluid. Therefore, it is imperative to thoroughly investigate the dynamics of this flow within the cavity and its consequential effect on the rate of heat dissipation. Figure 5 illustrates the temperature evolution in the cavity at various H’. At some heights (H’ = 50–100 mm), both the temperature profile and the velocity magnitude attain a symmetrical state relative to the vertical centerline of the cavity in a steady state.

As a result of the difference in density, the air within the enclosure rises above the heat source in a thermal plume. The plume is situated along the vertical axis at the center of the enclosure. The air then reaches the upper wall of the cavity, which is kept adiabatic. After that, the fluid follows the contour of the cavity, moving along the upper fence before descending and circulating along the side walls. This process creates isothermals, which contribute to cooling the hot air. Finally, the air rests against the bottom wall of the enclosure, moving horizontally until it returns to the heat sink, thereby restarting the cycle. As the cavity length extends, the air motion within intensifies, leading to heightened heat exchange.

However, this also introduces instabilities that disrupt the once symmetrical flow at heights greater than 100 mm (H’ > 100 mm). The central thermal plume starts to deflect, marking a transition to a flow state characterized by increased instability. Concurrently, the vortices, which denote the recirculation zones positioned on either side of the central region, shift in response to the elongation of the cavity. This phenomenon is explained by the momentum equation in the Navier–Stokes equations: increased airflow results in higher fluid velocities, enhancing heat exchange. However, the elongation induces instabilities in the flow field, modelled by the continuity and energy equations, disrupting symmetry, and causing the central thermal plume to deflect. The Rayleigh number, RaH, based on the cavity’s length, increases significantly, further amplifying the instability. This dimensionless parameter characterizes the relative importance of buoyancy forces to viscous forces.

Additionally, the vortices’ positions shift due to the elongation, as governed by the vorticity equation in the Navier–Stokes equations. Figure 6 and Figure 7 provide the temperature evolution and velocity magnitude within the cavity at different values of height (H) and boundary width (b), respectively. Across all variations of H and b, both the temperature profile and fluid flow achieve a symmetrical steady state relative to the vertical centerline of the cavity. This symmetrical state is a result of the difference in air density, leading to the formation of a thermal plume that rises above the heat source, situated along the vertical axis at the center of the enclosure.

Upon reaching the adiabatic upper wall of the cavity, the airflow follows the cavity’s contour, moving along the upper fence before descending and circulating along the side walls. This circulation pattern creates isothermal regions, aiding in the cooling of the hot air within the cavity. Finally, the air rests against the bottom wall of the enclosure, moving horizontally until it returns to the heat sink, completing the cycle and restarting the process of thermal management.

4.2. Control Parameters Optimization Using Taguchi Method

In order to create an effective model through rigorous statistical analysis, the Taguchi method is utilized. Its primary goal is to identify the best values for the control parameters and determine the most favorable arrangement of the heat sink design.

Table 8. Experimental design with S/N ratio.

Tests	Factors (mm)			Results
	H’	H	b	Temperature (K)	S/N Ratios
1	30	15	4	323.11	−50.19
2	30	15	12	321.10	−50.13
3	30	15	24	319.52	−50.09
4	30	45	4	316.31	−50.00
5	30	45	12	315.44	−49.98
6	30	45	24	314.69	−49.96
7	30	90	4	313.72	−49.93
8	30	90	12	313.13	−49.91
9	30	90	24	312.62	−49.90
10	90	15	4	317.45	−50.03
11	90	15	12	316.31	−50.00
12	90	15	24	315.41	−49.98
13	90	45	4	313.67	−49.93
14	90	45	12	312.91	−49.91
15	90	45	24	312.37	−49.89
16	90	90	4	311.74	−49.88
17	90	90	12	311.28	−49.86
18	90	90	24	310.91	−49.85
19	180	15	4	316.62	−50.01
20	180	15	12	315.36	−49.98
21	180	15	24	314.43	−49.95
22	180	45	4	312.56	−49.90
23	180	45	12	311.94	−49.88
24	180	45	24	311.43	−49.87
25	180	90	4	310.93	−49.85
26	180	90	12	310.50	−49.84
27	180	90	24	309.88	−49.82

shows the experimental configuration using the S/N ratios based on the Taguchi method and the temperature. In this design, every row represents a distinct combination of different levels.

In this experiment, columns are designated as control parameters, and temperature values for 27 tests along with their corresponding S/N ratios are calculated and assigned to the respective columns.

The experiment focuses on three control parameters, which are tested using an orthogonal array provided. The parameters tested are, as follows: (H’) distance between the upper walls of the cavity and the fins (ranging from 30 to 180 mm); (H) fin length (ranging from 20 to 90 mm); and (b) base of the heat sink thickness (ranging from 8 to 24 mm).

In Table 8, for the temperature, the fin length (H) corresponds to Rank 1 and reaches up to 51.80%, while the distance between the upper walls of the cavity and the fins (H’) corresponds to Rank 2 and influences up to 34.14% while the base of the heat sink thickness (b) has a CR of 14.06%.

Table 9. Response tables for S/N ratios of various control factors and their delta, rank, and contribution ratio for temperature.

Temperature
Level	H’	H	b
1	−50.01	−50.04	−49.97
2	−49.93	−49.92	−49.94
3	−49.90	−49.87	−49.92
Delta	0.11	0.17	0.05
Rank	2	1	3
CR (%)	34.14	51.80	14.06

displays the mean S/N ratios pertaining to various control parameters across three distinct levels. The table encompasses delta factors, rankings, and the contribution ratio (CR) linked to each parameter. Delta signifies the variance between the minimum and maximum S/N ratios corresponding to each parameter, while CR indicates the proportion of a parameter’s delta value in relation to the overall delta value. This table aids in discerning the pivotal factors that exert a significant influence on temperature.

As a result, the influence of the parameters on the temperature of the heat sink base is in the order of H, H’, and b. In addition, the outcome of the response indicates that the length of the fin has a significant impact on the temperature of the heat sink base.

The main effects of the S/N ratios are presented in Figure 8, based on the optimal control factors obtained from the response table. The ultimate goal of the study is to achieve the highest possible S/N ratio. As such, the optimal heat sink temperature combinations are H’3, H3, and b3.

4.3. Analysis of Variance (ANOVA) for the Study Case of the High-Power LED

Apart from the application of the Taguchi method, the analysis of variance (ANOVA) serves as an additional statistical approach to strengthen the credibility of the findings. This technique is utilized to ascertain the key parameters influencing the results and to validate the outcomes obtained through the Taguchi method. ANOVA plays a crucial role in recognizing the individual contributions of each parameter to the objective functions.

The outcomes of the ANOVA analysis for the levels of the S/N Ratio and control factors are presented in Table 9. The F-value is computed for each parameter, providing valuable insights into their respective significance.

As the value of a parameter represented by the F-value increases, so does its impact on the response value. The ANOVA F-value results were used to calculate the percentage of influence of each factor, which is presented at a 95.31% confidence level. Additionally, Figure 9 illustrates the contribution rate of control parameters to temperature.

The analysis results indicate, in

Table 10. ANOVA results of levels and control factors for the S/N ratio of temperature.

Source of Variation	Degree of Freedom	Sum of Squares	Mean of Square	F-Value
H’	2	78.89	39.44	60.49
H	2	173.90	86.95	133.34
b	2	12.29	6.14	9.42
Error	20	13.04	0.65	1.00
Total	26	278.12

, that the largest F-value of 133.34 is attributed to height (H), indicating that fin length (H) significantly affects temperature. This is followed by the distance between the upper walls of the cavity and the fins (H’), and finally, the impact of the heat sink thickness (b). It is clear that the factor trend and ranking align with the Taguchi method in Section 4.2.

4.4. Application Case on a Commercial 13 W High Power LED

In the previous paragraph, we validated the model with an input power of 1 W. In this paragraph, we will use the same heat sink structure (Figure 1 and Figure 2) from the previous paragraph but apply it to a concrete case study for a high-power commercial LED (XLAMP^® CXA1304) of 13 W in order to validate the results.

4.4.1. Thermal Analysis Applied on the Study Case for 13 W High Power LED

As for the thermal analysis, we used the best structure found in Section 4.1 for the 1 W LED, and we applied it to a high-power commercial LED. From the study of Figure 5, the best case obtained was H’ = 200 mm for a free space height above the heat sink. The best result obtained (Section 4.1, Figure 6) previously for 1 W was the one where the fin height was H = 80 mm and, in the results presented in Figure 7, had a fin thickness value of b = 25 mm. The applied results of the new thermal study using the commercial LED are presented in Figure 10, Figure 11 and Figure 12.

Figure 10, Figure 11 and Figure 12 illustrate how the geometry of a heat sink affects the thermal and airflow performance in the cavity for a concrete case of a luminaire with a power of 13 W. In the first case, with a height (H’) of 200 mm (see Figure 10), which represents the distance between the upper wall of the cavity and the fins, the thermal plume rises above the heat source due to the airflow driven by the buoyancy. However, this large distance allows for the development of flow instabilities, disturbing the symmetry and reducing the efficiency of heat dissipation. As a result, higher temperatures are observed.

In the second case, with a fin height (H) of 80 mm (see Figure 11), the shorter fins improve heat dissipation by promoting a more uniform temperature distribution and symmetrical airflow patterns. This leads to improved cooling efficiency, as the airflow remains stable and concentrated around the fins.

For a base thickness (b) of 25 mm (see Figure 12), the airflow is well distributed. However, the slightly elevated base temperatures suggest reduced heat transfer efficiency, probably caused by the narrower base, which limits both thermal conduction and convective cooling capacity.

4.4.2. Control Parameters Optimization Using Taguchi Method

We recreated the model presented in Section 4.2 using statistical analysis, in particular, the Taguchi method, with the same parameters tested in Table 8. The parameters examined include, as follows: H’, which is the distance between the cavity top walls and the fins (ranging from 30 to 180 mm); H, the length of the fins (ranging from 20 to 90 mm); and b, the thickness of the heat sink base (ranging from 8 to 24 mm).

In the temperature analysis presented in

Table 11. Experimental design with S/N ratios for the application case.

Tests	Results
	Temperature (K)	S/N Ratios
1	460.4	−53.26
2	444.1	−52.94
3	433.6	−52.74
4	414.6	−52.35
5	407.7	−52.20
6	401.7	−52.07
7	397	−51.97
8	392.6	−51.87
9	388.4	−51.78
10	418.9	−52.44
11	410	−52.25
12	404.1	−52.12
13	395	−51.93
14	392.9	−51.88
15	392.4	−51.87
16	391.8	−51.86
17	391.3	−51.85
18	391	−51.84
19	379.5	−51.58
20	375.5	−51.49
21	374	−51.45
22	368.2	−51.32
23	365.6	−51.26
24	362.9	−51.19
25	364	−51.22
26	361.7	−51.16
27	359.5	−51.11

, the distance between the upper walls of the cavity (H’) is the most important (Rank 1), contributing 56.16%. The length of the fins (H) is the second most influential factor (Rank 2), accounting for 33.5%, while the thickness of the base (b) contributes a smaller share, with a CR of 10.14%.

Table 12. Response tables for S/N ratios of various control factors and their delta, rank, and contribution ratio for temperature for the application case.

Temperature
Level	H’	H	b
1	−52.36	−52.26	−51.99
2	−52.01	−51.79	−51.88
3	−51.31	−51.63	−51.80
Delta	1.05	0.62	0.19
Rank	1	2	3
CR (%)	56.16	33.50	10.34

provides a summary of the average signal-to-noise (S/N) ratios for various control parameters over three levels applied to the case study. It also includes the delta values, rankings, and contribution ratio (CR) associated with each parameter.

Therefore, the influence of the parameters on the heat sink base temperature follows this order: H’ > H > b. Furthermore, the results indicate that the fin length (H) has a significant impact on the heat sink base temperature.

Figure 13 illustrates the main effects of S/N ratios, showing the optimal control factors derived from the response table. The objective of the study is to maximize the S/N ratio. Thus, the optimal combinations to minimize the heat sink temperature are H’3, H3, and b3.

4.4.3. ANOVA Applied for the Concrete Study Case 13 W High-Power LED Lighting

We reapplied Equations (16) to (20) to obtain the ANOVA results. The ANOVA results for the signal-to-noise ratio (SNR) levels and control factors are shown in

Table 13. ANOVA results of levels and control factors for the S/N ratio of temperature for the application case.

Source of Variation	Degree of Freedom	Sum of Squares	Mean of Square	F-Value
H’	2	10,517.85	5258.92	63.34
H	2	4176.38	2088.19	25.15
b	2	375.46	187.73	2.26
Error	20	1660.44	83.02	1
Total	26	16,730.15

. The F-value is calculated for each parameter, which provides valuable information about their significance. An increase in the F-value of a parameter indicates a greater influence on the response value. The ANOVA F-value results were used to determine the percentage influence of each factor, presented at a 95.67% confidence level. The analysis results shown in Table 13 reveal that the highest F-value of 63.34 is attributed to the height (H’), indicating that the distance between the upper cavity walls and the fins (H’) has a significant impact on the temperature. It is followed by fin length (H), while heat sink thickness (b) shows the weakest effect. The trend and ranking of factors are consistent with the results of the Taguchi method in Section 4.4.2. In addition, Figure 14 illustrates the contribution rate of the control parameters to the temperature.

As part of the validation process, the optimized design of the H’3H3b3 heat sink was implemented, with dimensions of 180 mm × 90 mm × 24 mm. This heat sink was specifically developed to improve the thermal performance of the XLAMP^® CXA1304 LED 13 W high-performance LED lighting system. During testing, the heat sink recorded a base temperature of 86.4 °C, which is well below the manufacturer’s maximum permissible operating case temperature of 135 °C [68].

These results confirm the effectiveness of the design in maintaining safe operating temperatures even under high thermal loads, ensuring both the reliability and longevity of the LED module. The optimized heat sink geometry not only reduces thermal resistance but also improves heat dissipation efficiency, as shown in Figure 15, making it suitable for demanding real-world applications. These results underline the success of the optimization process in addressing the thermal management challenges associated with high-power LED lighting.

5. Conclusions

If LED lighting leads to a considerable reduction in energy production, which serves as a lever for reducing the carbon footprint and developing smart cities, then thermal management is essential to maintain its energy efficiency and lifespan. However, heat sinks have a considerable number of parameters, particularly in terms of their dimensions. Finding simulation methods to quickly determine and optimize the best parameters for LED heat sinks is therefore essential for the development of this technology and contributes significantly to the lifespan of products and their energy efficiency. This makes them more sustainable, mainly in relation to SDGs 7 and 11.

By implementing the Taguchi method using the L27 orthogonal network, this research has identified the most effective parameters for optimizing heat transfer in an LED housing unit by improving the temperature distribution at the base of the heat sink. Several control parameters were analyzed using the Taguchi method and ANOVA to determine their contribution rates. This study rigorously determined the factors influencing the thermal performance of a 1 W COB (chip-on-board) LED package. The analysis revealed that the heat sink temperature is mainly influenced by the following parameters, in descending order of importance: the length of the heat sink fins (51.80%); the distance between the upper cavity walls and the fins (34.14%); and the thickness of the heat sink (14.06%).

For our application case, we applied the same methodology to the XLAMP^® CXA1304 high-power commercial LED (13 W) to validate the method with real values. Here too, several control parameters were analyzed using the Taguchi method and ANOVA to determine their contribution rates.

The results showed that the heat sink temperature is mainly affected by these parameters, ranked in descending order of importance: the distance between the cavity top walls and the fins (56.16%); the heat sink fin length (33.5%); and the heat sink thickness (10.34%). The high agreement between the ANOVA and Taguchi results underlines the robustness of the employed methodology. This potential was validated by confirmatory tests with a real LED lighting system using the XLAMP^® CXA1304 (13 W). The results showed a junction temperature of 86.4 °C, which is well below the maximum allowable operating case temperature of 135 °C, thanks to the optimized heat sink design. The comparison between the initial study (1 W LED) for model validation and the study for a real case (13 W LED) indicates that increasing the module input power changes the percentages and order of the parameters affecting the system.

This study can be used as a reference for real-world lighting applications in confined spaces, such as vehicle headlights, where the volume is a closed and reduced space and also in indoor applications such as recessed lighting in false ceilings.

By improving the thermal performance of LEDs, this study contributes to broader sustainability and energy-saving objectives (SDGs 7 and 11). Better thermal management results in better energy efficiency, longer LED life, and the greater reliability of LED luminaires.