Optimization of Microjet Location Using Surrogate Model Coupled with Particle Swarm Optimization Algorithm

mathematics

Article Optimization of Microjet Location Using Surrogate Model Coupled with Particle Swarm Optimization Algorithm

Mohammad Owais Qidwai 1,* , Irfan Anjum Badruddin 2,* , Noor Zaman Khan 3 , Mohammad Anas Khan 4 and Saad Alshahrani 2

1 Department of Mechanical Engineering, Delhi Skill and Entrepreneurship University, Okhla Campus-1, Okhla Industrial Estate, Phase-III, New Delhi 110020, India 2 Mechanical Engineering Department, College of Engineering, King Khalid University, Abha 61421, Saudi Arabia; [email protected] 3 Department of Mechanical Engineering, National Institute of Technology Srinagar, Hazratbal, Srinagar 190006, Jammu and Kashmir, India; [email protected] 4 Department of Mechanical Engineering, Jamia Millia Islamia, New Delhi 110025, India; [email protected] * Correspondence: [email protected] (M.O.Q.); [email protected] (I.A.B.)

Abstract: This study aimed to present the design methodology of microjet heat sinks with unequal jet spacing, using a machine learning technique which alleviates hot spots in heat sinks with non- uniform heat ﬂux conditions. Latin hypercube sampling was used to obtain 30 design sample points on which three-dimensional Computational Fluid Dynamics (CFD) solutions were calculated, which were used to train the machine learning model. Radial Basis Neural Network (RBNN) was used as a surrogate model coupled with Particle Swarm Optimization (PSO) to obtain the optimized Citation: Qidwai, M.O.; Badruddin, location of jets. The RBNN provides continuous space for searching the optimum values. At the I.A.; Khan, N.Z.; Khan, M.A.; predicted optimum values from the coupled model, the CFD solution was calculated for comparison. Alshahrani, S. Optimization of The percentage error for the target function was 0.56%, whereas for the accompanied function it Microjet Location Using Surrogate was 1.3%. The coupled algorithm has variable inputs at user discretion, including gaussian spread, Model Coupled with Particle Swarm number of search particles, and number of iterations. The sensitivity of each variable was obtained. Optimization Algorithm. Mathematics Analysis of Variance (ANOVA) was performed to investigate the effect of the input variable on 2021, 9, 2167. https://doi.org/ thermal resistance. ANOVA results revealed that gaussian spread is the dominant variable affecting 10.3390/math9172167 the thermal resistance.

Academic Editor: Carlos Llopis-Albert Keywords: optimization; surrogate model; Radial Basis Neural Network; Particle Swarm Optimiza- tion; thermal resistance; Computational Fluid Dynamics Received: 8 July 2021 Accepted: 31 August 2021 Published: 5 September 2021 1. Introduction Publisher’s Note: MDPI stays neutral A high-heat flux removal from microdevices is still a major challenge even after more with regard to jurisdictional claims in than four decades of research and development. The micro-cooling devices are being inves- published maps and institutional affil- tigated both experimentally and in combination with the computational study. The experi- iations. mental results are not synchronous, but the computational results appear to minimize the anomaly in the experimental analysis of micro-devices termed as “scaling effects,” which are difficult to capture in experimental investigation [1]. Various computation-intensive studies have been performed in the past exploring the effect of fluid flow characteristics Copyright: © 2021 by the authors. and the heat-transfer effects. The micro-cooling devices encompass the use of microchan- Licensee MDPI, Basel, Switzerland. nels [2–5], porous materials [6], and microjets [7–14] with a promising future in all types This article is an open access article of designs. However, each of them has its own set of challenges, from the design stage distributed under the terms and to the manufacturing stage. The substrate temperature non-uniformity along the length conditions of the Creative Commons of the microchannel heat sink is one of its inherent features. In order to counter it, Samal Attribution (CC BY) license (https:// and Moharana [15] provided a novel concept of recharging design of microchannel, which creativecommons.org/licenses/by/ improves temperature uniformity and reduces back axial conduction. The porous materials 4.0/).

Mathematics 2021, 9, 2167. https://doi.org/10.3390/math9172167 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 2167 2 of 18

have to tackle the challenge of the high-pressure drop [6]. The microjet impingement needs to deal with the problem of material erosion [16], but relaxes on the constraint of temperature non-uniformity and pressure drop. Most of the investigation of micro-cooling assumes uniform heat flux generation; however, in practical cases, microprocessor chips generate non-uniform heat flux [17]. The flexibility in microjet arrangement allows it to be directly embedded within the device to alleviate the hotspots and thermal stresses. Many investigations were conducted over different designs to improve the heat transfer effects in the heat sink within the constraints such as maximum pumping power, maximum pressure drop, a constant flow rate of cooling fluid, constant heat flux, constant interface area, or constant cross-sectional area. Hajmohammadi et al. [7] investigated to obtain minimum peak temperature in the substrate, for optimal location and size of heat sink attachments, with the total area of heat sink remaining constant. It was suggested that dividing the large heat sink into a group of smaller ones enhances thermal performance by reducing peak temperatures. Husain et al. [8] numerically investigated a hybrid design of microchannel comprising of a microchannel pillar and jet impingement technique, which enhances heat transfer rate as the stagnation effect under jet diminishes due to channel flow. Zhang [9] combined a slot jet with microchannel design and analyzed the hybrid design numerically. For the same cross-sectional area, three channel shapes were identified, such as circular, trapezoidal, and rectangular. Results showed that the module with a circular channel had the highest pressure drop while the trapezoidal channel shape has the least substrate temperature. The highest pressure drop in the circular shape is attributed to the strong vorticities formed perpendicular to the direction of flow. Peng et al. [10] performed a comparative analysis between traditional microchannel heat sink (TMC) and multi-jet microchannel (MJMC) heat sink, using numerical techniques. They found that temperature uniformity improves significantly with the help of an increasing number of jets. The cooling performance of MJMC surpasses TMC. Husain et al. [11] compared the effect of different flow spent schemes with and without extraction ports. They presented that fluid removed from edges in unconfined flows have higher temperature uniformities and lower pressure drop. Qidwai et al. [4] found that temperature uniformity is obtained at the trade-off with Nusselt number in their study of a diverging channel. Han et al. [12] combined the microchannel with the microjet to capture the merits of both designs. The full-length trenches take away the spent fluid, which helps to reduce the crossflow effect. The maximum temperature distribution remains within the range of 25 ◦C. Deng et al. [14] investigated slot jet array and hybrid microchannel slot jet array heat sinks in the range of Reynolds number 230 to 2760. They affirmed that a hybrid heat sink has better performance for the same flow rate. Xie et al. [18] presented an argument that a uniform heat flux condition is acceptable for chips to surface only, rather than for heat sink surface, as assumed in many studies. The study included a combination of flow arrangements and different positions of heat-generating chips. The investigation highlighted that chip position significantly affects the temperature distribution. Lelea [19] found that the position and number of jets strongly influence hydrothermal performance. For fixed pumping power, the minimum temperature is higher for heat sinks with multiple inlets, and the temperature difference across the substrate is reduced. Wiriyasart and Naphon [20] considered parametric study of different fin structures liquid jet impingement to provide guidelines for heat sink design for minimum thermal resistance. In most of the investigations, hybrid designs are analyzed for uniform heat flux conditions. For non-uniform heat flux, Yoon et al. [21], considered thermal resistance as an indicator to obtain the optimal position of partial heating for uniform and non- uniform heat flux conditions. They concluded that partial heating located at the centre behind the heat sink is the optimal position irrespective of the non-uniformity of heat flux. Sharma et al. [17] proposed a one-dimensional semi-empirical modeling approach for quick determination of initial design of microchannel heat sink targeting hot-spots. Design parameters in a microchannel heat sink were optimized by Hadad et al. [22] for non-uniform heat flux, using the RSM and JAYA algorithms together. The other work Mathematics 2021, 9, 2167 3 of 18

by Hadad et al. [23] investigated the shape optimization of a water-cooled impingement micro-channel heat sink, including manifolds.

1.1. Regression Based Surrogate Model Optimization Techniques Parno et al. [24] implemented a Design and Analysis of Computer Experiments (DACE) surrogate model with PSO to optimize the solution of the groundwater problem. The sample of DACE was obtained using Latin Hypercube sampling to provide wide distributions of initial particles to yield robust convergence. Further, different variations of surrogate-assisted PSO excelled in different applications such that among Local PSO, adaptive PSO, and Global PSO; Local PSO outperformed the other two for problems with more than 10 dimensions. To improve heat transfer characteristics in a microchannel heat sink with the least trade-off of friction factor, Seo et al. [25] put forward another method wherein the objective functions were formulated by two surrogate models: Response Surface Approximation and the Kriging model. The Pareto Optimal Front obtained using a multi-objective Genetic Algorithm for two surrogate models were compared based on convergence and divergence criteria. The results of the two models were combined using K-means clustering to obtain the best Pareto Optimal front. Many researchers proposed a design methodology intending to increase thermal efﬁ- ciency with low-pressure drops. Shi et al. [26] considered Response Surface Approximation (RSA) combined with the NSGA-II algorithm to obtain Pareto solutions to optimize the selected design parameters. They found that the ratio of secondary channel width to microchannel width is the most inﬂuential parameter. Park et al. [27] evaluated design parameters in the wavy channel with grooves with the help of a multi-objective Genetic algorithm coupled with RSA surrogate model. Kulkarni et al. [28] proposed a design methodology for the heat sink to obtain lower thermal resistance. They used RSA with a multi-objective genetic algorithm to obtain Pareto-optimal design parameters.

1.2. Machine Learning-Based Surrogate Model Optimization Techniques Machine learning algorithms evolve constantly with the aim for better prediction and the least amount of time and effort. They have been implemented in several fields such as intelligent communications [29], 3D printing technology [30], hydrological science [31], and many more. Many studies are conducted, in order to take advantage of machine learning, to improve heat transfer in heat sinks. Some investigations are focused on shape optimization, such Krzywanski [32], who introduced the design methodology of a bio- inspired falling-film heat exchanger using the combination of Genetic Algorithm (GA) and Artificial Neural Network (ANN). They reported that the AGENN model can successfully determine the required design parameters and operating conditions to generate the desired total heat transfer rate of the heat exchanger. Han et al. [33] used surrogate-based shape optimization of the wing-body of aircraft. They used Latin Hypercube sampling; however, to choose new sample points they used infill-criterion, so as to generate new designs based on the known designs. To explore the sensitivity of the change in design parameters, the investigation comprised of Maximum-Likelihood Estimation (MLE), with its only drawback that large sample points are needed to estimate the correlation. They suggested using surrogate-based models several times to reach the global optimum. Xi et al. [34] used the MATLAB function nonlinear fitting model ‘nlinfit’ to obtain a correlation of 90 experimental data points to determine the Nusselt number. Then, the Back Propagation Neural Network (BPNN) was used to train the model with a Genetic Algorithm (GA) to improve heat transfer performance of ribbed channels. In the combined GA-BPNN algorithm, GA was used to optimize the weights and threshold of BPNN so that, together, they could reach the global optimum solution. Singh [35] addressed the problem of turbine-blade cooling using a slot jet cooling mechanism. The location of slot jet impingement on a concave surface was optimized using hybrid feed-forward Artificial Neural Network (ANN) and Genetic Algorithm (GA). ANN was used to train nonlinear data on 175 cases in the design space. The predicted optimal solution of ANN- Mathematics 2021, 9, 2167 4 of 18

GA was simulated in CFD simulation, for which the difference in output was negligible. Shi et al. [36] proposed a methodology for modifying heat exchanger design using CFD coupled with the Radial Basis Neural Network surrogate model and genetic algorithm to achieve maximum ﬂow uniformity.

1.3. Optimization Techniques GA algorithms can be sub-divided into three operations: Mutation, Crossover, and Selection. Tan et al. [37] suggested that parameters in GA are chosen based on the rule of thumb. There is no standard to select a proper strategy or parameter or their combination, which may lead to poor convergence and trapping in local optima. Additionally, these algorithms need to be run a thousand times before reaching global optima. Samii [38] compared the two different approaches and concluded that PSO is simpler to use based on the fact that it has higher convergence compared with GA, since, when the individual possesses the same genetic material, the crossover effect is almost eliminated. For structural optimization Eagle strategy, the PSO algorithm combines the advantages of global search and intense local search [39]. The Eagle strategy uses the Lévy walk method to identify the search domain, then PSO is applied in the second stage. This efficient combination reduces the computation time and increases the probability of achieving global optima. Bello-Ochende et al. [40] used the Constructal theory in combination with CFD to reduce maximum wall temperature; however, the originator of the Constructal theory considers it to be a universally valid theory and not an optimality statement [41]. Entropy Generation minimization (EGM) coupled with CFD is another methodology, based on the first principle, applied in many investigations [42,43]. Some studies preferred to use the simplified conjugate gradient method [44,45], which uses sensitivity analysis to establish the conjugate direction. However, in this method, the probability of trapping in the local optima is quite high [46].

1.4. Hybrid Design Schemes Some studies with non-uniform heat flux focussed on minimizing the maximum substrate temperature. Cho et al. [47] performed an experimental and numerical analysis of the different combinations of channel geometries and header designs. Three types of thermal loads were chosen to obtain peak temperature and temperature distribution in the substrate. Their results showed that a reasonable trade-off between peak temperature and pressure drop is a must, as pressure drop is higher in the diverging channel with a trapezoidal header design with the lowest peak temperature and better temperature distribution. Abo-Zahhad et al. [48] implemented hybrid schemes with microjet and microchannel for thermal management in high concentrator photovoltaic solar cells. They used exergy efficiency (electrical and thermal) for the performance comparison among different layouts of cell structure. The lowest temperature non-uniformity was found to be 5.8 ◦C. The main objective of their study was to avoid exergy loss, i.e., to have uniform cooling at the lowest possible flow rates. Muszynski and Mikielewicz [49] investigated the arrangement of an array of jets on heat transfer in the heat sink. The fluid interaction of fully submerged jets is very complex; therefore, they preferred an equal-nozzle spacing design for the analysis.

1.5. Research Gap The non-uniformity in temperature distribution arises due to the non-uniform heat flux generated in the processors. However, due to the miniature design and high thermal conductivity of the substrate material, it is convenient to assume uniform heat flux boundary in most studies, but it is not an absolute true case. According to the literature review, most of the design strategies assumed uniform heat flux boundary conditions in the numerical analysis. However, in experimental conditions, the miniaturized heat sinks have to deal with non-uniform heat flux conditions. The microjet heat sink geometry dealing with hot-spots assumes equal jet spacing to eliminate the complexity of multi-jet interaction Mathematics 2021, 9, x FOR PEER REVIEW 5 of 19

The non-uniformity in temperature distribution arises due to the non-uniform heat flux generated in the processors. However, due to the miniature design and high thermal conductivity of the substrate material, it is convenient to assume uniform heat flux boundary in most studies, but it is not an absolute true case. According to the literature review, most of the design strategies assumed uniform heat flux boundary conditions in the numerical analysis. However, in experimental conditions, the miniaturized heat sinks have to deal with non-uniform heat flux conditions. The microjet heat sink geometry Mathematics 2021, 9, 2167 dealing with hot-spots assumes equal jet spacing to eliminate the complexity of multi-jet5 of 18 interaction in the fluid domain. To the best of the authors’ knowledge, no heat sink designed with non-uniform jet spacing is investigated to date. The objective of this study wasin the to fluid minimize domain. Tothermal the best resistance of the authors’ while knowledge, keeping a no check heat sink on designed temperature with non-uniformitynon-uniform jet spacing of the heat is investigated sink by providing to date. The a strategy objective for of this practical study design was to minimize of a water-cooledthermal resistance microjet whileheat sink, keeping which a check utilizes on temperature the significance non-uniformity of machine oflearning the heat tech- sink niquesby providing to empower a strategy the design for practical process design to yield of aa water-cooledbetter design microjetto tackle heat ever-increasing sink, which challenges.utilizes the significance of machine learning techniques to empower the design process to yield a better design to tackle ever-increasing challenges. 2. Computational Model 2.1.2. Computational Flow Configuration Model and Numerical Methods 2.1. Flow Configuration and Numerical Methods Figure 1 shows the configuration of the complete computational domain. The di- mensionsFigure of1 theshows heat the sink configuration are 4000 × 3000 of the × complete250 μm (L computational × W × H), with domain. a copper The substrate dimen- thicknesssions of the of 50 heat μm. sink The are inlet 4000 of ×fluid3000 takes× 250 placeµm from (L × theW holes× H), provided with a copper at the substratetop plate µ withthickness a fixed of 50diameterm. The of inlet 400 ofμm, fluid and takes the placespent fromfluid thecarries holes away provided from atone the side top only. plate µ Non-uniformwith a fixed diameterheat flux ofwas 400 appliedm, and at the the bottom spent fluidof the carries heat sink away as fromper the one power side map only. adoptedNon-uniform from the heat study flux wasof Sharma applied et atal. the [17]. bottom The design of the has heat inherent sink as perlimitations the power of fixed map geometricaladopted from location the study hot ofspots Sharma and fixed et al. [region17]. The of designinlet holes. has inherent Operational limitations limitations of fixed in- geometrical location hot spots and fixed region of inlet holes. Operational limitations clude fixed coolant inlet temperature, constant mass flow rate (0.00025 kg/s), and the include fixed coolant inlet temperature, constant mass flow rate (0.00025 kg/s), and the power dissipated at the substrate base. The thermo-physical properties of copper and power dissipated at the substrate base. The thermo-physical properties of copper and water are given in Table 1. water are given in Table1.

(a) (b)

FigureFigure 1. ((aa)) Physical Physical model model of of microjet microjet heat heat sink. sink. ( (bb)) A A non-uniform non-uniform power power map map adopted adopted from from Sharma et al. [17]. [17].

TableTable 1. 1.ThermophysicalThermophysical properties properties of of liquid liquid and and solid solid state state at at 300 300 K. K. 3 풄풑 (J/kg.K) 풌 (W/m. K) 흆 (kg/m ) 3 흁 (kg/m. s) cp (J/kg·K) k (W/m·K) ρ (kg/m ) µ (kg/m·s) Water 4181 0.614 997 2.414 × 10 × 10.⁄() [50] −5 247.8/(T−140) Copper 381 Water388 41818978 0.614 997 2.414 × 10 × 10 [50] Copper 381 388 8978 The following assumptions were made for the computational domain. 1. Continuum regime applies, in which Navier-Stokes equation remains valid. The following assumptions were made for the computational domain. 1. Continuum regime applies, in which Navier-Stokes equation remains valid. 2. The ﬂow is three-dimensional, incompressible, and steady-state for energy and

fluid flow. 3. Thermo-physical solid and fluid properties are constant, except fluid viscosity. 4. Buoyancy and viscous effects are negligible. The above assumptions guide the governing Equations (1)–(4) for both cooling fluid and solid substrate domain as follows: For coolant (liquid phase). Mathematics 2021, 9, 2167 6 of 18

Conservation of mass for coolant (liquid phase).

→ ∇ · V = 0 (1)

Momentum for coolant (liquid phase).

→ → → 2 ρ f (V · ∇)V = −∇p + µ f ∇ V (2)

Energy for coolant (liquid phase).

→ 2 ρ f cp, f (V · ∇)Tf = k f ∇ Tf (3)

Energy for substrate (solid phase).

→ 2 ρ f cp, f (V · ∇)Tf = k f ∇ Tf (4)

The Ansys Mesh modeler was used for discretization of the computational domain. The conservation equations of mass, momentum, and energy equations were solved using Ansys CFX. The transport equations were solved with the laminar model. The Rhie-Chow algorithm was used to couple pressure and velocity fields. The convergence criterion was set to be 10−6 for all the equations. Since there is no symmetry in the geometry, complete domain analysis is necessary. Non-uniform heat flux was applied at the bottom wall of the heat sink, as shown in Figure1b. All the other sides of the heat sink domain were considered insulated, and heat is convected away from the heat sink by fluid only. The mass flow inlet boundary condition was applied with a uniform temperature field across the inlet cross-sections, and the pressure outlet condition is applied at the outlet cross- section. At the interface between the heat sink and the coolant, a no-slip condition was chosen with heat transfer coupling between the two domains. The boundary conditions are summarized in Table2.

Table 2. Boundary condition of numerical method.

Boundary Condition Momentum Thermal

Fluid domain Mass ﬂow inlet/Pressure outlet Tin = 290 K → Heat sink domain at y = 0 − ∂T = V = 0 ks ∂η qk → b ∂T Heat sink domain at x = 0, x = W, y = H, z = 0, z = L V = 0 ∂η = 0 (adiabatic) → Tf ,Γ = Ts,Γ Interface between ﬂuid and substrate V = 0 (No slip condition) ∂Tf ∂T −k f = −ks ∂η Γ ∂η Γ

The Reynolds number was found to be within the laminar regime at both inlet and outlet cross-section for the given mass ﬂow rate. Based on the respective hydraulic diameter given in Equation (5), the Reynolds number was calculated as given in Equations (6) and (7).

4A d = c (5) h $

. mindh,in Rein = (6) Ac,inµ f . modh,o Reo = (7) Ac,oµ f Mathematics 20212021,, 99,, xx FORFOR PEERPEER REVIEWREVIEW 7 7 of of 19 19

푚̇̇ 푑,, 푅푒 = (6)(6) 퐴,,휇

Mathematics 2021, 9, 2167 푚̇̇ 푑,, 7 of 18 푅푒 = (7)(7) 퐴,,휇

2.2.2.2. GridGrid IndependenceIndependence TestTest InInIn orderorderorder tototo savesavesave computingcomputingcomputing timetimetime andandand memory,memory,memory, thethethe sizesizesize ofofof spatialspatialspatial girdgirdgird sizesizesize waswaswas de-de-de- terminedterminedtermined bybyby thethethe trial-and-error trial-and-errortrial-and-error method. method.method. TheTheThe girdgirdgird independence independenceindependence testtesttest waswaswas considered consideredconsidered for forfor thermalthermalthermal resistance resistanceresistance and andand mean meanmean substrate substratesubstrate temperature, temperature,temperature, as asas shown shownshown in inin Figure FigureFigure2 .2.2. The TheThe difference differencedifference ininin results resultsresults with withwith grid gridgrid systems systemssystems 6 66 million, million,million, 9.1 9.19.1 million, million,million, and andand 10.44 10.4410.44 million millionmillion elements elementselements is isis much muchmuch less lessless thanthanthan the thethe maximum maximummaximum allowed allowedallowed limit. limit.limit. Thus,Thus,Thus, a aa grid gridgrid with withwith 9.1 9.19.1 million millionmillion elements elementselements was waswas selected selectedselected for forfor furtherfurtherfurther calculations. calculations.calculations.

6.40 325.8 Rth 6.35 Tmean 325.6

6.30 325.4

6.25 325.2 Thermal Resistance Thermal Resistance Mean Temperture (K) 6.20 325.0 Mean Temperture (K) 3 4 6 9 10 No. of Cells (millions)

FigureFigure 2.2. TheThe results results of of grid grid independence independence test test based based on onthermal thermal resistance resistance and and mean mean substrate sub- stratetemperature.temperature. temperature.

2.3.2.3. ValidationValidation TheThe current current study’s study’s numerical numerical values values were were compared compared with with the the experimental experimental results results for aforfor single-phase aa single-phasesingle-phase microjet microjetmicrojet silicon siliconsilicon heat heatheat sink sink (ksink = 148(k(k ==W/m 148148 W/mW/m K) as K)K) reported asas reportedreported by Wang byby WangWang et al. etet [51 al.al.]. The[51].[51]. test TheThe cases testtest casescases with thewithwith number thethe numbernumber of nozzles ofof nozzlesnozzles 13 and 1313 4 andand with 44 with awith nozzle aa nozzlenozzle diameter diameterdiameter of 40 ofofµm 4040and μmμm 50andµm 50were μm were subjected subjected to ﬂow to flow rates rates of 2 of mL/min 2 mL/min and and 8 mL/min, 8 mL/min, for for a a power power range range of of 3.83.8 toto 23.8 23.8 W W,, respectively. respectively. AA direct direct comparison comparison between between experimentalexperimental andand numericalnumerical resultsresultsresults for forfor maximum maximummaximum temperature temperaturetemperature rise riserise in inin the thethe substrate substratesubstrate as asas a aa function functionfunction of ofof power powerpower is isis shown shownshown in inin FigureFigure3 3,, withwith a a deviation deviation within within 15%. 15%. ThisThis deviationdeviation is is attributed attributed to to several several factors factors such such asas simpliﬁcationsimplification of of the the physical physical model model and and experimental experimental uncertainties. uncertainties.

65 d/H=0.2 Wang et al. d/H=0.2 Num d/H=0.25 Wang et al. 45 d/H=0.25 Num

25 Temperature Rise (K) Temperature Rise (K) 5 0 5 10 15 20 25 Power (W) Power (W)

Figure 3. Comparison of the current numerical simulation result with the experimental results of Wang et al. [51] for a ﬂow rate of 8 mL/min.

2.4. Computational Complexity and Implementation Cost The initial cost of the CFD setup is very high, which includes a high-end computing machine and annual license fees along with the operating cost associated with manpower. Mathematics 2021, 9, 2167 8 of 18

However, this setup can be used to solve many problems associated with thermal and fluid interactions, which reduces its overall expenditure per problem. In the CFD simulation, the Navier-Stokes equation was discretized over a mesh, formulated separately for each case. This mesh may vary in the number of elements, based on which mesh can be categorized between “coarse” to “very fine”. The “coarse” mesh takes less time compared with “very fine” mesh for simulating the same case. However, the results are far from reality for coarse mesh, whereas results of “very fine” mesh approach values of exact solutions, at the cost of consumption of a large amount of computation power and time. Therefore, the number of elements in a mesh needs to be selected appropriately, and the same was carried out by performing a Grid Independence test. The system used for CFD simulation comprised an Intel Xeon Processor E5-1620 v4 (4 Core, 3.5 GHz, 3.8 GHz Turbo, 2400 MHz, 10 MB, 140 W) and RAM 32 GB (8 × 4 GB) 2400 MHz DDR4. On this machine, each case took more than 10 h of simulation, as the time taken for each case depends on the complexity of each case, the number of elements, and the combination of physical models used. Further, the result at discrete values of design variables can only be obtained one at a time. In this scenario, it is not feasible to search optimum values in continuous space with the help of the CFD technique alone. To tackle this challenge, different techniques were used, as discussed in the literature. To save a large amount of computation cost and time, previous researchers have used CFD in combination with AI models, which reduces time to reach optimum values within a few minutes. This is the huge advantage of using combinatorial algorithms.

3. Optimization Method 3.1. Design Space, Sample Points, and Objective Function The flow configuration remains fixed in earlier studies and assumes uniform base heat flux with modifications in geometry for parametric investigation of design variables without considering the local heat-transfer effects. For each new parameter of the design variable, a complicated numerical analysis must be performed to obtain its flow and heat transfer characteristics. The top plate was provided with 9 inlet holes in this numerical model, as shown in Figure4. Each hole remains completely in the region marked by section lines A, B, A’, and B’. From the lower-left corner, the A and B locations are 0.96 mm and 1.93 mm, while for A’ and B’, they were 1.3 mm and 2.6 mm, respectively. The center of each hole was placed such that at least one jet was available for cooling in the marked section. The design space was set as:

0.2 < x11, x21, x31 < 0.725; 0.2 < y11, y12, y13 < 0.975; 1.125 < x12, x22, x32 < 1.1775; 1.375 < y21, y22, y23 < 2.525; 2.175 < x13, x23, x33 < 2.7; 2.925 < y31, y32, y33 < 3.7; The overall thermal resistance and temperature uniformity, which is defined as the standard deviation from the mean substrate temperature [11], are given as in Equations (8) and (9).

Tb,max − Tf ,in f1 = (8) qAb s 2 Ne (Tk − Tmean) f2 = ∑ = (9) k 1 Ne

where, ‘k’ represents the different heat flux regions (q1, q2, q3, q4, q5). When considering two objective functions to be minimized, the weighted objective function may be defined. However, there is no basis for defining any weightage in the current case. Therefore, in this work, f1 will be the prime objective function and f2 will be checked simultaneously after optimization. With the help of the Latin hypercube sampling technique, different positions of the hole were obtained, which are given in Table3. Latin hypercube sampling (LHS) is a modified Monte Carlo sampling method that divides sample design variables with nonoverlapping intervals [52]. It provides a random sampling combination of each design Mathematics 2021, 9, x FOR PEER REVIEW 9 of 19

where, ‘k’ represents the different heat flux regions (푞,푞,푞,푞,푞). When considering two objective functions to be minimized, the weighted objective function may be defined. However, there is no basis for defining any weightage in the current case. Therefore, in this work, 푓 will be the prime objective function and 푓 will be checked simultaneously after optimization. Mathematics 2021, 9, 2167 With the help of the Latin hypercube sampling technique, different positions of the9 of 18 hole were obtained, which are given in Table 3. Latin hypercube sampling (LHS) is a modified Monte Carlo sampling method that divides sample design variables with nonoverlapping intervals [52]. It provides a random sampling combination of each de- signvariable variable in the in designthe design space. space. In theIn the present present case, case, 30 30 sample sample points points were were obtained obtained from fromsampling sampling to satisfactorily to satisfactorily train train the the surrogate surrogate model model using using CFD CFD results results [36 [36].].

FigureFigure 4. 4. SchematicSchematic diagram ofof inlet inlet holes holes on on the the top top plate plate with with position position constraints constraints in sections in sections A, B, A’, A, and B’. B, A’, and B’. Table 3. Sample of centre of inlet holes on the top plate of the substrate. 3.2. Surrogate Model 11 12RBFN is13 successfully 21 implemented 22 to train 23 models [53]. 31 The Radial 32 Basis Neural 33 Sample xyxyxyxyxyxyxyxyxyNetwork (RBNN) function is established using MATLAB 2015, with newrb function, whose maximum number of neurons needs to be provided as an input. It is a three-layer 1 0.46 0.73 1.74 0.53 2.53 0.49 0.60 1.55 1.21 2.52 2.50 2.16 0.55 3.50 1.58 3.62 2.36 3.46 network that includes the input layer, hidden layer, and output layer, as shown in Figure 2 0.64 0.24 1.76 0.93 2.35 0.62 0.54 2.21 1.19 1.68 2.46 1.42 0.72 3.05 1.71 3.03 2.42 3.23 3 0.66 0.39 1.255. 0.28It has 2.62a strong 0.87 ability 0.41 to 1.62predict 1.44 the objective 1.78 2.66 function 2.05 with 0.30 a 3.24rapid 1.42convergence 3.19 2.18 rate. 3.63 4 0.26 0.92 1.32The 0.73 distance 2.58 between 0.76 0.70 selected 1.83 points 1.48 2.10in the 2.39design 1.66 space 0.34 and 3.34the sample 1.39 3.69data points 2.27 2.93 5 0.51 0.89 1.63forms 0.86 the 2.38 basis 0.84 to obtain 0.48 the 2.04 influence 1.76 of 1.46 the data 2.18 points. 2.52 This 0.67 distance 3.37 is 1.35 used 3.26 for training 2.29 3.16 6 0.71 0.28 1.14of 0.89AI model 2.35 on 0.32 the sample 0.50 2.51 data 1.72points 2.26 to calculate 2.20 1.52the inertia 0.24 weight 3.57 1.15(w). 3.65 The γ 2.21 con- 3.56 7 0.37 0.30 1.36stant 0.62 determines 2.57 0.74 the 0.25 radial 1.93 influence 1.54 of 1.64 the Gaussian 2.62 1.98 function, 0.26 3.56which 1.33 is of 3.29less signifi- 2.52 3.01 8 0.24 0.83 1.65cance 0.60 in 2.40the case 0.54 of a 0.65 large 1.69sample 1.64 in the 1.99 design 2.70 space. 1.91 However, 0.54 3.62 for small 1.70 data 3.65 sets, 2.59 the 3.58 9 0.21 0.49 1.31value 0.77 of 2.70 γ needs 0.48 careful 0.42 consideration 1.88 1.30 1.73 before 2.56 any1.87 value 0.22 is assigned 3.31 1.25 to it. 3.33 The RBNN 2.34 3.21 10 0.35 0.85 1.68function 0.402.48 can be 0.21 summarized 0.72 1.84 in the 1.18 following 1.98 2.38Equation 2.35 (10) 0.28 [54]. 3.21 1.51 3.00 2.58 3.37 11 0.55 0.56 1.21 0.22 2.26 0.66 0.62 2.34 1.62 2.05 2.55 1.60 0.45 3.67 1.17 3.10 2.69 3.26 12 0.32 0.68 1.56 0.37 2.60 0.78 0.30 1.43 1.50 1.55Ɲ 2.29 1.93 0.51 3.01 1.67 3.44 2.37 3.07 ℎ(푥, 푦) = 푤 exp (−훾‖푿 − 휐‖) 13 0.59 0.42 1.28 0.50 2.51 0.27 0.28 2.47 1.73 1.83 2.65 2.46 0.37 3.43 1.77 3.24 2.67 3.30 (10) 14 0.49 0.48 1.60 0.31 2.28 0.41 0.20 2.39 1.60 1.49 2.61 1.99 0.44 3.14 1.38 3.56 2.39 3.50 15 0.45 0.61 1.36 0.92 2.33 0.92 0.35 2.44 1.71 2.28 2.32 1.38 0.67 3.69 1.32 2.96 2.23 2.97 16 0.72 0.78 1.29 0.26 2.66 0.37 0.32 2.01 1.16 1.88 2.44 1.76 0.42 3.48 1.75 3.13 2.22 3.47 17 0.27 0.80 1.54 0.35 2.31 0.94 0.33 2.10 1.53 2.15 2.68 2.32 0.52 3.32 1.66 3.38 2.56 3.11 18 0.38 0.33 1.62 0.44 2.22 0.53 0.43 1.47 1.23 2.42 2.34 1.75 0.32 3.60 1.61 3.49 2.63 3.35 19 0.52 0.62 1.42 0.68 2.43 0.85 0.39 1.59 1.41 2.12 2.36 2.11 0.40 3.53 1.46 3.46 2.61 3.40 20 0.58 0.46 1.69 0.83 2.18 0.45 0.63 2.25 1.35 1.84 2.23 2.07 0.63 3.46 1.54 3.18 2.32 3.60 21 0.61 0.21 1.47 0.66 2.45 0.68 0.48 2.32 1.40 2.21 2.31 1.63 0.69 3.04 1.58 3.54 2.65 3.14 22 0.22 0.97 1.15 0.82 2.41 0.25 0.68 1.52 1.68 1.60 2.27 2.44 0.49 2.93 1.45 3.02 2.41 3.33 23 0.63 0.71 1.51 0.56 2.64 0.34 0.57 2.16 1.32 2.40 2.53 1.56 0.64 2.99 1.64 3.52 2.53 3.65 24 0.43 0.56 1.44 0.46 2.47 0.81 0.56 1.75 1.13 2.48 2.23 2.26 0.59 3.28 1.26 3.08 2.44 3.04 25 0.55 0.95 1.49 0.77 2.23 0.57 0.66 1.66 1.57 2.32 2.48 1.80 0.31 3.08 1.20 3.23 2.30 2.98 26 0.29 0.76 1.18 0.41 2.50 0.61 0.26 2.28 1.31 1.38 2.58 2.19 0.21 2.95 1.48 2.93 2.48 3.52 27 0.43 0.65 1.71 0.24 2.30 0.72 0.23 1.77 1.36 1.93 2.25 1.68 0.60 3.17 1.52 3.34 2.51 3.27 28 0.32 0.37 1.21 0.97 2.67 0.96 0.46 1.96 1.26 1.70 2.52 2.40 0.47 3.13 1.28 3.11 2.25 3.69 29 0.40 0.53 1.39 0.71 2.20 0.28 0.37 1.41 1.46 1.43 2.44 2.28 0.39 3.19 1.23 3.40 2.47 3.09 30 0.69 0.26 1.53 0.57 2.55 0.39 0.52 2.13 1.66 2.35 2.42 1.47 0.58 3.41 1.16 3.59 2.65 3.43 Mathematics 2021, 9, x FOR PEER REVIEW 10 of 19

Table 3. Sample of centre of inlet holes on the top plate of the substrate. 11 12 13 21 22 23 31 32 33 Sample x y x y x y x y x y x y x y x y x y 1 0.46 0.73 1.74 0.53 2.53 0.49 0.60 1.55 1.21 2.52 2.50 2.16 0.55 3.50 1.58 3.62 2.36 3.46 2 0.64 0.24 1.76 0.93 2.35 0.62 0.54 2.21 1.19 1.68 2.46 1.42 0.72 3.05 1.71 3.03 2.42 3.23 3 0.66 0.39 1.25 0.28 2.62 0.87 0.41 1.62 1.44 1.78 2.66 2.05 0.30 3.24 1.42 3.19 2.18 3.63 4 0.26 0.92 1.32 0.73 2.58 0.76 0.70 1.83 1.48 2.10 2.39 1.66 0.34 3.34 1.39 3.69 2.27 2.93 5 0.51 0.89 1.63 0.86 2.38 0.84 0.48 2.04 1.76 1.46 2.18 2.52 0.67 3.37 1.35 3.26 2.29 3.16 6 0.71 0.28 1.14 0.89 2.35 0.32 0.50 2.51 1.72 2.26 2.20 1.52 0.24 3.57 1.15 3.65 2.21 3.56 7 0.37 0.30 1.36 0.62 2.57 0.74 0.25 1.93 1.54 1.64 2.62 1.98 0.26 3.56 1.33 3.29 2.52 3.01 Mathematics8 20210.24, 9 , 21670.83 1.65 0.60 2.40 0.54 0.65 1.69 1.64 1.99 2.70 1.91 0.54 3.62 1.70 3.65 2.59 103.58 of 18 9 0.21 0.49 1.31 0.77 2.70 0.48 0.42 1.88 1.30 1.73 2.56 1.87 0.22 3.31 1.25 3.33 2.34 3.21 10 0.35 0.85 1.68 0.40 2.48 0.21 0.72 1.84 1.18 1.98 2.38 2.35 0.28 3.21 1.51 3.00 2.58 3.37 11 0.55 0.56 1.21 0.22 2.26 0.66 0.62 2.34 1.62 2.05 2.55 1.60 0.45 3.67 1.17 3.10 2.69 3.26 12 0.32 0.68 1.56 0.373.2. Surrogate2.60 0.78 Model 0.30 1.43 1.50 1.55 2.29 1.93 0.51 3.01 1.67 3.44 2.37 3.07 13 0.59 0.42 1.28 0.50 RBFN2.51 0.27 is successfully 0.28 2.47 implemented 1.73 1.83 to2.65 train 2.46 models 0.37 [533.43]. The 1.77 Radial 3.24 Basis 2.67 Neural 3.30 14 0.49 0.48 1.60 0.31Network 2.28 (RBNN)0.41 0.20 function 2.39 is established1.60 1.49 using2.61 MATLAB1.99 0.44 2015, 3.14 with 1.38 newrb 3.56 function, 2.39 whose3.50 15 0.45 0.61 1.36 0.92maximum 2.33 number0.92 0.35 of neurons2.44 1.71 needs 2.28 to be provided2.32 1.38 as an0.67 input. 3.69 It is1.32 a three-layer 2.96 2.23 network 2.97 16 0.72 0.78 1.29 0.26that includes2.66 0.37 the input0.32 layer,2.01 hidden1.16 1.88 layer, 2.44 and output1.76 0.42 layer, as3.48 shown 1.75 in 3.13 Figure 2.225. It has3.47 a 17 0.27 0.80 1.54 0.35 2.31 0.94 0.33 2.10 1.53 2.15 2.68 2.32 0.52 3.32 1.66 3.38 2.56 3.11 strong ability to predict the objective function with a rapid convergence rate. The distance 18 0.38 0.33 1.62 0.44 2.22 0.53 0.43 1.47 1.23 2.42 2.34 1.75 0.32 3.60 1.61 3.49 2.63 3.35 between selected points in the design space and the sample data points forms the basis 19 0.52 0.62 1.42 0.68 2.43 0.85 0.39 1.59 1.41 2.12 2.36 2.11 0.40 3.53 1.46 3.46 2.61 3.40 to obtain the influence of the data points. This distance is used for training of AI model 20 0.58 0.46 1.69 0.83 2.18 0.45 0.63 2.25 1.35 1.84 2.23 2.07 0.63 0 3.46 1.54 3.18 2.32 3.60 21 0.61 0.21 1.47 0.66onthe 2.45 sample 0.68 data 0.48 points 2.32 to calculate1.40 2.21 the inertia2.31 1.63 weight 0.69 (w m3.04). The 1.58γ constant 3.54 determines2.65 3.14 22 0.22 0.97 1.15 0.82the radial2.41 influence0.25 0.68 of the1.52 Gaussian 1.68 1.60 function, 2.27 which2.44 is0.49 of less 2.93 significance 1.45 3.02 in the2.41 case 3.33 of a 23 0.63 0.71 1.51 0.56large 2.64 sample 0.34 in the0.57 design 2.16 space. 1.32 However, 2.40 2.53 for small1.56 data0.64 sets, 2.99 the value1.64 of3.52γ needs 2.53careful 3.65 24 0.43 0.56 1.44 0.46consideration 2.47 0.81 before 0.56 any 1.75 value 1.13 is assigned 2.48 2.23 to it. The2.26 RBNN 0.59 function3.28 1.26 can be3.08 summarized 2.44 3.04 in 25 0.55 0.95 1.49 0.77the following2.23 0.57 Equation 0.66 (10)1.66 [541.57]. 2.32 2.48 1.80 0.31 3.08 1.20 3.23 2.30 2.98 26 0.29 0.76 1.18 0.41 2.50 0.61 0.26 2.28 1.31 1.38 2.58 2.19 0.21 2.95 1.48 2.93 2.48 3.52 27 0.43 0.65 1.71 0.24 2.30 0.72 0.23 1.77( 1.36) = 1.93N 2.250 1.68(− k0.60− 3.17k2) 1.52 3.34 2.51 3.27 h x, y ∑m=1 wm exp γ X υ (10) 28 0.32 0.37 1.21 0.97 2.67 0.96 0.46 1.96 1.26 1.70 2.52 2.40| 0.47{z 3.13} 1.28 3.11 2.25 3.69 29 0.40 0.53 1.39 0.71 2.20 0.28 0.37 1.41 1.46 1.43 2.44 2.28 0.39Radial 3.19 1.23 3.40 2.47 3.09 | {z } 30 0.69 0.26 1.53 0.57 2.55 0.39 0.52 2.13 1.66 2.35 2.42 Basis1.47function 0.58 3.41 1.16 3.59 2.65 3.43

FigureFigure 5.5.Radial Radial BasisBasis NeuralNeural NetworkNetwork mechanism.mechanism.

3.3. Particle Swarm Optimization Algorithm (PSO) The Particle Swarm Optimization (PSO), ﬁrst proposed by Kennedy and Eberhart [55], is a metaheuristic algorithm inspired by the combined intelligence of a swarm of birds in search of food and shelter. This evolutionary algorithm initializes the optimization process with a population of random solutions, which is then passed on to each particle, based on which each particle updates its velocity and position. As soon as the new velocities and positions are updated, the new optimum solution appears in the design space. Again, based on the new optimum solution, the velocities and position of each particle are updated. This process continues until the convergence criterion is reached by using Equations (11) and (12):

Vt+1 = w0Vt + C rt (Pbest − Xt ) + c rt (gbest − X)t ij ij 1 1 ij ij 2 2 j ij (11)

t+1 t t+1 Xij = Xij + Vij (12) Mathematics 2021, 9, 2167 11 of 18

The procedure of PSO is summarized in Figure6. The initial solution is represented Mathematics 2021, 9, x FOR PEER REVIEWby the RBNN function’s parameters, including input weights, output weights, and values 12 of 19

of design parameters. For each iteration, the position and velocities of the particles were updated and for which the optimal solution was obtained, which is given by Equation (8).

Mathematics 2021, 9, x FOR PEER REVIEW 12 of 19

FigureFigure 6. 6. OptimizationOptimization procedure.procedure.

3.4. ParameterThe overall and Design methodology for RBNN can beand summarized PSO in the following steps: 1.In mostIdentify of thethe designstudies, and a response Bayesian parameters. approach is preferred for comparing the perfor- 2. Determine the constraint values of design parameters. mance of algorithms [56] and to check that the algorithm converges to a global minimum 3. Identify the ﬁxed parameters. solution.4. Obtain To ensure values that of design optimization parameters does using not a suitablestop far sampling away from method. the global minimum, another5. Calculatesimilar statistical the values technique, of response parametersAnalysis of at allvariance design points(ANOVA), using CFDwas techniques. used. The cou- pled6. algorithmUse design has and variable response inputs parameters that include to train Gaussian the Radial BasisSpread Neural (훾), NetworkNumber model. of particles 7. Search for optimal design parameters using the Particle Swarm Optimization algorithm. (P), and Number of iterations (ɨ) at user discretion. Each variable at three levels was used 8. The optimal design points are used in CFD analysis for verifying the result of in this study toFigure understand 6. Optimization the influence procedure. on the optimized results. Table 4 presents the process PSO parameters algorithm. with their levels. After performing simulations, thermal resistance and3.4.3.4. temperature Parameter Parameter and and uniformityDesign Design for for RBNN RBNNwere and measured, and PSO PSO which served the purpose of response variables.In NineIn most most experimental of of the the studies, studies, runs a BayesianBayesian with approach threeapproach replicates is is preferred preferred were for for comparingperformed comparing the asthe performance per perfor- the L9 (33) orthogonalmanceof algorithms of algorithms array [56 (OA)] and [56] and to and check presented to check that the that in algorithm theTable algorithm 5. converges converges to a global to a minimumglobal minimum solution. solution.To ensure To thatensure optimization that optimization does not does stop not far stop away far from away the from global the minimum,global minimum, another Tableanothersimilar 4. Input similar statistical parameters statistical technique, and technique, their Analysis levels Analys (L). ofis variance of variance (ANOVA), (ANOVA was), was used. used. The The coupled cou- pledalgorithm algorithm has has variable variable inputs inputs that that include include Gaussian Gaussian Spread Spread (γ ),(𝛾 Number), Number of of particles particles (P), (P),and and NumberFactor/Parameter Number of of iterations iterations ( ɨ )at at user user discretion. Symbol discretion. Each Each variable L-1 variable at three at three levels L-2 levels waswas used used in L-3 thisin thisstudyGaussian study to understand to understandSpread the theinfluence inﬂuence on γ onthe the optimized optimized 0.3 results. results. Table Table 0.6 4 4 presents presents the the 0.9 processprocess No.parameters parameters of Particles with with their their levels. AfterAfter P performingperforming simulations, 50simulations, thermal thermal 125 resistance resistance and 200 andtemperature temperatureNo. of uniformityIterations uniformity were were measured, measured,ɨwhich which served served100 the the purpose purpose of response 250of response variables. var- 500 iables. Nine experimental runs with three replicates were performed as per the L9 (33) orthogonal array (OA) and presented in Table 5. Table 5. Simulation plan according to L9 (33) Orthogonal array.

Table 4. InputRun parameters and their levels (L). γ P ɨ Factor/1Parameter Symbol 1 L-1 1 L-2 L-3 1 Gaussian2 Spread γ 1 0.3 2 0.6 0.9 2 No. of3 Particles P 1 50 3 125 200 3 ɨ No. of4 Iterations 2 100 1 250 500 3 5 2 2 1 Table 5. Simulation plan according to L9 (33) Orthogonal array. 6 2 3 2 Run γ P ɨ 7 3 1 2 1 1 1 1 2 1 2 2 3 1 3 3 4 2 1 3 5 2 2 1 6 2 3 2 7 3 1 2

Mathematics 2021, 9, x FOR PEER REVIEW 12 of 19

Mathematics 2021, 9, 2167 12 of 18

Figure 6. Optimization procedure. Nine experimental runs with three replicates were performed as per the L9 (33) orthogonal array3.4. (OA) Parameter and presented and Design in Table for RBNN5. and PSO In most of the studies, a Bayesian approach is preferred for comparing the perfor- Table 4. Input parameters and their levels (L). mance of algorithms [56] and to check that the algorithm converges to a global minimum solution.Factor/Parameter To ensure that optimization Symbol does not L-1 stop far away L-2 from the global L-3 minimum, anotherGaussian similar Spread statistical technique,γ Analysis0.3 of variance (ANOVA 0.6), was used. 0.9 The cou- pledNo. algorithm of Particles has variable inputs P that include 50 Gaussian Spread 125 (𝛾), Number 200 of particles (P),No. and of Iterations Number of iterations ɨ at user discretion.100 Each variable 250 at three levels 500 was used in this study to understand the influence on the optimized results. Table 4 presents the process parameters with their levels. After performing simulations, thermal resistance Table 5. Simulation plan according to L (33) Orthogonal array. and temperature uniformity were9 measured, which served the purpose of response variables.Run Nine experimental runsγ with three replicatesP were performed as per the L9 (33) orthogonal array (OA) and presented in Table 5. 1 1 1 1 2 1 2 2 Table 34. Input parameters and 1their levels (L). 3 3 4 2 1 3 Factor/Parameter Symbol L-1 L-2 L-3 5 2 2 1 6Gaussian Spread 2γ 30.3 0.6 2 0.9 7No. of Particles 3 P 1 50 125 2 200 8No. of Iterations 3ɨ 2100 250 3 500 9 3 3 1

Table 5. Simulation plan according to L9 (33) Orthogonal array. 4. Results and Discussion Run γ P ɨ The CFD approach simulates the heat transfer and ﬂow characteristics in the heat sink 1 1 1 1 with a cooling jet with laminar ﬂow. These results were validated with the experimental results [51]. The CFD2 results were used to1 train the surrogate2 model RBNN to form2 a continuous design space,3 on which the particles1 of PSO algorithm3 were allowed to3 search for optimum design4 parameters. Table6 shows2 the predicted values1 of an optimal solution3 obtained by the algorithm.5 Using these optimal2 solutions, the2 CFD case was prepared1 and simulated, for which6 the temperature contour2 was plotted3 (Figure 7) and in which2 the maximum temperature7 was found to be 343.383 K. As shown1 in Table7, for the predicted2 values and CFD-calculated values of thermal resistance and temperature uniformity, the percentage error was less than 2%. The metaheuristic algorithms are a powerful tool that provide the optimal solution rapidly. However, careful validation with the actual results is

necessary for these algorithms since other variables inﬂuence the optimal solution, which is obtained by the trial-and-error method. Therefore, multiple solutions must be obtained prior to validation.

Table 6. Predicted optimized position of inlet holes by the PSO algorithm.

11 12 13 21 22 23 31 32 33 xyxyxyxyxyxyxyxyxy 0.20 0.84 1.31 0.59 2.70 0.28 0.62 1.65 1.13 1.88 2.45 2.17 0.20 3.12 1.28 2.93 2.39 3.37

Table 7. Comparison of values predicted by the AI model and CFD results.

f1 f2 Predicted Value 6.205 8.019 Actual Value 6.170 7.914 %e 0.565 1.323 Mathematics 2021, 9, x FOR PEER REVIEW 14 of 19

Table 6. Predicted optimized position of inlet holes by the PSO algorithm. 11 12 13 21 22 23 31 32 33 x y x y x y x y x y x y x y x y x y 0.20 0.84 1.31 0.59 2.70 0.28 0.62 1.65 1.13 1.88 2.45 2.17 0.20 3.12 1.28 2.93 2.39 3.37

Table 7. Comparison of values predicted by the AI model and CFD results.

f1 f2 Predicted Value 6.205 8.019 Mathematics 2021, 9, 2167 13 of 18 Actual Value 6.170 7.914 %e 0.565 1.323

Mathematics 2021, 9, x FOR PEER REVIEW 16 of 19

5. Conclusions In this work, a 3-D numerical study of microjets’ position was conducted, based on the samples provided by Latin Hypercube sampling. The results obtained from physical simulation were used to train the surrogate model RBNN coupled with the PSO algorithm to obtain the optimized location of microjets which minimize substrate temperature for non-uniform heat flux distribution. The predicted values by surrogate models Mathematics 2021, 9, x FOR PEER REVIEW 12 of 19 were compared with the results of CFD simulation, for which the percentage error was Figure 7. Global range of substrate temperature contour0.53% of the for optimaloptimal the target solutionsolution function obtainedobtained thermal fromfrom resistance thethe (f1), and for temperature uniformity (f2) it CFD solution. was 1.3%. The advantage of this method is that earlier studies considered equal jet spacing, since fluid interaction in the multi-jet domain is complicated and difficult to quanti- 4.1. Effect of PSO Parameters fy. This method accommodates complications of fluid interaction within the design In Figure8, the value of the objective function thermal resistance is found to be In Figure 8, the value of the objective functionspace. thermal The Particle resistance Swarm is found Optimization to be re- is a metaheuristic algorithm which is used to reducing with each iteration of the PSO algorithm. The searching for optimal parameters in ducing with each iteration of the PSO algorithm. searchThe searching the design for spaceoptimal for parameters the optimal in solution s. The influence of three parameters in the the design space with infinite points would be computationally expensive if a metaheuristic the design space with infinite points would be computationallycoupled surrogate expensive model was if a analyzed.metaheu- Results from the coupled algorithm vary slightly algorithm would not have been applied. However, even with quick search and fast ristic algorithm would not have been applied. However,for each evencomplete with run.quick Therefore, search and three fast experiments were performed for each combina- convergence features, these algorithms have some issues such as unknown variables that convergence features, these algorithms have sometion issues of selected such as parametersunknown variables. ANOVA that was performed on the measured values of thermal significantly influence the solution. Additionally, there is a chance that solutions can be significantly influence the solution. Additionally,resistance, there is a andchance it was that found solutions that can gaussian be spread is the most dominant parameter af- trapped in local optima. In order to avoid trapping in local optima, it is necessary to trapped in local optima. In order to avoid trappingfecting in local the optima,thermal itresistance, is necessary followed to run by the number of iterations and number of pop- run multiple simulations. ANOVA was performed on the measured values of response multiple simulations. ANOVA was performed onulations the measured for the given values range of response of process var- parameters. variables,iables, as presented as presented in inTable Table 8.8 .This This was was carried carried out out to to understand understand the effecteffect ofof unknownunknown Mathematics 2021, 9, x FOR PEER REVIEWvariable inputs in the search algorithm and to obtain the most dominant parameter that 15 of 19 variable inputs in the search algorithm and to obtainAuthor the Contributions: most dominant Conceptualization: parameter that M.O.Q., N.Z.K.; methodology, M.O.Q., N.Z.K.; formal significantly affects the thermal resistance (f ). significantly affects the thermal resistance (f11 ). analyses: I.A.B., M.A.K., S.A. Investigation: M.O.Q., N.Z.K. Writing—original draft: M.O.Q., N.Z.K., I.A.B.; supervision and project administration and resources: M.A.K., S.A.; review and editing: I.A.B.; funding: I.A.B., S.A. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by King Khalid University, under grant number RGP.2/166/42. Institutional Review Board Statement: Not applicable Informed Consent Statement: Not applicable Data Availability Statement: Required data available in manuscript. Acknowledgments: The authors extend their appreciation to the Deanship of Scientific Research at Figure 6.King Optimization Khalid University procedure. for funding this work through the research groups program under grant number RGP.2/166/42.

3.4. Parameter and DesignConflicts for RBNN of Interest: and PSO The authors declare no conflict of interest. In most of the studies, a Bayesian approach is preferred for comparing the performance of algorithms [56]Nomenclature and to check that the algorithm converges to a global minimum solution. To ensure thatA optimizationArea (m does2) not stop far away from the global minimum, another similar statisticalÑ technique, Total numberAnalys ofis regionsof variance with (ANOVAdifferent heat), was flux used. condition The cou-

pled algorithm has variableƝ inputsNumber that ofinclude data sample Gaussian points Spread (𝛾), Number of particles FigureFigure 8.8.Best Best cost cost of of Objective Objective(P), functions functions and Number with withγ =ofγ 0.9,= iterations 0.9,ñ = ñ 100, = 100, ɨ at= Maximum ɨ 150,user= 150, P discretion. =P 120. =number 120. ofEach neurons variable at three levels was used in this study to understandP the influence Number of on swarm the particlesoptimized results. Table 4 presents the Table 8. Measurement of theprocess response parameters variable after withQ performing their Volumetriclevels. simulations After flow performing asrate per (m Taguchi3/s) simulations, L9 OA. thermal resistance and temperature uniformityT wereTemperature measured, (K) which served the purpose of response var- Run Exp1 Exp2 Exp3 Average iables. Nine experimental𝑉 runsVelocity with three of particle replicates i at time were t performed as per the L9 (33) No. ɨ 1 1 1 1 γ orthogonalP array (OA) 𝑋 and presentedfPosition in of fTable particle 5. i at f time t f 1 0.3 50 100 X 6.357Input vector 6.352 6.340 6.350 2 0.3 Table 125 4. Input parameters 250 c1 and their 6.312 Personal levels 6.522(L). acceleration 6.522 coefficient 6.452 c2 Social acceleration coefficient 3 0.3 200 500 6.464 6.501 6.522 6.496 Factor/Parameterdh HydraulicSymbol Diameter (m)L-1 L-2 L-3 4 0.6 50 Gaussian 500 Spread 𝑓 6.226Objective 6.226γ function 6.226thermal0.3 resistance 6.226 (K/W)0.6 0.9 5 0.6 125 No. of Particles 100 𝑓 6.232Objective 6.229 P function 6.227temperature 50 6.229 uniformity 125 200 6 0.6 200 No. of Iterations 250 h 6.226Gaussian 6.226 ɨfunction 6.226100 6.226 250 500 j Dimensions along which the vectors V and X are updated 7 0.9 50 250 6.225 6.222 6.215 6.221 Table 5. Simulation plank according Thermal to L9 (33 )Conductivity Orthogonal array.(W/m K) 8 0.9 125 500 6.211 6.206 6.211 6.209 9 0.9 200 Run 100 6.222 γ 6.207 6.212 P 6.214 ɨ 1 1 1 1 4.2. Analysis of Variance (ANOVA) 2 1 2 2 ANOVA (Table 9) was used to determine3 which parameter1 significantly3 influences 3 the thermal resistance. 4 2 1 3 ANOVA results revealed that gaussian5 spread is the most2 significant parameter2 to 1 influence the thermal resistance, followed6 by the number of2 iterations and 3 number of 2 populations. Gaussian spread affects the7 values of the output3 of nearby values1 of sample 2 data points, which happened to be the center of the normal distribution curve from the center, as determined by the RBNN function, and this distance is multiplied by the gaussian spread. Since the gaussian spread is at user discretion, careful consideration is a necessity for obtaining adequate heat transfer to obtain good thermal performance.

Table 9. ANOVA Results.

Degrees of Percentage Source of Variation Sum of Squares Mean Square F-Ratio p-Value Freedom Contribution Gaussian spread 2 0.169181 0.084591 20.53 0.046 89.10 No. of population 2 0.006097 0.003049 0.74 0.575 3.21 No. of iterations 2 0.006358 0.003179 0.77 0.564 3.35 Error 2 0.008241 0.004121 4.34 Total 8 0.189877 100

Mathematics 2021, 9, x FOR PEER REVIEW 12 of 19

Figure 6. Optimization procedure.

Mathematics 2021, 9, 2167 3.4. Parameter and Design for RBNN and PSO 14 of 18 In most of the studies, a Bayesian approach is preferred for comparing the performance of algorithms [56] and to check that the algorithm converges to a global minimum Table 8.solution.Measurement To ensure of the responsethat optimization variable after does performing not stop simulations far away from as per the Taguchi global L9 minimum, OA. another similar statistical technique, Analysis of variance (ANOVA), was used. The coupled algorithm has variable inputs thatExp1 include Gaussian Exp2 Spread Exp3 (𝛾), Number Average of particles Run No. (P), and Numberγ of Piterations ɨ at user discretion.f1 Eachf1 variablef1 at three levelsf1 was used in 1this study 0.3 to understand 50 the 100 influence 6.357 on the optimized 6.352 results. 6.340 Table 6.350 4 presents the 2process parameters 0.3 125 with their 250 levels. 6.312After performing 6.522 simulations, 6.522 thermal 6.452 resistance 3and temperature 0.3 uniformity 200 500were measured, 6.464 which 6.501 served 6.522the purpose of 6.496 response var- 4iables. Nine 0.6 experimental 50 runs 500 with three 6.226 replicates 6.226 were performed 6.226 as 6.226 per the L9 (33) 5orthogonal 0.6 array (OA) 125 and presented 100 6.232in Table 5. 6.229 6.227 6.229 6 0.6 200 250 6.226 6.226 6.226 6.226 7 0.9 50 250 6.225 6.222 6.215 6.221 Table 4. Input parameters and their levels (L). 8 0.9 125 500 6.211 6.206 6.211 6.209 9Factor 0.9/Parameter 200 100Symbol 6.222 6.207L-1 6.212 L-2 6.214 L-3 Gaussian Spread γ 0.3 0.6 0.9 4.2. Analysis of VarianceNo. of Particles (ANOVA) P 50 125 200 ANOVA (TableNo. of9) Iterations was used to determine whichɨ parameter100 significantly250 influences the500 thermal resistance. Table 5. Simulation plan according to L9 (33) Orthogonal array. Table 9. ANOVA Results. Run γ P ɨ Degrees of Sum of Percentage Source of Variation 1 Mean Square1 F-Ratio 1p -Value 1 Freedom Squares Contribution 2 1 2 2 Gaussian spread 2 0.169181 0.084591 20.53 0.046 89.10 3 1 3 3 No. of population 2 0.006097 0.003049 0.74 0.575 3.21 No. of iterations 2 0.0063584 0.0031792 0.771 0.564 3.35 3 Error 2 0.0082415 0.0041212 2 4.34 1 Total 8 0.1898776 2 3 100 2 7 3 1 2 ANOVA results revealed that gaussian spread is the most significant parameter to influence the thermal resistance, followed by the number of iterations and number of populations. Gaussian spread affects the values of the output of nearby values of sample data points, which happened to be the center of the normal distribution curve from the center, as determined by the RBNN function, and this distance is multiplied by the gaussian spread. Since the gaussian spread is at user discretion, careful consideration is a necessity for obtaining adequate heat transfer to obtain good thermal performance.

Author Contributions: Conceptualization: M.O.Q., N.Z.K.; methodology, M.O.Q., N.Z.K.; formal Authoranalyses: Contributions: I.A.B., M.A.K.,Conceptualization: S.A. Investigation: M.O.Q., M.O.Q., N.Z.K.; N.Z.K. methodology, Writing—original M.O.Q., N.Z.K.;draft: M.O.Q., formal N.Z.K.,analyses: I.A.B.; I.A.B., supervision M.A.K., S.A. and Investigation: project administration M.O.Q., N.Z.K. and Writing—originalresources: M.A.K., draft: S.A.; M.O.Q.,review and N.Z.K., editing:I.A.B.; I.A.B.; supervision funding: and I.A.B., project S.A. administration All authors andhave resources: read and M.A.K.,agreed to S.A.; the review published and editing:version I.A.B.;of the manuscript.funding: I.A.B., S.A. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by King Khalid University,University, underunder grantgrant numbernumber RGP.2/166/42.RGP.2/166/42. Institutional Review Board Statement: Not Not applicable. applicable Informed Consent Statement: Not Not applicable. applicable Data Availability Statement:Statement: Required Required data data available available in in manuscript. manuscript. Acknowledgments: The authors extend their appreciation to the Deanship of ScientificScientific Research at King Khalid University for funding this work through the research groups programprogram underunder grantgrant number RGP.2/166/42. ConflictsConflicts ofof Interest:Interest: TheThe authorsauthors declaredeclare nono conflictconflict ofof interest.interest.

Nomenclature A AreaArea (m (m2) 2) Ñ Total Total number number of regionsof regions with with different different heat heat flux flux condition condition Ɲ NumberNumber of of data data sample sample points points ñ Maximum Maximum number number of of neurons neurons P Number Number of of swarm swarm particles particles Q Volumetric Volumetric flow flow rate rate (m (m3/s)3/s) T TemperatureTemperature (K) (K) V𝑉t i t 𝑉i VelocityVelocity of particleof particleat i timeat time t t 𝑋X Position of particle i at time t 𝑋i Position of particle i at time t X InputInput vector vector c Personal acceleration coefficient c1 Personal acceleration coefficient c Social acceleration coefficient c2 Social acceleration coefficient d Hydraulic Diameter (m) dh Hydraulic Diameter (m) f1 Objective function thermal resistance (K/W) 𝑓 Objective function thermal resistance (K/W) f2 Objective function temperature uniformity 𝑓 Objective function temperature uniformity h Gaussian function h Gaussian function j Dimensions along which the vectors V and X are updated j Dimensions along which the vectors V and X are updated k Thermal Conductivity (W/m·K) k Thermal Conductivity (W/m K) n Dimensions of particles p Pressure (Pa) q Heat Flux (W/m2) t t r1, r2 Numbers generated by random function w Input weight w0 Inertia coefficient → V Velocity vector (m/s) . m Mass flow rate (kg/s) subscript b Base of Substrate f Fluid in Inlet o Outlet max Maximum s Solid k Different heat flux region c Cross-section Greek symbols ρ Density (kg/m3) θ Dimensionless Temperature µ Dynamic viscosity (kg/m·s) Mathematics 2021, 9, x FOR PEER REVIEW 12 of 19

Mathematics 2021, Figure9, 2167 6. Optimization procedure. 16 of 18

3.4. Parameter and Design for RBNN and PSO In most of the studies, a Bayesian approach is preferred for comparing the perfor- η Normal direction vector mance of algorithms [56] and to check that the algorithm converges to a global minimum Γ Interface solution. To ensure that optimization$ Perimeter does not (m) stop far away from the global minimum, another similar statistical technique,γ Gaussian Analysis function of variance spread (ANOVA), was used. The coupled algorithm has variable inputsυ thatWeighted include vector Gaussian Spread (𝛾), Number of particles (P), and Number of iterations ɨ at userMaximum discretion. number Each of variable PSO iterations at three levels was used in this study to understand the influence on the optimized results. Table 4 presents the Referencesprocess parameters with their levels. After performing simulations, thermal resistance 1.and temperatureMorini, G.L. 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