Dimensional optimization of a micro-channel heat sink using Jaya algorithm

Applied Thermal Engineering 103 (2016) 572–582

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Research Paper

Dimensional optimization of a micro-channel heat sink using Jaya algorithm R.V. Rao a,⇑, K.C. More a, J. Taler b, P. Ocłon´ b a b

Dept. of Mech. Engg., S.V. National Institute of Technology, Surat 395007, India ´ Energetycznych, Al. Jana Pawła II 37, 31-864 Kraków, Poland Politechnika Krakowska, Instytut Maszyn i Urza˛dzen

h i g h l i g h t s

g r a p h i c a l a b s t r a c t


A new optimization algorithm named

Flowchart of Jaya algorithm.

as ‘Jaya algorithm’ is proposed.
Two case studies of dimensional optimization of a micro-channel heat sink are presented.
Two objective functions of thermal resistance and pumping power are considered.
The results are compared with those obtained by MOEA, TLBO and numerical analysis.
The results obtained by Jaya algorithm are found better than the other approaches.

a r t i c l e

i n f o

Article history: Received 6 November 2015 Accepted 23 April 2016 Available online 26 April 2016 Keywords: Micro-channel heat sink Thermal resistance Pumping power Optimization Jaya algorithm TLBO algorithm Multi-objective evolutionary algorithm (MOEA)

a b s t r a c t In this paper, a recently proposed optimization algorithm named as ‘Jaya algorithm’ is used for the dimensional optimization of a micro-channel heat sink. Two case studies are considered and in both the case studies two objective functions related to the heat transfer and pressure drop, i.e. thermal resistance and pumping power, are formulated to examine the performance of the micro-channel heat sink. In the first case study, two non-dimensional design variables related to the micro-channel depth, width and fin width are chosen and their ranges are decided through preliminary calculations of three-dimensional Navier–Stokes and energy equations. In the second case study, three design variables related to the width of the micro-channel at the top and bottom, depth of the micro-channel and width of fin are considered. The objective functions in both the case studies are expressed in terms of the respective design variables and the two objectives are optimized simultaneously using the proposed Jaya algorithm. The results are compared with the results obtained by the TLBO algorithm and a hybrid multi-objective evolutionary algorithm (MOEA) and numerical analysis. The results obtained by the application of Jaya algorithm are found much better than those reported by the other approaches. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction

⇑ Corresponding author. E-mail address: [email protected] (R.V. Rao). http://dx.doi.org/10.1016/j.applthermaleng.2016.04.135 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.

Downscaling of micro-electro-mechanical systems (MEMS) devices and advances in micro-fabrication processes have helped to satisfy the increasing requirement for higher dissipation of heat flux from electronic devices. The progress in MEMS and power

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Nomenclature Ac cross-section area of micro-channel As surface area of substrate base trapezoidal micro-channel depth hc Hc rectangular micro-channel depth k thermal conductivity Lx, Ly and Lz length, width and height of heat sink, respectively, of a rectangular MCHS lx, ly and lz length, width and height of heat sink, respectively, of a trapezoidal MCHS n number of micro-channels P pumping power p pressure q heat flux Rth thermal resistance T temperature u liquid velocity in micro-channel wb trapezoidal micro-channel width at top trapezoidal micro-channel width at bottom wc Wc width of rectangular micro-channel ww fin width corresponding to trapezoidal micro-channel

required driving cooling devices integrated in electronics robotics, avionics, and medicine industry have opened the doors for optimization. Micro-channel heat sink, as an integrated part of silicon based electronic device, has been a possible micro cooling device having a high surface area to volume ratio. The flow in micro-channels mostly remains in the laminar region due to the pumping power limitations. In designing a micro-channel heat sink, the pressure drop and the pumping power should be kept low. At the same time the thermal performance is to be maximized and the thermal resistance is desired to be low. The pumping power is the limiting factor for the volumetric flow rate of the working fluid. For a given pumping power, thermal resistance could be reduced by 20% as a result of dimensional optimization. In many optimization studies, the pumping power was taken as a constraint and the thermal resistance was considered as the sole objective function. Micro-channel heat sink (MCHS) was proposed first by Tuckerman and Pease [1] who had designed and tested a very compact water-cooled integral MCHS for silicon integrated circuits. The MCHSs have many advantages over the conventional cooling techniques regarding high heat dissipation, compact dimension and volume for each heat load and minimum coolant necessity with lesser operational rate. Bowers and Mudawar [2] analyzed pressure drop of mini and micro channel heat sinks of 1 cm heated length using R113. The mini-channel’s performance was proved superior to the micro-channel due to pressure drops less than 0.01 bar for comparable critical heat flux values as well as the reduced likelihood of clogging and the relative ease in fabricating the mini-channel. Kim and Kim [3] reported analytical solutions for both velocity and temperature profiles in MCHS by modeling the MCHS as a fluid-saturated porous medium. The expression for the total thermal resistance, derived from the analytical solutions and the geometry of the micro-channel heat sink for which the thermal resistance of the heat sink was minimal, was obtained. Vafai and Zhu [4] analyzed design of a two-layered MCHS with flow direction on each layer opposite. The thermal performance and the temperature distribution for these types of micro-channels were analyzed and a procedure for optimizing the geometrical design parameters was presented. The results showed that the two-layered MCHS design is a considerable improvement over a conventional one-layered MCHS. Ryu et al. [5] developed 3-D analysis procedure for thermal performance of a manifold MCHS and attempted to optimize the

Ww x, y, z

fin width corresponding to rectangular micro-channel orthogonal coordinate system

Greek symbols / design variable, Ww/Hc l dynamic viscosity h design variable, Wc/Hc q density a design variable, wc/hc b design variable, ww/hc c design variable, wb/wc sij stress tensor Subscripts f fluid i inlet o outlet s substrate

MCHS design. The sensitivity of thermal performance on every design variable was examined. Among various design variables, the channel width and depth were explained as more crucial than others to the heat-sink performance. It was shown that the optimal dimensions and corresponding thermal resistance have a powerlaw dependence on the pumping power. Jeevan et al. [6] used a genetic algorithm and a box optimization method by considering fixed pumping powers for minimizing the thermal resistance by considering the optimum values of channel height, channel width and rib width. The authors had shown that the double-layer MCHS created a smaller amount of thermal resistance compared to the single-layer MCHS. Liu and Garimella [7] optimized the micro-channel geometry by comparing analytical models with a stronger 3-D numerical model. The modeling approaches of increasing levels of complexity for the analysis of convective heat transfer in micro-channels which offer adequate descriptions of the thermal performance, while allowing easier manipulation of micro-channel geometries for the purpose of design optimization of micro-channel heat sinks were presented. Cheng [8] simulated the mathematical model of a two layer stacked MCHS by using an improved combination of passive microstructure. It was shown that the stacked micro-channel with passive structures had better performance than the smooth microchannels. Tsai and Chein [9] analyzed micro-channel heat sink performance using carbon nanotube–water (CNT–H2O) and copper– water (Cu–H2O) nanofluids as coolants. It was found that using nanofluid can enhance the MCHS performance when the porosity and aspect ratio were less than the optimum porosity and aspect ratio. When the porosity and channel aspect ratio were higher than optimum porosity and aspect ratio, the nanofluid did not produce a significant change in MCHS thermal resistance. Husain and Kim [10] carried out the optimization of a MCHS with temperature dependent fluid properties using surrogate analysis and hybrid multi-objective evolutionary (MOE) algorithm. Two design variables related to the micro-channel depth, width and fin width were chosen and their ranges were decided through preliminary calculations of three-dimensional Navier–Stokes and energy equations. Objective functions related to the heat transfer and pressure drop i.e. thermal resistance and pumping power were formulated to analyze the performance of the heat sink. Using the

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numerically evaluated objective function, polynomial response surface was constructed for each objective function. An evolutionary algorithm for multi-objective optimization was performed to obtain global Pareto optimal solutions. Trade-off between objectives was found and analyzed with the design variables and flow constraints. In another work, Husain and Kim [11] optimized shape of MCHS using surrogate methods of Response Surface Approximations and 3-D numerical analysis by considering pumping power as a constraint. Furthermore, Husain and Kim [12] optimized a liquid flow micro-channel heat sink with the help of three-dimensional numerical analysis and multiple surrogate methods. Two objective functions, thermal resistance and pumping power were selected to assess the performance of the micro-channel heat sink. The design variables related to the width of the micro-channel at the top and bottom, depth of the micro-channel, and width of fin were identified and a three-level full factorial design was selected to exploit the design space. The numerical solutions obtained at these design points were utilized to construct surrogate models, namely Response Surface Approximations, Kriging and Radial Basis Neural Network. A hybrid multi-objective evolutionary algorithm coupled with surrogate models was applied to find out global Paretooptimal solutions. The accuracy of the surrogate models was discussed in view of their predictions and trade-off analysis was performed in view of conflicting nature of the two objectives. Hu and Xu [13] proposed minimum thermal resistance as an objective function and a nonlinear, single objective and multiconstrained optimization model was proposed for the microchannel heat sink in electronic chips cooling. The sequential quadratic programming (SQP) method was used to do the design optimization of the structure size of the micro-channel. The numerical simulation results showed that the heat transfer performance of micro-channel heat sink was affected intensively by its dimension. Chong et al. [14] developed a thermal resistance model and applied an algorithm called multiple variables constrained direct search for optimizing the performance of a single and double layer MCHSs at fixed pressure drops. The model proposed by Chong et al. [14] was proved to have higher performance for both single and double-layer MCHSs functioning in the laminar flow. Karathanasis et al. [15] used multi-objective optimization of MCHS for concentrating photovoltaic/thermal (CPVT) system by applying genetic algorithm. Two different micro-channel configurations were considered, fixed and stepwise variable-width microchannels respectively. The performance evaluation criteria comprised the thermal resistance of the heat sink and the cooling medium pressure drop through the heat sink. The overall analysis demonstrated that the micro-channel heat sinks achieve very low values of thermal resistance and that the use of variable-width channels can significantly reduce the pressure drop of the cooling fluid. Hung and Yan [16] proposed a method for rising the thermal functioning of a three-dimensional double-layered MCHS by using a nanofluid and changing the geometric parameters. Türkakar and Özyurt [17] minimized the total thermal resistance for dimensional optimization of silicon MCHS. The effects of the thermal and hydrodynamic entrance regions on heat transfer and flow were investigated. Hung et al. [18] used an optimization procedure consisting of a simplified conjugate-gradient method and a threedimensional fluid flow and heat transfer model to investigate the optimal geometric parameters of a double-layered micro-channel heat sink. The overall thermal resistance was the objective function to be minimized, and the number of channels, channel width ratio, lower channel aspect ratio, and upper channel aspect ratio were the considered as the design variables. It was found that the optimal thermal resistance decreases rapidly with the pumping power and then tends to approach an constant value. Wang et al. [19] analyzed the results of temperature dependent thermophysical properties for water-based Al2O3 nanofluid with

1% element volume by performing an inverse geometric optimization for nanofluid cooled MCHS under constant pumping power. Simplified-conjugate-gradient-method was used as an optimization tool. Xie et al. [20] designed three types of water-cooled MCHS, a rectangular straight, mono and bi-layer wavy MCHS and then evaluated using computational fluid dynamics (CFD). The authors had considered parameters such as amplitude of heat transfer, pressure drop and thermal resistance to monitor the effects on the MCHS. Results showed that for removing an identical heat load, the overall thermal resistance of the single-layer wavy micro-channel heat sink decreased with increasing volumetric flow rate, but the pressure drop was increased greatly. At the same flow rate, the double-layer wavy micro-channel heat sinks reduced not only the pressure drop but also the overall thermal resistance compared to the single-layer wavy micro-channel heat sinks. The CFD applications are increasing in different fields of fluid flow and heat transfer [21–25]. Lin et al. [26] developed a combined optimization procedure to look for optimal design for a water-cooled, silicon-based double layer MCHS. The geometry and flow rate distribution were considered for optimization. By using this method the authors had enhanced the performance of MCHS by optimizing the geometry and flow rate allocation for two-layer heat sink. A threedimensional solid–fluid conjugated model was coupled with a simplified conjugate-gradient method to optimize the flow and heat transfer in a water-cooled, silicon-based double-layer microchannel heat sink. Six design variables were optimized simultaneously to search for a minimum of global thermal resistance. However, the optimization carried out was proven effective only for the double-layer MCHS with a specific dimension. Ahmed and Ahmed [27] conducted 3-D numerical analysis to investigate the effect of geometrical parameters on forced convection and laminar flow in thermal design of triangular, trapezoidal and rectangular grooved MCHS. The design variables included were: depth, tip length; pitch and orientation of cavities for optimizing the performance of an aluminum heat sink. These geometric parameters could change the cavity shape from triangular to trapezoidal and then to rectangular shape. The governing and energy equations were solved using the finite volume method. From the literature review, it is observed that some of the researchers had considered only thermal resistance as the objective for optimization [3–9,13,14,16–19,26]. Few others had considered both thermal resistance and pumping power as the objectives [10,12,15]. Also, it is observed that not many researchers had applied the optimization techniques for dimensional optimization of a MCHS. The optimization methods used by researchers included SQP, GA, NSGA-II, MOEA, surrogate models, neural networks, 3-D numerical models, conjugate-gradient methods and CFD. The use of advanced engineering optimization algorithms is found limited in this area of research. Hence, attempts are made in the present work to see if there will be any improvement in performance of heat transfer in the MCHS by considering both the objectives of thermal resistance and pumping power. A very recently proposed advanced optimization algorithm named as Jaya algorithm is used for this purpose. The proposed Jaya algorithm differs from the other advanced optimization algorithms in that it is simple and easy to apply and it does not need any algorithm specific parameters to tune. The following section presents the description of the proposed Jaya algorithm.

2. Jaya algorithm Jaya algorithm is a very recently developed algorithm by Rao [28]. The algorithm is based on the concept that the solution obtained for a given problem which should move toward the best

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solution and must avoid the worst solution. The algorithm always tries to get closer to success (i.e. reaching the best solution) and then tries to avoid failure (i.e. moving away from the worst solution). The flowchart of Jaya algorithm is shown in Fig. 1 [28]. Let f(x) is the objective function to be minimized (or maximized). At any iteration i, assume that there are ‘m’ number of design variables (i.e. j = 1, 2, . . . , m), and ‘n’ number of candidate solutions (i.e. population size, k = 1, 2, . . . , n). Let the best candidate best obtains the best value of f(x) (i.e. f(x) best) in the entire candidate solutions and the worst candidate worst obtains the worst value of f(x) (i.e. f(x) worst) in the entire candidate solutions. If Xj,k,i is the value of the jth variable for the kth candidate during the ith iteration, then this value is modified as per the following Eq. (1).

X 0j;k;i ¼ X j;k;i þ r1;j;i ðX j;best;i  jX j;k;i jÞ  r2;j;i ðX j;worst;i  jX j;k;i jÞ

ð1Þ

where Xj,best,i is the value of the variable j for the best candidate and Xj,worst,i is the value of the variable j for the worst candidate. X 0j;k;i is the updated value of Xj,k,i and r1,j,i and r2,j,i are the two random numbers for the jth variable during the ith iteration in the range [0, 1]. The term ‘‘r1,j,i (Xj,best,i  |Xj,k,i|)” indicates the tendency of the solution to move closer to the best solution and the term ‘‘r2,j,i(Xj,worst,i  |Xj,k,i|)” indicates the tendency of the solution to avoid the worst solution. X 0j;k;i is accepted if it gives better function value. All the accepted function values at the end of iteration are maintained and these values become the input to the next iteration. The random numbers r1 and r2 ensure good exploration of the search space. The absolute value of the candidate solution (|Xj,k,i|) considered in Eq. (1) further enhances the exploration ability of the algorithm.

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The working of the proposed Jaya algorithm is illustrated by means of a benchmark function known as Rastrigin function in Appendix A. For additional details on Jaya algorithm one may refer to: https://sites.google.com/site/jayaalgorithm. In this paper the Jaya algorithm is applied for the dimensional optimization of MCHS. Two case studies from the literature are considered for demonstration and validation of the proposed algorithm. Thermal resistance and pumping power are considered as the objectives and both the objectives are optimized simultaneously. TLBO algorithm is another algorithm-specific parameter-less algorithm developed by Rao et al. in 2011 [29,30]. For additional details on TLBO algorithm one may refer to Rao [31] and https://sites.google.com/site/tlborao. However, the Jaya algorithm proposed in this paper is comparatively simpler and easier to understand. The TLBO algorithm is also considered in the present work to make performance comparison with the proposed Jaya algorithm. The following section presents the case studies. 3. Case studies 3.1. Case study 1 This case study was introduced by Husain and Kim [10]. Many industries such as medicine, avionics, electronics and robotics require MEMS chilling devices for minimization of heat diffusion. The MCHS considered by Husain and Kim [10] is shown in Fig. 2 and computational domain of MCHS is shown in Fig. 3. The following equations are used for the convective heat transfer in the MCHS [10].

Fig. 1. Flowchart of Jaya algorithm [28].

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Mass:

@ qf @ðqf ui Þ ¼0 þ @xi @t

ð2Þ

Momentum:

qf

@ui @ui @p @ ¼ þ qf g i þ þ qf uj @xi @xj @t @xi
 2 @ @u lf k  3 @xi @xk




lf


 @ui @ @u þ lf i @xj @xj @xi ð3Þ

Energy (for fluid):

qf

@h @h @p þ qf uj þ ¼ @t @xi @t




lf


 @T f @p @ @ui þ þ sij kf @xi @xi @xi @xj

ð4Þ

Energy (for the substrate conduction):


 @ @T s ks ¼0 @xi @xi

Water was used as coolant liquid and it flowed into the microchannel and left at the outlet. The silicon substrate occupied the remaining portion of heat sink. No slip conditions were assumed at the inner walls of the channel, i.e. u = 0. The thermal circumstance in the z-direction was given as:

ks



 @T s @T s ¼ q at z ¼ 0 and ks ¼ 0 at z ¼ Lz : @xi @xi

The design variables considered by Husain and Kim [10] were h = Wc/Hc and / = Ww/Hc, where Wc was the micro-channel width; Ww was the fin width and Hc was the micro-channel depth. Hc was kept 400 lm during the whole optimization procedure. In this case study, two objective functions were considered and those were (i) thermal resistance associated with heat transfer performance and (ii) the pumping power to drive the coolant or to pass the coolant through the micro-channel. Table 1 shows the ranges of values considered for these design variables h and /. The thermal resistance was given by:

RTH ¼ DT max =ðAs  qÞ

ð6Þ

where As was area of the substrate subjected to heat flux, and DTmax was the maximum temperature in MCHS, which is given as:

DT max ¼ T s;o  T f ;i

ð7Þ

The pumping power to move the coolant (water) through MCHS was calculated as:

P ¼ n  uav g  Ac  Dp

Fig. 3. Computational domain of MCHS [10].

ð5Þ

ð8Þ

where Dp was the pressure drop and uavg was the mean velocity.

Pumping power and thermal resistance compete with each other because a decrease in pumping power contributes to increase in thermal resistance. These two objectives are to be achieved simultaneously and the optimum values of the design variables are to be found out that satisfy both the objectives. Evolutionary algorithms which are used for solving the optimization problems require many evaluations for objective functions to search for the optimum solutions. Therefore, surrogate models were constructed by Husain and Kim [10] to evaluate the values of the objective functions to avoid experimental or numerical expenses and to save time. Husain and Kim [10] calculated the objectives by using Navier–Stokes and heat conduction equations at specific design points. The Response Surface Approximation (RSA) was then used to obtain the functional forms of the two objective functions. RSA is a methodology of fitting a polynomial function for discrete responses obtained from numerical calculations. The polynomial responses are expressed as: The Response Surface Approximation (RSA) was used to obtain the functional forms of the two objective functions.

Rth ¼ 0:0964 þ 0:3124  h  0:7005  /  1:1122  h  / þ 0:6044  h2 þ 4:8528  /2 P ¼ 0:9925  9:3955  h þ 3:5575  /  14:9250  h  / þ 22:9024  h2  0:0706  /2

ð9Þ ð10Þ

The design variables h and / are in terms of the ratios of the microchannel width to depth (i.e. Wc/Hc) and fin width to the microchannel depth (i.e. Ww/Hc), respectively. Thus, solving Eqs. (9) and (10) for h and / will give the optimum values of the dimensions of the micro-channel, i.e. Wc, Ww and Hc. The two design variables h and / have significant effect on the thermal performance of microchannel heat sink. Design and manufacturing constraints can be handled in a better way, and Pareto optimal solutions can be spread over the whole range of variables. The Pareto optimal analysis provides information about the active design space and relative sensitivity of the design variables to each objective function which is helpful in comprehensive design optimization. Thus, Eqs. (9) and (10) in terms of design variables h and / have the physical meaning. Table 1 Design variables and their ranges for case study 1 [2].

Fig. 2. Conventional diagram of rectangular MCHS geometry [10].

Design variable

Lower range

Upper range

h = Wc/Hc / = Ww/Hc

0.10 0.04

0.25 0.10

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Husain and Kim [10] used NSGA-II algorithm to obtain Pareto optimal solutions and the solutions were refined by selecting local optimal solutions for each objective functions using a Sequential Quadratic Programming (SQP) method with NSGA-II solutions as initial solutions. Then K-means clustering method was then used to group the global Pareto optimal solutions into five clusters. The whole procedure was termed as a hybrid multi-objective optimization evolutionary algorithm (MOEA). Now, the model considered by Husain and Kim is attempted using the proposed Jaya algorithm. The model has more than one objective and hence it comes under the category of multiobjective optimization problems. In this paper a priori approach of solving the multi-objective optimization problem is considered. In a priori approach, multi-objective optimization problem is transformed into a single objective optimization problem by assigning an appropriate weight to each objective. This ultimately leads to a unique optimum solution. The priori approach takes into consideration the preferences of the designer or decision maker. The solution obtained by a priori approach depends on the weights assigned to various objective functions by designer or decision maker. By changing the weights of importance of different objective functions a dense spread of the Pareto points can be obtained. Following a priori approach in the present work, the two objective functions are combined into a single objective function. The combined objective function, Z, is formed as:




Minimize; Z ¼ w1


 Z1 Z2 þ w2 Z1min: Z2min:

ð11Þ

Fig. 4. Conventional diagram of trapezoidal MCHS [12].

Water was used as coolant liquid and it flowed into the microchannel and left at the outlet. The silicon substrate occupied the remaining portion of heat sink. No slip condition was assumed at the inner walls of the channel, i.e. u = 0. The thermal condition in the z-direction was given as:

ks



 @T s @T s ¼ q at z ¼ 0 and ks ¼ 0 at z ¼ lz : @xi @xi

The design variables considered by Husain and Kim [12] were

a = wc/hc, b = ww/hc, and c = wb/wc, where wc was the micro-

Z1 ¼ Rth and Z2 ¼ P where w1 and w2 are the weighs assigned to the objective functions Z1 and Z2 respectively between 0 and 1. These weights can be assigned to the objective functions according to the designer’s/decision maker’s priorities. Z1min and Z2min are the optimum values of the Z1 and Z2 respectively, obtained by solving the optimization problem when only one objective is considered at a time and ignoring the other. In the present work, apart from the newly proposed Jaya algorithm, the TLBO algorithm is also used for solving this multiobjective optimization problem for comparison purpose.

channel width at bottom; wb was the micro-channel width at top; ww was the fin width and hc was the micro-channel depth. hc was kept 400 lm during the whole optimization procedure. In this case study, two objective functions were considered and those were (i) thermal resistance associated with heat transfer performance and (ii) the pumping power to drive the coolant or to pass the coolant through the micro-channel. Table 2 shows design variables a, b and c and their limits for both rectangular (wb/wc = 1) and trapezoidal (0.5 < wb/wc < 1) cross sections of MCHS. The two objective functions considered were, thermal resistance and pumping power. The thermal resistance was given by:

3.2. Case study 2

RTH ¼ DT max =ðAs  qÞ

This case study was introduced by Husain and Kim [12]. A 10 mm ⁄ 10 mm ⁄ 0.42 mm silicon based MCHS considered by Husain and Kim [12] is shown in Fig. 4. The following equations were used for the convective heat transfer in the MCHS. Mass:

@ui @p @ ¼ þ @xi @xj @xi

Energy (for fluid):

@h ¼ @xi

DT max ¼ T s;o  T f ;i

ð17Þ

The pumping power to move the coolant (water) through MCHS was calculated as:




lf

@p @xi



þ




lf

@ui @xj

 þ

@ @xj




lf

@ui @xi


 @T f @ @ui kf þ sij @xi @xi @xj

 

2 @ 3 @xi




lf

@uk @xk

ð13Þ

ð14Þ

Table 2 Design variables and their ranges for case study 2 [12].



Limits

Energy (for the substrate conduction):


 @ @T s ¼0 ks @xi @xi

ð18Þ

where Dp was the pressure drop and uavg was the mean velocity. Pumping power and thermal resistance compete with each other because a decrease in pumping power contributes to increase in thermal resistance. As explained in case study 1, Husain and Kim [12] calculated the objectives by using Navier–Stokes and heat conduction equations at specific design points. The Response Surface Approximation (RSA) was then used to obtain the functional

ð12Þ

Momentum:

qf uj

where As was area of the substrate subjected to heat flux, and DTmax was the maximum temperature in MCHS, which was given as:

P ¼ n  uav g  Ac  Dp

@ðqf ui Þ ¼0 @xi

qf uj

ð16Þ

ð15Þ

Upper Lower

Variables

a (wc/hc)

b (ww/hc)

c (wb/wc)

0.10 2.50

0.02 1.0

0.50 1.00

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forms of the two objective functions. The polynomial responses are expressed as:

Rth ¼ 0:096 þ 0:31  a  0:019  b  0:003  c  0:007  a  b þ 0:031  a  c  0:039  b  c þ 0:008  a2 þ 0:027  b2 þ 0:029  c2 P ¼ 0:94  1:695  a  0:387  b  0:649  c  0:35  a  b

ð19Þ

ð20Þ

The design variables a, b and c are in terms of the ratios of the micro-channel width at bottom to depth (i.e. wc/hc), fin width to the micro-channel depth (i.e. ww/hc), and micro-channel width at top to width at bottom (wb/wc) respectively. Thus, solving Eqs. (19) and (20) for a, b and c will give the optimum values of the dimensions of the micro-channel, i.e. wc, ww, wb and hc. The three design variables a, b and c have significant effect on the thermal performance of micro-channel heat sink. Design and manufacturing constraints can be handled in a better way, and Pareto optimal solutions can be spread over the whole range of variables. The Pareto optimal analysis provides information about the active design space and relative sensitivity of the design variables to each objective function which is helpful in comprehensive design optimization. Thus, Eqs. (19) and (20) have the physical meaning. Husain and Kim [12] used these surrogate models given in Eqs. (19) and (20) and a hybrid MOEA involving NSGA-II and Sequential Quadratic Programming (SQP) method to find out the Pareto optimal solutions. Husain and Kim [12] used NSGA-II algorithm to obtain Pareto optimal solutions and the solutions were refined by selecting local optimal solutions for each objective function using a Sequential Quadratic Programming (SQP) method with NSGA-II solutions as initial solutions. Then K-means clustering method was then used to group the global Pareto optimal solutions into five clusters. The whole procedure was termed as a hybrid multi-objective optimization evolutionary algorithm (MOEA). Now, the model considered by Husain and Kim [12] is attempted using the proposed Jaya algorithm. Following a priori approach, the combined objective function, Z, is formed as:


Minimize; Z ¼ w1






Z1 Z2 þ w2 Z1min: Z2min:

4. Results and discussions 4.1. Case study 1

þ 0:557  a  c  0:132  b  c þ 0:903  a2 þ 0:016  b2 þ 0:135  c2

The next section describes the results of application of the proposed Jaya algorithm for the multi-objective dimensional optimization of MCHS.

Husain and Kim [10] used a hybrid multi-objective optimization of evolutionary algorithm (MOEA) with population size of 250 and number of generations of 100. Thus the number of function evaluations used by Husain and Kim [10] was 25,000. Hence, for fair comparison the Jaya and TLBO algorithms are also used with the same number of function evaluations, i.e. 25,000. The results of application of the proposed Jaya algorithm, TLBO algorithm and hybrid MOEA with five representative clusters are presented in Table 3. Optimal results with various weighting factors for the objective functions are shown. The values of design variables and corresponding values of the objectives given by the algorithms are shown in Table 3. It can be observed that the proposed Jaya algorithm and TLBO algorithm give better results for both the objective functions than the hybrid MOEA proposed by Husain and Kim [10] for the bi-objective optimization problem considered. For w1 = 0.9 and w2 = 0.1 the values of the two objective functions are same for all the algorithms. The proposed Jaya algorithm and TLBO algorithm show equal performance for both the objectives except for the weights of w1 = 0.5 and w2 = 0.5 for which found Jaya algorithm is better than the TLBO algorithm. Fig. 5 shows Pareto fronts obtained by using Jaya algorithm, TLBO algorithm and hybrid MOE algorithm representing five clusters. Also, it can be observed that the proposed Jaya and TLBO algorithms provide better results than the hybrid MOEA proposed by Husain and Kim [10]. Every peak end of the Pareto curve represents the higher Case study 1 0.16

Thermal Resistance (ºC/W)

578

 ð21Þ

Z1 ¼ Rth and Z2 ¼ P where w1 and w2 are the weighs assigned to the objective functions Z1 and Z2 respectively between 0 and 1. These weights can be assigned to the objective functions according to the designer’s/decision maker’s priorities. Z1min and Z2min are the optimum values of the Z1 and Z2 respectively, obtained by solving the optimization problem when only one objective is considered at a time and ignoring the other.

0.15 0.14 0.13

TLBO

0.12

Hybrid MOEA JAYA Jaya

0.11 0.1

0

0.1

0.2

0.3

0.4

0.5

Pumping Power (W) Fig. 5. Pareto optimal solutions using Jaya, TLBO and Hybrid MOE algorithm representing 5-clusters for case study 1.

Table 3 Design variables and corresponding values of objective functions by using Jaya, TLBO and NSGA-II of MCHS for case study 1. Sr. no.

Weighting factors w1

1 2 3 4 5

0.9 0.7 0.5 0.3 0.1

w2

0.1 0.3 0.5 0.7 0.9

Design variables

Objective functions

h

/

Hybrid MOEA, 5-cluster [12]

TLBO

Jaya algorithm (present study)

Hybrid MOEA

TLBO

Jaya

Hybrid MOEA

TLBO

Jaya

P

RTH

P

RTH

P

RTH

0.994 0.459 0.096 0.000 0.000

0.2135 0.1963 0.1631 0.1341 0.110

0.2135 0.1963 0.164 0.1341 0.110

0.140 0.693 0.638 0.886 0.609

0.0475 0.0575 0.074 0.0719 0.0751

0.0475 0.0575 0.075 0.0719 0.0751

0.048 0.072 0.152 0.257 0.387

0.157 0.152 0.1252 0.113 0.103

0.048 0.066 0.151 0.256 0.379

0.157 0.144 0.124 0.113 0.103

0.048 0.066 0.150 0.256 0.379

0.157 0.1443 0.1248 0.1132 0.103

Bold value indicates the best results.

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0.24 0.22

Design Variables

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.1

0.11

0.12

0.13

0.14

0.15

0.16

Thermal Resistance (ºC/W) Fig. 6. Distribution of design variables h and / with the objective function of thermal resistance for case study 1.

0.24 0.22

value of one objective and lower value of another. Figs. 6 and 7 show the distribution of the design variables h and / with the objective function of thermal resistance and pumping power, respectively. It is observed that the h increases in the direction of increasing thermal resistance and decreases pumping power. The design variable / remains almost constant up to certain values of thermal resistance and then decreases with increase in the values of thermal resistance. Also, it is observed that / increases up to certain values of pumping power but decreases later. Fig. 8 shows the convergence of the combined objective functions obtained by TLBO and Jaya algorithm for equal weights of objectives (w1 = 0.5 and w2 = 0.5). The Jaya algorithm converges after 8th iteration and TLBO algorithm converges after 11th iteration for case study 1. The convergence rate of Hybrid MOEA was not given in Husain and Kim [10] and hence comparison is not possible. It can be observed that the Jaya algorithm converges faster than the TLBO algorithm. It can be said the Jaya and TLBO algorithms have performed better than the hybrid MOEA for the bi-objective dimensional optimization problem of MCHS.

Design Variables

0.2

4.2. Case study 2

0.18 0.16 0.14

θ ϕ Phi

0.12 0.1 0.08 0.06 0.04 0

0.1

0.2

0.3

0.4

Pumping Power (W) Fig. 7. Distribution of design variables h and / with the objective function of pumping power for case study 1.

Combined objective function

for w1=0.5 & w2=0.5 5.4 4.9 4.4 TLBO TLBO

3.9

JAYA Jaya

3.4 2.9 0

20

40

60

80

100

No. of generations Fig. 8. Convergence of the combined objective function obtained by TLBO and Jaya algorithms for the heat sink (for w1 = 0.5 and w2 = 0.5) for case study 1.

Husain and Kim [12] used a hybrid MOEA coupled with surrogate models to obtain the Pareto optimal solutions. A population size of 100 with number of generations of 250 for considering. Thus the number of function evaluations used by Husain and Kim [12] was 25,000. Hence for fair comparison, the proposed Jaya and TLBO algorithms used the same number of function evaluations, i.e. 25,000. Numerical analysis was also carried out by Husain and Kim [12]. The values of the design variables given by the Jaya algorithm, TLBO algorithm, hybrid MOEA and numerical analysis are shown in Table 4. Table 5 shows the results comparison of Jaya algorithm, TLBO algorithm, hybrid MOEA and numerical analysis. It is observed that the Jaya algorithm gives better results as compared with hybrid MOEA, Numerical analysis and TLBO algorithm for different weights of the objective functions for the bi-objective optimization problem considered. The performance of TLBO algorithm comes next to Jaya algorithm. Fig. 9 shows Pareto fronts obtained by using Jaya algorithm, TLBO algorithm and hybrid MOE algorithm representing five clusters. Also, it can be observed that the proposed Jaya and TLBO algorithm provide better results than the hybrid MOEA proposed by Husain and Kim [12]. Every peak end of the Pareto curve represents the higher value of one objective and lower value of another. Figs. 10 and 11 show the distribution of the design variables a, b and c with the objective function of thermal resistance and pumping power respectively for Jaya algorithm. Figs. 12 and 13 show the distribution of the design variables a, b and c with the objective function of thermal resistance and pumping power respectively for TLBO algorithm.

Table 4 Design variables of objective functions by using Jaya, TLBO and NSGA-II of MCHS for case study 2. Sr. no.

Weighting factors w1

1 2 3 4 5

0.9 0.7 0.5 0.3 0.1

w2

0.1 0.3 0.5 0.7 0.9

Design variables

a

c

b

TLBO

Jaya

Hybrid MOEA

TLBO

Jaya

Hybrid MOEA

TLBO

Jaya

Hybrid MOEA

0.7952 0.595 0.325 0.132 0.1067

0.448167 0.068346 0.324 0.0324 0.124

0.994 0.459 0.096 0.000 0.000

0.040 0.735 0.745 0.7601 0.69

0.879132 0.526466 0.721 0.857001 0.67

0.140 0.693 0.638 0.886 0.609

0.534 0.5912 0.601 0.6299 0.528

0.9999 0.55973 0.59701 0.9692 0.5692

0.528 0.991 0.982 0.971 0.456

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Table 5 Comparison of results of Jaya algorithm, TLBO algorithm, hybrid MOEA and numerical analysis of MCHS for case study 2. Sr. no.

Weighting factors

1 2 3 4 5

Hybrid MOEA [12]

Numerical analysis [12]

TLBO

Jaya (present study)

w1

w2

RTH

P

RTH

P

RTH

P

RTH

P

0.9 0.7 0.5 0.3 0.1

0.1 0.3 0.5 0.7 0.9

0.145 0.118 0.100 0.094 0.093

0.097 0.195 0.455 0.633 0.828

0.143 0.119 0.100 0.094 0.094

0.094 0.175 0.410 0.634 0.821

0.143 0.1172 0.1033 0.094 0.0927

0.0931 0.1933 0.4054 0.6282 0.6966

0.138308 0.116959 0.103345 0.093279 0.093779

0.086174 0.192433 0.402558 0.6282 0.6444

Bold values indicate the best results.

0.9

Case study 2 0.9

0.8 0.7

0.7 0.6 0.5

TLBO

0.4 0.3

Hybrid MOEA

0.2

Jaya

Design Variables

Pumping Power (W)

0.8

0.6 0.5

α β γ

0.4 0.3 0.2

0.1 0 0.09

0.1

0.11

0.12

0.13

0.14

0.1

0.15

0

Thermal resistance (º C/W)

0

Fig. 9. Pareto optimal solutions using Jaya, TLBO and Hybrid MOE algorithm representing 5-clusters for case study 2.

0.9

0.2

0.4

0.6

0.8

Pumping Power (W) Fig. 12. Distribution of design variables a, b and c with the objective function of pumping power for case study 2 using TLBO algorithm.

0.8

1.2

Design Variables

0.7 0.6

α β γ

0.4 0.3 0.2

Design Variables

1

0.5

0.8

α β γ

0.6 0.4

0.1 0 0.08

0.2 0.09

0.1

0.11

0.12

0.13

0.14

0.15

Thermal Resistance (ºC/W)

0 0

Fig. 10. Distribution of design variables a, b and c with the objective function of thermal resistance for case study 2 using TLBO algorithm.

0.1

0.2

0.3 0.4 0.5 Pumping Power (W)

0.6

0.7

Fig. 13. Distribution of design variables a, b and c with the objective function pumping power for case study 2 using Jaya algorithm.

1.2

Design Variables

1 0.8 0.6 0.4 0.2 0 0.09

0.1

0.11

0.12

0.13

0.14

0.15

Thermal Resistance (ºC/W) Fig. 11. Distribution of design variables a, b and c with the objective function of thermal resistance for case study 2 using Jaya algorithm.

It is observed that a increases in the direction of increasing thermal resistance. Further, a increases and decreases relatively in the direction of increasing pumping power using Jaya algorithm.

In the case of using TLBO algorithm, a increases in the direction of increasing thermal resistance and decreases pumping power. Also, it is observed that the design variable b and c increase and decrease alternately in the direction of increasing thermal resistance. Also, the design variables b and c decrease and increase alternately in the direction of increasing pumping power using Jaya algorithm. The values of b and c increase then remain almost constant up to certain values of thermal resistance and then decrease with increase in the values of thermal resistance. In the case of TLBO algorithm, it is observed that b and c increase and then decrease in the direction of increasing pumping power. Fig. 14 shows the convergence of the combined objective functions obtained by TLBO and Jaya algorithm for equal weights of objectives (w1 = 0.5 and w2 = 0.5). The Jaya algorithm converges after 6th iteration and TLBO algorithm converges after 9th iteration for case study 2. The convergence rate of Hybrid MOEA was not given in Husain and Kim [12] and hence comparison is not possible. It can be observed that the Jaya algorithm converges faster than the TLBO

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for w1=0.5 & w2=0.5

Table A1 Initial population.

Combined objective function

5

4.5

4

TLBO TLBO

Candidate

x1

x2

f(x)

Status

1 2 3 4 5

4.570261872 3.574220009 2.304524513 1.062187325 0.84373426

0.045197073 1.823157605 4.442417134 0.767182961 3.348170112

40.33111401 40.59470054 57.75775652 11.39297485 32.15169977

Worst Best

JAYA Jaya 3.5

3

Table A2 New values of the variables and the objective function during first iteration. 0

20

40

60

80

100

No. of generations Fig. 14. Convergence of the combined objective function obtained by TLBO and Jaya algorithms for the heat sink (for w1 = 0.5 and w2 = 0.5) for case study 2.

algorithm. It can be said the Jaya and TLBO algorithms have performed better than the hybrid MOEA for the considered dimensional optimization problem of micro-channel heat sink. 5. Conclusions The dimensional optimization of micro-channel heat sink (MCHS) is carried out with the help of a very recently proposed Jaya algorithm. Two case studies of MCHS are presented to compare the performance of Jaya algorithm with TLBO algorithm and a hybrid multi-objective evolutionary algorithm. Thermal resistance and pumping power are considered as the objective functions in both the case studies. The design variables involving micro-channel width, depth and fin width are considered in case study 1 and the design variables involving micro-channel depth, micro-channel width at top and bottom and fin width are considered in case study 2. These design variables affect the thermal resistance and pumping power. It is observed that in case study 1 the Jaya algorithm obtains equal or better optimal results as compared to the TLBO algorithm and better than those given by the hybrid MOE algorithm. The optimal results obtained by Jaya algorithm in case study 2 are better than those obtained by TLBO, hybrid MOE algorithm and numerical analysis. The TLBO algorithm has performed next to the Jaya algorithm in case study 2. The Pareto optimal solutions obtained by Jaya algorithm are better than those obtained by TLBO and hybrid MOE algorithms. The Pareto optimal analysis provides information about the active design space and relative sensitivity of the design variables to each objective function which is helpful in comprehensive design optimization and this helps in practical design of micro-channels. Also the convergence behavior of Jaya is better than TLBO algorithm. The Jaya algorithm has shown its ability to solve the bi-objective optimization problem of MCHS in the present work. However, the use of the proposed Jaya algorithm may be extended to solve many-objective optimization problems of different thermal engineering systems and devices.

Candidate

x1

x2

f(x)

1 2 3 4 5

1.142020267 6.574163612 0.149450027 0.857582225 0.982100472

2.856829068 1.843391565 0.350413633 3.979663034 0.974134436

16.97028899 70.01455716 20.1382099 20.39823624 2.108405971

Table A3 Updated values of the variables and the objective function based on fitness comparison at the end of first iteration. Candidate

x1

x2

f(x)

1 2 3 4 5

1.142020267 3.574220009 0.149450027 1.062187325 0.982100472

2.856829068 1.823157605 0.350413633 0.767182961 0.974134436

16.97028899 40.59470054 20.1382099 11.39297485 2.108405971

Status Worst

Best

Range of variables: 5.12 6 xi 6 5.12. The known solution to this benchmark function is 0 for all xi values of 0. Now to demonstrate the Jaya algorithm, let us assume a population size of 5 (i.e. candidate solutions), two design variables x1 and x2 and two iterations as the termination criterion. The initial population is randomly generated within the ranges of the variables and the corresponding values of the objective function are shown in Table A1. As it is a minimization function, the lowest value of the Rastrigin function is considered as the best solution and the highest value is considered as the worst solution. From Table A1 it can be seen that the best solution is corresponding the 4th candidate and the worst solution is corresponding to the 3rd candidate. Now assuming random numbers r1 = 0.38 and r2 = 0.81 for x1 and r1 = 0.92 and r2 = 0.49 for x2, the new values of the variables for x1 and x2 are calculated using Eq. (1) and are placed in Table A2. For example, for the 1st candidate, the new values of x1 and x2 during the first iteration are calculated as shown below.

X 01;1;1 ¼ X 1;1;1 þ r 1;1;1 ðX 1;4;1  jX 1;1;1 jÞ  r 2;1;1 ðX 1;3;1  jX 1;1;1 jÞ ¼ 4:570261872 þ 0:38ð1:062187325  j  4:570261872jÞ  0:81ð2:304524513  j  4:570261872jÞ ¼ 1:142020267 X 02;1;1 ¼ X 2;1;1 þ r1;2;1 ðX 2;4;1  jX 2;1;1 jÞ  r2;2;1 ðX 2;3;1  jX 2;1;1 jÞ ¼ 0:045197073 þ 0:92ð0:767182961  j0:045197073jÞ

Appendix A

 0:49ð4:442417134  j0:045197073jÞ ¼ 2:856829068 A.1. Demonstration of the working of Jaya algorithm To demonstrate the working of Jaya algorithm, a benchmark function of Rastrigin is considered. The objective function is to find out the values of xi that minimize the value of the Sphere function.

F min ¼

D X ½x2i  10 cosð2pxi Þ þ 10 i¼1

ðA:1Þ

Similarly, the new values of x1 and x2 for the other candidates are calculated. Table A2 shows the new values of x1 and x2 and the corresponding values of the objective function. Now, the values of f(x) of Tables A1 and A2 are compared and the best values of f(x) are considered and placed in Table A3. This completes the first iteration of the Jaya algorithm. From Table A3 it can be seen that the best solution is corresponding the 5th candidate and the worst solution is correspond-

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Table A4 New values of the variables and the objective function during second iteration. Candidate

x1

x2

f(x)

1 2 3 4 5

1.805374379 1.889342005 0.097024584 1.692011601 0.385912674

3.785421269 0.760188114 1.604839841 1.967429156 2.147476939

31.97203233 15.8290846 22.29400177 20.50579399 26.29413498

Table A5 Updated values of the variables and the objective function based on fitness comparison at the end of second iteration. Candidate

x1

x2

f(x)

Status

1 2 3 4 5

1.142020267 1.889342005 0.149450027 1.062187325 0.982100472

2.856829068 0.760188114 0.350413633 0.767182961 0.974134436

16.97028899 15.8290846 20.1382099 11.39297485 2.108405971

Worst Best

ing to the 2nd candidate. Now, during the second iteration, assuming random numbers r1 = 0.65 and r2 = 0.23 for x1 and r1 = 0.38 and r2 = 0.51 for x2, the new values of the variables for x1 and x2 are calculated using Eq. (1). Table A4 shows the new values of x1 and x2 and the corresponding values of the objective function during the second iteration. Now, the values of f(x) of Tables A3 and A4 are compared and the best values of f(x) are considered and placed in Table A5. This completes the second iteration of the Jaya algorithm. From Table A5 it can be seen that the best solution is corresponding the 5th candidate and the worst solution is corresponding to the 3rd candidate. It can also be observed that the value of the objective function is reduced from 11.39297485 to 2.108405971 in just two iterations. If we increase the number of iterations then the known value of the objective function (i.e. 0) can be obtained within next few iterations. In the case of maximization problems, the best value means the maximum value of the objective function and the calculations are to be proceeded accordingly. Thus, the proposed method can deal with both minimization and maximization problems. References [1] D.B. Tuckerman, R.F.W. Pease, High-performance heat sinking for VLSI, IEEE Electr. Dev. Lett. 2 (1981) 126–129. [2] M.B. Bowers, I. Mudawar, High flux boiling in low flow rate, low pressure drop mini-channel and micro-channel heat sinks, Int. J. Heat Mass Transfer 37 (2) (1994) 321–332. [3] S.J. Kim, D. Kim, Forced convection in microstructures for electronic equipment cooling, ASME J. Heat Transfer 121 (1999) 639–645. [4] K. Vafai, L. Zhu, Analysis of two-layered micro-channel heat sink concept in electronic cooling, Int. J. Heat Mass Transfer 42 (1999) 2287–2297. [5] J.H. Ryu, D.H. Choi, S.J. Kim, Three-dimensional numerical optimization of a manifold micro-channel heat sink, Int. J. Heat Mass Transfer 46 (2003) 1553– 1562. [6] K. Jeevan, I.A. Azid, K.N. Seetharamu, Optimization of double layer counter flow (DLCF) micro-channel heat sink used for cooling chips directly, in: Electronics Packaging Technology Conference (2004), Guilin, China, 2004, pp. 553–558.

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