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Numerical Heat Transfer, Part A:Applications: An International Journal ofComputation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unht20

Multiobjective Optimization of StaggeredElliptical Pin-Fin ArraysPing Li a & Kwang-Yong Kim aa Department of Mechanical Engineering , Inha University , Incheon,Republic of KoreaPublished online: 06 Nov 2007.

To cite this article: Ping Li & Kwang-Yong Kim (2008) Multiobjective Optimization of StaggeredElliptical Pin-Fin Arrays, Numerical Heat Transfer, Part A: Applications: An International Journal ofComputation and Methodology, 53:4, 418-431, DOI: 10.1080/10407780701632759

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MULTIOBJECTIVE OPTIMIZATION OF STAGGEREDELLIPTICAL PIN-FIN ARRAYS

Ping Li and Kwang-Yong KimDepartment of Mechanical Engineering, Inha University, Incheon, Republicof Korea

This work presents a numerical procedure for multiple objectives to optimize staggered

elliptic-shaped short pin-fin arrays. The multiobjective problem is to achieve an acceptable

compromise between augmentation of turbulent heat transfer and reduction in friction loss.

Four nondimensional variables, pin-fin height-to-channel height ratio, major axis length-to-

channel height ratio, minor-axis length-to-channel height ratio, and pin-fin pitch-to-channel

height ratio are chosen as design variables. The D-optimal method is used to determine the

training points. The response surface method is used to approximate the Pareto optimal

front with Reynolds-averaged Navier-Stokes analysis of the flow and heat transfer using

the shear stress transport (SST) turbulence model. The Pareto-optimal solutions are

obtained using a combination of an evolutionary algorithm and a local search.

1. INTRODUCTION

Finned surfaces are frequently used in heat exchange devices for the purpose ofincreasing the heat transfer between a primary surface and the surrounding fluid byincreasing the surface area and inducing turbulent mixing of flow. In the past twodecades, as reviewed by Ligrani et al. [1], heat transfer and friction characteristicsof pin-fin array systems have been the subject of extensive investigation because oftheir importance in a wide variety of engineering applications, such as compact heatexchangers and the cooling of advanced gas turbine blades and electronic devices.Various types of pin-fin arrays, ranging from relatively simple shapes such as rec-tangular, cylindrical, annular, and tapered to a combination of different geometries,have been used. In addition, the pin-fin may be positioned in arrays that are eitherstaggered or inline with respect to the flow direction.

Convective heat transfer of the finned surface is affected significantly by theflow properties, such as reattachment length of separated streamline and turbulenceintensities, as well as Reynolds number. Therefore, shape optimization of pin-finarrays for enhancement of turbulent heat transfer should be based on a preciseanalysis of the flow structure, such as a three-dimensional Reynolds-averagedNavier-Stokes (RANS) equations analysis.

Received 19 January 2007; accepted 14 July 2007.

This research was supported by a Korea Science & Engineering Foundation (KOSEF) grant funded

by the Korea government (MOST) (No. R01-2006-000-10039-0).

Address correspondence to Kwang-Yong Kim, Department of Mechanical Engineering, Inha

University, 253 Yonghyun-Dong, Nam-Gu, Incheon 402-751, Republic of Korea. E-mail: kykim@inha.ac.kr

418

Numerical Heat Transfer, Part A, 53: 418–431, 2008

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407780701632759

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Numerous experimental investigations [2–5] on the heat transfer characteristics inpin-fin arrays have been reported. However, in most of these studies, the optimum shapeof the pin-fins was determined based on only a few geometric parameters. As an attemptto find a globally optimum shape, by using experimental data, Chiang and Chang [6]recently applied the response surface method [7] to shape optimization of a heat sinkhaving an array of circular pin-fins in an effort to achieve high thermal performance.

A number of numerical studies [8–11] have also been carried out to analyze theflows through pin-fins. Matos et al. [11] attempted to find the optimum shape of cir-cular and elliptic pin-fins in the range of laminar flow by solving two-dimensionalgoverning equations without employing any optimization technique. They observeda relative heat transfer gain of up to 13% in the optimal elliptic arrangement com-pared to the optimal circular configuration. Park et al. [12] performed a numericaloptimization of a pin-fin array with square cross sections by using the BFGSmethod, which employs a gradient-based optimization algorithm. For ribs attachedin square channels, a series of investigations [13–15] identified the optimum shapes toenhance turbulent heat transfer by using a three-dimensional RANS analysis and anumerical optimization technique with response surface approximation. Kim andChoi [16] also tried to optimize the shape of a dimpled channel in the same manner.

Recently, response surface approximations [7] have been applied to many single-and multidisciplinary optimization problems [17] in order to substantially reduce thecomputational burden of performing optimization and generating a multitude oftrade-off solutions. Along with the development of numerical optimization techniques,some research has been directed toward multiobjective optimization problems, such asthe weighted-sum strategy and Pareto-optimal solutions methods. The weighted-sumstrategy is very popular and simple to implement [18]. However, it is difficult to decidethe value of weighting factors. Meanwhile, over the past decade, a number of multiob-jective evolutionary algorithms [18–21] have been suggested, their strong feature beingthe ability to find multiple Pareto-optimal solutions in a single simulation run. EachPareto-optimal solution represents a different compromise among design objectives.However, it is hard to determine optimum using only one performance as the delegatefor all these Pareto-optimal solutions. The main feature of the Pareto-optimal method is

NOMENCLATURE

a half ofmajor axis length of ellipse

b half of minor axis length of ellipse

cp specific heat

D channel height

Dh channel hydraulic diameter

f friction factor

f0 reference friction factor

H pin-fin height

k turbulent kinetic energy

Nu local Nusselt number

Nua average Nusselt number

Nuo reference local Nusselt number

p, Dp pressure and pressure drop in a

channel, respectively

Pi pin-fin pitch

Pr Prandtl number

qo wall heat flux

Re Reynolds number ð¼ UDh=nÞR2

adj adjusted R squared

T local mean temperature

Tw wall temperature

W width

x, y, z streamwise, spanwise, and cross-

streamwise coordinates, respectively

n molecular viscosity

q fluid density

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to show the relationship among objectives. Once many Pareto optimal solutions areobtained, the Pareto-optimal front can be represented by its own response surface,and the domain of application can be identified. The response surface approximationof the Pareto-optimal front allows the designer to visualize and assess trade-offs amongthe objectives, to explore compromise solutions, and to make decisions based onrealistic goals.

In this work, multiobjective optimization methods are applied to the design ofheat transfer surfaces of a three-dimensional channel with elliptic-shaped pin-finarrays. The major interests here are to suggest the optimal shape and to probe theinteractions and trade-offs between two objectives, i.e., enhancing heat transferand reducing pressure drop, as well as assessing correlation and competition betweenthem. Turbulent convective heat transfer is analyzed with a three-dimensionalRANS analysis. Response surface approximations are employed in two aspects,namely, to generate surrogates of computationally expensive simulation-based mod-els and to approximate the Pareto-optimal front by representing one objective interms of others. Except for the multiobjective optimization methods, the other meth-ods employed in this work for flow analysis and optimization are similar to thoseused in previous works [14–16].

2. NUMERICAL ANALYSIS

For the analysis of convective heat transfer in a three-dimensional pin-finarrays channel, the commercial computational fluid dynamics (CFD) code CFX-5.7 [22] was used in this work. This code solves the RANS equations with a high-resolution scheme for the advection term, and employs Rhie-Chew interpolationfor the pressure–velocity coupling. The linear solver uses an algebraic multigridmethod based on the additive correction multigrid strategy, and fully coupledmomentum and energy equations are solved simultaneously.

To adopt periodic boundary conditions, the source terms in the streamwisemomentum and energy equations have been modified in order to calibrate the grad-ual decrease and increase of pressure and temperature, respectively, as done by Kimand Kim [14].

The shear stress transport (SST) turbulence model with automatic wall treat-ment [23] is used as a turbulence closure. Basically, the SST model combines theadvantages of the k–e and k–x models with a blending function. The k–x modelis activated in the near-wall region and the k–e model is used in the remaining region.Bardina et al. [24] showed that the SST model captures separation under an adversepressure gradient well compared to other eddy viscosity models. It thus accuratelypredicts the near-wall turbulence, which plays a vital role in the accurate predictionof the turbulent heat transfer.

Geometric parameters and the computational domain are shown in Figure 1.The computational domain is composed of one pitch of pin-fin displacement withperiodic boundary conditions on the surfaces normal to the streamwise direction,and also with symmetric conditions on the surfaces normal to the cross-streamwisedirection. At the inlet and outlet of the domain, the following periodic conditions areadopted. For each dependent variable, the inlet value is the same as the outlet valueat the corresponding transverse location. Uniform turbulent kinetic energy and its

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dissipation rate are assumed at the inlet as initial boundary conditions. The uniformvalues are calculated as in previous works [14–16]. However, these uniform inlet con-ditions are used only at the initial stage of calculation, and are not activated from the

Figure 1. Geometric parameters and calculation domain.

Figure 2. Examples of grid system.

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second iteration. At all wall boundaries, the wall function based on the empiricalwall law for near-wall turbulence is adopted for mean axial velocity. Uniform heatflux is specified on all of the surfaces including the fin surfaces. Bulk velocity corre-sponds to the Reynolds number based on the hydraulic diameter, i.e., 5,000.

With the periodic conditions, it is difficult to assign a specific flow rate to thecalculated flow. Thus, an iterative procedure is inevitably employed. At the initialstage of the calculation, the pressure gradient in the streamwise direction is assumedthrough the source term in the streamwise momentum equation, and is continuallyupdated until the Reynolds number falls within 1% of the target Reynolds number.

In Figure 2, an example of a grid system is shown for the whole computationaldomain and around the pin-fin. In the present study, an unstructured tetrahedralgrid system is used with the hexahedral at the wall region to resolve the high velocitygradient. The first grid points are placed at 0.0002D from the walls to satisfy yþ lessthan 1.0.

3. OPTIMIZATION METHODS

The three-dimensional channel with pin-fin arrays is shown in Figure 1 with sixgeometric variables: height of the channel (D), height of the pin-fin (H), major axislength (2a) and minor axis length (2b) of the elliptic cross section, pitch (Pi), and dis-tance between two neighbored pin-fins (W). In the present optimization, W is set tobe Pi=2.0; thus, four dimensionless geometric variables, i.e., H=D, 2a=D, 2b=D, andPi=D, are selected as design variables.

To maximize the performance of the pin-fin arrays, the optimum shape shouldbe determined through a compromise between enhancement of heat transfer andreduction of friction loss. After the definition proposed by Gee and Webb [25], aver-age Nusselt number and 1=3 power of friction factor became indexes for representingthe thermal performance, as have been used in previous works [13–16].

The objective function for heat transfer performance is defined as an inverse ofthe Nusselt number,

FNu ¼1

Nuað1Þ

where

Nua ¼R

ANuNuo

dA

A

Nu ¼ q0Dh

kf ðTw � TÞ

Nuo ¼ 0:023 Re0:8Pr0:4

Nu is calculated on the finned surfaces, and Nuo is the Nusselt number obtainedfrom the Dittus-Boelter correlation, which is for fully developed turbulent flows ina smooth pipe, and the integration is performed over the heated surface (A).

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On the other hand, the objective function for friction loss-related performanceis defined as follows:

Ff ¼f

f0

� �1=3

ð2Þ

where

f ¼ DpDh

2qU2L

f0 ¼ 2ð2:236 ln Re� 4:639Þ�2

where f0 is a friction factor for a fully developed flow in a smooth pipe. It isobtained from the Petukhov empirical correlation [26], which is modified fromthe Karman-Nikuradse correlation for the best fit in the range 104 < Re < 106.

The optimization problem is defined as an appropriate compromise betweentwo objective functions, FNu(x) and Ff (x) with xi

l� xi � xiu, where x is a vector

of design variables and xil and xi

u are the lower and upper bounds of each designvariable, respectively.

Response Surface Approximation

In order to obtain the optimum shape of the pin-fin arrays in a three-dimensional channel, the response surface method [7] is used in this work as a numeri-cal optimization method. The response surface method is an optimization techniquethat performs a series of numerical analyses for a prescribed set of design pointsand constructs a response surface of the measured quantity over the design space.

The response surface is usually expressed as a second-order polynomial asfollows:

g ¼ c0 þXn

j¼1

cjxj þXn

j¼1

cjjx2j þ

XXi 6¼j

cijxixj ð3Þ

where g is the response function, xi are the design variables, and c’s are unknownpolynomial coefficients, which are determined by the least-square method withevaluation of the response function at design points.

Prescribed design points, so-called training points, are selected by D-optimaldesign [7]. D-optimal design is a useful and reliable way of constructing the responsesurface with a small number of design points, which is only 1.5–2.5 times the numberof polynomial coefficients.

Multiobjective Evolutionary Algorithms

The optimization problem with two objectives described above is solved by amultiobjective evolutionary algorithm (MOEA) to find the Pareto-optimal solutions.

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In this study, the real-coded NSGA-II developed by Deb et al. [21] is used as a MOEA.After running the evolutionary algorithm, a set of approximate Pareto-optimal pointsis obtained as the solution.

Local Search

A local search is conducted to further refine the solutions obtained by NSGA-II.In the current study, the posterior hybrid method [18] is used, and the sequentialquadratic programming method implementation in MATLAB [27] is used as the localoptimizer. When using a local search method, the multiobjective optimization problemshould be converted to a single-objective optimization problem. Two strategies, theweighted-sum strategy [18] and the e-constraint strategy, are normally used. Theweighted-sum strategy is very popular and simple to implement. However, as Goeland Deb [20] suggested that the e-constraint gives better results for two-objectiveoptimization problems than the weighted-sum strategy, the e-constraint strategy isused in this work.

The Pareto-optimal solution set obtained by NSGA-II combined with a localsearch consists of a large number of solutions—it is practically infeasible for adesigner to evaluate all of the designs for selecting the optimal design. To help thedesigner in making effective use of the data, two strategies are proposed, clustering[28] and the global Pareto-optimal response surface [20].

The methodology used in this work for generating the Pareto-front responsesurface approximation is illustrated in Figure 3.

Figure 3. Flowchart for Pareto-optimal front response surface approximation.

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4. RESULTS AND DISCUSSION

To find the optimal number of grids, four different numbers were tested inrelation to staggered pin-fin arrays in a three-dimensional channel, where the con-ditions correspond with those of an experiment reported by Li et al. [3] at a Reynoldsnumber of 5,000. The results are shown in Figure 4. From the results, 1.67� 105 isdetermined to be the optimal number of grids.

For validation of the present numerical solutions, the predicted Nusselt num-bers at different Reynolds numbers are compared with the experimental data of Liet al. [3], where heights of the channel (D), major axis length (2a), minor axis length(2b), and pitch (Pi) are 12.75, 16, 9, and 60 mm, respectively, with W=Pi ¼ 0.5 andH=D ¼ 1. In Figure 5, for a wide range of Reynolds numbers, averaged Nusseltnumbers normalized by the Nusselt number for a fully developed turbulent flowin a smooth pipe are in good agreement with the experimental data [3], especiallyin the range of Reynolds number from 2,000 to 5,000, thus supporting the validityof the results.

As the flow conditions for the present optimization, the Reynolds numberbased on hydraulic diameter and 25�C air is 5,000. Uniform heat flux imposed onboth walls is 600 W=m. For the optimization, response surface-based optimizationis used and 29 training points with four design variables are selected by D-optimaldesign to construct the response surface. Ranges of each design variable are listedin Table 1.

Numerical optimization is performed using a hybrid technique based on evol-utionary methods and local search methods to find the Pareto-optimal solutions. Tomeasure uncertainty in the set of coefficients in a polynomial, analysis of variance

Figure 4. Grid dependency test.

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(ANOVA) and regression analysis provided by t-statistic [7] are used, and the resultsare shown in Table 2. Guinta [29] suggested that the typical values of R2

adj are in therange of 0.9�R2

adj� 1.0 when the observed response values are accurately predictedby the response surface model. In this respect, the present response surfaces aredeemed reliable.

The two objective optimization problems were solved using a multiobjectiveevolutionary algorithm. The Pareto-optimal solutions obtained from NSGA-IIsimulations are shown in Figure 6.

In a local search with e-strategy, one objective (FNu) is taken as the objectivefunction and the other objective (Ff) is treated as an equality constraint. Theconstraint value is set equal to the corresponding objective value, as found byNSGA-II simulation. This constrained optimization problem is solved usingMATLAB (sequential quadratic programming) [27]. The process is repeated forall the individuals in the population, yielding a set of Pareto-optimal solutions,referred to as Set A in this study. Similarly, the Pareto optimal Set B is obtained withFf as the objective and FNu as constraints. At this stage, all the Pareto-optimal

Figure 5. Comparison of computational and experimental local Nusselt numbers.

Table 1. Design variables and ranges

Design variable Lower bound Upper bound

2a=D 0.706 1.568

2b=D 0.392 0.706

Pi=D 2.196 6.275

H=D 0.529 1.000

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solutions obtained thus far, i.e., Set A, Set B, and the set found using NSGA-II, arecombined. All the dominated solutions are discarded, and then all duplicatesolutions are removed. The true Pareto-optimal front is obtained as shown inFigure 6. It is clear from the graph that the hybridization improves the Pareto opti-mal solutions, and this set is considered as the global Pareto optimal solution set.

The Pareto-optimal solution set has 255 solutions. In order to illustrate alterna-tive representative design concepts, the K-mean clustering algorithm [28] is used. Ninedesigns are selected such that they represent different compromise solutions. Thevalues of the objective function and design variables for these designs are listed inTable 3. These nine Pareto-optimal solutions are also shown graphically in Figure 6.

It is clear from Figure 6 that solutions are selected uniformly along the Pareto-optimal front. Some useful results are observed from Table 3. For the same pin-finsection area shapes, longer pitch and shorter pin-fin height can yield better heattransfer performance. While the pitch and pin-fin height are fixed, it is found thatthe heat transfer rate is increased by decreasing both the major and minor pin-finaxis lengths. There is a strong effect of cross-interactions between different variables,

Table 2. Results of ANOVA and regression analysis

Model R R2 R2adj

FNu 0.977 0.954 0.907

Ff 0.997 0.995 0.989

F 0.979 0.958 0.916

Figure 6. Pareto-optimal solutions using NSGA-II, e-constraint, and clustering.

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which affects the optimal design. This prevents us from drawing any conclusionswith respect to the effect of individual parameters.

The global Pareto-optimal front is approximated by fitting the polynomial-based response surface to the available Pareto-optimal solutions. JMP [30] is usedto develop the response surface for the global Pareto optimal solutions. Thisresponse surface shows the interactions of the different objectives for the globaloptimal values.

It is clear from Figure 7 that the Pareto-optimal surface is a nonlinear andnonconvex curve. It is important to note that all the regions of the response surfacefunction do not represent the global Pareto-optimal front. The analysis of theresponse surface informs us that FNu increases with decreasing Ff. The gradient ofFf is decreased with an increase of FNu. One conclusion can be made, that is, only

Table 3. Objective functions and design variables for nine selected designs

Design variable

No. 2a=D 2b=D Pi=D H=D Ff Nua

1 0.7059 0.7059 4.5840 0.7188 2.9574 60.4482

2 0.7059 0.7059 4.7619 0.6789 3.3442 111.5395

3 1.1365 0.7059 4.6117 0.6863 2.5210 34.4923

4 0.9469 0.7059 4.6203 0.6863 2.8042 48.6361

5 0.8933 0.4195 2.1961 0.6863 3.1396 80.5374

6 0.7795 0.3688 2.1961 0.6863 3.5763 138.983

7 0.8435 0.7059 2.1961 0.6888 2.3728 25.7174

8 0.8435 0.7059 2.1961 0.6905 2.4204 29.6590

9 0.9831 0.5506 5.0960 0.6802 2.6542 44.3997

Figure 7. Global Pareto-optimal response surface.

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one optimal minimum value for Ff can be found for every value of FNu. It is alsoevident that for a good compromise among all the values, the values of Ff shouldbe in a range of 2.2–4.0, and the value of FNu in a range of 0.001–0.052. In showingthe trade-offs among all the objectives, useful information for designers is provided.

One (No. 9) of the Pareto-optimal solutions is selected from Table 3 as an opti-mum shape to compare the performance with those of the reference and circularshapes. The reference shape is that tested experimentally by Li et al. [3]. Cross-sectional area of the circular shape as well as Pi=D and H=D correspond with thoseof the optimum shape. A comparison of the optimum shape with the reference andthe circular shapes is provided in Table 4. The optimal shape gives average Nusseltnumbers that are 87.8% and 12.8% larger than those of the reference and the circu-lar shapes, respectively. This is consistent with the results obtained by Metzger et al.[31], suggesting that the heat transfer of oblong pin-fin arrays is greater than that ofcorresponding arrays with round pin-fin arrays. The same results were also obtainedby Li et al. [3]. On the other hand, with the optimization, the value of the friction-loss-related objective Ff also increased.

The computations were carried out using a personal computer with an IntelPentium IV CPU, 3.0 GHz. The computing time for a single flow analysis is in therange of 6–8 h, depending on the geometry considered and the convergence rate.

5. CONCLUSIONS

A multiobjective problem involving the shape optimization of pin-fin arraysin a three-dimensional channel has been solved by a hybrid technique based onevolutionary methods and local search methods of optimization coupled withReynolds-averaged Navier-Stokes analyses of flow and heat transfer. The aim wasto find an appropriate compromise between augmentation of heat transfer andreduction in friction loss. Response surface approximation was used to obtain thePareto-optimal front and to reduce the computational burden. The response surfaceobtained with 29 numerical experiments selected by D-optimal design for four designvariables was verified to be sufficiently reliable. For the same pin-fin section area,longer pitch and shorter pin-fin height yielded better heat transfer performance.On the other hand, for fixed pitch and pin-fin height, smaller major and minorpin-fin axis length delivered a higher heat transfer rate. The channel with an ellipticpin-fin showed better heat transfer characters than that with a circular pin-fin,while both shapes have the same section areas and other geometry parameters. Inshort, the multiobjective numerical optimization based on a combination of a

Table 4. Comparison of results between reference and optimum shapes

Design variable

2a=D 2b=D Pi=D H=D Nua Ff

Reference 1.255 0.706 4.706 1.00 23.64 1.91

Circle 0.706 0.706 5.096 0.680 39.35 1.98

Optimum (No. 9) 0.983 0.551 5.096 0.680 44.40 2.65

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weighted-sum strategy and evolutionary methods with local search methods using theresponse surface method combined with a RANS analysis provides a reliable andeconomic means of designing a heat-transfer channel with extruded pin-fin arrays.

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