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260 Marco Manzan, Enrico Nobile, Stefano Pieri and Francesco Pinto (a) (b) Fig. 8.22 Example of secondary vortices for the 3D NURBS channels: (a) extrusion angle of 20 ◦ ;(b) extrusion angle of 40 ◦ f/f 0 20 15 10 5 2D linear−piecewise 2D NURBS 3D linear−piecewise 0 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Nu/Nu 0 Fig. 8.23 Pareto fronts: 2D-3D linear piecewise and 2D smooth NURBS profile Therefore, a second, simpler optimization has been carried out using 5 design variables, by fixing W. Moreover the objective ∆T has been transformed to a constraint, in order to simplify the optimization and to favour module geometries with lower surface area and lower pressure drop. The optimization has been carried out for 332 designs. At this point the process was practically converged. The Pareto front of this optimization is reported in Fig. 8.24 where it can be seen that the designs with the same inclination angle θ are rather perfectly aligned. A decrease of the inclination angle leads to lower
8 Multi-objective Optimization in Convective Heat Transfer 261 Surface Area Variation (%) Surface Area Variation (%) −0.5 −1 −1.5 −2 Pareto Front −2.5 −25 −20 −15 −10 −5 0 −1.7 −1.75 −1.8 −1.85 −1.9 −1.95 Pressure Drop Variation (%) o θ =42 (a) o θ =43 337 173 338 334 130, 335 134 o θ =44 −2 −20 −19 −18 −17 Pressure Drop Variation (%) (b) Fig. 8.24 Pareto fronts for the CC channel: ♦ feasible designs, + unfeasible designs, ◦ Pareto front, (a) Paretofronts,(b) designnumbers pressure drop and surface area as shown in Fig. 8.24(b) while maintaining the same heat transfer characteristics. This trend is maintained up to an angle of 42 ◦ , while for lower values the same heat transfer characteristics could not be maintained. These correspond to unfeasible results, since the value of the mean Nusselt number is lower than that of the original design 0. Among the designs that belong to the Pareto front, the design 134 has been chosen for illustrative purposes. In fact it has less than 19% pressure gradient β, and a surface area 2% lower than that of the original one while
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Optimization and Computational Flui
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Editors: Prof. Dr.-Ing. Dominique T
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vi Preface Our first research proje
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viii Contents 2.4.1 GoverningEquati
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x Contents 6.2.3 Parameterization..
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xii Contents 9.4.1 Multi-objectiveO
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xiv List of Contributors Gábor JAN
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Part I Generalities and methods
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4 Dominique Thévenin 12 10 8 6 4 2
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6 Dominique Thévenin Objective 10
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8 Dominique Thévenin Objective 10
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10 Dominique Thévenin Parameter 2
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12 Dominique Thévenin three-dimens
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14 Dominique Thévenin Let us come
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16 Dominique Thévenin References 1
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18 Gábor Janiga 2.1 Introduction D
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20 Gábor Janiga y Tinlet = 293 K 0
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22 Gábor Janiga y [mm] 25 20 15 10
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24 Gábor Janiga Table 2.1 Number o
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26 Gábor Janiga A dominates the in
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28 Gábor Janiga be shown later, th
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30 Gábor Janiga INPUT FILE Gambit
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34 Gábor Janiga ∆T ∆P Position
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36 Gábor Janiga ∆P [mPa] 2.2 2.0
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38 Gábor Janiga Table 2.3 Speed-up
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40 Gábor Janiga For all computatio
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42 Gábor Janiga y [mm] 4 2 0 -1 0
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44 Gábor Janiga Air mass flow-rate
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46 Gábor Janiga Temperature variat
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50 Gábor Janiga The modified k-ω
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52 Gábor Janiga Error for Re∗ =
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54 Gábor Janiga Error for Re∗ 20
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56 Gábor Janiga References 1. Ali,
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58 Gábor Janiga 43. Janiga, G., Th
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Chapter 3 Mathematical Aspects of C
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3 Mathematical Aspects of CFD-based
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Chapter 4 Adjoint Methods for Shape
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4 Adjoint Methods for Shape Optimiz
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Part II Specific Applications of CF
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112 Nicolas R. Gauger Nomenclature
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114 Nicolas R. Gauger Fig. 5.1 Cost
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116 Nicolas R. Gauger and the four
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118 Nicolas R. Gauger CL := 1 � C
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120 Nicolas R. Gauger the cost func
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122 Nicolas R. Gauger Table 5.1 An
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124 Nicolas R. Gauger The main adva
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126 Nicolas R. Gauger Spalart-Allma
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128 Nicolas R. Gauger (a) (b) Fig.
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134 Nicolas R. Gauger Fig. 5.10 Plo
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136 Nicolas R. Gauger drag, lift, s
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138 Nicolas R. Gauger solution of t
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140 Nicolas R. Gauger the presence
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142 Nicolas R. Gauger Fig. 5.16 Con
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144 Nicolas R. Gauger optimization
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Chapter 6 Numerical Optimization fo
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6 Numerical Optimization for Advanc
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192 Laurent Dumas 7.1 Introducing A
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194 Laurent Dumas C Fig. 7.2 Wake f
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196 Laurent Dumas Table 7.1 Example
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198 Laurent Dumas Fig. 7.5 Numerica
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200 Laurent Dumas 7.3.1.1 Genetic A
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202 Laurent Dumas START Random init
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204 Laurent Dumas • if ˜ J(x ng
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206 Laurent Dumas Table 7.3 Compari
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208 Laurent Dumas Fig. 7.9 General
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