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222 Marco Manzan, Enrico Nobile, Stefano Pieri and Francesco Pinto the result section, all shape-design alternatives become visible at once for the design engineer, and the set of optimized solutions is obtained in one optimization run, in contrast to classical single-objective methods. The use of the Pareto dominance concept has been illustrated, for heat transfer optimization problems, by Park et al. [33, 34]. In both studies, there were two variables to be minimized: the pressure drop and the thermal resistance of a plate exchangers [34], and of a heat-sink [33]. However, the multiobjective optimization problems were treated as a single objective problem, using weight coefficients. By performing a series of optimizations with varying weighting factors, the authors obtained an approximate Pareto front. 8.3 Problem Statement The problems considered in this work are: I – the multi-objective shape optimization of two-dimensional (2D) and three-dimensional (3D) convective wavy channels; II – the multi-objective optimization of Cross-Corrugated channels. The former is of fundamental nature and is of interest for heat exchangers and other heat transfer devices. The second problem has a practical significance since it represents the building block of many gas turbine recuperators [26]. For both problems, the study is limited to a single module at fully developed flow and heat transfer conditions. In such a circumstance, channels of periodic cross section form can be considered periodic in the flow and thermal fields as well. Therefore, the computational domain of interest becomes a single periodic module of the entire geometry as depicted in Figs. 8.1 and 8.2 for the 2D and 3D channels, respectively. CC heat exchanger is formed by stacking undulated plates at different inclination angles as reported in Fig. 8.3(a). The geometry obtained is presented in Fig. 8.3(b) while the repeating module is depicted in Fig. 8.3(c). It can be noticed that, unlike other authors, both hot and cold fluid domains are included [26, 27]. We have limited the study to the steady, laminar flow regime which is found in many practical circumstances. The Reynolds number chosen for the simulation is Re = 200 for both two-dimensional channel and CC module which corresponds to the typical value found in microturbine recuperators, while the Prandtl number is assumed as Pr = 0.7, representative of air and other gases. For computational convenience, a lower value of the Reynolds number, Re = 100, has been selected for the 3D wavy-channels.
8 Multi-objective Optimizatio Fig. 8.1 Periodic module of the two-dimensional wavy channel Fig. 8.2 Periodic module of the three-dimensional wavy channel 8.3.1 Governing Equations The flow is assumed to be incompressible, Newtonian, and of constant thermophysical properties. Moreover, we refer to a stationary and laminar forced convection with negligible viscous dissipation and buoyancy contribution. In these conditions, continuity, Navier-Stokes, and energy equations are written as follows: ∇·u =0 (8.1) ∇·(ρuu)=∇·(μ∇u) −∇p (8.2) ∇·(ρcpuT)=∇·(λ∇T) . (8.3)
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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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9 CFD-based optimization for paperm
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292 Index heat exchanger, 222, 224,
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