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180 René A. Van den Braembussche Fig. 6.26 Penalty function for negative loading and loading unbalance in a compressor with splitter vanes loading and is proportional to the area between the suction and pressure side when the pressure side Mach number is higher than the suction side one (Fig. 6.26). Areas of negative loading are penalized because they result in extra friction losses without contribution to the pressure rise Pnegativeloading = � 1 0 max(Mps(s) − Mss(s), 0.0) · ds The second contribution to the Mach penalty increases with the load unbalance between main blade and splitter blade. This penalty compares the area between the suction and pressure side Mach number distribution of main blade Abl and splitter blade Asp, corrected for the difference in blade length (Fig. 6.26): Pload unbalance = � Abl − Asp Abl + Asp 6.6.3 Design Conditions and Results The computational domain starts at constant radius in the radial inlet and ends in the parallel vaneless diffuser at R/R2 = 1.5 (Fig. 6.22). Part of the hub surface at the inlet (for R
6 Numerical Optimization for Advanced Turbomachinery Design 181 Table 6.4 List of penalty parameters and weight factors stress penalty weight 24.0 mass flow difference weight 20.0 efficiency required 82.5 Mach negative loading weight hub 15.0 efficiency penalty weight 40. Mach negative loading weight shroud 24.0 mass flow required 20.0 g/s Mach loading unbalance weight 1.0 mass flow penalty weight 18.0 Mach loading unbalance weight 1.5 The disk thickness at the compressor rim is 1 mm. It is connected to a 8 mm diameter shaft with a fillet of 2 mm radius, all made of one piece. A fillet radius of 0.25 mm is applied at the blade hub to limit the local stress concentrations. The unshrouded impeller has a tip clearance of 0.1 mm, which is 10% of the exit blade height. This is typical for these small impellers and one of the reasons for the moderate efficiencies. The weights in the OF depend on the application and allow emphasis on performance or on mechanical integrity. They have been determined from the knowledge gained in previous optimizations and are listed in Table 6.4. Taking into account the difference in weight factors, an efficiency drop of 1% is as penalizing as an excess in stress limit of 40/24 = 1.66% (or 6.66 MPa). The optimization starts from a “baseline” impeller which is the result of a simple aerodynamic optimization without stress analysis. Although this geometry has a good efficiency, it cannot be used because an FEA predicts von Mises stresses in excess of 750 MPa. It serves as a reference for further optimizations. The TRAF3D solver [3] with an extension to calculate impellers with splitters is used to predict the aerodynamic performance of the radial impellers. Structured H-grids with 2 × 216 × 48 × 52 (1,090,000 cells) are used for all computations to guarantee a comparable accuracy for all the samples stored in the database. All computations are non-adiabatic with wall temperature fixed at 400 K, as found in a previous study on the heat transfer from the turbine to the compressor [19]. The commercial code SAMCEF [14] is used for the stress calculation. Quadratic tetrahedral elements are used as a compromise between element quality and automatic meshing. Similar grids with 250,000 nodes and 160,000 elements are used for all samples. The grid is refined in areas of stress concentrations. Periodic boundary conditions are applied, such that only a 1/7 th part of the geometry needs to be analyzed. The maximum allowable stress is a function of the material temperature. A large safety margin, to account for vibrations and possible local temperature peaks, results in a maximum allowable value of 400 MPa. Eight individual ANNs are used: one to predict the efficiency, two to predict the mass flow in the channels on each side of the splitter, 4 ANNs predict the Mach number distribution respectively at hub and shroud of the full and splitter blades and one predicts the maximum stress in the geometry. This
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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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Page 206 and 207: 192 Laurent Dumas 7.1 Introducing A
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Page 226 and 227: 212 Laurent Dumas Fig. 7.13 Blades
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8 Multi-objective Optimization in C
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Chapter 9 CFD-based Optimization fo
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9 CFD-based optimization for paperm
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292 Index heat exchanger, 222, 224,
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