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274 J. Hämäläinen, T. Hämäläinen, E. Madetoja, H. Ruotsalainen Angle (˚) CD Position (m) Measured Estimated Optimized Fig. 9.6 Measured, estimated (Step 1) and optimized (Step 2) fiber orientation profiles optimal control for the slice opening profile in order to obtain an even fiber orientation angle, that is, with angles equal to zero. When the optimal slice opening profile is found, it can be fed into actual slice opening control system of the paper machine. Nowadays, the HOCS Fiber simulator is installed on the papermaking engineer’s laptop, and hence the necessary control corrections can be calculated and implemented at the paper mill. Despite the model reductions, the accuracy of the HOCS Fiber has been confirmed: typically, for the first paper machine start-up, only one slice opening tuning proposed by the software is needed. HOCS Fiber has been successfully used at dozens of paper mills and one example of a real trouble shooting case is given in Fig. 9.6. As seen in the figure, the fiber orientation angle is much better after optimization than the original one measured at the mill. The original fiber orientation profile may cause problems in printing machines, but the optimized profile fulfills all the market requirements. 9.3.3 Depth-averaged Navier-Stokes Equations In order to be able to optimize the fiber orientation for trouble shooting purposes, certain model simplifications have to be carried out. First of all, the FOPD model cannot be used. Instead, the expected value of the FOPD is assumed to be determined by the mean velocity field. Furthermore, instead
9 CFD-based optimization for papermaking 275 of using the 3D geometry of the headbox slice channel, the depth-averaged Navier-Stokes equations are used. These equations are introduced next. Let the velocity vector U =(u,v,w) and the static pressure p be a solution of the 3D Navier-Stokes equations. The depth-averaged pressure P and the velocity components U and V are defined as averaged values in the vertical direction over the depth of the slice channel as follows U (x, z)= V (x, z)= P (x, z)= 1 D (x, z) 1 D (x, z) 1 D (x, z) � D(x,z) 0 � D(x,z) 0 � D(x,z) 0 u (x, y, z)dy v (x, y, z)dy p (x, y, z)dy (9.1) where D =(x, z) is the depth of the slice channel depending on a position (x, z). Then, by integrating the 3D continuity equation over the depth and by using the definitions (9.1), the following depth-averaged continuity equation is obtained ∇·(DV ) = 0 (9.2) where V =(U,V ) is the reduced 2D velocity vector. The depth-averaged momentum equation is derived similarly, and it is where − 1 D ∇·�2μDε (V ) � + ρC(m)V ·∇(DV ) � 4(m +1)μ + D2 � V + ∇P =0 C (m)= 2m +2 2m +1 (9.3) , m ≥ 2 (9.4) is a coefficient associated with the velocity profile in the depth direction. For example, m = 2 corresponds to a parabolic velocity profile, and m =7to a turbulent plug flow profile. See also [33] for depth-averaging of the k–ε turbulence model.
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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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210 Laurent Dumas Table 7.4 Four ex
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212 Laurent Dumas Fig. 7.13 Blades
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214 Laurent Dumas Table 7.5 Converg
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Chapter 8 Multi-objective Optimizat
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8 Multi-objective Optimization in C
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8 Multi-objective Optimization in C
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