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270 J. Hämäläinen, T. Hämäläinen, E. Madetoja, H. Ruotsalainen Header Fig. 9.2 Paper machine headbox Header Inflow Slice channel Fig. 9.3 Cross-directional flows inside a headbox due to uneven flow distribution coming from the tapered header water mixture coming from a pump is turned 90 ◦ into the machine direction and distributed inside the header as illustrated in Fig. 9.3. An uneven flow rate distribution may cause an uneven basis weight profile in the paper web produced. Inside the headbox, a non-optimal header may also be a potential source of cross-directional flows which in turn causes misaligned fibers in the paper since fibers approximately follow the velocity vectors at the outlet jet of the headbox. Optimization is utilized to minimize these disturbances. The design of the tapered header has been identified as an optimal shape design problem. A cost function describes how even the flow rate distribution after the header should be and design variables define the location of the back Slice channel Slice opening
9 CFD-based optimization for papermaking 271 Optimized Fig. 9.4 Initial and optimized back wall of the tapered header Initial wall of the header. The first numerical experiments in header optimization were reported in the early 1990’s [10, 11]. Modeling of the headbox flows includes certain special features. First, the tube bundles (see Figs. 9.2 and 9.3) consist of hundreds of small tubes. They cannot be included in detail in CFD but are taken into account as an effective porous medium. A three-dimensional (3D) CFD model would also be too time-consuming for optimization. Hence, specific two-dimensional (2D) models have been developed for the headbox applications [12]. The header model is derived from a 3D, incompressible, k–ε turbulence model by averaging the equations in the vertical direction, which results in a non-standard 2D flow model. A similar approach has also been studied in [33] for open-channel problems. In addition to model reductions, optimization methods also have a significant influence on computing efficiency. In general, gradient-based methods have proven to be more efficient than gradient-free methods for the optimization problems introduced in this paper. The most critical step in gradientbased optimization algorithms is the evaluation of the cost function gradient. Finite difference approximation is easy to obtain, even for complex models, but on the other hand, it is inefficient. The so-called adjoint state technique has also been studied for the tapered header [13] but only for laminar or algebraic turbulence models. Evaluation of the cost function gradient can be avoided by using genetic algorithms [4, 14]. Nevertheless, at least in papermaking applications, gradient-based methods are much faster than genetic algorithms, even when the gradient is approximated by finite differences. In genetic algorithms, a large set of uninteresting solutions has to be calculated in order to find the interesting ones. The designing software tool for optimization of the tapered header was introduced in the industry in 1995. The design procedure takes place only once a week, and thus, a CPU time of several hours is acceptable. It is also justified to utilize a two-equation turbulence model, together with finite difference approximation of the cost function gradient, instead of a fast solver based on an algebraic turbulence model and the adjoint state technique. Optimal shape design of the header has been utilized in the design process by the industry for a decade [15]. One example of the initial and optimized
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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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