Optimization and Computational Fluid Dynamics - Department of ...

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