Optimization and Computational Fluid Dynamics - Department of ...

284 J. Hämäläinen, T. Hämäläinen, E. Madetoja, H. Ruotsalainen
optimization algorithm. This kind of approach has been used in the following
kind of paper quality optimization examples where the statistical modeling
techniques have been involved [23].
9.4.3 Numerical Examples
Next we present two numerical examples, where the virtual papermaking line
was utilized. Both examples were solved using appropriate gradient-based optimizer
and gradients were evaluated with the help of the technique described
above.
9.4.3.1 Example 1
This example illustrates the advantages of multi-objective optimization compared
with trial-and-error simulation. We studied here a conflict between two
paper quality properties, formation and tensile strength ratio. Formation is
a small scale weight variation of a paper sheet and tensile strength ratio is
the machine-directional tensile strength divided by the cross-directional one.
Bothofthesepropertiesweretobeminimized, but because they were in conflict
they could not reach their optimum at the same time. These conflicting
targets were simulated using the above presented virtual papermaking line
combining dissimilar unit-process models from different disciplines. The virtual
papermaking line used included the CFD models, and there were also
models for moisture and heat transfer, and naturally statistical models for
paper quality properties were involved.
By doing trial-and-error simulations, a number of solutions were obtained
as can be seen on the left-hand side of Table 9.1. These solution were obtained
by using typical machine control values, which were determined using socalled
engineering knowledge. That is, a person with expertise in papermaking
provided the control variable values used in simulation. Based on his/her
knowledge, she/he tried to find a solution which would fulfill the preferences.
As can be seen in Table 9.1, the person made some simulations and obtained
different kinds of solutions, but she/he could not be sure if any of these
solutions was optimal. Trial-and-error simulations could have continued for
as long as the expert had time or a satisfactory solution was obtained.
Instead we used a multi-objective optimization method to search the optimal
control variable values. In this way, much better solutions were found because
the method obtained only Pareto optimal solutions. Figure 9.13 shows
all the solutions obtained (simulated and optimized). A few Pareto optimal
solutions are presented in Table 9.1 on the right-hand side. As one can
see, all the solutions were better than the solutions obtained by simulation.