Optimization and Computational Fluid Dynamics - Department of ...

9 CFD-based optimization for papermaking 281
9.4.2 Modeling and Optimizing the Complete
Papermaking Process
Although accurate simulation of the entire paper machine is still far ahead,
a virtual papermaking line, as we call it, has been developed [20, 22]. It can
be used for papermaking simulation as well as for tailored multi-objective
optimization. The virtual papermaking line combines dissimilar unit-process
models from different disciplines that include mathematical formulas ranging
from simple algebraic equations to CFD models. It also includes models for
moisture and heat transfer, and for paper quality properties. Simplifications
of computationally expensive models are used, for instance, in water removal
which means that an accurate multi-phase flow model of the dewatering phenomenon
is replaced with a statistical model based on data produced by the
accurate model. The latter models are especially useful in optimization, when
tens, hundreds or even thousands of simulation model evaluations are needed.
We define the virtual papermaking line model as follows
⎧
⎪⎨
⎪⎩
A1(p 1,q 1)=0
A2(p 2,q 1,q 2)=0
.
Anm(p nm,q 1,...,q nm)=0
(9.6)
where Ai for all i =1,...,nm stand for the unit-process models, p i ∈ R li
denotes a vector of the inputs, and q i ∈ R ki is a vector of the outputs (model
states) for the i-th unit-process model Ai. We assume here that all the unitprocess
models and their outputs are continuously differentiable or if there is
nonsmoothness involved, we assume at least H-differentiability [7].
Based on (9.6) multi-objective optimization problems related to papermaking
process can be determined. Instead of the problem formulation (9.5)
we use the following model-based optimization problem formulation
Optimize {f1(x,q 1,...,qnm),...,fnf(x,q 1,...,qnm)} x
�
(9.6)
subject to
x ∈ S
(9.7)
where x ∈ S is a vector of the continuous decision variables (also called
control or design variables) which is a selected set of the unit-process model
inputs p =(p 1 ,...,p nm ) T . The feasible set S is defined by the box constraints
of the decision variables and the linear and non-linear constraint functions
similarly to (9.5). In (9.7), we ignore those input parameters that are not chosen
as decision variables, because they are constants during the optimization
process. By f1(x, q 1,...,q nm),...,fnf(x, q 1,...,q nm) we denote the objectives
which are to be optimized, i.e., minimized or maximized. Similarly to