Paper Technology International 2020 - Journal - Page 76

PAPERTECHNOLOGYINTERNATIONAL
Conventional techniques of empirical modelling, however,
are linear statistical techniques. These tend to have serious
limitations because nothing in nature is very linear, and particularly
so in process engineering and materials science. It therefore makes
sense to use better techniques of empirical and semi-empirical
modelling which take nonlinearities into account.
Nonlinear modelling
There is hardly any material behaviour which is absolutely
linear. It is therefore wise to treat the nonlinearities rather than
ignore them. The proponents of linear techniques draw on their
simplicity and the possibility of adding nonlinear terms in linear
regression. Often this is not done, and is not efficient even if it is
done. Nature does not follow the simplicities that we try to fit it in,
using common linear techniques.
Nonlinear modelling is empirical or semi-empirical modelling
which takes at least some nonlinearities into account. Nonlinear
modelling can be carried out with a variety of methods. The older
techniques include polynomial regression, linear regression with
nonlinear terms and nonlinear regression. These techniques have
several disadvantages compared to the new techniques of nonlinear
modelling based on free-form nonlinearities.
The newer methods like feed-forward neural networks and
series of basis functions do not require a priori knowledge of the
nonlinearities in the relations. Among these new techniques, feedforward neural networks have turned out to be particularly valuable
in chemical engineering [3] and materials science. Feed-forward
neural networks have several features which make them better
tools for nonlinear empirical modelling. Besides their universal
approximation capability [4], it is usually possible to produce
nonlinear models with some extrapolation capabilities with feedforward neural networks.
There are many different types of neural networks, and
some of them have practical uses in process industries. Neural
networks have been in use in process industries for about 30 years.
The multilayer perceptron, a kind of a feed-forward neural network,
is the most common one. Most neural network applications in
industries are based on them [5].
Feed-forward neural networks resemble structurally and to
a smaller extent functionally the networks of neurons in biological
systems. Like the networks of neurons in the brains, artificial neural
networks also consist of neurons in layers directionally connected to
others in the adjacent layers (see Figure 3).
Figure 3: A typical feed-forward neural network.
74
In a feed-forward neural network of the kind shown in Figure 3, the
output of each neuron i in the feed-forward neural network is usually
given by

N
z i = σ ∑ wij x j

j =0
where σ is called the activation function, and is usually the logistic
sigmoid, given by
σ (a ) =
1
1 e −a
The incoming signals to the neuron are xj, and wij are the
weights for each connection from the incoming signals to the ith
neuron. The wi0 terms are called biases. Then the output is simply
calculated as the weighted sum of the outgoing signals zi from
the neurons in the hidden layer. This results in a set of algebraic
equations which relate the input variables to the output variables.
Thus, for each observation (a set of input and output variables), the
outputs can be predicted from these equations based on a given set
of weights. The training procedure aims at determining the weights
which result in the smallest sum of squares of prediction errors.
Today, most people use good optimisation methods for that purpose.
Nonlinear modelling in process engineering
Nonlinear modelling has been utilised successfully for various
industrial sectors including plastics and rubbers, metals, cement
and concrete, medical materials, semiconductors, ceramics,
mineral wools, glass, power generation, biotechnology, pulp and
paper, etc. Different processes have different characteristics different raw materials, different compositions, and are produced
by different batch, continuous or fed-batch processes. However,
some things are common to modelling of various kinds of processes.
Material properties or product properties, production rate and
production economics depend on composition variables (or feed
characteristics), process variables and dimension variables, as as
summarised in Figure 4.
For process development, one would like to determine
the best values of composition variables (or feed characteristics),
process variables and/or dimension variables such that the resulting
product properties will be within
desired limits, with a good production
rate or at a minimal production cost.
Sometimes, feed characteristics
or composition variables might be
constants. In more common situations,
the process variables may be constant
or dependent variables, and the
only degrees of freedom in materials
development may be the composition
of the feed, the amounts of raw
materials and possibly dimension
variables.

Paperturn flipbook