INTRODUCTION
The transfer function is a compact description of the
input-output relation linear time invariant dynamical system. It
provides information which specifies the behavior of the system, and is
used in the mathematical analysis of systems. Transfer functions appear
as relevant mathematical tools because they are usually easy for
modification and easy for solution. They need only dynamics
input–output data to represent even complex systems. One can say the
transform function is “a black box” that transforms input into output
signal. They can be applied to different systems however they are most
often employed to estimate the response of dynamical systems and
analysis of control systems. In the last field it was the primary tool
used in classical control engineering. In the mentioned areas the
concept of the transfer function is widely regarded as a powerful method
of dealing with complex systems.
Mathematically the transfer function is a function of complex variables.
It is obtained by inspection or by simple algebraic manipulations of the
differential equations that describe the systems. The transfer functions
can describe both simple one-dimensional systems and even infinite
dimensional systems governed by partial differential equations. A
complex model can be written very conveniently as an algebraic function
of transfer functions of subsystems. The model reconstruction comes down
to replacing one of the transfer function for subsystem by another one.
Multiplication of transfer function and input function and, next,
application of inverse Laplace transform algorithm results in the model
solution.
Many textbooks provide a detailed account of transfer function
derivation for simple, lumped parameters models. However chemical
engineering employs mainly distributed parameters models for fluid flow
and related processes description. Application of transfer function
concept for the cases was limited due to problems with derivation of
transfer function (differential equation should be solved) and with
inverse Laplace transform (classical methods cannot be applied). In
consequence few attempts have been made to investigate usefulness of the
transfer function in chemistry, additionally the attempts mainly based
on experimentally determined transfer functions what made possible to
omit the mentioned problems. In the widely understood chemical
engineering (without connections with control) there are really few
published works. Kupper et al.[1] analyzed the problem of mass
transport in discrete fracture networks. The authors developed a
transfer function approach to mass transport modeling as a basis for
comparing flowline routing or complete mixing models. The transfer
function for the single fracture was derived from an analytical solution
for a single fracture, while the intersection component was approximated
by either the complete mixing or the streamline routing model. Different
possible flow patterns at intersections were checked to develop an
acceptable model. The results indicated that there is possible to
develop a system analysis approach that could take the process of
diffusion into matrix into account for complex, two-dimensional fracture
networks. Marquez et al. [2] analyzed the heat processing of
products immersed in a low-viscosity medium (the experiments were done
with freshly harvested raspberries immersed in sucrose aqueous solution)
where convection is the main heat transfer mechanism. The transfer
function used in the paper had an assumed form (it was not a result of
mathematical manipulations). Transfer function coefficients were
calculated by simulating system evolution against a reference input
signal. The authors concluded that the method approximates very well all
practical situations of thermal treatment of raspberry preserves for the
commoner systems, it gives the possibility of adjusting thermal
treatments by allowing calculations and different sterilization policies
can be easily tested to find that leading to maximum quality parameters.
Sayyafzadeh et al.[3] proposed to use the transfer function to
provide an accurate forecast of hydrocarbon reservoir performance during
water flooding. They are proposed the method as a very good alternative
of another methods: (i) a method that requires less data to simulate
reservoirs but its outputs are not reliable in many cases, (ii) a
complex, high accuracy method, that is time-consuming and requires large
quantities of data. Different groups of transfer functions of subsystems
were checked to develop a combination which has physical meaning and
gives outputs with tolerable error. Four different cases were employed
to validate the derived model. So, in this work, the transfer function
concept was useful both for model development as well as simulations.
Similarly, to the last reference transfer function coefficients were
calculated based on experimental results. Kicsiny[4-5], Buzás and
Kicsiny[6] as a way to improve solar collectors to get full solar
systems efficiency as well as for better environmental protection
indicated mathematical modeling with the use of system transfer
functions. The transfer functions for solar heating systems with pipes
were determined based on a validated mathematical model. The transfer
functions were used for the dynamic analysis of a considered solar
heating system. Effect of inlet temperature as well as initial
temperature of collector were analyzed. The authors underline that the
worked-out transfer functions are relatively easy to apply for dynamic
analysis and stable control design. Ansorena and Di Scala[7],
Glavina et al.[8-9], presented transfer function approach for food
processing to estimate the thermal response of the system, e.g., to
analyze the evolution of temperature in the geometrical center of
potatoes during cooling and predict thermal histories of conductive
foodstuffs. The authors applied simple mathematics to describe the
processes behavior – only first or second order transfer functions
however they obtained good approximations between predicted and
experimental values. In the first paper the authors concluded that the
model provides a very good approximation to experimental data and the
main advantage of the transfer function is a much simpler mathematical
model for parameter estimation. In the second one, they pointed that the
model satisfactorily predicted the central temperature that is observed
as a result of different forcing signals in the surrounding temperature.
Moreover, the methodology could be applied to other fruits and
vegetables and to different cooling or heating processes. And finally,
in the last work authors analyzed both food and equipment thermal
responses during cooking and retorting operations. The results were very
satisfactory. The authors proposed this methodology as a potential
approach to the implementation of new strategies leading to more
efficient processes and to improve product quality. Application of
transfer function makes easier developing a transient model and this
property was explained the presented cases. Since the above examples
cover many various types of processes, therefore, it shows that the
concept of transfer function is not limited to specific applications
related to process control. However, the use of the transfer functions
has one more practical aspect – it compels us to use inverse Laplace
transform algorithms and, for more complex cases their numerical
versions. Numerical algorithms of inverse Laplace transform are a topic
with wide representation and many aspects. According to the literature
reports, many scientists have been recommended numerical algorithms of
inverse Laplace transform to find a solution in the time domain of a
specific type of problem. Particular algorithms are suitable for various
problems. A comprehensive review of earlier developed algorithms was
presented by Davies and Martin [10]. Other studies on inversion
methods based on precision of calculations and CPU time consumption are
given by Wójcik et al [11] and Escobar et al [12].
Derivation and validation of a high precision model of tracer flow in a
measurement system using transfer function is presented in this work. A
transfer function concept makes easier to build and re-build models of
complex systems and consequently allows for obtaining a model that
matches in the best way a physical system. Transfer function for a
system is based on theoretical model of gas flow. A system of partial
differential equations is converted into a transfer function and next, a
model solution is compared with the recorded outlet signal of a system.
The mentioned above problems with transfer function derivation and
solution of resulted Laplace domain model were minimized by application
of CAS-type program, as described below. From an optimal fit of
experimental and calculated signal curves dispersion coefficient is
obtained. Details of the method is presented in Wójcik [13]. The
method has an additional profit viz. - numerical inverse Laplace
transform algorithm - can be employed for solution of a model. Transfer
function models were solved using a relatively simple method i.e.
numerical inverse Laplace transform algorithm presented by den Iseger
[14]. Den Iseger method characterizes of a very broad scope for
application; recently it is one the most often cited method in
literature. The den Iseger’s algorithm was coded by authors in
Maple®. Besides of computing of inverse Laplace
transform, program Maple® was applied also for
derivation of transfer functions (mathematical capabilities of program
for analytic solutions of differential equation as well as integral
transform were used). For the problem under consideration application of
transfer function concept in combination with fast and precise algorithm
and CAS-type computer program resulted in development of high precision
mathematical model of the process and, consequently, determination of
precise gas flow parameters. The method is very precise and it is faster
comparing to methods previously described in literature.
Determination of dispersion coefficient is not mentioned as an
application of the system used (Micromeritics’ AutoChem 2950HP), so the
method under consideration is the simple way with no equipment design
changes and thus inexpensive technique to unlock the full potential of
yours measurement system. Flexibility of the method is its large
advantage – the model can be easily adjusted to other measurement
system as well as modified to determining of other parameters e.g.
effective diffusivity in the filled column etc. The method is a fast
alternative to computational fluid dynamics for high precision
calculations.