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.