An Approximate Transfer Function Model for a Double-Pipe Counter-Flow Heat Exchanger

The transfer functions G(s) for different types of heat exchangers obtained from their partial differential equations usually contain some irrational components which reflect quite well their spatio-temporal dynamic properties. However, such a relatively complex mathematical representation is often not suitable for various practical applications, and some kind of approximation of the original model would be more preferable. In this paper we discuss approximate rational transfer functions G^(s) for a typical thick-walled double-pipe heat exchanger operating in the counter-flow mode. Using the semi-analytical method of lines, we transform the original partial differential equations into a set of ordinary differential equations representing N spatial sections of the exchanger, where each nth section can be described by a simple rational transfer function matrix Gn(s), n=1,2,…,N. Their proper interconnection results in the overall approximation model expressed by a rational transfer function matrix G^(s) of high order. As compared to the previously analyzed approximation model for the double-pipe parallel-flow heat exchanger which took the form of a simple, cascade interconnection of the sections, here we obtain a different connection structure which requires the use of the so-called linear fractional transformation with the Redheffer star product. Based on the resulting rational transfer function matrix G^(s), the frequency and the steady-state responses of the approximate model are compared here with those obtained from the original irrational transfer function model G(s). The presented results show: (a) the advantage of the counter-flow regime over the parallel-flow one; (b) better approximation quality for the transfer function channels with dominating heat conduction effects, as compared to the channels characterized by the transport delay associated with the heat convection

A GENERAL TRANSFER FUNCTION REPRESENTATION FOR A CLASS OF HYPERBOLIC DISTRIBUTED PARAMETER SYSTEMS

Results of transfer function analysis for a class of distributed parameter systems described by dissipative hyperbolic partial differential equations defined on a one-dimensional spatial domain are presented. For the case of two boundary inputs, the closed-form expressions for the individual elements of the 2×2 transfer function matrix are derived both in the exponential and in the hyperbolic form, based on the decoupled canonical representation of the system. Some important properties of the transfer functions considered are pointed out based on the existing results of semigroup theory. The influence of the location of the boundary inputs on the transfer function representation is demonstrated. The pole-zero as well as frequency response analyses are also performed. The discussion is illustrated with a practical example of a shell and tube heat exchanger operating in parallel- and countercurrent-flow modes

Approximate state-space and transfer function models for 2x2 linear hyperbolic systems with collocated boundary inputs

Two approximate representations are proposed for distributed parameter systems described by two linear hyperbolic PDEs with two time- and space-dependent state variables and two collocated boundary inputs. Using the method of lines with the backward difference scheme, the original PDEs are transformed into a set of ODEs and expressed in the form of a finite number of dynamical subsystems (sections). Each section of the approximation model is described by state-space equations with matrix-valued state, input and output operators, or, equivalently, by a rational transfer function matrix. The cascade interconnection of a number of sections results in the overall approximation model expressed in finite-dimensional state-space or rational transfer function domains, respectively. The discussion is illustrated with a practical example of a parallel-flow double-pipe heat exchanger. Its steady-state, frequency and impulse responses obtained from the original infinite-dimensional representation are compared with those resulting from its approximate models of different orders. The results show better approximation quality for the “crossover” input–output channels where the in-domain effects prevail as compared with the “straightforward” channels, where the time-delay phenomena are dominating

Mathematical Modeling and Simulation Analysis of a Bioreactor with Forced Aeration

Proces kompostowania jest powszechnie stosowany w gospodarce odpadami jako metoda przekształcania lub stabilizacji odpadów organicznych. Ze względu na złożony, nieliniowy charakter zachodzących w nim zjawisk biologicznych oraz fizykochemicznych, jest on stosunkowo trudny z punktu widzenia predykcji oraz sterowania. Sterowanie procesem ma tu na celu uzyskiwanie w określonym horyzoncie czasowym produktu finalnego, czyli zwykle kompostu spełniającego określone wymagania jakościowe. W artykule zaprezentowano prosty model matematyczny procesu kompostowania z wymuszonym napowietrzaniem, potencjalnie umożliwiającym realizację wspomnianego celu sterowania. Opracowano model drugiego rzędu, z dwiema wielkościami wejściowymi reprezentującymi zewnętrzne oddziaływania na proces. Na podstawie modelu matematycznego przekształconego do postaci równań stanu, zbudowano w środowisku MATLAB/Simulink model komputerowy bioreaktora, który następnie wykorzystano do przeprowadzenia badań symulacyjnych. Pokazano, że możliwe jest oddziaływanie na proces za pomocą wymuszonego napowietrzania, bezpośrednio wpływającego na zmianę temperatury w bioreaktorze, a w konsekwencji również na czas otrzymania końcowego produktu reakcji. Wyniki analizy właściwości dynamicznych procesu, przeprowadzonej z wykorzystaniem modelu zlinearyzowanego wzdłuż wybranej, nominalnej trajektorii stanu, wskazują na zmienny charakter jego stabilności - począwszy od niestabilności w początkowych fazach reakcji, przez stabilizację w fazie pośredniej, aż do stabilności asymptotycznej, zakończonej osiągnięciem stanu równowagi.The composting process is commonly used in waste management as a method of converting or stabilizing organic waste. Due to the complex, non-linear nature of biological and physicochemical phenomena involved, this process is relatively difficult to predict and control. The control is usually aimed at obtaining the final product, that is, the compost that meets legal standards. The article presents a simple mathematical model of the composting process with forced aeration, which will potentially facilitate the control task. A second order model was developed, with two inputs signals. Based on the mathematical model in the form of the state equations, the computer model of the bioreactor was built in the MATLAB/Simulink environment, which was then used to conduct different simulation tests. It was shown that it is possible to control the process using forced aeration, directly influencing the temperature changes in the bioreactor, and consequently also the time of obtaining the final product of the reaction. The analysis of the dynamic properties of the process performed using its model linearized about some nominal state trajectory shows the changes in its internal stability - starting from the unstable character in the initial phases of the reaction, through stabilization in its intermediate phase, up to the asymptotic stability, ending in the stable equilibrium state

Key Performance Indicators as a Tool for Production Process Assessment – Part II: Industrial Research

Zastosowanie nowych technologii w Przemyśle 4.0 umożliwia lepszą organizację, monitorowanie, kontrolę oraz skuteczną optymalizację procesów produkcyjnych, szczególnie w zakresie wydajności. Prezentowane rozwiązanie opiera się na hierarchicznej analizie wskaźników efektywności, w tym głównie na kontroli wskaźnika ogólnej efektywności zasobów produkcyjnych OEE. Rosnąca liczba możliwych do uzyskania skwantyfikowanych sygnałów monitorujących pracę maszyn, temperaturę otoczenia czy częstotliwość drgań sprawia, że narzędzia wspomagające decyzje są coraz bardziej wyrafinowane i, poza prezentacją obecnego stanu zasobów, coraz częściej obejmują także analizę predykcyjną. Opisywane narzędzie PUPMT pozwala zidentyfikować kluczowe zdarzenia, które mają istotny wpływ na bieżącą lub przyszłą efektywność produkcji. Umożliwia także analizę typu what-if, dopuszczając symulację wpływu projektowanych zmian, a wyniki tej symulacji uzależnia od skutków podobnych zmian, które miały miejsce w przeszłości w danym przedsiębiorstwie. Dzięki automatycznej identyfikacji potencjalnych zależności rozwiązanie dostosowuje się do specyfiki firmy lub wybranej jednostki produkcyjnej. Początkowe rozdziały zawierają m.in. opis najważniejszych metod wykorzystywanych w rozwiązaniu PUPMT. W dalszej części przedstawiono wybrane wyniki badań przemysłowych, które przeprowadzono na kilkudziesięciu jednostkach produkcyjnych.The use of new technologies in Industry 4.0 enables better organization, monitoring, control and effective optimization of production processes, especially in terms of efficiency. The solution is based on a hierarchical analysis of key performance indicators, including mainly the control of Overall Equipment Effectiveness (OEE). The growing number of quantifiable signals monitoring machine operation, ambient temperature or even the frequency of vibrations makes decision support tools more and more sophisticated. Moreover, they also include predictive analysis in addition to presentations of the current state of resources. PUPMT tool allows identifying key events that have a significant impact on current or future production efficiency. It also allows the what-iftype analysis, running the simulation of the impact of the proposed changes, and the results of this simulation depend on the effects of similar changes that occurred in the past in a given enterprise. Thanks to the automatic identification of potential dependencies, the proposed solution adapts to the specifics of a given company or even a selected production unit. The paper in the first part contains a description of the essential methods used in the PUPMT tool. The second part presents selected results of industrial research, which were carried out on several dozen production units

Modeling heat distribution with the use of a non-integer order, state space model

A new, state space, non-integer order model for the heat transfer process is presented. The proposed model is based on a Feller semigroup one, the derivative with respect to time is expressed by the non-integer order Caputo operator, and the derivative with respect to length is described by the non-integer order Riesz operator. Elementary properties of the state operator are proven and a formula for the step response of the system is also given. The proposed model is applied to the modeling of temperature distribution in a one dimensional plant. Results of experiments show that the proposed model is more accurate than the analogical integer order model in the sense of the MSE cost function