Abstract
The dynamic models of elements of technological systems with perfect mixing and plug-flow hydrodynamics are based on the systems of algebraic and differential equations that describe a change in the basic technological parameters. The main difficulty in using such models in MathWorks Simulink™ computer simulation systems is the representation of ordinary differential equations (ODE) and partial differential equations (PDE) that describe the dynamics of a process as a MathWorks Simulink™ block set. The study was aimed at developing an approach to the synthesis of matrix dynamic models of elements of technological systems with perfect mixing and plug-flow hydrodynamics that allows for transition from PDE to an ODE system on the basis of matrix representation of discretization of coordinate derivatives. The process of synthesis of the dynamic matrix mathematical model was considered by the example of a sugar syrup cooler, the quality indicator of the finished product are selected as sucrose crystals and their portion in the total volume of caramel mass. Taking into account the dependence of syrup viscosity on temperature, thermal effects as a result of the process of crystallization of sucrose from syrup, design features of a typical caramel machine made it possible to clarify the dynamics of the process of syrup cooling. The model developed with this approach allows to obtain real-time estimates of temperatures at the outlet of the cooler, which makes it possible to study the dynamics of the technological process and synthesize the control system. The presented approach allows to implement mathematical models of ideal reactors in Simulink system and to move to matrix ordinary differential equations, which makes it possible to convert them into Simulink blocks. The approach is also applicable to other models of ideal reactors, which allows to form libraries of typical ideal reactors of Simulink system for synthesis of heat and mass exchange equipment. The proposed approach significantly simplifies the study and modernization of the current and the development of new technological equipment, as well as the synthesis of algorithms for controlling the processes therein.
Highlights
The dynamic models of elements of technological systems with perfect mixing and plug-flow hydrodynamics are based on the systems of algebraic and differential equations that describe a change in the basic technological parameters
The main difficulty in using such models in MathWorks SimulinkTM computer simulation systems is the representation of ordinary differential equations (ODE) and partial differential equations (PDE) that describe the dynamics of a process as a MathWorks SimulinkTM block set
The study was aimed at developing an approach to the synthesis of matrix dynamic models of elements of technological systems with perfect mixing and plug-flow hydrodynamics that allows for transition from PDE to an ODE system on the basis of matrix representation of discretization of coordinate derivatives
Summary
Хвостов А.А., Журавлев А.А., Шипилова Е.А., Сумина Р.С., Khvostov A.A., Zhuravlev A.A., Shipilova E.A., Sumina R.S., Магомедов Г.О., Хаустов И.А. В случае систем обыкновенных дифференциальных уравнений (ОДУ) используется блок интегрирования входного сигнала и библиотека численных методов MathWorksTM, а также методика представления ОДУ или системы ОДУ в виде структурной модели Simulink [5, 6]. В случае дифференциальных уравнений в частных производных (ДУЧП), которыми описываются, например, потоки идеального вытеснения, уравнение теплопроводности и т. Используемых при переходе от ДУЧП к ОДУ, заключается в применении интегральных преобразований, например, преобразований Фурье, Лапласа [7, 8] и избавлении от производных по пространственной координате. Один из подходов основан на дискретизации только пространственной переменной методом конечных разностей, при этом производные по времени остаются непрерывными и ДУЧП представляется в виде задачи Коши для системы ОДУ [10]. Реализация каждого уравнения системы ОДУ приводит к довольно громоздким структурным моделям в Simulink, и актуальным является компактное представление таких систем в матричном виде [11–13]. Актуальной задачей является разработка способа представления типовых идеальных реакторов с помощью матричных обыкновенных дифференциальных уравнений (МОДУ) и реализация с их помощью структурных моделей в Simulink
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