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Modelling and Optimization of Complex Technological Processes
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IFAC PapersOnLine 51-30 (2018) 646–649 Modelling and Optimization of Complex Technological Processes Modelling and Optimization of Complex Technological Processes Modelling of Technological Modelling and and Optimization Optimization of Complex Complex Technological Processes Processes Nataliya V. Mokrova* Modelling and Optimization of Complex Technological Processes Nataliya V. Mokrova*
Nataliya Nataliya V. V. Mokrova* Mokrova* * Moscow State University of Mokrova* Civil Engineering, Moscow, Nataliya V. * Moscow State University of Civil Engineering, Moscow, Russia (e-mail: [email protected]) ** Moscow University of Moscow State State of Civil Civil Engineering, Engineering, Moscow, Moscow, RussiaUniversity (e-mail: [email protected]) RussiaUniversity (e-mail: [email protected]) [email protected]) * Moscow State of Civil Engineering, Moscow, Russia (e-mail: Russiaanalysis (e-mail:for [email protected]) Abstract: The methodology of system mathematical model of the complex technological Abstract: The methodology of system analysis for to mathematical of theofcomplex technological process is considered. The mathematical description process finemodel purification cobalt solutions from Abstract: The methodology of system analysis for mathematical model of the complex technological Abstract: The methodology of system analysis for of mathematical model of of thethe technological process iscopper considered. Theinmathematical description to process fine purification ofcomplex cobalt solutions from iron and discussed this article. The results numerical integration system of differential process is The considered. The mathematical description process fine purification cobalt solutions from Abstract: methodology of system analysis for to mathematical model of theof technological process considered. The description to process fine purification ofcomplex cobalt from iron andiscopper discussed inmathematical thisoptimal article.concentrations The results of numerical integration of the systemsolutions of differential equations can be used to obtain of reagents. The introduced models can be applied iron and copper discussed in this article. The results of numerical integration of the system of differential process is considered. The mathematical description to process fine purification of cobalt solutions from iron and copper discussed in this article. The results of numerical integration of the system of differential equations can be usedand to obtain optimal concentrations of reagents. The introduced models can be applied for control problems predicting future states of sophisticated technological systems. equations can be used to obtain optimal concentrations of reagents. The introduced models can be applied iron and copper discussed in this article. The results of numerical integration of the system of differential equations be usedand to obtain optimal concentrations of reagents. The introduced models can be applied for controlcan problems predicting future states of sophisticated technological systems. for control problems and predicting future states technological systems. equations can be used to obtain optimal of reagents. The introduced canreserved. be applied © 2018, IFAC (International Federation ofconcentrations Automatic Control) Hosting by Elsevier Ltd.models Allprocess. rights for control problems and predicting future states of of sophisticated sophisticated technological systems. Keywords: system analysis, results of mathematical simulation, multistage purification for control problems and predicting future states of sophisticated technological systems.process. Keywords: system analysis, results of mathematical simulation, multistage purification Keywords: system analysis, analysis, results results of of mathematical mathematical simulation, simulation, multistage multistage purification purification process. Keywords: system process. Keywords: system analysis, results of mathematical simulation, multistage purification process. The methodology of system analysis revealed the levels of a 1. INTRODUCTION The methodology of system analysis revealed the levels of a hierarchical structure of the complex technological system. 1. INTRODUCTION The methodology of system analysis revealed the levels of The methodology of system analysis revealed the levels of aa 1. INTRODUCTION hierarchical structure of the complex technological system. These levels structure detailing from the molecular one and ending with 1. INTRODUCTION The most of technological process of industrial production hierarchical of the complex technological system. The methodology of system analysis revealed the levels of a hierarchical structure of complex system. 1. These levels estimates detailing from the molecular one interrelationship and ending with The most of technological process of industrial production the integral takethe into accounttechnological the couldmost be considered as INTRODUCTION a complex process full of variables and These levels detailing from the molecular one and ending with The of technological process of industrial production hierarchical structure of the complex technological system. These levels detailing from the molecular one and ending with The most of Scientific technological process of industrial production integral estimateslevels take into account the each interrelationship could be considered as a research complex process full of variables and the between individual of molecular the system, of them is connections. involves studying a related integral estimates take into account the could be considered as complex process full of and These levels detailing from one interrelationship and ending with The most of technological process of industrial production the integral estimates take the into account the interrelationship could beinterrelated considered as aa research complex process full of variables variables and the between individual levels ofmathematical the system, each of themAn is connections. Scientific involves studying a related characterized by relevant description. class of processes under the common regularities. individual of the system, of them is connections. Scientific studying aa related the integral estimateslevels take into account the each interrelationship could be considered as a research complex involves process full of variables and between between individual levels ofmathematical the system, each ofdetect themthe is connections. Scientific research involves studying related characterized by relevant description. An class of interrelated processes under thesimulation common regularities. adequate mathematical description allows us to The process of mathematical and models of by relevant description. class of interrelated processes under the common regularities. between individual levels ofmathematical the system, each of themAn is connections. Scientific research involves studying a related characterized characterized by relevant mathematical description. An class of interrelated processes under the common regularities. mathematical description allows us toand detect the The process of must mathematical and simulation models of adequate ways howmathematical to manage processes moreallows efficiently address complex systems be iterated numerous times,regularities. and at each adequate description us to detect the The process of mathematical and simulation models of characterized by relevant mathematical description. An class of interrelated processes under the common description us toand detect the The process of must mathematical and simulation models of adequate ways howmathematical to issues. manage processes moreallows efficiently address complex systems be iterated numerous times, and at each optimization stage its attributes are refined. The priorities of modelling are how to manage processes more efficiently address complex systems be iterated numerous times, and at each adequate mathematical description allows us toand detect the The process of must mathematical and simulation models of ways ways how to manage processes more efficiently and address complex systems must be iterated numerous times, and at each optimization issues. stage itsthe attributes are refined. The priorities of modelling are set and ways to build newlyThe built objectstimes, aremodelling synthesized. optimization issues. stage its attributes are refined. priorities of are ways how to manage processes more efficiently and address complex systems must be iterated numerous and at each issues. ANALYSIS METHODOLOGY stage itsthe attributes refined. priorities of are optimization set and ways toare build newlyThe built objectsreflects aremodelling synthesized. The scheme of mathematical description the most 2. SYSTEM set and ways to build newly built objects are synthesized. issues. stage itsthe attributes are refined. The priorities of modelling are optimization set and the ways to build newly built objects are synthesized. The scheme of mathematical description reflects the most 2. SYSTEM ANALYSIS METHODOLOGY essential features of the newly object or objects process andsynthesized. allows to The scheme of mathematical description reflects the most 2. SYSTEM ANALYSIS METHODOLOGY set and the ways to build built are The scheme of mathematical description reflects the most 2. SYSTEM ANALYSIS METHODOLOGY The basis of the system analysis is decomposition of a complex essential features of the object or process and allows to understand theof mechanism of the phenomenon andallows make it essential features of the object or process and to The scheme mathematical description reflects the most 2. SYSTEM ANALYSIS METHODOLOGY The basis of the system analysis is decomposition ofquantitative a complex essential features of the object or process and allows to system into subsystems and establishment of understand the mechanism of the phenomenon and trend. make it The possible to forecast changes in it and its development basis of system is of aa complex The basisinto of the the system analysis analysis is decomposition decomposition ofquantitative complex understand the mechanism of the phenomenon make it essential features of the object or process andand allows to system subsystems and establishment of understand the mechanism of the phenomenon and make it relationships between them. The subsystems are possible to forecast changes in it and its development trend. system subsystems and establishment of The basisinto of the system analysis is decomposition ofquantitative a allocated complex possible to forecast changes in it and its development trend. system into subsystems and establishment of quantitative understand the mechanism of the phenomenon and make it relationships between them. ofThe subsystems are allocated possible to forecastdescription changes in scheme it and itsobtained development The mathematical in thetrend. course relationships depending the complexity the subsystems object under investigation between them. allocated system intoon subsystems and The establishment ofare quantitative relationships between them. The subsystems are allocated possible to forecast changes in it and its development trend. The mathematical description scheme obtained in the course depending on the complexity ofprocess the object underand investigation of modelling together with the parameters identified as acourse result depending and on theon previous study ofof physical chemical The mathematical description scheme obtained in the complexity the object under investigation relationships between them. The are allocated The mathematical description scheme obtained inofthe the the complexity the subsystems object under investigation modelling together with the parameters identified as acourse result depending and on theon previous study or ofofprocess physical and chemicala of interpolation, reflect the most essential features the object regularities, availability possibility of obtaining of modelling together with the parameters identified as a result and on the previous study of process physical and chemical The mathematical description scheme obtained in the course depending on the complexity of the object under investigation of modelling together the parameters identified result and on the previous study of of process physical chemicala of interpolation, reflectwith thetomost essentialthe features of as theaobject regularities, availability or possibility of and obtaining or interpolation, process and allow usthe understand mechanism of the regularities, mathematical description each stage. The independent of reflect most essential features of the availability possibility of obtaining modelling together with the parameters identified as aobject result and on the previous study or of process physical and chemicalaa of interpolation, reflect the most essential features of the object regularities, availability or possibility of obtaining or process and allow us to understand the mechanism of the mathematical description of each stage. The independent phenomenon and make ittopossible to forecast changes in the mathematical consideration of each subsystem with its I/O flows appearsa or process and allow us understand the mechanism of description of each stage. The independent of interpolation, reflect the most essential features of the object regularities, availability or possibility of obtaining or process andand allow us understand the Different mechanism of description of each stage. The independent phenomenon make ittopossible totrend. forecast changes in the the mathematical consideration of each subsystem with itsand I/O flows appears and its development models, possible to estimate the potential of levels subsystems and phenomenon make to forecast changes in consideration of each subsystem its I/O appears or process andand allow us it topossible understand the mechanism of the mathematical description of eachwith stage. Theflows independent phenomenon and make it possible to forecast changes in the consideration of each subsystem with its I/O flows appears phenomenon and its development trend. Different models, possible to estimate the potential of levels and subsystems and connections and relationships between individual components identify loss sources and ways to minimize wastage phenomenon and its Different models, to estimate the potential of levels and and make it possible totrend. forecast changes in the possible consideration of each itsand I/Osubsystems flows appears phenomenon and its development development trend. Different models, possible to estimate thesubsystem potential ofwith and subsystems and connections and relationships between individual components identify loss sources and ways tolevels minimize wastage and or subsystems, which are considered most relevant for this determine the reserves for increasing the efficiency of connections between individual components loss sources ways minimize wastage and phenomenonand andrelationships its development trend. Different models, identify possible to estimate theand potential ofto levels and subsystems connections and relationships between individual components identify loss sources and ways to minimize wastage and or subsystems, which are considered most relevant for this determine the reserves for increasing the efficiency of research, can correspond to the same process. They are written individual subsystems and the entire system. or subsystems, are considered most for this determine the reserves increasing the of connections andwhich relationships between individual components identify loss andforways to minimize wastage and or subsystems, which are most relevant relevant this determine thesources reserves increasing the efficiency efficiency of research, can mathematical correspond to considered the same process. Theychecked arefor written individual subsystems and for the entire system. down using relations, which are for individual research, can correspond to the same process. They are written subsystems and the entire system. or subsystems, which are considered most relevant for this determine the reserves for increasing the efficiency of research, can correspond to the same process. They are written individual subsystems and the entire system. The scheme of system analysis methodology applied in the down using mathematical relations, which and, are checked for adequacy at mathematical the final stage ofsame modelling, finally, the down using relations, which are checked for research, can correspond to the process. They are written individual subsystems and the entire system. The scheme ofof system analysis methodology applied in the down using mathematical relations, which for are accuracy checked for The development a mathematical model of the generalized adequacy at the final stage of ismodelling, and, finally, and the developed mathematical model validated scheme system analysis methodology applied in scheme of of analysis methodology applied in the the adequacy at the final stage of modelling, finally, the down using mathematical relations, which and, are checked for The development of system a mathematical model1),ofwhere the generalized adequacy at the final stage of modelling, and, finally, the technological process is shown on (Fig. an example developed mathematical model is validated for accuracy and usability. development of a mathematical model of the generalized The scheme of system analysis methodology applied in the development of a mathematical model of the generalized developed mathematical model is validated for accuracy and adequacy at the final stage of modelling, and, finally, the technological process is shown on (Fig. 1), where an example developed mathematical model is validated for accuracy and technological of the plasma-chemical process is.1),The usability. process is on an of a mathematical model ofwhere themathematical generalized technological process is shown shown on (Fig. (Fig. where ana example example usability. developed mathematical model are is validated forand accuracy and development of the plasma-chemical process is.1),The mathematical usability. All the technological processes continuous stochastic; description of the plasma-chemical process from systems of the plasma-chemical process is. The mathematical technological process is shown on (Fig. 1), where an example of the plasma-chemical process is. The mathematical usability. All the technological processes are continuous and stochastic; description of the plasma-chemical process from a systems chemical transformations and heat and mass transfer depend analysis perspective allows you to confirm importance of the All the technological processes are continuous and stochastic; description of plasma-chemical aa systems of the plasma-chemical process process is. Thefrom mathematical All theon technological processes are continuous and stochastic; of the theand plasma-chemical process from systems chemical transformations and heat and mass transfer depend description analysis perspective allows you to confirm importance of the both the internal state of an object and external quenching rate initial temperature decrease time for chemical transformations and heat and mass transfer depend perspective allows you to confirm of the All the technological processes are continuous and stochastic; description of the plasma-chemical processimportance from a systems chemical andoperation heat mass transfer depend analysis analysis perspective you to of confirm importance ofThe the both on transformations the The internal state of and an object andindividual external quenching ratemaximum and allows initial temperature decrease time for environment. optimal mode of producing the amount the target product. both on the internal state of an object and external rate and initial temperature time for chemical and heat mass transfer depend quenching analysis perspective you to confirmdecrease importance of the both on transformations the internal state of and an condition object and external quenching ratemaximum and allows initial temperature decrease time for environment. The optimal operation mode of individual producing the amount of the target product. The processes and stability of external provide the mathematical description of fine purification of cobalt environment. optimal mode of the amount of the target product. both on the The internal stateoperation of an object andindividual external producing quenching ratemaximum and initial temperature decrease time The for environment. The optimal operation mode of individual producing the maximum amount of the target product. The processes and stability of The external conditionof provide the mathematical description of isfine purification of cobalt efficiency of The production. complexity aindividual specific solutions from iron and copper discussed in this article. processes and stability of external condition provide the mathematical description of fine purification of cobalt environment. optimal operation mode of producing the maximum amount of the target product. The processes and stability external condition the mathematical description of isfine purification of cobalt efficiency of process; production. The complexity of provide a specific solutions from iron and copper discussed in this article. technological theof need forcomplexity complex processing of raw efficiency of production. The of aa specific from iron and copper discussed in this article. processes and stability of external condition provide the solutions mathematical description of is fine purification of cobalt efficiency of production. The complexity of specific solutions from iron and copper is discussed in this article. technological process; the need for complex processing of raw materials; the requirement forfor rational useprocessing of energy and solutions from iron and copper is discussed in this article. technological process; the need complex of raw efficiency of production. The complexity of a specific technological the need complex offlows raw materials; theprocess; requirement forfor rational useprocessing ofenergy energy and materials and recovery of side-line material and materials; the requirement for rational use of energy and technological process; the need for complex processing of raw materials; the requirement for rational use of energy and materials and recovery of side-line material and energy flows require consideration of complex multi-stage materials and of material flows materials;systematic therecovery requirement for rational useand ofenergy energy and materials and recovery ofbyside-line side-line material and energy flows require systematic consideration of complex multi-stage processes characterized homogeneous and heterogeneous require systematic consideration of complex multi-stage materials and recovery of side-line material and energy flows require systematic consideration of complex multi-stage processes characterized by homogeneous and heterogeneous stages in systematic different hi-tech processes characterized byindustries. homogeneous and require consideration of complex multi-stage processes characterized homogeneous and heterogeneous heterogeneous stages in different hi-techbyindustries. stages in different hi-tech industries. processes characterized by homogeneous and heterogeneous stages in different hi-tech industries. stages in different hi-tech industries.
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IFAC TECIS 2018 Baku, Azerbaidschan, Sept 13-15, 2018 Nataliya V. Mokrova et al. / IFAC PapersOnLine 51-30 (2018) 646–649
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Main regularities of the process Relationships between phenomena Analytical information and experiment Fig. 2. Structural scheme of the technological process yk – output of k-th subsystem, (k = 1,..,3); xk(i) – input of (k + 1)-th subsystem, (k = 1,2).
Mathematical model
Adjustment of the mathematical description
The starting reagents are soluble salts: Fe2(SO4)3, CuSO4, NiSO4, CoSO4, H2SO4, caustic ash Na2CO3 and H2O. The reaction products are insoluble salts: Fe2(OH)4SO4, CuCO3, NiCO3, CoCO3, soluble salts Na2SO4 and CO2. [H+] is the concentration of hydrogen ions [H+] = 10–pH mol/L. Concentrations of ions Fe3+, Cu2+, Ni2+, Co2+ measured with a spectrophotometer.
Rating the adequacy of the model
A simplified mathematical model of the purification process is constructed:
Fig. 1. System analysis methodology applied in the development of a mathematical model.
r1
𝐹𝐹𝑒𝑒2 (𝑆𝑆𝑂𝑂4 )3 + 2𝑁𝑁𝑎𝑎2 𝐶𝐶𝑂𝑂3 + 2𝐻𝐻2 𝑂𝑂 = = 𝐹𝐹𝑒𝑒2 (𝑂𝑂𝐻𝐻4 )𝑆𝑆𝑆𝑆4 + 2𝐶𝐶𝑂𝑂3 + 2𝑁𝑁𝑁𝑁2 𝑆𝑆𝑂𝑂4 ,
3. TECHNOLOGICAL PROCESS MODELLING Let us now turn to the description of the problem of modelling a multi-stage chemical and technological process of cobalt production. A cascade of sequential action devices at the cobalt solution purification stage can be considered as a multidimensional control object with distributed parameters and a complex structure of relationships between variables within the subsystems and an internal source of random interference. In accordance with the system approach, we considered the ways for building a mathematical model of cascade devices. We applied a sequential approximation method to identify the characteristics of subsystems, when building a mathematical model aimed to identify the most essential characteristics of the object of research.
r2
𝐶𝐶𝐶𝐶𝐶𝐶𝑂𝑂4 + 𝑁𝑁𝑎𝑎2 𝐶𝐶𝑂𝑂3 = 𝐶𝐶𝐶𝐶𝐶𝐶𝑂𝑂3 + 𝑁𝑁𝑎𝑎2 𝑆𝑆𝑆𝑆4 , r3
𝑁𝑁𝑁𝑁𝑁𝑁𝑂𝑂4 + 𝑁𝑁𝑎𝑎2 𝐶𝐶𝑂𝑂3 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑂𝑂3 + 𝑁𝑁𝑎𝑎2 𝑆𝑆𝑆𝑆4 ,
(1)
r4
𝐶𝐶𝐶𝐶𝐶𝐶𝑂𝑂4 + 𝑁𝑁𝑎𝑎2 𝐶𝐶𝑂𝑂3 = 𝐶𝐶𝐶𝐶𝐶𝐶𝑂𝑂3 + 𝑁𝑁𝑎𝑎2 𝑆𝑆𝑆𝑆4 , r5
𝐻𝐻2 𝑆𝑆𝑂𝑂4 + 𝑁𝑁𝑎𝑎2 𝐶𝐶𝑂𝑂3 = 𝐶𝐶𝑂𝑂2 + 𝐻𝐻2 𝑂𝑂 + 𝑁𝑁𝑎𝑎2 𝑆𝑆𝑆𝑆4 , where rj – rate of j-th reaction (j = 1,..,5). The main method for purifying cobalt solutions from impurities of Fe, Ni, Cu, Al is the method of precipitation of basic salts and hydroxides of these metals, based on the different pH of their release.
The mathematical modelling problem is solved from the position of the system approach; the purification mathematical model is developed using the building-block concept. The generalized cascade model consists of models of reactors arranged in series and connected by material flows of reagents. A reactor model consists of blocks of chemical kinetics, hydrodynamics and material balances. Chemical kinetics equations have a complex structure and describe the formation of both stable basic substances and highly active intermediate compounds. The interconnected blocks of equations reflecting the elementary processes in each device coincide to the nearest coefficient.
The main objective of control is to achieve the maximum yield of the target product while remaining within the limits on the impurities content. The 1st stage of the research is devoted to building a mathematical model of the process statics on the basis of experimental data on the ion concentration measured with a spectrophotometer. When building a simplified mathematical model, we derived the equations of reaction mechanism without taking into account chemical transformations that enable the formation of intermediate and unstable compounds (1). To determine the loss of cobalt, we solved the problem of parametric identification of a mathematical model.
Cobalt solutions are purified in a cascade of 3 continuous reactors; primary products pass through the devices to react with caustic ash (main agent) with subsequent precipitation of iron and copper impurities and inevitable cobalt decline.
When building a mathematical model of dynamics of the fine purification process, we used experimental data and coefficients obtained in the analysis of the process statics.
The general structural diagram of the technological process (Fig. 2) shows the relationship between the variable individual processes.
Taking into account the hydrodynamics model and assuming the ideal mixing conditions, we derived the following equations reflecting the change in the content of key
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components for each of the three cascade reactors in dynamic mode:
𝑑𝑑𝐶𝐶𝐶𝐶𝐶𝐶 0 = 𝑔𝑔𝐶𝐶𝐶𝐶𝐶𝐶 − (𝑔𝑔 + 𝑔𝑔𝑐𝑐 )𝐶𝐶𝐶𝐶𝐶𝐶 − 𝑣𝑣𝑣𝑣1(𝑝𝑝𝑝𝑝)𝐶𝐶𝐶𝐶𝐶𝐶 , 𝑑𝑑𝑑𝑑 𝑑𝑑𝐶𝐶𝐶𝐶𝐶𝐶 0 𝑣𝑣 = 𝑔𝑔𝐶𝐶𝐶𝐶𝐶𝐶 − (𝑔𝑔 + 𝑔𝑔𝑐𝑐 )𝐶𝐶𝐶𝐶𝐶𝐶 − 𝑣𝑣𝑣𝑣2(𝑝𝑝𝑝𝑝)𝐶𝐶𝐶𝐶𝐶𝐶 , (2) 𝑑𝑑𝑑𝑑 𝑑𝑑𝐶𝐶𝐹𝐹𝐹𝐹 0 𝑣𝑣 = 𝑔𝑔𝐶𝐶𝐹𝐹𝐹𝐹 − (𝑔𝑔 + 𝑔𝑔𝑐𝑐 )𝐶𝐶𝐹𝐹𝐹𝐹 − 𝑣𝑣𝑣𝑣3(𝑝𝑝𝑝𝑝)𝐶𝐶𝐹𝐹𝐹𝐹 , 𝑑𝑑𝑑𝑑 𝑣𝑣
where g – consumption of the solution; gc – consumption of caustic ash; v – volume of reactors; k* – reaction rate constants for a component (depending on pH); pH – acidity of the solution; С*0 – concentration in the original solution (for cobalt, copper, iron); С* – concentration at the output of the reactor cascade (for cobalt, copper, iron);
𝑣𝑣
𝑑𝑑𝐶𝐶𝑐𝑐 = 𝑔𝑔𝑐𝑐 𝐶𝐶𝑐𝑐0 − 𝑣𝑣(2𝑘𝑘1(𝑝𝑝𝑝𝑝)𝐶𝐶𝐶𝐶𝐶𝐶 + 𝑑𝑑𝑑𝑑 +𝑘𝑘2(𝑝𝑝𝑝𝑝)𝐶𝐶𝐶𝐶𝐶𝐶 + 𝑘𝑘3(𝑝𝑝𝑝𝑝)𝐶𝐶𝐹𝐹𝐹𝐹 + 𝑘𝑘4(𝑝𝑝𝑝𝑝)𝐶𝐶𝑐𝑐 ),
a)
(3)
where Cc – concentration of caustic ash. The system of differential equations (2, 3) that characterize the dynamics of fine purification (decontamination) of cobalt solutions is described by the quartic Runge-Kutta method at a constant step. Let us investigate the dependencies obtained in addressing the problem. Figure 3 (a) shows the change in cobalt concentration in the course of the chemical substitution reaction that occurs when solution passes consequently through the cascade reactors. 𝑡𝑡𝑘𝑘 variable denotes the process lead time in the 𝑘𝑘-th reactor. By analyzing the curve, it can be concluded that the values of the target product concentration in each of the reactors reach an equilibrium value, while cobalt losses are commensurable for all three devices at the same level of reagent consumption.
b) Fig. 3. Dependencies of changes in the target product and impurities in the cascade of reactors. 4. TECHNOLOGICAL PROCESS OPTIMIZATION The problem of optimal control over the purification process is formulated with a strict restriction with regard to the concentration of impurities; the third reactor can provide the desired degree of decontamination of the solution in the presence of significant interference, which is mainly associated with changes in the composition of raw materials and other subjective factors. Formulation of the optimal control problem:
For comparison, let us consider the change in the concentration of one of the impurities (copper) (Fig. 3b). It is noted that the nature of the change in the concentration of impurities (iron and copper) differs from the change in the concentration of the target product with the maximum precipitation of impurities occurring in the first reactor, whereas only a slight decrease in impurities can be observed in the subsequent reactors with a significant loss of the target product. The analysis of dynamic processes in the cascade of reactors allows us to conclude that it is important to solve control tasks by means of the last reactor in the cascade as this helps control the set acidity value, because even a slight change in this value will lead to a dramatic decrease in the efficiency of the process in terms of the yield of the target product.
0 𝑖𝑖 𝐶𝐶𝐶𝐶𝐶𝐶 − ∑ ∆𝐶𝐶𝐶𝐶𝐶𝐶 ⇒ 𝑚𝑚𝑚𝑚𝑚𝑚 , 𝑖𝑖
𝐶𝐶𝐶𝐶𝐶𝐶 ≤
𝐶𝐶1∗ , 𝐶𝐶𝐹𝐹𝐹𝐹
𝑈𝑈
≤ 𝐶𝐶2∗ ,
(4)
0 𝑖𝑖 – concentration cobalt in the original solution; ∆𝐶𝐶𝐶𝐶𝐶𝐶 where 𝐶𝐶𝐶𝐶𝐶𝐶 – cobalt losses in each of three reactors (i = 1, 2, 3); 𝑈𝑈 = 𝑔𝑔𝑐𝑐𝑐𝑐 – the selected control – consumption of caustic ash in each of reactors; 𝐶𝐶1∗ , 𝐶𝐶2∗ – the set values of concentrations of copper and iron are 0.01 g/l and 0.3 g/l, respectively.
In general, the description of dynamic processes occurring in the cascade of continuous reactors does not contradict the previous strategy of control adopted for the chemical and technological purification system. According to the system approach chosen, methods for identifying a mathematical model of a cascade of reactors at the fine purification stage were proposed and a successive approximation method was adopted to identify the characteristics of the subsystems. Such models are applicable both for solving control problems and predicting future states of systems.
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The problems of optimal control over the cobalt solution fine purification process on the basis of mathematical modelling of static and dynamic modes, as well as the control problem, were solved using the decomposition approach. A great attention was devoted to both adequate mathematical modelling of each reactor and solving control tasks at each stage, because when under-researched, individual phenomena and processes will not allow us to have a completely mathematically formalized description of the object; this also applies to the highlighting of hierarchy levels and establishing relations between phenomena. An important aspect in the implementation of the system approach is the use of analytical data, experimental data and observations.
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Mokrova, N. V. (2018). Methodology and Results of Mathematical Modelling of Complex Technological Processes. IOP Conference Series: Materials Science and Engineering, 317, 012038.
5. CONCLUSIONS The system analysis methodology allowed to select the levels of the hierarchical structure of the high-technology systems; we examined their interrelations, determined the aspects of their mutual influence, elucidated physical and chemical phenomena in each device and took into account characteristics of main state variables and controlling and disturbing influences. The mathematical description of the stages helped us find more rational ways of running the process and solve optimization problems. The models of multistage technological processes are applicable for solving control problems and predicting future states of sophisticated high-technology systems. Subsystems are segregate depending on the complexity of the object on the basis study of process physical and chemical regularities, availability or possibility of obtaining a mathematical description of each stage. This allows you to get a synergistic effect. The potential of flows and subsystems detected and identify loss sources and ways to minimize that. The capability for increasing the efficiency of individual devices and the entire system was studied and that is very important in technological industries. A quantitative and quality system analysis and mathematical modelling of complex technological processes employing state-of-the-art computing facilities is necessary both for the development of new processes and intensification of existing ones. REFERENCES Artamonov, A. G., Volodin, V. M. and Avdeev, V. G. (1989). Mathematical Modelling and Optimization of Plasmachemical Processes. Moscow, Himia. Lasdon, L. S. (2011). Optimization Theory for Large Systems. New York, Dover Publications. Mesarovic, M. D. and Takahara, Y. (1975). General Systems Theory: Mathematical Foundations. Amsterdam, Elsevier Science. Mokrova, N. V. and Volodin, V. M. (2006). Substantiation of the choice of methods for solving the problem of optimal control of complex processes. Bulletin of TSTU, 12, 22– 28. Mokrova, N. V. and Volodin, V. M. (2007). Modeling of decomposition control of multistage processes. Chemical and Oil and Gas Engineering, 2, 17–19. 649
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IFAC PapersOnLine 51-30 (2018) 646–649 Modelling and Optimization of Complex Technological Processes Modelling and Optimization of Complex Technological Processes Modelling of Technological Modelling and and Optimization Optimization of Complex Complex Technological Processes Processes Nataliya V. Mokrova* Modelling and Optimization of Complex Technological Processes Nataliya V. Mokrova*
Nataliya Nataliya V. V. Mokrova* Mokrova* * Moscow State University of Mokrova* Civil Engineering, Moscow, Nataliya V. * Moscow State University of Civil Engineering, Moscow, Russia (e-mail: [email protected]) ** Moscow University of Moscow State State of Civil Civil Engineering, Engineering, Moscow, Moscow, RussiaUniversity (e-mail: [email protected]) RussiaUniversity (e-mail: [email protected]) [email protected]) * Moscow State of Civil Engineering, Moscow, Russia (e-mail: Russiaanalysis (e-mail:for [email protected]) Abstract: The methodology of system mathematical model of the complex technological Abstract: The methodology of system analysis for to mathematical of theofcomplex technological process is considered. The mathematical description process finemodel purification cobalt solutions from Abstract: The methodology of system analysis for mathematical model of the complex technological Abstract: The methodology of system analysis for of mathematical model of of thethe technological process iscopper considered. Theinmathematical description to process fine purification ofcomplex cobalt solutions from iron and discussed this article. The results numerical integration system of differential process is The considered. The mathematical description process fine purification cobalt solutions from Abstract: methodology of system analysis for to mathematical model of theof technological process considered. The description to process fine purification ofcomplex cobalt from iron andiscopper discussed inmathematical thisoptimal article.concentrations The results of numerical integration of the systemsolutions of differential equations can be used to obtain of reagents. The introduced models can be applied iron and copper discussed in this article. The results of numerical integration of the system of differential process is considered. The mathematical description to process fine purification of cobalt solutions from iron and copper discussed in this article. The results of numerical integration of the system of differential equations can be usedand to obtain optimal concentrations of reagents. The introduced models can be applied for control problems predicting future states of sophisticated technological systems. equations can be used to obtain optimal concentrations of reagents. The introduced models can be applied iron and copper discussed in this article. The results of numerical integration of the system of differential equations be usedand to obtain optimal concentrations of reagents. The introduced models can be applied for controlcan problems predicting future states of sophisticated technological systems. for control problems and predicting future states technological systems. equations can be used to obtain optimal of reagents. The introduced canreserved. be applied © 2018, IFAC (International Federation ofconcentrations Automatic Control) Hosting by Elsevier Ltd.models Allprocess. rights for control problems and predicting future states of of sophisticated sophisticated technological systems. Keywords: system analysis, results of mathematical simulation, multistage purification for control problems and predicting future states of sophisticated technological systems.process. Keywords: system analysis, results of mathematical simulation, multistage purification Keywords: system analysis, analysis, results results of of mathematical mathematical simulation, simulation, multistage multistage purification purification process. Keywords: system process. Keywords: system analysis, results of mathematical simulation, multistage purification process. The methodology of system analysis revealed the levels of a 1. INTRODUCTION The methodology of system analysis revealed the levels of a hierarchical structure of the complex technological system. 1. INTRODUCTION The methodology of system analysis revealed the levels of The methodology of system analysis revealed the levels of aa 1. INTRODUCTION hierarchical structure of the complex technological system. These levels structure detailing from the molecular one and ending with 1. INTRODUCTION The most of technological process of industrial production hierarchical of the complex technological system. The methodology of system analysis revealed the levels of a hierarchical structure of complex system. 1. These levels estimates detailing from the molecular one interrelationship and ending with The most of technological process of industrial production the integral takethe into accounttechnological the couldmost be considered as INTRODUCTION a complex process full of variables and These levels detailing from the molecular one and ending with The of technological process of industrial production hierarchical structure of the complex technological system. These levels detailing from the molecular one and ending with The most of Scientific technological process of industrial production integral estimateslevels take into account the each interrelationship could be considered as a research complex process full of variables and the between individual of molecular the system, of them is connections. involves studying a related integral estimates take into account the could be considered as complex process full of and These levels detailing from one interrelationship and ending with The most of technological process of industrial production the integral estimates take the into account the interrelationship could beinterrelated considered as aa research complex process full of variables variables and the between individual levels ofmathematical the system, each of themAn is connections. Scientific involves studying a related characterized by relevant description. class of processes under the common regularities. individual of the system, of them is connections. Scientific studying aa related the integral estimateslevels take into account the each interrelationship could be considered as a research complex involves process full of variables and between between individual levels ofmathematical the system, each ofdetect themthe is connections. Scientific research involves studying related characterized by relevant description. An class of interrelated processes under thesimulation common regularities. adequate mathematical description allows us to The process of mathematical and models of by relevant description. class of interrelated processes under the common regularities. between individual levels ofmathematical the system, each of themAn is connections. Scientific research involves studying a related characterized characterized by relevant mathematical description. An class of interrelated processes under the common regularities. mathematical description allows us toand detect the The process of must mathematical and simulation models of adequate ways howmathematical to manage processes moreallows efficiently address complex systems be iterated numerous times,regularities. and at each adequate description us to detect the The process of mathematical and simulation models of characterized by relevant mathematical description. An class of interrelated processes under the common description us toand detect the The process of must mathematical and simulation models of adequate ways howmathematical to issues. manage processes moreallows efficiently address complex systems be iterated numerous times, and at each optimization stage its attributes are refined. The priorities of modelling are how to manage processes more efficiently address complex systems be iterated numerous times, and at each adequate mathematical description allows us toand detect the The process of must mathematical and simulation models of ways ways how to manage processes more efficiently and address complex systems must be iterated numerous times, and at each optimization issues. stage itsthe attributes are refined. The priorities of modelling are set and ways to build newlyThe built objectstimes, aremodelling synthesized. optimization issues. stage its attributes are refined. priorities of are ways how to manage processes more efficiently and address complex systems must be iterated numerous and at each issues. ANALYSIS METHODOLOGY stage itsthe attributes refined. priorities of are optimization set and ways toare build newlyThe built objectsreflects aremodelling synthesized. The scheme of mathematical description the most 2. SYSTEM set and ways to build newly built objects are synthesized. issues. stage itsthe attributes are refined. The priorities of modelling are optimization set and the ways to build newly built objects are synthesized. The scheme of mathematical description reflects the most 2. SYSTEM ANALYSIS METHODOLOGY essential features of the newly object or objects process andsynthesized. allows to The scheme of mathematical description reflects the most 2. SYSTEM ANALYSIS METHODOLOGY set and the ways to build built are The scheme of mathematical description reflects the most 2. SYSTEM ANALYSIS METHODOLOGY The basis of the system analysis is decomposition of a complex essential features of the object or process and allows to understand theof mechanism of the phenomenon andallows make it essential features of the object or process and to The scheme mathematical description reflects the most 2. SYSTEM ANALYSIS METHODOLOGY The basis of the system analysis is decomposition ofquantitative a complex essential features of the object or process and allows to system into subsystems and establishment of understand the mechanism of the phenomenon and trend. make it The possible to forecast changes in it and its development basis of system is of aa complex The basisinto of the the system analysis analysis is decomposition decomposition ofquantitative complex understand the mechanism of the phenomenon make it essential features of the object or process andand allows to system subsystems and establishment of understand the mechanism of the phenomenon and make it relationships between them. The subsystems are possible to forecast changes in it and its development trend. system subsystems and establishment of The basisinto of the system analysis is decomposition ofquantitative a allocated complex possible to forecast changes in it and its development trend. system into subsystems and establishment of quantitative understand the mechanism of the phenomenon and make it relationships between them. ofThe subsystems are allocated possible to forecastdescription changes in scheme it and itsobtained development The mathematical in thetrend. course relationships depending the complexity the subsystems object under investigation between them. allocated system intoon subsystems and The establishment ofare quantitative relationships between them. The subsystems are allocated possible to forecast changes in it and its development trend. The mathematical description scheme obtained in the course depending on the complexity ofprocess the object underand investigation of modelling together with the parameters identified as acourse result depending and on theon previous study ofof physical chemical The mathematical description scheme obtained in the complexity the object under investigation relationships between them. The are allocated The mathematical description scheme obtained inofthe the the complexity the subsystems object under investigation modelling together with the parameters identified as acourse result depending and on theon previous study or ofofprocess physical and chemicala of interpolation, reflect the most essential features the object regularities, availability possibility of obtaining of modelling together with the parameters identified as a result and on the previous study of process physical and chemical The mathematical description scheme obtained in the course depending on the complexity of the object under investigation of modelling together the parameters identified result and on the previous study of of process physical chemicala of interpolation, reflectwith thetomost essentialthe features of as theaobject regularities, availability or possibility of and obtaining or interpolation, process and allow usthe understand mechanism of the regularities, mathematical description each stage. The independent of reflect most essential features of the availability possibility of obtaining modelling together with the parameters identified as aobject result and on the previous study or of process physical and chemicalaa of interpolation, reflect the most essential features of the object regularities, availability or possibility of obtaining or process and allow us to understand the mechanism of the mathematical description of each stage. The independent phenomenon and make ittopossible to forecast changes in the mathematical consideration of each subsystem with its I/O flows appearsa or process and allow us understand the mechanism of description of each stage. The independent of interpolation, reflect the most essential features of the object regularities, availability or possibility of obtaining or process andand allow us understand the Different mechanism of description of each stage. The independent phenomenon make ittopossible totrend. forecast changes in the the mathematical consideration of each subsystem with itsand I/O flows appears and its development models, possible to estimate the potential of levels subsystems and phenomenon make to forecast changes in consideration of each subsystem its I/O appears or process andand allow us it topossible understand the mechanism of the mathematical description of eachwith stage. Theflows independent phenomenon and make it possible to forecast changes in the consideration of each subsystem with its I/O flows appears phenomenon and its development trend. Different models, possible to estimate the potential of levels and subsystems and connections and relationships between individual components identify loss sources and ways to minimize wastage phenomenon and its Different models, to estimate the potential of levels and and make it possible totrend. forecast changes in the possible consideration of each itsand I/Osubsystems flows appears phenomenon and its development development trend. Different models, possible to estimate thesubsystem potential ofwith and subsystems and connections and relationships between individual components identify loss sources and ways tolevels minimize wastage and or subsystems, which are considered most relevant for this determine the reserves for increasing the efficiency of connections between individual components loss sources ways minimize wastage and phenomenonand andrelationships its development trend. Different models, identify possible to estimate theand potential ofto levels and subsystems connections and relationships between individual components identify loss sources and ways to minimize wastage and or subsystems, which are considered most relevant for this determine the reserves for increasing the efficiency of research, can correspond to the same process. They are written individual subsystems and the entire system. or subsystems, are considered most for this determine the reserves increasing the of connections andwhich relationships between individual components identify loss andforways to minimize wastage and or subsystems, which are most relevant relevant this determine thesources reserves increasing the efficiency efficiency of research, can mathematical correspond to considered the same process. Theychecked arefor written individual subsystems and for the entire system. down using relations, which are for individual research, can correspond to the same process. They are written subsystems and the entire system. or subsystems, which are considered most relevant for this determine the reserves for increasing the efficiency of research, can correspond to the same process. They are written individual subsystems and the entire system. The scheme of system analysis methodology applied in the down using mathematical relations, which and, are checked for adequacy at mathematical the final stage ofsame modelling, finally, the down using relations, which are checked for research, can correspond to the process. They are written individual subsystems and the entire system. The scheme ofof system analysis methodology applied in the down using mathematical relations, which for are accuracy checked for The development a mathematical model of the generalized adequacy at the final stage of ismodelling, and, finally, and the developed mathematical model validated scheme system analysis methodology applied in scheme of of analysis methodology applied in the the adequacy at the final stage of modelling, finally, the down using mathematical relations, which and, are checked for The development of system a mathematical model1),ofwhere the generalized adequacy at the final stage of modelling, and, finally, the technological process is shown on (Fig. an example developed mathematical model is validated for accuracy and usability. development of a mathematical model of the generalized The scheme of system analysis methodology applied in the development of a mathematical model of the generalized developed mathematical model is validated for accuracy and adequacy at the final stage of modelling, and, finally, the technological process is shown on (Fig. 1), where an example developed mathematical model is validated for accuracy and technological of the plasma-chemical process is.1),The usability. process is on an of a mathematical model ofwhere themathematical generalized technological process is shown shown on (Fig. (Fig. where ana example example usability. developed mathematical model are is validated forand accuracy and development of the plasma-chemical process is.1),The mathematical usability. All the technological processes continuous stochastic; description of the plasma-chemical process from systems of the plasma-chemical process is. The mathematical technological process is shown on (Fig. 1), where an example of the plasma-chemical process is. The mathematical usability. All the technological processes are continuous and stochastic; description of the plasma-chemical process from a systems chemical transformations and heat and mass transfer depend analysis perspective allows you to confirm importance of the All the technological processes are continuous and stochastic; description of plasma-chemical aa systems of the plasma-chemical process process is. Thefrom mathematical All theon technological processes are continuous and stochastic; of the theand plasma-chemical process from systems chemical transformations and heat and mass transfer depend description analysis perspective allows you to confirm importance of the both the internal state of an object and external quenching rate initial temperature decrease time for chemical transformations and heat and mass transfer depend perspective allows you to confirm of the All the technological processes are continuous and stochastic; description of the plasma-chemical processimportance from a systems chemical andoperation heat mass transfer depend analysis analysis perspective you to of confirm importance ofThe the both on transformations the The internal state of and an object andindividual external quenching ratemaximum and allows initial temperature decrease time for environment. optimal mode of producing the amount the target product. both on the internal state of an object and external rate and initial temperature time for chemical and heat mass transfer depend quenching analysis perspective you to confirmdecrease importance of the both on transformations the internal state of and an condition object and external quenching ratemaximum and allows initial temperature decrease time for environment. The optimal operation mode of individual producing the amount of the target product. The processes and stability of external provide the mathematical description of fine purification of cobalt environment. optimal mode of the amount of the target product. both on the The internal stateoperation of an object andindividual external producing quenching ratemaximum and initial temperature decrease time The for environment. The optimal operation mode of individual producing the maximum amount of the target product. The processes and stability of The external conditionof provide the mathematical description of isfine purification of cobalt efficiency of The production. complexity aindividual specific solutions from iron and copper discussed in this article. processes and stability of external condition provide the mathematical description of fine purification of cobalt environment. optimal operation mode of producing the maximum amount of the target product. The processes and stability external condition the mathematical description of isfine purification of cobalt efficiency of process; production. The complexity of provide a specific solutions from iron and copper discussed in this article. technological theof need forcomplexity complex processing of raw efficiency of production. The of aa specific from iron and copper discussed in this article. processes and stability of external condition provide the solutions mathematical description of is fine purification of cobalt efficiency of production. The complexity of specific solutions from iron and copper is discussed in this article. technological process; the need for complex processing of raw materials; the requirement forfor rational useprocessing of energy and solutions from iron and copper is discussed in this article. technological process; the need complex of raw efficiency of production. The complexity of a specific technological the need complex offlows raw materials; theprocess; requirement forfor rational useprocessing ofenergy energy and materials and recovery of side-line material and materials; the requirement for rational use of energy and technological process; the need for complex processing of raw materials; the requirement for rational use of energy and materials and recovery of side-line material and energy flows require consideration of complex multi-stage materials and of material flows materials;systematic therecovery requirement for rational useand ofenergy energy and materials and recovery ofbyside-line side-line material and energy flows require systematic consideration of complex multi-stage processes characterized homogeneous and heterogeneous require systematic consideration of complex multi-stage materials and recovery of side-line material and energy flows require systematic consideration of complex multi-stage processes characterized by homogeneous and heterogeneous stages in systematic different hi-tech processes characterized byindustries. homogeneous and require consideration of complex multi-stage processes characterized homogeneous and heterogeneous heterogeneous stages in different hi-techbyindustries. stages in different hi-tech industries. processes characterized by homogeneous and heterogeneous stages in different hi-tech industries. stages in different hi-tech industries.
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Main regularities of the process Relationships between phenomena Analytical information and experiment Fig. 2. Structural scheme of the technological process yk – output of k-th subsystem, (k = 1,..,3); xk(i) – input of (k + 1)-th subsystem, (k = 1,2).
Mathematical model
Adjustment of the mathematical description
The starting reagents are soluble salts: Fe2(SO4)3, CuSO4, NiSO4, CoSO4, H2SO4, caustic ash Na2CO3 and H2O. The reaction products are insoluble salts: Fe2(OH)4SO4, CuCO3, NiCO3, CoCO3, soluble salts Na2SO4 and CO2. [H+] is the concentration of hydrogen ions [H+] = 10–pH mol/L. Concentrations of ions Fe3+, Cu2+, Ni2+, Co2+ measured with a spectrophotometer.
Rating the adequacy of the model
A simplified mathematical model of the purification process is constructed:
Fig. 1. System analysis methodology applied in the development of a mathematical model.
r1
𝐹𝐹𝑒𝑒2 (𝑆𝑆𝑂𝑂4 )3 + 2𝑁𝑁𝑎𝑎2 𝐶𝐶𝑂𝑂3 + 2𝐻𝐻2 𝑂𝑂 = = 𝐹𝐹𝑒𝑒2 (𝑂𝑂𝐻𝐻4 )𝑆𝑆𝑆𝑆4 + 2𝐶𝐶𝑂𝑂3 + 2𝑁𝑁𝑁𝑁2 𝑆𝑆𝑂𝑂4 ,
3. TECHNOLOGICAL PROCESS MODELLING Let us now turn to the description of the problem of modelling a multi-stage chemical and technological process of cobalt production. A cascade of sequential action devices at the cobalt solution purification stage can be considered as a multidimensional control object with distributed parameters and a complex structure of relationships between variables within the subsystems and an internal source of random interference. In accordance with the system approach, we considered the ways for building a mathematical model of cascade devices. We applied a sequential approximation method to identify the characteristics of subsystems, when building a mathematical model aimed to identify the most essential characteristics of the object of research.
r2
𝐶𝐶𝐶𝐶𝐶𝐶𝑂𝑂4 + 𝑁𝑁𝑎𝑎2 𝐶𝐶𝑂𝑂3 = 𝐶𝐶𝐶𝐶𝐶𝐶𝑂𝑂3 + 𝑁𝑁𝑎𝑎2 𝑆𝑆𝑆𝑆4 , r3
𝑁𝑁𝑁𝑁𝑁𝑁𝑂𝑂4 + 𝑁𝑁𝑎𝑎2 𝐶𝐶𝑂𝑂3 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑂𝑂3 + 𝑁𝑁𝑎𝑎2 𝑆𝑆𝑆𝑆4 ,
(1)
r4
𝐶𝐶𝐶𝐶𝐶𝐶𝑂𝑂4 + 𝑁𝑁𝑎𝑎2 𝐶𝐶𝑂𝑂3 = 𝐶𝐶𝐶𝐶𝐶𝐶𝑂𝑂3 + 𝑁𝑁𝑎𝑎2 𝑆𝑆𝑆𝑆4 , r5
𝐻𝐻2 𝑆𝑆𝑂𝑂4 + 𝑁𝑁𝑎𝑎2 𝐶𝐶𝑂𝑂3 = 𝐶𝐶𝑂𝑂2 + 𝐻𝐻2 𝑂𝑂 + 𝑁𝑁𝑎𝑎2 𝑆𝑆𝑆𝑆4 , where rj – rate of j-th reaction (j = 1,..,5). The main method for purifying cobalt solutions from impurities of Fe, Ni, Cu, Al is the method of precipitation of basic salts and hydroxides of these metals, based on the different pH of their release.
The mathematical modelling problem is solved from the position of the system approach; the purification mathematical model is developed using the building-block concept. The generalized cascade model consists of models of reactors arranged in series and connected by material flows of reagents. A reactor model consists of blocks of chemical kinetics, hydrodynamics and material balances. Chemical kinetics equations have a complex structure and describe the formation of both stable basic substances and highly active intermediate compounds. The interconnected blocks of equations reflecting the elementary processes in each device coincide to the nearest coefficient.
The main objective of control is to achieve the maximum yield of the target product while remaining within the limits on the impurities content. The 1st stage of the research is devoted to building a mathematical model of the process statics on the basis of experimental data on the ion concentration measured with a spectrophotometer. When building a simplified mathematical model, we derived the equations of reaction mechanism without taking into account chemical transformations that enable the formation of intermediate and unstable compounds (1). To determine the loss of cobalt, we solved the problem of parametric identification of a mathematical model.
Cobalt solutions are purified in a cascade of 3 continuous reactors; primary products pass through the devices to react with caustic ash (main agent) with subsequent precipitation of iron and copper impurities and inevitable cobalt decline.
When building a mathematical model of dynamics of the fine purification process, we used experimental data and coefficients obtained in the analysis of the process statics.
The general structural diagram of the technological process (Fig. 2) shows the relationship between the variable individual processes.
Taking into account the hydrodynamics model and assuming the ideal mixing conditions, we derived the following equations reflecting the change in the content of key
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components for each of the three cascade reactors in dynamic mode:
𝑑𝑑𝐶𝐶𝐶𝐶𝐶𝐶 0 = 𝑔𝑔𝐶𝐶𝐶𝐶𝐶𝐶 − (𝑔𝑔 + 𝑔𝑔𝑐𝑐 )𝐶𝐶𝐶𝐶𝐶𝐶 − 𝑣𝑣𝑣𝑣1(𝑝𝑝𝑝𝑝)𝐶𝐶𝐶𝐶𝐶𝐶 , 𝑑𝑑𝑑𝑑 𝑑𝑑𝐶𝐶𝐶𝐶𝐶𝐶 0 𝑣𝑣 = 𝑔𝑔𝐶𝐶𝐶𝐶𝐶𝐶 − (𝑔𝑔 + 𝑔𝑔𝑐𝑐 )𝐶𝐶𝐶𝐶𝐶𝐶 − 𝑣𝑣𝑣𝑣2(𝑝𝑝𝑝𝑝)𝐶𝐶𝐶𝐶𝐶𝐶 , (2) 𝑑𝑑𝑑𝑑 𝑑𝑑𝐶𝐶𝐹𝐹𝐹𝐹 0 𝑣𝑣 = 𝑔𝑔𝐶𝐶𝐹𝐹𝐹𝐹 − (𝑔𝑔 + 𝑔𝑔𝑐𝑐 )𝐶𝐶𝐹𝐹𝐹𝐹 − 𝑣𝑣𝑣𝑣3(𝑝𝑝𝑝𝑝)𝐶𝐶𝐹𝐹𝐹𝐹 , 𝑑𝑑𝑑𝑑 𝑣𝑣
where g – consumption of the solution; gc – consumption of caustic ash; v – volume of reactors; k* – reaction rate constants for a component (depending on pH); pH – acidity of the solution; С*0 – concentration in the original solution (for cobalt, copper, iron); С* – concentration at the output of the reactor cascade (for cobalt, copper, iron);
𝑣𝑣
𝑑𝑑𝐶𝐶𝑐𝑐 = 𝑔𝑔𝑐𝑐 𝐶𝐶𝑐𝑐0 − 𝑣𝑣(2𝑘𝑘1(𝑝𝑝𝑝𝑝)𝐶𝐶𝐶𝐶𝐶𝐶 + 𝑑𝑑𝑑𝑑 +𝑘𝑘2(𝑝𝑝𝑝𝑝)𝐶𝐶𝐶𝐶𝐶𝐶 + 𝑘𝑘3(𝑝𝑝𝑝𝑝)𝐶𝐶𝐹𝐹𝐹𝐹 + 𝑘𝑘4(𝑝𝑝𝑝𝑝)𝐶𝐶𝑐𝑐 ),
a)
(3)
where Cc – concentration of caustic ash. The system of differential equations (2, 3) that characterize the dynamics of fine purification (decontamination) of cobalt solutions is described by the quartic Runge-Kutta method at a constant step. Let us investigate the dependencies obtained in addressing the problem. Figure 3 (a) shows the change in cobalt concentration in the course of the chemical substitution reaction that occurs when solution passes consequently through the cascade reactors. 𝑡𝑡𝑘𝑘 variable denotes the process lead time in the 𝑘𝑘-th reactor. By analyzing the curve, it can be concluded that the values of the target product concentration in each of the reactors reach an equilibrium value, while cobalt losses are commensurable for all three devices at the same level of reagent consumption.
b) Fig. 3. Dependencies of changes in the target product and impurities in the cascade of reactors. 4. TECHNOLOGICAL PROCESS OPTIMIZATION The problem of optimal control over the purification process is formulated with a strict restriction with regard to the concentration of impurities; the third reactor can provide the desired degree of decontamination of the solution in the presence of significant interference, which is mainly associated with changes in the composition of raw materials and other subjective factors. Formulation of the optimal control problem:
For comparison, let us consider the change in the concentration of one of the impurities (copper) (Fig. 3b). It is noted that the nature of the change in the concentration of impurities (iron and copper) differs from the change in the concentration of the target product with the maximum precipitation of impurities occurring in the first reactor, whereas only a slight decrease in impurities can be observed in the subsequent reactors with a significant loss of the target product. The analysis of dynamic processes in the cascade of reactors allows us to conclude that it is important to solve control tasks by means of the last reactor in the cascade as this helps control the set acidity value, because even a slight change in this value will lead to a dramatic decrease in the efficiency of the process in terms of the yield of the target product.
0 𝑖𝑖 𝐶𝐶𝐶𝐶𝐶𝐶 − ∑ ∆𝐶𝐶𝐶𝐶𝐶𝐶 ⇒ 𝑚𝑚𝑚𝑚𝑚𝑚 , 𝑖𝑖
𝐶𝐶𝐶𝐶𝐶𝐶 ≤
𝐶𝐶1∗ , 𝐶𝐶𝐹𝐹𝐹𝐹
𝑈𝑈
≤ 𝐶𝐶2∗ ,
(4)
0 𝑖𝑖 – concentration cobalt in the original solution; ∆𝐶𝐶𝐶𝐶𝐶𝐶 where 𝐶𝐶𝐶𝐶𝐶𝐶 – cobalt losses in each of three reactors (i = 1, 2, 3); 𝑈𝑈 = 𝑔𝑔𝑐𝑐𝑐𝑐 – the selected control – consumption of caustic ash in each of reactors; 𝐶𝐶1∗ , 𝐶𝐶2∗ – the set values of concentrations of copper and iron are 0.01 g/l and 0.3 g/l, respectively.
In general, the description of dynamic processes occurring in the cascade of continuous reactors does not contradict the previous strategy of control adopted for the chemical and technological purification system. According to the system approach chosen, methods for identifying a mathematical model of a cascade of reactors at the fine purification stage were proposed and a successive approximation method was adopted to identify the characteristics of the subsystems. Such models are applicable both for solving control problems and predicting future states of systems.
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IFAC TECIS 2018 Baku, Azerbaidschan, Sept 13-15, 2018 Nataliya V. Mokrova et al. / IFAC PapersOnLine 51-30 (2018) 646–649
The problems of optimal control over the cobalt solution fine purification process on the basis of mathematical modelling of static and dynamic modes, as well as the control problem, were solved using the decomposition approach. A great attention was devoted to both adequate mathematical modelling of each reactor and solving control tasks at each stage, because when under-researched, individual phenomena and processes will not allow us to have a completely mathematically formalized description of the object; this also applies to the highlighting of hierarchy levels and establishing relations between phenomena. An important aspect in the implementation of the system approach is the use of analytical data, experimental data and observations.
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Mokrova, N. V. (2018). Methodology and Results of Mathematical Modelling of Complex Technological Processes. IOP Conference Series: Materials Science and Engineering, 317, 012038.
5. CONCLUSIONS The system analysis methodology allowed to select the levels of the hierarchical structure of the high-technology systems; we examined their interrelations, determined the aspects of their mutual influence, elucidated physical and chemical phenomena in each device and took into account characteristics of main state variables and controlling and disturbing influences. The mathematical description of the stages helped us find more rational ways of running the process and solve optimization problems. The models of multistage technological processes are applicable for solving control problems and predicting future states of sophisticated high-technology systems. Subsystems are segregate depending on the complexity of the object on the basis study of process physical and chemical regularities, availability or possibility of obtaining a mathematical description of each stage. This allows you to get a synergistic effect. The potential of flows and subsystems detected and identify loss sources and ways to minimize that. The capability for increasing the efficiency of individual devices and the entire system was studied and that is very important in technological industries. A quantitative and quality system analysis and mathematical modelling of complex technological processes employing state-of-the-art computing facilities is necessary both for the development of new processes and intensification of existing ones. REFERENCES Artamonov, A. G., Volodin, V. M. and Avdeev, V. G. (1989). Mathematical Modelling and Optimization of Plasmachemical Processes. Moscow, Himia. Lasdon, L. S. (2011). Optimization Theory for Large Systems. New York, Dover Publications. Mesarovic, M. D. and Takahara, Y. (1975). General Systems Theory: Mathematical Foundations. Amsterdam, Elsevier Science. Mokrova, N. V. and Volodin, V. M. (2006). Substantiation of the choice of methods for solving the problem of optimal control of complex processes. Bulletin of TSTU, 12, 22– 28. Mokrova, N. V. and Volodin, V. M. (2007). Modeling of decomposition control of multistage processes. Chemical and Oil and Gas Engineering, 2, 17–19. 649