Scientific base of application conceptual, design, technological and materials science solutions for all stages of the creation of machines and structures should be the principles and methods of physical and mathematical modeling.
Physical and mathematical modeling in mechanical engineering, it is based on general approaches developed on the basis of fundamental sciences, primarily mathematics, physics, chemistry, etc. Mathematical modeling and computational experiment are becoming a new method for analyzing complex machines, work processes, and the machine-man-environment system. Physical and mathematical modeling is carried out in several stages.
Simulation starts from setting and refining the problem, considering physical aspects, determining the degree of influence on the simulated processes of various factors in the programmed conditions for the functioning of the simulated systems or process. On this basis, a physical model is built. Then, on its basis, a mathematical model is built, which includes a mathematical description of the simulated process or mechanical system in accordance with the laws of kinematics and dynamics, the behavior of materials under the influence of loads and temperatures, etc. The model is studied in such areas as compliance with the task, existence solutions, etc.
At the next stage a computational algorithm for solving the modeling problem is chosen. Modern numerical methods make it possible to remove restrictions on the degree of complexity of mathematical models.
The next step is programming computational algorithm for computers. At the same time, problem-oriented application packages are created, which make it possible to create complex programs on their basis for a comprehensive description of processes, machines, and machine systems.
At the next stage, calculations are performed on a computer according to the developed programs. In this case, a rational presentation of the final results is essential. The final stage involves the analysis of the obtained results, their comparison with the data of physical experiments on full-scale samples of products. If necessary, the task is to refine the selected mathematical model with subsequent repetition of the above steps.
After completion of work on physical and mathematical modeling, a general conclusion and conclusions are formed on design, technological and operational measures related to the creation of new materials and technologies, ensuring the conditions for reliable and safe operation of machines, meeting the requirements of ergonomics and ecology. The creation of new machines and structures with an increased level of operating parameters, environmental and ergonomic requirements is a complex complex problem, the effective solution of which is based on physical and mathematical modeling. The general scheme for using simulation at various stages of creating machines is shown in the picture below.
The development of a draft design provides building physical models based on the experience of creating prototypes. Mathematical models include new knowledge about the analysis and synthesis of structural and kinematic schemes, about the dynamic characteristics of the interaction between the main elements, taking into account working environments and processes. At the same stage, issues of ecology and ergonomics are formed and solved in general terms.
When developing a technical project there should be a transition to physical models of the main units tested in the laboratory. The mathematical support of the technical project includes computer-aided design systems.
Creation of fundamentally new machines (cars of the future) requires the improvement of methods of mathematical modeling and the construction of new models. This largely applies to the unique objects of new technology. (nuclear and thermonuclear power engineering, rocket, aviation and cryogenic technology), as well as to new technological, transport vehicles and devices (laser and pulse technological installations, magnetic suspension systems, deep-sea vehicles, adiabatic internal combustion engines, etc.). At the same time, super-powerful computers and expensive programs are needed to implement the tasks of mathematical modeling.
At the stage of detailed design, physical modeling involves the creation of mock-ups and test benches to verify design solutions. The mathematical side of this stage is connected with the development of automated systems for the preparation of technical documentation. Mathematical models are refined as the detailing and refinement of the boundary conditions of design problems.
Simultaneously with the design the design and technological problems of choosing materials, assigning manufacturing and control technologies are solved. In the field of structural materials science, the experimental determination of physical and mechanical properties on laboratory samples is used both in standard tests and in tests under conditions simulating operational ones. In the manufacture of highly responsible parts and assemblies from new materials (high-strength corrosion and radiation resistant, clad, composite, etc.) it is necessary to carry out specialized tests to determine the limit states and damage criteria. Mathematical modeling is used to build simulation models of the mechanical behavior of materials under various loading conditions, taking into account the technology for obtaining materials and shaping machine parts. Simulation models are used to perform complex mathematical analysis of thermal, diffusion, electromagnetic and other phenomena associated with new technologies.
Based on physical and simulation models get a complex set of physical and mechanical properties, the characteristics of which should be used when creating computer-based data banks on modern and promising materials.
At the stage of developing the technology for manufacturing parts, assemblies and machines as a whole, physical modeling is used in laboratory and pilot testing of technological processes as traditional (machining, casting, etc.), and new (laser processing, plasma, explosive, magnetic-pulse, etc.).
Parallel to technological processes physical models are developed, as well as the principles of control and flaw detection of materials and finished products. Mathematical models of technological processes allow solving complex problems of thermal conductivity, thermoelasticity, superplasticity, wave and other phenomena in order to rationally choose effective methods and processing parameters for these parts.
At the stage of creating machines and structures When fine-tuning and testing of prototypes and pilot batches is carried out, physical modeling provides for bench and full-scale tests. Bench tests provide high information content and reduce the time for finishing prototypes of products of mass and large-scale production. Full-scale tests* are necessary to assess the performance and reliability of unique products in extreme conditions. At the same time, algorithms and test management programs become the tasks of mathematical modeling. The analysis of the obtained experimental information should be carried out on a computer in real time.
When operating machines physical modeling is used to diagnose the condition and justify the extension of the safe operation life. Mathematical modeling at this stage aims to build models of operational damage according to a set of criteria adopted in the design: The development of such models is currently being carried out for objects of nuclear and thermal power engineering, rocket and aviation technology and other objects.
Mathematical modeling allows to automate the control of operating modes with the help of a computer according to specified programs, to ensure optimal control of transient processes and to exclude, with the help of automated protection systems, the achievement of limiting situations leading to emergency failures.
Under object modeling understand any subject, process or phenomenon that is studied by modeling. When studying an object, only those properties that are necessary to achieve the goal are taken into account. The choice of object properties when building a model is an important task at the first stages of modeling.
Object Model - this is:
1) such a mentally representable or materially realized system that, displaying or reproducing the object of study, is able to replace it in such a way that its study provides new information about the object.
2) an object is a substitute that takes into account the real properties of the object necessary to achieve the goal.
The main function of the model is not only the description of the object, but also obtaining information about it.
There are physical and mathematical modeling.
Physical modeling- a method of experimental study of various physical phenomena, based on their physical likeness. The method is applied under the following conditions:
- An exhaustively accurate mathematical description of the phenomenon at this level of development of science does not exist, or such a description is too cumbersome and requires a large amount of initial data for calculations, which is difficult to obtain.
- Reproduction of the studied physical phenomenon for the purpose of experiment on a real scale is impossible, undesirable or too expensive (for example, a tsunami).
In a broad sense, any laboratory physical experiment is a simulation, since a specific case of a phenomenon is observed in the experiment under particular conditions, and it is required to obtain general patterns for the entire class of similar phenomena in a wide range of conditions. The art of the experimenter is to achieve physical likeness between a phenomenon observed in laboratory conditions and the entire class of phenomena under study.
Math modeling, in a broad sense, includes research not only with the help of purely mathematical models. Informational, logical, simulation and other models and their combinations are also used here. In this case, the mathematical model is an algorithm that includes determining the relationship between the characteristics, parameters and calculation criteria, the conditions for the process of the system functioning, etc. This structure can become a model of a phenomenon if it adequately reflects its physical essence, correctly describes the relationship of properties, and is confirmed by the test results. The use of mathematical models and computer technology implements one of the most effective methods of scientific research - a computational experiment that makes it possible to study the behavior of complex systems that are difficult to physically model. Often this is due to the great complexity and cost of objects, and in some cases the inability to reproduce the experiment in real conditions.
The effectiveness of the use of information systems in the field of education. Tasks solved by IS in the field of education. The specifics of the information needs of teaching and management personnel in the field of education. The main indicators of the quality of information support in the field of education and the justification of the requirements for their quantitative values
In modern society, the use of information technology in all spheres of life has become a mandatory accompanying component. A particularly important role is assigned to its application in the field of cognition, study, i.e. in the field of education. IT technologies occupy one of the leading places in the intellectualization of a person and society as a whole, raising the cultural and educational level of every citizen.
Recently, in the field of education, information technologies based on the latest computer and audiovisual achievements of science and technology are increasingly being used. One of the effective directions for the implementation of educational services is the use of various forms of education based on information and training technologies.
In addition, the desire to actively apply modern information technologies in the field of education should be focused on improving the level and quality of training. Every year the number of organizations and enterprises applying to the market of educational services is growing. In this regard, the most favorable conditions are those educational institutions that include pre-university, university and postgraduate education using new educational technologies.
At present, the role of information and social technologies in education is increasing, which provide universal computerization of students and teachers at a level that allows solving at least three main tasks:
- providing access to the Internet for each participant in the educational process, and, preferably, at any time and from various places of stay;
- development of a single information space of educational industries and the presence in it at different times and independently of each other of all participants in the educational and creative process;
– creation, development and effective use of managed information educational resources, including personal user databases and data and knowledge banks of students and teachers with the possibility of widespread access to work with them.
The main advantages of modern information technologies are: visibility, the ability to use combined forms of information presentation - data, stereo sound, graphics, animation, processing and storage of large amounts of information, access to world information resources, which should become the basis for supporting the education process.
The need to strengthen the role of independent work of the student requires significant changes in the structure and organization of the educational process, improving the efficiency and quality of training, enhancing the motivation of cognitive activity in the course of studying theoretical and practical educational material in a particular discipline.
In the process of informatization of education, it must be borne in mind that the main principle of using a computer is to focus on those cases when a person cannot complete the pedagogical task. For example, a teacher cannot visually demonstrate most of the physical processes without computer simulation.
On the other hand, a computer should help develop the creative abilities of students, promote learning new professional skills and abilities, and develop logical thinking. The learning process should not be aimed at the ability to work with certain software tools, but at improving the technology of working with various information: audio and video, graphics, text, tables.
Modern multimedia technologies and tools make it possible to implement the whole range of computer training programs. However, their use requires highly qualified users from teachers.
Scientific research related to the creation of new machines
The main areas of scientific research related to improving the quality, reliability and safety of machinery and equipment are:
fundamental research in the field of new work processes, resource-saving technologies and new construction materials;
creation, development and implementation of modern methods of machine design, substantiation of their optimal operating parameters, design forms;
obtaining new materials, developing parts, assemblies and assemblies in compliance with the requirements for technological parameters;
development of new metrological methods, systems and means;
conducting accelerated and conventional tests for reliability and service life of models and full-scale products;
organization of operation of machines with a given degree of reliability, safety, efficiency, while observing the requirements of ergonomics and ecology.
Of paramount importance in modern mechanical engineering are the problems of reliability and safety of equipment, taking into account the role of the human factor.
The scientific basis for the application of conceptual, design, technological and materials science solutions for all stages of the creation of machines and structures should be the principles and methods of physical and mathematical modeling.
Physical and mathematical modeling in mechanical engineering is based on general approaches developed on the basis of fundamental sciences, primarily mathematics, physics, chemistry, etc.
Mathematical modeling and computational experiment are becoming a new method for the analysis of complex machines, workflows and the machine-man-environment system. Physical and mathematical modeling is carried out in several stages.
Modeling begins with the formulation and refinement of the problem, consideration of physical aspects, determination of the degree of influence on the simulated processes of various factors in the programmed conditions for the functioning of the simulated systems or process. On this basis, a physical model is built.
Then, on its basis, a mathematical model is built, which includes a mathematical description of the simulated process or mechanical system in accordance with the laws of kinematics and dynamics, the behavior of materials under the influence of loads and temperatures, etc. The model is studied in such areas as compliance with the task, existence solutions, etc.
At the third stage, a computational algorithm for solving the modeling problem is selected. Modern numerical methods make it possible to remove restrictions on the degree of complexity of mathematical models.
Further, using modern mathematical software packages, such as MathCad, Matlab, which have a large set of capabilities and functions and allow solving problems both analytically and numerically, conduct computational experiments.
When carrying out calculations and obtaining results, special attention must be paid to the literacy and correctness of the presentation of solutions.
The final stage involves the analysis of the obtained results, their comparison with the data of physical experiments on full-scale samples of products. If necessary, the task is to refine the selected mathematical model with subsequent repetition of the above steps.
After completion of work on physical and mathematical modeling, a general conclusion and conclusions are formed on design, technological and operational measures related to the creation of new materials and technologies, ensuring the conditions for reliable and safe operation of machines, meeting the requirements of ergonomics and ecology.
Recently, purely mathematical modeling is extremely rare in the design and construction of mechanisms and parts. Traditional mathematical modeling in the design of modern mechanisms and parts is replaced by computer modeling. The main method used by modern software products is the finite element method. Such modeling, in addition to calculation accuracy and a visual representation of the behavior of the object of study under given conditions, speeds up the design process and reduces the cost of conducting research with physical models.
The creation of new machines and structures with an increased level of operating parameters, environmental and ergonomic requirements is a complex complex problem, the effective solution of which is based on physical and mathematical modeling.
The development of a preliminary design involves the construction of physical models based on the experience of creating prototypes. Mathematical models include new knowledge about the analysis and synthesis of structural and kinematic schemes, about the dynamic characteristics of the interaction between the main elements, taking into account working environments and processes. At the same stage, issues of ecology and ergonomics are formed and solved in general terms.
When developing a technical project, a transition should be made to physical models of the main components tested in laboratory conditions. The mathematical support of the technical project includes computer-aided design systems.
The creation of fundamentally new machines (machines of the future) requires the improvement of mathematical modeling methods and the construction of new models. This applies to a large extent to the unique objects of new technology (nuclear and thermonuclear power engineering, rocket, aviation and cryogenic technology), as well as to new technological, transport vehicles and devices (laser and pulse technological installations, magnetic suspension systems, deep-sea vehicles, adiabatic internal combustion engines, etc.).
At the stage of detailed design, physical modeling involves the creation of mock-ups and test benches to verify design solutions. The mathematical side of this stage is connected with the development of automated systems for the preparation of technical documentation. Mathematical models are refined as the detailing and refinement of the boundary conditions of design problems.
Simultaneously with the design, the design and technological problems of choosing materials, assigning manufacturing and control technologies are solved. In the field of structural materials science, the experimental determination of physical and mechanical properties on laboratory samples is used both in standard tests and in tests under conditions simulating operational ones. In the manufacture of highly critical parts and assemblies from new materials (high-strength corrosion and radiation resistant, clad, composite, etc.), it is necessary to carry out specialized tests to determine the limit states and damage criteria. Mathematical modeling is used to build simulation models of the mechanical behavior of materials under various loading conditions, taking into account the technology for obtaining materials and shaping machine parts. Simulation models are used to perform complex mathematical analysis of thermal, diffusion, electromagnetic and other phenomena associated with new technologies.
On the basis of physical and simulation models, a complex set of physical and mechanical properties is obtained, the characteristics of which should be used when creating computer-based data banks on modern and promising materials.
At the stage of developing the technology for manufacturing parts, assemblies and machines as a whole, physical modeling is used in laboratory and pilot testing of technological processes, both traditional (machining, casting, etc.) and new ones (laser processing, plasma, explosive, magnetic-pulse and etc.).
In parallel with the technological processes, physical models are being developed, as well as "the principles of control and flaw detection of materials and finished products. Mathematical models of technological processes allow solving complex problems of thermal conductivity, thermoelasticity, superplasticity, wave and other phenomena in order to rationally choose effective methods and parameters for these parts processing.
At the stage of creating machines and structures, when fine-tuning and testing of prototypes and pilot batches is carried out, physical modeling provides for bench and full-scale tests. Bench tests provide high information content and reduce the time for finishing prototypes of products of mass and large-scale production. Full-scale tests are necessary to assess the performance and reliability of unique products in extreme conditions. At the same time, algorithms and test management programs become the tasks of mathematical modeling. The analysis of the obtained experimental information should be carried out on a computer in real time.
During the operation of machines, physical modeling is used to diagnose the condition and justify the extension of the safe operation life. Mathematical (computer) modeling at this stage aims to build models of operational damage according to a set of criteria adopted in the design: The development of such models is currently being carried out for objects of nuclear and thermal power engineering, rocket and aviation technology and other objects.
Poets know that everything is similar to everything. The creation of metaphors is based on this position:
In the garden, a fire of red rowan is burning,
But he cannot warm anyone.
Modeling is based on the same position. Modeling is the construction and exploration of models. In turn, a model is a certain system, by examining which one obtains information about another system.
At first glance, this seems like nonsense. Is it possible to look at one object and get an idea of another object? Where is the sea, and where is that cottage?
Meanwhile, to look at ourselves from the outside, we use a mirror. At the same time, we identify our reflection in the mirror glass with ourselves. Although our reflection is somewhat different from the original. For example, right and left in the mirror are reversed. But we almost automatically make allowance for this difference, which is not essential in this case, and use the mirror to our advantage and greater convenience. All boys leave the mirror clean and combed. And the girls are absolutely beautiful!
The model, metaphorically speaking, is such a mirror attached to the subject under study.
By creating a model, we decide which properties of the system under study are important to us, and which are secondary. For example, when studying the wings of aircraft in a wind tunnel, we are interested in their shape and the material from which they are made. The color of the wings in this case is not significant. Although when calculating the visibility of an aircraft, the color of its planes will probably be the most important information.
Having decided on the main and not the main properties of the system or object being modeled, we establish certain relationships between the properties of the system and its model. For example, if the size of a house model is half the size of a real house, the volume, and therefore the weight of the model, will be eight times less than the real one.
Then we begin the study of the model and determine the various relationships between the parameters of interest to us. For example, at what speed of the air flow will the wing vibrate. This is a formulation of the problem of flutter, oscillations of an aircraft that suddenly arise at certain values of the speed of the air flow around the wing. Without a solution to this problem, aircraft would not be able to fly at high speeds. To solve it, it was necessary to observe the destruction of a large number of wing models in a wind tunnel. Here we immediately see the advantages of modeling. We test for strength not an expensive aircraft, but a cheap model, recalculating the properties of the model into the properties of a simulated real aircraft. Cost savings, and most importantly, test pilots should not risk their lives.
Another area of application of models is the strength of materials and structural mechanics. How strong should steel be for a bridge? How thick should the supporting columns be made so that the building does not collapse? Is it possible to build a skyscraper out of bricks? Here, the real material model is a sample being tested on special test benches. The strength characteristics obtained from the test results are converted into the strength characteristics of real parts of machines or buildings.
And when “settlement” of a new building, modeling is also indispensable. In order to optimally arrange the furniture in the rooms, no one drags heavy tables and bulky refrigerators back and forth. All objects are modeled by small paper rectangles that move along the surface of a paper sheet with a floor plan depicted on it.
Yes, and in medicine, we can not do without modeling. No person is exactly like the other. At the same time, all human organisms have enough similarities, both in "details" and in "functions". A physician studies anatomy from a single skeleton, and sometimes even from a model skeleton, and understands how all people are arranged. A psychologist studies how a particular person reacts to certain stimuli, and then draws general conclusions about the behavior of all people.
Modeling is of two types - mathematical and physical. In mathematical modeling, systems of relationships are studied that describe the processes occurring in the object being modeled. Relationships can be described by equations, often quite complex, which are derived on the basis of a theoretical model of the process or system under study. But mathematical models can also be probabilistic. In such models, changes in the input parameters determine the behavior of the output parameters not rigidly, but with a certain degree of probability.
A mathematical model is always a compromise between the real complexity of the system under study and the simplicity required to describe it. There are not always "qualitative" theories that allow you to accurately calculate what happens, for example, when there is a voltage drop in large electrical networks. Yes, even the behavior of the flow of water flushed into the toilet, depending on its shape, is a serious theoretical problem.
In physical modeling, the properties of models are studied, which are similar in physical properties to the originals. For example, in car crash tests, the set of cars being crashed simulates the behavior of any car that will eventually be put on the road.
The study of physical models is carried out on real installations or test benches. Test results are translated into real results using calculations based on a special mathematical apparatus, which is called the theory of similarity. An example of the testing of physical models is the already described testing of aircraft models in a wind tunnel. Or the calculation of a hydroelectric dam. The disadvantage of physical modeling is the relative laboriousness of creating and testing models and the lower universality of the physical modeling method.
But in any case, physical and mathematical modeling, complementing each other, allow us to change our world in the desired direction.
TYPES OF CHEMICAL REACTORS
A chemical reactor is a device designed to carry out chemical transformations.
Chemical reactor - a generalized concept, refers to reactors, columns, towers, autoclaves, chambers, furnaces, contact devices, polymerizers, hydrogenators, oxidizers and other devices, the names of which come from their purpose or even appearance. A general view of the reactor and diagrams of some of them are shown in fig. 4.1.
The capacitive reactor / is equipped with a stirrer that mixes the reagents (usually liquids, suspensions) placed inside the apparatus. The temperature regime is maintained by means of a coolant circulating in the reactor jacket or in a heat exchanger built into it. After the reaction, the products are unloaded, and after cleaning the reactor, the cycle is repeated. The process is periodic.
capacitive reactor 2 is flowing, because reagents (usually gas, liquid, suspension) continuously pass through it. The gas bubbles through the liquid.
column reactor 3 characterized by the ratio of height to diameter. which for industrial reactors is 4-6 (in capacitive reactors this ratio is about 1). The interaction of gas and liquid is the same as in reactor 2
Packed reactor 4 is equipped with Raschig rings or other small elements - packing. Gas and liquid interact. The liquid flows down the nozzle, and the gas moves between the elements of the nozzle.
Reactors 5-8 mainly use the days of interaction of gas with a solid reagent.
In the reactor 5, the solid reactant is immobile, the gaseous or liquid reactant passes continuously through it. The process is periodic in solid matter.
Reactors 6~ 8 modified in such a way that the process is continuous with respect to the solid reagent. The solid reagent moves along the rotating obliquely mounted round reactor, and wakes up through the reactor 7. In the reactor 8 gas will be supplied from below under high pressure so that the solid particles are in suspension, forming a fluidized, or boiling, layer, which has some of the properties of a liquid.
Tubular reactor 9 similar in appearance to a shell-and-tube heat exchanger. Gaseous or liquid reagents pass through the tubes in which the reaction proceeds. Typically, the tubes are loaded with a catalyst. The temperature regime is provided by the circulation of the coolant in the annulus.
Reactors 5 and 9 also used for processes on a solid catalyst.
Tubular reactor 10 often used to carry out high-temperature homogeneous reactions, including in a viscous liquid (for example, pyrolysis of heavy hydrocarbons). Often such reactors are called furnaces.
Multilayer reactor 11 equipped with a system that allows you to cool or heat the reagent located between several layers of a solid substance that acts as, for example, a catalyst. The figure shows the cooling of the initial gaseous substance by cold gas introduced between the upper layers of the catalyst and the coolant through a system of heat exchangers placed between other layers of the catalyst.
Multilayer reactor 12 designed for carrying out gas-liquid processes in it.
Shown in fig. 4.1 diagrams show only a part of the reactors used in industry. However, the further systematization of reactor designs and ongoing processes makes it possible to understand and conduct research in any of them.
All reactors are characterized by common structural elements presented in the reactor in Fig. 4.2, similar 11 -mu in fig. 4.1.
reaction zone 7, in which a chemical reaction takes place, represent several layers of catalyst. It is in all reactors: in reactors 1-3 in fig. 4.1 is a liquid layer, in reactors 4, 5, 7 - packing layer or solid component, in reactors 6, 8 - part of the volume of the reactor with a solid component, in reactors 9, 10 is the internal volume of the tubes where the reaction takes place.
The initial reaction mixture is fed through the top fitting. A flow distributor is installed to ensure evenly distributed gas passage through the reaction zone, causing a uniform contact of the reactants. Ego - input device 2. in the reactor 2 in fig. 4.1 The gas distributor is a bubbler, in the reactor 4 - sprinkler.
Between the first from above and the second layers, the two streams are mixed in mixer 3. Between the second and third layers heat exchanger 4. These structural elements are designed to change the composition and temperature of the flow between the reaction zones. Heat exchange with the reaction zone (removal of heat released as a result of exothermic reactions or heating of the reacting mixture) is carried out through the surface of the built-in heat exchangers.
kov or through the inner surface of the reactor jacket (apparatus 1 in fig. 4.1), or through the walls of pipes in reactors P, 10. The reactor can be equipped with flow separation devices.
Products are shipped via output device 5.
In heat exchangers and devices for input, output, mixing, separation, distribution of flows, physical processes take place. Chemical reactions are carried out mainly in the reaction zones, which will be a further object of study. The process taking place in the reaction zone is a set of partial stages, which are schematically shown in Fig. 4.3 for catalytic and gas-liquid interaction.
Rice. 4.3, a represents a scheme of the reaction process with the participation of a catalyst, through the fixed bed of which a common
(convective) flow of gaseous reactants (7). Reagents diffuse to the grain surface (2) and penetrate into the pores of the catalyst ( 3 ), on the inner surface of which the reaction ( 4 ). The resulting reaction products are discharged back into the stream. The heat released as a result of chemical transformation is transferred through the layer (5) due to thermal conductivity, and from the layer through the wall to the refrigerant (b). The emerging concentration and temperature gradients cause additional flows of heat and matter (7) to the main convective movement of the reagents in the layer.
On fig. 4.3, b a process in a liquid layer through which a gas bubbles is presented. Between bubbles (/) of gas and liquid there is a mass exchange of reagents ( 2 ). Fluid dynamics is composed of motion around bubbles (.?) and circulation at the layer scale (4). The first is similar to turbulent diffusion, the second is similar to the circulation convective movement of liquid through the reaction zone. In a liquid and, in the general case, in a gas, a chemical transformation occurs (5).
The given examples show the complex structure of the processes taking place in the reaction zone. If we take into account the many schemes and designs of existing reactors, then the variety of processes in them increases many times. "A scientific method is needed to systematize this diversity, find commonality in it, develop a system of ideas about the patterns of phenomena and relationships between them, i.e. create a theory of chemical processes and reactors.Such a scientific method is discussed below.
4. Use of methods and principles of systematic research in the development of CTS
4.2. MATH MODELING
AS A METHOD FOR STUDYING CHEMICAL PROCESSES AND REACTORS
Model and simulation. Modeling - the method of studying an object (phenomenon, process, device) on a model has long been used in various fields of science and technology in order to study the object itself by studying its model. The resulting model properties are transferred to the properties of the modeled object.
Model- an object of any nature specially created for study, simpler than the one under study, in all properties, except for those that need to be studied, and capable of replacing the object under study in such a way as to obtain new information about it.
The phenomena and parameters taken into account in each model are called constituents models.
To study the different properties of an object, several models can be created, each of which meets a specific goal of the study, however, one model can provide the necessary information about several studied parameters, then we can talk about the unity of the “goal-model”. If the model reflects more (or fewer) properties, then it is called wide(or narrow). The sometimes used concept of "general model" as reflecting all the properties of an object is meaningless in essence.
To achieve the goal, the model under study must be influenced by the same factors as the object. The components and parameters of the process that affect the properties under study are called essential components models. Changing some parameters may have very little effect on the properties of the object. Such components and parameters are called insignificant, and they can be ignored in the construction of the model. Respectively, simple the model contains only essential components, otherwise the model will be excess therefore, a simple model is not simple in appearance (for example, simple in structure or construction). But if the model does not include all the components that significantly affect the properties under study, then it will be incomplete, and the results of its study may not accurately predict the behavior of a real object. This is the essence of creativity and scientific approach to building a model - to single out exactly those phenomena and take into account exactly those parameters that are essential for the properties under study.
In addition to predicting given properties, the model must provide information about the unknown properties of the object. This can only be achieved if the model is simple and complete, then new properties can appear in it.
Physical and mathematical modeling
An example of physical modeling is the study of air flow around an aircraft on a model in a wind tunnel.
In this research method, the similarity of phenomena (processes) in objects of different scales is established, based on the quantitative relationship between the quantities characterizing these phenomena. These quantities are: the geometric characteristics of the object (shape and dimensions); mechanical, thermophysical and physicochemical properties of the working medium (movement speed, density, heat capacity, viscosity, thermal conductivity, etc.); process parameters (hydraulic resistance, heat transfer coefficients, mass transfer, etc.). The developed similarity theory establishes certain relationships between them, called similarity criteria. Usually they are denoted by the initial letters of the names of famous scientists and researchers (for example, Re - Reynolds criterion, Nu - Nusselt criterion, Ag - Archimedes criterion). To characterize any phenomenon (heat transfer, mass transfer, etc.), dependencies between similarity criteria are established - criterion equations.
Physical modeling and similarity theory have found wide application in chemical technology in the study of thermal and diffusion processes. Criteria equations for calculating some parameters of heat and mass transfer will be used below.
Attempts to use the theory of similarity for chemical processes and reactors have been unsuccessful due to its limited application. The reasons are as follows. Chemical transformation depends on the phenomena of heat and substance transfer, since they create the appropriate temperature and concentration conditions at the site of the reaction. In turn, a chemical reaction changes the composition and heat content (and, accordingly, the temperature) of the reacting mixture, which changes the transfer of heat and matter. Thus, the chemical (transformation of substances) and physical (transfer) components are involved in the reaction technological process. In a small apparatus, the released heat of reaction is easily lost and has little effect on the conversion rate; therefore, the main contribution to the results of the process is made by the chemical component. In a large apparatus, the released heat is "locked" in the reactor, significantly changing the temperature field and, consequently, the rate and result of the reaction. Consequently
the chemical and physical components of the reaction process are generally scale dependent.
Another reason is the incompatibility of day similarity conditions for the chemical and physical components of the process in reactors of different sizes. For example, the conversion of reagents depends on their residence time in the reactor, which is equal to the ratio of the size of the apparatus to the flow rate. The conditions of heat and mass transfer, as follows from the similarity theory, depend on the Reynolds criterion, which is proportional to the product of the size of the apparatus and the flow rate. It is impossible to make both the ratio and the product of two quantities (in this example, size and speed) the same in devices of different scales.
The difficulties of a scale transition of an object to a model for reaction processes can be overcome using mathematical modeling, in which the model and the object have a different physical nature, but the same properties. For example, a mechanical pendulum and a closed electrical circuit consisting of a capacitor and an inductor have a different physical nature, but the same property: oscillation (mechanical and electrical, respectively).
The properties of these devices are described by the same oscillation equation:
.
Hence the name of the type of modeling - mathematical. Device parameters (lM /g - for the pendulum and LC for the electrical circuit), can be selected so that the oscillations in frequency are the same. Then the electric oscillatory circuit will be a pendulum model. You can also explore the solution of the above equation and predict the properties of the pendulum. Accordingly, mathematical models are divided into real, represented by some physical device, and iconic, represented by mathematical equations. The classification of models is shown in fig. 4.4.
To build a real mathematical model, you must first create a sign model, and usually a mathematical model is identified with the equations that describe the object. A universal real mathematical model is an electronic computing
machine (computer). According to the equations describing the object, the computer is "tuned" (programmed), and its "behavior" will be described by these equations. Further, it is the sign mathematical model that we will call the mathematical model of the process.
On the similarity of mathematical models of different processes. As has already been shown, the processes of movement of a mechanical pendulum and changes in the current strength in an electric circuit can be represented by the same mathematical models, i.e. be described by the same second-order differential equation. The solution of this equation is the function x(/), which indicates the oscillatory type of motion of these objects of different nature. From the solution of the equation, it is also possible to determine the change in time of the position of the pendulum relative to the vertical axis or the change in time of the direction of the current and its magnitude. This is the interpretation of the properties of the mathematical model for the indicators of the objects under study. 13 this reveals a very useful feature of mathematical modeling. Different processes can be described by similar mathematical models. Such "universality" of the mathematical model is manifested in the study, for example, of processes in a capacitive J and tubular 9 reactors in fig. 4.1 (see Section 4.1), studying the interaction of a gaseous reagent with a solid particle and a heterogeneous catalytic process (Sections 4.5.2 and 4.5.3), considering critical phenomena on a single catalyst grain and in the reactor volume
Mathematical modeling of chemical processes and reactors. AT
In general, mathematical modeling of reactors can be represented in the form of a diagram shown in Fig. 4.5. Since the influence of physical and chemical components (phenomena) on the reaction process is different in reaction processes of various scales, the identification of these phenomena and their interaction is analysis- the most essential moment in mathematical modeling of chemical processes and reactors. The next step is to determine the thermodynamic and kinetic patterns for chemical transformations (chemical phenomena), transport phenomena parameters (physical phenomena) and them interaction. To do this, use the data of experimental studies, mathematical modeling does not exclude the experiment, but actively uses it, but the experiment is precision, aimed at studying the patterns of individual components of the process. The results of the analysis of the process and the study of its components make it possible to build a mathematical model of the process (stage synthesis on fig. 4.5) are the equations describing it. The model is created on the basis of the fundamental laws of nature, for example, the conservation of mass and energy, the obtained information about individual phenomena and the established interactions between them. Model study is directed to the study of its properties, while using the mathematical apparatus of qualitative analysis and computational methods, or, as they say, a computational experiment is carried out. The resulting model properties are as follows interpret as properties of the object under study, which in this case is a chemical reactor. For example, mathematical dependence y( m) must be presented as a change in the concentration of substances along the length of the reactor or over time, and several roots of the equation should be interpreted as ambiguity of modes, etc.
Nevertheless, even the approximate scheme of the process in the catalyst bed (Fig. 4.3) includes quite a lot of components; accordingly, the process model will be quite complex, and the CC analysis is unnecessarily complicated. For a complex object (process), a special approach to building a model is used, which consists in dividing it into a number of simpler operations that differ in scale. For example, in the catalytic process, the following stand out: the reaction on the surface of the grain, the process on a single catalyst grain, and the process in the catalyst bed.
catalytic reaction- a complex multi-stage process occurring on the scale of molecular size. The reaction rate is determined by the conditions of its occurrence (concentration and temperature) and does not depend on where such conditions are created: in a small or large reactor, i.e. does not depend on scale the whole process. Izu
The study of the complex mechanism of the reaction makes it possible to construct its kinetic model, i.e., the equation for the dependence of the reaction rate on the conditions of its occurrence. It is clear that this model will be much simpler than the system of equations for all stages of the reaction, and its study will be informative.
Process on a single catalyst grain, a few millimeters in size, includes the reaction represented by the ss kinetic model, and the transfer of matter and heat in the pores of the grain and between its outer surface and the flow around. The transformation on the grain is determined by the conditions of the process - the composition, temperature and velocity of the flow and does not depend on where such conditions are created - in a small or large reactor, i.e. does not depend on scale the whole process. The analysis of the obtained model makes it possible to obtain the properties of the process, for example, the rate of transformation in the form of a dependence only on the conditions of its occurrence - the observed rate of transformation.
Catalyst bed process includes the process on grain, for which patterns have already been identified, and the transfer of heat and matter at the layer scale.
Isolation of simple stages in a complex process, which differ in the scale of flow, allows us to build hierarchical system of models, each of which has its own scale and, most importantly, the properties of such a system do not depend on the scale of the entire process (scale-invariant).
In general, a model of the reaction process, built according to a hierarchical principle, can be represented by a diagram (Fig. 4.6).
Chemical reaction, consisting of elementary steps, proceeds on a molecular scale. Its properties (for example, speed) do not depend on the scale of the reactor; the reaction rate depends only on the conditions of its occurrence, regardless of how or where they are created. The result of research at this level is a kinetic model of a chemical reaction - the dependence of the reaction rate on conditions. The next big level - chemical process- a combination of chemical reaction and transfer phenomena, such as: diffusion and heat conduction. At this stage, the kinetic model of the reaction is one of the components of the process, and the volume in which the chemical process is considered is chosen with such conditions that the patterns of its flow do not depend on the size of the reactor. For example, it may be the catalyst grain discussed above. Further, the resulting model of the chemical process, as one of the constituent elements, in turn, is included in the next scale level - the reaction zone which also includes the structural regularities of the flow, and the phenomena of transfer in the cc scale. AND,
finally to scale reactor the components of the process include the reaction zone, mixing units, heat exchange units, etc. Thus, the mathematical model of the process in the reactor is represented by a system of mathematical models of different scales.
The hierarchical structure of the mathematical model of the process in the reactor allows:
7) fully describe the properties of the process through a detailed study of its main stages of different scales;
8) to study a complex process in parts, applying to each of them specific, precision research methods, which increases the accuracy and reliability of the results;
9) establish connections between individual parts and find out their role in the operation of the reactor as a whole;
10) facilitate the study of the process at higher levels;
11) solve the problems of large-scale transition.
In the further presentation of the material, the study of the process in a chemical reactor will be carried out using mathematical modeling.
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