Representation of a mathematical model of chemical-technological processes with acceleration through a continuous function

. This article discusses the construction of a mathematical model of accelerated chemical-technological processes. Usually, in practice, the problem of building models of multi-stage processes, as a rule, is complicated by the universality, uncertainty and nonlinearity of the objects being modeled, the complete or partial lack of expert experience and an analytical description of dependencies. Mathematical models describing chemical technologies are considered to break down the state into discrete stages. The study below allows us to recommend stage processes to be considered as continuous functions.


Introduction
Mathematical modeling has always been an important activity in science and engineering. The formulation of qualitative questions about an observed phenomenon as mathematical problems was the motivation for and an integral part of the development of mathematics from the very beginning. Although problem solving has been practised for a very long time, the use of mathematics as a very effective tool in problem solving has gained prominence in the last 50 years, mainly due to rapid developments in computing. Computational power is particularly important in modeling chemical engineering systems, as the physical and chemical laws governing these processes are complex. Besides heat, mass, and momentum transfer, these processes may also include chemical reactions, reaction heat, adsorption, desorption, phase transition, multiphase flow, etc. This makes modeling challenging but also necessary to understand complex interactions. All models are abstractions of real systems and processes. Nevertheless, they serve as tools for engineers and scientists to develop an understanding of important systems and processes using mathematical equations. In a chemical engineering context, mathematical modeling is a prerequisite for:  design and scale-up;  process control;  optimization;  mechanistic understanding;  evaluation/planning of experiments;  trouble shooting and diagnostics;  determining quantities that cannot be measured directly;  simulation instead of costly experiments in the development lab;  feasibility studies to determine potential before building prototype equipment or devices.
A typical problem in chemical engineering concerns scale-up from laboratory to fullscale equipment. To be able to scale-up with some certainty, the fundamental mechanisms have to be evaluated and formulated in mathematical terms. This involves careful experimental work in close connection to the theoretical development. [1][2][3][4][5] Modern production is a complex of chemical and technological stages, including automated control and communication systems, financial, economic and marketing services, scientific and technological and design centers. As a result of the intensive development of digital technology and information processing tools, the functional significance of information management systems and information transfer processes in chemical, petrochemical and their internal complexes is increasing. When solving problems of analysis and synthesis of chemical-technological processes, as well as in matters of building control systems in these processes, it is advisable to apply methods of mathematical modeling [2]. Mathematical modeling is an effective weapon in determining the optimal control parameters, especially in cases where the laws of physical and chemical processes have been sufficiently studied. Based on this, in a wide range of external influences, the control parameters are determined by calculating the mathematical model of the object [3,[6][7][8][9].
Therefore, an important step in mathematical modeling is the creation of a mathematical model that would adequately describe the process under consideration. Usually, mathematical models of individual devices are created based on models of processes occurring in these devices, and then technological schemes are modeled that link these devices into a single technological process. Depending on the complexity of the process itself and the possibilities of obtaining experimental information about its passage, when developing mathematical models, either a deterministic approach is used, which is based on fundamental laws, or an empirical one, which is based on statistical processing of experimental information. Since mathematical models can be represented by linear, nonlinear, differential equations, partial differential equations and their systems, depending on the complexity of the phenomena being modeled, it is necessary to know and be able to apply numerical methods to solve them. In order for the solution of optimization problems to be feasible, it is necessary to correctly determine the optimality criteria, present the goal function, set restrictions on the optimizing parameters, and correctly choose the optimization method. [10][11][12][13][14][15][16][17] The results of experimental studies of various reactions have shown that the rate is influenced not only by the factors that determine the state of chemical equilibrium (temperature, pressure, composition of the reaction system), but also by other factors, such as the presence or absence of foreign substances that do not undergo changes as a result of the reaction, conditions for the physical transportation of reagents to reaction centers, etc. Factors that affect the rate of chemical transformation are usually divided into two groups: purely kinetic (microkinetic), which determine the rate of interaction at the molecular level, and macrokinetic, which determine the effect on the reaction rate of the conditions of transport of reagents to the reaction zone, the presence or absence of mixing, the geometric dimensions of the reactor. The laws of chemical kinetics are based on two simple principles, first established in the study of reactions in solutions: -the rate of a homogeneous reaction is proportional to the concentrations of the reactants; -the total rate of several successive transformations, which differ widely in rate, is determined by the rate of the slowest stage. The functional dependence of the rate of a chemical reaction on the concentrations of the components of the reaction mixture is called the kinetic equation of the reaction.

= ( , … )
In chemical kinetics, it is customary to divide reactions into elementary and nonelementary (complex). Elementary (one-stage) are called reactions, the implementation of which is associated with overcoming one energy barrier during the transition from one state of the reaction system to another. Kinetic equation of an irreversible elementary reaction aA + bB → products in accordance with the law of mass action has the form,

= • •
where k -is the proportionality factor included in the kinetic equation, it is called the rate constant of a chemical reaction.

Methods
Chemical reactions are the second stage of the chemical-technological process. In the reacting system, several consecutive (and sometimes parallel) chemical reactions usually occur, leading to the formation of the main product, as well as a number of side reactions between the main starting materials and impurities, the presence of which in the starting material is inevitable. As a result, in addition to the main product, by-products (materials of national economic importance) or waste and production waste are formed, i.e. reaction products that do not have significant value and do not find sufficient application. By-products and production wastes can be formed during the main reaction along with the target product, as well as due to side reactions between the main starting materials and impurities. Usually, when analyzing production processes, not all reactions are taken into account, but only those that have a decisive influence on the quantity and quality of the resulting target products.
When constructing a mathematical model of such processes, it is considered that in the initial period the process has the property of acceleration, and during the transition to the normal state, the property of speed ( ( ) = ). Usually, when constructing a mathematical model of the acceleration period, the values of the velocity at the initial ( (0) = ) and end points ( ( 0 ) = ) are used, based on the equation of the straight line passing through these points, i.e., the acceleration is expressed as: The general mathematical model of the process speed is as follows: The branched function ( ) has no derivative at = 0 because 0 approaches the point from the left lim − → 0 − = ( 0 ) (3) and when approaching from the right lim + → 0 ( ) = ( 0 ) = (4) (where: − , + tends to be less than and greater than 0 respectively.) takes on different values, and therefore has no derivative at that point. When constructing a mathematical model of accelerated chemical-technological processes with a derivative at all points, attention is paid to: -expression of acceleration in the form of a linear function through a non-linear function; -Ensuring the fulfillment of all boundary conditions; -Expression of the speed of the process through a single function. To solve the above problems, instead of the linear acceleration function ( ( )) in the initial period of time, it is proposed to use trigonometric functions, such as Similarly, the limit of formulas (3) and (4)  The graph of the process rate function was created using the MathCAD application software package. The logical expression of the speed function takes the following form:

Results and discussion
As an example, the graph of the process speed with initial values 0 = 15 seconds, = 0, = 100 km/h is shown in Fig. 1. It can be seen from the graph that the acceleration continues until 0 = 15 seconds, during which the speed increases from = 0 to = 100 km/h, and then at a constant speed 100 km/h =m(100 km/h)=27, 7 m/s. continues The speed v1(t) in the case when the acceleration is linear is as follows Since the acceleration u(t) is equal to the product of the velocity, it can be written as and is divisible by a continuous function for all ∈ ( 0 ; ). This circumstance contributes to a more perfect study of the processes [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34].

Conclusions
To use computer technology and solve a unique problem associated with modeling a specific chemical process, you need to know any of the modern programming languages and be able to work in the appropriate environment, creating a user-friendly interface. Of course, mathematicians can be involved to solve the problems of choosing the most appropriate numerical method, professional programmers can be involved to create a program with a user-friendly interface, but the mathematical model itself should be created by subject matter specialists, i.e. specialists competent in the field of chemical technology and industrial implementation of chemical and petrochemical industries.
Mathematical models describing chemical technologies are considered breaking down the state into discrete stages. The above study allows us to recommend that stage processes be considered as continuous functions, although it is difficult to describe. [5]