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Mathematical modeling of the kinetics of chemical reactions. Modeling the kinetics of heterogeneous chemical reactions

Introduction........................................................
1. Chemical kinetics....................................................
1.1. Stoichiometry and material balance in chemical reactions
1.2. Rate of homogeneous chemical reaction......
1.3. Methodology of kinetic research......................
1.4. Differential method processing experimental data.........................................................
1.5. Integral method for processing experimental data
2. Kinetic modeling and chemical reaction mechanism
2.1. Classification of elementary reactions...................
2.2. Construction of kinetic models of the sequence of elementary stages....................................................
2.2.1. Chain reactions...................................
2.2.2. Homogeneous catalysis...................................
2.2.3. Basics of the kinetics of polymer formation..........
Conclusion................................................

Introduction

A chemical process is a complex phenomenon in which the occurrence of a chemical reaction is accompanied by the transfer of heat and matter. The chemical reaction, heat and mass transfer occurring together mutually influence each other, therefore only knowledge of the laws of the chemical process as a whole is the basis for creating highly efficient chemical production and a basis for constructing mathematical models that allow optimizing the conduct of chemical technological processes. To solve these problems, it is necessary to find out in what ratios the reagents (starting substances and reaction products) interact with each other, how temperature, pressure, the composition of the reaction mixture and the phase state of these reagents affect the speed of the process and the distribution of its products. In addition, it is necessary to establish in what type of apparatus and under what hydrodynamic conditions the process is most intense relative to the selected output indicators (degrees of conversion of starting substances, yield of reaction products, with the greatest selectivity for any product or intermediate substance, etc.) . Many of these questions can be answered by studying the stoichiometric, thermodynamic and kinetic patterns of processes.

Stoichiometric patterns show the ratios in which reactants interact with each other. On their basis, material balances are compiled, and recalculations are made between the amounts of reacted and formed substances. Based on their results, process efficiency indicators are calculated.

Knowledge of thermodynamic laws makes it possible to estimate the probability of a process by changing the Gibbs or Helmholtz energies, to calculate the composition of the equilibrium mixture for reversible reactions and the thermal effects of ongoing reactions.

The study of kinetic patterns makes it possible to determine the most probable reaction mechanisms and obtain equations that describe the intensity of consumption of starting substances or the formation of reaction products.

Apparatuses in which chemical processes (reactors) are carried out are classified according to various criteria: according to the structure of the flow, according to the time and temperature conditions of the process, etc. The text will always indicate the type of device and these conditions.

This manual will only consider methods for studying the kinetic patterns of homogeneous chemical reactions, but they often turn out to be inextricably linked with the study of stoichiometric relationships and thermodynamics of reactions. In this regard, the manual will pay attention to the issues of the material balance of simple and complex chemical reactions, as well as their thermodynamic aspects.

Goal of the work

1. Familiarize yourself with methods for constructing kinetic models of heterogeneous chemical reactions.

2. Create a kinetic model of a heterogeneous chemical reaction in accordance with a given mechanism.

3. Select a numerical method and develop a calculation program.

4. Investigate the dynamics of changes in the concentrations of the reactants of the reaction and intermediate compounds.

Kinetics of heterogeneous chemical reactions

The foundations of heterogeneous chemical kinetics were laid in the works of Langmuir, Temkin and others. In these works, the concept of an ideal adsorbed layer was formulated, based on an analogy with the concepts of homogeneous kinetics. This model makes the following assumptions:

1) the equivalence of all areas of the catalyst surface and the independence of chemisorption energy from the degree of surface filling with various adsorbents;

2) the immutability of the catalyst and the independence of its properties from the composition of the reaction mixture and its effect on the catalyst;

equilibrium energy distribution.

A formal analogue of the kinetic law of mass action for elementary processes on solid surfaces is law of acting surfaces (ZDP) .

According to its original formulation, the rate of a chemical reaction is proportional to the product of the surface concentrations of the reacting substances in powers equal to the stoichiometric ratios in which they interact (1.38).

Let a basic chemical reaction take place

In this case, all substances interact from the adsorbed state. Let's denote z i– fraction of surface occupied i-th adsorbed substance. Then, in accordance with the law of acting surfaces, the rate of irreversible reaction (1.38) can be written as

Where W– rate of chemical reaction;

k– rate constant;

–proportion of surface occupied i th adsorbed particle;

–fraction of free surface;

 i– stoichiometric coefficients of stages;

– change in the number of moles during a chemical reaction.

If not all substances interact from the adsorbed state, but react directly from the gas phase, then more general view The expression for the law of acting surfaces is written as follows:

Where – partial pressures (concentrations) -substances reacting from the gas phase;

–stoichiometric coefficients;

n,m– the amount of substances adsorbed on the surface of the catalyst and reacting from the gas phase.

Example: let hydrogen adsorption occur on the active site of catalyst Z with the formation of an adsorbed surface compound ZH 2:

At first, the factor of displacement, the “struggle” of the components of the reaction mixture for places on the catalyst surface, was considered as the main factor determining the kinetic dependences. In this case, an additional assumption was made about the high rate of adsorption and desorption stages in comparison with the chemical transformations themselves.

Subsequent studies have shown significant limitations to these assumptions. Nevertheless, Hinshelwood, Schwab, Hougen, Watson and others, based on them, obtained equations that satisfactorily describe the kinetic experiment in a certain range of parameters.

The typical formula for the kinetic equation corresponding to these assumptions had the form

Where k– rate constant;

WITH i – concentration i-th gaseous medium reagent;

– equilibrium constant of the adsorption stage i-th component;

 i– stoichiometric coefficient.

The most general description of kinetics complex reactions given in the Horiuchi–Temkin theory of stationary reactions.

Juro Horiuchi introduced the following concepts: independent intermediates, stoichiometric number, reaction route, independent reaction routes.

Stoichiometric numbers are numbers chosen in such a way that after multiplication chemical equations each stage by the corresponding stoichiometric number and subsequent addition of the equations, all intermediate substances are reduced. The resulting equation is called gross equation. Each set of stoichiometric numbers leading to the elimination of intermediates is called reaction route.

In the theory of stationary reactions, the concepts of “stage run,” “route run,” and “reaction speed along the basic route” are introduced. The number of stages of a stage is understood as the difference in the number of acts of an elementary reaction in the forward and reverse directions. Then the rate of a simple reaction is equal to the number of its runs per unit time in a unit reaction space. One run along a route means that as many runs of each stage have occurred as its stoichiometric number for a given route. In the case when the formation of a molecule of an intermediate substance in one of the stages is compensated by the consumption of this molecule in another stage, a stationary reaction mode is realized. If during this stage not the final product is formed, but a new intermediate substance, then it must also be consumed in another stage. Full compensation for the formation and consumption of intermediate substances means the completion of a run along any of the routes.

Thus, the speed of a stationary reaction is determined by individual runs along all possible routes. As a result, all stage runs for a given time will be uniquely determined through the basic routes. Reaction speed along the basic route is the number of runs along the base route per unit time in a unit reaction space, provided that all the runs of the stages are distributed along the routes of a given basis. The reaction rate as a whole is determined by the speeds along the basic routes.

The stationarity condition for the elementary stages of chemical reactions can be written as follows:

Where
– speeds of elementary stages ( s th, direct and reverse);

– speed along the route R;

s-th stage, route R.

Based on equation (1.44), we obtain an equation called equation of stationary reactions:

Where
,
... – speeds along the routes;

, . .. – speeds of elementary stages in forward and reverse directions;

–stoichiometric coefficient i- 1st stage j-th route.

Using this equation, it is easier to derive kinetic equations for heterogeneous chemical reactions in explicit form - for linear mechanisms and, in some cases, for nonlinear ones.

Study of the kinetics of a heterogeneous chemical reaction

Let's consider an example of a complex heterogeneous chemical reaction - the hydrocracking reaction of toluene.

The initial data are:

detailed mechanism of a heterogeneous chemical reaction, where Z is the active centers on the surface of the catalyst; Z H 2, etc. – adsorbed intermediates:

1. H 2 + ZZH 2

2. ZH 2 + C 7 H 8 ZC 7 H 8 ∙H 2

3. ZC 7 H 8 ∙H 2  Z + C 6 H 6 +CH 4

C 7 H 8 + H 2  CH 4 + C 6 H 6

initial concentrations of substances and rate constants, which are equal

C H2 (0) = 0.6 mol. shares;

C C7H8 (0) = 0.4 mol. shares;

;

;
;

;

;
.

Let us write down the speeds of the elementary stages of the mechanism according to the law of acting surfaces:

;
;

;

; .

The mathematical model of this chemical process will be a system differential equations, expressing changes in the concentrations of observed substances and intermediate compounds over time:

When solving the system of differential equations (1.48), you can use the numerical methods of Euler and Runge-Kutta. Examples of the results of calculations of the kinetics of a heterogeneous chemical reaction are shown in Fig. 1.5, 1.6. The program for calculating the kinetics of chemical reactions is given in Appendix B.

Modeling the kinetics of homogeneous chemical reactions

Stages of development of chemical kinetics

Chemical kinetics as the science of the rates of chemical reactions began to take shape in the 50-70s. XIX century

In 1862-1867. Norwegian scientists Guldberg and Waage gave the initial formulation of the law of mass action: during a chemical reaction:

Concentration of the i-th substance, ;

Rate constant;

Stoichiometric coefficients.

Chemical kinetics was fully formulated in the works of Van't Hoff and Arrhenius in the 80s. XIX century; the meaning of reaction orders was clarified and the concept of activation energy was introduced. Van't Hoff introduced the concepts of mono-, bi- and polymolecular reactions:

where n is the order of the reaction.

Van't Hoff and Arrhenius, who developed his ideas, argued that temperature is not the cause of the reaction, temperature is the cause of the change in the rate of reaction:

(Arrhenius in 1889), (1.3)

where A is the pre-exponential factor;

E - activation energy;

R - gas constant;

T - temperature.

Since 1890, the activation energy has become a universal measure reactivity converting substances. Thus, in the period 1860-1910. formal kinetics was created. The clarity and sparseness of the basic postulates distinguish the chemical kinetics of the Van't Hoff and Arrhenius period.

Subsequently, the original integrity is lost, and many “kinetics” appear: the kinetics of gas-phase and liquid-phase reactions, catalytic, enzymatic, topochemical, etc.

However, for a chemist, two concepts remain the most important to this day:

The law of mass action as the law of simple reaction.

The complexity of the chemical reaction mechanism.

Basic concepts of chemical kinetics

Kinetics of homogeneous chemical reactions

The rate of a chemical reaction is the change in the number of moles of reactants as a result of chemical interaction per unit time per unit volume (for homogeneous reactions) or per unit surface (for heterogeneous processes):

where W is the rate of the chemical reaction, ;

V - volume, m3;

N is the number of moles;

t - time, s. According to equation (1.4), introducing the concentration, we get

where C is concentration, mol/m3,

For reactions occurring at constant volume, the second term in equation (1.6) is equal to zero and, therefore,

One of the basic laws of chemical kinetics, which determines the quantitative laws of the rates of elementary reactions, is the law of mass action.

According to kinetic law acting masses, the rate of an elementary reaction at a given temperature is proportional to the concentrations of the reacting substances in powers indicating the number of particles interacting:

where W is the rate of the chemical reaction;

Rate constant;

Concentrations of starting substances, ;

Corresponding stoichiometric coefficients in the gross equation of a chemical reaction.

Equation (1.8) is valid for elementary reactions. For complex reactions, the exponents in equation (1.8) are called reaction orders and can take not only integer values.

The rate constant of a chemical reaction is a function of temperature, and the dependence on temperature is expressed by the Arrhenius law:

where is the pre-exponential factor;

E - activation energy, ;

T - temperature, K;

R - gas constant, .

Consider a homogeneous reaction

where a, b, c, d are stoichiometric coefficients.

According to the law of mass action (1.8), the rate of this reaction will be written as follows:

There is a relationship between the reaction rates for individual components (let’s denote them WA, WB, WC, WD) and the overall reaction rate W

The following expressions follow from this:

To apply the law of mass action to a complex chemical reaction, it is necessary to represent it in the form of elementary stages and apply this law to each stage separately.

Kinetic equations

Kinetic equations relate the rate of a reaction to the parameters on which it depends. The most important of these parameters are concentration, temperature, pressure, and catalyst activity.

For batch reactors, in which the concentrations of reactants at each point in the reaction volume during a reaction change continuously over time, the rate of a chemical reaction is the number of moles of a given substance reacting per unit time in a unit volume:

or per unit surface area, for heterogeneous catalytic reactions

where Wi is the rate of chemical reaction, mol/m3s;

Ni is the current amount of the i-th component of the reaction mixture, mol;

V is the volume of the reaction mixture or catalyst layer (reactor volume), m3;

S - catalyst surface, m2;

0 - specific surface area of the catalyst, m2/m3;

t - time, s.

For continuous displacement reactors, in which, under steady-state conditions, the concentration of a substance continuously changes along the length of the apparatus, the rate of a chemical reaction is the number of moles of a substance passing through the reactor per unit time, reacting per unit volume:

where ni is the molar flow rate of the i-th component of the reaction mixture, mol/s;

Feed rate of the reaction mixture, m3/s;

Contact time, s.

For a continuous fully mixed reactor, at steady state,

where ni0 is the initial amount of the i-th component of the reaction mixture, mol/s.

In practice, the rate of change in the molar concentration of Ci (mol/m3; mol/l) is usually measured.

For a batch reactor

For a continuous reactor

Where - volumetric speed supply of the reaction mixture, m3/s.

If the reaction is not accompanied by a change in volume, then for a plug-flow reactor

For continuous stirring reactor

where xi is the degree of conversion, ;

Average residence time, = V/, s.

Methods for solving kinetic equations

Kinetic models are systems of ordinary differential equations, the solution of which is a function of the concentrations of reactants on an independent argument of time.

To solve differential equations - integration - the following are used:

· tabular method (using tables of integrals) - used for the simplest differential equations;

· analytical methods used to solve first order differential equations;

· numerical methods, the most universal, allowing to solve systems of differential equations of any complexity, are the basis of computer methods for analyzing chemical technological processes.

Numerical methods

The simplest numerical method for solving ordinary differential equations is the Euler method. This method is based on the approximation of the derivative for small changes in the argument.

For example, the rate equation for a chemical reaction is described by the equation

where CA is the concentration of the substance, mol/l;

Time, s.

For small t we can approximately assume that

the quantity is called the integration step. Solving equation (1.23), we obtain the general Euler formula

where is the right side of the differential equation (for example,

Having asked initial conditions: at t = 0 С = С0, the value of the integration step h, as well as the parameters of the equation, using formula (1.24) you can carry out a step-by-step calculation and obtain a solution to this equation (Fig. 1.1).

Rice. 1.1. Graphic illustration of Euler's method

By organizing cyclic calculations according to equation (1.24), we obtain for the kinetic model the change in the concentrations of reacting substances over time.

The size of the integration step is selected based on achieving the minimum calculation time and the smallest calculation error.

General presentation of one-step methods for solving ordinary differential equations

Let there be a differential equation

satisfying the initial condition

It is required to find a solution to problem (1.25), (1.26) on the segment . Let's split the segment with points

This set of points is called a grid, and the points xi (i = i, n) are called grid nodes.

One-step numerical methods provide approximations yn to the exact solution values y(xn) at each grid node xn based on the known approximation yn-1 to the solution at the previous node xn-1. In general, they can be represented as follows:

For explicit one-step methods, the function F does not depend on yn+1.

Designating

We will also write explicit one-step methods in the form

Explicit Runge-Kutta type methods

The idea of this method is based on calculating the approximate solution y1 at the node x0 + h in the form linear combination with constant coefficients:

The numbers are chosen so that the expansion of expression (1.29) in powers of h coincides with the Taylor series expansion:

This is equivalent to the following. If you introduce an auxiliary function

then its expansion in powers of h should begin with the maximum possible degree:

If these constants can be determined so that the expansion has the form (1.32), then formula (1.29) with the chosen coefficients is said to have order of accuracy s.

Magnitude

is called the error of the method at the step, or the local error of the method, and the first term in expression (1.32)

is called the leading member of the local error of the method.

It has been proven that if q = 1, 2, 3, 4, then it is always possible to choose the coefficients so as to obtain a Runge-Kutta type method of order of accuracy q. When q = 5, it is impossible to construct a Runge-Kutta type method (1.29) of the fifth order of accuracy; it is necessary to take more than five terms in combination (1.29).

Study of the kinetics of homogeneous chemical reactions

The study of the kinetic patterns of a chemical reaction using the method of mathematical modeling involves determining changes in the concentrations of reacting substances over time at a given temperature.

Let chemical reactions take place

Based on the law of mass action, we write down the equations for the rates of chemical reactions and create a kinetic model:

where CA, CB, CC, CD are the concentrations of substances, mol/l;

ki - rate constant i-th chemical first order reactions, s-1; (for second-order reactions, the dimension of a constant; for third-order reactions, the dimension of a constant);

Wi- speed i-th chemical reaction, mol/hp; t - reaction time, s.

Rice. 1.2. Flowchart for calculating the kinetics of a homogeneous chemical reaction using the Euler method

The system of first-order ordinary differential equations (1.35) can be solved using Euler's numerical method, the algorithm of which is written according to equation (1.24).

A flow diagram for calculating the kinetics of a homogeneous chemical reaction using the Euler method is shown in Fig. 1.2.

Examples of programs for calculating the kinetics of homogeneous chemical reactions are given in Appendix A. The results of a study using a mathematical model (1.35) of the effect of temperature on the degree of conversion of the initial reagent and on the concentration of substances are presented in Fig. 1.3, 1.4.

The results obtained allow us to draw a conclusion about the optimal time for carrying out the process in order to obtain the target product. The mathematical model (1.35) also allows one to study the influence of the composition of the raw material on the yield of reaction products.

It is necessary to take into account that the rate of a chemical reaction depends on temperature, therefore, in order to use the kinetic model (1.35) to study the process at different temperatures, it is necessary to introduce the dependence of the rate constant of a chemical reaction on temperature according to the Arrhenius equation (1.9).

The algorithm of the fourth order Runge-Kutta method can be written as follows:

where ai are the Runge-Kutta coefficients, which are calculated using the following formulas:

Literature

kinetics chemical homogeneous

1. Panchenkov G. M., Lebedev V. P. Chemical kinetics and catalysis. - M.: Chemistry, 1985. - 589 p.

2. Yablonsky G. S., Bykov V. I., Gorban A. I. Kinetic models of catalytic reactions. - Novosibirsk: Science, 1983. - 254 p.

3. Kafarov V.V. Methods of cybernetics in chemistry and chemical technology. - M.: Chemistry, 1988. - 489 p.

4. Kravtsov A.V., Novikov A.A., Koval P.I. Methods for analyzing chemical and technological processes. - Tomsk: TPU publishing house, 1994. - 76 p.

5. Kafarov V.V., Glebov M.V. Mathematical modeling of the main processes of chemical production. - M.: Higher. school, 1991. - 400 p.

6. Moises O. E., Koval P. I., Bazhenov D. A., Kuzmenko E. A. Informatics: textbook. allowance. In 2 parts - Tomsk, 1999. - 150 p.

7. Turchak L. I. Fundamentals numerical methods. - M.: Nauka, 1987. - 320 p.

8. Ofitserov D.V., Starykh V.A. Programming in the integrated Turbo-Pascal environment. - Minsk: Belarus, 1992. - 240 p.

9. Beskov V. S., Flor K. V. Modeling of catalytic processes and reactors. - M.: Chemistry, 1991. - 252 p.

10. Rood R., Praustnitz J., Sherwood T. Properties of gases and liquids

/ ed. B.I. Sokolova. - L.: Chemistry, 1982. - 591 p.

11. Tanatarov M. A. et al. Technological calculations of oil processing plants. - M.: Chemistry, 1987. - 350 p.

12. Zhorov Yu. M. Thermodynamics chemical processes. - M.: Chemistry, 1985

13. Calculations of the main processes and apparatus of oil refining: reference book / ed. E. N. Sudakova. - M.: Chemistry, 1979. - 568 p.

14. Kafarov V.V. Separation of multicomponent systems in chemical technology. Calculation methods. - M.: Moscow Institute of Chemical Technology, 1987. - 84 p.

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Modeling the kinetics of homogeneous chemical reactions

Stages of development of chemical kinetics

Chemical kinetics as the science of the rates of chemical reactions began to take shape in the 50-70s. XIX century

In 1862-1867. Norwegian scientists Guldberg and Waage gave the initial formulation of the law of mass action: during a chemical reaction:

Concentration of the i-th substance, ;

Rate constant;

Stoichiometric coefficients.

where n is the order of the reaction.

Van't Hoff and Arrhenius, who developed his ideas, argued that temperature is not the cause of the reaction, temperature is the cause of the change in the rate of reaction:

(Arrhenius in 1889), (1.3)

where A is the pre-exponential factor;

E - activation energy;

R - gas constant;

T - temperature.

Since 1890, the activation energy has become a universal measure of the reactivity of transforming substances. Thus, in the period 1860-1910. formal kinetics was created. The clarity and sparseness of the basic postulates distinguish the chemical kinetics of the Van't Hoff and Arrhenius period.

Subsequently, the original integrity is lost, and many “kinetics” appear: the kinetics of gas-phase and liquid-phase reactions, catalytic, enzymatic, topochemical, etc.

However, for a chemist, two concepts remain the most important to this day:

The law of mass action as the law of simple reaction.

The complexity of the chemical reaction mechanism.

Basic concepts of chemical kinetics

Kinetics of homogeneous chemical reactions

where W is the rate of the chemical reaction, ;

V - volume, m3;

N is the number of moles;

t - time, s. According to equation (1.4), introducing the concentration, we get

where C is concentration, mol/m3,

For reactions occurring at constant volume, the second term in equation (1.6) is equal to zero and, therefore,

One of the basic laws of chemical kinetics, which determines the quantitative laws of the rates of elementary reactions, is the law of mass action.

According to the kinetic law of mass action, the rate of an elementary reaction at a given temperature is proportional to the concentrations of the reactants in powers indicating the number of particles interacting:

where W is the rate of the chemical reaction;

Rate constant;

Concentrations of starting substances, ;

Corresponding stoichiometric coefficients in the gross equation of a chemical reaction.

Equation (1.8) is valid for elementary reactions. For complex reactions, the exponents in equation (1.8) are called reaction orders and can take not only integer values.

The rate constant of a chemical reaction is a function of temperature, and the dependence on temperature is expressed by the Arrhenius law:

where is the pre-exponential factor;

E - activation energy, ;

T - temperature, K;

R - gas constant, .

Consider a homogeneous reaction

where a, b, c, d are stoichiometric coefficients.

According to the law of mass action (1.8), the rate of this reaction will be written as follows:

There is a relationship between the reaction rates for individual components (let’s denote them WA, WB, WC, WD) and the overall reaction rate W

The following expressions follow from this:

To apply the law of mass action to a complex chemical reaction, it is necessary to represent it in the form of elementary stages and apply this law to each stage separately.

Kinetic equations

Kinetic equations relate the rate of a reaction to the parameters on which it depends. The most important of these parameters are concentration, temperature, pressure, and catalyst activity.

or per unit surface area, for heterogeneous catalytic reactions

where Wi is the rate of chemical reaction, mol/m3s;

Ni is the current amount of the i-th component of the reaction mixture, mol;

V is the volume of the reaction mixture or catalyst layer (reactor volume), m3;

S - catalyst surface, m2;

0 - specific surface area of the catalyst, m2/m3;

t - time, s.

where ni is the molar flow rate of the i-th component of the reaction mixture, mol/s;

Feed rate of the reaction mixture, m3/s;

Contact time, s.

For a continuous fully mixed reactor, at steady state,

where ni0 is the initial amount of the i-th component of the reaction mixture, mol/s.

In practice, the rate of change in the molar concentration of Ci (mol/m3; mol/l) is usually measured.

For a batch reactor

For a continuous reactor

where is the volumetric flow rate of the reaction mixture, m3/s.

If the reaction is not accompanied by a change in volume, then for a plug-flow reactor

For continuous stirring reactor

where xi is the degree of conversion, ;

Average residence time, = V/, s.

Methods for solving kinetic equations

Kinetic models are systems of ordinary differential equations, the solution of which is a function of the concentrations of reactants on an independent argument of time.

To solve differential equations - integration - the following are used:

· tabular method (using tables of integrals) - used for the simplest differential equations;

· analytical methods are used to solve first-order differential equations;

· numerical methods, the most universal, allowing to solve systems of differential equations of any complexity, are the basis of computer methods for analyzing chemical technological processes.

Numerical methods

The simplest numerical method for solving ordinary differential equations is the Euler method. This method is based on the approximation of the derivative for small changes in the argument.

For example, the rate equation for a chemical reaction is described by the equation

where CA is the concentration of the substance, mol/l;

Time, s.

For small t we can approximately assume that

the quantity is called the integration step. Solving equation (1.23), we obtain the general Euler formula

where is the right side of the differential equation (for example,

Having set the initial conditions: at t = 0 С = С0, the value of the integration step h, as well as the parameters of the equation, using formula (1.24) you can carry out a step-by-step calculation and obtain a solution to this equation (Fig. 1.1).

Rice. 1.1. Graphic illustration of Euler's method

By organizing cyclic calculations according to equation (1.24), we obtain for the kinetic model the change in the concentrations of reacting substances over time.

The size of the integration step is selected based on achieving the minimum calculation time and the smallest calculation error.

General presentation of one-step methods for solving ordinary differential equations

Let there be a differential equation

satisfying the initial condition

It is required to find a solution to problem (1.25), (1.26) on the segment . Let's split the segment with points

This set of points is called a grid, and the points xi (i = i, n) are called grid nodes.

For explicit one-step methods, the function F does not depend on yn+1.

Designating

We will also write explicit one-step methods in the form

Explicit Runge-Kutta type methods

The idea of this method is based on calculating an approximate solution y1 at the node x0 + h in the form of a linear combination with constant coefficients:

The numbers are chosen so that the expansion of expression (1.29) in powers of h coincides with the Taylor series expansion:

This is equivalent to the following. If you introduce an auxiliary function

then its expansion in powers of h should begin with the maximum possible degree:

If these constants can be determined so that the expansion has the form (1.32), then formula (1.29) with the chosen coefficients is said to have order of accuracy s.

Magnitude

is called the error of the method at the step, or the local error of the method, and the first term in expression (1.32)

is called the leading member of the local error of the method.

Study of the kinetics of homogeneous chemical reactions

Let chemical reactions take place

Based on the law of mass action, we write down the equations for the rates of chemical reactions and create a kinetic model:

where CA, CB, CC, CD are the concentrations of substances, mol/l;

ki is the rate constant of the i-th chemical reaction of the first order, s-1; (for second-order reactions, the dimension of a constant; for third-order reactions, the dimension of a constant);

Wi is the rate of the i-th chemical reaction, mol/hp; t - reaction time, s.

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Rice. 1.2. Flowchart for calculating the kinetics of a homogeneous chemical reaction using the Euler method

The system of first-order ordinary differential equations (1.35) can be solved using Euler's numerical method, the algorithm of which is written according to equation (1.24).

A flow diagram for calculating the kinetics of a homogeneous chemical reaction using the Euler method is shown in Fig. 1.2.

The algorithm of the fourth order Runge-Kutta method can be written as follows:

where ai are the Runge-Kutta coefficients, which are calculated using the following formulas:

Literature

kinetics chemical homogeneous

1. Panchenkov G. M., Lebedev V. P. Chemical kinetics and catalysis. - M.: Chemistry, 1985. - 589 p.

2. Yablonsky G. S., Bykov V. I., Gorban A. I. Kinetic models of catalytic reactions. - Novosibirsk: Science, 1983. - 254 p.

3. Kafarov V.V. Methods of cybernetics in chemistry and chemical technology. - M.: Chemistry, 1988. - 489 p.

4. Kravtsov A.V., Novikov A.A., Koval P.I. Methods for analyzing chemical and technological processes. - Tomsk: TPU publishing house, 1994. - 76 p.

5. Kafarov V.V., Glebov M.V. Mathematical modeling of the main processes of chemical production. - M.: Higher. school, 1991. - 400 p.

6. Moises O. E., Koval P. I., Bazhenov D. A., Kuzmenko E. A. Informatics: textbook. allowance. In 2 parts - Tomsk, 1999. - 150 p.

7. Turchak L.I. Fundamentals of numerical methods. - M.: Nauka, 1987. - 320 p.

8. Ofitserov D.V., Starykh V.A. Programming in the integrated Turbo-Pascal environment. - Minsk: Belarus, 1992. - 240 p.

9. Beskov V. S., Flor K. V. Modeling of catalytic processes and reactors. - M.: Chemistry, 1991. - 252 p.

10. Rood R., Praustnitz J., Sherwood T. Properties of gases and liquids

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Modeling stages

The process of both theoretical and experimental modeling consists of the following steps:

1. Building a model.

2. Study of the model.

3. Extrapolation - transfer of the obtained data to the area of knowledge about the original object.

At the first stage, upon realizing the impossibility or inappropriateness of direct study of the object, its model is created. The purpose of this stage is to create conditions for the full replacement of the original with an intermediary object that reproduces its necessary parameters.

At the second stage, the model itself is studied - as detailed as required to solve a specific cognitive problem. Here the researcher can observe the behavior of the model, conduct experiments on it, measure or describe its characteristics - depending on the specifics of the model itself and the initial cognitive task. The purpose of the second stage is to obtain the required information about the model.

The third stage (extrapolation) represents a “return” to the original object, i.e. interpretation of the acquired knowledge about the model, assessment of its acceptability and, accordingly, its application to the original, allowing, if successful, to solve the original cognitive problem.

These steps implement a kind of modeling cycle, during which the model and the original are related to each other (Fig. 1).

Rice. 1. Modeling stages

Modeling in chemistry

Simulation of molecules, chemical processes and reactions

Material (experimental) modeling is widely used in chemistry to understand and study the structure of substances and the characteristics of chemical reactions, to identify optimal conditions for chemical technological processes, etc.

In biochemistry and pharmacology modeling plays very big role. The progress of pharmacology is characterized by the continuous search and creation of new, more advanced drugs. IN last years When creating new drugs, the basis is not the biologically active substance, as was done previously, but the substrate with which it interacts (receptor, enzyme, etc.). Such studies require the most detailed data on the three-dimensional structure of those macromolecules that are the main target for the drug. Currently, there is a bank of such data, including a significant number of enzymes and nucleic acids. A number of factors have contributed to progress in this direction. First of all, X-ray diffraction analysis was improved, and spectroscopy based on nuclear magnetic resonance was developed. The latter method opened up fundamentally new possibilities, since it made it possible to establish the three-dimensional structure of substances in solution, i.e. in a non-crystalline state. Another significant point was that, with the help of genetic engineering, it was possible to obtain a sufficient amount of substrates for detailed chemical and physicochemical studies.

Using available data on the properties of many macromolecules, it is possible to simulate their structure using computers. This gives a clear idea of the geometry of not only the entire molecule, but also its active centers that interact with ligands. The features of the topography of the substrate surface, the nature of its structural elements and possible types of interatomic interactions with endogenous substances or xenobiotics. On the other hand, computer modeling of molecules, using graphics systems and related statistical methods, allows you to get a fairly complete picture of the three-dimensional structure of pharmacological substances and the distribution of their electronic fields. Such summary information about physiological active substances and substrate should facilitate the efficient design of potential ligands with high complementarity and affinity. Until now, such opportunities could only be dreamed of - now they are becoming a reality.

Computer modeling of molecules is based on numerous approximations and assumptions. Thus, it is assumed that the energy of molecules is determined only by the coordinates of their atoms in space. But in reality, molecules are not stationary, and energy calculations on a computer are carried out on static molecules. Molecular dynamics methods are now being developed that make it possible to take into account the thermal motion of molecules, but there are still no approaches that would reliably take into account the entropy component of energy. In addition, within a reasonable time it is possible to calculate the lifetime of the system on the order of several picoseconds.

Studying the three-dimensional structure of proteins presents great difficulties. To date, there are no methods that can accurately predict the three-dimensional structure of a protein based on its amino acid sequence. Although the method of analogies is used, when it is assumed that identical amino acid sections of different proteins are stacked in a similar way. Experimental acquisition of three-dimensional images is associated with many difficulties: X-ray diffraction analysis requires protein crystallization (which is only possible for soluble proteins), and the capabilities of nuclear magnetic resonance are limited by the molecular size of proteins.

The role of molecular modeling for both fundamental and applied research in the field of molecular biology and biochemistry is steadily growing. This is due to the improvement of the mathematical apparatus, and with the increase in the productivity of computer technology, and the accumulation of a huge amount of factual material requiring analysis.

Simulation of chemical reactors used to predict the results of chemical technological processes when given conditions in devices of any size. Attempts to carry out a large-scale transition from a small-sized reactor to an industrial reactor using physical modeling were unsuccessful due to the incompatibility of the conditions for the similarity of the chemical and physical components of the process (the influence of physical factors on the rate of chemical transformation in reactors of different sizes is significantly different). Therefore, for a large-scale transition, they mainly used empirical methods: processes were studied in successively larger reactors (laboratory, large-scale, pilot, semi-industrial installations, industrial reactor).

Mathematical modeling made it possible to study the reactor as a whole and carry out a large-scale transition. The process in the reactor consists of large number chemical and physical interactions at various structural levels - molecule, macroregion, reactor element, reactor. In accordance with the structural levels of the process, a multi-stage mathematical model reactor. The first level (the chemical transformation itself) corresponds to a kinetic model, the equations of which describe the dependence of the reaction rate on the concentration of reactants, temperature and pressure in the entire range of their changes, covering the practical conditions of the process. The nature of the following structural levels depends on the type of reactor. For example, for a reactor with a fixed bed of catalyst, the second level is a process occurring on one catalyst grain, when the transfer of substance and heat transfer in the porous grain are significant. Each subsequent structural level includes all previous ones as component parts, for example, a mathematical description of the process on one catalyst grain includes both transport and kinetic equations. The third level model also includes equations for the transfer of matter, heat and momentum in the catalyst layer, etc. Models of other types of reactors (fluidized bed, column type with suspended catalyst, etc.) also have a hierarchical structure.

Using mathematical modeling, the optimal conditions for the process are selected, the required amount of catalyst, the size and shape of the reactor, the parametric sensitivity of the process to initial and boundary conditions, transient conditions are determined, and the stability of the process is also studied. In a number of cases, theoretical optimization is first carried out - optimal conditions are determined under which the yield of useful product is greatest, regardless of whether they can be implemented, and then, at the second stage, an engineering solution is selected that allows the best way to get closer to the theoretical optimal regime, taking into account economic and other indicators. To implement the found modes and normal operation of the reactor, it is necessary to ensure uniform distribution of the reaction mixture over the cross section of the reactor and complete mixing of flows that differ in composition and temperature. These problems are solved by physical (aerohydrodynamic) modeling of the selected reactor design.

To study various processes in which phase and chemical transformations occur, thermodynamic modeling methods.

Thermodynamic modeling of phase-chemical transformations is based, on the one hand, on the laws and methods of chemical thermodynamics, on the other, on the mathematical apparatus for solving extremal problems. A complete combination of these two approaches makes it possible to implement a calculation technique that has no fundamental restrictions on the nature and component nature of the systems under study.

To study various practical and theoretical problems associated with phase and chemical transformations, it is necessary to conduct a deep and detailed study of the physical and chemical essence of the process, to identify the patterns of phase and chemical transformations occurring during this process, and the influence of state parameters (temperature, pressure, composition of the reaction mixture, etc.).

The complexity of most real physical and chemical processes does not allow the described problems to be solved exclusively experimentally. Analysis of possible approaches shows the effectiveness of attracting modern theories and methods of physicochemical and mathematical modeling and calculation using thermodynamic concepts. Using these methods, it is possible to conduct a detailed study of phase and chemical transformations.

Theoretical modeling

The role of theoretical modeling in development chemical science is especially significant since the world of atoms and molecules is hidden from the direct observation of the researcher. Therefore, cognition is carried out by constructing models of invisible objects based on indirect data.

Rice. 2. Building and modifying models

The process of theoretical modeling, as mentioned above, is carried out in stages: building a model, studying the model and extrapolation. At each stage, you can identify certain actions necessary for its implementation. (Figure 2). Models can be supplemented, changed and even replaced by new models. Such processes occur if researchers encounter new facts that contradict the constructed model. The new model is the result of rethinking the contradictions of the old model and newly obtained data.

Let us consider the specifics of the cognition process during theoretical modeling.

Ideal modeling is one of the methods of theoretical knowledge. Thus, such structural components of theoretical knowledge as problem, hypothesis and theory should form the basis of theoretical modeling.

After the accumulation of factual material and its analysis, the problem is identified and formulated. A problem is a form of theoretical knowledge, the content of which is what has not yet been known by man, but what needs to be known. In other words, this is knowledge about ignorance, a question that arose in the course of cognition and requires an answer. A problem is not a frozen form of knowledge, but a process that includes two main points (stages of the movement of knowledge) - its formulation and solution. Correct derivation of problematic knowledge from previous facts and generalizations, the ability to correctly pose a problem is a necessary prerequisite for its successful solution. "The formulation of a problem is often more significant than its solution, which can only be a matter of mathematical or experimental art. The raising of new questions, the development of new possibilities, the consideration of old problems from a new angle require creative imagination and reflect real success in science."

V. Heisenberg noted that when posing and solving scientific problems, the following is necessary: a) a certain system of concepts with the help of which the researcher will record certain phenomena; b) a system of methods chosen taking into account the objectives of the research and the nature of the problems being solved; c) reliance on scientific traditions, since, according to Heisenberg, “in the matter of choosing a problem, tradition, the course historical development play significant role", although, of course, the interests and inclinations of the scientist himself are of a certain importance.

According to K. Popper, science begins not with observations, but with problems, and its development is a transition from one problem to another - from less profound to more profound. Problems arise, in his opinion, either as a consequence of a contradiction in a particular theory, or when two different theories collide, or as a result of a collision between a theory and observations.

Thereby scientific problem is expressed in the presence of a contradictory situation (appearing in the form of opposing positions), which requires appropriate resolution. The determining influence on the way of posing and solving a problem is, firstly, the nature of thinking of the era in which the problem is formulated, and, secondly, the level of knowledge about those objects that concern the problem that has arisen. Each historical era has its own characteristic forms of problem situations.

To solve the identified problem, the scientist formulates a hypothesis. A hypothesis is a form of theoretical knowledge containing an assumption formulated on the basis of a number of facts, the true meaning of which is uncertain and requires proof. Hypothetical knowledge is probable, not reliable, and requires verification and justification. In the course of proving the put forward hypotheses, some of them become a true theory, others are modified, clarified and specified, others are discarded and turn into delusions if the test gives a negative result. Proposing a new hypothesis, as a rule, is based on the results of testing the old one, even if these results were negative.

So, for example, the quantum hypothesis put forward by Planck, after testing, became a scientific theory, and the hypotheses about the existence of “caloric”, “phlogiston”, “ether”, etc., without finding confirmation, were refuted and turned into delusions. The open D.I. also passed the hypothesis stage. Mendeleev's periodic law.

DI. Mendeleev believed that in organizing a purposeful, systematic study of phenomena, nothing can replace the construction of hypotheses. “They,” wrote the great Russian chemist, “are necessary for science and especially its study. They provide harmony and simplicity, which is difficult to achieve without their assumption. The entire history of science shows this. And therefore we can safely say: it is better to adhere to such a hypothesis that can time to become more faithful than none."

According to Mendeleev, hypothesis is a necessary element natural science knowledge, which necessarily includes: a) collecting, describing, systematizing and studying facts; b) drawing up a hypothesis or assumption about the causal relationship of phenomena; c) experimental testing of logical consequences from hypotheses; d) turning hypotheses into reliable theories or discarding a previously accepted hypothesis and putting forward a new one. DI. Mendeleev clearly understood that without a hypothesis there can be no reliable theory: “By observing, depicting and describing what is visible and subject to direct observation - with the help of the senses, we can, when studying, hope that first hypotheses will appear, and then theories of what is now happening to form the basis of what is being studied."

Thus, a hypothesis can exist only as long as it does not contradict reliable facts of experience, otherwise it becomes simply a fiction. It is checked (verified) by relevant experimental facts (especially experiment), obtaining the character of truth. A hypothesis is fruitful if it can lead to new knowledge and new methods of cognition, to an explanation of a wide range of phenomena.

A hypothesis as a method of developing scientific and theoretical knowledge in its application goes through the following main stages.

1. An attempt to explain the phenomenon being studied on the basis of known facts and laws and theories already existing in science. If this attempt fails, then a further step is taken.

2. Making conjectures, assumptions about the causes and patterns of a given phenomenon, its properties, connections and relationships, its occurrence and development, etc. At this stage of cognition, the proposition put forward represents probable knowledge, not yet proven logically and not so confirmed by experience as to be considered reliable. Most often, several assumptions are put forward to explain the same phenomenon.

3. Assessing the validity and effectiveness of the hypotheses put forward and selecting the most probable from among them based on the above-mentioned conditions for the validity of the hypothesis.

4. Deployment of the put forward assumption into an integral system of knowledge and deductive derivation of consequences from it for the purpose of their subsequent empirical verification.

5. Experienced, experimental verification of the consequences put forward from the hypothesis. As a result of this test, the hypothesis either “goes to the rank” scientific theory, or refuted, “leaves the scientific scene.” However, it should be borne in mind that empirical confirmation of the consequences of a hypothesis does not fully guarantee its truth, and the refutation of one of the consequences does not clearly indicate its falsity as a whole. This situation is especially characteristic of scientific revolutions, when fundamental concepts and methods are radically disrupted and fundamentally new ideas emerge.

Thus, the decisive test of the truth of a hypothesis is ultimately practice in all its forms, but the logical (theoretical) criterion of truth also plays a certain (auxiliary) role in proving or refuting hypothetical knowledge. A tested and proven hypothesis becomes a reliable truth and becomes a scientific theory.

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