Bashkir State UniverityThe Chair of Mathematical Modeling

450074, Ufa, Zaki Validi str. 32Phone: 7-347-2299635, 7-9174448611

email: s.spivak@bashnet.ru

Institute of Petrochemistry and CatalysisRussia Academy of Science

The Laboratory of Mathematical Chemistry450075, Ufa, October Avenue 141

SEMYEN I. SPIVAK

THE INFORMATIVENESS OF THE KINETICS EXPERIMENT IN INVERSE PROBLEMS OF

CHEMICAL

2

k+

k-

n

jjij

n

jjij AA

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- number of stages in the reaction mechanism,

- the characters involved in the reactions of substances;

- total number of substances;

and stoichiometric coefficients of the initial reactants and reaction products, respectively;

and - rate constants of stages in the forward and reverse directions respectively.

Numbers and characterize the reaction order

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n

jij 3

n

1jij

m

NAAA 1

n

ijij

k k

ij ij

mi 1

mi 1

COMPLEX CHEMICAL REACTIONS MECHANISMTHE SET OF ELEMENTARY STAGES

3

iii

The Low of Mass Action:

n

1jji1

ijak

n

jjiijak

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naa 1 nAA1 – concentrations of substances

mi 1

REACTION RATE

4

m

iiij

j

dt

da

1 nj 1

The general formula for the element of the stoichiometric matrix Г

ijijij ij

ТГdt

da

The chemical composition of substances reflect the molecular matrix A. Element number of atoms of q-th element contained in the molecule of the j-th agent.

ГА = 0

The most important condition - the existence of linear integrals (linear relations between the concentrations)/ The number of independence integrals are the number of

different types of chemical elements that make up the system.

0aA aT

qja

THE SYSTEMS OF DIFFERENTIAL EQUATIONSOF CHEMICAL KINETICS

– transpose Г. ТГ

5)y...y(y 2n1

)y;x(А

)x...x(x n11 )y...y(y n21

nnn 21

X – vector of the measured substances, precursors and reaction products;

У – vector of unmeasured substances, intermediates (radicals, catalysts and their complexes, substances on the surface of the catalyst, enzymes, etc.).

Consequence of the lack of experimental data – ambiguity of the solution

  the inverse problemidentification of the reaction mechanism.

PROBLEMABSENSE OF MEASUREMENTS PART OF SUBSTANCES

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)k...k(k s1Identify

which when substituted in the system of differential equations of chemical kinetics, reproduce the experimentally measured X

Informativity of the experiment -the number of independent parametric functions of the constants that allow unambiguous estimation based on the available information.

What kind of individual constants, or their non-linear parametric combinations?

1. Nonstationary)k;y;x(f

dt

dx1 )k;y;x(f

dt

dy2

2. Quasistationary )k;y;x(f

dt

dx1 02 )k;y;x(f

Bodenshtein – Semenov Quasistationary PrincipleChoriuti – Temkin stationary reactions theory

3. Equilibrium01 )k;y;x(f 02 )k;y;x(f

Zeldovich function

STRUCTURE OF THE EXPERIMENT

INVERSE PROBLEMS OF CHEMICAL KINETICS

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The number of independent columns

k

xJ

equals the number of unknown constants s.

The investigation noninformayivity of measurements

sJrang

The problem of non-uniqueness of solutions of principle.

The main problem -informativity analysis of the measurements.

Establishment of appropriate methods and algorithms mathematical software.

UNIQUENESS CRITERION(Calman and oth.)

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Consequence of Jacobi matrix degeneration

kA )s(s dimension is existence

0JА

The system of independence nonlinear parametric functions is existence - the independence parameters of model

kkk 1

rang

The equations system for independence parameters

0А

k

The system of linear partial differential equations of first degree with variable coefficients

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The main theorem

There is the dimension of the matrix 1nsU

k

f

y

f

y

f

k

fU

2

1

211

00 UA)JА(

Instead of the Jacobian matrix is sufficient to consider the matrix U,structure is completely determined by the type of right-hand sides of the

original system of equations, therefore, the reaction mechanism

We prove the solvability of analytic systems of differential equations

to determine theindependent parametric functions.

A constructive procedure to determine them.Implemented as a system of analytical calculations

QUASISTIONARY

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Non-uniqueness of solutions )k;t(y;k;tX)k;t(y;k;tX *********

*** kk *** yy

There is a transformation

)k(k *** )k;y;t(y ****

invariant with respect to X.

Transformations allowed by the original system, form a group of transformations.

The number of generators of the group -number of independent parameters of the model.

The explicit form of generators -  expressions for the independent parameters.

For quasi-stationary approximationproved to be equivalent to the approach based on an analysis of the matrix U.

NONSTAIONARY

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)k;y;x(fdt

dx1 )k;y;x(f

dt

dy2

Derivation of explicit expressionsfor the independent model parameters.

The system ofdesign algorithms for exceptions

based on the methods of computer algebra.The software is developed.

Direct exclusion of Y by moving to systems of differential equationshigh dimension of X.

EXCEPTION UNMEASURED CONCENTRATIONS

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)k;x(rr ii Ni 1

N – number of measurements

)k;x(rr iii )k;x(rr iii

mink

Normal distribution of measurement error –the sum of squared deviations (the minimal squares method)

Laplace distribution of measurement error - sum of absolute deviations

The uniform law of distribution of measurement error - the module maximum deviation (Chebyshev method for alignment)

"Everyone believes in the normal distribution in its own way a mathematician thinks he should be out of the experiment, the experimenter believes that it is strictly proved mathematic”. Puankare.

CRITERIA FOR A CALCULATION OF MEASUREMENTS

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Leonid V. Kantorovich. On new approaches to problems processing observations. // Siberian mathematical journal, 1962, v.3, №5, p. 701-709.

The basic idea -Correlation the measurement error

with an error in the determination of parameters of

mathematical models.

The main result -The absence of the hypothesis

that the distribution of measurement error.

Calculate the domain within which each point describes a

measurement within the margin of error of measurement

There is no minimizing the criterion of conformity error measurements.

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Calculation - experiment:

iii )k;x(rr

- vector of the maximum permissible error of measurement

We determine ss kmaxkmin

under the conditions of compliance

We decide 2s problems of mathematical programming.

For some s

0skmin

No information provided by a constant.

The problem -Planning of measurements in order to reduce the uncertainty.

The possibility of using solutions dual tasks.

)...( N1

UNCERTAINTY INTERVALS FOR THE PARAMETERS

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The methods and algorithms for calculating the domains of uncertainty

of kinetic parameters in constructing mathematical models

of complex chemical reactions developed and implemented.

SCHEME OF STUDY

Analytical analysis of information content.

The ideology of the Gauss relation of the parameters kinetic models of complex chemical reactionswith the characteristics of the measurements.

Kantorovich method.

CONCLUSION

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Thank You for Your Attention!