Download presentation

THE INFORMATIVENESS OF THE KINETICS
EXPERIMENT IN INVERSE PROBLEMS OF
CHEMICAL
SEMYEN I. SPIVAK
Bashkir State Univerity
The Chair of Mathematical Modeling
450074, Ufa, Zaki Validi str. 32
Phone: 7-347-2299635, 7-9174448611
email: [email protected]
Institute of Petrochemistry and Catalysis
Russia Academy of Science
The Laboratory of Mathematical Chemistry
450075, Ufa, October Avenue 141
COMPLEX CHEMICAL REACTIONS MECHANISM
THE SET OF ELEMENTARY STAGES
k+
n

j 1
ij
n
Aj   ij Aj
k-
j 1
i  1m
m - number of stages in the reaction mechanism,
A   A1  AN  - the characters involved in the reactions of substances;
n- total number of substances;
 ij and  ij stoichiometric coefficients of the initial reactants and
reaction products, respectively;
k  and k - rate constants of stages in the forward and reverse directions
respectively.
Numbers  ij and  ij characterize the reaction order
n

j 1
ij
3
n

j1
ij
3
i  1m
2
REACTION RATE
i  i  i
i  1m
The Low of Mass Action:
 k

1

1
 k

i

i
n
a j
 ij
j1
n
 a j ij

j 1
a1 an – concentrations of substances A1  An
3
THE SYSTEMS OF DIFFERENTIAL EQUATIONS
OF CHEMICAL KINETICS
da j
dt
m
   ij i
i 1
The general formula for the element
j  1n
 ij of the stoichiometric matrix Г
 ij  ij   ij
da
 ГТ
dt
Г Т – transpose Г.
The chemical composition of substances reflect the molecular matrix A. Element aqj
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.
AT a  a0
4
y  ( y1 ...y n 2 )
PROBLEM
5
ABSENSE OF MEASUREMENTS PART OF SUBSTANCES
А  ( x; y )
x  ( x1 ...xn1 )
y  ( y1 ...yn2 )
n1  n2  n
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 problem
identification of the reaction mechanism.
INVERSE PROBLEMS OF CHEMICAL KINETICS
Identify
k  ( k1 ...ks )
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?
STRUCTURE OF THE EXPERIMENT
1. Nonstationary
dx
 f1( x; y ; k )
dt
dy
 f 2 ( x; y ; k )
dt
2. Quasistationary
dx
f 2 ( x; y; k )  0
 f1( x; y ; k )
dt
Bodenshtein – Semenov Quasistationary Principle
Choriuti – Temkin stationary reactions theory
3. Equilibrium
f 2 ( x; y; k )  0
f1( x; y; k )  0
Zeldovich function
6
UNIQUENESS CRITERION
(Calman and oth.)
The number of independent columns
 x 
J  
 k 
equals the number of unknown constants s.
The investigation noninformayivity of measurements
rang J    s
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.
7
8
Consequence of Jacobi matrix degeneration
Ak  dimension s  ( s   ) is existence
JА  0
The system of independence nonlinear parametric functions is existence
- the independence parameters of model
k   1 k  k 
rang   
The equations system for independence parameters

А0
k
The system of linear partial differential equations
of first degree with variable coefficients
9
QUASISTIONARY
The main theorem
There is the dimension of the matrix U

s  n1
1
f f  f  f 2
U  1  1  2 
k y  y  k
( JА  0 )  UA  0
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 the
independent parametric functions.
A constructive procedure to determine them.
Implemented as a system of analytical calculations
10
NONSTAIONARY
Non-uniqueness of solutions
X t ; k * ; y* ( t ; k * )  X t ; k ** ; y** ( t ; k ** )


k **  k *


y**  y*
There is a transformation
**
*
*
k **  ( k * ) y  ( t ; y ; k )
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.
EXCEPTION UNMEASURED CONCENTRATIONS
dx
 f1( x; y ; k )
dt
dy
 f 2 ( x; y ; k )
dt
Direct exclusion of Y by moving to systems of differential equations
high dimension of X.
Derivation of explicit expressions
for the independent model parameters.
The system of
design algorithms for exceptions
based on the methods of computer algebra.
The software is developed.
11
CRITERIA FOR A CALCULATION OF MEASUREMENTS
i  1 N 
ri  ri ( x; k )
N – number of measurements
i  ri  ri ( x; k )
i  ri  ri ( x; k )
min 
k
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.
12
13
The basic idea Correlation the measurement error
with an error
in the determination of parameters of
mathematical models.
Leonid V. Kantorovich. On new
approaches to problems processing
observations. // Siberian
mathematical journal, 1962, v.3, №5,
p. 701-709.
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.
UNCERTAINTY INTERVALS FOR THE PARAMETERS
Calculation - experiment:
ri  ri ( x; k )  i
  (1 ... N )
- vector of the maximum permissible error of measurement
We determine
min ks
max ks
under the conditions of compliance
We decide 2s problems of mathematical programming.
For some s
min ks  0
No information provided by a constant.
The problem Planning of measurements in order to reduce the uncertainty.
The possibility of using solutions dual tasks.
14
CONCLUSION
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 reactions
with the characteristics of the measurements.
Kantorovich method.
15
16
Thank You
for Your Attention!