=Paper=
{{Paper
|id=Vol-2076/paper-09
|storemode=property
|title=New Information Technology on the Basis of Interval Analysis: Estimation of Aluminum Corrosion Parameters in Real Electrochemical Process
|pdfUrl=https://ceur-ws.org/Vol-2076/paper-09.pdf
|volume=Vol-2076
|authors=Sergey I. Kumkov,Milan Hladík,Ludmila A. Yolshina,Varvara A. Yolshina
}}
==New Information Technology on the Basis of Interval Analysis: Estimation of Aluminum Corrosion Parameters in Real Electrochemical Process==
https://ceur-ws.org/Vol-2076/paper-09.pdf
New Information Technology on the Basis of
Interval Analysis:
Estimation of Aluminum Corrosion Parameters
in Real Electrochemical Process
Sergey I. Kumkov1,4 , Milan Hladík2 , Ludmila A. Yolshina3 ,
and Varvara A. Yolshina3,4
1
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of Russian
Academy of Sciences, Ekaterinburg, Russia
2
Charles University, Faculty of Mathematics and Physics, Department of Applied
Mathematics, Prague, Czech Republic
3
Institute of High-Temperature Electrochemistry of Ural Branch of Russian Academy
of Sciences, Ekaterinburg, Russia
4
Ural Federal University, Ekaterinburg, Russia
kumkov@imm.uran.ru
Abstract. Practical applications of computer technologies are consid-
ered. Estimation of parameters of the aluminum corrosion process is in-
vestigated. In practice, estimation of parameters of experimental chem-
ical processes is very actual and important for researches for necessary
organization of corresponding technological processes by correct choice
of the parameters. But usually, determination of the process parameters
is hampered by their complicated internal structure and incompleteness
of input information for processing. The paper considers application of
the interval analysis procedures to estimation of parameters of an exper-
imental chemical process under conditions of corruption, uncertainty of
errors (corruption) probability characteristics, and short sample of mea-
surements. For these reasons, the standard statistical approach can be
applied only formally; especially, it becomes impossible to determine ac-
curately the confidential intervals for parameters of the process. Under
such conditions, namely methods of the interval analysis can work reli-
ably and give exact set of the admissible values of the parameters to be
estimated. In this paper, existing interval analysis procedures were ad-
vanced and arranged for a concrete experimental process and its data. As
a comparison, approximate estimations of parameters have been calcu-
lated by the standard statistical approach. It is shown that the standard
approach gives very rough and, often, practically senseless estimations
of the process parameters.
Keywords: Interval analysis, metal corrosion, uncertainty conditions, es-
timation, parameters.
1 Introduction
In the paper, the corrosion process and transformations of its components is con-
sidered. Components (N aN O3 , Al, O2 , Al2 O3 ) are described by a kinetic system
77
of ordinary differential equations of the first approximation and analytical func-
tions.
The vector of parameters, i.e., the activity coefficients, comprises: KN T,AL of
transformation of nitrate in the direct reaction with aluminum, KN T,T of the own
nitrate thermal decomposition, and KAL,Ox of aluminum transformation into the
aluminum oxide by free oxygen. Moreover, aluminum oxide partially (with the
coefficient KF ) precipitates as a film on the aluminum surface but residuary
part of oxide sediments in the molten environment and does not participate
further in reactions. The coefficients of the reagents activity (of creation and
transformations) and redistribution of the aluminum oxide between the oxide
film and the powder sediment have to be found and the latter coefficient KF is
also of interest.
But estimation of these coefficients is strongly hampered by complexity of
the process description, fatally short length of measurement samples, and com-
plete absence of any probabilistic characteristics of errors in the measurements.
Measuring is implemented only at the beginning t = 0 of the process (the initial
concentration of nitrates and mass of metallic aluminum before the corrosion
test) and at its termination t = tf (the final concentration of nitrates and final
mass of the metallic specimen with the oxide sediment on it).
Estimation of the set of admissible values of the coefficients is performed in
the following way. On the basis of the measurements, the uncertainty intervals
of the direct or indirect measurements (both at the beginning and the end of
the process) of the components are built using the prescribed bounds on the
measuring errors. The sought-for set is a totality only of such parameters values,
for which the integral curves of the process components pass through the men-
tioned intervals at the beginning and termination instants of the experiment.
This excludes application of the standard statistical methods [3] to estimation
of parameters of the corrosion process. But under such conditions, methods of
the Interval Analysis [7, 8] work efficiently. On their basis, applied procedures
and algorithms of estimation were elaborated [9–11]. These algorithms were suc-
cessfully used for processing the data of chemical experiments.
The goal of this work is in further development of a new approach for process-
ing the mentioned experimental data. The paper has the following structure. In
Section 1, the process model is described and the estimation problem is formu-
lated. Section 2 is devoted to the main Interval Analysis ideas applied. Section 3
shortly describes structure of the estimation algorithm. Section 4 presents results
of numerical simulation of the estimation problem.
2 Model of the process and problem formulation
Earlier, the corrosion process was investigated in [1, 2]. The process is imple-
mented in a molten eutectic mixture of cesium and sodium chlorides that con-
tained up to 30 wt% of the sodium nitrate in the argon atmosphere in the
temperature interval 793-903 K. Here, the corrosion is stipulated by both the
sodium nitrite and free oxygen appeared as a result of the thermal decomposi-
tion of the sodium nitrate. The corrosion process is followed by creation of the
aluminum oxide that partially deposits as the oxide film on the surface of the
78
metallic aluminum electrode and partially leaves the process in the form of the
powder sediment.
The original description of chemical reactions (of the first approximation) is
a) 2N aN 03 + 2Al − N a2 O + Al2 O3 + 2N O, b) 2N aN 03 − 2N aN O2 + O2 ;
c) 4Al + 3O2 − 2Al2 O3 .
On the basis of this description, the following mathematical model (the first
approximation) of the process was built:
t ∈ [0, tf ], (1)
˙
N T = −KN T,AL N T AL − KN T,T N T, (2)
˙
AL = −KN T,AL N T AL − KAL,Ox AL.Ox, (3)
Ox˙ = KN T,T N T − KAL,Ox AL.Ox, (4)
OxAl(t) = AL(0) − AL(t), (5)
F lm(t) = KF OxAl(t), (6)
M (t) = AL(t) + F lm(t), (7)
where, t is time, the independent argument with termination instant tf ; N T (t)
is the nitrate current concentration, wt%; AL(t) is the current mass of the alu-
minum electrode, grams; OxAl(t) is the current total mass of the oxide, grams;
Ox(t) is the current oxygen mass; F lm(t) is the current mass of the film precip-
itated on the surface of the metal aluminum electrode, this value is a part of the
whole oxide mass; M (t) is the auxiliary variable representing the current mass of
metal AL and the oxide Al2 O3 film precipitated on the metal surface; KN T,AL ,
KN T,T , and KAL,Ox are the activity coefficients (of corresponding physical di-
mensions; here and later, their dimensions are omitted for simplicity of descrip-
tion); KF is a deal coefficient of the film mass. The oxide part OxAl(t) − F lm(t)
goes out of the reaction as a sediment and does not participate in the further
processes.
It was revealed [1, 2] that corresponding to equations (a)–(c), a large deal of
oxides can appear during the process. These oxides precipitate as the film on
the metal surface or accumulate as sediment in the volume of the molten salt
electrolyte. Such redistribution of the oxides is stipulated by adhesion of oxide
layers and velocity of the metal interaction with the melt.
That is why for adequate description of the process in the mathematical
model (1)–(7), a special important variable M (t) is introduced that describes
the total current mass of the aluminum electrode and film on its surface. Value
of this variable can be measured at the termination instant.
Remark 1: Since of description of the process in the differential form (2)–
(4), values of the components during the process and at the termination instant
can be presented only by numerical integration for each value of the coefficients
KN T,AL , KN T,T , and KAL,Ox .
Measured data. Measurements at the initial (t=0) instant: N Tmes,0 with
bounded additive error emax,N T , ALmes,0 with bounded additive error emax,AL ,
Oxmes,0 = 0, OxAlmes,0 = 0, F lmmes,0 = 0, Mmes,0 = ALmes,0 .
Measurements at the termination (final t = tf ) instant: N Tmes,f with
bounded additive error emax,N T , Mmes,f with bounded additive error emax,AL .
79
Values of AL(tf ), Ox(tf ), OxAl(tf ), and F lm(tf ) are not measured. No
probabilistic information on measuring errors is known. There are no a priori
bounds on possible values of the coefficients.
The problem is formulated as follows: to built the set (i.e., the Infor-
mation Set or Set-membership) of admissible values of the activity coefficients
KN T,AL , KN T,T , KAL,Ox , and to estimate a collection of intervals for the coef-
ficient KF consistent with the described data.
Note once more that because of incomplete observability of the process phase
coordinates, fatally short length of the measurements sample (only two mea-
surements), absence of probabilistic characteristics of errors, and measurements
uncertainty, it is impossible to use standard statistical methods (see, [3]).
3 Interval approach to solving the problem
Since probabilistic information on errors is unknown, only uncertainty intervals
of measurements can be constructed:
H N T,0 = [N Tmes,0 − emax,N T , N Tmes,0 + emax,N T ], (8)
H AL,0 = [ALmes,0 − emax,AL , ALmes,0 + emax,AL ], (9)
H N T,f = [N Tmes,f − emax,N T , N Tmes,f + emax,N T ], (10)
H M,f = [Mmes,f emax,AL , Mmes,f + emax,AL ]. (11)
Note: here and in the sequel, notations of the interval variables recommended
by the Standard [12] are used.
After termination of the process, an interval of the coefficient KF is calcu-
lated as
KF = (H M,f − ALf )/OxALf , (12)
where ALf and OxALf are values of the variables at the terminal instant tf
computed by integration of the describing differential equations system (1)-(7).
Ideas and methods of the Interval Analysis Theory and Applications arose
from the fundamental, pioneering work by L.V. Kantorovich [4]. See, also, pio-
neering works in USA [5, 6]. Nowadays, very effective developments of the theory
and computational methods were elaborated by many researchers both in Russia
and abroad [7, 8]. Special interval algorithms have been created for estimating
parameters of experimental chemical processes [9–11].
In application to the problem under consideration, essence of Interval Anal-
ysis Methods consists in estimation (or identification) of a process parameters
under bounded measuring errors in the input information and under total ab-
sence of probabilistic characteristics of the errors.
The estimation results are represented in the form of so-called the Informa-
tion Set or Set-membership that approximate (see, for instance, [7, 8, 10, 11])
the estimated desirable totality of admissible values of the parameters from the
outer side or from the inner side.
In practice and, especially, in the case of small dimension of the parameters
vector to be estimated, the inner grid approach is more preferable since its
computational simplicity and smaller computations in comparison with both
the outer peeling and inner–box approach.
80
In our problem, it is just the case: vector of the main parameters is only
three-dimensional: KN T,AL , KN T,T , and KAL,Ox . So, the inner grid approach
was used.
Definition. A point (KN T,AL , KN T,T , KAL,Ox ) of the parameters space
KN T,AL ×KN T,T ×KAL,Ox is admissible and an initial point (N T0 , AL0 ) from the
box HN T,0 ×HAL,0 is admissible if the corresponding computed point (N Tf , ALf )
at the termination instant belongs to the box HN T,f × HAL,f .
Definition allows one to sift out directly such points (KN T,AL , KN T,T , KAL,Ox )
and (N T0 , AL0 ) that are not consistent with the given experimental data.
4 The main algorithms
Structure of the main algorithms and steps of its performing are illustrated in
Fig. 1. The algorithms are implemented in the following steps:
Step1: finding the initial guess by Step 2: building the outer box
Kj Kj around the initial guess (Kj,...,Ki)ing
the inverse problem of transfer
(NTmes,0, ALmes,0) (NTmes,f, ALmes,f)
King,j King,j
true initial true initial
point guess point guess
King,i Ki King,i Ki
Step 3: introducing a three-dimen- Step 4: building the inner
Kj sional grid “with outer reserve”
Kj grid approximation of I(K)
.............................. .. .. ......... ...
King,j
.............................
...... ..
..
........ .. King,j
. ......
..............................
...... ..
.. .. ..............................
........
.............................. ..
..... ...........................
. ...
........
..............................
......
........
... ........ .. . .
King,i Ki King,i Ki
Fig. 1. Structure of the main algorithms and steps of its performing; values of the
initial guess coefficients are marked by the ing–index; parameters shown on the axes
are illustrative
Step 1: finding the initial guess (KN T,AL , KN T,T , KAL,Ox )ing , i.e., the initial
admissible point of coefficients inside the information set;
Step 2: constructing the auxiliary outer guaranteeing box of coefficients;
Step 3: introducing the three–dimensional grid on the auxiliary box of coeffi-
cients;
Step 4: building the inner grid approximation of the information set I(KN T,AL ,
KN T,T , KAL,Ox ) of coefficients.
5 Computation results
Numerical results are presented in Figs. 2 and 3. Computation was performed
∗ ∗
with the model prescribed true coefficients of values KN T,AL = 0.001, KN T,T =
∗
0.001, KAL,Ox = 0.001, and KF∗ = 0.1 and with the initial values N Tmes,0 , ALmes,0 .
The true model values N T0∗ = 10 and AL∗0 = 0.3 are pointed with arrows.
81
The general picture (Fig. 2) shows decreasing the nitrate concentration N T (t)
(dash-two-points curve). Also, the initial N Tmes,0 , ALmes,0 corrupted measure-
ments and ones N Tmes,f , Mmes,f at the termination instant are marked by
crosses.
Processes on the components AL(t), M (t), ALOx(t), Ox(t), and F lm(t) are
given in the lower part in Fig. 3 and in zoomed images in Fig. 5. It is seen both
active decreasing the aluminum mass (since corrosion) and increasing the oxide
and film masses.
NT, wt%
10 NT(t)
NT*
0 NTmes,0
9
8 NTmes,f
7
Model true coefficients:
6 * ,AL = 0.001 K*
KNT NT,T = 0.001
* ,Ox = 0.001 K*
KAL F = 0.1
5
4
3
2
ALmes,0 = Mmes,0 ALOx(t) {AL(t), M(t)} Flm(t) Ox(t)
1 t,
hour
{AL*0} 1 2 3 4 Mmes,f 5
Fig. 2. Results of computations; general picture of the process
AL, ALOx, M, Ox, Flm
Model true coefficients:
ALmes,0 = Mmes,0 * ,AL = 0.001 K*NT,T = 0.001
KNT
0.3 Ox(t) * ,Ox = 0.001 K*F = 0.1
KAL
AL*0
0.2 ALOx(t)
M(t)
0.1 AL(t)
Mmes,f
Flm(t) t, hour
1 2 3 4 5
Fig. 3. Results of computations; picture in small components (zoomed)
Picture of the measured data is given in Fig. 4 and 5. Here, the true model
values are marked, the measurements and calculated values N Tmes,f , Mmes,f
(at the termination instant) are pointed by arrows; uncertainty boxes (sets)
82
H N T,0 × H AL,0 and H N T,f × H M,f of measurements are drawn by rectangles.
Integration of system (1)-(7) with true coefficients gives the values N Tf , Mf .
Note that this point lies inside the uncertainty box H N T,f × H M,f that
proves admissibility of taken values of the coefficients and the initial position
N Tmes,0 , ALmes,0 . Consider steps of the suggested algorithm.
AL and M true point t=0
0.3 HNT,0 HAL,0
measurement NTmes,0, ALmes,0
and its uncertainty box
0.20 example of integration result
with true values of coefficients
NTf , Mf
t = tf
HNT,f HM,f
0.1 true point measurement NTmes,f , Mmes,f and its uncertainty box
9 10 11 NT, wt%
Fig. 4. Picture of the measured data
Step 1. The initial guess of admissible values of coefficients
(KN T,AL = 0.001046, KN T,T = 0.001163, KAL,Ox = 0.001)ing (13)
are found as a solution of an usual inverse problem of dynamics with the initial
(left) value of the process N Tmes,0 , Mmes,0 and the termination (right) value
N Tmes,f , Mmes,f .
true point
AL and M
0.3 t=0
measurement NTmes,0 , ALmes,0
and its uncertainty box
0.20 inverse problem gives
the initial guess coefficients (“ing”)
KNT,AL = 0.001046, KNT,T = 0.001163,
NTf , Mf KAL,Ox = 0.001, and [KFlm] = [0, 0.174]
t = tf
0.1 true point measurement NTmes,f , Mmes,f and its uncertainty box
9 10 11 NT, w t%
Fig. 5. Step 1; finding the initial guess (KN T,AL , KN T,T , KAL,Ox )ing of coefficients
KN T,AL , KN T,T , KAL,Ox
Step 2. Constructing an auxiliary outer box around the initial guess (KN T,AL , KN T,T ,
KAL,Ox )ing . The procedure consists of:
83
– coordinate-wise variation of each coefficient under fixed values of two other
ones with verification of admissibility each changed value of the coefficient un-
der variation;
– variations are implemented for two directions to find the minimal and maximal
values on each coefficient, i.e., to find approximate intervals of each coefficient.
Results of this procedure are shown in Fig. 6. The maximal upper points are
marked by white squares; the minimal ones are represented by black squares.
The following marginal boundary values were found:
KN T,AL = [0.000896, 0.00125], KN T,T = [0.000618, 0.001738],
(14)
KAL,Ox = [0, 0.004225].
Remark 2: Because of independent coordinate-wise variations, the box found
can have inadmissible points, for example, the box apices or, even, parts of the
edges.
KAL,Ox
0.004225
Outer box w.r.t.
the initial guess
coefficients
0.0010
0.000618
0.000896 0 0.0010
0.0010 0.001163
0.001046
KNT,T
0.00125
(KNT,AL,KNT,T, 0.001738
true point KAL,Ox)ing
KNT,AL
is inside
Fig. 6. Step 2; the auxiliary outer box of parameters KN T,AL , KN T,T , KAL,Ox around
the initial guess point (KN T,AL , KN T,T , KAL,Ox )ing
Step 3. Having boundaries (14), introduce grid in each of these intervals
with some reasonable values of the grid step on each parameter. In model com-
putations, the number of nodes in each grid was sufficiently given as 51.
Step 4. The concluding operation is in simple direct verification of each node
of the introduced three-dimensional grid. Only admissible nodes are included
further into the inner part of the information set.
Results of constructing a cross-section of the inner approximation of the
information set I(KN T,AL , KN T,T , KAL,Ox ) for the fixed value of the icoefficient
KN T,T = 0.001163 is shown in Fig. 7.
The similar pictures are obtained for presentations of the initial grid and
final inner approximation of the cross-sections of the information set I(KN T,AL ,
KN T,T , KAL,Ox ) for fixed values of the coefficient KAL,Ox , and the initial grid
and final inner approximation of the cross-section of the information set I(KN T,AL ,
KN T,T , KAL,Ox ) for fixed valuesof the coefficient KN T,AL .
By the similar procedures, the whole informational set I(KN T,AL , KN T,T ,
KAL,Ox ) for all points from the initial uncertainty box H N T,0 × H AL,0 is built
84
KAL,Ox cross-section for fixed value KNT,T = 0.001163
0.004225
0 KNT,AL
0.000896 true point 0.00125
Fig. 7. Step 4; the initial grid and final inner approximation of the cross-section of the
information set I(KN T,AL , KN T,T , KAL,Ox ) for fixed value of the initial guess coeffi-
cient KN T,T = 0.001163; circles denote admissible nodes; points denote inadmissible
ones
KNT,T
0.00195
Information set outer
marginal approximation box
pointwise
cross-section
for
KNT,AL = 0.00133
Initial
cross-section
for KNT,AL = 0
0.000575 0 KAL,Ox
true
point 0 0.04
is inside
information set as a collection of cross-sections
for admissible values of the coefficient KNT,AL
KNT,AL
0.00133 Intervals of information set outer approximation:
KNT,AL : [0, 0.00133], KAL,Ox : [0, 0.04], KNT,T : [0.000575, 0.00195]
Fig. 8. The three-dimensional image of the information set I(KN T,AL , KN T,T , KAL,Ox )
for fixed KF = 0.1
as a totality (see ideology and techniques of such representation in [9–11]) of all
admissible cross-sections; its spatial image is shown in Fig. 8.
Underline that the results represented in Figs. 7, 8 have the guaranteed char-
∗ ∗
acter since the true values of the coefficients KN T,AL = 0.001, KN T,T = 0.001,
∗
and KAL,Ox = 0.001 obligatory (by construction) lie inside the final inner ap-
proximation of the information set I(KN T,AL , KN T,T , KAL,Ox ) for all initial
values N T (t = 0), AL(t = 0) from the initial uncertainty box H N T,0 × H AL,0 .
Analysis of experimental data has shown that for practical applications, the
mentioned point-wise initial guess of type (13) with approximate estimating
intervals of type (14) are sufficient and acceptable.
85
6 Conclusion
In contrast to now existing approaches to estimation of experimental process
parameters under mentioned conditions of uncertainty, the interval approach al-
lows one:
– to built verified inner approximation of the information set for the activity
coefficients;
– to find verified interval estimations of the important coefficient of the oxide
film precipitated on the aluminum surface for various values of admissible coef-
ficients from the information set;
– if necessary, to enhance the box (set) of the initial (or termination) measure-
ments for some fixed prescribed values of coefficients.
7 Acknowledgment
The work was supported by the Russian Foundation for Basic Research, Project 18-
01-00410 and by the Ural Federal University’s Center of Excellence in ”Quantum and
Video Information Technologies: from Computer Vision to Video Analytics” (accord-
ing to the Act 211 Government of the Russian Federation, contract 02.A03.21.0006).
M. Hladík was supported by the Czech Science Foundation Grant P402/13-10660S.
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