APPLICATION OF DIGITAL CELLULAR RADIO FOR MOBILE LOCATION ESTIMATION


IIUM Engineering Journal, Vol. 21, No. 1, 2020 Mustafina et al. 
https://doi.org/10.31436/iiumej.v21i1.1180 

NUMERICAL ALGORITHM FOR FINDING 
OPTIMAL INITIAL CONCENTRATIONS OF 

CHEMICAL REACTIONS 

SVETLANA MUSTAFINA1*, ANDREY ANTIPIN1, EVGENIA ANTIPINA1, 
ELENA ODINOKOVA2, LARISA TUCHKINA2, KONSTANTIN KOLYAZOV2, 

AND SOFIA MUSTAFINA1
1Dept. Mathematics and Information Technology, Bashkir State University, Ufa, Russia 
2Dept. Information Technology and Management Systems, Moscow State University of 

Technologies and Management, Moscow, Russia.  

*Corresponding author: Mustafina_SA@mail.ru 

(Received: 16th June 2019; Accepted: 19th October 2019; Published on-line: 20th  January 2020) 

ABSTRACT: This article is devoted to the problem of developing methods for 
mathematical modelling in the sphere of optimal planning in a chemical experiment. In 
the article, the problem of finding the optimal ratio for initial concentrations of 
substances is formulated in general terms and an algorithm for solving this problem is 
constructed basing on the method of artificial immune systems. The developed algorithm 
for finding the optimal initial concentrations of substances allows solving the problem of 
experiment planning in chemistry at the computational experiment stage. In this case, the 
solution of the optimization problem found with its help does not depend on the choice 
of the initial approximation. The algorithm was tested for the industrially meaningful 
process of benzilidenebenzilamine  synthesis for which the optimum values of the initial 
concentrations were calculated in order to obtain maximum yield of the reaction product.  

ABSTRAK: Artikel ini ditujukan untuk masalah mengembangkan kaedah pemodelan 
matematik dalam bidang perancangan optimum dalam percubaan kimia. Dalam artikel 
itu, masalah mencari nisbah optimum untuk konsentrasi awal bahan digubal secara 
umum dan algoritma untuk menyelesaikan masalah ini dibina berdasarkan kaedah sistem 
imun buatan. Algoritma yang dibangunkan untuk mencari kandungan konsentrasi awal 
yang optimum membolehkan menyelesaikan masalah perancangan percubaan dalam 
kimia pada peringkat percubaan pengkomputeran. Dalam kes ini, penyelesaian masalah 
pengoptimuman yang didapati dengan bantuannya tidak bergantung pada pilihan 
permulaan awal. Algoritma ini telah diuji untuk proses sintesis benzilidenebenzilamin 
yang bermakna industri yang mana nilai-nilai optimum kepekatan awal dikira untuk 
mendapatkan hasil maksimum produk reaksin. 

KEYWORDS: chemical kinetics; artificial immune systems; initial concentration 

1. INTRODUCTION
The solution to many problems in chemistry and chemical technology is associated

with complex and expensive experiments. In the study of the chemical process at the 
computational experiment stage, it is advisable to identify its main regularities because of 
the complexity of the initial raw materials’ compositions and the low content of valuable 
components in them. In this regard, the actual task is to develop methods for optimal 
planning of the experiment, which allow saving time and material resources for 
conducting a full-scale experiment. 

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https://doi.org/10.31436/iiumej.v21i1.1180 

The development of computer technology and information technology allows the use 
of mathematical methods in the planning of the chemical experiments. The methods of 
mathematical modelling allow the determination of optimal concentrations of reagents to 
obtain the best yield of the final product and significantly reduce the cost of substance 
synthesis. 

The calculation of the initial concentrations of the reactants in a chemical reaction is 
made by searching for an extremum of the function for calculating the dynamics of the 
substance concentrations during the chemical experiment. The significant drawbacks of 
most numerical methods for finding the extremum of functions are the difficulties in 
achieving convergence of the process, which depends on the choice of the initial 
approximation. 

Currently, metaheuristic optimization methods are widely used, which make it 
possible to efficiently find a global optimum in a reasonable time [1-3]. Their important 
feature is the ability to overcome the local extremum point of the target function in the 
search process. Therefore, they allow finding better solutions in comparison with classical 
heuristics [4-7]. 

Among the metaheuristic algorithms, the method of artificial immune systems is 
distinguished. Its advantage is the presence of a memory mechanism. The memory 
mechanism makes it possible to use information about previously found best solutions, 
thereby increasing the efficiency of global search. An important feature of the method of 
artificial immune systems is the independence of the optimization problem solution from 
the initial approximation [8-10]. 

2. FORMULATION OF THE PROBLEM
Let us formulate in general terms the problem of finding the optimal ratio for initial

concentrations of substances. A mathematical model for the dynamics of substance 
concentrations is a system of ordinary differential equations 

( , ( ), ),
dx

f t x t T
dt

(1) 

with initial conditions 
0(0) , 1, ,i ix x i n (2) 

where 1 2( ) ( ( ), ( ), , ( ))nx t x t x t x t is the vector of the concentrations of the reacting 
substances,  f (t, x(t), T) is continuous together with their partial derivatives of the 
function, ],0[ 2tt  is the time of reaction, and T  is the reaction temperature [11]. 

At the initial moment of time, substances are interconnected by some ratio: 

1 1(0) : : (0) : : .n nx x (3) 

It is required to find the ratio for initial concentrations of substances, which ensures 
the extremum of optimality criterion 

2
1

( ) ( ) .
n

i i
i

Q x x t extr (4) 

The optimality criterion expresses the maximum yield of the target products or the 
minimum impurity content depending on the coefficients i (positive or negative). You 

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https://doi.org/10.31436/iiumej.v21i1.1180 

can also specify some economic criterion, then the coefficients i are the cost coefficients 
of the reagents. 

3. THE ALGORITHM FOR THE METHOD OF ARTIFICIAL
IMMUNE SYSTEMS TO CALCULATE OPTIMAL INITIAL
CONCENTRATIONS OF SUBSTANCES
The method of artificial immune systems simulates the functioning of a living

organism immune system, designed to destroy foreign bodies and improve the fight 
against them. Consider the main concepts of the method. 

An antigen is an alien substance from which the body tries to protect itself with 
antibodies. An antibody is a substance that recognizes and destroys an antigen [12]. A 
memory cell is an immune cell that accumulates data about new antibodies that can 
recognize an antigen. The target function of the optimization problem is called the fitness 
function [13,14]. 

As a function of fitness, we used the maximum yield of the target reaction product at 
the final moment of the reaction. This required the solution of a direct kinetic problem 
with a given ratio for the initial concentrations of the initial substances, that is, the solution 
of a system of differential equations (Eq. (1)) with initial conditions (Eq. (2)). 

We modified the algorithm to solve the problem of finding the optimal initial 
concentrations of the components in the reacting mixture. As a population of immune 
cells, we introduced sets of initial concentrations of substances 

1 2
(0) ( (0), (0), , (0)), 1, ,j j j jnx x x x j k where k is the number of immune cells in the 

population. 

To calculate the value of the fitness function ( )Q x , it is necessary to know the 
concentration of reagents at the final point in time. To do this, we introduced a block for 
solving a direct kinetic problem into the algorithm. 

The problem of finding the minimum of the target function ( )Q x was reduced to the 
problem of finding the maximum by replacing the sign in front of the function with the 
opposite one: *( ) min ( ) max [ ( )].Q x Q x Q x If Q(x) was positive then Q(x*)<0.

We formulated this algorithm. 

Step 1. The creation of an initial population for the starting substance concentrations. 

Set the number of immune cells in the population k, the number of parental cells for 
the selection r_sel, the number of cells in the population with the worst value of the 
fitness function n_bad, the maximum number of iterations i_max, the clone operator 
parameter n_cl, the mutation parameter m, the iteration counter num = 0. 

Randomly set the values of k cells of the initial population (0), 1, ,jx j k
0 (0) 1.jx

Step 2. Calculation of fitness function values. 

For sets (0)jx  calculate the value of fitness function ( )jQ x , by solving a direct 
kinetic problem for each immune cell. That is, at this step it is necessary to find a 
numerical solution to the system of differential equations (Eq. (1)) with the initial 

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IIUM Engineering Journal, Vol. 21, No. 1, 2020 Mustafina et al. 
https://doi.org/10.31436/iiumej.v21i1.1180 

conditions (Eq. (2)) by an explicit or implicit method. The initial conditions of the 
system (Eq. (2)) are the immune cell. 

Step 3. Cloning. 

Select r_sel parent cells from the population (a set of antibodies), which correspond to 
the best values of fitness function ( )Q x . Generate n_cl clones for each cell. 

Step 4. Mutation. 

For each parent cell generate random numbers [0,1],l 1 [0,1 (0)],
j

ih x  

2 [0, (0)],
j

ih x  1, .i n  Perform a mutation of its clones by the rule: if 0, 5,l  then 

1(0) : (0) ,
s s
i ix x h m else 2(0) : (0) ,

s s
i ix x h m where 1, _ ,s n cl m – mutation

parameter. 

Step 5. Calculation of fitness function values. 

For each mutant clone calculate the value of the fitness function by Eq. (4). 

Step 6. Selection and transition to a new population. 

For each parent cell, find the mutant clone with the lowest value of the fitness 
function. Compare its value with the value of the fitness function of the parent cell. In 
the new population, leave the parent cell or replace it with a mutant clone with the 
best fitness function value. 

Step 7. Renewal of a population. 

In the population, find n_bad cells with the lowest value of fitness function ( )Q x . 
Replace these cells with new ones, randomly generated in the interval [0,1), and 
calculate the value of the fitness function ( )Q x  for them. 

Increase the iteration counter: num= num + 1.  

Step 8. Checking the conditions for the end of search. 

If num<i_max, then go to step 3, else finish the search and go to step 9. 

Step 9. Selection of optimal concentrations of substances from the population. 

As a solution to the problem of finding the optimal concentrations of substances, 
select the cell with the highest fitness function value from the last population. 

The developed algorithm was implemented as a software tool in the visual 
programming environment Delphi. 

4. COMPUTATIONAL EXPERIMENT
Based on the formulated algorithm, we calculated the optimal ratio of the initial

concentrations for the reaction of the benzylidenebenzylamine synthesis in order to obtain 
the maximum yield of the reaction product. The mechanism of the chemical reaction for 
the synthesis of benzylidenebenzylamine in the presence of a catalyst 3 26FeCl H O is 
represented by a set of stages [15]: 

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1 2 3 4

3 5 6

5 1 7 8

8 6 9

,
,

,
,

X X X X
X X X
X X X X
X X X

(5) 

where X1 – benzylamine (C7H9N), X2 – carbon tetrachloride (CCl4), X3 – 
chlorobenzylamine (C7H8NCl), X4 – chloroform (CHCl3), X5 – 1-phenylmethanimine 
(C7H7N), X6 – hydrogen chloride (HCl), X7 – benzylidenebenzylamine (C14H13N), X8 – 
ammonium (NH3), X9 – ammonium chloride (NH4Cl). 

The speed of each stage is determined according to the law of mass action: 

1 1 1 2

2 2 3

3 3 5 1

4 4 8 6

,
,

,
,

k x x
k x
k x x
k x x

(6) 

where ix is the concentration of the i-th substance ( 1, 9i ) (mole fraction), jk is the rate

constant of the j-th reaction stage 1, 4 ,j  calculated according to the Arrhenius
equation 

0( ) exp ,
j

j j

E
k T k

RT
 

where jЕ is the energy of activation of the j-th stage (kJ/mol), T is the reaction temperature 
(К), and R is the universal gas constant (8,31 J/(mol∙К)). 

The matrix of stoichiometric coefficients of substances ( )ij  ( 1, 4,i 1, 9j ) is given in 
Table 1. 

Table 1: Matrix of stoichiometric coefficients of the reaction of synthesis of 
benzylidenebenzylamine 

ω1 ω2 ω3 ω4 
X1 -1 0 -1 0 
X2 -1 0 0 0 
X3 1 -1 0 0 
X4 1 0 0 0 
X5 0 1 -1 0 
X6 0 1 0 -1
X7 0 0 1 0 
X8 0 0 1 -1
X9 0 0 0 1 
δ 0 1 0 -1

Since there are nonzero elements among the elements of the last line , the reaction 
proceeds with a change in the reaction volume [16, 17]. Then the kinetic model of the 
reaction is a system of differential equations: 

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IIUM Engineering Journal, Vol. 21, No. 1, 2020 Mustafina et al. 
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),,(

,9,1,
),(),(

TxF
dt
dN

i
N

TxFxTxF
dt
dx

n

niii

 (7) 

with initial conditions: 
0(0) , 1, 9, (0) 1,i ix x i N (8) 

where N is the variable reaction volume, 
4

1
, 1, 9,i ij j

j
F W i

4 9

1 1
,n j ij

j i
F W

0

j
jW C

is the values of rate of reaction stages ( 1, 4j ) (h-1), and С0 is the initial total 

concentration of substances (mol/l). 

The functions ( , ),nF x T ( , )iF x T ( 1,9)i  have the form: 

1 1 3

2 1

3 1 2

4 1

5 2 3

6 2 4

7 3

8 3 4

9 4

2

( , ) ( , ) ( , ),
( , ) ( , ),
( , ) ( , ) ( , ),
( , ) ( , ),
( , ) ( , ) ( , ),
( , ) ( , ) ( , ),
( , ) ( , ),
( , ) ( , ) ( , ),
( , ) ( , ),
( , ) ( ,n

F x T W x T W x T
F x T W x T
F x T W x T W x T
F x T W x T
F x T W x T W x T
F x T W x T W x T
F x T W x T
F x T W x T W x T
F x T W x T
F x T W x T 4) ( , ).W x T

(9) 

The initial substances of the reaction are benzylamine (X1) and carbon tetrachloride 
(X2), the concentrations of which are related by the ratio: 1 2(0) (0) 1.x x

For other substances (0) 0,ix 3, 9.i

Since the target substance of the reaction scheme is benzylideneaniline (X7), the 
optimization problem is formulated as follows: find the ratio for the initial concentrations 

1 2(0) : (0)x x  of the source substances X1 and X2, at which at the final time of the reaction 
the maximum yield of benzylidenebenzylamine X7 is reached, i.e. 7 2( ) ( ) max.Q x x t

The solution of this problem was obtained at the temperature 23T C  and the time 
of the reaction was 2 4t h. The solution of the system (Eq. (7)) with initial conditions 
(Eq. (8)) was obtained by the implicit Euler method. 

The result of the calculation obtained the optimal ratio for the initial concentrations of 
the source substances was: 

1 2(0) : (0) 0,51: 0, 49.x x (10) 

The maximum yield of the reaction (benzylideneaniline) was 0,013 mole fractions. 

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Figure 1 shows the dynamics of the benzylidenebenzylamine concentration at 
optimal initial concentrations. 

Fig. 1: Dynamics of the optimal concentration of benzylidenebenzylamine. 

5. CONCLUSIONS
Consider the solution of a direct kinetic problem for the reaction scheme (Eq. (5))

with arbitrary sets for initial concentrations of substances. 

According to Table 2, the maximum concentration of benzylidenebenzylamine is 
observed at a specific ratio of the source substances (Eq. (10)). This confirms the correct 
operation of the algorithm. 

Table 2: Dependence of the concentration of benzylidenebenzylamine 7X from the 
ratio of source substances 1X  and 2X

x1, mole 
fraction 

x2, mole 
fraction 

x7, mole 
fraction 

0.1 0.9 0.004 
0.2 0.8 0.008 
0.3 0.7 0.010 
0.4 0.6 0.012 

0.51 0.49 0.013 
0.7 0.3 0.011 
0.8 0.2 0.009 
0.9 0.1 0.005 

Thus, the developed algorithm for finding the optimal initial concentrations of the 
initial substances makes it possible to solve the problem of planning an experiment in 
chemistry at the stage of a computational experiment. The solution of the optimization 
problem will be found for any set of values of the initial concentrations of substances, 
since the work of the artificial immune systems method does not depend on the initial 
approximation. 

ACKNOWLEDGEMENT 
The reported study was funded by RFBR according to the research project No 17-47-
020068 and project No. 13.5143.2017 / 8.9. 

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    /ITA (Utilizzare queste impostazioni per creare documenti Adobe PDF adatti per visualizzare e stampare documenti aziendali in modo affidabile. I documenti PDF creati possono essere aperti con Acrobat e Adobe Reader 6.0 e versioni successive.)
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    /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken waarmee zakelijke documenten betrouwbaar kunnen worden weergegeven en afgedrukt. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 6.0 en hoger.)
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    /ENU (Use these settings to create Adobe PDF documents suitable for reliable viewing and printing of business documents.  Created PDF documents can be opened with Acrobat and Adobe Reader 6.0 and later.)
  >>
>> setdistillerparams
<<
  /HWResolution [600 600]
  /PageSize [595.440 841.680]
>> setpagedevice