MATHEMATICAL MODELING OF THE CURING PROCESS OF
NEWLY CREATED MATERIALS BASED ON POLYESTER
RESIN
Abdullaev O.G.
1
., Toshmirzaev Yu.U.
1
., Mirzaev T.S.
1
,
Umarov A.V.
2,3
, Kasimova G.A.
3
1
Namangan State University,Uychi Str.316v Namangan region 160119, Uzbekistan,
2
University of Tashkent for Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan,
3
Tashkent State Transport University, Adilkhojaev Str. 1, Tashkent 100067, Uzbekistan.
https://doi.org/10.5281/zenodo.10471282
Keywords
:
auto putty, unsaturated polyester resin, accelerator, hardener, mathematical modeling, curing process.
Abstract:
In this article, using mathematical modeling of the curing rate, the influence of factors directly affecting the
curing processes of created materials prepared on the basis of unsaturated polyester resin is studied: the
concentration of the accelerator, hardener and temperature. When modeling the process, the concentrations
of the accelerator, hardener and temperature were taken as the initial (input) parameters, and the curing rate
of the autofill at different temperatures was taken as the final (output) parameter.
1 INTRODUCTION
Paints and varnishes are multicomponent
compositions that, in a liquid state, are applied to the
surface of products and, after drying, form a film. this
film is held by adhesion forces to the base and is
called paint coatings.
Paint and varnish coatings are widely used to
cover vehicles, performing two functions - protective
and decorative, i.e. technical and aesthetic. Paint and
varnish coatings have significant advantages over
other types of coatings, since they are easier to apply
to the surface, which reduces the cost of painting
work. Paint and varnish coatings have great
durability; they can be applied to metal, plastics and
other materials [1].
Paints and varnishes are classified according to
various criteria, for example, varnishes, enamels,
primers, putties, and each class has certain
requirements [2].
The main paint and varnish materials used in
vehicles include primers (primers), putties and
enamels.
This article is devoted to the study of the curing
process of auto putty prepared on the basis of
unsaturated polyester resin [3-6].
As you know, putties are used to level the surface
to be painted and are a thick, viscous, pasty mass
consisting of a mixture of fillers and pigments in a
film former. Putties have a rather higher viscosity
than other paints and varnishes due to the higher
content of solid components [7].
Currently, the main volume of produced auto
putty falls on the share of a polymer composition
produced on the basis of unsaturated polyester resin
[8, 9]. One of the important parameters of auto putty
is the curing speed after applying it to the surface.
As noted above, unsaturated polyester resin is the
binding component of car putty of various brands
from many manufacturers.
Automotive putties, as one of the representatives
of paints and varnishes, are multicomponent
composite materials. To harden such a composition,
an accelerator is used, which is added to the
composition of the auto putty, and a hardener, which
is used before its use. Typically, the instructions
supplied with the putty indicate the required amount
of hardener for the working composition at a certain
temperature. The amount of hardener for the curing
reaction of the auto putty and accelerator, which is
introduced into the composition in advance, is also
important.
When creating an auto putty, the optimal working
concentration of the accelerator in the composition of
the composite material and hardener is selected
experimentally. Moreover, the paint and varnish
industry produces car putties of various
compositions, depending on the region where they
will be used.
In this regard, it becomes absolutely necessary to
use a method of mathematical modeling of the curing
process of auto putty, since the rate of curing of auto
putty depends on the concentration of the accelerator,
hardener and the ambient temperature where it is
used.
2 Main part
The object of the study is a car putty created by
the authors on the basis of an unsaturated polyester
resin (brand PN-1), where cobalt octoate is used as an
accelerator, introduced into the composition, and
methyl ethyl ketone peroxide, introduced before
applying the car putty, is used as a hardener.
Experimental data on the curing time of the
auto putty were analyzed at various concentrations of
accelerator, hardener and temperatures, analyzed
using the method of mathematical modeling using a
multivariate linear regression equation [10, 11].
In Fig. 1 shows the results of experimental
studies of the dependence of the curing time of auto
putty at various concentrations of accelerator and
hardener on temperature.
Fig. 1. Dependence of the curing time of the auto putty on
temperature, where the concentrations of cobalt octoate and
methyl ethyl ketone peroxide: 1 - 1.5%, 1.8%; 2- 0.8%,
1.6%; 3- 0.4%, 1.2%.
Based on the obtained experimental data, a
mathematical model was built for the problem under
consideration of determining the optimal amount of
accelerator and hardener to optimize the curing time
of the auto filler at different temperatures.
Let's construct a linear regression equation for the
dependence of time (t) on temperature (T),
concentration of accelerator (x) and hardener (y)
using the least squares method.
The relationship between time, temperature,
accelerator and hardener is determined using a matrix
of paired correlation coefficients using the MS Excel
computer program system (2-table).
2-table. Results of dependencies between factors
of the studied processes
Factors
𝑡
𝑇
𝑥
𝑦
𝑡
1
𝑇
-0.81
1
𝑥
-0.29 -0.31
1
𝑦
-0.27 -0.32 0.94
1
Analysis of the results shows that there is a strong
inverse dependence of the curing time of the auto
putty on temperature and a weak inverse dependence
on the accelerator and hardener, which is apparently
due to the occurrence of complex chemical reactions
in the composite system during the curing of the auto
putty, in the presence of initiating components.
Let us assume that the multivariate linear
regression equation has the form
𝑡(𝑇, 𝑥, 𝑦) = 𝑎𝑇 + 𝑏𝑥 + 𝑐𝑦 + 𝜀
(1)
To estimate the unknown parameters, we find the
minimum
value of the following sum over a, b, c, ε.
𝐹(𝑎, 𝑏, 𝑐, 𝜀) = ∑(𝑡
𝑖
− (𝑎𝑇
𝑖
+ 𝑏𝑥
𝑖
+ 𝑐𝑦
𝑖
+ ε))
2
𝑛
𝑘=0
(2)
To determine the minimum of the function
F(a,b,c,ε), we find its partial derivatives with respect
to a, b, c, ε, equate them to zero and obtain the
following normal system of equations:
Based on the experimental results from Fig. 1,
we find the following auxiliary table (3-table) and
using a system of normal linear equations we
compose a system of linear equations with unknown
parameters a, b, c, ε.
3-table. An auxiliary table based on experimental
data
Based on the above table and the system of normal
equations (3), we obtain:
{
7721𝑎 + 207𝑏 + 368𝑐 + 245𝜀 = 4997
207𝑎 + 9.15𝑏 + 13.38𝑐 + 8.1𝜀 = 182.1
368𝑎 + 13.38𝑏 + 21.72𝑐 + 13.8𝜀 = 327
245𝑎 + 8.1𝑏 + 13.8𝑐 + 9𝜀 = 218
(4)
Having calculated this system of equations using
the Mathcad program, we obtain the following
results:
𝐴 = (
7721
207
207
9.15
368
245
13.38
8.1
368 13.38
245
8.1
21.72
13.8
13.8
9
),
𝐸 = (
4997
182.1
327
218
)
𝐴
−1
= (
1.057𝑥10
−3
1.762𝑥10
−3
1.762𝑥10
−3
4.67
0.011
−0.048
−7.981
7.987
0.011
−7.981
−0.048
7.987
15.624
−17.086
−17.086
20.426
),
𝐴
−1
𝐸 = (
−1.099
−9.498
−11.739
80.689
)
The last matrix displays the parameter values:
𝑎 = −1.099, 𝑏 = −9.498, 𝑐 = −11.739, 𝜀
= 80.689
According to the results found above, if we
replace the parameters of the multivariate linear
regression equation of type (1) with the
corresponding values, we obtain a function of the
following form:
𝑡(𝑇, 𝑥, 𝑦) = −1.099𝑇 − 9.498𝑥 − 11.739𝑦 +
80.689
(5)
We enter the values of the corresponding factors
in Fig. 1 into this function, calculate the differences
and use them to determine the average error
approximation in order to get a general idea of the
quality of the model based on relative deviations for
each observation, and based on statistical data (5) we
plot the function:
𝐴 =
1
𝑛
∑ |
𝑡 − 𝑡(𝑇, 𝑥, 𝑦)
𝑡
| ∗ 100% = 5.968%
Below is a graph of the dependence of the curing
reaction rate of the auto putty on the concentration of
the accelerator, hardener and temperature.
Fig.2. Dependence of curing time on parameters x, y and T,
1-according to experimental data, 2-according to calculated
data.
CONCLUSIONS
1. Mathematical modeling of the autofill curing
process was used to determine the functional
dependence of the curing time (reaction rate) of the
created composite material on the operating
temperature and on the concentration of the
components.
2. Using this functional dependence on the
concentration of the accelerator and hardener
(based on known x and y) included in the
composition, it is possible to determine the rate of
𝜕𝐹
𝜕𝑎
=
−
2
∑ 𝑡
𝑖
𝑇
𝑖
𝑛
𝑖
=1
− 𝑎 ∑ 𝑇
𝑖
2
𝑛
𝑖
=1
− 𝑏 ∑ 𝑥
𝑖
𝑇
𝑖
𝑛
𝑖
=1
− 𝑐 ∑ 𝑦
𝑖
𝑇
𝑖
𝑛
𝑖
=1
− 𝜀 ∑ 𝑇
𝑖
𝑛
𝑖
=1
= 0
𝜕𝐹
𝜕𝑏
=
−
2
∑ 𝑡
𝑖
𝑥
𝑖
𝑛
𝑖
=1
− 𝑎 ∑ 𝑥
𝑖
𝑇
𝑖
𝑛
𝑖
=1
− 𝑏 ∑ 𝑥
𝑖
2
𝑛
𝑖
=1
− 𝑐 ∑ 𝑥
𝑖
𝑦
𝑖
𝑛
𝑖
=1
− 𝜀 ∑ 𝑥
𝑖
𝑛
𝑖
=1
= 0
𝜕𝐹
𝜕𝑐
=
−
2
∑ 𝑡
𝑖
𝑦
𝑖
𝑛
𝑖
=1
− 𝑎 ∑ 𝑦
𝑖
𝑇
𝑖
𝑛
𝑖
=1
− 𝑏 ∑ 𝑥
𝑖
𝑦
𝑖
𝑛
𝑖
=1
− 𝑐 ∑ 𝑦
𝑖
2
𝑛
𝑖
=1
− 𝜀 ∑ 𝑦
𝑖
𝑛
𝑖
=1
= 0
𝜕𝐹
𝜕𝜀
=
−
2
∑ 𝑡
𝑖
𝑛
𝑖
=1
− 𝑎 ∑ 𝑇
𝑖
𝑛
𝑖
=1
− 𝑏 ∑ 𝑥
𝑖
𝑛
𝑖
=1
− 𝑐 ∑ 𝑦
𝑖
𝑛
𝑖
=1
− ∑ 𝜀
𝑛
𝑖
=1
= 0
(3)
its curing at the operating temperature of the auto
putty or, conversely, determine the optimal
temperature at a certain curing rate.
3. In addition, using this function, you can pre-
determine the amount of hardener (or accelerator)
that should be added at the temperature of applying
the auto filler to the surface with a predetermined
curing rate, in accordance with the concentration
of the accelerator (or hardener) included in the
unsaturated polyester composition (when one of
the parameters x or y is unknown).
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