Математичне моделювання кристалізації розчинів полімерів
The processes of homogenization and crystallization of polymer solutions in cylindrical pipes are considered, which are described by the convective-diffusion equation with respect to the solution temperature and kinetic equations with respect to homogenization and crystallization of the polymer know...
Gespeichert in:
| Datum: | 2023 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2023
|
| Schlagworte: | |
| Online Zugang: | https://journal.iasa.kpi.ua/article/view/279802 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | System research and information technologies |
| Завантажити файл: |  |
Institution
System research and information technologies| _version_ | 1867334434684928000 |
|---|---|
| author | Zelensky, Kyryl |
| author_facet | Zelensky, Kyryl |
| author_institution_txt_mv | [
{
"author": "Kyryl Zelensky",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
}
] |
| author_sort | Zelensky, Kyryl |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2023-05-24T21:28:17Z |
| description | The processes of homogenization and crystallization of polymer solutions in cylindrical pipes are considered, which are described by the convective-diffusion equation with respect to the solution temperature and kinetic equations with respect to homogenization and crystallization of the polymer known as the thermokinetic nonlinear boundary value problem. A numerical-analytical iterative method for solving this problem is proposed, which consists of stepwise obtaining solutions of kinetic equations with respect to homogenization and crystallization of polymer solutions depending on the solution temperature and obtaining a solution of the convective-diffusion problem with respect to melt temperature. The accuracy of the obtained solution is determined by the norm of the difference between two adjacent iterations. The value of the crystallization coefficient, which is close to unity, determines the length of the dosing zone and the transition to the next zone – the flow of homogenized polymer into the distribution head of the extruder. The results of mathematical modelling are given. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2023.1.12 |
| first_indexed | 2025-07-17T10:28:10Z |
| format | Article |
| fulltext |
K. Zelensky, 2023
Системні дослідження та інформаційні технології, 2023, № 1 141
UDC 621.3
DOI: 10.20535/SRIT.2308-8893.2023.1.12
MATHEMATICAL MODELLING OF CRYSTALLIZATION
OF POLYMER SOLUTIONS
K. ZELENSKY
Abstract. The processes of homogenization and crystallization of polymer solutions
in cylindrical pipes are considered, which are described by the convective-diffusion
equation with respect to the solution temperature and kinetic equations with respect
to homogenization and crystallization of the polymer known as the thermokinetic
nonlinear boundary value problem. A numerical-analytical iterative method for
solving this problem is proposed, which consists of stepwise obtaining solutions of
kinetic equations with respect to homogenization and crystallization of polymer
solutions depending on the solution temperature and obtaining a solution of the
convective-diffusion problem with respect to melt temperature. The accuracy of the
obtained solution is determined by the norm of the difference between two adjacent
iterations. The value of the crystallization coefficient, which is close to unity,
determines the length of the dosing zone and the transition to the next zone – the
flow of homogenized polymer into the distribution head of the extruder. The results
of mathematical modelling are given.
Keywords: homogenization, integral transformations, iterative method, crystallization,
mathematical model, nonlinear boundary value problem.
INTRODUCTION
The solution of the problem of polymer crystallization can be carried out on the
basis of a mathematical model that describes the heat transfer processes of
polymer melts and the cooling agent.
The main difference between the crystallization of polymers and the
crystalliza\-tion of other types of materials (eg, metals) is that the crystallization
of polymer melts is carried out in the space-time domain. With regard to the
crystallization of polymer solutions in extrusion devices, the situation is
complicated by the movement not only of the crystallization front, but also by the
motion of the solid mixture heated to the melting temperature under the rotational
motion of the screw worm. The crystallization process is phase, accompanied by
the release of heat and leads to stresses. Determining the values of the coordinate
z L , in which the degree of crystallization becomes close to unity, determines
the length of the dosing zone of the polymer, after which the polymer enters the
extruder head. This factor plays a crucial role in ensuring the formation of a
quality source product (for example, the quality of the insulating coating of the
cable or the quality of the polyethylene film) at the outlet of the extruder.
PROBLEM STATEMENT
The goal of solving the problem of crystallization of polymer solutions is to
develop an optimal strategy for cooling polymer solutions in order to reduce
processing time and limit the use of excessively low cooling temperatures.
K. Zelensky
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 142
Overview of approaches to building models
A significant number of scientific and applied works [1, 6–8,10–15] are devoted
to the construction of mathematical models of crystallization for various purposes.
Mathematical models of polymer crystallization in the form of a thin disk
(one-dimensional model) are considered in [1, 12, 13]. The model consists of two
nonlinear differential equations for the degree of crystallinity ),( try , defined as
the average volume fraction of the space occupied by crystals, and the
temperature field ),( trT , combined using the norm of the nucleation and growth
function )(TbN and )(TbG , the nucleation initiation function and the aggregation
21=)( yyk and saturation function of nuclei )(1=)( yyy :
;)),(()),(()),(()),((= 0 trTbtrykvtrTbtry
t
y
NG
)).,(()),((
1
=
2
2
trTbtrya
r
T
rr
T
t
T
GG
In the given mathematical model of crystallization of the polymer solution,
there are no connections of the determining components Gb , Nb from the
temperature of the solution.
In [4], a mathematical model is considered, in which the dependence of the
crystallization coefficient on the temperature of the polymer melt is specified
explicitly.
In [8], the mathematical model takes into account the clear dependence of
not only the crystallization coefficient, but also the polymerization coefficient,
since these characteristics are interconnected.
Two macrokinetic parameters and , which determine the specific
contribution of the polymer and crystalline phases, respectively, are introduced.
The degree of polymerization )(t determines the degree of completion of the
polymerization process and can take values from 0 (the polymer content is 0) to 1
(all the monomer has turned into a polymer). The degree of crystallization )(t
determines the degree of completion of the crystallization process and can take
values from 0 (the content of the crystalline phase is 0) to 1 (the entire amorphous
phase of the polymer has turned into a crystalline one). To quantitatively
characterize the degree of completion of the phase transformation [0,1] , a
parameter called the relative degree of crystallinity is used.
Determination of temperature fields ),( zrT and the degree of crystallinity
),( zr in insulation consists in the joint solution of the heat conduction equation
and the kinetic equation:
;)(0,),,(,
11
= LzRRrQ
z
T
zr
T
r
rrcz
T
V izgv
(1)
;])()[(1
)(
exp
1
= 00
TC
TTT
T
RT
U
K
Vz p
iz
iz (2)
Mathematical modelling of crystallization of polymer solutions
Системні дослідження та інформаційні технології, 2023, № 1 143
.=),(0,),,(
zc
VQ
QLzRRr m
vizg
The problem is solved under appropriate boundary and initial conditions.
Note that the equations (1), (2) are written relative to the processes in the
insulation, and the boundary conditions contain the conditions for the current-
conducting core. In addition, the equation for the degree of crystallization is
written relative to the variable , that is, a stationary problem is considered.
The presented mathematical model of polymer crystallization refers,
according to the authors, to the processes of applying polymer insulation to a
conductive core. However, the process of crystallization of the polymer solution
is carried out even before the insulation is applied to the core, that is, in the
dosing zone of the extruder. Therefore, such an approach to the study of the
crystallization process of polymer solutions cannot be considered to correspond to
the essence of the matter. For this reason, the mathematical model does not take
into account the convective transfer of the polymer melt during the cooling
process in the dosing zone.
Formulation of the problem
Assuming that technological stresses do not affect the temperature and flow of the
crystallization process, it is necessary to solve the problem of determining the
temperature fields and the degree of homogenization and crystallization, or the
thermokinetic problem.
The problem of determining temperature fields and the degree of
polymerization and crystallization is described by the following boundary value
problem:
z
T
v
r
T
v
t
txT
TTc zr
),(
)()( ;
;;)),(grad)((div= Vx
dt
d
Q
dt
d
QtxTT
kinetic equation of polymerization
;))((1=
)(
0
cK
dt
td
(3)
kinetic equation of crystallization
;))()()((=
)(
0
TATK
dt
td
p (4)
;exp=;exp=
TT
T
RT
E
kK
RT
U
kK
p
p (5)
.=,/= 010 bbREb
Initial and boundary conditions:
;=,0)(0,=,0)(0;=,0)( 0TxTxx (6)
;0,=),(grad);),((=),(grad)( 2SxtxTnTtxThtxTnT av (7)
K. Zelensky
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 144
,);,(=),( 1
* SxtxTtxT (8)
where c is the specific heat capacity; — density; — thermal conductivity
coefficient; QQ , — intensities of heat sources due to polymerization and
crystallization, respectively; R — universal gas table; 10,,,,, cckkEU —
kinetic steels determined experimentally from calorimetric measurements;
melting point; p — equilibrium degree of crystallization;
).()(
0,26
)/(10,52= 44
1
44
2
4 TTETT
T
TT pp
p
pb
Thermophysical properties of low density polymer (PENG):
2,4=c )C/( kgkJ ); 0,182= )C(/ mW ;
C170=
pT ; 1080= kg/m
3
; .C60=
cT
Taking into account the fact that at the melting temperature of the polymer,
the thermophysical coefficients can be considered constant, the differential
equations describing the processes of heat release in the dosing zone, taking into
account the smallness of the gradients relative to the axial component of the speed
of movement of the polymer solution, take the form:
;
1
= zze
z
z
z
r
z gv
z
p
z
v
v
r
v
v
t
v
(9)
.
)(
=
dt
td
Q
dt
d
QT
z
T
v
r
T
v
t
T
c zr
(10)
Initial and boundary conditions:
;=|),,(0;=|),,(1,=|),,( 0=0=0= pttt TtzrTtzrtzr
0.=|)(0;=|)(
201100 RR TTh
r
T
TTh
r
T
Cooling channel:
0.=|0,=|);(=| 0=0=20= r
z
r
r
zz z
v
r
v
rfv
On hard surfaces for the components of the agent that cools, meet the
conditions of adhesion and impenetrability.
In the equations, the following are indicated: — thermal conductivity
coefficient; Q — thermal effect of polymerization; Q — the thermal effect of
crystallization (the rate of specific heat release during PENG crystallization);
kk , — polymerization and crystallization rate constants; U — activation
energy of the polymerization process; — characteristic temperature of the
polymer; R — universal gas table; 10 , hh — coefficients of heat exchange with
the environment; 415=pT K is the equilibrium melting temperature.
Mathematical modelling of crystallization of polymer solutions
Системні дослідження та інформаційні технології, 2023, № 1 145
PROBLEM SOLVING
So, we have a system of differential equations (9), (10), (3), (4) with respect to the
temperature of the polymer melt and the degree of homogenization and crys-
tallization with the corresponding initial and boundary conditions (6)–(8).
This system of equations is nonlinear, its solution will be sought by an
iterative scheme by analogy with the previous sections.
Let’s consider the equations regarding the degree of polymerization and the
degree of crystallization . At the same time, at the first stage, we consider,
kK , kK .
Since these equations contain a nonlinear component relative to ),,( tzrT ,
the exponent must be approximated, for example, by a fractional-rational
expression [16]. Let’s denote RUe /=0 . Then we will have
;=1)(
2
321
2
321
2
321
0
TeTee
TeTee
k
TeTee
Te
kTK
1.=/2,=/12,= 302
2
01 eeeee
By analogy, we approximate K
in (6) REb /=0 , 01 = bb ).
.
)(
exp=
4
4
3
3
2
210
4
4
3
3
2
21010
TcTcTcTcc
TdTdTdTdd
k
TTT
Tbb
kK
p
(11)
Consider the equation (3) and denote
.))((1)()(
1
= 022
1
ckTe
dt
d
TeT
e
N
Then we will have
.),())((1= 0 TNck
dt
d
(12)
We get the solution of the equation (12) without taking into account
),( TN :
0.=(0);=,= 0
2
(1)
ckkkkk
dt
d
The solution of this equation in the first approximation:
.]11)(2[
1
=)( 2
0
(1) ktkt eekt
c
t
Further iterations in determining the expressions for the solutions in this
form leads to a significant increase in the exponential terms in the solution, which
actually makes it impossible to use the iterative procedure.
Therefore, it is worth now to apply the simplification algorithm and obtain
the following solution of the equation (12):
.)]()([)( 21102
(1) tfatfateAt
(13)
K. Zelensky
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 146
Now further iterations to determine the expression for the polymerization
coefficient can be implemented according to the standard algorithmic
procedure [16].
Let’s go back to the expression (12). It contains an expression for the
melting temperature of the polymer:
);()()(=),,( ,
1=1=
tvrzZrRtzrv knk
N
k
n
M
n
r )()()(=),,( ,
1=1=
tvzzZrRtzrv knk
N
k
n
M
n
z ;
)()()(=),,( ,
1=1=
tttzZrRtzrT knkm
N
k
M
m
;
.))1(1)1(1(1=)( 5
,2
2
,
5
,1
1
,
4
,10
,, ttftttft
t
etttt knknknkn
knt
knkn
(14)
This expression is obtained in [17]. Application of integral transformations
by spatial variables gives:
.),,()()()()(=)(
11
0
1
2
2
(1) drdztzrNzZzZ
L
rRrR
r
tN kkkk
z
nn
r
Substituting the expression for the melt temperature in the image space by
the spatial variables in gives:
.)(,)(,)(,
,
=)(
(1)
211
(1)
21111
11
(1)
dt
d
etkttetktttktt
k
tN nnn
n
Substituting (13) and (14) into this expression, performing the corresponding
transformations gives
)()( (1)
2
43
102(1) tN
ppNN
pNN
p
N
pN
).()(( 522511
4
0 tnafnatnafnatena na
Taking into account (13)
.))()((=)()(=)( (2)
52
(2)
2
(2)
51
(2)
1
(2)
4(2)
0
(1)(1)(2) tAfAtAfA
t
eAtNtt
A
In Fig. 1, 2 graphs of the distribution of the proportion of melt homogenization
at fixed time values are given. It can be seen that the distribution of the
homogenization (and crystallization) process is evened out due to the decrease in
the temperature of the polymer melt.
In the space of originals by spatial variables:
)()(),,(
,
zZrRtzr kknn
kn
×
.)))(())((( ,2
2
,,1
1
,
,)(0
,
tfAtfA
t
eA knknknkn
knx
kn (15)
Mathematical modelling of crystallization of polymer solutions
Системні дослідження та інформаційні технології, 2023, № 1 147
Taking (11) into account, the equation (4) for determining the degree of
crystallization takes the form
dt
td
TcTcTcTcc
)(
)( 4
4
3
3
2
210
=
;])()())[)((= 44
10
4
4
3
3
2
210 TTEtETdTdTdTdd p
.0,26/= 2
1 pTE
);,())((=
)( 2
21000
TNeeed
dt
td
c (16)
))((=),( 2
210
3
4
2
321 eeeTdTdTddTTN .
.
)(
)( 3
4
2
321 dt
td
TcTcTccT
(17)
By analogy with (13) we obtain the solution of the equation (16)
)].()([)( 22110
(1) tfbtfb
t
ebt
In the next step, we will add to the obtained solution the dependence on the
temperature of the melt ),( txT , perform the transformation for ),( tTN
according to (17). Due to the algorithms of equivalent simplification, the solution
of the equation (16) can be written in the form
Fig. 1. Graph of distribution of polymer melt homogenization at the 1st iteration
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
rz
Gomogen.: t=2
1
–s
t i
te
r
z t
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
rz
Gomogen.: t=2
1
–s
t i
te
r
z t
Fig. 2. Distribution graph of polymer melt homogenization at the 3rd iteration
K. Zelensky
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 148
)()(=),,(
,
zZrRtzr kknn
kn
×
.)))(())(((
)
,2
2
,,1
1
,
,(0
,
tfBtfB
t
eB knknknkn
kn
kn
Now we need to define an expression for
dt
td
Q
dt
td
QL crpT
)()(
=
.
Taking into account the obtained solutions regarding the degree of polymerization
and crystallization and the calculation of the time-dependent ones, we have:
))(())((= 1010 tAAtAAteQL pT
+
)].)(()])(([ 1010 tBBtBB
t
eQcr
We have
)].()([=)( ,,
2
,,
1
,
,
0, tftf
t
eftF kn
f
knkn
f
kn
kn
fknT
kn
Therefore, the temperature field of the polymer in the cooling zone is
determined by the expression:
).()()(),,(=),,( ,
1=1=
pol tFzZrRtzrTtzrT T
knkn
M
k
M
n
The coefficients in these expressions are calculated using the appropriate C
program.
Calculations were performed for the following values of the problem
parameters:
;)CckJ/(m102,14848=;kg/m101,1=;)CJ/(kg57,3= 2
0
33 kc
;1/0,041=;1/0,83=;)J/(sm8,3552=;)J/(sm4,297= 33 ckckQQ krn
kgkJERAkgkJU /0,034=;C45=8,314;=;10=;/0,0443= 2 .
In fig. 3 shows the graph of the distribution of the fraction of crystallization
for a fixed value of time.
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
rz
Crist: t=2
1
–s
t i
te
r
z t
Fig. 3. Distribution graph of polymer melt crystallization on the 3rd iteration
Mathematical modelling of crystallization of polymer solutions
Системні дослідження та інформаційні технології, 2023, № 1 149
CONCLUSIONS
1. The purpose of mathematical modeling of polymerization and
crystallization processes was to determine the equilibrium state of these processes
and to determine the length of the dosing zone.
2. The problem of determining the degree of polymerization and
crystallization of the polymer melt in the dosing zone of the extruder is
formulated in the form of a system of nonlinear differential equations with respect
to the temperature of the melt (mass and heat transfer equations), taking into
account the cooling boundary conditions and nonlinear kinetic equations of the
degree of polymerization and crystallization of the polymer melt.
3. The solution of the system of equations of motion of the polymer melt
(thermal conductivity) and kinetic equations of the degree of polymerization and
crystallization is carried out by a numerical-analytical iterative method, which
made it possible to obtain a solution in quadrature.
4. The distribution of the temperature of the melt crystallizing in the dosing
zone, the degree of polymerization and crystallization of the polymer are
obtained.
5. The proposed numerical analytical method for solving nonlinear differen-
tial equations made it possible to automate the software implementation of the
iterative procedure and obtain an approximate solution for the distribution of the
crystallization field of polymer solutions. Four iterations were performed to
achieve a relative solution error of 5%.
6. The proposed iterative numerical analytical method proved the effective-
ness of its application to a wide range of problems described by systems of non-
linear equations of the parabolic type.
REFERENCES
1. N.A. Belyaeva, Mathematical modeling of deformation of viscoelastic structured
polymer (composite) systems: Author. dr. diss. Chelyabinsk, 2008, 35 p.
2. K.Kh. Zelensky, “A numerical-analytical method for solving space-time problems
with moving boundaries,” Interv. science and technology coll. “Adaptive automatic
control systems”, no. 14 (34), pp. 107–117, 2009.
3. K.Kh. Zelensky, Mathematical modeling of nonlinear polymer materials in
extruders: Author.diss... of Doctor of Technical Sciences. Kyiv, 2021, 43 p.
4. R.R. Zinatullin and N.M. Trufanova, “Numerical modeling of technological stresses
during the production of plastic wire insulation,” Computational Mechanics of Solid
Media, vol. 2, no. 1, pp. 38–53, 2009.
5. T.G. Kulikova and N.A. Trufanov, “Numerical solution of the boundary value
problem of thermomechanics for a crystallizing viscoelastic polymer,” Ex.
Mechanics of continuous media, vol. 1, no. 2, pp. 38–52, 2008.
6. V.N. Mitroshin, “Structural modeling of the cooling process of an insulated cable
core when it is produced on an extrusion line,” Vestn. Samar Mr. technical Univ.
Ser.: “Technical Sciences”, no. 40, pp. 22–33, 2006.
7. O.S. Sakharov, V.I. Sivetskyi, and O.L. Sokolskyi, Modeling of processes of melting
and homogenization of polymer compositions in worm equipment: monograph.
Kyiv: VP “Edelweiss”, 2012, 120 p.
8. I.N. Shardakov and N.A. Trufanov, “Modeling of thermomechanics of crystallizing
polymers,” Izv. Tula State University. Natural sciences, vol. 2, pp. 117–123, 2008.
K. Zelensky
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 150
9. D. Andreucci, A. Fasano, and M. Primicerio, “Numerical simulation of polymer
crystallization,”Mathematical Models and Methods in Applied Sciences, November
2011, 11 p.
10. V. Capasso, H. Engl, J. Periaux (Eds.), “Computational Mathematics Driven by
Industrial Problems,” in Mathematical models for polymer crystallization processes,
Springer, Berlin/Heidelberg, 2000, pp. 39–67.
11. V. Capasso, R. Escobedo, and C. Salani, “Moving Bands and Moving Boundaries in
a Hybrid Model for the Crystallization of Polymers,” in Free Boundary Problems
Theory and Applications, vol. 147, ed. by P. Colli, C. Verdi, A. Visintin, Birkhauser,
2004, pp. 75–86.
12. R. Escobedo and L. Fernandez, “Free boundary problems (FBP) and optimal control
of axisymmetric polymer crystallization processes,” Computers and Mathematics
with applications, 68, pp. 27–43, 2014.
13. R. Escobedo and L. Fernandez, “Optimal control of chemical birth and growth
processes in a deterministic model,” J. Math. Chem., 48, pp. 118–127, 2010.
14. A. Friedman and A. Velazquez, “A free boundary problem associated with
crystallization of polymers in a temperature field,” Indiana Univ. Math. J., 50,
pp. 1609–1649, 2001.
15. J. Ramos, “Propagation and interaction of moving fronts in polymer crystallization,”
Appl. Math. Comput., 189, pp. 780–795, 2007.
16. O. Trofymchuk, K. Zelensky, V. Pavlov, and K. Bovsunovska, “Modeling of heat
and mass transfer processes in the melting zone of polymer,” System Research &
Information Technologies, no. 4, pp. 68–80, 2021.
17. N. Zelenskaya and K. Zelensky, “Approximation of Bessel functions by rational
functions,” Electronics and control systems, no. 2 (44), pp. 123–129, 2015.
Received 03.08.2022
INFORMATION ON THE ARTICLE
Kyryl Kh. Zelensky, ORCID: 0000-0003-1501-8214, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: zelensky126@ukr.net
МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ КРИСТАЛІЗАЦІЇ РОЗЧИНІВ ПОЛІМЕРІВ
/ К.Х. Зеленський
Анотація. Розглянуто процеси гомогенізації та кристалізації розчинів
полімерів у циліндричних трубах, які описуються рівнянням конвективно-
дифузійної залежності від температури розчину та кінетичними рівняннями з
гомогенізації та кристалізації полімера, відомими як термокінетична нелінійна
крайова задача. Запропоновано числово-аналітичний ітераційний метод
розв’язування цієї задачі, який полягає в поетапному отриманні розв’язків
кінетичних рівнянь з гомогенізації та кристалізації розчинів полімерів залежно
від температури розчину та отримання розв’язку конвективно-дифузійної
задачі щодо температури розплаву. Точність отриманого розв’язку
визначається нормою різниці двох сусідніх ітерацій. Значення коефіцієнта
кристалізації, близьке до одиниці, визначає довжину зони дозування і перехід
до наступної зони – потоку гомогенізованого полімера в розподільну головку
екструдера. Наведено результати математичного моделювання.
Ключові слова: усереднення, інтегральні перетворення, ітераційний метод,
кристалізація, математична модель, нелінійна крайова задача.
|
| id | journaliasakpiua-article-279802 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:10Z |
| publishDate | 2023 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/c2/dd7ffcfab045000814a2ca9a6d0e0bc2.pdf |
| spelling | journaliasakpiua-article-2798022023-05-24T21:28:17Z Mathematical modelling of crystallization of polymer solutions Математичне моделювання кристалізації розчинів полімерів Zelensky, Kyryl усереднення інтегральні перетворення ітераційний метод кристалізація математична модель нелінійна крайова задача homogenization integral transformations iterative method crystallization mathematical model nonlinear boundary value problem The processes of homogenization and crystallization of polymer solutions in cylindrical pipes are considered, which are described by the convective-diffusion equation with respect to the solution temperature and kinetic equations with respect to homogenization and crystallization of the polymer known as the thermokinetic nonlinear boundary value problem. A numerical-analytical iterative method for solving this problem is proposed, which consists of stepwise obtaining solutions of kinetic equations with respect to homogenization and crystallization of polymer solutions depending on the solution temperature and obtaining a solution of the convective-diffusion problem with respect to melt temperature. The accuracy of the obtained solution is determined by the norm of the difference between two adjacent iterations. The value of the crystallization coefficient, which is close to unity, determines the length of the dosing zone and the transition to the next zone – the flow of homogenized polymer into the distribution head of the extruder. The results of mathematical modelling are given. Розглянуто процеси гомогенізації та кристалізації розчинів полімерів у циліндричних трубах, які описуються рівнянням конвективно-дифузійної залежності від температури розчину та кінетичними рівняннями з гомогенізації та кристалізації полімера, відомими як термокінетична нелінійна крайова задача. Запропоновано числово-аналітичний ітераційний метод розв’язування цієї задачі, який полягає в поетапному отриманні розв’язків кінетичних рівнянь з гомогенізації та кристалізації розчинів полімерів залежно від температури розчину та отримання розв’язку конвективно-дифузійної задачі щодо температури розплаву. Точність отриманого розв’язку визначається нормою різниці двох сусідніх ітерацій. Значення коефіцієнта кристалізації, близьке до одиниці, визначає довжину зони дозування і перехід до наступної зони – потоку гомогенізованого полімера в розподільну головку екструдера. Наведено результати математичного моделювання. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2023-03-30 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/279802 10.20535/SRIT.2308-8893.2023.1.12 System research and information technologies; No. 1 (2023); 141-150 Системные исследования и информационные технологии; № 1 (2023); 141-150 Системні дослідження та інформаційні технології; № 1 (2023); 141-150 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/279802/274396 |
| spellingShingle | усереднення інтегральні перетворення ітераційний метод кристалізація математична модель нелінійна крайова задача Zelensky, Kyryl Математичне моделювання кристалізації розчинів полімерів |
| title | Математичне моделювання кристалізації розчинів полімерів |
| title_alt | Mathematical modelling of crystallization of polymer solutions |
| title_full | Математичне моделювання кристалізації розчинів полімерів |
| title_fullStr | Математичне моделювання кристалізації розчинів полімерів |
| title_full_unstemmed | Математичне моделювання кристалізації розчинів полімерів |
| title_short | Математичне моделювання кристалізації розчинів полімерів |
| title_sort | математичне моделювання кристалізації розчинів полімерів |
| topic | усереднення інтегральні перетворення ітераційний метод кристалізація математична модель нелінійна крайова задача |
| topic_facet | усереднення інтегральні перетворення ітераційний метод кристалізація математична модель нелінійна крайова задача homogenization integral transformations iterative method crystallization mathematical model nonlinear boundary value problem |
| url | https://journal.iasa.kpi.ua/article/view/279802 |
| work_keys_str_mv | AT zelenskykyryl mathematicalmodellingofcrystallizationofpolymersolutions AT zelenskykyryl matematičnemodelûvannâkristalízacíírozčinívpolímerív |