Моделювання процесів тепломасоперенесення у зоні плавлення полімерів
Issues related to the melting of the polymer mixture in cylindrical channels are considered. The peculiarity of the problem solved in this work is the presence in the channel of two-phase flows of a solid-liquid mixture under the influence of heating the outer surface of the cylinder and dissipative...
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2021
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System research and information technologies| _version_ | 1866302795335860224 |
|---|---|
| author | Trofymchuk, Olexander Zelensky, Kyryl Pavlov, Vladimir Bovsunovska, Katerina |
| author_facet | Trofymchuk, Olexander Zelensky, Kyryl Pavlov, Vladimir Bovsunovska, Katerina |
| author_sort | Trofymchuk, Olexander |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2022-06-20T14:19:48Z |
| description | Issues related to the melting of the polymer mixture in cylindrical channels are considered. The peculiarity of the problem solved in this work is the presence in the channel of two-phase flows of a solid-liquid mixture under the influence of heating the outer surface of the cylinder and dissipative forces arising from friction of the mixture against the walls of this channel. The problems of heating the mixture, melting the polymer and determining the velocity of the phase transition boundary in the conditions of a mobile solid mixture due to the rotational motion of the mechanism that feeds the mixture into a cylindrical channel are considered. Mathematical models of certain processes are systems of differential equations with partial derivatives of mathematical physics; the corresponding boundary value problems are solved using numerical-analytical methods, which made it possible to obtain solutions in quadratures. The results of computer modeling of the developed algorithms are given. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2021.4.05 |
| first_indexed | 2025-07-17T10:27:44Z |
| format | Article |
| fulltext |
O. Trofymchuk, K. Zelensky, V. Pavlov, K. Bovsunovska, 2021
68 ISSN 1681–6048 System Research & Information Technologies, 2021, № 4
UDC 621.3
DOI: 10.20535/SRIT.2308-8893.2021.4.05
MODELING OF HEAT AND MASS TRANSFER PROCESSES
IN THE MELTING ZONE OF POLYMERS
O. TROFYMCHUK, K. ZELENSKY, V. PAVLOV, K. BOVSUNOVSKA
Abstract. Issues related to the melting of the polymer mixture in cylindrical
channels are considered. The peculiarity of the problem solved in this work is the
presence in the channel of two-phase flows of a solid-liquid mixture under the
influence of heating the outer surface of the cylinder and dissipative forces arising
from friction of the mixture against the walls of this channel. The problems of
heating the mixture, melting the polymer and determining the velocity of the phase
transition boundary in the conditions of a mobile solid mixture due to the rotational
motion of the mechanism that feeds the mixture into a cylindrical channel are
considered. Mathematical models of certain processes are systems of differential
equations with partial derivatives of mathematical physics; the corresponding
boundary value problems are solved using numerical-analytical methods, which
made it possible to obtain solutions in quadratures. The results of computer
modeling of the developed algorithms are given.
Keywords: equivalent simplification, Stefan type problem, integral transformations,
polymer mixture, Navier–Stokes equation, phase transition, Bessel functions.
INTRODUCTION
Single-screw extrusion is the most important method of processing polymeric
materials, which explains the significant amount of available theoretical and
experimental work in this area (for example [8–12]).
The process of processing the material in it is divided into four zones: the
loading zone of the polymer mixture, in which the mixture is heated by an
inductor to a certain temperature and dry friction against the wall of the extruder
housing is heated to a temperature close to the melting point of the polymer;
melting zones, where the polymer melt is formed (taking into account the phase
transition solid mixture — liquid phase); the dosing zone in which the
homogenization and crystallization of the melt; and areas of application of
polymer insulation on the conductive core in the cable head. Given the significant
cost of materials and equipment for polymer processing, an approach based on
mathematical modeling of mass and heat transfer processes is preferred.
Mathematical modeling of mass and heat transfer processes in single-screw
extruders provides an opportunity to determine the processing conditions of
polymeric materials, determine the optimal operating conditions of existing
extrusion lines for processing polymeric materials and create conditions for the
development of such lines for a wide class of polymer products (insulating coating
of cables for ultra-high voltages, production of wide range of polymer films).
In this case, taking into account the nonlinear properties of polymeric
materials becomes essential during mathematical modeling of these processes,
which affects the accuracy of determining the main parameters of both the
Modeling of heat and mass transfer processes in the melting zone of polymers
Системні дослідження та інформаційні технології, 2021, № 4 69
polymer mixture in the loading zone and in the melting process when determining
the melt temperature field of polymeric materials.
PROBLEM STATEMENT
The processes in the melting zone of the polymer mixture are crucital in terms of
ensuring the quality of the original product.
The problem of modeling the processes of melting the polymer mixture is to
solve the problem in the two-phase region of the solid phase — liquid phase,
known as the Stefan type problem. The formulation of the problem of a phase
transition in a two-phase region, formulated in [14, p. 50–53].
Many works are devoted to the description of problems such as Stefan and
methods of their solution, for example [1 –5]. The vast majority consider one- and
two-phase problems for one-dimensional models.
Overview of mathematical models
The solution of the corresponding boundary value problem, which describes the
heating of the plug to the melting temperature, is similar to the solution of the
corresponding boundary value problem in the loading zone. The difference is that
in the two-phase heating zone–melting conditions at the phase boundary are
functions of time (time movement of the boundary between the solid phase —
liquid phase).
In most works on modeling the processes of melting the polymer mixture,
boundary value problems are formulated, which are described by systems of mass
and heat transfer equations in a cylindrical coordinate system [1–8], which are
considered “classical”.
Taking into account the cylindrical geometry of the extruder, the equation of
motion of the polymer melt and the temperature field of the melt can be described
by the following system of equations:
continuity equation:
0=
)(1)(
r
vr
rz
v rz
;
motion equation:
;
1
=
2
r
z
re
r
z
r
r
r g
zr
v
v
r
p
z
v
v
r
v
v
t
v
(1)
;
111
=
2
z
rr
ze
z
z
z
r
z g
r
v
zrzr
v
r
v
z
p
z
v
v
r
v
v
t
v
(2)
energy equation:
;
1
=
2
2
2
2
T
v
zr q
c
A
z
T
r
T
rr
T
k
z
T
v
r
T
v
t
T
(3)
z
v
r
rv
rT
p
Tq zr
T
)(1
|=
O. Trofymchuk, K. Zelensky, V. Pavlov, K. Bovsunovska
ISSN 1681–6048 System Research & Information Technologies, 2021, № 4 70
,2
2222
z
v
z
v
r
v
z
v
r
v rzzr
e
where A — thermal equivalent of work; vc — specific heat of the liquid at a
constant volume; e — effective shear viscosity; — kinematic viscosity;
,)/2=2=,)/2)(= 0(
20
0(2
1
20
TT
e
TT
n
e eIneI
,
)(1
=)(
z
v
r
rv
r
v zr
where 0 — initial viscosity; 2I — the second invariant of the strain rate tensor;
n — viscosity anomaly index; zr vvv ,, — components of the velocity vector;
ij ( zrji ,,=, ) — components of the stress tensor; T — temperature; —
temperature coefficient; 0T — initial temperature (the temperature in left
boundary of melting zone).
The expression |)/( Tp is determined from the state equation. In the case
of a constant value we have RTp = it is equal to R .
The initial conditions for the equations (1) – (3) are obtained, taking into
account the solution of the heat and mass transfer problem at the exit from the
melting delay zone:
).,(=),,(),(=),,( 00 zrvztzrvzrvztzrv zzrr
The above system of equations does not take into account the dependence of
the viscosity coefficient on the melt temperature, does not take into account the
phase transition solid mixture — melt, which is a determining factor that affects
the quality of the final product. There are no boundary conditions for heating the
extruder body in the melting zone of the polymer and conditions on the left
boundary of the polymer mixture — melt. These factors encourage the formula-
tion of a nonlinear boundary value problem taking into account the above.
PROBLEM SOLVING
The problem of determining the velocity of the boundary between the solid phase
and the liquid phase consists of the following steps.
Heat transfer of solid mixture
The boundary value problem of heating the solid mixture in the melting zone is
similar to the problem of heating the “plug” in the loading zone. The difference is
that the conditions change at the boundary of the phase transition solid
phase — liquid phase in the sense that this boundary is mobile. The iterative
procedure is similar to that described in the previous section, taking into
account the need to calculate the integral transformations within the variable
limits tbba zrrrr )](=,[ :
Modeling of heat and mass transfer processes in the melting zone of polymers
Системні дослідження та інформаційні технології, 2021, № 4 71
.
1
=)(
2
2
sn k
pp
p
ppzpr
snppvp q
z
T
r
T
r
rrt
T
z
T
V
r
T
VTc
(4)
For polyethylene coefficient
pvc significantly depends on the heating
temperature. On the base of experimental investigations this dependence
approximated [3] as
.105,70,0242,5= 242
210 TTTcTccc
pv
This expression corresponds to the original (in the space of integral
transformations by spatial variables)
.0>,ch 2sh 1
;0<,cos2sin1
=)(
1=1,=
(0)
nknknknknk
nknknknknknk
M
kn
nk tete
tetetetD
Then the solution in the first approximation will take the form
.)()(1)()(=),,( (0)
1=1=
(1)
tDte
G
zZrRtzrT nk
nk
nk
nk
kn
M
k
M
n
Further approximations are implemented according to a similar scheme (Fig. 1):
.)()(1)()(=),,( 1)(
1=1=
)(
tDte
G
zZrRtzrT m
nk
nk
nk
nk
kn
M
k
M
n
m
In Fig. 1 shows the temperature distribution of sold polymer after 3-d
iteration.
Therefore, the algorithm for solving the equation (4) differs from the one
given in the previous section in the part of calculating integrals from Bessel
functions of the second kind with respect to variable limits. Integral transformations
of the nonlinear component of the thermal conductivity equation (4):
Fig. 1. Temperature of polimer at 2t
)2,,( tzrT of solid polimer — 3-d iter
0.2
0.4
0.6
0.8
1
0
0.5
1
0
50
100
150
zrr z,
,
,
,,
O. Trofymchuk, K. Zelensky, V. Pavlov, K. Bovsunovska
ISSN 1681–6048 System Research & Information Technologies, 2021, № 4 72
.)()()(
)(
)(
1
=,, rdrrBrBrB
zr
rB
nr lnm
b
arbmm
lnm
Naturally, the upper bound of the integrals of r from similar to (4)
expressions as the “plug” moves along the z axis, it will decrease and move to
the lower limit of ar . The point L , at which ab rr will determine the length of
the melting zone and the transition to the zone of homogenization and
crystallization of the polymer melt.
Solving the problem in the liquid phase
Since the equation with respect to the velocity of the polymer solution contains
pressure gradients in the solution, we begin solving the problem by solving the
boundary value problem with respect to temperature. Taking into account the
temperature dependence of the heat capacity coefficient vc , as well as the
presence of the convective component and the dissipative component Tq we have
.=)](0[ *
TzrV qT
z
T
v
r
T
v
t
T
TTLc
Let us separate the linear part in this equation
,)]([
1
0
= *
TT
V
q
t
T
TTLCT
ct
T
(5)
where is an abrupt change in the thermal conductivity at the interface between
the solid phase and the liquid phase, TC denotes the convective component in the
heat transfer equation
].,,[=,= TvvF
z
F
v
r
F
vC zrzrF
(6)
Determination of the temperature of the liquid phase in the linear
approximation. This problem is solved in the previous section, because in the
linear approximation (excluding the convective components and the dissipative
component in the right part of the equation (5) it coincides with the heat transfer
equation in the tube):
].10)[()(=),,( ,
,,
,
,
(0) t
eTTzZrRtzrT km
kmkmkm
NM
kn
(7)
The presence of a pressure gradient in the equations relative to the velocity
of the melt is taken into account according to the equation RTp = , i.e.
rTRrp /=/ , zTRrp /=/ taking into account (7).
Then the equation of motion can be written as follows:
;=
2(0)
rv
z
re
r C
zr
v
v
r
T
R
t
v
(8)
.
11
=
2(0)
zv
rr
ze
z C
r
v
zrzr
v
r
v
z
T
R
t
v
(9)
Modeling of heat and mass transfer processes in the melting zone of polymers
Системні дослідження та інформаційні технології, 2021, № 4 73
Denote, as before, the linear part for the components of the velocity of the
melt through ),,((0) tzrvr , ),,((0) tzrvz application of equations (8), (9) integral
transformations on spatial variables gives for 1,2,=m :
;)(=)(= ,
1)(1)()(
,
)(
rvkn
m
rr
mm
r
rv
kn
m
r CtFtTRv
dt
vd
.)(=)(= ,
1)()(
,
)(
zvknzz
mm
z
zv
kn
m
z CtFtTRv
dt
vd
Taking into account the expression (6) after the application of integral
transformations of spatial variables in the “linear approximation” we will have
KM
km
T
km
kmkm
kkmmkm
rv
mk
km
r
p
T
p
T
zdrvr
p
pV
, 10
112,
,(0)
11 11
11111
=)()( ,
where
1,2 mmr ,
1,kkzd are the coefficients of integral transformations for the
variables r and z , respectively.
Summation in these expressions using the algorithm of equivalent
simplification gives:
;)()(
2
543
102,(0)
pvrpvrvr
pvrvr
p
vr
pV km
r
.)()(
2
543
102,(0)
pvzpvzvz
pvzvz
p
vz
pV km
z
In these expressions indicated: 0, = vrzrvr kmkm , 0, = vzzrvz kmkm ;
,)()(=;)()(=
1
1
0
1,1
1
1, dzz
z
Z
zZ
z
zddrr
r
R
rR
r
rd k
k
kk
l
z
kkm
m
mm
b
ar
mm
where 00 , vzvr — initial values of velocity — the movement of the “traffic jam”
in the longitudinal and radial directions.
The originals of these expressions are:
);()()(==),,( (0)
,
1=1=
(0) tVrtzZrRtzrvr kmk
N
k
mm
M
m
(10)
;))(cos)(sin(=)( 21
,
0
(0)
, tVrttVrt
t
eVrttVrt
rvrv
rv
km
km
(11)
)()()(=),,( (0)
,
1=1=
(0) tVztzZrRtzrvz kmk
N
k
mm
M
m
; (12)
))(cos)(sin(=)( 21
,
0
(0)
, tVzttVzt
t
eVzttVzt
zvzv
zv
km
km
. (13)
If there are expressions for the components of the melt velocity in the linear
approximation, it is possible to determine the convective components in the
equations for velocities and temperature. Substitute (7), (10), (11) to (6). Here is
the expression for
rvC . The expression for
zvC is similar.
O. Trofymchuk, K. Zelensky, V. Pavlov, K. Bovsunovska
ISSN 1681–6048 System Research & Information Technologies, 2021, № 4 74
)()(
)(
)()()(= )0(
,
)0(
,
,,
)0(
2222
22
111111
2121
tVrtzZ
dr
rdR
tVrtzZrRC kmkk
mm
kmkkmm
K
kk
M
mm
vr
).(
)(
)()()()( )0()0(
,,
22
22
22111111
2121
tVrt
dz
zdZ
rRtVztzZrR km
kk
mmkmkkmm
K
kk
M
mm
Application of integral transformations by spatial variables to this expression
leads to expressions of the form
);(= ,)0(
2121
2121
tFC km
kkmm
K
kk
M
mm
rv
).()(,2)()(,1=)(, )0()0()0()0(
22112121221121212121
tVzttVztkmFtVrttVrtkmFtkmF kmkmkkmmkmkmkkmmkkmm
Summation by indices 2121 ,,, kkmm using the algorithm of equivalent
simplification [15], gives:
;)(
2,
5
,
4
,
3
,
1
,
0
,
2,
pvrpvrvr
pvrvr
p
vr
pC
kmkmkm
kmkmkm
km
r
(14)
By a similar algorithm we obtain the expression for (12) whith (13):
;)(
2,
5
,
4
,
3
,
1
,
0
,
2,
pvzpvzvz
pvzvz
p
vz
pC
kmkmkm
kmkmkm
km
z
(15)
.)(
2,
5
,
4
,
3
,
1
,
0
,
2,
ptptt
ptt
p
t
pC
kmkmkm
kmkmkm
km
T
(16)
The coefficients km
jvr , , km
jvz , , km
jt , in convective components are determined
using programs that implement the appropriate algorithms. Because the
dissipative component in the equation for temperature can be written as
,
0
2
=
2
z
v
r
v
c
A
q zr
V
e
T
taking into account (14)–(16) we will have
.
0
2
)(
2
543
102,
pTqpTqTq
pTqTq
p
Tq
c
A
pQ
V
ekm
T
Then in the first approximation for the temperature field in the image space
we will have:
.
111
11
)()()(=)(
25
,
4
,
3
,
1
,
0
,
,
,
,
(0)
,
(1)
ptptt
ptt
pqpCpTpT
kmkmkm
kmkm
kmT
km
Tkmkm
Or in the space of the originals
)()()(=),,( (1)
,
1=1=
(1) tTTzZrRtzrT knkm
N
k
M
m
;
Modeling of heat and mass transfer processes in the melting zone of polymers
Системні дослідження та інформаційні технології, 2021, № 4 75
),1(1)1(1(1=)( (5)
,2
(2)
,
(5)
,1
(1)
,
(4)
,1(0)
,
(1)
, ttftttft
t
ettTT kmkmkmkm
kmt
kmkn
where )(),( 21 atfatf — originals from chains of the second order.
The calculation of these integrals according to the algorithms given in the
section is performed once for all iterations. As a result of the solution for the
components of the velocity of the melt and the temperature field acquire the
following form:
.)()()(=),,(;)()()(=),,( )(
,
1=1=
)()(
,
1=1=
)( tvzzZrRtzrvtvrzZrRtzrv m
knk
N
k
n
M
n
m
z
m
knk
N
k
n
M
n
m
r
In Fig. 2 shows the graphs of the velocity components ),,()( tzrv m
r :
In Fig. 3 shows the graphs of the velocity components ),,()( tzrv m
z :
Fig. 2. Radial velocity field (sm/min)
Radial velocity )1,,( tzrVr of melt sm/min
0.2
0.4
0.6
0.8
1
1
1.5
2
2
4
6
8
10
zrr z,
,
,,
,
Fig. 3. Axial velocity field (sm/mi)
Axial velocity )1,,( tzrVz of melt —sm/min
0.2
0.4
0.6
0.8
1
1
1.5
2
0
5
10
15
20
zrr z,
,,
,
,
,
O. Trofymchuk, K. Zelensky, V. Pavlov, K. Bovsunovska
ISSN 1681–6048 System Research & Information Technologies, 2021, № 4 76
The application of the iterative procedure for solving the nonlinear thermal
equation can be written for the m -th approximation (Fig. 4):
)()()(=),,( )(
,
1=1=
)( tTTzZrRtzrT m
knkm
N
k
M
m
m ; (17)
)1(1)1(1(1=)( )(5,
,2
)(2,
,
)(5,
,1
)(1,
,
)(4,
,1)(0,
,
)(
, ttftttft
t
ettTT m
km
m
km
m
km
m
km
m
kmtm
km
m
kn
. (18)
Here are the graphs of the temperature fields obtained in the first 3 iterations
(Fig. 5).
According to the available approximate solution (17) whith (18) for the
temperature field of the polymer melt in the first approximation we can write an
expression for the pressure field
Fig. 4. Temperature field in linear approximation
0.2
0.4
0.6
0.8
1
1
1.5
2
60
80
100
120
140
zr
)2,,( tzrT of melt linear approximation
,
,
,,
,
zr
,
Fig. 5. Temperature field in the second approximation
)2,,( tzrT of melt
0.2
0.4
0.6
0.8
1
1
1.5
2
130
140
150
160
170
zrr z,
,,
,
Modeling of heat and mass transfer processes in the melting zone of polymers
Системні дослідження та інформаційні технології, 2021, № 4 77
)()()(=),,(=),,( )(
,
1=1=
)()( tTTzZrRtzrRTtzrp m
knkn
N
k
M
n
mm
taking into account the expression (18) for )()(
, tTT m
kn .
Since we are interested in the value of the pressure at the outlet of the
melting zone of the polymer, it is appropriate to determine the value of the
pressure at the boundary of the melting zone L and the dosing zone:
)()()(=),,(=),,( )(
,
1=1=
)()( tTTLZrRtLrRTtLrp m
knkn
N
k
M
n
mm .
The obtained solution of the problem of heating the “plug”— solid polymer
mixture and the process of forming a thin film of melt at the end of the melting
delay zone can be considered as the first step in determining the boundary of the
phase transition solid phase — liquid phase.
The formation of the liquid phase is accompanied, of course, by a decrease
in the proportion of solid phase. The melting process leads to the formation of a
boundary ( ],[ ba rrr ) for the solid phase and ],[ ab rrr ) for the liquid
phase.
A significant difference between the problem of phase transition in the process
of polymer melting in contrast to the classical problems such as Stefan is that:
1) in the process of melting the polymer, the solid mixture moves in the
direction of the axis z with a speed of zV ;
2) this leads to the dependence of the variable boundary on the longitudinal
coordinate z , ie )(= z .
Defining the phase transition boundary
At the boundary of the phase transition, the conditions of heat conservation at the
free boundary separating the solid phase and the soluble one must be fulfilled:
,=|
l
l
l
l
s
s
s
s
nV n
T
n
T
VL
where dzdVn /= is the velocity of the phase transition boundary, VL is the heat
of the phase transition (determined experimentally).
The expression for the temperature field of the polymer mixture is obtained
in the form:
.0>,ch2sh1
;0<,cos2sin1
=)(
1=,
(0)
nknknknknk
nknknknknknk
M
kn
nk tete
tetetetD
Then the solution in the first approximation will take the form
.)()(1)()(=),,( (0)
1=1=
(1)
tDte
G
zZrRtzrT nk
nk
nk
nk
kn
M
k
M
n
Further approximations are implemented according to a similar scheme:
.)()(1)()(=),,( 1)(
1=1=
)(
tDte
G
zZrRtzrT m
nk
nk
nk
nk
kn
M
k
M
n
m
O. Trofymchuk, K. Zelensky, V. Pavlov, K. Bovsunovska
ISSN 1681–6048 System Research & Information Technologies, 2021, № 4 78
For solid and liquid phases one can write:
)()(
)(
= )(
,
1==1=
tttzZ
r
rB
r
T
n
T sl
kmk
K
kbrr
m
M
m
ss
;
.)()(
)(
= )(
,
=1==1
tttzZ
r
rB
r
T
n
T pl
kmk
K
karr
m
M
m
ll
Given that the density of the polymer material almost does not change with
the transition from solid to liquid state, we have the following equation to
determine the velocity of the phase interface:
kk
K
kV
n CzZ
Ldz
d
V )(
1
==
1=
;
.])()([)(
)(1
= )(
,
)(
,
1==1=
ttttttzZ
r
rB
VL pl
kml
sl
kmsk
K
krr
m
M
mpl
nV
Next, in order to obtain an expression for the phase separation boundary, we
assume that the process of melting the polymer in the extruder with the formation
of the phase transition boundary can be considered quasi-stationary. Then one can
write the equation to determine the boundary of the phase transition as follows:
](0)(0)[
)(1
= )(
,
)(
,
=1=
pl
kml
sl
kms
rr
m
M
mpl
k tttt
r
rB
C
.
The initial condition for this equation:
.0)(
1
=(0)
1=
kk
K
kV
CZ
L
As a result in Fig. 6 a graph of the velocity of the solid-liquid interface
during melting of the polymer is given.
Fig. 6. Velocity of move of boundary solid–liquid (sm/min)
Velocity of boundary — sm/min
0.2
0.4
0.6
0.8
1
1
1.5
2
0.2
0.4
0.6
0.8
1
zrr z,
,
,,
,
,
,
,
Modeling of heat and mass transfer processes in the melting zone of polymers
Системні дослідження та інформаційні технології, 2021, № 4 79
CONCLUSIONS
1. The problem of modeling the processes of polymer melting in the
extruder is formulated. The mathematical model of the process is a system of
differential equations that takes into account the convective transfer of fluid, the
nonlinear temperature dependence of the thermophysical parameters of the
polymer and the dissipative component.
2. The solution of this system of equations is obtained using the iterative
numerical-analytical method of solving nonlinear boundary value problems.
3. The mathematical model of heat and mass transfer as a problem with
variable limits is formulated — a problem like Stefan. A method for solving the
problem of determining the moving boundary of phase separation solid phase —
liquid phase is proposed.
4. The obtained expression for the moving boundary can be used to set the
problem of optimal control of the melting process of polymers.
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production of plastic insulation”, Izvestiya Tomskogo polit. un-ta, vol. 320, no. 4,
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Experiments and 3D Finite Element Simulations”, Int. Polym. Proc., 26, pp. 182–196, 2011.
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PA, USA, 2019.
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screw extruder”, Polymer Engineering & Science, vol. 44, no. 11, pp. 2148–2157, 2004.
14. A. Samarsky and P. Vaibishevich, Computational heat transfer. Moscow, 2003, 785 p.
15. N. Zelenskaya and K. Zelensky, “Approximation of Bessel functions by rational
functions”, Electronics and control systems, no. 2 (44), pp. 123–129, 2015.
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O. Trofymchuk, K. Zelensky, V. Pavlov, K. Bovsunovska
ISSN 1681–6048 System Research & Information Technologies, 2021, № 4 80
17. K. Zelensky, “Numerical-analytical method for solving space-time problems with
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Received 04.08.2021
INFORMATION ON THE ARTICLE
Olexander N. Trofymchuk, ORCID: 0000-0003-3358-6274, Institute of Telecommuni-
cations and Global Information Space of the National Academy of Sciences of Ukraine,
Ukraine, e-mail: trofymchuk@nas.gov.ua
Kyryl Kh. Zelensky, ORCID: 0000-0003-1501-8214, Igor Sikorsky Kyiv Polytechnic
Institute, Ukraine, e-mail: zelensky126@ukr.net
Vladimir A. Pavlov, ORCID: 0000-0002-6234-113X, Igor Sikorsky Kyiv Polytechnic
Institute, Ukraine, e-mail: pavlov.vadimir264@gmail.com
Katerina S. Bovsunovska, ORCID: 0000-0003-0936-2246, Igor Sikorsky Kyiv Poly-
technic Institute, Ukraine, e-mail: period0@ukr.net
МОДЕЛЮВАННЯ ПРОЦЕСІВ ТЕПЛОМАСОПЕРЕНЕСЕННЯ У ЗОНІ
ПЛАВЛЕННЯ ПОЛІМЕРІВ / О.М. Трофимчук, К.Х. Зеленський, В.А. Павлов,
К.С. Бовсуновська
Анотація. Досліджено питання тепломасотеплоперенесення полімерної
суміші у циліндричних каналах. Особливість задачі, що розв’язується в роботі,
полягає у наявності у каналі двофазних потоків тверда суміш — рідина під
впливом нагрівання зовнішньої поверхні циліндра та дисипативних сил, які
виникають унаслідок тертя суміші об стінки цього каналу. Розглянуто задачі
нагрівання суміші, плавлення полімеру та визначення швидкості руху межі
фазового переходу в умовах рухомої твердої суміші за рахунок обертального
руху механізму, що подає суміш у циліндричний канал. Математичні моделі
визначених процесів являють собою системи диференціальних рівнянь із час-
тинними похідними математичної фізики; розв’язання відповідних крайових
задач виконано із застосуванням числово-аналітичних методів, що дало змогу
отримати розв’язки у квадратурах. Наведено результати комп’ютерного моде-
лювання розроблених алгоритмів.
Ключові слова: еквівалентне спрощення, задача типу Стефана, інтегральні
перетворення, полімерна суміш, рівняння Нав’є–Стокса, фазовий перехід,
функції Бесселя.
МОДЕЛИРОВАНИЕ ПРОЦЕССОВ ТЕПЛОМАССОПЕРЕНОСА В ЗОНЕ
ПЛАВЛЕНИЯ ПОЛИМЕРОВ / А.Н. Трофимчук, К.Х. Зеленский, В.А. Павлов,
К.С. Бовсуновская
Аннотация. Исследованы вопросы тепломасопереноса полимерной смеси в
цилиндрических каналах. Особенность задачи, решаемой в работе, заключает-
ся в наличии в канале двухфазных потоков твердая смесь — жидкость под
воздействием нагревания наружной поверхности цилиндра и диссипативных
сил, возникающих вследствие трения смеси в стенки этого канала. Рассмотре-
ны задачи нагрева смеси, плавления полимера и определения скорости движе-
ния границы фазового перехода в условиях подвижной твердой смеси за счет
вращательного движения подающего смесь в цилиндрический канал. Матема-
тические модели определенных процессов представляют собой системы диф-
ференциальных уравнений с частными производными математической физи-
ки; решение соответствующих краевых задач выполнено с применением
численно-аналитических методов, что позволило получить решения в квадра-
турах. Приведены результаты компьютерного моделирования разработанных
алгоритмов.
Ключевые слова: задача типа Стефана, интегральные преобразования, поли-
мерная смесь, уравнения Навье–Стокса, фазовый переход, функции Бесселя,
эквивалентное упрощение.
|
| id | journaliasakpiua-article-252174 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:27:44Z |
| publishDate | 2021 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/f8/3ac2056e0a45b7106d24074a49c78ff8.pdf |
| spelling | journaliasakpiua-article-2521742022-06-20T14:19:48Z Modeling of heat and mass transfer processes in the melting zone of polymers Моделирование процессов тепломассопереноса в зоне плавления полимеров Моделювання процесів тепломасоперенесення у зоні плавлення полімерів Trofymchuk, Olexander Zelensky, Kyryl Pavlov, Vladimir Bovsunovska, Katerina equivalent simplification Stefan type problem integral transformations polymer mixture Navier–Stokes equation phase transition Bessel functions эквивалентное упрощение задача типа Стефана интегральные преобразования полимерная смесь уравнения Навье–Стокса фазовый переход функции Бесселя еквівалентне спрощення задача типу Стефана інтегральні перетворення полімерна суміш рівняння Нав’є–Стокса фазовий перехід функції Бесселя Issues related to the melting of the polymer mixture in cylindrical channels are considered. The peculiarity of the problem solved in this work is the presence in the channel of two-phase flows of a solid-liquid mixture under the influence of heating the outer surface of the cylinder and dissipative forces arising from friction of the mixture against the walls of this channel. The problems of heating the mixture, melting the polymer and determining the velocity of the phase transition boundary in the conditions of a mobile solid mixture due to the rotational motion of the mechanism that feeds the mixture into a cylindrical channel are considered. Mathematical models of certain processes are systems of differential equations with partial derivatives of mathematical physics; the corresponding boundary value problems are solved using numerical-analytical methods, which made it possible to obtain solutions in quadratures. The results of computer modeling of the developed algorithms are given. Исследованы вопросы тепломасопереноса полимерной смеси в цилиндрических каналах. Особенность задачи, решаемой в работе, заключается в наличии в канале двухфазных потоков твердая смесь — жидкость под воздействием нагревания наружной поверхности цилиндра и диссипативных сил, возникающих вследствие трения смеси в стенки этого канала. Рассмотрены задачи нагрева смеси, плавления полимера и определения скорости движения границы фазового перехода в условиях подвижной твердой смеси за счет вращательного движения подающего смесь в цилиндрический канал. Математические модели определенных процессов представляют собой системы дифференциальных уравнений с частными производными математической физики; решение соответствующих краевых задач выполнено с применением численно-аналитических методов, что позволило получить решения в квадратурах. Приведены результаты компьютерного моделирования разработанных алгоритмов. Досліджено питання тепломасотеплоперенесення полімерної суміші у циліндричних каналах. Особливість задачі, що розв’язується в роботі, полягає у наявності у каналі двофазних потоків тверда суміш — рідина під впливом нагрівання зовнішньої поверхні циліндра та дисипативних сил, які виникають унаслідок тертя суміші об стінки цього каналу. Розглянуто задачі нагрівання суміші, плавлення полімеру та визначення швидкості руху межі фазового переходу в умовах рухомої твердої суміші за рахунок обертального руху механізму, що подає суміш у циліндричний канал. Математичні моделі визначених процесів являють собою системи диференціальних рівнянь із частинними похідними математичної фізики; розв’язання відповідних крайових задач виконано із застосуванням числово-аналітичних методів, що дало змогу отримати розв’язки у квадратурах. Наведено результати комп’ютерного моделювання розроблених алгоритмів. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2021-12-22 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/252174 10.20535/SRIT.2308-8893.2021.4.05 System research and information technologies; No. 4 (2021); 68-80 Системные исследования и информационные технологии; № 4 (2021); 68-80 Системні дослідження та інформаційні технології; № 4 (2021); 68-80 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/252174/249509 |
| spellingShingle | еквівалентне спрощення задача типу Стефана інтегральні перетворення полімерна суміш рівняння Нав’є–Стокса фазовий перехід функції Бесселя Trofymchuk, Olexander Zelensky, Kyryl Pavlov, Vladimir Bovsunovska, Katerina Моделювання процесів тепломасоперенесення у зоні плавлення полімерів |
| title | Моделювання процесів тепломасоперенесення у зоні плавлення полімерів |
| title_alt | Modeling of heat and mass transfer processes in the melting zone of polymers Моделирование процессов тепломассопереноса в зоне плавления полимеров |
| title_full | Моделювання процесів тепломасоперенесення у зоні плавлення полімерів |
| title_fullStr | Моделювання процесів тепломасоперенесення у зоні плавлення полімерів |
| title_full_unstemmed | Моделювання процесів тепломасоперенесення у зоні плавлення полімерів |
| title_short | Моделювання процесів тепломасоперенесення у зоні плавлення полімерів |
| title_sort | моделювання процесів тепломасоперенесення у зоні плавлення полімерів |
| topic | еквівалентне спрощення задача типу Стефана інтегральні перетворення полімерна суміш рівняння Нав’є–Стокса фазовий перехід функції Бесселя |
| topic_facet | equivalent simplification Stefan type problem integral transformations polymer mixture Navier–Stokes equation phase transition Bessel functions эквивалентное упрощение задача типа Стефана интегральные преобразования полимерная смесь уравнения Навье–Стокса фазовый переход функции Бесселя еквівалентне спрощення задача типу Стефана інтегральні перетворення полімерна суміш рівняння Нав’є–Стокса фазовий перехід функції Бесселя |
| url | https://journal.iasa.kpi.ua/article/view/252174 |
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