=Paper=
{{Paper
|id=Vol-2762/paper10
|storemode=property
|title=Thermodynamic Fundamentals of Cellular Automata Model of the Process of Solidification of Metals and Alloys Considering the Phase Transition
|pdfUrl=https://ceur-ws.org/Vol-2762/paper10.pdf
|volume=Vol-2762
|authors=Tatyana Selivorstova,Vadim Selivorstov,Anton Guda,Katerina Ostrovska
|dblpUrl=https://dblp.org/rec/conf/ictes/SelivorstovaSGO20
}}
==Thermodynamic Fundamentals of Cellular Automata Model of the Process of Solidification of Metals and Alloys Considering the Phase Transition==
<pdf width="1500px">https://ceur-ws.org/Vol-2762/paper10.pdf</pdf>
<pre>
Thermodynamic Fundamentals of Cellular Automata
Model of the Process of Solidification of Metals and
Alloys Considering the Phase Transition
Tatyana Selivorstovaa , Vadim Selivorstova , Anton Gudaa and Katerina Ostrovskaa
a
    The National Metallurgical Academy of Ukraine, Gagarina avenue 4, Dnipro, Ukraine, 49005


                                         Abstract
                                         We considered the mathematical apparatus that can be used to calculate the thermal problem associated
                                         with the features of solidification of metals and alloys. Thermodynamic laws are presented, that made
                                         it possible to correctly take into account the phase transition, a feature of which is to release latent heat.
                                         A one-dimensional problem and its generalization to a cylindrical coordinate system are considered.

                                         Keywords
                                         Cellular automata, thermodynamic model, temperature, amount of liquid phase, solidification, phase
                                         transition




1. Introduction
In industrial process control systems, including foundries, embedded systems based on microcon-
trollers are widely used. Unfortunately, direct simulation on microcontrollers is impossible due
to limited computing resources. Using cellular automata approach allows simplified calculations
on embedded systems.
   Description of real metallurgical processes associated with solidification [1, 2] leads to signif-
icant differences from the classical boundary value problem due to many factors [3, 4], the most
important of which are due to: heat release during the phase transition [5], which is described
by a complex diagram states [6]; heat transfer due to convective flows in the liquid phase [7, 8];
taking into account the real boundary conditions of heat transfer [9]; inhomogeneity of alloy
properties [10, 11]; the dependence of thermophysical parameters on temperature [12]; the
complexity of the geometry of the product [13, 14].
   The analysis of methods for modelling metallurgical processes showed that the existing
analytical solutions of the solidification problem provide a high accuracy of the solution [15, 16],
but can be used only in some simple cases and are rather methodological in nature [17, 18]. The
description of the solidification problem in the form of partial differential equations leads to
the use of numerical solution methods [19, 20], for which a wide range of modifications has
been developed [19]. They are united by the presence of artificial methods, such as catching
ICT&ES-2020: Information-Communication Technologies & Embedded Systems, November 12, 2020, Mykolaiv, Ukraine
" tatyanamikhaylovskaya@gmail.com (T. Selivorstova); seliverstovvy@gmail.com (V. Selivorstov);
atu.guda@gmail.com (A. Guda); kuostrovskaya@gmail.com (K. Ostrovska)
 0000-0002-2470-6986 (T. Selivorstova); 0000-0002-1916-625X (V. Selivorstov); 0000-0003-1139-1580 (A. Guda);
0000-0002-9375-4121 (K. Ostrovska)
                                       © 2020 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
    CEUR
    Workshop
    Proceedings
                  http://ceur-ws.org
                  ISSN 1613-0073
                                       CEUR Workshop Proceedings (CEUR-WS.org)
the front at the grid node, straightening the fronts, successively approximating the position of
the transition phase, smoothing the phase transition, introducing the effective heat capacity
[21] and then determining the boundaries of the two-phase zone by interpolation, which lead
to a distortion of the original physical setting of the considered problem [22, 23, 24]. The
use of thermodynamic models [25] presupposes the existence of a large-scale database of
experimental data [26] on the redistribution of components between phases, the dependence of
the composition and number of precipitated phases on temperature changes under conditions
of equilibrium and nonequilibrium crystallization [27].
   Thus, the development of mathematical models of metallurgical processes with phase tran-
sitions is an urgent task, which is caused by the necessity of improvements of the quality of
products of metallurgical production [28, 29, 30] and their cost reduction [31, 32]. The use
of cellular automata in the development of mathematical models of solidification [33, 34, 35]
makes it possible to describe various nonlinear processes immediately in a discrete language
[36, 37] and has a number of advantages associated with the possibility of organizing high-speed
parallel computations [38, 39, 40], the obviousness of algorithms, the possibility of using them
to describe processes that are difficult or even impossible to describe by partial differential
equations.


2. Cellular automata solidification model
Solidification of metals and alloys is accompanied by complex physical and chemical processes
in the melt. The most important of them are due to: heat release during the phase transition;
heat transfer due to convective flows in the liquid phase; the complexity of the real boundary
conditions of heat transfer; heterogeneity of alloy properties; dependence of thermophysical
parameters on temperature; the complexity of the geometry of the product [41].
   Thermodynamic phenomena accompanying the process of solidification of metals is described
by the Fourier equation [42]

                                                𝑘𝑆 △ 𝑇
                                        △𝑄 =           𝜏,                                    (1)
                                                  ℎ
   where △𝑄 [𝐽 ] – the amount of heat passed through the surface 𝑆 [𝑚2 ] for the time 𝜏 [𝑠] with
temperature differences △𝑇 [𝐾 ], 𝑘 [𝑊 /(𝑚𝐾 )] – – coefficient of thermal conductivity, ℎ [𝑚] –
layer thickness.
   Equilibrium Specific Energy 𝑞 [𝐽 /𝑘𝑔] is uniquely related to the temperature and phase state
of the unit volume
                                                 𝑄
                                           𝑞=       ,                                        (2)
                                                𝜌ℎ𝑆
  where 𝜌 [𝑘𝑔/𝑐𝑚3 ] – density.

2.1. One-dimensional cellular automata thermodynamic model
The cellular automaton mathematical model assumes the description of nonlinear processes in
a discrete language at once and allows avoiding the description of thermodynamic processes in
the form of partial differential equations [43].
  To construct a one-dimensional thermodynamic model of solidification, taking into account
the phase transition, it is necessary to determine the rules of heat transfer between cells.
Consider a cell with a Neumann neighbourhood for the one-dimensional case (Fig. 1).




Figure 1: Schematic representation of a group of cells and symbols for the one-dimensional case


   Fig. 1 notation accepted: 𝑇𝑖−1 , 𝑇𝑖 , 𝑇𝑖+1 – temperatures in 𝑖 − 1, 𝑖, 𝑖 + 1 adjacent cells, respectively,
ℎ – distance between cell centers, 𝑄𝑖−1,𝑖 , 𝑄𝑖+1,𝑖 – energy entering the cell from 𝑖 − 1 and 𝑖 + 1
respectively.
   Energy entering the cell 𝑖 from 𝑖 − 1 cell, is determined based on the replacement in (1) of the
differential by the difference

                                            𝑘𝑖,𝑖−1 𝑆(𝑇𝑖−1 − 𝑇𝑖 )𝜏
                                       𝑄𝑖−1,𝑖 =                   ,                         (3)
                                                      ℎ
  where 𝑘𝑖,𝑖−1 – effective coefficient of thermal conductivity, determined from the equality of
heat flows at the border of neighboring cells.
  A schematic representation of the heat flow motion for the one-dimensional case is shown
on Fig.2.
Figure 2: Diagram of heat flow movement between cells


   Heat flow 𝑃 [𝐽 /𝑠] (fig. 2), corresponding to the amount of energy transmitted through the
isothermal surface per unit of time is determined by the ratio

                                                𝑘𝑆𝑇
                                                𝑃=  .                                       (4)
                                                  ℎ
  Heat flow between 𝑖 and 𝑖 − 1 cells is equal to

                                          𝑘𝑖−1,𝑖 𝑆(𝑇𝑖 − 𝑇𝑖−1 )
                                     𝑃𝑖−1,𝑖 =                  .
                                                   ℎ
  Heat flow to the border that separates cells:

    • from 𝑖 − 1 side
                                                       𝑘𝑖−1 𝑆(𝑇 − 𝑇𝑖−1 )
                                          𝑃𝑖−1 =                         ,
                                                              ℎ/2
    • from 𝑖 cell side
                                                        𝑘𝑖 𝑆(𝑇𝑖 − 𝑇 )
                                                𝑃𝑖 =                  ,
                                                             ℎ/2
  where 𝑇 – the current temperature at the border of the considered cells.
  Based on the equality of heat flows, at the border of the cells 𝑃𝑖−1 = 𝑃𝑖 , we can find 𝑇 –
temperature on the border between them:

                                                𝑘𝑖−1 𝑇𝑖−1 + 𝑘𝑖 𝑇𝑖
                                        𝑇 =                       .
                                                   𝑘𝑖−1 + 𝑘𝑖
  Equating heat flows 𝑃𝑖−1 = 𝑃𝑖 , and we obtain the value of the effective thermal conductivity
coefficient

                                                    2𝑘𝑖 − 1𝑘𝑖
                                        𝑘𝑖−1,𝑖 =                .                           (5)
                                                    𝑘𝑖 − 1 + 𝑘𝑖
  So the change in energy △𝑄 for 𝑖 cells, based on (3), is determined by the relation
                               𝜏𝑆
                           △𝑄 =   (𝑘𝑖−1,𝑖 (𝑇𝑖−1 − 𝑇𝑖 ) + 𝑘𝑖,𝑖+1 (𝑇𝑖+1 − 𝑇𝑖 )).            (6)
                               ℎ
  The above approach for calculating the change in cell energy can be extended to the case of
two-dimensional and three-dimensional problems in Cartesian coordinates.

2.2. Three-dimensional cylindrical cellular automata thermodynamic model
A significant part of technical objects has a cylindrical shape, that allows using the property of
axial symmetry to reduce the dimension of the model and reduce the amount of computation
[44].
   Let’s consider the motion of heat flows for a group of cells with a Neumann neighbourhood
in the presence of axial symmetry of the simulation object.
   Fig. 3 schematically shows a cell in a cylindrical coordinate system. Before determining the
change in cell energy, it is necessary to determine the areas of the lateral surfaces of the cell.




Figure 3: Three-dimensional image of a cell


   Let’s introduce the notation:
𝑖 – cell number in the radial direction, 𝑖 = 1...𝑖𝑛 ;
𝑗 – sectoral cell number, 𝑗 = 1...𝑗𝑛 ;
𝑘 – cell number along the axis of symmetry, 𝑘 = 1...𝑘𝑛 ;
ℎ𝑖 – radial cell length;
ℎ𝑗 – the sectoral width of the cell depends on the position of the cell in the radial direction;
ℎ𝑘 – cell height along the axis of symmetry;
𝑛𝑖 – number of cells in the radial direction;
𝑛𝑗 – sectoral number of cells;
𝑛𝑘 – number of cells along the axis of symmetry.
   Fig. 4, 5, 6 show a schematic representation of a cell (𝑖, 𝑗, 𝑘) and notations of its lateral sides.




Figure 4: Schematic representation of a cell and notations of lateral sides


   The sectoral width of the cell depends on the position of the cell in the radial direction, i.e.
from 𝑖:

                                                       2𝜋ℎ𝑖 𝑖
                                            ℎ𝑗 (𝑖) =          .
                                                        𝑛𝑗
   The lateral sides of a cell are expressed in terms of the cell number, cell length and the number
of cells along the axes:

                                               𝑆𝑖 = ℎ𝑖 ℎ𝑘 ,

                                                      2𝜋ℎ𝑖 𝑖
                                           𝑆𝑗 (𝑖) =          ℎ𝑘 ,
                                                       𝑛𝑗

                                                    𝜋ℎ2𝑖 (2𝑖 − 1)
                                         𝑆𝑘 (𝑖) =                 .
                                                          𝑛𝑗
Figure 5: Schematic representation of cells in a cylindrical coordinate system (top view)




Figure 6: Schematic representation of cells in a cylindrical coordinate system (side view)


  The change in cell energy for a cylindrical coordinate system consists of the sums of energy
changes in the radial, sectoral directions and along the axis of symmetry of the body.
  Thus, the change in energy for 𝑖, 𝑗, 𝑘–th cell

    • in radial direction:
                                                𝑗,𝑘
                                       𝑗,𝑘     𝑘𝑖−1,𝑖 𝑆𝑗 (𝑖) [𝑇𝑖−1 − 𝑇𝑖 ] 𝜏
                                      𝑄𝑖−1,𝑖 =                              ,
                                                             ℎ𝑖
                                              𝑗,𝑘
                                     𝑗,𝑘     𝑘𝑖+1,𝑖 𝑆𝑗 (𝑖 + 1) [𝑇𝑖+1 − 𝑇𝑖 ] 𝜏
                                    𝑄𝑖+1,𝑖 =                                  ;
                                                             ℎ𝑖
    • in the sectoral direction:
                                                    𝑖,𝑘
                                        𝑖,𝑘        𝑘𝑗−1,𝑗 𝑆𝑗 (𝑖) [𝑇𝑗−1 − 𝑇𝑗 ] 𝜏
                                       𝑄𝑗−1,𝑗 =                                 ,
                                                                ℎ𝑗 (𝑖)
                                                  𝑖,𝑘
                                      𝑖,𝑘        𝑘𝑗+1,𝑗 𝑆𝑗 (𝑖 + 1) [𝑇𝑖+1 − 𝑇𝑖 ] 𝜏
                                     𝑄𝑗+1,𝑗 =                                     ;
                                                               ℎ𝑗 (𝑖)

    • along the axis of symmetry:
                                                   𝑖,𝑗
                                      𝑖,𝑗         𝑘𝑘−1,𝑘 𝑆𝑘 (𝑖) [𝑇𝑘−1 − 𝑇𝑘 ] 𝜏
                                     𝑄𝑘−1,𝑘 =                                       ,
                                                                ℎ𝑘
                                                 𝑖,𝑗
                                    𝑖,𝑗         𝑘𝑘+1,𝑘 𝑆𝑘 (𝑖 + 1) [𝑇𝑘+1 − 𝑇𝑘 ] 𝜏
                                   𝑄𝑘+1,𝑘 =                                             .
                                                                ℎ𝑘

  In the above formulas, superscripts indicate the same coordinates for neighbouring cells, and
subscripts indicate the direction of the considered interaction.
  A cell energy change is the sum of energy changes in all directions:

                                                  ∑ 𝑄𝑖,𝑗,𝑘 =                                  (7)
    In the above formulas, the unknown parameter is the effective thermal conductivity, that is
calculated from the equality of heat flows between cells for a cylindrical coordinate system (Fig.
7).




Figure 7: Heat flows between cells for radial heat distribution


  Heat flow between cells 𝑖 and 𝑖 + 1 is determined by ratio

                                              𝑘𝑖,𝑖+1 (𝑇𝑖 − 𝑇𝑖+1 ) 2𝜋ℎ𝑖 ℎ𝑘 𝑖
                                   𝑃𝑖,𝑖+1 =                                 .
                                                       ℎ𝑖           𝑛𝑗
  Heat flow to the border that separates cells:
    • from 𝑖 cell’s side
                                             𝑘𝑖 (𝑇𝑖 − 𝑇 ) 2𝜋ℎ𝑖 ℎ𝑘 (𝑖 − 0.25)
                                     𝑃𝑖 =                                    ,
                                                 ℎ𝑖 /2           𝑛𝑗
    • from 𝑖 + 1 cell’s side
                                            𝑘𝑖+1 (𝑇 − 𝑇𝑖+1 ) 2𝜋ℎ𝑖 ℎ𝑘 (𝑖 − 0.25)
                                  𝑃𝑖+1 =                                        .
                                                  ℎ𝑖 /2             𝑛𝑗
  Based on their equality of heat flows at the cell border 𝑃𝑖 = 𝑃𝑖+1 we can find 𝑇 – cell border
temperature:

                                 𝑘𝑖 𝑇𝑖 (𝑖 − 0.25) + 𝑘𝑖+1 𝑇𝑖+1 (𝑖 + 0.25)
                               𝑇 =                                       .                    (8)
                                     𝑘𝑖 (𝑖 − 0.25) + 𝑘𝑖+1 (𝑖 + 0.25)
  Equate 𝑃𝑖,𝑖+1 = 𝑃𝑖 and we obtain the value of the effective coefficient of thermal conductivity
for radial heat propagation

                                        2𝑘𝑖 (𝑖 − 0.25)𝑘𝑖+1 (𝑖 + 0.25)
                               𝑘𝑖−1,𝑖 =                                  .                 (9)
                                      𝑖(𝑘𝑖 (𝑖 − 0.25) + 𝑘𝑖+1 (𝑖 + 0.25))
   When calculating the change in energy along the axis of symmetry and along the circum-
ference, the effective coefficient of thermal conductivity is calculated similarly to the one-
dimensional case using the formula (5).


3. Conclusions
The presented mathematical model can be integrated into an automated production control
system, the components of which communicate in order to optimise the parameters of the
production cycle.
  We have given a description of a cellular-automaton thermodynamic model of solidification,
taking into account the phase transition for the Cartesian and cylindrical coordinate systems,
which differs from the existing ones by correctly considering the thermodynamic features of
the solidification process.


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