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MATHEMATICAL MODELING OF PHASE TRANSITIONS IN THE METAL ELECTRODE THROUGH "ENTHALPY" METHOD
Arutyunyan Robert Vladimirovich,
, candidate of physical and mathematical
sciences, docent, Moscow Technical University of Communication
and Informatics (MTUCI),Russia, Moscow,
Keywords:: metal electrode, a high current pulse electric field, temperature field calculation through method.
The article investigates the thermal and electrical processes in heating the metal electrode a high current pulse. The aim is to clarify the nature of the influence of non-linearities with thermal parameters of phase transitions of melting and evaporation, the type of boundary conditions in the current spot. To solve the problem formulated a mathematical model, and developed finite-difference method and a computer program to effectively carry out computer simulations of thermal and electrical processes under the influence of a high pulse on the metal electrodes. The solution of the Stefan problem is using through "enthalpy" method. Calculation of the electric field is carried out by Seidel iteration. Computer error is monitored by thermal and current balance and comparing the results of model problems. It carried out a series of calculations for the case of iron informative. A significant influence of non-linearities of thermal parameters of the phase transitions of melting and evaporation, the type of boundary conditions on the values of the temperature and electric fields, especially in the area of the current spot. The presence of high current density and temperature, respectively, in the vicinity of the edge of the current spot confirms the well-known hypothesis about the causes of contact welding on the edges of the contact area. It was found that the impact of the loss to radiation and convection cooling is negligible. The article continues and complements the well-known research in the theory of electrical contacts and welding processes based on a detailed account of the properties of the electrode material nonlineari-ties and type of boundary conditions for temperature and electric fields. The results can be used in the practice of research and design of electrical machines and other electrical devices. The study revealed the need to improve the enthalpy finite difference method for solving the Stefan problem in order to improve its accuracy, in particular in the calculation of the phase transition front. Of interest is a further detailed analysis of the thermal and electrical processes in the boiling point, determination of the degree of adequacy of model of quiet evaporation, additional accounting gas, hydro and electro factors.
Для цитирования:
Арутюнян Р.В. Математическое моделирование фазовых переходов в металлическом электроде через метод "энтальпии" // T-Comm: Телекоммуникации и транспорт. - 2015. - Том 9. - №11. - С. 62-67.
For citation:
Arutyunyan R.V. Mathematical modeling of phase transitions in the metal electrode through "enthalpy" method. T-Comm. 2015. Vol. 9. No.11, рр. 62-67.
Introduction
In theory, electrical contacts, plasma torches, electric welding, etc. is an urgent problem for electric and temperature fields, taking into account the non-linearities of electric and thermal properties of the material, phase transitions and other factors [1-4].
In this paper the results of selected studies are developed on the basis of through "enthalpy" method to effectively take into account the thermal characteristics of the material nonlinearity, phase transitions (melting, evaporation), the radiative and convective cooling of the surface material.
1. Mathematical model
The influence of short-term high-current pulse to the electrodeis investigated. Calculation of thermal and electrical processes in the matter is based on the respective non-stationary spatial axisymmetric model. It is assumed that the electrode covers a half-space z > 0, z - space coordinate (applicate) in a cylindrical coordinate system. The current value /0 conducted through the spot with the current radius R0 on the surface of the electrode (z = 0).
In mathematical modeling of three stages of the thermal process: heating the material up to the melting point (solid phase); and further heating the melt penetration solid piece of material (liquid phase); began intensive evaporation and boiling material (phase evaporation and boiling).
In an article for the purposes of computer simulation used a numerical method through computation [6,7], based on the conversion of the multi-phase Stefan problem for "enthalpy" referring to the concentrated heat. The corresponding boundary value problem is:
i d ( dh\ d2A . n n ...
— + qv,r>0,z>0,t>0, (1)
u
A(w) = J'X{v)dv, A(w) - thermal conductivity^nder
«0
such a system of notation: Xu = A , t - time/ r - the polar radius, a - Stefan-Boltzmann constant/ k -factor of convective heat transfer, W0 - Energy, and tF - the current pulse time, qv - density Joule sources:
q^ =y\V<p\2, where y = electrical conductivity,
(p - electrical potential is a solution of the problem:
' dz
^,r<R0,z = 0,
7TR,
0,r>tf0,z = 0.
<p->
0, л/г2 +z2 -^oo.
(5)
(6)
where it is assumed that the current in the current spot evenly distributed. qvap - Specific losses to evaporation
from the surface of the material qvap = p2{u)rvapvvap, where r - latent heat of vaporization, v - the speed
vap r ' vap r
of the front vacuum evaporation, determined Hertz-Knudsen equation:
= I M p„(ut)
vap
dQ dt 8A
r dr
dr) dz1
( ^ UJJUIII^UUI I \J I 11
u -u0 )+k(u-uo)-q^(u),z = 0,r>0,t>0, i2> ^ on temperature:
2jtRus p2(us)
wherepw - saturated vapor pressure corresponding to the temperature of the surface u$ = £</-,0,0, M- mass of moles and R- gas constant. The vapor pressure Is derived from the Clausius-Ciapeyron, which is integrated in the assumption of independence of the heat of evaporation
&
u = u0,r > 0,z > 0,t = 0; u
yjr2 +z2 ->00 . Where Q— enthalpy:
*0'
(3)
PsvM = PQexp
f
Mr_ Mr.
^
vap
Ru,
vap
Ru..
, where pQ - pres-
a
Q(u) = jp(v)c(v)dv + pJ (umel,)rmell\(u -umeb)-
"o
l(w)- Heaviside unit function, u - temperature, umelt -melting pointy r u - latent heat of fusion, w0 - the initial value of body temperature («„ < umelt), p - p{u) - Bulk density, px (u) - density of the solid phase, respectively, p2{u)~ the density of the melt, c - c(zv) - heat capacity of the electrode material, A - heat potential,
h "-"g y
sure of the gas surrounding the body of the medium/ ub - boiling point at a pressure p0.
The thickness of the evaporated layer z (t) was
defined as an integral:
vap
Finite-difference method for solving
The domain of integration - the infinite, because it is advisable to use a non-uniform grid. Differential ratio considered problem is approximately replaced by a system of finite-difference equations:
Q!?-Qb_ 1 ( A^'.-Aft1 лн'-лс'Л
г, h .
( Г,|
wj /,/ - - -и ГМ-;--П—-
ь' -у
к>
\ V./+1
+1
Ч/+1 J N,/ ' Ч/ ' t
^Aft!, - Aft1 Aft1 - Afî1 4
ч
Kl у
г, J
J bjlKïL; uf zJ 2 4
<
k "o SW* < »
]
r&j
ч+if m j , M î,j ,
"rj+1 "rj J
h, s
yP Z±lLÏ2 LhL yP v '•-' v'-/"1
^ y
= 0,
i = -1; y = 1.....«, -1; p = 0,1,..,;
W + vp
/y'1 P F" F2 P
/,_/' -,/J ■• rJJ
P
I I,/
/>
* J 1
л; i
/ = 1,.-1; 7 = 1,...,/7.-1; p = 0,1,...;
wherein the time step, t — pi; p- number sacrificial layer; h ¡ — grid coordinate r; /-node number grid spatial coordinate r; h_ - gridcoordlnate z; j - node
jq h-h
number grid spatial coordinate z; h ——--,
2/,„, ' ** 2h7j
Finite-difference approximation of the boundary conditions on the boundaries of the region:
-yP Ti,0
Ki
7ГЯ
grid function, ufj xu(ritZj,tp),i=0,...,nr;j- 0,...,nz;p = 0,1,..
The initial condition can be written as: ufj=u0>
i = 0,j = 0,...,nz-
The boundary condition on the surface of the body:
hzA
Conditions on the axis of symmetry: Qq =Q\ ->
j = -1;/? = 0,1,....
Conditions at the boundaries of the grid area, meet the conditions at infinity: Q? - 0, j = 1,...,n_-1;
QL=0J = 0.....nr;p = 0,l.....
The system of finite-difference equations approximating the boundary problem for the electric potential:
0 ,rf>R0.
z = 0„.,,»r;/> = 0,l,...;
Kj = Kj> J = ■ ~ UP = V■ 4 =0, / = 0,= 0,1,...;
<j=0, y'-l,.-l;p-0,l,... .
The size of the spatial coordinate values of integration steps are selected on the basis of the requirements of the accuracy of the finite-difference solutions.
Finite-difference method is implicit and is characterized by high stability.
At each time step system of finite-difference equations solved by Seidel's method. First, carry out calculations of the electric field, allowing to calculate the density of Joule sources. The next step for each internal node of the grid system was solved by Newton's method with respect to the
grid enthalpy Q^1 if known / = ],.-1;
j - 1,- 1, these values are calculated temperature in the same grid nodes: uff = Q~] (Qff ), p = 0,1,...;
Q~l -the inverse of the function Q(it) is calculated as Newton's method as a solution of a nonlinear equation with respect to the form: Q(y} = z.
The value uj'^ of in finite difference derivative boundary condition at z-0 is expressed through the rest of expression and also in the process by the method of Seidel iteration.
Control of error of numerical solution was carried out by the values of the thermal balance (accuracy of about 5%).
2. Simulation results
As the main object of the study examined the iron. This is due to the fact that iron-based materials - steel, are most useful metal In the industry in general, and in various techniques (e.g., welding of steel plates, etc.). This iron is characterized by strong non-linearity of thermophysical characteristics [5].
The main thermal parameters iron included in the calculation: Density (at normal conditions) - 7874 kg/m3; melting point - 1812 K (1538,85 °C); Boiling point - 3134 K; the heat of fusion - 247,1 kj/kg; Specific heat of
vaporization - 6088 kJ/kg; heat at normal conditions -444 J/(kg K); thermal conductivity - (300 K) 80.4 W/(m K); molar mass - 55.847 g/mol.
Specific heat of iron, like any other element, its structure is determined and varies depending on the temperature; the average value of the specific heat of iron at O-iOOO °C is equal to 640.57 J/(kg K). The dependence of the specific heat of the iron temperature is characterized by a pronounced nonlinearity (http://steelcast.ru/iron_heat_capacity).
Table 1
The values of the heat capacity of iron for different temperature ranges
T,K T, °c Type Of grating СЮ/ (kg ■ mol -deg) С Ю/ (kg- deg)
273...1033 0...760 a 17,50082 + 24,78586 ■ 10 3 T 0,31335+ 0,4438'1D"3 T
1033...1181 760...908 ß 37.6812 0,6747
1181...1674 908... 1401 V 7,70371 + 19,51049 ■ 10'3 T 0,1397 + 0,3493 - 10'3 T
1674,,, 1810 1401... 1537 5 43,96140 0,7870
1810...1873 1537...1600 Liquid 41,868 0,7495
Figure 1 shows in relative terms the basic thermal parameters of iron (c- specific heat,(?- enthalpy, A - heat potential, a. - thermal conductivity, a- thermal diffusivity).
Fig. 1. Thermal parameters of iron (in relative terms)
In the phase transition characteristics of iron undergo jumps. The density of the molten iron at a temperature of 1535 °C is equal to 6900 kg/m3, ie 12% lower than at normal conditions.Near the melting temperature have the following parameters jumps iron heat capacity - from 450 to 710 J/(kg K), thermal conductivity - from 17 to 29 W/(m K), the thermal diffusivity of about 3 x 10"6 to 8x 10"5 m3/s [5].
The characteristic size of the computational domain, the duration and the magnitude of the current pulse was chosen for the typical electrical contacts:
Lr = 10~V L. = 10"3m, tF a 2,5-10"3 5; /„=500,4.
The square and the radius of the current spot on the electrode surface:
S0 = \0'7m2, R^yfs^hr.
Options finite difference method chosen in accordance with the requirements of accuracy: nr -30, n. -70,
T = 10"6.
The area in the vicinity of the conductive patches covered with mesh sizes Lr = 5-10 4 m, L. = 5-10 4 m, smaller increments in the number of units: nr0 = 20,
Fig. 2 presents a series of graphs of the temperature field based on the spatial coordinates (with a time step 2,5*10 4s ), rand zare measured in mm.
Fig. 2. Graphs of the temperature profiles of the field (1 - depending on zfor a fixed r, 2- depending on rfor fixed z)
Fig. 3 shows graphs (in arbitrary units), depending on the time of origin of the fronts of phase transitions. The highest values of the coordinates of the phase transition front: Xmetk^Fi = 80 microns; ;tv^fr) = 1,7 microns. The velocities of the evaporation front: = 0,012 m/s.
The stepped nature of the graph number 4 associated with Stefan's nonlinearity. Interpolation values of xva£() carried by the grid values of the temperature field, whose gradient discontinuities of the first kind. Using the grid for the interpolation enthalpy results in partial smoothing of the graph.
Fig. 4 are graphs describing the nature of the electric field in the electrode.
The paper also studied the loss to radiation and convection cooling, which make up about 1%, and are negligible.
Т-Сотт Том 9. #11-2015
МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ФАЗОВЫХ ПЕРЕХОДОВ В МЕТАЛЛИЧЕСКОМ ЭЛЕКТРОДЕ ЧЕРЕЗ МЕТОД "ЭНТАЛЬПИИ"
Арутюнян Роберт Владимирович, кандидат физико-математических наук, доцент, Московский технический университет связи и информатики (МТУСИ), Россия, Москва, [email protected]
Аннотация
Исследуются термические и электрические процессы в нагревании металлический электрод высокий импульс тока. Цель состоит в выяснении характера влияния нелинейностей с тепловых параметров фазовых переходов плавления и испарения, типа граничных условий в текущем месте. Чтобы решить эту проблему сформулировал математическую модель, и разработали метод конечных разностей и компьютерную программу, чтобы эффективно осуществлять компьютерное моделирование тепловых и электрических процессов под влиянием высокой импульса на металлических электродов. Решение задачи Стефана использует метод "через энтальпии". Расчет электрического поля осуществляют Зейделя. Компьютер ошибки контролируется тепловой и текущего баланса и сравнения результатов модельных задач. Это осуществляется ряд расчетов для случая железа информативным. Выявлено существенное влияние нели-нейностей тепловых параметров фазовых переходов плавления и испарения, типа граничных условий на значения полей температуры и электрических, особенно в области текущем месте. Наличие высокой плотности тока и температуре, соответственно, в непосредственной близости от кромки текущем месте подтверждает известную гипотезу о причинах контактной сварки по краям площади контакта. Было обнаружено, что воздействие потери излучения и конвекционного охлаждения является незначительным. Статья продолжает и дополняет известную исследования в теории электрических контактов и процессов сварки на основе детального учета свойств материала электродов нелиней-ностями и типа граничных условий для температуры и электрических полей. Полученные результаты могут быть использованы в практике научных исследований и проектирования электрических машин и других электрических устройств. Исследование показало, необходимость повышения энтальпии метода конечных разностей для решения задачи Стефана для того, чтобы улучшить ее точность, в частности, в расчете фронта фазового перехода. Представляет интерес дальнейшее подробный анализ тепловых и электрических процессов в точке кипения, определение степени адекватности модели тихом испарения, дополнительного учета газа, гидро- и электро факторов.
Ключевые слова: металлический электрод, высокий импульс тока электрическое поле, расчет температурного поля через метод.
Литература
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3. Абрамов Н.Р. и др. The characteristics of penetration of the walls of metal objects when exposed to lightning // Электричество. I986. №II. С. 22-27.
4. Некрасов C.A. Математическое моделирование процессов тепло- , весо- и электромиграции. Дис... канд. техн. наук. -Новочеркасск, I992.
5. Thermal properties of meltshttp://files.lib.sfu-kras.ru/ebibl/umkd/Mamina/u_lectures.pdf.
6. Samarsky A.A., Moiseenko B.D. Cheap Difference scheme for solving of Stephen problem // Journal calculated mathematics and math. physics, I96S. T.S. № S. С. 8I6-82.
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