Revista
de la
Universidad
del Zulia
Fundada en 1947
por el Dr. Jesús Enrique Lossada
DEPÓSITO LEGAL ZU2020000153
ISSN 0041-8811
E-ISSN 2665-0428
Ciencias del
Agro
Ingeniería
y Tecnología
Año 12 N° 32
Enero - Abril 2021
Tercera Época
Maracaibo-Venezuela
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127
Simulation of thermal processes on the electrode of a miniature
protective spark gap
Igor A. Murog *
Valery F. Gnido **
Elena V. Tinina ***
Igor A. Ilchuk ****
Tatiana A. Asayeva *****
ABSTRACT
The article discusses the issues that arise when determining the temperature in the region
of the cathode spot in miniature protective spark gaps. The modeling principle is used to
study the temperature field on the spark gap electrode. A mathematical model of the
process is compiled on the basis of the balance of power entering the cathode spot and its
removal inside the cathode due to thermal conductivity. A numerical solution of the
obtained nonlinear heat equation with inhomogeneous boundary conditions by the finite-
difference method is presented. The authors compared the found temperatures in the
cathode spot for metals of the fourth and fifth groups of the Mendeleev's Periodic Table
with the corresponding melting points of the selected metals. A complete correlation was
obtained between these temperatures. Simulation of thermal processes in the region of the
cathode spot on the electrode made of 42NA-VI alloy has been carried out. The results are
presented in the form of diagrams.
KEYWORDS: mathematical modeling, the electrode, electronic device (spark gap), thermal
processes, thermal conductivity, cathode spot, metals, temperature, time, energy
accumulation.
*Doctor of Technical Sciences, Professor, Director of the Ryazan Institute, Ryazan Institute (branch) of
Moscow Polytechnic University Russia, Ryazan. E-mail: alen-pa14@yandex.ru
** Candidate of Technical Science, Assistant professor, Associate Professor, Department of Mechanics
and Technology, Ryazan Institute (branch) of Moscow Polytechnic University Russia, Ryazan
*** Candidate of Technical Science, Assistant professor, Associate Professor, Department of
Informatics and Information Technology, Ryazan Institute (branch) of Moscow Polytechnic University
Russia, Ryazan.
**** Candidate of Technical Science, Assistant professor, Associate Professor, Department of
Mechanics and Technology, Ryazan Institute (branch) of Moscow Polytechnic University Russia,
Ryazan.
***** Candidate of physical and mathematical Sciences, Assistant professor, head of the Department of
Informatics and information technologies, Ryazan Institute (branch) of Moscow Polytechnic
University, Russia, Ryazan.
Recibido: 16/10/2020 Aceptado: 10/12/2020
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Simulación de procesos térmicos en el electrodo de una brecha de
chispa protectora en miniatura
RESUMEN
El artículo analiza los problemas que surgen al determinar la temperatura en la región del
punto de cátodo en las brechas de chispa protectoras en miniatura. El principio de
modelado se utiliza para estudiar el campo de temperatura en el electrodo de separación de
chispas. Un modelo matemático del proceso se compila sobre la base del equilibrio de
potencia que entra en el punto del cátodo y su eliminación dentro del cátodo debido a la
conductividad térmica. Se presenta una solución numérica de la ecuación de calor no lineal
obtenida con condiciones límite inhomogéneas por el método de diferencia finita. Los
autores compararon las temperaturas encontradas en el punto cátodo para los metales de
los grupos cuarto y quinto de la Tabla Periódica de Mendeleev con los correspondientes
puntos de fusión de los metales seleccionados. Se obtuvo una correlación completa entre
estas temperaturas. Se ha realizado la simulación de procesos térmicos en la región del
punto cátodo en el electrodo de aleación 42NA-VI. Los resultados se presentan en forma de
diagramas.
PALABRAS CLAVE: modelado matemático, el electrodo, dispositivo electrónico (brecha de
chispas), procesos térmicos, conductividad térmica, punto de cátodo, metales, temperatura,
tiempo, acumulación de energía.
Introduction
Miniature uncontrolled spark gaps are widely used in equipment for railway
transport and communications (Kiselev, 1988; Anisimov, Belsky, Kiselev and Yashkova,
2001). One of the parameters of the spark gap is the current amplitude in the pulse, which
can be on the order of tens of kiloamperes or more, and the pulse duration of tens or more
microseconds. The spark gap consists of a ceramic body, two electrodes made of 42NA-VI
alloy, and a cathode with a certain emission composition applied to one of these electrodes.
The cathode is connected to the electrode by soldering or applied to its surface by vacuum
deposition. One of the regularities discovered during the development of spark gaps is that
the parameters of the spark gap are significantly affected by the thermal regime of the
cathode (Anisimov, and Kiselev, 1990; Anisimov and Kiselev, 1995). This is due to the fact
that during the switching process a cathode spot is formed at the cathode of the spark gap:
a small brightly luminous region on the cathode surface. The cathode spot has a high
temperature, at which the necessary emission from the cathode occurs, which provides the
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current through the spark gap for the successful protection of equipment against dangerous
overvoltages.
Cathode spots appear on the electrodes in all powerful switches. The physical
processes occurring in the cathode spot on the electrodes of powerful switches are
considered in (Rakhovsky, 1970; Kesaev, 1972).
The physics of current switching in spark gaps does not have experimental data on
the temperature in the region of the cathode spot. This is due to a number of reasons. The
interelectrode distance in the spark gap is 2 ... 5 mm, the current is highly localized, heat is
released in a very small volume. In this case, protective spark gaps operate in a single
switching mode, the depth of penetration of the thermal field into the electrode during the
switching time is less than the thickness of the electrode, therefore, the measurement of the
temperature in the region of the cathode spot under such conditions is experimentally
difficult.
To analyze the temperature field in the region of the cathode spot, numerical
calculation methods can be used, with the help of which analyzes of many physical
phenomena are already carried out (Deniskin and Nekrasova, 1982).
When solving problems of non-stationary thermal conductivity, which include the
problem of temperature field distribution in the cathode spot region, the method of finite
time intervals is widely used (Kalitkin, 1978). This method allows us to build a
mathematical model of physical processes in the cathode spot, and modern computer
technologies provide a successful solution of these types of problems by conducting
temperature estimates in the region with a small volume.
The purpose of this paper is to present the practical application of the finite-
difference method for modeling thermal processes occurring in a cathode spot on an
electrode made of 42NA-VI alloy. The results obtained will contribute to the optimization
of existing spark gaps, in which the main cathode is this alloy, as well as the development of
new devices with cathodes made of other materials.
The authors simulated thermal processes in the cathode spot on the 42NA-VI alloy
electrode in modes that are close in their parameters to the conditions that occur in the
cathode spot on the cathode of the spark gap during switching.
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1. Basic relations
A cathode spot is formed on the electrode under the action of a high-intensity heat
flux. The temperature in the region of the cathode spot can reach the melting temperature
of the electrode material and higher. Let us assume that the main process determining heat
loss in the cathode spot is its removal into the electrode due to thermal conductivity.
To find the temperature field at the cathode, it is necessary to solve the nonlinear
equation of heat conduction with inhomogeneous boundary conditions.
The scheme for calculating the thermal regime in the region of the cathode spot is
shown in Figure 1.
Figure 1. Scheme for calculating the thermal regime in the area of the cathode spot: F
+
is falling heat flux; F
–
is the heat flux that removes heat from the cathode spot into the
cathode; r
0
is the radius of the cathode spot
Let us assume that the conditions for the propagation of heat in the electrode have
cylindrical symmetry; therefore, the initial expression describing the conditions for the
propagation of heat through the thickness of the electrode for the unsteady mode can be
written as follows
,
1
2
2
2
2
z
T
r
T
r
T
r
a
t
T
(1)
where T is the temperature at this point;
r is the radius measured from the center of the cathode spot;
z is coordinate in the direction perpendicular to the cathode plane;
t is the time counted from the moment of the cathode spot formation.
Here
T
c
a
is the thermal diffusivity; λ is coefficient of thermal conductivity; с
T
is
the heat capacity coefficient; ρ is the density of the substance.
To calculate the temperature field using the finite-difference method, equation (1)
can be represented in the form [8].
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131
kji
kjikjikjikjikjikjikjikji
kjikji
T
TTT
h
TTT
h
TT
jh
tTaT
,,
1,,,,1,,
2
,1,,,,1,,,,1,
,,,,
22
1
)(
.
(2)
Here t is time, j is the temperature distribution along the radius, k is the temperature
distribution along the electrode depth. To calculate temperatures by expression (2), a
program has been developed. Variables that determine the state of the parameters of the
mathematical model were introduced into the program: the number of iterations that
determine the duration of the modeling process, the power of the heat flux released in the
cathode spot, the radius of the cathode spot, the initial temperature, the values of the heat
capacity coefficients с
T
, the density of the cathode material ρ, and the thermal conductivity
λ.
2. Simulation results and their discussion
First, let us estimate the temperature of the cathode spot on electrodes made of
different metals. The thermophysical properties of the electrode material depend on
temperature. First of all, this concerns the coefficient of thermal conductivity: with an
increase in temperature, its value decreases.
The thermal conductivity coefficient for most metals decreases in the first
approximation according to a linear law when the temperature changes from 0
0
С C to the
melting point. In the range of several hundred degrees, changes in the thermal conductivity
are of the order of 10%. Therefore, the maximum temperature in the cathode spot during the
simulation will be set to no more than 400
0
С.
Table 1 shows the metals of the fourth and fifth groups of the Mendeleev's Periodic
Table, for which the temperature of the cathode spot was estimated. Simulation
parameters: the duration of the heat flow is 500 μs, the power of the heat flow is 1200 W.
The third column of Table 1 shows the values of the melting temperatures of these metals. It
was found that the values of the cathode spot temperature decrease monotonically with an
increase in the ordinal number of metals. The metals are arranged in the same sequence
according to the melting point.
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This pattern is determined by the peculiarities of the formation of the heat-
conducting properties of metals.
Table 1. Comparison of cathode spot temperatures and melting temperatures for
different materials
Material
Cathode spot temperature,
0
С
Melting temperature,
0
С
Iron
370
1535
Cobalt
355
1490
Nickel
335
1445
Copper
315
1083
Silver
290
960
As is known (Kalitkin, 1978) heat transfer in metals is carried out by electron λ
е
and
phonon λ
ф
thermal conductivity. With electronic thermal conductivity, energy transfer is
carried out by conduction electrons, with phonon thermal conductivity - by lattice
vibrations. In accordance with the quantum model of a solid for pure metals, the ratio of
electronic and phonon thermal conductivity is about 100. Therefore, in most cases, it can be
assumed that heat transfer for metals is due to the electronic component of thermal
conductivity.
The electronic thermal conductivity is determined (Worth and Thomson, 1966).
e
.
2
f
f
E
lTn
(3)
Here: n is the number of free electrons in a unit volume; к is Boltzmann's constant; l
is the average free path; T is the temperature; E
f
is the Fermi energy; ν
f
is the speed of
electrons with Fermi energy.
As follows from equation (3), the electronic component of thermal conductivity is
proportional to the number of free electrons - conduction electrons. At the same time, the
number of conduction electrons is determined by the position of the atom in the
Mendeleev's Periodic Table. A gradual increase in conduction electrons from iron to copper
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and silver leads to an increase in the λ
е
coefficient and, accordingly, to a decrease in the
temperature on the surface of the cathode spot.
The obtained connection between the temperatures testifies to the correctness of the
constructed model and the possibility of its use for studying thermal processes at the
electrodes of spark gaps made of different metals, including alloys.
Let us consider the kinetics of the temperature field on an electrode made of 42NA-
VI alloy, which, as already mentioned earlier, is widely used in protective miniature spark
gaps. The values of the parameters used in the simulation are shown in Table 2.
Table 2. Values of parameters used in modeling
The dimensions of the electrode corresponded to the real geometry of the miniature
protective spark gap. The electrode surface, which limits the working area of the device, has
a diameter of 6 mm, and the electrode thickness is 1.5 mm. We assume that a cathode spot
is formed on the electrode under the action of a power pulse.
The diagram of the distribution of the temperature field in the region of the cathode
spot is shown in Figure 2. “n” along the axes in the figures is the number of steps in the
program array. One step in “n” is 33 μm. The calculated field in the program is 30n by 70n.
The simulation was carried out under the following conditions: the pulse duration
was 500 μs, the pulse power was 180 W. Figure 2 shows the temperature field displayed
after 100 μs from the moment of the pulse application. Similar diagrams were obtained after
300 and 500 μs, respectively. Since the heat flux removed due to thermal conductivity
Parameter
Designation
Dimension
42NA-VI
Thermal
conductivity
λ
cal/(cm·s·K)
0,11
Density
ρ
cal/(cm·s·K)
8,2
Heat capacity
с
cal/(cm·s·K)
0,5
Melting
temperature
Т
0
С
К
1723
1996
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inside the cathode is less than the flux entering the cathode spot, there is a rise in
temperature in the region of the cathode spot. During the simulation, it was found that the
temperature of the cathode spot during a time of 100 μs changes from room temperature of
293 K to a value of about 520 K. At subsequent moments of time, it rises to 630 K after 300
μs and to 680 K after 500 μs, respectively.
The temperature rise, in comparison with the boundaries of the computational
domain, at the selected times are also different. Heat entering the cathode spot, due to
thermal conductivity, is removed inside the electrode, gradually increasing its temperature.
If the excess of 293 K at the boundary of the computational domain relative to room
temperature is practically absent at a time instant of 100 μs, then after 300 μs the excess is
about 80 K, and after 500 μs it reaches 150 K.
The temperature field inside the electrode is a concentric semicircle. With distance
from the surface of the cathode spot, the temperature gradually decreases; therefore, the
isotherms are semicircles with an increasing radius. The temperature over the surface of the
cathode spot will be variable. The largest value is observed in the center of the cathode
spot: the isotherm radius is minimal.
a
0
20
40
60
0
10
20
293.26
318.18
343.09
368.01
392.92
417.83
442.75
467.66
492.58
517.5
542.41
Figure 2. The temperature field under the influence of a heat flux with a duration of
500 μs and a power of 180 W at a time of 100 μs
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The temperature fields were calculated for various modes. The main condition in the
simulation was chosen so that at a given power supplied to the cathode, phase changes in
the cathode material did not occur. The simulation results for a pulse power of Р=3300 W
with different pulse durations are shown in Figure 3.
The data were obtained: Figure 3, a - the duration of the heat flux 244 ns, Figure 3, b
- 4.2 μs. An increase in the pulse power to 3300 W causes a sharp rise in temperature in the
region of the cathode spot.
With a pulse duration of 244 ns, the temperature reaches 900 K, and with a duration
of 4.2 μs, it is already close to the melting temperature. At the same time, the depth of heat
propagation inside the electrode during the pulse action significantly decreases, which
follows from a comparison of the results presented in Figures 2 and 3. The excess of the
temperature of the cathode spot in relation to the “cold” part of the electrode is about 1426
K (Figure 3, b).
a)
a
0
20
40
60
0
10
20
292.57
353.69
414.82
475.94
537.06
598.18
659.31
720.43
781.55
842.68
903.8
b)
a
0
20
40
60
0
10
20
291.35
434.88
578.4
721.93
865.45
1008.98
1152.51
1296.03
1439.56
1583.08
1726.61
Figure 3. Temperature fields in the cathode spot region
at a pulse power P=3300 W and for different pulse durations:
a) 244 ns; b) 4.2 μs
The temperature fields for different points in time during the cooling process are
shown in Figure 4. The simulation results correspond to the case with a pulse duration of
4.2 µs. Since a small amount of heat accumulates in the cathode spot during the pulse
duration, cooling occurs almost instantaneously. During 8 μs, the temperature of the
nмn
T, K
T, K
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cathode spot decreases almost twice and is about 770 K (Figure 4, a). Subsequently, the
cooling rate slows down. After 31 μs, counted from the moment the pulse is removed, the
surface of the cathode spot is cooled to a temperature of about 450 K (Figure 4, b). At
subsequent moments in time, due to the process of thermal conductivity, the heat further
spreads in the electrode material, increasing the temperature in it within the calculated
region (Figure 4, c). The limiting distribution of the calculated temperature field (at the
moment of time 100 μs) is shown in Figure 4, d. The temperature at the boundary of the
computational domain is about 315 K.
Therefore, with an increase in the heat flux power, the arising high-temperature field
in the region of the cathode spot will penetrate less and less deep into the electrode, being
within a few micrometers on the electrode surface. This shows that the thickness of the
cathodes, which are deposited on the electrode, can be several micrometers. The practice of
developing cathodes for miniature protective spark gaps confirms the found patterns.
It is known that in real designs of protective miniature spark gaps, materials with a
high emissivity are applied to the 42NA-VI alloy electrode (Anisimov, 1966; Kiselev, Gnido,
Anisimov and Tinina, 2001). Cathodes made of these materials have a low work function,
which ensures a low arc maintenance voltage in the device during switching, and,
accordingly, reduces the power released on the electrode as a whole. In this case, the
thickness of the cathodes is a few micrometers. This, as studies have shown, was sufficient
to stabilize the parameters during operation (Kiselev, Gnido, Anisimov and Tinina, 2001).
The erosion zone at the cathode is within 1 ... 1.5 microns. These values correspond in order
of magnitude to the dimensions of the high-temperature field obtained on the 42NA-VI
electrode during simulation.
Conclusion
Modeling the temperature field on electrodes made of metals of the fourth and fifth
groups of the Mendeleev's Periodic Table showed a complete correlation in the temperature
of the cathode spot and the melting temperature of metals. The results show that with an
increase in the serial number of the element, the temperature of the cathode spot decreases:
from iron to copper and silver.
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a
0
20
40
60
0
10
20
291.15
339.82
388.48
437.15
485.82
534.49
583.15
631.82
680.49
729.15
777.82
a
0
20
40
60
0
10
20
292.33
308.35
324.37
340.39
356.41
372.42
388.44
404.46
420.48
436.5
452.52
a) b)
a
0
20
40
60
0
10
20
0
35.03
70.05
105.08
140.11
175.13
210.16
245.19
280.22
315.24
350.27
a
0
20
40
60
0
10
20
0
35.03
70.05
105.08
140.11
175.13
210.16
245.19
280.22
315.24
350.27
c) d)
Figure 4. Fields of cathode temperatures during cooling at different times: a) 8 μs; b)
31 μs c) 60 μs d) 100 μs
Numerical modeling of the kinetics of the temperature field on the electrode made of
42NA-VI alloy made it possible to establish:
1. Under conditions of powerful heat flux, the temperature of the cathode spot can
reach the melting temperature.
2. The time of energy accumulation in the region of the cathode spot, when its
surface reaches the melting temperature, is microseconds.
3. The thickness of the high-temperature field in the region of the cathode spot
during the time of the heat flux can be micrometers.
It is shown that with the help of modeling, taking into account heat losses in the
cathode spot only due to thermal conductivity, results can be obtained that sufficiently
г
)
n
n
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fully determine the thermal regime in the cathode spot, when in the studied range of heat
flow power, material erosion will practically not occur or will be minimal.
In order to further study the process of erosion of the cathode material in spark gaps,
it is necessary to further study and model the temperature field, taking into account the
heat losses due to evaporation and melting, which can actually be observed in the cathode
spot of the spark gap.
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