CALCULATION OF THE THERMOEMISSION CATHODE PROCESSES OF
THE ARC PLASMA TORCHES
S. Nguyen-Kuok
Department of Physics and Fusion Synthesis of Moscow Power Engineering Institute (Technical University).
Krasnokazarmennaya St. 14, Moscow 111250, Russia. Email: [email protected]
Abstract – Work gives model and results of calculating the cathode processes of the Arc plasma torches at
atmospheric pressure with the thermoemission cathodes. Calculations were performed on the basis of the method
of integral balances for the cathode made of the pure tungsten W and the thoriated tungsten W(2% ThO2 )) over a
wide range of temperatures of cathodes (Tk = 3000 – 4400 K for W and Tk = 2000 – 3200 K for W(2% ThO2 )) and
argon plasma (Te = 20000 – 40000 K).
Keywords: Thermoemission cathode, the Arc plasma torches, the cathode processes, the region near the
cathode, the layer of space charge, ionizing layer, two-layered model, the method of the integral balances.
1. Introduction
Arc plasma torches work in the inert gases,
argon, nitrogen, hydrogen and their mixtures widely
are used thermo-emission cathodes on the basis of
tungsten. In oxygen, nitrogen and carbon-containing
gases adapt the cathodes on the basis of the metals
(zirconium, hafnium, molybdenum, and other),
which with interaction with the plasma-forming
gases form the films of connections (oxides, nitrides,
carbides), which possess the high emission
properties and thermal resistance.
The processes near the electrode play the
significant role in the formation of the property of
arc stream and are the closed system of the physical
processes, which take place simultaneously in the
near-electrode region, in electrode and on its surface.
The simplified picture of phenomena in the cathode
region of arc can be presented with the two-layered
model [1-5].
The first layer, if the counting of news from
the surface of cathode is called the layer of space
charge, has an extent of the order of a Debye radius
rD and less than the mean free path of electrons and
ions lei. For the argon plasma of atmospheric
pressure with the characteristic temperature range Te
= 10000 – 40000 K, we have rD ~ (0.05 – 0.02) µm,
lei ~ (4 – 1) µm. In this layer as a result of a relatively
larger fraction of ion current and smaller ion
mobility is formed excess positive about the space
charge, which causes potential jump on the cathode
surface (a cathode drop in the potential).
The second layer divides the first layer and
arc stream it is called ionizing. In it is satisfied the
condition of quasi-neutrality and occurs the intensive
generation of the charged particles due to the energy,
acquired by electrons in the first layer. From the
second layer into the first move not only the ions,
but also opposite electrons. Only the small part of
the electrons, which possess the energy, sufficient
for overcoming the barrier, reaches because of the
field the surface of cathode braking influences. Ions,
being accelerated by field, reach the surface of
cathode and they are neutralized the electron of
metal. The electrons emitted by surface and atoms
move from the cathode. Atoms, after passing
without the collisions the first layer, fall the
secondly, reducing the degree of ionization. At the
same time the electrons, accelerated in the first layer
and after falling into the second, gradually increase
the degree of ionization and the concentration of the
charged particles, reaching on the Border with the
arc stream of the value, close to the equilibrium
value.
2. Model of the thermoemission cathode
processes
A space charge layer. In the region of the
space charge practically there are no collisions and
the flow of different charged particles through this
region can be considered constant. Potential
distribution in this layer is described by Poisson's
equation:
d 2V
ε 0 2 = e ( ne − Zni )
dz
(1)
Where the V - electrostatic potential, connected with
the density of electric field as E = − dV / dz , z axis
is directed perpendicular to the surface of cathode; Z
- charge number of ions; ε0 - dielectric constant of
vacuum.
The solution of this equation can be obtained
by direct integration, if is known the distribution of
electrons and ions on the velocities on the Border of
region. At present strict solutions exist only for the
case, when the displacement of ions they prevail
above the collisions with the atoms with a change in
the charge [3,4]. As usually it is relied, the
distribution of electrons in the field of space charge
is described by the distribution of Boltzmann. Zero
electrostatic potential infinity is selected on the
Border space charge, and ionic density is determined
from the solution of the equation of the steady state
of Boltzmann for ions. As a result the density of
electric field on the surface of cathode can be found
from the known equation Makkoun:
E2 =
4
ε0
me
2e


mi
− je  U d
 ji

Zme


(2)
Where ji, je - current densities of ions and electrons;
mi, me - mass of ions and electrons; UD - voltage
drop in the space layer of charge.
Expression (2) does not consider fluctuation
of electric field, connected with the discretion of ion
charges. It is shown in [6] that under arc conditions
for atmospheric pressure (j ≤ 105 A/cm2, E < 107
V/cm) a quantity of additionally emitted by cathode
electrons due to the fluctuation of electric field to
one ion is substantially lower than one.
Balance of currents on the surface of
cathode. With the thermo-electron’s emission it is
necessary to preliminarily accomplish a heat supply
to cathode, due to what occurs the population of
energy states higher than the Fermi level. Some of
these states have energy, which exceeds the height of
potential barrier, and electrons from these levels can
leave the metal of cathode. The current density of
thermo-emission is described by the formula of
Richardson-Schottky [1-3]:
 eϕ − ∆( eϕ ) 
jem = ATk2 exp  −

kTk


(3)
where A – Richardson's constant is determined
experimentally; Tk - temperature of the cathode
surface; eϕ - the work function of the material of
cathode; k - Boltzmann constant; ∆(eϕ) - reduction
in the work function due to the high electric field on
the cathode surface - the Schottky effect:
∆ ( eϕ ) =
eE
4πε 0
(4)
The Schottky effect becomes essential with
the density of electric field E ≥ 104 В/см. On the
cold cathodes with the very high electric fields
potential barrier is made by narrow and that
permeated for the electrons due to the tunnel effect,
the autoelectronic mechanism of emission realizes
(autoelectric emission). The connection between the
current density of autoelectronic emission and the
density of electric field is determined by the
equation of Fowler-Nordgeym [7]:
jem = 1,54.10-6

ϕ 3/ 2 
exp  −6 ,810
. −7

Ek 
ϕ

E
(5)
It follows that autoelectronic emission (5)
becomes essential only with Е > 107 V/cm. The
carried out analysis of experimental data according
to the refractory arc cathodes and the results of
further calculation shows that with the realizable
values of current density on the cathode of the arc
plasma torches j = 102 – 104 A/cm2, the density of
the electric field Е < 107 V/cm. The analysis, carried
out into [2,3] also shows that with conditions of the
arc discharge the secondary emission of electrons
due to the ion bombardment, by excited atoms and
photoemission can be disregarded. Therefore for the
thermal cathodes of electric arcs usually they are in
practice limited only to the emission, described by
the formula of Richardson- Schottky (3). The
velocity of the ions on the Border of the layer of the
space charge and ionizing layer vis is determined
from the criterion of Bohm [1,3,4]:
vis = k (Ti + ZTe ) / mi
(6)
The density of ion current to the cathode is
equal:
ji = Zenis vis
(7)
where ion concentration nis on the Border of the field
of the space charge layer and ionizing layer is
possible to be determined according to the
correlation formula [3,4], obtained from the analysis
of the results of the numerical calculation of multiliquid equation for the ionizing layer with a constant
electron temperature and the temperature of atoms
and ions.
In order to determine the density of opposite
electronic current of jec, which diffuse from the
quasi-neutral plasma to the cathode it is possible to
assume that the velocity of propagation of the
electrons on the Border of the region of space charge
is close to the average velocity in the Maxwellian
distribution at a temperature of electrons Te [1-4],
then:
 eU 
enes ve
exp  − d 
4
 kTe 
ve = 8kTe / π me ; nes = Znis
jec =
where
(8)
-
the
concentration of the electrons on the Border of the
region of space charge and ionizing layer.
Total current density on the surface of
cathode is equal:
j = ji + jem − jec = ji + je
(9)
Баланс энергии на поверхности катода.
The energy flow of the ionic bombardment of the
cathode surface can be obtained with the aid of the
Maxwellian distribution of ions in the space charge
layer on the velocities [3]:
qik =
ji
 k ( 2Ti + ZTe / 2 ) + ZeU d 
Ze 
(10)
Energy flow carried by opposite electrons to
the cathode is equal:
jec
(11)
( 2kTe + eϕeff )
e
= eϕ − ∆( eϕ ) - the effective work
qec =
where eϕeff
function.
The energy flow of the electrons, which
leave the surface of cathode, can be evaluated with
the aid of the approximation of the distribution of
their velocity by haft-Maxwellian distribution with
the temperature of surface Tk and it will be equal
2 jem kTk / e . It follows:
qem =
jem
( 2kTk + eϕeff
e
)
(12)
It is analogous, the energy flow of the atoms,
which leave cathode surface after they came out as a
result to the neutralization of the ions from the
plasma, will be equal:
qa = ji
2kTk
Ze
(13)
The energy flow, which releases on the
cathode as a result of the neutralization of ions, is
equal:
qii =
ji
( Ei − Zeϕeff
Ze
)
(14)
where Ei – the ionization energy.
The flow of thermal conductivity into the
body of cathode is equal:
qλ = λ
dTk
dz

ji  
ZTe

− 2Tk  + ZeU d + Ei − Zeϕeff 
 k  2Ti +
Ze  
2


j
j
+ ec ( 2kTe + eϕeff ) − em ( 2kTk + eϕeff ) − hm
e
e
qλ =
(17)
In the equation of energy balance (17) are
not included the thermal radiant fluxes of plasma for
the cathode and the emissions from the surface of
cathode, or the flows of autoemission and secondary
emission of electrons due to the ion bombardment.
Our estimations shown that by the general
contribution of these flows (in comparison with the
rest) to the cathode of arc plasma torches it is
possible to be disregard.
Ionizing layer. The thickness of ionizing
layer is of the order of the length of recombination.
For the dense plasma in the arc discharge at
atmospheric pressure the basic mechanism of the
loss of the charged particles is triple (shockradiation) recombination with the participation of the
third body of electron [8]. In this case the length of
recombination can be evaluated as d =
(16)
Da
,
K r ne2
where Da - the coefficient of the ambipolar diffusion;
Kr – the coefficient of the triple recombination [811]. For Te = 10000 – 40000 K, d = 100 – 1000 µm.
In equation the energy balance of this layer
are included the following flows:
+ The energy flow into this layer by the
emissive electrons, accelerated in the layer of space
charge, is equal
jem
( 2kTk + eU d ) .
e
+ The work of the electric field on electrons
(15)
where λ - the thermal conductivity of the cathode
material.
The energy flow for the evaporation of the
cathode surface is equal:
qh = hm
where h – the latent heat of evaporation of the
- the mass flow rate of the
cathode material; m
evaporation of the cathode.
For the steady-state operating conditions of
the cathode, when the temperature of the cathode
and of the plasma do not change during this time
interval, energy balance on the surface of cathode
can be written down as:
inside the layer is equal
Ui =
jem − jec + j
U i . Where
2
kTe ne∞
ln
[3] - a voltage drop in the ionizing
e
nes
layer; ne∞ - the electron concentration on the outer
boundary of layer with the arc stream.
+ The energy flow of electrons into the layer
of space charge from ionizing layer, is equal
j ec
( 2kTe + eU d ) .
e
+ The energy flow of electrons into the post
of the arc discharge of plasma from ionizing layer is
equal 3, 2 jkTe / e . Coefficient 3,2 is sum of 5/2,
which considers the transfer of enthalpy because of
the electric current and the coefficient of thermal
diffusion, calculated for the strongly ionized plasma
[3].
+ Energy losses by electrons to the ionization
of atoms are equal
respectively for W and W (2% ThO2), Te = 20000 40000 K) the density of electric field E is ~ 105 - 106
V/cm (Fig. 1), that is justified the assumption about
the low current of autoemission and secondary
emission of electrons due to the ion bombardment.
ji Ei
; we disregard energy losses
Ze
by the elastic collisions with the heavy particles
inside the layer.
Thus, the equation of energy balance for the
ionizing layer takes the following form:

 n

jem eU d + 2kTk + kTe  ln e∞ − 3.2   =
 nes



 n

jec eU d + kTe  ln e∞ − 1.2   +
 nes


(18)
E

n 
+ ji  i + kTe  3.2 − 0.5ln e∞  
nes  

Z
3. Results and discussion
We present the results of the calculations of
the cathode region for the tungsten W and the
thoriated-tungsten W (2% ThO2) of the majority for
the arc plasma torches in argon [12]. Pressure in
both cases P = 1 atm.
The density of electric field on the cathode
is represented in Fig. 1 and it decreases with an
increase in the temperature of cathode and plasma.
This explains by the fact that with an increase in the
temperature of cathode (to the larger degree) or
plasma (to a lesser degree) increases the
thermoemission property of cathode, that for the
borrowing of the arc current it is not necessary of
large field strengthening.
Because of the improved property of
thermoemission the thoriated-tungsten filament has
this value of the density of electric field as in clean
tungsten cathode at much a lower temperature of
cathode (Tk = 2000 – 3200 K in comparison with Tk
= 3000 – 4400 K respectively). In the working
temperature range of the hot cathodes of arc plasma
torches (Tk = 3000 – 4400 K and Tk = 2000 – 3200 K
Fig. 1. Density of electric field on the surface of
cathode.
A voltage drop in the cathode region is the
important parameter with the selection of cathode
unit. The dependence of a voltage drop in the layer
of space charge Ud, in the ionizing layer Ui and in
the cathode region Uk from the temperature of the
cathode surface is given in Fig. 3. We can see that in
both cases for the hot cathodes a voltage drop in the
cathode region decreases with an increase in the
temperature and comprises order ~ 10-30 V, which
coincides with the data of the experiments of
different works [1,2]. For the majority of Arc plasma
torches at a temperature of cathode, close to the
melting point, a small voltage drop in the region near
the cathode can ensure a good current of the
thermoemission of arc. At a temperature of the
cathode of close one to the melting point, the
guarantee of a thermoemission current from the
cathode surface does not need a large voltage drop.
Fig. 2. Voltage drop in the layer of space charge Ud,
in the ionizing layer Ui and in the cathode region Uk..
Is extended opinion among the different
researchers about the fact that a voltage drop in the
cathode region, in essence in the space layer of
charge because of the high density of electric field
on the cathode. However, the results of our
calculation show that this is correct only at a
relatively low temperature of cathode (Fig. 1, 2). For
the temperature of the cathode of close one to the
melting point, a voltage drop in the ionizing layer
can be more than a voltage drop in the space layer of
charge. A voltage drop in the ionizing layer Ui
depends, in essence, from the temperature of plasma;
therefore in Fig. 1 it has constant value.
Fig. 3. Balance of current density on the cathode
surface: jem - current density of thermoemission from
the cathode; jec - current density of opposite
electrons to the cathode; ji - current density of ions to
the cathode; j - summary current density to the
cathode.
The balance of current density on the
surface of cathode is represented in Fig. 3. In the
working temperature range of both cathodes the
current density on the cathode composes ~ 102-104
A/cm2, which corresponds with the real work of arc
plasma torches. We can see that the current of
thermoemission has high portion in comparison with
the current of opposite electrons and the ion current.
With an increase in the temperature of cathode a
voltage drop in the layer of space charge decreases
and the current of opposite electrons rapidly grows.
For the melting point of the cathode Tk = 3700 К and
2920 К respectively for W and W (2% ThO2), the
current of opposite electrons composes 40-50% of
current of thermoemission. In the region of the small
temperature of cathode (for W, Tk < 3000 К and for
W (2% ThO2), Tk < 2200 К) the currents of
thermoemission and opposite electrons are small in
comparison with the ion current. This is possible
because of the high density of electric field on the
cathode (E > 106 V/cm) and to a large voltage drop
into the space charge layer (U > 60 V) (Fig. 1,2).
For this temperature range the cathode changes
gradually to the mode of operation of autoelectric
emission with the current density j < 100 A/cm2.
Fig. 4. Share of electron and ion currents on the
cathode.
The high share of ion current on the cathode
surface is a working regime of the arc plasma
torches of technological designation. The share of
electron (thermoemission and opposite electrons)
and ion current depending on the temperature of the
cathode surface with the different values of
temperatures of electrons is represented in Fig. 4. As
shows the balance of currents on the cathode surface
(Fig. 3) and the data about the shares of currents in
Fig. 4, for the range of small temperature (cold
cathode) the share of ion current even predominates
above electron current. In the working temperature
ranges of cathode, ion current can comprise to 30% 50% of total current on the cathode.
Fig. 5. Energy balance on the cathode surface: qem –
the heat flux of thermoemission from the cathode;
qec – the heat flux of opposite electrons to the
cathode; qi - summary of the ion heat flux to the
cathode, including energy of the ion bombardment,
energy of the neutralization of ions on the cathode
and energy of the atoms, which leave the cathode
surfaces; q - summary of the heat flux to the cathode.
Change in the share of currents from the
temperature of plasma (from 20000 to 40000 K) do
not exceed 20%, and, the more temperature of
plasma, the greater the share of electron current.
Thus, a good thermoemission of electrons from the
surface of electrode is ensured by high temperatures
of the cathode (closely to the melting point) and the
plasma (from 20000 to 40000 K) it is the working
condition of the cathodes of the arc plasma torches
for technological designation.
Energy balance on the cathode surface is
represented in Fig. 5. The result of calculation shows
that the heat flux to the cathode in essence depends
on the temperature of cathode and to a lesser degree
from the temperature of plasma. General heat flux to
the cathode in the working temperature range
composes ~ 4-7 kW/cm2, which coincides with the
experimental data of works [1, 2].
Should be noted the large role of the heat
flux of opposite electrons in the energy balance on
the cathode. The result shows that in the working
temperature range of the cathode, when the
temperature of cathode is more than the melting
point of material, energy of opposite electrons can be
more than energy of ion bombardment and energy of
thermoemission (Fig. 5). This explains by the fact
that with the introduction “hot” electrons of plasma
into the lattices of the cathode material are given off
their work function (eϕeff.) and kinetic energy (2kTe).
As a result, the general heat flux of the cathode
surface can be much more than the flux of ion
bombardment (Fig. 5).
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[2]
[3]
[4]
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[6]
[7]
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Fig. 6. Total power into cathode and radius of
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= 0; 3 – 0.6 cm; 4 – 1.2 cm; 5 – 6 cm [14], lc –length
of the water-cooled region of cathode.
In Fig. 6 are represented data of total power
into the cathode and a radius of cathode spot
depending on arc current. A good agreement of
calculated and experimental data makes it possible
grow prettier the applicability of this model of the
cathode region and method of integral balances for
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