22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Boundary conditions and heat flux to the wall for modeling the anode
attachment in two temperature thermal plasma approximation
L. Pekker and N. Hussary
Victor Technologies, West Lebanon, NH, U.S.A.
Abstract: A new set of boundary conditions at the wall for the electron and heavy particle
energy equations and the electrical potential in 2T thermal plasma modeling are derived,
which take into account the plasma sheath formed at the wall and the thermionic electron
emission at the hot electrodes. These boundary conditions allow the self consistent
calculation of the heat flux to the walls and consequently the thermionic electron current.
Keywords: thermal plasmas, boundary conditions, thermionic electron emission
1. Introduction
The formation of the plasma sheath at walls plays a
fundamental role in the heat flux to the wall from the
plasma, thermionic emission, the structure of the cathode
spot and anode attachment, electrode erosion process, and
other electrode processes. As has been pointed out in
review [1], the two temperature hydrodynamic thermal
plasma models reasonably well describe the arcs, however
one fundamental question remains open: what boundary
conditions should be used at the electrodes which would
take into account the plasma sheath formed at the wall?
In previous studies of high-pressure plasmas (the plasma
pressure is as large as or larger than atmospheric pressure)
different sets of boundary conditions at the wall were
constructed disregarding the sheath formed at the wall,
see [2, 3] and references in [4], or taking the sheath into
account using imposed numerical conditions and
procedures, see for example [5].
In [4], the authors constructed the sets of boundary
conditions for the electron and heavy particle energy
equations at the floating and biased walls for the case of
cold walls with no thermionic electron emission or
erosion of the wall. These boundary conditions consider
the plasma sheath formed at the wall as the interface
between the plasma and the wall (Godyak's collisional
sheath model was employed [6, 7]). In [4] it was
demonstrated that using theses boundary conditions in
modeling plasma cutting arcs may lead to much larger
heat fluxes to the wall and to significantly cooler arcs
compared to the models that ignore the sheath at the wall.
Using the method [4], a set of boundary conditions for
the case of hot walls with thermionic electron emission
was constructed [8]. In the derived boundary conditions,
the walls are assumed to be made from refractory metals
and that the erosion of the wall is small and, therefore, is
not taken into account in the model. The derived
boundary conditions allow the calculation of the heat flux
to the walls from the plasma, and consequently the
thermionic electron current, that makes the two
temperature thermal model with the suggested boundary
conditions self consistent. In [8] we also consider the
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case of the virtual cathode that forms at super high
thermionic electron emission rates and the obtained
boundary conditions were applied for zero dimension
model of the cathode spot.
The obtained results
confirmed the significance of incorporating the Schottky
correction factor for calculating thermionic electron
emission in modeling plasma cutting arcs.
In the present work we apply the boundary conditions
derived in [8] to a zero dimensional model of anode
attachment assuming that the wall is cold and therefore
there is no thermionic electron emission or erosion of the
wall. This is done for a singly ionized argon plasma.
2. The anode sheath electrical potential drop
As in [4, 8], we consider the plasma sheath as the
interface between the wall and the plasma and assume
that 𝑇𝑒 ≫ 𝑇ℎ = 𝑇𝑠𝑠𝑠 , where 𝑇𝑒 and 𝑇ℎ are the temperature
of electrons and heavy particles in the vicinity of the wall,
and 𝑇𝑠𝑠𝑠 is the temperature of the wall surface. Because
the ions recombine with electrons at the wall and make
their way back to plasma as neutrals where they are
immediately ionized by electrons, the temperature of
heavy particles at the plasma wall interface (at the sheath)
was taken as 𝑇ℎ = 𝑇𝑠𝑠𝑠 .
Following [4], the anode potential drop across the
anode sheath is
𝜑𝑎𝑎𝑎𝑎𝑎 = −
𝑘𝐵 𝑇𝑒
2𝑒
𝑙𝑙 �
2𝜋𝑚𝑒
𝜋𝑟𝐷𝐷
𝑀�1+
�
2𝜆𝑖−𝑚𝑚𝑚
𝑗 2
�1 + � �,
𝑗𝑖
(1)
where the ion current density in the sheath [5, 6] is
𝑘𝐵 𝑇𝑒
𝑗𝑖 = 𝑒𝑛𝑝 𝑉𝑠 = 𝑒𝑛𝑝 �
𝑀
/ �1 +
𝜋𝑟𝐷𝐷
2𝜆𝑖−𝑚𝑚𝑚
�,
(2)
𝑛𝑝 is the plasma number density, 𝑉𝑠 is the ion velocity at
which the ions enter the sheath [6, 7], 𝜆𝑖−𝑚𝑚𝑚 = 1/𝑛𝑛 𝜎𝑖,𝑛
is the ion transport mean free path, and 𝜎𝑖,𝑛 is the chargeexchange cross-section (the dominated ion-neutral
momentum transfer process in the sheath [6, 7]), 𝑛𝑛 is the
number density of neutrals, 𝑗 is the total current density
1
at the wall, 𝑟𝐷𝐷 = �𝜀0 𝑘𝐵 𝑇𝑒 /𝑛𝑝 𝑒 2 is the electron Debye
radius, and 𝑀 is the mass of a heavy particles. In
derivation of Eqs. (1) and (2) it was assumed that
exp(−𝑒𝜑𝑎𝑎𝑎𝑎𝑎 /𝑘𝐵 𝑇𝑒 ) ≪ 1.
3. A boundary condition for the anode attachment
The enthalpy flux from the plasma to the wall due to
the charge particles [4, 8] can be written as
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑞𝑐ℎ𝑎𝑎𝑎𝑎𝑎 = 𝑒𝑛𝑝 𝑉𝑠 �𝐼𝑖𝑖𝑖𝑖𝑖 + 𝜑𝑎𝑎𝑎𝑎𝑎 +
+2𝑘𝐵 𝑇𝑒 𝑛𝑝 𝑒𝑒𝑒 �−
𝑒𝑒𝑎𝑎𝑎𝑎𝑎
𝑘𝐵 𝑇𝑒
��
𝑘𝐵 𝑇𝑒
𝑀𝑉𝑠 2
,
2𝜋𝑚𝑒
2𝑒
�+
(3)
where 𝐼𝑖𝑖𝑖𝑖𝑖 is the ionization potential of gas atoms.
Since 𝑘𝐵 𝑇ℎ ≪ 𝑒𝜑𝑎𝑎𝑎𝑎𝑎 , in Eq. (3) we have neglected the
ion thermal heat flux to the wall. Taking into account the
energy flux that the anode gains due to the "condensation"
energy of electrons at the wall, we obtain that the total
heat flux to the wall due to the charge particles [4] can be
written as
𝑝𝑎𝑟𝑟𝑟𝑟𝑟𝑟𝑟
𝑄𝑐ℎ𝑎𝑎𝑎𝑎𝑎
=
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑞𝑐ℎ𝑎𝑎𝑎𝑎𝑎
+ 𝑗�𝜑𝑤𝑤 − ∆𝜑𝑆𝑆ℎ𝑜𝑜 �,
(4)
where 𝜑𝑤𝑤 is the work function of the anode material,
∆𝜑𝑆𝑆ℎ𝑜𝑜 = (−𝑒𝐸𝑠𝑠𝑠 /4𝜋𝜀0 )1/2 ,
(5)
is the Schottky correction factor, and
𝐸𝑠𝑠𝑠 =
+
2𝑀𝑉𝑠2
𝑘𝐵 𝑇𝑒
𝑘𝐵 𝑇𝑒
𝑒𝑟𝐷𝐷
�2𝑒
��1 +
𝑒𝜑𝑎𝑎𝑎𝑎𝑎
𝑘𝐵 𝑇𝑒
−
2𝑒𝜑𝑎𝑎𝑎𝑎𝑎
𝑀𝑉𝑠2
+
1/2
− 1� − 1�
.
(6)
is the electric field at the surface of the wall [8]. Thus, at
a given plasma composition, 𝑛𝑝 and 𝑛𝑛 , 𝑇𝑒 , and 𝑗 the heat
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
flux to the wall due to the charge particles 𝑄𝑐ℎ𝑎𝑎𝑎𝑎𝑎 , Eq.
(4), can be calculated.
Because of the continuity of the enthalpy flux through
the sheath, the boundary conditions for the heat flux at the
anode can be written as
−𝜅𝑛
𝜕𝑇ℎ
𝜕𝜕
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
+ 𝑄𝑐ℎ𝑎𝑎𝑎𝑎𝑎 = −𝜅𝑤
𝜕𝑇𝑤
𝜕𝜕
− 𝑅𝑅𝑅,
(7)
where 𝜕𝜕/𝜕𝜕 is the space derivative of 𝑇 normal to the
wall. The first term in the left hand side of Eq. (7) is the
heat flux of the neutral gas molecules to the wall, the first
term on the right hand side of this equation is the heat flux
in the anode, 𝑅𝑅𝑅 is net radiation heat flux of the wall,
and 𝜅𝑛 and 𝜅𝑤 are the thermal conductivities of the
neutrals and the anode material respectively.
In Section 4 we will analyse the anode attachment in
terms of heat transfer to the anode due to the charged
particles assuming that the electron plasma temperature,
the plasma pressure, and the anode surface temperature
2
are given.
The thermal heat flux of the neutral particles to the
anode, the first term in the left hand side of Eq. (7),
cannot be calculated in the frame of the present model,
and therefore is not considered here. According to [2] the
heat flux contribution from neutrals can range from 20%
to 60% depending on the anode current density and
geometry.
4. Numerical results
Let us consider the anode attachment for a free burning
arc in atmospheric pressure singly ionized argon, Fig. 1.
Following [9], the plasma composition is determined by
solving the Saha equation with 𝑇𝑒 at given plasma
pressure 𝑃 and given temperature of heavy particles 𝑇ℎ :
𝑛𝑒 2
𝑛𝑛
= 2�
2𝜋𝑚𝑒 𝑘𝐵 𝑇𝑒 3/2 𝑄𝐴𝐴+ (𝑇𝑒 )
𝑒𝐼
�
𝑒𝑒𝑒 �− 𝑖𝑖𝑖𝑖𝑖�
ℎ2
𝑘𝐵 𝑇𝑒
𝑄𝐴𝐴 (𝑇𝑒 )
= 2.89 × 1022 𝑇𝑒 3/2 𝑒𝑒𝑒 �−
1.827×105
𝑇𝑒
𝑃 = 𝑘𝐵 (𝑛𝑒 + 𝑛𝑛 )𝑇ℎ + 𝑘𝐵 𝑛𝑒 𝑇𝑒 ,
�,
=
(8)
(9)
where 𝑛𝑛 is the density of neutral argon atoms, 𝑛𝑒 is the
electron number density which is equal to 𝑛𝑝 , and
𝑄𝐴𝐴 + (𝑇𝑒 ) and 𝑄𝐴𝐴 (𝑇𝑒 ) are the statistical sums of partition
functions of argon ions and argon neutral atoms
respectively. Two assumptions were made in Eqs. (8) and
(9): (1) the contributions of the excited states to the
statistical sums 𝑄𝐴𝐴 + and 𝑄𝐴𝐴 are less than 5 percent [10],
and therefore, have been neglected in Eq. (8); (2) because
the number densities of multi-charged ions are many
orders of magnitude smaller than the number density of
singly ionized argon, multi-charged ions are ignored in
this model.
The results of simulations of the above model are
presented in Figs. 1 - 7. In this simulation we chose
𝜑𝑤𝑤 = 4.53𝑒𝑒 that corresponds to the copper anode
material, 𝑃 = 105 𝑃𝑃, 𝑇ℎ = 𝑇𝑠𝑠𝑠 = 103 𝐾, the 𝑇𝑒 range of
8000 - 14000K, and the 𝑗 range of 5⋅105 - 107 A/m2 which
are typical for free burning arcs [11].
𝑛𝑝
𝑛𝑝 +𝑛𝑛
𝑇𝑒 [𝐾]
Fig. 1. Ionization ratio.
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The ionization ratio, 𝑛𝑝 /(𝑛𝑝 + 𝑛𝑛 ), and the collision
factor, 1 + 𝜋𝑟𝐷𝐷 /2𝜆𝑖−𝑚𝑚𝑚 , are shown in Figs. 1 and 2,
respectively. As follows from Fig. 1 with an increase in
𝑇𝑒 the plasma becomes more ionized as expected. This
leads to a decrease in 𝑟𝐷𝐷 and an increase 𝜆𝑖−𝑚𝑚𝑚 and
consequently to a decrease in the collision factor: for
small electron temperatures the plasma is collisional and
for large collisionless.
2.5
2.0
1.5
1.0
8000
10000
12000
Fig. 2. Collisional factor 1 +
𝜋𝑟𝐷𝐷
2𝜆𝑖−𝑚𝑚𝑚
Figs. 3 and 4 show the ratio of 𝑗/𝑗𝑖 and 𝑒𝜑𝑎𝑎𝑎𝑎𝑎 /𝑘𝐵 𝑇𝑒
vs. 𝑇𝑒 for different anode current densities. As follows
from Fig. 3 the ratio of 𝑗/𝑗𝑖 decreases with an increase in
𝑇𝑒 that was expected because 𝑗𝑖 is dependent of 𝑇𝑒 only,
Eq. (2). Consequently, this leads to an increase in the
anode sheath drop, Eq. (1), with an increase in 𝑇𝑒 and a
decrease in 𝑗. The slight decrease in 𝑒𝜑𝑎𝑎𝑎𝑎𝑎 /𝑘𝐵 𝑇𝑒 with
an increase in 𝑇𝑒 for small current densities is explained
by a decrease in collisional factor with an increase in 𝑇𝑒 ,
Fig. 2, which becomes more profound when 𝑗𝑖 ≫ 𝑗, Eq.
(1).
Fig. 5 presents the Schottky decrease in the work
function of the copper anode ∆𝜑𝑆𝑆ℎ𝑜𝑜 due to the strong
electric field at the wall surface 𝐸𝑠𝑠𝑠 , Eq. (5). As was
expected, with an increase in the plasma electron
temperature the electron Debye length decreases that
leads to a decrease in the thickness of the sheath and as a
result to an increase in 𝐸𝑠𝑠𝑠 and ∆𝜑𝑆𝑆ℎ𝑜𝑜 . As one can see
from Fig. 4, ∆𝜑𝑆𝑆ℎ𝑜𝑜 reaches up to 0.5 eV at high plasma
electron temperatures.
14000
vs. 𝑇𝑒 [K]
Red
𝑗 = 107 𝐴/𝑚2
Purple 𝑗 = 5 ∙ 106 𝐴/𝑚2
Green 𝑗 = 106 𝐴/𝑚2
Blue
𝑗 = 5 ∙ 105 𝐴/𝑚2
Red
𝑗 = 107 𝐴/𝑚2
Purple 𝑗 = 5 ∙ 106 𝐴/𝑚2
Green 𝑗 = 106 𝐴/𝑚2
Blue
𝑗 = 5 ∙ 105 𝐴/𝑚2
Fig. 5. ∆𝜑𝑆𝑆ℎ𝑜𝑜 [eV] versus 𝑇𝑒 [K] for different 𝑗.
Fig. 3. 𝑗/𝑗𝑖 versus 𝑇𝑒 [K] for different 𝑗.
Red
𝑗 = 107 𝐴/𝑚2
Purple 𝑗 = 5 ∙ 106 𝐴/𝑚2
Green 𝑗 = 106 𝐴/𝑚2
Blue
𝑗 = 5 ∙ 105 𝐴/𝑚2
Fig.4. 𝑒𝜑𝑎𝑎𝑎𝑎𝑎 /𝑘𝐵 𝑇𝑒 versus 𝑇𝑒 [K] for different 𝑗.
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Fig. 6 presents the contributions of ion, electron, and
condensation energy fluxes in the total energy flux
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑄𝑐ℎ𝑎𝑎𝑎𝑎𝑎 : the first and the second terms in the right hand
side of Eq. (3) and the second term in the right hand side
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
of Eq. (4) respectively; 𝑄𝑐ℎ𝑎𝑎𝑎𝑎𝑎 is shown in Fig. 7.
As follows from Fig. 6 with an increase in 𝑇𝑒 the
contribution of ion energy flux to the total energy flux to
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
the wall, 𝑄𝑐ℎ𝑎𝑎𝑎𝑎𝑎 , increases asymptotically to ~ 0.85.
This can be explained by the fact that with an increase in
𝑇𝑒 the plasma becomes more ionized and the ion current
to the wall increases very rapidly, Eq. (2). As plasma
becomes well (almost full) ionized, the rate of increase of
𝑛𝑝 with temperature decreases and becomes negative for
large 𝑇𝑒 because the plasma pressure in the model is
assumed to be constant, Eq. (9).
Although, the energy flux of electrons to the wall
increases with an increase in 𝑇𝑒 , its relative contribution
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
to 𝑄𝑐ℎ𝑎𝑎𝑎𝑎𝑎 , Fig. 6, decreases because the ion energy flux
to the wall increases very rapidly with an increase in 𝑇𝑒 .
3
Red
𝑗 = 107 𝐴/𝑚2
Purple 𝑗 = 5 ∙ 106 𝐴/𝑚2
Green 𝑗 = 106 𝐴/𝑚2
Blue
𝑗 = 5 ∙ 105 𝐴/𝑚2
Fig. 6. Contributions of electron (broken lines), ion (thick
solid lines) and condensation (thin solid lines) energy
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
fluxes in the heat flux to the anode 𝑄𝑐ℎ𝑎𝑎𝑎𝑎𝑎 [𝑊/𝑚2 ]
versus 𝑇𝑒 [K] for different 𝑗.
1.E+10
1.E+09
Red
𝑗 = 107 𝐴/𝑚2
Purple 𝑗 = 5 ∙ 106 𝐴/𝑚2
Green 𝑗 = 106 𝐴/𝑚2
Blue
𝑗 = 5 ∙ 105 𝐴/𝑚2
1.E+08
1.E+07
8000
10000
12000
14000
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
Fig. 7. 𝑄𝑐ℎ𝑎𝑎𝑎𝑎𝑎 [𝑊/𝑚2 ] versus 𝑇𝑒 [K] for different 𝑗.
The condensation energy flux to the wall, 𝑗�𝜑𝑤𝑤 −
∆𝜑𝑆𝑆ℎ𝑜𝑜 � decreases with an increase in the plasma
electron temperature because ∆𝜑𝑆𝑆ℎ𝑜𝑜 increases with an
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
increase in 𝑇𝑒 , Fig. 5. Its contribution to 𝑄𝑐ℎ𝑎𝑎𝑎𝑎𝑎 at
small 𝑇𝑒 is relatively large, Fig. 6, because the ion current
density is much smaller than the total current density at
the anode, Fig. 3. With an increase in 𝑇𝑒 the ratio 𝑗/
𝑗𝑖 decreases, Fig. 3, leading to a decrease in the
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
contribution of the condensation energy flux to 𝑄𝑐ℎ𝑎𝑎𝑎𝑎𝑎 .
As follows from Fig. 7, the heat flux to the anode
profoundly increases with an increase in the total anode
current density at relatively small electron temperatures,
however, at high values of 𝑇𝑒 , it becomes almost
independent on 𝑗. As it has been mentioned above, this
observation is explained by a significant contribution of
the condensation energy flux to the total energy flux to
the wall, Eq. (4), at relatively small 𝑇𝑒 and the decrease of
this contribution with an increase in 𝑇𝑒 where the ion
current density becomes larger than 𝑗.
4
5. Conclusion
The boundary conditions in [4, 8] are applied to the
anode attachment. This allows for calculating the
contribution of electron, ion, and condensation heat flux
to the cold anode with no thermionic electron emission or
erosion of the wall. It was demonstrated that for
relatively small anode current densities the ions play
major role in heating the anode, while in the case of high
anode current densities the condensation heat flux to wall
can play a significant role in heating of the anode for
relatively small electron plasma temperatures.
The model also allows to self consistently calculate the
Schottky decrease in anode material work function.
Combining the suggested anode attachment model with
a model of heat transfer in the anode along with a two
temperature thermal plasma model will allow to self
consistently calculate the heat transfer in the anode for
real arc geometries.
6. Acknowledgments
The authors would like to express their gratitude to
J. Nowak for discussions of results and his kind help in
preparation the text of this paper.
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