22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Numerical simulation of heat transfer mechanisms in RF-ICP torches running
at low argon flow rates
A. Kashani1, J. Mostaghimi2, H. Badiei3 and K. Kahen3
1
2
Simulent Inc., Toronto, Ontario, Canada
MIE Department, University of Toronto, Toronto, Ontario, Canada
3
PerkinElmer Inc., Woodbridge, Ontario, Canada
Abstract: While in most ICP systems the plasma field is generated by passing RF current
through helical copper coils, we discuss the possibility of replacing coils with aluminium
flat discs that have many advantages over the traditional systems. In this paper, we present
new formulation, simulation and experimental results for the flat disc technology and study
possible torch devitrification associated with running such systems at low argon flow rates.
Keywords: RF plasma, LTE simulation, current induction disc, torch devitrification
1. Introduction
In conventional ICP systems, the plasma is generated
by a plasma gas flowing inside three concentric quartz or
fused silica tubes and passing RF currents through coils
wound around the torch (Fig. 1).
increase the devitrification process of quartz tubes and
negate the advantage of the new design.
To study the devitrification process and the new design
of the induction coils, the mathematical formulation of the
plasma must be revisited and the appropriate boundary
conditions have to be developed for the quartz torch.
2. Mathematical Formulation
The mathematical formulation of this work is based on
the work of Mostaghimi and Boulos [1] where plasma
flow assumed to be in LTE conditions, optically thin,
laminar, steady state and with negligible viscous
dissipation.
The flow and energy equations in vector are given by:
�⃗ = 0
∇. 𝑉
�⃗. ∇𝑉
�⃗ = −∇𝑝 + ∇. 𝜇∇𝑉
�⃗ + 𝐽⃗ × 𝐵
�⃗
𝜌𝑉
�⃗. ∇ℎ = ∇. �𝑘/𝑐𝑝 �∇ℎ + 𝐽⃗. 𝐸�⃗ − 𝑅̇
𝜌𝑉
Fig. 1. Schematic diagram of the induction plasma torch.
It is possible however to replace the copper coils with
aluminium flat discs and achieve comparable analytical
results. This replacement offers many advantages over
helical coil configuration. For instance, the large surface
area of flat discs is very efficient in dissipating heat
without the need for external cooling and as a result
requires considerably less maintenance. In addition, the
new design generates perfectly symmetrical induction
fields that are perpendicular to the flow minimizing the
common upward lift caused by traditional helical coils.
But perhaps the most interesting feature of the new
technology is that it may operate at low argon flow rates
(8-9 L.min-1) and reduces the operational costs
dramatically. However, the reduced flow rates could
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(1)
(2)
(3)
These equations are coupled with electromagnetic
equations through terms J×B and J.E.As shown in [1], a
vector potential formulation can be used to combine all
the Maxwell equations into a single equation such that:
∇2 𝐴𝜃 − 𝑖𝜇 0 𝜎𝜎𝐴𝜃 = 0
𝐸𝜃 = −𝑖𝑖𝐴𝜃
1𝜕
(𝑟𝐴𝜃 )
𝑟 𝜕𝜕
𝜕
𝜇0 𝐻𝑟 = − (𝐴𝜃 )
𝜕𝜕
𝜇0 𝐻𝑧 =
(4)
(5)
(6)
(7)
where A θ , E θ , H z , H r, ω=2πf , σ are vector potential,
electric and magnetic field intensities, frequency and
electrical conductivity respectively.
In Eqs. 4-7, A θ is a complex variable which requires to
be expanded into its real and imaginary parts for
numerical simulation.
1
𝐴𝜃 = 𝐴𝑅 + 𝑖𝐴𝐼
(8)
Eq. (1) requires boundary conditions for 2D numerical
domain. The boundary conditions are given by:
𝐴𝑟 = 𝐴𝐼 = 0;
𝜕𝐴𝑅 𝜕𝐴𝐼
=
;
𝜕𝜕
𝜕𝜕
𝜕𝐴𝑅 𝜕𝐴𝐼
=
;
𝜕𝜕
𝜕𝜕
𝑟=0
(9)
𝑧=0
(10)
𝑧 = 𝐿𝑡
(11)
However, at the torch walls where r=R 0 , the vector
potential is determined by superposition of induction coil
and current carrying regions of the plasma:
𝐴𝜃 = 𝐴𝜃−𝑐𝑐𝑐𝑐𝑐 + 𝐴𝜃−𝑝𝑝𝑝𝑝𝑝𝑝 ;
𝑟 = 𝑅0
(12)
The vector potential of coils can be approximated from
[2] by:
𝐴𝜃 (𝑅0 , 𝑧) =
𝑐𝑐𝑐𝑐
𝜇0 𝑅𝑐
𝐼� � 𝐺(𝑘𝑖 )
2𝜋 𝑅0
(13)
𝑖=1
where G(k) definition is given in [1].
Eq. (13) can be used to develop a new formulation for
current carrying disc whose inner and outer radius is R c2
Assuming that the
and R c1 respectively (Fig. 2).
differential vector potential is generated by a differential
current passing through the differential cross-section
(dR c ×δ) where δ is the skin depth we have:
𝑑𝐴𝜃 =
𝜇0
2𝜋
𝑅
𝑑𝐼 � 𝑐 G(k)
𝑟
𝑑𝐼 = 𝐽(𝑅𝑐 )𝛿𝑑𝑅𝑐
(14)
(15)
Assuming that a disc is made of current loops all
connected to the same voltage difference then:
𝐽⃗
∆𝑉 = − � . ���⃗
𝑑 = 𝑐𝑐𝑐𝑐𝑐
𝜎 𝑙
𝐼
𝐽(𝑅𝑐 ) =
𝑅
𝑅𝑐 𝛿 ln � 𝑐2 �
𝑅𝑐1
(16)
(17)
By substituting Eq. [17] and [15] into Eq. (14) and
integrating over the disc radius, the vector potential
expression for a current carrying disc is given as:
𝐴𝜃 (𝑅0 , 𝑧) =
𝑅𝑐2 𝑅
𝜇0
𝑐
𝐼 � � 𝐺(𝑘)
2𝜋 𝑅𝑐1 𝑅0
𝑑𝑅𝑐
𝑅
𝑅𝑐 ln � 𝑐2 �
𝑅𝑐1
(18)
Large industrial torches are externally cooled in most
applications, thus the assumption of constant wall
temperature on the outer surface of such torches is valid.
However, smaller torches especially those typically used
in ICP-MS and ICP-OES are not externally cooled. This
becomes critical especially at lower plasma gas flow rates
(e.g., <12 L.min-1).
As the torch wall temperature increases with reduced
plasma gas flow rate, heat transfer through radiation
becomes significant. Thus the constant wall temperature
equation should be revised to consider the radiation
effect.
ℎ𝑡𝑡𝑡 𝐴𝑠 (𝑇 − 𝑇∞ ) = −𝑘
ℎ𝑡𝑡𝑡 = ℎ𝑛𝑛 + ℎ𝑟𝑟𝑟
𝜕𝜕
;
𝜕𝜕
𝑟 = 𝑅0
(19)
(20)
Here, h nc and h rad correspond to natural convection and
radiation heat transfer coefficient respectively.
The average natural heat transfer coefficient is
approximated by the correlation for the free convection
flow over a horizontal cylinder [3]. The radiation heat
transfer is however an order of magnitude larger than the
natural convection mechanism as will be shown later.
3.
Fig. 2. Schematic diagram of a current carrying disc.
The task is therefore reduced to find an expression for
the distribution of current density on the induction disc.
2
Numerical Simulation and Results
Torch failure due to over-heating and devitrification can
be costly and will have both functional and performance
consequences. Hence, it is necessary to understand the
physical phenomena and be able to predict the onset of
devitrification particularly at reduced argon flow rates.
The numerical computation were performed on a two
dimensional domain with 48 and 60 non-uniform control
volumes in R and Z directions respectively.
The geometrical dimensions of the torch are
R 1 = 1.25 mm, R 2 = 2.25 mm, R 3 = 8 mm, R 0 = 9 mm,
L t = 37 mm, torch thickness = 2 mm, R c1 = 11.1 mm,
R c2 = 21.3 mm.
Disc thickness = 2 mm and two flat discs are placed at 4
and 14.85 mm from the torch inlet (Figs. 1 and 2). The
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injector and auxiliary flow rates were kept at 0.2 and
0.7 L.min-1 respectively, however the plasma flow rate
was changed from 9 to 15 L.min-1 to study the effect of
flow rate on the torch wall temperatures. All simulations
were performed at atmospheric pressure and at two
plasma power levels (1.0 and 1.5 kW) and frequencies (27
and 40 MHz).
The numerical simulation can provide detailed
information of the plasma flow, temperature and
electromagnetics field inside the torch. For instance,
Fig. 3 depicts the results of simulated temperature and
velocity profiles inside the torch. Based on the results,
under given conditions, plasma temperatures reach as
high as 10400 K. Plasma temperatures before the first
disk (i.e., z < 4 mm) are relatively low; however, plasma
temperature increases rapidly in the region between the
two disks (z > 4 mm). The gas velocity at its maximum
reaches ~29 m.s-1 but the flow remains laminar due to
high gas temperatures and lower gas density. Fig. 4
presents the electric and the axial magnetic field
intensities inside the torch. The field intensities are the
strongest in between the two disks and also closer to the
torch walls. The field intensities, however, decrease as
they near the torch centreline.
0.01
10000
80
500
0
3000
00
1000
0.005
R
10342.8
0
28
6
2
8
16
-0.005
0
0.02
0.015
0.01
0.005
22
24
26
18
12
0.025
0.03
the torch. Fig. 5 compares the effective ratio of the
radiation to natural convection heat transfer coefficient
along the torch outer surface. Based on the simulation
results, the ratio is initially small at the inlet and decreases
slightly up to z =20 mm, but then the ratio grows larger
until it reaches a maximum at around z = 26 mm and
decreases again beyond that point. Overall, the radiation
heat transfer is on average an order of magnitude larger
than the natural convection heat transfer.
The effect of input power and plasma flow rates on the
temperature of the inner and outer wall of the torch is
shown in Fig. 6 for a plasma frequency of 40 MHz. As
can be seen from the results, the difference between outer
and inner torch temperatures grows larger by increasing
the input power (red versus purple lines) and (blue versus.
green lines) for the same flow rates. However this
difference is more pronounced if the plasma flow rate is
reduced from 15 to 9 L.min-1 (red versus blue lines) and
(purple vs. green lines). In addition, for 15 L.min-1 flow
rate, the devitrified length corresponding to temperatures
above devitrification temperature (1423 K) is relatively
small and negligible while this length reaches to 9-16 mm
for input power of 1.0 and 1.5 kW respectively.
Similarly, Fig. 7 compares the effect of plasma frequency
on torch temperature profiles. While lowering the plasma
flow rate increases the difference between inner and outer
torch wall temperatures, changing the frequency has
virtually now effect on this temperature difference.
However, a temperature rise of approximately 100 K can
be seen for inner and outer torch wall temperatures by
increasing frequency from 27 to 40 MHz. Moreover, the
change in plasma frequency has very little effect on the
devitrification length shown in Fig. 7.
0.035
Z
Fig. 3. Temperature in [K] (up) and velocity magnitude
in [m.s-1] (down) inside a torch running at P = 1.5 kW,
f = 40 MHz and Qplasma = 9 L.min-1.
25
20
hrad/hnc
0.01
3500
3000
2500
2000
0
150
00
10
0.005
R
50
15
0
10
0
1500
10
50
0
00
2500
5
-0.005
3500
3000
0
0.005
4000
0.01
4500
0.015
0.02
0.025
0.03
0.035
Z
Fig. 4. Electric field intensity in [V/m] (up) and axial
component of magnetic field intensity in [A/m] (down)
for the same flow conditions as Fig. 3.
In the absence of external cooling systems, the only
heat dissipation mechanisms on the torch outer surface are
radiation and natural convection heat transfer. However
the two mechanisms are not equally efficient in cooling
P-II-12-8
0.01
0.02
0.03
z [m]
Fig. 5. Ratio of radiation to natural convection heat
transfer coefficient along the torch outer surface for
f = 40 MHz, P = 1.5 kW and argon flow rate of
Q plasma = 9 L.min-1.
3
5. Acknowledgement
This work has been financially and technically
supported by PerkinElmer Inc. The authors gratefully
acknowledge this collaboration and support.
Ti, P=1.5 kW, Qaux=15 lpm
To, P=1.5 kW, Qaux=15 lpm
Ti, P=1.5 kW, Qaux=9 lpm
To, P=1.5 kW, Qaux=9 lpm
Ti, P=1.0 kW, Qaux=15 lpm
To, P=1.0 kW, Qaux=15 lpm
Ti, P=1.0 kW, Qaux=9 lpm
To, P=1.0 kW, Qaux=9 lpm
Devitrification Limit
2200
2000
1800
T [K]
1600
1400
1200
f=40 MHz
1000
Qinj = 0.7 lpm
Qp = 0.2 lpm
Devitrified length
800
600
400
0.005
0.01
0.015
0.02
0.025
0.03
References
[1] J. Mostaghimi and M.I. Boulos. “Two Dimensional
Electromagnetic Field Effects in Induction Plasma
Modeling”. Plasma Chem. Plasma Phys., 9 (1989)
[2] J.D. Jackson. Classical Electrodynamics. (New
York: Wiley) (1962) / Part II (P.W.J.M.
Boumans; Ed.) (New York: Wiley) (1987)
[3] S.W. Churchill and H.H.S. Chu. “Correlating
Equations for Laminar and Turbulence Free
Convection from a Horizontal Cylinder”. Int. J.
Heat Mass Transfer, 18 (1975)
0.035
z [m]
Fig. 6. Effect of input power and plasma flow rates on
inner and outer surface temperatures of a torch running at
f = 40 MHz.
2200
Ti, f=40 MHz, Qaux=15 lpm
To, f=40 MHz, Qaux=15 lpm
Ti, f=27 MHz, Qaux=15 lpm
To, f=27 MHz, Qaux=15 lpm
Ti, f=40 MHz, Qaux=9 lpm
To, f=40 MHz, Qaux=9 lpm
Ti, f=27 MHz, Qaux=9 lpm
To, f=40 MHz, Qaux=9 lpm
Devitrification Limit
2000
1800
T [K]
1600
1400
1200
1000
P = 1.5 kW
Devitrified length
Qinj = 0.7 lpm
Qp = 0.2 lpm
800
600
400
0.005
0.01
0.015
0.02
0.025
0.03
0.035
z [m]
Fig. 7. Effect frequency on inner and outer surface
temperatures of a torch running at P = 1.5 kW.
4. Conclusion
In this paper, we developed the necessary mathematical
formulation for current carrying induction discs and
revisited the boundary conditions on the torch outer
surface to account for radiation and natural convection
cooling effects.
The numerical simulation results
indicated that the radiation heat transfer mechanism is
more efficient in dissipating heat from the torch outer
surface by up to 26 times.
The inner and outer surface temperatures are
significantly affected by reducing the plasma flow rate
and input power whereas the effect of the plasma
frequency was evidently negligible.
4
P-II-12-8