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MODELLING OF A HIGH-POWER TRANSFERRED ARC. PART I : THE PLASMA JET
J. Gonzalez, A. Gleizes, S. Vacquie, P. Brunelot
To cite this version:
J. Gonzalez, A. Gleizes, S. Vacquie, P. Brunelot. MODELLING OF A HIGH-POWER TRANS-
FERRED ARC. PART I : THE PLASMA JET. Journal de Physique Colloques, 1990, 51 (C5), pp.C5-
221-C5-228. �10.1051/jphyscol:1990527�. �jpa-00230834�
MODELLING OF A HIGH-POWER TRANSFERRED ARC. PART I : THE PLASMA JET
J.J. GONZALEZ, A. GLEIZES, S. VACQUIE and P. BRUNELO?'
Laboratoire DBcharges dans les Gaz, U R A n o 277, CPAT, Universite Paul Cabatier, 118 route de Narbonne. 31062 Toulouse Cedex, F ~ d n C e
SocibtB ABrospatiale, Etablissement dlAquitaine, BP 11, 33165 Saint-MBdard en Jalles Cedex, France
RCsumC
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Nous presentons une modklisation globale de la zone de jet cathodique d'un arc transfkre d'une puissance de l'ordre de 1 MW, dans l'air. Les r6ultats montrent l'influence de plusieurs parambtres extkrieurs sur les caractkristiques du jet d'arc : intensitk du courant entre 500 et 1500 A ; dCbit de gaz entre 10 et 50 g/$ ; injection en vortex. Le comportement de l'arc est analysC en fonction des phknomknes physiques tels que le rayonnement et la turbulence.Abstract
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A physical modelling of the cathode jet zone of a 1 MW transferred arc, in air, is presented. We show the influence of several parameters on plasma jet properties : current intensity between 500 and 1500 A ; gas mass flow rate between (0 and 50 gfs) ; vortex injection. Through a few examples we explain the role of various physical mechanisms (radiation, turbulence) on arc characteristics.1
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INTRODUCTIONThe company "Aerospatiale" is working on a device based on a transferred arc with a power of one MW. In parallel we have developed a physical modelling of this arc. Schematically the arc can be divided into three zones : -just after the upstream electrode (generally the cathode) there is a water-cooled tube of fixed length which stabilizes the arc (constricted arc) ;
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following this tube there is a region of arc without walls, governed by the cathode jet, which is several tens of centimeters long ;-
finally, near the anode the arc is influenced by both the flow from the cathode and by the presence of the anode itself.From the modelling point of view, the equations and boundary conditions in the first two zones give rise to a system of parabolic equations : the characteristics of the arc at point z only depend on the values of the variables at point z - h . This is no longer true in the third zone where numerical modelling requires a so-called "elliptical" code with a fixed grid of the region to be studied and multiple iterations in the resolution of the equations. Rather than setting up a general elliptical code valid for all the regions of the transferred arc, we chose to divide the modelling into 3 distinct successive codes.
In the present paper we shall study the modelling of the first two zones of the arc. The injccted gas is air and the jet of plasma at the outlet of the stabilization tube reaches the air of the atmosphere where cold gas is drawn into the plasma and mixes with it. In comparison with studies of the litterature (see for example 11-24, our modelling is applied to an arc of total length 1 m burning in air with a strong gas flow (several tens of g/s) and high power (about 1 MW). We are going to analyse the properties of the arc in function of the physical phenomena and of the macroscopic parameters such as the gas flow and currcnt intensity.
It is assumed that the plasma is in thermal equilibrium and that the arc is cylindrically symmetrical. The unknowns are the pressure p, enthalpy h and the 3 components of the velocity : radial U, azimuthal v and axial W. The conservation equations (ignoring the terms of the axial gradients compared to those of the radial gradients and assuming that U << W) are written :
mass :
radial momentum :
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990527
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azimuthal momentum :
axial momentum :
energy : the total enthalpy is the sum of the enthalpy of the resting fluid and of the kinetic energy :
where p is the mass density, p' the total viscosity, sum of the normal larninar viscosity p and of the turbulent viscosity pt, G the electric conductivity, E the electric field, $ the thermal potential, qr the radiation radial flux and qt the turbulent energy radial flux. The magnetic pressure term is not taken into account because the limit conditions near the cathode are given by a radial enthalpy profile. The thermal potential $ is defined by :
where K is the larninar thermal conductivity and T(r) the temperature at a given point r.
Turbulence is taken into account using Prandtl's approximation. We define a mixing length 1 and then the turbulent viscosity E is given by :
The turbulent energy flux is :
where Prt is the Prandtl number. Variables 1 and P* describe the turbulent flow and are calculated from semi- empirical relationships which come from the paper by Nicolet et a1 131.
In the constricted arc, if R is the radius of the tube, we have :
in the central region, whereas near the wall, the mixing length is a function of the roughness of the wall 131. The Prandtl number also depends on the radial distance r and varies from 0.5 in the axis to 3 near the wall.
In the free arc jet the radius of the jet R is defined such that the enthalpy inside the jet is greater than a chosen value. The mixing length is given by Eq. (9). For r < R the Prandtl number varies between 0.5 and 0.95. For r >
R, P n is fixed at 0.95.
To determine the radiation flux qr, we used the same procedure as Nicolet and al/3/. It is hypothesized that the radiation can be expressed as a continuum. Also, for fixed temperature and pressure, it is assumed that the absorption coefficient can only take certain values depending on the wavelength : a certain number of frequency (or wavelength) ranges were therefore defined and in each range the absorption coefficient remained constant.
The radiative transfer equation is resolved for every range to obtain the flux q ~ v over this range which is a complicated function of the optical depth.
upstream pressure and the diameter of the tube in the constricted zone. The discretization method used was similar to that developed by Nicolet et al/3/ and generalized to take vortex injection into account 141. The method is based on finite differences over a radial grid with constant step Ar except between the first two and the last two points where the step has the value Ar
2-
The material functions for air are taken from the results of Nicolet et all31 and Bacri et a1 151 and are functions of temperature and pressure. For radiation a specific study of the absorption coefficient in the air was performed and is presented in ch. IV. Finally, to link the current intensity I to the electric field E we used Ohm's law assuming that E is constant over one section of the arc :
I = 2 n E r o d r = E.G
0
G being the lineic conductance of the arc.
The axial step Az is not constant. It is very small near the upstream limit section (for every zone) and becomes liarger until it reaches a maximum value fixed by the code. This procedure stabilizes the solution for the first axial iterations and optimizes computation time. Two discretization methods were developed to account for the specific properties of the arc.
1
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Constricted arcIn this region the method used is that reported by Nicolet et all31 : the axial blowing is quite strong and it is assumed that the axial convection terms are dominant in transfer of mass, of momentum and of energy. The terms with pw are therefore used to calculate the variables at distance z as a function of the values obtained for distance z
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Az' (the other terms are evaluated from the values calculated at z-
Az).The boundary conditions for the modelling of this zone are the following :
at z = 0 (upstream section), we give an arbitrary enthalpy profile h (r, z = 0) and analytical forms of the axial velocity profiles w(r) = WO. f(r) and azimuthal velocity v(r) = v. g(r). Amplitudes W, and v, are calculated from the given value of the initial mass flow mte Do :
R
Do=27c
j
r p w d r0
and the initial swirl coefficient So defined by : R
C 2 x r2 D v W d r l
d
S O = K R (12)
j 2 n p r w 2 d r
0
The pressure at the wall P(R, z = 0) is given a fixed value which is slightly greatcs than atmospheric pressure.
The profiles of pressure and radial velocity at z = 0 are calculated by resolving Eqs. 1 and 2.
ar
awa t r = O w e h a ~ e ~ = ~ = u = O a t r = R w e s e t e n t h a l p y h w a n d w = u = O . The calculation procedure is the following :
i) Let assume we know the values of all the variables within the arc volume, at the axial distances between 0 and z
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Az.ii) The enthalpy profile h (r, z) is calculated resolving Eq. 5.
iii) Eq. 3 is resolved to compute the azimuthal velocity v (r, z).
iv) The pressure p(r, z) is calculated using Eq. 2 giving an artibrary value to the variation of pressure Ap, at the wall, between the distances z - Az and z.
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v) The axial velocity is then calculated using Eq. 4.
vi) The new flow rate D is determined from Eq. 11.
If the flow is maintained, the calculation at distance z is finished and we go back to step i) at the distance z
+
Az.If not a new value is chosen for Ap and steps iv) to vi) are repeated until convergence on the value of D.
2
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Free arc jetIn the central part of the arc where the axial velocity is high, the same method can be used as for the constricted arc. However, outside the hot core of the arc the gas moves down with a low axial velocity and thus does not allow this method to be used. So, we discretized all the terms of Eqs. (3-5) at distance z. Thus for a given variable (v, W of h) the value at point (r, z) depends on the values of the variables at distance z
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Az and also on the values at the neighbouring radial points at distance z.In this zone the mass flow rate is not conservated , hence there are no longer any iterations for the calculation of p(r) or w(r). The boundary conditions are modified in the following way :
the conditions of the upstream section of this zone are the final conditions calculated for the first zone, f o r r < R . F o r r 2 R w e s e t w = u = v = O ; h = h o ( e n t h a l p y a t 3 0 0 K ) a n d p = p o .
as there is no longer a real wall, the discretization
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domain is large. The maximum radial distance is R'.IV
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GENERAL RESULTSFirrure 1 illustrates some typical results : they are the temperature variations in the free jet zone calculated for a current intensity of 1000 A, an initial flow rate of 35 gjs and an initial swirl coefficient of 0.40. The temperature is deduced from the enthalpy and the pressure using the equilibrium thermodynamic properties of air. In this calculation the step Ar is 2 mm
.
The axial temperature high in the constricted part of the arc, decreases quite rapidly in the free jet zone where outside gases are drawn in and the radial temperature gradients are decreased.Firmre 2 represents the variations of axial ve!ocity in the same conditions.
The same kinds of results have been obtained using a radial step Ar = 1 mm. The differences between the two sets of results remain weak. For example the value of the axial temperature varies of less than 5 % when changing this step. All the results given here correspond to Ar = 2 mm.
Finally, one of the general characteristics of the transferred arc studied is that it draws in external gas through the action of the plasma jet flow. This causes an increase in the gas flow rate D along the axial direction. The variation of D with the axial distance is represented in
m
: z = 0 is the end of the constricted zone. Two test conditions were simulated for an initial tube diameter d of 5 cm and non-vortex injection of gas. The variation of the flow rate was similar to that of a jet : for z = 10 d. theoretical works predict a flow rate of 2.5 to 4.5 times the initial flow 161.In the initial study by Nicolet et all31 the spectral domain was divided into two frequency ranges. In both ranges the absorption coefficient Kv was constant (for fixed pressure and temperature). In order to check the influence of radiation on the arc characteristics we used another set of frequency ranges. We divided the spectrum into 4 ranges
1" band
o
< v 14.35.1014 HZ2"t band 4.35 1014 Hz c v I 6.281014 Hz grd band 6.28 1014 Hz < v I 1.015 1015 Hz 4fh band 1.015 1015 Hz c v 5 2.42 1016 Hz
To determine the variations with temperature and pressure of the mean absorption coefficient we used the values for equilibrium plasma composition /S/ and the values of K,, calculated by Wilson and Nicolet /7/ and by Churchill et a1 /8f. For the same test conditions we present in Figure4 the temperature profile in the constricted arc (z = 10 cm) and in the free arc (z = 50 cm) calculated with the 2-band and the 4-band radiation models. It is in the hottest zone (constricted arc) that radiation has the most important role. With the 4-band model the emission coefficient is greater than with the 2-band model when T 1 11 000 K (h > 6 105 Jlkg). This leads to greater radiative energy losses with a 4-band model and hence to a decrease of the axial temperature, a temperature profile broadening and an electric field increase.
gas flow rate and vortex injection. During this study we shall define the role of various phcnomena on the local variables of the plasma.
V - l
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Initial flow rate (Del-
When the initial flow rate increases, the temperature profile calculated in the free jet zone (z = 40 cm) changes shape as shown in figure 5. Note first that for a given flow rate, the profile T(r) shows two changes in gradient : at about 7000 and 4000 K corresponding to the dissociation of nitrogen and oxygen molecules respectively. The thermal conductivity of air is therefore important for these two temperatures firstly because of larninar conductivity but also because of turbulent conductivity through the terme :
dh dT
~ " C P
dr
(13)C p Q presents a very pronounced maximum when the temperature approaches the dissociation temperature 151. In zones where thermal conductivity is high, the temperature gradient decreases explaining the shape of the T(r) profiles in Fig. 5.
It can be seen, from Fig. 5, that the increase in the flow rate leads to a contraction of the hot core of the arc. We can say that the rise in flow rate produces better shielding of the arc by the gas. In fact the rise in Do causes more cold gas to be drawn in and a rise in energy exchanges through turbulence (see eqs. 7-8). These effects tend to decrease temperature along the edges of the arc. Moreover, the increase in eneqy loss must be compensated by an increase in energy sources i.e. of electric field E, for a given current intensity. According to Ohm's low (Eq. 10) if E increases, the conductance G decreases. The fall in G must cause contracting of the hot conducting core of the arc.
V-2
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Current inte*The effects of current intensity on the temperature profile is presented in figure 6. The increase in intensity does not cause an increase in axis temperature, as is the case in wall-stabilized arcs, but causes the broadening of the T(r) profile. The calculations show that the local electric field decreases when current increases, in accordance with the relationship E = f(1) in non stabilized arcs.
The gas can be injected as a vortex from the outset. The swirl coefficient So (Eq. 12), which takcs this effect into account, is one of the parameters of the model. After about 10 cm in the constricted part of the arc, the swirl coefficient has decreased by a factor of 10. In spite of this large decrease of the vortex effect in the constricted part, the effect is still felt in the free part of the arc as can be seen in figures 7 and 8. An increase in the swirl coefficient increases the shielding in the arc and the amount of cold gas which is drawn in and mixes with the plasma. The vortex therefore plays a similar role to that of the gas flow rate : when it increases the axial velocity and thus the total flow rate increase. Hence the electric ficld and the total voltage increase with the swirl coefficient.
V1
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CONCLUSIONThe next stage of our work will firstly be to model the zone near the anode which requires a discretization method different from that developed for the prcsent study. Then the results of the models should be validated by experiment with a transferred arc device. The experimental study will be necessary to improve the prediction abilities of the model in particular by comparing the theoretical influences of the flow rate and of gas vortex injection with the experimental features.
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REFERENCES :
I11 Hsu, K.B. Etemadi, K and Plender, E. J., Appl. Phys.
54
(1983) 1293.121 Kovitya, P. and Cram, L.E., Am. Weld. 3 . 5 5 (1986) 34.
131 Nicolet, W.E., Shepard, C.E., Clark, K.J., Balakrishnan, A., Kesselring ; J.P., Suchsland, K.E. and Reese Jr, J.J., "Analytical and design study for a high-pressure, high enthalpy constricted arc heater". Rep. AEDC-'IR-75-47 (1975).
I41 Bacri, J, Gleizes, A. and Raffanel, S., "Etude thtorique des tchanges entre un arc Clectrique et l'air dans un tube .cylindrique". Unpublished Rep., C.P.A. Toulouse (1985).
I51 Bacri, J. and Raffanel, S., Plasma Chem. Plasma Proc. 2 (1987) 53.
/6/Pateyron, B, Th5se d'etat, Universite de Limoges, no 21-1987 (1987).
D/ Wilson, K.H. and Nicolet, W.E., J. Quant. Spectrosc. Radiat. Transfer 2 (1967) 891.
181 Churchill, D.R., Armstrong, B.H., Johnston, R.R. and Muller, K.G., J. Quant. Spectrosc. Radiat.
Transfer,
d
(1966) 371.: Temperature field in the free jet zone (I = 1000 A, D, = 35 g/s).
: Axial velocity field in the free jet zone (I = 1000 A ; Do = 35 @S).
Fig. 5 : Influence of the initial flow rate D, on temperature (I = IOW A).
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F A 6 : Influence of current intensity on temperature (D, = 35 gls ; S , = 0.4)
: Influence of the initial swirl coefficient S , on axial velocity (1000 A ; 35 gls).
: Influence of So on temperature.
