Some Issues in the Formation of a Free High-Frequency Torch Discharge
Abstract Generated abstract
This paper analyzes mechanisms responsible for the formation of plasma flow in a free high-frequency torch discharge, whose channel and envelope have markedly different dimensions and temperatures. It develops two approximate models: one based on electrodynamic compression and pressure gradients associated with the discharge current, and another based on convective flow generated by strong temperature and density gradients near the electrode and channel. Order-of-magnitude estimates for typical channel parameters suggest that, for currents up to about 10 A, convective excitation gives larger flow velocities than the electrodynamic mechanism, while at higher currents the pinch effect may become significant.
Full Text
UDC 532.7
SOME ISSUES IN THE FORMATION OF A FREE HIGH-FREQUENCY TORCH DISCHARGE
Vl. L. Teploukhov, G. N. Kolmakov, Vik. L. Teploukhov
(Presented by the scientific seminar of the Faculty of Physics and Technology)
The paper examines questions concerning the formation of a free high-frequency torch discharge. Two models are proposed for the excitation of a plasma flow in the discharge channel, and estimates of absolute velocities are made. It is shown that, for discharge currents up to 10 amperes, the convective mechanism of excitation of plasma flows in the high-frequency torch-discharge channel predominates; at higher currents the pinch effect begins to play an important role.
Tables 2, references 6.
A free high-frequency torch discharge in a gas is a plasma formation with a clearly distinguished channel and envelope; moreover, the diameter of the channel is an order of magnitude smaller than the diameter of the envelope. The temperature in the discharge channel, if its structure is not disturbed by walls, is determined by the power introduced into the discharge, the frequency of the feeding field, and the type of surrounding gas; in the envelope the temperature is substantially lower. Thus, in a discharge in air, the temperature in the channel, according to the measurement results of many authors [1—3], lies within the range 4000–4500°K, while in the envelope it is approximately 2000°K [2].
According to the electrodynamic model of the high-frequency torch discharge currently in existence [4], it is accepted that an electromagnetic wave propagates along the discharge channel, its energy being absorbed in the form of ohmic losses. Because of the low conductivity of the envelope, ohmic losses in it are neglected, and it is assumed that heating of the envelope occurs owing to diffusion of heat from the surface of the channel. Thus, the heat sources are concentrated in the discharge channel. The structure of a free high-frequency torch discharge is often compared with the structure of a candle flame; however, in a flame, as is known, the region of heat sources has a diffuse, blurred structure. But the similarity in structure may be caused by a common phenomenon in its formation. Such a formative action is provided by convective flows [5]; in the case of a high-frequency torch discharge, the occurrence of a directed flow due to electrodynamic compression of the plasma is possible.
In order to compare the magnitudes of the flows in the discharge channel, let us carry out the following estimates. Consider, by analogy with the problem in [6], an element of the plasma of the discharge channel in its own magnetic field, assuming the direction of the current in the channel to be constant. Through an element \(dx\,dy\,dz\) along \(Z\) there flows a current \(dI = jdxdy\). According to Lorentz’s law, the force acting on it is
\[ d\vec{F}_1 = \mu_0 [d\vec{I}\vec{H}]\,dZ, \tag{1} \]
directed along \(x\) toward the center of the column, where \(H\) is the field strength inside the element; \(\mu_0\) is the magnetic permeability.
With a pressure difference \(dP\) at the boundaries of the element perpendicular to \(x\), the element will also be acted upon by the force
\[ dF_z = dP\,dy\,dz . \tag{2} \]
As a result, the gas of the element with density \(\rho\) is set in motion:
\[ \rho\,dx\,dy\,dz\,\frac{dv_x}{dt} = \mu_0 Hj\,dx\,dy\,dz - dp\,dy\,dz \tag{3} \]
or
\[ \rho\,\frac{dv_x}{dt}=\mu_0 Hj-\frac{dP}{dx}. \]
In the stationary case \(\dfrac{dv_x}{dt}=0\), hence
\[ \frac{dP}{dx}=\mu_0 Hj . \tag{4} \]
Consequently, along the \(x\) axis a pressure field is created due to magnetic compression. In the case of axial symmetry, expression (4) can be written as
\[ P=\frac{\mu_0 Ij}{4\pi}\left(1-\frac{x^2}{r_c^2}\right), \tag{5} \]
where \(j=\dfrac{I}{\pi r_c^2}\); \(r_c\) is the channel radius.
Since in an H.F. torch discharge the displacement current decreases with height [4], let us assume that \(I\) changes along the length of the channel from \(I_1\) to 0. Then along the height of the channel there will exist a pressure difference
\[ \Delta P=\frac{\mu_0 I^2}{4\pi S}, \tag{6} \]
where \(S=\pi r_c^2\).
Due to the pressure gradient along \(Z\), a plasma element \(dZ\) will receive an increment of kinetic energy
\[ \frac{1}{2}\rho\,dv^2=-\,dP. \]
If we assume that \(v_0=0\), then
\[ \frac{1}{2}\rho v_{\max}^2=\nabla P_{\max}. \tag{7} \]
Table 1
| No. | \(I\) | \(\Delta P\) | \(v_{\max}\) | Note |
|---|---|---|---|---|
| A | atm | m/s | ||
| 1 | 1.0 | \(1.4\cdot10^{-6}\) | 0.2 | Since viscosity was not taken into account in deriving equations 1–7, the actual values will be substantially lower |
| 2 | 2.0 | \(5.6\cdot10^{-6}\) | 0.8 | Since viscosity was not taken into account in deriving equations 1–7, the actual values will be substantially lower |
| 3 | 5.0 | \(3.5\cdot10^{-5}\) | 5.0 | Since viscosity was not taken into account in deriving equations 1–7, the actual values will be substantially lower |
| 4 | 10.0 | \(1.4\cdot10^{-4}\) | 20.0 | Since viscosity was not taken into account in deriving equations 1–7, the actual values will be substantially lower |
For an approximate calculation we take: the current \(I\) in the channel varies within the limits \(1\div 10\) A, \(r_c=3\cdot10^{-3}\ \text{m}\); \(\mu_0=1.2567\cdot10^{-6}\ \dfrac{\text{H}}{\text{m}}\); \(T_k=4500^\circ\text{K}\), \(P=1\) atm. The results of the approximate calculation are presented in Table 1.
On the other hand, the velocity of the gas flow in the h.f. torch channel can be estimated from the conditions for the excitation of convective flows. We shall represent the model of their appearance as follows: the temperature in the discharge channel is high, while the temperature of the water-cooled electrode hardly exceeds \(500^\circ\mathrm{K}\) even in the electrode spot; therefore in the near-electrode region there will exist a large temperature gradient and, correspondingly, a density gradient. Since the gas density (neglecting dissociation and ionization processes)
\[ \rho=\frac{mP}{\kappa T}, \tag{8} \]
where \(m\) is the molecular mass, \(\kappa\) is Boltzmann’s constant, \(P\) is the absolute pressure, \(T\) is the absolute temperature, and pressure equalization in space (along \(x, y, z\)) proceeds at the speed of sound, powerful convective flows will be excited along the \(z\)-axis through the channel. Let us estimate their magnitude.
Suppose that in the channel the temperature reaches \(4500^\circ\mathrm{K}\), and the gas density is \(\rho_2\); the temperature of the surrounding gas is \(500^\circ\mathrm{K}\) and its density is \(\rho_1\). Then the magnitude of the convective buoyant force, as is known, is determined as \((\rho_1-\rho_2)g\), where \(g\) is the acceleration of gravity. Under the action of this force, an element of gas in the discharge channel will move with acceleration
\[ \frac{\rho_1-\rho_2}{\rho_2}g . \]
The final velocity \(v_k\) can be determined as \(\dfrac{\rho_1-\rho_2}{\rho_2}g\cdot t_{\mathrm{p}}\), where \(t_{\mathrm{p}}\) is the time of transfer of charged particles along the discharge channel. The time \(t_{\mathrm{p}}\) can be determined from the condition of charge transfer along the \(Z\) and \(X\) axes. The point is that energy transfer from the discharge channel into the envelope is effected through diffusion of both charged particles and molecules, which are then entrained by convective flows or by forced gas blowing. The channel parameters will be homogeneous only in the case when the rate of charge transfer along the \(Z\)-axis is greater than the rate of their diffusion in the radial direction, i.e.
\[ t_{\mathrm{p}}\ll t_D, \tag{9} \]
where
\[ t_{\mathrm{p}}=\frac{2l}{v_k}, \qquad t_D=\frac{r_c^2}{6D}. \]
Here \(D\) is the diffusion coefficient, and \(l\) is the channel length.
Table 2
| No. | \(D\), \(\mathrm{cm^2/sec}\) | \(t_D\), sec | \(v_{\mathrm{conv}}\), \(\mathrm{cm/sec}\) | \(l\), cm | Note |
|---|---|---|---|---|---|
| 1 | 0.1 | 0.15 | 2000 | 100 | for air |
| 2 | 0.2 | 0.075 | 1000 | 2.6 | \(\overline{v}=\sqrt{\dfrac{8\kappa T}{\pi m}}\approx 6\cdot10^4\ \mathrm{cm/sec}\) |
| 3 | 0.3 | 0.05 | 700 | 17 | \(\lambda\approx10^{-5}\ \mathrm{cm}\) [5] |
| 4 | 0.5 | 0.03 | 400 | 10 |
We shall carry out approximate calculations according to the indicated relations for the channel of an h.f. torch discharge with the parameters: \(r_c=0.3\ \mathrm{cm}\), atmospheric pressure, and ratio
\[ \frac{\rho_1}{\rho_2}=15;\qquad \left(\frac{T_2}{T_1}=\frac{4500}{300}\right). \]
This gives the following orders of magnitude (Table 2). These values of \(v_{\max}\) and \(l\) agree with those observed in the experiments. From a comparison of the results of calculating the velocities, whose values were obtained in the model of electrodynamic excitation of a flow in the channel and in the model of convective excitation, it follows that the convective excitation mechanism predominates over the electrodynamic one at currents of approximately up to 10 amperes. At higher currents, apparently, an important role in the mechanism by which plasma flows arise in the channel of, in particular, a torch discharge will be played by the pinch effect.
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