Physical and Mathematical Modeling of non Transferred Plasma Torches

Physical and Mathematical Modeling of non Transferred Plasma Torches

R. C. Favalli, R. N. Szente^*

Departamento de Física Aplicada

Instituto de Física da Universidade de São Paulo

P.O. Box 66318 - Cep 05315-900, São Paulo, Brazil

Received August 14, 1997

The main objective of this work is to obtain temperature and velocity profiles of the plasma jet, including the electric arc region, generated by non transferred plasma torches. The profiles are obtained from a numerical solution of the conservation equations wich were used to describe the plasma flow. This modelling approach will help the development and optimization of plasma torches, saving time and reducing costs of an alternative empirical development; it could also give some insight on the phenomena occuring inside the torch. Fluid mechanics models for laminar and turbulent flows were adopted to simulate the plasma inside and outside the torch. Patankar's control volume method was chosen to solve the resulting coupled differential equations. The method is very stable and requires less computational time than higher order methods, although it can be less accurate for some applications. A computer code was developed to simulate the jet flow of a plasma torch. The results obtained from this program compared very well with published ones, corroborating the assumptionsn of the present model and the numerical method. Temperature and velocity profiles for a plasma torch with dimentions and operating conditions similar to the ones used in indusrtial applications of spraying were generated and analyzed. The plasma torch simulated had an eletric arc of 100A, plasma gas flow rate of 20 l / min, cross section of 5.2 mm and anode length of 13mm.

I. Introduction

Thermal plasmas have been employed in an increasing number of industrial applications since their first development in the early 50's. Typical applications of this technology include materials processing (spraying, production of ceramic parts, synthesis of compounds such as titaniun dioxide, silicon carbide, silicon nitride, superconductors) and metallurgy (blast furnaces, production of special steel and ferroalloys, cutting, recovery of metal fines). In the last five years, thermal plasmas have been extensively employed to treat residues or remediate enviromental problems such as hospital waste, destruction of toxic materials, vitrification of asbestos, remelting of incineration ashes, treatment of galvanic sludges, mercury contaminated soils and many others [1-3].

Thermal plasmas are normally generated in an equipment called plasma torch which consists essentially of two electrodes (cathode and anode); an electric arc is mantained between the electrodes. A certain amount of gas is injected into the torch (in principle any gas can be used) receiving energy from the arc and generating the thermal plasma.

A specific torch design and operating conditions are normally needed for the different industrial application of this technology. This normally implies an empirical development of the equipment, resulting in high costs and long development period for the construction, modification and tests of the specific equipment.

A possible alternative for reducing the development costs of a plasma torch is to understand the phenomena involved in the torch operation and, using mathematical models and numerical schemes, solve the resulting equations. Not only will this approach reduce the time spent for developing the torch, but will reduce the direct costs involved in an empirical development. The resulting physical and mathematical models of the torch will hopefully be used for different applications, simulating different sets of operating conditions and torch geometries.

In this work we present a physical and mathematical modeling of a non transferred plasma torch. The geometry of the plasma torch used in the simulations is described below. The models adopted to simulate the jet flow in the plasma torch and the results obtained from the computer program are shown after the description of the torch. The last section of the paper presents a summary of the main discussed aspects.

II. Non Transferred Plasma Torch

There are different types and geometries of plasma torches. The two main classes of plasma torches are the so called transferred and non transferred torches. In the former, the electric arc used to generate the plasma is mantained between one electrode of the torch (normally the cathode) and a piece of metal (or another conducting material) that one wants to cut or melt, located outside the torch. Those torches are typically used for metallurgical processes such as ferroalloy production or tundish heating. The other type, the non transferred plasma torches, employs the two electrodes of the torch in order to mantain the electric arc, i.e., the electric arc strikes between the two electrodes of the torch and it is kept inside the torch. These torches are normally used for spraying, production of advanced materials, and the treatment of hazardous wastes. In this work we studied the non transferred plasma torch although the computer code developed can be easily adapted to simulating transferred torches.

The plasma torch simulated in this work is based on a previous torch developed for the processing of materials and for treating industrial residues [4]. The former torch had a central injection of particles (through the cathode) and an external magnetic field was used to rotate the arc. We modeled here a more generic and simpler plasma torch, similar to the ones used in spraying (coating), without the central injection of particles and the external magnetic field. However the computer code developed here presents great flexibility, allowing easy and fast modifications in order to simulate other types of plasma torches as mentioned before.

A schematic diagram of the plasma torch modeled here is given in Fig.1. An electric arc is mantained between the two electrodes (cathode and anode); the gas passes through the space between the electrodes and it is heated by the arc, forming the plasma jet.

Figure 1. Schematic representation of the plasma torch modeled.

Typical power levels of plasma torches used for spraying are between 10 and 100kW. Argon is normally employed as the plasma gas since it is an easily ionized gas and it does not react with the material being treated or sprayed. The overall length of those torches varies between 5 and 40cm, with an external diameter between 1 and 10cm; dimensions of the torch are normally function of the power level of the equipment.

III. Theory

The physical and the mathematical models developed in this work as well as the numerical method chosen to solve the set of the resulting governing equations are presented below, followed by the computational domain and boundary conditions necessary for solving the equations and simulating the plasma jet.

III.1 Physical and Mathematical Modeling

Physical modeling of plasma torches commonly consider only the jet exiting the torch, assuming temperature and velocity profiles at the exit of the torch as boundary conditions [5,6]. However, characterization of the plasma jet temperature and velocity inside the torch can provide important information on the phenomena occurring in this region with consequences on the whole jet development. The physical model developed in this work takes into account the cold gas at the torch entrance region, the electric arc region and the free boundary region (the expansion region of the plasma jet). This model is based on the following assumptions:

• The plasma is optically thin, i.e., the black body radiation emmited is not reabsorbed by the plasma;

• The arc is in local thermodynamic equilibrium (L.T.E.), i.e., all plasma components (electrons, ions and neutrons) have the same temperature and this condition prevails all over the plasma column, from the cathode to the anode;

• The heat dissipation due to viscosity is negligible because it is composed by second order terms;

• The gravity effects are negligible due to the high Froude number ( » 3×10

  ⁶);

• The arc is assumed to be steady and rotationally symmetric.

These assumptions are commonly employed in plasma torch simulations [7-10].

The plasma jet is considered here to be a fluid, instead of a statistical collection of particles (Boltzmann approach) and therefore the behavior of the plasma can be described using equations of state and conservation of mass, momentun, energy and current.

The set of differential equations used to describe the physical model can be represented by a general transport equation.

where J is the dependent variable, G_J is the corresponding diffusion coefficient, S_J is the source term, u and v are the axial and radial velocity components, r is the mass density, z and r are the coordinates at axial and radial directions. The left hand side is the convective part and the right hand side represents the diffusive part and the sources.

As mentioned before, equation 1 can represent the conservation of mass, momentum and energy of the plasma jet. The parameters of the governing equations for the plasma jet are given in Table 1.

Table 1:
Equations used for the model

In Table 1, p is the pressure, h is the specific enthalphy, C_p is the specific heat, the thermal and electrical conductivities are k and s, respectively; n is the effective viscosity, j_z and j_r are the axial and radial current densities, B_q is the azimutal component of the magnetic field and S_r is the radiation heat loss. The last two equations in the Table 1 are used for the turbulence model K-e [11], where K is the kinetic turbulent energy and e its dissipation rate; m and m_t are the laminar and turbulent viscosities; k_t is the turbulent thermal conductivity. C₁, C₂, Pr_k and Pr_e are constants of the K-e model (see Table below) and G, given

is the product of the eddy viscosity and viscous dissipation terms.

The turbulent viscosity (m_t) and the thermal conductivity (k_t) are defined using the values of K and e obtained from the solution of their differential equations,

where C_m and Pr_t are also constants of the turbulent model adopted in this work. These constants are presented in Table 2.

Table 2 Constant values of K-e model

Those values were suggested by Launder & Spalding [11]. According to those authors for axisymmetric jets C_m and C₂ should be modified as follows,

where , d is the jet diameter, u₀ is the axial velocity at the center line and Du is the difference between the center line and the free boundary velocities.

The K-e model was chosen among several turbulence models normally used to describe turbulent flows since e appears directly as an unknown in the K equation [12] as well as it represents nicely the characteristic of plasma flows.

The thermodynamic and transport properties of the plasma such as viscosity, density and electrical conductivity were taken initially from the literature [13] and later modified [14]. Those properties were considered to be only temperature dependent.

In the energy equation the term for the energy transfer due the to electron flow was omitted since it is considerably smaller than the other terms of the equation [15]. Enthalpy was chosen instead of temperature to guarantee the conservation of energy in the calculations [15,16]. The temperature values, used for calculating the plasma transport and thermodynamic properties were obtained using the variation of enthalpy as a function of temperature,

It is necessary to take into account the current densities and the azimuthal magnetic field of the arc, considered as source terms in the governing equations. Those parameters were obtained from the current conservation equation and one of the Maxwell's equations,

where m₀ is the permeability of the freespace, F is the electrical potential and s is the electrical conductivity. The current densities are given by

Equation 6 does not present the conservative form of equation 1 but it was solved by the same algorithm used for solving the governing equations shown in Table 1. The Maxwell equation, for simplicity, was solved using the finite difference method. As mentioned before, s depends only on the plasma temperature.

III.2 Numerical Method

The control volume is a special version of the weighted residuals method. This method consists on dividing the calculation domain in control volumes around grid points. The set of governing equations, or a single differential equation, is integrated over each control volume. This method guarantees the total conservation of quantities such as mass, momentum and energy.

The governing equations shown in Table 1 were solved iteratively using Patankar's control volume approach [17-19]. Although Patankar's method can present numerical diffusion and is less accurate than higher-order methods, it is highly stable and converges easily while the higher order methods tend to have convergence problems and use more computational time.

The numerical iteration procedure is described below:

1. Initial guess values for

   p,

   u,

   v,

   T,

   K and e;

2. Plasma transport and thermodynamic properties are calculated for each grid point;

3. After solving equation 6, the current densities

   j_z and

   j_r can be obtained from equation 8. The azimuthal magnetic field is taken from the solution of equation 7;

4. Updated values of

   p,

   u,

   v,

   h (and consequently

   T),

   K and e are obtained from the solution of their respectives equations presented in the

   Table 1;

5. Return to the second step and repeat the procedure until convergence is reached.

III.3 Computational Domain and Boundary Conditions

The computational domain used to solve the set of governing equations is shown in Fig.2. It is the upper half of a cross section of the plasma torch shown in Fig.1.

Figure 2. Computational domain of the plasma torch.

The computational domain is a simplified version of the plasma torch (Fig. 1), but it contains all the essential elements of that torch. Simplifications were conducted for computational reasons. The dimensions of the torch simulated in this work were chosen in order to represent a commonly used spraying torch.

The gas is injected at FG. It flows through the torch until the electric arc region AA'EE', where the gas is heated. EBCD represents the region of free-boundary (outside the torch) and the line AB is the simmetry axis.

The boundary conditions assumed for the computational domain described in Fig.2 are the following.

1. Velocity.

   The axial velocity (

   u) at the entrance of the torch is determined specifing the gas flow rate. The radial velocity (

   v) is zero there. No-slip conditions are assumed in the electrodes region, i.e.,

   u =

   v = 0. At the centre line AB,

   . Free-boundary conditions for BC and CD are assumed,

   and

   respectively.

2. Enthalpy.

   The boundary conditions for enthalpy are given in terms of temperature and later converted into the enthalpy of the respective gas using equation 5. The temperature at the entrance of the torch and at CD are assumed to be equal to the room temperature

   . The surface of the cathode and of the anode are assumed to be at 3,000

   K and 1,000

   K, respectively (taken into consideration the interaction of the electric arc). In the centre line

   , and at

   

3. Turbulence.

   The plasma flow is laminar at FG since the gas has not been heated by the electric arc yet; then K and e are given by [16]

where i_t = 0.003 is the turbulence intensity and R is the radius of the inlet to the nozzle. In the line AB, and at the free-boundaries BC and CD, and K = e = 0, respectively. Near the anode region, a wall-function is used to obtain the values of K and e (the laminar effects are quite significant there),

where k₀ = 0.41 is the Von Kármán constant, U_t is the resultant frictional velocity given by the law of the wall,

where U_p is the parallel velocity to the wall, E = 9.973 is the roughness parameter for smooth walls, and y_p is the distance from point P to the wall.

1. Electrical Potential.

   The boundary condition for the electrical potential is determined assuming a current profile at line AA' [16,20],

   

   where j₀ and r_c are constants which depend on the current. For a blunter cathode tip and I = 200A, j₀ = 100Amm^-2 and r_c = 1.5mm; for I = 400A, j₀ = 130Amm^-2 and r_c = 2.5mm. These values are not the same if a sharp cathode is used: j₀ is slightly higher and r_c smaller. In the line AB, in ED and F = 0 at the anode.

A summary of the used boundary conditions is given in Table 3.

Table 2: Boundary conditions used for the model

IV. Results

The computer program was successfully tested in terms of iteration number and grid points. After 3,000 iterations for a grid with 100 points in the axial direction and 60 points in the radial direction, the temperature and velocity profiles remained unchanged even if a greater number of iterations or grid points were used. After the preliminary tests with the code, the results obtained in this work were compared with published ones. Examples of the temperature and velocity profiles, as well as the electrical potential and axial current density profiles, obtained using the computer code described previously and the torch shown in Fig. 2, are presented after the comparison with results from the literature.

IV.1 Comparisons

The results obtained from the computer code developed here were compared with the results obtained from other researchers. Unfortunately, experimental results for plasma velocities and temperatures inside the torch are extremelly difficult to be obtained due to the high temperatures and velocities found there.

The results of this work were compared with results obtained from numerical simulations. It is quite difficult to find numerical simulations that include the electric arc region. It is generally assumed a velocity and temperature profile at the nozzle exit, and using those profiles simulations for the behaviour of the plasma jet are conducted. The shortback of that method is obvious; the velocity and temperature profiles obtained depend entirely on the profiles initially assumed.

The only published results found by the authors that could be compared to the present work are the ones of Bauchire et al [21]. The plasma torch simulated in that work was similar to that of Fig.2; the dimensions of that torch were 9.1mm for the anode length and 3.4mm for the cross section of the torch; the operating conditions were I = 100A, gas flow rate of 3.4 l/min and argon was considered for both the plasma and the surrounding gas. Those were the same conditions and torch geometry reported by Bauchire et al [21], using a similar numerical method as the one used here.

The results obtained by Bauchire et al and in this work compare well, corroborating the assumption of the present model and the numerical method. Fig.3 shows the temperature and velocity profiles at the centre line obtained from the computer code developed here and from Bauchire et al [21]. The 0mm corresponds to the beginning of the computational domain represented by the point O in Fig.2.

Figure 3. Temperatures and velocities at the centre line obtained from the present work (full line) and for Bauchire et al (dotted line).

The temperatures, as well as the plasma jet velocities, reach their maximum at the vicinity of the cathode ( » 3mm) due to the high current density existing there. The differences between the results of the two studies can be attributed to different assumptions such as the energy transported by the electrons travelling to the anode, considered to be negligible compared with other terms of the energy equation in this work. Another reason for the differences could be the use of different values for the constants j₀ and r_c since they were not available in Bauchire et al [21].

IV.2 Turbulence

In Fig.4 we show the temperature and velocity profiles at the centre line for laminar and turbulent flow as obtained in this work.

Figure 4. Temperatures (a) and velocities (b) at the centre line obtained for laminar (full line) and turbulent (dotted line) jet flow.

It can be seen that turbulence effects were negligible. This conclusion is valid when using low current ( < 300A), low gas flow rate ( < 50l/min) and when the surrounding and the plasma gas are the same. Similar conclusions were also reached by Bauchire et al [21].

IV.3 Simulations of the non Transferred Plasma Torch

Examples of results obtained in this study are presented below. The gas was considered to be Argon for both the plasma and the surrounding gas. The plasma gas flow rate was 20 l/min and an electric arc of 100A was used for the examples given here. The cross section of the torch and the anode length are given in the computational domain. The temperature and velocity contours at the entrance of the torch, electric arc region and free boundary region for the conditions given above are shown in Fig.5. The outermost contour is T = 2,000K, the innermost contour is T = 18,000K and the countour steps are 2,000K; the outermost contour of the velocity is 20m/s and the innermost is 240m/s; the contour steps are 20m/s. We can see from Fig. 5 that the hotest and fastest region of the plasma jet is inside the torch. Someone using the plasma torch for spraying or treating residues can take advantage of the high temperatures and velocities of the plasma jet found inside the torch by injecting the particles with the working gas.

Figure 5. Velocity (a) and temperature (b) contours of the plasma flow in the torch. The outermost contours are 2,000K and 20m/s and the innermost are 18,000K and 240m/s. The contour steps are 2,000K and 20m/s.

The highest temperature and velocity occur in the axis of the torch. The centre line profiles of the velocities and temperatures of the plasma jet are shown in Fig.6.

Figure 6. Temperature and velocity profiles at the centre line for a plasma torch of 20 l/min of gas flow rate and I = 100A simulated in this work.

Near the cathode tip, the temperature reaches its maximum around 21,000K; at the torch exit ( » 13mm ) T = 13,500K and after 50mm, the temperature drops to 9,500K. The maximum velocity is reached near the cathode tip similarly to what was observed for the temperature; the value of the velocity is around 260m/s there and at the torch exit, the plasma jet velocity at the center line is 240m/s; at the end of the computational domain the velocity is aproximately 160m/s.

The temperature decays faster inside than outside the torch. This behaviour is probably due to the electrodes cooling (inside the torch) while in the free boundary region (outside the torch) the plasma jet cools off influenced only by the surrounding gas.

Fig. 7 presents the radial temperature distribution at different axial positions. The temperature decays very quickly in this direction reaching values near the room temperature at r = 5mm (nozzle diameter).

Figure 7.
Radial profile of temperature of the torch in different axial positions. z = 14.5mm (full line), z = 30.4mm (dashed line) and z = 50mm (dot-dashed line).

It is observed that the temperature lines for different axial positions in Fig.7 cross at a radial distance of approximately 3.5mm. This behaviour is probably caused by the failure of the L.T.E. assumption for temperatures below 9,000K and has been reported by other researchers [10,22,23].

The computer code developed in this work calculates also the electrical potential of the plasma. The value of the electrical potential for the operating conditions given above was around 13V, not including the anode and cathode falls. The electrodes fall should add to the electrical potential a value between 10 and 15V [7-9]. Plasma torches using Argon, with geometries and operating conditions similar to the ones simulated here, have overall voltages between 20 and 30V, in agreement with the results obtained here.

The axial coordinate in Fig.8 starts at a distance of 2.5mm which coincides with the cathode tip. The axial current density shows an exponential behaviour and its maximum value occurs exactly at the cathode tip resulting on the increase of the plasma temperature to its maximum value as mentioned before.




V. Conclusion

We have developed physical and mathematical models to simulate the plasma flow through the torch, including the regions of the gas entrance and electric arc besides the free boundary region. The equations of the model were numerically solved using the control volume formulation of Patankar.

The computer code was tested and showed great flexibility, allowing easy modification of the torch geometry and operating conditions such as arc current and gas flow rate.

The results obtained from the computer program were initially compared with published data showing good agreement. Examples of temperature and velocity profiles for the torch simulated in this work were given, indicating that the highest temperature occurs inside the torch and its value is around 21,000K. The velocity profile shows a similar behaviour observed for the temperature, with a maximum value of 260m/s. It was also seen that those profiles (temperature and velocity) decay faster inside the torch; this fact is probably due to the influence of the cooling of the electrodes.

Acknowledgement

Financial support from CNPq and FAPESP through a scholarship for R. C. Favalli is gratefully acknowledged.

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^*

To whom correspondence should be addressed

Publication Dates

• Publication in this collection
  28 Apr 1999
• Date of issue
  Mar 1998

History

• Received
  14 Aug 1997