**Numerical Modelling of a Cutting Arc Torch**

Beatriz Mancinelli, F. O. Minotti, Leandro Prevosto and Héctor Kelly

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/57045

**1. Introduction**

### **1.1. Thermal plasmas**

Thermal plasmas are partially or strongly ionized gases, usually developed at atmospheric pressure. In fact, thermal plasmas can be generated by many methods, such as dc electrical discharges at current intensities higher than a few A and up to 105 A: free burning arcs, transferred arcs, or non-transferred plasma torches; ac or transient arcs (e.g., discharge lamps), circuit-breakers, or pulsed arcs; rf and microwave discharges at near-atmospheric pressure; and laser induced plasmas after the ignition and expansion phases (Gleizes et al., 2005; Boulos et al., 1994).

In this kind of plasma, the electric field remains rather weak (less than a few kVm−1) and the electron number density is rather high (more than 1020 m−3) so that the velocity distribution functions of all the types of particles can be considered as Maxwellian. The energy delivered to the plasma, in general by the Joule effect, is first picked up by the electrons because of their high mobility. The electrons transfer part of this energy to the heavy particles by elastic collisions. Due to the high electron number density, elastic collision frequencies are very high, so energy transfer is important and leads to an almost even distribution of the energy: the mean heavy particle energy is the same as the electron energy and there is thermal equilibrium among all the kinds of particles, allowing a single temperature to be defined for the plasma at a given position. In the hottest regions of thermal plasmas this mean kinetic energy is of the order of 1 eV, which corresponds to a temperature of the order of 104 K.

In thermal plasmas, the electrons are also most responsible for inelastic collisions, such as ionization, recombination, excitation, de-excitation, attachment, and detachment. Due to the large value of the electron density, the inelastic collision frequencies are high, and they tend

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to establish a statistical equilibrium among all kinds of particles; i.e. the populations of the excited atoms, molecules and ions tend to obey equilibrium laws, such as the Boltzmann and Saha laws. This situation requires one condition: the influence of radiation on the populating mechanisms of the various species must be negligible in comparison to the inelastic electron collisions. This condition is, in general, verified in the hottest regions of thermal plasmas, corresponding to the regions with high electron density values, but not in the outer regions (Ghorui et al., 2007). In spite of the weak role of radiation in the population, the experimental evidence shows that thermal plasmas emit strong amounts of radiation, particularly in the UV and visible parts of the spectrum, and they cannot be considered as black body emitters. This radiation emission, together with the presence of temperature and number density gradients within the plasmas inducing diffusion effects, shows that thermal plasmas are not in a state of thermodynamic equilibrium. Nevertheless, locally and as a first approximation, thermal plasmas can be considered to be in a state of local thermodynamic equilibrium (LTE), meaning that the particle number densities are related by equilibrium laws (because of high collision frequencies), whereas radiation is not in equilibrium with the particle distribution (Planck's law is not valid). More specifically, for thermal plasma to be considered in LTE, it has to accomplish the following requirements:

plasmas. In the last years numerical plasma modelling has reached a state advanced enough to be of practical use in the study of cutting-arc processes (e.g., Gleizes et al., 2005).

Numerical Modelling of a Cutting Arc Torch http://dx.doi.org/10.5772/57045 67

Plasma modelling by numerical simulation in cutting torches is a powerful tool to predict the values of the fundamental physical quantities, namely the plasma temperature, the particles concentration and the fluid velocity. These numerical codes are employed to understand the relevant physical processes ruling the plasma behavior in order to inter‐ pret the experimental results of several plasma diagnostics, and ultimately to obtain optimized designs of such devices (Colombo et al., 2008). However, no complete predic‐ tive power has been possible as yet due to the complexity and variety of the processes involved. In particular, the practical use of cutting torch codes requires the introduction of some numerical coefficient whose value has to be obtained from a comparison between the model predictions and the experiment. For these reasons, the experimental validation of

On the other hand, experimental data of cutting arcs are hard to obtain due to the harsh ambient conditions, and also because much of the plasma flow takes place in relatively small, bounded regions inaccessible to experimental probing (Nemchinsky & Severance, 2006). In practice, most of the available experimental data are related to spectroscopic (Pardo et al., 1999; Girard et al., 2006) and probe (Prevosto et al., 2008a, 2008b, 2009a) measurements in the external plasma region (the outer region between the nozzle exit and the anode, where the pressure has relatively small variations about the atmospheric value), giving information of only part of the variables involved, mostly temperatures and species concentrations. Although data on flow velocities are commonly reported for low-energy density (subsonic flow) non-transferred arc torches (e.g., Singh et al., 2000), measurement of flow velocities in cutting torches have been

Because of the above quoted reasons, the experimental validation of the existing cutting torch models has been restricted to the temperature distribution in the nozzle-anode gap, together with other global parameters easy to obtain as the arc voltage and the arc chamber pressure. That experimental validation consists in obtaining a good matching with the experimental temperature values within the experimental uncertainty, which typically amounts to 10 % of the measured value (Peters et al., 2007). However, since in a cutting torch the plasma pressure is close to the atmospheric pressure, the plasma temperature is mainly determined by the energy equation (with the dominant energy loses being due to inelastic electron collisions that produce a sort of "anchorage" in the temperature), and so it is almost decoupled from the hydrodynamic flow. Hence, the calculated temperature value does not result in practice quite sensitive to changes in the numerical coefficients of the models at least within the temperature

In high-energy density cutting torches it appears that the flow velocity is a more sensitive variable than the temperature to the modelling details. This can be understood considering the following argument. The acceleration of the flow is driven by a mostly axial pressure gradient along the torch nozzle, established between the pre-nozzle chamber and the nozzle exit. So, given the pressure profile, the velocity of each fluid element at different positions

such models is of primary importance.

experimental uncertainty.

reported only recently by our Group (Prevosto et al., 2009b).


### **1.2. Plasma cutting torch modelling**

Plasma cutting is a process of metal cutting at atmospheric pressure by an arc plasma jet, where a transferred arc is generated between a cathode and a work-piece (the metal to be cut) acting as the anode (Ramakrishnan et al., 1997). Small nozzle bore, extremely high enthalpy and operation at relatively low arc current (≈ 10 ÷ 200) A are a few of the primary features of these torches (Nemchinsky & Severance, 2006). The physics involved in such arcs is very complicated. The conversion of electric energy into heat within small vol‐ umes causes high temperatures and steep gradients. Dissociation, ionization, large heat transfer rates (including losses by radiation), fluid turbulence and electromagnetic phenom‐ ena are involved. In addition, wide variations of physical properties, such as density, thermal conductivity, electric conductivity and viscosity have to be taken into account. These factors make hopeless the possibility of an analytical solution for such thermal plasmas. In the last years numerical plasma modelling has reached a state advanced enough to be of practical use in the study of cutting-arc processes (e.g., Gleizes et al., 2005).

to establish a statistical equilibrium among all kinds of particles; i.e. the populations of the excited atoms, molecules and ions tend to obey equilibrium laws, such as the Boltzmann and Saha laws. This situation requires one condition: the influence of radiation on the populating mechanisms of the various species must be negligible in comparison to the inelastic electron collisions. This condition is, in general, verified in the hottest regions of thermal plasmas, corresponding to the regions with high electron density values, but not in the outer regions (Ghorui et al., 2007). In spite of the weak role of radiation in the population, the experimental evidence shows that thermal plasmas emit strong amounts of radiation, particularly in the UV and visible parts of the spectrum, and they cannot be considered as black body emitters. This radiation emission, together with the presence of temperature and number density gradients within the plasmas inducing diffusion effects, shows that thermal plasmas are not in a state of thermodynamic equilibrium. Nevertheless, locally and as a first approximation, thermal plasmas can be considered to be in a state of local thermodynamic equilibrium (LTE), meaning that the particle number densities are related by equilibrium laws (because of high collision frequencies), whereas radiation is not in equilibrium with the particle distribution (Planck's law is not valid). More specifically, for thermal plasma to be considered in LTE, it has to

**a.** The different species of the plasma (atoms, ions, electrons, molecules) share a single

**b.** The ratio of the electrostatic energy density to pressure has to be small enough, and the temperature must be high enough, so charge carriers equilibrate through collisions the

**c.** Collisions (but not radiation) are the dominating mechanisms for ionization and excita‐ tion, and there must be micro-reversibility among collisions. Hence, Saha equilibrium and

**d.** Spatial variations of the plasma properties are enough smooth, so a given particle that

Plasma cutting is a process of metal cutting at atmospheric pressure by an arc plasma jet, where a transferred arc is generated between a cathode and a work-piece (the metal to be cut) acting as the anode (Ramakrishnan et al., 1997). Small nozzle bore, extremely high enthalpy and operation at relatively low arc current (≈ 10 ÷ 200) A are a few of the primary features of these torches (Nemchinsky & Severance, 2006). The physics involved in such arcs is very complicated. The conversion of electric energy into heat within small vol‐ umes causes high temperatures and steep gradients. Dissociation, ionization, large heat transfer rates (including losses by radiation), fluid turbulence and electromagnetic phenom‐ ena are involved. In addition, wide variations of physical properties, such as density, thermal conductivity, electric conductivity and viscosity have to be taken into account. These factors make hopeless the possibility of an analytical solution for such thermal

diffuses from one location to another has sufficient time to equilibrate.

Maxwellian distribution, characterized by a single temperature.

accomplish the following requirements:

66 Computational and Numerical Simulations

energy gained from the electric field.

Boltzmann distribution laws are valid.

**1.2. Plasma cutting torch modelling**

Plasma modelling by numerical simulation in cutting torches is a powerful tool to predict the values of the fundamental physical quantities, namely the plasma temperature, the particles concentration and the fluid velocity. These numerical codes are employed to understand the relevant physical processes ruling the plasma behavior in order to inter‐ pret the experimental results of several plasma diagnostics, and ultimately to obtain optimized designs of such devices (Colombo et al., 2008). However, no complete predic‐ tive power has been possible as yet due to the complexity and variety of the processes involved. In particular, the practical use of cutting torch codes requires the introduction of some numerical coefficient whose value has to be obtained from a comparison between the model predictions and the experiment. For these reasons, the experimental validation of such models is of primary importance.

On the other hand, experimental data of cutting arcs are hard to obtain due to the harsh ambient conditions, and also because much of the plasma flow takes place in relatively small, bounded regions inaccessible to experimental probing (Nemchinsky & Severance, 2006). In practice, most of the available experimental data are related to spectroscopic (Pardo et al., 1999; Girard et al., 2006) and probe (Prevosto et al., 2008a, 2008b, 2009a) measurements in the external plasma region (the outer region between the nozzle exit and the anode, where the pressure has relatively small variations about the atmospheric value), giving information of only part of the variables involved, mostly temperatures and species concentrations. Although data on flow velocities are commonly reported for low-energy density (subsonic flow) non-transferred arc torches (e.g., Singh et al., 2000), measurement of flow velocities in cutting torches have been reported only recently by our Group (Prevosto et al., 2009b).

Because of the above quoted reasons, the experimental validation of the existing cutting torch models has been restricted to the temperature distribution in the nozzle-anode gap, together with other global parameters easy to obtain as the arc voltage and the arc chamber pressure. That experimental validation consists in obtaining a good matching with the experimental temperature values within the experimental uncertainty, which typically amounts to 10 % of the measured value (Peters et al., 2007). However, since in a cutting torch the plasma pressure is close to the atmospheric pressure, the plasma temperature is mainly determined by the energy equation (with the dominant energy loses being due to inelastic electron collisions that produce a sort of "anchorage" in the temperature), and so it is almost decoupled from the hydrodynamic flow. Hence, the calculated temperature value does not result in practice quite sensitive to changes in the numerical coefficients of the models at least within the temperature experimental uncertainty.

In high-energy density cutting torches it appears that the flow velocity is a more sensitive variable than the temperature to the modelling details. This can be understood considering the following argument. The acceleration of the flow is driven by a mostly axial pressure gradient along the torch nozzle, established between the pre-nozzle chamber and the nozzle exit. So, given the pressure profile, the velocity of each fluid element at different positions inside the nozzle depends strongly on its density, which in turn depends on the temperature and, in the high velocity, compressible regime, on the velocity itself. As inside the nozzle the radial temperature profile is very sharp, small variations of its shape, for instance its peak width, lead to appreciable changes in the radial mass distribution that reflect directly on the velocity. All this is more clearly seen and quantified using simple 1-D, two zone models in which only axial dependence is considered of an inner hot zone and an outer cold region. By assuming sonic flow at the nozzle exit, these simple 1-D models are able to predict quite reasonably the measured temperatures (within the experimental uncertainty), but give very imprecise values (100% off) of the flow velocity (Kelly et al., 2004).

The purpose of this work is to present a validation of the most frequently used plasma cutting torch models employing not only temperature but also velocity values as the experimental data to be confronted with. In order to do this, a 2-D model (similar to those proposed in the literature) was developed and applied to the same 30 A oxygen cutting torch that was used in a previous velocity measurement experiment. In this paper the plasma torch, model assump‐ tions, governing equations, boundary conditions and the physical details of the model are presented. The calculated distributions of temperature and velocity and its comparison with the experimental data are shown.

**3. Mathematical model**

**Figure 1.** Schematic of the Cutting torch.

**3.1. Computational domain**

tan (13º) ≈ 0.23 was used at the torch inlet CD.

**Figure 2.** Cutting torch computational domain.

single temperature *T*. Other assumptions are:

**3.2. Model assumptions**

The schematic of the modelled domain for the simulation is presented in Fig. 2. AC is the cathode part, including a 0.75 mm radius hafnium insert (AB). DE represents the copper nozzle. The edge of the domain EF is located at a radius of 10 *RN* (Gonzalez-Aguilar at al., 1999). FG represents the anode located at 6 mm from the nozzle exit. A mass flow rate of 0.71 g s-1 with a vortex injection that leads to a ratio of the azimuthal to the axial inlet velocity of

Numerical Modelling of a Cutting Arc Torch http://dx.doi.org/10.5772/57045 69

The most frequently used cutting torch models use the LTE approximation, with the plasma flow in chemical equilibrium, and the internal energy of the fluid being characterized by a
