**4. Validation of the model**

potential value of the nozzle DE was calculated so as to preserve the zero current balance at

Finally, at the interface between the plasma and the anode, in order to maintain the conserva‐ tion of the energy flux and current intensity at this boundary, the following relations (neglect‐

( ) <sup>5</sup> , <sup>2</sup>

*<sup>B</sup> <sup>x</sup> anode A w*

,

<sup>∂</sup> *<sup>y</sup>* =0 300 K <sup>∂</sup> <sup>ϕ</sup>

∂ *T*

∂ *T*

<sup>∂</sup> *<sup>x</sup>* <sup>=</sup> <sup>0</sup> <sup>ϕ</sup> <sup>=</sup> <sup>0</sup>

<sup>∂</sup> *<sup>y</sup>* <sup>=</sup> <sup>0</sup> <sup>∂</sup> <sup>ϕ</sup>

é ùé ù æö æö ¶ ¶ - =- ê úê ú ç÷ ç÷ ë ûë û èø èø ¶ ¶ *anode plasma x x* (17)

*<sup>T</sup> <sup>T</sup> <sup>k</sup> J TT <sup>x</sup> <sup>x</sup> <sup>e</sup>* (16)

f

s

here *φA*, *φw* and *Tanode* are the anode voltage drop, the anode work function and the anode

The unsteady form of (1) was solved using a time-marching method (Ferziger &Períc, 2002; Fletcher, 1991). Consequently, initial conditions had to be supplied in order to complete the

For the internal flow calculations the initial pressure distribution was prescribed as a linearly decreasing function of the axial position, from a specific value at the torch inlet to the atmos‐ pheric one at the exit. The fluid temperature was set to the ambient value everywhere, while the initial fluid velocity was given taking into account the mass flow conservation. The total

*p u T* **Φ**

AB \_ 0 3500 K \_ BC \_ 0 500 K \_ CD \_ mass flow 0.71 g s-1 300 K <sup>∂</sup> <sup>ϕ</sup>

DE \_ 0 500 K \_

<sup>∂</sup> *<sup>y</sup>* <sup>=</sup> <sup>∂</sup> *uz*

<sup>∂</sup> *<sup>x</sup>* <sup>=</sup> <sup>∂</sup> *uz* <sup>∂</sup> *<sup>x</sup>* =0

<sup>∂</sup> *<sup>y</sup>* <sup>=</sup> 0, *uy* <sup>=</sup> *uz* <sup>=</sup> <sup>0</sup>

j j

<sup>∂</sup> *<sup>x</sup>* <sup>=</sup> <sup>0</sup>

<sup>∂</sup> *<sup>y</sup>* <sup>=</sup> <sup>0</sup>

<sup>∂</sup> *<sup>y</sup>* <sup>=</sup> <sup>0</sup>

ing radiation) were used to calculate the local thermal and electric conductivities

é ùé ù æö æö æ ¶ ¶ <sup>ö</sup> - = - <sup>+</sup> - ++ ê úê ú ç ÷ ç÷ ç <sup>÷</sup> ë ûë û è ø ¶ èø è ¶ <sup>ø</sup>

its surface (i.e., the nozzle is electrically floating).

k

s

∂ *ux* <sup>∂</sup> *<sup>y</sup>* <sup>=</sup> <sup>∂</sup> *uy*

<sup>∂</sup> *<sup>x</sup>* <sup>=</sup> <sup>∂</sup> *uy*

temperature; respectively (Gleizes et al; 2005).

*anode plasma*

f

k

74 Computational and Numerical Simulations

EF 1 atm

FG \_ <sup>∂</sup> *ux*

GA \_ <sup>∂</sup> *ux*

**Table 1.** Model boundary conditions

formulation of the problem.

**3.7. Numerical aspects**

This section is devoted to the torch model validation (for the temperature and the velocity) by comparing the model results with experimental data on these quantities obtained in our Laboratory for the same cutting torch.

For the temperature, its radial profile at 3.5 mm from the nozzle exit was used. This profile was derived from electrostatic probe measurements (Prevosto et al, 2008a, 2008b, 2009a) by using a rotating Langmuir probe system; and from a Schlieren technique (Prevosto et al, 2010) by using a Z-type optical configuration with a laser as light source. Both techniques give an experimental uncertainty of ≈ 10 % in the temperature values. For the axial velocity, two axial distributions (with an experimental uncertainty of ≈ 10 %) derived from a time-of-flight technique were employed. One of the distribution corresponded to light emitted from the arc central core, while the other corresponded to an averaged emission over the whole emitting section of the arc (Prevosto et al., 2009b). The arc core velocity was obtained from spectrally filtered light fluctuations measurements using a band-pass filter (475 nm) to detect light emission fluctuations emitted only from the arc axis.

Figure 3 presents the radial profiles of the calculated temperature at 3.5 mm from the nozzle exit for *c* = 0 (i.e., laminar flow), *c* = 0.08 and *c* = 0.20. As shown, all the temperature profiles are similar, their differences being smaller than the experimental temperature uncertainty. As an example, Fig. 4 shows the comparison among the theoretical profile corresponding to *c* = 0.08 and the experimentally derived temperature profiles. It can be seen from Fig. 4 that the model results are in good agreement with the experimental data.

**Figure 4.** Radial profile of the plasma temperature predicted by the model at 3.5 mm from the nozzle exit for *c* = 0.08 together with the experimental data derived from electrostatic probes and Schlieren techniques. Taken from Manci‐

Numerical Modelling of a Cutting Arc Torch http://dx.doi.org/10.5772/57045 77

**Figure 5.** Measured values of the axial velocity of the arc (*z* corresponds to the axial coordinate measured from the

Figure 7 shows the averaged theoretical axial plasma velocity over the emitting arc cross section, defined from the arc core (axis) to the ≈ 4000 K temperature line (Prevosto et al., 2009a) for the same *c* values presented in Fig. 3. As before, the experimental values of *ux* corresponding to an averaged emission over the whole emitting section of the arc are also

nelli et al., 2011.

nozzle exit). Taken from Prevosto et al., 2009b.

**Figure 3.** Radial profile of the calculated plasma temperature for *c* = 0, *c* = 0.08 and *c* = 0.20. Taken from Mancinelli et al., 2011.

The measured plasma flow velocity obtained for different axial positions (*z*) with and without band-pass filter, together with a visible photograph of the arc is shown in Fig. 5. The theoretical distributions of the axial velocity on the axis for the same *c* values present‐ ed in Fig. 3 are shown in Fig. 6. For comparison purposes, the measured values of the axial velocity corresponding to light emitted from the arc central core are also included in Fig. 6. It can be seen that the theoretical profiles are close among them at the vicinities of the nozzle exit (reflecting the well known fact of the little importance of the turbulence inside the nozzle (Gleizes et al., 2005) but soon after the scatter in the *ux* values is larger than those found for the temperature, reaching about 100 % at the middle of the gap. Hence, it can be concluded that the fluid velocity strongly depends on the particular value of the turbulent parameter *c*. On the other hand, the theoretical profile presenting the best matching with the experimental data is that corresponding to *c* = 0.08.

technique were employed. One of the distribution corresponded to light emitted from the arc central core, while the other corresponded to an averaged emission over the whole emitting section of the arc (Prevosto et al., 2009b). The arc core velocity was obtained from spectrally filtered light fluctuations measurements using a band-pass filter (475 nm) to detect light

Figure 3 presents the radial profiles of the calculated temperature at 3.5 mm from the nozzle exit for *c* = 0 (i.e., laminar flow), *c* = 0.08 and *c* = 0.20. As shown, all the temperature profiles are similar, their differences being smaller than the experimental temperature uncertainty. As an example, Fig. 4 shows the comparison among the theoretical profile corresponding to *c* = 0.08 and the experimentally derived temperature profiles. It can be seen from Fig. 4 that the

**Figure 3.** Radial profile of the calculated plasma temperature for *c* = 0, *c* = 0.08 and *c* = 0.20. Taken from Mancinelli et

The measured plasma flow velocity obtained for different axial positions (*z*) with and without band-pass filter, together with a visible photograph of the arc is shown in Fig. 5. The theoretical distributions of the axial velocity on the axis for the same *c* values present‐ ed in Fig. 3 are shown in Fig. 6. For comparison purposes, the measured values of the axial velocity corresponding to light emitted from the arc central core are also included in Fig. 6. It can be seen that the theoretical profiles are close among them at the vicinities of the nozzle exit (reflecting the well known fact of the little importance of the turbulence inside the nozzle (Gleizes et al., 2005) but soon after the scatter in the *ux* values is larger than those found for the temperature, reaching about 100 % at the middle of the gap. Hence, it can be concluded that the fluid velocity strongly depends on the particular value of the turbulent parameter *c*. On the other hand, the theoretical profile presenting the best

matching with the experimental data is that corresponding to *c* = 0.08.

emission fluctuations emitted only from the arc axis.

76 Computational and Numerical Simulations

al., 2011.

model results are in good agreement with the experimental data.

**Figure 4.** Radial profile of the plasma temperature predicted by the model at 3.5 mm from the nozzle exit for *c* = 0.08 together with the experimental data derived from electrostatic probes and Schlieren techniques. Taken from Manci‐ nelli et al., 2011.

**Figure 5.** Measured values of the axial velocity of the arc (*z* corresponds to the axial coordinate measured from the nozzle exit). Taken from Prevosto et al., 2009b.

Figure 7 shows the averaged theoretical axial plasma velocity over the emitting arc cross section, defined from the arc core (axis) to the ≈ 4000 K temperature line (Prevosto et al., 2009a) for the same *c* values presented in Fig. 3. As before, the experimental values of *ux* corresponding to an averaged emission over the whole emitting section of the arc are also included in this figure. It can be seen that the theoretical profile presenting the best matching with the experimental data is that corresponding to *c* = 0.08.

**5. Conclusions**

of time and length scales.

The modelling of dc arc plasma torches is quite challenging because the plasma flow is highly nonlinear, presents strong quantity gradients and is characterized by a wide range

Numerical Modelling of a Cutting Arc Torch http://dx.doi.org/10.5772/57045 79

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. However, a self-consistent descrip‐ tion of the plasma starting from only macroscopic parameters (such as the geometry, current intensity, nature of the gas and type of employed materials, mass flow rate and/or some boundary conditions) has not yet been possible because of the lack of precise knowledge of some phenomena (electrode phenomena, radiation, turbulence, wall ablation, etc), which impose simplifications on the models. 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,

Assuming LTE conditions, the properties of the plasma that must be computed are the temperature, pressure and velocity fields. Numerical models for the plasma generated in cutting torches published during the last ten years have been validated using tempera‐ ture data derived from spectroscopic measurements in the nozzle-anode gap. It has been shown in this work that the plasma temperature is not the most appropriate quantity to validate numerical codes since it is not quite sensitive to changes in the model numerical parameters. Instead, it has been shown that the plasma velocity appears to be a more

In order to realize this validation to such a sensitive variable as the plasma velocity, a 2- D model similar to those proposed in the literature was developed and applied to the same 30 A high-energy density cutting torch that was used in the velocity measurements recently published by some of the authors. Within the experimental uncertainties, it was found that a Prandtl mixing length turbulent parameter *c* = 0.08 allows to reproduce both the experimental data of velocity and temperature. However, this value has to be taken with caution, since that *c* value depends on the actual torch geometry, gas type and arc current. It can also be concluded that the simple Prandtl mixing length model is appropriated to

This work was supported by grants from the CONICET (PIP 112/200901/00219) and Univer‐

the experimental validation of such models is of primary importance.

predict the plasma characteristics in low-current cutting torches.

adequate quantity to perform such validation.

**Acknowledgements**

sidad Tecnológica Nacional (PID 1389).

**Figure 6.** Comparison between calculated plasma velocity values at the axis for *c* = 0, *c* = 0.08 and *c* = 0.20, and the measured values of the axial velocity corresponding to light emitted from the arc central core. Taken from Mancinelli et al., 2011.

**Figure 7.** Averaged theoretical axial plasma velocity over the emitting arc cross section, defined from the arc core (ax‐ is) to the ≈ 4000 K temperature line for the same *c* values presented in Fig. 3. The experimental values corresponding to an averaged emission over the whole emitting section of the arc are also shown. Taken from Mancinelli et al., 2011.
