# An Improved Local Thermal Equilibrium Model of DC Arc Plasma Torch

## The Influence of Spray Conditions on Deposition Characteristics of Aluminum Coatings in Cold Spraying

R. Huang and H.Fukanuma,Plasma Giken Co., Ltd., Tokyo, JapanY.

Uesugi andY. Tanaka,Kanazawa University, Kanazawa City, Japan

## Abstract

Most of the simulation models about arc plasma arebasedon the hypothesis of Local Thermal Equilibrium (LTE).The non-equilibriummodel isverycomplicatedduetothe calculationofelectron temperature.In this study, animproved LTEmodel is developed andappliedto the three-dimensional simulation of the flowpatternsinside a non-transferred DC arc plasma torch.Numerical calculations onthe distribution of gas temperature and velocity intheplasma torch were carried out using argon astheplasma gas. The electriccurrentdensity and potentialarealso discussed. The results indicate that the temperature and velocitydistributionsofarc are almostaxisymmetrical. Theresults of voltage drop agree wellwith the experiment observations.It seems that anodeerosionislocated ontheinternalsurface oftheanode,wherethelargest numberofelectronsareinjected.

IndexTerms-plasma spraying, plasma torch,local thermalequilibrium, three-dimensional modeling.

### Model Assumptions

The model developed in this study is based on the following main assumptions for simulating the heat transfer and flow patterns inside a plasma torch.

(1) The continuum assumption is valid and the plasma can be considered as a compressible gas in the state of Local Thermodynamic Equilibrium (LTE).

(2) The plasma is optically thin.

(3) Gravitational effect and viscous dissipation are considered negligible.

(4) The induced electric field is negligible in comparison with the applied electric field intensity in the plasma arc region.

(5) The variation of gas pressure inside the torch is so little that the effects of pressure on the thermodynamic and transport properties of plasma are negligible. Based on the LTE assumption, the thermodynamic and transport properties of plasma gas (such as c p , μ and λ) are determined by the gas temperature excluding the electrical conductivity (σ) as mentioned above. The electrical conductivity of plasma gas is determined by the nominal electron temperature derived from gas temperature, corrected by the electric field.

## I.INTRODUCTION

Plasma spraying is widely used in industrial fields to providecoatings for protection of materials against wear, erosion,corrosion and thermal loads based ona high heat source withtemperature over 10000K enough to melt any material atatmospheric pressure. Since the appearance of industrial DC arcplasma spray torches in the 1960s, the research of this fieldthrough both measurements and modeling has been extensivelyconducted[1]. In recent years, although a number of robust, userfriendly particle diagnostic tools have become available forplasma spray processes to assess the in-flight particletemperature, velocity, trajectory and particle diameterdistributions, it is still difficult to observe the complex propertiesof arc inside a plasma torch[2].

A conventionalDC non-transferredplasma torch (representingmore than 90% of industrial torches) with a stick-type cathode isshown schematically in figure 1[3-4]. After the working gasenters the torch, it is heated by an electric arc formed between anozzle-shaped anode and a conical cathode, and ejected as a jet.Particles to be plasma sprayed are fed into the particle inlet,heated and accelerated within the plasma jet by the working gasvia the plasma arc.The experimental research performed on theplasma arc is mainly concentrated on the measurement of arcvoltage[3-6],arc behavior at the torch exit and observation ofthe jetformed by the plasma torch[3].S. Goutier et al.alsomeasured the particle temperature fluctuations[7]. Few of thereports about the arc behavior inside the torch presentexperimental results, due to theequipmentlimitations.

Fortunately, numerical calculation provides a valid way tounderstand arc beior inside the plasma torch.The modelingof DC arc plasma torches is an extremely challenging taskbecause the plasma flow is highly nonlinear and presents strongproperty gradients. It is characterized by a wide range of timeand length scales, and often includes chemical and thermodynamic non-equilibrium effects, especially near itsboundaries[8]. Despite of the complexity of the subject, over thepast few decades, many papersconcerning numerical studies ofthe characteristics of DC arc plasma torches have been published[8-23]. At the initial stage, two-dimensional (2D) modelingmethod was employed in the research to predict the heat transferand flow patterns inside the plasma torch[9-13]. The predictedarc voltage of the torch in the turbulent regime is much higherthan the measured value; in addition the predicted axial locationof the arc attachment at the anode surface is also much fartherdownstream than that observed in experiments[14]. With therapid development of computer technology, the calculation ofheat transfer and fluid flow for a 3D thermal plasma torch withaxisymmetrical geometriesbecome feasible[14-23]. The modelsmost frequently used for simulations ofplasma spray torches relyon the LTE approximation, andregardthe plasma flow as aproperty-varying electromagnetic reactive fluid in chemicalequilibrium state, in which the internal energy of the fluid ischaracterized by the single parameter of gas temperature[14-22].Selvanet al.developed a steady 3D LTE modelto describe thetemperature and velocity distributions inside a DC plasma torch.Moreover the arc length and radius were also discussed. But themodel overestimated the plasma gas temperature near thearc-root due to the assumption that all the electric currenttransferred to the anode only through a fixed arc-root[15-16].Klinger L.et al.also developed a steady 3D LTE modelsimulation of the plasma arc inside a DC plasma torch. However,the position of the arc-root was determined arbitrarily[17].A.Vardelle and J. P. Trellesdeveloped a time-dependent 3D LTEmodel representing the fluctuationsof plasma arc[18-22]. Thevoltage drop for the LTE model was larger compared with theexperimental ones due to the hypothesis of LTE. Anon-equilibrium (NLTE)model was developed for thenon-transferred arc plasma torch, which showedbetter agreementwith theexperimental results[23]. However, to solve the NLTEmodel is extremely difficult due to the fact that thetwo-temperature chemical equilibrium needs to be consideredcompared with the LTE mode.

Due to the LTE assumptions for the conventional LTE mode,the value of electron temperature is equal to heavy particlestemperature, which is low near the electrodes, especially near theanode surface. Hence the equilibrium electrical conductivity,being a function of electron temperature, is extremely low, whichlimits the flow of electrical current through the electrodes. Toalleviate this, some additional assumptions are necessary toachieve high electrical conductivity near the electrodes. In thecurrent study, a nominal electron temperature was proposed, thatwas derived fromthe plasma gas temperature and adjusted by theelectrical field strength, to amend the electrical conductivity ofplasma gas. Therefore, nomore additional assumptions arenecessary to ensure the electrical current path between thecathode and anode if the electrical conductivity of plasma gas isdetermined by the nominal electron temperature instead of thegas temperature.

In order to differentiate from the conventional LTE model,themodel using this study was named “improved LTE model”.Withthe improved LTE model,the plasma gas temperature andvelocity distributions inside a DC plasma torch were calculated,and the distributions ofelectrical potentialand current densitywere investigated. The results show that the total voltage dropand the location of anode erosion obtained by the improved LTEmodel are well consistent with experimental observations.

## II. DESCRIPTION OF THE MATHEMATICAL MODEL

### A. Model Assumptions

The model developed in this study is based on the following main assumptions for simulating the heat transfer and flow patterns inside a plasma torch.

(1) The continuum assumption is valid and the plasma can be considered as a compressible gas in the state of Local Thermodynamic Equilibrium (LTE).

(2) The plasma is optically thin.

(3) Gravitational effect and viscous dissipation are considered negligible.

(4) The induced electric field is negligible in comparison with the applied electric field intensity in the plasma arc region.

(5) The variation of gas pressure inside the torch is so little that the effects of pressure on the thermodynamic and transport properties of plasma are negligible. Based on the LTE assumption, the thermodynamic and transport properties of plasma gas (such as c p , μ and λ) are determined by the gas temperature excluding the electrical conductivity (σ) as mentioned above. The electrical conductivity of plasma gas is determined by the nominal electron temperature derived from gas temperature, corrected by the electric field.

### B. Governing Equations

Based on the forgoing assumptions, the governing equationsfora3D time-dependentflowof arc plasmacan be written asfollows:

Conservation of mass:

In order to determine the electrical conductivity of plasma gas, the electron temperature must be calculated. It is difficult to solve the electron energy conservation equation to get the electron temperature because of strong nonlinearity. Therefore, a nominal electron temperature was calculated from the equation as follows [24]:

### C. Computational Domain and Boundary Conditions

The geometry used in the current study corresponds to theSG-100 plasma torch from Praxair. The cross sectionaldimension of SG-100 torch used in this study is shown in figure2(a), and the 3D computational domain is shown in figure 2 (b).The computational domain is meshed using 217600 structuredhexahedral cells with 224567 nodes as shown in the figure 2 (b).

As seen in figure 2, the boundary of the computational domainis divided into 4 different faces to allow thespecification ofboundary conditions. Table 1 shows the boundary conditionsused in the simulation, wherePinrepresents the inlet pressureequal to 111325 Pa (10 kPa overpressure),hwthe convective heattransfer coefficient at the anode wall equal to2×104W.m-2.K-1[19-23], andTwa reference cooling water temperature of 500 K.The electrical current density and temperature of the cathode wasdefined by:

whereris radial distance from the torch axis (d8352=d8352+d8352), andJcath0andncare parameters that specify the shape of the currentdensity profile. TheRcis calculated to ensure that integration ofj(r)over the cathode equals the total applied current. Accordingto references 20 and 23, the values of the shape parameters usedin the current study are shown in table 2.

Argon gas was employed as the plasma gas in this study. Theconditions forsimulation are shown in table 3.

The gas flow insidethe plasma torchwas calculated byFLUENT, commercial CFD software,with the SIMPLEalgorithm. For gas flow calculations, the K-εmodel is employedin this study.

Base on the assumption above, the electrical conductivity ofplasma gas depends on gas electron temperature. The calculatedmethod of electrical conductivity was presented in Ref.23 and25 to 27. The electrical conductivity of argon plasma gas inrelationto the electron temperature is shown in figure 3. Theother thermodynamic and transport properties of the plasma gasthat only rely on the gas temperature, such as heat specific,thermal conductivity,viscosityand volumetricradiation, aretaken from Ref. 28.

## III.RESULTS AND DISCUSSIONS

### A.Nominal electron temperature

Based on equation 10, the calculatednominal electrontemperatureof the argon gas in relation to different electric fieldstrengths at the pressure of 1 atm is shown in figure 4. It’srevealed that the high electric field strength prevents theestablishment of an equilibrium state in which the gastemperature is equalto the electron temperature. Therefore, thenominal electron temperature is much higher than the gas temperature especially under the low gas temperature conditionswhile the electric field strength is strong. In contrast, the nominalelectron temperatureis similar to the gas temperature while theelectric field strength is low, due to few ionizations occurring.When the gas temperature is high enough to obtain sufficientcollisions between the heavy particles and the electrons,thenominal electron temperature is also similar to the gastemperature with little dependency on the electric field strengthas shown in the figure 4.

### B.Flow fields inside the torch

Figure 5 (a) shows the voltage drop of the plasma columncalculated by the improved LTE model. The amplitude ofvoltage is between 22 and 25 V with a fluctuation frequency of13.9 kHz. An experiment was carried out to measure the plasmasystem power source voltage underthe same conditions used inthe current simulation,600 A electrical current and50 SLM gasflow rate. Limited by the experimental equipment, only theaverage voltage of power source was obtained. Theexperimentally observed voltage of the power source is 29.5 V.There are two reasons that the voltage drop calculated is lowerthan theexperimental results. One is thatthe voltage drop of theplasma columnshould be less than voltage drop of plasma torchbecause the sheath voltage drop of the electrode was not takeninto account in the simulation. The other is that the experimentalvoltage of the power source should be higher than the voltagedrop of plasma torch owing to the voltage drop on the electricalcables. Figure 5 (b) shows the voltage drop of a SG-100 plasmatorch researched by Trelles et al under the conditions of 700Aelectrical current and 60 SLM gas flow rate[23]. It seems thatthe experimental voltage drop in the reference 23 is a little lowerthan the ones in this study in spite of the higher electrical currentapplied. This is caused by the cable voltage drop in the currentstudy because the voltage in reference 23 is the voltage drop ofplasma torch instead of power source. The research reveals thatvoltage drop calculated with the LTE model is more than twicethat of the experimental results. It seems that the improved LTEmode can obtain comparable accuracy to the NLTE mode.

The instantaneous temperature distributions inside the torch at600 and 630μs, two representative times for observing theconditions at maximumand minimum voltage drop, arepresented in figure 6. It can be observed thatthe distributions arealmost axisymmetrical and the temperature of the plasma core is near 36500K at the current simulation conditions. The gastemperature distribution changes slightly with the elapse of time.Figure 7 shows the instantaneous velocitydistribution inside thetorch. Similar to the temperature distribution, the distribution ofgas velocity is almost axisymmetrical. The maximum velocity ofinside the torch is near 1740 m/s and the variation of velocitywith the elapse of time is significant compared with the gastemperature.

Figure 8 and 9 show the gas temperature and velocitydistributions of different axialcross sections inside the torch. Itreveals that the distribution of gas temperature and velocity arealmost axisymmetrical at all the axial cross sections.

The calculated nominal electron temperature distributioninside the torch is presented in figure 10. Compared to the gastemperature distribution in figure 6 (b), it is obviously higherthan the gas temperature with greatly varying degrees depending on theregions. As figure 4 shows, in the high temperature regionnear the plasma core, the nominal electron temperature is similarto the gas temperature owing to sufficient collisions betweenelectrons and heavy particles under the condition of higher gastemperature. However, a significant difference between thenominal electron temperature and gas temperature is observed inthe region far from the plasma core, especially the regions nearthe corner of the anode internal surface, where the nominalelectron temperature is more than 10000 K. This is on account ofthe insufficient collisions between the electrons and the heavyparticles in the lower gas temperature regions resulting in a greattemperature difference. Consequently, the electrical current canreachthe anode in spite of the lower gas temperature near theanode boundary because of the higher electrical conductivitydetermined by the nominal electron temperature. Therefore, theproblem of lower electric conductivitynear the anode boundaryin the conventional LTE model can be solved in this study bycalculating nominal electron temperature. Using the improvedmodel, the simulation can be executed without making furtherassumptions about the electric conductivity.

Figure 11 shows the electric potential distribution inside thetorch. It seems that different electrical potentialis obtained alongthe cathode boundary although a uniform electric potential isloaded on the cathode. The minimumelectrical potentialof theplasma gas is observed atthe cathodetipand theelectricalpotentialshould decrease with the increase of distance from thecathode tip. The situation is caused by the sheath voltage of thecathode, not consideredin the current study.Therefore, themagnitude of the gas electrical potential at the cathode boundaryshould be the value of the plasma voltage subtracting the sheathvoltage drop. According to the studies ofBenilov and Zhou,higher temperature and electric current density lead to a lowersheath voltage drop[29-30]. Therefore,different electricalpotential is observed at the cathode boundary because of thedifferent sheath voltage drop.

### C. Gas flow at the torch exit

Figure 12 and 13 show therespectivegas temperature andvelocity distributionsat the torch exit.Thedistributions oftemperature and velocity are bothaxisymmetrical, thetemperature atthesymmetrical center of the torch exit is near13000 K, and the velocity there is about 1000 m/s. It can be seenthat the gas temperature and velocity of the torch exit at the timeof 600μs are a little higher than the ones at the time of 630μsowing to the higher voltage drop of arc column as shown in Fig.5 (a) resulting in a higher plasma power. This is evidence that thearc length is longer at the time of 600μs than that of 630μs.

Figure 14 shows the velocity and gas flow rate at thesymmetrical center of the torch exit. It reveals that the gasvelocity of the symmetrical center at the torch exit fluctuatedperiodically from 650 to 1000 m/s. Thefluctuation of gasvelocityat the symmetrical center of the torch exitis caused bythe plasma arc fluctuation because the frequency is 13.9 kHz,similar to the fluctuation frequency of plasma voltage. The gas velocity fluctuation leads the gas flow rate at the torch exit tofluctuatearound 50 SLM, the inlet flow rate, as shown in thefigure 14 (b).

### D.Erosion ofthe anode

An anode erosion test was carried out to ascertain location oferosion. In order to accelerate the rate of erosion, the conditionsfor the anode erosion test are somewhat different from thesimulation conditions, as shown in table 4. The appliedelectricalcurrent was increased to 750 A, and helium was mixed with argon gas to raise the plasma voltage. Figure 15 shows two crosssections of used SG-100 torch’s anodes. In Fig. 15 (a), the anodewas used for 30 hours, and some slight erosion occurredat theanode internal surface.After extended usage, the anode wasseverely eroded as shown in figure 15 (b). It seems that anodeerosion always occurs at the location close to the corner of theanode internal surface. Extended usage only leads to a spreadofthe erosion range.

Figure 16 shows the electric current density distributionsinside the torch. It can be seen that the arc length at the time of600μsis longer than the one at the time of 630μs. For anon-transferred DC plasma torch, the arc length depends on thebalance of flow drag force and electromagnetic force. If the flowdrag force dominates over electromagnetic force, the arc lengthbecomes longer and theelectromagnetic force increases. If theopposite occurs, the arc length will become shorter (ref 20). Thearc length change leads to arc voltage fluctuation because alonger arc can gain high plasma voltage as shown in Fig 5 (a).Even though the arc length changes with time as shown in Fig. 16 (a), the similar location in the internal surface of anode isobtained, wherethe main electrical current passes through. Itseems that the location is well consistent with the place oferosion. Overlapping the electrical current density distributionand the eroded anode reveals that the location of anode erosion iswhere the main electric current “path” passes through as shownin figure 16 (b).

IV.CONCLUSIONS

An improved LTE model has been developed and applied tothe three-dimensional and time-dependent simulation of the flowinside a DC arc plasma torch. The important change comparedwith the conventional LTE model is to adjust the electricalconductivity of plasma gas with a nominal electron temperatureinsteadof the gas temperature. The temperature and velocitydistribution of arc gas inside the torch were calculated, and theflow will fluctuate with the elapse of time. A gas temperature ofabout 13000K and velocity of about 1000 m/s were obtained atthe torchexit. The voltage drop of arc column calculatedmatches well with the value measured. The location of anodeerosion can also be predicted correctlyusing this model by thecalculation of electric current density distribution.

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