Introduction

Thermal plasmas are often used in metallurgical applications such as welding, cutting or plasma spraying (see other papers in this special issue). Obviously in these conditions, metallic materials are in contact with thermal plasmas whose temperature is of the order of 10,000 K leading to an injection of metallic vapours within the plasma. Furthermore the electrodes of electric arcs are in general ablated, particularly at high current, even if the arc duration is short as it is the case for the circuit-breakers arcs. Hence the presence of metallic vapours is often observed in thermal plasmas.

The composition, the thermodynamic properties and the transport coefficients of a plasma depend strongly on the nature of the gas or of the mixture of gases constituting the plasma, especially when the temperature is lower than 15,000 K [1]. Let us note that at higher temperature the plasma is strongly ionized and its properties depend slowly on the nature of gases. So, when the proportion of a gas in a mixture is important, all the material properties of a plasma are affected by the presence of this gas. On the contrary when this proportion is of the order of 1% the plasma properties are (in general) almost unchanged. But this is not true for metallic vapours. Indeed as it can be seen in Fig. 1 showing the evolution of the electron number density in Ar–metal plasmas [2], the presence of a small amount of metal vapour increases this function at low or intermediate temperature (T ≤ 10,000 K). This kind of result has been obtained performing the equilibrium composition calculation whose bases are well known [1]. The increase of the electron number density n e due to the presence of the metal can be explained by the low ionization energies of these species (see Table 1 showing this energy for various neutral species) favouring the ionization of the gas following the Saha’s law [1]. Figure 1 shows clearly the important effect of the metal at low temperature even for proportions lower than 1%. As expected, the effect increases with the metal proportion, but tends to disappear at high temperature (T > 12,000 K) because all the species become rather strongly ionized. It must be added that the influence of the metal vapour on n e , is less important in air plasma than in argon plasma because the difference between the ionization energies of the metals and those of the gas components is smaller with air (N, O) than with argon, as it can be seen in Table 1.

Fig. 1
figure 1

Influence of the presence of metal vapours (mass proportions) on the variations of the electron number density, for an argon plasma (from [2]) at atmospheric pressure (P = 1.013 × 105 Pa) (© IOP Publishing. Reproduced with permission. All rights reserved)

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Table 1 Ionisation energy (in eV) of various neutral species
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As a consequence the effect of metal vapours on n e at low temperature tends to modify specific properties in this temperature range. Among these properties, the radiation emission is very sensitive to the nature of the gas or vapour, reason why we present this paper on the influence of metal on the radiative transfer in thermal plasmas, influence that can be decisive even for small proportion of metal.

The main assumption used in this study and in literature cited in this paper is that the plasma is in Local Thermodynamic Equilibrium (LTE). This assumption allows first using the equilibrium composition giving the population of the number densities of the plasma components as illustrated in Fig. 2 for air–Al plasma [3]. The populations of the excited states can then be deduced using the Boltzmann’s law and the partition functions previously calculated (see [1–3] for the methods). The second relation available in LTE conditions is the Kirchhoff’s law, very important for calculating the radiative transfer and that will be defined in “Bases of Radiation Phenomena and Transport” section.

Fig. 2
figure 2

Equilibrium composition of a 90%air–10%Al (mass proportions) plasma (from [3]) (P = 105 Pa)

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As this paper takes part of a special issue in homage of Prof. Pfender, it must be noted that E. Pfender was a pioneer in the field of thermal plasmas research and development and that among the many works he performed we must emphasis the study of energy transfer [4, 5] and specially the radiative transfer [6, 7] even in presence of metallic vapours [8, 9]. These two last references will be commented in “Net Emission Coefficient: Influence of Metal Vapours” and “Role of Metal Radiation on General Thermal Plasmas Properties” sections.

“Bases of Radiation Phenomena and Transport” section of this paper is devoted to a rather brief description of the bases of radiation transport and phenomena. The most popular and simplest method to compute the radiation transfer in thermal plasmas in presence of metallic vapours is the net emission coefficient (NEC). This method and the most characteristics results are detailed in “Net Emission Coefficient: Influence of Metal Vapours” section. Some recent results concerning other methods of radiative transfer calculation based on the use of mean absorption coefficients (MAC) are also presented in this section. “Role of Metal Radiation on General Thermal Plasmas Properties” section deals with the study of the influence of the metal vapours on the behaviour of thermal plasmas systems showing the important role of radiation on the plasma parameters (temperature for example) and on the global energy balance. The main part of this paper corresponds to a review from literature but some new and original results are included in “Net Emission Coefficient: Influence of Metal Vapours” section and in the presentation of the MAC.

It has to be noted that metallic species and their eventual by-products (oxides, carbides, nitrides, …) are in general in condensed form at low temperature (liquid or solid). As we are concerned with plasmas in gaseous phase, we will present results on radiation properties and radiative transfer in presence of metal vapours, for temperature values higher than roughly 2000 K.

Bases of Radiation Phenomena and Transport

General Laws and Definitions

The radiation intensity at a point P and in the direction \(\vec{s}\) (see Fig. 3) is the power crossing the elementary surface area \(dS_{ \bot }\) (apparent surface perpendicular to the direction \(\vec{s}\)) and per solid angle unit. The spectral intensity I λ refers to radiation at a given wavelength λ (in fact in an interval dλ around λ) while the total intensity I is an integration of the previous one, over the total spectrum, i.e. including all wavelengths.

Fig. 3
figure 3

Definition of the radiation intensity

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Let us remember that the intensity of a blackbody noted I bλ is isotropic. It is given by the Planck’s law and the form depends on the spectral variable. For the frequency ν:

$$I_{b\nu } (T) = \frac{{2h\nu^{3} n^{2} }}{{c_{o}^{2} \cdot \left( {e^{{{{h\nu } \mathord{\left/ {\vphantom {{h\nu } {kT}}} \right. \kern-0pt} {kT}}}} - 1} \right)}}$$
(1)

where T is the temperature, k the Boltzmann constant and n the refractive index, very near from 1 for a plasma and often omitted. The expressions of the blackbody intensity for the other spectral variables are deduced from Eqs. (1) and from the relationship:

$$I_{b\nu } d\nu = I_{b\lambda } d\lambda = I_{b\eta } d\eta$$
(2)

We must distinguish the radiation intensity defined on the basis of projected area, and the emissive power which is the power emitted in a given direction per unit of actual (unprojected) surface area. From Fig. 1, and defining the angle θ between the directions \(\vec{s}\) and \(\vec{n}\), the emissive power E (see Modest [10] and Siegel and Howell [11] for a detailed analysis) is given by:

$$E_{\lambda } = I_{\lambda } \cos \theta$$
(3)

Hence, for the blackbody isotropic radiation, the hemispherical spectral emissive power E bλ is the power emitted by a black surface per unit area and per unit wavelength, integrating over all the solid angles of the hemisphere:

$$E_{b\lambda } = \pi \cdot I_{b\lambda } \quad {\text{or}}\quad E_{b\nu } = \pi \cdot I_{b\nu }$$
(4)

The Planck’s law can be written on the form of the blackbody hemispherical spectral emissive power. As an example for the frequency:

$$E_{b\nu } (T) = \frac{{2\pi {\kern 1pt} h\nu^{3} n^{2} }}{{c_{o}^{2} \cdot \left( {e^{{{{h\nu } \mathord{\left/ {\vphantom {{h\nu } {kT}}} \right. \kern-0pt} {kT}}}} - 1} \right)}}$$
(5)

The total blackbody emissive power or intensity can be calculated by integrating over all the frequencies (or wavelengths). The total hemispherical emissive power of a blackbody is then given by the well-known Stefan–Boltzmann’s law:

$$E_{b} (T) = \sigma_{S} T^{4}$$
(6)

where \(\sigma_{S}\) is the Stefan–Boltzmann constant equal to 5.6704 × 10−8 W m−2 K−4.

Obviously thermal plasma is not a blackbody and the radiation intensity within discharge thermal plasma is far from the Planck intensity. Nevertheless, as a good approximation, thermal plasmas are often considered in the state of local thermodynamic equilibrium (LTE) which allows using the Kirchhoff’s law. To introduce this law two other functions must be defined, the absorption and the emission coefficients.

The absorption coefficient K corresponds to the attenuation of a radiation beam along a distance x, following the expression:

$$I = I_{o} \exp \left( { - K \cdot x} \right)$$
(7)

I 0 being the incident intensity. K is the inverse of a length (unit m−1). This coefficient is a property of the gas or plasma and it depends strongly on the nature of the gases, on the physical conditions (temperature, pressure) and on the spectral variable.

The emission of an isothermal volume element is the radiation power emitted by this volume and can be defined as 4πε, ε being the emission coefficient and the term 4π representing the integration of the isotropic intensity over the total solid angle. The unit of 4πε is W m−3 for the total radiation (integrated on the spectral variable) or W m−3 s for the spectral radiation (with frequency here).

Considering the microreversibility between emission and absorption in equilibrium conditions [10, 11], it can be demonstrated that the emission and absorption coefficients are linked with the blackbody intensity, by the following relation.

$$\varepsilon_{\nu } = K(\nu ) \cdot I_{b\nu }$$
(8)

This form of Kirchhoff’s law is very important for studying thermal plasmas because it is valid in LTE conditions and not only in complete equilibrium. Finally let us introduce the induced emission which corresponds to the emission of a photon stimulated by a photon from the radiation field. It acts as a negative absorption. For this reason, the true absorption coefficient \(K(\nu )\) is corrected in order to take into account the induced emission. In LTE conditions, the corrected absorption coefficient \(K^{{\prime }} (\nu )\) is written:

$$K'(\nu ) = K(\nu ) \cdot \left( {1 - \exp \left( { - \frac{h\nu }{kT}} \right)} \right)$$
(9)

Emission and Absorption Phenomena in Thermal Plasmas

We present here some generalities on these phenomena, details can be found in the corresponding references. In thermal plasma radiation can be emitted or absorbed following various kinds of mechanisms that can be classified in 4 categories:

Atomic Continuum

The reactions involve free electrons and atomic particles (neutral or ions).

Bremsstrahlung

A free electron in the field of an ion or an atom may be slowed down and therefore loses energy under the form of radiation. Approximate expressions are given in [12, 13]. As the mean kinetic energy of the electrons is of the order of 1–2 eV, the photons emitted by these mechanisms have in general a limited energy, of the order or less than 1 eV, so that this radiation is mainly located in the infrared (IR) part of the spectrum.

Radiative Recombination

It corresponds to the interaction of a free electron with a positive atomic ion. The emission coefficient of recombination is given by Cabannes and Chapelle [12] and Gleizes et al. [13]. The non-hydrogenic structure of the elements is taken into account with the Biberman-Schlüter factor determined for a lot of species [14–16]. Data for metallic species can be found in [2, 3, 18]. A large amount of data on photoionisation cross sections is available from the Opacity Project and TOPbase site [17].

Radiative Attachment

This phenomenon only occurs in presence of an electro-negative species and leads to a continuum emission, the emitted photons having energy higher than the attachment energy. The total emission due to this mechanism is always small in comparison with recombination.

Molecular Continuum

Due to the high temperature in central regions, the molecules, if they exist, are in general located in the external regions, and their role in the continuum component is mainly to absorb radiation through two mechanisms: photodissociation and photoionisation. For a given molecule these reactions are grouped and represented by a global absorption coefficient deduced from a global cross section [19, 20]. More complex examples are given in [21].

Atomic Lines

They correspond to radiative transitions between excited levels of a same atomic species and they often constitute the most important part of radiation transfer in thermal plasmas. The local total intensity of a line is simple to compute because it is proportional to the population of the upper state number density (deduced from the equilibrium composition) and of the spontaneous emission coefficient whose values can be found in literature, for a lot of neutral and ionized atoms. In particular a very large compilation and useful data from experimental and theoretical works are given by the NIST Institute that can be completed by Kurucz’s database [22, 23].

But the main problem in the treatment of the lines is the line profile: because of the exponential form of the radiation transmission (Eq. 7) the central part of the line profile can be strongly attenuated. Hence the study of line broadenings is essential and various experimental or theoretical approximations must be used from literature (see specific references in [2, 3, 8, 18–20, 24]).

Molecular Bands

The molecular structure is much more complicated than the atomic one because more degrees of freedom are concerned. For example, for a diatomic molecule, vibration and rotation of the atoms are added to the electronic excitation involved in atomic structure. All the additional phenomena are quantified, and the allowed transitions between these energy levels lead to a great number of lines that are grouped in “bands”. The fine description of this molecular structure is out of the scope of this paper and is described in special books, the most famous are those written by Herzberg [25].

For diatomic molecules some codes or databases useful to compute the spectral properties are available for specific species, based on scientific articles [26, 27]. It must be added that several hundreds of papers have been published on the spectral description of radiation properties of atoms and molecules.

Total Spectra

The most useful description of radiation for a gas or plasma is the spectral variation of the absorption coefficient for a given general composition. The best method to take into account the atomic lines and the molecular bands is called the “line by line” method. This kind of calculation has been made for plasmas in atomic gases such as argon and metallic vapor [8, 9] but the results are more spectacular with molecular gases as it is the case for air plasmas [19, 20, 28].

The influence of the presence of copper vapour on the total spectrum of an air plasma is illustrated in Fig. 4 from [29]: this influence is marked at low and intermediate temperature due in particular to resonance lines. At high temperature the relative influence is not so important. We can see in Fig. 5 from [30] that the quantitative influence of the metal depends on the nature of this metal. In this example the spectrum of tungsten contains much more lines than that of copper. We have seen in “Introduction” section that the ionisation potential is another parameter explaining the role of vapour radiation.

Fig. 4
figure 4

Absorption coefficient of pure air (left hand side) and air with 20mass% copper vapour (right hand side) as the function of frequency and temperature (from [29])

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Fig. 5
figure 5

Total absorption coefficient for the air plasma with admixtures of Cu and W at temperatures of 7000 K, from [30] (© IOP Publishing. Reproduced with permission. All rights reserved)

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Net Emission Coefficient: Influence of Metal Vapours

Definitions

Assuming a non-scattering medium, the equation of radiative transfer along the direction s, can be written [11]:

$$\frac{{dI_{\lambda } }}{ds} = - K(\lambda ,s) \cdot I_{\lambda } (s) + K(\lambda ) \cdot I_{b\lambda } (s)$$
(10)

This equation has been written here for a monochromatic radiation and assuming that the refractive index of the plasma is 1. K(λ) is the absorption coefficient corrected with the induced emission: for simplifying K′ is replaced by K. Let us consider now the radiative term in the energy conservation equation: in an elementary volume dV per time unit, this term is the difference between the power deposited in dV and the power emitted by this volume (“net” power). The radiative flux vector \(\vec{q}_{r}\) can be defined in the direction \(\vec{s}\) across an elementary area dA taking into account all the directions of the incident intensities [11]:

$$\vec{q}_{r} = \int_{\omega = o}^{4\pi } {I(x,y,z) \cdot \vec{s} \cdot d\omega }$$
(11)

Hence the radiation energy term required in the energy conservation equation is the net radiative balance which is the divergence of the radiative flux that can be expressed by:

$$- \vec{\nabla } \cdot \vec{q}_{r} = \int_{\lambda = o}^{\infty } {K(\lambda )\left( {\int_{\omega = o}^{4\pi } {I_{\lambda } (\omega ) \cdot d\omega } } \right) \cdot d\lambda } - 4\pi \int_{\lambda = o}^{\infty } {K(\lambda ) \cdot I_{b\lambda } \cdot d\lambda }$$
(12)

The calculation of the divergence of the radiation flux is very complex in practical conditions. This calculation can be performed assuming severe added assumptions. Two kinds of treatment can be schematically defined: the first one is a simplification of the spatial or geometrical conditions; the second one is a simplification in the spectral description of radiation.

A very simple but useful treatment of the radiation term in thermal plasma modeling is called the “net emission coefficient” (NEC) introduced by Lowke [31]. The main assumption is that the plasma is considered isothermal and homogeneous (in case of mixtures). The second condition is a simplified geometry. Let us consider a sphere of radius R p , the NEC ε N is defined as the divergence of the radiative flux per solid angle unit that means that the radiation term in the energy equation will be −4πε N . The minus sign is due to the fact that in the hottest regions of the plasma the radiation is an energy loss phenomenon.

From Eq. 12, considering an isothermal spherical plasma, the NEC is written:

$$\varepsilon_{N} (T,R_{p} ) \equiv \frac{{\vec{\nabla } \cdot \vec{q}_{r} }}{4\pi } = \int_{\lambda = o}^{\infty } {K(\lambda ) \cdot I_{b\lambda } \cdot d\lambda } - \int_{\lambda = o}^{\infty } {K(\lambda ) \cdot I_{b\lambda } \cdot \left( {1 - e^{{ - K(\lambda ) \cdot R_{p} }} } \right) \cdot d\lambda } = \int_{\lambda = o}^{\infty } {K(\lambda ) \cdot I_{b\lambda } \cdot e^{{ - K(\lambda ) \cdot R_{p} }} \cdot d\lambda }$$
(13)

The interests of the use of the NEC in modelling are its simplicity and its good accuracy for evaluating the radiative losses from the hottest regions. Its computation using Eq. 13 is rather long but it can be performed independently of the general modelling and the NEC can be tabulated as a function of the temperature T and of the radius R p for a given gas (or mixture of gases) at a given pressure.

Properties of the Net Emission Coefficient

The NEC can be calculated using two methods:

  • Direct integration (Eq. 13) with a very small wavelength step. It takes between 200,000 spectral points (for Ar) to more than one million points (for molecular plasmas with diatomic bands) to have a fine description (“line by line”) of the absorption coefficient corresponding to Figs. 4 and 5.

  • The second is a method requiring much less cpu time for the computation and assuming that the line overlapping has no influence on the radiative transfer [2–24], using an escape factor for each line.

The use of escape factors tends to overestimate the NEC because the overlapping of the lines not taken into account through the escape factors, tends to attenuate the transferred radiation. At atmospheric pressure and for a plasma size of the order of a few mm, the error is acceptable, but it increases at very high pressure [32] and for high values of R p [8, 32].

As the object of this paper is to show the influence of metal vapors on radiation properties we will present the variations of the NEC in presence of these vapors. Figure 6 from [30] shows the variations of the NEC for a mixture 90%air–10%Cu (mole or volume proportions). The case R p  = 0 is a fictitious case assuming that radiation is not absorbed (exponential term in (12) is equal to 1). With this Fig. 6, we can see two general properties of the NEC whatever the gases or vapours involved: fast increase of the NEC with temperature up to 10–12 kK, this threshold depending on the nature of the species; strong absorption of radiation, even in the first tenth of mm.

Fig. 6
figure 6

Net emission coefficient of a mixture 90%air–10%Cu (molar proportions) for various values of plasma radius (from [30]) (P = 105 Pa) (© IOP Publishing. Reproduced with permission. All rights reserved)

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The influence of the presence of metal vapour on the NEC is illustrated in Figs. 7 and 8, showing that the radiation of a plasma (here argon or air) is strongly enhanced by the metal vapour at low and intermediate temperature. This is due to the low ionisation potentials of the metals (see Table 1) leading to an increase of the number densities of the electrons and of the atomic (neutral or ions) excited levels at a given temperature in comparison with those of the pure gas.

Fig. 7
figure 7

Net emission coefficient for Ar–Al mixtures (mass proportions) [2] (© IOP Publishing. Reproduced with permission. All rights reserved)

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Fig. 8
figure 8

Net emission coefficient for air–metal mixtures (mass proportions) [18] (© IOP Publishing. Reproduced with permission. All rights reserved)

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Apart from the quantitative values of the NEC, Figs. 7 and 8 give some general information:

  • The NEC of the metal does not only depend on the ionisation potential, but also on the atomic structure: Fe is more emissive than Ag or Cu because of its rich line spectrum.

  • The relative influence of the metal depends also on the initial gas. For example, air is more emissive than Ar (at a given temperature) so that the same amount of a metal vapour has a relative effect greater with Ar than with air.

  • At low temperature where the NEC of the pure gas is negligible in front of the metal NEC, the NEC of the mixture is roughly proportional to the metal partial pressure (this is not an exact law but it could be quite convenient in practical applications).

Figure 9a shows the influence of pressure on the variations of the NEC given by Aubrecht et al. [30]. The presence of metal does not change the general behaviour observed for pure gases. In the low and intermediate temperature range (T < 15 kK) the NEC is rather proportional to pressure which is not exactly the case at higher temperature (easy to see at 20 kK on Fig. 9). This evolution is connected to the variations of the equilibrium composition with pressure, and in particular to the variations of the electron number density.

Fig. 9
figure 9

a Influence of pressure on the NEC (from [30]) (in units of bar with 1 bar = 105 Pa) (© IOP Publishing. Reproduced with permission. All rights reserved). b Influence of pressure on the NEC for air–Fe mixture

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We present on Fig. 9b some original unpublished results showing the influence of pressure on the NEC of an air–5%iron mixture (mass proportion). They have been obtained following the bases of paper [18], and they present a similar behaviour.

Several teams have calculated the NEC for metal containing thermal plasmas and we have indicated in Table 2 the corresponding references classifying them following the metal nature. In general the temperature range lies between 2 and 30 kK.

Table 2 Bibliography on the calculation of net emission coefficients for metal containing thermal plasmas
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Role of Metal Radiation on General Thermal Plasmas Properties

Radiative Transfer

The NEC allows evaluating the radiation power escaping the hottest regions of the plasma but it is not able to determine the total radiative transfer, and in particular not able to calculate the part of radiation absorbed in the external regions. This knowledge is often not necessary because in general radiation does not play an essential role in the energy balance of the external regions, but in certain cases it is important to calculate this absorption (high voltage circuit-breaker for example) and/or to estimate the radiation flux escaping the plasma, responsible for material ablation rather far from the plasma. This problem has been solved for several molecular gases such as air and SF6 using in particular P1 method or Discrete Ordinates Method (DOM) (other methods have been explored for thermal plasmas but are less efficient). But the study of radiative transfer in presence of metal vapour has not been strongly developed. This is mainly due to the fact that the presence of metal enhanced the emission from the hottest regions but changes only weakly the absorption of radiation in the other zones, and the use of the NEC is often sufficient to take this effect into account. Nevertheless we may mention first, the work proposed by the team of profs. Pfender and Heberlein (Menart et al. [9]) using a line by line method rigorously applied to a pure argon arc and applied to an Ar–Cu arc with simplifications (same temperature field and uniform repartition of Cu).

In iterative procedure used in a general modelling, the computation of the radiative flux and of its divergence cannot be performed for all the spectral points (of the order of 1 million points for a given temperature!). Hence a simplified spectral description must be used and in thermal plasmas it consists to calculate the Mean Absorption Coefficient (MAC) on a few spectral intervals, which is not trivial [21, 32, 51, 52]. These values of the MAC can then be applied to compute the radiative transfer using the P1-method as it was the case for metallic halogenates lamp [39] in a general modelling. For mixtures with metals others than that studied in [39] only a few variation of the MAC have been published (and sometimes associated with the P1-model in simplified cases of imposed temperature field) such as Ar–Fe [53], SF6–Cu [54] and complex mixtures with salted water [55].

We present in Figs. 10 and 11 some new results relative to the MAC we have calculated for air–iron mixtures following paper [18] for the bases of radiation properties calculation. We have used the Planck MAC defined in a given spectral interval [λ 1 − λ 2] by:

$$\bar{K} = \frac{{\int_{{\lambda_{1} }}^{{\lambda_{2} }} {K_{\lambda } I_{b\lambda } d\lambda } }}{{\int_{{\lambda_{1} }}^{{\lambda_{2} }} {I_{b\lambda } d\lambda } }}$$
(14)

where I bλ is the blackbody intensity. We have determined 6 spectral intervals noted 1–6 in the figures with limits given in Table 3.

Fig. 10
figure 10

Influence of iron on the mean absorption coefficient of air for 6 spectral intervals (see Table 3 for the intervals). Solid line: pure air; dashed lines: 95%air–5%Fe (in mass)

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Fig. 11
figure 11

Influence of the mass proportion of iron on the MAC of air–Fe mixtures (solid 1%Fe; dash 5%Fe; dot 20%Fe)

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Table 3 Spectral intervals used for calculating MAC represented in Figs. 10 and 11
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We can see from Fig. 10 that for the two first intervals corresponding to the highest photon energies, the values of the MAC are not influenced by the presence of iron vapour. This is not the case for the other intervals in particular for the large interval number 5 covering all the visible part and the non-absorbed UV part, where is emitted a very large number of iron lines. The relative influence of iron proportion for intervals 3–6 is detailed in Fig. 11.

Cooling Effect

As seen in Figs. 7 and 8 a weak amount of metal vapour may greatly enhance the radiation emission. But in practical conditions such as arcs in metallurgy the theoretical curves of Figs. 7 and 8 must be coupled with the true temperature field because in general metal vapour tends to cool down the plasma for a fixed current intensity (if an arc) or a fixed input power. In fact two mechanisms are involved in the cooling. The first one is the increase of radiation power losses leading directly to the cooling. The second phenomenon is indirect and is correlated to the increase of the electrical conductivity (due to the low ionization potential of the metal) which tends to enhance the conduction radius of the arc and then to decrease the temperature in the hottest regions. These effects have been largely studied in literature (see [56] for example) and the relative importance of these effects in arc devices depends on the current intensity and on the nature of the gas as it has been demonstrated in [57] illustrated by Fig. 12 showing theoretical axis temperature for a wall-stabilized arc. Solid curves correspond to temperature computed with the correct values of electrical conductivity and of the NEC for the various gases or mixtures. Dashed curves were computed using the electrical conductivity of the mixture and the NEC of the pure gas whereas dotted curves were obtained using the radiation of the mixture and the conductivity of the gas. From Ar–Cu curves we can deduce that the effect of the electrical conductivity is important at low current intensity and negligible at high current whereas the contrary is observed for radiation. For N2–Cu curves the same tendency is observed but the crossing of the fictitious curves is produced at higher current. This is due to the fact that radiation losses in nitrogen plasmas are more important than those of a pure argon plasma, and thus the relative influence of copper vapour on the NEC is lower in nitrogen than in argon.

Fig. 12
figure 12

Theoretical temperature profiles calculated for a wall-stabilized arc (5 mm diameter) with Ar–Cu (left) and N2–Cu (right) plasmas. Solid curves correspond to temperature profiles computed with the correct values of electrical conductivity and radiation whereas the other curves are fictitious using mixed properties (from [57])

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Hence the presence of metal vapour tends to increase the radiation losses but not as strongly as Figs. 7 and 8 could suggest it because of the effective decrease of temperature. It must be added that some resonance metallic lines are emitted in the UV range but not in the VUV range and hence are not absorbed by the cold surrounding gas such as air, which tends to increase the radiation escaping the arc.

Global Energy Balance

Various authors have calculated or measured the radiation losses in electric arcs and compare them to the global power balance. In fact the comparison should be done with the column power balance, subtracting the electrode voltage fall (rather independent on the metal vapour presence) from the total arc voltage. For current intensity of the order of 100–200 A the radiation losses may represent 50% of the input column energy [9, 58], whereas the same authors have indicated that without metal this part is of the order of 25%. The strongest emission of radiation is created by long and powerful arcs seeded by metallic vapour. Strachan [59] measured radiation power in air–Cu arc plasmas of the order of 50% of the input power. For very long (2 m) and high current (up to 40 kA) air arcs with strong electrode ablation, Cressault et al. [60] measured the total radiation received 10 m from the arc. The main results relative to the radiated power are presented in Fig. 13 as a function of the AC intensity and of the nature of the electrode (copper, steel or aluminum). For copper electrodes the results are consistent with the previous ones of Strachan [59]. With Al and Fe vapors the radiated power may reach 80% of the total input arc energy; this result is quite surprising for Al but the authors indicate that a chemical energy (combustion of Al) is added to the electrical energy.

Fig. 13
figure 13

Proportion of radiative energy relative to the total arc energy. The 4 diagrams correspond respectively (from left to right) to 4 values of mean rms current intensity: 4, 10, 20 and 40 kA (from [60]) (© IOP Publishing. Reproduced with permission. All rights reserved)

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Heating of Particles in Plasma

In classical plasma spraying metallic or ceramics particulates are injected in a thermal plasma jet and are heated by various mechanisms. Radiation takes part in the global energy balance due to the interaction of the particulate with the plasma. In a lot of studies, this radiation process is not taken into account because it does seem to be an important term of the balance. Furthermore there is some compensation between the radiation from the plasma to the particle and the radiation emitted by the heated particle (grey body). In pioneer works [6, 7] Pfender and co-authors have shown that the role of radiation emitted by the particulate depends on various parameters.

But in certain conditions the presence of a cloud of metal vapour around the particle may increase greatly the radiation from the plasma to the particle. For example using some simplifications on the radiative transfer, Essoltani et al. [61, 62] have shown that for iron particles in an argon plasma, the properties of the particles and of the surrounding plasma jet can be strongly influenced by the radiation of the mixture plasma whereas radiation of pure argon plasma has no influence. Of course this case of Ar–Fe is extreme because the thermal conductivity of this mixture is rather low whereas the radiation is strongly enhanced by the iron vapour.

Conclusion

In this paper we have first presented and explained the well-known effect of metallic vapours in thermal plasmas, consisting in increasing the radiation emission at low and intermediate temperature ranges for a lot of gas–metal mixtures. Let us note that the calculation of radiation properties for a new metal would need a specific work because the NEC depends on the electronic structure of the metal. For a new mixture of gas and metal and when the NEC of the pure species are known, one can use mixture laws to obtain a first estimation of the mixture radiation [63]. We have also described the influence of metal radiation on global properties showing in particular its influence on the total energy balance and comparing its role with that of the electrical conductivity on the plasma cooling.

All the presented results assume that the plasma is in LTE. This hypothesis allowing the use of the Kirchhoff’s law is largely used and validated in thermal plasmas studies. Departures from equilibrium may occur in the external regions of the plasmas where the electron number density n e is lower than a few 1021 m−3. The presence of metallic vapour increases the validity of LTE for two reasons:

  • This vapour tends to enhance the value of n e in these regions, because the ionisation potential of the metals is lower than that of the usual gases (see Table 1);

  • At very low temperature, metal radiation does not play any role in the plasma because the metal (pure metal or its by-products such as oxides, nitrides, …) condensates.

Finally the NEC is largely used in presence of metals but the real radiative transfer is in general not computed. Is it necessary to treat it using for example mean absorption coefficients and then a classical method such as P1 or Discrete Ordinate methods? In general the spectral region where radiation is strongly absorbed corresponds roughly to the VUV region (λ < 200 nm) and the most important part of the neutral metal radiation is emitted and reabsorbed at λ > 200 nm, even the resonance lines. Hence as a first approximation, a good solution could be a hybrid method coupling the radiative transfer in the VUV region if necessary (for air or SF6 for example) and the use of a NEC (without the VUV part) in the other spectral ranges. Of course the reality is more complex in particular if the temperature in the central region is higher than 12,000 K, the most emissive metal species being the ions, with resonance lines in the VUV range. But even in this case the use of P1 or DOM method is not recommended because they are enable to correctly treat the real line radiative transfer. If such treatment is needed, in particular to interpret some experimental spectra showing a very marked absorption of resonance lines, with non-homogeneous distribution of metal vapour (see for example [64]) a special calculation should be developed.