Evaluation of the composition, the pressure, the

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The pressure, the composition, the internal energy, the heat capacity and ... temperature range 5000–25 000 K at thermodynamic equilibrium. ... to explain the changes in the physical quantities throughout the .... present in the mixture, R is the perfect gas constant, T is the .... The heat capacity at constant volume shown in.
INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 34 (2001) 1657–1664

www.iop.org/Journals/jd

PII: S0022-3727(01)17917-9

Evaluation of the composition, the pressure, the thermodynamic properties and the monatomic spectral lines at fixed volume for a SiO2–Ag plasma in the temperature range 5000–25 000 K W Bussi`ere and P Andr´e Laboratoire Arc Electrique et Plasmas Thermiques, Universit´e Blaise Pascal–CNRS, Phys. Bˆat. 5, 24 Avenue des Landais, F 63177 Aubiere Cedex, France

Received 11 October 2000 Abstract The pressure, the composition, the internal energy, the heat capacity and several monatomic spectral line intensities are calculated at constant volume for a plasma composed of SiO2 and Ag for several initial densities and in the temperature range 5000–25 000 K at thermodynamic equilibrium. We show that with a small quantity of material in the plasma we obtain a high pressure. From the heat capacity and composition calculation, we deduce that the main reactions are the ionization of Ag, the dissociation of SiO2 to SiO with further dissociation and ionization of Si and O in the considered temperature range. Furthermore, with the monatomic spectral line calculation, we deduce that the oxygen spectral line has a behaviour rather different from those emitted by Ag and Si. 1. Introduction High-breaking-capacity (HBC) fuses are widely used to ensure a specific current cutting. Such fuses are classically made up of four principal elements: the fuse element, the arc quenching material (usually silica sand), two metal electrodes and the insulating cartridge. This paper is concerned with the establishment and the extinction of the arc plasma which leads to the current cutting in the fuse. Because of the dynamic evolution, it is quite difficult to explain the changes in the physical quantities throughout the phenomenon. The arc plasma is initiated because of the current flow through the fuse element, generally in silver; in addition to the fusion of silver, some metallic vapours are produced together with silicon vapours because the arc continuously interacts with the surrounding insulator [1,2] which consists of silica sand. The physical quantities, such as the temperature, the electron density and the pressure of the inner part of the arc plasma, depend on the material amount removed in the two physical processes (melting and vaporization). The very high pressure produced by the creation of the arc plasma causes the liquid and the vapours to spread towards the surrounding filler between the sand grains. Thus, the arc plasma is sustained by a continuous contribution from the surroundings until the 0022-3727/01/111657+08$30.00

© 2001 IOP Publishing Ltd

dissipated energy is not sufficient enough. Once the erosion of the fuse element is finished, all the phenomena are made at constant volume. This is the reason why it is attractive to calculate the composition of the plasma and the ensuing physical properties for several mixtures. The physical mechanisms responsible for the arc plasma evolution during the fuse operation are of different types: (i) overheating of the silver fuse element leading to the fusion and the partial vaporization of the metal, the necessary energy being supplied by the electric current—this defines the pre-arcing period; (ii) creation of the gap in which the arc is immediately initiated—this defines the beginning of the arcing period; (iii) fusion and vaporization of the silica sand; (iv) development of the arc plasma column which is enclosed within a tube of molten silica; (v) energy dissipation in the form of radial and axial emission from the arc column to the surroundings. Some of the major processes are: transmission of the energy from the arc plasma to the silica tube wall, flow of the fused material (especially fused silica) in the interstices between the solid sand grains under the effect of the pressure increase in the arc plasma column.

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1657

W Bussi`ere and P Andr´e

The variety of physical processes makes it very difficult to build up an accurate model for a description throughout the breaking phenomenon in a fuse. The pre-arcing period, the arc plasma column growth, the energy dissipation (convection, conduction, radiation), the fluid material diffusion, the physical properties of the arc plasma which define the characteristics of the energy supply in the energy dissipation mechanism, all have to be treated separately. Some interesting physical quantities can be evaluated from the experiments: the rise of the pressure [3,4], the evolution of the temperature and of the electronic density during the fuse working [5–8], and the energy dissipated in the fuse element and in the fuse filler. The experiments leading to these results are difficult to create because of the presence of the insulator in a granular form, and because of the temporal and spatial evolution of the arc plasma. The experimental results are useful to partially explain the influence of some parameters on the arc plasma evolution, such as the silica sand properties [9]; such results also help to give an approximate description of the physical mechanisms responsible for the arc extinction and for the energy dissipation. Moreover, the experimental results are widely used in the numerical simulations of the electrical and physical quantities of the arc plasma. The calculations published until now can be separated into two categories, as follows. • The breaking process is carried out using an equivalent electrical circuit including the fuse in the global powersupply circuit, the description changing with the overload current level and with the fuse type. Dolegowski [10] has given some electrical and design parameter calculations (the current flowing in the circuit, the voltage across the fuse, the arc energy, the fuse element dimensions) and the possibility of evaluating the fuse length for a given short-circuit condition, which is of great importance for manufacturers. The relationships between the physical and electrical parameters are deduced from the experiments and are presented in the form of empirical laws depending on the current densities used in the tests. Turner and Turner [11] deduced the fuse element erosion rate calculation from the experimental study of the erosion loss rate of a pair of metallic contacts. In the same way Wilkins and Gnanalingam [12] determine the equation describing the burn-back velocity of silver strip fuse elements in quartz-filled cartridges from a series of tests. • The fuse arcing phenomenon, or a part of it, is carried out in the form of a digital simulation including empirical relationships between electrical and physical parameters and some simplifying assumptions. Wright and Beaumont [13] in particular consider that the electrode material which is vaporized represents 40% of the total mass fused and this assumption is made for the two electrodes; such amounts of vaporized metal should imply very high pressure levels. Moreover, in the calculation of the electrical conductivity of the positive column, Wright and Beaumont [13] have assumed that the temperature and the electron density are relatively uniform on the one hand, and that the atoms (ions) and the electrons are characterized by the same temperature on the other hand. These latter assumptions seem to be difficult to explain in so far as the material which sustains the arc plasma is injected from the surrounding layers of silica sand. 1658

From all the above remarks, it will be useful to have the SiO2 –Ag plasma composition and thermodynamic properties. From the bibliographic study, it seems that there has been no publication of these physical properties in the case of SiO2 –Ag plasma. Furthermore, the authors have usually made the calculation at constant pressure [14–16]. At thermodynamic equilibrium, the two main methods of calculation are the mass action laws (Saha, Guldberg and Waage) [14] and the Gibbs free-energy minimization [15, 16]. In these papers, we chose to use the method based on the Helmoltz free energy (similar to the Gibbs free-energy minimization). Furthermore, since the pressure reached in fuses is high (0.5 MPa to 20 MPa) the calculation is made assuming the thermodynamic equilibrium (LTE). In the first part of this paper we give the theoretical formulation used to characterize the chemical composition of the arc plasma, the internal energy, the pressure, the heat capacity and the monoatomic spectral line intensities. All these properties are deduced assuming LTE from the expression of the total Helmoltz free energy. This allows us to perform the calculations considering that the pressure is the variable parameter. The results will be given and then discussed in the last part of the paper.

2. Theoretical formulation If the temperature T and the volume V are maintained constant, the thermodynamic equilibrium is reached when the total Helmholtz free energy is minimum. The Helmoltz free energy can be written as a function of the chemical potential: A=

N  i

    ni RT 0 + e ni µ0i + RT ln − RT i VP0

(1)

where ni is the molar density, N is the total number of species present in the mixture, R is the perfect gas constant, T is the temperature, V is the volume occupied by the mixture, P 0 is the reference pressure (105 Pa), and µ0i is the chemical potential of the i species calculated at the reference pressure. To minimize the relation (1), we need to write the conservation of nuclei number and the quasi electrical neutrality: N 

cij ni = bj

j = 1, . . . , m

(2)

i=1

where m is the number of different nuclei; cij represents the nucleus number of type j in particle i; bj is the total number of nucleus of type j that is determined from the initial density; ci1 represents the number of electronic charges for the particle i; the electrical neutrality thus imposes b1 = 0. In the calculation, we take into account: • 16 monatomic species—Ag− , Ag, Ag+ , Ag++ , Ag+++ , e− , O, O− , O+ , O++ , O+++ , Si, Si− , Si+ , Si++ , Si+++ ; + • 6 diatomic species—O2 , O− 2 , O2 , Si2 , SiO, Ag2 ; • 3 polyatomic species—O3 , SiO2 and Si3 . The calculations of monatomic and diatomic thermodynamic properties are described in the appendix. As regards the polyatomic molecules, all the data are taken from the thermochemical tables given in [17].

Thermodynamic properties in SiO2 –Ag plasma

To determine the pressure, we use the Dalton law: P − P =

N 

ni RT

8

10

(3)

7

10

i=1 Pressure (Pa)

3

N  T 2 q i=1 i ni

3

0.5 kg/m

6

10

3

0.2 kg/m

0.1 kg/m

3

5

10

(4)

4

10 5000

10000

15000

1

20000

25000

Temperature (K)

2

(a)

(5) 8

10

where ε0 is the vacuum permittivity. We do not take into account the effect of high pressure on the equation of state because we consider that the temperature is high enough [19], and furthermore the experimental pressure is hard to measure precisely [4]. Concerning the internal energy, the following formulation is used:   N  U= ni hi − H − P V /Md. (6)

5 kg/ m 2 kg/m 7

10

3

3

3

1 kg/m

Pressure (Pa)

ld = ε0 R

3

1 kg/m

with the Debye length ld defined by 

3

2 kg/m

where P is the pressure correction due to Coulombic interaction between the charged particles. This pressure correction is written as [18]: N  1 q 2 ni P = − 24πε0 ld i=1 i

5 kg/ m

0.5 kg/m

3

3

0.2 kg/m 6

0.1 kg/m

10

3

i=1 5

10 5000

In parentheses we recognize the specific enthalpy of each species i, where hi is the specific enthalpy calculated from the partition functions, Md is the massic volume, and H is the enthalpy correction due to Coulombic interaction between the charged particles [18]:

15000

20000

25000

Temperature (K)

(b) 8

10

3

3

SiO2 : 1 kg/m Ag : 1 kg/m

N V  q 2 ni . 6πε0 ld i=1 i

3

SiO2 : 2.5 kg/m Ag : 2.5 kg/m

3

SiO2 : 0.5 kg/m Ag : 0.5 kg/m

3

3

7

10

By numerical derivation of the internal energy we obtain the heat capacity at constant volume:   ∂U . (7) Cν = ∂T ν

Pressure (Pa)

H = −

10000

6

10

3

SiO2 : 0.25 kg/m Ag : 0.25 kg/m 3

3

SiO2 : 0.05 kg/m Ag : 0.05 kg/m

3

SiO2 : 0.1 kg/m Ag: 0.1 kg/m

3

3

5

From the composition calculation, we can determine the spectral line intensities of the monatomic species. We use the Boltzmann distribution at temperature T to calculate intensities: ni 1 hc Amn gm e−Em /kT I (λmn ) = 4π λmn zint (T )

10

4

10 5000

10000

15000

20000

25000

Temperature (K)

(8)

where Amn is the transition probability, λmn is the wavelength between the upper level m and lower level n, gm is the statistical weight, Em is the energy of the upper level, ni is the total concentration of the species i, and zint (T ) is the internal partition function calculated at temperature T . All the necessary data are given in table 1.

3. Results In figure 1, we give the pressure obtained for various densities of Ag and SiO2 , and for a mixture of 50% of Ag and 50%

(c) Figure 1. Pressure calculated for several initial densities versus temperature for: (a) Ag; (b) SiO2 ; (c) Ag and SiO2 .

of SiO2 (the proportion is given in weight percentage). We notice that, for the same density, the pressure of the SiO2 plasma is higher than the pressure of the Ag plasma. As a matter of fact, the molar mass of Ag is higher than that of SiO2 . Indeed, in each case, the pressure increases when the temperature increases. Respectively, in figures 2 and 3, we show the electronic concentration and the internal energy obtained for various 1659

W Bussi`ere and P Andr´e 7

26

4x10

10

5 kg/m

3

2 kg/m

25

10

1 kg/m

0.1 kg/m 7

3x10

3

Internal Energy (J/kg)

0.5 kg/m

3

3

24

-3

[e-] (m )

10

0.1 kg/m

3

0.2 kg/m

3

23

10

0.2 kg/m 0.5 kg/m

3

3

3

7

2x10

1 kg/m 5 kg/m

7

2 kg/m

3

3

3

1x10

22

10

21

10

5000

10000

15000

20000

0 5000

25000

10000

Temperature (K)

15000

20000

25000

Temperature (K)

(a)

(a) 8

26

1.5x10

10

1 kg/m

2 kg/m

3

5 kg/m

3

3

25

10

Internal Energy (J/kg)

0.5 kg/m

24

-3

[e-] (m )

10

0.1 kg/m

3

0.2 kg/m

3

0.5 kg/m

3

23

10

8

1.0x10

0.2 kg/m 0.1 kg/m

3

3

3

1 kg/m 2 kg/m

8

0.5x10

5 kg/m

3

3

3

22

10

0 5000

21

10

5000

10000

15000

20000

25000

10000

15000

26

10

3

3

24 3

-3

[e-] (m )

SiO2 : 0.5 kg/m Ag : 0.5 kg/m 3 3 SiO2 : 0.25 kg/m Ag : 0.25 kg/m 3

SiO2 : 0.1 kg/m Ag: 0.1 kg/m 23

3

0.8x10

8

3

10

10

8

SiO2 : 0.25 kg/m Ag : 0.25 kg/m

SiO2 : 0.05 kg/m Ag : 0.05 kg/m

3

3

3

Internal Energy (J/kg)

3

1.0x10 3

25

10

SiO2 : 2.5 kg/m Ag : 2.5 kg/m

25000

(b)

(b)

SiO2 : 1 kg/m Ag : 1 kg/m

20000

Temperature (K)

Temperature (K)

0.6x10

8

0.4x10

8

3

SiO2 : 0.1 kg/m Ag: 0.1 kg/m

3

SiO2 : 0.05 kg/m Ag : 0.05 kg/m

22

10

0.2x10

3

3

3

SiO2 : 2.5 kg/m Ag : 2.5 kg/m

3

3

SiO2 : 1 kg/m Ag : 1 kg/m

8 3

SiO2 : 0.5 kg/m Ag : 0.5 kg/m

3

3

3

21

10

5000

10000

15000

20000

25000

Temperature (K)

(c)

5000

10000

15000

20000

25000

Temperature (K)

(c)

Figure 2. Electronic concentration calculated for several initial densities versus temperature for: (a) Ag; (b) SiO2 ; (c) Ag and SiO2 .

Figure 3. Internal energy calculated for several initial densities versus temperature for: (a) Ag; (b) SiO2 ; (c) Ag and SiO2 .

densities of Ag and SiO2 , and for a mixture of 50% of Ag and 50% of SiO2 (weight percentage). We notice that the higher the initial density, the higher the electronic concentration and the internal energy. The internal energy increases with the temperature because the dissociation and ionization processes appear when the temperature increases. Consequently, the energy contained in the plasma increases as the temperature rises. The energy

necessary for the ionization leads to a variation of the internal energy of the plasma. Consequently, the sharp increase in the internal energy corresponds to a chemical reaction. The heat capacity at constant volume shown in figures 4(a)–(c) comes from the derivation of the internal energy. So the peaks can be associated with some chemical reactions. By comparison with the composition (figure 5),

1660

Thermodynamic properties in SiO2 –Ag plasma

Table 1. Spectroscopic properties of the spectral lines studied in this paper [20, 21]. Statistical weight

Upper energy level (cm−1 )

Transition probability (108 s−1 )

Ag Ag Si Si+ Si+ Si+ Si+ Si+ Si+ Si+ Si+ O

5209.068 5465.498 3905.5232 3853.665 3856.018 3862.595 4128.05 4130.87 4130.89 6347.11 6371.37 7771.94

4 6 3 4 4 2 6 6 8 4 2 7

48744.0 48764.22 40991.884 81251.32 81251.32 81191.34 103556.16 103556.16 103556.03 81251.32 81191.34 86631.454

7.50E−01 8.60E−01 1.18E−01 2.80E−02 2.50E−01 2.80E−01 1.32E +00 9.40E−02 1.42E +00 7.00E−01 6.90E−01 3.40E−01

Heat Capacity at Constant Volume

Species

Wavelength (λ)

3000

2000

0.1 kg/m

3

0.2 kg/m

3

0.5 kg/m

3

1000 2 kg/m 1 kg/m 5 kg/m

0 5000

3

3

3

10000

15000

20000

25000

20000

25000

Temperature (K)

(a) 0.1 kg/m

3

Heat Capacity at Constant Volume

0.2 kg/m

7500

3

0.5 kg/m

3

1 kg/m

3

2 kg/m

3

5 kg/m

3

5000

2500 5000

10000

15000 Temperature (K)

(b) 6000 3

SiO2 : 0.05 kg/m Ag : 0.05 kg/m

5000 Heat Capacity at Constant Volume

we can determine the main chemical reaction produced in a specific temperature range. For a pure Ag plasma, the first peak in the temperature range 5000–15 000 K, corresponds to the ionization of Ag; the second peak corresponds to the ionization of Ag+ . For the SiO2 plasma, the first peak in the temperature range 5000–10 000 K corresponds to the dissociation of SiO in Si and O, and the second peak between 8000–15 000 K corresponds to the ionization of Si; the third peak between 13 000–25 000 K corresponds to the ionization of O. In the case of the Ag and SiO2 mixture, because of the choice of the weight percentage, the main species are those produced by SiO2 . So we find the same peaks as in the case of SiO2 . In figure 5, we give the concentration versus the temperature for an initial density of 1 kg m−3 . In figure 5(a), for silver plasma we show that the main species in the temperature range 5000–12 550 K is the monatomic species Ag; and for the temperature range 12 550–25 000 K the main species are the electrons and the monatomic species Ag+ . In the considered temperature range, the electrical neutrality is mainly maintained by Ag+ and e− . In figure 5(b), we represent the composition of a SiO2 plasma. The main species between 5100–22 150 K is the monatomic species oxygen and then until 25 000 K, electrons. The electrical neutrality is fulfilled between Si+ and e− in the temperature range 5000–15 000 K, and then between electron, O+ and Si+ . Indeed, that follows the ionization potential of Si (8.15 eV) and O (13.61 eV). In figure 5(c), we represent the composition of a SiO2 –Ag plasma for an initial density of 1 kg m−3 . This initial density value is chosen because the calculated electron density and the experimental electron density are similar [9]. The weight percentage chosen is 50% Ag and 50% SiO2 . Furthermore, the molar weight is higher for Ag than for SiO2 , thus the number of particles stemming from the dissociation of SiO2 is higher than those produced by Ag. So the main species has to be the same as in the case of SiO2 ; this is actually observed by comparing figures 5(b) and 5(c). The electrical neutrality is mainly made between e− and Ag+ until the temperature of 9800 K, and between e− and Si+ for the temperature range 9800–21 800 K, and then between e− and O+ . Obviously that follows the ionization potential of Ag (7.57 eV), Si (8.15 eV) and O (13.61 eV).

10000

4000

3

SiO2 : 0.1 kg/m Ag: 0.1 kg/m

3

SiO2 : 0.25 kg/m Ag : 0.25 kg/m

3

3

3

3000

3

SiO2 : 1 kg/m Ag : 1 kg/m

2000

1000 5000

3

SiO2 : 0.5 kg/m Ag : 0.5 kg/m 3 3 SiO2 : 2.5 kg/m Ag : 2.5 kg/m

10000

15000

20000

3

3

25000

Temperature (K)

(c) Figure 4. Heat capacity at constant volume for several initial densities versus temperature for: (a) Ag; (b) SiO2 ; (c) a mixture of 50% Ag and 50% SiO2 in weight percentage.

In figure 6, we represent the Ag, Si, Si+ spectral lines and one (O 7771.94 A) of the oxygen triplet. We have chosen these spectral lines because they have been readily observed in experiments [7]. We point out the fact that, for Ag+ spectral 1661

W Bussi`ere and P Andr´e 8

10

Ag : 5465.5 Å

Ag

e-

24

10

Ag+

Ag : 5209.07 Å 7

10

10

Intensity (W/m /sr)

-3

Concentration (m )

++

Ag 22

-

3

Ag

Ag2

6

10

20

10

5

10

18

10

5000

10000

15000

20000

4

25000

10 5000

Temperature (K)

10000

15000

(a)

25000

(a)

26

10

8

10 O

+

Si : 3856.02 Å + Si : 4128.05 Å

Si e-

7

10

+

Si

24

10

Si : 3905.52 Å

+ ++

O2

22

+

Si : 4130.89 Å

+

Si : 6371.37 Å

3

Si

SiO SiO2

Intensity (W/m /sr)

O

-3

Concentration (m )

20000

Temperature (K)

-

O

Si2

10

+

+

Si : 6347.11 Å

6

10

Si : 3862.59 Å

O: 7771.94 Å

-

+

Si

Si : 3853.67 Å

5

10

+ O2

+

Si : 4130.87 Å

20

10

5000

10000

15000

20000

25000

4

10 5000

Temperature (K)

10000

15000

20000

25000

Temperature (K)

(b)

(b)

26

10

8

10

Ag : 5465.5 Å

O -

e

Si

Si

Ag : 5209.07 Å

7

+

Ag

24

10

Ag

+

Si : 4128.05 Å

+

Si : 3856.02 Å

10

Si : 3905.52 Å

Intensity (W/m /sr)

-3

Concentration (m )

+

+

SiO

3

O

++

Si

O2

++

Ag

-

22

10

O

SiO2

+

Si : 4130.89 Å

6

10

+

Si : 6371.37 Å +

Si : 3862.59 Å +

Si : 6347.11 Å

Ag2

Si2

+

Si : 3853.67 Å

5

-

Si

-

Ag

++

10

O

O: 7771.94 Å +

+

Si : 4130.87 Å

O2

20

10

5000

10000

15000

20000

25000

Temperature (K)

(c) Figure 5. Concentration versus temperature of (a) Ag and (b) SiO2 plasmas with an initial density of 1 kg m−3 . (c) Concentration versus temperature of SiO2 –Ag plasma with an initial density of 0.5 kg m−3 for SiO2 and 0.5 kg m−3 for Ag.

lines, we did not find any available data, and furthermore no author has indicated that they have observed these lines in their experiments. So, in figure 6(a), we indicate only two Ag spectral lines. We notice that the intensity decreases when the ionization of Ag appears (figure 5(a)). In figures 5(a) and 5(b), when the ionization appears, the intensity of the spectral lines of neutral monatomic (Ag, Si) species decreases, unlike 1662

4

10 5000

10000

15000

20000

25000

Temperature (K)

(c) Figure 6. Spectral lines intensities for (a) Ag and (b) SiO2 plasmas with an initial density of 1 kg m−3 . (c) Spectral lines intensities for SiO2 –Ag plasma with an initial density of 0.5 kg m−3 for Ag and 0.5 kg m−3 for SiO2 .

the neutral oxygen spectral line whose intensity increases in the temperature range observed.

Thermodynamic properties in SiO2 –Ag plasma

4. Conclusion

Appendix A. Calculation of the monatomic and diatomic chemical potential

All the equations necessary to determine the composition and the thermodynamic properties of a plasma which consists of Ag and SiO2 vapours, at constant volume and thermodynamic equilibrium, have been given. These data can be helpful in the understanding of the fuse working. As a matter of fact, only a few numerical calculations on the physical properties have been made until now; interest has mainly been devoted to the electrical properties [22].

For the i species at thermal equilibrium the chemical potential is written as:   µ0i = −RT ln zitr zint + e0

From the calculation of the pressure we have shown that only few materials can lead to a high pressure in the plasma, and the higher the initial density the higher the pressure. Furthermore the value of the pressure differs according to the initial composition (pure Ag or SiO2 ). Nevertheless the values are quite similar: 20 × 105 Pa for Ag, 70 × 105 Pa for SiO2 at 15 000 K. We have given several electronic concentrations versus the temperature for several initial densities (0.1–5 kg m−3 ) and for three kinds of plasma: Ag, SiO2 , SiO2 –Ag (50%, 50% weight). These curves can be useful for the interpretation of the electronic concentration obtained, for example, from the Stark broadening of the silicon spectral lines [7]. The internal energy can be associated with the energy input in the plasma owing to the electrical supply. We have shown that the higher the initial density in the plasma, the higher the energy needed to obtain the same temperature. By studying the heat capacity at constant volume and the plasma composition, we have shown that in the temperature range 5000–25 000 K, the main chemical reactions are the ionization of Ag, the dissociation of SiO, and the ionization of Si and O. Finally, we have given the intensities of the spectral lines versus temperature. From these results we have pointed out that the intensity of the Ag and Si monatomic spectral lines decreases rapidly with temperature when the ionization occurs, unlike the intensity of the oxygen spectral lines that increases in the temperature range considered (5000–25 000 K). The complete process of arc extinction in a silica-filled HBC fuse-link is accomplished within the first loop of alternating current so that the arc duration is typically of the order of 4 ms. In this time it is unlikely that thermodynamic equilibrium would be attained. However, the calculations based on an assumption of thermodynamic equilibrium made in this paper should give valuable clues to the interpretation of future experimental results. This will be the subject of another paper.

Acknowledgments We thank, both for their financial support and for their help through many discussions, M R Barrault of Schneider Electric, R Dides and S Melquiond of Alstom, M R Rambaud of Ferraz S A, J C V´erit´e of Electricity of France.

where e0 is a reference energy taken equal to the enthalpy of formation at the reference pressure P 0 [17, 23, 24]. The translation partition function is written as   2π mi kT kT zitr = h2 P0 where mi is the mass of the i species, h is the Planck constant and k is the Boltzmann constant. The internal partition function zint for the monatomic species is written as:    at g(n, l, s)e− E(n,l,s)/kTex zint = n,l,s

where g(n, l, s) and E(n, l, s) are, respectively, the degeneracy and the energy of the state with the principal quantum number n, the azimuthal number l and the spin s [20, 25]. The internal partition function zint for the diatomic species is written as [26]: zint =

 e

ge e−L(Te /T )

(e) νmax  ν=0

e−L(Ge (ν)/T )

(ν) jmax  j =0

2j + 1 −L(Fν (j )/T ) e σ

where L = hc/k, h is the Planck constant, k is the Boltzmann constant, and c is the velocity of light. The other terms are defined as follows. • ge is the statistical weight of the electronic excitation level e that is equal to the multiplicity for the molecule having zero orbital angular momentum (- = 0), and twice the multiplicity for the other states (- = 0). The summation is made over the whole of the excitation state whose electronic spectral term Te is given in tables in [27–29]. • Ge (ν) is the vibrational spectral term of the excitation state e. The summation is performed for all vibrational quantum numbers between 0 and νmax (e). The quantum number νmax (e), which depends on the excitation state e, is determined by comparing the energy of dissociation and the energy of vibration. To avoid any problem of calculation due to inaccurate spectroscopic constants, another step of calculation is added: Ge (ν + 1) > Ge (ν)∀ν. • Fν (J ) is the rotational spectral term for the quantum number ν of excitation. The summation is made for all rotational quantum numbers J between 0 and jmax (ν). The latter is the maximum rotational quantum number of the vibration level ν. It is determined by the Herzberg molecular effective potential method [26,27]. σ is a factor of symmetry equal to 1 when the atoms are different, and equal to 2 when the atoms are identical.

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