Abstract

The transport coefficients of Air - PMMA mixtures thermal plasmas is important to estimate the performances of cutting of the electrical arc in this gas by a circuit breaker. In this paper, Air - PMMA mixtures thermal plasmas dynamical viscosity, thermal and electrical conductivities coefficients are calculated in a temperature range from 5000K to 30000K. The calculations are made by supposing thermodynamic equilibrium at pressure of 1 bar to 10 bar. The results of the calculations show the influence of the initial proportion of PMMA but also that of the pressure on the mixture plasma transport coefficients. As the efficiency of the breaking of the electric current by the circuit breaker depends closely on the thermal and electrical characteristics of the extinguishing medium, this extinguishing medium should have a high thermal conductivity and an electrical conductivity varying rapidly with temperature. The plasma of the mixture constituted by 80% of air and 20% of PMMA presents at first sight best characteristics for the breaking of electric current. It has the highest thermal conductivity peak among the plasmas of the mixtures studied.

1. Introduction

The electrical arc which is generally defined as a high current electrical discharge has a wide variety of applications these days. Due to the wide range of its applications, it has given rise to numerous research works both on experimental and theoretical level ^{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}. Among its applications, we can cite lighting, welding, cutting, waste treatment and cutting of electric current. Indeed, the interruption of the electric current in certain switchgear, in particular the circuit breaker, is done from the electrical arc. To cut the electrical current, the opening of the circuit breaker contacts creates an electric arc which interacts with the surrounding environment. If in most applications, the arc must be maintained for as long as possible, its rapid extinction is sought in breaking devices. In fact, they are designed to provide manual or automatic control of electrical circuits (contactors) as well as protection of installations against short circuits (circuit breaker).

Certain arc electric cutting techniques are based on the rapid elongation of the arc between the contacts and others, on a rolling of the arc between two insulating parts. Numerous theoretical ^{3, 4, 13, 18, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} and experimental ^{3, 10} studies have already been carried out on polymers. From studies carried out on laminated arcs with ablative walls ^{2, 3}, it appears that the PMMA polymer has interesting characteristics for breaking the electric current.

Our study is part of the interruption of the electric current by means of the electric arc used as energy dissipater by heat exchanges with the surrounding environment. As the electric current cutting technique in air and compressed air ³⁷ is widely used in low and medium voltage circuit breakers, in this work, we want to study the influence of the PMMA polymer on air breaking performance. The breaking performance is partly linked to certain characteristics of the plasma formed in the circuit breaker at the time of breaking. This article is therefore devoted to the calculation of the transport coefficients of Air - PMMA mixtures thermal plasmas at thermodynamic equilibrium. In addition to pure air and pure PMMA, the mixtures concerned are: 80% Air - 20% PMMA; 50% Air - 50% PMMA; 20% Air - 80% PMMA. The temperature range goes from 5000K to 30000K for pressures of 1 bar; 2 bar; 5 bar; 8 bar and 10 bar. Transport properties such as dynamic viscosity, thermal and electrical conductivities are calculated using expressions deduced from Chapman Enskog theory. Determining any characteristic of a plasma begins with knowing the equilibrium composition of the plasma. The different chemical species that we have taken into account in the composition of plasmas are: e, C, H, N, O, C₂, H₂, N₂, O₂, CH, CO, CN, OH, HN, NO, C⁺, H⁺, O⁺, N⁺, CH⁺, CO⁺, CN⁺, OH⁺, HN⁺, NO⁺, C₂⁺, H₂⁺, N₂⁺, O₂⁺, C²⁺, N²⁺ and O^{2+ 28}.

2. Transport Coefficients

The electrical and thermal conductivities and the coefficient of dynamical viscosity are the transport coefficients retained in this study. The theoretical study of these coefficients is based on the resolution of the Boltzmann equation by the Chapman-Enskog ³⁸ method. Devoto developed exact theoretical expressions which allow the calculation of transport coefficients of gas mixtures at ν components. But for gases constituted by numerous particles as ours, the calculations become complicated from the third-order approximation of Chapman-Enskog method. However, it is shown that we can bring simplifications to the expressions of the transport coefficients of mixture at ν components when one of the chemical species has a very low mass with regard to that of the other components. Indeed, electrons have an unimportant mass with regard to that of the heavy particles. Thus the transport properties of electrons and heavy particles are independently treated.

The following relation gives the total thermal conductivity of the plasma:

(1)

and are respectively the translation thermal conductivities of electrons and heavy particles. is the internal thermal conductivity bound to the existence of freedom degrees due to the molecules vibrations and the rotations. is the thermal conductivity due to the reactions of ionization, recombination and dissociation. and can be written respectively from the second and third-order approximation of Chapman-Enskog method as follows:

(2)

(3)

Where x_i is the molar fraction of the particle i. The term L_{i j} is a function of the heavy particles collision integrals, masses molars and the plasma composition ³⁸. Because of the existence of chemical reactions in the plasma, the transfer of energy can be made under the shape of chemical enthalpy. The thermal conductivity of reaction reports processes not elastics in the plasma. It is defined at the first order approximation by:

(4)

where ΔH_i represents the variation of enthalpy due to the i^th chemical reaction. R is the constant of perfect gases. The transfer of energy in internal movements (rotation and vibration) of molecules is expressed by the internal conductivity. By extension of the theory of Euken for a pure gas, we have:

(5)

is the intern thermal conductivity of the i^th component.

are defined by the relation:

(6)

By neglecting the contribution of the ions, Devoto obtained a simplified theoretical expression for the calculation of the electrical conductivity coefficient:

(7)

n_e is the electrons numerical density and m_e the mass of the electron.

q^ij depends on numerical densities and on average effective sections of particles collision.

The transport of the impulse in the plasma is characterized by the coefficient of dynamical viscosity. It is independent from properties of electrons because of their low mass. A good approximation of this coefficient is obtained by:

(8)

where m_i and x_i are respectively the mass and the molar fraction of the species i; is the coefficient of viscosity of a pure gas defined by Hirschfelder like:

(9)

3. Collision Integrals

Determining the transport coefficients requires knowledge of the collision integrals between the particles present in the plasma. The values of these transport coefficients therefore depend closely on the values of the collision integrals corresponding to the different interactions present in the plasma. The calculation of the plasma transport coefficients can only be carried out once the equilibrium composition of the medium has been determined.

The knowledge of the collision integral is determining for the calculation of the transport coefficients. It is thus important to choose the types of interaction which are appropriate with the nature of the particles populating studied plasma. The calculation of the collision integrals is made by considering four categories of interaction: Neutral - neutral interaction; neutral - ion interaction; electron - neutral interaction and charged - charged interaction.

Two methods were applied for the determination of the collision integrals: the method of smoothing of the published results and the method of the reduced temperatures.

Interaction N-N: We used the values published by Capitelli et al ³ from the Morse potential.

Interaction N-N₂ and N₂-N_2: These interactions are represented by a repulsive exponential potential. The corresponding collision integrals are proposed by Capitelli et al ³.

Interaction H-H, H-H₂ and H₂-H_2: We retained the results of Vanderslice et al ³ but for the collision integral Ω ^{(2, 2)} of the H-H interaction, we chose the values proposed by Belov ³. This author considers an interval of much wider temperature covering our temperature range.

Interaction O-O, O-O₂ and O₂-O₂: We used the values obtained by Mexmain ³ who made a detailed study of these interactions. By the method of the reduced temperatures, we use the reduced collision integrals published by Hirschfelder ³⁸ according to the reduced temperature T^*. For other interactions between not polar neutral species, the used potential is the one of Lennard-Jones.

The reduced temperature is given by:

(10)

(11)

where T_b is the boiling point of the particle, ε is the maximum of attractive energy for an interaction between similar species.

We used the collision integrals published by Spencer et al ³ for e - H, e - O, e - H₂, e - O₂, e - N₂, e - CO, e - OH and e - NO interactions and the method of the stiff spheres for the interactions e - C₂, e - CH, e - CN and e - HN. For the interaction e - N, we retained the values proposed by Capitelli et al ³. The collision integrals of e - C interaction are calculated by Robinson et al ³.

It is the collision integrals calculated by Capitelli et al ³ that we used for the interactions H - H⁺; N - N⁺ and those calculated by Eymard ³ for the interaction N₂ - N₂⁺. For the interactions O - O⁺ and O₂ - O₂⁺, we used the values calculated by Mexmain ³. Other neutral - ion interactions were considered purely elastic and the corresponding integrals of collision was determined from Maxwell's potential.

We used the collision integrals of charged - charged particles calculated by Devoto ³.

In general, the collision integrals published in the references cited previously is given for and . Thus to obtain the others we used the formula of recurrence established by Hirschfelder ³⁸:

(12)

4. Results and Discussion

In this part we present the results of the calculations of the transport coefficients of the studied mixtures plasmas. An analysis of the influence of the PMMA percentage in the mixture as well as that of the pressure on the plasma transport coefficients is made.

Figure 1 shows the evolution of thermal conductivity λ of plasma as a function of the PMMA percentage in the mixture, at a pressure of 1 bar. With the exception of pure air, we observe few differences between the values of the thermal conductivity of the various mixtures plasmas. The peaks of λ observed, at T = 7000K, correspond to the dissociation of the particles CO and N₂. The peaks due to the dissociation of the other diatomic particles of the plasmas of the Air - PMMA mixtures, are located at temperatures below 5000K. The thermal conductivity highest peak at T = 7000K is that of pure air plasma.

For high temperatures, ie for T ≥ 20000K, the plasmas are almost completely ionized. The thermal conductivity is mainly due to the translational thermal conductivity of the electrons. The equilibrium composition of the different plasmas shows that they all have almost the same electron numerical density, for high temperatures ²⁸. This explains the similarity of the thermal conductivity evolution of the various mixtures plasmas beyond 20000K. The influence of the ionization of simple atoms C, H, O and N, appears from temperatures between 13000K and 16000K. In this temperature range, it is the 80% Air - 20% PMMA mixture plasma which gives the most significant thermal conductivity peak. The thermal reaction conductivity constitutes the main component of the total thermal conductivity, for temperatures corresponding to the dissociation and ionization temperatures of the various particles. The thermal conductivity λ plays a very important role on the arc time constant τ. Consequently, the shape of the curve λ (T) gives information on the quality of the mixture considered for cutting the electric arcs. The importance of this thermal conductivity manifests itself particularly through its reaction component λ_R. From the study of several gases, Hertz ³ proposes an empirical relation linking the arc time constant τ to the maximum of the plasma thermal reaction conductivity coefficient: τ.λ_R^max = C^te. This relation clearly shows the primordial role that the thermal conductivity of reaction plays in the speed of extinction of arcs. If the peak of λ_R is high, then the extinction time of the arc is short.

To show the influence of pressure, we have only presented results for the case of the plasma of the 50% Air -50% PMMA mixture to give the most significant results. The remarks made on the influence of pressure on the thermal conductivity of this plasma are valid for the other plasmas. In Figure 2, we have represented the curves, as a function of the temperature, of the 50% Air - 50% PMMA mixture plasma thermal conductivity for the different values of the pressure. The direction of the arrows in the figure indicates the increasing direction of pressure.

It is noted that when the pressure increases, the peaks of the thermal conductivity λ move towards the high temperatures. The translation of the peaks is linked to the displacements of the chemical reaction equilibria that we observed and explained when studying the composition of the plasma. This translation of the ionization and dissociation peaks is accompanied by a slight increase in the value of the thermal conductivity.

The evolution of the electrical conductivity of the mixtures plasmas studied is shown in Figure 3. For the high temperatures, the various mixtures plasmas have almost identical electrical conductivities. At these temperatures, all plasmas are almost completely ionized, and have the same electron numerical density. The electrical conductivity then becomes independent of the nature of the mixture considered.

In the temperature range of 7000K to 13000K, the air plasma has the weakest electrical characteristic. From 13000K to 22000K, it has an electrical conductivity slightly higher than that of other plasmas. Beyond 22000K its electrical conductivity is significantly lower than that of all other plasmas. On the other hand, the electrical properties of 80% Air - 20% PMMA, 50% Air - 50% PMMA and 20% Air - 80% PMMA mixtures plasmas are identical over almost the entire temperature range.

To show the influence of pressure, we have only presented results for the case of the plasma of the 50% Air -50% PMMA mixture to give the most significant results. The remarks made on the influence of pressure on the electrical conductivity of this plasma are valid for the other plasmas studied. Figure 4 gives the curves, as a function of the temperature, of the electrical conductivity of 50% Air - 50% PMMA mixture plasma for the different values of the pressure. In this figure, the arrow indicates the increasing direction of pressure.

It is found that the plasma becomes less conductive when the pressure increases at low temperatures (T ≤ 12000K). The increase in pressure leads to an increase in the concentrations of molecular species. Collisions e- heavy particles are then more important and more frequent. This causes the electrons to brake, and leads to a decrease in the electrical conductivity. From 13000K, an inversion of this evolution of the electrical conductivity is observed; it increases when the pressure increases. At high temperatures, all molecular species are dissociated. Collisions e-heavy particles become negligible. As at these temperatures the plasma is characterized by a high electronic density, this leads to high electrical conductivity values. Furthermore, we note that the influence of pressure on electrical conductivity is identical for all the plasmas studied.

The evolution of the dynamic viscosity of the five plasmas of the mixtures, as a function of the temperature, is given in Figure 5. This figure shows that the most viscous plasmas are those containing a high percentage of oxygen and a low percentage of hydrogen and carbon.

The arrow shows the increasing direction of the PMMA percentage in the mixture which gave rise to plasma. Pure air plasma has the highest dynamic viscosity and pure PMMA plasma the lowest dynamic viscosity. Air plasma is characterized by the presence of heavy particles, especially NO at relatively high temperatures, and this is what contributes to making it fairly viscous. But this presence of the NO species also leads to an increase in the electron density, at low temperature. For all plasmas, there is a similar variation in dynamic viscosity as a function of temperature. It increases at first time to reach its peak between 10000K and 120000K, then decreases rapidly in a second time between 12000K and 20000K and finally tends towards a limit value between 20000K and 30000K.

To show the influence of pressure, we have only presented results for the case of the plasma of the 50% Air -50% PMMA mixture to give the most significant results. The influence of pressure on the dynamical viscosity of this plasma are valid for the other plasmas. Figure 6 shows the evolution of the dynamic viscosity of the 50% Air - 50% PMMA mixture plasma, as a function of the temperature for the different pressure values. At low temperature, the dynamic viscosity is independent of the pressure. From T = 9000K, we notice that the maximum of the dynamic viscosity moves towards high temperatures when the pressure increases. This evolution of the dynamic viscosity with pressure is explained by the fact that heavy particles disappear at higher temperatures when the pressure is high.

5. Conclusion

In this work, Air - PMMA mixtures thermal plasmas dynamical viscosity, thermal and electrical conductivities are calculated in a temperature range from 5000K to 30000K. The calculations are made by supposing thermodynamic equilibrium at pressure of 1 bar to 10 bar. The calculations of these transport coefficients is based on the resolution of the Boltzmann equation by the Chapman-Enskog method. The results obtained show that the dynamic viscosity of the plasma varies similarly as a function of temperature whatever the percentage of PMMA in the mixture. It increases at first time to reach its peak between 10000K and 120000K, then decreases rapidly in a second time between 12000K and 20000K and finally tends towards a limit value between 20000K and 30000K. Pure air plasma has the highest dynamic viscosity and pure PMMA plasma the lowest dynamic viscosity. At low temperature, the dynamic viscosity of the plasma is independent of the pressure. But, from T = 9000K, we notice that its peak moves towards high temperatures when the pressure increases.

For high temperatures, the various mixtures plasmas have almost identical electrical conductivities. In the temperature range of 7000K to 13000K, the air plasma has the weakest electrical characteristic. But from 13000K to 22000K, it has an electrical conductivity slightly higher than that of other plasmas. On the other hand, the electrical properties of 80% Air - 20% PMMA, 50% Air - 50% PMMA and 20% Air - 80% PMMA mixtures plasmas are identical over almost the entire temperature range. It is found that the plasma becomes less electrically conductive when the pressure increases at low temperatures (T ≤ 12000K).

From 13000K, an inversion of this evolution of the electrical conductivity is observed; it increases when the pressure increases. The shape of the thermal conductivity curve λ (T) gives information on the quality of the mixture considered for breaking electric arcs.

It is noted that when the pressure increases, the peaks of the thermal conductivity λ move towards the high temperatures. This translation of the peaks is accompanied by an increase of the value of the thermal conductivity. In all the temperature range considered, it is the 80% Air - 20% PMMA mixture plasma which has the highest thermal conductivity peak. This mixture could therefore have performances for the cut of the electric current superior to those of pure air and pure PMMA with regard to its thermal conductivity in the temperature range going from 13000K to 16000K. However, other properties of the plasma such as thermodynamic properties and other parameters should be the subject of further studies to complement this study.

References