West Africa is dominated by desert aerosols and biomass fires. In recent years, it has suffered from overexploitation of the soil resulting in a reduction in plant cover and an increase of 30% to 50% in desert dust emissions.

Studies in this sub-region have shown that the chemical composition of aerosols in West Africa is mainly dominated by oxides of silicon, calcium, aluminum, and iron [1] - [9]. In addition, carbonaceous aerosols are produced from biomass and human activities [10] [11] [12] [13]. The collection of aerosols is generally performed on filters by aspiration of a volume of air measured by a sensor (the data are expressed in μg∙m⁻³) and the chemical composition is determined by spectrometric analysis.

The process of lifting aerosols is a threshold phenomenon which depends essentially on the nature of the soil and the force of the wind direction. The threshold of rising desert aerosols is estimated between 6 m∙s⁻¹ and 20 m∙s⁻¹ depending on the nature of the soil. Once injected into the atmosphere, the particles are transported over great distances and form the body of sandstorms. In the case of West Africa, the meeting between the monsoon flow and the harmattan flow favors the high-altitude transport of aerosols: as the harmattan is less dense, it passes over the monsoon and promotes the high-altitude transport of desert aerosols in the rainy season. However, in the dry season, harmattan is the main wind blowing over almost all of West Africa. It continuously causes dust clouds to move [14] [15] [16] [17]. Aerosols are deposited mainly in two ways: dry deposition by sedimentation under the action of gravity and wet deposition by rain.

West African countries like Burkina Faso, therefore, have atmospheres continuously polluted by aerosols. Electrical devices, particularly air circuit breakers, are generally immersed in a dusty atmosphere and operate without specific protection and without maintenance. Aerosols can enter the circuit breaker through the vent holes, settle and form a more or less thick layer on the gasifier cheeks [18]. The medium will therefore contain, in addition to the air composed mainly of nitrogen and oxygen, other chemical elements induced by the aerosols which will modify the composition of the plasma in the circuit breaker during the interruption of the electric current.

This can result in abnormal behavior and failed power cuts of air circuit breakers during their operation [18]. The objective of this work is to show the influence of aerosols on the chemical composition of the air plasma at local thermodynamic equilibrium (ETL) and at atmospheric pressure (1 bar), in a temperature range from 1000 K to 6000 K. We will describe the chosen calculation method, present the results obtained and analyze them.

The study is made at atmospheric pressure. We decided to take into account only the gas phase. We therefore take a temperature range from 1000 K to 6000 K. Several methods are used to determine the composition of plasma: we can quote the law of mass action, the pseudo-kinetic method, the collisional-radiative method, the minimization of the free enthalpy. In this study, we used Gibbs’ free energy minimization method to calculate the composition of plasma at local thermodynamic equilibrium and atmospheric pressure (1 bar). This method was compiled on Matlab software. The Gibbs’ free energy minimization method is very efficient and adapted to the type of plasma studied because it only takes into account the chemical nature of the species present in the mixture but not the reaction scheme that can describe the evolution of the mixture.

For air, Fe₂O₃, SiO₂, Al₂O₃ and CO mixtures, we consider the following:

31 monatomic species: C, O, N, Si, Al, Fe, C⁺, O⁺, N⁺, Si⁺, Al⁺, Fe⁺, C⁻, O⁻, N⁻, Si⁻, Al⁻, Fe⁻, C²⁺, O²⁺, N²⁺, Si²⁺, Al²⁺, Fe²⁺, C³⁺, O³⁺, N³⁺, Si³⁺, Al³⁺, Fe³⁺, and electrons;

28 diatomic species: C₂, O₂, N₂, Si₂, Al₂, Fe₂, CO, CN, SiC, AlC, NO, SiO, AlO, SiN, FeO, AlN, C 2 + , O 2 + , N 2 + , CO⁺, CN⁺, NO⁺, AlO⁺, N 2 − , C 2 − , O 2 − , CN⁻ and AlO⁻;

40 polyatomic species: CO₂, C₃, CCN, CNC, CNN, C₂O, O₃, N₃, NCO, NO₂, N₂O, NCN, SiC₂, SiO₂, Si₂C, Si₂N, Si₃, AlC₂, AlO₂, Al₂O, OCCN, C₂N₂, CNCOCN, C₃O₂, C₄, NO₃, N₂O₃, N₂O₄, N₂O₅, Al₂C₂, Al₂O₂, Al₂O₃, Fe(CO)₅, N₂O⁺, CO 2 + , Al₂O⁺, Al 2 O 2 + , AlO 2 − , NO 3 − , NO 2 −

We decided to only take into account the gas phase. We consider that plasma contains eigthy eigth (N = 99) chemical species. Determining the composition at equilibrium requires prior knowledge of the specific chemical potentials of all the particles inhabiting the plasma [19].

For electrons, atomic species and molecules, their chemical potential can be determined using specific enthalpy and specific entropy data tabulated by National Institute of Standards and Technology (NIST), Bonnie and Bendjebbar [20] [21] [22] [23]. We mainly use the thermodynamic data tabulated by Bonnie in the calculation of specific thermodynamic properties. The specific heat capacity, the specific enthalpy and the specific entropy are obtained by the following relations [22]:

C p 0 ( T ) R = a 1 T − 2 + a 2 T − 1 + a 3 + a 4 T + a 5 T 2 + a 6 T 3 + a 7 T 4 (1)

H 0 ( T ) R T = − a 1 T − 2 + a 2 ln ( T ) T + a 3 + a 4 T 2 + a 5 T 2 3 + a 6 T 3 4 + a 7 T 4 5 + b 1 T (2)

S 0 ( T ) R = − a 1 T − 2 2 − a 2 T − 1 + a 3 ln ( T ) + a 4 T + a 5 T 2 2 + a 6 T 3 3 + a 7 T 4 4 + b 2 (3)

R is the ideal gas constant and T the temperature in Kelvin.

Thus the chemical potential as a function of the temperature is given by the following formula:

μ 0 = H 0 − T S 0 (4)

To determine the concentrations of different chemical species, we use Gibbs’ free energy minimization calculation method [24] - [31]. We first determine the numbers of moles and from these one we calculate the concentrations.

Let Y ( y 1 , y 2 , ⋯ , y n ) be the number of moles of species. The numbers of moles must satisfy the electric neutrality and the conservation of the number of nuclei in the plasma. These conditions are obtained by:

∑ a i j y i = b j ( j = 1 , 2 , ⋯ , m ) (5)

In this equation, N is the number of chemical species; m is the number of types of nuclei in the mixture including electrons; a i j the number of j type nucleus in molecule i (the a i j corresponds to the coefficients of the decomposition matrix of the different species) and b j represents the number of initial j type nuclei.

We have seven different types of nuclei in our plasma (e⁻, C, O, N, Si, Al, Fe) so m = 7. b j = 0 reflects the electric neutrality in the electric arc plasma. The Gibbs free energy for plasmas at several temperatures is given by the following equation [24] [32]:

E ( Y ) = ∑ i = 1 N y i ( μ i 0 + R T i ln ( P P 0 ) + R T i ln ( y i T i ∑ k = 1 N y k T k ) ) (6)

P₀ is the reference pressure (P₀ = 1E5 Pa) and P the pressure at a given temperature; T_i is the translation temperature of particle i.

A simplified expression of Equation (6) is given by:

E ( Y ) = ∑ i = 1 N y i ( C i + R T i ln ( y i T i ∑ k = 1 N y k T k ) ) (7)

with

C i = μ i 0 + R T i ln ( P P 0 ) (8)

The y i are the numbers of moles. We must find the point Y ( y 1 , y 2 , ⋯ , y n ) which minimizes the function E (Y) and for which the coordinates y i satisfy the following conditions:

· the mole numbers must be positive y i ≥ 0 ∀ i ;

· the coordinates y i must satisfy the conservation of the number of nuclei and the electric-neutrality.

After a Taylor series expansion of order 2 around point Y ( y 1 , y 2 , ⋯ , y n ) , we get:

Q ( X ) = E ( Y ) + ∑ i = 1 N ( ∂ E ∂ x i ) X = Y ( x i − y i ) + 1 2 ∑ i = 1 N ∑ k = 1 N ( ∂ 2 E ∂ x i ∂ y k ) X = Y ( x i − y i ) ( x k − y k ) (9)

Which leads to:

Q ( X ) = E ( Y ) + ∑ i = 1 N ( x i − y i ) ( C i + R T i ln ( y i T i ∑ k = 1 N y k T k ) ) + 1 2 ( ∑ i = 1 N y i R T i ( x i − y i y i − ∑ k ( x k − y k ) ∑ k = 1 N y k T k ) ) (10)

Taking into account the physical conditions (Equation (5)), we introduce the Lagrange multipliers π i . Thus, we get the function ξ ( X ) .

ξ ( X ) = Q ( X ) + ∑ j π j ( − ∑ a i j x i + b j ) (11)

E ( X ) is minimum when [33] ∂ ξ ( X ) ∂ x i = 0

Which leads to:

f i y i = R T i [ x i y i − ∑ k T k x k ∑ k T k y k ] − ∑ i = 1 m a i j = 0 (12)

with

f i = y i ( C i + R T i ln ( y i T i ∑ k = 1 N y k T k ) ) (13)

Using the Newton-Raphson method, we obtain a system of equations [21] [23] [34]:

( R T 1 y 1 ⋯ 0 a 11 ⋯ a 1 m ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ 0 ⋯ R T N y N a N 1 ⋯ a N m a 11 ⋯ a 1 m 0 ⋯ 0 ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ a N 1 ⋯ a N m 0 ⋯ 0 ) = ( − μ 1 0 − R T 1 ln ( T 1 y 1 ∑ k = 1 N y k T k ) − R T 1 ln ( P P 0 ) − ∑ j = 1 m π j a i j ⋮ − μ N 0 − R T N ln ( T N y N ∑ k = 1 N y k T k ) − R T N ln ( P P 0 ) − ∑ j = 1 m π j a i j − ∑ j = 1 m y i a i 1 + b 1 ⋮ − ∑ j = 1 m y i a i m + b m ) (14)

In the case of our study, we took into account ninety-ninie (99) chemical species, so N = 99. The Equation (14) is therefore a system of 106 equations with 103 unknowns, which are the number of moles of the 99 chemical species and seven (7) Lagrange multipliers. The temperature of the medium was maintained at T i = T . The principle of this numerical method consists of initially and arbitrarily assigning values to the number of moles i ∈ [ 1 ,   99 ] and the Lagrange multipliers j ∈ [ 1 ,     7 ] . The number of moles must satisfy Condition (5).

The new values of the number of moles ( n i ) and Lagrange multipliers ( π j ) were obtained by solving Equation (14). This leads to the following system.

{ n i = y i + λ Δ n i   ∀   i ∈ 〚 1 ,   N = 99 〛 π j = π j + λ Δ π j   ∀   j ∈ 〚 1 , m = 7 〛 (15)

The parameter λ ∈ [ 0 , 1 ] is the correction coefficient. This helps to avoid negative values of the mole numbers that may appear when moving away from the solution. It must satisfy the condition of d E ( λ ) d λ ≺ 0 to avoid going beyond

the point of convergence. This corresponded to the largest value between 0 and 1.

n i = y i + λ Δ n i ≻ 0   ∀   i ∈ 〚 1 , N 〛 (16)

The new values of the number of moles and Lagrange multipliers were used for a new calculation cycle. The criterion for interrupting iterations was set by the following condition given by Cayet [33]:

G ( n i ) − G ( y i ) G ( n i ) ≺ 10 − 14 (17)

The numerical densities ( x i ) were obtained using Dalton’s law [24].

X i = n i P − Δ P k T ∑ i N n i (18)

k is the Boltzmann constant and ∆P is the lowering of the pressure given by the following equation:

Δ P = − k T 24 π λ d 3 (19)

λ d the Debye length is given by Equation (20):

λ d = 1 N A e ( ∑ i = 1 N ε 0 R T x e ) 1 / 2 (20)

where ε 0 is the permittivity of vacuum, x e is the mole fraction of electrons, e is the elementary charge, and N A Avogadro’s number.

Tables 1-3 present the thermodynamics data coefficients using in the calculation of equilibrium composition.

In this section, we present the composition of the dry air plasma of an air circuit breaker operating under normal conditions and the compositions of the air plasmas of an air circuit breaker that can be contaminated by aerosols. The study was conducted at local thermodynamic equilibrium.

To validate our calculation program, we compare the concentrations of the particles of hydrogen plasma for a thermal imbalance θ of 2 for a temperature of the heavy species of 6000 K which was calculated by Pascal, with the concentrations given by our program (Table 4). We notice that the two results concord with a maximum deviation of 27.45% observed for the H₂ molecule. The differences observed can be explained by the data used. In general, we can say that our program is valid since our values concord with those of the literature.

We represent in Figures 1-5 respectively the concentration of chemical species

versus temperature at atmospheric pressure and local thermodynamic equilibrium (LTE) of the plasmas formed of 100% air, 99.5% air - 0.5% impurites, 99% air - 1% impurites, 95% air - 5% impurites, 90% air - 10% impurites.

In general, we noticed that the numerical densities of the chemical species of the different plasmas evolve in a similar way in two (2) phases.

For air dry (Figure 1), the first phase concerns temperature less than 3000 K (T < 3000 K). This phase is mainly composed by N₂O, NO₂, N₂, O₂, and NO. The second phase concerns the temperature range of 3000 K to 6000 K (3000 K < T < 6000 K). In this interval, the main chemical species are: N₂, O₂, NO N, O, NO⁺, O⁺, N⁺, e⁻, N 2 + and O 2 + . Densities of N 2 + , O 2 + and O⁻ species appear with low concentration. In this temperature range, densities of N₂, O₂, NO, N₂O and NO₂ decrease with temperature while N, O, NO⁺, O⁺, N⁺, e⁻ increase with temperature. This decrease of N₂, O₂, NO, N₂O and NO₂ is due to their low dissociation energy: O₂ dissociates to form atomic oxygen at approximately 3300 K. N₂ also dissociates to form nitrogen. Dissociation of NO leads to the formation of oxygen and nitrogen.

Electrical neutrality was rigorously established between electrons and NO⁺ ions over the entire temperature range.

To estimate the effect of the percentage of Fe₂O₃, SiO₂, Al₂O₃ and CO on the equilibrium composition, we have show in Figures 2-5 the numerical densities of the majority chemical species. We noticed the same phases of evolution as in the case of air plasma. For temperatures less than 3000 K (T < 3000 K), most of the species were N₂, O₂, CO₂, FeO, Al₂O₂, Al₂O₃, NO, N₂O, NO₂, O, Fe, SiO, Al₂O, AlO₂, AlO, CO, and SiO₂. For temperatures most than 3000 K, he main chemical species are: N₂, O₂, NO, SiO, Si, AlO, SiO₂, Al₂O₂, CO, AlN, SiN, CN, N, O, Al, Fe, NO⁺, AlO⁺, Si, Al⁺, Fe⁺, Si⁺, e⁻, AlO⁻, O⁻, N⁻, Al⁻ and Fe⁻. N₂, O₂, CO₂, FeO, Al₂O₂, Al₂O₃, NO, N₂O, NO₂, O, Fe, SiO, Al₂O, AlO₂, AlO, CO, and SiO₂ decrease with temperature because, they dissociate to give mono-atomic species. The main reason for these dissociations at low temperatures is due their low dissociation energy. For example, Carbon dioxide dissociates into carbon monoxide at approximately 3000 K. The dissociation of Al₂O, AlO₂, Al₂O₂, and Al₂O₃ at temperatures below 3000 K led to an increase in the number density of Al and O. Moreover, SiO and SiO₂ dissociated at temperatures below 2500 K. This was the origin of the increase in the density of Si in the plasma. Moreover, FeO dissociated at temperatures below 3000 K. We noticed that polyatomic species from aerosols dissociated at approximately 3000 K or below 3000 K. In general, dissociation of polyatomic species leads to the emergence of atoms. These atoms, in turn, ionize to form ions and electrons. We also noticed that the increase in the amount of Fe₂O₃, SiO₂, Al₂O₃ and CO in the plasma led to an increase in the different species, except for nitrogen and dinitrogen. Electro neutrality is strictly established for temperatures below 3500 K between electrons, AlO⁻, Al⁺ and Fe⁺ and beyond 3500 K, between electrons, Al⁺ and Fe⁺. because the ionization energy of the particle Al, Fe Si is low than those of N, C and O (Ei_Al = 5986 eV, Ei_Si = 8151 eV, Ei_Fe = 7902 eV, Ei_N =14.55 eV, Ei_O =13.628 eV and Ei_C =11.26 eV). So, Al⁺, Fe⁺, O⁻, Al₂O⁺, AlO 2 + , and AlO⁻ ions appeared between 2000 and 3000 K at low densities. Si+ ions appeared at approximately 4200 K.

To determine the influence of aerosols on the electron density of the air plasma, we compared the electron densities of the different mixtures (Figure 6). We observed that electron density increased with the aerosol rate of the mixture. This increase is justified by the fact that C, Si, Al, and Fe ionize at low temperatures by releasing electrons. This helps increase the electron density. Therefore, the electrical conductivity can be high at low temperatures because it depends on the electron density.

This study investigated the chemical composition of plasmas of air and aerosol mixtures in air circuit breakers at local thermodynamic equilibrium and at atmospheric pressure for temperatures of 6000 K. The calculation method used was the minimization of the Gibbs free energy. The numerical densities obtained made it possible to show the influence of aerosols on the chemical composition of the studied plasmas. The higher the percentage of aerosols is, the higher the density of the species is originating from these aerosols in the plasma. The increase in the electron density could be explained by the low ionization energies

of the Si, Al, and Fe species at low temperatures. This can lead to an increase in the electrical conductivity of the plasma. Therefore, the calculation of the transport coefficients is necessary to confirm this hypothesis. Thus, the slight presence of dust in the interrupting chamber of an air circuit breaker placed in a polluted atmosphere can completely modify its operation, that is, arc re-ignition and contact heating.

The authors declare no conflicts of interest regarding the publication of this paper.

Charles, Y.W., Eric, K., Rachid, K.A., Zacharie, K. and François, Z. (2022) Study of the Composition of a Plasma of Dry Air Contaminated by Al₂O₃, CO, Fe₂O₃ and SiO₂. Advances in Materials Physics and Chemistry, 12, 177-193. https://doi.org/10.4236/ampc.2022.128013