Calculation of 2-temperature plasma thermo-physical properties considering condensed phases: application to CO₂–CH₄ plasma: part 1. Composition and thermodynamic properties

The determination of the plasma composition is a prerequisite for the calculation of the thermodynamic and the transport properties. In a non-LTE plasma system, due to the inefficiency of electron–heavy-species collisions, the electron temperature differs from that of heavy species. A parameter of thermal non-equilibrium can be defined as θ  =  T_e/T_h, where T_e and T_h are respectively the electron and heavy-species temperatures.

The classic composition calculations for 2-T gaseous system were based on the laws of mass action (Saha law and Guldberg–Waage law), the conservation of the chemical elements, Dalton's law and electrical quasi-neutrality [24, 25]. For the mass action law, the derivation proposed by van de Sanden [17] is widely used. In this method, the dissociation reaction ab ↔ a  +  b and ionization reaction a^r+ ↔ a^(r+1)+  +  e⁻ are described as follow:

$$\begin{eqnarray}&&\frac{{{n}_{a}}{{n}_{b}}}{{{n}_{ab}}}=\frac{Q_{a}^{\operatorname{int}}Q_{b}^{\operatorname{int}}}{Q_{ab}^{\operatorname{int}}}{{\left(\frac{2\pi k{{T}_{\text{h}}}}{{{h}^{2}}}\right)}^{3/2}}{{\left(\frac{{{m}_{a}}{{m}_{b}}}{{{m}_{ab}}}\right)}^{3/2}}\exp \left(-\frac{{{E}_{\text{d}}}}{k{{T}_{\text{ex}}}}\right)\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\frac{{{n}_{\text{e}}}{{n}_{{{a}^{(r+1)+}}}}}{{{n}_{{{a}^{r+}}}}}=2\frac{Q_{{{a}^{(r+1)+}}}^{\operatorname{int}}}{Q_{{{a}^{r+}}}^{\operatorname{int}}}{{\left(\frac{2\pi {{m}_{\text{e}}}k{{T}_{\text{e}}}}{{{h}^{2}}}\right)}^{3/2}}\exp \left(-\frac{{{E}_{I,r+1}}-\delta {{E}_{I,r+1}}}{k{{T}_{\text{ex}}}}\right)\end{eqnarray} \tag{ 2 }$$

where $Q_{i}^{\operatorname{int}}$, n_i and m_i are the internal partition function, species number density and mass of the ith species, respectively; E_d and E_I,r+1 are the dissociation energy of molecule ab and ionization energy of species a^r+, respectively; δE_I,r+1 is the lowering of the ionization energy due to the Coulomb interactions, and k and h are the Boltzmann and Planck constants, respectively, and T_ex is the excitation temperature. In this paper, for monatomic species ionization, which is governed by electron, we let T_ex  =  T_e. On the other hand, for the molecular ionization, which occurs in the temperature range where there is basically no electrons in the system, we let T_ex  =  T_h. This definition of T_ex consists with previous research work [26, 27].

However, the phase transition process cannot be considered by this method and therefore this method could be inaccurate for plasma system where the condensing effect is not negligible [13]. According to the second law of thermodynamics, when a multi-component system reaches phase equilibrium, the chemical potential of condensed particle equals to that of corresponding gaseous particle, which is given by [28]

$$\begin{eqnarray}&&{{\mu}_{i-\text{gas}}}={{\mu}_{i-\text{cond}}}\end{eqnarray} \tag{ 3 }$$

where μ_i-gas and μ_i-cond are chemical potential of gaseous particle and corresponding condensed particle respectively. By using this equation, the composition of condensed species is calculable.

Calculating chemical potential of condensed particle and its corresponding gaseous particle is the crux. The chemical potential of gaseous particle is calculated by [11, 29]

$$\begin{eqnarray}&&{{\mu}_{i-\text{gas}}}=\mu _{i}^{o}+R{{T}_{\text{h}}}\ln \left(\,p\text{/}{{p}_{\text{ref}}}\right)+R{{T}_{\text{h}}}\ln {{x}_{i}}\end{eqnarray} \tag{ 4 }$$

where x_i is molar fraction of ith particle in the gaseous phase. p_ref is reference pressure which is 1 atm in current calculation. $\mu _{i}^{o}$ is the chemical potential of the ith particle in the standard state, which is given by

$$\begin{eqnarray}&&\mu _{i}^{o}=G_{i}^{o}={{ \Delta }_{f}}H_{i,{{T}_{\text{ref}}}}^{o}-T \Phi _{i}^{o}\end{eqnarray} \tag{ 5 }$$

where $\Phi _{i}^{o}$ is molar weighted Gibbs energy, which can be calculated by internal partition function $Q_{i}^{\operatorname{int}}$ and translational partition function $Q_{i}^{\text{tran}}$ [13, 29]. For monatomic species, the internal partition function is only governed by electron temperature [13] and therefore the $\mu _{i}^{o}$ is given by

$$\begin{eqnarray}&&\mu _{i}^{o}={{ \Delta }_{f}}H_{i,{{T}_{\text{ref}}}}^{o}-{{T}_{\text{e}}}R\ln Q_{i}^{\operatorname{int}}-{{T}_{\text{h}}}R\ln Q_{i}^{\text{tran}}\end{eqnarray} \tag{ 6 }$$

The calculation method of internal partition function of gaseous species is presented in the next part.

The chemical potential of condensed species cannot be established by calculation with sufficient accuracy and therefore the data are derived from measurement data. In the present calculation, we use the fitting data of molar tempered Gibbs energy presented by Coufal [29] on the assumption that the chemical potential of condensed particle (like graphite in this paper) is governed by T_h.

Figure 1 shows the calculation flow of composition of multiphase 2-T plasma system containing graphite. By knowing the chemical potential of graphite at given T_h, the chemical potential of gaseous C is obtained by equation (3) according to the second law of thermodynamics. With the knowledge of $\mu _{i}^{o}$ of gaseous C, which is derived from formation enthalpy and partition function, the molar fraction of gaseous species at phase equilibrium can be obtained by equation (4). If the molar fraction of gaseous C in pure gaseous system is higher than that at phase equilibrium, it means the phase transition process is not completed at given condition. By using the molar fraction of gaseous C at phase equilibrium, mass action law, the conservation of the chemical elements, Dalton's law and electrical quasi-neutrality with consideration of the Debye–Huckel corrections, the molar fraction of other species in the multiphase system is obtained. Otherwise, if the molar fraction of gaseous C in pure gaseous system is lower than that at phase equilibrium, which means the graphite sublimation is completed in this given condition, the composition result of pure gaseous system is adopted.

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The calculation of the partition function is a prerequisite for calculating the plasma composition and thermodynamic properties. Following the recommendation of Gleizes and some other previous work [4, 6, 25, 27, 30–32], we assumed that translational, rotational and vibrational motions of heavy species were governed by T_h while electronic excitation (of both heavy species and electrons) and translational motion of electrons were governed by T_e. The calculation method of partition functions are summarized as follows.

For monatomic species, the internal partition function is given by

$$\begin{eqnarray}&&{{Q}_{i}}=Q_{i}^{\text{el}}\left({{T}_{\text{e}}}\right)=\underset{n}{\overset{{{\varepsilon}_{n}}<{{E}_{i ~\text{eff}}}}{\sum}}\,{{g}_{n}}\exp \left(\frac{-{{\varepsilon}_{n}}}{k{{T}_{\text{e}}}}\right)\end{eqnarray} \tag{ 7 }$$

where $Q_{i}^{\text{el}}$(T_e) is the electronic internal partition function of species i, g_n and ε_n are the degeneracy and energy of the nth electronic levels, and E_{i eff} is the effective ionization limit considering the Debey correction at which the sum over electronic levels is cut off. The data of energy levels and degeneracy are mainly taken from NIST database [33]. The data of high energy levels with high principal quantum number, which are not given by Moore or NIST database, was estimated using the hydrogenic approximation based on the last principal quantum number [34].

For diatomic and polyatomic species, the internal partition function is the product of electronic, vibration and rotation partition functions. The expression is given by

$$\begin{eqnarray}&&{{Q}_{i}}=Q_{i}^{\text{el}}\left({{T}_{\text{e}}}\right)Q_{i}^{\text{vib}}\left({{T}_{\text{h}}}\right)Q_{i}^{\text{rot}}\left({{T}_{\text{h}}}\right)\\ &&\,\,\,\,\,\,\,\,=\underset{n}{\sum}\,\left\{{{g}_{n}}\exp \left(\frac{-{{\varepsilon}_{n}}}{k{{T}_{\text{e}}}}\right)\underset{v}{\overset{{{v}_{\max}}}{\sum}}\,\left({{g}_{n,v}}\exp \left(\frac{-{{\varepsilon}_{n,v}}}{k{{T}_{\text{h}}}}\right)\underset{J}{\overset{{{J}_{\max}}}{\sum}}\,\left({{g}_{n,v,J}}\exp \left(\frac{-{{\varepsilon}_{n,v,J}}}{k{{T}_{\text{h}}}}\right)\right)\right)\right\}\end{eqnarray} \tag{ 8 }$$

where $Q_{i}^{\text{vib}}$(T_h) and $Q_{i}^{\text{rot}}$(T_h) are respectively the vibrational and rotational partition functions of species i. The subscript n, v indicates the vth vibration energy level of the nth electron energy level. And similarly, the subscript n, v, J indicates the Jth rotation energy level which belongs to vth vibration energy level of the nth electron energy level. The Morse potential minimization method has been used to limit the calculation on the vibrational and rotational levels [35]. All the data needed have been taken from the most recent data of NIST [33] and the JANAF thermochemical tables [36].

For polyatomic species, due to the lack of accurate spectroscopic data, the internal partition functions were calculated by a simplified method, in which the discrete summations over rotational quantum numbers are replaced by an integral form [34]

$$\begin{eqnarray}&&{{Q}_{i}}=\underset{e}{\sum}\,{{g}_{\text{e}}}\exp \left(-{{\varepsilon}_{\text{el}}}\text{/}k{{T}_{\text{e}}}\right)\underset{i}{\prod}\,{{\left[\frac{\exp \left(-{{\omega}_{i}}/2k{{T}_{\text{h}}}\right)}{1-\exp \left(-{{\omega}_{i}}/k{{T}_{\text{h}}}\right)}\right]}^{{{g}_{i}}}}{{\left({{T}_{\text{h}}}\text{/}{{\theta}_{\text{rot}}}\right)}^{3/2}}\end{eqnarray} \tag{ 9 }$$

where ${{\theta}_{rot}}$ is the rotational constant. For the polyatomic species, the rotational energy can be expressed as a function of the three principal moments of inertia I_A, I_B and I_C:

$$\begin{eqnarray}&&{{\left(\frac{{{T}_{\text{h}}}}{{{\theta}_{\text{rot}}}}\right)}^{3/2}}=\frac{{{\pi}^{1/2}}}{{{\sigma}_{c}}{{h}^{3}}}{{\left({{I}_{A}}{{I}_{B}}{{I}_{C}}\right)}^{1/2}}{{\left(8k{{T}_{\text{h}}}\right)}^{3/2}}\end{eqnarray} \tag{ 10 }$$

After obtaining the composition, the thermodynamic properties can be evaluated through a classical statistical mechanics approach. The method of thermodynamic property calculation presented by Coufal [29] is applied in this paper.

The mass density is given by

$$\begin{eqnarray}&&\rho =\frac{\underset{i}{\sum}\,{{c}_{i}}{{m}_{i}}}{{{V}_{\text{g}}}}\end{eqnarray} \tag{ 11 }$$

where c_i and m_i are the amount and mass of species i in the system, respectively. V_g is the volume of the gaseous phase. The volume of condensed phase is neglected.

The specific enthalpy of system is summation of specific heat of every single species. For gaseous species, the translational contributions, and reactive and internal contributions due to electronic excitation are taken into account. For monatomic species, the specific enthalpy is given by

$$\begin{eqnarray}&&{{h}_{i}}=\frac{5}{2}\frac{k}{\rho}{{n}_{i}}{{T}_{i}}\,\,+\,\frac{1}{\rho}{{n}_{i}}{{E}_{i}}+\frac{k}{\rho}{{n}_{i}}\left(T_{\text{e}}^{2}\frac{\partial \ln Q_{i}^{\text{el}}}{\partial {{T}_{\text{e}}}}\right)\end{eqnarray} \tag{ 12 }$$

where n_i is the number density of species i and E_i is the formation energy of species i. For molecular species, the specific enthalpy is given by

$$\begin{eqnarray}&&{{h}_{i}}=\frac{5}{2}\frac{k}{\rho}{{n}_{i}}{{T}_{i}}+\frac{1}{\rho}{{n}_{i}}{{E}_{i}}+\frac{k}{\rho}{{n}_{i}}\left[T_{\text{e}}^{2}\frac{\partial \ln Q_{i}^{\text{el}}}{\partial {{T}_{\text{e}}}}+T_{\text{h}}^{2}\frac{\partial \ln \left(Q_{i}^{\text{vib}}Q_{i}^{\text{rot}}\right)}{\partial {{T}_{\text{h}}}}\right]\end{eqnarray} \tag{ 13 }$$

The specific heat for condensed species can be calculated through standard thermodynamic function as proposed by Coufal [29].

$$\begin{eqnarray}&&{{h}_{i}}=\frac{1}{\underset{i}{\sum}\,{{c}_{i}}{{m}_{i}}}\times {{c}_{i}}H_{i,{{T}_{\text{ref}}}}^{o}=\frac{1}{\underset{i}{\sum}\,{{c}_{i}}{{m}_{i}}}\times {{c}_{i}}\times \left[{{ \Delta }_{f}}H_{i,{{T}_{\text{ref}}}}^{o}+ \Delta H_{i,{{T}_{\text{ref}}}}^{o}(T)\right]\end{eqnarray} \tag{ 14 }$$

where $H_{i,{{T}_{\text{ref}}}}^{o}$ is the summation of standard molar enthalpy change Δ$H_{i,{{T}_{\text{ref}}}}^{o}$(T) and standard molar formation enthalpy ▵_f$H_{i,{{T}_{\text{ref}}}}^{o}$. All the data needed have been taken from NIST Atomic Spectra Database [33] and the JANAF thermochemical tables [36]. According to the JANAF table, the graphite, O₂ and H₂ are chosen as the reference state particles.

Taking total pressure p, electron temperature T_e, and non-equilibrium parameter θ  =  T_e/T_h as independent variables, differentiation of the total enthalpy gives [31, 37]

$$\begin{eqnarray}&&\text{d}h={{\left(\frac{\partial h}{\partial p}\right)}_{{{T}_{\text{e}}},\theta}}\text{d}p+{{\left(\frac{\partial h}{\partial {{T}_{\text{e}}}}\right)}_{p,\theta}}\text{d}{{T}_{\text{e}}}+{{\left(\frac{\partial h}{\partial \theta}\right)}_{p,{{T}_{\text{e}}}}}\text{d}\theta \end{eqnarray} \tag{ 15 }$$

The second term of this equation can be interpreted as the total specific heat at constant pressure and constant non-equilibrium parameter.

CO₂–CH₄ mixture is now being considered as a possible substitution for SF₆ due to its environmental friendliness and good electrical properties. In this part, the composition and thermodynamic properties of CO₂–CH₄ mixture are calculated using the method presented above. Totally 48 species have been considered in the present calculation as given in table 1. The species C(c) is the condensed carbon, namely graphite in this calculation. The chemical potential of graphite is derived from standard molar tempered Gibbs energy using the measurement results which is fitted by Coufal [29]

$$\begin{eqnarray}&&{{ \Phi }^{o}}(T)={{\varphi}_{0}}+\varphi \ln (t)+{{\varphi}_{-2}}\text{/}{{t}^{2}}+{{\varphi}_{-1}}\text{/}t+{{\varphi}_{1}}t+{{\varphi}_{2}}{{t}^{2}}+{{\varphi}_{3}}{{t}^{3}}\end{eqnarray} \tag{ 16 }$$

where t  =  T/(10 000 K). The coefficients ${{\varphi}_{0}}$, $\varphi$, ${{\varphi}_{-2}}$, ${{\varphi}_{-1}}$, ${{\varphi}_{1}}$, ${{\varphi}_{2}}$ and ${{\varphi}_{3}}$ of graphite are given in table 2.

Table 1. Species in the calculation of CO₂–CH₄ plasma composition.

Elements	Species
C	C3, C2, C, C+, C2+, C3+, C(c)
H	H2, H, $\text{H}_{2}^{+}$, $\text{H}_{2}^{-}$, H−, H+
O	O3, O2, O, $\text{O}_{2}^{-}$, $\text{O}_{2}^{+}$, O2−, O−, O+, O2+, O3+
CH	C2H6, C2H4, C2H2, C2H, CH4, CH3, CH2, CH
CO	CO2, C2O, CO, CO+
HO	H2O2, H2O, HO2, OH, OH+, OH−
Others	e−, COOH, H2CO, HCOOH, CH3OH, HCO, C2H4O, C2H5OH, HCO+

Table 2. Coefficients of graphite in equation (16).

Th (K)	${{\varphi}_{0}}$	$\varphi$	${{\varphi}_{-2}}$	${{\varphi}_{-1}}$	${{\varphi}_{1}}$	${{\varphi}_{2}}$	${{\varphi}_{3}}$
298–1600	−4.1845	0.697 29	−0.000 8680	0.128 911	195.3163	−369.2265	359.1611
1600–5000	61.9368	28.351 58	−0.021 45	1.854 69	−16.2256	17.3267	−0.6276
5000–6000	78.8933	27	0	−9.064 66	0	0	0

As presented by many previous published papers [12, 19, 21], the presence of condensed species can significantly influence the plasma composition and further influence the plasma properties under LTE. Figure 2 shows the molar fraction of 50%CO₂–50%CH₄ mixture at atmospheric pressure under LTE assumption for temperature lower than 3000 K with and without the consideration of condensed graphite. For a pure gaseous system, as shown in figure 2(a), the CH₄ and CO₂ dominate the mixture before CO₂ dissociation which occurs at about 800 K. However, once condensed carbon, namely graphite, is taken into account, the composition at low temperature is different. As shown in figure 2(b), due to the presence of graphite, the reaction CO₂  +  CH₄ ↔ 2H₂O  +  2C(c) makes H₂O and graphite being the dominant species before the graphite sublimation at around 1300 K. This discrepancy in composition, which can make difference in thermo-physical properties, shows the importance of considering condensed species in plasma simulation.

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Figure 3 shows the comparison of specific enthalpy and total specific heat between the system with and without considering graphite. For mass density, the presence of graphite makes no difference because the reaction CO₂  +  CH₄ ↔ 2H₂O  +  2C(c) introduced by graphite does not change the system volume or total mass.

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The presence of graphite decreases the specific enthalpy as shown in figure 3(a), which is also due to the difference of composition and formation enthalpy. In the system with graphite, the composition consists of 50%H₂O–50%C(c) before graphite sublimation at 1300 K due to the reaction CO₂  +  CH₄ ↔ 2H₂O  +  2C(c). The lower formation enthalpy of 50%H₂O–50%C(c) mixture comparing to 50%CO₂–50%CH₄ mixture leads to lower value of specific enthalpy.

The presence of graphite changes the peaks of specific heat which are closely associated with chemical reactions, as shown in figure 3(b). In a pure gaseous system without graphite, there are two peaks in specific heat corresponding to CO₂ dissociation at 800 K and CH₄ and H₂O dissociation at around 1300 K. In a multiphase system considering graphite, the sharp peak at 1200 K is due to the graphite sublimation. The inflection at 1300 K is corresponding to the fully sublimation point where phase transition completes. For temperature higher than that point, the mixture is no longer a multiphase system but a gaseous system.

Figure 4 shows the temperature dependence of 50%CO₂–50%CH₄ mixture composition with different non-equilibrium parameter. In order to make the figure more readable, only the species whose molar fraction is higher than 10⁻³ are presented. For LTE system, as shown in figure 4(a), the reaction CO₂  +  CH₄ ↔ 2H₂O  +  2C(c) makes H₂O and graphite being the dominant species at low temperature (lower than 1000 K). After the sublimation of graphite at around 1200 K, the reaction CO₂  +  CH₄ ↔ 2CO  +  2H₂ leads the domination of CO and H₂. At 3200 K and 6800 K, H₂ and CO dissociate respectively. The first order ionization of C, H and O occur at 12 500 K, 14 700 K and 15 700 K respectively.

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In a two-temperature CO₂–CH₄ mixture, the non-LTE effect may significantly affect both the dissociation and ionization process and further affect the thermo-physical properties. In a non-LTE system, the excitation temperature of dissociation is T_h while the excitation temperature of monatomic particle ionization is T_e as given in equations (1) and (2). Therefore from the perspective of T_e, the dissociation reactions are shifted to higher electron temperature in non-LTE system [9, 10], namely 'delay of dissociation reaction'. When θ  =  3 (figure 4(b)), the reaction of CO₂  +  CH₄ ↔ 2CO  +  2H₂ occurs at electron temperature range from 2500 K to 3000 K with H₂O as intermediate product. H₂ dissociates into H atoms at T_e  =  10 000 K. Due to the delay of CO dissociation which occurs at T_e  =  17 000 K, H is almost the only monatomic neutral species in the T_e range from 6000 K to 15 000 K and therefore the system ionization is dominated by ionization of H atoms at temperature below 14 000 K. This phenomenon is different from the LTE system in which the system ionization is dominated by C⁺ in low temperature range. At T_e  =  17 000 K, the dissociation of CO and ionization of C and O atoms occur at same time due to the high electron temperature, which causes the low molar fraction of C and O atoms in this system.

As the non-equilibrium parameter increases further (see the θ  =  5 in figure 4(c)), the producing of CO and H₂ are shifted to higher electron temperature (around T_e  =  4000 K to 5000 K). The dissociation of H₂ and ionization of H atom occur at same time due to the delay of dissociation reactions (at around T_e  =  15 000 K). The H⁺ dominates the system ionization process until T_e  =  20 000 K at which CO dissociates and C and O atoms ionize.

On the other hand, from the perspective of T_h, the influence of non-LTE effect on dissociation reactions can be different or even opposite, as shown in figure 5. In non-LTE system, the high electron temperature promotes the ionization of dissociation product and therefore promotes the dissociation process. For instance, the reaction CO ↔ C  +  O occurs at 6800 K at LTE system, while in non-LTE system this reaction is shifted to T_h  =  5700 K when θ  =  3 and T_h  =  4600 K when θ  =  5. This is because the ionization of C and O consumes the dissociation product and further promotes the dissociation process. Similar phenomenon can be found in H₂ dissociation which occurs at T  =  3300 K when LTE and at T_h  =  3100 K when θ  =  5. However, when θ  =  3 the T_h of H₂ dissociation is basically same as that when LTE because the H ionization does not happen at this temperature.

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Besides the chemical reactions, non-LTE effect can also affect the phase transition process, as shown in figure 6. At same T_e, the T_h leads to higher chemical potential of graphite μ_i-cond and further increases the gaseous carbon molar fraction and lowers the graphite molar fraction. At the meantime, similar to the dissociation reaction, the non-LTE effect shifted the sublimation process to higher T_e, as shown in the figure 6(a). On the other aspect, at same T_h, the high electron temperature increases the internal partition function and therefore decreases the $\mu _{i}^{o}$ in equation (4), which further increases the gaseous carbon molar fraction and lowers the graphite molar fraction. As a consequence, in non-LTE system, the molar fraction of graphite is lower than that in LTE system and the sublimation process occurs in lower heavy species temperature, as shown in figure 6(b). This trend is consistent with results of condensed metal species presented by André et al [13] who found the heavy species temperature of sublimation decreases as θ increases.

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Figure 7 shows the thermodynamic properties of 50%CO₂–50%CH₄ mixture at atmospheric pressure with different non-equilibrium parameter. As θ increases, the sharp decrease of the mass density, which is due to the successive dissociations of polyatomic species, is shifted to higher electron temperature, because of the delay of dissociation reactions.

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The non-LTE effect also affects the enthalpy and specific heat, as shown in figures 7(b) and (c). The enthalpy increases strongly in the temperature ranges in which chemical reactions occur. This is associated with the peaks in the specific heat. Under LTE, the first peak at 1200 K is due to the reaction phase transition and CO₂  +  CH₄ ↔ 2CO  +  2H₂. The other peaks at 3300 K, 6800 K, 15 000 K and 30 000 K are due to H₂ dissociation, CO dissociation, first and second order ionization respectively. In a two-temperature plasma system, the delay of dissociation reaction can change the presence temperature of monatomic species and therefore change the ionization order of system [9]. As a consequence, two-temperature CO₂–CH₄ plasmas can show different properties from plasmas in LTE. For θ  =  3, as shown in blue dot curve in figure 7(c), the first peak due to CO₂  +  CH₄ ↔ 2CO  +  2H₂ reaction is shifted to T_e  =  3000 K. The gentle peak at around T_e  =  12 000 K is due to the combination of H₂ dissociation at 10 000 K and H ionization at T_e  =  14 000 K. Besides, the CO dissociation reactions are shifted to higher electron temperature, higher than that required for C and O ionization reactions. Once CO dissociation takes place, the ionization of C and O occur at the same time, which leads to sharp peaks in specific heat at T_e  =  17 000 K. As the degree of non-equilibrium increases further (see the θ  =  5 curve), the electron temperature at which H₂ dissociates is higher than that of H ionization. It leads to a sharp peak at T_e  =  16 000 K rather than a gentle peak shown in θ  =  3 curve.

On the other hand, from the perspective of T_h, the influence of non-LTE effect on thermodynamic properties can be different, as shown in figure 8. In non-LTE system, the high electron temperature promotes the ionization of dissociation product and therefore promotes the dissociation process (see figure 5). Therefore, the peaks caused by dissociation reactions can be shifted to lower T_h in non-LTE system. For the peaks caused by CO dissociation, when θ  ⩽  2, the CO dissociation occurs at T_h  =  6800 K while the T_e is not high enough to trigger the C and O ionization. However, when θ  ⩾  3, the extremely high T_e promotes the ionization of dissociation product and shifts the peaks caused by CO dissociation to lower T_h. For H₂ dissociation, this phenomenon occurs only when θ  ⩾  5, due to its relatively low dissociation temperature.

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As shown in figure 6, the molar fraction of graphite decreases as non-equilibrium parameter increases. Therefore, the influence of condensed phase species, which is closely related to the condensed species composition, is negligible in high non-LTE system, see figure 9. For the non-LTE system whose θ is higher than 3, the molar fraction of graphite is too small to make difference in thermodynamic properties.

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The mixture ratio can significantly affect chemical reaction and composition, especially the condensed species. Figure 10 shows the composition of 10%CO₂–90%CH₄ mixture at atmospheric pressure under LTE. Similar to 50%CO₂–50%CH₄ mixture, at low temperature (lower than 1000 K), the reaction CO₂  +  CH₄ ↔ 2H₂O  +  2C(c) produces graphite and H₂O. However, due to the low ratio of CO₂, CH₄ is the dominant species in this temperature range. As temperature increases, the CH₄ dissociates into graphite and H₂ and therefore there is an increase of graphite molar fraction at 1300 K. At around 4100 K, the graphite fully sublimates and phase transition process completes.

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As shown in figure 11, the composition of graphite is drastically affected by mixture ratio in CO₂–CH₄ plasma. For pure CH₄, which is relatively stable for temperature lower than 1000 K, there is barely any graphite in low temperature. As temperature increases, the molar fraction of graphite increases sharply and remains relatively high due to CH₄ dissociation. At around 4100 K, the graphite fully sublimates and phase transition process completes.

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For CO₂–CH₄ mixture, there are two gentle peaks in graphite molar fraction. The first peak at temperature lower than 1000 K is caused by reaction CO₂  +  CH₄ ↔ 2H₂O  +  2 C(c). The largest value of this peak can be found in 50%CO₂–50%CH₄ mixture (see green curve in figure 11(a)) in which the ratio between H element and O element is 2 : 1. The second peak in temperature range from 1400 K to 4100 K, which is due to the CH₄ dissociation, can only be found in the mixture whose CH₄ proportion higher than 50%. For the mixture whose CH₄ proportion lower than 50%, there are enough O elements for C to produce CO which is more stable than graphite in this temperature range.

In non-LTE system, which is able to contain more gaseous species than LTE system at same electron temperature and pressure, the molar fraction of graphite is significantly decreases (see figure 11(b)). The first peak of graphite molar fraction is lower than 3‰. The second peak of graphite is still high in mixture with high CH₄ proportion. The peak values are 0.1 in pure CH₄ and 0.02 in 10%CO₂–90%CH₄ with θ  =  5 respectively.

Figure 12 shows the influence of mixture ratio on specific enthalpy in LTE and non-LTE system. Generally the specific enthalpy increases as CH₄ proportion due to its low molar mass. The inflection at 4200 K in mixture with CH₄ proportion higher than 50% is due to the graphite sublimation. Our enthalpy results for pure CO₂ under LTE show good agreement with that of Yang et al [21]. In a non-LTE system, there is no visible inflection in specific enthalpy even in mixture with high CH₄ proportion because the multiphase effect is weak in non-LTE system as discussed above.

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Figure 13 shows the influence of mixture ratio on specific enthalpy in LTE and non-LTE system. Due to the complicate reaction process in the CO₂–CH₄ mixture at low temperature, the first two peaks under LTE at around 1000 K and 4000 K are the consequence of combination of several reactions (see figure 13(a)). The first peak at around 1000 K can only be found in pure CH₄ and CO₂–CH₄ mixture. For the pure CH₄ system, this peak is the cause of CH₄ dissociation. For CO₂–CH₄ mixture, this peak is due to the reaction CO₂  +  CH₄ ↔ 2CO  +  2H₂. The peaks at around 4000 K in different mixtures are due to the different chemical process. For the pure CO₂, this peak is caused by CO₂ dissociation. For CO₂–CH₄ mixture whose CO₂ proportion is higher than 50%, this peak is due to the combination of CO₂ and H₂ dissociation. For 50% CO₂–50%CH₄ mixture, in which nearly all the CO₂ is consumed by reaction CO₂  +  CH₄ ↔ 2CO  +  2H₂ and there is no CO₂ at 4000 K, this peak is the consequence of H₂ dissociation only. For the pure CH₄ and CO₂–CH₄ mixture whose CH₄ proportion is higher than 50%, the dramatic sharp peak at 4000 K corresponding to the sharp increase in specific enthalpy is due to the H₂ dissociation and graphite sublimation. The third peak which cannot be seen in pure CH₄ is due to the CO dissociation. The highest peak value can be found in 50%CO₂–50%CH₄ mixture in which the molar fraction of CO is highest because of the reaction CO₂  +  CH₄ ↔ 2CO  +  2H₂. The rest two peaks at around 14 000 K and 30 000 K are caused by first and second order ionization.

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In non-LTE system, as shown in figure 13(b), the influence of condensed can only be found in pure CH₄ at T_e  =  13 000 K. The three peaks at 4000 K, 16 000 K and 23 000 K are caused by CO₂  +  CH₄ ↔ 2CO  +  2H₂, H₂ ↔ 2 H⁺  +  2e⁻ and CO ↔ C⁺  +  O⁺  +  2e⁻, respectively.

The multiphase effect on thermodynamic properties is significant especially in CO₂–CH₄ mixture with high CH₄ proportion due to the large production of graphite during CH₄ dissociation process, as shown in figure 14. When CH₄ proportion is lower than 50%, the influence of condensed phase is weak and can be neglected in thermodynamic property calculation, especially in non-LTE system.

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As shown in figure 15, the influence of pressure on phase transition shows in two ways. Firstly, according to Le Chatelier's law, the increase of the pressure opposes changes to the original state of equilibrium, so that reactions, including phase transition, are shifted to higher electron temperature as the pressure increases. Secondly, the increase of pressure enlarges the $\mu _{i}^{o}$ in equation (4). Since the pressure does not influence chemical potential of condensed phase which is equal to chemical potential of corresponding gaseous species μ_i-gas, the increase of $\mu _{i}^{o}$ leads to decrease of gaseous carbon and therefore increase the molar fraction of graphite according to equations (3) and (4). This trend is consistent with results of condensed metal species presented by André et al [13].

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According to Le Chatelier's law, the reactions, including dissociation, ionization and phase transition, are shifted to higher electron temperature as the pressure increases. Therefore, the peaks in specific heat which are closely related to reactions are shifted to high temperature in high pressure system, as shown in figure 16. The multiphase effect on thermodynamic properties is strong in high pressure system, consistent with that on composition.