Calculation of 2-temperature plasma thermo-physical properties considering condensed phases: application to CO₂–CH₄ plasma: part 2. Transport coefficients

The transport coefficients, namely viscosity, thermal conductivity and electrical conductivity, were calculated using the classical Chapman–Enskog method [13, 14], which assumes that the particle distribution function is a first-order perturbation to the Maxwellian distribution. The perturbations are expressed in series of Sonine polynomials, finally leading to a system of linear equations that can be suitably solved to obtain different transport properties. In a non-LTE system, 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.

Different approaches have been proposed to deal with the 2-T transport properties in partially-ionized plasmas. Among these, two main methods are frequently used: Devoto [1, 2, 15–17] developed a simplified theory neglecting the collisional coupling between heavy species and electrons, whereas Rat et al [3] developed a method that retains the coupling. It should be noted that coupling between electrons and heavy species does not lead to significant changes in the predicted non-equilibrium plasma transport properties, except for certain ordinary diffusion coefficients [4]. In the current work, the simplified approach of Devoto has been used, with a third-order approximation for diffusion coefficients, thermal conductivity and electrical conductivity. The same method has been used for calculating the viscosity, for which the second-order approximation has been adopted.

Among these transport coefficients, special attention had to be paid to the thermal conductivity. Thermal conductivity depends on the translation of electrons (λ_tre), the translation of heavy species (λ_trh), internal energy changes (λ_in) and chemical reactions (λ_reac). The total thermal conductivities for electrons and heavy species can be evaluated by

$$\begin{eqnarray}&&{{\lambda}_{\text{tot}}}={{\lambda}_{\text{tr}\,\text{e}}}+{{\lambda}_{\text{tr}\,\text{h}}}+{{\lambda}_{\text{in}}}+{{\lambda}_{\text{reac}}}.\end{eqnarray} \tag{ 1 }$$

The translational thermal conductivities (λ_{tr e} and λ_{tr h}) were calculated according to the method of Devoto [1, 15] in the third approximation of Sonine polynomial terms:

$$\begin{eqnarray}&&{{\lambda}_{\text{tr}\,\text{e}}}=\frac{75n_{\text{e}}^{2}k}{8}\sqrt{\frac{2\pi k{{T}_{\text{e}}}}{{{m}_{\text{e}}}}}{{\left({{q}^{11}}-{{\left({{q}^{12}}\right)}^{2}}\text{/}{{q}^{22}}\right)}^{-1}}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}{{\lambda}_{\text{tr}\,\text{h}}}=-\frac{75k}{8}\frac{\sqrt{2\pi k{{T}_{\text{h}}}}}{|q|}|\begin{array}{c} q_{ij}^{00} & q_{ij}^{01} & 0 \\ q_{ij}^{10} & q_{ij}^{11} & {{n}_{i}} \\ 0 & {{n}_{j}}\text{/}\sqrt{{{m}_{j}}} & 0 \end{array}|\end{eqnarray} \tag{ 3 }$$

where the q^mp and $q_{ij}^{mp}$ are computed from associated collision integrals [1, 15].

The presence of internal degrees of freedom can affect the heat flux vector and therefore gives rise to an internal thermal conductivity, which is derived using the Hirschfelder–Eucken approximation [14]:

$$\begin{eqnarray}&&{{\lambda}_{\text{in}}}=\underset{i\ne e}{\overset{w}{\sum}}\,{{x}_{i}}D_{ii}^{b}k\left({{C}_{pi}}\text{/}R-2.5\right)\underset{j\ne e}{\overset{w}{\sum}}\,\,{{\left(\frac{{{x}_{j}}}{{{x}_{i}}}\frac{D_{ii}^{b}}{D_{ij}^{b}}\right)}^{-1}}\end{eqnarray} \tag{ 4 }$$

where C_pi is the specific heat at constant pressure of the species i, $D_{ii}^{b}$ and $D_{ij}^{b}$ are respectively the self-diffusion coefficient and the binary diffusion coefficient between species i and species j, and R, x_i and w respectively denote the ideal gas constant, the mole fraction of species i, and the total number of species.

Chemical reactions lead to an additional heat flux, which introduces an additional reactive thermal conductivity component. A general expression of total reactive thermal conductivity with good applicability to monatomic, diatomic and polyatomic gases in terms of a two temperature model is utilized [18]:

$$\begin{eqnarray}&&{{\lambda}_{\text{reac}}}=\underset{r=1}{\overset{v}{\sum}}\, \Delta h_{r}^{\ast}\frac{|{{H}_{r}}|}{|H|}\end{eqnarray} \tag{ 5 }$$

where v and Δh_r are, respectively, the number of chemical reactions and reaction enthalpy change of reaction r. The H_r and H are matrix derived by generalized Saha equations, Guldberg–Waage equations, the species diffusion velocities and Van't Hoff's equation modified for the two-temperature plasmas [18, 19].

The viscosity and electrical conductivity are obtained following the derivation presented by Devoto [15], which are given by

$$\begin{eqnarray}\mu =-\frac{5\sqrt{2\pi {{k}_{\text{B}}}{{T}_{h}}}}{2}|\begin{array}{c} \hat{q}_{ij}^{00} & {{n}_{j}}\sqrt{{{m}_{j}}} \\ {{n}_{j}} & 0 \end{array}|{{\left(|\hat{q}_{ij}^{00}|\right)}^{-1}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}\sigma =\left(\frac{{{e}^{2}}n{{m}_{\text{e}}}{{n}_{\text{e}}}}{\rho {{k}_{\text{B}}}{{T}_{\text{e}}}}\right)\,\centerdot \left(\frac{3{{n}_{\text{e}}}\rho}{2n{{m}_{\text{e}}}}\right)\,\centerdot \sqrt{\frac{2\pi {{k}_{\text{B}}}{{T}_{\text{e}}}}{{{m}_{\text{e}}}}}|\begin{array}{c} {{q}^{11}} & {{q}^{12}} \\ {{q}^{21}} & {{q}^{22}} \end{array}|\\ \,\,\,\times \,{{\left(|\begin{array}{c} {{q}^{00}} & {{q}^{01}} & {{q}^{02}} \\ {{q}^{10}} & {{q}^{11}} & {{q}^{12}} \\ {{q}^{20}} & {{q}^{21}} & {{q}^{22}} \end{array}|\right)}^{-1}}.\end{eqnarray} \tag{ 7 }$$

The coefficients $\hat{q}_{ij}^{00}$ and q^mp are given in the previously published paper [15].

Expressions for transport properties contain determinants that depend on collision integrals, which are averages over a Maxwellian distribution of the collision cross–sections for the binary interactions between species [13, 14]. Collision integrals for the interaction between species i and j are defined as

$$\begin{eqnarray} &&\Omega _{ij}^{(l,s)}=\sqrt{\frac{kT_{ij}^{\ast}}{2\pi {{\mu}_{ij}}}}{\int}_{0}^{\infty}\exp \left(-\gamma _{ij}^{2}\right)\gamma _{ij}^{2s+3}Q_{ij}^{l}(g)\text{d}{{\gamma}_{ij}}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&T_{ij}^{\ast}=\frac{{{m}_{i}}{{T}_{j}}+{{m}_{j}}{{T}_{i}}}{{{m}_{i}}+{{m}_{j}}}\end{eqnarray} \tag{ 9 }$$

where γ is the reduced initial speed of the colliding molecules i and j. g_ij and μ_ij are the initial relative speed and reduced mass (given by $\mu _{ij}^{-1}$  =  $m_{i}^{-1}$  +  $m_{j}^{-1}$) of the colliding species i and j. For non-equilibrium plasmas, collision integrals have to be calculated using a reduced temperature $T_{ij}^{\ast}$, given in equation (9). $Q_{ij}^{l}$ is the transport cross section, which can be derived from interaction potentials.

Each encounter between particles can be classified in to one of four types: (A) neutral–neutral encounters; (B) neutral–ion encounters; (C) electron–neutral encounters; (D) encounters between charged particles. For different types of encounter, the different interaction potentials have been adopted.

2.2.1. Neutral–neutral interactions.

2.2.2. Ion–neutral interactions.

For neutral–ion interactions, two kinds of processes should be taken into account, which are purely elastic collisions and the resonant charge–exchange inelastic process. For l odd (l  =  1 or 3), we followed the work of Murphy [27] and estimate the total collision integrals from the elastic and inelastic contributions with the empirical mixing rule:

$$\begin{eqnarray}&&{{ \Omega }^{(l,s)}}=\sqrt{{{\left(\Omega _{\text{in}}^{(l,s)}\right)}^{2}}+{{\left(\Omega _{\text{el}}^{(l,s)}\right)}^{2}}}\end{eqnarray} \tag{ 10 }$$

where $\Omega _{\text{in}}^{(l,s)}$ is the inelastic collision integral and $\Omega _{\text{el}}^{(l,s)}$ is the elastic interactions. The elastic collision integrals between neutral species and first order ions are obtained by adopting the phenomenological potential [24, 25]. The ions polarizability is taken from the NIST Computational Chemistry Comparison and Benchmark Database [26] and presented in table 2. For the elastic collision integrals between neutral species and high order ions, the polarization potential is adopted.

Table 2. Polarizability of first order ions.

Species	α (Å)	Species	α (Å)
C+	0.899	O+	0.388
$\text{H}_{2}^{+}$	0.447	CO+	1.337
$\text{H}_{2}^{-}$	2.836	OH+	0.599
$\text{O}_{2}^{-}$	3.223	OH−	3.944
$\text{O}_{2}^{+}$	0.97	HCO+	1.44
O−	2.164

As given in equation (10), the inelastic interaction is taken into account in collision integral calculation. The charge-transfer cross section is approximated in the form

$$\begin{eqnarray}&&{{Q}_{\text{ex}}}={{(A-B\ln E)}^{2}}\end{eqnarray} \tag{ 11 }$$

where E is the collision energy. The constants A and B can be obtained from experimental data or theoretical calculations. The data of interactions between C–C⁺, H–H⁺ and O–O⁺ are taken from the work by Copeland and Crothers [28], which agrees well with the data presented by Laricchiuta [29] and Eletskii [30]. For the charge exchange interactions between neutral atom and its negative ion, no accurate experimental data or theoretical calculations were found in the literature. The constants A and B have been determined using an empirical formula [31].

Laricchiuta [29] showed that charge exchange cross section of neutral-multiply charged ion interaction could be relevant especially for divalent ion and trivalent ion. However, this effect is not taken into account in the present paper, due to the small possibility of simultaneous presence of neutral species and multiply charged ions. Besides, the charge exchange cross sections for collisions between unlike species are small compared to the elastic collision integrals and are neglected in this paper.

The internal excited state may affect the many kinds of collision integrals and then may affect the transport coefficients [32, 33]. However, according to [32, 33], the influence of internal excited state on transport coefficients is relatively small when pressure is lower than 10 atm. Therefore, we believe that the results presented in this paper, which are lower than 10 atm, are not perfectly accurate but still reliable. To our knowledge, little papers of thermo-physical properties calculation considered the effect of internal excited state on collision integrals and transport coefficients. The importance of this effect is underestimated and needs more attention. In our further research work, we will focus on the effect of internal excited state on the collision integrals and transport properties under different condition.

2.2.3. Electron–neutral interactions.

2.2.4. Charged species interactions.

As presented in [23], the collision integrals for this interaction are known either in tabular form by Hahn [35] or approximated with closed forms presented by Liboff [36]. Devoto's work [37] shows that different definition of Debye length can significantly affect the charged–charged interactions and further affect the transport coefficients. In the present work, we used the method presented by Liboff with the Debyle length without consideration of ions, following the recommendation of Murphy [38] who found that without consideration of ions in Debye length calculation, the calculated argon electrical conductivity agrees better with the measured value. The Debye length without consideration of ions is given by

$$\begin{eqnarray}&&{{\lambda}_{d}}=\sqrt{\frac{{{\varepsilon}_{0}}k{{T}_{\text{e}}}}{{{e}^{2}}{{n}_{\text{e}}}}}\end{eqnarray} \tag{ 12 }$$

where ε₀ is the vacuum electric permittivity and e is the elementary charge.

Considering the close interrelationship between composition and transport coefficients, a brief analysis of reactions and composition in CO₂–CH₄ mixture is necessary before researching the transport coefficients. Figure 1(a) shows the composition of 50%CO₂–50%CH₄ mixture composition under LTE at atmospheric pressure. 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. The second order ionization of C occurs at 27 000 K.

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The mixture ratio can significantly affect chemical reaction and composition, especially the condensed species. Figure 1(b) 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.

Figure 2 shows the electrical conductivity of CO₂–CH₄ mixtures with different ratio under LTE at atmospheric pressure. The mixture ratio can affect electrical conductivity especially at temperature lower than 15 000 K and higher than 23 000 K. Due to the low temperature of C ionization (12 500 K), the ionization process is dominated by C⁺ in low temperature. Therefore the increase of CO₂ ratio causes increase of electron number density and further increases the electrical conductivity. When temperature is higher than 23 000 K, the high order ionization of C and O enhances the coulomb interaction and therefore lowers the electron mobility degree. Due to this phenomenon, the presence of CO₂ in CO₂–CH₄ mixtures decreases the electrical conductivity. Our results show good agreement with that of Aubreton [39] and Yang [10].

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The total thermal conductivity consists of four different parts namely heavy species translation thermal conductivity, electron translation thermal conductivity, internal thermal conductivity and reactive thermal conductivity. Figure 3 shows the components of total thermal conductivity of 50%CO₂–50%CH₄ plasmas at atmospheric pressure under LTE. The sharp peaks in total thermal conductivity are mainly caused by chemical reactions. The first peak at 1200 K is due to the phase transition and reaction CO₂  +  CH₄ ↔ 2CO  +  2H₂. The other peaks at 3200 K, 6800 K and 15 000 K are due to H₂ dissociation, CO dissociation, first order ionization respectively. The total thermal conductivity is dominated mainly by reactive and heavy species translational thermal conductivity at temperature under 10 000 K. In temperature range from 10 000 to 20 000 K, the total thermal conductivity is affected by both electron translational and reactive thermal conductivity. When the temperature is above 20 000 K, the total thermal conductivity is completely dominated by the electron translational component.

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The mixture ratio can significantly affect the thermal conductivity, as shown in figure 4. The first peak at around 1000 K, which cannot be found in pure CO₂ plasma, is due to the CH₄ dissociation and 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 CO₂–CH₄ mixture whose CH₄ proportion is higher than 50%, the dramatic sharp peak at 4000 K 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. Our results for pure CO₂ consist well with Yang et al [10]. However, our results are significantly lower than Aubreton's results [39]. This discrepancy may due to the different value of polarizability we used. Considering the new polarizability we used and the good agreement with newly published paper [10], our results should be more reliable.

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The influence of mixture ratio on viscosity under LTE is shown in figure 5. Due to the positive correlation between molar weight and viscosity, the increase of CO₂ proportion in CO₂–CH₄ mixture significantly increases the viscosity especially in low temperature. Our results of pure CO₂ show good agreement with Yang et al [10] while our results of 50% CO₂–50%CH₄ is slightly lower than that of Aubreton [39]. This may be also due to the different use of polarizability.

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Another phenomenon shown in viscosity of pure CH₄ in figure 5 should be noted. Due to the reaction CH₄ ↔ 2H₂  +  C(c), the transport process is dominated by gaseous H₂ and H in temperature range from 1200 to 4100 K, since the condensed graphite does not contribute to the transport process. Due to the low polarizability and low molar weight of H₂ and H, the viscosity of pure CH₄ lowers sharply in this temperature range. The slight fluctuation in viscosity of 20% CO₂–80%CH₄ is also caused by this reason.

As shown in figures 5 and 6, the presence of condensed phase can significantly influence the composition and further affect the transport coefficients, especially in the CO₂–CH₄ mixture with CH₄ proportion higher than 50%. For the viscosity, the reaction CH₄ ↔ 2H₂  +  C(c) causes sharp decrease in temperature from 1200 to 4100 K. This phenomenon is significant in pure CH₄ system, in which the system is dominated by H₂ in temperature from 1500 to 4000 K. The viscosity in this temperature range agrees well with that of pure H₂ plasma presented by Capitelli et al [40]. For the total thermal conductivity, the presence of condensed phase increases the molar fraction of H₂ and therefore increases its dissociation temperature. For the mixture with CH₄ proportion lower than 50%, the influence of condensed phase is negligible at 1 atm under LTE. The influence of condensed phase on electrical conductivity is negligible because at this temperature range (lower than 4000 K) there are barely electrons and ions in system and therefore the electrical conductivity is relatively low. Therefore the comparison is not presented.

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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. For the electrical conductivity, see figure 7(a), the delay of dissociation reactions in non-LTE system limits the number density of monatomic species at low electron temperature [41, 42]. Therefore the increase of non-equilibrium parameter θ decreases the electrical conductivity at low temperature. Once the dissociation reaction occurs, the ionization can be extremely rapid due to the high electron temperature, which causes the sharp increase in electrical conductivity.

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The delay of dissociation reactions in non-LTE system can significantly affect the total thermal conductivity mainly by influence the reactive thermal conductivity. Under LTE, the first peak at 1200 K is due to the phase transition and reaction CO₂  +  CH₄ ↔ 2CO  +  2H₂. The other peaks at 3200 K, 6800 K and 15 000 K are due to H₂ dissociation, CO dissociation and first order ionization respectively. For θ  =  3, as shown in blue curve in figure 7(b), 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 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 a sharp peak 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.

The non-LTE effect can also affect the viscosity, as shown in figure 7(c). As given in equation (6), the viscosity is proportional to $T_{\text{h}}^{1/2}$. At same T_e, as the increase of θ, the $T_{\text{h}}^{1/2}$ decreases therefore the viscosity is limited to a lower value. Ionization is also delayed, as discussed above, nevertheless the peak value of viscosity that is reached before ionization is much lower than for the LTE case. Besides, when θ is higher than 4, the high dissociation temperature of CO leads to a stepwise decrease in viscosity.

On the other hand, from the perspective of T_h, the influence of non-LTE effect on thermal conductivity can be different or even opposite, 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. 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. As a result, the peaks in thermal conductivity caused by CO dissociation are shifted to lower T_h in non-LTE system. For H₂ dissociation, this phenomenon occurs only when θ  ⩾  5, due to its relatively low dissociation temperature where the ionization of H does not occur in low θ system.

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As shown in the first article [5], the molar fraction of graphite decreases as non-equilibrium parameter increases no matter from the T_e scale or T_h scale. 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 thermal conductivity.

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In a non-LTE system, due to the delay of dissociation reactions which may change the sequence of reactions, the influence of mixture ratio is different with that in LTE system. For the electrical conductivity, the difference can be found in low temperature range. Under LTE system, the low ionization temperature of C makes C⁺ dominates the system ionization in low temperature. However, in non-LTE system, the delay of CO dissociation leads to low proportion of C atom and therefore limits the C ionization. On the other hand, the early dissociation of H₂ makes H the only monatomic species in low temperature. Therefore in non-LTE system, the ionization process at low temperature is dominated by H⁺. As a consequence, in 2-T system, the electrical conductivity increases as the CH₄ ratio in whole temperature range, as shown in figure 10(a). Besides, a sharp increase at around 24 000 K can be found in pure CO₂ and CO₂–CH₄ mixture, which is due to the dissociation of CO and ionization of C and O atom.

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In non-LTE system, the delay of dissociation reaction makes dissociation and ionization reactions occur at same temperature and leads to high peaks in thermal conductivity. As shown in figure 10(b), the first peak, which can only be found in CO₂–CH₄ mixture, is due to the dissociation of H₂O. The peak at around 14 000 K in high CH₄-content system is due to the successive dissociation of CH₄. The peaks at around 16 000 K and 23 000 K are respectively caused by H₂ and CO dissociation. The largest peak can be found in 50%CO₂–50%CH₄ mixture due to the high molar fraction of H₂ caused by reaction CO₂  +  CH₄ ↔ 2CO  +  2H₂.

Similar to the LTE system, the viscosity in non-LTE CO₂–CH₄ system increases as CO₂ proportion, as shown in figure 10(c). Besides, the stepwise decrease caused by high dissociation temperature of CO dissociation can be found in pure CO₂ and CO₂–CH₄ mixture.

The influence of condensed phase in non-LTE CO₂–CH₄ mixture is much weaker than that in LTE system due to the low molar fraction of graphite [5], as shown in figure 11. The slight difference between multiphase system and pure gaseous system can be found in pure CH₄ system as shown in black solid and dashed curve in figure 11. For CO₂–CH₄ mixture and pure CO₂ system, the multiphase effect is negligible in non-LTE assumption.

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Figure 12 shows the influence of pressure on the specific heat and transport coefficients for pressures 0.1 atm, 1 atm, 5 atm and 10 atm for θ  =  5. According to Le Chatelier's law, the increase of the pressure opposes changes to the original state of equilibrium, so that dissociation and ionization reactions are shifted to higher electron temperature as the pressure increases. As a result, the sharp increase in electrical conductivity and peaks in thermal conductivity and viscosity are shifted to higher electron temperature in high pressure system.

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