Two-temperature model of the non-thermal chemical dissociation of CO₂

Abstract

A two-temperature model with a vibrational temperature different from the gas temperature is presented for the decomposition of CO₂. All vibrational modes of CO₂, CO and O₂ are included, and a novel procedure to calculate the vibrational–dissociation reaction rate constant is proposed. In all cases, a Boltzmann distribution is assumed for the vibrational states and the effect of the activation barrier on the chemical reactions is taken into account using the Fridman–Macheret expression. The non-thermal reaction rate constants are therefore functions of gas and vibrational temperatures. The results show that relatively higher vibrational temperatures benefit CO₂ conversion. This work provides more insights into the dissociation and recombination reactions of CO₂ conversion under non-thermal conditions and offers the opportunity for multi-dimensional non-thermal modelling in the future.

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1 Introduction

The accumulation of anthropogenic greenhouse gases has resulted in significant impacts on human life, making carbon capture, utilisation and storage (CCUS) an increasingly popular topic worldwide.¹ CO₂ is an inherently stable molecule and a significant amount of energy is required for its dissociation (5.453 eV). By producing value-added compounds from CO₂ feedstock, CCUS presents a pragmatic and financially viable avenue towards realising the commercial deployment of technologies aimed at abating carbon dioxide emissions.^2,3 Development and optimisation of CO₂ processing technologies are attracting more and more attention.^4,5

Thermal and non-thermal dissociation are both popular approaches for CO₂ conversion. CO₂ dissociation using microwave (MW) plasma is considered to be a promising technology, as recent experimental results show that the maximum energy efficiency (∼48%) in our group is very close to the theoretical thermal efficiency limit (∼50%) (Fig. 1).^6–9 Compared to thermal dissociation, more efficient energy utilisation in the non-thermal state appears promising for CO₂ dissociation,^10,11 which might explain the high efficiencies reached. In the non-thermal plasma, vibrational energy decreases or even eliminates the need to heat all degrees of freedom, potentially improving the energy efficiency.¹² However, since a great number of vibrational levels are involved in the vibrational kinetics, it is challenging to simulate the non-thermal dissociation process, especially for polyatomic molecules that have multiple vibrational modes.¹³

Fig. 1 Experimental data for CO₂ splitting from DIFFER, showing the energy efficiency as a function of the pressure.^6–9

Recently, a state-to-state (STS) model has been widely used to investigate vibrational kinetics in CO₂ plasma.¹⁴ In the STS model, vibrational states of the molecules are treated as independent species and their interactions are considered as separate reactions, providing the possibility to resolve the non-Boltzmann distribution of internal energy. However, numerous rate coefficients for all the excitation and dissociation processes of every vibrational level are required, which makes the STS model computationally expensive and resource-demanding. Due to this limitation of the STS approach, the number of vibrational levels of CO₂ molecules used in the model has to be reduced.¹⁵ Based on the assumption that vibrational relaxation of symmetric and asymmetric modes has a significantly different relaxation time, the PLASMANT group developed a set of STS models, including 25 vibrational levels of CO₂ (4 lumped symmetric mode levels + 21 asymmetric mode levels, up to the dissociation limit) with a focus on the asymmetric mode. This allows the model to reveal CO₂ vibrational kinetics under different conditions such as reduced electric field, temperature, pressure, etc., which are applied to dielectric barrier discharges (DBD) and MW plasma investigation.^10,16–18 However, the temperature measurement experiments by Biondo et al.¹⁹ and Klarenaar et al.²⁰ indicate that the vibrational temperature of the bending and symmetric modes is not always equal to the gas temperature, though it is closer to gas temperature compared to the vibrational temperature of the asymmetric mode. It implies that the vibrational energy in the bending and symmetric modes still can enhance CO₂ dissociation, thereby increasing the energy efficiency. The group from IST Lisbon University also proposed several models, covering 72 CO₂ vibrational levels with vibrational energies up to about 2 eV, and paid great attention to the symmetric and bending modes, which have good agreement with the experiments of direct current (DC) glow discharge.^13,21–23 Despite many efforts to reduce the computational cost,^24–28 it is currently not possible to couple the STS approach with computational fluid dynamics (CFD) to represent a more realistic fluid-dynamic situation with a multidimensional model.

A multi-temperature model, another approach to depict the non-thermal effect, can decrease the computational cost and raise the numerical efficiency. Park²⁹ proposed a two-temperature model to account for the effects of vibrational excitation in the dissociation rates of N₂ and O₂, governed by the geometric mean of the translational–rotational temperature (T_g) and the vibrational temperature (T_v).^30,31 However, the model is not consistent with microscopic reversibility and also has other limitations.³¹ A more complicated multi-temperature model is the Treanor–Marrone model, adjusted by a semi-empirical parameter U, which is suitable for computing dissociation rate coefficients for molecule–molecule and molecule–atom collisions.³² Due to the dependence on U, the Treanor–Marrone model with the same value of U fails to calculate all the dissociation rates at different vibrational levels in the wide temperature range.³³ Compared with Park and Treanor–Marrone models, the Fridman–Macheret model^34–36 has no limitation to the vibrational and gas temperatures, and provides the most accurate results.³⁷

This work proposes a method which aims to quantify the impact of non-thermal effects and to provide a benchmark for the simple case of a two-temperature system governed by a gas temperature different from the vibrational temperature. The Boltzmann distribution is postulated to govern the vibrational states with a vibrational temperature, which is different from gas temperature. The non-thermal model presented in this study employs a weighted algebraic expression to calculate the reaction rate constants. Using the proposed model, the energy efficiencies are calculated at different combinations of vibrational and gas temperatures, which can serve as a benchmark for future experiments under non-thermal conditions, and give insight into underlying mechanisms. This approach significantly reduces computational costs and facilitates the establishment of multi-dimensional non-thermal CO₂ models for comprehensive analysis in the future.

2 Model description

Numerical modelling becomes increasingly important as it is difficult to measure all the parameters in experiments in sufficient detail.¹² However, establishing a specific reactor model with complex CO₂ molecule properties is an arduous task.³⁸ To date, many zero-dimensional (0D) models have been used to describe the reaction mechanism in CO₂ dissociation under various conditions.^24,38 Although these models have limitations, the results still show good agreement with experimental data.^7,39 Therefore, a 0D continuous stirred tank reactor (CSTR) model is simulated, where the gas composition (CO₂, CO, C, O, O₂, and C₂), and other characteristics (gas temperature, pressure, etc.) are assumed to be spatially homogeneous through perfect mixing. For this reactor model, the pressure is held constant (100 mbar).

The model is based on Cantera, which is an open-source Python package that solves chemical kinetics, thermodynamics, and transport processes.⁴⁰ Thermodynamic data, encompassing enthalpy, entropy, heat capacity, and other relevant parameters for all species, are obtained from the NASA database.⁴¹ Because translational–rotational equilibrium is much faster than vibrational excitation, translational and rotational modes are lumped together, governed by T_g. All the vibrational modes are assumed to have Boltzmann distribution with the single vibrational temperature T_v. In this paper, we consider vibrational–vibrational relaxation to be infinitely faster than vibrational–translational relaxation, to separate T_g and T_v from each other.

2.1 Vibrational levels and their energies

When chemical conversion is driven by gas heating, the supplied energy is distributed across all molecular degrees of freedom. However, the effectiveness in promoting reactivity is generally not expected to be the same for all internal modes, as proposed in the field of mode-selective chemistry.^42,43 Since vibrational motion involves changes in the relative positions of atoms, it plays a crucial role in promoting the cleavage of specific bonds. Selective excitation of vibrational modes can guide the chemical reaction toward the desired product and lower the energy barrier for the reaction. Consequently, vibrational activation of chemical reactions can reduce gas temperature requirements, thereby enhancing energy efficiency.

The non-thermal vibrational distribution depends on the model of the energy spectrum. Vibrational energies of CO₂ molecule are calculated by anharmonic oscillator approximation in our model.¹⁹ The vibrational energy is limited by the dissociation energy of the molecule. For CO₂, we use the limit of 5.453 eV which is the dissociation reaction enthalpy at the temperature of 0 K. As a linear triatomic molecule, CO₂ has three vibrational modes, including symmetric, doubly degenerated bending, and asymmetric modes, that need to be considered in the simulation. The energies of the CO₂ levels are calculated using the following anharmonic oscillator approximation:³⁸

where ω_i, x_ij, r_ijk are spectroscopic constants, which are taken from Biondo et al.,¹⁹v_i is the vibrational level in each vibrational mode, c and h are the speed of light and the Planck constant, and d_i characterises the degeneration. It should be noted that in eqn (1), Fermi resonance is not taken into account.

For the diatomic molecules, i.e. CO and O₂, the vibrational energies for each level are directly taken from Laporta et al.⁴⁴ and Esposito et al.,⁴⁵ respectively. Due to the very limited amount of C₂, we do not consider its vibrational excitation to save computing time. Our comprehensive set of vibrational levels for CO₂, O₂, and CO molecules includes 10 810, 47, and 81 levels, respectively. More details about the vibrational levels of different molecules can be found in the ESI† material.

2.2 Reaction scheme

The evolution of the species densities is defined by the following equation:where n_i [in m⁻³] is the number density of species i, a^R_im and a^L_im are the right-hand side and left-hand side stoichiometric coefficients of species i in the reaction m, K_m [in m³ s⁻¹] is the rate constant of reaction m, and n_l [in m⁻³] is the density of the lth reactant.¹⁵

For strongly endothermic reactions, such as reaction N₁ in Table 1, the backward reactions are barrierless and cannot be vibrationally enhanced. However, it should be emphasised that for the reaction where the activation energies of the forward and reverse reactions are comparable, such as the reaction N₄, vibrational excitation is considered for both forward and reverse reactions. Fig. 2 illustrates the effect of vibrational energy on the activation barrier.

Table 1 List of thermal chemistry reactions. All the rate constants of the backward reactions are calculated by detailed balance

No.	Reaction	A (m^3 s^−1)	b	E a (eV)	α	Cite	Note
a Multiply A by 2.0, and 1.0 for M = atoms and molecules, respectively. b Multiply A by 1.5, and 1.0 for M = atoms and molecules, respectively. c Multiply A by 5, and 1.0 for M = atoms and molecules, respectively.
N^f1	CO2 + M ↔ O + CO + M	1.15 × 10^−8	−1.5	5.453	1.00	46
N^f2	CO + M ↔ C + O + M	1.99 × 10^−9	−1.0	11.128	1.00	46
N^f3	O2 + M ↔ 2O + M	6.20 × 10^−12	−0.56	5.210	0.99	45
N^f4	O + CO2 ↔ O2 + CO	4.50 × 10^−16	0	2.913	0.54	47
N^f5	C + CO ↔ C2 + O	3.98 × 10^−13	−1.00	4.998	1.00	48
N^f6	O + CO ↔ C + O2	6.48 × 10^−17	−0.18	5.963	1.00	48

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Fig. 2 Schematic illustration of the impact of vibrational energy on the activation barrier of reversible reaction CO₂ + O.

Compared with the thermal state, the vibrational energy in CO₂ molecules can facilitate the dissociation of CO₂ chemical bond by reducing the energy barrier, dramatically lowering the need for thermal energy. The non-thermal reaction rate constant is calculated by:

where v is the vibrational level, and for CO₂, it denotes the triple of v₁, v₂ and v₃. According to the Fridman–Macheret α-model approach,⁴⁹α is the so-called “coefficient of vibrational energy utilisation” ranging from 0 to 1, which is used to estimate the effect of the vibrational excitation on overcoming the activation barrier of the chemical reaction. Values of α for different reactions are displayed in Table 1. is the effective vibrational energy of molecules, which is always ≤ E_a/α. For the effective vibrational energy higher than the activation barrier, the reactions are barrierless. R₀ is the pre-exponential factor and is considered to be vibrational level independent.

The overall reaction rate constant, K^non_total [in m³ s⁻¹], can be calculated by multiplying the rate constant of each vibrational level with its corresponding population fraction:

where f_vi is the population fraction in the vibrational level i. To simplify the model, a Boltzmann distribution is assumed for the population of all vibrational states.⁵⁰ For the diatomic molecular CO and O₂, the population fraction can be calculated by:where E_vi is the vibrational energy, and Z(T_v) is the vibrational partition function, which is expressed by the following equation:Due to the doubly degenerated bending mode, the Boltzmann distribution (f_v(1,2,3)) and partition function of CO₂ molecules (Z(T_v)) are changed to:³⁸where 1, 2 and 3 represent symmetric, doubly degenerated bending and asymmetric modes respectively. For convenience, the following eqn (11)–(15) are derived for diatomic molecules.

The main difference between the present method and the theoretical-informational approach lies in the calculation of the pre-exponential factor R₀ [in m³ s⁻¹]. In the theoretical-informational approach, the value of R₀ is equivalent to the pre-exponential factor A (Table 1).^51,52 However, this value calculated by the theoretical-informational approach differs from that calculated using the thermal Arrhenius expression when T_v = T_g. In this method, R₀ can be calculated by matching the non-thermal rate and the thermal reaction rate (i.e. T_v = T_g):

Using the expressions for R₀, the overall reaction rate constant is computed as follows:

It should be noted that eqn (10) assumes that only one reactant has vibrational energy. For the reaction N^b₄, both CO and O₂ have vibrational energy. Therefore, the population of vibrational level is described by the following expression:

where and are the vibrational energy of O₂ and CO molecule, respectively.

The pre-factor R₀ and overall reaction rate constant becomes:

where, (and for vibrational energies higher than E_a/α, ).

2.3 Post-processing of the results

For each simulation step, we can obtain the composition of the gas mixture. Here, we define some characteristics which are important for the following section.

The conversion (γ) and energy efficiency (η) [both in %] are the primary global performance metrics for evaluating the dissociation process. The conversion is given as:

where X_CO, X_C and X_CO2 are molar fraction [in %] of CO, C and CO₂ of the converted mixture, respectively.⁵³ The energy efficiency is defined based on the enthalpy ΔH_CO = 2.93 eV of the dissociation process at the inlet temperature of 300 K (CO₂ → CO + 1/2O₂), which can be expressed as:where ΔH_sum [in eV] is the total enthalpy change of the converted mixture, i.e. the sum of the enthalpy change of thermal energy ΔH_therm, dissociation energy ΔH_disso, and extra vibrational energy ΔH_vib. An important additional aspect to be taken into account, is the surplus energy needed to elevate the vibrational temperature over the gas temperature. The enthalpy change of extra vibrational energy of each molecule with a vibrational temperature higher than the gas temperature can be described by:

Except for the vibrational energy, the enthalpy change of thermal energy for heating the gas from room temperature (T_room) to a higher temperature (T_g) (mole fractions are unchanged) can be expressed by:⁵³

ΔH_therm = H(T_g, CO₂) − H(T_room, CO₂) (17)

where H [in eV] is the enthalpy. The enthalpy change of dissociation energy is defined by:

ΔH_disso = H(T_g, X) − H(T_g, CO₂) (18)

where X [in %] is the mole fraction components at the end of the state.

3 Results and discussion

According to the Fridman–Macheret model, reactions with a relatively low activation barrier are enhanced less by vibrational excitation. Therefore, we assumed that the contribution of vibrational energy towards backward reactions N^b₁, N^b₂, N^b₃, N^b₅, and N^b₆ can be neglected. For the other reactions, the reaction rate constants are influenced by the higher vibrational temperature.

As previously mentioned, in eqn (9), R₀ is primarily influenced by the gas temperature and independent of the vibrational temperature. When α is equal or close to 1 (i.e. N^f₁, N^f₂, N^f₃, N^f₅ and N^f₆), R₀ increases with gas temperature (Fig. 3), which follows the same trend as the partition function (Fig. 4). For reactions N^f₄ and N^b₄, the low activation barrier limits the effect of vibrational excitation on the pre-factor, leading to an opposite trend with respect to gas temperature. Overall, except for the pre-factor of reaction N^f₁, it can be found that the pre-factors are not sensitive to the gas temperature (Fig. 3).

Fig. 3 Pre-factor R₀ of related reactions as a function of gas temperature.

Fig. 4 The vibrational partition functions of CO₂, CO and O₂ molecules as a function of gas temperature, which are calculated by the eqn (6) and (8).

Fig. 5 explores the influence of gas temperature as well as the impact of varying the vibrational temperature on the overall rate constant of reaction N^f₁. The other overall reaction rate constants have similar trends as functions of vibrational and gas temperature (see ESI† material). With the help of the higher vibrational temperature, all the overall reaction rate constants increase dramatically, especially at low gas temperatures, offering the potential to achieve higher energy efficiency in the non-thermal state. In many STS models, because of the massive computing cost, most of the attention is paid to the CO₂ asymmetric mode.^16,24,27 In order to elucidate the importance of the other two vibrational modes, i.e. symmetric and bending modes, on the overall rate constants, a comparison is made between the case when only the asymmetric mode is vibrationally excited (dotted) and when all modes are vibrationally excited (solid) (Fig. 5).

Fig. 5 The overall rate constants of N^f₁ as a function of gas and vibrational temperatures. The dotted lines represent the case when only the asymmetric mode of CO₂ is vibrationally excited (T₁₂ = T_g, T₃ = T_v); the dashed line represents the thermal state (T₁₂ = T₃ = T_g) and the solid lines represent the case when three vibrational modes of CO₂ are vibrationally excited (T₁₂ = T₃ = T_v).

To simplify, the symmetric and bending modes are assumed to share a common vibrational temperature denoted as T₁₂, while the asymmetric mode is characterised by T₃.⁵⁴ Due to the difference in vibrational temperatures between T₁₂ and T₃, the calculation of the CO₂ vibrational energy set must exclude the influence of the third term in eqn (1). Furthermore, in order to prevent the rate constant difference caused by using different vibrational energy sets, the rate constant calculation in Fig. 5 also ignores the third term effect of eqn (1) when T₁₂ = T₃. In reality, since the symmetric and bending modes have shorter equilibration times, T₁₂ is lower than T₃, as demonstrated in the works of Biondo et al.¹⁹ and Klarenaar et al.²⁰ As a result, the real dissociation rate constants fall between the solid and dotted lines in Fig. 5. The excitation of all vibrational modes with higher vibrational temperature yields a pronounced enhancement in the overall rate constant of the reaction N^f₁, particularly evident at low gas temperatures ranging from 300 to 1000 K.

The slope of the rate constant with respect to the gas temperature is similar between the different fixed vibrational temperatures (i.e. the trend is not sensitive to the vibrational temperature) at low gas temperature until 1000 K (Fig. 5). The behaviour can be explained by the dominance of the energy level closest to the dissociation limit or levels with energies exceeding E_a/α in case it is less than the dissociation limit, as the minimum difference between the effective vibrational energy and activation barrier exerts a substantial influence on the overall reaction rate, particularly at relatively low gas temperatures.^37,55 As an example, for the reaction N^f₃, the vibrational energy of the highest level is closest to the E_a/α. Although the population is the lowest for the highest vibrational level (Boltzmann distribution), the contribution of this level to the overall reaction rate is still the most significant, especially at low gas temperature (Fig. 6). However, with increasing gas temperature, low vibrational levels play a more and more significant role in the overall rate constant, until the thermal state is reached where the contribution of each level to the overall rate constant is the same (purple line in Fig. 6).

Fig. 6 The contribution of each vibrational level to the overall rate constant of reaction N^f₃ at a set vibrational temperature of 4000 K.

To validate the accuracy of our method, our obtained reaction rate constants are compared to the theoretical-informational approach, the forced harmonic oscillator (FHO) method and the quasi-classical trajectory (QCT) method for different vibrational levels (Fig. 7).^45,56,57

Fig. 7 Rate constants at different vibrational levels by present work (solid line), by theoretical-informational approach (dashed line), by FHO method⁵⁷ (dotted line) and by QCT calculations (solid dots circle) from ref. 45 and 56 as a function of gas temperature.

The QCT method is considered as an excellent way of determining non-equilibrium dissociation rate constants and can be used as a reference.⁴⁵ However, the QCT method is extremely computationally expensive and requires accurate potential energy surfaces.⁵⁶ Furthermore, the QCT data related to CO₂ dissociation is still lacking.⁵⁸ Compared with the QCT method, the FHO method requires much less computing time but still supplies relatively accurate results.⁵⁹ The FHO results are taken directly from the STELLAR database,⁵⁷ and QCT results are fitted by Andrienko et al.⁵⁶ Since the vibrational energy sets for O₂ across these three methods are nearly identical (i.e., the number of vibrational levels is the same, and the values for each level are very close), the calculation difference of rate constants due to using different vibrational energy sets can be ignored. The results obtained by our method are in much better agreement with QCT results for different temperatures, particularly at elevated vibrational levels, which exert the most significant influence on the overall rate constant (Fig. 6). Our approach offers greater flexibility and computational efficiency compared to the QCT method. In comparison with the FHO method, our results are not only closer to the QCT results, demonstrating higher accuracy, but also our method is 2–3 orders of magnitude faster than the FHO calculation.

Fig. 8 depicts the energy utilisation fractions for a residence time of 1 ms (i.e. how much of the energy is associated with additional CO₂, CO or O₂ vibrational energy, dissociation and thermal energy). The thermal energy is only determined by the gas temperature. In the thermal case (Fig. 8(a)), temperatures below 2200 K only heat the gas (i.e. no significant dissociation takes place). At T_g > 2200 K, more and more energy is used for dissociation as T_g increases further until T_g = 4200 K, at which point the dissociation energy fraction hits its maximum (66%). It should be pointed out that this does not directly link to energy efficiency, as the dissociation energy fraction also encompasses the energy required for dissociating CO and O₂. For the relatively low vibrational temperature T_v = 3000 K (Fig. 8(b)), the relatively low vibrational energy in CO₂ molecules does not increase the dissociation significantly. Most energy is still used to heat CO₂ molecules.

Fig. 8 The energy utilisation fraction as a function of the gas temperature at a residence time of 1 ms, and a vibrational temperature of a) thermal (T_g = T_v); b) 3000 K; c) 4000 K; and d) 5000 K. There is a white region in b) and c), which represent the situation in which the gas temperature is higher than the vibrational temperature, a region not explored in this work.

As the vibrational temperature rises to 4000 K (Fig. 8(c)), a significant amount of CO₂ molecules are dissociated even at the lowest gas temperatures. Due to significant dissociation, vibrational energy is also stored in the dissociation products CO and O₂. The extra vibrational energy in CO molecules always exceeds the vibrational energy stored in the O₂ molecules, as the production of CO molecules is greater than O₂ molecules (stoichiometry). On the other hand, CO has a higher dissociation barrier, requiring more energy to break its chemical bond. Full dissociation of CO₂ at T_g = 3500 K is responsible for the disappearance of the extra vibrational energy of CO₂ molecules (Fig. 8(c)). At the highest vibrational temperature (T_v = 5000 K) in Fig. 8(d), more CO₂ molecules are pumped into higher vibrational states, significantly accelerating the dissociation process. Consequently, almost all CO₂ and O₂ molecules can be split even at a relatively low gas temperature (T_g = 1200 K), and the dissociation fraction exceeds 85%. Then, the thermal energy fraction increases with the gas temperature, consequently reducing the efficiency. The energy utilisation fractions as a function of the vibrational temperature at different gas temperatures can be found in the ESI† material.

Similar to the thermal model, the residence time also plays a vital role in the non-thermal state (Fig. 9). When the residence time is 10⁻³ ms, nearly all CO₂ molecules remain undissociated no matter how high the gas temperature is. Although CO₂ molecules have enough vibrational energy to support dissociation, the reaction rates are limited by kinetics. However, for longer residence times, the high vibrational energy enables the dissociation even at the gas temperature of 300 K. Eventually, 55% energy is used for dissociation, and about 35% vibrational energy is still stored in the CO₂ molecules (Fig. 9(a)). With the gas temperature increasing to 1000 K, the energy utilisation fraction of dissociation also rises to 85% in the end (Fig. 9(b)).

Fig. 9 The energy utilisation fractions as a function of residence time at a vibrational temperature of 5000 K and a gas temperature of a) 300 K; b) 1000 K; c) 2000 K; and d) 3000 K.

Then, continually raising the gas temperature to 2000 K or even 3000 K increases the energy utilisation fraction of thermal energy, which reduces the utilisation for dissociation and efficiency accordingly. It should be noted that, at the same vibrational temperature, the chemical equilibration time, when all the energy utilisation fractions and concentrations of species remain stable (i.e. the chemical equilibrium is reached), shortens with increasing gas temperature. Therefore, it is essential to figure out the characteristic time at different vibrational and gas temperatures in the non-thermal state.

One of the factors limiting efficiency is the chemical equilibration time, which is typically longer than the residence time in experiments.^7,53 In this work, the chemical equilibration time is defined as the time when relative differences of all densities from their values at the infinite time become less than 10⁻⁴ (Fig. 10).

Fig. 10 The chemical equilibration time as a function of the gas temperature at different vibrational temperatures.

For the thermal state (green line), due to enhanced reaction rates at higher gas temperatures, the equilibration time decreases in the temperature range from 2000 K to 4500 K. However, beyond 4500 K, the high temperature offers enough energy to decompose CO after CO₂ dissociation, requiring more time to reach the chemical equilibrium. Therefore, even though the reaction rates increase with gas temperature, the chemical equilibration time still extends because of the process of CO dissociation. In the non-thermal state, the time constant follows a similar trend to that in the thermal state. If the total energy (the sum of thermal and vibrational energy) is insufficient for CO dissociation after CO₂ dissociation, the higher vibrational or gas temperatures increase the reaction rates, leading to a shorter equilibration time. Otherwise, although CO₂ and O₂ dissociate faster at high gas or vibrational temperatures, the extra CO dissociation process, due to the high internal energy, still prolongs the equilibration time. The extra vibrational energy also lowers the gas temperature of the inflexion point (i.e. the point where the trend of the equilibration time with gas temperature changes).

In order to better understand the benefit of higher vibrational temperature on the efficiency, Fig. 11 presents the variations in efficiency for different residence times. At a residence time of 0.1 ms (Fig. 11(a)), the thermal model attains a maximum efficiency of 36% at T_g = 4700 K. In contrast, the non-thermal efficiency reaches 61% at 6431 K of vibrational temperature and 300 K of gas temperature at the same residence time (0.1 ms). It is the most energy-efficient dissociation process, i.e. pure vibrationally-induced dissociation. As the residence time extends to 1 ms (Fig. 11(b)), the region of efficiency above 40% expands, and the maximum efficiency reaches 63% at T_v = 5280 K and T_g = 382 K. The region of efficiency above 30% reduces, at high vibrational and gas temperatures. This reduction occurs because a portion of the energy continues to be utilised for O₂ and CO dissociation with a longer residence time, leading to a narrower T_g and T_v window of relatively high efficiency. With a residence time of 10 ms, the non-thermal state offers a large range of temperature conditions to achieve efficiencies over 48% (i.e. close to the maximum we observed in our previous experimental work (cf.Fig. 1)^6–9).

Fig. 11 Energy efficiency as a function of the gas and vibrational temperature with different residence times: a): 0.1 ms; b): 1 ms; c) 10 ms; d) infinite time.

As the gas reaches chemical equilibrium (Fig. 11(d)), the maximum non-thermal efficiency is 64% at T_v = 5280 K and T_g = 382 K. Under thermal conditions, the maximum energy efficiency is ∼50% at T_g = 2950 K. This is significantly less than the non-thermal case, due to energy wasted for gas heating. It should be noted that higher vibrational temperature does not always result in high efficiency. For instance, Fig. 12 presents the mole fractions of species and energy efficiency for a range of gas temperatures in the thermal and non-thermal states at the chemical equilibrium (T_v = 5000 K). At the lowest gas temperature (T_g = 300 K), nearly 45% of CO₂ is decomposed in the non-thermal state. As gas temperature increases to 400 K, the efficiency and mole fraction of O₂ reach the maximum at the same time in the non-thermal state.

Fig. 12 The mole fractions and energy efficiency change as a function of gas temperature at the chemical equilibrium: non-thermal state with T_v = 5000 K (solid line) and thermal state (dash-dotted line).

Although CO₂ is not fully dissociated (the conversion is around 65%), more energy is wasted on the O₂ dissociation and the gas heating at higher gas temperatures. With the gas temperature increasing to 2000 K, all the CO₂ and O₂ are dissociated, and the components remain steady. When the gas temperature rises to 2500 K, compared with the thermal model, a larger part of the energy is used for O₂ dissociation, which results in low efficiency in the non-thermal model. At T_g = 2950 K, the efficiency reaches its maximum in the thermal state, and the difference in efficiency between the thermal and non-thermal states is also at its highest. The discrepancy persists until the gas temperature reaches 5000 K, at which the non-thermal state transforms into the thermal state. The results under thermal chemical equilibrium are in good agreement with the data calculated by the NASA Computer model CEA based on the same conditions.⁷

Besides the dissociation process in the hot zone, the quenching process also significantly influences conversion and efficiency. Rapid quenching can prevent recombination reactions, safeguarding the targeted product (CO).^60,61 Under the conditions of optimal efficiency in the hot region, regardless of how fast the cooling rate is, a part of CO is always destroyed, decreasing the efficiency after quenching. However, a faster quenching rate can reduce the CO loss in the quenching process. In this study, we simulate the quenching region after the hot region, with the gas temperature linearly decreasing (i.e. the cooling rate is constant) over time until both room temperature and chemical equilibrium are reached. The optimal efficiency is calculated by scanning the residence time in the hot zone (ranging from 10⁻⁷ s to 100 s) and the cooling rate (ranging from 10³ K s⁻¹ to 10⁹ K s⁻¹) in the quenching process at a specific temperature (T_hot) and pressure in the CSTR. Fig. 13 illustrates the optimal efficiency at different gas temperatures and pressures. The optimal efficiency (45%) at different pressures and temperatures is lower than our highest experimental efficiency. Therefore, the thermal model fails to explain the experimental results (cf.Fig. 1).

Fig. 13 The optimal efficiency under the thermal condition with scanning residence time and cooling rate as a function of gas temperature in the hot region and pressure.

In the non-thermal quenching model, we assume the same linear cooling rate for gas and vibrational temperatures. Fig. 14 depicts the highest efficiency as a function of vibrational and gas temperatures at 100 mbar. Despite the fact that the quenching process also decreases efficiency in the non-thermal state, the higher efficiency achieved in the hot zone still allows efficiencies to exceed the experimentally observed 48% after the cooling process. The optimal efficiency after the quenching process is 56% at T_g = 300 K, T_v = 5698 K and a cooling rate of 10⁸ K s⁻¹. However, it should be stressed that the quenching process in the non-thermal state overlooks the enhancement of vibrational energy due to recombination, which could also increase CO production after quenching. Consequently, the efficiency is underestimated at high vibrational and gas temperatures, where more O atoms and CO are dissociated in the hot region and more recombination energy is released in the quenching process. More details about intra- and inter-mode energy transitions in the quenching process of CO₂ plasma will be included in future work.

Fig. 14 The optimal efficiency of non-thermal state (at different residence times in the hot region and cooling rates in the quenching region) as a function of gas and vibrational temperatures in the hot zone.

4 Conclusions

The CO₂ dissociation in the non-thermal state is described using a 0D CSTR model with a vibrational temperature exceeding the gas temperature. The model includes 10 810 vibrational levels of CO₂, 81 vibrational levels of CO and 47 vibrational levels of O₂. A new approach to calculate the vibrational–dissociation rate constants is presented. Compared with the rate constant calculated by the theoretical-informational method and the FHO method, it shows better agreement with the results of the accurate QCT method, especially at high vibrational levels. Furthermore, the present method is 2–3 orders of magnitude faster than the FHO calculation. Based on the new approach, the rate constants are calculated as a function of the gas and vibrational temperatures. Furthermore, the model reveals the energy utilisation fractions of dissociation, thermal energy, and extra vibrational energy stored in different molecules, for different temperatures and residence times. This work illustrates the impact of gas temperature, vibrational temperatures, and residence time on efficiency. The results show a maximum efficiency of 63% with 1 ms of residence time and 64% with infinite residence time (i.e. gas reaches chemical equilibrium) at T_v = 5280 K and T_g = 382 K. The chemical equilibration time changes as a function of T_v and T_g in the thermal and non-thermal states. Due to the dissociation of CO molecules, the chemical equilibration time does not monotonically decrease with increasing T_g.

Finally, accounting for the impact of the quenching process, the optimal efficiency of the thermal dissociation predicted by our model under various parameters, including pressure, cooling rates, and temperature, is limited to 45%, which is clearly in disagreement with the maximum experimental efficiency of ∼48% (cf.Fig. 1). The highest non-thermal efficiency after the quenching process is 56%, demonstrating the potential to achieve maximum experimental efficiency.

The model described in this paper can serve as a benchmark for the CO₂ dissociation process in the non-thermal two-temperature state, offering insights for the development of more sustainable and environmentally friendly chemical conversion processes. Compared to traditional STS models, the proposed model delivers enhanced computational efficiency. However, it is important to note that the model overlooks the intricate internal structure of polyatomic molecules, neglecting their interplay. Future research should delve into the impact of various microscopic collision processes, including VV and VT vibrational energy transitions, surface and gas phase recombination, as well as dissociation and deactivation processes at the reactor surface.

Data availability

The model is based on Cantera, which is an open-source Python package that solves chemical kinetics, thermodynamics, and transport processes. The source code will be made publicly available upon the publication of the paper.

Conflicts of interest

The authors state that there are no conflicts to declare.

Acknowledgements

This work is financially supported by the China Scholarship Council Grant No. CSC202106240037. The work of A. Pikalev is financially supported by the European Space Agency under Project I-2021-03399.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4re00300d

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