Diomede, P.

Longo, S.

M. C. M. van de Sanden

Viegas, P.

English

DIFFER: Publications

Self-consistent diffusion approach to CO2 vibrational kinetics  P. Viegas1, M. C. M. van de Sanden1, S. Longo2 and P. Diomede1  1Center for Computational Energy Research, DIFFER - Dutch Institute for Fundamental Energy Research, De Zaale 20, 5612 AJ Eindhoven, the Netherlands 2Dipartimento di Chimica, Università degli Studi di Bari, via Orabona 4, 70126 Bari, Italy   Abstract: The vibrational kinetics of the CO2 asymmetric stretching mode is analysed numerically through two different methods: a full State-To-State (STS) model and a simplified STS model for the first vibrational levels consistently coupled to the solution of the Fokker-Planck (FP) equation in the vibrational energy space. Results obtained with the two methods are compared and it is shown that consistent results can be obtained with higher computational efficiency using the FP approach.  Keywords: CO2 dissociation, vibrational kinetics, Fokker-Planck equation  1. Introduction In recent years, much attention has been dedicated to low-temperature plasmas to convert greenhouse CO2 into new carbon-neutral fuels or useful chemicals [1]. Vibrational excitation of the CO2 molecule plays an important role in energy-efficient non-equilibrium dissociation kinetics. The main tool used to study vibrational kinetics is the state-to-state (STS) finite rate method, based on the numerical solution of a master equation (ME). This last is actually a stiff system of nonlinear Ordinary Differential Equations (ODEs), as many as the number of vibrational levels. When polyatomic molecules are involved, with a high number of vibrational levels and transitions, the computational efficiency of models describing reactors with spatial resolution and coupling different physical phenomena is compromised. Such models require new methods allowing a significant reduction in computational cost. In our previous publications [2, 3], we have proposed, as an alternative, a numerical method based on the diffusion formalism developed in the past for analytical studies [4]. The vibrational distribution function (VDF) representing the populations of the asymmetric stretching mode of CO2 on a continuous internal energy scale has been calculated through the resolution of the drift-diffusion Fokker−Planck (FP) equation with fixed input vibrational temperature (Tv). The numerical solution has been accomplished through the time-dependent diffusion Monte Carlo method in [2] and in stationary condition by the flux-matching approach in [3]. In this work, the FP equation is solved for Tv self-consistently calculated from a simplified STS model. Results are compared to the ones obtained through a full STS model. 2. Computational methods a. Full State-To-State (STS) approach The temporal evolution of the number densities of each species i is calculated by the solution of the rate equations: 𝑑𝑛𝑖𝑑𝑡= ∑ 𝑆𝑖𝑗(𝑡)𝑗     (1) where ni is the number density of species i and Sij is the source/loss term associated to reaction j. A total of 24 species is considered in a pure CO2 plasma: all the states of the vibrational asymmetric stretching mode of CO2 (CO2(v=n), with n the vibrational quantum number from 0 to 21), together with electrons and the positive ion CO2+. The vibrational kinetics is described by a total of 253 reactions, listed in Table 1, and with rate coefficients from [5]. In Table 1, eV indicates vibrational excitation/de-excitation by electron impact, VV vibrational-vibrational energy exchange, VT vibrational-translational energy exchange. Table 1. List of processes in the full STS model. Process name Reaction Ionization e + CO2(v=0) → CO2+ + 2 e Recombination e + CO2+ → CO2(v=0) (CO + O, C + O2) eV e + CO2(v=0) ↔ CO2(v=1-21) + e VV CO2(v=n) + CO2(v=1) ↔ CO2(v=n+1) + CO2(v=0) CO2(v=n) + CO2(v=n) ↔ CO2(v=n+1) + CO2(v=n-1) VT CO2(v=n) + CO2 ↔ CO2(v=n-1) + CO2 Dissociation CO2(v=21) + CO2(v=1) → (CO + O) + CO2(v=0) CO2(v=21) + CO2(v=21) → (CO + O) + CO2(v=20)  We consider a pressure p = 2660 Pa, fixed gas temperature Tg = 300 K and gas density N = 6.43 × 1017 cm-3. We assume a constant power density Pdep transferred to the electrons and thus the temporal evolution of the electron mean energy ε is described by the electron energy equation 𝑑(𝑛𝑒𝜖)𝑑𝑡= 𝑃𝑑𝑒𝑝(𝑡) −𝑃𝑒𝑙𝑁(𝜀) × 𝑁 × 𝑛𝑒  − 𝑃𝑖𝑛𝑒𝑙𝑁(𝜀) × 𝑁 × 𝑛𝑒   (2) where ne is the electron number density and Pel and Pinel are the components of the power lost by electrons through elastic and inelastic collisions, respectively. The electron-impact rate coefficients and the electron power losses are obtained as function of ε from the Electron Energy Distribution Function (EEDF) solution from the Electron Boltzmann Equation (EBE) solver BOLSIG+ [6], using the electron-impact cross sections from the Phelps database in LXcat [7]. The set of ODEs is solved until reaching a stationary condition with the solver RADAU5, an implicit Runge-Kutta method of order 5 [8]. We take as initial conditions ne = [CO2+] = 4.0 × 1010 cm-3 and ε = 1.0 eV. An example of temporal evolutions of species number densities obtained with this model is presented in Fig. 1, using as input power density Pdep = 100 Wcm-3.  Fig. 1. Temporal evolution of species number densities using the full STS model with Pdep = 100 Wcm-3.  b. Fokker-Planck (FP) approach The FP equation allows one to describe the evolution of the VDF f as a transport process in the space of energy 𝜀𝑣 of the vibrational asymmetric stretching mode, as follows: 𝑑𝑓𝑑𝑡=𝑑𝐽𝑑𝜀𝑣; 𝐽 = 𝑎𝑓 − 𝑏𝑑𝑓𝑑𝜀𝑣   (3) where 𝐽(𝜀, 𝑡) is the flux in energy space and 𝑎 and 𝑏 are, respectively, the drift and diffusion coefficients, functions of Tg, Tv and the VV and VT rate coefficients, as described in [3]. In this work, Tg is fixed and the rate coefficients are the same as those used in the STS model presented in the previous section. The new approach we present in this work consists in solving a simplified STS model that allows to calculate the temporal evolution of the vibrational temperature Tv until reaching a stationary condition, coupled to the resolution of the FP equation for df/dt = 0 through the flux-matching method presented in [3]. The flux 𝐽 (which is a constant under stationary conditions) is determined by the VV dissociation processes of Table 1 with rate coefficients 𝑘𝑑1,21 and 𝑘𝑑21,21: 𝐽 = 𝑓(𝜀𝑚𝑎𝑥)1𝑁(𝑘𝑑1,21𝑛1 + 𝑘𝑑21,21𝑓(𝜀𝑚𝑎𝑥)) (4) where 𝜀𝑚𝑎𝑥 = 5.467 eV, 𝑓(𝜀𝑚𝑎𝑥) is the number density at the dissociation energy and 𝑛1 is the number density of level v = 1. The simplified STS model contains the same ionization and recombination processes as the full STS model (Table 1), as well as the eV, VV and VT processes for the vibrational levels v ≤ 2. Hence, it contains 5 species and 15 reactions, much less than in the full STS model. Tv is calculated in both full and reduced STS models assuming a Boltzmann distribution as: 𝑇𝑣 = −𝐸1𝑘𝐵 log (𝑛1/𝑛0)  (5) where E1 is the energy of the vibrational level v = 1 and 𝑛1 and 𝑛0 are the number densities of levels v = 1 and v = 0 respectively.  3. Model comparison a. Base case For the particular case with transferred power density Pdep = 100 Wcm-3, the full STS calculation has reached a stationary condition at t = 1.003 ms with electron density and temperature 𝑛𝑒 = 1.2 × 1011 cm-3 and 𝑇𝑒 = 0.83 eV. With the model using a simplified STS calculation, the stationary condition has been reached at t = 1.320 ms with the same 𝑛𝑒 and 𝑇𝑒. Concerning vibrational kinetics, the results using the full and simplified STS calculations are only slightly different. Fig. 2 shows the temporal evolution of 𝑇𝑣 calculated through Eq. (5) with both models.  Fig. 2. Temporal evolution of Tv using both full and reduced STS models with Pdep = 100 Wcm-3.  The temporal evolution of Tv has the same trend with both models. When the stationary condition is reached, Tv is 2335 K in the full STS model and 2148 K in the reduced STS model. The difference may be due to truncating the vibrational kinetics scheme at v = 2 in the reduced STS calculation. Once the stationary condition is reached, the FP equation is solved for Tv = 2148 K. The resulting VDF is presented in Fig. 3 and compared to the VDF obtained with the full STS model in stationary condition. Fig. 3 shows that the VDFs obtained by the two methods are close, even though there are visible discrepancies in the VDF tail. Moreover, the dissociation rate due to the VV processes listed in Table 1 is calculated in the full STS model as 4.76 × 1013 cm-3s-1. Using the FP approach, the rate of dissociation through the same processes is calculated from Eq. 4 as 𝐽 × 𝑁 and is less than an order of magnitude higher: 2.28 × 1014 cm-3s-1. This difference is a result of the discrepancies in the VDF tail. In a more complete model, the dissociation rate is then used to calculate the densities of other species and ultimately to calculate the conversion degree in the plasma. This result shows that an accurate trend can be obtained using the FP approach self-consistently coupled with a reduced STS model.  Fig. 3. Vibrational distribution function in stationary condition, using the full STS model (blue curve with crosses) and the FP equation (orange curve). It is worth noticing that with the full STS model, the calculation time until reaching the stationary condition is of 377.7 s (~ 6 min). Although acceptable in zero-dimensions, we should notice that in multidimensional fluid models this calculation time would be multiplied by thousands or millions, and thus the total calculation time could compromise the computational cost of such models. On the other hand, with the model coupling a simplified STS calculation with the FP equation, the stationary condition is reached with a calculation time of only 52.9 s (52.8 s for the simplified STS calculation + 0.1 s for the resolution of the FP equation), seven times lower than the full STS calculation. The difference is fundamentally due to the fast semi-analytical solution which is possible for the FP equation and shows the computational gain that can be obtained through the FP approach. b. Variation of input power density By changing the input parameter Pdep, we can assess the validity of the FP approach for different conditions by comparing the results with the two models. In Table 2 are listed the resulting Tv obtained with the full and reduced STS calculations in stationary condition for transferred power densities Pdep between 40 and 140 Wcm-3.    Table 2. Vibrational temperatures obtained in stationary condition with the full and reduced STS schemes for different Pdep. Pdep [Wcm-3] Full STS Tv [K] Reduced STS Tv [K] 40 1781 1523 60 2024 1762 80 2204 1967 100 2335 2148 120 2429 2312 140 2500 2463  As expected, with higher Pdep there is more electron production and vibrational excitation, thus increasing Tv. Tv obtained with the reduced STS calculation is consistently lower than the one calculated with the full STS scheme, the difference being below 300 K.  Fig. 4. Vibrational distribution functions in stationary condition, using the full STS approach for different Pdep.  Fig. 5. Vibrational distribution functions in stationary condition, using the FP approach for different Pdep. The effect of Pdep variation on the VDF obtained through the full STS approach is shown in Fig. 4 and the one calculated through the FP approach can be observed in Fig. 5. We can notice that relatively small variations of Tv can significantly change the VDF. In particular, using the FP approach, 𝑓(𝜀𝑚𝑎𝑥) increases around 9 orders of magnitude from Pdep = 40 Wcm-3 until Pdep = 140 Wcm-3. With the STS approach, the variation of density of level v = 21 is of only 3 orders of magnitude. The FP method produces VDFs overall very similar to the STS ones, also reproducing the features, variations and trends while changing the power. The changes on the VDF obviously impact the VV dissociation rate. This influence can be observed in Table 3 as function of Pdep. Table 3. Dissociation rates in stationary condition with the full STS and FP approaches for different Pdep. Pdep [Wcm-3] Diss. rate STS [cm-3s-1] Diss. rate FP [cm-3s-1] 40 2.99 × 1012 5.67 × 109 60 6.60 × 1012 6.51 × 1011 80 1.52 × 1013 1.68 × 1013 100 4.76 × 1013 2.28 × 1014 120 1.59 × 1014 4.61 × 1015 140 4.78 × 1014 1.25 × 1019  The dissociation rates in Table 3 reflect the dependence of the VDF on Pdep observed in Figs. 4 and 5. We should notice that, while the difference between the dissociation rates calculated with the two models remains lower than one order of magnitude between Pdep = 60 Wcm-3 and Pdep = 100 Wcm-3, it becomes clearly excessive for the lowest and highest values of Pdep considered here. Progresses in the determination of the transport coefficients from chemical rate coefficients should reduce these differences while keeping the clear advantage in terms of computational cost. 4. Conclusions The vibrational kinetics of the CO2 asymmetric stretching mode has been studied numerically in conditions of fixed gas temperature and transferred power density in a pure CO2 plasma. The solution has been obtained with the classical STS approach and with a new approach introduced here: the resolution of a simplified STS model coupled to the calculation of the VDF in stationary conditions by solving the FP equation in the vibrational energy space. The results obtained through the two methods have been compared and it has been shown that Tv can be reasonably estimated from a reduced STS scheme with only vibrational levels v ≤ 2. Hence, the FP approach can be used in self-consistent plasma models. The accuracy of the FP approach, i.e. the agreement between the FP and STS approaches on the resulting VDFs and dissociation rates, has been shown to be dependent on the choice of the input deposited power density and thus on Tv. The difference in the resulting dissociation rates is lower than one order of magnitude for Pdep between 60 Wcm-3 and 100 Wcm-3. In the cases under study, the use of the FP approach is about seven times more computationally efficient than the full STS approach. The result on computational efficiency is very promising for the development of more complete multidimensional fluid models to describe discharge reactors. In the near future, we plan to correct the discrepancies in the VDF between the two approaches. The present set of transport coefficients used in the FP equation is preliminary and can be significantly improved. Moreover, we will include more complete chemistry in the CO2 plasma model and couple it with the EEDF solution for different mixtures.  5. Acknowledgements This work is part of the European project KEROGREEN, which has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement 763909. This work is also part of the Shell-NWO/FOM initiative ‘Computational sciences for energy research’ of Shell and Chemical Sciences, Earth and Life Sciences, Physical Sciences, FOM and STW. This work was also carried out under TTW open technology project (grant nr. 15325) in collaboration with Gasunie, Stedin, DNVGL and Ampleon. 6. References [1] A. Goede and M. C. M. van de Sanden, Europhys. News 47 (2016), 22−26 [2] P. Diomede et al., J. Phys. Chem. C 121 (2017), 19568-19576 [3] P. Diomede et al., J. Phys. Chem. A 122 (2018), 7918-7923 [4] V. D. Rusanov et al., Sov. Phys. Usp. 24 (1981), 447-474 [5] T. Kozák and A. Bogaerts, Plasma Sources Sci. Technol. 23 (2014), 045004 [6] G. J. M. Hagelaar and L. C. Pitchford, Plasma Sources Sci. Technol. 14 (2005), 722-733 [7] Phelps database, www.lxcat.net, retrieved on October 5, 2018 [8] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer-Verlag (1996)  

Self-consistent diffusion approach to CO2 vibrational kinetics

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Self-consistent diffusion approach to CO2 vibrational kinetics

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