Diomede, P.

Longo, S.

M. C. M. van de Sanden

Viegas, P.

English

DIFFER: Publications

DIFFER is part ofSelf-consistent Fokker-Planck approachto CO2 vibrational kineticsP. Viegas1, M. C. M. van de Sanden1, S. Longo2 and P. Diomede11 CPPC group, DIFFER, Eindhoven, the Netherlands2 Dipartimento di Chimica, Bari, ItalyJuly 16th 2019juli 11, 2019Motivation - CO2 conversionUnderstanding and optimizing CO2 conversion in MW plasma reactors requires multidimensional modellingForward vortex Reverse vortexSupersonic expansionEfficient conversion in plasma reactors at DIFFER: 2/18juli 11, 20193/18• The description of plasma reactors requiresmultidimensional modelsthat couple microwave, flow, heat and plasma.• Vibrational non-equilibrium has an impact on dissociation mechanisms in CO2 discharge reactors.Modelling CO2 plasma reactorsKozak and Bogaerts,PSST 2014juli 11, 20194/18Modelling CO2 plasma reactors• The description of plasma reactors requires multidimensional models that couplemicrowave, flow, heat and plasma.• Vibrational non-equilibrium has an impact on dissociation mechanisms in CO2discharge reactors.• Usual State-To-State (STS) approach:Resolution of conservation equations for dozens of vibrational levels,including all their reactions.• In reactor models, the STS approach can compromise computational efficiency. juli 11, 20195/18Modelling CO2 plasma reactors• Alternatively, we can consider only a few species and effective rate coefficients that take into account vibrational non-equilibrium• But it still requires the population of vibrationally excited states:Vibrational Distribution Function (VDF)• Need fast solution of the VDF that can be coupled with plasma models and preserves accurate results juli 11, 20196/18Alternative approach for vibrational kineticsMultilevel kinetics → Brownian motionSystem of ODEs → Fokker-Planck EquationCollisional interactions → Drift and diffusion in energy spacejuli 11, 20197/18From STS to Fokker-Planck (FP) approachGain-loss equation for population of states 𝑖:𝑑 𝑛𝑖𝑑𝑡= Σ𝑗 𝑆𝑖𝑗 𝑡• Assumption of continuum in energy space of asymmetric stretching vibrational mode:Energy between states << dissociation energy • Assumption of only small energy transitions:Monoquantum transitions References:Van Kampen,North-Holland (1981)Biberman et al., Consultants Bureau (1987)𝑛𝑖 = number density of species 𝑖𝑆𝑖𝑗 = source term associated to reaction jjuli 11, 20198/18Fokker-Planck (FP) approach𝑑𝑓(𝜀)𝑑𝑡= −𝑑𝐽 𝜀𝑑𝜀𝐽 𝜀 = 𝐴 𝜀 𝑓 𝜀 − 𝐵 𝜀𝑑𝑓(𝜀)𝑑𝜀Transport coefficients𝐴 and 𝐵 as f(Tg, Tv, k) derived for each process inViegas et al., submitted𝑓 = vibrational distribution function𝐽 = flux in energy space𝐴 = drift coefficient𝐵 = diffusion coefficientRusanov, Fridman and Sholin,Sov. Phys. Usp. 24 (1981), 447juli 11, 20199/18Numerical FP approachIn stationary conditions,𝑑𝑓(𝜀)𝑑𝑡= −𝑑𝐽𝑑𝜀= 0J is uniform and can be defined asdissociation rate at 𝜀 = 𝜀𝑑𝑖𝑠𝑠Finding 𝐽 and 𝑓 becomes a flux-matchingproblem that can be solved fastDiomede et al., J. Phys. Chem. C 121 (2017), 19568Diomede et al., J. Phys. Chem. A 122 (2018), 7918Need to implement in self-consistent plasma model and verify its correctness𝑓 = vibrational distribution function𝐽 = flux in energy spacejuli 11, 201910/18Benchmarking the FP approach Comparison of STS and FP approaches, assuming:• Fixed pressure p = 100 mbar and gas temperature Tg = 300 K.• Constant input power density Pdep.• Almost pure CO2 plasma; low dissociation.• Dissociation only through asymmetric stretching vibrational channel.• Same kinetic data in STS and FP models.juli 11, 201911/18• Electron energy 𝜀𝑚 equation• Species densities rate equations:𝑑 𝑛𝑖𝑑𝑡= Σ𝑗 𝑆𝑖𝑗 𝑡• Solved until stationary condition is reached• k(𝜀𝑚) from EEDF solution from Bolsig+ for pure CO2• Kinetic data fromKozak and Bogaerts, Plasma Sources Sci. Technol. 23 (2014), 045004Zero-dimensional STS model𝑛𝑖 = number density of species i𝑆𝑖𝑗 = source term associated to reaction jjuli 11, 201912/18Total of 261 reactions and 28 ODEs to solve:εm, e, CO2(v=0-21), CO2(va), CO2(vb), CO2(vc), CO2(vd)1. Ionization: e + CO2(v=0-3,va,vb,vc) → CO2+ + 2 e2. Recombination: e + CO2+ → CO2 (CO+O)3. e-V: e + CO2(v=0-2,va,vb,vc) ←→ CO2(v=1-3,va,vb,vc) + e4. V-V: 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)5. V-V dissociation: CO2(v=21) + CO2(v=1) → CO2 (CO+O) + CO2(v=0)CO2(v=21) + CO2(v=21) → CO2 (CO+O) + CO2(v=20)6. V-V’: CO2(v=n) + CO2(v=0) ←→ CO2(v=n-1) + CO2(va,vb)7. V-T: CO2(v=n) + CO2 ←→ CO2(v=n-1) + CO2CO2(va,vb,vc ,vd) + CO2 ←→ CO2(v=0,va,vb,vc) + CO2Zero-dimensional STS modeljuli 11, 201913/18• Use a simplified STS model to calculate Tv in stationary condition:Same energy equation and electron rate equation as full STS model;V-V, V-V’ and V-T in asymmetric stretching mode truncated at v=3.• Total of 77 reactions and 9 ODEs to solve:εm, e, CO2(v=0-3), CO2(va), CO2(vb), CO2(vc)• Solve the stationary FP equation to obtain the VDF and the V-V dissociation rate.Replaces STS V-V, V-V’ and V-T in asymmetric stretching mode.Reduced STS + FP modeljuli 11, 201914/18Full STS model:• 𝑛𝑒 = 1.9 × 1011 cm-3 and 𝑇𝑒 = 1.52 eV• 𝑇𝑣 = 2101 K• V-V dissociation rate = 1.11 × 1012 cm-3s-1• Calculation time 423.87 sReduced STS + FP model:• 𝑛𝑒 = 1.9 × 1011 cm-3 and 𝑇𝑒 = 1.52 eV• 𝑇𝑣 = 2169 K• V−V dissociation rate = 1.24 × 1012 cm-3s-1• Calculation time 69.61 s (69.58 + 0.03 s)• Very good agreement on VDFBenchmarking the FP approach (Pdep = 1000 Wcm-3)juli 11, 201915/18• Tv slightly higher with reduced STS – difference increases with Pdep• Very good agreement on VDFBenchmarking the FP approach (variation of Pdep)juli 11, 201916/18• Tv slightly higher with reduced STS – difference increases with Pdep• Very good agreement on VDF for Tv < 2500 K• Correctness of FP approach is verified for Tg = 300 K and Tv < 2500 KBenchmarking the FP approach (variation of Pdep)juli 11, 201917/18Conclusions➢ The correctness of the FP approach has been verified for use inself-consistent plasma models.➢ The FP approach is much more computationally efficient forstationary solution than the full STS approach.➢ The computational efficiency obtained by the FP approach isvery promising for the development of multidimensional models.juli 11, 201918/18Next steps➢ Understand the influence of further chemical processes on VDF,and vice-versa, using the FP approach.➢ Study of Tg dependence of FP coefficients and VDF.➢ Use the self-consistent FP model in conditions of experimentswith CO2 plasma reactors.juli 11, 2019Thank you for your attentionAcknowledgements:European Union project KEROGREENTTW open technology project (grant nr. 15325)Shell-NWO/FOM initiative ‘Computational sciences for energy research’

Self-consistent Fokker-Planck approach to CO2 vibrational kinetics

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Self-consistent Fokker-Planck approach to CO2 vibrational kinetics

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