A Simulation of the Effects of Varying Repetition Rate and PulseWidth of Nanosecond Discharges on Premixed LeanMethane-Air Combustion

Two-dimensional kinetic simulation has been carried out to investigate the effects of repetition rate and pulse width of nanosecond repetitively pulsed discharges on stabilizing premixed lean methane-air combustion. The repetition rate and pulse width are varied from 10 kHz to 50 kHz and from 9 ns to 2 ns while the total power is kept constant. The lower repetition rates provide larger amounts of radicals such as O, H, and OH. However, the effect on stabilization is found to be the same for all of the tested repetition rates. The shorter pulse width is found to favor the production of species in higher electronic states, but the varying effects on stabilization are also found to be small. Our results indicate that the total deposited power is the critical element that determines the extent of stabilization over this range of discharge properties studied.


Introduction
Nanosecond repetitively pulsed discharges have been studied as possible sources to stabilize combustion at fuel-lean and blow-off conditions [1].The stabilization has been attributed to the significant radical production and gas heating within the discharges [1,2].The relative importance of these two outcomes on the stabilization has been debated.However, the effects of these two discharge outcomes on stabilization are difficult to separate because the two mechanisms stem from the same process of collisional quenching of electronically excited species (produced by direct electron impacts) [2,3].Also, the influence of radicals such as O and H is more pronounced at high gas temperature where the branching and propagation reactions can compete with the termination reactions.It is notable that Pilla et al. [4] reported that they were able to stabilize combustion when the discharge is at filamentary mode (e.g., streamers) rather than at glow mode.Deminsky et al. [5] also reported that an acceleration of the ignition process is seen in premixed supersonic flows with filamentary discharge modes.This suggests that the gas heating that is often more pronounced in the filamentary mode is necessary for significant radical production and that the elevated gas temperature results in the persistence of radicals during the time between pulses.
A few tens of microjoule of energy per pulse, tens of kilohertz repetition rates, and a few kilovolt peak voltages are typical operation conditions of nanosecond repetitive discharges used in combustion stabilization.Under these conditions, the average electron number density that can be sustained is about 10 11 cm −3 peaking as high as 10 15 cm −3 with a lower power budget compared to other types of discharges [6].There have been several studies carried out of the kinetics responsible for plasma-assisted stabilization of combustion [1], but to our knowledge, there are few, if any, experimental studies investigating the effect of repetition rate and pulse width (for this type of discharge) on stabilization.This is because the power that is deposited into the discharge region is difficult to control and accurately characterized.
In this paper, we examine the effect of variations in the repetition rate and the pulse width on premixed methaneair combustion using a 2-D kinetic simulation.In these simulations we set the total power to be constant and vary the repetition rate between 10 kHz and 50 kHz and the pulse width from 9 ns to 2 ns.In this way, the primary variables are the energy per pulse and therefore the amount of produced radicals and their ability to survive between pulses.The kinetics within the discharge region and the quasi-steady contours between the cases are compared in detail.

Simulation Description
The simulated domain, marked as a red-dotted box, is shown in Figure 1.Axisymmetric coordinates are used in this simulation, and the size of the domain is set to 12 mm in height (z-axis) and 5.6 mm in radius (r-axis), spanning the discharge region.The center of the discharge region is located 4 mm above the lower computational boundary, on the axis, and has size of 1 mm in height and 0.35 mm in diameter.This diameter is chosen to agree with the measured diameter in the similar experiments of Pai et al. [7].A uniform grid spacing (0.333 mm along the z-axis and 0.175 mm along the r-axis) is used in these simulations.Species considered include ground and electronic excited states of N 2 (X, A, B, a , C), the ground electronic states of O , and free electrons (e), amongst others.These species are added to the reduced reaction mechanism, DRM19 [8], that is often used for simulating methane/air combustion.DRM19 has been tested against the more detailed GRI-Mech mechanism [9] for computing ignition delay times and laminar flame speeds.Thermodynamic properties for neutral species are calculated based on the use of NASA polynomials as tabulated in the GRI-Mech package.The thermodynamic properties for electrons and ions are taken from Burcat [10].Reactions considered are electron-impact excitation and ionization of N 2 , electron-impact dissociation and ionization of O 2 and CH 4 , electron-impact ionization of H 2 O and CO 2 , ion conversion, recombination of electron and positive ions, quenching of electronically excited nitrogen (N 2 * ) by N 2 , dissociative quenching of N 2 * by O 2 and CH 4 , and chemical transformation of neutral species typical in methane/air combustion reaction mechanisms.The reactions pertinent to the plasma kinetics and associated rates are provided in our previous paper [2].Rate coefficients for reactions between neutral species and ions are adapted from previous literature, whereas those for reactions involving electrons are calculated as functions of reduced electric field (E/n) based on the solution of the Boltzmann equation, which is facilitated using the commercial software, BOLSIG+ [11].Such an approach is necessary since the reactions are coupled to the electron energy distribution function.It is noteworthy that only elastic and inelastic cross-sections associated with the major species N 2 , O 2 , CH 4 , H 2 O, and CO 2 are considered in establishing the electron energy distribution.The species conservation equation used in our simulations has the form The species equation is solved simultaneously with the energy equation, to compute the species number densities and gas temperature at each grid point.The species diffusion velocity V D, j is composed of the diffusion-induced convection velocity V c , the ordinary diffusion velocity, and the thermal diffusion velocity, the latter of which is accounted for light species only, having molecular weight less than five: Here, V c is determined to satisfy the condition In the above equations, n j , X j , and ν j are the number density, mole fraction, and stoichiometric coefficient of species j, k f and k b are the forward and reverse rate coefficients, T e and T gas are the electron and gas temperatures, D jm is the mixture average diffusion coefficient of species j, Θ jm is the thermal diffusion ratio of species j, and V * adv is the local advection velocity, which is scaled as T gas /T gas,i (T gas,i is the initial inlet gas temperature) assuming negligible radial velocity and constant pressure, to account for the flow acceleration caused by heat release during combustion.We assume that electrons and ions exist only in the discharge region and that the binary diffusion coefficients for electronically excited N 2 are equal to those of the ground state N 2 .In the energy equation, μ e is the electron mobility (a function of E/n), c v, j is the heat capacities at constant volume of species j, h f , j and h sens, j are the formation and sensible enthalpies of species j, and λ is the mixture-averaged thermal conductivity.The mixture diffusion coefficient of species j, thermal diffusion ratio of species j, and mixture-averaged thermal conductivity are computed at each grid points according to (6), that is, where X j and Y j are the mole and mass fractions of species j, respectively, D k j is the binary diffusion coefficient between species k and j, θ jk is the binary thermal diffusion ratio for species j into species k, and λ j is the pure thermal conductivity of species j.In solving the system of partial differential equations, the diffusion processes (species and thermal conduction) are discretized by a central difference scheme, and the convection and advection processes (species and enthalpy) are discretized by an upwind scheme.The system of ordinary differential equations is then solved implicitly for each (adaptive) time step based on a backward difference formula (BDF).In the computations, the domain is divided into smaller subdomains, which are allocated to separate processes.These processes are computed in parallel, synchronizing their boundary values via the message passing interface, MPI.Sundials CVODE [12] with MPI support is used as a solver and Open MPI [13] is used for MPI-2 standard implementation.Each temporal solution is computed iteratively using the Generalized Minimal Residual method (GMRES) [14].A Dirichlet condition is used for the lower domain boundary, whereas Neumann conditions are used for the sides and top of the computational boundary.
A Gaussian-shaped (in time) reduced electric field with given full-width half maximum is applied at the discharge region.The reduced electric field is varied to provide the total power of 0.4323 W while varying the repetition rate and the pulse width.The methane-air equivalence ratio is set to be 0.45, conditions at which the flame is not self-sustaining and the combustion that is initiated at the discharge region quenches at downstream flow locations.This low value is deliberately chosen; otherwise self-sustained combustion makes it difficult to resolve the effect of the discharge.The initial advection speed is 42.5 cm/s.The initial inlet gas temperature and pressure are set to 296 K and 1 atm, respectively.

Results for Repetition
Rates Ranging between 10 kHz and 50 kHz.Simulations were carried out for different repetition rates ranging from 10 kHz to 50 kHz, under conditions of constant average power.Under these constraints, the energy per pulse is higher for the lower repetition rates.The discharge pulse energies are 43 μJ, 22 μJ, 14 μJ, 11 μJ, and 9 μJ for 10 kHz, 20 kHz, 30 kHz, 40 kHz, and 50 kHz, respectively.The corresponding effective reduced electric fields are determined to be 345 Td, 335 Td, 328 Td, 323 Td, and 318 Td.The kinetics within the discharge region is shown in Figure 2. Because of the higher electric fields for the lower repetition rates, the electron number density (Figure 2(b)) peaks at higher values and the population of excited electronic states of N 2 (Figure 2(a)) are significantly larger.As a result, the gas temperature rises (Figure 2(c)) and the produced amount of radicals such as O, H, and OH (Figure 2(d)) are also significantly greater for the low repetition rates.This larger amount of the produced radicals after the pulse converts CH 4 further to CO and H 2 , and finally to CO 2 and H 2 O (Figures 2(e) and 2(f)).It is noteworthy that with this type of discharge the methane concentration remains low within the discharge region after the very first discharge pulse because the first discharge ignites the methane-air mixture and the repetition timescales tested are shorter than the times for species diffusion and advection.Although the lower repetition rates provide for higher values for the temperature and radical concentrations, their quasi-steady state levels are found to be lower because of the longer time between pulses for thermal conduction, species diffusion, and radical recombination processes.According to these simulations, the kinetic evolution is found to be more drastic for the lower repetition rates.
The quasi-steady state contours of the major and minor combustion species for different repetition rates between 10 kHz and 50 kHz are shown in Figures 3 and 4 , and 4(c), respectively.Interestingly, in spite of their different quasi-steady state values and degree of temporal evolutions for the gas temperature and the radicals, the results for these cases are almost exactly the same.Our results indicate that, for this range of repetition rate and average power, the discharge is able to maintain a sufficiently high level of    excited state species population required to keep the discharge region combusted (Figures 3(a), 3(b), and 3(c)), while the peak levels of the produced radicals (Figure 4(c)) are less important because they decay quickly to the thermally equilibrated values in the post-discharge region through radical recombination reactions.These recombination reactions eventually release heat energy to the stream.In essence, while radicals play some role in the kinetics, we find that the average power, irrespective of the mechanism through which heating takes place, is the critical factor on stabilizing combustion.

Results for Different Pulse Width Ranging from 9 ns to 2 ns.
Simulations for different pulse widths corresponding to 9 ns, 4 ns, and 2 ns are compared, while maintaining constant average discharge power.The energy per pulse is 14 μJ because the repetition rate is also kept constant, but the temporal energy density during the pulse is higher for the shorter pulse width.This is reflected in the higher reduced electric fields for shorter pulses.The fields are determined to be 328 Td, 421 Td, and 535 Td for 9 ns, 4 ns, and 2 ns pulse widths, respectively.The detail kinetics within the discharge region is shown in Figure 5.Because of the higher temporal energy density for the shorter pulses, the shorter pulses result in larger peak electron number densities (Figure 5(b)).For excited electronic states of N 2 (Figure 5(a)), more energized electrons during the shorter pulses populate the higher energy states.N 2 C is produced most when the pulse width is the shortest.However, this different degree of population between these states does not appear to lead to significant differences in the amounts of produced radicals such as O, H, and OH (Figure 5(d)).The results are very similar for all of the tested pulse widths, and the kinetic evolutions for minor species (H 2 and CO, Figure 5(f)) and major species (CH 4 , CO 2 , and H 2 O, Figure 5(e)) are found to be the same.This finding indicates that shorter pulses populate higher energy states with more preference but do not lead to a noticeably larger radical amount.The quasi-steady states contours of major and minor combustion species are shown in Figures 6 and 7 As seen in the figures, the contours are almost exactly the same for the three different pulse widths.This is expected since the discharge kinetics describing radical production (Figure 5) were also very similar.This again suggests that the average power is the defining factor on the chemistry, and little benefit is achieved by shortening the pulse into the nanosecond range.

Conclusion
The effect of repetition rate and pulse width on combustion stabilization for nanosecond repetitively pulsed discharges was investigated by computational simulations.In these simulations, the total average discharge power is kept constant.Since the lower repetition rates have larger pulse energy and a corresponding longer time between pulses, the gas temperature rises and the produced radicals were greater but their quasi-steady values were correspondingly lower than cases of higher repetition rate.However, in spite of this different degree of kinetic evolution, the contours for major and minor combustion species were found to be almost exactly the same and independent of repetition rate.Shortening the pulse widths while maintaining a constant average discharge power produced a higher peak population of excited electronic state species but the overall quasi-steady amounts were similar to all pulse widths and therefore the contours of the combustion products were also similar.From these simulations, we conclude that the average discharge power is found to be the determining factor on combustion stabilization and little if any benefit is obtained by varying the operation parameters such as the repetition rate and pulse width of nanosecond pulsed discharges over the range of conditions studied here.

Figure 2 :
Figure 2: Kinetic evolution of (a) excited electronic states of N 2 (A, B, C, a ), (b) electron, (c) gas temperature, (d) radicals (O, H, and OH), (e) combustion major species (CH 4 , CO 2 , and H 2 O), and (f) minor species (H 2 and CO) within the discharge region during a single period when the pulse width is 9 ns and the repetition rates are 10 kHz, 20 kHz, 30 kHz, 40 kHz, and 50 kHz.

Figure 6 :
Figure 6: Quasi-steady state spatial distribution of (a) CH 4 , (b) CO 2 , and (c) H 2 O for three different pulse widths of 9 ns, 4 ns, and 2 ns at 30 kHz repetition rate.

Figure 7 :
Figure 7: Quasi-steady state spatial distribution of (a) CO, (b) H 2 , and (c) O for three different pulse widths of 9 ns, 4 ns, and 2 ns at 30 kHz repetition rate.