Solving the electron Boltzmann equation in microwave-enhanced combustion

Abstract

In low-temperature plasma devices, free electrons are the main conduit for converting electromagnetic field energy into the chemical excitation and heating of neutral gases. This non-equilibrium process is described by the Boltzmann equation (EBE), whose solution provides rate data essential for multi-physics simulations that couple traditional combustion and fluid dynamics with electromagnetic fields.

This work details a new approach for solving the EBE for electrons exposed to time-dependent high-frequency AC fields in dynamic non-equilibrium gases, called the Multi-Term Multi-Harmonic Boltzmann equation (MTMH-BE). The traditional numerical solution uses a series of orthogonal polynomials to represent the periodic and angular velocity of the electron energy distribution function (EEDF), with the result being a system of quasi-linear ODEs. This work simplifies the problem by reducing the numerical Jacobian to a single matrix multiplication between a time-independent sparse matrix and an array of time-dependent coefficients. The resulting solver was tested against a range of benchmarks and performance tests, demonstrating excellent agreement with existing work and significant performance improvements compared to existing methods in problems with non-equilibrium state-specific gases. This enables the possibility of direct coupling of the EBE to the plasma kinetic rate equations.

The principles of solving the EBE were then applied to the problem of microwave enhancement of alkali-seeded combustion, both in energetic materials and detonation waves.

First, experimental measurement of microwaved-enhanced light emission of alkali-metal pyrotechnics was investigated using a preliminary 1-D model of microwave-propagation, alkali-excitation, and sodium light emission. This model was then used to explain enhancement across three alkali metals, providing an explanation for the unexpected trend of increasing enhancement with lower molecular-weight alkali metals.

Second, the MTMH-BE was applied directly to analysis and experiments with alkali-seeded detonation waves. The MTMH-BE solver was used to map the theoretical dependence of absorption of freely propagating microwaves, highlighting the importance of multi-harmonic effects at low frequencies. This was complemented by a detailed study of microwave power loss, which advanced existing techniques by including super-elastic heating. Additionally, a second model of microwave power loss specific to solution within a TE10 mode microwave cavity was developed to measure plasma conductivity, along with an effort to predict sodium density from the collision-broadened self-reversed sodium doublet.