Modeling Trace Gas Sensors with the Coupled Pressure-Temperature Equations

Quartz enhanced photoacoustic spectroscopy (QEPAS) is a technique for detecting trace gases which relies on a quartz tuning fork resonator to amplify and measure the weak acoustic pressure waves that are generated when a laser heat source periodically interacts with a gas sample. At low ambient pressures, the same tuning fork can also detect the thermal diffusion waves generated by the laser-gas interaction in a process called resonant optothermoacoustic detection (ROTADE). In this dissertation we study a series of progressively more sophisticated models for trace gas sensors. As a result, we develop a unified computational model for both ROTADE and QEPAS sensors that can be used to improve the performance of these sensors. Our first model is based on a coupled system of Helmholtz equations for pressure and temperature in a fluid domain surrounding the tuning fork. The standard heat equation is used to solve for temperature in the tuning fork itself. We employ the perfectly matched layer (PML) approach to absorb outgoing waves and prevent reflections off of the boundary of the computational domain. The resulting linear system is highly ill-conditioned, but Krylov subspace solvers can be used to solve the system effectively if one employs an appropriate parallel block preconditioner. This method reduces the problem to that of solving a scalar Helmholtz problem with PML, which we precondition by coupling an algebraic multigrid solver in the interior of the computational domain to a direct solver in the PML region. Numerical results indicate that the preconditioner for the scalar Helmholtz problem with PML is both scalable and mesh-independent. Simulations show that the coupled pressuretemperature waves can strongly differ from the solution to the acoustic wave equation at low ambient pressures. Next, we develop a one-way coupled three-stage model for the vibration of the tuning fork that is excited by the coupled pressure and temperature waves. Our approach is to first determine the resonant numerical eigenfrequency of the tuning fork and then solve for the pressuretemperature wave generated by the laser using the model above. Finally, we determine the thermoelastic deformation of the tuning fork driven by the pressure forces acting on its walls and internal stress due to thermal heating. Using this model, we obtain quantitative agreement with QEPAS data and good qualitative agreement with the ROTADE results. However, the approach we use relies on damping parameters which cannot be predicted by analytic means and must be measured experimentally. Finally, we study a two-way coupled model which incorporates the viscous damping of the tuning fork due to its motion through the surrounding gas. We develop a system where we model temperature, pressure, fluid velocity and tuning fork displacement where damping is realized through coupling the motion of the tuning fork to the motion of the fluid. The numerical results show that our approach is qualitatively capable of simulating viscous damping. Furthermore, the dependence of the frequency of the simulated resonance peaks on ambient pressure agrees well with previous experimental data.