Mathematical Modeling, Analysis, and Simulation of Trace Gas Sensors

Sensor systems with the ability to identify trace gases with sensitivities in the parts-per-million range are becoming essential tools for environmental monitoring, medical diagnostics, and homeland security. The development of the next generation of sensors is based on new trace gas sensing techniques such as quartz-enhanced photoacoustic spectroscopy (QEPAS) and resonant optothermoacoustic detection (ROTADE). QEPAS and ROTADE are complementary techniques which combine a laser source and a quartz tuning fork (QTF) detector to measure the concentration of a gas. The parameters of QEPAS and ROTADE sensors, such as the QTF geometry and the laser source position, play an important role in their performance. In this thesis, we describe and validate models for three QTF-based trace gas sensors: a QEPAS sensor without a microresonator, a QEPAS sensor with a microresonator, which is added to enhance the sensitivity of the sensor, and a ROTADE sensor. For a QEPAS sensor, when the laser source interacts with a trace gas, the optical energy absorbed by the gas results in periodic thermal expansion. This heat disturbance gives rise to a weak acoustic pressure wave which excites a resonant vibration of the tuning fork thereby generating an electrical signal via the piezoelectric effect. An analytic model of a QEPAS sensor without a microresonator that couples pressure wave propagation and deformation allows us to study the effect of the laser source position and modulation frequency on the signal strength. Simulation results show that the optimal position of the laser beam is about 4/5 of the way up the QTF which agrees with the experimental data. In addition, a comparison of the signal strength obtained with a standard 32.8 kHz QTF and a 4.25 kHz QTF shows that the 32.8 kHz QTF produces a signal that is 2-3 times higher than the one obtained with a 4.25 kHz QTF. For a QEPAS sensor with a microresonator the model includes solution of the forced time-harmonic acoustic wave equation in an unbounded domain. The boundary integral equation formulation for the inhomogeneous Helmholtz equation leads to a problem on a bounded domain of one degree lower dimension which we solve using the Galerkin Boundary Element Method (BEM). The costly part of the method is the evaluation of a particular solution to the inhomogeneous Helmholtz equation. Numerical experiments for a test problem demonstrate the correctness of the implementation. The application of the method to a QEPAS sensor with a microresonator, which consists of two open-ended tubes placed on either side of the QTF, leads to results which show that the pressure is highest at the center of the tubes. A ROTADE sensor employs optothermoacoustic detection to measure the concentration of a gas. A finite element (FEM) based 3D computational model that couples heat diffusion and linear elastic deformation accurately predicts the source location which provides the maximum signal. Moreover, this model allows for the formulation of a design optimization problem that maximizes the ROTADE signal. To avoid computation of the gradient the derivative-free Mesh Adaptive Direct Search (MADS) algorithm is applied. Numerical results presented for a frequency-constrained and a frequency-unconstrained problem show that, while the frequency-constrained problem gives a signal that is 3 times larger than the one obtained with the standard 32.8 kHz QTF used in experiments, when the frequency can vary, the optimal solution is 24 times greater. These results suggest that a custom made QTF could increase the sensitivity of ROTADE sensors