# Modeling and Sensitivity Analysis for Trace Gas Sensors

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## Abstract

Trace gas sensors can detect very low concentrations of gases such as methane and sulfur dioxide. An important class of trace gas sensors are quartz-enhanced photoacoustic spectroscopy (QEPAS) sensors, which employ a quartz tuning fork (QTF) and modulated laser to detect trace gases. Existing models of QEPAS sensors employ one-way coupling from the fluid to the structure that requires prior experimental measurements of the damping of the QTF due to its motion in the viscous fluid. We study an improved two-way coupled model that is based on a Helmholtz system of thermo- visco-acoustic equations in the fluid, together with a system of equations for the temperature and the displacement of the structure. These two subsystems are coupled across the fluid-structure interface via several conditions. With this model, the user specifies the geometry of the structure and the viscous and thermal parameters of the fluid, and the model outputs an effective damping parameter and a signal strength that is proportional to the concentration of the trace gas. We derive analytic solutions of the two-way coupled model in the special case that the QTF is replaced by an annular structure. This simplification of the geometry allows the pressure, temperature of the fluid, and the displacement of the structure to be expressed in terms of Bessel functions. These solutions show reasonable agreement between the one-way and two-way coupled models at higher ambient pressures. However, at low ambient pressure the one-way coupled model does not adequately capture thermo-viscous effects. For the two-way coupled model, excellent agreement is obtained between the analytical results and simulations performed using a finite element formulation of the model. Computational models for trace gas sensors involve a large number of parameters. If one wants to quantify uncertainty of the output quantities of interest (for example, pressure, temperature or displacement of the tuning fork tines) then one must estimate statistical distributions that describe these quantities. However, statistical studies require that one runs the physical simulator for hundreds or thousands of different input parameter values. This process is computationally prohibitive unless one can first identify which parameters influence the model output. We use the active sub- space method to identify a subset of parameters that is influential for the output. The application of the active subspace method to the pressure-temperature subsystem in the special case of cylindrical symmetry identifies one influential parameter for the fluid temperature and three for the pressure from the five dimensional parameter space. Similarly, for the two-way coupled model with annular geometry the active subspace method reduces the 13 dimensional parameter space to a 5 dimensional subspace by identifying 4 influential parameters for the fluid temperature and 5 influential parameters for the remaining quantities of interest. Finally, for the one-way coupled model with tuning fork geometry, the active subspace method identifies 5 influential parameters for the fluid pressure, temperature, and displacement of the QTF reducing a 10 dimensional parameter space to a 5 dimensional subspace. These results also show an excellent agreement with the results obtained using kernel density estimation and a simple sensitivity study.