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An analytic solution to the coupled pressure–temperature equations for modeling of photoacoustic trace gas sensors

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Abstract

Trace gas sensors have a wide range of applications including air quality monitoring, industrial process control, and medical diagnosis via breath biomarkers. Quartz-enhanced photoacoustic spectroscopy and resonant optothermoacoustic detection are two techniques with several promising advantages. Both methods use a quartz tuning fork and modulated laser source to detect trace gases. To date, these complementary methods have been modeled independently and have not accounted for the damping of the tuning fork in a principled manner. In this paper, we discuss a coupled system of equations derived by Morse and Ingard for the pressure, temperature, and velocity of a fluid, which accounts for both thermal effects and viscous damping, and which can be used to model both types of trace gas sensors simultaneously. As a first step toward the development of a more realistic model of these trace gas sensors, we derive an analytic solution to a pressure–temperature subsystem of the Morse–Ingard equations in the special case of cylindrical symmetry. We solve for the pressure and temperature in an infinitely long cylindrical fluid domain with a source function given by a constant-width Gaussian beam that is aligned with the axis of the cylinder. In addition, we surround this cylinder with an infinitely long annular solid domain, and we couple the pressure and temperature in the fluid domain to the temperature in the solid. We show that the temperature in the solid near the fluid–solid interface can be at least an order of magnitude larger than that computed using a simpler model in which the temperature in the fluid is governed by the heat equation rather than by the Morse–Ingard equations. In addition, we verify that the temperature solution of the coupled system exhibits a thermal boundary layer. These results strongly suggest that for computational modeling of resonant optothermoacoustic detection sensors, the temperature in the fluid should be computed by solving the Morse–Ingard equations rather than the heat equation.

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Notes

  1. These formulae were obtained by solving the incompressible linear Navier–Stokes equation for viscous fluid flow around a sphere.

  2. A theorem in the text of Chorin and Marsden [31] states that for a vector field, \(\mathbf {v}\), on a domain \(\varOmega \subset {\mathbb {R}^{3}}\), the Helmholtz decomposition, \(\mathbf v = \mathbf v _\ell + \mathbf v _\mathrm{t}\), is unique, provided that we also assume that \(\mathbf v _\mathrm{t}\) is tangential to the boundary surface, \(\partial \varOmega \).

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Authors and Affiliations

  1. Department of Mathematical Sciences, The University of Texas at Dallas, 800 West Campbell Road, Richardson, TX, 75080-3021, USA

    Jordan Kaderli, John Zweck, Artur Safin & Susan E. Minkoff

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  1. Jordan Kaderli
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  2. John Zweck
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  3. Artur Safin
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  4. Susan E. Minkoff
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Correspondence to Jordan Kaderli.

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Kaderli, J., Zweck, J., Safin, A. et al. An analytic solution to the coupled pressure–temperature equations for modeling of photoacoustic trace gas sensors. J Eng Math 103, 173–193 (2017). https://doi.org/10.1007/s10665-016-9867-5

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  • DOI: https://doi.org/10.1007/s10665-016-9867-5

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