Modeling of Trace Gas Sensors

Transcription

1 Modeling of Trace Gas Sensors Susan E. Minkoff 1, Noémi Petra 2, John Zweck 1, Anatoliy Kosterev 3, and James Doty 3 1 Department of Mathematical Sciences, University of Texas at Dallas 2 Institute for Computational Engineering and Sciences, University of Texas at Austin 3 Department of Electrical and Computer Engineering, Rice University IMA Special Workshop: Career Options for Women in Mathematical Sciences March 3, 2013 Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 1 / 24

2 Outline 1 Applications of Trace Gas Sensors 2 Description of How Sensors Work 3 Modeling and Numerical Simulation of a Resonant Optothermoacoustic (ROTADE) Sensor 4 Design Optimization of Tuning Forks for Resonant Optothermoacoustic (ROTADE) Sensors Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 2 / 24

3 Applications of Trace Gas Sensing Medicine and Life Sciences Non-invasive disease diagnosis (e.g. lung cancer) using breath biomarkers. Environmental Monitoring Monitoring of atmospheric carbon dioxide levels Volcanic emissions Portable Breath Analyzers Urban and Industrial Emission Measurements Detection of harmful gases Automobile, aircraft and marine emissions Homeland Security Susan E. Minkoff (UTD) view over downtown Houston, TX. Trace Gas Sensors IMA Women in Math 3 / 24

4 The physics of trace gas sensing The detection of trace gases is based on the interaction between optical radiation: a laser source gas molecules: absorb light only at certain wavelengths Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 4 / 24

5 How do we detect the acoustic and thermal waves? Detection using Quartz Tuning Forks (QTFs): Quartz crystals have piezo- and pyroelectric properties. Piezoelectricity is the ability of some materials to generate electricity in response to applied stress. Pyroelectricity is the ability of some materials to generate electricity when heated. Quartz Tuning Fork Quartz-Enhanced PhotoAcoustic Spectroscopy (QEPAS) acoustic pressure wave mechanical resonance in a QTF Resonant OptoThermoAcoustic DEtection (ROTADE) diffusion (heat) wave mechanical resonance in a QTF Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 5 / 24

6 Characteristics of QEPAS and ROTADE Sensors QEPAS and ROTADE are complementary techniques. QEPAS ROTADE Ambient Pressure 50 Torr ( 7kPa) 50 Torr ( 7 kpa) Laser Source Position Top of QTF Bottom of QTF QEPAS and ROTADE Characteristics: allow for the analysis of very small concentration of gas (< 1mm 3 in volume); offer immunity to environmental acoustic noise; high sensitivity; compact, low cost; potential for trace gas sensor networks. Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 6 / 24

7 Experimental configurations of QEPAS and ROTADE sensors QEPAS without a microresonator QEPAS with a microresonator ROTADE Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 7 / 24

8 Optothermal Detection of a Trace Gas Optical radiation is focused between the tines of a tuning fork. Trace gases absorb optical energy at characteristic frequencies. A diffusion (heat) wave can be generated by modulating this interaction. Resonant mechanical vibration is excited by the diffusion wave. The mechanical vibration is converted to an electrical current. The concentration of the trace gas is proportional to the signal strength. Experimental Configuration for ROTADE First Principal Stress Eigenmode of vibration Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 8 / 24

9 Mathematical Modeling of a ROTADE Sensor Modeling Goal: to determine how the signal strength depends on system parameters such as the optimal placement of the laser beam source, and numerical optimization of the tuning fork geometry. The mathematical model of a ROTADE sensor includes three parts: I. Modeling heat transfer II. Modeling the vibration of the tuning fork III. Modeling the conversion of mechanical stresses into electrical signal (as before). Modeling Assumptions: ignore the QEPAS signal; model heat diffusion via the Heat equation. Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 9 / 24

10 I. Heat Transfer Model We compute the temperature distribution via the heat equation where c pρ T t (K T ) = H, T = temperature t = time H = time-harmonic heat source c p = specific heat ρ = density K = thermal conductivity tensor. Considering only time-harmonic waves of the form T (x, t) = T (x)e iωt, the heat equation reduces to a Helmholtz type equation (k T ) + iωt = H, where k = K /ρc p is thermal diffusivity, ω = is the laser modulation frequency, and H is the spatial part of the heat source. Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 10 / 24

11 II. Model for the vibration of the tuning fork Thermoelastic deformation problem: 8 < C[ u] + (ρω 2 ibω)u = C[α tt ], in Ω TF u = 0, on Γ 1 : C[ u]n = C[α tt ]n, on Γ 2, u = displacement field ρ = density C = elasticity tensor ω = laser modulation frequency T = temperature b = damping constant α t = thermal expansion tensor n = is the outward unit normal vector to Γ 2. Eigenfrequency Analysis: 8 < C[ u] + ρω 2 u = 0, in Ω TF, u = 0, on Γ 1, : C[ u]n = 0, on Γ 2. Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 11 / 24

12 Numerical Solution of the Heat Problem The heat transfer from the exterior to the interior of the tuning fork gas QTF gas QTF Temperature Temperature x (mm) x (mm) Left: 1D slice in the x-direction of the temperature. Right: A semilog plot of the temperature shown in the left figure. Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 12 / 24

13 Numerical Solution of the Deformation Problem The magnitude of the piezoelectric current as a function of frequency. 4 x Amplitude of the piezoelectric signal (pa) Frequency (Hz) The first principal stress (left) and the fourth eigenmode of the QTF corresponding to the 32.8 khz eigenfrequency Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 13 / 24

14 ROTADE Simulation Results Left: Schematic diagram of laser positions with respect to tuning fork. Theoretical piezoelectric signal (center) and the phase (right) as functions of the vertical position of the laser beam. Nomalized signal strength S1(pA) 0.15 mm, center S2(pA) 0.03 mm S3(pA) mm Phase (degrees) Phase mm, center Phase mm Phase mm z (mm) z (mm) The results show that the output is largest when the source is focused near the base of the QTF. Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 14 / 24

15 ROTADE Simulation Results Left: Map of experimental ROTADE signal as a function of laser position. Right: The first principal stress of the QTF at the resonance frequency. Signal strength is largest near where the stress is largest. Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 15 / 24

16 Comparison of Model with Experiments 10 7 Theory Experiment 180 Theory Experiment Normalized signal strength Phase z (mm) z (mm) Comparison of the theoretical and experimental normalized amplitude (left) and phase (center) of the ROTADE signal as a function of the vertical position of the laser source for C 2 H 2 :N 2. Right: Experimental signal map obtained at 20 Torr for pure CO 2. The right figure shows an interference between ROTADE and QEPAS signals: at the dark spots on the center the QEPAS and ROTADE amplitudes are equal and the phase is opposite. Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 16 / 24

17 Comparison of Model with Experiments Normalized signal strength Theory Experiment z (mm) Phase (degrees) Theory Experiment z (mm) Comparison of the theoretical (blue dotted line) and experimental (blue circles) normalized amplitude (left) and phase (right) of the ROTADE signal as a function of the vertical position of the laser source for CO 2. These results are obtained at an ambient pressure of 20 Torr. The slopes of the initial part of the experimental and theoretical ROTADE signal phase agree well. Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 17 / 24

18 Design Optimization of Tuning Forks for ROTADE sensors Since the thermal wave decays rapidly, optimization of sensor geometry is important for ROTADE sensors. Optimization Problem: minimize J(p) subject to: p X, J : X R, J(p) = v L (p, f (p)), p = (l, w, g, l b, t) X, X R 5, i.e. X = X u or X = X c, where X u = [l l, l u ] [w l, w u ] [g l, g u ] [l l b, lu b ] [tl, t u ] (frequency-unconstrained) X c = {p X u/ f l < f (p) < f u } (frequency-constrained), f : X R is the resonance frequency of the QTF. Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 18 / 24

19 Tuning fork optimization for ROTADE sensors via NOMADm NOMADm is optimization software developed by Abramson et al. - it is a MATLAB implementation of the class of Mesh-Adaptive Direct Search (MADS) algorithms; intended for solving nonlinear and mixed variable optimization problems with general nonlinear constraints; expected to perform well when the dimension of the search space is 10. References: C. Audet and J. E. Dennis, Jr., Mesh Adaptive Direct Search Algorithms for Constrained Optimization, SIAM J. Optim., vol. 17, pp , Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 19 / 24

20 Optimization Simulation Results The first principal stress of the 32.8 khz (top left), 30 khz (top right), and 3 khz (bottom left) quartz tuning forks, respectively (at the resonance frequency). Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 20 / 24

21 Optimization Simulation Results (cont d) Optimization results for tuning forks with and without constrained resonance frequency: f (numerical resonance frequency), l (length), w (width), g (gap), l b (length of the base), t (thickness), v L (the velocity of the tines of the QTF). Parameter Standard Frequency- Frequency- SI unit 32.8 khz QTF constrained unconstrained Optimum QTF Optimum QTF f khz l mm w mm g mm l b mm t mm v L m/s The frequency-constrained problem gives a signal that is 3 times larger than the one obtained with the standard 32.8 khz QTF. When the frequency can vary, the optimal solution is 24 times greater. Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 21 / 24

22 Conclusions and Future Work We validated experimental results which show that the ROTADE signal is largest when the source is focused near the base of the quartz tuning fork. We found that the optimally-shaped quartz tuning fork (with the resonance frequency constrained to about 30 khz) is almost 3 times larger than the signal obtained with the standard 32.8 khz tuning fork. The frequency-unconstrained formulation provided a ROTADE signal that is 24 times larger than the signal obtained with the standard 32.8 khz tuning fork. Ongoing and Future Work: Model the molecular interactions of trace gases. Develop a method to automatically compute the damping of the QTF in terms of the geometry. Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 22 / 24

23 Acknowledgements Funding was provided by the National Science Foundation through the MIRTHE-ERC program (grant no. EEC ). Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 23 / 24

24 For Further Reading See: 1 Petra, N., Zweck, J., Minkoff, S., Kosterev, A., and Doty, J., Validation of a Model of a Resonant Optothermoacoustic Trace Gas Sensor, Proceedings of the CLEO/QELS: 2011 Laser Science to Photonic Applications Conference, Optical Society of America, 2011, #JTuI Petra, N., Zweck, J., Minkoff, S., Kosterev, A., and Doty, J., Modeling and Design Optimization of a Resonant Optothermoacoustic Trace Gas Sensor, SIAM Journal on Applied Mathematics, 71, pp , N. Petra, A. A. Kosterev, J. Zweck, S. E. Minkoff, and J. H. Doty III, Numerical and Experimental Investigation for a Resonant Optothermoacoustic Sensor, in Conference on Lasers and Electro-Optics, Optical Society of America, 2010, p. CMJ6. 4 Petra, N., Zweck, J., Kosterev, A., Minkoff, S., and Thomazy, D., Theoretical Analysis of a Quartz-Enhanced Photoacoustic Spectroscopy Sensor, Applied Physics B: Lasers and Optics, 94, pp , 2009: DOI: /s Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 24 / 24