Modeling of Trace Gas Sensors - Institute for Mathematics ... · Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 10 / 24. II. Model for the vibration of the tuning fork

Modeling of Trace Gas Sensors

Susan E. Minkoff1, Noemi Petra2, John Zweck1, Anatoliy Kosterev3, and JamesDoty3

1Department of Mathematical Sciences, University of Texas at Dallas

2Institute for Computational Engineering and Sciences, University of Texas at Austin

3Department of Electrical and Computer Engineering, Rice University

IMA Special Workshop: Career Options for Women in Mathematical SciencesMarch 3, 2013

Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 1 / 24

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


Applications of Trace Gas Sensing

Medicine and Life Sciences

Non-invasive disease diagnosis (e.g. lung cancer) usingbreath biomarkers.

Environmental Monitoring

Monitoring of atmospheric carbon dioxide levelsVolcanic emissions

Urban and Industrial Emission Measurements

Detection of harmful gasesAutomobile, aircraft and marine emissions

Homeland Security

Portable Breath Analyzers

view over downtownHouston, TX.


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 onlyat certain wavelengths


How do we detect the acoustic and thermal waves?

Detection using Quartz Tuning Forks(QTFs):

Quartz crystals have piezo- and pyroelectricproperties.

Piezoelectricity is the ability of some materials togenerate electricity in response to applied stress.

Pyroelectricity is the ability of some materials togenerate 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


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 (< 1mm3 in volume);

offer immunity to environmental acoustic noise;

high sensitivity;

compact, low cost;

potential for trace gas sensor networks.


Experimental configurations of QEPAS and ROTADE sensors

QEPAS withouta microresonator

QEPAS witha microresonator

ROTADE


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 Configurationfor ROTADE

First PrincipalStress

Eigenmodeof vibration


Mathematical Modeling of a ROTADE Sensor

Modeling Goal: to determine how the signal strength depends on systemparameters 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.


I. Heat Transfer Model

We compute the temperature distribution via the heat equation

cpρ∂T∂t−∇ · (K∇T ) = H,

whereT = temperature t = time H = time-harmonic heat sourcecp = specific heat ρ = density K = thermal conductivity tensor.

Considering only time-harmonic waves of the form

T (x, t) = T (x)eiωt ,

the heat equation reduces to a Helmholtz type equation

−∇ · (k∇T ) + iωT = H,

where k = K/ρcp is thermal diffusivity, ω = is the laser modulation frequency, andH is the spatial part of the heat source.


II. Model for the vibration of the tuning fork

Thermoelastic deformation problem:8<: ∇ · C[∇u] + (ρω2 − ibω)u = ∇ · C[αtT ], in ΩTFu = 0, on Γ1

C[∇u]n = C[αtT ]n, on Γ2,

u = displacement field ρ = densityC = elasticity tensor ω = laser modulation frequencyT = temperature b = damping constantαt = thermal expansion tensor n = is the outward unit normal vector to Γ2.

Eigenfrequency Analysis:8<: ∇ · C[∇u] + ρω2u = 0, in ΩTF,u = 0, on Γ1,

C[∇u]n = 0, on Γ2.


Numerical Solution of the Heat Problem

The heat transfer from the exterior to the interior of the tuning fork.

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

x (mm)

Tem

pera

ture

gas QTF

0 0.05 0.1 0.15 0.2

10−6

10−4

10−2

100

x (mm)T

empe

ratu

re

gas QTF

Left: 1D slice in the x-direction of the temperature.Right: A semilog plot of the temperature shown in the left figure.


Numerical Solution of the Deformation Problem

The magnitude of the piezoelectric current as a function of frequency.

33805.83 33807.33 33808.83 33810.33 33811.830

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

Frequency (Hz)

Am

plitu

de o

f the

piez

oele

ctric

sig

nal (

pA)

The first principal stress (left) and the fourth eigenmode of the QTF correspondingto the 32.8 kHz eigenfrequency


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.

−0.2 0.2 0.6 1

1

2

3

4

z (mm)

Nom

aliz

ed s

igna

l str

engt

h

S1(pA) − 0.15 mm, centerS2(pA) − 0.03 mmS3(pA) − 0.013 mm

−0.2 0.2 0.6 1

−60

0

60

z (mm)

Pha

se (

degr

ees)

Phase1 − 0.15 mm, centerPhase2 − 0.03 mmPhase3 − 0.013 mm

The results show that the output is largest when the source isfocused near the base of the QTF.


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.


Comparison of Model with Experiments

0 0.4 0.810

4

105

106

107

z (mm)

Nor

mal

ized

sig

nal s

tren

gth

TheoryExperiment

0 0.4

100

140

180

z (mm)P

hase

TheoryExperiment

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 C2H2:N2.

Right: Experimental signal map obtained at 20 Torr for pure CO2.

The right figure shows an interference between ROTADE and QEPAS signals: atthe dark spots on the center the QEPAS and ROTADE amplitudes are equal and

the phase is opposite.


Comparison of Model with Experiments

0 0.4 0.810

1

102

103

z (mm)

Nor

mal

ized

sig

nal s

tren

gth

TheoryExperiment

0 0.4 0.8

−150

−50

z (mm)

Pha

se (

degr

ees)

TheoryExperiment

Comparison of the theoretical (blue dotted line) and experimental (blue circles) normalized amplitude (left) andphase (right) of the ROTADE signal as a function of the vertical position of the laser source for CO2. These

results are obtained at an ambient pressure of 20 Torr.

The slopes of the initial part of the experimental and theoreticalROTADE signal phase agree well.


Design Optimization of Tuning Forks for ROTADE sensors

Since the thermal wave decays rapidly, optimization of sensor geometryis important for ROTADE sensors.

Optimization Problem:

minimize J(p)subject to: p ∈ X ,

J : X → R, J(p) = −vL(p, f (p)),

p = (l, w, g, lb , t) ∈ X ,

X ⊂ R5, i.e. X = Xu or X = Xc , where

Xu = [ll , lu ]×[wl , wu ]×[gl , gu ]×[llb, lub ]×[tl , tu ] (frequency-unconstrained)

Xc = p ∈ Xu/ f l < f (p) < f u (frequency-constrained),

f : X → R is the resonance frequency of the QTF.


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 withgeneral nonlinear constraints;

expected to perform well when the dimension of the search space is . 10.

References:

http://www.gerad.ca/NOMAD/Abramson/NOMADm.html

C. Audet and J. E. Dennis, Jr., “Mesh Adaptive Direct Search Algorithms for Constrained Optimization”,SIAM J. Optim., vol. 17, pp. 88-217, 2006.


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).


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), lb (length of the base), t

(thickness), vL (the velocity of the tines of the QTF).

Parameter Standard Frequency- Frequency- SI unit32.8 kHz QTF constrained unconstrained

Optimum QTF Optimum QTFf 33.77 30.04 3.03 kHzl 3.75 3.11 7.99 mm

w 0.60 0.34 0.20 mmg 0.30 0.20 0.19 mmlb 2.33 1.76 2.04 mmt 0.34 0.21 0.59 mm

vL 0.06 0.18 1.47 m/s

The frequency-constrained problem gives a signal that is 3 times larger thanthe one obtained with the standard 32.8 kHz QTF.

When the frequency can vary, the optimal solution is 24 times greater.


Conclusions and Future Work

We validated experimental results which show that the ROTADE signal is largestwhen the source is focused near the base of the quartz tuning fork.

We found that the optimally-shaped quartz tuning fork (with the resonancefrequency constrained to about 30 kHz) is almost 3 times larger than the signalobtained with the standard 32.8 kHz tuning fork.

The frequency-unconstrained formulation provided a ROTADE signal that is 24times 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 ofthe geometry.


Acknowledgements

Funding was provided by the National Science Foundation through the MIRTHE-ERCprogram (grant no. EEC–0540832).


For Further Reading See:

1 Petra, N., Zweck, J., Minkoff, S., Kosterev, A., and Doty, J., “Validation of a Model of a ResonantOptothermoacoustic Trace Gas Sensor,” Proceedings of the CLEO/QELS: 2011 Laser Science toPhotonic Applications Conference, Optical Society of America, 2011, #JTuI115.

2 Petra, N., Zweck, J., Minkoff, S., Kosterev, A., and Doty, J., “Modeling and Design Optimization of aResonant Optothermoacoustic Trace Gas Sensor,” SIAM Journal on Applied Mathematics, 71, pp.309-332, 2011.

3 N. Petra, A. A. Kosterev, J. Zweck, S. E. Minkoff, and J. H. Doty III, “Numerical and ExperimentalInvestigation 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 aQuartz-Enhanced Photoacoustic Spectroscopy Sensor,” Applied Physics B: Lasers and Optics, 94, pp.673–680, 2009: DOI:10.1007/s00340-009-3379-1.


Modeling of Trace Gas Sensors - Institute for Mathematics ... · Susan E. Minkoff (UTD) Trace Gas Sensors IMA Women in Math 10 / 24. II. Model for the vibration of the tuning fork

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