Quantitative Finite Element Modelling of Compact Photoacoustic Gas Sensors

We report on the demonstration of quantitative simulation of compact photoacoustic cells signals. The finite element method is used to calculate the response of differential Helmholtz resonator cells of two different sizes. Simulated quality factors and resonance frequencies are compared with experimental ones. Taking account of the gas sample absorption coefficient and the laser intensity, cell constants are also evaluated and compared to experimental values showing a very good agreement. For compact sensors, where sub-millimeter features are present, the resolution of the pressure acoustics equation is not sufficient to reproduce experimental results and a viscothermal model must be used.


Introduction
Laser absorption spectrometry has proved to be a powerful tool for trace gas sensing providing high selectivity of the monitored species and low detection limits. In addition, semiconductor lasers can emit in the mid-infrared region where the fundamental absorption lines of most gases lie. Among laser spectrometry techniques, photoacoustic (PA) spectrometry measures the sound generated in the gas sample cell when exciting the gas with laser radiation [1]. Thanks to resonant cells and differential measurement techniques, PA sensors have shown excellent sensitivities at part-per-billion (ppb) level for most atmospheric gases. These results are comparable to those of other apparatus, such as multipass cells, but with a generally simpler set-up.
The Groupe de Spectrométrie Moléculaire et Atmosphérique of Reims (GSMA, Reims France) has developed since 1997 a photoacoustic sensor with a specific cell design based on Helmholtz resonance [2]. This instrument was originally dedicated to the methane detection with near-infrared diode lasers, then Quantum Cascade Lasers (QCL) appeared as a new opportunity for improvements in gas detection as the PA response is linear with power [3]. More recently the sensor was implemented with a commercial external cavity-QCL emitting at 10.5 μm and allowed the possibility to detect small and complex molecules such as carbon dioxide and butane [4]. For these works, the geometry of the Helmholtz cell was not modified but improvements in the microphone sensitivity and the laser specifications allow decreasing the detection limit down to the ppb level. Detection limits were estimated knowing the cell response, the microphone response, and the gas absorption coefficient at the laser emitting wavenumber. These predictions were then experimentally verified. The cell response was usually compared to simulation using an electric analog circuit method.
In 2006, Baumann et al. proposed to simulate the PA cell responses and resonance frequency using a Finite Element Method (FEM) [5,6]. Our team used the same method to demonstrate for the first time the quantitative modelling of photoacoustic signals including resonance frequency and signal levels quantification [7]. More recently, we also demonstrated the complete optimization and characterization of the sensor for the detection of methane [8] and the possibility to detect methane in large concentrations from 370 ppb up to at least 8% in volume i.e., more than five orders of magnitude with the same Helmholtz sensor [9]. In these papers the experimental results were in very good agreement with the FEM resolution of pressure acoustics equation.
For trace gas measurements, the PA signal is directly proportional to the absorption coefficient of the molecules, to the radiation power and to the cell response. The latter is inversely proportional to the cell volume. This favorable scaling behavior has provoked in recent years a growing interest in the miniaturization of PA cells for gas sensing. The first technique developed to reduce the size of the PA sensors is called Quartz-Enhanced PhotoAcoustic Spectroscopy (QEPAS) [10]. An alternative technique consists in the miniaturization of standard macro-scale PA cells. One can particularly cite the work of Firebaugh [11], later carried on by Pellegrino and Holthoff [12], the realization of 3Dprinted PA cells [13] and the miniaturization of the Helmholtz resonator [14]. However, at the sub-millimeter scale of the chambers and capillaries diameters, the thermal and viscous boundary layers constitute a non-negligible part of the cell volume. Thus, the pressure acoustics description is no more sufficient and a complete viscothermal model taking into account the dissipation effects that must be used to design compact photoacoustic sensors. In this paper, we present the results obtained for devices of various dimensions using two different kinds of FEM simulations based either on the classical pressure acoustics equations or on viscothermal formulation. Both kinds of simulations were performed using a FEM analysis of commercial software [15].

Modelling Pressure Acoustics Model (PAM)
The PAM is commonly used to simulate the PA device behavior. This model is derived from the first principles governing equations, namely the mass, momentum and energy conservation laws supplemented with a thermodynamic equation of state. Assuming adiabatic propagation in an ideal lossless gas and after some manipulation, a single inhomogeneous Helmholtz equation for the unknown pressure is obtained [16]. If harmonic excitation is assumed the equation takes the following form: where p is the Fourier transform of the acoustic pressure, k = ω/c with c is the sound velocity and ω the angular frequency.
Following the reasoning described for instance by Kreuzer [17], and previously used by Baumann et al. [6], it is possible to partly include dissipation effects with the following three steps method:

ThermoAcoustics Model (TAM)
The TAM is especially developed for accurate simula-

Experimental set-up PA cells
In [7] we demonstrated for the first time the possibility to quantitatively simulate photoacoustic signals using FEM software and PAM. Quality factors and resonance frequencies were compared with experimental ones demonstrating a very good agreement. We also go further at using the FEM model with PA cells of different shapes [8] and also measurement conditions where the repartition of the laser intensity across PA cell in the FEM model must be taken into account [9], for these studies the method has proved to be robust as well. To explain the remaining differences between FEM and experimental results, some technical characteristics were impossible to precisely determine and where p~, T~ and u~ are respectively the pressure, temperature and velocity fields in the gas. p 0 ,T 0 and ρ 0 are the mean values of the pressure, temperature and density fields. l and μ are the bulk and shear viscosity. The bulk viscosity is set to 0.6 μ [18]. κ, C p and Q are respectively the thermal conductivity, the heat capacity at constant pressure and the volume heat source. I is the identity matrix. The previous PDE system is complemented by a set of boundary conditions. First, at the interface between the cell walls, considered as rigid, and the gas, a no-slip condition is used, imposing a zero gas velocity. Second, as the glass thermal conductivity is much larger than that of the gas, it is assumed that the walls are isothermal, leading to a zero temperature perturbation at the walls. implement in the model such as potential surface defects due to the manufacturing process that are not represented in the perfect 3D geometry of the FEM model, leading to a decrease in the response. Moreover, modeling of PA cells geometry and associated microphones leads to inevitable approximations in terms of shape. In the present study, we especially designed simplified cells permitting to obtain simple shapes and to adapt simple electret microphones (Knowles EK-23133 with a sensitivity of 22 mV/Pa @ 1000 Hz) thus limiting the effect of microphone volumes that are negligible in comparison with the cell volume. A photograph of a cell is given in Figure. 2. This cell is made of glass in order to enhance the surface quality roughness of the resonator walls and thereby limit losses. Two different cells were designed. The dimensions are given in

Test set-up
Characteristics of the photoacoustic cells are investigated using the setup presented in Figure 3. The laser source is a

Simulation of Helmholtz cells
For both models, the calculations mainly rely on the physical properties of the gas inside the cell and on the shape of the acoustical resonator. We assume that the cell will be used for the detection of a small amount of absorbing gas diluted in air and use the properties defined for air at T 0 = 293 K and P 0 = 101,325 Pa in the COMSOL material library. These parameters are presented in   Most of the dissipation (viscous dissipation and heat conduction) processes occur in a thin region located near the cell walls. The thickness of this region can be estimated to approximately 0.06 mm for a working frequency around 1 kHz, in air at standard temperature and pressure [6]. Except for the highest mesh quality, this region cannot be correctly described by the tetrahedral mesh (only one element in the boundary region).
In order to improve the description of dissipation processes an adapted boundary-layer mesh is used, that refines the mesh near the cell walls and typically adds 4 or 5 meshing elements in the 60 µm region. Total calculation time and memory requirement both increase with the mesh quality and with model complexity.
For the simple pressure acoustics model, all mesh qualities were solved whereas for viscothermal acoustic model the highest mesh qualities (2 and 1) computations were not carried out due to memory requirements (Out of memory during LU factorization on a 256 GB workstation).

Cell #1
Calculated resonance frequency, quality factor and cell constant for cell #1 are presented in Figure 4 as a function of the mesh quality and compared to experimental results. The PAM results are less sensitive to mesh quality. The resonance frequency is close to the experimental one (shifted from approximatively 10 Hz, that is a 2.5 % difference). The agreement is very good for quality factor (less than 1 % difference) except for the finest mesh (without explanation). The cell response is clearly more sensitive to mesh quality and for coarser meshes, there may be a factor 2 difference. The value is close to the experimental one only for high-quality meshes. On the contrary, the TAM results are highly dependent on this parameter. As expected, it appears that the TAM using standard tetrahedral mesh (TAM-T) was not adapted even if the TAM with tetrahedral mesh seems to converge towards experimental values when the mesh is refined. Results are closer to the experimental ones when using the boundary-layers mesh (TAM-BL) even if, in this case, the finest mesh qualities cannot be solved, due to memory limitations. The resonance frequency is shifted from 1-2 Hz, i.e. a very good agreement is found for this study. The quality factor agreement is good (10 % difference) and seems to improve when the mesh refines. When using the TAM-BL, both parameters show only a weak dependence on the bulk tetrahedral mesh quality confirming that losses mechanism are mainly located at the boundary layer. The simulated cell response remains close to the experimental one with the TAM. The difference is always lower than 25 % (for the coarsest mesh quality) and diminishes down to approximately 5 % when the mesh refines. The discrepancy between experiment and simulation is greater for cell response than for quality factor.
Q This higher discrepancy is probably due to insufficient mesh quality on the path of the laser beam leading to an incorrect value of the absorbed energy. In conclusion, the PAM with the finest mesh gives good results and the TAM with adapted mesh improves the validity of the model. The TAM-BL simulation results closest to the experimental data are compared with the experimental PA signal for microphones A and B in Figure 5. The agreement is excellent, thus confirming previous results [7] and the possibility to qualitatively simulate the PA signals of macroscopic PA cells, even in case of complex designs. Note that the PAM and TAM differences remain small and we choose to represent only simulations using the TAM-BL on this graph. This excellent agreement is confirmed in Figure 6, where the experimental cell response is compared to simulation results.

Cell #2
Calculated resonance frequency, quality factor and cell constant for cell #2 are presented in Figure 7 as a function of the mesh quality and compared to experimental results. The PAM results are almost insensitive to mesh quality even for the cell response. The resonance frequency is shifted from the experimental one from approx. 100 Hz. The agreement is correct  Localized defect: Due to the fusing process during cells manufacturing, the actual cells may present localized defects like the narrowing of capillaries near the walls (See Figure 2). The effect of such narrowing in one or both capillaries was numerically studied but did not allow to reproduce the observed dependences of signals A, B and |A-B| with frequency: partial obstruction leads to a decrease of R c associated to a shift of resonance frequency. No sensible effect on the asymmetry between A and B was observed. Moreover, as for the previous case, the underestimation of A near 1000 Hz was not correctly described.  Submit your manuscript at http://www.jscholaronline.org/submit-manuscript.php Submit your manuscript to a JScholar journal and benefit from: ¶ Convenient online submission ¶ Rigorous peer review ¶ Immediate publication on acceptance ¶ Open access: articles freely available online ¶ High visibility within the field ¶ Better discount for your subsequent articles differential Helmholtz resonators for photoacoustic trace gas detection", Sensors and Actuators, B: