# Impact of cavity shape on thermal accelerometer parameters

Valentin Toulme, Soukaina Ben Salk, Marc Faucher, Philippe Combette, Alain Giani

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Valentin Toulme, Soukaina Ben Salk, Marc Faucher, Philippe Combette, Alain Giani. Impact of cavity shape on thermal accelerometer parameters. Micro and Nano Engineering, 2025, 29, pp.100331. 10.1016/j.mne.2025.100331. hal-05438065

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# Impact of cavity shape on thermal accelerometer parameters

Toulmé Valentin<sup>a,*</sup>, Ben Salk Soukaina<sup>b</sup>, Faucher Marc<sup>b</sup>, Combette Philippe<sup>a</sup>, Giani Alain<sup>a</sup>

## ARTICLE INFO

**Keywords:**

Accelerometer
Thermal
Convection
MEMS
Hot wire

## ABSTRACT

This publication examines CFD (Computational Fluid Dynamics) analyses of a uniaxial accelerometer's behavior, focusing on convective heat transfer. A thermal convective accelerometer is a sensor that employs hot wire technology to quantify acceleration. The analytical modelling of sensitivity is inadequate for predicting sensitivity evolution as a function of acceleration in complex scenarios, particularly in temporal investigations; hence, CFD simulations have been employed. The sensitivity dependent on the sensor's form has been examined through simulations for accelerations exceeding 100,000 g. Furthermore, the bandwidth of the thermal accelerometer has been studied under various setups. The outcomes of these simulations will enhance sensor performance (cavity and wire dimensions, spacing between detectors and heater) to attain a measurement range of 100,000 g and a bandwidth of 10 kHz.

## 1. Introduction

For many years, various types of accelerometers have been developed to measure vibration, shock and inertial motion. The first accelerometers were developed at the end of the 20th century to measure acceleration and vibration for industrial and defense applications. The first principle was based on mechanical system such as springs and moving masse. In the 1970–1980, the first electromechanical accelerometers were invented. These accelerometers used electrical systems instead of mechanical systems to measure the acceleration. These sensors were used in defense, space vehicle or automotive industry, such as first air bag crash detection [1]. The most widely used principles of operation are based on piezoelectric or piezoresistive effects or capacitive changes [2]. Nevertheless, the existence of the solid proof mass in the accelerometers brings some disadvantages, such as low shock survival rating and a complex fabrication process.

The first thermal accelerometer was studied at the end of the 20th century. Fig. 1 shows the evolution of thermal accelerometer studies. The first one was uniaxial thermal accelerometer, studied by Leung and al. This accelerometer can measure the acceleration on one axis. [3]

Then different kinds of accelerometer were studied such as dual-axis and tri-axis sensors. These sensors are able to measure acceleration on two or three axis at the same time. Since the beginning of the 21st century, the performance of thermal accelerometer has improved to 10,000 g measurement range and 1KHz bandwidth. These

improvements are made on one axis sensor [4] [5] [6].

Thermal accelerometer is a good alternative to solve the moving parts problem on common accelerometers. This kind of sensors works on thermal convection with no moving parts, allowing them to withstand very high shock levels, up to tens to thousands of g.

This paper will show how the measurement range of acceleration and the bandwidth of the sensor depend on the size of the cavity and wires. In previous studies, it was found that sensors with a bandwidth of 100 Hz and a maximum range of 100 g could be made. Improvements in industry and transport systems mean that the new accelerometers need to be able to measure very high accelerations and shocks. These shocks of up to 50,000 g are difficult to test, which is why the CFD simulations are a good way to perform study on high-level acceleration sensors.

The present studies are about sensors with more than 100,000 g measurement range and 10KHz of bandwidth. Such high levels of acceleration can be applied in several environments, including automotive industry (e.g., crash testing and airbag deployment), shock measurement in oil drilling, aeronautic field or automation in monitoring machine condition.

## 2. Thermal accelerometer principle and analytical modelling

The thermal accelerometer, described in Fig. 2 (a), is composed of three wires positioned in the middle of a cavity containing a gas. This technology is based on natural free convection inside a closed cavity.

<footnote><sup>a</sup> IES, University of Montpellier, CNRS, France</footnote>

<footnote><sup>b</sup> IEMN, CNRS, Villeneuve d'Ascq, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: valentin.toulme@umontpellier.fr (T. Valentin).</footnote>
The heating resistor (central wire in Fig. 2 (a)-(b)) creates a thermal gradient in the gas by supplying an electric current. Thermal accelerometer is also composed of two smaller wires to measure the temperature at a particular points in the cavity.

When the heating resistor is powered (around 250 °C), it creates a symmetrical temperature profile inducing a thermal balance inside the cavity. Under the influence of an acceleration Γ, the temperature profile shifts and the balance is modified creating a temperature difference on the detectors equal to δT, proportional to the applied acceleration, giving an equivalent sensitivity S = δT/Γ (°C/g) (Fig. 2c).

This is due to the difference of the gas density between cold and hot gas. Under the influence of acceleration, cold gas moves to the opposite direction of acceleration, and hottest gas, created by the heater, moves in direction of acceleration.

This gas movement causes one of the detectors to heat up and the other to cool down. This temperature difference δT is correlated with the acceleration through the sensitivity S, according to the following relation [7]:

$$S [C \cdot g^{-1}] = \frac{\delta T}{\Gamma} \qquad (1)$$

In addition to the intensity, we can know the direction of the acceleration by detecting which of the two detectors warms up and which one cools down.

To study the sensor's sensitivity, we have used three models which we then compared. The first one is the most simplified model, it consists of two concentric cylinders. This model was first proposed by Mack and Bishop [8], it is composed of two cylinders inside each other (Fig. 3). The outer cylinder represents the substrate and the inner cylinder the heater. In this model, only hot wire is represented, and the detectors are not present. The temperature profile between the two cylinders can be modelled by Rayleigh number (Ra) and a function depending of geometrical parameters f(r,r₀,rᵢ) where rᵢ is the radius of the inner cylinder, r₀ the radius of outer cylinder and r the variation of radius between the two cylinders [9].

Therefore, the difference of temperature between the two detectors is given by:

$$\delta T = 2.Ra.f(r, r_i, r_0) \qquad (2)$$

With Rayleigh number:

$$Ra = \frac{\rho^2 \cdot \beta \cdot C_p \cdot \Gamma \cdot \Delta T \cdot L_0^3}{\mu \cdot \lambda} \qquad (3)$$

Using the previous eq. (1), the sensitivity of the accelerometer can be

calculated as follows:

$$S = \frac{\delta T}{\Gamma} = 2 \frac{\rho^2 \cdot \beta \cdot C_p}{\mu \cdot \lambda} \cdot (T_i - T_0) \cdot r_i^3 \cdot f(r, r_i, r_0) \qquad (4)$$

For more details and to validate the first model, two other models have been created. These two models have an hexagonal cavity as the real shape of sensor. The first hexagonal model has no detectors. For the second one, the detectors are modelized by rectangles as in their realistic representation (Fig. 4). For both hexagonal models, the heater is rectangular as the real 2D representation. These models will be used to validate the sensitivity eq. (4), and to compare the impact of the cavity shape and the presence of detectors on it.

The equation of sensitivity (4) has been made with some simplification such as an incompressible gas in steady state. Therefore, this equation can't be used to determine the sensitivity as a function of the acceleration. To perform a study base on acceleration, we need to use CFD simulations.

### 3. 2D modelling of thermal accelerometer for CFD simulations

Due to a few simplifications, the analytical equation of sensitivity (4) can't be used to determine the sensitivity as a function of acceleration. The CFD software Fluent® was used to solve the Navier Stokes equations and to simulate the displacement of thermal gradient in the cavity when an acceleration is applied [10] [11]. These simulations have been performed on the three models.

The density, momentum and energy equations were spatially discretized by means of a fully implicit third order finite volume method conceived from the Second Order Upwind. The pressure discretization was performed by Second Order equations. The gas Nitrogen have been used as fluid model, because this is the main component of ambient air and the different properties as a function of temperature are well known [12]. The variation of specific-heat and thermal conductivity according to the temperature using a polynomial profile where coefficients were found using NIST (National Institute of Standards and Technology) data. The viscosity is constant and the density is configured as incompressible ideal gas.

As regard solid materials, the wire are composed of Platinum (for the conductive part) and Silicon Nitride to isolate the conductive wire to semi-conductive substrate. The Platinum density is configured as a constant parameter fixed at 21460 kg/m³. The specific heat and thermal conductivity are entered as a piecewise linear function that depends on the temperature. The density and thermal conductivity of Silicon Nitride are also set as a fixed value with 3194 Kg/m³ and 32.9 W/m⁻¹ K⁻¹. And

<figure><img src="image_4.png" /><figcaption>Fig. 1. Chronology of the evolution of micromachined thermal accelerometer.</figcaption></figure>
the specific heat is a temperature dependent piecewise linear function. All of the solid parameters have been created using the Fluent database.

The cavity walls are considered isothermal and are fixed at an ambient temperature of 298 K. Due to the difference in mass between the gas and the substrate, and the low power injected into the heater, the sensor substrate remains unaffected by the thermal gradient.

For all hexagonal geometries, the width is equal to the height. This is an ideal geometry because when the cavity is etched, the depth can be more or less in-depth than the width. These studies are made for different size of cavity. The reference geometry, named “Ref Dim” has 1500 µm of cavity size (diameter for cylindric model and width and height for hexagonal models), and 100 µm for heater width. The other geometries are made by homothetical reducing of reference geometry.

To reduce computation time and data volume, the symmetrical axis was used to split the geometries. For both cylindric and hexagonal models, there is a symmetry on the acceleration axis (Fig. 5). This is due to the fact that the gravitational acceleration (–9.81 m.s⁻² on ⃗y axis) is neglected in front of high acceleration on ⃗x axis.

A mesh convergence study has been conducted for the three models. The evolution of the results as a function of the number of cells enabled us to determine the order of magnitude of required number of cells. The Fig. 6 show the evolution of sensitivity as function of cells number for both models cylindric and hexagonal. This study has been conducted for each model with reference dimension (Ref Dim). For the other dimension, the number of cells has been adapted according to time, data and convergence calculations.

For the cylindric model, the number of mech element is around 50,000 cells with an inflation from inner radius to outer radius. Both hexagonal models have around 200,000 cells, as for the first model, there is a large concentration of cells around heater and detectors, and the density of cells decrease with distance from the middle of cavity (Fig. 7).

The first simulations performed on the three models aimed to find the best position within the cavity for maximum sensitivity.

By plotting the temperature difference along the acceleration axis, It can be possible to determine the best distance from the middle where this difference is greatest (Fig. 8). This specific point is used to determine where the temperature should be measured and where the detectors should be placed because the sensitivity is maximum where the temperature difference is maximum.

The simulations were monitored with temperatures at specific points of maximum sensitivity for each iteration. The simulations were considered complete when both temperatures at the two points converged. For the model with detectors, this is the average temperature for all cells in detectors that have been monitored. The following Fig. 9 shows the temperature distribution with and without acceleration.

On the graph above (Fig. 8), we can see that there is a difference between cylinder model and hexagonal model for the “ref dim”. However, the maximum level of sensitivity is the same for both models. This graph also show that the position of maximum sensitivity is almost

<figure><img src="image_5.png" /><figcaption>Fig. 3. Simplified model of thermal accelerometer, with hot wire in red and cavity limit in blue.</figcaption></figure>

identical for all three models. However, the position of the hexagonal models differs slightly from that of the cylinder model by 25% and 30%.

## 4. Sensitivity simulation results

Fig. 11 shows the results of the 2D CFD simulations for the three models. Firstly, this graph shows the limit of the acceleration range as a function of cavity size. For the “ref Dim”, the linear range is until 1.10²g, after that, the sensitivity is no longer in the linear zone. To reach at least 1.10⁵g, the cavity.

Secondly, the graphs (Fig. 11 and Fig. 10) compare the results of the three models. As we can see, the main difference is between the cylinder and the hexagonal models. This is due to the difference in maximum sensitivity position. However, there is no significant difference between the two hexagonal models. We can conclude that it is mainly the shape of the cavity that makes the difference and the presence of the detection wires is almost transparent. As we have seen, the sensitivity is influenced by the dimension of the cavity. We can plot the sensitivity as function of the cavity diameter. In the graph (Fig. 10), the slope of the curve is +3, which corresponds to the cubic term of the sensitivity's eq. (4). [7] [13].

To complete the simulation results, some measured data has been added to the graphs. With this data, we can see that the simulations are valid because the sensitivity values are the same for the CFD simulations and experimental measurements.

## 5. Time response and cut-off frequency simulation

As seen before, the sensitivities simulations have been performed in steady state. All we need to calculate the sensitivity is the temperature difference between the two detectors when the thermal gradient is stable. To perform a study on the cut-off frequency, the simulations must be in transient mode. In this case, it's the gradient stabilisation time we are interested in.

First, the fluid cut-off frequency has been performed with the 2D hexagonal model without detectors. As for sensitivity simulations, the temperatures have been measured at the specific points of maximum sensitivity. Then the hexagonal model with detectors have been

<figure><img src="image_6.png" /><figcaption>Fig. 2. (a) Schematic thermal accelerometer. (b) Schematic temperature distribution created by the heating resistor, with and without acceleration. (c) 2D CFD result of temperature distribution displacement due to an acceleration.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 4. Schematic of hexagonal model without detectors (a), and with detectors (b).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. CAO of sensor cavity without detectors. (a) Full geometry, (b) 1/2 geometry with symmetry axis.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Mech convergence study for cylinder and hexagonal model.</figcaption></figure>

performed. With these simulations, we can know the impact of cavity shape on the cavity cut-off frequency. We can also see how the detectors affect the sensor's bandwidth. Before presenting the results, the transient simulation protocol should be outlined.

The frequency simulations are made in two steps. First, the simulations are made without acceleration to calculate the thermal gradient when the sensor is not subject to an acceleration. Then the calculation is continued with the acceleration until the temperature at the two detectors becomes constant (Fig. 12).

For these simulations, the acceleration used is in the linear zone of sensitivity. To obtain good results, the time step which has been chosen

is $1.10^{-7}$ s and for each time-step, around 100 iterations have been performed. This configuration is a good compromise between time simulation and results precisions.

The thermal response of sensor can be assimilate to electrical system, in this case, the cut-off frequency can be calculated as a first order system and the relation between cut-off frequency and time response depends on time constant $τ$ (5) as:

$$ F_c [Hz] = \frac{1}{2 \cdot \pi \cdot \tau [s]} \qquad (5) $$

We can demonstrate that the cavity's cut-off frequency depend on a
<figure><img src="image_10.png" /><figcaption>Fig. 7. Mesh inflation for concentric cylinder and hexagonal model. (a) inflation around inner cylinder. (b) inflation around detector in hexagonal model.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Sensitivity difference along the cavity radius for different geometries without detectors.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. CFD results of gradient distortion under acceleration.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 10. Results of CFD studies about sensitivity as function of acceleration for the three models and experimental data.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 11. Sensibility's evolution as function of cavity diameter for the three models and measured data (Δ: concentric cylinder, □: hexagonal without detectors, ●: hexagonal with detectors, Ι: experimental data).</figcaption></figure>

term “a” (thermal diffusivity of fluid) divided by the square of distance between heater and cavity “r₀²” [7].

$$F_c \propto \frac{1}{\tau} \propto \frac{a}{r_0^2} = \frac{\lambda}{\rho \cdot C_p} \frac{1}{r_0^2} \quad (6)$$

Fig. 13 represents the cut-off frequency as a function of the cavity size and, detectors size. The blue points represent the results for cut-off frequency of cavity only. The curve has a slope of -2, which correspond to the theory because the cut-off frequency is inversely proportional to the square of the cavity radius.

When the detectors are added to the time response simulations, the results show how the detector's size and the size of the cavity affect each other. For the detectors under 0.3 µm, it is possible to achieve a bandwidth of 10KHz if the cavity size is less than 100 µm. 0.3 µm of detectors size, this bandwidth can't be achieved regardless of the cavity.

## 6. Conclusion

This work has been incorporated in a main objective of improving

the thermal accelerometer to measure very high accelerations (up to 100,000 g) and 10KHz bandwidth. The sensitivity of sensors has been investigated with CFD code for three 2D models in steady-state mode. Then, this work has investigated the frequency response of sensors, with and without detectors. The results of steady state simulations have shown that there is no significant difference between the most simplified model cylinder concentric and the real model hexagonal. These simulations have also shown that there is no difference between hexagonal models without and with detectors because the time is not a factor in these simulations, and the final temperature of detectors is simply necessary. To simplify the steady-state simulations, this is the hexagonal model without detectors that it will be used for the next studies. In case of transient simulations, the hexagonal model with detectors must be used because the volume of the detectors and the materials have an effect on the temporal response of the sensor.

To measure an acceleration of up to 100,000 g, the simulations have shown that the cavity width must be less than 100 µm. Above this size, the sensitivity is too high and the acceleration range decreases.

The second part of this study was about the frequency response of the
<figure><img src="image_15.png" /><figcaption>Fig. 12. Temperature of two detectors as function of time for transient simulations.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13. Study of cut-off frequency as function of cavity size. ☐ cavity without detectors, ● Detector 0.3 µm, + Detector 1.5 µm.</figcaption></figure>

accelerometer. With studying cut-off frequency of cavity without detectors, the concordance of the theory has been shown. This study highlights the influence of the detectors on the frequency response of sensors. With detector size below 0,3 µm, and cavity width of 100 µm, it is possible to achieve a bandwidth of 10KHz.

The development of the thermal accelerometer is carried out in a clean room using specialized tools to create microstructures. The cavity shape and microwire with dimensions larger than 100 µm and 1 µm are fabricated by photolithography and the cavity depth is etching using a KOH solution. For smaller dimensions, electron-beam lithography is required, and the cavity is etched using a reactive gas method. This process is more challenging and time-consuming, this is why we need to explore different way to improve the time response of accelerometer, with the thickness for example.

In order to improve the results and the studies, we need to carry out the sensitivity and temporal response in 3D. In this case, it will be possible to study the effect of the cavity length on the performance of the sensor. The 3D sensitivity simulations will be performed with detectors,

but the frequency response simulations will be performed without detectors to study the effect of the cavity length on the fluid cut-off frequency. The same simulations are then performed with the detectors.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This research was conducted with the support of the IES Laboratory at the University of Montpellier. I would like to express my sincere gratitude to the CTM team - particularly M. Frédéric Pichot, M. Renaud Félix and M. Jean-Marie Peiris - for their invaluable technical expertise and guidance on my research. This study was funded by the national programme PEPR Electronic RESIST number ANR-22-PEEL-0002,
Equipex Nanofutur Project and Renatech +.

## Data availability

Data will be made available on request.

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