1. Introduction
Sensor technology and development has been an important breakthrough in industrial science and engineering. Over the years, research has been conducted for the design and manufacturing of efficient, handy, and cost-effective sensors. The development of micro and nanotechnology has further enabled the miniaturization of these sensors while maintaining performance. Sensor applications have increased a lot over the years and multiple sensors are incorporated into engineering devices. The manufacturing and repairing cost of such a large number of sensor-installment is expensive and time-consuming. For that reason, a new idea has been proposed here, which would allow the measurement of multiple physical quantities by employing a smaller number of sensors. This may provide a new cost-effective and time-saving approach.
Accelerometers have diverse applications in the automotive, consumer electronics, and biomedical industries. Conventional accelerometers convert accelerations into electrical signals using several mechanisms, including piezoresistivity, piezoelectricity, and capacitive type. However, the use of a solid-proof mass in such devices imposes mechanical limitations on the amount of shock and the sensing range they can undergo. In contrast, the convective nature of a thermal accelerometer with no moving parts can increase the sensing range of such a device.
The concept of a thermal accelerometer is based on free-convection heat transfer. A heating source generates a consistent temperature profile that is altered by applied acceleration; thus, this difference in temperature Δ
T is related to the change in acceleration, as shown in
Figure 1. In this figure, the thermal sensor consists of a single heater source that heats the surrounding gas, creating a symmetrical temperature profile (solid line in (b)). When no acceleration is applied, the equally spaced sensors placed on the sides of the heater detect the same temperature. However, when acceleration is applied, the difference in temperature Δ
T is modified, creating an asymmetrical temperature profile around the sensor (dashed line in (b)).
Figure 2 depicts a clearer image of the concept in terms of the temperature contours and a side view of a cylindrical cavity with two heating sources. The red part denotes the highest temperature at the heating sources, and that around the boundaries (dark blue) is at room temperature (300 K). When no acceleration is applied, the thermal bubble is consistent around the heating sources. However, when acceleration is applied, the thermal bubble shifts in the direction of the applied acceleration. In this way, the pair of temperature sensors, placed equidistant from the heater, as shown in
Figure 1, detect a difference in temperature, which is correlated with the change in acceleration.
One problem that has been pointed out recently is associated with the incorporation of multiple sensors into engineering devices. For micro- and insect-scaled unmanned aerial vehicles (UAVs), installing multiple sensors to measure each physical quantity not only imposes higher manufacturing and repair costs but can also be time-consuming. Therefore, to solve this problem, in our study, we propose the novel idea that in addition to acceleration, Δ
T can also be correlated with other physical quantities, such as rotational speed, amplitude, and frequency of vibration. This technique does not involve creating a completely new device, but rather modifying existing motion sensors, such as thermal accelerometers. This is accomplished by considering the cross-axis sensitivity, which is the sensitivity observed in the plane perpendicular to the measuring direction relative to the measuring direction. Ogami [
1] suggested that cross-axis sensitivity should not be removed but rather exploited. In this way, if multiple motion types are applied on a single axis, with sensitivities observed in other axes, the input physical quantities will have a relationship with the output sensitivities. In this study, a thermal motion sensor is considered; using cross-axis sensitivity, we can measure the acceleration and rotational speed simultaneously.
Because our thermal motion sensor involves physics similar to that of a regular thermal accelerometer, the performance parameters are also the same. The sensitivity and frequency bandwidth of any sensor device play important roles in defining its performance. Sensitivity is the quotient of the change in an indication of a measuring system and the corresponding change in a value of a quantity being measured. Frequency bandwidth is a measure of how quickly the sensor can respond to changes in input physical parameters. Over the years, various analytical studies and computational simulations have been conducted to predict the sensitivity and frequency bandwidth of microelectromechanical systems (MEMS) thermal accelerometers. The thermal accelerometer was first reported by Albert [
2] in 1997. Goustouridis [
3] developed a conductive thermal accelerometer comprising a polysilicon heater and two thermopiles. This device uses electrical energy as a parameter related to thermal energy. However, the literature has mentioned that obtaining the temperature profile using temperature sensors and relating it to the input thermal energy generates better results. Brahim [
4] developed a 3D model for finite element method (FEM) simulations using a derived analytical model to study the conductive behavior of MEMS thermal accelerometers.
Researchers have developed theoretical, computational, and experimental models for improving the performance of thermal accelerometers. It has been observed that a high heating power and a large device size lead to an increase in sensitivity [
5]. However, an increase in the pressure of the air medium results in a decrease in the frequency bandwidth. Additionally, gas media with high densities and low viscosities appear to result in better sensitivity [
6]. Leung [
2,
7] demonstrated that the sensitivity of a thermal accelerometer is linearly proportional to the Grashof number (Gr):
where g, ρ, β, L, ∆
T, µ, and α are the applied acceleration, gas density, coefficient of volumetric expansion, characteristic size (generally denotes the cavity size; see
Figure 3), temperature difference between the heater and the boundary of the sensor, kinematic viscosity, and thermal diffusivity, respectively. These parameters can be used to predict the device performance. From Equations (1) and (2), it can be observed that the sensitivity of the device can be significantly increased by using a high-density and low-viscosity fluid. The properties of some fluids and their calculated Gr and Prandtl (Pr) numbers are listed in
Table 1 and
Table 2, respectively. These parameters can be modified and optimized to increase the sensitivity of a thermal accelerometer.
As shown in
Table 1 and
Table 2, CO
2 has the highest density and lowest kinematic viscosity; consequently, it has the highest Grashof number. In comparison, air has a lower density-to-kinematic viscosity ratio and hence a lower
Gr value compared with CO
2. High viscosity yields higher resistance to gas flow and, in return, lower sensitivity.
The size and power of our target model were determined as follows: in the literature, different scales of UAVs have been studied to achieve lighter weight and lower lifting and sensing power. Kevin [
9] built an 80 mg, insect-scale, flapping-wind robot with a power consumption of 19 mW. Another study specifies the lifting and sensing power to be 100 mW for an insect-scale UAV with a mass of 100 mg [
10]. Therefore, keeping this in view, our target model consists of a device with dimensions of 1 cm (height) and 2 cm (diameter) with a heating power of 70 mW for application in small-scale UAVs and robots.
The design of the computational study is illustrated in
Figure 3. Four sets of heaters were considered, and two sets of sensors were placed equidistant from the heaters in
x and
y directions (therefore, four sets of sensors for each heater). The changes in the input physical quantities correlate with the altered temperature distribution. Because the fluid under study is governed by the conservation of mass, momentum, and energy, the commercially available software ANSYS FLUENT 18.2, which is a reliable and accurate fluid simulation software, was used for the analysis. After the temperature responses corresponding to the input physical quantities were obtained by computational simulation, the next step was to find an inverse function, and the results were plotted on a 3D surface using MATLAB, a programming, and numerical computing platform. This inverse function is installed in the computing unit of a real thermal motion sensor so that the sensor can calculate the values of the input physical quantities from the measured temperature values.
2. Materials and Methods
A computational simulation was performed to observe natural convection, and changes in the temperature profile as acceleration and rotational speed were simultaneously applied to the computational model. The governing equations for predicting the temperature profile of our thermal motion sensor are based on the principle of conservation of mass, momentum, and energy, which are as follows:
where u is the flow velocity vector field,
is the spatial divergence operator, p is the pressure, I is the total stress tensor, and f denotes the body forces acting on the fluid. The parameters
Cp,
, and k are the specific heat, density, and thermal conductivity, respectively, of the fluid in the cavity.
As shown in
Figure 3, for our computational study, four temperature sensors (black dots) adjacent to the four edges of rectangular heaters (red rectangle) were considered.
The computational fluid dynamics (CFD) package FLUENT was employed for the analysis using the finite-difference method to discretize the governing Equations (3)–(5). A parametric study can be conducted because the software accommodates changing flow (initial and boundary) and geometrical conditions. As shown in
Figure 2, the thermal bubble around the heater changes with changing acceleration. In this study, we observed that with the application of rotation, the thermal bubble around the heater also changed in the direction of rotatory motion.
An optimal grid design is required to obtain accurate results with a reduced computational time. To achieve this, we employed the grid resolution method proposed by Minhyung [
11] and created a computational grid, as shown in
Figure 4, using the meshing software ICEM in ANSYS 18.2. The number of elements and nodes of this structure are 310,947 and 1,283,732, respectively.
As seen in
Figure 4, a simple cylindrical geometry for our device has been created with a height of 1 cm height and a diameter of 2 cm, as explained above. Heaters and temperature sensors are incorporated into this design using user-defined functions (UDFs). In the UDFs, the locations of heaters and sensors are defined and tracked using their cell IDs, which are unique for every cell, even when the geometry is moving. Instead of tracing the coordinates of heating sources, the cell IDs of cells containing the heating sources are traced to determine the centroid location of these coordinates. In addition, using the ‘DEFINE_SOURCE’ UDF, heat is applied to the cells where the heat sources exist. Furthermore, temperature values are extracted by looping over the entire cells and locating the cell IDs of sensors.
In FLUENT, a pressure-based transient solver was used with the energy model because the flow was not highly compressible with a lower Mach number. As explained above, a heating power of 70 mW was applied in response to the device’s intended applicability to small-scale UAVs that require low sensing and lifting powers. The DEFINE_CG_MOTION UDF was used to define linear and rotational motion alterations.
For the material of the computational domain surrounding the cavity, polyvinylidene fluoride (PVDF) and polyimide were chosen to be best suited because of their good thermal heat resistance with high values of specific heat capacity (
Cp) and low values of thermal conductivity (k), as listed in
Table 3.
4. Discussion
In this study, we present a novel technique that can be implemented to measure multiple physical quantities simultaneously using a thermal motion sensor. We successfully implemented and demonstrated the simultaneous determination of acceleration and rotation.
The general method is described as follows. The idea is to define a relationship between multiple physical quantities that we are interested in and multiple outputs by computational simulations, as generalized below:
for each value of a physical quantity, there is a corresponding set of outputs (ΔT
max values in this case). However, the number of outputs should be the same as the number of physical input quantities. Therefore, the cross-axis sensitivity is useful. Once this relation has been obtained, the next step is to determine the inverse function of this relationship as:
Using this relation, we can easily obtain a graphical inverse function for a specific range of physical quantities. Within this range, any value can be extracted based on the output. In this study, the maximum temperature values ΔTmax around the X and Y sensors were obtained as two outputs corresponding to acceleration in the x-direction and rotation around the z-axis. The data of the inverse function can then be installed in the computing unit of a real thermal motion sensor so that the sensor can calculate the acceleration and rotation speed from the measured maximum temperature values ΔTmax around the X and Y sensors.
Increasing the aerial system size with similar heater power of the thermal motion sensor should generate similar graph trends and performance parameters like sensitivity. As described in
Section 1, sensitivity is related to the Grashof number, and in fluid mechanics, such groups are considered dimensionless groups such as Reynold’s number and Prandtl number. Therefore, temperature values and hence the inverse function need not be modified even if the size of the aerial system is increased or decreased. It is dependent on the applied physical quantities. The range of acceleration and rotation considered in the paper, however, is limited and needs to be further extended. If the range of the physical quantities is further increased, the modified inverse function can be added and programmed into the computing unit of the thermal motion sensor.
While using the concept of cross-axis sensitivity for a 3-axis thermal accelerometer, it has been mentioned by Nguyen [
12] and Ogami [
1] that “the same response can be observed for two accelerations with different magnitudes and opposite signs”, which means that two combinations of temperature outputs can determine a single acceleration. In our study, we observed acceleration in the
x-direction and rotation around the
z-direction, but even by considering X- and Y-sensor responses, because of cross-axis sensitivity, results were observed to have a unique combination of temperature outputs corresponding to accelerations of 1
g–4
g and rotational speeds of 6.28–15.71 rad/s. We define this characteristic of the results as the
uniqueness of the solution. For distances of 0.1254 cm and 0.1433 cm between the heater and sensors, we observed areas of multiple solutions for changing rotational speeds. Therefore, for a defined range of input physical quantities, it is necessary to verify the results by simulations and find parameters that generate unique solutions.
In
Section 3, we observed that a shorter distance between the heater and sensors results in better sensitivity and resolution. Six different sensor positions were considered, among which the shortest distance, 0.0179 cm, gave the most favorable results, while the largest one, 0.1433 cm, gave the least favorable results. This means that reducing this distance to the least possible value would generate better device performance. However, this distance is measured from the center of the heater to the center of the sensors; therefore, considering the practical aspects, there is a limit to how much this can be decreased. Therefore, a compromise is needed to choose the ideal distance between the heater and sensors.
In addition to sensitivity and resolution, another important factor that determines the quality of results is the frequency response. This is a measure of how quickly a device can respond to changes in acceleration and rotation speeds. To obtain a fast and wide frequency response, it is necessary to change the thermal physical properties of the gas medium. A large thermal conductivity (α) and small density (ρ) will accelerate thermal diffusion, which consequently facilitates heat balance in the cavity and provides a fast frequency response to the device. Multiple studies have been conducted on the frequency response of thermal accelerometers. Garraud [
13] investigated the frequency response using analytical and CFD models. This could also be accomplished for a thermal motion sensor that measures multiple physical quantities, which will be our future work.
For our thermal motion sensor material, we used PVDF, which is known for its piezoelectric behavior. This implies that it can generate an electric charge in response to applied mechanical stress. Using this to our advantage, we may also define a new variable concerning pressure changes around the body. Using this application of the material, we can simultaneously measure an additional physical quantity, such as applied mechanical stress. However, PVDF has a high coefficient of thermal expansion (α), which limits its usage at higher temperatures. The properties and responsiveness of PVDF with respect to incorporation in a thermal motion sensor for pressure measurement will be investigated in our future work.
5. Conclusions
In this study, we presented a new method for simultaneously measuring multiple physical quantities using computational fluid dynamics. For any acceleration between 9.81 m/s2 and 39.24 m/s2 (1g–4g) and rotational speed of 6.28 rad/s–15.71 rad/s, this device can measure both quantities at the same time. This range can be expanded by additional computational simulations. The inverse function can be installed in the computing unit of a real thermal motion sensor so that the sensor can calculate the acceleration and rotation speed from the measured temperature values.
We also studied the effects of various parameters on the performance of the device, and it was observed that by reducing the distance between the heater and sensor positions, the sensitivity and resolution may be improved. In addition, using a gas medium with a high density and low viscosity may improve the sensitivity and resolution of the device.
In the future, using the same technique, other physical quantities, such as vibrations, can also be analyzed. Here, we studied a thermal motion sensor, and by modifying it and using cross-axis sensitivity, we were able to measure multiple physical quantities simultaneously. However, we believe that the idea presented here can also be applied to other sensors, such as stress sensors or vibration sensors. In addition to the performance parameters mentioned in this study, i.e., sensitivity, resolution, and uniqueness of the solution, a frequency response analysis should also be performed to verify the quality of the results. The computational model stated here should also be validated experimentally.