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Abstract
A fiber optical anemometer using dual Fabry-Perot sensors with sealed cavity is proposed for high-speed airflow measurement. The airflow velocity is measured based on principle of differential pressure, and temperature compensation is realized by reference F-P sensor to improve measurement accuracy and environmental adaptability. The location of dual F-P sensors in the airflow field and quadratic functional relation between differential pressure and airflow velocity are obtained by the simulation of turbulence model. F-P sensors in this experiment can be employed to measure pressure from 100kPa to 107kPa and temperature from 5°C to 50°C. The full-scale error of F-P sensors is less than 0.53% by calibration. It is demonstrated experimentally this fiber optical anemometer is qualified for measuring air velocity in the range of 7.9-81m/s with velocity error less than 0.69%. The device has the potential to measure high speed airflow in various applications.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Flow measurement is widely applied in fields of chemical industry, biochemistry, atmospheric environmental monitoring, etc. Fiber optic sensors based on various principles have been developed for flow measurement [1–3] due to their unique advantages such as small size, high accuracy and immunity to electromagnetic interference. “Hot-wire” anemometry combined with fiber optic sensor is a common method for sensing airflow velocity [4,5]. High power dissipation and complex system are the main drawbacks of this method. Fiber Bragg gratings (FBG) [6,7] can be fixed in the pipeline to measure the flow rate by detecting the strain of the sensor caused by the fluid. However, it is not suitable for the measurement in open-space. Besides, the cross-sensitivity of FBG is also a problem must be overcome.
The fiber Fabry-Perot (F-P) sensor is highly developed with the continual improvement process of Micro-electro-mechanical system (MEMS) and vacuum coating technology [8,9] in recent years. It has been employed to measure pressure, temperature, acoustic vibration and refractive index in many industrial and scientific research fields, including biochemical sensing [10], biomedical field [11], safety monitoring [12] and liquid level measurement [13]. Some researchers also focus on fluid sensing to expand the application fields of F-P sensors [14,15]. M. Gander et al. [16] embedded F-P pressure sensors in the trailing edge of the nozzle guide vane to measure the unsteady pressure. C. Lee [17] presented an air-gap fiber F-P interferometer to measure distance change between end faces of two optical fibers induced by minute airflow velocity. A. Cipullo et al. [18] employed the fiber optic ferrule-top cantilever to measure laminar air flow velocity range of 0-8m/s in the wind tunnel. However, they did not consider the impact of temperature which cannot be ignored in high speed airflow field.
Some researchers measure pressure and temperature simultaneously using F-P sensors of different types, such as hybrid sensor fabricated with intrinsic and extrinsic F-P interferometers [19,20], dual-cavity F-P interferometer sliced with hollow-core fibers [21]. In this paper, we present a fiber optical anemometer based on dual F-P sensors with sealed cavity, which can achieve temperature self-compensation in the high-speed airflow field. F-P sensor placed at the front surface of the holder is employed to measure stagnation pressure, while the sensor placed on the upper surface can realize temperature compensation. The shape of sensor holder and the position of dual F-P sensors in the airflow field are determined and optimized according to the simulation of turbulence model. This device can measure airflow velocity in the range of 7.9-81m/s with measuring error less than 0.69%, which is demonstrated in experiment.
2. Principle and simulation
The schematic diagram of anemometer composed of dual F-P sensors is shown as Fig. 1. Sensor A is placed at the front surface, which is perpendicular to the direction of the airstream. Sensor B is installed at the side wall of the sensor holder. Sensitive diaphragms of sensor A and B are located on the surface of the sensor holder. The stagnation point with the flow rate 0 is generated on the surface of sensor A when the air flow is blocked. Stagnation pressure Ps is induced on sensor A because kinetic energy is converted into pressure potential energy at the stagnation point. Moreover, sensor B’s diaphragm is located at the side wall to measure the reference pressure Pr. Ps, Pr and air velocity u obey the ideal fluid Bernoulli equation (Eq. (1)) when the viscosity and compressibility is ignored.
where ρ is the density of air, and g is the gravitational constant. The airflow velocity can be calculated with Ps and Pr according to Eq. (2).Actually, the viscosity of air cannot be ignored if the airflow velocity is high. There is a boundary layer clings to the surface of the sensor holder due to the friction between airflow and surface. The velocity of the airflow is reduced to almost 0 in this layer. The kinetic energy of the fluid is converted into internal energy, which increases the surface temperature of sensor holder. The temperature of sensor A and B will also rise in the fluid field and reduce the sensing precision. In order to compensate temperature variation, two problems need to be solved: the locations and the response characteristics of dual F-P sensors.
Sensor B is placed on the position where the pressure is equal to that of the ambient atmosphere. On this position, the elastic deformation of diaphragm on sensor B is only induced by the variation of internal energy, which reflects the temperature of the airflow. The simulation of the airflow field is carried out to design F-P sensor holder and find the optimum location for F-P sensors. The Finite Volume Method (FVM) of Computational Fluid Dynamics (CFD) is suitable to realize the simulation. We employ the standard k-omega turbulence model to achieve numerical analysis of pressure distribution and streamlines. The k-omega turbulent model, which takes low Reynolds number, compressibility and shear flow into considerations, is suitable for the simulation of airflow around the sensor holder.
Figure 2 shows streamlines around the sensor holder when the velocity of airflow is 60m/s. It can be seen the airflow stagnates at the front surface, and then flows around the side wall. In the simulation, time step size is 0.2 second, and the number of time steps is 200. The fluid is air and the material of sensor holder is aluminum. In order to compare the effect of different shapes on the high-speed airflow field, the upper surface of the sensor holder is streamlined while the lower surface is rectangular. It is worth noting that the airstream flows smoothly over the upper surface of the sensor holder but breaks away from the lower surface and form a vortex. In the near-wall region, the closer to the surface, the lower the airflow velocity will be until it reaches 0 in the boundary layer due to dynamic viscosity of air. The surface temperature will increase because the kinetic energy is converted to internal energy. In order to sufficiently contact with steady airflow, the F-P sensor B for temperature sensing and compensation should be located at upper surface.
Figure 3 shows the pressure distribution around the sensor holder when the airflow velocity is 60m/s. It also denotes that the forebody shape has a significant impact on the surrounding flow field. There is a much larger negative pressure area at the bottom of the sensor holder than the upper surface due to the non-streamlined boundary. D is the side length of the sensor holder, and the upper surface of the holder is streamlined string of ellipse, so the vortex formed by the boundary is weakened. It is reasonable to mount pressure sensor A at the front surface to measure stagnation pressure, and sensor B at the upper surface for avoiding the swirling area. The optimal location of sensor B is 3.5D from front surface based on the simulation.
It should be noted that, the optimal location of 3.5D is deduced with the existing sensor holder. Considering that the distance between sensor A and B determines the accuracy of temperature compensation, it can be further improved by optimizing the shape and size of the sensor holder. For instance, the location of sensor B is related to the side length of the sensor holder (D), and D is related to the size of F-P sensor diaphragm, therefore, the distance between dual sensors can be shortened by reducing the size of diaphragm. Moreover, the distance can also be reduced by optimizing the shape of sensor holder based on the impact of forebody shape to the pressure distribution as shown in Fig. 3. Therefore, with a detailed design for the arc function of upper surface, such as increasing the semi-major axis of the elliptic curve, the effect of the vortex can be further reduced and thus sensor B can be located closer to the front surface.
Figure 4 shows correlation curves between airflow velocity and Ps, Pr, differential pressure (DP) through simulation. Among them, Ps is stagnation pressure measured by sensor A, Pr is reference pressure measured by sensor B which is located 3.5D from the front surface, and DP is differential pressure between Ps and Pr. The velocity range of air flow is from 10m/s to 80m/s, and the velocity step is 10m/s. With the increasing of airflow velocity, Ps increases rapidly but the change of Pr is almost negligible. Thus, the change of F-P sensor B’s signal is mainly caused by temperature rather than pressure, and sensor B can be employed to achieve temperature compensation. The simulation results show that airflow velocity and DP satisfy a quadratic function, which is in accord with Eq. (2).
MEMS based F-P sensor is suitable for measuring pressure and temperature in the airflow field because its diaphragm is made of silicon wafer, which is sensitive to external pressure, and the gas inside the sealed F-P cavity is sensitive to temperature. Anodic bonding is employed to realize vacuum-bonding of silicon wafer and Pyrex substratum (Fig. 5) [22]. The center of Pyrex substrate is corroded to fabricate a cylindrical micro cavity. The low-finesse F-P interferometer consists of the internal surface of silicon wafer and reflectance coating at the bottom of the cavity on Pyrex substrate. Both incident and reflected light are transmitted by multimode optical fiber (MMF).
In the process of vacuum-bonding, to enable the sensor to measure temperature, vacuum degree is controlled at 10−3 mbar, and minute amount of gas is sealed in the micro cavity, which obeys the well-known ideal gas law
where PI is the internal pressure, V is the volume of the sealed microcavity, n is the amount of substance, Rg is the gas constant and T is the thermodynamic temperature. As the gas is sealed in the cavity, nRg is a constant. The volume of the cavity V can be expressed bywhere h is the initial length of F-P cavity, def (r) is the deformation quantity, R is the radius of F-P microcavity and r is the distance from the central point.The length of the F-P cavity will be changed because of silicon wafers bending along with the change of external pressure PE and internal pressure PI. The deformation quantity of distance r from the central point can be expressed as deformation of circular plate with clamped edges [23]
where def (0) is the deformation quantity of central point, t is the thickness of the silicon wafer, E is the Young modulus and v is the Poisson's ratio.Combining Eqs. (4) and (5), the volume can be expressed as
Combining Eqs. (3) and (7), Eqs. (8) and (9) are obtained as below
Equation (9) denotes that PI can be expressed with PE and T. Therefore, combining Eqs. (6) and (9), the central point deformation of the silicon wafer def (0) is the binary function of PE and T. For a single sensor, if def (0) and PE is known, T can be solved; similarly, if def (0) and T is known, PE can be solved. Thus, the F-P sensor is suitable for measure pressure or temperature if one of them is known.
3. Experiment and discussion
In order to verify the analysis in the second part, experiment is carried out to investigate temperature and pressure characteristics of F-P sensors. Figure 6 shows the schematic diagram of the system which is based on low-coherence interferometry. The white light from broadband LED sources propagates in the MMF, and then enters into F-P sensor via 3dB couplers. The reflected lights from F-P sensor passes through 3dB couplers and then guided into the polarized low-coherence interferometry demodulation section. The light passes through the spatial light path composed of convergent lens, polarizer, optical wedge, analyzer and linear CCD in sequence. The birefringent wedge can generate continuous optical path difference (OPD) induced by ordinary and extraordinary light. When the OPD generated by birefringent wedge is the same as the OPD caused by F-P sensor, the low-coherence interference fringes appear on the linear CCD. The signal of linear CCD is transferred to the data acquisition (DAQ) and then processed with PC. The absolute phase demodulation algorithm [24] is employed to transform fringe spatial position signal into absolute phase (AP), which is positively related to the F-P cavity length.
The F-P sensors in this paper are of the same size. The Young’s modulus E and Poisson’s ratio v are 130Gpa and 0.2782, respectively. The thickness t of the silicon wafer is 30μm, the side length of the square wafer is 5mm, and the radius R of the airtight F-P cavity is 1.2mm. F-P sensors are put inside the pressure vessel, and the vessel is then put inside the temperature test chamber. During the calibration process, the pressure range is from 100kPa to 107kPa, the pressure interval is 0.2kPa, and the pressure stabilization time is 5 minutes. The temperature range is from 5°C to 50°C, the temperature interval is 2.5°C, and the temperature stabilization time is 120 minutes.
Relations of PE, T and AP can be obtained by polynomial fitting as below
where i and j are the orders of T and AP respectively, m and n are the max values of i and j respectively, and pij & qij are the multinomial coefficients which can be solved by polynomial fitting. Figure 7 shows the fitting surfaces of PE and AP at different ambient temperatures. For every sensor, relation curves between PE and AP at different temperatures almost have the same slope rate. Because the diaphragms of sensor A and B are of the same type, the slopes of the PE-AP curve of these two sensors are almost the same. The absolute PE error of sensor A and B can be calculated. It is 0.034kPa for sensor A and 0.037kPa for sensor B in the range of 100kPa to 107kPa, so the corresponding full range accuracy is 0.48% and 0.53% respectively.
Fig. 7 Fitting surfaces of pressure, temperature and absolute phase. (a) Fitting surface of sensor A employed to calculate pressure with temperature and absolute phase. (b) Fitting surface of sensor B employed to calculate temperature with pressure and absolute phase.
Based on the analysis of temperature and pressure characteristics of F-P sensors, we design a F-P sensor holder, which is made of aluminium alloy. This holder is manufactured by 3-D printing, which can be found in Fig. 8. The shape of the sensor holder is cubic, and the cross section of it is a square with side length of 10 mm. The upper surface is designed to be streamlined according to the simulation results shown in Fig. 3. F-P sensors are glued to the sensor holder with heat-conducting silicone grease (X-23-7868-2D, ShinEtsu) to make the sensor temperature consistent with the holder. The silicon diaphragm is sitting flush with the structure surface. Sensor A is fixed at the front surface of the holder to measure the stagnation pressure, and sensor B is fixed at the upper surface to achieve temperature compensation. In order to verify the conclusion of simulation, the central point of sensor B and b is 35 and 25mm from the front surface, respectively. The airflow is generated by air blower (HG-1500, YASHIBA Motor), and the air speed regulation is realized by variable-frequency drive (VFD). The velocity of airflow is measured with pitot tube anemometer (FLUKE 922) as the reference value.
Figure 9 shows the variation of AP for all sensors in the fluid field of airflow velocity 25m/s and 60m/s. At the beginning of the experiment, AP signal of F-P sensors are stable. As sensor A and B are in the same experimental environment, ambient pressure and temperature can be calculated with Eq. (10), which are 102.556KPa and 25.3°C. This result agrees with the actual situation. The dramatical decreasing of sensor A's AP denotes airflow start to contact with the diaphragm. At this time, the length of F-P cavity is shortened by the increasing of air pressure. The decline in AP value of sensor A is positively correlated with airflow velocity. From 20 second to 70 second, the gradual increasing of sensor A and B’s AP shows the kinetic energy of airflow is converted to internal energy which rises the temperature of sensors. After 50 seconds, the AP signal becomes stable. It can be seen that the stagnation pressure of sensor A fluctuates more severely than reference pressure of sensor B. This phenomenon is because the silicon diaphragm of sensor A suffers a stronger impact from pulsating flow, which is more obvious when the airflow velocity is 60m/s. When the airflow velocity is 25m/s, AP change of sensor B is introduced by temperature variation, so the temperature of flow field can be calculated as 27.6°C, and Ps of sensor A can be calculated as 102.874kPa. Thus, DP between Ps and Pr is 0.319kPa. When the airflow velocity is 60m/s, the temperature of airflow is 33.0°C and DP is 1.910kPa. Temperature and DP increase with the increase of airflow velocity.
The enlarged view in the insert panel of Fig. 9 shows the signal difference of sensor B and b. It can be seen AP of sensor b has slight decrease the moment airstream acting on the diaphragm, which indicates the pressure of sensor b has a tiny increasement, because sensor b is positioned closer to the front surface than sensor B, and it is in the sphere of vortex. As a result, the pressure signal of sensor b is affected by vortex and this position is not suitable for temperature compensation. In contrast, the AP change of sensor B is continuous and with no jump. On the premise of avoiding the vortex area, the F-P sensor used for temperature compensation should be located as close as possible to the front surface to measure the temperature more accurately. Thus, 35mm away from the front surface is the most suitable location for temperature compensation.
It is not suitable for measuring airflow velocity with single F-P sensor, for example, sensor A. The initial phase of sensor A is affected by ambient pressure and temperature, so the environmental parameters cannot be calculated with only one F-P sensor. AP change of sensor A is positively correlated with airflow velocity as shown in Fig. 9, but this transformation is transient and unstable, which is not reliable for measuring velocity for long time.
Figure 9 shows that the AP difference of sensor A and B is positively related to the airflow velocity. And also, we can calculate DP between sensor A and B with Eqs. (10) and (11). We compare these two methods by fitting curves. The fitting curve of airflow velocity and DP between sensor A and B is shown in Fig. 10(a), while the fitting curve of airflow velocity and AP difference between sensor A and B is shown as Fig. 10(b). The range of airflow velocity is from 7.9m/s to 81m/s. For the fitting curve of DP, the coefficient of determination is 0.998, which is higher than that of AP difference. Besides, the root-mean-square error (RMSE) of fitting curve in Fig. 10(a) is 0.047 which is much less than 0.137 of fitting curve in Fig. 10(b). These results indicate that the curve fitting of DP has higher accuracy. The reason is that the ambient temperature and pressure vary inevitably in the process of experiments. As a result, measuring airflow velocity according to the AP difference of sensor A and B directly will induce larger error, which can be reduced by the calculation of temperature compensation with Eqs. (10) and (11). Fitting curve denotes that DP is a quadratic function of air speed in the range of 7.9-81m/s, which is in accord with the simulation result. Theoretically, the device can be employed to measure higher airflow velocity, but 81 m/s is the upper limit of the referential pitot tube anemometer.
Fig. 10 Fitting curves of air velocity and (a) DP between sensor A and B, (b) AP difference between sensor A and B.
The velocity error curves of different airflow velocity are shown in Fig. 11. The velocity error is between −0.36m/s and + 0.35m/s on condition that airflow velocity is 81.0m/s, between −0.25m/s and 0.29m/s for airflow velocity of 47.5m/s and between −0.56m/s and 0.49m/s for airflow velocity of 7.9m/s, so the measuring error is less than 0.69% full scale (F.S.). The device shows good repeatability and accuracy to measure high speed airflow up to 81m/s despite fluctuations of ambient temperature.
4. Conclusion
In conclusion, we present a sensing device using dual F-P sensors to measure velocity of airflow. The airflow velocity is measured by the differential pressure between two sensors. The sealed cavity structure allows the device to achieve temperature self-compensation. The sensor holder is manufactured with 3-D printing technology. Through calibration experiments, the F-P sensors can be employed in the pressure range 100kPa to 107kPa and temperature range 5°C to 50°C with the F.S. error less than 0.53%. The fiber optical anemometer is proven to be useful for measuring airflow velocity in the range of 7.9-81m/s with the measurement error less than 0.69% by airflow experiment. This method is expected to play an important role in the field of meteorology, hydraulics and aerodynamics in future. Our further research will focus on optimizing the shape and size of sensor holder with refined simulation and experiment, thereby further improving the measurement accuracy of temperature compensation and airflow velocity.
Funding
National key research and development plan of China (Grant 2016YFC0401902), National Natural Science Foundation of China (Grant U1833104, 61735011, 61675152, 61405139).
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