Flow measurement is one of the components of metrology science and technology, closely related to national economy, defense construction, and scientific research^[1–4]. The flowmeter plays an important role in ensuring product quality, improving production efficiency, and promoting technological development. In the era of the energy crisis and increasing automation of industrial production, the position of flowmeters in the national economy is becoming more and more important. According to different working principles, flowmeter technologies can be grouped into the electromagnetic flowmeter^[5], ultrasonic flowmeter^[6], acoustics Doppler current meter^[7], differential pressure type flowmeter^[8], and fiber Bragg grating (FBG) flowmeter^[9–14]. Each of them plays an important role in their respective fields.

Fiber Bragg grating (FBG) has the advantages of anti-electromagnetic interference, wavelength coding, strong ability of multiplexing, and compact size^[15]. FBG has attracted more attention in recent years, and various kinds of flowmeters based on FBG with different principles have been reported. For example, differential pressure-based flowmeters quantify flow using the Bernoulli equation to measure the pressure difference before and after the throttling set, such as the orifice valve and Venturi tube^[9]. However, measurement ranges of such flowmeters are usually small, of which the range ratio is commonly 3:1. Furthermore, application is also limited because of its relatively bulky size. Flowmeters based on a vortex monitor flow through the measuring frequency that is linear with flow rate, when a flow passes through a vortex generator^[10]. However, micro-particles or micro-bubbles mixed in the flow can introduce obvious interference during measurements, which leads to low accuracy. Finally, target-type FBG flowmeters have been reported with many kinds of forms^[11–14]. But most of them have some disadvantages, including complicated structures, low measurement accuracy, and no real-time temperature compensation. Reference [12] gives an FBG sensor for simultaneously discriminating measurement of the flow rate and temperature based on target and cantilever beams, while the spectrum of the FBG glued on the rectangular beam will be distorted for the uneven distribution of strain under the force of flow^[12]. Reference [13] proposed a target-type FBG flow velocity sensor based on the equal-strength cantilever. The force that has a corresponding relationship with velocity is induced by the fluid on the target, and then the lever is adopted to transform the force into the Bragg wavelength shift. However, the measurement accuracy will be reduced with the existence of the force transmission.

In this paper, we propose a target-type FBG flowmeter based on a cascaded cantilever beam with constant strength. Temperature- and pressure-independent flow rate sensing can be performed by monitoring the wavelength separation change of the FBGs glued on the surface of the beam. Experimental results show that a sensing accuracy of 0.015m3/h and a measurement range from 0 to 1m3/h are achieved.

Figure 1 illustrates a schematic diagram of the proposed target-type FBG flowmeter, based on a cascaded cantilever beam with constant strength. The cascaded cantilever beam constitutes two cantilever beams with constant strength but different sizes, where a target is located at the end of the beam 2 [Fig. 1(a)]. The whole cantilever beam and target are fabricated from the same piece of steel plate. The cantilever beam is fixed on the top of a tank body, while the centers of the target and the water inlet are coaxial [Fig. 1(b)]. The diameter of the target should be a little bigger than that of the water inlet, which ensures a larger force area of the flow. For the convenience of processing, the thickness of the target is the same as the cantilever beam, which is closely related to the sensitivity of the flowmeter. Two FBGs connected in series are glued on the surface of the two cantilever beams.

Beam bending occurs when the target is rushed by a flow from the water inlet, which leads to strain changes on the surfaces of two cantilever beams and further results in wavelength shifts of the two FBGs. There is a big difference between the strain changes on the two beam surfaces, due to the largely different sizes of the cantilever beams [Fig. 1(a)]. Consequently, it means that the strain-induced wavelength shifts of the two FBGs are different. Moreover, the FBG responses to ambient temperature and pressure variations are similar due to their similar bonded bases and working environments. As a result, a wavelength separation of reflection peaks between two FBGs, which is insensitive to ambient temperature and pressure changes, can be used for flow sensing, where the flow rate can be obtained from a differential wavelength shift with certain algorithms.

To build a mathematical relationship between the differential wavelength shift of two FBGs and the flow rate, starting from the theory of fluid mechanics, a force applied on the target that is caused by a fluid flow can be described as^[16]F=ξρAν2/2,(1)where F is the force applied on the target, ξ is the resistance coefficient of the flow, ρ is the density of the flow, A is the area of the target, and v is the flow rate.

Next, based on the theory of the cantilever beam with constant strength, the force-induced strains on the two cantilever beams can be expressed as ε1=6FLEb1h2,(2)ε2=6FL2Eb2h2,(3)where ε1 and ε2 are the force-induced strains on beam 1 and beam 2, respectively, L is the total length of the cascaded beam, L2 is the length of cantilever beam 2, E is the elastic modulus of the beam, b1 and b2 are the widths of beam 1 and beam 2, respectively, and h is the thickness of two beams. More details about the geometry parameters of the cascaded cantilever beam can be found in Fig. 2. Then, assuming that the strains can be fully transferred from the cantilever beams to the attached FBGs, the strain/force-induced relative shifts of the central wavelengths of the two FBGs (i.e., Δλ/λ) can be expressed as^[17]Δλ1λ1=(1−Pe)ε1=(1−Pe)6FLEb1h2,(4)Δλ2λ2=(1−Pe)ε2=(1−Pe)6FL2Eb2h2,(5)where Δλ1 and Δλ2 are the wavelength shifts of the two FBGs, respectively, λ1 and λ2 are the central wavelengths of the two FBGs, respectively, and Pe is the effective photo-elastic constant. Thus, from Eqs. (4) and (5), a differential wavelength shift between the two FBGs can be derived as Δλ1λ1−Δλ2λ2=(1−Pe)6FEh2(Lb1−L2b2)=kv2,(6)where k=(1−Pe)3FEh2(Lb1−L2b2)ξρA. Considering that Δλ1 and Δλ2 are orders of magnitude smaller than λ1 and λ2, Eq. (6) can be simplified as Δλ1λ1−Δλ2λ2≈Δλ1−Δλ2λ1=kν2.(7)

Therefore, according to Eq. (7), Δλ1−Δλ2 is in proportion to v2 so that we can obtain the flow rate by measuring the differential wavelength shift of the two FBGs. Finally, the sensitivity of the flowmeter can be easily enhanced by adjusting the values of L, L2, b1, and b2 (for example, the flow sensitivity is increased with the term Lb1−L2b2).

Solidworks and Ansys were used to simulate strain distributions on the cascaded cantilever beam with constant strength under a certain force. The beam is manufactured by one kind of stainless steel with the following parameters: E=2.034×1011pa, ρ=7.87×103kg/m3, μ=0.3, L=88mm, L1=64mm, L2=20mm, b1=20mm, b2=12mm, and h=1mm. As shown in Fig. 3, the constraint is applied to the end of the beam, and a force of 1 N is applied to the center of the target to simulate the rush of a flow on the target.

From the simulation result shown in Fig. 3, the strains of the cantilever beam 1 and beam 2 appear differently under the force applied to the target. The distributions of strain on each cantilever beam center axis are basically even (shown in Fig. 4), which avoids chirping FBGs glued on the central axis of the cantilever beams. The simulated strains of cantilever beam 1 and beam 2 are 100 and 36  µε, respectively. The former (beam 1) is about 2.8 times the latter (beam 2), which indicates that the strain-induced wavelength shifts of two FBGs can differ by 2.8 times. Thus, we can achieve a high strain sensitivity by measuring the differential wavelength shift of two FBGs, where contaminations of ambient temperature and pressure variations on flow measurements can be removed.

The parameters of the FBGs used in the experiment are as follows: the length of the grating is 10 mm, the side-mode suppression ratio is larger than 15 dB, the reflectivity is 85%, and the center wavelengths of FBG1 and FBG2 are 1540 and 1530 nm, respectively. It should be noted that the grating periods of the used FBGs are uniform (i.e., no chirped FBG), and the grating planes are straight to the fiber axis (i.e., no tilted angle).

According to the simulation result, the distribution of strain on the surface of the cantilever beams is basically even except for the area of the connection section of two equal-strength cantilever beams. So, each FBG should be glued on the surface of each cantilever beam with even distribution of strain, and the central axis of the cantilever beams is the best place to be glued on. An auxiliary line was drawn to ensure that the FBGs were located in the centerline of the equal-strength cantilever beam. During the gluing operation, we should make sure that the glue has excellent liquidity and is daubed equally on the grating area of the FBGs, which ensures high bonding reliability. Pretension with a certain size was applied before the FBGs were glued on the surface of the beam. As shown in Fig. 5, a soft layer of silicone rubber was daubed to protect the FBGs after the glue cued. An FBG sensing interrogator (MOI-si155, Luna) is used to measure the FBG spectrum in this work, where the main parameters include a wavelength accuracy of 1 pm, a sampling rate of 1 kHz, a wavelength repeatability of 1 pm, and a dynamic range of 40 dB.

In the experiment, small irons with different weights were hung on the target to simulate the rush of the flow (shown in Fig. 6). Test results are shown in Fig. 7. The Bragg wavelengths of two FBGs shift toward longer wavelengths with the increased weight, where the wavelength shifts of FBG1 are obviously larger than those of FBG2 [Fig. 7(a)]. It also shows no signs of broadening of the Bragg peaks during the loading process, which means that both FBGs were glued with good conditions without any local stress concentration.

The relationships between the Bragg wavelength shifts of the two FBGs and the gravity of the weight are in good linearity [Fig. 7(b)], which benefits flow measurement with high accuracy. Furthermore, the estimated sensitivities of FBG1 and FBG2 are 0.1782 and 0.0642 nm/N, respectively. The former is about 2.8 times the latter, which well agrees with the simulation results. Meanwhile, Δλ1−Δλ2 is in a good linear relationship with the gravity of the weight, and the latter is in proportion to v2 with theory [Eq. (1)]. So a good linear relationship also can be achieved between Δλ1−Δλ2 and v2, which is consistent with the theory [Eq. (7)].

The cascaded cantilever beam with constant strength was placed in a thermostatic water bath to test the temperature responses of FBG wavelengths. As shown in Fig. 8(a), both wavelengths of the two FBGs are in good linearity with the ambient temperature, where the fitted sensitivities are approximately the same. Moreover, the time courses of wavelength variations of two FBGs almost overlap during the continuous temperature increase. The differential wavelength shift fluctuates around 0 without any uptrend or downtrend [Fig. 8(b)]. Therefore, the differential wavelength shift of two FBGs can be used for eliminating the temperature cross-sensitive issue.

In actual operation, pressure always exists in the pipeline to be measured, so the pressure influence on the flow test must be studied. During the experiment, the packaged cascaded cantilever beam with constant strength was placed in a pressure tank to simulate the ambient pressure. The pressure was increased up to 5 MPa by certain intervals, and the wavelengths of the two FBGs were recorded simultaneously. The two FBGs are contracted under pressure, which results in wavelength shifts toward short wavelengths. As shown in Fig. 9(a), the wavelengths of the two FBGs are basically in linearity with the pressure, where the fitted sensitivities are close. The wavelength variations of the two FBGs are almost synchronous during the fast pressure increase. The differential wavelength shift also fluctuates around 0 without any uptrend or downtrend [Fig. 9(b)]. Accordingly, during the temperature compensation process, we can also eliminate the pressure influence by measuring the differential wavelength shifts of two FBGs.

The cascaded cantilever beam with constant strength was installed at a specific location inside the tank body of the flowmeter. The tail fiber was connected to the demodulator after passing through the lid of the tank body and sealing. Two joints were fixed at the water inlet and outlet, which were used to connect with the pipeline that was to be measured. A photograph of the flowmeter is shown in Fig. 10.

In the flow test (shown in Fig. 11), the flowmeter was linked up to a water pipe, and the flow rate could be adjusted by a screwed valve. An electrical flowmeter and a mechanical flowmeter were also installed in the pipe, where the measured flow rate was used as reference values after being made the average. During the test, we obtained the flow data synchronously to make sure that all the flowmeters have the same flow. The flow rate was gradually increased by adjusting the screwed valve.

Beam bending occurs when the target is rushed by the flow from the water inlet, which leads to strain changes on the surfaces of two cantilever beams and further results in wavelength shifts of the two FBGs. Figure 12(a) shows the spectral shifts of the reflection peaks of two FBGs under different flow rates. Similar to the static loading test [Fig. 7(a)], the Bragg wavelengths of two FBGs shift toward long wavelengths along with the increased flow rate, and the shift of FBG1 is obviously larger than that of FBG2. During the increasing process, we recorded the reference value of the flow rate and the wavelengths of the two FBGs with equal sampling intervals. As shown in Fig. 12(b), the wavelength separation of the two FBGs is almost in a linear relationship with the square of the flow rate, which is consistent with the theory [Eq. (7)]. During the measurement, we will obtain a fitting curve by calibrating the relationship between the wavelength separation of the two FBGs and the square of the flow rate in advance. Then we derive the algorithm of the flow based on the fitting curve. Finally, we will measure the flow using the algorithm. Then, the measurement accuracy of the flow rate is estimated, and the result is shown in Table 1. Finally, the measurement accuracy of 0.015m3/h is achieved in the range of 0–1 m³/h.

In this paper, a target-type FBG flowmeter based on a cascaded cantilever beam with constant strength is proposed. The cascaded cantilever beam consists of two sections with constant strength but different dimensions. Two FBGs connected in series are glued on the surface of the two sections. Bending occurs when the target of the beam is rushed by a flow, which results in different strain-induced wavelength shifts for the two FBGs. A differential wavelength shift that is insensitive to the ambient influences can be simply realized by monitoring the wavelength separation change of the FBGs because the two FBGs have the same responses to ambient temperature and pressure changes. Various tests in detail are enforced before the cascaded cantilever beam is installed into the tank body of the flowmeter. Test results show that wavelengths of FBG1 and FBG2 are in good linearity with load, temperature, and pressure. Furthermore, the wavelength difference of FBG1 and FBG2 is not affected by changes in ambient temperature and pressure. After the flowmeter is assembled, its flow performance is tested. Experimental results show that a sensing accuracy of 0.015m3/h and a measurement range from 0 to 1m3/h are achieved.

Meanwhile, the proposed flowmeter has its limitations, such as the reliability of the sensor because the FBGs are always immersed in the fluid and the measurement fluctuation due to the rush of the flow. More studies will be carried out to promote the packaging process of the sensor and the algorithm of the flow calculation.