VOLUME FLOWMETER AND METHOD FOR DETERMINING A VOLUMETRIC FLOW RATE

Abstract

An aspect of the present invention pertains to a method for determining the volumetric flow rate of a compressible fluid flow flowing through a volume flowmeter having a flow inlet, a wheel downstream of the flow inlet and a constriction downstream of the flow inlet and upstream of the wheel, the compressible fluid flowing through the flow inlet and actuating the wheel. The method comprises measuring the rotational speed of the wheel and determining the permanent pressure loss across the wheel based on the measured rotational speed. The method further comprises measuring the fluid pressure of the compressible fluid flow at the flow inlet and determining whether the compressible fluid flow in the volume flowmeter is in the subsonic or in the supersonic regime based on the determined permanent pressure loss and the measured fluid pressure. The method also comprises measuring the fluid temperature of the compressible fluid flow at the flow inlet and determining the volumetric flow rate of the compressible fluid flow based on the determined permanent pressure loss, the measured fluid pressure, the regime of the compressible fluid flow and the measured fluid temperature. Other aspects of the present invention pertain to volume flowmeter for determining the volumetric flow rate of a compressible fluid flow, a data processing device for controlling a volume flowmeter, a computer program for the controller and a computer-readable medium having stored thereon the computer program.

Claims

1. A method for determining the volumetric flow rate of a compressible fluid flow flowing through a volume flowmeter having a flow inlet, and a wheel downstream of the flow inlet, the compressible fluid flowing through the flow inlet and actuating the wheel, comprising: measuring a rotational speed of the wheel; determining a permanent pressure loss across the wheel based on the measured rotational speed; measuring a fluid pressure of the compressible fluid flow at the flow inlet; determining whether the compressible fluid flow in the volume flowmeter is in a subsonic or in a supersonic regime based on the determined permanent pressure loss and the measured fluid pressure; measuring a fluid temperature of the compressible fluid flow at the flow inlet; and determining the volumetric flow rate of the compressible fluid flow based on the determined permanent pressure loss, the measured fluid pressure, the regime of the compressible fluid flow and the measured fluid temperature.

2. The method according to claim 1, comprising powering the volume flowmeter by at least one of a battery and a supercapacitor.

3. The method according to claim 1, comprising powering the volume flowmeter with energy harvested from the compressible fluid flow by a turbine, the turbine comprising the wheel and a generator, the wheel being operatively connected to the generator.

4. (canceled)

5. The method according to claim 1, comprising transmitting at least one of the measured rotational speed of the wheel, the measured fluid pressure and the measured fluid temperature.

6. The method according to claim 1, comprising transmitting at least one of the measured rotational speed of the wheel, the determined volumetric flow rate, the measured fluid pressure, the regime of the compressible fluid flow, the measured fluid temperature and the cumulative volume through the volume flowmeter, and time series thereof.

7. (canceled)

8. (canceled)

9. (canceled)

10. (canceled)

11. A volume flowmeter for determining the volumetric flow rate of a compressible fluid flow comprising: a flow inlet for the compressible fluid flow, the flow inlet comprising a pressure sensor and a temperature sensor for measuring the fluid pressure and the fluid temperature, respectively, of the compressible fluid flow; a wheel, in fluid communication with the flow inlet, the wheel being configured for being actuated by the compressible fluid flow; a rotational speed sensor for measuring the rotational speed of the wheel; and a controller configured for determining: a permanent pressure loss across the wheel based on a rotational speed measured by the rotational speed sensor; whether the compressible fluid flow in the volume flowmeter is in a subsonic or in a supersonic regime based on the determined permanent pressure loss and the measured fluid pressure by the pressure sensor; and a volumetric flow rate of the compressible fluid flow based on the determined permanent pressure loss, the measured fluid pressure by the pressure sensor, the determined regime of the compressible fluid flow and the measured fluid temperature by the temperature sensor.

12. The volume flowmeter according to claim 11, comprising a wheel bypass arrangement for bypassing the wheel.

13. The volume flowmeter according to claim 11, comprising a turbine for powering the volume flowmeter, the turbine comprising the wheel and a generator, the wheel being configured to be operatively connected to the generator.

14. The volume flowmeter according to claim 11, comprising at least one of a battery and a supercapacitor configured for powering the volume flowmeter.

15. The volume flowmeter according to claim 11, comprising a power management system.

16. The volume flowmeter according to claim 11, comprising a wireless communication system.

17. The volume flowmeter according to claim 11, comprising a GNSS receiver for providing at least one of geolocation and time.

18. The volume flowmeter according to claim 11, wherein the rotational speed sensor comprises at least one of a phase-locked loop (PLL) control system and/or a comparator with a digital counter for measuring the rotational speed of the wheel.

19. (canceled)

20. A data processing device for controlling a volume flowmeter, comprising: one or more signal input terminals for receiving fluid pressure and fluid temperature signals of a compressible fluid flow at a flow inlet of the volume flowmeter as well as a rotational speed signal of a wheel of the volume flowmeter; and a controller configured for determining: the a permanent pressure loss across the wheel based on the rotational speed signal; whether the compressible fluid flow in the volume flowmeter is in the a subsonic or in the a supersonic regime based on the determined permanent pressure loss and the fluid pressure signal; and a volumetric flow rate of the compressible fluid flow based on the determined permanent pressure loss, the fluid pressure signal, the determined regime of the compressible fluid flow and the fluid temperature signal.

21. (canceled)

22. A computer-readable medium with a non-volatile memory having stored therein a computer program comprising instructions to cause a controller for a volume flowmeter according to claim 11 to execute the steps of: determining the permanent pressure loss across the wheel based on the measured rotational speed; determining whether the compressible fluid flow in the volume flowmeter is in the subsonic or in the supersonic regime based on the determined permanent pressure loss and the measured fluid pressure at the flow inlet; and determining the volumetric flow rate of the compressible fluid flow based on the determined permanent pressure loss, the measured fluid pressure, the regime of the flow compressible fluid flow and the measured fluid temperature.

23. The volume flowmeter according to claim 11, comprising a constriction arranged downstream of the flow inlet and upstream of the wheel, the wheel being in fluid communication with the flow inlet via the constriction.

24. The volume flowmeter according to claim 23, wherein the constriction is a nozzle.

25. The volume flowmeter according to claim 12, wherein the wheel bypass arrangement comprises a valve for selectively opening and closing the bypass.

26. The volume flowmeter according to claim 16, wherein the wireless communication system comprises at least one of a Bluetooth, a ZigBee, a Z-wave and a Wi-Fi communication system.

27. The volume flowmeter according to claim 16, wherein the wireless communication system comprises a Bluetooth Low Energy communication system.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0064] By way of example, preferred, non-limiting embodiments of the disclosure will now be described in detail with reference to the accompanying drawings, in which:

[0065] FIG. 1: is a schematic representation of a self-powered flowmeter, according to a preferred embodiment of the disclosure;

[0066] FIG. 2: is a schematic cutaway perspective drawing of the turbine according to a preferred embodiment of the disclosure;

[0067] FIG. 3: is a schematic cutaway perspective drawing of a detail of the turbine of FIG. 2 according to a preferred embodiment of the disclosure;

[0068] FIG. 4: is a schematic cutaway perspective drawing of a detail of the turbine of FIG. 2 according to a preferred embodiment of the disclosure;

[0069] FIG. 5: is a schematic representation of a test bench for testing the turbine;

[0070] FIG. 6: shows a comparison between flow rates measured by the test bench and predicted by the subsonic or supersonic model described hereinbelow, the flow rate being maintained at 5 L/min;

[0071] FIG. 7: shows a comparison between flow rates measured by the test bench and predicted by the subsonic or supersonic model described hereinbelow, the flow rate being maintained at 7 L/min;

[0072] FIG. 8: shows a comparison between flow rates measured by the test bench and predicted by the subsonic or supersonic model described hereinbelow, the rotational speed of the rotor being maintained at 100 Hz;

[0073] FIG. 9: shows a comparison between flow rates measured by the test bench and predicted by the subsonic or supersonic model described hereinbelow, the rotational speed of the rotor being maintained at 700 Hz;

[0074] FIG. 10: shows the relation between the measured and predicted rotational speed of the rotor as a function of the permanent pressure loss;

[0075] FIG. 11: shows the RMS power produced by the turbine as a function of the rotational speed of the rotor; and

[0076] FIG. 12: is a schematic representation of an exemplary implementation of comparator with a digital counter.

[0077] The reader's attention is drawn to the fact that the drawings are not to scale. Furthermore, for the sake of clarity, proportions between height, length and/or width may not have been represented correctly.

DETAILED DESCRIPTION

[0078] FIG. 1 is a schematic representation of a self-powered flowmeter 2, according to a preferred embodiment of the disclosure. The flowmeter 2 comprises a turbine 4 having a flow inlet 6 and a flow outlet 8 configured to be connected to a pipe housing a fluid flow. The fluid flow flows through the turbine 4 from the flow inlet 6 to the flow outlet 8 when the flowmeter 2 is connected to the pipe. The turbine 4 harvests kinetic energy from the fluid flow and transforms it into electrical energy that powers the flowmeter 2. The turbine 4 therefore acts as a power generator. The flow inlet 6 comprises a pressure sensor 10 (e.g. a low consumption pressure gauge, e.g. a MS5541C from TE Connectivity, Servoflo Corporation) and a temperature sensor 12 (e.g. the temperature sensor may be embedded in the low consumption pressure gauge which provides temperature measurement capabilities, e.g. a MS5541C, or the temperature sensor may be a separate low consumption temperature sensor, e.g. a PCT2202 from NXP) for measuring the fluid pressure and the fluid temperature, respectively, of the fluid flow. The turbine 4 comprises a first and a second electrical terminals 13, 14 providing an AC current for powering the flowmeter 2.

[0079] The flowmeter 2 further comprises a Power Management System 16 (PMS) to which the terminals 13, 14 of the turbine 4 are connected. The PMS 16 is in charge of maintaining an optimal impedance matching for power transfer between the terminals 13, 14 of the generator of the turbine 4 and the electronic components of the flowmeter 2. The PMS 16 is also connected to a battery 18 for storing electrical energy. In another embodiment, the battery may be replaced or supplemented by a supercapacitor. The PMS 16 is in charge of controlling the electrical system of the flowmeter 2. The PMS 16 determines whether the overall electrical power consumption of the flowmeter 2 is greater or lower than the electrical power generated by the turbine 4. In the first case, the power management system complements the generated electrical power by electrical power stored in the battery 18 for optimal operation of the flowmeter 2. In the second case, the PMS 16 redirects the generated electrical power to the battery 18 for later use. The PMS 16 is also configured to selectively power on or off (uncritical) components of the flowmeter 2, or reduce/increase the duty cycles thereof, based on the power consumption of said components (the power management system may e.g. power off a display or the wireless communication unit of the flowmeter 2 in case of low generated and/or stored electrical power). The PMS 16 provides a power line 20 for (eventually selectively) powering components of the flowmeter 2 such as the pressure sensor 10 and the temperature sensor 12.

[0080] The flowmeter 2 also comprises a high input impedance frequency analyzer 22 (e.g. a low consumption PLL chip, e.g. LMC568 from Texas Instruments). The frequency analyzer 22 is connected to the output terminals 13, 14 of the turbine 4 for determining the frequency of the AC current. Alternatively or additionally, a comparator with a digital counter for pre-processing the signal may be provided by either terminal 13 or terminal 14. An example of implementation of such a comparator with digital counter is provided in FIG. 12, wherein the comparator is powered by the power line 20. The comparator provided at its output terminal a digital square signal.

[0081] The flowmeter 2 further comprises a microcontroller 24 (μC) which can be implemented as an application-specific integrated circuit (ASIC), as a digital signal processor (DSP) and/or as a field-programmable gate array (FPGA).

[0082] The microcontroller 24 is connected to the pressure sensor 10, the temperature sensor 12 and the frequency analyzer 22. The pressure sensor 10, the temperature sensor 12 and the frequency analyzer 22 provide a pressure signal, a temperature signal and a frequency signal, respectively, to the microcontroller 24 through e.g. one or more of its input terminal(s). The microcontroller 24 is configured to determine the permanent pressure loss across the flowmeter 2 based on the frequency signal. The microcontroller 24 is further configured to determine whether the fluid flow in the flowmeter 2 is in the subsonic or in the supersonic regime based on the determined permanent pressure loss and the pressure signal. In addition, the microcontroller 24 is configured to determine the volumetric flow rate of the fluid flow based on the determined permanent pressure loss, the pressure signal, the regime of the flow fluid flow and the temperature signal.

[0083] It follows that the electrical power harvested from the fluid flow not only allows providing electrical power for operating the flowmeter 2 but also allows measuring the frequency of the AC current of the turbine 4. This, in turn, allows determining the permanent pressure loss across the flowmeter 2 (see below).

[0084] A data logger 26, connected to the microcontroller 24, is also provided for logging time series of the pressure signal, the temperature signal, the permanent pressure loss, the regime of the flow, the volumetric flow rate, and/or the incremental consumption in volume of the fluid.

[0085] Furthermore, a display 28 (e.g. a low consumption display (LCD, e-ink or OLED)), connected to the microcontroller 24, is also provided for direct monitoring of the pressure, the temperature, the permanent pressure loss, the regime of the flow and/or the volumetric flow rate by a user.

[0086] The pressure signal, the temperature signal, the permanent pressure loss, the regime of the flow and/or the volumetric flow rate, or time series thereof, may be transmitted by a Bluetooth (preferably Low Energy) wireless communication system 30, connected to the microcontroller 24. In other embodiments, the wireless communication system may be a Wi-Fi, a ZigBee or a Z-wave communication system.

[0087] The structure of the turbine 4 according to a preferred embodiment of the disclosure is illustrated in FIGS. 2-4. The turbine 4 comprises a bucket wheel 32 with a multipolar magnetic rotor 33, a double basket structured stator (34a, 34b) and a coil 36. The generator comprises elements 33, 34a and 34b, and 36 as an electromagnetic generator. It should be noted that, in other embodiments, the wheel 32 may be replaced by a paddle wheel, a bucket wheel, a helix, or any other type of wheel. The bucket wheel 32 is housed in the double basket structured stator (34a, 34b) which is connected to a coil 36. The coil 36 provides the AC current at the electrical terminals 13, 14 of turbine 4. The bucket wheel 32 is in fluid communication with the flow inlet 6 and the flow outlet 8. Fluid entering through the flow inlet 6 of the flowmeter 2 is directed to the bucket wheel 32 and induces a rotation thereof. The turbine 4 is configured so that the fluid flow flows around half of turn of the bucket wheel 32 before exiting the bucket wheel 32 and flowing to the flow outlet 8. The rotation of the bucket wheel 32 creates induced current in the coil 36 thereby generating the AC current provided at the electrical terminals 13, 14. It follows that the frequency of AC current determined by frequency analyzer 22 is the rotational frequency of the rotor (i.e. the rotational speed of the bucket wheel 32), or a known (possibly integer) multiple thereof. The frequency analyzer 22 thus senses the rotational speed of the bucket wheel 32. It follows that a separate rotational speed sensor is not necessary for determining the rotational speed of the bucket wheel 32.

[0088] The flow inlet 6 and the flow outlet 8 have a diameter preferably comprised in the interval from 2 mm to 15 mm, preferably from 3 mm to 10 mm, more preferably from 4 mm to 7 mm, even more preferably from 4 mm to 6 mm. The flow inlet 6 and the flow outlet 8 may have the same or different diameters.

[0089] In an embodiment, no inlet or outlet other than the flow inlet 6 and the flow outlet 8 is arranged in the turbine 4 between the flow inlet 6 and the flow outlet 8. For example, no exhaust holes, venting holes or tapping points (e.g. for measuring the temperature or pressure) are arranged in the turbine 4 between the flow inlet 6 and the flow outlet 8. In other words, the turbine 4 is fluid-tight.

[0090] The turbine 4 further comprises a nozzle 37 having a small circular orifice plate. The orifice has a diameter comprised in the interval from 0.1 mm to 1 mm, preferably from 0.2 mm to 0.8 mm, more preferably from 0.4 mm to 0.6 mm, even more preferably of 0.5 mm In other embodiments, the nozzle 37 may have different shape, such as, e.g., a fine throat or a beveled orifice with a well-defined angle.

[0091] A cross sectional diameter change in the pipe causes the velocity of the flowing fluid to change. As the flowing fluid passes through the nozzle 37, the restriction (constriction) causes an increase of fluid velocity and a decrease of fluid pressure.

[0092] A general expression for computing the volumetric flow rate of a fluid flow flowing through a nozzle is given by:

[00001]$\begin{array}{cc}Q={\mathrm{CYA}}_2\sqrt{\frac{2g_c\left(p_1-p_2\right)}{\left(1-{\beta }^4\right){\rho }_1}},& \left(\mathrm{Eq}.1\right)\end{array}$

where Q is the volumetric flow rate, C is the discharge coefficient (˜0.6 for a circular orifice plate), Y is the expansion factor (1 for incompressible fluids flows, or ≠1 for compressible fluid flows), A.sub.2 is the cross-section surface of the pipe, g.sub.c is a dimensionless constant (1 in SI), p.sub.1 is the fluid pressure upstream of the nozzle, p.sub.2 is the fluid pressure at the vena-contracta, downstream of the nozzle, β is the ratio between nozzle diameter and the pipe diameter upstream of the nozzle, and ρ.sub.1 is the fluid density upstream of the nozzle.

[0093] It is well known that the nozzle induces a change of the fluid pressure, fluid temperature and fluid velocity of the fluid and may induce a subsonic to supersonic transition. Also, the nozzle causes a permanent pressure loss. The subsonic to supersonic transition is a point of paramount importance to consider in order to evaluate accurately the volumetric flow rate of a compressible fluid flow.

[0094] It is also well-known that increasing the pressure difference of a given subsonic flow between the region upstream and downstream of a constriction (e.g. nozzle) will increase the Mach number of the flow, in particular at the constriction. When the pressure difference increased in such a way that the flow is just at M=1 at the constriction, the flow upstream and downstream of the constriction is at M<1. The flow is called choked since the flow remains at M=1 at the constriction even when further increasing the pressure difference. Further increasing the pressure difference creates a flow in a supersonic regime just downstream of the constriction. It should be noted that the supersonic regime is often called choked regime since the flow remains choked at the constriction.

[0095] According to the Standard provided by The International Society of Automation “Flow Equations for Sizing Control Valves” (ISA-75.01.01-2007, 60534-2-1 Mod), the volumetric flow rate Q.sub.sub for a subsonic flow may be written as

[00002]$\begin{array}{cc}{Q}_{sub}\left(\frac{L}{\min }\right)=\frac{4.17}{60}1000C_v{p}_1\left(1-\frac{\Delta p}{3p_1{F}_{\gamma }{x}_T}\right)\sqrt{\frac{\Delta p}{p_1\left(T_a+273.15\right)}}.& \left(\mathrm{Eq}.2\right)\end{array}$

Also according to the same reference, the volumetric flow rate Q.sub.sup for a supersonic flow may be written as

[00003]$\begin{array}{cc}{Q}_{\sup }\left(\frac{L}{\min }\right)=\frac{4.17}{60}10000.667C_v{p}_1\sqrt{\frac{F_{\gamma }{x}_T}{T_a+273.15}}.& \left(\mathrm{Eq}.3\right)\end{array}$

[0096] For determining whether the fluid is in the subsonic or in the supersonic regime, the following flow transition criteria is used:

[00004]$\begin{array}{cc}\frac{\Delta p}{p_1}-F_{\gamma }{x}_T\{\begin{array}{c}<0\mathrm{for}\mathrm{subsonic}\mathrm{regime},\\ \ge 0\mathrm{for}\mathrm{supersonic}\mathrm{regime}\end{array}& \left(\mathrm{Eq}.4\right)\end{array}$

with T.sub.a the fluid temperature in ° C., Δp=p.sub.1−p.sub.2F.sub.γ is the specific heat ratio factor of the fluid (e.g. 1.401 for air at room temperature), x.sub.T is the pressure differential ratio factor of a control valve without attached fittings at choked flow and C.sub.v is the flow coefficient.

[0097] The flow coefficient C.sub.v may be determined according to Lohm's definition. For a circular orifice plate of diameter d, one has:

[00005]$\begin{array}{cc}{C}_v={C\left(\frac{d}{4.654}\right)}^2.& \left(\mathrm{Eq}.5\right)\end{array}$

[0098] According to D. W. Green “Perry's Chemical Engineers' Handbook” (McGraw-Hill, 2008), in particular in Sec. 10 “Transport and Storage of Fluids”, Δp is related to the permanent pressure loss p.sub.1−p.sub.3, where p.sub.3 is the fluid pressure measured downstream, far away from the flowmeter 2, in the following way:

[00006]$\begin{array}{cc}\Delta p=p_1-p_2=\frac{p_1-p_3}{1-{\beta }^2}.& \left(\mathrm{Eq}.6\right)\end{array}$

[0099] Turning now to the dynamics of the turbine, more particularly of the rotor, the angular equation of motion can be written as (A. Napolitano et al. “A wide range (up to 1010 P) rotating cylinder viscometer”, J. Res. Nat. Bur. Stand. -A. Phys. and Chem. 69A(5), p. 449 (1965)):

[00007]$\begin{array}{cc}I\frac{d^2\theta }{{\mathrm{dt}}^2}+\eta K_1\frac{d\theta }{\mathrm{dt}}+K_2\sin \left(m\theta \right)=K_3\left(p_1-p_3\right),& \left(\mathrm{Eq}.7\right)\end{array}$

where I is the moment of inertia, θ≡θ(t) is the instantaneous angular position (in radians) of the rotor relatively to the stator, η.sub.I is the dynamic viscosity of the fluid, K.sub.1 is the “electromagnetic” viscosity (originating from eddy currents), K.sub.2 is the cogging torque factor, m is an even integer defining the periodicity of the magnetic cogging torque equals to the number of pair poles of the rotor (33) and stator (34a, 34b) and K.sub.3 is the driving torque factor. In order to be easily analytically solved, the equation of motion can be simplified to:

[00008]$\begin{array}{cc}I\frac{d^2\theta }{{\mathrm{dt}}^2}+\eta K_1\frac{d\theta }{\mathrm{dt}}+K_2=K_3\left(p_1-p_3\right),& \left(\mathrm{Eq}.8\right)\end{array}$

where the periodic cogging term K.sub.2 sin(mθ) is replaced by a continuous torque K.sub.2 opposite to the rotation. This assumption is justified by considering the dynamic steady state of the constant rotation speed of the generator in the turbine 4, and not the transitory state.

[0100] The closed-form solution for the instantaneous rotational speed of the rotor ω(t)=dθ/dt, can be written as:

[00009]$\begin{array}{cc}\omega \left(t\right)=\frac{K_3}{\eta K_1}\left[\left(p_1-p_3\right)-\Delta p_{\mathrm{th}}\right]\left[1-\exp \left(-\frac{\eta K_1}{I}t\right)\right],& \left(\mathrm{Eq}.9\right)\end{array}$

where Δp.sub.th=K.sub.2/K.sub.3 is a constant for a threshold differential pressure to achieve by the driving flow to initiate the rotation of the rotor.

[0101] For t.fwdarw.∞, the instantaneous rotational speed of the rotor ω(t.fwdarw.∞) tends to

[00010]$\begin{array}{cc}\omega \left(t.fwdarw.\infty \right)=\frac{K_3}{\eta K_1}\left[\left(p_1-p_3\right)-\Delta p_{\mathrm{th}}\right].& \left(\mathrm{Eq}.9\right)\end{array}$

It follows that a linear relationship exists between (p.sub.1−p.sub.3) and the rotational speed ω=ω(t.fwdarw.∞) of the rotor:

[00011]$\begin{array}{cc}{p}_1-p_3=\omega \frac{\eta K_1}{K_3}+\Delta p_{\mathrm{th}}.& \left(\mathrm{Eq}.10\right)\end{array}$

[0102] The volumetric flow rate for a subsonic flow (see Eq. 2) may therefore be rewritten as

[00012]$\begin{array}{cc}{Q}_{\mathrm{sub}}\left(\frac{L}{\min }\right)=\frac{4.17}{60}1000{C\left(\frac{d}{4.654}\right)}^2{p}_1\left(1-\frac{\omega \frac{\eta K_1}{K_3}+\Delta p_{\mathrm{th}}}{3\left(1-{\beta }^2\right)p_1{F}_{\gamma }{x}_T}\right)\sqrt{\frac{\omega \frac{\eta K_1}{K_3}+\Delta p_{\mathrm{th}}}{\left(1-{\beta }^2\right)p_1\left(T_a+273.15\right)}},& \left(\mathrm{Eq}.11\right)\end{array}$

and the volumetric flow rate for a supersonic flow (see Eq. 3) may also be rewritten as

[00013]$\begin{array}{cc}{Q}_{\sup }\left(\frac{L}{\min }\right)=\frac{4.17}{60}1000\text{.0}\text{.667}{C\left(\frac{d}{4.654}\right)}^2{p}_1\sqrt{\frac{F_{\gamma }{x}_T}{T_a+273.15}}.& \left(\mathrm{Eq}.12\right)\end{array}$

Experimental Confirmation of the Models of Eqs. 2, 3

[0103] With reference to FIG. 5, the turbine 4 is placed on a test bench 39 comprising a pressure sensor 38 for measuring the fluid pressure p.sub.3. The test bench 39 further comprises a micro leak valve 40, arranged downstream from the pressure sensor 38, and a flowmeter 42 for measuring the flow rate Q. The micro leak valve 40 allows for accurately setting the volumetric flow rate. The turbine 4 also comprises, at its flow inlet, a temperature sensor which is not represented in FIG. 5. The fluid is air with T.sub.a=20° C.

[0104] FIGS. 6, 7 show the measured volumetric flow rates, set constant by the mean of the manual micro leak valve (Q=5 L/min for FIG. 6 and Q=7 L/min for FIG. 7), as a function of fluid pressure p.sub.1 at the inlet. The black squares are the experimental data. The volumetric flow rate is also computed according to the subsonic model (Eq. 2—dotted curve, circle symbols) and the supersonic model (Eq. 3—dashed curve, triangle symbols). In addition, the flow transition criteria (Eq. 4) is also shown (solid curve, hollow square symbols) in the lower panel of each graph for visualizing the corresponding subsonic/supersonic transition. The subsonic model (Eq. 2) allow predicting accurately the volumetric flow rate Q based on p.sub.1, p.sub.3 and T.sub.a. The supersonic model (Eq. 3) also allows predicting the volumetric flow rate Q in the case of a supersonic flow.

[0105] FIGS. 8, 9 show the measured volumetric flow rates as a function of fluid pressure p.sub.1 at the inlet for a constant rotational speed of the rotor. The rotational speed of the rotor is determined in the same way as described in the embodiment illustrated in FIG. 1. The rotational speed of the rotor is maintained constant by the means of the manual micro leak valve. The black squares are the experimental data. The volumetric flow rate is also computed according to the subsonic model (Eq. 2—dotted curve, circle symbols) and the supersonic model (Eq. 3—dashed curve, triangle symbols). In addition, the flow transition criteria (Eq. 4) is also shown (solid curve, hollow square symbols) in the lower panel of each graph for visualizing the corresponding subsonic/supersonic transition. The same conclusions as in FIGS. 6, 7 can be drawn for FIGS. 8, 9, i.e. that the subsonic model (Eq. 2) allows predicting accurately the volumetric flow rate Q based on p.sub.1, p.sub.3 and T.sub.a. The supersonic model (Eq. 3) also allows predicting the volumetric flow rate Q in the case of a supersonic flow.

[0106] To sum up, the analytical models of Eqs. 2, 3 accurately reproduce the experimental data.

[0107] Experimental confirmation of the model of Eq. 9 and Eq. 10

[0108] The turbine 4 is placed on the test bench 39. FIG. 10 shows the measured rotational speed of the rotor as a function of permanent pressure loss p.sub.1-p.sub.3 for constant volumetric flow rates Q. The fluid is air with T.sub.a=20° C. The flow transition criteria (Eq. 4) is also shown in the lower panel in order to highlight the subsonic to supersonic transition of the compressible fluid flow. A linear relationship between the permanent pressure loss and the rotational speed of the rotor is obtained for the subsonic regime and for any volumetric flow rate. The data points were fitted by equation (9) which allowed to determine that Δp.sub.th≈0.3 bar and that K.sub.3/(ηK.sub.1)≈3950.93 rad.s.sup.−1bar.sup.−1.

Electrical Power Delivered by the Turbine

[0109] FIG. 11 shows the electrical power delivered by the turbine 4 as a function of the rotational speed (pulsation (RPM), also noted ω) of the rotor. The figure shows that an electrical power close to 200 milliWatts (rms) can be provided for rotational speed of 20 kiloRPM. Such electrical power would be sufficient to power a flowmeter as illustrated in FIG. 1.

[0110] While specific embodiments have been described herein in detail, those skilled in the art will appreciate that various modifications and alternatives to those details could be developed in light of the overall teachings of the disclosure. Accordingly, the particular arrangements disclosed are meant to be illustrative only and not limiting as to the scope of the disclosure, which is to be given the full breadth of the appended claims and any and all equivalents thereof.