Source: https://patents.google.com/patent/US20040069069
Timestamp: 2018-07-18 19:59:33
Document Index: 336047019

Matched Legal Cases: ['Application No. 60', 'Application No. 60', 'Application No. 60', 'Application No. 60', 'Application No. 60', 'Application No. 60', 'Application No. 60', 'Application No. 60', 'Application No. 60']

This application claims the benefit of U.S. Provisional Application No. 60/371,606 (Cidra Docket No. CC-0466) filed Apr. 10, 2002, U.S. Provisional Application No. 60/427,964 (Cidra Docket No. CC-0515) filed Nov. 20, 2002, and U.S. Provisional Application No. 60/451,375 (Cidra Docket No. CC-0598) filed Feb. 28, 2003; and is a continuation-in-part of U.S. patent application Ser. No. 10/376,427 (Cidra Docket No. CC-0596) filed Feb. 26, 2003, which claimed the benefit of U.S. Provisional Application No. 60/359,785 (Cidra Docket No. CC-0403), filed Feb. 26,2002; and is a continuation-in-part of U.S. patent application Ser. No. 10/349,716 (Cidra Docket No. CC-0579), filed Jan. 23, 2003, which claims the benefit of U.S. Provisional Application No. 60/351,232 (Cidra Docket No. CC-0410), filed Jan. 23, 2002; U.S. Provisional Application No. 60/359,785 (Cidra Docket No. CC-0403), filed Feb. 26, 2002; U.S. Provisional Application No. 60/375,847 (Cidra Docket No. CC-0468), filed Apr. 24, 2002; U.S. Provisional Application No. 60/425,436 (Cidra Docket No. CC-0538), filed Nov. 12, 2002; and U.S. Provisional Application No. 60/426,724 (Cidra Docket No. CC-0554), filed Nov. 15, 2002, all of which are incorporated herein by reference in their entirety.
[0016]FIG. 1 is a representative phase diagram for water.
[0017]FIG. 2 is a schematic illustration of a probe in accordance with the present invention.
[0018]FIG. 3 is a perspective view of a probe in accordance with the present invention.
[0019]FIG. 4a is a perspective view of a probe embodying the present invention mounted within a pipe having circular cross-section in accordance with the present invention.
[0020]FIG. 4b is a perspective view of a probe embodying the present invention mounted within a duct having rectangular cross-section in accordance with the present invention.
[0021]FIG. 5 is a cross-sectional view of a plurality of probes disposed within a pipe for characterizing the flow pattern of the flow passing through the pipe in accordance with the present invention.
[0022]FIG. 6 is a side view of a steam turbine having a plurality of probes disposed at different stages of the turbine and a different depths within each stage in accordance with the present invention.
[0023]FIG. 7 is a cross-sectional view of a probe in accordance with the present invention.
[0024]FIG. 8 is a side elevational view of the pipe and pressure sensors of a steam probe in accordance with the present invention.
[0025]FIG. 9 is a cross-sectional view of a piezoelectric film sensor in accordance with the present invention.
[0026]FIG. 10 is a top plan view of a piezoelectric film sensor in accordance with the present invention.
[0027]FIG. 11 is a side elevational view of a plurality of pressure sensors, having PVDF, clamped to the outer surface of the pipe, in accordance with the present invention.
[0028]FIG. 12 is a side view of another embodiment of a probe, having ported pressure sensors, in accordance with the present invention.
[0032]FIG. 23 is a block diagram of a probe for measuring the speed of sound propagating through a saturated vapor/liquid mixture flowing within a pipe, in accordance with the present invention.
[0033]FIG. 24 is a plot showing the standard deviation of sound speed versus frequency for various arrays of saturate vapor/liquid mixture parameter measurement system, in accordance with the present invention.
[0034]FIG. 25 is a plot of sound speed as a function of frequency for vapor/liquid mixtures with fixed droplet size (50 mm) and varying vapor-to-liquid mass ratio in accordance with the present invention.
[0035]FIG. 26 is a plot of sound speed as a function of frequency for vapor/liquid mixtures with varying particle size where the vapor-to-liquid mass ratio is equal to 1.8 in accordance with the present invention.
[0036]FIG. 27 is a plot of sound speed as a function of frequency for vapor/liquid mixtures with varying particle size, in accordance with the present invention.
[0037]FIG. 28 is a flow diagram of an optimization procedure employed to determine vapor-to-liquid ratio and droplet size from analytical model and experimentally determined dispersive speed of sound data in accordance with the present invention.
[0038]FIG. 29 is a plot of the speed of sound propagating through a saturated vapor/liquid mixture having varying temperature and pressures versus quality of the mixture, in accordance with the present invention.
[0039]FIG. 30 is a plot of the volumetric vapor phase fraction for vapor/liquid mixtures having varying temperature and pressures versus quality of the mixture, in accordance with the present invention.
[0040]FIG. 31 is a plot of the enthalpy/volume for vapor/liquid mixtures having varying temperature and pressures versus the speed of sound propagating through the mixture, in accordance with the present invention.
[0041]FIG. 32 is a plot of the enthalpy/volume for vapor/liquid mixtures having varying temperature and pressures versus quality of the mixture, in accordance with the present invention.
[0042]FIG. 33 is a kω plot of data processed from an array of pressure sensors use to measure the speed of sound propagating through a saturated vapor/liquid mixture flowing in a pipe, in accordance with the present invention.
[0043]FIG. 34 is a block diagram of a probe for measuring the vortical field of a saturated vapor/liquid mixture flowing within a pipe, in accordance with the present invention.
[0044]FIG. 35 is a cross-sectional view of a pipe showing a turbulent pipe flow velocity profile.
[0045]FIG. 36 is a side elevational view of another embodiment of a probe for measuring the vortical disturbances in a pipe, in accordance with the present invention.
[0046]FIG. 37 is a plot of the pressure signals measured by a pair of pressure sensors of the probe of FIG. 36.
[0047]FIG. 38 is a plot of the cross-correlation of the pressure signals plotted in FIG. 37.
[0048]FIG. 39 is a kω plot of data processed from a probe embodying the present invention that illustrates slope of the convective ridge, and a plot of the optimization function of the convective ridge, in accordance with the present invention.
[0049]FIG. 40 is a schematic diagram of another embodiment of a probe embodying the present invention.
[0050]FIG. 41 is a graph of resonant frequency versus axial Mach number for a resonant cavity having a 6-inch diameter in accordance with the present invention.
[0051]FIG. 42 is a graph of resonant frequency versus axial Mach number for a resonant cavity having a 12-inch diameter in accordance with the present invention.
[0052]FIG. 43 is a side view in partial cross section of a resonant cavity speed of sound probe in accordance with the present invention.
[0053]FIG. 44 is a graphical representation of an acoustic model for steam particles in accordance with the present invention.
[0054]FIG. 45 is a schematic representation of a resonant cavity speed of sound system incorporating a probe in accordance with the present invention.
Referring to FIGS. 2 and 3, a probe, generally shown as 10, is provided to sense and determine specific characteristics or parameters of a single phase fluid 12 and/or a multi-phase mixture 12 flowing through a pipe (conduit) or in an unconfined space. The multi-phase mixture may be a two-phase liquid/vapor mixture, a solid/vapor mixture or a solid/liquid mixture, or even a three-phase mixture. One example of a multiphase mixture that can be measured is a saturated vapor/liquid mixture, such as steam. To simplify the description of the present invention, the probe 10 will be described as an apparatus for measuring the parameters of a steam mixture, however, one will appreciate that the probe may be used to measure specific characteristics of any single phase fluid (i.e. vapor or liquid) or any multiphase mixture. As will be described in greater detail, the probe measures the speed of sound propagating through the fluid or multiphase mixture flow to determine any one of a plurality of parameters of the flow, such as the steam quality or “wetness”, vapor/mass ratio, liquid/solid ratio, the volumetric flow rate, the mass flow rate, the size of the suspended particles, and the enthalpy of the flow. Additionally, the probe '0 is capable of measuring the unsteady pressure disturbances (e.g., vortical effects, density changes) of the flow passing through the probe to determine the volumetric flow rate of the flow.
[0056]FIG. 2 illustrates a schematic drawing of the probe 10 that includes a sensing device 16 comprising an inner tube 14 and an array of pressure sensors (or transducers) 18-21 spaced axially along the outer surface 22 of the tube 14. The pressure sensors measure the unsteady pressures produced by acoustical and/or vortical disturbances within the tube, which are indicative of a parameter of the single phase fluid or multiphase mixture 12. The output signals (P1-P4) of the pressure sensors 18-21 are provided to a processing unit 24, which processes the pressure measurement data and determines at least one parameter of the mixture. Specifically, the characteristics and parameters determined may include the volumetric flow of the flow, the consistency or composition of the flow, the density of the mixture, the Mach number of the flow, the size of particle flowing through the mixture, the air/mass ratio of the mixture and/or the percentage of entrained air within the mixture.
The second technique measures the velocities associated with unsteady flow fields and/or pressure disturbances created by vortical disturbances or “eddies” to determine the velocity of the flow 12. The pressure sensors 18-21 measure the unsteady pressures P1-P4 created by the vortical disturbances as these disturbances convect within the flow 12 through the probe 10 in a known manner, as shown in FIG. 34. Therefore, the velocity of these vortical disturbances is related to the velocity of the mixture and hence the volumetric flow rate may be determined, as will be described in greater detail hereinafter.
The probe 10 may be used a number of different ways. For example as shown in FIGS. 4a and 4 b, the sensing device 16 of the probe may be mounted within a pipe 19 or duct 23, respective to measure the fluid flow or mixture passing therethrough. The probe 10 is particularly useful for large diameter pipes or ducts having a large cross-sectional area, such as smokestacks, exhaust ducts or HVAC systems. The utility of the probe is especially evident for measuring the flow of a single phase fluid or multiphase mixture 12 that is not confined within piping or ducting. For example, the probe may be mounted within a gas turbine to measure the steam “wetness” or other parameters of the steam exiting the exhaust duct of the steam or LPT turbine. Other applications or uses of the sensing device 16 of the probe 10 include mounting the probe to the exterior of a vehicle such as an automobile, airplane and a train to measure parameters of the air or velocity of the vehicle. Further, the probe may be mounted to the bottom of a ship to measure the SOS propagating through the probe, or mounted to the outer hull of a submarine to measure the speed of sound at different depths in the ocean, as well as other parameters. Generally, the probe may be used in any application that one may use a pitot-static probe. The probe may also be used to measure parameters of a river flow, an open conduit or partially filled pipe.
Similar to that described in U.S. patent application Ser. No. (Cidra's Docket No. CC-0187), which is incorporated herein by reference, the space 66 between the tube 14 and the housing 50 may be evacuated to provide “vacuum backed” sensors 18-21. Evacuating the space 66 provides additional insulation/isolation to prevent external acoustic and/or unsteady pressure disturbances from affecting the sensors 18-21.
As best shown in FIGS. 9 and 10, the piezoelectric sensors 30 include a piezoelectric material or film 32 to generate an electrical signal proportional to the degree that the material is mechanically deformed or stressed. The piezoelectric sensing element is typically conformed to allow complete or nearly complete circumferential measurement of induced strain. The sensors can be formed from PVDF films, co-polymer films, or flexible PZT sensors, similar to that described in “Piezo Film Sensors Technical Manual” provided by Measurement Specialties, Inc., which is incorporated herein by reference. A piezoelectric film sensor that may be used for the present invention is part number 1-1002405-0, LDT4-028K, manufactured by Measurement Specialties, Inc.
Piezoelectric film (“piezofilm”), like piezoelectric material, is a dynamic material that develops an electrical charge proportional to a change in mechanical stress. Consequently, the piezoelectric material measures the strain induced within the inner tube 14 due to unsteady pressure variations (e.g., vortical and/or acoustical) within the process mixture 12. Strain within the tube is transduced to an output voltage or current by the attached piezoelectric sensor. The piezoelectrical material or film may be formed of a polymer, such as polarized fluoropolymer, polyvinylidene fluoride (PVDF).
[0077]FIGS. 9 and 10 illustrate a piezoelectric film sensor (similar to the sensor 18 of FIG. 1), wherein the piezoelectric film 32 is disposed between and pair of conductive coatings 34,35, such as silver ink. The piezoelectric film 32 and conductive coatings 34,35 are coated onto a protective sheet 36 (e.g., mylar) with a protective coating 38 disposed on the opposing side of the upper conductive coating. A pair of conductors 40,42 is attached to a respective conductive coating 34,35.
The piezoelectric film sensors may be mounted directly onto the outer diameter of the tube 14 by epoxy, glue or other adhesive. Alternatively, the piezoelectric film sensors 30 may be adhered to a strap 70 which is then clamped onto the outer surface of the tube 14, as shown in FIG. 11, similar to that described in U.S. Provisional Application No. (Cidra's Docket No. CC-0554).
As described hereinbefore, the probe 10,170 of the present invention may be configured and programmed to measure and process the detected unsteady pressures P1(t)-PN(t) created by acoustic waves and/or vortical disturbances, respectively, propagating through the mixture to determine parameters of the mixture flow 12. One such probe 10 is shown in FIG. 3 that measures the speed of sound (SOS) of one-dimensional sound waves propagating through the vapor/liquid mixture to determine the composition the mixture, namely the “wetness” or steam quality of the mixture. The probe is also capable of determining the average size of the droplets, velocity of the mixture, enthalpy, mass flow, steam quality or wetness, density, and the volumetric flow rate of the mixture. It is known that sound propagates through various mediums at various speeds in such fields as SONAR and RADAR fields. The speed of sound of a mixture within the inner tube 14 may be determined using a number of known techniques, such as those set forth in U.S. patent application Ser. No. 09/344,094, entitled “Fluid Parameter Measurement in Pipes Using Acoustic Pressures”, filed Jun. 25, 1999, and U.S. patent application Ser. No. 10/007,749, entitled “Fluid Parameter Measurement in Pipes Using Acoustic Pressures”, filed Nov. 7, 2001, each of which are incorporated herein by reference. The present invention utilizes at least one probe 10 to determine various parameters of the saturated vapor/liquid mixture, wherein one of the parameters is the speed at which sound travels within in the flow, as will be more fully described herein below.
The frequency signals P1(ω)-PN(ω) are fed to amix-Mx Calculation Logic 138 which provides a signal to line 40 indicative of the speed of sound of the vapor/liquid mixture amix (discussed more hereinafter). The amix signal is provided to map (or equation) logic 142, which converts amix to a percent composition of the vapor/liquid mixture and provides a %Comp signal to line 44 indicative thereof (as discussed hereinafter). Also, if the Mach number Mx is not negligible and is desired, the calculation logic 40 may also provide a signal Mx to line 46 indicative of the Mach number Mx.
where A,B are the frequency-based complex amplitudes of the right and left traveling waves, respectively, x is the pressure measurement location along a tube 14, ω is frequency (in rad/sec, where ω=2πf), and kr,ki are wave numbers for the right and left traveling waves, respectively, which are defined as: k r ≡ ( ω a mix )  1 1 + M x and k l ≡ ( ω a mix )  1 1 - M x Eq .  2
where amix is the speed of sound of the mixture in the tube, ω is frequency (in rad/sec), and Mx is the axial Mach number of the flow of the mixture within the tube, where: M x ≡ V mix a mix Eq .  3
Alternatively, to minimize any error effects (and the need for the corresponding calibration) caused by tube compliance, the axial test section 150 of the tube 14 along where the sensors 115-118 are located may be made as rigid as possible. To achieve the desired rigidity, the thickness of the wall of the test section 150 may be made to have a predetermined thickness, or the test section 150 may be made of a very rigid material, e.g., steel, titanium, Kevlar®, ceramic, or other material with a high modulus.
For one-dimensional waves propagating within a vacuum backed tube 14 ( or a tube immersed in large volume of low impedance fluid such as air at atmospheric conditions), the compliance introduced by the tube ( in this case a circular tube of modulus E, radius R and wall thickness t) reduces the measured sound speed from the infinite dimensional sound speed. The effect of the conduit is given by the following relationship: 1 ρ mix  c measured 2 = 1 ρ mix  c mix 2 + σ where σ ≡ 2  R Et
In the above relation, the fluid SOS, density (ρ) and viscosity (ø) are those of the pure phase fluid, vp is the volume of individual droplets and φp is the volumetric phase fraction of the droplets in the mixture.
In particular FIG. 25 shows the predicted behavior for nominally 50 μm size liquid droplets in vapor for a range of liquid-to-vapor ratios. As shown, the effect of liquid-to-vapor ratio is well defined in the low frequency limit. However, the effect of the liquid-to-vapor ratio becomes indistinguishable at higher frequencies, approaching the sound speed of the pure air at high frequencies (above ˜100 Hz).
[0139]FIGS. 25 and 26 illustrate an important aspect of the present invention. Namely, that the dispersive properties of mixtures of droplets suspended in a continuous vapor can be broadly classified into three frequency regimes: low frequency range, high frequency range and a transitional frequency range. Although the effect of droplet size and liquid-to-vapor ratio are inter-related, the predominant effect of liquid-to-vapor ratio is to determine the low frequency limit of the sound speed to be measured and the predominate effect of droplet size is to determine the frequency range of the transitional regions. As droplet size increases, the frequency at which the dispersive properties appear decreases. For typical steam applications, this transitional region begins at fairly low frequencies, ˜2 Hz for 50 μm size particles.
Referring to FIG. 28 there is shown an optimization procedure in accordance with the present invention in which the free parameters of an analytical model are optimized to minimize an error function. For illustration purposes, the error function utilized is the sum of the differences of the sound speeds between an analytical model and the experimentally determined sound speed as a function of frequency: err = ∑ f = f low f = f high  ( a  ( f ) model - a  ( f ) measured ) 2
As is known in the art, the relationship between quality of a vapor/liquid mixture, a mass ratio, and the volumetric phase fraction of the vapor phase is dependent on the properties of the vapor and liquid phases. For steam the relationship is shown in FIGS. 29 and 30. According to an empirical flow model, the assumption of well mixed, mist-like flows are typically applicable for process mixtures having vapor volumetric phase fractions greater than 0.83 and with mixture velocities exceeding 3.5*sqrt(D*g), where D is the tube 14 diameter and g is the acceleration due to gravity. For example, an 18 inch diameter steam tube translates to mixture velocities greater than ˜8 m/s (˜26 ft/sec).
Therefore, by measuring the propagation velocity of acoustic waves in both directions relative to the stationary tube 14 as described hereinbefore, the mean flow velocity can be calculated by multiplying the mean flow velocity by the cross-sectional area of the tube 14.
For the sound speed measurement, the probe 10 utilizes similar processing algorithms as those employed herein before, and described in greater detail hereinafter. The temporal and spatial frequency content of sound propagating within the tube 14 is related through a dispersion relationship. ω = k a mix
[0162]FIG. 33 shows a k-ω plot generated for acoustic sound field of a vapor/liquid mixture flowing through a tube. Two acoustic ridges are clearly evident. Each of the slopes of the two depicted acoustic ridges respectively defines the speed of sound traveling with and against the mean flow.
Further, FIG. 33 illustrates the ability of the present invention to determine the velocity of a fluid moving in a pipe. The figures are plots of data from an actual test run of a probe 10 in accordance with the invention as described herein above. FIG. 33 shows a wavenumber-frequency plot (k-w plot) of unsteady pressure data collected with a probe 10 of the present invention comprising a 4-sensor axial array in an atmospheric pressure loop flowing air at a velocity of approximately 40 ft/sec. The color contours represent the relative signal power at all combinations of frequency and wavenumber. The highest power “ridges” represent the acoustic wave with slope of the ridges equal to the propagation speed. Note that the acoustic ridges “wrap” to the opposite side of the plot at the spatial Nyquist wavenumber equal to ±3.14 in this case (i.e. the acoustic ridge that slopes up and to the right starting at the bottom of the plot, the right-side ridge, wraps to the left side of the plot at approximately 550 Hz and continues sloping up and to the right). The dashed lines show the best-fit two-variable maximization of the power with the two variables being sound speed and flow velocity. The right-side ridge represents the acoustic wave traveling in the same direction as the bulk flow and therefore its slope is steeper than the left-side ridge that represents the acoustic wave traveling in the opposite direction of the bulk flow. This indicates that the acoustic wave traveling in the same direction of the flow is traveling faster than the acoustic wave traveling in the opposite direction of the bulk flow relative to the stationary sensors located on the probe.
Turbulent tube flows are highly complex flows. Predicting the details of any turbulent flow is problematic, however, much is known regarding the statistical properties of the flow. For instance, turbulent flows contain self-generating, coherent vortical structures often termed “turbulent eddies”. The maximum length scale of these eddies is set by the diameter of the tube 14. These structures remain coherent for several tube diameters downstream, eventually breaking down into progressively smaller eddies until the energy is dissipated by viscous effects.
The probe 170 of FIG. 34 determines the convection velocity of the vortical disturbances within the vapor/liquid mixture by cross correlating unsteady pressure variations using an array of unsteady pressure sensors, similar to that shown in U.S. patent application Ser. No. 10/007,736, filed Nov. 8, 2001, entitled “Flow Rate Measurement Using Unsteady Pressures”, which is incorporated herein by reference.
The vortical flow fields 188 are, in general, comprised of pressure disturbances having a wide variation in length scales and which have a variety of coherence length scales such as that described in the reference “Sound and Sources of Sound”, A. P. Dowling et al, Halsted Press, 1983, which is incorporated by reference to the extend of understanding the invention. Certain of these vortical flow fields 188 convect at or near, or related to the mean velocity of at least one of the elements within a mixture flowing through the inner tube 14 of the probe 170. The vortical pressure disturbances 188 that contain information regarding convection velocity have temporal and spatial length scales as well as coherence length scales that differ from other disturbances in the flow. The present invention utilizes these properties to preferentially select disturbances of a desired axial length scale and coherence length scale as will be more fully described hereinafter. For illustrative purposes, the terms vortical flow field and vortical pressure field will be used to describe the above-described group of unsteady pressure fields having temporal and spatial length and coherence scales described herein.
[0203]FIG. 39 shows an example of a k-ω plot generated from a phased array of pressure sensors. The power contours show a well-defined convective ridge. A parametric optimization method was used to determine the “best” line representing the slope of the convective ridge 200. For this case, a slope of 14.2 ft/sec was determined. The intermediate result of the optimization procedure is displayed in the insert, showing that optimized value is a unique and well-defined optima.
The pressure sensors 18-21 of FIG. 2 described herein may be any type of pressure sensor, capable of measuring the unsteady (or ac or dynamic ) pressures within a tube 14, such as piezoelectric, optical, capacitive, resistive (e.g., Wheatstone bridge), accelerometers (or geophones), velocity measuring devices, displacement measuring devices, etc. If optical pressure sensors are used, the sensors 18-21 may be Bragg grating based pressure sensors, such as that described in U.S. patent application, Ser. No. 08/925,598, entitled “High Sensitivity Fiber Optic Pressure Sensor For Use In Harsh Environments”, filed Sep. 8, 1997, now U.S. Pat. No. 6,016,702, and in U.S. patent application, Ser. No. 10/224,821, entitled “Non-Intrusive Fiber Optic Pressure Sensor for Measuring Unsteady Pressures within a Pipe”, which are incorporated herein by reference. Alternatively, the sensors 14 may be electrical or optical strain gages attached to or embedded in the outer or inner wall of the tube which measure tube wall strain, including microphones, hydrophones, or any other sensor capable of measuring the unsteady pressures within the tube 14. In an embodiment of the present invention that utilizes fiber optics as the pressure sensors 14 they may be connected individually or may be multiplexed along one or more optical fibers using wavelength division multiplexing (WDM), time division multiplexing (TDM), or any other optical multiplexing techniques.
It is reasonable to model the 1-Dimensional acoustic field of a flowing fluid within a duct with the following partial differential equation (Acoustic of Ducts and Mufflers, M. L. Munjal, John Wiley and Sons, page 18): 1 a mix 2  ∂ 2  P ∂ t 2 - 2  M x a  ∂ 2  P ∂ x  ∂ t + ( M x 2 - 1 )  ∂ 2  P ∂ x 2 = 0
The governing equations has propagating wave solutions given as follows:
Where k r = ω a mix  ( 1 + M x ) k l = ω a mix  ( 1 - M x ) ;
and Mx is the axial Mach number and amix is the mixture sound speed.
Acoustics in ducts have a so-called cut-on frequency, below which only one-dimensional acoustic waves propagate. Since this probe is based on extracting information for the resonant behavior of the one-dimensional acoustics, it is prudent to design the probe to operate at frequencies below the cut-on frequency. For circular ducts, the cut-on frequency is given by (Munjal, p12): f cut  -  on = 1.84 D   π  a mix
For a 1 inch diameter circular tube in a fluid with speed of sound of 1000 ft/sec, the cut-on frequency is ˜7000 Hz.
Combining the above equations and solving for the conditions for which the system admits non-trivial solutions results in the following transcendental solution for the eigenvalues of the system:  -    ω a mix  ( 1 + M x )  L -     ω a mix  ( 1 - M x )  L = 0
Thus, for a tube 302 of known length, the sound speed of the fluid, the axial mach number of the fluid, and the natural frequency of the system are linked through the solution of the above equation. Provided an accurate method and apparatus are available for determining the natural frequency of the tube suspended in a duct, the natural frequency measurement can be used to determine the speed of sound of the flow 12 in duct. For ducts with vanishingly small axial Mach numbers, Mx<<1, there is a direct relationship between resonant frequency and sound speed. f = n   a mix 2  L
The transfer function between the input to the acoustic source 304 to the output of the microphone and be expressed parametrically as follows: H  ( s ) = Num Den = ∑ n = 1 N zeros   s - a n ∑ n = 1 N poles   s - b n
Thus, the procedure for determining the natural frequency of the probe 300 involves determining the transfer function from speaker to microphone, fitting the transfer function with poles and zeros, and determining the natural frequency from the location of the poles. Note, best practices in system identification techniques (known by those skilled in the art) should be employed to assure accurate determination of the poles of the transfer function.
The following relation represents a model for the dispersive behavior of an idealized fluid particle mixture 12. a mix  ( ω ) = a fluid * 1 1 + ϕ p  ρ p ρ fluid  ( 1 + ω 2  ρ p 2  v p 2 12  π   μ   D )
[0237]FIG. 44 illustrates importance of particle size in determining the dispersive characteristics of steam. As shown, both the low frequency and high frequency limits of the sound speed are independent of particle size. The high frequency limit determines the sound speed of vapor phase, and the low frequency limit determines the quality of the steam. a mix ( ω  > 0 ) = a fluid * 1 1 + ϕ p  ρ p ρ fluid a mix ( ω  > ∞ ) = a fluid
For steam mixtures, the quality of the steam is given by the squared ratio of the quasi-steady sound speed and the pure phase vapor sound speed. Quality = m vapor m vapor + m liquid = 1 1 + ϕ p  ρ p ρ fluid a mix ( ω  > 0 ) = a vapor * Quality Quality = ( a mix ( ω  > 0 ) a vapor ) 2
[0240]FIG. 45 outlines a flow chart describing the method for using a resonant cavity sound speed probe to characterize dispersive mixtures. The steps of the method includes measuring the frequency response acoustical source to the microphone. The natural frequencies are identified and the poles and zeroes are fit to the transfer function of the acoustical source to the pressure transducer (e.g., microphone ). The speed of sound propagating through the probe is determined as a function of frequency. The SOS vs frequency function is used to determine by fitting the dispersion Model yields to determine a parameter of the fluid or mixture, such as the particle size and steam quality.