System and method for employing an imaginary difference signal component to compensate for boundary condition effects on a Coriolis mass flow meter

A system for, and method of, compensating for a boundary condition effect on a Coriolis meter having (at least) two sensors for generating preliminary signals that are a function of fluid flow through the meter and a Coriolis meter employing the system or the method. In one embodiment, the system includes: (1) signal combination circuitry, couplable to the (at least) two sensors, that develops an imaginary difference signal based on the preliminary signals and (2) boundary effect compensation circuitry, coupled to the signal combination circuitry, that calculates a boundary effect compensation factor based on the imaginary difference signal.

TECHNICAL FIELD OF THE INVENTION
 The present invention is directed, in general, to Coriolis mass flow meters
 and, more specifically, to a system and method for compensating for
 boundary condition effects on a Coriolis mass flow meter based on an
 imaginary difference signal component and a Coriolis mass flow meter
 incorporating the system or the method.
 BACKGROUND OF THE INVENTION
 In the field of flow meters, Coriolis flow meters are unique in that they
 can directly measure the mass flow rate of a fluid with little or no
 intrusion into the fluid stream. Because of this, they have become
 increasingly popular and currently account for the fastest growing segment
 of the overall flow meter market.
 Over the last 15 years, there has been a rapid evolution of developments in
 the field of Coriolis flow meters. These developments have concentrated on
 improving performance by optimizing flow conduit shapes and introducing
 improved signal processing techniques and different modes of vibration.
 This evolutionary process began with the introduction of the first
 commercially-viable Coriolis mass flow meter using a U-shaped flow conduit
 vibrated in its first bending mode of vibration. The signal processing
 scheme employed was a time delay measurement between inlet and outlet
 motion signals. This method could give useful results, however, it was
 understood at that time that the elastic modulus of the vibrating portion
 of the flow conduit was itself a function of temperature, and that any
 changes therein change the sensitivity of the device. The temperature of
 the flow conduit had to be measured; then, the effect of temperature upon
 the elastic modulus of the flow conduit had to be characterized and a
 compensation value added to the flow signal to minimize the effects of
 changes in the elastic modulus of the flow conduit.
 For example, 316L stainless steel is commonly used for the flow conduit
 material in these devices, yielding a theoretical tensile elastic modulus
 vs. temperature relationship of about -2.2% per 100.degree. F. increase
 (in the range between 0.degree. F. and 350.degree. F.) and nearly linear
 for that material. Therefore, the compensation value is commonly applied
 in a linear relationship to account for the effects of temperature on
 tensile elastic modulus. It should be noted here that some meter designs
 depend upon the shear modulus rather than the tensile modulus, or a
 combination thereof, and a corresponding compensation value exists
 thereto.
 While the prior art compensation method was simple, it was also known that
 316L elastic modulus became increasingly non-linear as the temperature
 became colder or hotter, and in general, for most common conduit
 materials, the elastic modulus versus temperature curves are non-linear.
 This fact therefore necessitated adding more complex temperature
 compensation methods to account for a wider range of materials and
 non-linear temperature relationships.
 As more Coriolis flow meters of different designs were put into service, it
 was found that not only temperature, but fluid density and pressure could
 also effect the sensitivity of the device. This realization prompted the
 same type of response from manufacturers as did the temperature problem
 earlier described in that the effects were required to be characterized
 and compensated for.
 In the case of density effects, many types of Coriolis flow meters can
 calculate the density by virtue of the natural frequency of the conduit
 thereby yielding a signal proportional to density that can be used to
 compensate for density effects on sensitivity.
 In the case of pressure effects, it was found that by restricting the
 conduit geometry to certain design relationships, pressure effects could
 be minimized. In either case however, the result was either more
 compensation circuit complexity or geometric design restrictions.
 Flow meters with straight flow conduits were later introduced into the
 market. These meters are subject to temperature gradients between the flow
 conduit and the surrounding support structure that cause stresses in the
 flow conduit that can alter the sensitivity and zero of the device.
 Several methods were therefore introduced to accommodate this added
 problem, such as measuring the difference in temperature between the flow
 conduit and its support and calculating what the stress should be and
 deriving a compensation value based on that difference. Methods employing
 strain gages have also been employed for the purpose of determining the
 stress level and deriving the requisite compensation value, again adding
 more complexity to the circuit and necessitating greater understanding of
 the complex relationships between stress and the change in the sensitivity
 of a given device.
 While the prior discussion has dealt primarily with effects on the
 sensitivity of a flow meter, another important flow measurement parameter
 is the zero. Since Coriolis flow meters are highly linear devices (or are
 made to have linear outputs) relative to mass flow rate, the two most
 important mathematical factors allowing their use as flow measurement
 devices are therefore (a) the slope of the output signal versus the mass
 flow rate therein (here defined as the "sensitivity" or "K-factor"), and
 (b) the value of the output signal at the intercept of the line with a
 zero mass flow point (herein defined as the "zero").
 The zero has been a much more elusive parameter for manufacturers to
 control because zero shifts are not usually caused by predictable changes
 in material constants, etc., but can be caused by a number of subtle and
 interrelated problems in both the mechanics of the flow conduit, and in
 the electronics, both by design or by imperfections therein. These zero
 shifts are normally encountered along with changes in fluid or ambient
 conditions on the device similar to those just described for sensitivity
 effects, e.g., changes in temperature, pressure, density, frequency,
 viscosity or conduit stress.
 To summarize the history, as Coriolis flow meter manufacturers have
 discovered effects on their devices that cause errors or changes in the
 sensitivity of their devices, they have generally chosen to characterize,
 measure and compensate for each effect individually, thereby creating
 complex compensation methods that are more expensive and less accurate
 than the method disclosed herein. A similar progression has taken place
 toward zero effects as well.
 Although these various means and methods just described (and others not
 described) are employed to measure, and compensate for parameters that
 effect Coriolis flow meter sensitivity and zero, the primary and
 fundamental goal of all of these have been simply to determine the
 sensitivity and/or zero of the device to fluid flow, and then compensate
 for any changes therein. What is needed in the art is a way of avoiding
 the need to measure and compensate. What is needed is improved systems and
 methods for directly determining sensitivity or zero characteristics, or
 both, of a Coriolis flow measurement device, thereby allowing overall
 compensation for any changes in sensitivity or zero characteristics,
 regardless of source.
 SUMMARY OF THE INVENTION
 U.S. Pat. No. 5,827,979 deals primarily with apparatus and methods of
 sensing and signal processing for a Coriolis meter and, more particularly,
 for distinguishing between mass flow effects and boundary condition
 effects to produce an output signal that is substantially free from zero
 shifts due to boundary condition effects. The present invention improves
 upon U.S. Pat. No. 5,827,979 by basing adjustments to raw flow rate
 signals on a difference in imaginary components of preliminary sensor
 signals.
 For purposes of the present invention, a "real difference" is a difference
 between real components of signals produced by two sensors, and a "real
 sum" is a sum of real components of signals produced by two sensors. The
 "real components" are the portions of the signals that are substantially
 in-phase with the drive forces that produce flow conduit vibrations.
 Likewise, an "imaginary difference" is a difference between imaginary
 components of signals produced by two sensors, and an "imaginary sum" is a
 sum of imaginary components of signals produced by two sensors. The
 "imaginary components" are the portions of the signals that are
 substantially 90.degree. out-of-phase with the drive forces that produce
 flow conduit vibrations. Boundary conditions are ambient or fluid
 conditions that may introduce error in flow rate measurements and include
 one or more of: fluid temperature, fluid pressure, fluid density or flow
 conduit stress or strain caused by mounting.
 Therefore, the present invention provides a system for, and method of,
 compensating for a boundary condition effect on a Coriolis meter having
 (at least) two sensors for generating preliminary signals that are a
 function of fluid flow through the meter and a Coriolis meter employing
 the system or the method. In one embodiment, the system includes: (1)
 signal combination circuitry, couplable to the (at least) two sensors,
 that develops an imaginary difference signal based on the preliminary
 signals and (2) boundary effect compensation circuitry, coupled to the
 signal combination circuitry, that calculates a boundary effect
 compensation factor based on the imaginary difference signal.
 The present invention therefore introduces the broad concept of employing
 an imaginary difference signal, to determine the degree to which boundary
 condition effects distort the measurements made by a Coriolis meter. The
 present invention is founded on the novel recognition that the imaginary
 difference varies far more as a function of changes in boundary condition
 than it does as a function of changes in fluid flow rate.
 In one embodiment of the present invention, the boundary effect
 compensation circuitry scales the imaginary difference signal to calculate
 the boundary effect compensation factor. The imaginary difference signal
 may be manipulated in any appropriate manner to obtain the desired
 boundary effect compensation factor. Of course, no scaling, skewing or
 other manipulation is necessary to the broad scope of the present
 invention.
 In one embodiment of the present invention, the sensors are selected from
 the group consisting of: (1) strain gages, (2) magnet/coil pairs and (3)
 accelerometers. Those skilled in the art will recognize, however, that any
 sensor that measures a characteristic associated with flow conduit
 movement is within the broad scope of the present invention.
 In one embodiment of the present invention, the boundary effect
 compensation factor is employed to skew a mass flow rate signal derived
 from the meter. In a related embodiment, the boundary effect compensation
 factor is employed to adjust a sensitivity of a mass flow rate signal
 derived from the meter. In an embodiment to be illustrated and described,
 the boundary effect compensation factor may be involved in real-time
 compensation and zero determination.
 In one embodiment of the present invention, the meter operates in a mode
 selected from the group consisting of: (1) a bending mode of vibration and
 (2) a radial mode of vibration. Thus, the present invention is not limited
 to a particular mode of vibration.
 The foregoing has outlined, rather broadly, preferred and alternative
 features of the present invention so that those skilled in the art may
 better understand the detailed description of the invention that follows.
 Additional features of the invention will be described hereinafter that
 form the subject of the claims of the invention. Those skilled in the art
 should appreciate that they can readily use the disclosed conception and
 specific embodiment as a basis for designing or modifying other structures
 for carrying out the same purposes of the present invention. Those skilled
 in the art should also realize that such equivalent constructions do not
 depart from the spirit and scope of the invention in its broadest form.

DETAILED DESCRIPTION
 The following terms are defined for purposes of the present discussion:
 Sensitivity--The slope of the output signal level versus mass flow rate
 relationship of a Coriolis mass flow meter. A typical unit of measurement
 is output signal level per unit mass flow rate (e.g., milliamps/kg/min).
 Zero--The output intercept of the output signal level versus mass flow rate
 relationship of a Coriolis mass flow meter. A typical unit of measurement
 is output signal level indicated (e.g., milliamps, or kg/min) when the
 actual flow rate is zero.
 Zero Shift--Any change in the indicated output signal level of the meter
 not caused by a change in mass flow rate.
 Flow Conduit--The device interacting between the fluid to be measured and
 the sensor or sensors measuring motion, usually a conduit or tube through
 which or around which fluid is caused to flow, but broadly including any
 arbitrary surface over, under or through which fluid flows.
 Mass Flow Effects (Coriolis Effects)--Effects on the flow conduit due to
 Coriolis forces acting to alter the amplitude and or phase relationship of
 the motion at a given location on the flow conduit.
 Boundary Conditions (also "BCs")--The physical properties associated with
 the ends of the active portion of the flow conduit, including properties
 such as stiffness, mass and damping.
 Boundary Condition Effects--Effects on the flow conduit due to changes in
 the boundary conditions in combination with the driven mode of vibration,
 including changes such as stiffness, mass and damping. Typically, these
 effects alter the amplitude or phase relationship of the driven motion at
 a given location on the flow conduit, that can be interpreted as a mass
 flow related signal using traditional signal processing techniques.
 Driven Mode (also "Dm")--The mode of vibration of the flow conduit that is
 intentionally excited as necessary to cause Coriolis forces.
 Coriolis Mode (also "Cm")--The mode of vibration of the flow conduit that
 is a response to Coriolis forces.
 Boundary Condition Mode (also "BCm")--The mode of vibration of the flow
 conduit that is a response to the driven mode of vibration in combination
 with boundary condition effects.
 Referring initially to FIG. 1, illustrated is a cross sectional view of a
 Coriolis meter, generally designated 100, which provides an environment
 within which a system for compensating for a boundary condition effect
 that is constructed according to the present invention can operate. A flow
 conduit 110 is illustrated as being a single straight tubular member made
 of strong, resilient material, such as stainless steel or titanium. The
 flow conduit 110 is fixedly attached at both ends to an inlet manifold 120
 and an outlet manifold 130 by, for example, welding or brazing. The inlet
 and outlet manifolds 120, 130 act to terminate the active portion of the
 flow conduit 110 and to interconnect with the user's pipe fittings (not
 shown) and with a bracket 140 and a case 150. The bracket 140 may be
 fixedly attached to both manifolds 120, 130 by welding or brazing, and
 acts to hold wiring and force drivers 160, 170. The force drivers 160, 170
 are illustrated as being magnet/coil pairs, the magnets of which are
 fixedly attached to the flow conduit 110 and the coils of which are
 fixedly attached to the bracket 140. The force drivers 160, 170 act to
 excite and maintain the driven mode of vibration of the flow conduit 110,
 and to apply reference excitations for sensitivity determination as
 explained in Ser. No. 08/569,967 included herein by reference.
 The bracket 140 is preferably designed to resonate in conjunction with the
 flow conduit 110 to achieve at least a partial state of "balance" so as to
 minimize the energy necessary to maintain the driven mode vibration. The
 resonance of the bracket 140 is not a necessary condition; however, it is
 normally-accepted design practice to minimize the power necessary to run
 the device. Since the bracket 140 is not affected by changes in fluid
 parameters, the state of balance herein achieved is imperfect. The remnant
 imbalance that normally causes zero drift problems with changing boundary
 conditions is acceptable in the present invention.
 The case 150 acts to protect the elements contained therein, enclosing them
 (in the illustrated embodiment) in a pressure-tight case capable of
 maintaining a prescribed amount of pressure or vacuum. Preferably, the
 annular space inside the case 150 and outside the flow conduit 110 is
 maintained at a vacuum or filled with an inert gas, such as helium.
 A feed-through 180 is fixedly attached to the case 150 by welding or
 brazing and acts to convey signals (electrical, optical or of other type)
 between the sensor components and electronic processing circuitry (to be
 described).
 The flow conduit 110 is instrumented with at least two and preferably 5
 motion sensors, such as the first, second, third, fourth and fifth motion
 sensors 190a, 190b, 190c, 190d, 190e that detect some type of measurement
 parameter, such as displacement, velocity, acceleration, strain or stress.
 In the preferred embodiment, each of the first, second, third, fourth and
 fifth motion sensors 190a, 190b, 190c, 190d, 190e comprises four strain
 gages arranged circumferentially around the flow conduit 110 and
 interconnected in a bridge circuit configuration to measure the strain of
 the flow conduit 110 at respective locations.
 The first motion sensor 190a is mounted near the inlet manifold end of the
 flow conduit 110 to measure the strain of the flow conduit 110 at that
 location. So mounted, the first motion sensor 190a can measure a large
 portion of boundary condition effects and a smaller portion of Coriolis
 effects associated with the inlet manifold end of the flow conduit 110.
 The second motion sensor 190b is mounted part way between the inlet
 manifold end of the flow conduit 110 and the center of the flow conduit
 110 to measure the strain at that location. So mounted, the second motion
 sensor 190b can measure a large portion of Coriolis mode effects and a
 lesser portion of boundary condition effects.
 The third motion sensor 190c is mounted near the center of the flow conduit
 110 to measure the strain at that location. So mounted, the third motion
 sensor 190c measures a large portion of the driven mode of vibration and a
 lesser portion of Coriolis mode effects and boundary condition effects.
 Similarly, the fourth motion sensor 190d is mounted part way between the
 center of the flow conduit 110 and the outlet manifold 130 to measure the
 strain at that location. So mounted, the motion sensor 190d can measure a
 large portion of the Coriolis mode effects and a lesser portion of
 boundary condition effects.
 The fifth motion sensor 190e is mounted near the outlet manifold end of the
 flow conduit 110 to measure the strain at that location. So mounted, the
 fifth motion sensor 190e can measure a large portion of the boundary
 condition effects and a smaller portion of Coriolis effects associated
 with the outlet manifold end of the flow conduit 110.
 As stated above, in the illustrated embodiment, the first, second, third,
 fourth and fifth motion sensors l90a, 190b, 190c, 190d, 190e are
 illustrated as being 4-leg strain gages arranged in bridge circuits on the
 preferred embodiment, however alternate embodiments can use alternate
 numbers of strain gages arranged in different configurations. Similarly,
 the first, second, third, fourth and fifth motion sensors 190a, 190b,
 190c, 190d, 190e can alternately be velocity sensors, such as magnets and
 coils, accelerometers or displacement sensors. There is an advantage in
 using strain gages as described, since the bracket 140 may be designed to
 resonate in conjunction with the flow conduit 110, albeit with imperfect
 balance. Sensors that are not referenced to the motion of the bracket 140
 are thereby advantageous, such as the strain gages described or
 accelerometers or inertial reference sensors of any type.
 A first temperature sensor 195a is mounted in conjunction with the flow
 conduit 110 to measure its temperature. Similarly, a second temperature
 sensor 195b is mounted in conjunction with the bracket 140 to measure the
 temperature of the bracket 140 and the force drivers 160, 170.
 Turning now to FIG. 2, illustrated is a graphical representation of typical
 real and imaginary components of sum and difference signals produced by
 symmetrically opposite pairs of sensors along the length of flowtube 110.
 Since in the preferred embodiment, strain sensors are utilized, this
 figure represents strain values as a function of tube position along the
 length of flowtube 110. Adding and subtracting signals from symmetrically
 opposite pairs of motion sensors to form sum and difference components is
 not necessary to employ the present invention, however it simplifies later
 signal processing and is therefore used in the preferred embodiment.
 Symmetrically opposite pairs (190A, 190E) and (190B, 190D) would be
 combined to form the sum and difference signals herein described.
 Signal 200 represents the real component of a sum signal that would be
 acquired by symmetrically opposite motion sensor pairs along the length of
 flowtube 110. Signal 210 represents the real component of a difference
 signal that would be acquired by symmetrically opposite motion sensor
 pairs along the length of flowtube 110. Signal 220 represents the
 imaginary component of a sum signal that would be acquired by
 symmetrically opposite motion sensor pairs along the length of flowtube
 110. Signal 230 represents the imaginary component of a difference signal
 that would be acquired by symmetrically opposite motion sensor pairs along
 the length of flowtube 110.
 In traditional Coriolis meters in which some or all of these signals 200,
 210, 220, 230 are derived, the real difference signal 210 is divided by
 the imaginary sum signal 220, tube frequency and other parameters, to
 yield a resulting signal (not shown) that is proportional to mass flow
 rate. However, the resulting signal is subject to variations in
 proportionality (K-factor) and zero shift subject to changing boundary
 conditions.
 Turning now to FIG. 3, illustrated is a graphical representation of
 variation of the imaginary difference signal 230 of FIG. 2 as a function
 of position along the flowtube 110, where five data sets are presented
 representing four different boundary conditions.
 The present invention is based on the realization that, while the imaginary
 difference signal 230 varies as a function of boundary condition effects,
 the variation is substantially proportional to the variation experienced
 in the imaginary sum signal 220 and substantially independent of mass
 flowrate. When the imaginary difference signal 230 is divided by the
 imaginary sum signal 220, the variations substantially cancel one another,
 yielding a resulting signal (not shown) that is less subject to boundary
 condition effects.
 Finite element analysis (FEA) results show the proportional relationship
 between boundary condition changes and the magnitude of the imaginary
 difference signal. FIG. 3 compares five sets of FEA data. While FIG. 3
 appears to illustrate only three curves, it should be understood that FIG.
 3 in fact illustrates five curves. The two, non-linear curves actually
 embody four superimposed data sets. The third, linear curve embodies a
 fifth data set. Each data set represents the imaginary difference strain
 signal that would be acquired from strain gages positioned along the flow
 conduit 110 at positions shown on the x-axis (from 0" to 20" along a 20"
 flow conduit). The different boundary conditions represented are:
 1. Fixed/free, with flow, asymmetrical design (represented by a data set
 310);
 2. Fixed/free, no flow, asymmetrical design (represented by a data set
 320);
 3. Fixed/fixed, with flow, asymmetrical design (represented by a data set
 330);
 4. Fixed/fixed, no flow, asymmetrical design (represented by a data set
 340); and
 5. Fixed/fixed, no flow, perfect symmetry (represented by a data set 350).
 The data sets 310, 320, 330, 340, 350 on the graph show that each imaginary
 difference signal is greatly affected by different boundary conditions
 and, at most, insignificantly by mass flow rate, since each data set with
 flow is the same as that same boundary condition data set without flow.
 Further, the data set 350 represents no flow and perfect boundary
 conditions and therefore shows virtually no signal at all.
 Since the imaginary difference signal is substantially proportional to
 boundary condition changes, and not substantially proportional to flow
 rate, and since boundary condition changes cause errors on the real
 difference signals (causing errors on the resulting calculated mass flow
 rate), a scaled version of this imaginary difference signal can be applied
 to the mass flow rate calculation algorithm to compensate errors in both
 sensitivity and zero offset.
 There are many different methods of signal processing employed in Coriolis
 mass flow meters, and the present invention is not limited to any
 particular signal processing method. The fact that the imaginary
 difference signal is substantially proportional to boundary condition
 changes and not to flow rate makes it useful for compensation of
 sensitivity and zero errors caused by such changes, regardless of the
 particular method of signal processing or equations used. In the preferred
 embodiment, five motion sensors of the strain gage type are used to
 determine strain value at their locations, and those values are then used
 in a later described curve fitting routine to determine the true mass flow
 rate independent of errors due to boundary condition changes. Simpler
 signal processing methods using only two motion sensors and simple
 equations are also described and may have adequate accuracy in many
 situation. This simpler method is first described.
 The following equation may therefore be used in the illustrated embodiment
 to calculate mass flow rate from the signals derived from the Coriolis
 meter. Compensation is included for boundary condition changes using
 scaled versions of the imaginary difference signals (Idif.). The scale
 factors C1, C2 and C3 are determined by calibration. For purposes of the
 equation, two sensors of the strain gage type are assumed, for example,
 motion sensors 190A and 190E of FIG. 1.
EQU Mdot.tbd.[Rdifel-(C1*Idifel)]/[(Isumel*(C2*SREF)*.OMEGA.)] (1)
 Where:
 Mdot=mass flow rate
 SREF=reference sensitivity=ref_strain /ref_force
 R=real component (component in-phase with drive forces)
 I=imaginary component (component 90.degree. out-of-phase with drive forces)
 sum=summation of (inlet+outlet) symmetrical positions
 dif=difference of (inlet-outlet) symmetrical positions
 e1=no reference excitation
 e2=with reference excitation
 nf=no flow
 .OMEGA.=driven frequency
 C1=constant of proportionality, boundary conditions vs. zero shift
 C2=constant of proportionality, SREF vs. SCOR
 C3=constant of proportionality, boundary conditions vs. sensitivity shift
 Using the simpler two motion sensor method, and looking at FIG. 4, motion
 sensors 190a and 190e, are strain gages arranged in four-leg bridge
 circuits that are conditioned by components 410, and 450 respectively. The
 output signals from the components 410, and 450 are converted to digital
 values by a converter component 460. Component 470 then demodulates the
 strain signals into their real and imaginary components using demodulation
 references from component 480. The real and imaginary components are then
 passed to solver 510 for summing and differencing, thereby creating the
 real difference, the imaginary sum, and the imaginary difference signals
 necessary to implement equations 1 & 2 above. Solver 510 then implements
 said equations 1 & 2 to determine mass flow rate.
 Additional motion sensors along the length of flowtube 110 will yield
 additional data and resolution of the flowtube motion and will result in
 higher accuracy. Therefore the preferred embodiment employs five motion
 sensors. Turning now to FIG. 4, illustrated is a block diagram of signal
 processing circuitry containing a system for compensating for a boundary
 condition effect constructed according to the principles of the present
 invention. In keeping with FIG. 1, the sensors 190a, 190b, 190c, 190d,
 190e are strain gages arranged in four-leg bridge circuits that are
 conditioned by components 410, 420, 430, 440, 450 respectively. The output
 signals from the components 410, 420, 430, 440, 450 are converted to
 digital values by a converter component 460. It is often advantageous to
 reference the measurements taken from the flow conduit 110 relative to the
 motion at a particular location that is least affected by other
 disturbances, such as Coriolis effects or boundary condition effects. In
 the preferred embodiment in which the driven mode of vibration is a first
 order bending mode, the center location of the third sensor 190c is the
 best location for a reference. Therefore, the motion information from
 sensors 190a, 190b, 190c, 190d, 190e is synchronously demodulated in the
 circuit component 470, using the motion of the sensor 190c as a reference.
 The motion of the sensor 190c is conveyed to the component 480 that
 transforms that motion into a reference signal (or a plurality of
 reference signals) for the synchronous demodulator 470. The exact type of
 transformation depends on the type of motion sensors used and the desired
 phase relationship of the reference signal. By selecting a 90.degree.
 phase transformation for a reference signal in the component 480, the
 synchronous demodulator component 470 extracts imaginary strain amplitude
 values containing both driven mode vibration information and boundary
 condition effect information. The synchronous demodulator 470 preferably
 uses a plurality of demodulators, a second one of which then demodulates
 the Coriolis mode information from the signals from sensors 190a, 190b,
 190c, 190d, 190e using a second reference signal from the component 480
 that is in phase with (real) drive forces. This drive mode information Dm
 490 is then used as feedback to the drive and reference exciter component
 500. A temperature sense component 580 conditions temperature signals from
 temperature sensors 195a, 195b for use in temperature compensation within
 the signal processing circuitry.
 A solver 510 performs the function of summing and differencing
 symmetrically opposite pairs of motion sensor signals (190a, 190e) and
 (190b, 190d) thereby creating the aforementioned real difference,
 imaginary sum, and imaginary difference signals necessary to extract true
 mass flowrate information independent of boundary conditions. Solver 510
 also performs the function of curve-fitting the data and thereby solving
 for any or all of the magnitudes of (a) drive mode component, (b) Coriolis
 mode component and (c) boundary condition mode component. For this
 example, strain amplitude data are taken during a vibration cycle of the
 flow conduit 110.
 With these preliminary signals determined, the solver 510 fits the data to
 the anticipated function curves of the driven mode curve Dm (imaginary
 sum), the Coriolis mode curve Cm (real difference) and the boundary
 condition mode Bcm (imaginary difference). Using traditional curve fitting
 techniques, the magnitude of each of these three components (Dm, Cm, Bcm)
 can be accurately established. These components (Dm, Cm, Bcm) then
 correspond to the imaginary sum, the real difference, and the imaginary
 difference, components respectively. Once established, the true mass
 flowrate can then be determined by substituting these values into
 equations (1) and (2) above, in solver 510.
 Many curve-fitting methods are known and well documented and available for
 use to determine the magnitude of one or more characterized components
 (e.g., the Cm component in Equation (4), above) that may be present in a
 data set. These methods include simultaneous solutions of non linear
 equations, curve-fitting routines, and application of particular
 algorithms to the data to separate the Coriolis mode component from the
 boundary condition component, or other error components as hereinafter
 described.
 The highest accuracy can be attained by accurately characterizing the
 curves of all the possible components that can occur in the data (e.g.,
 the Coriolis mode component, the drive mode component, the boundary
 condition mode component, and any other known component). With all the
 possible components known and characterized, numerical solutions can
 accurately determine the relative magnitudes of each component that may be
 present in a given data set. Therefore in the preferred embodiment, this
 is the method used and all known components are characterized and
 submitted to the solver for solution.
 In lieu of characterizing all the possible components, as a minimum the
 Coriolis mode component can be characterized, and submitted to a
 curve-fitting routine for determination of the "best fit" of the Coriolis
 mode shape to a given data set. The closeness of the fit to the data can
 be determined therefrom usually in the form of the "root of the mean
 square deviation" or "IR" value as is commonly used in curve-fitting
 terminology. This R value can also be employed as a compensation value to
 the Coriolis mode component value since it represents the magnitude of the
 deviation of the data from the anticipated characteristic shape of the
 Coriolis mode shape. The use of the R value is not the preferred method
 however, since both random noise in the data, and boundary condition mode
 in the data can alter the R value. By accurate characterization of all the
 anticipated components, the numerical solution accurately discerns between
 the similar shapes of the Coriolis mode component and the boundary
 condition mode component.
 Therefore, the preferred method is that of curve-fitting the data to the
 anticipated characteristic curves to determine (as a minimum) the
 magnitude of the Coriolis mode component. The preferred method of
 curve-fitting involves the use of the Gauss-Newton method as described in
 "C-Curve Fitting and Modeling For Scientists And Engineers" by Dr. Jens
 Georg Reich, McGraw Hill, ISBN 0-07-051761-4. This method involves the use
 of the anticipated characteristic curves involved. These functions are the
 mathematical representations of the expected Driven mode motion, the
 Coriolis mode motion and the boundary condition mode motion. If strain
 gages are used for motion sensors (as in the case of the preferred
 embodiment), then mathematical representations of the strain curves may be
 used for the anticipated functions. Finally, other components 520, 530,
 540 provide signals 550, 560, 570 of use in determining mass flow through
 the flow conduit 110.
 Although the present invention has been described in detail, those skilled
 in the art should understand that they can make various changes,
 substitutions and alterations herein without departing from the spirit and
 scope of the invention in its broadest form.