CA1257784A - Apparatus for mass flow rate and density measurement - Google Patents

Abstract

ABSTRACT

A Coriolis mass flow rate mesuring apparatus including at least one elongated and helically con-figured loop 30 of conduit, a drive mechanism 42 for causing oscillatory movement of a portion of the loop in a direction approximately normal to the direction of flow through the conduit portion and approximately parallel to the central axis of the loop, and one or more detectors 46, 48, 56 for detecting the resulting motion caused by the Coriolis forces exerted on other portions of the loop as a result of the mass flow therethrough and the oscillatory translation thereof.

Description

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Specification "Improved Apparatus For Mass Flow Rate And Density Measurement"

Related Applications.

This application is related to the copending U.S. application of Erik B. Dahlin entitled "Apparatus For Mass Flow Rate And Density Measurement", Serial No. 616,808, filed June 4, 1984 and assigned to the assignee of this application, now U.S. Patent No. 4,711,132.

BACKGROUND OF THE INVENTION
Field of the Invention The present invention relates generally to mass flow rate and density measuring apparatus and more particularly to an improved means for measuring the mass flow rake of a flowing mass using the effects of Coriolis forces and centrifugal forces upon an oscillatorally translated or deflected portion of one or more loops of conduit through which the mass flow is caused to pass.

Description of the Prior Art There has been a continuing need for more accurate and more efficient devices for determining the mass flow rate and density of fluids and flowing solids conveyed through pipe lines and other var.ious types of conduit. Prior art flow meters similar to the present invention have in the past been embodied as gyroscopic mass flow meters or Coriolis type mass , ~

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flow meters.
~ ne such device which utilizes Coriolis forces to measure mass Elow is disclosed in U.S. P~tent ~Jo.
4,109,524 entitled "Method and ~pparatus for Mass ~low Rate Measuremenl-", issued Aogllst 29, 1978 to ~James E.
Smith. In this patent an aPparat.US is lisclose~
wherein a mechanically reciprocating force is applied to first and second sections of a linear conduit by means of a beam that is dispossed parallel to the first anfl second sections and has its ends mechani-cally linked to the adjacent ends of the two conduit sections. The adjacent ends o~ the first and second conduit sections are connected together by means Oe a short segment of conduit or flexible couplings and the opposite ends of each conduit section is separ~tely supported in cantilever fashion to a base structure. The reciprocating forces applie-l to the conduit are resisted by separate Coriolis forces in the first and second conduit sections which act in opposite directions and induce a force moment about the center of the beam which is measured by a torque sensor. By measuring the force moment induced in the con~uits (and transferred to the beam) by the Coriolis reactant forces, measurement of the mass flow through the conduit may be made. ~owever, the measurement is complicated because of the need to avoid spurious measurements of the forces resulting from seismic or other vibrational forces transmitted through the support structure. Other similar devices are dis-closed in the ~.S. patents to Wiley et al, No.
3,g80,750; Sipin, 3,218,851;Souriau, 3,3g6,579; and Sipin, 3,329,019.
Rathee than use linear sections of conduit that are pivoted at opposite ends and reciprocated at ~ ~577~

the adjacent ends, a U-tube or similar con~iguration is more commonly employed in mass flow measurement In such cases the inlet and out1et ends Oe the leys o~
the U-shaped tube are fixedly mountecl to ~ base and the bight end o~ the U-tube is reciprocated. 'rhe differential displacement of corresponding portions of the U-tube legs caused by Coriolis influence on the flow is then measured as an indicator of mass flow rate. Such a technique and apparatus is su~gested in the above-mentioned Smith patent and is illustrat2d in U.S. Patent No. 4,187,721 for "Method and Structure for Flow Measurement" issued February 12, l9~ to James E. Smith, now RE 31,450. As disclosed in the referenced patent, a U-shaped conduit is mounted in a cantilevered manner at the leg ends thereof and an oscillating means is mounted on a spring arm having a natural frequency substantially equal to that of the U-shaped conduit and is used to provide up ancl down motion to the center of the bight end thereof.
Measuring sensors (flags and photodetectors) are pro-vided which detect the leading and trailing portions of the legs of the U-shaped conduit as they pass through a plane deined by the U-shaped conduit at substantially the mid-point of its oscillation. The time diEferential of passage of the legs through the midplane is measured as an indication of mass flow rate. Essentially, the same structure is used in the subsequent Smith Patent no. 4,422,338 referenced below except that in the latter, a pair of velocity sensors are substituted for the photodetectors, and condition-ing electronics.are provided or cleveloping signal.s corresponding to the passage of the side legs through the midplane.
In U.S. Patent No. 4,127,028 entitled "Coriolis ~Z~

- ~ -Mass Flow Rate Metering Means" issued Novernber ~, 1978 to Bruce M. Cox, et al, a pair of vibrating generally U-shape~ tubes are fixedly mo~Jnted at the inlet and outlet ends thereof, in parallely disposed, spaced apart cantilevered ~ashion so that the bight ends oE the respective tubes are free to move relative to each other. An oscillatory drive mechanism is connected between the bight ends of the respoctive tubes and actuated to provide opposing reciprocation thereof such that the U-shaped members act as the tines of a tuning fork. The frequency of the oscillation oE the tube is adjusted until the tubes vibrate a fixed displacement when a known material is flowing therethrough. The power needed to vibrate the tubes the known dispiacement at a fixed frequency determines the density of an unknown fluent material flowing the U-shaped tubes. Mass flow rate is detected by photodetectors positioned to operate in the same manner as taught by Smith ~or a single tube embodiment. Cox also suggests that strain gages or velocity sensors could be substituted for the photodetectors, and acknowledges that it is known in the prior art that there will be a phase shift between the output-s or ~he two sensors which is proportional to the Coriolis force couple.
The principle teaching of this re~erence is the narrowing of the separation of the legs of each U-shaped tube proximate the support ends thereof so as to improve the freedom of torsional twist that may be imposed upon the respective tubes by the Coriolis reactance Eorces. This reference also illustrates a looped tube configuration in Fig. 5 thereof, but fails to teach or suggest how such configuration mi~ht be used to provide enhanced flow measurement. It is ~2577~'~

therefore not believed to anticipat,e the present invention.
Other prior art known to the present inventors may be found in the U.S. Patents to Barnaby et al,

2,752,173, Roth 2,865,20l and 3,0~9,919; Sipin

3,355,9~4; ,Sipin 3,485,09~; Catheral,l 3,955,401 and Shiota ~1,381,b~jt, and the EP~ application o~ Smith, 1 Publication No. EP O 083 144 Al which corresponds to U.S. Patent No. 4,422,338. A listing of prior art utilizing the Coriolis principle may be found in the above-referenced Smith patent RE 3l,450.
A disadvantage of the Smith and Cox type of flow measuring devices, as w9ll as those of others in the prior art, is that they are highly sensitive to to external vibrations which cause the measuring tube or tubes to be subjected to twisting forces other than those imparted by Coriolis reaction forces, and such forces interfere with the actual measurement of mass flow.
Another disadvantage of the prior art U-tube type devices is that they require right angle bends ', outside the measuring sections of the conduit leading ,' to an excessively large pressure drop.
Another disadvantage pertaining to the preferred embodiments in the Smith Reissue RE 31,450 and Smith

4,42~,338 patents is that the proposed methods of time differential measurement at the midplane of the U-tube will produce flow measurement errors when the fluid i,, density is changing. ~, Yet another disadvantage of the prior art ', Coriolis type devices is that they are not capable Of b;
providing accurate flow data over a wide range of flow due to limitations in sensitivity in the flow struc- .J,;
ture used.

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S~ill another disadvantage oE the prior art devices is that they are not provided w;th dynclmic damping means to reduce the sensitivity to externa]
vibrations.
Yet another disadvantafJe of the prior art Coriolis type devices is that they utilize a directly proportional relationship between mass flow rate and differential phase angle or differential time measure-ments.
Yet another disadvantage of the prior art Coriolis type devices is that they have substantial errors in mass flow rate if the temperature of the sensing structure ch~nges.

SUMMA Y OF THE PRESENT I~VENTION
It is therefore a primary objective of the present invention to provide a new and improved apparatus oE the Coriolis type for measuring the mass flow of a fluid or fluent solids, or mixtures oE
these passing through a conduit.
It is another object of the present invention to provide a new an improved apparatus for measuring the density of a mass flowing through a conduit.
A further object of the present invention is to provide means for measuring mass flow rate and density of a mass Elowing through a conduit without introducing perturbing objects or mechanisms in the 5 fluid flow path.
~riefly, a preferred embodiment Oe the present invention inc]udes at least one helically conEigured loop of condult, means for causing oscillatory translation of a portion of the loop in a direction approximately normal to the direction of flow through the conduit portion and approximately parallel to the .~;

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central ~xis of the loop, and means for detecting the effects of Coriolis forces e~erted on other portions of the loop as a reslJlt o~ the mass flow therethrough and the oscillatory translation thereof. ~ strain gage and associated processing electronics are also used in combination with the mass flo~ information to determine the density of the flow.
An important advantage oE the present invention is that it enables true mass flow measurement to be made independent o-f variations of the physical proper-ties of the material flowing through the measuring apparatus.
These and other advantages of the present inven-tion will no doubt become apparent to those skilled in the art after having read the following detailed description of the perferred embodiments illustrated in the several figures of the drawings.

IN THE DRAWINGS
Fig. 1 is a schematic diagram used to illustrate theoretical operation of the present invention;
Fig~ 2 is a diagram schematically illustrating a simplified embodiment of one form of the present invention;
Fig. 3 is a diagram schematically illustrating a first alternative embodiment of the present invention;
Fig. 4 i~ a diagram schematically illustrating a serial multi-loop embodiment in accordance with the 3 present invention;
Figs. 5-8 illustrate parallel flow multi-looped embodiments of the present invention;
Fig. 9a is a diagram schematically illustratlng one method of applying oscillatory energy to the loop i' or loops in accordance with the present invention;

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Fig. 9b is a diagram schematically illustrating a damping technique used in accordanc0 with the present invention;
Fig. 9c is a diagrarn schematically illust:ratincJ
a methoA of detecting the Coriolis ineluence on multiple loops in accordance with the present invention;
Fig. l~ is a set of waveforms illustrating operatior. Qf he present invention;
Fig. 11 is a diagram illustrating an alternative embodiment of a sensor for providing increased sensitivity to measurement;
Fig. 12 is a partial cross section kaken along the line 12-12 oE Fig. 11;
Fig. 13 illustrates an alternative method of detecting the effects of Coriolis forces in accordance with the present invention;
Fig. 14 illustrates an embodiment including a strain gage and electronic processing apparatus in accordance with the present invention; and Fig. lS is a block diagram illustrating appara-tus for computing density of a mass flow in accordance with the present invention.

GENERAL T~IEORY OF OPERATION
~ he present invention is based upon the princi-ple that ~ mass flowing through a looped tube or other straight or curved conduit and experiencing a velocity gradient transverse to the flow path will interact with the wall of the conduit with a ~orce directly related to the transverse velocity gradient and the mass flow rate. When the velocity gradient is caused by the transverse motion of the loop or rotation of the loop about an axis other than the ., .
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central axis of the loop, the reaction i.s known as a Coriolis force. The rnagnitude and direc~ion of the reaction force depends upon the magnitude and direc-tion of the mass flow. If two portions of the loop have the same transverse velocity ~Jradient but have opposite directions of flow, a force couple consisting of equal and opposite reaction forces will result. In accordance with the present invention the result of this force couple is measured as a means of deterrnin-ing the mass flow rate through the conduit.
Referring now to Fig. 1 of the drawings, for purposes of illustation there is shown an example o~
a generali~ed helical loop of conduit 10, with its crossed ends mounted to a base structure 12 and 1~.
The following general theory of the present invention applies for any shape of helical structure and any shape of tube cross-section.
The loop 10 may thus be considered to lie sub-stantially in the X-Y plane only for purposes of sim-plified mathematical ana]ysis; deviation o~ the tube in the Z direction (normal to the X-Y plane) required to permit crossover is ignored.
Accordingly, for a flow tube that is essentially symmetric in the X-Y plane and around the X-axis, the flow and total Coriolis force Pl/2 loop acting upon each half section 16 and 18 respectively, is given by the expression Pl/2 loop Fmass VD (1) where VD is the velocity in the Z direction (normal to the X-Y plane) at the drive-point of the loop, such as point 20 in Fig. l, and ~
Fmass is the fluid mass flow rate. f;
This equation can be used wlth anothe~ equation , .

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to be presented be]ow to describe the dynamics of a loop with the following approximations; narnely, (1) the Cor;olis Eorces are assurned to be lumped in a single point for each half loop "center oE
action point" instead of beiny distributed with varying magnitude along the tube, (2) the ma.ss of fluid and tube material are assumed to be lumped in a single mass point for each half loop instead oE being distributed along the half-loop, and (3) the motion which is different for the different points on the loop is represented by the motion oF the two respec-tive mass points.
At the "center of action point" of the Coriolis force, e~ch half-loop has a certain "participating mass". The center of action is the point where the resultant force of the distributed force for a half-loop is applied and can be computed from th~ particu-lar tube geometry and the general Coriolis force formula for individual mass elements. "Participating mass" is approximately the weight of the tubing and the fluid in each half-loop but ignores the section between the X-axis and the suspension point. This concept takes into account that the motion is not uniform for different points on the half-loop. The participating mass can be experimentally determined by measuring the natural frequency of the bending mode of oscillation around the X-axis and by comparing it with the theoretical natural frequency of the diEEerential equation to be presented. The participating mass is ~;
determine~ so that the two natural Erequencies agree.
The differential equation describ;ng inertial force, damping Eorce and spring action force is: ~
Mpd (Z2zn) ~ A d~t n) ~ s(Z-ZD) = P1/2 loop (2) ?

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--ll--where ~ is the damping factor including both natural damping in the rnaterial and damping introduced by a damping coil as de3cribed hereinbelow;
~ is the spring Eactor describing the restoring force from the spring action due to fixed attachment of the ends of the tube 10;
Mp is the participating mass for one hal~ of the loop;
t is time;
2D is the motion of the center of action point due to drive motion alone; and Z is the motion due to drive motion and Cori-olis force at the center of action point.
The natural frequency of equation tl) above expressed in radians/unit time is Wth = ~ (3) The mode of natural oscillation defined by equa-tions 2 and 3 will be referred to as the "Coriolis mode".
The spring factor B can be determined by static application of a force couple at the center of action points (working in opposite directions approximately at the points 24 and 26 of the two sides of the loop 10) in the Z direction, and measuring the deflection of the center of action points.
If the loop 10 is excited at point 20 by an oscillatory force in the Z direction which varies sinusoidally with an angular frequency of w, the Coriolis force~Pl/2 (at constant flow rate) will be 7' a sinusoidal signal having the same frequency.
Equation (1) above determines approximately the ;
magnitude of the Coriolis Eorce where vD has a , ,i.

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sinusoidal time variation.
The phase shift between Pl~ and (Z-zD) in accordance with equation (2) ls well understood as published in the literture. For example see Grabbe, Ramo, Woodridge, "~landbook of Automation Computation an~ Control" volumn 1, pages 20-59. Defining the damping coefficient z as z = (1/2Wth)(~/Mp) (~) if, for example, the drive frequency W is chosen as 0.5 times the natural frequency, Wth , and the damping coefficient z is chosen as 0.1, then using equations (2) and (3) above, the phase shift can be found to be about -0.8 degrees.
In this example, from equation (4) it can be determined that the ratio of damping factor A to mass Mp is _ A _ = 0.02Wth The amount of damping in this example results from the application of a proper amount of damping force to the loop as will be explained further below.
With a different amount of damping, or a different selection of drive frequency w, but the same natural frequency in the Coriolis mode, a different amount of phase shift would occur.
If the fluid density changes, the natural fre-quency oE the Coriolis mode will change and the phase shift at the drive frequency will also change some-what. For normal density changes of a fluid and for the purpose of calculating an app~oximate phase shift for a given fluid, and for implimenting an approximate compensation for the phase shift by a particular circuit to be described, the density change can be .
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ignored.
A method to compensate approximately for the phase shift between the Coriolis Eorce couple and the structural position in a sinyle or multi-loop ernbodi-ment will be described. [t is especia]ly useful for digital signa] analysis of the waveforms such as described herein. It is also applicablé for different embodiments of the motion sensing devices as presented herein, The structual deflection g, where g is proportional to the magnitude of the Coriolis force couple with a factor predetermined by calibration. A
linear combination of the time derivative g of g, and the integral of the same variable g (for example, performed by an analog integration as will be shown below) may be designated G where G = Klg + K2~gdt = Klg + K2 g (6) where g is a symbol defined by g = dg/dt (7) After L~nlace transformation G may be expressed as G(s) = KlS g(s) ~ K2g(s) By selection of the ratio Kl/K2 , an arbitrary positive phase shift between 0 degrees and 90 degrees can be introduced relating the new function G to the measurement of g. This ratio is selected so that it creates a positive phase shift equal to the negative phase shift resulting from the inertia of movement as approximately described by the differential equation (2). By the proper selection of Kl/K2 there 'r.' ~L2~
~14-will be no significant phase s~ift between the cornpu~
ted variable G and the drive point velocity vD in equation (1) as loncJ as the drive velocity is apprGx-imatel~ a si~ soidal ~unc~ion. Ilo~ever, ~ome deviation from a sin~le sinewave ~harrnonic free) wave-form is permissable; for example, as caused by disturbing mechanical vibrations. Indeed, the insensitivity to this type of disturbance is a strong advantage of the present invention.
In perEorming signal analysis, the variable G and the velocity of the drive point vD are sampled period-ically. The rate would typically be 20 times for each full cycle of application of the drive force. However, in the presence of high frequency disturbing vibration of strong magnitude, a much higher frequency would be more suitable. A lower sampling rate speeds up the signal analysis and may be desirable if the flow meter is used for fast acting flow control.
The samples for G and vD are called Gi and Vi respectively where i = 1,2,3...N, and N is the number of sample pairs used for each measurement. The static relationship between a static force coupling acting as the Coriolis force couple and the static structure measurement "g stat" is g 3 1/2 loop-static ~ ) For dynamic Coriolis forces, using equation (6) the function G describing dynamic gap changes may be expressed as G = K1'R3g -~ K2 K3g (10) where Kl'K3 = Kl and K2'K3 = K2 in equation (6).
For simplicity, it may be assumed that K2 is selected as equal to l/K3 and ~7~
~15-g (Kl /K 2)g (Il) This expression illustrates that G is essen-tially the differential position, velocity or acceler-ation measurement modified by a derivative term ko correct for the phase lag defined by equation (2) above.
Since the variables Pl/2l00pand G are approxi-mately in phase due to the compensation defined by equation (ll), one can for a dynamic system use the equation Pl/2 loop= (1/K3)G (l2) Similar to equation (9), using thi.s expression in equation (1) and solving for G one obtains G = 2K F v (13) 3 mass D
where G and vD are nearly in phase. In a digital system, the variables G and vD are sampled and the sampled pair, i, is called Gi and vi.
Defining ~ = 2K3F (14) J
we then have from equation (13) G ~ vD (14a) may then be determined by linear regression analysis of sample population of Gi and vi . The solution to this expression is for one of the two regression llnes .
related to equation (19a) oC = (~ GiVi)/(;~vi2 ) (15) i-l i=l ,.
One can also use the other regression line which is defined by ~, 57'~

1/~ ( ) (~ Giv~ Gi ) ~l6) The line which divide.s the angle between the two regression lines in half is given by ~ ( g = tan [1/2(arcTan ~ -~ arcTar ~ )1 (17) and the estimate of the mass Elow from this line is obtained from equation (14) as FmaSs = (1/2K3) ~ ~V9) (18) One may, of course, use either one of the regression lines instead of the middle line. ~n illu-stration of FmaSs computed using equations (14) and (15) is given below and in Fig. 15.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
Referring now to Fig. 2 of the drawing, a simplified embodiment of the present invention is illustrated. Tn this embodiment a circular loop 30 of conduit is mounted to a base 32 by means of two standards 34 and 36. Note that loop 3~ is deformed upwardly ~way from the pipe line axis 37 at 38 and down~ardly at 90 to provide clearance at the cross-over point 41. Alternatively, the pipe line matching deformations could have taken place o~tside the standards 39 and 36.
Opposite cross-over point 41 a loop actuating mechanism 42 is mounted to base 32 and has its Eorce applying armature, or the like, electromagnetically coupled to loop 30 at 44. Actuating means 42 i5 of any suitable type which is capahle of causing reciprocating motion of the engaged loop portion along the Z-axis as defined in equation (2) above.
Disposed on each side of the loop 30 are ,~.

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suitable sens~s 46 and 4~ which sirnultan~ously detect the motion ~positions or any tirne derivative or time integral thereof such as velocity OK accell-eration) o~ the loop points 50 and 52 relatlve to base 32 and communicate such measurement to a suitable indicator means 54 which will provide a tneasurement of the effects of the Coriolis forces and thus the mass flow though loop 30. A suitable circuit would be one which first calculates the difference between the properly weighted signals produced by 46 and 48. The weighting factors can be determined so that the difference is zero at zero flow. Alternatively, indicator 54 could be coupled to a suitable means 56 coupled to the crossing portions of loop 30 at the cross-over point 41 and operative to output a signal indicative of the loop separation, relative velocity or relative acceleration; such signal also serving to cause indicator 54 to indicate the mass flow through t~ f~ 30.
~ lthough the present invention as illustrated in the embodiment of ~ig. 2 is a substantial improvement over other prior art devices, it does have the disadvantage that it requires a rigid base 32 for supporting the standards 34 and 36 as well as the actuating mechanism ~2 so as to prevent any vibration in the pipe line from being transmitted to the apparatus in a manner which would influence the measurement obtained thereby. It will be appreciated that in this embodiment, because of the rigid base, any vibrational motion transmltted from the pipe line to the base 32 will likewise be transmitted to the drive mechanism 42 and the loop position detectors 46 and 48. Accordingly, vibrational disturbances will not normally affect the accuracy of the me~surement.

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-]8-However, it will also be appreciated that seismic disturbances may cause the loop 30 to move relakive to the base and thus effect the accuracy of rneasuretnent.
But, if the loop detection source is the detector 56, a large degree of isolation against seismic disturbance is afforded due to the fact that seismic motion in the Z direction will be equally applied to the upper and lower loop portions at the cross-over point 41, and the two will be deflected in the same direction with equal intensity.
In Fig. 3 of the drawing, a modification of the Fig. 2 embodiment is illustrated wherein instead of mounting the drive means directly to the base, the drive means 60 is mounted to a counter-balancing structure 62 which is rigidly attached to the standards 64 and 66. In this embodiment the counter-balancing arm 62 is configured to have the same natural frequency about its support axis as that of the loop 68, and is further provided with an adjustment slide weight 7~ for allowing it to be adjusted to match different densities of the fluid expected to flow in loop 68. Accordingly, in this embodiment, even though rigid end mounts are required, the base does not need to be vibration resistant since the drive mechanism 60 is not attached directly to the base. Using this alternative, another possible source of error is also avoided in that vibrations generated by the flow meter drive 60 are effectively prevented r' from influencing the attached pipe line which might reflect energy back into the subject apparatus.
A feature of the present invention that should be noted from the embodiments of Figs. 2 and 3 is that stresses induced in the tube at its attachment points to standards 34(64) and 36(66) due to actua-;~
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tion by the drive means 42(60), i.e., drive modestresses, are torsional rather than bending. On the other hand, stresses at such points caused by Coriol i5 forces, i.e., Coriolis mo~3e stre.qses, are primarily bending in nature.
In order to increase the sensitivity of the present invention to Coriolis forces, serial and parallel combinations of cross-over loops such as are illustrated in Figs. 4-6 of the drawing may be utilized. In the case the serial double-cross-over loop of Fig. 4, a drive force might be applied between the two loop$ 70 and 72 at 74 causing deflection oE
the two loops in opposite directions. Means provided at 76 and 78 could detect changes in separation, i.e., relative position, ve]ocity or acceleration between the loops, with the d~namic difference in separation being used for computation of the mass flow rate.
~lternatively, detecting of the loop separations at 80 and 82, or the diEference therebetween, could be used as mass flow rate indicators. Similarly, detec-tion of separation between the loops at 84 could like-wise be utilized.
It will oE course also be appreciated that the relati-ve posit.ons of corresponding portions of the individual loop above a base or other reference could also be detected as depicted in Fig. 2 of the draw-ings, and the differences therebetween used to deter-mine mass flow rate. The serial double-cross-over loop of Fig. 4 has excellent flow sensitivity and is especially suited for measuring low flow rates.
However, the structure is somewhat sensitive to outside vibrations and may require the use of damping schemes as described below.
In the parallel loop embodiment illustrated in 7~

Fig. 5, both loops are wound spirally in the same direction, while in the Fig. 6 embodiment, the ~pper loop is wound spirally advancin-~ downwardly whil~ the lower loop is wound spirally advancing upwardly. With actuating Eorces applied between the loops at 86 in Fig. 5, and 88 in Fig. 6, in addition to the loop separation differential measurements (position, velocity, acceleration or other time derivatives or integrals~ discussed relative to the Fig. 2 embodiment, measurements could be taken of the top gap 1~0 (114 in Fig. 6) or the bottom gap 102 (ll6), or the difference between the top gap l00 (11~) and bottom gap 102 (116) could be measured. Simllarly, the differential loop characteristics of separation at 108 (118), 110 (120), 112 (122), could be taken as indicators, as could the differences between 110 and 112, (120 and 122). Measuring the position difference or velocity of relative motion or acceleration between the upper an(l lower loops on opposite sides of the loops, and then calct~lating the differences between these distances (or velocities or accelerations) is the measurement mode which is believed to be the most sensitive to mass -Elow rate. The mass flow rate measurement sensitivity of the devices depicted in Fig. 5 and Fig. 6 can be further improved by configurin~ the loops in such a fashion that the ratio of ~he length L to the height 1l is greater than unity, i~
i.e., L/l-l>l as shown in Fig. 8.
One such embodiment is illustrate~3 in Fig. 7 and 8 and includes a pair of axially elongated loops !~
125 and 127. The shape of the loops need not be ~.
precisely oval or rectangular but can be of arbitary shape so long as L/H>l, where L is the loop length in axis flow direction and 1l is the loop height in the ~I

1, ,, transverse flow direction. Loops of this confiyura-tion exhibit higher sensitivity to measuriny mass flo~
in comparison with loops that have L/fJ - l or L/fl < 1.
In general, this higher sensitivity results from the fact that loops having L/~l > I exhibit a drive mode natural fr?quel,cy Wth closer to the natural frequency in the Coriolis mode, and therefore execute a larger vibrational amplitude, i.e., a higher dynamic ampli-fication factor, in the Coriolis mode for a given mass flow rate and Coriolis force. Loops of this general configuration exhibit significantly larger measurement sensitivity and signal-to-noise ratios compared to prior art devices. In fact, the sensitivity of the loop, i.e., the desired degree of dynamic ampliEica-tion, can be specified and selected by a suitable choice of the L/H ratio. The ability to enhance the measurement sensitivity of the loops in this fashion is of particular interest when optimizing their ability to measure small mass flow rates associated with the flow of highly viscou~ fluids or dense gases.
In order to increase the sensitivity of some of the devices heretofore described, one possible modi-fication is to incorporate structural linkages at the cross-over points in the loop configurations depicted in Figs. 5, 7 and 8. These linkages are depicted in dashed lines at 128 and 129 in Figs. 7 and 8 wherein one link 129 connects the outermost legs of the loops and another link 128 interconnects the innermost legs at the loop cross-over points. These "cross-links"
are rigidly affixed to the flow tubes and would typically be we~lded thereto. The width or thickness oE the links is not cruicial to their performance.
The cross-links 128 and 129 enhance measurement sensitivity to mass flow rate as revealed by the fol-I

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lowing analysis. Careful examination of the structure with cross-links, as shown in Fig. 7, indicates that the structural sti~fness of the loop pair has been substantially increased for vibrations in the drive mode, i.e., response to forces applied approximately a]ong the line of arrow D. However, the structures stiffness for vibrational motion in the Coriolis mode indicated by the arrow C remains substantially unchanged from that when the linkages are absent.
Thus, the incorporation of these cross-links has the effect of increasing the natural frequency of the drive mode relative to the natural frequency of the Coriolis mode, thereby increasing the dynamic amplification factor of the structure and, hence, its sensitivity to mass flow.
~ s an alternative embodiment of difference mea-surement, the difference in phase angle O measured at a given signal level for the position, velocity or acceleration signals produced by sensors g6 and ~8 in Fig. 2 can be used. If velocity sensors are used, F would be calculated from the equation tan 2 (19) mass 2KRwD(sin~ - cos 0 tan ~ ~ ) where ae is the phase angle difference between the outputs of the two velocity sensors, = ~r arctan( ~ z-) (20) 2 Wc -wD
wc R = 2 ~ (WC2-WD2~2+4Zc2 WD
K = loop "flexibility constant" definin~ the change in loop position at one of the velocity sensor :' ' ~X7~,a, locations per unit Coriolis force on the corresponding half-loop. This constant K related to (~ouny's Modu]~s) depencls on the temperature as well as material and geometrical, dirnensions Oe t~le loop. As a special embodiment, one can, durinq flowtneter usage in a process, measure the ternperature of the tuhular wall and input the value o-E the variable into a computing circuit or digital computer and calculate the current value of K. The change in K
with temperature is tabulated in literature or can be determined experimentally. The value of K at a given reference temper~ture is determined by calibration for each flowmeter design or each individual unit.
Wc is the actual frequency of the mode excited by the Coriolis forces (the Coriolis mode) and corre-sponds to the theoretical value Wthin equation (3);
and Zc is the actual damping coefficient for the Coriolis mode (corresponds to the theoretical value z in equation (4)).
Note that the phase angle differencea ~ is equal to the drive frequency wD times the time difference between the waveforms developed by the detectors operating at left and right positions such as 46 and 48 respectively in Fig. 2, 76 and 78 in Fig. 4, or ll0 and 112 in Fig. 5. See Electronics and Radio Engineering by Frederick E. Terman, McGraw-Hill (1955).
Equation (193 is different from and more accur-ate than the equations presented and implemented in the prior art.
Using differential phase angle and drive frequency measurement, it is desirable to drive the device at its resonant frequency because the waveform .:i ~,~

L~
--2~--developed by detectors responding to Coriolis forces will be free of harmonics. Using position or acceler-ation sensors one can easily derive similar formulas.
Well known prior art apparatus capable of detecting the phase angle diff~erence.s is disc]osed in ~pplications ~andbook of Precision Phase Measurement (1975) by Dranet~ Engineering Laboratories, Inc. of South Plainfield, New Jersey, and 11 lett-Packard A~lication Note 2~0-3 (1974) entitled "Precision ~ .
Time Interval Measurement Using an Electronic Counter".
'rurning now to Fig. 9a of the drawing, one possible implementation of a drive system of the type depicted at 42 in Fig. 2 is illustrated in detail.
Mounted to one tube 130 by means of a standoff 131 i 5 a permanent magnet 132. Attached to the loop directly above, as illustrated by the tube segment l34, is a double coil mechanism 133 including an upper winding 136 and a lower winding 138 which are mechanically attached to each other by a member 140, but are electrically isolated from each other. The assembly 133 is mounted relative to magnet 132 such that as current is input to winding 136 a motive force will be applied the magnet which will tend to drive the loop segment 130 relative to the loop segment 132. As the magnet 132 moves within the lower coil 138, a signal will be induced thereln which is proportional to the velocity of the magnet with respect to the velocity of the coil as it moves along the axis of the assembly 133. This signal, illustrated a~ the current il is proportional to the velocity diE~erence. The voltage creatéd by il in the resistor Rl is input to a diEferential applifier ~1, which in turn will gen-erate a voltage signal Vl that is also proportional to : ~, . ,. . ,, .~;

;7~

the velocity diEEerence.
The voltage Vl is then subtracted by an analog computing device 142 from an input voltage V0 gener-ated by an oscillator 1~ that generates a suitable periodic voltage signal Vo in a form such as a sine wave. The difference V0-Vl =V2 is then fed into an amplifier ~2 that generates a drive current i2 which, on passing through the drive coil 136, produces a magnetic fie1d that creates a magnetic force which acts on permanent magnet 132 causing it to osc;llate up and d~wn wi.hin the winding 136, thereby causing loop 130 to be move~ up and down relative to loop 134.
The purpcse of the velocity feedback loop, including winding 138 and amplifier A, is to control the amplitude of the tube oscillation at a desirable magnitude, even if the oscillator is operated at or near the natural frequency of the tube. Without this velocity feedback, or some other means for applying damping force to the tubes, should the oscillatory frequency be set at the natural frequency of the tubes it would cause the tubes to be driven to their elastic limit and perhaps failure.
The behavior of the closed feedback loop illu-strated in Fig. 9a can be approximately described by the transfer function X = K2K3 (21) V0 ms2+ s [a + KKl ~ K3 K4]
where K = velocity feedback loop gain (V/Vl) X = the separation between the tubes at points 161 and 165 in Fig. 9c.
Kl is the gain factor (Vl/il), K2 is the gain factor (i2/V2), r!
K3 is the force between drive coil l36, Fig. 7, 5:
and magnet 132 per unit current i2, ;i j7 ~

K4 is the maynitude of current il per unit velocity difference between drive coil 136 and permanent magnet 13~, s is the r~aplace oper~tor symbol, m is the mass of both tubes 130 and 13~ and the fluid contained therein including only the cir-cular portion of the loops, a is the damping factor of the structure for \ the drive motion, and b is the spring constant of the structure for the drive motion.
The expression a ~ KKlK2K3K4 shows that the nor-mally small damping constant "a" without the velocity ~eedback loop has been enhanced. By selection of appropriate gain factors, damping can be chosen to make the drive amplitude and response c~ignal Vl follow the oscillation signal VO in a desirable fashion.
Any flow tube will exhibit a natural mode of vibration with low damping. Artificial damping and control of the drive may of course be achieved as illustrated in Fig. 9a. ~owever, to accomplish damping without velocity and amplitude control, a similar system, such as is illustrated in Fig. 9b, can also be used. Such a device would be a valuable addition to any type of Coriolis flow sensor, but would be an important improvement over the apparatus shown in the above-referenced Smith patents.
In this embodiment, a permanent magnet 150 is attached to the center loop portion 152 of a double serial loop device that is driven by an actuating assembly 15~ such as was previously clescribed at 133 '' in Fig. 9a. Th~ magnet 150 is disposed to move verti-cally within a damping coil 156 which is rigidly attached to a device base 15a. Connected acr~s~ the l;

,, ~, .

winding of coil 156 is a variable load resistor R.
The current induced in coil 156 by motion of the magnet 150 therewithin creates a current whih passes through resistor R and which i5 proportional to the velocity of the motion of the maynet relative to coil 156. Energy generated by motion of the magnet within coil 156 will be absorbed by energy dissipated in the resistor R. Accordingly, by selection of coil size, number of turns, permanent magnet strength, and the value of resistor R, the extent of damping achieved by such device can be selected to accomodate a particular application.
~ s another alternative which is shown in ~ig. 9c damping coi]s 160 and 162 can be physically tied to velocity sense coils 164 and 166 in a double loop serial (~ig. 4) or parallel (Figs. 5-8) device configuration. In this particular embodiment, the velocity sense coils 164 and 166 are wound in opposite directions and connected together serially so that when both gaps close at the same rate, the total induced EMF is 0. The output currents developed by coils 16~ and 166 in this embodiment are passed through a resistor R to develope a voltage that is fed into a differential amplifier 168 which in turn generates an output signal S that is proportional to the velocity difference between the relative motions of tube portions 161 and 165, and 163 and 167 respec-tively. lhe amplifiers 170 and 172, and the potentio-meters Pl and P2 perform a phase-shieting function to compensate for the phaseshift between the Coriolis and the related motion oE the flow tubes as described mathematically by the equations (2)-(8). This compen-sation is an alternative design eeature which is especially useful with digital signal analysis schemes ~;
i`

7~

-2~-such as described by equations (9)-(18). The signal Sl is fed through the variab~e potentiometer Pl to generate a proportional voltage that is fed into one side of the difference amplieier 170. Sim~ltaneously, Sl is also fed through the integrator 172 to develop a corresponding position signal SO, and this signal is passed through the second potentiometer P2 to generate a proportional voltage that is Eed into the nther side of amplifier l70. The resultant output signal S2 generated is described by equation (~) where the coefficients Kl and K2 correspond to the settings of the pol-?ntiometèrs Pl and P2 respectively.
Position information is obtained in this embodiment by integration oE velocity (or double integration of acceleration if such sensors are used), and difference in position can be computed by integration of the velocity difference (or double integration of the acceleration difference), as shown in Fig. 9c, velocity data is equivalent to position information for the purpose of measurement. The effect of initial conditions associated with inte~ration disappear quickly since the analysis of amplitude is made during many cycles of periodic excitation of the loop for each point of measurement of mass flow rate.
In the illustrated embodiment, the sense coils l6~ and 166 are connected serially so that, as suggested above, in-phase motion of the tube portions 165-167 will produce opposing currents in the resistor j~
R, thus resulting in a net voltage drop ac~oss R of 0 Z
Under influence of ~low through the tubes, induced Coriolis forces in the portions 165 and 167 will cause relative movement of these tube sections in opposite p directions, and a net current resulting from the sum 3~

3l~
.

of the induced voltages in the sense coils 15~ and l66 will be delivered through the resistor R As indica-ted above, the voltage developed across resistor T~ is then fed into the differential amplifier L68 and the output thereof is integrated by 172 to convert the signal Sl, which represents the velocity difference between the relative motion between thé tubes 161 and 165 and the tubes 163 and 167, to a separation difference signal SO.
In Fig. 10 the relationship between steady state signals with sinusoidal drive velocity is illustrated.
More particularly, in part (a) of Fig. 10, the réla-tive drive position of the tubes 130-13~ is shown by the solid line 174, while the relative velocity curve corresponding thereto is shown by the dashed line 175~
It will be noted that the velocity is the derivative of the driving motion and is therefore 90 degree out-of-phase therewith. If there were no flow in the illustrated tubes, it will be appreciated that the position of the tube segments 165 and 167 would be in phase with the position of tube segment 130. These positions are illustrated by the drive component curves 176 in part (c) and 177 in part (b) of ~ig. 10.
Similarly, it will be appreciated that any Coriolis induced deflection will be nearly in-phase with the velocity component of the drive motion, and will be positive with respect to tube 5egment 167 and negative F
with respect to tube segment 165. ~ccordingly, by summing the Coriolis components and the drive compon-ents, the Coriolis induced positional displacements of tube segments 167 and 165 can be derived, and such `
displacements are respectively illustrated in parts (b) and (c) by the indicated waveforms 178 and 179.
In part (d), the difference between the position .

!, ::;

~57~7~

of curves 165 and 167 is depicted as 18~ with no visible phase lag between drive velocity and Coriolis force. In actuality, it will be noted that khere is a small phase ]ag of approximately minus one deyree depending upon drive, "Coriolis rnode", n~tural fre-quency ratio and damping. Curve 181 illustrates this with exaggerated magnitude. Note also that the magni-I tude of the Coriolis component 178 or 179 is very small compared with the drive components 171 and 176, and that Fig. 10 shows exaggerated size of the Coriolis component for ease of illustration.
For effective signal analysis using digital sam-pling of the drive velocity and the response to the Coriolis force, it is desirable that these signals be synchronized. By suitable selection of the settings of potentimeters Pl and P2 of the circuit of Fig. 9c appropriate compensation can be made such that the signal S2 is caused to be exactly in phase with the drive velocity signal.
A shortcoming of prior art velocity sensors of the type disclosed in the above-identified EPO appli-cation of J.E. Smith, is that they are prone to mass flow measurement errors resulting from static deflec-tions caused by thermal changes and variations in static pressure within the flow tube structure. This results from ~',e fact that the velocity sense coil of the said prior art device is moving in a fringing and s~atially non-uniform magnetic field. This results in several undesirable effects all Oe which contrib~te to mass flow measurement error.
The spatially non-uniform field within which the sense coil moves results in unequal induced current contributions (due to non~uniform flux concentrations) in the upper and lower legs of the ~.
r:!
~.

~5~7~3.~

--3l--coil. This can result in variations oE the induced EMF and unwanted harmonic distortions in the induced EMF that are spatially dependent and that also change with static defections of the magnet and coil eyuili-brium positions resulting in flow errors and ~ero flow offset errors. Additionally, velocity sensors o~ the type depicted in the above referred to reference are more sensitive to relative motion of the magnet and coil in directions other than that of preEerential interest in maasuring mass flo~/. This resul ts in more (unacceptable) sens i ti vi ty to unwanted vibrations that degrade measurement accuracy and signal-to-noise ratio.
The above-mentioned shortcomings can be alle-vitated by configuring the sensor as shown in Figs.
11 and 12. In this embodiment, a permanent magnet 182 carried by ei ther another loop or a base structure (not shown) Eorms a gap 184 into which i5 placed, for vertical movement as indicated by the arrow 183, of slectrical conductor 186 wrapped about a spool 187. Spool 187 is rigidly connected by a bracket 188 to the loop conduit shown in broken part at 189. ~s is well known by those skilled in the art, the fl~x field created is the gap between the pole faces of a magnet such as that illustrated, is quite uniform, and a straight conductor (or bundle of conductors) cutting through the flux field will have induced therein an EMF that is directly proportional i to its movement across the field so long as such movement remains ~ ithin the conEines oE the gap.
~ccordingly, the EMF induced in the winding 186 will be directly proportional to the vertical movement (velocity) to loop 189 relative to the magnet 182 and i~
will not be subject to the disadvantages oE the prior 577~'~

art mentioned above.
In Fig. l3 an alternative sensing arrangernent is illustrated wherein instead oE utilizing an electro-magnetic sensing means to sense relative motion between adjacent tube segments, or a tube segment and a base, a fiberoptic sensor may be utilized. In this case a fiberoptic bundle 190 is attached to the upper tube l92 and a reflective plate 195 is attached to the lowe~ tube 194 (or to a base). The fiberoptic sensor apparatus 196 then causes a beam of light to pass through a portion of the optical bundle 190, be reflected by the surface 195 and be returned through a different portion of the bundle 190 to the sensor 196 to effect positional detection. It will of course be understood that any other suitable means of detecting relative position or relative velocity or accellera-tion information may also be used in accordance with the present invention.
The inclusion of density measurement as part of this invention is illustrated in Fig. 14 wherein for simplicity the Coriolis measuring apparatus is not shown. This feature utilizes the fact that the cen-trifugal forces acting upon each element of the fluid flowing in the looped portion 202 of tube 200 is directed from the center of curvature for the element perpendicular to the tube section the element is in.
For the purpose oE technical analysis, it is assumed that the centrifugal forces for the elements are not far removed from the plane of the drawing. Thus, the centrifugal force is inversely proportional to the radius of curvacure for the element, proportional to the mass within the element and proportional to the square of the fluid velocity. Moreover, the centrifugal forces on the output half 20~ of the loop ;1 3_2~j~7~1~

will cause a pul] to the left at point 2~5 and the forces on the input half 206 of the loop will cause a pull to the right at point 205, The oppo~ing forces at 205 will thus cause a strain in the material that is related to the aggregate of the centrifugal forces on the whole test section. Since the Coriolis forces cause no strain at the point 205, measurement of the strain at that point 20~ may be accomplishecl using a simple strain gauge 2U8. Moreover, a strain gauge measurement taken anywhere along the loop will furnish inEormation permitting the accomplishment of a density measurement in conjunction with the Coriolis mass flow rate measurement although the calibration rela-tionship will be more complex.
- For the illustrated confiyuration, the ratio of the total centrifugal force ~or the half-]oop 20 acting in the horizontal direction at the point 205, and the total Coriolis force acting upon the same half-loop is directly proportional to the velocity of the material in the fluid and is independent of all other characteristics of the fluid.
The density of the material in the conduit is directly proportional to the square of the measured mass flow divided by the centrifugual force acting upon the half-loop. One circuit by which such information may be developed is illustrated in Fig. 15 t and includes a means 210 for sampling the drive signal Vl (from Fig. 9a) and signal S2 (from Fig. 9c), and strain gauge signal C (Erom Fig. I4), a means 212 for computing F~ssfrom the sampled signals Vl and S2 , a means 214 Eor calculating the centri~ugal ~orce Pcentr ~l from the strain gauge signal C, and a means 216 for ~l calculating the density from Fma~S and Pcent. I~, Although the present invention has been described il !.~;,~
~' 7~ h -3~-above by referring to several examples illustrated in the drawing, it is to be understood that such embo-li-ments are presentqcl for ilLustration onl.y and are r,ot intended to in any way be limiting. It is intended that the appenZed claims be interpreted as covering all embodiments, alterations and modifications as fall within the true spirit and scope of the invention.

Claims (19)

1. A flow meter for measuring the mass flow rate in a stream of flowing material, comprising:
tubular conduit means arranged to form at least one elongated loop having an elongated entrance portion, an elongated exit portion crossing said entrance portion in spaced apart generally parallel relationship, an actuated portion, a first detection portion coupling said entrance portion and said actuated portion, and a second detection portion coupling said actuated portion and said exit portion, said loop having a dimension L in the direction generally parallel to the direction of flow entering and exiting said conduit means and a loop dimension H
transverse thereto such that the ratio L/H is greater than unity, said conduit means being adapted to receive, conduct and then emit said stream of flowing material;
actuating means for causing said actuated portion to oscillate between positions on opposite sides of a rest position;
detection means for detecting motive differences occurring between said entrance portion and said exit portion caused by Coriolis forces induced in said first detection portion and said second detection portion as said actuating means causes said actuated portion to oscillate; and computing and indicator means responsive to said detection means and operative to indicate the mass flow rate of material flowing through said tubular conduit means.

2. A flow meter as recited in claim 1 wherein said detection means includes a first sensor means for detecting the dynamic position of said first detection portion relative to a reference and for generating a first position signal, and second sensor means for detecting the dynamic position of said second detec-tion portion relative to said reference and for generating a second position signal, said computing and indicator means utilizing the difference between said first and second position signals to provide an indication of the mass flow rate of material flowing through said tubular conduit means.

3. A flow meter as recited in claim 1 wherein said tubular conduit means includes two loops having corresponding parts disposed in spaced apart relation-ship, and wherein said actuating means is coupled between the actuated portions of each loop and serves to apply simultaneous equal and opposite actuating forces to each loop when energized.

4. A flow meter as recited in claim 3 wherein said first and second loops form a serial flow path for said flowing materials.

5. A flow meter as recited in claim 3 wherein said first and second loops form parallel flow paths for said flowing materials.

6. A flow meter as recited in claim 5 and further comprising first and second linking means rigidly linking together corresponding portions of said first and second loops at points diametrically opposite the points at which said actuating means apply said actuating forces to said loops.

7. A flow meter as recited in claim 6 wherein said detection means includes a first sensor means coupled between the first detection portion of said first loop and the first detection portion of said second loop, and a second sensor means coupled between the second detection portion of said first loop and the second detection portion of said second loop said first and second sensors being operative to develope output signals indicative of the motive relationships of the corresponding portions of said first and second loops.

8. A flow meter as recited in claim l wherein said detection means detects a characteristic of the change in the spacing between said entrance portion and said exit portion at the crossover point.

9. A flow meter as recited in claim 1 wherein the said motive differences are the differences in arrival times of said entrance portion and said exit portion at one or more predetermined reference points of position, velocity or acceleration.

10. A flow meter as recited in claim 5 wherein said detection means detects a characteristic of the change in the spacing between said entrance portions and said exit portions at the crossover point.

11. A flow meter as recited in claim 6 wherein the said motive differences are the differences in arrival times of said entrance portions and said exit portions at one or more predetermined reference points of position, velocity or acceleration.

12. A flow meter as recited in claim 1 wherein said tubular conduit means includes two loops of the type recited in claim 1 connected to form parallel flow paths and having corresponding parts disposed in spaced apart parallel relationship, and wherein said actuating means is coupled between the actuated portions of each loop and serves to apply simultaneous equal and oppositely directed actuating forces to each loop.

13. A flow meter as recited in claim 12 wherein said detection means includes a first sensor means coupled between the first detection portion of said first loop and the first detection portion of said second loop, and a second sensor means coupled between the second detection portion said first loop and the second detection portion of said second loop, said first and second sensors being operative to develop output signals indicative of the motive relationships of the corresponding portions of said first and second loops.

14. A flow meter as recited in claim 13 wherein said first and second sensor means each include a pair of coaxially disposed electromagnetic windings and a permanent magnetic element magnetically coupled to both said windings, said windings being rigidly secured to one of said loops, and said magnetic element being rigidly secured to the other of said loop, and resistive impedance means connected to one winding of each said pair of windings for dissipating energy induced therein by movement of said magnet relative to said windings, and means connecting the other winding of each said pair of windings to said computing and indicator means.

15. A flow meter as recited in claim 13 wherein said first and second sensor means are velocity sensors and said computer and indicator means calcu-lates mass flow rate Fmass according to the formula where .DELTA. ? is the phase angle difference between the output signals developed by said first and second velocity sensor means;
K is a constant or alternatively, a variable computed from measurement of the flow tube tempera-ture and related to Young's Modulus;

WD is the natural frequency of the tubes in the drive mode;
WC is the actual natural frequency of tubes in the Coriolis mode; and ZC is the damping coefficient for the Coriolis mode.

16. A flow meter for measuring the mass flow rate in a stream of flowing materials, comprising:
tubular conduit means arranged to form first and second loops disposed to form parallel flow paths with each loop having an entrance portion crossing an exit portion, an actuated portion, a first detection por-tion coupling said entrance portion and said actuated portion, and a second detection portion coupling said actuated portion and said exit portion, said conduit means being adapted to receive, conduct and then emit portions of said stream of flowing material;
actuating means for causing said actuated por-tions to oscillate relative to one another;
first and second linking means rigidly joining corresponding portions of said first and second loops together at points diametrically opposite said accuating means;
detection means for detecting motive differences between said entrance portions and said exit portions caused by Coriolis forces exerted thereon as said actuating means causes said actuated portions to oscilliate; and computing and indicator means responsive to said detection means and operative to indicate the mass flow rate of the fluid flowing through said tubular conduit means.

17. A flow meter as recited in claim 16 wherein said detection means includes a velocity sensor comprised of a permanent magnet having a gap formed between opposing pole faces, and a sense winding affixed to and carried by said conduit means and having a straight portion of at least one con-ductor of said winding adapted to move within said gap as said conduit means moves relative to said magnet.

18. A flow meter as recited in claim 16 wherein said detection means includes first and second sensor means that are velocity sensors and said computer and indicator means calculates mass flow rate Fmassaccord-ing to the formula where .DELTA. ? is the phase angle difference between the output signals developed by said first and second velocity sensor means;
K is a constant or alternatively, a variable computed from measurement of the flow tube tempera-ture, related to Young's Modulus;

WD is the natural frequency of the tubes in the drive mode;
WC is the natural frequency of tubes in the Coriolis mode; and ZC is the damping coefficient for the Coriolis mode.

19. A flow meter as recited in claim 1 wherein said detection means includes a velocity sensor comprised of a permanent magnet having a gap formed between opposing pole faces, and a sense winding affixed to and carried by said conduit means and having a straight portion of at least one con-ductor of said winding adapted to move within said gap as said conduit means moves relative to said magnet .
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