FUNDAMENTALS of ULTRASONIC FLOW METERS Keven Conrad and Larry Lynnworth Panametrics, Inc

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FUNDAMENTALS OF ULTRASONIC FLOW METERS Keven Conrad and Larry Lynnworth Panametrics, Inc. 7255 Langtry, Houston, TX 77040-6626 and 221 Crescent Street, Waltham, MA 02453-3497

ABSTRACT Depending on the uncertainty in flow profile, the velocity along the path or paths can be converted to an area- Ultrasonic contrapropagation methods have been used averaged velocity VAVG. For a single path it is common to to measure the flow of natural gas since the 1970s, flare relate the path and area-averaged velocities by a meter gases since the 1980s, and smokestack gases in cem factor K defined by K = VAVG/VPATH. The actual volumetric (continuous emissions monitoring) since the 1990s. Since × flowrate Q = VAVG A where A = area of the conduit. This the early 2000s, ultrasonic clamp-on flow measurements, means Q = KVPATH. In certain multipath flowmeters the previously restricted mainly to liquids, were found paths and weights assigned to the paths are such that effective in measuring in standard steel pipes, the flow the resulting integration of individual path measurements of steam, natural gas and other gases and vapors, is largely independent of profile details. Of course, as including air, as long as the flow velocity was not so high the flow departs from ideal conditions, even a quadrature as to cause excessive beam drift or excessive turbulence integration method becomes less accurate, but in many (in other words, below about Mach 0.1), and provided practical situations, accuracies better than 0.5% are the acoustic impedance of the gas was equivalent to air routinely obtained. above about six bar and no important molecular absorption or scattering mechanisms were present. Since the early 2000s, ultrasonic clamp-on flow Although the flow of gases by ultrasonics has long been measurements, previously restricted mainly to liquids, thought to be more difficult to measure than liquids, in were found effective in measuring in standard steel pipes, fact the measurement is easier in two important respects. the flow of steam, natural gas and other gases and One is, for the contrapropagation method, the upstream vapors, including air, as long as the flow velocity was - downstream time difference is generally much greater not so high as to cause excessive beam drift or excessive for gases, as a consequence of the much lower sound turbulence (in other words, below about Mach 0.1), and speeds in gases compared to liquids. The other provided the acoustic impedance of the gas was significant factor that becomes important in mass flow equivalent to air above about six bar and no important metering (including scfm output) is the existence of molecular absorption or scattering mechanisms were theoretical and/or empirical relationships between present. The caveats mean, avoid carbon dioxide; avoid ultrasonic propagation and density, where either of such some or all halogen vapors; and beware of mist or relationships is easier to exploit for gases than for liquids. particulate-laden gases. Another limit: the gas (or steam) To provide an idea of the scope of applications temperature T cannot exceed the T limit of the transducer addressable with ultrasonic technology that is or couplant, whichever is lower. [1, 12] commercially available now or likely to be available in the near future, this paper starts with an analysis from Although the flow of gases by ultrasonics has long been the point of view of acoustic impedance; considers thought to be more difficult to measure than liquids, in designs as a function of the number of nozzles, from fact the measurement is easier in two important respects. zero to a dozen; and lists factors conducive to high One is, for the contrapropagation method, the upstream accuracy versus factors detrimental to high accuracy, - downstream time difference is generally much greater i.e., conducive to uncertainty. for gases, as a consequence of the much lower sound speeds in gases compared to liquids. (Exception: INTRODUCTION hydrogen gas; its sound speed at 100˚C ~ speed of sound in ordinary water.) The other significant factor that Ultrasonic contrapropagation methods have been used becomes important in mass flow metering (including scfm to measure the flow of natural gas since the 1970s, flare output) is the existence of theoretical and/or empirical gases since the 1980s, and smokestack gases in cem relationships between ultrasonic propagation (sound (continuous emissions monitoring) since the 1990s. speed c and/or attenuation coefficient α) and molecular Contrapropagation means sound waves are timed in a weight or density, where such relationships may be easier direction with the flow and later or simultaneously, against to exploit for gases than for liquids. Examples supporting the flow. At low Mach number, <<1, the time difference the determination of gas density from ultrasonic

is directly proportional to the flow velocity VPATH along measurements, after T compensation, include: the the path. Even at Mach 0.1 the time difference is very amplitude of the received signal in still gas is nearly nearly proportional to the velocity along the path. In any proportional to gas pressure; the sound speed c is event, by timing upstream and downstream, the correct inversely proportional to the square root of MW velocity can be computed along the path. (molecular weight), and gas density is proportional to

PAGE 52 2002 PROCEEDINGS AMERICAN SCHOOL OF GAS MEASUREMENT TECHNOLOGY molecular weight times pressure. At high molecular 2LV / c2 ∆ 2 2 4 weight and/or high pressure, these simple (linear) t = = (2LV / c ) (1 + MN + MN + ...) (5) approximations are inadequate. Virial equations and 1 – MN supercompressibility provide a remedy. At sufficiently small Mach numbers, the following To provide an idea of the scope of applications approximations are valid: addressable with ultrasonic technology that is commercially available now or likely to be available in the V ≈ c2∆t/2L and ∆t ≈ 2LV/c2 (6 & 6a) near future, this paper starts with a short theoretical section in which the contrapropagation equations are derived. This 2. Profile Considerations is followed by profile considerations. Then we go to acoustic impedance. Lastly, we consider designs as a As is well known [2], flow of any fluid in a pipe is lower function of the number of nozzles, from zero to a dozen; near the wall and higher near the center. Disturbances and list factors conducive to high accuracy versus factors upstream or downstream perturb the profile and most detrimental to high accuracy, i.e., conducive to small or real profiles are not symmetrical about the axis. Gas flow large uncertainty. There are other “fundamentals” besides profiles can be more complicated than simple liquid these. However, to keep this paper to reasonable length, profiles for various reasons including: (a) gases are only these topics are treated. The references, particularly compressible; (b) at high Mach number, MN 0.3, the the 2002 paper by Walters et al. [15] deal with aspects flow itself becomes “compressible flow;” (c) condensate beyond the scope of this paper. perturbs the boundary conditions and can materially affect the duct area A available for gas flow. See Table 2. DISCUSSION Liquids. To deal with complex liquid flow patterns in such Theory a way that an accurate measure of total flow is obtained despite the complexity, manufacturers of ultrasonic 1. Equations for a Contrapropagation Flowmeter flowmeters found that one diameter path did not suffice. Thus we find a progression from one diameter traverse In the contrapropagation method, ultrasonic (or in [14] to crossed diameter paths in [13], and quadrature sometimes audible) waves are transmitted upstream and multipath solutions in [2(b) or 7]. Essentially the same downstream. From the transit times t1 and t2 in each quadrature multipaths were utilized for high-accuracy direction, and knowledge of the path and flow profile, (low-uncertainty) gas flow measurement. Midradius paths the average flow velocity VA is determined. A rather are also used, but unlike the history for liquids, their first simple derivation of the basic flow-sensing equation is use (1975) by Roger C. Baker appears to be in air [cited possible if one imagines a fluid of sound speed c flowing in 7 & 15 ] , later in liquids and most recently, in natural at a uniform velocity V < c in a duct of cross-sectional gas by Jan G. Drenthen and his colleagues. [See 2(b) or area A, interrogated by two point sensors on the axis 15 for these references]. and spaced a distance L apart. The transit times in the upstream and downstream directions, respectively, are 3. Acoustic Impedance Z

ρ ρ t1 = L/(c – V) and t2 = L/(c + V). (1 & 1a) The characteristic acoustic impedance Z = c where = gas density and c = sound speed in the gas. Why is Z The reciprocals of these transit times, when multiplied important? Because it determines the fraction of available by the axial projection of the axial interaction path L, are ultrasonic energy from the transducer transmitted into the gas, and vice versa. Figure 1(a) plots gases, liquids ρ L/t1 = c – V and L/t2 = c + V. (2 & 2a) and solids as a function of their and c. As this is a log- log plot, lines having a slope of –1 are lines of constant Accordingly, Z. The Z for most solids, say steel, is orders of magnitude greater than for methane or any other gas at ordinary L 11 L ∆t conditions of temperature T and pressure P. Solids are V = = (3) 2 [ t t ] 2 [ t t ] in the >10 megarayl range. Gases are in the 0.1 to 1 2 1 1 2 kilorayl range. [Plots for gases only are given in Figs. and 1(b-d). When c yields average MW (molecular weight) or density, mass flowrate M is not far behind.] L 11 Lt F c = = (4) 2 [ t t ] 2 [ t t ] Entering a gas on such a chart requires that the gas’ 2 1 1 2 sound speed and density are known. Entries are usually associated with STP (0˚ C, 760 mm Hg). The locus for The upstream-downstream time difference can be methane, if examined carefully, reveals a slight increase obtained from Equation (1) as ∆t = 2LV/(c2-V2). This can in c as P increases, as well as the expected nearly ρ be expressed in terms of the Mach number MN=V/c for proportionate increase in density as P increases. MN << 1:

2002 PROCEEDINGS PAGE 53 AMERICAN SCHOOL OF GAS MEASUREMENT TECHNOLOGY The departure from linearity between density and 4. Designs As a Function of the Number of Nozzles pressure is often treated by a supercompressibility factor. See for example, pages 29-52 in the chapter by Bill Zero Nozzles. This means clamp-on (Figure 3). For Buzzard in David W. Spitzer’s 2001 book cited in [2]. details see [1 & 9] or www.panametrics.com, PCI or PCI R&D pages, Ultrasonic Report UR-264. Accuracies One of the ways of overcoming some of the large around 2% have been achieved with one path. Crossed mismatch in impedances for gases at or near atmospheric diameters can improve the accuracy, as in [13]. As of pressure is to build the transducer with a quarterwave mid-2002, applications were restricted in general to metal impedance matcher between the solid piezoelectric (of pipes of diameter >3 inches (>75 mm) but less than about high Z) and the gas. Figure 2(a) shows a T7 transducer 2 ft (600 mm). These are guidelines. At high pressure, if built this way. To preserve the transducer from unwanted V is not too high, gas flows in pipes as large as 30 inches effects of the “gas,” a Ti, SS or other thin metal layer have been measured by clamp-on. Bad pipe, heavy wall, surrounds the piezo and matcher, in these designs [8; 8(b)]. noisy environment, and/or access limited to one side, Sometimes, as in cem (continuous emissions monitoring), can frustrate the method. Success depends on a number one transducer is not enough, so arrays are used [Fig. 2(b)]. of factors, including the state of the equipment, e.g., its Fiberacoustic “BWT” bundled waveguides [6] have been software version, as well as pipe and flow conditions. used in many gas applications (up to ~ 500˚C) without any matcher [Fig. 2 (c), (d)]. As has been the practice for liquids since the 1960s and 1970s [11, 13, 14], measurements in ⊥ planes (X in end This matching idea works if the transducer is “wetted,” view) yield better accuracy than measurements in one i.e., in contact with the gas as in Figure 2(a). In clamp-on plane only. Unlike liquids [Fig. 3(a), parts (1) & (7)], the (Figure 3) the transducer wedge contacts the steel (or vee path, with both transducers on the same side of the other material), not the gas, so another solution is pipe, is hard to use when the fluid is a gas, because of necessary to overcome the mismatch. Clamp & too much acoustic crosstalk. Sometimes crosstalk is installation details are given in Fig. 3(b-d). called acoustic short circuit noise.

One might say the low Z of a gas is mostly due to its low One Nozzle. Insertion probes can go in oblique or density, compared to steel or other elastic solids. We can normal. Some ultrasonic solutions use an intrusive also ask, are there special consequences of the low c of a reflector but the transducers themselves are flush to the gas, say methane, compared to steel? One consequence ID or recessed by up to a foot (300 mm) or so if the gas is, in clamp-on, the angle of the sound beam refracted is hot. The advantages of the one-port compared to >1 into the gas is small, say 5 to 8 deg, according to a port are economy and ease of installation. Drawbacks calculation using Snell’s Law of Refraction. This implies, are accuracy, to the extent the profile is uncertain. The a V measurement along a path only a few degrees off one-port samples only a small chord segment. If the normal, is going to be very sensitive to crossflow. If reflector is located about 60% of the radius in from the crossflow is significant, crossed clamp-on paths as shown wall, and to the extent the profile obeys a Nikuradse in Figure 3(f) may be necessary. A second consequence power law, then the flow averaged from the wall to that

of low c and small refracted angle is that paths off the reflector nearly equals VAVG. However, the length of the diameter (as seen in the end view) are not reachable with chord segment, in the end view, is only 0.3D where today’s technology. This means a clamp-on quadrature D = pipe inside diameter. This small sample may be is out of reach. Exception: hybrid [Figure 4(g)]. contrasted with multipath sampling for a 12-nozzle spoolpiece referred to at the end of this section. One- The “good” news associated with low c is the time ports are shown in [15]. difference ∆t between upstream and downstream interrogations is larger, typically in the many (tens of) Two Nozzles. Examples from the “flare gas world” are microseconds range, compared to liquids, where ∆t may given in [10 & 15]. The bias 90 arrangement samples a be just a few microsences and must be resolved to 1 ns short chord segment but seems to provide adequate or less. On the other hand, the attenuation coefficient in accuracy. It is probably the most common hot-tap gases is usually much higher than in liquids, and along geometry for flare gas flow measurements. with impedance mismatch, constrains the ultrasonic frequency to be < 1 MHz for gases, whereas for liquids ƒ Twelve Nozzles. Custody transfer spoolpieces shown > 1 MHz is common. in Figure 4(d) use crossed paths (hence four transducers per plane) in each of three parallel planes (hence 12 How long have we waited for a practical clamp-on transducers total). ultrasonic flowmeter for gases to become available commercially? One answer is thirty-nine years. This Accuracies (by which we mean uncertainties) are 0.5% answer is based on (a) the GC868 announcement in [9], or better, 0.3% in some tests. By using transducers and on (b) the last paragraph in [14]. That 1966 patent is shown in Figure 4(e), tests can be run in air at atmospheric one of the earliest (perhaps the earliest) clamp-on U.S. pressure. Air calibration allows relatively economical patents for liquid flow, and claims a Japanese priority testing under various disturbed-flow conditions such as date of 1962. upstream elbows, diameter changes, partly open valve,

PAGE 54 2002 PROCEEDINGS AMERICAN SCHOOL OF GAS MEASUREMENT TECHNOLOGY or their combinations. [3] Quadrature integration (Figure measurement of secondary flow, especially circulation, 4) is intended to suppress the effects of flow profile. The is dealt with in [5]. TABLE 1. Factors Conducive to High Accuracy Versus Factors Detrimental to High Accuracy, i.e., Conducive to Uncertainty High accuracy if: Low accuracy or uncertain results if: Long straight run, steady flow, single phase Nearby upstream disturbances; intermittently two- phase or multiphase Quiet environment, no electrical interference Noisy environment; crosstalk unavoidable Good pipe (concentric ID & OD), roughness known Rough or scaled pipe, roughness unknown and subject and not changing to changes over time; gas fouls transducers Gas is known and does not exhibit molecular Highly attenuative, e.g., carbon dioxide, halogens; absorption; T & P allow good transmission; P too low or T too high, such that absorption due to equipment provides adequate reciprocity (signals classical viscosity and thermal conductivity effects from A → B look like signals B → A) is high Flow velocity is in a range such that the ∆t falls into V is too low or too high to be accurately and reliably an easily measured range; no cycle skip; no packet measured by the available sensors and electronics skip; no ambiguity about arrival time

FIGURE 1. Some basic acoustic ideas and facts for gases. (a) Impedance nomogram for gases, liquids and solids. (b) In this ρc plot for gases, the c’s range from ~100 to 1300 ms-1, the MW’s from 2 to 240, but γ, the ratio of specific heats, lies between 1 and 1.67. The effect of gas absolute temperature T is shown on cAr when argon is heated at constant density, 1/2 drawn assuming cAr increases in proportion to T . (c) Flare gas: empirical relation between c at 38C˚ and average molecular weight (MW) for 2≤ MW≤ 58. (d) Air density ρ vs temperature T. Sound speed c vs T. ➂ ρ vs c. (P = 760 mm for graphs -➂.) Note that even if RH (relative humidity) is not known, between 0˚C and 60˚C, c yields with small uncertainty. For details see Matson, J., Mariano, C. F., Khrakovsky, O., and Lynnworth, L. C., Ultrasonic Mass Flowmeters Using Clamp-On or Wetted Transducers Proc of the 5th International Symposium on Fluid Flow Measurement (April 7-10, 2002) or refer to the website of the authors’ firm, where Ultrasonic Report UR-240 may be posted. © 2002 Panametrics.

2002 PROCEEDINGS PAGE 55 AMERICAN SCHOOL OF GAS MEASUREMENT TECHNOLOGY TABLE 2. Diameter, midradii, their combination, tomographic and quadrature arrangements. Complex flow patterns, high-accuracy, motivate multipaths on or off the diameter.

PAGE 56 2002 PROCEEDINGS AMERICAN SCHOOL OF GAS MEASUREMENT TECHNOLOGY FIGURE 2. Some transducer fundamentals. (a) T7 “air transducer” for 100-kHz operation has a flange OD = 19 mm and is internally impedance-matched [8]. (b) Array of sources as used in cem (continuous emissions monitoring) applications [4]. (c) Left: BWT® bundle waveguide icon represents a fiberacoustic waveguide. Right: Close-up photo of a BWT transducer. (d) Details of an early BWT transducer from the mid-1990s [6].

2002 PROCEEDINGS PAGE 57 AMERICAN SCHOOL OF GAS MEASUREMENT TECHNOLOGY θ FIGURE 3. (a) Refraction with clamp-on, calculated assuming a refracted angle 2 of 60˚ for a shear wave in the steel pipe, depends on sound speed c3 in the fluid. Referring to items (1)-(3) commercial contrapropagation clamp-on flowmeters available since the early 1990s include the PT868 and 6068 for measuring the flow of liquids. In water, the vee path usually θ θ works, and 3 is about 25˚ at room temperature. In air, 3.AIR is only about 6˚ and the transducers usually need to be placed on opposite sides of the pipe. (5) For liquid clamp-on, the vee path [shown in (1)] tends to cancel crossflow as well as double the sensitivity to flow compared to a single traverse. For gases, odd numbers of traverses are preferred, to reduce crosstalk. This means, if crossflow is significant, crossed paths are recommended. The velocities measured along the legs of the X should be averaged. Best solution: find a long straight run far from disturbances and joints. For gases [diagrams (4) & (5)] the flowmeter instrument (6) introduced in 2001 is the GC868 (Ao, 1999; Ao, et al. (2002); Lynnworth θ ≈ 2001). Diagram (7), drawn for LOX or LN2, shows the refracted angle 3 16˚, nearly midway between water (25˚) and air (6˚). (b) Clamp and gas paths, schematic. (c) Clamps and instrument for clamp-on gas flow measurement [1, 12]. (d) Application on steam. © 2001 Panametrics.

PAGE 58 2002 PROCEEDINGS AMERICAN SCHOOL OF GAS MEASUREMENT TECHNOLOGY FIGURE 4. (a) Example of a solid PanAdapta® precision plug for liquids. This plug is not to be removed when the line is pressurized as it is a permanent part of the pressure boundary. Its outer surface is prepared to receive a removable transducer. Frequency range: 0.5 to 5 MHz. (b) Similar to preceding case except the plug for gases consists of a bundle of thin waveguides welded within a sleeve. The bundle plug can be as short as one inch or as long as several feet (25.4 mm to ~1 m). This sleeved sealed waveguide construction allows the removable piezoelectric transducer assembly to be separated from the buffer bundle. (c) Schematics show planes of measurement in end view, and the crossed paths in a three-dimensional SolidWorks rendition. (d) Photos of a spoolpiece manufactured by RMG and corresponding to the schematics in (c). (e) Example of a T11 transducer. It is Ti-housed, internally quarter-wave matched, and twelve of them are used in (d). (f) Plugged version of (c). (g) Liquid version of the concept shown in (f).

2002 PROCEEDINGS PAGE 59 AMERICAN SCHOOL OF GAS MEASUREMENT TECHNOLOGY FIGURE 5. Plastic pipe, air flow at atmospheric pressure. These measurements, made by our colleague Oleg Khrakovsky, used ordinary liquid flowmeter clamp-on equipment, which was sufficient because the pipe was plastic, not steel.

Schematic

FIGURE 6. FIGURE 7. Short, thick-walled N-path spoolpiece whose OD matches Hybrid example: “cow” = clamp-on + wetted flange raised face dimensions. The “flanged transducer” transducers. is the T7 air transducer [ 8(c)] shown in Figure 2(a).

ACKNOWLEDGMENTS

The work reported here includes important contributions UR-273 and -274.www. Saul Jacobson, Toan Nguyen, from Shirley Ao, Jim Hill, and their and the authors’ David Hesketh, Jed Matson, Paul Ceglia and others at colleagues. The authors acknowledge Panametrics’ Panametrics and Hans J. Kastner, Andreas Weber and permission to reproduce passages, tables and others at RMG contributed to the equipment in Figure 4(d). illustrations from its copyrighted reports including The manuscript was prepared by Lin L. Leeming.

PAGE 60 2002 PROCEEDINGS AMERICAN SCHOOL OF GAS MEASUREMENT TECHNOLOGY BIBLIOGRAPHY AND REFERENCES 8. Lynnworth, L. C., Patch, D. R. and Mellish, W. C., Impedance-Matched Metallurgically Sealed 1. Ao, X., Matson, J., Kucmas, P., Khrakovsky, O., and Transducers, IEEE Transactions on Sonics and Li, X. S., Ultrasonic Clamp-On Flow Measurement Ultrasonics, SU-31 (2) pp. 101-104 (March 1984); of Natural Gas, Steam and Compressed Air, (b) Lynnworth, L. C., Fowler, K. A. and Patch, D. R., Proceedings of the 5th ISFFM (5th International Sealed, Matched Piezoelectric Transducer, U. S. Symposium, Fluid Flow Measurement) (April 7-10, Patent 4,297,607 (Oct. 27, 1981); (c) LCL, Ultrasonic 2002). Transducer System with Crosstalk Isolation, U. S. Patent 5,515,733 (May 14, 1996). 2. Brown, A. E., Ultrasonic Flowmeters, in Spitzer, D. W. (Editor): Flow Measurement, pp. 415-442, ISA 9.Scelzo, M., A Clamp-on Ultrasonic Flowmeter for (1991); (b) Brown, A., and Lynnworth, L., Ultrasonic Gases, Flow Control 7 (9) pp. 34-37 (Sept. 2001). Flowmeters, in Spitzer, D. W. (Editor): Flow Measurement - Practical Guides for Measurement 10. Smalling, J. W., Braswell, L. D., Lynnworth, L. C. and Control, 2nd Edition, Ch. 20, pp. 517-575, ISA (2001). Wallace, D. R., Flare Gas Ultrasonic Flow Meter, Proceedings 39th Texas A&M Annual Symposium on 3. Hill, J., Weber, A., and Koyama, T., Qualification of Instrumentation for the Process Industries, pp. 27- Ultrasonic Flowmeters for Custody Transfer of 38 (January 17-20, 1984); (b) Smalling, J. W., Natural Gas Using Atmospheric Air Calibration Braswell, L. D. and Lynnworth, L. C., Apparatus and Facilities, submitted for Proc. 20th North Sea Flow Methods for Measuring Fluid Flow Parameters, U. Measurement Workshop, St. Andrews, Scotland S. Patent 4,596,133 (June 24, 1986); (c) U. S. Patent (Oct. 22-25, 2002). 4,754,650 (July 5, 1988); (d) U. S. Patent 4,856,321 (August 15, 1989). 4. Jacobson, S. A., Flow Measurement System Including Ultrasonic Transducers, U. S. Patent 11. Suzuki, H., Nakabori, H., and Yamamoto, M., 5,460,047 (October 24, 1995). “Ultrasonic Method of Flow Measurement in Large Conduits and Open Channels,” p. 11538 in C. G. 5. Johari, H., and Durgin, W. W., Direct Measurement Clayton (Ed.), Modern Developments in Flow of Circulation Using Ultrasound, Experiments in Measurement, Peregrinus, London (1972). Fluids 25 pp. 445-454 (September 1998). (Later and earlier related references are in [15].) 12. Ting, V. C. and Ao, X., Evaluation of Clamp-On Ultrasonic Gas Flowmeters for Natural Gas 6. Liu, Y., Lynnworth, L. C. and Zimmerman, M. A., Applications, Proc. 20th North Sea Flow Buffer Waveguides for Flow Measurement in Hot Measurement Workshop, St. Andrews, Scotland Fluids, Ultrasonics, 36 (1-5) pp. 305-315 (February (Oct. 22-25, 2002). 1998); (b) Liu, Y., and Lynnworth, L.C., U. S. Patent 5,962,790 (October 5, 1999); (c) CIP (continuation- 13. Yamamoto, M. and Amano, A., Ultrasonic Flow in-part) U. S. Patent 6,343,511 (February 5, 2002). Quantity Measuring System, U. S. Patent 3,555,899 (Jan. 19, 1971). 7. Lynnworth, L. C., Ultrasonic Flowmeters, Chap. 5, in Physical Acoustics - Principles and Methods, 14. Yamamoto, M., and Ito, K., Ultrasonic Flowmeter Mason, W. P., and Thurston, R. N., (Eds.), 14 pp. System, U. S. Patent 3,237,453 (March 1, 1966). 407-525, Academic Press (1979); (b) Ultrasonic Measurements for Process Control Theory, 15. Walters, J., Smalling, J. W., Ao, S., Hill, J. and Techniques, Applications, Academic Press (1989); Lynnworth, L. C., Transit-time Ultrasonic Flowmeters (c) with Magori, V., Industrial Process Control Sensors for Gases, presented at and published in the Proc. and Systems, Chapter 4, pp. 275-470 in: Papadakis, 41st Annual CGA (Canadian Gas Association) Gas E. P., (Guest Editor), Ultrasonic Instruments and Measurement School, Grand Okanagan, Kelowna Devices: Reference for Modern Instrumentation, BC, Canada (June 4-6, 2002). See also, Techniques, and Technology, 23 in the series Physical UR-274.www, a Panametrics website version of this Acoustics, Academic Press (1999). paper.

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