Young’s modulus E, as many engineers will remember, is the slope of the stress-strain curve. Shear modulus G is the analogous slope when stress is applied as a twist or torque. Not so well known is the fact that these moduli are often measured not with the tensile testing machines and specially prepared specimens we learned about in engineering classes, but instead using ultrasonic pulse techniques on materials of all sorts of shapes and sizes.
The basis for the ultrasonic measurement of moduli is that sound waves travel in solids at a speed that is proportional to the square root of the elastic modulus that contributes to the specimen’s stiffness and inversely proportional to the square root of the specimen density ρ. Poisson’s ratio σ is sometimes involved too [9, 16]. If ρ is known, sound speeds yield the moduli, as will be shown in a moment. The type of ultrasonic wave (e.g. longitudinal and/or shear) determines which modulus will be determined. Two different wave types yield the two moduli E and G and also, Poisson’s ratio, since σ = E/2G – 1. Ultrasonic measurements of elastic moduli have been applied to solid specimens that are in the shape of bars, cubes, wafers, thin rods, strips of sheet metal, short whiskers, etc. Equations are simplest if the material is isotropic. In an isotropic bar of large diameter compared to wavelength, bulk longitudinal waves propagate at a speed cL = [E(1 – σ)/ρ(1 + σ)(1 – 2σ)]1/2. Shear waves travel at a lower speed, cS = [G/ρ]1/2. The ratio of these speeds yields Poisson’s ratio: σ = [1 – 2(cS/cL)2]/[2 – 2(cS/cL)2]. In other words, by measuring cL and cS in a thick bar of known density ρ, one can compute E, G and σ. If the bar, instead of being thick, is thin compared to wavelength, then compressional waves travel at extensional speed cEXT = [E/ρ]1/2.
What does all this have to do with ultrasonic flowmeters? Can there be a connection between historical academic studies of ultrasonic moduli in rods or wires at high temperature and today’s challenges in ultrasonic measurements of fluid flow at temperature extremes in industrial environments?
|Figure 1. The technical problems addressed by 1947 and reported in  were aimed at determining Young’s and shear moduli in specimens comprising the end of, or coupled to the end of, a lead-in rod. Ultrasonic pulse techniques were used from cryogenic temperatures (liquid nitrogen, -196 C) to high temperatures (815 C). Decades later, some aspects of those solutions found their way into ultrasonic flowmeters, both clamp-on and wetted types. “Ultrasonic” flowmeters include contrapropagation and other types [9, 14].|
The year 2007 marks the 50th and 60th anniversaries of two publications in ultrasonics, which probably did not receive much attention when published. One was a doctoral dissertation at the University of Michigan  by a graduate student, J.R. Frederick, who later became a professor there. The other, by a professor in the United Kingdom, was published in Philosophical Magazine . Both publications described ultrasonic pulse methods, new at the time, to measure elastic properties of materials. Neither publication dealt with flow; apparently neither author anticipated flow applications would be derived half a century later from ideas contained in their research. The first paper used pulses in the MHz range. The other used pulses in the 100-kHz range. Some readers will recognize that such frequencies are often used in today’s ultrasonic flowmeters for liquids and gases, respectively. A third paper , published 41 years ago, also should be mentioned, as it too is part of this story.
|Figure 2. In 1947, Frederick’s work  was aimed at measuring ultrasonic transit-time in a short specimen (4), which was heated or cooled to temperatures in the range of -196 C to 815 C. The question mark refers to the unknown elastic moduli. He thermally separated the ultrasonic quartz transducer (1) from the high (or low) temperature using a buffer rod, shown here as threaded (2). Coupling (3) was achieved by means of gold foil, silver solder, or other means. Coupling nut is not shown. Threading was one of the solutions he used to break up the sidewall reflections and mode conversions that accompany a smooth-walled rod as depicted in Figure 1. In a 2007 high-temperature contrapropagation flow measurement application, the depicted goal (the question mark) is to measure transit time in a hot fluid (7). The transducer (6) nowadays is probably a ferroelectric material other than quartz. A short “buffer” may be used to construct a removable transducer assembly, coupled to buffer rod (2). Figures 3 and 4 help illuminate the path leading from solid buffer to bundle.|
In brief, these papers described new and improved ways of transmitting ultrasound efficiently in a solid specimen from Point A to Point B, in a way that allows accurate measurement of transit time. In modulus studies, the path is the length of the specimen. In flow applications, which are the focus of our attention here, the path includes the fluid. In contrapropagation flowmeters, time is measured with the flow and against the flow.
Old publications from 1947, 1957, and 1966 might seem unlikely places to search for hints on how to make an ultrasonic flowmeter for accurate measurement of gas or liquid flow, e.g. gas flow at very high temperature and pressure, or liquid flow at cryogenic conditions. However, these three publications could be said to comprise important parts of the scientific and engineering foundation for the buffer waveguide described below and widely used since the late 1990s. By late 2001 more than 1,000 high-temperature ultrasonic gas flowmeters incorporating these buffers had been installed , and the total is likely several thousand today.
Measuring Elastic Moduli via Ultrasonic Pulses
|Figure 3. In 1957 Bell avoided sidewall echoes by arranging for the lead-in and specimen diameters to be small compared to wavelength . Diagram adapted from .|
No doubt musicians through the centuries have known that the pitch of their instruments varied with temperature. After this effect was quantitatively associated with elastic moduli, resonance methods evolved for determining moduli as a function of temperature. Resonance methods require measuring resonant frequencies, providing a solution in the frequency domain. (Interference from moisture and other variables needs to be eliminated. This is typically achieved by testing the specimen in a controlled environment, e.g., in vacuum.)
|Figure 4(a). Dispersion curve: group velocity versus frequency times diameter (fd), calculated for Ti rod, courtesy of J. L. Rose, reproduced with permission. Calculations used the following longitudinal and shear sound speeds for Ti: cL = 6.1 km/s, cS = 3.12 km/s, respectively; numerical values for sound speeds are the same in units of km/s or mm/μs. Details on dispersion curves are found in . Titanium resists fouling, and its acoustic impedance is lower than in stainless steel alloys such as SS316L. The latter feature is an advantage when the fluid itself is of low impedance. Accordingly, Ti is often used for making wetted ultrasonic flowmeter transducer buffer rods. In a bundle, when the diameter of each element is sufficiently small compared to wavelength (small fd), there is no dispersion.|
Since no method has all the advantages, it is natural to inquire, what might be the advantages of the corresponding solution in the time domain? For our purposes we can simplify the problem into determining the speed of sound in a short specimen attached to or coupled to a lead-in rod in such a manner that the specimen’s sound speed can be measured at all temperatures of interest, cryogenic, ordinary, or high. Following this line of inquiry in the mid-1940s, Frederick considered the situation depicted in Figure 1. The elastic “lead-in” rod diameter is not small compared to wavelength. This little detail led to problems, which in turn led to solutions that were utilized years later in ultrasonic flowmeters.
Mode Conversion: Longitudinal waves striking the rod’s periphery at grazing incidence generate shear waves by a mode conversion process. Shear waves propagate diagonally at a speed about half that of longitudinal waves towards the opposite side of the rod. Here, they partly reflect as shear and, again due to mode conversion, reflect partly as longitudinal waves. The diagram illustrates the multiple mode conversion process in the lead-in, with the initial longitudinal wave A0 yielding a first reflected longitudinal wave A1 followed by delayed longitudinal waves A2 and A3. Each of the latter two results from shear-to-longitudinal conversions. Angles for the reflected shear wave and the partition of energy between longitudinal and shear depend on incidence conditions, Snell’s Law and σ, analyzed in 1948 by Arenberg . Arenberg’s amplitude ratio graphs are reproduced on pages 226 and 227 in . The net result of the many extra echoes is noise that can obscure the signal. In other words, the reflections, mode conversions, and subsequent reflections create multiple sidewall echoes. These can interfere with timing the signal echoes of interest from a specimen comprising the rightmost portion of this waveguide system, unless, as shown by Frederick , the periphery is altered, for example, by grooving or threading, or properly proportioning the specimen’s dimensions. In  the specimen was sometimes created by notching the lead-in near the end opposite the transducer.
|Figure 4(b). Dispersion can be utilized to create a chirped pulse. In this oscillogram, chirping was produced intentionally with one Ti waveguide, diameter 3.2 mm by length 914 mm, tested in through-transmission. The transducers were 1-MHz NDT types, held by hand and gel coupled. Excitation was a spike from a conventional pulser/receiver. The first arriving cycle is of relatively low frequency, followed by higher-frequency components. This way of generating a chirped pulse might be used to reduce the chance of a large timing error due to “cycle skip.” After , © 2005 IEEE, reproduced with permission.|
Frederick’s solutions included pressure-coupling using gold foil. Pressure coupling was achieved by clamping the threaded lead-in and specimen together using a nut. In flowmeter parlance we might say he demonstrated by 1947 a high-temperature, clamp-on technique. In many of his experiments the metal specimen temperature reached 815 C. Threaded buffer rods and gold foil coupling reappeared in various high-temperature ultrasonic flowmeter studies in the 1970s. Threaded pressure-coupling, sometimes aided by grease, has been used in thousands of flowmeter applications where a permanent pipe plug is installed in a pipe or spool in a way that accommodates a removable transducer. The “nut” can be part of the plug, adapted to the removable transducer. (Caution must be exercised to safely remove the transducer from the wetted plug, such that the pressure boundary is always maintained leak-tight.)
Adapting ideas and techniques from the magnetostrictive delay line art, Bell avoided sidewall echoes by arranging for the lead-in and specimen diameters to be small compared to wavelength . In this thin-rod case the “longitudinal” wave is often referred to as an extensional wave, and the equation relating its speed to Young’s modulus E is simpler than for ultrasonic waves in thick rods [15, 16]. Bell’s work  led first to process control applications other than flow: temperature sensing starting in the 1960s and liquid level measurement in the 1970s . Although a bundle buffer possibility was recognized and demonstrated by Gelles in 1966 , flow applications did not develop on a large-scale basis until the mid-to-late 1990s .
Out of Many Rods, One Bundle
|Figure 4(c). Four-cycle tone burst echo shows little dispersion. Diagram represents many rods packed in a metallurgically sealed enclosure. After , © 2005 IEEE, reproduced with permission.|
The motto imprinted on U.S. coins, E pluribus unum, means “out of many, one.” These words might also be interpreted as a suggestion to make one bundle out of many waveguides. Another principle taught by a close-packed array of pennies (Figure 5) is that when arranged as shown, each penny is contacted by others only at six or fewer points. Most of the perimeter is free.
In 1966, Gelles  demonstrated pulse echo operation of a fiberoptic bundle and suggested several potential applications. But it was not until the mid-1990s that a practical way was found of packing hundreds of rods whose diameters were sufficiently small compared to wavelength to avoid dispersion into rigid metallurgically sealed enclosures [7, 8]. These designs started with simple welded enclosures, such that radiation was from a flat end. The end was perpendicular to the bundle axis. Later enclosures chamfered the radiating face once or twice, achieving radiation oblique to the bundle axis (Figure 6). Motivation was to allow installation in a nozzle normal to the main pipe or spool axis, with the buffer remaining nonintrusive, i.e., not protruding beyond the inside wall, yet radiating obliquely.
|Figure 5. Closely packed array of coins reminds one that most of the surface of each cylindrical rod in a bundle is free. This suggests that the dispersion characteristics of an individual rod, including absence of dispersion for rods sufficiently thin compared to wavelength, will be mimicked by a bundle of such rods, even hundreds packed in a tubular enclosure.|
There are other ways to avoid the noise associated with unwanted echoes from sidewall reflections and delayed mode conversions. Approximately 10 years ago a clad waveguide solution became available . This solution, which is one of several clad solutions due to C.-K. Jen and colleagues, may be explained in terms of a similarity to the SOFAR channel in the ocean. The sound speed in the ocean initially decreases with depth because the ocean is colder as one goes down. However, the temperature cannot decrease indefinitely. But as one goes deeper, the pressure continues to increase in proportion to depth, and this leads to an increase in sound speed. Therefore, there is a depth, ~1 km below the ocean surface in middle and equatorial latitudes, at which the sound speed is minimum, with higher speeds above and below . Because of refraction, sound waves bend towards the region of minimum sound velocity. Sound launched in the low-speed SOFAR channel is trapped. Similarly, sound in the clad rod’s slow core is trapped and can’t reach the periphery composed of a fast cladding, so the unwanted reflections and mode conversions shown in Figure 1 are avoided. The clad rod, like the bundle, avoids energy losses due to diffraction or beam spread. The threaded or grooved buffer rod does not avoid these losses, which explains why, if the buffer is long, the SNR (signal to noise ratio) is expected to be better for the clad or the bundle solutions.
|Figure 6. (a) Bundles comprised of rods of different diameters (1.6 to 4.8 mm in this mockup) and different materials. Major face is approximately perpendicular to axis of bundle. One rod in each bundle has been brought forward to show diameter of all the identical elements in that bundle. Coin is a U.S. penny. (b) Radiating face chamfered or double-chamfered; radiating faces in two parallel planes create differential path in bundle as well as in fluid; one bundle remains simple [9, 13]. (c) Closeup of chamfered bundle and differential-path bundle. (d) Bundle comprised of two sets of rods, one set thinner than the other. By using more sets, arranged so that the element diameters change in a controlled way across the bundle, the elements, operated in their dispersive region, can act like a phased array to spread or steer the radiated beam as the frequency is hopped or swept. A buffer of this type, radiating off-axis, can be installed perpendicular to a pipe or spoolpiece with its face flush to the inside wall of the pipe (Figure 7). This keeps the port cavity volume small, thereby reducing flow disturbances by the “wetted flow sensor.” Above mockups, corresponding to illustrations in [8, 13] convey ideas of how rods might be arranged within a bundle. No details are included here on safely enclosing the bundle within a sealed metal structure while preserving ultrasonic performance. Nonproprietary aspects of such information appear elsewhere [7-13]. For information on the commercial availability of the design possibilities mentioned here, in sizes and materials appropriate to a given application, contact the manufacturer associated with bundle assemblies shown in [7, 13].|
Returning to the bundle operated in its dispersive region: the normal (perpendicular) installation idea is represented in Figure 7. One design problem is that as the sound speed c3 in the fluid changes, the sound speed c1 in the bundle must be adjusted to maintain a fixed ratio for c3 /c1, or the beam’s refracted angle in the fluid, Θ3, would change. Figure 4(a) suggests how adjusting the frequency (or wavelength within the rods) can be utilized to adjust c1. Of course, if the nozzle is oblique, the simple nondispersive bundle with radiating flat end perpendicular to its axis may suffice despite the small fluid-filled cavity near the inside wall of the pipe or spool.
In November 2001, it was reported in  that over 1,000 gas flow applications at temperature extremes had been solved using bundled waveguides. In some cases, e.g., at a refinery in the Netherlands, the design temperature was about 500 C. It seems reasonable to estimate that since then, BWT™ bundle waveguide technology buffers have been installed in thousands of additional installations worldwide, for measuring flow of fluids such as air, steam, natural gas, and other gases [13, 14]. Liquids have included hydrocarbons, chemicals, and water, and occasionally, two-phase or multi-phase mixtures. Some liquids are of unusually high viscosity. Liquids are often interrogated using ultrasonic frequencies of 0.5 or 1 MHz. For gas or steam the frequency usually is lower, 200 kHz. The wetted material comprising the radiating portion of the buffer’s enclosure, as well as other parts, is most often SS316L, and optionally, commercially pure Ti. Special metals or alloys have been used when necessary for compatibility with particularly aggressive fluids or to meet NACE or other requirements.
|Figure 7. Extensional waves travel faster in rods that are thin. If rod diameters increase monotonically from left to right, the bundle can be operated like a phased array, steering the radiated beam obliquely into the fluid. In this diagram the radiating face is perpendicular to the bundle’s axis and flush with the inside wall of the pipe.|
Generally speaking, in bundle waveguide applications encountered over the past ten years, the temperatures ranged from -196 C (liquid nitrogen’s boiling point) to 315 C (short buffer spec; with extended buffer, the maximum spec temperature for the fluid is 600 C). Pressures ranged up to the maximum allowable flange operating pressure or 3,480 PSIG (240 bar) for flanged designs. Flow velocities ranged from 0.03 to 12 m/s in liquids, and from 0.03 to nominally 46 m/s in gases or steam. However, in gases, the maximum flow velocity in a given application depends on factors such as gas sound speed, acoustic impedance, and path length. The lower ultrasonic frequency that is used for gases and steam is better able to handle the higher Mach number and attenuation often associated with these relatively low-density fluids. Higher frequencies are more suitable for liquids, in general. (Reminder: water’s density is about 775 times greater than the density of ordinary air.) Readers are advised to consult the latest specs of the manufacturer of bundle waveguide products.
|Figure 8. Application of threaded assembly to achieve removable transducer from a plug permanently installed to maintain the pressure boundary. In this example the long portion is a bundle. After , © 2005 IEEE, reproduced with permission.|
Almost all bundles used in industry have been in contrapropagation flowmeter systems. However, in principle, the bundle buffer may be used with other ultrasonic flow measuring technologies too. Examples: tag , stroboscopic scattering [3, 9], and vortex shedding [9, 14, 18]. Applications in the future might occur with measurands other than flow.
Larry Lynnworth’s work in ultrasonics includes R&D in NDT, elastic moduli measurements at high temperature, and process control, primarily temperature, liquid level, and flow. Mr. Lynnworth has published many papers, seven chapters, and one book . He has 48 U.S. patents, two of which relate to the bundle technology described in this article. After retiring from GE Sensing, Mr. Lynnworth founded Lynnworth Technical Services, a consulting firm. Mr. Lynnworth can be reached at firstname.lastname@example.org or 781 894-2309.
Entire contents of this article are © 2007 Lynnworth Technical Services. This article appeared in the March 2007 print issue of Flow Control magazine, Vol. XIII No. 3, pages 36-43.
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3. Brown, A.E. and Lynnworth, L., Ultrasonic flowmeters, chap. 20, 515-573 in Spitzer, D.W. (Ed.), Flow Measurement, 2nd Ed., ISA (2001).
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16. Rose, J.L., Ultrasonic Waves in Solid Media, Chapter 11, Cambridge University Press (1999).
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18. Spitzer, D.W. (Ed.), Flow Measurement, 2nd Edition, ISA (2001), esp. Chapter 20, pp. 530-537.
Professor Joseph L. Rose of Pennsylvania State University supplied the dispersion curve [Fig. 4(a)] for this article.