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.