N UMERICAL S IMULATION OF U LTRASONIC T RANSIT-T IME F LOWMETER P ERFORMANCE IN H IGH T EMPERATURE G AS F LOWS Mario Kupnik∗ , Paul O’Leary∗ , Andreas Schr¨oder† , and Ivan Rungger† ∗ Christian-Doppler
Laboratory for Sensory Measurement c/o Institute for Automation, University of Leoben, Austria, A-8700 † Institute of General Physics, Vienna University of Technology, Austria, A-1040
Abstract - The use of electrostatic transducers avoids the limitations associated with piezoelectric transducers in gas flowmeters, such as their restricted maximum allowable gas temperatures and attainable measuring repetition rates. The measurement at high temperature (up to 600◦ C) has necessitated the development of a new and innovative electrostatic transducer. In this work a structured thermally oxidized silicon back plate covered with a bulk conducting 3 µm titanium foil as membrane is used instead of e.g. a metallized polymer film. This configuration enables the application of transit-time flowmeters to the measurement of hot pulsating gas flows (up to 3 kHz). Knowledge of the influence of the temperature and velocity profiles on the wave propagation in this high temperature range is essential for design improvements and operating and accuracy limits of the gas flowmeter. This paper presents a numerical 3-D procedure based on Ray-tracing to simulate the sound refraction and drift due to different temperature and velocity profiles for several transducer arrangements. The limits and dynamics of the used profiles are taken from measured data acquired on an exhaust train of an automotive combustion engine, as an exemplary application under extreme operating conditions. The wave propagation is modelled for a heatable double path flowmeter with transducers of finite surface. Due to the high dynamics of the temperature variations and the thermal inertia of the flowmeter, negative temperature gradients must be taken into account. Since they result in focusing the wave front and a reversed refraction direction. The physical limits of transit-time flowmetering in such hot gases can be determined. The up and down stream travel times and the bestcase-normalized transmitter-receiver pressure ratios are presented for different working conditions and measuring set-ups. These results have been used to optimize the design of the measurement cell and to find the temperature-induced correction of the usually used equation to calculate the gas flow velocity. Additionally, clear 3-D visualizations of the wave fronts and their temporal propagation through the gas are generated.
I. I NTRODUCTION Transit time ultrasonic flowmeters are commonplace for metering liquid and gas flows. They offer little or no obstruction with no pressure drop to the fluid flow. Generally they utilize piezoelectric transducers to generate and detect the ultrasound. These types of transducers limit the application of ultrasonic flowmetering to fluid temperatures up to 250◦ C [1] and have low attainable measuring repetition rates, due to their small frequency bandwidth. In many applications their performance also suffers due to an impedance mismatch to gas flows. For this reason ultrasonic flowmetering was not fully available for many applications up to now, e.g. the measurement of the mass flow in an exhaust gas train of a combustion engine. Such demanding applications, with extreme operating conditions (high gas temperatures, strong temperature gradients, and high pulsation frequencies), necessitate a high temperature resistant
0-7803-7922-5/03/$17.00 (c) 2003 IEEE
ultrasonic transducer with good bandwidth characteristics. A transducer based on an electrostatic principle satisfies the bandwidth requirements. In [2,3] it is shown, that electrostatic transducers are well suited to ultrasonic transit time gas flowmetering. A general overview of recent developments of capacitive transducers is given in [4]. This paper begins with a description of a capacitive transducer configuration, which also satisfies higher temperature requirements [5]. Only using such high temperature resistant capacitive transducers is the principle of ultrasonic transit-time flowmetering applicable to hot pulsating gas flows. This new field of application requires an investigation of the wave propagation inside an ultrasonic flowmeter under extreme operating conditions. After reviewing the basics of ray theory (Section III) to describe the wave propagation in an inhomogeneous moving medium, an exemplary application, flowmetering in an exhaust gas train of a combustion engine, is introduced (Section IV). Using this application realistic limits of the temperature and velocity profiles, with an emphasis on their dynamics, are defined for the 3-D simulation procedure used for a heatable double path flowmeter. The results are presented in Section V. II. H IGH TEMPERATURE RESISTANT CAPACITIVE TRANSDUCER
A commonly used electrostatic transducer configuration (e.g. [6]) uses two electrodes, one a structured bulk conducting back plate, and the other e.g. a metallized polymer film, this restricts the maximum allowable gas temperature due to a low melting point (Fig. 1a). An AC voltage, as a signal source, is superposed on top of a DC bias voltage for proper operation. The electrostatic force pulls the membrane, which is balanced due to the mechanical restoring force, toward the back plate. In [5] an innovative capacitive transducer configuration is introduced, which is resistant to gas temperatures of 600◦ C in continuous operation (Fig. 1b). Instead of an insulating membrane a bulk conducting 3 µm foil, consisting of a metastable beta titanium alloy (Ti-15Mo-3Nb-3Al-.2Si), is used as the moving membrane. Here the back plate, instead of the membrane, is coated with an insulating layer. For example, a doped silicon waver, which is first thermally oxidized to grow an insulating silicon dioxide layer, and then structured in a second step, is a good choice to get a temperature resistant back plate material (Fig. 1c). The coefficients of thermal expansion of the titanium alloy and silicon dioxide are similar,
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DC AC
~ (a)
DC AC
Ti SiO2 Si
~ (b)
SiO2 Si 2 m (c) Fig. 1. Schematic drawings of a typical capacitive transducer with metallized polymer film (a) in comparison to a high temperature resistant realization (b), using a thermally oxidized silicon back plate (c). (The photograph is taken from [5].)
which also is advantageous in high temperature applications with strong and fast variations in temperature. Two different methods can be used to fabricate the back plate: The grooves of the back plate can be fabricated either in the insulating layer, as shown here in Fig. 1c, or the back plate can be structured before it is thermally oxidized. III. R AY THEORY IN AN INHOMOGENEOUS MOVING MEDIUM
In transit-time ultrasonic flowmeters frequencies below 200 kHz are rarely used, due to accuracy requirements. Using the transducers described in Section II gives an usable frequency range from approximately 100 kHz up to 600 kHz. In the case of flowmetering in an exhaust gas train of an automobile combustion engine the smallest used pipe diameters usually start from 50 mm. This ensures that the wavelength λ = c/f is small compared with the dimensions of the measuring tube. This first prerequisite leads to the well known method of geometrical acoustics, also known as ray acoustics, to describe the wave propagation in the flowmeter. Ray acoustics is an approximation method and it’s solution will be a high-frequency solution of the acoustic wave equation. Having an initial position and direction of propagation of an acoustic wave front enables tracing of a point on that wave front through the flowmeter. This process is known as “ray tracing.” In [7] an adequate ray tracing system for the case of a non-zero ambient velocity v in the medium and a varying speed of sound c, due to a temperature profile, is
derived. This is what is required in the case of a transit-time flowmeter to investigate the influence of a temperature and velocity profile. If the medium is moving with velocity v the wavefront velocity becomes v + cn in a coordinate system at rest, due to a convection effect [8]. Here n is the unit vector normal to the wavefront. Due to the lateral component of the velocity v the direction of the wave front propagation cannot be the same as the unit vector. A point P on the wavefront will always lie on the wavefront if its velocity fulfills dxp = v (xp , t) + n (xp , t) c (xp , t) , (1) dt where the ambient velocity v and the speed of sound c may vary with position and time. The Point P lying on the wavefront traces out a line in space, which is described by xp (t). The goal is to determine the function xp (t). In [7] a parametric variable, a “slowness”-vector s, is used to describe the propagation direction of the point P on the wavefront. The label “slowness” is used because the reciprocal of |s| is the speed c + v · n. Hence instead of dealing with n directly, the vector s, which is always parallel to n, is used. This idea leads to six coupled, nonlinear differential equations of first order, i.e. the ray-tracing equations in three dimensions. The first three Equations (2) describe the motion of the point P and the second three Equations (3) describe the evolution of the slowness-vector s: c2 si dxi = + vi , (2) dt Ω 3 ∂vj Ω ∂c dsi = − − sj , (3) dt c ∂xi j=1 ∂xi where Ω = 1 − v · s.
(4)
It is important to notice, that these equations do not depend on the spatial derivatives of s, and so if the speed of sound c (x, t) and the velocity v (x, t) are specified (in space and time), and if an initial value for x and s exists, Equations (2) and (3) can be integrated in time to determine x and s at any subsequent instant. Standard numerical techniques of integration can be used. No information concerning neighboring rays is required, so one ray after the other can be calculated. The term vi in Equation (2) describes the drift of the sound ray due to the velocity of the medium. The influence of this velocity to the slowness-vector s is considered through Ω in the first term. The first term in Equation (3) describes the refraction due to a gradient of the speed of sound (temperature profile) in the medium. The second term describes the refraction due to a gradient of the velocity of the medium. Using these ray-tracing equations a set of rays with different starting positions, passing from the transmitter position to the receiver position, can be computed. These rays form a so called ray-tube. Next to the travel-time of the wave front, travelling from one position to another inside this ray-tube, the variation of the acoustic pressure amplitude p along a ray path is an important quantity concerning the flowmeter
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performance. Neglecting the internal loss of the medium the Blokhintzev invariant [7], which says that the acoustic power inside a ray-tube is constant, i.e.:
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p |v + nc| A = const, (1 − v · s) ρ c2
(5)
where A is the cross-sectional area and ρ is the density of the medium, can be used to calculate the best-case-normalized transmitter-receiver pressure ratio. As mentioned previously, ray-theory is an approximation method, hence a validity condition must be considered. The validity of ray acoustics is given when the acoustic wavelength λ is small in comparison to the range in which the refractive index changes significantly. In the case of a moving medium the validity condition is [9]: 2π |∇ (c + v · n) · n| . (6) c λ Since the slowness vector s is parallel to n, Equation (6) can be brought into a more useful form to check the validity of the ray-theory for a given temperature and flow profile during the simulation. Using Equation (4) and the two intermediate results cs (7) n= Ω and c , (8) Ω= c+v·n taken from the derivation of the ray-tracing system [7], the validity condition (6) can be written as c c s ∇ 2π Ω · Ω . (9) c λ This inequality must hold for any sound ray at any starting position inside the flowmeter. IV. V ELOCITY, T EMPERATURE , AND ACOUSTIC - VELOCITY D ISTRIBUTIONS IN THE F LOWMETER As an exemplary application of a transit-time flowmeter under extreme operating conditions (temperature limits and dynamics) an exhaust gas train of a combustion engine in a test bench environment is considered. This type of application is demanding for the flowmeter, due to the different temperature distributions which may occur. For example, after a long time of strong loading the combustion engine by a generator in producer operation mode, the generator can be changed to motor-driven mode. Thus, the exhaust gas train is heated up by the hot exhaust gas first, and then it is cooled down again by the flowing “induction air” in the exhaust gas train. Due to the thermal inertia of both the measuring tube (Fig. 4) and the exhaust gas train the value for the pipe wall temperature Tw is then greater than the pipe core temperature Tc . Depending on the different temporal operating conditions of the combustion engine many different temperature combinations (Tw < Tc , Tw = Tc and Tw > Tc ) inside a relevant temperature range (20 . . . 600◦ C) must be taken into account for the simulation of the flowmeter performance. The temperature range is well
R
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Fig. 4. Schematic drawing of the considered heatable double path transit-time flowmeter with circular measuring tube (R = 25 mm, α = 30◦ , membrane diameter = 9 mm, transducer port diameter = 13 mm). The diagonally oppositely arranged transmitters (T) and receivers (R) require cavities in the pipe wall. The assumed radially symmetric velocity and temperature distributions (v(r), T (r)), the up and downstream travel times (tup , tdown ), and the wall and core temperatures (Tw , Tc ) are outlined.
known from temperature measurements in the exhaust gas train. To use the ray tracing system (Equations (2) and (3)) one needs the velocity distribution v(x, t) and the distribution of the acoustic-velocity field c(x, t), which is linked to the fluid temperature field T (x, t). The speed of sound c for a gas mixture (e.g. exhaust gas) consisting of n components with the mass concentration xi , i = 1 . . . n, can be calculated using the ideal gas equation [10]: n xi cpi R T (x, t) i=1 (10) c (x, t) =
, n n R xi Mi xi cpi − M i i=1
i=1
where cP i is the specific heat capacity of the ith component of the gas mixture at constant pressure P , R is the molar gas constant, and Mi is the molecular weight of the ith component. The second fraction in Equation (10) is the adiabatic exponent κ, which also depends on the gas temperature. The velocity and temperature distributions over the tube cross-section vary with position along the tube, with flow rate and with time. To aggravate the situation the transducer port cavities (Fig. 4) also have an effect on these distributions in the relevant area inside the measuring tube, where the acoustic wavefronts are moving. For a slightly different configuration, for the protrusions caused by the mounted transducers, this influence was investigated in [2] using a computational fluid dynamics software (CFD). For the configuration considered in this work several CFD-simulations were used to investigate the influence of the transducer port cavities on the velocity and temperature distributions. The turbulence model used is the k − ε model. These transient CFD-simulations also have shown that in applications, such as flowmetering in an exhaust gas train, one can assume that the flow field is “frozen”, due to the short travel times of the acoustic wave compared to any time scale in the velocity and temperature fields. The
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norm. Temp. Θ− = ( Τ(r) − Τc ) / ( Τw - Tc )
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Fig. 2. CFD-Simulation results for different mass flow values Qm in comparison to model equation results for the axial velocity distribution inside a 50 mm measuring tube, including the transducer port cavity (r > 25 mm), for a cold (a) and hot (b) wall-temperature situation (m = 6.129, n = 6.853).
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Fig. 3. CFD-Simulation results for different mass flow values Qm in comparison to model equation results for the axial temperature distribution for a cold (a) and hot (b) wall-temperature situation (m = 2.971, n = 4.3125).
main goal of the CFD-simulations was to check the possibility to use simple adequate model equations [11,8] for the raytracing simulation to describe the velocity and temperature distributions. In Fig. 2 a comparison of the CFD-simulation results to the model equation results for the velocity profile along the path r (Fig. 4) is shown. The two temperature situations (Tc > Tw in Fig. 2a, and Tc < Tw in Fig. 2b) do not show a significant difference for the velocity distribution. Inside each transducer port cavity there exists a vortex, but the velocity values of the vortex are small in comparison to the velocity values of the main flow inside the measuring tube (r < 25 mm). To investigate the influence of this vortex two piecewise model equations (M odel1 , M odel2 in Fig. 2), i.e. with the assumption that the velocity is zero inside the cavity, were used to compute several single sound rays with different starting positions. Another simplification, which is implicitly made when using one of these model equations, is neglecting the x and y-components (Fig. 4) of the velocity field, due to their small values in comparison to the z-component. The same rays calculated by using the CFD-data directly only showed a difference of the coordinates at the receiver position of about < 1 %. Concerning the velocity profile itself, the average of
these two model equations showed the best agreement with the CFD-data along the path r in the relevant mass flow range of this application. Hence the average of these two model equations was used in the ray-tracing simulation to describe the velocity distribution. In Fig. 3 the calculated temperature distributions along the path r are shown. The CFD-simulations showed an almost constant value for the temperature over the whole transducer port cavities, which is identical to the wall temperature Tw . The simple piecewise model
2m T (r) = (Tc − Tw ) 1 − |r| + Tw R
|r| ≤ R
Tw
|r| > R
(11) was used to model the temperature profile for the ray-tracing simulation, since it showed good agreement with the results obtained from the CFD-simulation. A curve fitting method was used to obtain the parameters n and m for the model equations (velocity, temperature). Using the CFD-data directly for a complete ray-tracing simulation would be too time consuming, so the model equations were used instead.
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(d) Tw = 300◦ C, Tc = 100◦ C, vmax = 33 m/s, tup = 164.066 µs, tdown = 155.253 µs, Pup = 0.087, Pdown = 0.749.
Fig. 5. 3-D visualizations of the wave fronts and their temporal propagation (the wavefronts are visualized every 10 µs) through the gas for exemplary working conditions of the flowmeter. The measuring tube itself, the transducer port holes and the transducer membranes are outlined schematically. The corresponding geometry is presented in Fig. 4. The up and downstream travel times (tup , tdown ) and the best-case-normalized transmitter-receiver pressure ratios (Pup , Pdown ) are represented.
V. S IMULATION RESULTS AND DISCUSSION The simulation program starts with a recursive triangulation of the two transmitter membranes (Fig. 4). Stepwise increasing the recursive level has shown that 1659 sound rays for each path of the flowmeter are sufficient, due to a convergent behaviour of the results. Every node of this grid is a starting point of a sound ray. The beam profile generated by the transducer driven at a given frequency (f = 350 kHz) is considered by using the 10 dB opening angle of the theoretical distribution for an ideal piston-shaped transducer (farfield region). The half width of the beam in the far field is directly proportional to the wavelength λ [12]. In [5] the comparison of the calculated and measured results for the beam profile of the capacitive transducer used showed good agreement. The input values for the simulation program are the wall and core temperatures (Tw , Tc ), the maximum gas velocity vmax , the sound frequency f and the geometrical parameters of the flowmeter configuration (Fig. 4). As shown in Fig. 5, next to the sound drift, the temperature gradient between the pipe wall and the center of the pipe plays a major role for the physical limits of the flowmeter. Fig. 5(a) and 5(b) show that at higher gas temperatures the
influence of the sound drift is smaller. The reasons are the wider opening angle of the transducer main lobe and the shorter time spent of the wavefronts inside the pipe. If there exists a positive temperature gradient (e.g. 200◦ C in Fig. 5(c)) the wavefronts are extended in area and they are refracted toward the pipe wall. In the case of the up stream traveling wavefronts these effects partially compensate the sound drift effect. For the down stream traveling wavefronts the sound refraction is superposed onto the sound drift. In the case of a negative temperature gradient (e.g. −200◦ C in Fig. 5(d)) the wavefronts are focused and they are refracted toward the center of the pipe. The influence of the sound drift and sound refraction effects on the transmitter-receiver pressure ratios for different temperature conditions depending on the maximum gas velocity vmax are shown in Fig. 6. Each subfigure corresponds to those in Fig. 5. The acoustic pressure ratios are best-case normalized, i.e. they are normalized to the lowest considered gas temperature (20◦ C) with no temperature gradient and with zero gas velocity. The steeper (α = 20◦ ) transducer configuration shows a better averaged pressure ratio performance (+24 %), but also a lower averaged travel time difference (−27 %) over
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Fig. 6. Best-case-normalized transmitter-receiver pressure ratios for different exemplary working conditions and two different flowmeter geometries (α = 30◦ and α = 20◦ ). (a) and (b) show the situations with no temperature gradient at 20◦ C and at 300◦ C. (c) and (d) show the situations for positive and negative temperature gradients.
Meter Factor k
real world application a flowmeter in an exhaust gas train of a combustion engine has been taken into account. This paper has described the investigation of the temperature influence, caused by a positive or a negative temperature gradient, and the sound drift on the flowmeter performance. A 3-D procedure, based on Ray-tracing theory, has been used to simulate the sound drift and refraction effects inside a double path flowmeter. It was demonstrated that model equations with underlying simplifications, to describe the velocity and temperature profiles inside the flowmeter, can be used for numerically efficient simulations. The calculated travel-times for different temperature situations have been used to calculate the temperature-dependent meter factor. As future work, these obtained data for the meter factor form the basis for correcting the equation usually used to calculate the flow rate.
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Tc-Tw [°C] Fig. 7. Calculated meter factors k depending on the temperature gradient for different wall temperatures and vmax = 16.5 m/s using the simulation results (α = 30◦ ).
the considered working range of the flowmeter. A transit-time flowmeter delivers the mean velocity averaged along the path the wavefronts move along. However, to determine the volume flow or the mass flow, the mean velocity averaged over the tube cross-section A is needed. The connection between these two velocities is usually considered by a correction factor k, also termed meter factor [1]. Fig. 7 shows the influence of the temperature difference (Tc − Tw ) on the meter factor k for different wall temperatures.
R EFERENCES [1] L. C. Lynnworth, Ultrasonic Measurements for Process Control; Theory, Techniques, Applications. San Diego: Academic Press, Inc., 1989. [2] I. O’Sullivan and W. Wright, “Ultrasonic measurement of gas flow using electrostatic transducers,” Elsevier Ultrasonics, vol. 40, pp. 407–411, 2002. [3] P. Brassier, B. Hosten, and F. Vulovic, “High-frequency transducers and correlation method to enhance ultrasonic gas flow metering,” Elsevier Flow Measurement and Instrumentation, vol. 12, pp. 201–211, 2001. [4] A. Schroeder, M. Kupnik, R. Reitinger, M. Groeschl, and E. Benes, “Ultrasound Generation in Air with Focus On Recent Developments of Capacitance Ultrasonic Transducers,” in Congress of Alps Adria Acoustics Association, Portoroz, 2003. [5] A. Schroeder, S. Harasek, M. Kupnik, M. Wiesinger, E. Gornik, E. Benes, and M. Groeschl, “A Capacitance Ultrasonic Transducer for High Temperature Applications,” paper in preparation for IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2003. [6] G. Caliano, F. Galanello, A. Caronti, R. Carotenuto, M. Pappalardo, V. Foglietti, and N. Lamberti, “Micromachined Ultrasonic Transducers Using Silicon Nitride Membrane Fabricated in PECVD Technology,” in IEEE Ultrasonics Symposium, 2000, pp. 963–967. [7] A. D. Pierce, Acoustics: An Introduction to Its Physical Principles and Applications. New York: Acoustical Society of America, 1994. [8] J. Gaetke, Akustische Stroemungs- und Durchflussmessung. AkademieVerlag Berlin, 1991. [9] M. M. Boone and E. A. Vermaas, “A new ray-tracing algorithm for arbitrary inhomogeneous and moving media, including caustics,” Acoustical Society of America, vol. 90, no. 4, pp. 2109–217, 1991. [10] L. Zipser, “Acoustic gas sensor for extreme process conditions,” in IEEE Ultrasonics Symposium, 1997, pp. 445–448. [11] H. J. Dane, “Ultrasonic measurement of unsteady gas flow,” Elsevier Flow Measurement and Intrumentation, vol. 8, no. 3/4, pp. 183–190, 1997. [12] J. Allin and P. Cawley, “Design and construction of a low frequency wide band non-resonant transducer,” Elsevier Ultrasonics, vol. 41, pp. 147–155, 2003. ∗
Mario Kupnik, mailto:
[email protected]
VI. C ONCLUSION Using a new high-temperature resistant capacitive transducer enables flowmetering in hot pulsating gas flows. As a
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