Ray Tracing Models of Ultrasonic Flow Meters

One of the major sources of error in clamp-on flow measurement is misplacement of transducers in the field. When performing measurements on a metal walled pipe, the calculation of the appropriate separation between the transducers is relatively simple, because the angle of the transducer wedge is typically chosen such that only a shear wave is generated in the pipe wall. This is possible because the polymer transducer wedge and pipe wall have very different sound speeds, and so the critical angle for the longitudinal wave in the pipe wall can be exceeded. When metering on a plastic pipe, there is no critical angle because the pipe wall and transducer wedge are acoustically very similar. This leads to a situation where the compression wave generated in the transducer wedge generates both a longitudinal and a shear wave in the pipe wall which propagate in different directions, and a single ray can no longer be traced through the system showing where the transducers should be situated. The situation is complicated further by most polymers attenuating shear waves more than compression waves. It is, therefore, necessary to model all of the possible paths through the flow meter to determine which path most of the ultrasonic energy takes, enabling operators to calculate the correct transducer separation and minimise the error on the flow measurements.

A ray-tracing model was programmed to trace potential ultrasonic paths through the system and keep track of the amplitudes and phases of those waves when they reach the reception transducer. The geometry of the model is shown below.

Rays are configured to originate along the left transducers' piezoelectric element at height where they have an amplitude of unity and a specified bandwidth. Waveforms are generated and associated with each. As the rays are traced through the system, the following effects are accounted for by modifying each ray and its associated waveform in the frequency domain:

• Transits though different materials - time delays applied based on the material and wave type (longitudinal or shear).
• Mode conversion at material interfaces - affects the phase, amplitude, wave type and propagation direction. A single ray can split into 2 rays if both longitudinal and shear waves are transmitted across the boundary.
• Attenuation - affects the amplitude and is frequency, wave type, and material dependent.

Additionally, beam spread is accounted for by allowing a small range of angles for all of the starting rays at the generation piezoelectric element as shown in the image below. Note that the number of rays shown is much less than the number modelled for clarity in the figure. Blue lines indicate longitudinal waves and red lines indicate shear waves.

In the transducer wedges, only longitudinal waves are considered because the piezoelectric elements typically used in flow measurement are operated in their through-thickness resonance, which generates and is most sensitive to compression waves. In the fluid, no shear stress is supported and so only the longitudinal waves can be present. This leaves the pipe walls, in which both longitudinal and shear waves could exist. There are two transits through the top wall, one just after generation and one just before detection, but the model also allows for transmission into the far pipe wall, which can generate either longitudinal or shear waves. There are, therefore, a maximum of four transits through the pipe wall, and a minimum of two transits if the ultrasound reflects off the inner surface of the pipe.

When all rays have been traced through the system, the waveforms of all rays which are incident on the piezoelectric element can be summed to simulate the received signal. Additionally, all rays that have taken a particular path through the system (e.g. all rays which are longitudinal in all wall transits) can be included in a sum which indicates the fraction of received energy that takes that path. Repeating this for a range of transducer separations then allows us to see which paths energy takes through the meter as a function of the transducer position.

The results are shown above for two UPVC pipes which have exterior diameters of 63 mm. The left panel is for a PN10 pipe, which has a 3 mm wall thickness, and the PN16 pipe in the right panel has a 4.7 mm wall thickness. The legends indicate the wave mode in each wall transit, in the order encountered by the wave (L=longitudinal, S=shear, N=no transit). For example, LNNL refers to the path which includes longitudinal waves in both top wall transits, and no transits in the bottom wall, i.e., the waves reflect off the inner surface of the pipe. The vertical lines indicate the spacing that would be calculated using each path with a simple, single-ray model.

The key conclusions are:

1. Longitudinal waves dominate in the walls of polymer pipes. Although both wave modes are generated due to the similar acoustic properties of the transducer and pipe, the shear waves are preferentially attenuated and so they do not significantly contribute to the received signal.
2. Waves that are transmitted into the far wall and reflect off the outer surface have a significant contribution to the received signal in polymer pipes. The acoustic impedance mismatch between the water and pipe wall is much smaller than for metal pipes, so a significant amount of energy is transmitted into the far wall.

Point 2 above has implications for the correct placement of transducers. It means that a simple, single-ray model no longer adequately describes the ultrasonic behaviour, and the received wave is due to the interference of two beams: one that has reflected off the inner surface of the pipe, and one that reflected off its outer surface. The transducer separation at which the maximum amplitude is detected therefore no longer corresponds to any single ray path through the system. Since the transducer position for maximum amplitude is dependent on the interference between the two beams, it is dependent on the phase difference between them and their locations, which depend on:

• The wall thickness changes the positions of the two beams, the amplitude of the beam that has transited through the far wall, and their relative phases.
• The wall material changes the attenuation of the beam that travels through the far wall, and the refraction angles into it which affects their relative positions.
• The transducer wedge angle changes all of the ray angles through the system, which affects all of the mode conversion calculations, and lengths of transits in different materials.

This behaviour is much more complex than for a metal pipe, where the high acoustic impedance mismatch between the pipe and water/transducers means that only shear waves are present in the walls and the fraction of energy which is transmitted into the far wall is negligible. It is, therefore, recommended to perform modelling to determine optimal transducer placement when performing clamp-on flow measurement on polymer pipes. Note that it is shown in our article linked below, that simply scanning a transducer until the maximum received amplitude is reached can lead to errors in the flow measurement if the ultrasonic behaviour is not properly understood and accounted for.

This page is based on a journal article published here, from which the figures have been reproduced. The MATLAB model used to perform the calculations is available here.