Besides the time measurement error caused by the quality of the ultrasonic echo signal, the time measurement method is also a significant factor affecting accuracy. The main time measurement methods include pulse counting and cross-correlation. In pulse counting, the resolution of the measurement time is a direct factor leading to measurement error. Since the timing pulse always has a width, the measurement error will occur within one pulse period.
The cross-correlation method calculates the cross-correlation between the signals propagating between two points to obtain the time delay between the two signals. Assuming the ultrasonic signal detected at one point is z(t), and the signal that propagates to another point after the delay time is y(t), the time delay between the two signals can be calculated by cross-correlation.
The cross-correlation calculation formula is as follows:
Furthermore, ultrasonic flow meters also suffer from time delays caused by cables, transducers, mounting slots, and electronic components. These time delays introduce systematic errors. These errors can be overcome through delay testing, zero-flow calibration correction, and temperature and pressure compensation.
The propagation trajectory of ultrasonic signals within a pipe changes with flow velocity, ultimately leading to inaccuracies in time measurements. The ultrasonic wave propagation trajectory within the fluid is not linear, and the trajectories differ depending on whether the flow is upstream or downstream. This propagation trajectory can be described using the channel curve tracing method in geometric acoustics. Based on the derivation by Boone and Vermaas, assuming the sound velocity C is constant, the channel curve tracing equation is:
In the formula: p(r) is the angle between the sound channel at a distance r from the centerline and the axis direction in the pipe; V is the average surface velocity of the sound channel at a distance r from the centerline and the axis section in the pipe.
Once the sensor's location is determined, the channel curve can be established. Since V is not constant, the channel exhibits a curved shape, and the channel angle is non-constant. The Reynolds number Re and Mach number Ma determine the channel curvature, which changes with velocity distribution and increases in Ma.
The Mach number in the pipe is defined as：
Ma=V/C
In the formula: C is the speed of sound; V is the average volume velocity of the fluid in the pipe. When Ma < 0.1, the ultrasonic wave propagation trajectory is approximately a straight line; the larger Ma is, the greater the curvature of the propagation trajectory.
As shown in the figure, when ultrasonic waves propagate in the fluid within the pipe, their propagation paths differ depending on whether they travel with or against the flow, resulting in different distances traveled and thus inaccurate measurement times.
From a signal perspective, energy loss occurs at both the transmitting and receiving transducers, and also during propagation in both directions. The greater the curvature of the propagation trajectory, the weaker the received echo signal. If the ultrasonic transducer's emission angle is inaccurate, the echo signal may not even be detected at the receiving transducer.
Zero-crossing detection detects the zero-crossing point by setting a threshold voltage. However, in practice, detection only begins after a delay following signal arrival, and this delay introduces a systematic error. Furthermore, when the signal amplitude fluctuates, the starting time obtained from zero-crossing detection with a fixed value will change. The ultrasonic signal propagating in gas within a pipe is shown in the figure. When the fluid velocity increases, the received signal amplitude decreases, which may cause the secondary instrument to detect different propagation times.
