# Effects of Velocity Profiles on Measuring Accuracy of Transit-Time Ultrasonic Flowmeter
## 2 Methodology  
In transit-time type ultrasonic flowmeter, the two transducers can be arranged in W-type, V-type or Z-type, as shown in Figure 1. The difference between any of the two types is the propagation distance of ultrasonic wave. The W-type and V-type arrangement have longer distance, which will result in a longer time to be detected. So that the measurement accuracy could be improved. However, their measuring principle is the same, that is, one transducer works as transmitter and the other as a receiver, and the transmit time from one transducer to the other will reflect the flowing velocity of the fluid. Therefore, in order to make it easier for understanding, the Z-type is discussed in this paper. Additionally, the final results can be used for W-type and V-type.  
The basic principle of the transit-time ultrasonic flowmeter is shown in Figure 2, two ultrasonic transducers, upstream P\({}_{1}\) and downstream P\({}_{2}\), send and detect a short sound pulse with an oblique propagation direction (angle \(\theta\) with the pipe axis).  
Figure 1: Arrangement of ultrasonic transducers. (**a**) W-type; (**b**) V-type; and (**c**) Z-type.  
When the ultrasonic wave propagates downstream from transducer P1, its propagation speed will be coupled with the velocity projection of fluid in the direction of propagation. Therefore, the propagation time for downstream signal from P1 to P2 will be  
\[T_{1}=\frac{L}{c+ucos\theta} \tag{1}\]  
where \(L\) is the propagation distance of ultrasonic wave from transducer P1 to P2, \(c\) is the sound speed in the fluid, u is the velocity of the fluid, and \(\theta\) is the angle between the directions of fluid flowing and wave propagation.  
Similarly, when ultrasonic wave propagates upstream from transducer P2, its propagation speed will subtract the velocity projection of fluid in the direction of propagation. Therefore, the propagation time for upstream signal from P2 to P1 will be  
\[T_{2}=\frac{L}{c-ucos\theta} \tag{2}\]  
Eliminating the propagation speed of ultrasonic wave from Equations (1) and (2), the velocity of fluid can be expressed as  
\[\mathrm{u}=\frac{L}{2\cos\theta}\cdot\frac{T_{2}-T_{1}}{T_{1}\cdot T_{2}} \tag{3}\]  
If the flow is axially uniform, the flowrate in the pipe can be estimated by measuring the transit times of downstream signal \(T_{1}\) and upstream signal \(T_{2}\), expressed as  
\[\Delta T=T_{2}-T_{1}=\frac{L}{c-u\cos\theta}-\frac{L}{c+u\cos\theta} \tag{4}\]  
However, the velocity distribution across the pipe is not uniform. The variation in velocity will have effects on the propagation of ultrasonic wave. Thus, Equation (4) can be used only within a very thin layer of fluid. Then,  
\[dT_{2}-dT_{1}=\frac{dL}{c-u\cos\theta}-\frac{dL}{c+u\cos\theta} \tag{5}\]  
where \(dL\) is the propagation distance of ultrasonic wave across the thin layer.  
According to geometric relation, we have  
\[\mathrm{d}L=\frac{2 dr}{\sin\theta} \tag{6}\]  
where, \(dr\) is the thickness of the thin layer.  
Figure 2: Principle of the transit-time ultrasonic flowmeter.  
Substituting Equation (6) into Equation (5), the transit times must be expressed as  
\[\Delta T=T_{2}-T_{1}=\frac{2}{\sin\theta}\int_{0}^{R}\big{[}\frac{1}{c-u\cos \theta}-\frac{1}{c+u\cos\theta}\big{]}d\tau \tag{7}\]  
where \(R\) is the radius of the pipe.  
Introducing a dimensionless radius \(s=\frac{r}{R}\), then  
\[\Delta T=\frac{2R}{\sin\theta}\int_{0}^{1}\big{[}\frac{1}{c-u\cos\theta}-\frac {1}{c+u\cos\theta}\big{]}ds \tag{8}\]  
Re-written as  
\[\Delta T=\frac{2R}{u_{m}\sin\theta\cos\theta}\int_{0}^{1}\big{[}\frac{1}{K-u/u _{m}}-\frac{1}{K+u/u_{m}}\big{]}ds \tag{9}\]  
where \(c\) is the sound speed; \(K=\frac{c}{u_{m}\cos\theta}\).