Figure 1: Transit-time ultrasonic flowmeter.
The flowmeter measures fluid velocity by transmitting acoustic signals between the two transducers alternatively, first in the opposing direction of fluid flow, and then in the direction of flow. The transit time of the ultrasonic signal from the downstream transducer to the upstream transducer (\(t_{21}\)) and that in the opposite direction (\(t_{12}\)) are computable using Eq. (1) and (2) as follows.
\(t_{21}=\frac{L}{\left(c+vcos\theta\right)}\) …… (1)\(t_{12}=\frac{L}{\left(c-vcos\theta\right)}\) …… (2)
where \(c\) is the ultrasound speed in water, \(L\) is the path travelled by ultrasound, and θ is the incident angle.
Ultrasonic meters are velocity meters by nature. The fluid flow velocity\(v\) is calculated from the differences between the transit times of the signals which are directly proportional to fluid velocity. The equation for the flow velocity is:
\(v=\frac{L}{2cos\theta}\left(\frac{1}{t_{12}}-\frac{1}{t_{21}}\right)\)…… (3)
In a single-path flow meter, the ultrasonic beam path is diagonal, and the fluid flow velocity \(v\) is computed along this path. However, the velocity needed for computing volume flow rate is the average fluid velocity \(\overset{\overline{}}{v}\) across the pipe cross sectional area. Therefore, to convert velocity \(v\) to average fluid velocity\(\overset{\overline{}}{v}\), an average velocity correction factor is used that is shown as \(k_{c}\) according to Eq. (4). This quantity is generally a function of the Reynolds’ number.
\(\overset{\overline{}}{v}=k_{c}v\) …… (4)
Meter manufacturers have differing methodologies for computing velocity correction factor \(k_{c}\). Some derive it by using propriety algorithms. This requires knowledge about the velocity profile patterns at different Reynolds’ numbers. Nonetheless, the flow meter is still subjected to a certain degree of uncertainty. For example, fluid velocity profiles in the pipeline are not always uniform. Often there is swirl and asymmetrical flow profile within the meter (Lansing et al. 2003).