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).