where fs and fr are the frequencies in the transmitting and receiving crystals, respec- tively, and the plus/minus signs indicate different directions of flow. An important conclusion from the above equation is that the differential frequency is directly pro- portional to the flow velocity. Obviously, the crystals must have much smaller sizes than the clearance of the tube of flow. Hence, the measured velocity is not the av- erage but rather a localized velocity of flow. In practical systems, it is desirable to calibrate ultrasonic sensors with actual fluids over the useful temperature range, so that contribution of a fluid viscosity is taken into account.
An ultrasonic piezoelectric sensors/transducer can be fabricated of small ceramic disks encapsulated into a flowmeter body. The surface of the crystal can be protected by a suitable material, (e.g., silicone rubber). An obvious advantage of an ultrasonic sensor is in its ability to measure flow without a direct contact with the fluid.