An ultrasonic Doppler flowmeter is to be used to measure the volume flow rate of a slurry in a steel pipe of diameter 0.2 m. Two piezoelectric crystals, each having a natural frequency of 1 MHz, are positioned, a few millimetres apart, on the outside of the pipe to form an ultrasonic transmission link. The transmitting crystal directs an ultrasonic beam into the pipe so that the beam is moving in an opposite direction to the flowstream. The angle between the ultrasonic beam and the direction of flow is 60°. On average 10% of the ultrasonic power reaching each solid particle is scattered back in the direction of the receiving crystal. Assume that the slurry has the same density and sound velocity as water and a power attenuation coefficient of 1.0 m−1.

(a) Find the difference between the frequencies of the transmitted and received beams when the flow rate is 1.13 × 103 m3 h−1.

(b) Estimate the ultrasonic power incident on the receiving crystal for each watt of ultrasonic power leaving the transmitting crystal. State any assumptions made in your calculation