A deflected ray model of the transit time difference in ultrasonic flow meters applied to non-ideal flow profiles

Transit time difference (TTD) ultrasonic meters have widespread applications due to their high dynamic range, lack of moving parts and low maintenance requirements. In earlier work, the TTD is considered to arise due to the fluid flow changing the effective speed of sound along a fixed ray path, but recent publications have introduced a new way of viewing how the TTD arises. The flow velocity in the axial direction does not affect the speed of sound in the direction normal to the pipe wall. Therefore, in a clamp-on meter with the fluid at constant temperature and pressure, the transit time in the liquid must be constant with changing flow rate. The TTD then arises due to the ray path in the water being deflected, and the received wavefront travelling some extra distance along the pipe axis, which results in a different transit through the transducer wedge than for zero flow conditions. In a meter with wetted transducers, the mechanism behind the TTD phenomena is slightly different. The deflected ray travels a different distance in the fluid along the direction normal to the pipe walls due to the sloped transducer surface, and this is responsible for the TTD that is measured. Previous work has focused on calculating the magnitude of the ray deflection effect in clamp-on meters for ideal flow conditions, where the flow velocity does not depend on the position in the pipe. In this article, the ray deflection is calculated for a series of analytic profiles, and it is shown that the two models result in the same calculated TTD for both clamp-on meters and meters with wetted transducers at low flow speeds. The hydraulic correction factors calculated using the model which considers a change in the effective speed of sound are recovered using the ray deflection models. The resulting equations for beam deflection are the same as those for the conventional fixed ray path model, but the beam deflection model is more physically realistic and presents opportunities for calibration methods utilising carefully controlled movement of the transducers. The conventional model typically has a correction factor which is allowed to vary in cases where the flows are expected to approach the speed of sound to account for the effect of ray deflection. However, the ray deflection model automatically takes this into account, so is expected to be useful in these circumstances.