Calculation of acoustic flow-rate measurements error with the help of Fermat’s principle

Abstract

The establishment of dispatching and automatic control systems for mine ventilation is impossible without the availability of perfect air flow rate sensors. Existing anemometers (tachometer, heat) do not meet these requirements. The error of average in cross section velocity measurements with such sensors reaches 15-20, sometimes 30%. The reason - the speed measured at one point is interpreted as the average over the cross section. The reliability of the sensors is small, because they are exposed to the damaging effect of a dusty atmosphere. Stationary installed anemometers clutter cross section, which is not always allowed. Fermat’s variational principle is used for derivation of the formula for the time of propagation of a sonic signal between two set points A and B in a steady three-dimensional flow of a fluid or gas. It is shown that the fluid flow changes the time of signal reception by a value proportional to the flow rate independently of the velocity profile. The time difference in the reception of the signals from point B to point A and vice versa is proportional with a high accuracy to the flow rate. It is shown that the relative error of the formula does not exceed the square of the largest Mach number. This makes it possible to measure the flow rate of a fluid or gas with an arbitrary steady subsonic velocity field