Mathematical model describing air flow dynamics in a turbine spirometer

Diseases of the respiratory system are currently quite common, so the development of new effective ways to diagnose them is relevant. In this paper, we describe a mathematical model developed by us for the interaction of air flows with moving parts of a device for a recently created turbine spirometer of a new type. It has a number of technical features that must be taken into account when modeling. Among others, this is quite substantial inertia of the turbine and low friction. The model is based on the moment equation and contains several empirical parameters. Since the friction in the system is small, the main relations are considered in the linear approximation. Experimental verification of the model was carried out in two modes of operation of the spirometer. Firstly, the inertial motion of the turbine after turning off the external air source was investigated. Secondly, the dependence of the angular velocity of rotation of the turbine on the speed of an external constant air flow was analyzed. The calculations showed that in this two modes, the developed mathematical model describes the experimental results quite well. Also in this paper a simple method is given for determining empirical parameters at the device calibration stage. It is based on the use of the least squares method and does not require the involvement of large computational powers. This is an important circumstance, since the spirometer under investigation is intended for use not only in specialized medical institutions, but also in home conditions. On the base of the relations of the developed mathematical model, a numerical method is proposed for finding the velocity of the incoming air flow. This allows, basing on the readings of the device, to obtain clinically relevant information about the state of the respiratory system.