An Exact Analytical Model for the Relationship Between Flow Resistance and Geometric Properties of Tubes Used in Semi-occluded Vocal Tract Exercises

Abstract

Objectives: The purpose of this work was twofold. First, we aimed to develop an exact analytical model for tubes used in semi-occluded vocal tract exercises to gain a quantitative insight into the relationship between flow resistance and the tube's geometric parameters. The second goal was to provide an isometric resistance chart that can be used by clinicians to indicate either the diameter or the length of a tube, so as to provide the same flow resistance of a reference tube.

Methods: The theory for confined flows based on the Darcy-Weisbach equation was used to derive the analytical model, in which the friction factor was obtained by an explicit approximation with error smaller than 0.4%. The isometric resistance chart was generated with the analytical model, assuming the volume flow to be constant and equal to 0.4 L/s.

Results: The results obtained from the model agreed very well with both experimental and theoretical results from the literature, particularly for tubes longer than 6 cm and with inner diameters greater than 4.1 mm. In general, the analytical model slightly underestimates the back pressure for shorter and thinner tubes. For these cases, the maximum difference between analytical and experimental results corresponds to ≈5%.

Conclusions: Analysis of the equation terms indicated that the flow resistance is significantly more sensitive to variations in the inner diameter than to variations on the tube's length, which agrees with experimental results found in the literature. Moreover, the effect of the mouth configuration is negligible for tubes whose length are one order of magnitude greater the the inner diameter. Nevertheless, for short tubes the mouth configuration becomes a significant parameter and can be described by the ratio between the tube's inner diameter and the effective diameter of the patient's oral cavity. The analytical model presented in this work can be applied to the entire set of tube geometries found in clinical practice and constitutes a straightforward tool for designing tubes for semi-occluded vocal tract exercises with specific therapeutic purposes or patient needs.

Keywords: Back pressure; Flow resistance; Resonance tube phonation; Semi-occluded vocal tract exercises; Voice therapy.