Mathematical formalism of an internal air flow through a vortex tube machine of Hilsch-Ranque type: An analytical representation for velocity field

Abstract

In this paper, we will attempt to derive an analytical representation of velocity field for the concept of an internal air flow inside a vortex tube machine of Hilsch-Ranque type. The basic assumption here is that the flux field is assumed to be steady and incompressible. Τhe vector which denotes instantaneous velocity, for the description of the flux field via Lagrangian stand point, will be expressed in a cylindrical coordinate system with orthonormal basis in the center line of the cross section of this vortex tube. We also assume here, that the aforementioned flux field varies only in radial direction, hence velocity vector is actually an one-dimensional damped circular helix trajectory with variable pitch.