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

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.


INTRODUCTION
This vortex tube machine was firstly invented in 1931 by a Frenchman Physicist named Georges J. Ranque when he was studying processes in a dust separated cyclone (Fulton, 1950;Reynolds, 1962). The patent and idea was abandoned for several years until 1947, when a German engineer Rudolf Hilsch modified the design of the tube (De Vera, 2010;Eiamsa-ard and Promvonge, 2007). Particularly, R. Hilsch improved the initial design of this device by constructing a number of tubes and published experimental data with respect to their operation (Guillaume and Jolly, 2001). Since then, many researchers have tried to find ways to optimize its efficiency.
The vortex tube is actually a mechanical device that separates a compressed gas into hot and cold streams. It has no moving parts (Hilsch, 1947). Pressurized gas is injected tangentially into a swirl chamber and accelerated to a high rate of rotation. Due to the conical nozzle at the end of the tube, only the outer shell of the compressed gas is allowed to escape at that end. The remainder of the gas is forced to return in an inner vortex of reduced diameter within the outer vortex (Gutsol, 1997;Hilsch, 1947). There are different explanations for the effect and there is debate on which explanation is best or correct.
What is usually agreed upon is that the air in the tube experiences mostly "solid body rotation" which means that angular velocity of the inner gas is the same as that of the outer gas. This is different from what most consider standard vortex behavior -where inner fluid spins at a higher rate than outer fluid. The behavior of "solid body rotation" is probably due to the long length of time during which each parcel of air remains in the vortexallowing friction between the inner parcels and outer parcels to have a notable effect (Sibulkin, 1961). Besides during the vortex stretching procedure the angular momentum of the fluid matter stays invariant.
This vortex tube machine was firstly invented in 1931 by a Frenchman Physicist named Georges J. Ranque when he was studying processes in a dust separated cyclone (Fulton, 1950;Reynolds, 1962). The patent and idea was abandoned for several years until 1947, when a German engineer Rudolf Hilsch modified the design of the tube (De Vera, 2010;Eiamsa-ard and Promvonge, 2007). Particularly, R. Hilsch improved the initial design of this device by constructing a number of tubes and published experimental data with respect to their operation (Guillaume and Jolly, 2001). Since then, many researchers have tried to find ways to optimize its efficiency.
The vortex tube is actually a mechanical device that separates a compressed gas into hot and cold streams. It has no moving parts (Hilsch, 1947). Pressurized gas is injected tangentially into a swirl chamber and accelerated to a high rate of rotation. Due to the conical nozzle at the end of the tube, only the outer shell of the compressed gas is allowed to escape at that end. The remainder of the gas is forced to return in an inner vortex of reduced diameter within the outer vortex (Gutsol, 1997;Hilsch, 1947). There are different explanations for the effect and there is debate on which explanation is best or correct.
What is usually agreed upon is that the air in the tube experiences mostly "solid body rotation" which means that angular velocity of the inner gas is the same as that of the outer gas. This is different from what most consider standard vortex behavior -where inner fluid spins at a higher rate than outer fluid. The behavior of "solid body rotation" is probably due to the long length of time during which each parcel of air remains in the vortexallowing friction between the inner parcels and outer parcels to have a notable effect (Sibulkin, 1961). Besides during the vortex stretching procedure the angular momentum of the fluid matter stays invariant.
The chief characteristic of this peculiar device is separation of the introduced air into hot and cold fractions. His interest lay in its potentialities as a refrigerating unit, but apparently he was unable to develop it satisfactorily (Eckert and Hartnett, 1955;Lewellen, 1962;Linderstrom-Lang, 1967). His work attracted rather widespread interest and, as a result, a number of relevant publications have appeared.
Some further investigations by Physicists and Engineers have been actually of particular significance, because they reported measurable mass separation gas mixtures and discussed theories of operation (Ahlborn and Gordon, 2000;De Vera, 2010;Piralishili et al., 2005). However, no explanations have been attempted. Most of other reports have mainly dealt with the application of the tube to refrigeration (Ahlborn et al., 1996;Gao et al., 2005;Eckert and Hartnett, 1955;Khodorkov et al., 2003).

Venetis and Sideridis 61
Until today, there is no single theory that explains the radial temperature separation. However, the operating cost of this machine is generally low and its applications are indeed various (Tassios, 2006). The cross sectional area with a plane parallel to axis zz΄ of this aforementioned hook-up is represented in Figure 1.
As concerns the instantaneous velocity distribution, the following relationship holds: where  e  denotes the unit vector being collinear to the component: V  at a Cylindrical Coordinate System and in fact is able to be written out equivalently with respect to its own projections at the Cartesian Coordinate System Οxyz, as follows: Moreover, the unit vector which is collinear to the radial component: can be also written out as Hence, it follows: The cross sectional area with a plane parallel to axis zz΄ of this aforementioned hookup is represented in figure 1: Since the flux field that we examine is primarily assumed to vary only in the radial direction, it implies that velocity component  V is identically zero (Figure 2). Nevertheless, the generic transform relations read Hence, the one-dimensional instantaneous velocity at the vortex tube emerges via the following representation: In this paper, we will make a specific implementation for a characteristic rational function, particularly for the homographic one, that is where: It is also known from elementary Calculus, that Equation (6) can be exhibited equivalently in the following form: So we can deduce: The corresponding boundary conditions for the particular original problem are: Concurrently, the position vector components in Cartesian Ccoordinates are: constitutes an arbitrary single -valued function.
In the sequel, according to Lagrangian formalism we deduce: Hence, with the latter two equations in hand in accordance with the set of Equation (9) we can eventually infer

DESCRIPTION OF THE STREAM AND VORTEX LINES
It is known from elementary Vector Calculus that the following invariant vector identity holds true, for an infinitesimal part of an arbitrary stream line s d The corresponding arc length being traced by an arbitrary trip of position vector is given by the integral: It is also known from Differential Geometry that all curves which have both the same curvature function and same torsion function are congruent.
Hence the following conjunction holds On the other hand, according to Lagrangian formalism for the fluid motion along an arbitrary stream line of the flux field that we study the following equality holds: By operating the material derivatives at both members of Equation (22) We emphasize that the above relationship holds along any stream line of the investigated flux field. Besides, the following implication also holds: Alongside, from Vector Calculus the following identity holds: where V denotes the algebraic rate of vector V  regardless of its circumstantial orientation.
Hence, by considering an element of length s d  along an arbitrary stream line and operating the dot product in Equation (28) The arc length whose endpoints define this orientated segment is traced by a trip of position vector ) (t r and is given by the integral: We also observe that Equation (30)  denoting indeed the orientation of each arc whose length gets rates from the range of the scalar function: Apparently, Equation (32) holds for any stream lines network of incompressible flow patterns (internal and wake ones, as well as free turbulence).
Apart from these, according to Lagrangian formalism for a fluid moving particle in a cylindrical coordinate system the following formulas for instantaneous velocity and acceleration also hold true: Equations (34) and (28) are indeed identically equivalent to one another as the interface of two phenomenological points of view for the inceptive original problem, (that is, the approach of a bundle of moving particles and the approach of a continuum medium) and must be assessed in association with Equation (17).
Since we have focused on steady flux fields, by considering the geometry of the vortex tube as a succession of its cross sectional areas with planes parallel to axis ' zz we can neglect the vorticity transfer in space and therefore the initial original problem substantially reduces to a superposition of twodimensional flux fields.