Equilibrium orbit

Theoretically, for a particle of a given size that moves in the highly rotating fluid flow in a cyclone, a particular radial orbit position may be found in every horizontal plane of the cyclone where the outward centrifugal force is just balanced by the drag exerted on the particle by the radial inward fluid flow. If Stokes s law (13.16) is assumed, then the position of the equilibrium orbit on each horizontal plane of the cyclone may be obtained and is given by... 

Particles inside the equilibrium line penetrate the cyclone with the inward fluid flow, while the particles outside the equilibrium line, and cannot reach the equilibrium line, are collected by the downward fluid flow. When the particles do reach the equilibrium line, their fate is determined by the direction of the vertical fluid velocity at this point (i.e., whether it is upward or downward at the equilibrium line). If a particle is in equilibrium in an upward-moving stream, it will penetrate the cyclone otherwise it will be collected. It is evident that small particles find their equilibrium orbit at small radii and therefore are more likely to join the upward... 

Some of the chemical concepts with little or no quantum-mechanical meaning outside the Bohmian formulation but, well explained in terms of the new interpretation, include electronegativity, the valence state, chemical potential, metallization, chemical bonding, isomerism, chemical equilibrium, orbital angular momentum, bond strength, molecular shape, phase transformation, chirality and barriers to rotation. In addition, atomic stability is explained in terms of a simple physical model. The central new concepts in Bohmian mechanics are quantum potential and quantum torque. 

Eliminating the velocity, v, we obtain the equation determining the radius of the equilibrium orbit, r = ro F, B) ... 

In cyclic accelerators, the beam takes many turns and the particles should keep together in a bunch both longitudinally and transversaHy. By the action of focusing forces, the particles perform synchrotron and betatron osdDations around an imaginary synchronous particle moving on a stable/smooth equilibrium orbit. ... 

A positive value of n implies a focusing effect. The value of n varies with the radius r (it is zero at the center of the magnet) and it can be used to calculate the motion of particles in the radial and axial directions at a particular radius. Ignoring the detailed discussion, particles displaced axially or radially from their equilibrium orbit will sense a force which is proportional to the displacement (like a simple harmonic oscillator) and will perform stable axial and radial oscillations, which are known as betatron oscillations. These oscillations are stable when n lies between 0 and 1. Any ion leaving the ion source in a direction not in the median plane and not on the equilibrium orbit will make axial and radial betatron oscillation. 

The beam consists of a large number of particles. Most of them move along orbits that differ from the equilibrium orbit (e.g., the axis of a beam line). This behavior is characterized by the property called the emittance of the beam. This parameter is derived from considerations of the particle motion in the phase space, which is a six-dimensional hyperspace composed of the particle coordinates (x,y, z) and the three momentum components (p, py, pj (Banford 1966). 

Each theory in this category offers a relatively simple correlation for the static pressure drop and the cut size of a hydrocyclone described by a few (but often not all) dimensions. The theories fall into two main groups the equilibrium orbit theory and the residence time theory. 

The equilibrium orbit theory is based on the concept of the equilibrium radius, originally proposed by Dries sen" and Criner. According to this concept, particles of a given size attain an equilibrium radial orbit position in the cyclone where their terminal settling velocity is equal to the radial velocity of the liquid. Particles are therefore elutriated by the inward radial flow according to the balance of the centrifugal and drag forces, and Stokes law is usually assumed. 

Probably the best known and most credible approach to the equilibrium orbit theory is that due to Bradley and Pulling. This is based on the discovery of the mantle by the same authors, i.e. an area in the region immediately below the vortex finder where there is no inward radial velocity. Consequently, the authors only used a conical surface below the mantle, as shown... 

The equilibrium orbit theory in all its various forms suggested by various authors, can be criticized on the grounds that it takes no account of the residence time of the particles in the cyclone. Not all particles may be able to attain equilibrium orbits within their residence time. The theory also takes no account of turbulence as it might affect particle separation. Despite the above disadvantages, many of the various forms of the equilibrium orbit theory (as reviewed more fully by Svarovsky ) give reasonable predictions of cyclone performance at low feed solids concentrations, particularly if used under similar conditions and with similar cyclone designs and sizes as in the original work of their respective proposers. 

The residence-time theory, despite its very different approach and assumptions, often leads to correlations of very similar form to those from the equilibrium orbit theory. Either of the two theories will work better for the respective geometries to which they were tailored and applied by their authors. ... 

Under the action of these forces the particle moves inwards or outwards until the forces are balanced and the particle assumes its equilibrium orbit. At this point. 

If we assume that all particles with an equilibrium orbit radius greater than or equal to the cyclone body radius will be collected, then substituting r = R in Equation (9.19) we derive the expression below for the critical particle diameter for separation, Xan. ... 

A number of physical models have been proposed for the separation process in a hydrocyclone (Driessen MG, 1951 Bradley and Pulling, 1959 Fahlstrom, 1960 Kelsall, 1952 Rietema, 1961 and Schubert and Neesse, 1980). Among these, different phenomenological approaches have led to the development of two basic theories the equilibrium orbit theory and the residence time theory. 

Equilibrium Orbit Theory. The general concept that particles of a given size reach an equilibrium radial orbit position in the hydrocyclone forms the basis of equilibrium orbit theory. The fine particles reach equilibrium at small radii where the flow is moving upwards and transports fines to the overflow, while the coarse particles find equilibrium position at large radii where the flow is moving downwards and carries these particles to the underflow outlet (apex). The dividing surface is the locus of zero vertical velocity (LZW). The size of the particles that find equilibrium radius on LZW will be the cut size that has an equal chance to finish in either overflow or underflow. 

In developing the equilibrium orbit theory, a key assumption made by Bradley and Pulling (1959) is the existence of a mantel in the hydrocyclone, which precludes inward radial velocity in the region immediately below the vortex finder. Furthermore, the LZW is assumed to be in the form of an imaginary cone whose apex coincides with the apex of the hydrocyclone and whose base is at the bottom of the mantle. Based on these assumptions, the equilibrium orbit theory has led to the development of empirical correlations for determining the cut size and pressure drop in hydrocyclone operation. 

The major deficiency of the equilibrium orbit theory lies in its lack of consideration of the effect of turbulence flow on particle separation and the residence time of the particles in the hydrocyclone (as not all particles are able to find equilibrium orbits within their residence time). In spite of such weaknesses, it proves to be a reasonable approach for determining the hydro-... 

There are basically two modeling concepts for cyclone separation efficiency in the literature the equilibrium-orbit models and time-of-ffight models. 

The model of Barth (1956) (see also Muschelknautz (1972)) is the original equilibrium-orbit model. 

In conclusion, we can say that the time-of-flight modeling concept is entirely different in nature from the equilibrium-orbit concept. It is peculiar that these two very different concepts should result in efficiency models that agree well, both absolutely and in trends, over a wide range of cyclone designs... 

In Appendix 5.B, we give the model equations for one more cyclone performance model that of Mothes and Ldffler. This model is built on the approach of Dietz, and is hybrid between the equilibrium-orbit and the time-of-flight models. 

This type of model considers both particle interchange between the outer and the inner vortices across CS and particle migration to the wall. It is in this sense that these models can be said to be hybrids between the other two types. In practice the X50 predicted by these models often lie close to the X50 predicted by the equilibrium-orbit models. 

Barth Efficiency model valid for all cyclones and swirl tubes (but see Table 4.5.1). Equilibrium-orbit model. Calculates the cut size, then fits an empirical grade-efficiency curve through it. Time-of-flight model. Derivation considers particle motion, but the final model relates cut size to pressure drop. 

The models of Dietz (1981) and of Mothes and Loffler (1988) consider both the exchange of dust between the inner and outer part of the swirl and the migration of particles to the wall. They can therefore be considered as hybrids between the equilibrium-orbit and the time-of-flight models. 

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