Inner vortex spin velocity

In order to compute certain key cyclone characteristics, such as the internal spin velocity, vocs, or the particle cut size in the inner vortex core, X50, it is necessary to first compute the gas-phase and total gas-plus-solids wall friction factors, fair and /, respectively. Gas-phase wall friction factors for both cylindrical and conical cyclones as a function of body Reynolds number and relative wall roughness are presented in Fig. 6.1.3. Muschelknautz and Trefz define the cyclone body Reynolds number (compare with Eq. 4.2.8) as ... 

We notice first that the end of the vortex is precessing around the inner wall of the separator at some precessional frequency / and precessional velocity vector Vp. For the case illustrated, this motion is counterclockwise (ccw) as viewed from above. However, superimposed on this motion is the vortex core spin or rotational velocity vector, vcs- This core spin velocity adds to the precessional velocity Vp at its top-most position (position A in illustration) producing the resultant velocity v. On the other hand, the core spin vector opposes the precessional velocity at the bottom-most position (position B). 

As one might expect, the frequency of rotation of the vortex core around the inner walls of the cyclone (or swirl tube) has been observed to be directly proportional to the gas flow rate. Interestingly, for any given gas flow rate, this precession frequency is also foimd to be approximately equal to the core s maximum spin velocity divided by the circmnference of the inner wall at the plane of attachment. We will illustrate this below. 

In Fig. 9.2.7 we plot the velocity at which the vortex end or core trans-verses around the inside wall of the hopper (from Eq. 9.2.1) versus the estimated maximum tangential velocity of the vortex core (from Eq. 9.2.2). This plot strongly suggests that the precessional velocity is directly related to the maximum spin velocity. Thus, as stated earlier, precession frequency can be estimated by simply dividing the core s maximum spin velocity by the circumference of the inner wall to which it is attached. If this is true, then the end of the vortex acts much like a rubber wheel that rotates around the inner walls at a velocity equal to the maximum spin velocity of the vortex core. 

As gas moves from the outer to the inner part of the vortex in the cyclone body, it is accelerated in accordance with the principle of conservation of moment-of-momentum or, as some would call it, conservation of angular momentum (Chap. 2). In addition, its static pressure decreases (see also Eq. 2.A.12). We can say that the vortex transforms static pressure into dynamic pressure. For a given wall velocity, (governed largely by the inlet velocity), the less the frictional loss, the more intense the vortex, the more efhcient is this conversion, and the lower is the central static pressure with which the gas enters the vortex finder. The limit, which is never obtained in practice, is a frictionless vortex, where Eqs. (2.1.2) and (2.1.3) are valid. Smooth walled and aerodynamically clean cyclones therefore produce the highest spin in the vortex and the greatest decrease in static pressure within the core, other factors being equal. 

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