Editing Cyclonic separation (section)

== Cyclone theory ==
As the cyclone is essentially a two phase particle-fluid system, [[fluid mechanics]] and particle transport equations can be used to describe the behaviour of a cyclone. The air in a cyclone is initially introduced tangentially into the cyclone with an inlet velocity <math>V_{in}</math>. Assuming that the particle is spherical, a simple analysis to calculate critical separation particle sizes can be established.

If one considers an isolated particle circling in the upper cylindrical component of the cyclone at a rotational radius of <math>r</math> from the cyclone's central axis, the particle is therefore subjected to [[Drag (physics)|drag]], [[Centrifugal force|centrifugal]], and [[buoyancy|buoyant]] forces.  Given that the fluid velocity is moving in a spiral the gas velocity can be broken into two component velocities: a tangential component, <math>V_t</math>, and an outward radial velocity component <math>V_r</math>. Assuming [[Stokes' law]], the drag force in the outward radial direction that is opposing the outward velocity on any particle in the inlet stream is:

:<math> F_d = -6 \pi r_p \mu V_{r} .</math>

Using <math>\rho_p</math> as the particle's density, the centrifugal component in the outward radial direction is:

:<math> F_c= m \frac{V_t^2}{r} </math>
::<math> = \frac{4}{3} \pi \rho_p r_p^3 \frac{V_t^2 }{r} .</math>

The buoyant force component is in the inward radial direction.  It is in the opposite direction to the particle's centrifugal force because it is on a volume of fluid that is missing compared to the surrounding fluid.  Using <math>\rho_f</math> for the density of the fluid, the buoyant force is:

:<math> F_b = -V_p\rho_f \frac{V_t^2}{r} </math>
::<math> = -\frac{4 \pi r_p^3}{3} \frac{V_t^2}{r}\rho_f .</math>

In this case, <math> V_p </math> is equal to the volume of the particle (as opposed to the velocity).  Determining the outward radial motion of each particle is found by setting Newton's second law of motion equal to the sum of these forces:

:<math> m \frac{dV_r}{dt} = F_d + F_c + F_b </math>

To simplify this, we can assume the particle under consideration has reached "terminal velocity", i.e., that its acceleration <math>\frac{dV_r}{dt}</math> is zero. This occurs when the radial velocity has caused enough drag force to counter the centrifugal and buoyancy forces. This simplification changes our equation to:

<math>F_d + F_c + F_b = 0 </math>

Which expands to:

:<math> -6\pi r_p \mu V_r + \frac{4}{3}\pi r_p^3 \frac{V_t^2}{r}\rho_p -\frac{4}{3}\pi r_p^3 \frac{V_t^2}{r}\rho_f =0 </math>

Solving for <math>V_r</math> we have

:<math> V_r = \frac{2}{9} \frac{r_p^2}{\mu} \frac{V_t^2}{r} (\rho _p - \rho _f)</math>.

Notice that if the density of the fluid is greater than the density of the particle, the motion is (-), toward the center of rotation and if the particle is denser than the fluid, the motion is (+), away from the center. In most cases, this solution is used as guidance in designing a separator, while actual performance is evaluated and modified empirically.

In non-equilibrium conditions when radial acceleration is not zero, the general equation from above must be solved.  Rearranging terms we obtain

:<math> \frac{dV_r}{dt} +  \frac{9}{2} \frac{\mu}{\rho_p r_p^2}V_r - \left(1-\frac{\rho_f}{\rho_p}\right) \frac{V_t^2}{r} = 0</math>

Since <math>V_r</math> is distance per time, this is a 2nd order differential equation of the form <math>x''+c_1 x'+c_2=0</math>.

Experimentally it is found that the velocity component of rotational flow is proportional to <math>r^2</math>,<ref name="Rhodes">{{cite book|author=Rhodes M.| title=Introduction to particle technology |year=1998| publisher=John Wiley and Sons | isbn=978-0-471-98483-2}}</ref> therefore:

:<math>V_t \propto r^2 .</math>

This means that the established feed velocity controls the vortex rate inside the cyclone, and the velocity at an arbitrary radius is therefore:

:<math> U_r = U_{in}\frac{r}{R_{in}} .</math>

Subsequently, given a value for <math>V_t</math>, possibly based upon the injection angle, and a cutoff radius, a characteristic particle filtering radius can be estimated, above which particles will be removed from the gas stream.

=== Alternative models ===
The above equations are limited in many regards. For example, the geometry of the separator is not considered, the particles are assumed to achieve a steady state and the effect of the vortex inversion at the base of the cyclone is also ignored, all behaviours which are unlikely to be achieved in a cyclone at real operating conditions.

More complete models exist, as many authors have studied the behaviour of cyclone separators.<ref>{{cite thesis|title=PhD thesis: Experimental and Analytical Study of the Vortex in the Cyclone Separator|author=Smith, J. L. Jr.| url=http://dspace.mit.edu/handle/1721.1/11792|year=1959|publisher=Massachusetts Institute of Technology |hdl=1721.1/11792|type=Thesis }}</ref> Simplified models allowing a quick calculation of the cyclone, with some limitations, have been developed for common applications in process industries.<ref>{{Cite web|url=https://powderprocess.net/Equipments%20html/Cyclone_Design.html|title=Cyclone design - Step by step guide - Powderprocess.net|accessdate=26 March 2023}}</ref> Numerical modelling using [[computational fluid dynamics]] has also been used extensively in the study of cyclonic behaviour.<ref>{{cite journal|doi= 10.1590/S0104-66322007000100008|last1= Martignoni|first1= W. P.|last2= Bernardo|first2= S.|last3= Quintani|first3= C. L.|title= Evaluation of cyclone geometry and its influence on performance parameters by computational fluid dynamics (CFD)|journal=Brazilian Journal of Chemical Engineering|volume= 24|pages= 83–94|year= 2007|doi-access= free}}</ref><ref>{{cite book | url=http://archiv.tu-chemnitz.de/pub/2007/0013/data/diss.pdf | title=PhD Thesis: On the Potential of Large Eddy Simulation to Simulate Cyclone Separators | access-date=2009-06-20 | archive-url=https://web.archive.org/web/20070709215452/http://archiv.tu-chemnitz.de/pub/2007/0013/data/diss.pdf | archive-date=2007-07-09 | url-status=dead }}</ref><ref>{{cite book | url=http://www.wtb.tue.nl/woc/ptc/lit/kroes2012.pdf | title=PhD Thesis: Droplet collection in a scaled-up rotating separator }}{{Dead link|date=November 2019 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>  A major limitation of any fluid mechanics model for cyclone separators is the inability to predict the [[Particle aggregation|agglomeration]] of fine particles with larger particles, which has a great impact on cyclone collection efficiency.<ref>D. Benoni, C.L. Briens, T. Baron, E. Duchesne and T.M. Knowlton, 1994, "A procedure to determine particle agglomeration in a fluidized bed and its effect on entrainment", Powder Technology, 78, 33-42.</ref>