The forced vortex

In a forced vortex the angular velocity of the liquid is maintained constant by mechanical means, such as by an agitator rotating in the liquid or by rotation in the basket of a centrifuge. 

If the z-coordinate is Za at the point on the axis of rotation which coincides with the free surface of the liquid (or the extension of the free surface), then the coiresponding pressure Pq must be that of the atmosphere in contact with the liquid. 

For any constant pressure P, equation 2.78 is the equation of a parabola, and therefore all surfaces of constant pressure are paraboloids of revolution. The free surface of the liquid is everywhere at the pressure Pq of the surrounding atmosphere mid therefore is itself a paraboloid of revolution. Putting P = Pq in equation 2.78 for the free surface 

Thus the greater the speed of rotation co, the steeper is the slope. If rgco g, dzo/dr oo and the surface is nearly vertical, and if roco g, dzo/dro 0 and the surface is almost horizontal. 

The total energy per unit mass of fluid is given by equation 2.44  

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Thus, the energy per unit mass increases with radius r and is independent of depth In the absence of an agitator or mechanical means of rotation energy transfer will take place to equalise j/ between all elements of fluid. Thus the forced vortex tends to decay into a free vortex (where energy per unit mass is independent of radius). 

Rotational flow in the forced vortex within the cyclone body gives rise to a radial pressure gradient. This pressure gradient, combined with the frictional pressure losses at the gas inlet and outlet and losses due to changes in flow direction, make up the total pressure drop. This pressure drop, measured between the inlet and the gas outlet, is usually proportional to the square of gas flow rate through the cyclone. A resistance coefficient, the Euler number Eu, relates the cyclone pressure drop Ap to a characteristic velocity v ... 

For a forced vortex, the angular speed is constant and the liquid revolves as a solid body. Disregarding friction losses, Stepanoff (1993) claims that no power would be needed to maintain the vortex. The pressure distribution of this ideal solid body rotation is a parabolic function of the radius. When the forced vortex is superimposed on a radial outflow, the motion takes the form of a spiral. This is the type of flow encountered in a centrifugal pump. Particles at the periphery are said to carry the total amount of energy applied to the liquid. 

Cyclone Separators Finer feed sohds, from 0.04 to 0.0005 m (1.5 in to 28 mesh), may be treated in dynamic separators of the Dutch State Mines cyclone type (Fig. 19-36). In cyclone separators, the medium and the feed enter the separator together tangentially at the feed inlet (1) the short cyhndiical section (2) carries the central vortex finder (3), which prevents short circuiting within the cyclone. Separation is made in the cone-shaped part of the cyclone (4) by the action of centrifugal and centripetal forces. The heavier portion of the feed leaves the cyclone at the apex opening (5), and the hghter portion leaves at the overflow top orifice (6). 

Forced-vortex prewhirl. This type is shown as Vg lr = constant. This prewhirl distribution is also shown in Figure 6-14. Vg is at a maximum at the inducer inlet shroud radius, contributing to a decrease in the inlet relative Mach number. 

For Nr, > 1000, the properly baffled tank is turbulent throughout. Nq and P, are independent of Nr,. If the tank is not baffled, a forced vortex dominates the flow in the vessel. 

If it is assumed that there is no slip between the liquid and the basket, a> is constant and a forced vortex is created. 

Before discussing about the flame speed along a vortex core, it is first necessary to be familiar with the flames in various vortex flows. To date, four types of vortex flows have been used to study the flame behaviors. They are (1) a swirl flow in a tube [1,10], (2) vortex ring [2,3,12,13,16], (3) a forced vortex flow in a rotating tube [11], and (4) line vortex [22]. 

When a tube is rotated, a very simple, forced vortex flow can be obtained, where the rotational velocity is constant along the axis of rotation. However, the flame behavior becomes very complicated because the space is confined by the wall. 

For Gr, < 920, mass transfer could be represented by the forced-convection correlation and for Gr, > 920, by the free-convection correlation ofFenech and Tobias (F3). Tobias and Hickman (T2) also inferred the existence of cellular vortex flow near the electrode from deposition patterns, the induction length for this behavior agreeing with Eq. (44). 

Most studies of hydrocyclone performance for particle classification have been carried out at particle concentrations of about 1 per cent by volume. The simplest theory for the classification of particles is based on the concept that particles will tend to orbit at the radius at which the centrifugal force is exactly balanced by the fluid friction force on the particles. Thus, the orbits will be of increasing radius as the particle size increases. Unfortunately, there is scant information on how the radial velocity component varies with location. In general, a particle will be conveyed in the secondary vortex to the overflow, if its orbital radius is less than the radius of that vortex. Alternatively, if the orbital radius would have been greater than the diameter of the shell at a particular height, the particle will be deposited on the walls and will be drawn downwards to the bottom outlet. 

Thus, as might have been be expected, the primary vortex in the hydrocyclone is more akin to a free (n = 1) than to a forced (// = —1) vortex. 

As Re increases, skin friction becomes proportionately less and, at values greater than about 20, flow separation occurs with the formation of vortices in the wake of the sphere. At high Reynolds numbers, the size of the vortices progressively increases until, at values of between 100 and 200, instabilities in the flow give rise to vortex shedding. The effect of these changes in the nature of the flow on the force exerted on the particle is now considered. 

The soot formation and its control was studied in an annular diffusion flame using laser diagnostics and hot wire anemometry [17, 18]. Air and fuel were independently acoustically forced. The forcing altered the mean and turbulent flow field and introduced coherent vortices into the flow. This allowed complete control of fuel injection into the incipient vortex shedding process. The experiments showed that soot formation in the flame was controlled by changing the timing of fuel injection relative to air vortex roll-up. When fuel was injected into a fully developed vortex, islands of unmixed fuel inside the air-vortex core led to... 

As discussed earlier, the particle/droplet dynamics can be significantly modified by timing the fuel injection to be in- or out-of-phase with the large-scale vortex structures. To explore if timed fuel injection could alter the stability characteristics, the flow was forced at the quarter-wave mode of the inlet and droplet injection was timed to be in- or out-of-phase with the forcing. Results from these simulations show that the pressure fluctuations at the quarter-wave mode of the inlet can indeed be amplified or attenuated depending on the phasing of the droplet injection. 

With forcing, the air vortices are reinforced. In Fig. 20.3a, one can see the growth of an air vortex at four instances of time. At time 0, the vortex begins to form at the nozzle exit. At times 7t/2, tt, and 37t/2, the air vortex continues its roll-up until it is fully developed. 

Multiplexed diode-laser sensors were applied for measurement and control of gas temperature and species concentrations in a large-scale (50-kilowatt) forced-vortex combustor at NAWC to prove the viability of the techniques and the robustness of the equipment for realistic combustion and process-control applications [11]. The scheme employed was similar to that for measurements and control in the forced combustor and for fast extractive sampling of exhaust gases above a flat-flame burner at Stanford University (described previously). 

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