Free vortex

The rotor blades shall be of the tapered/twisted, free vortex, reaetion design (Figure 5-21). The blades shall be speeifieally designed for the dirty FCC flue gas applieation. Seller shall provide a Goodman Diagram for the highest stressed loeation... 

Free-vortex prewhirl. This type is represented by r Ve = constant with respect to the inducer inlet radius. This prewhirl distribution is shown in Figure 6-13. Vg is at a minimum at the inducer inlet shroud radius. Therefore, it is not effective in decreasing the relative Mach number in this manner. 

The Spin Velocity Under the inviscid flow assumption, where all fluid that enters the cyclone does so with approximately the same amount of momentum, a free vortex may be predicted for the spin velocity distribution as... 

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). 

In a free vortex the energy per unit mass of fluid is constant, and thus a free vortex is inherently stable. The variation of pressure with radius is obtained by differentiating equation 2.44 with respect to radius at constant depth z to give ... 

In all of these cases the free vortex may be modified by the frictional effect exerted by the external walls. 

For a uniform angular velocity ( > = constant, i.e., a solid body rotation ), n = — 1, whereas for a uniform tangential velocity ( plug flow ) n = 0, and for inviscid free vortex flow co = c/r2, i.e., n = 1. Empirically, the exponent n has been found to be typically between 0.5 and 0.9. The maximum value of Ve occurs in the vicinity of the outlet or exit duct (vortex finder) at r = De/2. 

Because the rotating motion of the gas in the cyclone separator arises from its tangential entry and no additional energy is imparted within the separator body, a free vortex is established. The energy per unit mass of gas is then independent of its radius of rotation and the velocity distribution in the gas may be calculated approximately by methods discussed in Volume 1, Chapter 2. 

For a free vortex, it is shown in Volume 1, Chapter 2, that the product of the tangential velocity and the radius of rotation is a constant. Because of fluid friction effects, this relation does not hold exactly in a cyclone separator where it is found experimentally that the tangential velocity is more nearly inversely proportional to the square root of radius,... 

The practical consequence of Eq. (2) for rotational fluid flow can be quite dramatic, as demonstrated by the high wind speeds that may be generated near the center of free vortex flows—e.g., tornadoes and typhoons. 

Equation (10.41) shows that H is the asymptote approached by (p/w) + z as r approaches infinity and V approaches zero. On the other hand, as r approaches zero, V approaches infinity, and (p/w) + z approaches minus infinity. Since this is physically impossible, the free vortex cannot extend to the axis of rotation. In reality, as high velocities are attained as the axis is approached, the friction losses, which vary as the square of the velocity, become of increasing importance and are no longer negligible. Hence the assumption that H is constant no longer holds. The core of the vortex tends to rotate as a solid body as in the central part of a pump impeller. 

The flow around the bend of a channel provides an application of the fundamentals of flow in a curved path. As there is no torque applied to the fluid, the flow should follow the laws of the free vortex—and indeed it would, were it not for the effect of friction on the walls and bottom of the channel. 

The existence of spiral flow is observed near the short radius of the bend. The water surface is superelevated at the outside wall for the cylindrical free vortex. The element EF is subjected to a centrifugal force mV2/r, which is balanced by an increased hydrostatic force on the left side due to the superelevation of the water surface at C above that at D. The element GH has exactly the same hydrostatic force inward, but the centrifugal force outward is much less because the velocity is decreased by friction near the bottom. This results in a cross flow inward along the bottom of the channel, which is balanced by an outward flow near the water surface, hence the spiral. This spiral flow is largely responsible for the commonly observed erosion of the outside bank of a river bend, with consequent deposition and building of a sand bar near the inside bank. 

It is generally not possible to perceive a convex surface profile, given by the free vortex theory. For most practical purposes the water surface may be supposed to be a straight line from A to B, raised at the outside wall and depressed at the inside, with the slope given by the ordinary superelevation formula used for highway curves, tan 0 = Vz/gr, where r is the radius of the curve to the center of the channel. 

The classification of product leaving the mill depends on a balance between centrifugal forces and drag forces in the flow field around the mill outlet. Mill types A and C create a free vortex at the outlet, while jet mill D makes use of gravity. Type B has an integrated rotor. The final product quality is largely determined by the success of classification. 

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