The free vortex

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

Hence the angular momentum of the liquid is everywhere constant. 

Substituting from equations 2.74 and 2.85 into equation 2.76 and integrating  

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

Chemical Engineering Thermodynamics (McGraw-Hill, New York, 1944). 

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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. 

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. 

In addition, for the free-vortex region, the tangential velocity at radius Rx, which we have called vecs elsewhere in this book is, according to Eq. (2.1.2). 

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). 

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). 

When an open tank with a free surface is stirred with an impeller, a vortex will form around the shaft. It is important to prevent this vortex from reaching the impeller, because entrainment of air in the liquid tends to cause foaming. The shape of the free surface depends upon (among other things) the fluid properties, the speed and size of the impeller, the size of the tank, and the depth of the impeller below the free surface. 

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. 

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. 

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,... 

At Re = 130, a weak long-period oscillation appears in the tip of the wake (T2). Its amplitude increases with Re, but the flow behind the attached wake remains laminar to Re above 200. The amplitude of oscillation at the tip reaches 10% of the sphere diameter at Re = 270 (GIO). At about this Re, large vortices, associated with pulsations of the fluid circulating in the wake, periodically form and move downstream (S6). Vortex shedding appears to result from flow instability, originating in the free surface layer and moving downstream to affect the position of the wake tip (Rll, R12, S6). 

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