Solid-body rotation flow

The rotating flow is the combination of a solid-body rotation flow and a vortex flow. As will be seen in Chapter 17, this case corresponds to many configurations for which centrifugal separation is implemented. Theoretically speaking, such flows verify the properly AS = 0. This properly greatly simplifies the BBOT equations. It also makes it easier to identify the terms and mechanisms responsible for centrifugal separation. 

Figure 17.3. Profile of the free surface for a solid-body rotation flow in a rotating tank...

The smaller the radius a, the more localized the pressure variations at the vicinity of the Oz axis. This is an observation that can be made by looking at a bathmb being emptied. When a rotation sets in, the flow is of vortex type. It is common to observe a deep depression in the free snrface entering the drain orifice, which is characteristic of a vortex flow and qnite different from the parabolic shape observed for solid-body rotation flow (Fignre 17.3). 

The Giesekus criterion for local flow character, defined as

extensional flow, 0 in simple shear flow and — 1 in solid body rotation [126]. The mapping of J> across the flow domain provides probably the best description of flow field homogeneity current calculations in that direction are being performed in the authors laboratory. 

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. 

Equation 26 is accurate only when the liquids rotate at the same angular velocity as the bowl. As the liquids move radially inward or outward these must be accelerated or decelerated as needed to maintain solid-body rotation. The radius of the interface, r, is also affected by the radial height of the liquid crest as it passes over the dischaige dams, and these crests must be considered at higher flow rates. 

The flow field in Eq. (Al-7) is really just a solid-body rotation which rotates, but does not deform, the fluid element. As a result, the rate-of-strain tensor D is the zero tensor, and the Finger strain tensor is the unit tensor. 

The characteristic solid body rotation is the primary flow pattern in un-bafiled tanks. To avoid these phenomena, baffles are installed in the tank. On wall baffles are sketched in Fig 7.1. Generally, baffles are placed in the tank to modify the flow and surface destroy vortices. Baffles mounted at the tank wall are most common, but also bottom baffles, floating surface baffles and disk baffles at the impeller shaft are possible. Often tank wall baffles are mounted a certain distance from the wall, as illustrated in Fig 7.3. This creates a different flow pattern in the tank. The purpose of installing baffles away from the wall is to avoid dead zones where liquid is seldom exchanged and where impurities accumulate. Experiments have confirmed that the flow patterns in baffled agitated tanks are different from the flow patterns in unbaffled agitated tanks. In baffled tanks the discharge flow dissipates partly in the bulk... 

For solid-body rotation at constant angular velocity, the vorticity vector, defined by I (V X v), is equivalent to the angular velocity vector of the solid. For two-dimensional flow in cylindrical coordinates, with Vr(r,0) and V0 r,9), the volume-averaged vorticity vector. 

At k = —1, the flow is spinning, obeying the law of circulation constancy (potential rotation) at k = 0, the spin angle is constant, i.e. does not depend on radius and at k = 1, the rotation obeys the laws of solid body rotation (a quasi-solid rotation). To ensure that rotation obeys (19.8), the blades of the swirler at the exit should have a certain dependence of the spin angle on the radius. The question... 

The effect of polymer solutions, ejected at very small flow rates at the tip of a finite span hydrofoil, on the inception and extent of tip vortex cavitation is investigated. The results show that the dessinent cavitation numbers are significantly reduced by polymer additives as compared with those in pure water. Moreover, for operating conditions well below critical, the presence of polymer in the vortex core results in the nearly complete elimination of the vapour cavity. These effects can be interpreted on the basis of a modification of the velocity field due to the viscoelastic properties of the polymer solutions. L.D.A. measurements indicate a large decrease of the tangential velocity component in the transition region between the inner solid body rotation co.e and the outer potential vortex flow. Also, a change of the slope of the velocity within the core is observed. 

This relationship was then used to calculate the tangential velocities from static pressure measurements in different places within hydrocyclones run with clean liquids. Driessen and many others following him thus deduced the general expression for tangential velocity profiles in the outer vortex given previously in equation 6.1, where n is an empirical exponent, usually from 0.6 to 0.9. Note that for a free vortex in inviscid flow n =, while in a forced vortex (solid body rotation) = 1. 

The bulk flow is given by the incompressible Navier-Stokes equations which follow. Note that eventhough we assume that the flow variables are independent of 9, we will allow for a non-zero azimuthal velocity u(r,. 2, t). This can arise from a solid body rotation, for example and can be used to model the effects of centrifugal forces on capillary instability. The momentum and continuity equations (in component) form are ... 

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. 

It is not straightforward to determine the transient solution that leads to the solid-body rotation of the fluid starting from rest The solution of the Navier-Stokes equations in the form ur=Q,ug r,z,t),u =0), for which the azimuthal component ug evolves in time nnder the effect of a process of vorticity diffusion from the lateral circular wall toward the tank axis, does not portray reality. A secondaiy flow Ur 0 and 0) actually occurs, which carries vorticity from the boundary layer on the horizontal bottom wall into the water layer. Boundary layers play a key part in rotating flows they allow steady-state flows to be established more rapidly. ... 

It must be pointed out that assuming solid-body rotation in the channels between the vanes implies that angular momentum is not conserved. Conservation of angular momentum would make the fluid and the particles increase their rotational speed as they get closer to the rotation axis. However, a secondary flow is produced by the boundary layers on the cones, which counteracts the increase in rotational speed in the spaces between the vanes. This mechanism explains how solid-body rotation can be maintained within the chaimels, when the spacing is small (e solid particles are determined by integrating the differential equation ... 

As in a centrrfirge decanter, the rotatiorral flow is a solid-body rotation ue r) = cor). The filtration process is axisyrrrmetric. At any time, the flow rate through any cylinder of radius r is identical. The radial filtration velocity is related to the permeate flow rate by 0 = InHrurir), and [17.60] becomes ... 

From Example 2.2. Ic we see that the velocity gradient tensor is not zero for a solid body rotation. From the statement of Newton s viscosity law, we would not expect stress to be generated as a result of the flow in solid body rotation because there is no relative separation of points. Like the deformation gradient F, the velocity gradient tensor L contains rate of rotation as well as stretching. We need a way to remove this rotation. 

From these examples we see that physically W gives the angular rotation in a material at any point. For a solid body rotation we have only W. D characterizes the rate of stretching at a point. We see that for uniaxial extension there is only stretching W s 0. From Example 2.2.2b we see that shear flow is a mixture of both stretching and rotation. 

Now imagine first that the swirling fluid has an infinite viscosity (behaves like a solid body). Hence, no shearing motion exists between fluid layers at different radii. In this case fluid elements at all radial positions are forced to have the same angular velocity. The angular velocity, f2, is measured in radians per unit of time, usually seconds, and therefore has units s . It equals ve/r, with v the tangential velocity, measured in m/s. Swirl with constant O is called forced vortex flow or solid-body rotation ... 

A real swirling flow normally has a core of near solid-body rotation surrounded by a region of near loss-free rotation as sketched in Fig. 2.1.3. This is called a Rankine vortex . 

To the right in Fig. 3.1.1 the radial profiles of the axial and tangential gas velocity components are sketched. The former shows the outer region of downwardly directed axial flow and the inner one of upwardly directed flow. As mentioned, the downward velocity at the wall is the primary mechanism for particle transport out the dust outlet. The axial velocity often shows a dip aroimd the center hne. Sometimes this is so severe that the flow there is downwardly directed. The tangential velocity profile resembles a Rankine vortex a near loss-free swirl surrounding a core of near solid-body rotation. 

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