Ideal vortex

Figure 4. Schematics of orientation field r for three different values of aster (95 = 0) generic vortex (0 < ( < 7t/2) and ideal vortex ip = 7t), see Eq. (3.15).

It is shown that the mixing process in an ideal vortex is described by the similarity variableThe departure from this result due to viscous stresses is described by a new dimensionless group /c E the ratio of diffusive to viscous spreading rates. The mixing is strongly reduced < 6 fox u x o the mixing is identical with that in an... 

There is little error in replacing i/,(r) in the wall region by v,w for m = 1 (ideal vortex) in general for cyclones, the vortex index m is between 0.5 and 1. We can therefore write... 

Fig. 2.1.3. Sketch showing the two ideal vortex flows, and the tangential velocity distribution in a real vortex...

A Ideal Vortex Laws from the Navier-Stokes Equations... 

This is one ideal vortex motion, where, as mentioned in the main text, the angular velocity f2 is constant. 

One very simple method of calculating the swirl velocity in the cyclone body, first introduced by Alexander (1949), is to assume that the vortex follows a modified form of the ideal vortex flows introduced in Chap. 2 ... 

Principles and Characteristics The principle of solid-fluid-vortex extraction, a recent development [152], is based on the creation of a relatively high filtration pressure as a result of cooling off a vapour chamber in a boiler vessel in such a way that there is (ideally) complete condensation and the extractive fluid is forced through a filter and/or extraction material at nearly one atmosphere in the case of open extractor systems and at more than one atmosphere in the case of closed extractor systems (cf. hydrostatic pressures up to 0.01 bar in Soxhlet). 

Evidence for the contribution of the CIO + BrO interaction is found in the detection and measurement of OCIO that is formed as a major product of this reaction, reaction (31a). This species has a very characteristic banded absorption structure in the UV and visible regions, which makes it an ideal candidate for measurement using differential optical absorption spectrometry (see Chapter 11). With this technique, enhanced levels of OCIO have been measured in both the Antarctic and the Arctic (e.g., Solomon et al., 1987, 1988 Wahner and Schiller, 1992 Sanders et al., 1993). From such measurements, it was estimated that about 20-30% of the total ozone loss observed at McMurdo during September 1987 and 1991 was due to the CIO + BrO cycle, with the remainder primarily due to the formation and photolysis of the CIO dimer (Sanders et al., 1993). The formation of OCIO from the CIO + BrO reaction has also been observed outside the polar vortex and attributed to enhanced contributions from bromine chemistry due to the heterogeneous activation of BrONOz on aerosol particles (e.g., Erie et al., 1998). 

The details of the transitions and the vortex behavior depend on the actual channel dimensions and wall-temperature distributions. In general, however, for an application like a horizontal-channel chemical-vapor-deposition reactor, the system is designed to avoid these complex flows. Thus the ideal boundary-layer analysis discussed here is applicable. Nevertheless, one must exercise caution to be sure that the underlying assumptions of one s model are valid. 

A stainless steel mixing vessel, fabricated from 316 grade stainless steel with some form of cover that allows access for ingredient addition, is an ideal unit for mixing dilutables. The vessel is normally fitted with a stirrer, the power and design of which take account of whether sugar is to be added as a crystalline solid (and thus dissolved) or added as a syrup. Either a top-mounted propeller stirrer or a side-entry unit will mix components adequately, especially if the inside surface of the vessel is fitted with fixed baffles. The use of a stirrer that creates a sufficient vortex to draw in air should be avoided. 

Aref (12, 13) was the first to investigate such systems. He analyzed an idealized system consisting of an incompressible, inviscid, two-point alternating vortex flow, as shown in Fig. 7.7. The flow field is assumed to be steady whenever a vortex is activated. The fluid particle moves along one streamline and when the vortex is switched, it embarks on another one. 

Weigh 0.92 mmol of Fmoc-amino acid (i.e., a fourfold excess of amino acid relative to the substitution level of the support) into clean and dry plastic tubes (Sarstedt, Germany) tubes with a 10-mL volume capacity are ideal. Add an equimolar amount of HOBT and HBTU relative to the amount of amino acid to 2 mL of DMF and a sixfold excess of DIPEA over the substitution level of the solid phase support. Dissolve fully by vortex and sonication. 

In an ideal fluid with a free surface and potential flow, a vortex ring nearing the surface will tend to dilate. This result can be rationalized by the mathematical construction of an image vortex ring reflected about the interface and approaching from the opposite direction. Fig. 5b depicts the experimental observation of a physical case similar to this type of behaviour. On the other hand, when a no-slip or intermediate-slip interface is... 

A vortex tube has certain advantages as a chemical reactor, especially if the reactions are endothermic, the reaction pathways are temperature dependent, and the products are temperature sensitive. With low temperature differences, the vortex reactor can transmit enormous heat fluxes to a process stream containing entrained solids. This reactor is ideally suited for the production of pyrolysis oils from biomass at low pressures and residence times to produce about 10 wt % char, 13% water, 7% gas, and 70% oxygenated primary oil vapors based on mass balances. This product distribution was verified by carbon, hydrogen, and oxygen elemental balances. The oil production appears to form by fragmenting all of the major constituents of the biomass. 

High outlet velocity of the heavy and light phases can also adversely affect separation if a vortex is formed and nonseparated dispersion band material is drawn out in the vortex. The exit flow of each separated phase should be drawn from the separator at velocities not more than 10 times the average velocity in the separator. The ideal outlet consists of overflow and underflow weirs extending across the separator. If standard nozzles are used at the top or bottom of a horizontal vessel, vortex breaker plates four times the nozzle diameter inside the vessel can prevent vortexing. 

Fig. 2.2 Schematic of the/i-vortex formation and definition scotch Initially the vorticity is organized spanwise at it is uniformly distributed in flow direction. By an instability process, the vorticity starts to wrap up into a vortex which by stretching takes the idealized form of a A composed mainly of two side-vortex-rods with an internal flow behaviour as described by eq.(2.2). This process ends when the vorticity cannot be concentrated anymore and viscous diffusion processes start to dominate the flow field in the event, which starts from thereon to decay. Since this model also has to account for the non-slip condition, the wall near flow can be described by a viscous tornado. Therefore the question arises whether by incorporating the model of the viscous tornado into the 1-vortex model it would be possible to describe the flow field completely.

In studies of vortex centrifugal devices primarily include spatial flow. Similarly it is conducted in models based on the h5rpothesis of a plane vortex. Motion of the gas is described by the Navier-Stokes equations. The equation is introduced with the closure of the fluctuating components of the h5q)othesis on the path of displacement. The values of the tangential and radial velocity are taken close to each other. Axial velocity is very small. Exclusion from consideration of the axial movement of the gas is greatly reduced, the (idealized). This is not consistent with the physical picture, in which a large place occupied by forward and backward axial currents. The results of these studies are of interest to determine the effects of the vortex flow asymmetry with respect to the axis of the camera. 

As illustrated in Figure 10.7, a cyclone consists of a vertical cylinder with a conical bottom, a tangential inlet near the top, and outlets at the top and the bottom, respectively. The top outlet pipe protrudes into the conical part of the cyclone in order to produce a vortex when a dust-laden gas (normally air) is pumped tangentially into the cyclone body. Such a vortex develops centrifugal force and, because the particles are much denser than the gas, they are projected outward to the wall flowing downward in a thin layer along this in a helical path. They are eventually collected at the bottom of the cyclone and separated. The inlet gas stream flows downward in an annular vortex, reverses itself as it finds a reduction in the rotation space due to the conical shape, creates an upward inner vortex in the center of the cyclone, and then exits through the top of the cyclone. In an ideal operation in the upward flow... 

In an ideal PFR, fluid elements do not mix in the axial direction (i.e. flow direction). However, in an actual tubular reactor, some amount of axial mixing of fluid elements may occur due to a number of reasons (such as vortex formation at tube inlet). A mathematical model called axial dispersion model was proposed by P. V. Danckwarts to account for axial mixing of fluid elemenfs in the tubular (plug flow) reactor. 

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