Tangential velocity distribution

It follows, therefore, that tangential velocity distributions at low solids concentrations can be estimated from simple measurements of radial static pressure. This was the idea behind the early studies of tangential velocity distributions in clean liquid flow. Driessen was the first to derive an expression relating the tangential velocity, Vt, to the radial pressure distribution by assuming the radial velocity component negligible in relation to the tangential component, so that... 

We derive the equations for the tangential velocity distribution in two types of ideal swirling flows ... 

The tangential velocity distribution in real swirling flows is intermediate between these two extremes. 

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

Figure 1.39. Typical velocity distributions in a hydrocyclone. (a) axial (h) radial (c) tangential (broken line LZVV is the locus of zero axial velocity)...

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. 

Carlos and Latif both fluidised glass particles in dimethyl phthalate. Data on the movement of the tracer particle, in the form of spatial co-ordinates as a function of time, were used as direct input to a computer programmed to calculate vertical, radial, tangential and radial velocities of the particle as a function of location. When plotted as a histogram, the total velocity distribution was found to be of the same form as that predicted by the kinetic theory for the molecules in a gas. A typical result is shown in Figure 6.11(41 Effective diffusion or mixing coefficients for the particles were then calculated from the product of the mean velocity and mean free path of the particles, using the simple kinetic theory. 

Figure 7.5. Tangential velocity and pressure distributions based on Rankine s combined vortex model.

Figure 7.7. Calculated and experimental results for radial variations of the axial velocity, tangential velocity, and pressure in a cyclone (from Zhou and Soo, 1990) (a) Axial velocity, m/s (b) Tangential velocity, m/s (c) Pressure distributions, Pa.

The SDF, like the RTD functions, can be calculated from the velocity distribution in the system that is, a certain flow pattern determines both functions. The reverse, however, does not necessarily apply. The calculation of the SDF requires a complete description of the flow pattern, whereas RTD functions often can be calculated from a less than complete flow pattern. For example, the RTD of axial annular flow between two rotating concentric cylinders (helical flow) of a Newtonian fluid depends only on the axial velocity, whereas the SDF depends on both the axial and the tangential velocity... 

The conditions necessary for equality of particle size distribution were determined under ambient conditions. The nozzles used in the investigation can be classified as swirl-spray pressure nozzles. They accommodate a swirl insert which imparts a tangential velocity to the exiting fluid and results in a conical spray pattern. These nozzles were sufficiently different from conventional swirl nozzles (see Putnam et al., Ref. 6) to require an experimental study of particle size distribution. 

The analysis of this section is typical of all lubrication problems. First, the equations of motion are solved to obtain a profile for the tangential velocity component, which is always locally similar in form to the profile for unidirectional flow between parallel plane boundaries, but with the streamwise pressure gradient unknown. The continuity equation is then integrated to obtain the normal velocity component, but this requires only one of the two boundary conditions for the normal velocity. The second condition then yields a DE (known as the Reynolds equation) that can be used to determine the pressure distribution. 

In other words, the pressure distribution in the boundary-layer is completely determined at this level of approximation by the limiting form of the pressure distribution impressed at its outer edge by the potential flow. It is convenient to express this distribution in terms of the potential-flow velocity distribution. In particular, let us define the tangential velocity function ue(x) as... 

There is a direct and an indirect effect of bubble surface retardation on the tangential particle velocity. The direct influence is caused by the dependence of the bubble hydrodynamic fields on the velocity distribution along its surface. The indirect influence is caused by the effect of the inertia path of a reflected particle on its tangential velocity and by the dependence of the path on the bubble surface retardation. The directions of the two effects are opposite. At the transition from a free to a retarded surface, the liquid tangential velocity diminishes at any point and the inertia path grows, which results in an increase in the tangential particle velocity. 

Integration of Eq. (10.28) along the cross-section of the hydrodynamic layer allows us to check whether within its limits the radial velocity component is proportional to the tangential derivative of the velocity distribution along the bubble surface, which differs slightly from the potential distribution. The effect of a boundary layer on the normal velocity component and on inertia-free deposition of particles should be therefore very small. The formula for the collision efficiency given by Mileva as an inertia-free approximation is thus VRc times less than the collision efficiency according to Sutherland, which is definitely erroneous. 

Mathur and Maccallum [18] have reported results of their experimental work done on hubless and annular vane swirlers for pressure drop across swirler, axial static pressure, and axial and tangential velocity components distribution. It was concluded that the swirling jets experienced a sudden expansion downstream of the swirl generators. An internal recirculation zone (IRZ)... 

The solution of Eq. (9.26) with conditions (9.27) can be obtained by the same method, as the solution of the problem on the Stokes flow over a sphere [51]. As a result, we will find the pressure and velocity distributions at the surface of the sphere, and the tangential stress at the sphere. Therefore the hydrodynamic force acting on the sphere, is equal to... 

Of the main interest is the velocity distribution near the sphere in the diffusion boundary layer. Introduce local system of coordinates y,9), where the y-axis is perpendicular and 6 is tangential to the corresponding area element of the surface. Then r/a = 1 + y/a. Considering the case y/a 1, expand (10.69) as a power series in y/a. In a result, we obtain... 

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