Velocity radial

The previous section gave a methodology for calculating V ir) given fi (r) and p(r). It will also be true that both /x and p will be functions of z. This will cause no dilficulty provided the changes in the axial direction are slow. 

FIGURE 8.8 Elongated velocity profile resulting from a factor of 50 increase in viscosity across the tube radius. 

The fomulation of Equation (8.68) gives the fully developed velocity profile, Fz(r), which corresponds to the local values of ix(r) and p(r) without regard to upstream or downstream conditions. Changes in Fz(r) must be gradual enough that the adjustment from one axial velocity profile to another requires only small velocities in the radial direction. We have assumed Vy to be small enough that it does not affect the equation of motion for V. This does not mean that Vr is zero. Instead, it can be calculated from the fluid continuity equation, 

Radial motion of fluid can have a significant, cumulative effect on the convective diffusion equations even when Vr has a negligible effect on the equation of motion for V. Thus, Equation (8.68) can give an accurate approximation for even though Equations (8.12) and (8.52) need to be modified to account for radial convection. The extended versions of these equations are 

The boundary conditions are unchanged. The method of lines solution continues to use a second-order approximation for dajdr and merely adds a Vr term to the coefficients for the points at r Ar. 

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In general, V For laminar Newtonian flow the radial velocity profile is paraboHc and /5 = 3/4. For fully developed turbulent flow the radial... 

Fig. 11. Computer-simulated recirculating patterns in a mixing tank with full baffles (a) elevation view shows circulation patterns generated by turbine blades (b) plane view shows the effect of the baffle on the radial velocity vectors above the turbine blades.

A particle entering the cyclone finds a poiat where its velocity with respect to the fluid is equal to the radial velocity, and hence is at rest with respect to radial movement to or from the wall. Because the radial velocity changes with radius, the particle spirals down the cone section, foUowiag a path defined... 

FIG. 1 7-37 Variation of tangential velocity and radial velocity at different points in a cyclone. [Ter Linden, Inst. Mech. Eng. J., 160, 235 (1949).]... 

Radial velocity below feedwell <90% of downward velocity. 

Baffles are responsible for restricting the tangential velocity component, u, and augment the vertical component, while simultaneously increasing the radial velocity, U,. The net result is that the liquid discharges from the impeller in a wider flow radius. 

Separation Rates in Tubular-and Solid-Bowl Centrifuges To evaluate the radial velocity of a particle moving toward a centrifuge wall, the expression for particle settling in a gravitational field is applied ... 

For the same case of n = 1200 rpm and r = 0.5, we obtain u,/Ug = 800, whereas for the turbulent regime the ratio was only 28. This example demonstrates that the centrifugal process is more effective in the separation of small particles than of large ones. Note that after the radial velocity u, is determined, it is necessary to check whether the laminar condition. Re < 2, is fulfilled. For the transition regime, 2 < Re < 500, the sedimentation velocity in the gravity field is ... 

The expression for particle radial velocity toward the wall is ... 

If the particle s density is lower than that of the liquid, the path of the liquid will be centripetal, as illustrated in Figure 14. Settling will occur when u (centripetal) is higher than the radial velocity Uf (centrifugal). 

Figure 10-13. The radial velocity at the inner screen of the reactor does not vary much with screen resistance.

Figure 10-15. The lines show radial velocity at the inner reactor screen for different modification proposals.

The point sink can approximate airflow near a hood with round or square/rectangular shape. The point sink will draw air equally from all directions (Fig. 7.83). The radial velocity (mys) at a distance r (m) from the sink can be calculated as a volume rate of exhaust airflow q (mVs) divided by the surface area of an imaginary sphere of radius r ... 

Restriction of the airflow by surfaces decreases the area through which the air flows toward the sink, which results in increased radial velocity. For cases with a restricted airflow created by the sink, Eq, (7.210) can be modified to... 

Cy is the absolute radial velocity component at the entrance is the absolute radial velocity component at the exit... 

In this section, the rotational velocity is directly proportional to the rotational velocity n according to the equation u — irDn. The impeller blade angles remain the same regardless of the rotational velocity of the impeller. Hence, the inlet and exit velocity triangles have the same form. The axial velocity of an axial fan changes directly proportionally to the circumference velocity u. This is also valid for the radial velocity at the outer circumference of a radial impeller fan. These velocities are directly proportional to the fan flow volume hence. 

Point Sink A point sink is defined as a pcjint in space at which the fluid is continuously and uniformly drawn off. The radial velocity into the sink at a distance r from the sink is, in spherical coordinates,... 

If the fluid enters the pipe from a duct of larger cross-section, the existence of a radial velocity component gives rise to the formation of a vena contractu near the entry to the pipe but this has been neglected here. 

Before the slit. Motion of the image delivered by the telescope with respect to the slit causes both a loss of throughput and an error in the barycentre of the spectral lines recorded on the detector, unless the object uniformly fills the slit (which implies low throughput). This can cause errors in measurement of radial velocities. For MOS, there is the particular problem of variations in the image scale or rotations of the mask. These can cause errors which depend on position in the field resulting in spurious radial trends in the data. Fibre systems are almost immune to this problem because the fibres scramble posifional information. 

Figure 17. Steps in the constmction of a datacube of the nucleus of NGC1068 from observations with the integral held unit of the Gemini Multiobject Spectrograph installed on the Gemini-north telescope. The datacube is illustrated by a few spectra distributed over the field and equivalently by a few slices at a given radial velocity in the light of the [OIII]5007 emission line. Only a few percent of the total data content is shown.

The radial velocity component Vr is negligible compared with the axial component Fz. 

This mass balance equation shows that material that is initially at radial position rin will move to radial position r for some downstream location, >0. A worked example of radial velocities and curved streamlines is given in Chapter 13, Example 13.10. 

McLaughlin, H. S., Mallikarjun, R., and Nauman, E. B., The Effect of Radial Velocities on Laminar Flow, Tubular Reactor Models, AIChE J., 32, 419-425 (1986). 

Include the radial velocity term in the convective diffusion equation and plot streamlines in the reactor. 

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