Special Velocity Profiles

This section considers three special cases. The first is a flat velocity profile that can result from an extreme form of fluid rheology. The second is a linear profile that results from relative motion between adjacent solid surfaces. The third special case is for motionless mixers where the velocity profile is very complex, but its net effects can sometimes be approximated for reaction engineering purposes. 

Unfortunately, the thinness of most liquid films makes it difficult to measure the velocity profiles experimentally, since it is practically impossible to introduce any of the usual fluid-velocity probes into a film which may be less than 1 mm. thick without grossly distorting the flow patterns. Nevertheless, film velocity profile measurements have been reported for a few special cases. 

CFD has also been applied to analyze the flow patterns in a special counter-current solvent extraction column (Angelov et al, 1990). They used a singlephase flow representation and a k- turbulence model to compute the flow patterns in a periodic structure of the column. Validation of the computational results was achieved by applying LDA to obtain experimental data on the velocity profiles. CFD is a very useful tool here because the optimization of the performance of the extraction column from a geometrical point of view can be achieved with relative ease in comparison with a pure empirical strategy. 

In order to handle the rapid variations near a w all, one must either use a fine computational mesh in this region or else employ a special treatment. The variation in shear stress is, to a first approximation, small across this region, and the law of the wall is known to be followed by the mean-velocity profile very near the wall for most turbulent boundary layers. One simple approach is therefore to patch the numerical solution at the first computation point away from the wall to the empirical wall law,... 

In addition to the laboratory-scale reactors described here, there are numerous more specialized reactors in use. However, as mentioned previously, the performance of these reactors must lie somewhere between the mixing limits of the PFR and the CSTR. Additionally, when using small laboratory reactors, it is often difficult to maintain ideal mixing conditions, and the state of mixing should always be verified (see Chapter 8 for more details) prior to use. A common problem is that flow rates sufficiently large to achieve PFR behavior cannot be obtained in a small laboratory system, and the flow is laminar rather than turbulent (necessary for PFR behavior). If such is the case, the velocity profile across the reactor diameter is parabolic rather than constant. 

It is important to keep in mind that one has assumed that the velocity profile is linear when this expression for a, (3.16), and Eq. (3.15) are derived. This is true when the strain rate and the Reynolds number is low. At higher strain rates this assumption is no longer true and special kinds of thermostats have to be applied [25]. 

The false-transient method can be applied to convective diffusion equations in a manner similar to that used for velocity profiles. Finite-difference approximations are written for the spatial derivatives. Second-order approximations can be used for first derivatives since they involve only the same five points needed for the second derivatives. The result is a set of simultaneous ODEs with (false) time as the independent variable. The computational template of Figure 16.3 is unchanged. The next two examples illustrate its application to problems where axial diffusion is negligible. Such problems are also readily solved by the method of lines as described in Chapter 8. Cases with significant axial diffusion are troublesome for the method of lines and require special boundary conditions for the method of false transients. They are treated in Section 16.2.4. 

Wall jets are bounded by a solid surface, the wall, on one side while the outer region of the flow is in contact with the ambient fluid. Wall jets find many applications such as cooling of surfaces, boundary layer control, building ventilation, energy dissipation etc. Jets emanating from sluices and other hydraulic structures close to the bed of channels need special attention to provide the necessary protective works. Prediction of the velocity profiles in the case of a wall jet is more difficult compared to the free jet. After a short distance downstream of the outlet, all jets tend to behave in a similar fashion. The schematic diagrams of a plane wall-jet and the boundary conditions used in the simulation are shown in Figure 1. 

In order to determine the velocity profile and the flow rate, the value of X has to be known. This involves solving Eq. 7.252 or Eq. 7.254. This is normally done by using a numerical technique, e.g., Newton-Raphson. Exact analytical solutions are only possible for the special case when s is a positive integer. However, by rewriting the equations and by performing a series expansion, a closed-form solution can be obtained. If a new variable x is introduced in Eq. 7.252 with x = X-0.5, then the equation can be rewritten as ... 

Booy (1963) has considered the effects of curvature and helicity of the screw and derived a model known as the curved channel model. This model was further generalized by Yu and Hu (1998) to calculate the velocity profile and flow rates. The results were compared with the parallel plate models and found to be very similar in drag and pressure flows. However, these models were more accurate in the case of deeper screw channels such as twin-screw extruders or some special single-screw extruders. The details of modeling of polymer flows in various stages of the screw are given by Tadmor and Gogos (1979). 

Analytical solutions of (49) have been presented for special configurations where the velocity profile is well established, e.g., the rotating disk electrode or laminar flow in a chaimel. An important simplification has been proposed by Levich, who recognized that electrochemical systems are typically characterized by a concentration boundary layer that is much thinner than the corresponding velocity boundary layer. Levich recognized that in such systems it is... 

Special emphasis is put on the fit to the He I line at 4471 A, which appears as an unshifted absorption profile of quasi "photospheric" origin. A striking result is the very high rotational velocity which is indicated by its profile (Fig. 3). 

---