Nonuniform flows

Difficulties of Moving-Bed Operation. The use of a moving bed iatroduces the problem of mechanical erosion of the adsorbent. Obtaining uniform flow of both soHd and Hquid ia beds of large diameter is also difficult. The performance of this type of operation can be gready impaired by nonuniform flow of either phase. 

Detention efficiency. Conversion from the ideal basin sized by detention-time procedures to an actual clarifier requires the inclusion of an efficiency factor to account for the effects of turbulence and nonuniform flow. Efficiencies vaiy greatly, being dependent not only on the relative dimensions of the clarifier and the means of feeding but also on the characteristics of the particles. The cui ve shown in Fig. 18-83 can be used to scale up laboratoiy data in sizing circular clarifiers. The static detention time determined from a test to produce a specific effluent sohds concentration is divided by the efficiency (expressed as a fraction) to determine the nominal detention time, which represents the volume of the clarifier above the settled pulp interface divided by the overflow rate. Different diameter-depth combinations are considered by using the corresponding efficiency factor. In most cases, area may be determined by factors other than the bulksettling rate, such as practical tank-depth limitations. 

High airflow rates may result in unintended fracturing leading to nonuniform flow or short-circuiting of injected air in the subsurface, or may result in unintended mobilization of contaminants as nonaqueous phase liquids (NAPL), dissolved in groundwater, or in soil gas. 

The hydraulic performance of sewer pipes can be described at different levels. In the case of nonstationary, nonuniform flow, the Saint Venant Equations should be applied. However, under dry-weather conditions, the Manning Equation is an adequate description of the wastewater flow in a gravity sewer pipe when considering the prediction of wastewater quality changes under transport. There are no grounds for using advanced hydraulic models because of the uncertainties in the prediction of the microbial transformations of the wastewater. 

In FFF, retention and separation are achieved by a direct concert of the nonuniform flow profile in a liquid carrier with a force field applied at the right angle to the flow in the channel. Because... 

In FFF, separation is determined by the combined action of the nonuniform flow profile and transverse field effects. The classical configuration assumes the FFF channel as two infinite parallel plates (see Figure 12.4), of which the accumulation wall lies at x=0, where x is the cross-channel axis (directed upward from the accumulation wall). Inside the channel, the carrier fluid, assumed to have a constant viscosity, has a velocity profile u(x) that takes the form... 

Bayer et al. (36) studied the efficiency of preparative columns and showed that, when using the van Deemter equation (Section 2.2.2), an additional term had to be introduced to account for the nonuniform flow distribution. This can be represented by... 

Shafer, J. M. (1987). Reverse pathline calculation of time-related capture zones in nonuniform flow. Ground Water, 25(3), 283-289. 

On each of these, random and structured reactors behave quite differently. In terms of costs and catalyst loading, random packed-bed reactors usually are most favorable. So why would one use structured reactors As will become clear, in many of the concerns listed, structured reactors are to be preferred. Precision in catalytic processes is the basis for process improvement. It does not make sense to develop the best possible catalyst and to use it in an unsatisfactory reactor. Both the catalyst and the reactor should be close to perfect. Random packed beds do not fulfill this requirement. They are not homogeneous, because maldistributions always occur at the reactor wall these are unavoidable, originating form the looser packing there. These maldistributions lead to nonuniform flow and concentration profiles, and even hot spots can arise (1). A similar analysis holds for slurry reactors. For instance, in a mechanically stirred tank reactor the mixing intensity is highly non-uniform and conditions exist where only a relatively small annulus around the tip of the stirrer is an effective reaction space. 

In the case of artificial channels that are free from the irregularities found in natural streams, it is possible to apply analytical methods to the various problems of nonuniform flow. In many instances, however, the formulas developed are merely approximations, and we must often resort to trial solutions and even purely empirical methods. For the treatment of many types of flow, see Bakhmeteff [35],... 

In the case of pressure conduits, we have dealt with uniform and nonuniform flow without drawing much distinction between them. This can be done because in a closed pipe the area of the water section, and hence the mean velocity, is fixed at every point. But in an open channel these conditions are not fixed, and the stream adjusts itself to the size of cross section that the slope of the hydraulic gradient requires. 

The Manning equation for uniform flow, Eq. (10.125), can be applied to nonuniform flow with an accuracy that is dependent on the length of the reach taken. Thus a long stream will be divided into several reaches, such that the change in depth is roughly the same within each reach. Then, within a reach, the Manning formula gives... 

A channel laid on a certain slope will carry a certain rate of discharge at critical velocity in uniform flow. The concepts of critical velocity and depth and alternate stages of flow, however, are of greater significance in cases of nonuniform flow because of the relation between the critical velocity and the celerity or velocity of propagation of the gravity wave. 

The qualitative analysis of backwater scenarios has been restricted to rectangular sections of great width. These equations are, however, applicable to any channel of uniform cross section, if y0 is the depth for uniform flow andyc is the depth that satisfies Eq. (10.118). The surface profiles can even be used qualitatively in the analysis of natural stream surfaces as well, provided that local variations in slope, shape, roughness of cross section, etc., are taken into account. The step-by-step integration method for the solution of nonuniform flow problems is not restricted to uniform channels and is therefore suited to backwater computations for any stream whatever. 

By far the most important of the local nonuniform flow phenomena is that which occurs when supercritical flow has its velocity reduced to subcritical. We have seen in these example scenarios that there is no ordinary means of changing from lower- to upper-stage flow with a smooth transition, because the theory calls for a vertical slope of the water surface. The result, then, is a marked discontinuity in the surface, characterized by a steep upward slope of the profile, broken throughout with violent turbulence, and known universally as the hydraulic jump. 

The die lip, or final die opening area, which is designed to give the proper cross-sectional shape to the product and to allow the melt to forget the generally nonuniform flow experience in regions 1 and 2. 

As noted earlier, our principal interest will be in an open, flowing CVD system. In order to correctly interpret the phenomena occurring in such systems, it will be necessary to study chemically reacting gas flows with nonuniform flow and temperature fields. And, of course, we will have to understand the surface reactions that lead to the solid film formation. Within the reacting gas, we will consider homogeneous gas phase reactions. At the surface, we have... 

The most commonly used devices for sample retrieval in blending homogeneity testing are thief probes. Several different sampling thieves have been developed and used for many years in the pharmaceutical industry.72 A major problem with most thieves is that the retrieved sample is not representative of the true concentration at the location from which the sample is supposed to be obtained. Contamination with powders from other locations in the mixture during probe insertion causes these sampling errors. Also, nonuniform flow of different components into the sampling cavity can skew the sample concentration. 

Figure S. Isotherms of moving reaction zones calculated with nonuniform flow.

---