Layered flow

Equations (5,61) and (5.62) can be used to derive a pressure potential equation applicable to thin-layer flow between curved surfaces using the following procedure. In a thin-layer flow, the following velocity boundary conditions are prescribed ... 

The comparison of flow conductivity coefficients obtained from Equation (5.76) with their counterparts, found assuming flat boundary surfaces in a thin-layer flow, provides a quantitative estimate for the error involved in ignoring the cui"vature of the layer. For highly viscous flows, the derived pressure potential equation should be solved in conjunction with an energy equation, obtained using an asymptotic expansion similar to the outlined procedure. This derivation is routine and to avoid repetition is not given here. 

A thorough description of the internal flow stmcture inside a swid atomizer requires information on velocity and pressure distributions. Unfortunately, this information is still not completely available as of this writing (1996). Useful iasights on the boundary layer flow through the swid chamber are available (9—11). Because of the existence of an air core, the flow stmcture iaside a swid atomizer is difficult to analyze because it iavolves the solution of a free-surface problem. If the location and surface pressure of the Hquid boundary are known, however, the equations of motion of the Hquid phase can be appHed to reveal the detailed distributions of the pressure and velocity. 

Slide Coating. A slide coatei, iRustiated in Figure 15, can coat an unlimited number of layers simultaneously. Each layer flows out onto the... 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

Sometimes the domain is semi-infinite, as in boundaiy layer flow. The domain can be transformed from the x domain (O-oo) to the T domain (1-0) using the transformation T = exp ( ). Another approach is to use a variable mesh, perhaps with the same transformation. For example, use T = exp (— x) and a constant mesh size in T the value of is found experimentally. Still another approach is to solve on a finite mesh in which the last point is far enough away that its location does not influence the solution (Ref. 59). A location that is far enough away must be found by trial and error. 

Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the sohd surface is called the boundary layer The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundaiy layer thickness is conventionally t en to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

The author has a patent for a dynamic surge-detection system, using a boundary-layer probe, presently undergoing field tests. This system consists of specially mounted probes in the compressor to detect boundary-layer flow... 

Intermittent (I) - In this flow pattern the liquid inventory in the pipe is non-uniformly distributed axially. Plugs or slugs of liquid that fill the pipe are separated by gas zones that contain a stratified liquid layer flowing along the bottom of the... 

In a turbulent boundary layer, flow takes place in the direction perpendicular to the surface over which the flow occurs. 

Considering the case of Eq. (4.244), it is normal to describe a real mass transfer case by taking into consideration the boundary layer flows and the turbulence by using a mass transfer factor which is defined by... 

Equations (4.279) and (4.280) are similar. Figure 4.37 shows a two-dimensional boundary layer flow over a plane. Ignoring any chemical reactions and considering steady-state conditions, Eqs. (4.279) and (4.280) give... 

The dimensionless form of the continuity equation (4.278) ( , = 0) in Uv o-dimensional boundary layer flow is... 

The preceding discussion has attempted to formulate the situation for laminar boundary layer flow as accurately as possible and to obtain precise correlation between the heat transfer and mass transfer factors. 

It the main reason for the stratification strategy is contaminant control in [he occupied zone, the same strategy should be applied in winter conditions, too. Thus, the selected hearing method has to fulfill two requirements to siip-pttrt the creation of the vertical stratification and not to create disturbing airflows into the hall. In this case one option would be the floor heating method see Fig. 8.16c. Additionally, one should consider the prevention of boundary layer flows along the outer walls using, for example, passive methods. -... 

Chien, K. Y. Predictions of channel and boundary layer flows with a low-Reynolds-nuraber turbulence model. AIAA J., vol, 20, pp. 33-18, 1982. 

Draw clearly schematic diagrams of a thin-layer flow detector utilizing (1) single working electrode, and (2) dual electrode. Explain how the latter improves the power and information content. 

For the common problem of heat transfer between a fluid and a tube wall, the boundary layers are limited in thickness to the radius of the pipe and, furthermore, the effective area for heat flow decreases with distance from the surface. The problem can conveniently be divided into two parts. Firstly, heat transfer in the entry length in which the boundary layers are developing, and, secondly, heat transfer under conditions of fully developed flow. Boundary layer flow is discussed in Chapter 11. 

The internal layer flow distribution, or micromanifold, is substantially more challenging an example is shown in Figure 11.2b. A single flow enters a layer that then feeds tens to thousands of parallel channels. Design variables in the micromanifold region before the coimecting process chaimels are an option to control flow distribution [19-23]. 

The main part of the report describes the results of systematic investigations into the hydrodynamic stress on particles in stirred tanks, reactors with dominating boundary-layer flow, shake flasks, viscosimeters, bubble columns and gas-operated loop reactors. These results for model and biological particle systems permit fundamental conclusions on particle stress and the dimensions and selection of suitable bioreactors according to the criterion of particle stress. 

Reactors with Dominating Boundary-Layer Flow. 59... 

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

If boundary-layer flow plays an important role in reactors, as is the case e.g. in unbaffled stirred tanks or in agitation with a smooth disc (reactors see Table 5),... 

As in boundary-layer flows, smaller reference floe diameters are found with gas sparging than with the same average power input in a baffled stirred tank 127] or [44,45]. This can be explained if it is assumed that the floes come into close contact with the gas phase and find their way into the zones of higher stress. 

This result makes it clear that particle stress is strongly dependent on the interaction between the particles and the interface, so that electrostatic and also hydrophobic and hydrophilic interactions with the phase boundary are particularly important. This means that the stress caused by gas sparging and also by boundary-layer flows, as opposed to reactors with free turbulent flow (reactors with impellers and baffles), may depend on the particle system and therefore applicability to other material systems is limited. 

The stress caused by gas sparging and also by boundary-layer flows, as opposed to reactors with free turbulent flow (reactors with impellers and bafQes), may depend on the particle system. 

The presented results for systematic studies on hydrodynamic stress in shake flasks, baffled stirred tanks, reactors in which boundary layer flow predominates (e.g. stirred tank with a smooth disc or unbaffled stirred tank), viscosi-... 

For reactors with free turbulent flow without dominant boundary layer flows or gas/hquid interfaces (due to rising gas bubbles) such as stirred reactors with bafQes, all used model particle systems and also many biological systems produce similar results, and it may therefore be assumed that these results are also applicable to other particle systems. For stirred tanks in particular, the stress produced by impellers of various types can be predicted with the aid of a geometrical function (Eq. (20)) derived from the results of the measurements. Impellers with a large blade area in relation to the tank dimensions produce less shear, because of their uniform power input, in contrast to small and especially axial-flow impellers, such as propellers, and all kinds of inclined-blade impellers. 

Cherry and Papoutsakis [33] refer to the exposure to the collision between microcarriers and influence of turbulent eddies. Three different flow regions were defined bulk turbulent flow, bulk laminar flow and boundary-layer flow. They postulate the primary mechanism coming from direct interactions between microcarriers and turbulent eddies. Microcarriers are small beads of several hundred micrometers diameter. Eddies of the size of the microcarrier or smaller may cause high shear stresses on the cells. The size of the smallest eddies can be estimated by the Kolmogorov length scale L, as given by... 

The group of Stickney reported the first deposits of mercury selenide (HgSe) formed via ECALE, using a thin layer flow system [54, 55]. The solutions used were HgO (pH 2) and Se02 (pH 3) with Na2S04 as a supporting electrolyte, and the... 

Reactor 14 [R14] Chip with Bi-/Tri-layer Flow Configuration Using Y-type Contact... 

Reactor type Chip with bi/tri-layer flow configuration using Y-type Outer dimensions 30 X 15 X 1.7 mm ... 

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