Influence of the Wall

The previous section showed that the one-dimensional numerical model can predict the average breakthrough times but is not able to explain the differences between the temperature measurements in the radial centre and the zone close to the wall of the bed. The difference in temperature profiles between these locations is particularly pronounced for the cooling step. The temperature development close to the wall is very disperse, which cannot be predicted by the one-dimensional model. 

It is very well possible that the axial heat conduction in the wall has an influence on the temperature in the region close to the wall. A detailed description of the temperature in the bed as a function of the radial and axial positions can be obtained by extending the model to two dimensions. However, it is also possible to get more insights into the behaviour by adjusting the one-dimensional model. Instead of accounting for the heat capacity of the wall in the accumulation term, an extra energy balance for the wall is included in the model, in which heat is exchanged with the packed bed  

The energy balance for the bed (both the gas and the solid phases) will therefore also be extended with an extra contribution due to the heat exchange with the wall  

The wall will also influence the temperature development during the capture and recovery steps. It is very well possible that CO2 will deposit at the tube wall (which is not accounted for in the extended model). This could well explain the slower removal of CO2 during the recovery step close to the wall. Locally there will be more COj deposited, due to the heat capacity of the wall, and, therefore, it will take longer to remove solid CO2 from those zones. An additional effect may be that the pressure drop will locally increase due to the higher amount of deposited CO2, resulting in lower flow rates and therefore even slower solid CO2 removal rates. 

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The correct treatment of boundaries and boundary effects is crucial to simulation methods because it enables macroscopic properties to be calculated from simulations using relatively small numbers of particles. The importance of boundary effects can be illustrated by considering the following simple example. Suppose we have a cube of volume 1 litre which is filled with water at room temperature. The cube contains approximately 3.3 X 10 molecules. Interactions with the walls can extend up to 10 molecular diameters into the fluid. The diameter of the water molecule is approximately 2.8 A and so the number of water molecules that are interacting with the boundary is about 2 x 10. So only about one in 1.5 million water molecules is influenced by interactions with the walls of the container. The number of particles in a Monte Carlo or molecular dynamics simulation is far fewer than 10 -10 and is frequently less than 1000. In a system of 1000 water molecules most, if not all of them, would be within the influence of the walls of the boundary. Clecirly, a simulation of 1000 water molecules in a vessel would not be an appropriate way to derive bulk properties. The alternative is to dispense with the container altogether. Now, approximately three-quarters of the molecules would be at the surface of the sample rather than being in the bulk. Such a situation would be relevcUit to studies of liquid drops, but not to studies of bulk phenomena. 

FIGURE 7.29 Correction parameter reflecting the influence of the wall (a) compact jets (experimental data from V. Mitkalinny, A. Abdushev, V. Baharev and L. Fedorov) (b) linear jets (experimental data from W. Kerka and Z. Sakipov) (c) radial jets (experimental data from N. Gelman). Reproduced from Grimitlyn, ... 

Elutriation differs from sedimentation in that fluid moves vertically upwards and thereby carries with it all particles whose settling velocity by gravity is less than the fluid velocity. In practice, complications are introduced by such factors as the non-uniformity of the fluid velocity across a section of an elutriating tube, the influence of the walls of the tube, and the effect of eddies in the flow. In consequence, any assumption that the separated particle size corresponds to the mean velocity of fluid flow is only approximately true it also requires an infinite time to effect complete separation. This method is predicated on the assumption that Stokes law relating the free-falling velocity of a spherical particle to its density and diameter, and to the density and viscosity of the medium is valid... 

Investigate the influence of the wall mass and the heat transfer coefficients on the maximum yield of B. 

Also, Schier s experiments revealed the necessity to consider the directional influence of the walls of the filter bed on the bed porosity e. This effect is very important especially for small sized beds that are usually used in laboratories for investigations. For beds made of spherically shaped collectors several correlations exist describing the e(y/dc) function where y denotes the distance from the wall in the radial direction. However, for relative large bed diameters DB/dc ranging from 5 to 25 it proved to be sufficient to use an averaged e in Eq. (3.2.5), as proposed by Jeschar [6],... 

In the course of the investigation detailed porosity measurements helped to improve the description of the influence of the wall on the porosity. 

In beds with very high N, the central structure will dominate throughout the bed. In low-7V beds the wall-induced structure will dominate as the influence of the wall on the structure penetrates relatively deep into the bed. As an example, consider a tube with a diameter of 100, in arbitrary units, and two different spherical packing materials with diameters of 10 (N — 10) and 1 (N — 100) further assume that the wall-induced structure is recognizable as such for four layers of spheres from the wall. In the N — 10 bed the first four layers at the wall occupy 96 volume % of the bed. In the N — 100 bed, the wall-influenced region only makes up 15.6 volume % of the bed. 

All the spheres in a layer were supported by two spheres of the layer below and the column wall, creating a stable packing structure. As the tube-to-particle diameter ratio of the bed was only four, the entire packing structure was controlled by the influence of the wall. Nevertheless, the packing was divided into an immediate wall layer and a central section, but this should not be taken to imply that the central structure was not wall influenced. Although a three-sphere planar structure would almost fit within the nine-sphere wall layer, there was just not enough room at the same axial coordinate. When, however, the... 

We can see that for these conditions, the temperature fields inside the wall particles are far from symmetric. Significant temperature incursions appear inside the spheres, and the influence of the wall is strong. The spheres are hotter close to the tube wall than on the side facing the center of the segment. The interior particles appear to be more symmetrical in temperature. It is noticeable that the particles are considerably lower in temperature than the surrounding... 

If the ratio of the diameter of the vessel to the diameter of the particle is greater than about 100, the walls of the container appear to have no effect on the rate of sedimentation. For smaller values, the sedimentation rate may be reduced because of the retarding influence of the walls. 

The critical limit may be an upper limit or a lower limit. This comes about in the following way. Chains are interrupted either by deactivation at the wall or in the gas phase. The greater the concentration of the gases, the smaller relatively is the influence of the wall factor. When this is the chief means by which the chains are broken there may be a transition from slow reaction to explosion as the pressure increases. When, on the other hand,... 

The lower limit at B would be the point at which the deactivating influence of the walls of the vessel is just great enough to keep the chains from developing explosively. 

In macroscopic reactors, knowledge of the velocity profile in the channel cross-section is a necessary and sufficient prerequisite to describe the material transport. In microscopic dimensions down to a few micrometers, diffusion also has to be considered. In fact, without the influence of diffusion, extremely broad residence time distributions would be found because of the laminar flow conditions. Superposition of convection and diffusion is called dispersion. Taylor [91] was among the first to notice this strong dominating effect in laminar flow. It is possible to transfer his deduction to rectangular channels. A complete fluid dynamic description has been given of the flow, including effects such as the influence of the wall, the aspect ratio and a chemical wall reaction on the concentration field in the cross-section [37]. 

It seems to be necessary, in order to account for the influence of the wall on the first explosion limit, to postulate termination of chains on the wall. From other evidence (which we shall discuss later in a few specific cases) on the slow rates outside the explosion peninsula it appears equally necessary to postulate chain initiation at the walls. Where surfaces such as the walls play an important role in chemical reactions, we must expect to find that in a certain range of reaction conditions the diffusion to and from such surfaces may exert a limiting effect on the rates of chemical reactions. If the reactions arc taking place in the volume of the vessel in competition with reactions at the wall, then we may expect to find concentration gradients within the volume of the vessel. ... 

When the two steps of plasma treatment are carried out successively in the same reactor without copious cleaning of the reactor between the two treatments, the influence of the wall surface on the second plasma treatment also becomes an important factor. Table 10.2 reveals the following important factors ... 

FIGURE 1.3 Fluid flow characteristics and profiles of fluid flow in pipes (a) At low Reynolds numbers, where streamline flow is obtained throughout the cross section, (b) At high Reynolds numbers, where turbulent flow is obtained for most of pipe volume. Streamline flow is only obtained in a thin boundary layer adjacent to the pipe wall where the influence of the wall and viscous forces control turbulence. 

Brenner [61], by using the reflection method, derived the following relation, which allows one to correct the Stokes drag law by taking into account the influence of the walls ... 

The behavior of a flowing fluid depends strongly on whether or not the fluid is under the influence of solid boundaries. In the region where the influence of the wall is small, the shear stress may be negligible and the fluid behavior may approach that of an ideal fluid, one that is incompressible and has zero viscosity. The flow of such an ideal fluid is called potential flow and is completely described by the principles of newtonian mechanics and conservation of mass. The mathematical theory of potential flow is highly developed but is outside the scope of this book. Potential flow has two important characteristics (1) neither circulations nor eddies can form within the stream, so that potential flow is also called irrotational flow, and (2) friction cannot develop, so that there is no dissipation of mechanical energy into heat. 

An extension of FFF to the separation of nonspherical particles and the influences of the wall effect have been studied both theoretically and experimentally by Gajdos and Brenner [72]. High-resolution polymer separations have been achieved with a TFFF channel of a new construction [73]. Retention and zone spreading equations in SFFF have been verified [74]. Recent studies of electroretention of proteins [75,76] have resulted in more detailed understanding of previously observed retention anomalies in EFFF. Several papers have demonstrated many possibilities of application of the FFF subtechniques in various fields of chemistry, biology and technology [77-79]. 

Eqn. 1 shows clearly that the smaller the cross-sectional area (proportional to the square of the inner diameter) the higher is the concentration of the solute at the end of the column. Therefore, in trace analysis, high-efficiency, short and small-diameter columns are to be preferred. The smallest inner diameter of columns that can be packed efficiently is about Z mm (ref. 7). Smaller diameters are less efficient because of an increasing influence of the wall effects and the relative increase in the external peak broadening versus the dispersion due to the column. 

The gas-liquid mixture arriving at the separator entrance contains a small volume concentration of the liquid phase (Wo 1). It means that the liquid phase has practically no influence on the velocity distribution inside the flow. It is also possible to neglect the mutual interaction of drops, i.e. the constrained character of their motion. Suppose that the velocity profile Mo(y) is given at the entrance to the separator. Direct the x-axis along the axis of the separator, and the y-axis perpendicular to the separator axis. For simplicity s sake, let us take a separator with a rectangular cross section. The influence of the walls curvature in the case of the circular cross section will be studied later. The equations of motion for a drop of radius R in the approximation that neglects inertia have the form ... 

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