Increasing distance from the wall

The simplest possible case of a gas-solid interaction for physical adsorption is that of a molecule interacting with a smooth hard wall. The wall can be planar, as for a free surface or a slit pore, or it can be cylindrical or some other shape for a pore. These cases have been extensively studied by Monte Carlo and molecular dynamics with results that show that such a gas-solid interaction gives a strongly structured fluid that can be best described as a series of layers that follow the contour of the wall. The sharpness of the density variations that define these layers increases as the overall density of the adsorbed film increases and decreases with increasing distance from the wall. 

In most rotational viscometers the rate of shear varies with the distance from a wall or the axis of rotation. However, in a cone—plate viscometer the rate of shear across the conical gap is essentially constant because the linear velocity and the gap between the cone and the plate both increase with increasing distance from the axis. No tedious correction calculations are required for non-Newtonian fluids. The relevant equations for viscosity, shear stress, and shear rate at small angles a of Newtonian fluids are equations 29, 30, and 31, respectively, where M is the torque, R the radius of the cone, v the linear velocity, and rthe distance from the axis. 

In streamline flow, E is very small and approaches zero, so that xj p determines the shear stress. In turbulent flow, E is negligible at the wall and increases very rapidly with distance from the wall. LAUFER(7), using very small hot-wire anemometers, measured the velocity fluctuations and gave a valuable account of the structure of turbulent flow. In the operations of mass, heat, and momentum transfer, the transfer has to be effected through the laminar layer near the wall, and it is here that the greatest resistance to transfer lies. 

The thickness of the downflowing layers at the wall of the CFB is typically defined as the distance from the wall to the position of zero vertical solid flux. Measurements of the layer thickness were made on a 12 MW and 165 MW CFB boiler by Zhang, Johnsson and Leckner (1995). They found that the thickness increased for the larger bed. They related data from many different beds (Fig. 19), with the equivalent bed diameter, taken as the hydraulic diameter, using the following form... 

When a fluid flows past a solid surface, the velocity of the fluid in contact with the wall is zero, as must be the case if the fluid is to be treated as a continuum. If the velocity at the solid boundary were not zero, the velocity gradient there would be infinite and by Newton s law of viscosity, equation 1.44, the shear stress would have to be infinite. If a turbulent stream of fluid flows past an isolated surface, such as an aircraft wing in a large wind tunnel, the velocity of the fluid is zero at the surface but rises with increasing distance from the surface and eventually approaches the velocity of the bulk of the stream. It is found that almost all the change in velocity occurs in a very thin layer of fluid adjacent to the solid surface ... 

From equation 1.41, the total shear stress varies linearly from a maximum fw at the wall to zero at the centre of the pipe. As the wall is approached, the turbulent component of the shear stress tends to zero, that is the whole of the shear stress is due to the viscous component at the wall. The turbulent contribution increases rapidly with distance from the wall and is the dominant component at all locations except in the wall region. Both components of the mean shear stress necessarily decline to zero at the centre-line. (The mean velocity gradient is zero at the centre so the mean viscous shear stress must be zero, but in addition the velocity fluctuations are uncorrelated so the turbulent component must be zero.)... 

The dimensionless distance y+ has the form of a Reynolds number. Equation 2.58 fits the experimental data in the range 0 y+ 5. In the viscous sublayer, the velocity increases linearly with distance from the wall. 

Fig. 12 shows the radial profile of the liquid velocity at different axial positions. Due of the baffles, the liquid is redistributed in the radial direction and the turbulent intensity is increased. The radial profile of the liquid velocity is almost uniform after passing the internal. Liquid velocity is lower at the center and higher near the wall as compared with that below the internal. With increasing distance from the internal, the turbulence intensity diminishes and the wall effect becomes more apparent, that is, the liquid velocity increases at the center and decreases near the wall. The radial profile obtained at the position of 114 cm from the internal is similar to that obtained below the internal and is the same as that at the position of 144 cm. 

Different types of distributions correspond to different operating modes. The two most frequently used operating modes are normal mode and steric and hyperlayer mode (Reschiglian et al., 2005). The normal mode of separation is active for particles < 1 pm and the steric and hyperlayer modes are applicable to particles > 1 pm. In the normal mode as macromolecules or particles that constitute the sample are driven by the field toward the accumulation wall, their concentration increases with decreasing distance from the wall. This creates a concentration gradient that causes sample diffusion away from the wall. Retention time in the normal FFF is therefore... 

The size of the turbulent eddies near the wall is determined by the distance from the wall i.e., near the wall it is to be expected that their size will increase linearly with distance from the wall. Now, the mixing length is related to the scale of the turbulence, i.e., to the size of the eddies, and it is to be expected therefore that near the wall ... 

Many experiments have established that, as mentioned before, there is a region near the wall where the local turbulent shear stress depends on the wall shear stress and the distance from the wall alone and is largely independent of the nature of the rest of the flow. In this region, the mixing length increases linearly with distance from the wall except that near the wall there is a damping of the turbulence due to viscosity. In the wall region, it is assumed therefore that ... 

As compared to the interphases detected in the case of the Cu/epoxy systems, the interphase measured in the case of the C-fibre/PPS system exhibits a negative stiffness gradient, i.e. decreasing local stiffness of the thermoplastic PPS with increasing distance from the C-fibre surface. The mean width 3/c 107 nm of the stiffness profile is 2.6 times smaller than that of the stiffness profile measured on the Cu/epoxy replica sample. Taking into consideration the spatial constraints imposed on polymer chains due to the presence of the nearby hard wall represented by the surface of the C-fibres, the observed increase in local stiffness can be ascribed to the respective loss in chain flexibility and mobility. 

Positronium can pick-off an electron during a collision with a pore wall and annihilate into two photons. Between collisions, only three photon annihilations occur, just as in vacuum. Quantum mechanically, the overlap with the wall-electron wave functions decreases with the distance from the wall and pick-off (two photons) becomes less likely. With increasing pore size collisions become less frequent. The ratio of 3 photon annihilations to 2 photons probes the combination of pore size and total pore volume as well as their link to the sample surface, and can be measured by examining the energy distribution of annihilation photons. This 3-to-2 photon ratio can be calibrated to absolute fractions of positronium in the annihilation spectrum [16, 17]. 

Figure 4-34. Wall effects in capillary electrophoresis. The capillary walls, which are usually made of fused silica contain a small proportion of dissociated silanyl groups which bind positively charged counterions. In the region close to the wall, defining the Stern layer, these ions are relatively immobile mobility increases beyond this layer in a region which defines the Guoy-Chapman layer. The distribution of positively charged ions is termed the Zeta-potential it is fairly constant in the Stern layer, but falls off sharply with distance from the wall in the Guoy-Chapman layer.

The results of measurements by the microscopic method show that the electrophoretic mobility of the particles varies with the distance from the wall of the cell particles close to the wall move in a direction opposite to that in which those in the center migrate. In any event, the results show an increase in velocity from the walls to the center of the cell. The explanation of this fact lies in the electro-osmotic movement of the liquid a double layer is set up between the liquid and the walls of the cell and under the influence of the applied field the former exhibits electro-osmotic flow. For the purpose of obtaining the true electrophoretic velocity of the suspended particles it is neceasary to observe particles at about one-fifth the distance from one wall to the other. A more accurate procedure is to make a series of measurements at different distances from the side of the cell and to apply a correction for the electro-osmotic flow. The algebraic difference of the corrected electrophoretic velocity and the speed of the particles near the walls gives the electro-osmotic mobility of the liquid in the particular cell. If the solution contains a protein which is adsorbed on the surface of the walls of the vessel and on the particles, it is possible to compare the electrophoretic and electro-osmotic mobilities in one experiment reference to the significance of such a comparison was made on page 532. 

When a fluid flows past a stationary wall, the fluid adheres to the wall at the interface between the solid and the fluid. Therefore, the local velocity v of the fluid at the interface is zero. At some distance y normal to and displaced from the wall, the velocity of the fluid is finite. Therefore, there is a velocity variation from point to point in the flowing fluid. This causes a velocity field in which the velocity is a function of the normal distance from the wall, i.e., v = /(y). If y = 0 at the wall, u = 0, and v increases with y. The rate of change of velocity with respect to distance is the velocity gradient ... 

In solid-liquid systems the size and shape of the baffles are important design parameters. The standard baffling is illustrated in Fig 7.1. As the solid concentration increases and the viscosity becomes high, narrower baffles (approximately 1/24T) placed a distance from the wall, should be used. This design is normally employed to avoid permanent settling of particles in the low velocity zones. In some processes such fillets (settled particles) can nevertheless be advantageous for the power consumption. 

The final result in equation (6.5.28) shows that the concentration falls from its initial value of Cq at the wall in a manner determined by the error function and the diffusion coefficient D. Typical concentration profiles estimated at various values of t are shown in fig. 6.3. Obviously, the concentration increases at a given distance from the wall as time increases. 

Qualitatively, the picture of stabilized turbulent flow in a plane channel is similar to that in a circular tube. Indeed, in the viscous sublayer adjacent to the channel walls, the velocity distribution increases linearly with the distance from the wall V(Y)/U = y+. In the logarithmic layer, the average velocity profile can be described by the expression [289]... 

VELOCITY GRADIENT AND RATE OF SHEAR. Consider the steady one-dimensional laminar flow of an incompressible fluid along a solid plane surface. Figure 3.1a shows the velocity profile for such a stream. The abscissa u is the velocity, and the ordinate y is the distance measured perpendicular from the wall and therefore at right angles to the direction of the velocity. At y = 0, u = 0, and u increases with distance from the wall but at a decreasing rate. Focus attention on the velocities on two nearby planes, plane A and plane B, a distance Ay apart. 

SEPARATION FROM VELOCITY DECREASE Boundary-layer separation can occur even where there is no sudden change in cross section if the cross section is continuously enlarged. For example, consider the flow of a fluid stream through the trumpet-shaped expander shown in Fig. 5.16. Because of the increase of cross section in the direction of flow, the velocity of the fluid decreases, and by the Bernoulli equation, the pressure must increase. Consider two stream filaments, one, aa, very near the wall, and the other, bb, a short distance from the wall. The pressure increase over a definite length of conduit is the same for both filaments, because the pressure throughout any single cross section is uniform. The loss in velocity head is, then, the same for both filaments. The initial velocity head of filament aa is less than that of filament bb, however, because filament aa is nearer... 

Within the viscous sublayer heat flows mainly by conduction, but eddies are not Completely excluded from this zone, and some convection does occur. The relative importance of turbulent heat flux compared with conductive heat flux increases rapidly with distance from the wall. In ordinary fluids, having Prandtl numbers above about 0.6, conduction is entirely negligible in the turbulent core, but it may be significant in the buffer zone when the Prandtl number is in the neighborhood of unity. Conduction is negligible in this zone when the Prandtl number is large. 

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