Layer inner diffusion boundary

In the inner problems of the convective mass transfer for kv - 0(1) as Pe —> oo, the concentration is leveled out along each streamline. The mean Sherwood number, by virtue of the estimate (5.4.8), is bounded above uniformly with respect to the Peclet number Sh < const kw. This means that the inner diffusion boundary layer cannot be formed by increasing the circulation intensity alone (i.e., by increasing the fluid velocity, which corresponds to Pe - oo) for moderate values of kv. This property of the mean Sherwood number is typical of all inner problems. For outer problems of mass transfer, the behavior of this quantity is essentially different here a thin diffusion boundary layer is usually... 

Intermediate stage of the process. At this stage the diffusion boundary layer still exists near the interface, and the concentration within the drop core is constant and equal to the initial value. However, as was mentioned above, the concentration at the entrance to the boundary layer is nonuniform one can find it from the conditions of matching with the concentration field in the inner diffusion layer. The problem is complicated by the fact that this concentration, in turn, depends on the concentration distribution in the diffusion boundary layer. In view of these effects, an integral equation was derived in [ 151,355] for Pep > 104 in order to obtain the onflow conditions at the entrance to the boundary layer, which leads to a nonself-similar solution. 

At the intermediate stage of the process, the diffusion wake interacts with the boundary layer and strongly erodes it, producing an increase in the boundary layer thickness (here the boundary layers for the inner and outer problems differ considerably). Gradually, as a result of absorption of the substance dissolved in the liquid on the interface, the diffusion boundary layer spreads over the entire drop and begins to decay. 

The diffusion boundary layer of any given drop in chain interacts with the diffusion wake of the previous drop (located upstream). The concentration field in it is substantially nonuniform and is depleted because of the absorption of the solute at the surfaces of all preceding drops. By virtue of such interaction, the inner mass exchange will be appreciably retarded (the shielding phenomenon) compared with the case of isolated drops. 

In the case of laminar flow, the velocity of the gas at the deposition surface (the inner wall of the tube) is zero. The boundary is that region in which the flow velocity changes from zero at the wall to essentially that of the bulk gas away from the wall. This boundary layer starts at the inlet of the tube and increases in thickness until the flow becomes stabilized as shown in Fig. 2.4b. The reactant gases flowing above the boundary layer have to diffuse through this layer to reach the deposition surface as is shown in Fig. 2.3. 

The hydrodynamic boundary layer has an inner part where the vertical velocity increases to a maximum determined by a balance of viscous and buoyancy forces. In fluids of high Schmidt number, the concentration diffusion layer thickness is of the same order of magnitude as this inner part of the hydrodynamic boundary layer. In the outer part of the hydrodynamic boundary layer, where the vertical velocity decays, the buoyancy force is unimportant. The profile of the vertical velocity component near the electrode can be shown to be parabolic. 

Outside the Stern surface the double layer continues to be described by Equation (63) or one of its approximations. The only modifications of the analysis of the diffuse double layer required by the introduction of the Stern surface are that x be measured from 6 rather than from the wall and that 06 be used instead of 0O as the potential at the inner boundary of the diffuse layer. 

It is the outer portion of the double layer that interests us most as far as colloidal stability is concerned. The existence of a Stern layer does not invalidate the expressions for the diffuse part of the double layer. As a matter of fact, by lowering the potential at the inner boundary of the diffuse double layer, we enhance the validity of low-potential approximations. The only problem is that specific adsorption effects make it difficult to decide what value to use for J/6. 

Figure 4.14 illustrates the transient solution to a problem in which an inner shaft suddenly begins to rotate with angular speed 2. The fluid is initially at rest, and the outer wall is fixed. Clearly, a momentum boundary layer diffuses outward from the rotating shaft toward the outer wall. In this problem there is a steady-state solution as indicated by the profile at t = oo. The curvature in the steady-state velocity profile is a function of gap thickness, or the parameter rj/Ar. As the gap becomes thinner relative to the shaft diameter, the profile becomes more linear. This is because the geometry tends toward a planar situation. 

Find the steady-state concentration profile during the radial diffusion of a diffusant through a bilayer cylindrical shell of inner radius, Rm, where each layer has thickness AR/2 and the constant diffusivities in the inner and outer layers are Dm and Dout. The boundary conditions are c(r = Rln) = cin and c(r = Rm + Ai ) = cout. Will the total diffusion current through the cylinder be the same if the materials that make up the inner and outer shells are exchanged Assume that the concentration of the diffusant is the same in the inner and outer layers at the bilayer interface. 

The treatment of the diffuse double layer outlined in the last section is based on an assumption of point charges in the electrolyte medium. The finite size of the ions will, however, limit the inner boundary of the diffuse part of the double layer, since the centre of an ion can only... 

Water vapor that evaporates from cell walls of mesophyll cells or the inner side of leaf epidermal cells (Fig. 1-2) diffuses through the intercellular air spaces to the stomata and then into the outside air. We have already introduced the four components involved—two are strictly anatomical (intercellular air spaces and cuticle), one depends on anatomy and yet responds to metabolic as well as environmental factors (stomata), and one depends on leaf morphology and wind speed (boundary layer). Figure 8-5 summarizes the symbols and arranges them into an electrical circuit. We will analyze resistances and conductances for these components, some of which occur in series (i.e., in a sequence) and some in parallel (i.e., as alternatives). 

The membrane reactor shown in Fig. 6.5 consists of a tubular shell containing a tubular porous membrane. It defines two compartments, the inner and the outer (shell) compartments. The reactants are fed into the inner compartment where the reaction takes place. We can observe that when the reactants flow along the reactor, one or more of the reaction participants can diffuse through the porous membrane to the outer side. In this case, we assume that only one participant presents a radial diffusion. This process affects the local concentration state and the reaction rate that determine the state of the main reactant conversion. The rate of reaction of the wall diffusing species is influenced by the transfer resistance of the boundary layer (1/lq.) and by the wall thickness resistance (S/Dp). 

Transport of the reactants by diffusion and convection out of the bulk gas stream, through a laminar boundary layer, to the outer surface of the catalyst particles, and further through the pore system to the inner surface (pore walls)... 

For the turbulent part of the inner boundary layer close to solid walls we usually adopt another closure for the Reynolds stresses and neglect the molecular diffusion term. The starting point for the boundary layer analysis, to be discussed shortly, is thus the fc-equation in the following form ... 

Helmholtz Double Layer A simplistic description of the electric double layer as a condenser (the Helmholtz condenser) in which the condenser plate separation distance is the Debye length. The Helmholtz layer is divided into an inner Helmholtz plane (IHP) of adsorbed, dehydrated ions immediately next to a surface, and an outer Helmholtz plane (OHP) at the center of a next layer of hydrated, adsorbed ions just inside the imaginary boundary where the diffuse double layer begins. That is, both Helmholtz planes are within the Stern layer. 

Sorption processes of significant importance have been demonstrated repeatedly in laboratory investigations. They are essential for the understanding of emission processes in buildings. Most inner surfaces may store chemicals, and equilibrium with the surrounding air will be approached by desorption or adsorption of airborne chemicals. These processes are typically fast and controlled by mass transport in the boundary layer of air. However, adsorbed pollutants may diffuse into the materials, and their reemission is typically slower and also influenced by the speed of dilfusion back to the surface. 

The treatment given above of the diffuse double layer is based on the assumption that the ions in the electrolyte are treated as point charges. The ions are, however, of finite size, and this limits the inner boundary of the diffuse part of the double layer, since the center of an ion can only approach the surface to within its hydrated radius without becoming specifically adsorbed (Fig. 6.4.2). To take this effect into account, we introduce an inner part of the double layer next to the surface, the outer boundary of which is approximately a hydrated ion radius from the surface. This inner layer is called the Stern layer, and the plane separating the inner layer and outer diffuse layer is called the Stern plane (Fig. 6.4.2). As indicated in Fig. 6.4.2, the potential at this plane is close to the electrokinetic potential or zeta ( ) potential, which is defined as the potential at the shear surface between the charge surface and the electrolyte solution. The shear surface itself is somewhat arbitrary but characterized as the plane at which the mobile portion of the diffuse layer can slip or flow past the charged surface. 

About 100 types of identified dryers are used in the world at present Boundary layer, inner diffusion, and boundary layer with inner diffusion Powder, granules, foil, film, and plate... 

The liquid layer surrounding the particle exists as two parts (i) an inner region (Stern layer) where the ions are strongly bound and (ii) an outer (diffuse) region where they are less firmly associated. Within the diffuse layer there is a notional boundary inside which the ions and particles form a stable entity. When a particle moves (e.g., due to gravity), ions within the boundary move with it The ions beyond the boundary do not travel with the particle. The potential at this boundary (surface of hydrodynamic shear) is the zeta potential (Fig. 8.5). 

Step 1 The solute in the aqueous phase (inner side of the hollow fiber) diffuses from the bulk to the aqueous-organic interface through the aqueous boundary layer. 

---