Boundary layer, diffusion walls

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. 

Consider a long cylindrical shell whose interior is filled with an incompressible fluid. If the fluid is initially at rest when the cylinder begins to rotate, a boundary layer develops as the momentum diffuses inward toward the center of the cylinder. The fluid s circumferential velocity vu comes to the cylinder-wall velocity immediately, owing to the no-slip condition. At very early time, however, the interior fluid will be only weakly affected by the rotation, with the influence increasing as the boundary layer diffuses inward. If the shell continues to rotate at a constant angular velocity, the fluid inside will eventually come to rotate as a solid body. 

We can obtain the concentration profiles simply by rescaling the velocity profiles, using Eqs. (4.5.14) and (4.5.15) therefore we do not explicitly discuss the similarity solution of the boundary layer diffusion equation. However, in terms of the similarity variable given above and from the series expressions for the velocity profile near the wall, the equation is reducible to the linear form... 

Convection mass transfer coefficients are often used as convective boundary conditions for gas diffusion in a stationary media. However, while applying mass transfer correlations to describe mass species transport from the electrode-gas diffusion layer to gas flow stream in the channel, it is assumed that species mass transport rate at the wall is small and does not alter the hydrod5mamic, thermal, and concentration boundary layers like in boundary layers with wall suction or blowing. 

A simplified model usiag a stagnant boundary layer assumption and the one-dimension diffusion—convection equation has been used to calculate wall concentration ia an RO module. The iategrated form of this equation, the widely appHed film theory (41), is given ia equation 8. 

Blade loading or diffusion loss. This loss is due to the type of loading in an impeller. The inerease in momentum loss eomes from the rapid inerease in boundary-layer growth when the veloeity elose to the wall is redueed. This loss varies from around 7% at a high-flow setting to about 12% at a low-flow setting. 

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. 

Fluid flow and reaction engineering problems represent a rich spectrum of examples of multiple and disparate scales. In chemical kinetics such problems involve high values of Thiele modulus (diffusion-reaction problems), Damkohler and Peclet numbers (diffusion-convection-reaction problems). For fluid flow problems a large value of the Mach number, which represents the ratio of flow velocity to the speed of sound, indicates the possibility of shock waves a large value of the Reynolds number causes boundary layers to be formed near solid walls and a large value of the Prandtl number gives rise to thermal boundary layers. Evidently, the inherently disparate scales for fluid flow, heat transfer and chemical reaction are responsible for the presence of thin regions or "fronts in the solution. 

Dispersion in packed tubes with wall effects was part of the CFD study by Magnico (2003), for N — 5.96 and N — 7.8, so the author was able to focus on mass transfer mechanisms near the tube wall. After establishing a steady-state flow, a Lagrangian approach was used in which particles were followed along the trajectories, with molecular diffusion suppressed, to single out the connection between flow and radial mass transport. The results showed the ratio of longitudinal to transverse dispersion coefficients to be smaller than in the literature, which may have been connected to the wall effects. The flow structure near the wall was probed by the tracer technique, and it was observed that there was a boundary layer near the wall of width about Jp/4 (at Ret — 7) in which there was no radial velocity component, so that mass transfer across the layer... 

COSILAB Combustion Simulation Software is a set of commercial software tools for simulating a variety of laminar flames including unstrained, premixed freely propagating flames, unstrained, premixed burner-stabilized flames, strained premixed flames, strained diffusion flames, strained partially premixed flames cylindrical and spherical symmetrical flames. The code can simulate transient spherically expanding and converging flames, droplets and streams of droplets in flames, sprays, tubular flames, combustion and/or evaporation of single spherical drops of liquid fuel, reactions in plug flow and perfectly stirred reactors, and problems of reactive boundary layers, such as open or enclosed jet flames, or flames in a wall boundary layer. The codes were developed from RUN-1DL, described below, and are now maintained and distributed by SoftPredict. Refer to the website http //www.softpredict.com/cms/ softpredict-home.html for more information. 

The velocity of the gases is high but it is always laminar flow. Over the susceptor there will be a boundary, or stagnant, layer where the velocity gradient decreases to zero. As the gases are heated, the silane and hydrocarbon will decompose and the species will diffuse through the boundary layer to grow on the reactor walls or on the substrate. A comprehensive study of this may be found in a paper by M. Leys and H. Veenvliet [41]. 

It is interesting to compare equations (6.32) and (6.33) with those for a fully developed laminar flow, equations (6.29) and (6.30). In Example 5.1, we showed that eddy diffusion coefficient in a turbulent boundary layer was linearly dependent on distance from the wall and on the wall shear velocity. If we replace the diffusion coefficient in equation (6.30) with an eddy diffusion coefficient that is proportional to hu, we get... 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

When fluid is pumped through a cell such as that shown in Fig. 12, transport of dissolved molecules from the cell inlet to the IRE by convection and diffusion is an important issue. The ATR method probes only the volume just above the IRE, which is well within the stagnant boundary layer where diffusion prevails. Figure 13 shows this situation schematically for a diffusion model and a convection-diffusion model (65). The former model assumes that a stagnant boundary layer exists above the IRE, within which mass transport occurs solely by diffusion and that there are no concentration gradients in the convection flow. A more realistic model of the flow-through cell accounts for both convection and diffusion. As a consequence of the relatively narrow gap between the cell walls, the convection leads to a laminar flow profile and consequently to concentration gradients between the cell walls. 

Since the electro-osinotic flow is induced by the interaction of the externally applied electric field with the space charge of the diffuse electric double layers at the channel walls, we shall concentrate in our further analysis on one of these 0 1 2) thick boundary layers, say, for definiteness, at... 

Flow dynamics predict that flow through a pipe is nonuniform with regard to velocity across the diameter of a pipe, for instance. The flow at pipe walls is assumed to be zero. In our idealized biochemical reactor, this concept is represented by a boundary layer in contact with the biofilm. It does not have, of course, a discrete dimension. Rather, it is represented as an area in the structure that has reduced flow and therefore different kinetics than what we would assume exist in a bulk liquid. The boundary layer is affected by turbulence and temperature and this is unavoidable to a degree. Diffusion within the boundary layers is controlled by the chemical potential difference based on concennation. Thus the rate of transfer of pollutant to the organisms is controlled by at least two physical chemical principles, and these principles differentiate an attached growth bioreactor from a suspended growth bioreactor. 

In the preceding section, the sediment surface was described as an intermedia wall boundary. Thereby we tacitly assumed that the diffusion wall, that is, the location where diffusivity drops from DB, to DA coincides with the interface between the two media. As shown in Fig. 19.86, the transition from a turbulent to a stagnant media includes a boundary layer in the former in which diffusivity drops in a characteristic manner. 

Figure 19.10 Schematic view of concentration profile across a wall boundary between different media with a boundary layer of thickness 8 on the B-side of the interface. Diffusivities are DB —> in the...

The entry-length region is characterized by a diffusive process wherein the flow must adjust to the zero-velocity no-slip condition on the wall. A momentum boundary layer grows out from the wall, with velocities near the wall being retarded relative to the uniform inlet velocity and velocities near the centerline being accelerated to maintain mass continuity. In steady state, this behavior is described by the coupled effects of the mass continuity and axial momentum equations. For a constant-viscosity fluid,... 

Porous Membrane DS Devices. The applicability of a simple tubular DS based on a porous hydrophobic PTFE membrane tube was demonstrated for the collection of S02 (dilute H202 was used as the scrubber liquid, and conductometric detection was used) (46). The parameters of available tubular membranes that are important in determining the overall behavior of such a device include the following First, the fractional surface porosity, which is typically between 0.4 and 0.7 and represents the probability of an analyte gas molecule entering a pore in the event of a collision with the wall. Second, wall thickness, which is typically between 25 and 1000 xm and determines, together with the pore tortuosity (a measure of how convoluted the path is from one side of the membrane to the other), the overall diffusion distance from one side of the wall to the other. If uptake probability at the air-liquid interface in the pore is not the controlling factor, then items 1 and 2 together determine the collection efficiency. The transport of the analyte gas molecule takes place within the pores, in the gas phase. This process is far faster than the situation with a hydrophilic membrane the relaxation time is well below 100 ms, and the overall response time may in fact be determined by liquid-phase diffusion in the boundary layer within the lumen of the membrane tube, by liquid-phase dispersion within the... 

Boundary layers also contribute to the effect of intestinal fluid hydrodynamics on drug absorption by both diffusional- and carrier-mediated processes. In a well-defined isolated in situ model such as perfused intestine of the rat, a good estimate of the gut wall permeability, which is the vector of convective diffusive mass transfer, passive diffusion and carrier-mediated transport, can be accomplished [99,100]. 

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