The boundary layer concept

The diameter of drug particles and hence the surface specific length L is much smaller than the pipe diameter. For this reason, particle-liquid Reynolds numbers characterizing the flow at the particle surface are considerably lower than the corresponding bulk Reynolds numbers. Particle-liquid Reynolds numbers for particle sizes below 250 pm were calculated to be below Re 1 for flow rates up to 100 mL/min. However, this circumstance does not limit the applicability of the boundary layer concept, since in aqueous hydrodynamic... 

In developing the boundary layer concept, Prandtl suggested an order-of-magnitude evaluation of the terms in the Navier-Stokes equations, which provides an expression (with an unknown proportionality constant) for the... 

The boundary layer concept is attributed to Ludwig Prandtl (1874—1953). His manuscript, published in 1904, formed the basis for the future work on skin friction, heat transfer, and fluid separation. He later made original contributions to finite wing theory and compressibility effects. Theodore von Karman and Max Munk were among his many famous students. 

Heat transfer by convection is a complex process but the analysis is simplified by the boundary layer concept. All resistance to heat transfer on the fluid side of a hot surface is supposed to be concentrated in a thin film of fluid close to the solid surface. Transfer within the film is by conduction. The temperature profile for such a process is shown in Figure 7.27. The thickness of the thermal boundary layer is not generally equal to that of the hydrodynamic boundary layer. The heat flux could be expressed as... 

The solutions to Navier-Stokes equations are typically very difficult to arrive at. This fact is attested to by the extraordinary development of numerical computation in fluid mechanics. Only a few exact analytical solutions are known for Navier-Stokes equations. We present in this chapter some laminar flow solutions whose interpretation per se is essential in this regard. We then introduce the boundary layer concept. We conclude the chapter with a discussion on the uniqueness of solutions to Navier-Stokes equations, with special reference to the phenomenon of turbulence. 

The boundary layer concept is introduced when dealing with a flow where the effect of viscosity is confined to the vicinity of solid walls. Such case is obtained when the flow s Reynolds number (introduced in Chapter 3) is sufficiently large. 

The boundary layer concept is formalized when the thickness 5 of the boundary layer is small compared to the geometrical dimensions of the zone of the flow that lies outside the boundary layer. Apart from the case (Figure 1.6(d)) where the boundary layer separates from the wall, the drawings of Figure 1.6 are distorted representations where the boundary layer thickness is exaggerated to show the inside of the boundary layer. The actual boundary layer thickness is very small compared... 

The boundary layer concept is needed to explain the following paradoxes associated with fluid flow ... 

In applying the correlation, use is made of the concept of logarithmic mean temperature difference across the boundary layer. For a boiler section, or pass, this is given by ... 

This relation for the thickness of the boundary layer has been obtained on the assumption that the velocity profile can be described by a polynomial of the form of equation 11.10 and that the main stream velocity is reached at a distance 8 from the surface, whereas, in fact, the stream velocity is approached asymptotically. Although equation 11.11 gives the velocity ux accurately as a function of v, it does not provide a means of calculating accurately the distance from the surface at which ux has a particular value when ux is near us, because 3ux/dy is then small. The thickness of the boundary layer as calculated is therefore a function of the particular approximate relation which is taken to represent the velocity profile. This difficulty cat be overcome by introducing a new concept, the displacement thickness 8. ... 

Explain the concepts of momentum thickness" and displacement thickness for the boundary layer formed during flow over a plane surface. Develop a similar concept to displacement thickness in relation to heat flux across the surface for laminar flow and heat transfer by thermal conduction, for the case where the surface has a constant temperature and the thermal boundary layer is always thinner than the velocity boundary layer. Obtain an expression for this thermal thickness in terms of the thicknesses of the velocity and temperature boundary layers. 

Originally, the concept of the Prandtl boundary layer was developed for hydraulically even bodies. It is assumed that any characteristic length L on the particle surface is much greater than the thickness (<5hl) of the boundary layer itself (L > Ojil) Provided this assumption is fulfilled, the concept can be adapted to curved bodies and spheres, including real drug particles. Furthermore, the classical ( macroscopic ) concept of the hydrodynamic boundary layer is valid solely for high Reynolds numbers of Re>104 (14,15). This constraint was overcome for the microscopic hydrodynamics of dissolving particles by the convective diffusion theory (9). 

The convective diffusion theory was developed by V.G. Levich to solve specific problems in electrochemistry encountered with the rotating disc electrode. Later, he applied the classical concept of the boundary layer to a variety of practical tasks and challenges, such as particle-liquid hydrodynamics and liquid-gas interfacial problems. The conceptual transfer of the hydrodynamic boundary layer is applicable to the hydrodynamics of dissolving particles if the Peclet number (Pe) is greater than unity (Pe > 1) (9). The dimensionless Peclet number describes the relationship between convection and diffusion-driven mass transfer ... 

The Schmidt number is the ratio of kinematic viscosity to molecular diffusivity. Considering liquids in general and dissolution media in particular, the values for the kinematic viscosity usually exceed those for diffusion coefficients by a factor of 103 to 104. Thus, Prandtl or Schmidt numbers of about 103 are usually obtained. Subsequently, and in contrast to the classical concept of the boundary layer, Re numbers of magnitude of about Re > 0.01 are sufficient to generate Peclet numbers greater than 1 and to justify the hydrodynamic boundary layer concept for particle-liquid dissolution systems (Re Pr = Pe). It can be shown that [(9), term 10.15, nomenclature adapted]... 

Figure 4-21 The concept of boundary layer and boundary layer thickness 5. (a) Compositional boundary layer surrounding a falling and dissolving spherical crystal. The arrow represents the direction of crystal motion. The shaded circle represents the spherical particle. The region between the solid circle and the dashed oval represents the boundary layer. For clarity, the thickness of the boundary layer is exaggerated, (b) Definition of boundary layer thickness 5. The compositional profile shown is "averaged" over all directions. From the average profile, the "effective" boundary layer thickness is obtained by drawing a tangent at x = 0 (r=a) to the concentration curve. The 5 is the distance between the interface (x = 0) and the point where the tangent line intercepts the bulk concentration.

If one assumes that S/8h is independent of x, one arrives again at Eq. (58). If a particular form is chosen for the function F, as one proceeds in the method of polynomials, the calculation of the constants A, B, and E becomes possible. While in this particular problem one can follow a parallelism between the algebraic method and the method of polynomials, the same parallelism can no longer be identified in the other examples examined. It is worth emphasizing that the use of the boundary layer thickness concept in the algebraic method does not imply the existence of a similarity solution. In general, the algebraic method interpolates between the two similarity solutions which are valid in the two asymptotic cases. 

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 chapter we pointed to electrical conductivity as one of the physical properties of semiconductors which is changed by a chemisorption process and is accessable to measurement. A further possibility for investigating the mechanism of chemisorption is the relation between the work function and the external electric field of the semiconductor as influenced by chemisorption. These effects have been used for the interpretation of the mechanism of chemisorption and heterogeneous catalysis by Suhrmann (42), and have been experimentally demonstrated in chemisorption processes by Ljaschenko and Stepko. These effects shall here be correlated with our concept of the boundary layer formed in the presence of oxygen and hydrogen. 

In many respects, similar to the diffusion layer concept, there is that of the hydrodynamic boundary layer, <5H. The concept was due originally to Prandtl [16] and is defined as the region within which all velocity gradients occur. In practice, there has to be a compromise since all flow functions tend to asymptotic limits at infinite distance this is, to some extent, subjective. Thus for the rotating disc electrode, Levich [3] defines 5H as the distance where the radial and tangential velocity components are within 5% of their bulk values, whereas Riddiford [7] takes a figure of 10% (see below). It has been shown that... 

We also consider first the case where fe, is large, such that the reaction layer concept is useful. For the transport limited current, imax, we have the further boundary condition... 

Keulegan (K13) applied the semiempirical boundary-layer concepts of Prandtl and von K arm an to the case of turbulent flow in open channels, taking into account the effects of channel cross-sectional shape, roughness of the wetted walls, and the free surface. Most of the results are applicable mainly to deep rough channels and bear little relation to the flow of thin films. 

Constant-Stress Layer in Flowing Fluids. In the boundary layer of a fluid flowing over a solid wall. Ihe shear stress varies with distance from Ihe wall bul ii may be considered nearly constant within a small fraction of the layer thickness. The concept is of particular importance in turbulent flow where it leads lo a theoretical derivation of the law of ihe wall," the logarithmic distribution of mean velocity. The constant stress layer is ihe best-known example of the equilibrium flow s near a wall. 

The experimental results described in this review support the concept that, in certain reactions of the redox type, the interaction between catalysts and supports and its effect on catalytic activity are determined by the electronic properties of metals and semiconductors, taking into account the electronic effects in the boundary layer. In particular, it has been shown that electronic effects on the activity of the catalysts, as expressed by changes of activation energies, are much larger for inverse mixed catalysts (semiconductors supported and/or promoted by metals) than for the more conventional and widely used normal mixed catalysts (metals promoted by semiconductors). The effects are in the order of a few electron volts with inverse systems as opposed to a few tenths of an electron volt with normal systems. This difference is readily understandable in terms of the different magnitude of, and impacts on electron concentrations in metals versus semiconductors. 

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