Boundary layer, theory

Boundary layer theory, just like film theory, is also based on the concept that mass transfer takes place in a thin him next to the wall as shown in Fig. 1.48. It differs from the him theory in that the concentration and velocity can vary not only in the y-direction but also along the other coordinate axes. However, as the change in the concentration prohle in this thin him is larger in the y-direction than any of the other coordinates, it is sufficient to just consider diffusion in the direction of the y-axis. This simplihes the differential equations for the concentration signihcantly. The concentration prohle is obtained as a result of this simplihcation, and from this the mass transfer coefficient [3 can be calculated according to the dehnition in (1.179). In practice it is normally enough to use the mean mass transfer coefficient 

This can be found, for forced how from equations of the form 

The determination of heat transfer coefficients with the assistance of dimensionless numbers has already been explained in section 1.1.4. This method can also be used for mass transfer, and as an example we will take the mean Nusselt number Num = amL/ in forced flow, which can be represented by an expression of the form 

196) the quantities c,n,m still depend on the type of flow, laminar or turbulent, and the shape of the surface or the channel over or through which the fluid flows. Correspondingly, the mean Sherwood number can be written as 

The average mass transfer coefficient can be calculated from (1.197). It can also be found by following the procedure outlined for average heat transfer coefficients, and dividing (1.197) by (1.196). This then gives a relationship between the heat and mass transfer coefficients, 

The boundary layer theory was first developed by Ludwig Prandtl in 1904. The theory points out that the fluid flow along the surface of a body is divided into two regions the boundary layer and the flow outside the boundary layer. 

The boundary layer is a thin fluid film that forms on the surface of a solid body moving through a viscous fluid. The majority of the drag force experienced by the moving body which is immersed in a fluid is due to viscous shear and inertial forces within the boundary layer  

The flow outside the boundary layer is determined by inertial forces only. Shear forces and the viscosity of the fluid within the boundary layer can be neglected without significant effects on the solution. 

The boundary layer theory reconciles the important contradictions between experimental hydraulics and theoretical hydrodynamics. It effectively combines both approaches into an integrated theory. 

The velocity boundary layer concept can be extended to define the temperature and concentration of a fluid. The temperature boundary layer thickness is the distance from the body to a layer at which the temperature is 99% of the temperature from an inviscid solution. The boundary layer thickness for the fluid concentration has the same definition. Their relationships are expressed by [29] 

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A quantitative treatment for the depletive adsorption of iogenic species on semiconductors is that known as the boundary layer theory [84,184], in which it is assumed that, as a result of adsorption, a charged layer is formed. Doublelayer theory is applied, and it turns out that the change in surface potential due to adsorption of such a species is proportional to the square of the amount adsorbed. The important point is that very little adsorption, e.g., a 0 of about 0.003, can produce a volt or more potential change. See Ref. 185 for a review. 

Schlichting, H., 1968. Boundary-Layer Theory, McGraw-Hill, New York. 

Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the sohd surface is called the boundary layer The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundaiy layer thickness is conventionally t en to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

Schlichting, H. 1969. Boundary layer theory, la Handbook of Fluid Dynamics. New York. Schlunder, IJ. 1971. Ober die ausbreitung turbulenter freistrahlen. Z. Plugwiss, vol. 19, no.. 1, pp. 108-113. 

H. Schlichting. Boundary Layer Theory. 7th ed. New York McGraw-Hill, 1 979. 

BOUNDARY LAYER THEORY APPLIED TO PIPE FLOW 11.5.1. Entry conditions... 

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

Schlichting, H. Boundary Layer Theory (trails, by KESTIN, J.) 6th edn (McGraw-Hill, New York, 1968). White, F. M. Viscous Fluid Flow (McGraw-Hill, New York, 1974). 

The equation should be compared with equation 11.49 obtained using PrandtTs simplified approach to boundary layer theory which also disregards the existence of the buffer layer ... 

Schhcting H, Gersten K (2000) Boundary layer theory, 8th rev and England edn. Springer, Berlin Heidelberg New York... 

Another device that finds frequent use is the stirred cell shown in Fig. 20-54. This device uses a membrane coupon at the bottom of the reservoir with a magnetic stir bar. Stirred cells use low fluid volumes and can be used in screening and R D studies to evaluate membrane types and membrane properties. The velocity profiles have been well defined (Schlichting, Boundary Layer Theory, 6th ed., McGraw-Hill, New York, 1968, pp. 93-99). 

A number of theoretical works have been devoted to the study of the hydrogen-deuterium exchange reaction. Hauffe (25) examined this reaction from the standpoint of the boundary layer theory of chemisorption. Dowden and co-workers (26) undertook a theoretical investigation of the hydrogen-deuterium exchange reaction from the viewpoint of the theory of crystalline fields. 

This boundary-layer theory applies to mass-transfer controlled systems where the membrane permeation rate is independent of pressure, for there is no pressure term in the model. In such cases it has been proposed that, as the concentration at the membrane increases, the solute eventually precipitates on the membrane surface. This layer of precipitated solute is known as the gel-layer, and the theory has thus become known as the gel-polarisation model proposed by Micii i i.si 0). Under such conditions C, in equation 8.15 becomes replaced by a constant Cq the concentration of solute in the gel-layer, and ... 

Sections IX,A-C have been devoted to expressions for m(z), (z), and Kyy based on atmospheric boundary layer theory. Because of the rather complicated dependence of u and K onz, Eq. (9.36) must generally be solved numerically (see, for example, Nieuwstadt and van Ulden, 1978 van Ulden, 1978). However, if they can be found, analytical solutions are advantageous for studying the behavior of the predicted mean concentration. 

The removal of an electron from an acceptor level or a hole from a donor level denotes, as we have seen, not the desorption of the chemisorbed particle but merely its transition from a state of strong to a state of weak bonding with the surface. The neglect of this weak form of chemisorption (i.e., electrically neutral form) which is characteristic of all papers on the boundary-layer theory of adsorption makes it quite impossible to depict the chemisorbed particle in terms of an energy level, i.e., to apply the energy band scheme depicted in Fig. 10 and used in these papers. ... 

When there is no weak bonding at all, one returns within the frame of the boundary-layer theory. In this case, however, the chemisorbed particles do not produce any levels in the crystal energy spectrum. 

Let us now consider another mechanism by which the imperfections affect the adsorptive and catalytic properties of the surface. This is based on their participation in the adsorption process as adsorption centers. The problem of chemisorption on an atomic imperfection has been treated quantum-mechanically by Bonch-Bruevich 98) it was discussed from the viewpoint of the boundary-layer theory by Hauffe 99) and has been investigated recently by Kogan and Sandomirsky 95). 

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