Mass boundary layer theory

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 ... 

Analytic solutions for flow around and transfer from rigid and fluid spheres are effectively limited to Re < 1 as discussed in Chapter 3. Phenomena occurring at Reynolds numbers beyond this range are discussed in the present chapter. In the absence of analytic results, sources of information include experimental observations, numerical solutions, and boundary-layer approximations. At intermediate Reynolds numbers when flow is steady and axisym-metric, numerical solutions give more information than can be obtained experimentally. Once flow becomes unsteady, complete calculation of the flow field and of the resistance to heat and mass transfer is no longer feasible. Description is then based primarily on experimental results, with additional information from boundary layer theory. 

Several theories have been developed to describe the rate of interphase mass transfer. These include film theory, boundary layer theory, penetration theory, and surface renewal theory. In this chapter we will review the first two, along with an overview of empirical correlations that are used to describe mass transfer. A more thorough overview of mass transfer theories can be found in Bird, Stewart and Lightfoot [48], Clark [49], Logan [50], and Weber and DiGiano [51]. 

The boundary layer theory is based on the system in Fig. 11, where mass transfer is occurring from a flat plate in the presence of laminar flow. 

Experimental data from various sources (C5, K2, G4, S16) were taken for comparison. Kauh (K2) determined the drying schedules for balsa wood slabs of various thicknesses (, j, f in.) at different wind velocities (100-124 ft/min). It was not possible to apply boundary-layer theory to calculate heat- and mass-transfer coefficients because the length of the slabs was not recorded. 

Garud s (G4) data on the drying of welding electrodes show agreement within 15%, (Fig. 12) although the critical moisture content was not known accurately. Whenever data were not sufficient to calculate heat-and mass-transfer coefficients by boundary-layer theory, initial drying rate data was used for the purpose. 

Heat transfer between a solid wall and a fluid, e.g. in a heated tube with a cold gas flowing inside it, is of special technical interest. The fluid layer close to the wall has the greatest effect on the amount of heat transferred. It is known as the boundary layer and boundary layer theory founded by L. Prandtl2 in 1904 is the area of fluid dynamics that is most important for heat and mass transfer. In the boundary layer the velocity component parallel to the wall changes, over a small distance, from zero at the wall to almost the maximum value occurring in the core fluid, Fig. 1.6. The temperature in the boundary layer also changes from that at the wall w to at some distance from the wall. 

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... 

Equations (1.198) and (1.199) are also known as Lewis equations. The mass transfer coefficients f3m calculated using this equation are only valid, according to the definition, for insignificant convective currents, fn the event of convection being important they must be corrected. The correction factors C, = /3 n/l3m for transverse flow over a plate, under the boundary layer theory assumptions are shown in Fig. 1.50. They are larger than those in film theory for a convective flow out of the phase, but smaller for a convective flow into the phase. 

The film and boundary layer theories presuppose steady transport, and can therefore not be used in situations where material collects in a volume element, thus leading to a change in the concentration with time. In many mass transfer apparatus fluids come into contact with each other or with a solid material for such a short period of time that a steady state cannot be reached. When air bubbles, for example, rise in water, the water will only evaporate into the bubbles where it is contact with them. The contact time with water which surrounds the bubble is roughly the same as that required for the bubble to move one diameter further. Therefore at a certain position mass is transferred momentarily. The penetration theory was developed by Higbie in 1935 [1.31] for the scenario described here of momentary mass transfer. He showed that the mass transfer coefficient is inversely proportional to the square root of the contact (residence) time and is given by... 

An apparent weakness of the film model is that it suggests that the mass transfer coefficient is directly proportional to the diffusion coefficient raised to the first power. This result is in conflict with most experimental data, as well as with more elaborate models of mass transfer [surface renewal theory considered in the next chapter, e.g., or boundary layer theory (Bird et al., I960)]. However, if we substitute the film theory expression for the mass transfer coefficient (Eq. 8.2.12) into Eq. 8.8.1 for the Sherwood number we find... 

Atmospheric turbulence near the earth s surface is generally much higher (Table I) than found in most wind tunnels (up to about 2%). Unfortunately, very few heat or mass transfer tests have been performed under natural outdoor conditions. Surface roughnesses of practical structures of interest may also deviate from laboratory conditions, although boundary layer theory may be used to compute critical roughness sizes and maximum permissible roughnesses, below... 

But before the virtues of the results and the approach are extolled, the method must be described in detail. Let us therefore return to a systematic development of the ideas necessary to solve transport (heat or mass transfer) problems (and ultimately also fluid flow problems) in the strong-convection limit. To do this, we begin again with the already-familiar problem of heat transfer from a solid sphere in a uniform streaming flow at sufficiently low Reynolds number that the velocity field in the domain of interest can be approximated adequately by Stokes solution of the creeping-flow problem. In the present case we consider the limit Pe I. The resulting analysis will introduce us to the main ideas of thermal (or mass transfer) boundary-layer theory. 

Chapter 2 includes a section on boundary-layer theory and an example on simultaneous mass and heat transfer during air humidification. 

Al) 1.075(Ape)c " <1 — Liquid-solid mass transfer Boundary layer theory and empiricism... 

Hence, the local mass transfer coefficient scales as the two-thirds power of a, mix for boundary layer theory adjacent to a solid-liquid interface, and the one-half power of A, mix for boundary layer theory adjacent to a gas-liquid interface, as well as unsteady state penetration theory without convective transport. By analogy, the local heat transfer coefficient follows the same scaling laws if one replaces a, mix in the previous equation by the thermal conductivity. 

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