Mass transfer coefficient boundary layer theory

Ka can be defined as a gas-phase transfer coefficient, independent of the liquid layer, when the boundary concentration of the gas is fixed and independent of the average gas-phase concentration. In this case, the average and local gas-phase mass-transfer coefficients for such gases as sulfur dioxide, nitrogen dioxide, and ozone can be estimated from theoretical and experimental data for deposition of diffusion-range particles. This is done by extending the theory of particle diffusion in a boundary layer to the case in which the dimensionless Schmidt number, v/D, approaches 1 v is the kinematic viscosity of the gas, and D is the molecular diffusivity of the pollutant). Bell s results in a tubular bifurcation model predict that the transfer coefficient depends directly on the... 

From the viewpoint of the chemical engineer, the application of the fundamentals just discussed takes the form of mass-transfer theory, and the constants Kj are called mass-transfer coefficients. However, some care must be exercised in determining the potential difference between two phases. One cannot simply take the difference between the thermodynamic activities in two phases without ensuring that they are adjusted to the same reference state. The point is illustrated with a brief consideration of the boundary-layer model. 

For sufficiently large electrodes with a small vibration amplitude, aid < 1, a solution of the hydrodynamic problem is possible [58, 59]. As well as the periodic flow pattern, a steady secondary flow is induced as a consequence of the interaction of viscous and inertial effects in the boundary layer [13] as shown in Fig. 10.10. It is this flow which causes the enhancement of mass-transfer. The theory developed by Schlichting [13] and Jameson [58] applies when the time of oscillation, w l is small in comparison with the time taken for a species to diffuse across the hydrodynamic boundary layer (thickness SH= (v/a>)ln diffusion timescale 8h/D), i.e., when v/D t> 1. Re needs to be sufficiently high for the calculation to converge but sufficiently low such that the flow does not become turbulent. Experiment shows that, for large diameter wires (radius, r, — 1 cm), the condition is Re 2000. The solution Sh = 0.746Re1/2 Sc1/3(a/r)1/6, where Sh (the Sherwood number) = kmr/D and km is the mass-transfer coefficient,... 

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. 

The calculation of the mass transfer coefficient can be carried out in different ways. Therefore a decision has to be made as to the type of problem, and which mass transfer theory is applicable to the solution of this problem, and therefore the determination of the mass transfer coefficient. The most important are the him, boundary layer and penetration theories. The essentials of these three theories will be introduced here. 

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... 

Many numerical and series solutions for the laminar boundary layer model of mass transfer are available for situations such as flow in coeduits under conditions of fully developed or developing concentration or velocity profiles. Skellaed31 provides a particularly good summary of these results. The laminar boundary layer model has been extended to predict tha effects of high mass transfer flux on the mass transfer coefficient from a flat plate. The results of this work ate shown in Fig. 2.4-2 and. in com rest to the other theories, iedicate a Schmith number dependence of Ihe correction factor. 

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. 

Now, it is necessary to discuss the mass transfer coefficient for component j in the boundary layer on the vapor side of the gas-liquid interface, fc ,gas, with units of mol/(area-time). The final expression for gas is based on results from the steady-state film theory of interphase mass transfer across a flat interface. The only mass transfer mechanism accounted for in this extremely simple derivation is one-dimensional diffusion perpendicular to the gas-liquid interface. There is essentially no chemical reaction in the gas-phase boundary layer, and convection normal to the interface is neglected. This problem corresponds to a Sherwood number (i.e., Sh) of 1 or 2, depending on characteristic length scale that is used to define Sh. Remember that the Sherwood number is a dimensionless mass transfer coefficient for interphase transport. In other words, Sh is a ratio of the actual mass transfer coefficient divided by the simplest mass transfer coefficient when the only important mass transfer mechanism is one-dimensional diffusion normal to the interface. For each component j in the gas mixture. 

In the previous sections, stagnant films were assumed to exist on each side of the interface, and the normal mass transfer coefficients were assumed proportional to the first power of the molecular diffusivity. In many mass transfer operations, the rate of transfer varies with only a fractional power of the diffusivity because of flow in the boundary layer or because of the short lifetime of surface elements. The penetration theory is a model for short contact times that has often been applied to mass transfer from bubbles, drops, or moving liquid films. The equations for unsteady-state diffusion show that the concentration profile near a newly created interface becomes less steep with time, and the average coefficient varies with the square root of (D/t) [4] ... 

This is a regime in which the diffusion coefficients of A and B in the liquid are the controlling parameters, and chemical reaction plays practically no part. Thus it has frequently been used to compare various theories of mass transfer to and from solid surfaces. The main conclusion is that the value of the exponent p in D /Dpy is different for different theories. The value of n for the boundary layer theory is 2/3. Recalling the values for the film and penetration theories,... 

Angelo [7] has shown that during periods of continual surface renewal, the actual mass transfer coefficient may be fifteen times as large as that predicted by boundary layer theory. Thus, the unsteady state absorption during surface renewal is a more complex situation not covered by these theories. In order to describe a fog or mist formation>it is necessary to study droplet growth by condensation with no Internal turbulence. Bogaevskii [2] reported water droplets growing by water vapor condensation in a mine shaft to absorb about six times more sulfur dioxide than that predicted by steady state absorption. 

Mass transfer coefficient (fe) A measure of the solute s mobility due to forced or natural convection in the system. Analogous to a heat transfer coefficient, it is measured as the ratio of the mass flux to the driving force. In membrane processes the driving force is the difference in solute concentration at the membrane surface and at some arbitrarily defined point in the bulk fluid. When lasing the film theory to model mass transfer, k is also defined as D/S, where D is solute diffusivity and d is the thickness of the concentration boundary layer. 

Sherwood number (Sh) A dimensionless measure of the ratio of convective mass transfer to molecular mass transfer. If the mass transfer coefficient k is defined in terms of the film theory, then Sh is a measure of the ratio of hydraulic diameter to the thickness of the boundary layer. See Section 6.5. 

Mass transfer coefficients are the basis for models where the dissolved species are transported by a combination of diffusive and advective processes. The diffusive mass transfer coefficient ko, m/sec) is based on boundary layer theory. The basic premise of boundary layer theory is that, for laminar ffow, the ffuid velocity adjacent to a solid surface is zero (the no slip condition ) and the velocity increases as a parabolic function of distance away from the surface until it matches the velocity of the bulk fluid (Figure 7.5). This means that there is a thin layer of fluid with a thickness of 5d (m) adjacent to the surface that is effectively static. The rate of mass transport through this layer is limited by the diffusion rate of the dissolved species. The diffusional boundary layer is much thinner than the velocity boundary layer. For laminar flow past a flat surface, the thickness of the diffusional boundary layer is related to the thickness of the velocity boundary layer (Sy) by the Schmidt number, which compares the fluid viscosity to the diffusivity (Probstein, 1989). 

Boundary-layer theory. The boundary-layer theory has been discussed in detail in Section 7.9 and is useful in predicting and correlating data for fluids flowing past solid surfaces. For laminar flow and turbulent flow the mass-transfer coefficient fc oc D g. This has been experimentally verified for many cases. 

High mass transfer rates will influence not only the mass transfer coefficient but also the heat transfer coefficients and friction factor. Analysis of film theory penetration theory and boundary layer theory (21) show that the relation of the various coefficients at high (k ) and low mass transfer (kj ) can be given by 0 s ... 

Fig. 9.5-2. Correction factors for rapid mass transfer. This figure gives the mass transfer coefficient A as a function of the interfacial convection v . In dilute solution, is small and k approaches the slow mass transfer limit k°. In concentrated solution, k may reach a new value, although estimates of this value from different theories are about the same. (The boundary layer theory shown is for a Schmidt number of 1,000.)...

The eddy diffusion coefficients that we introduced in Chapter 5 were steady quantities, using mean turbulence quantities (e.g., the temporal mean of u C). This temporal mean character of eddy diffusion coefficients can be misleading in determining the thickness of a diffusive boundary layer because of the importance of unsteady characteristics. We will review some conceptual theories of mass transfer that have been put forward to describe the interaction of the diffusive boundary layer and turbulence. 

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