Mass transfer coefficients from boundary layer theory

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. 

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. 

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. 

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... 

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,... 

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). 

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.)...

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