STEADY-STATE MOLECULAR DIFFUSION IN FLUIDS

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

In the two-film model (Figure 13), it is assumed that all of the resistance to mass transfer is concentrated in thin stagnant films adjacent to the phase interface and that transfer occurs within these films by steady-state molecular diffusion alone. Outside the films, in the bulk fluid phases, the level of mixing is so high that there is no composition gradient at all. This means that in the film region, only one-dimensional diffusion transport normal to the interface takes place. 

Single Statens. As developed in Section 2.3, steady-state molecular diffusion to a single sphere in a stagnant fluid provides the asymptotic limit of... 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

The modeling of mass transport from the bulk fluid to the interface in capillary flow typically applies an empirical mass transfer coefficient approach. The mass transfer coefficient is defined in terms of the flux and driving force J = kc(cbuik-c). For non-reactive steady state laminar flow in a square conduit with constant molecular diffusion D, the mass balance in the fluid takes the form... 

The mechanisms of mass transport can be divided into convective and molecular flow processes. Convective flow is either forced flow, for example, in pipes and packed beds, or natural convection induced by temperature differences in a fluid. For diffusive flow we have to distinguish whether we have molecular diffusion in a free fluid phase or a more complicated effective diffusion in porous solids. Like heat transport, diffusion may be steady-state or transient. 

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

The transfer across these films is regarded as a steady-state process of molecular diffusion. The turbulence in the bulk fluid is considered to die out at the interface of the films. Although it does not closely reproduce the conditions in most practical equipment, it gives expressions which can be applied to the experimental data which are generally available, and for this reason it is still extensively used. 

This model, developed by Whitman [16] in 1923, is still used. The assumption is that adjacent to the interface there exists a thin film that is laminar in character and through which transfer is by molecular diffusion only. In this film, the entire concentration driving force exists, i.e., outside the film, in the bulk fluid, the concentration is constant at some steady state. Flow outside the film may be laminar or turbulent, but inside the film, flow conditions are completely laminar. Figure... 

The feed is assumed to contain a low-molecular-weight solute A, a solute of intermediate size B, and a colloid C. There are concentration boundary layers on both sides of the membrane, and these may contribute significantly to the overall resistance if the membrane is thinner than the boundary layers. The gradient for A or B in the membrane is steeper than in the boundary layers, because the effective diffusivity is less than the bulk value, and at steady state, the flux through the membrane equals that through the boundary layers. The values of Ca and Cg in the membrane are the concentrations in the pore fluid and not... 

Transport of solute from a fluid phase to a spherical or nearly spherical shape is important in a vari of separation operations such as liquid-liquid extraction, crystallization from solution, and ion exchange. The situation depicted in Fig. 2.3-12 assumes that there is no forced or natural convection in the fluid about the particle so that transport is governed entirely by molecular diffusion. A steady-state solution can be obtained for the case of a sphere of fixed radius with a constant concentration at the interface as well as in the bulk fluid. Such a model will be useful for crystallization from vaqxtrs and dilute solutions (slow-moving boundary) or for ion exchange with rapid irreversible reaction. Bankoff has reviewed moving-boundary problems and Chapters 11 and 12 deal with adsorption and ion exchange. 

---