Steady state diffusion fluid

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

The constants A and B can be determined from the solutions available for the simple cases in which either = 0 or uw = 0. The first limiting case corresponds to the dissolution of a planar wall in a semiinfinite liquid in steady flow (Blasius flow), while the second limit corresponds to steady-state diffusion with chemical reaction in a stagnant fluid. For Blasius flow at large Schmidt numbers [2],... 

To illustrate the application of the film model for nonideal fluid mixtures we consider steady-state diffusion in the system glycerol(l)-water(2)-acetone(3). This system is partially miscible (see Krishna et al., 1985). Determine the fluxes Ap A2, and A3 in the glycerol-rich phase if the bulk liquid composition is... 

Film model is undoubtedly the most widespread approach for the rate-based mass and heat transfer through an interface (Wesselingh and Krishna, 2000). It effectively combines species diffusion and fluid flows and is based on the assumption that the resistance to mass and heat transfer is exclusively concentrated in a thin film where steady state diffusion and mass and heat convection take place (figure 2.5). 

If the density of the fluid decreases with height, gravity forces do not lead to free convection. This situation typically is found for metal deposition on a horizontal electrode, positioned face down, or for dissolution of an electrode positioned face up. In these cases, the transport of reactants and products is due to non steady state diffusion, and the current therefore never reaches a constant value. Equation (4.116) indicates the variation of the limiting current density as a function of the reaction time t, for a reaction carried out at constant potential and controlled by the diffusion of a species B towards the electrode. 

It is also possible that the reaction is so rapid that it takes place at the interface only. In section 5.4.2.1 this was called an instantaneous reaction. However, eq. (5.43) describing similar reactions in fluid phases is not applicable here, since there is no flow and consequently no "diffusion layer". Only a non-steady state condition is conceivable. When the reaction product remains "dissolved" in the solid phases, the concentration profiles of the reactants in both phases can be described by the well known Fourier equations, that apply for non-steady state diffusion (with constant diffusivity) for given boundary conditions. For relatively low degrees of conversion die rate of conversion of reactant A can be approximated by the penetration theory ... 

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 ... 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

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 Chapman-Enskog method has been used to solve for steady state tracer diffusion (. ). According to the method the singlet distribution function for the diffusing species 1, present In a trace amount n nj, 1 1) In an otherwise equilibrium fluid. Is approximated by... 

Fluid density and component brownian diffusivity D are also assumed constant. A steady-state component mass balance can be written for component concentration c ... 

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... 

In order to design such an efficient and effective device, one must understand the mechanisms by which drug is transported in the ocular interior. One issue debated in the literature for some time has been the relative importance of transport by passive diffusion versus that facilitated by the flow of fluid in the vitreous (see, e.g., Ref. 226). To predict the geometric distribution even at steady state of drug released from an implant or an intravitreal injection, one must appreciate which of these mechanisms is at work or, as appropriate, their relative balance. 

The steady-state continuity equations which describe mass balance over a fluid volume element for the species in the stagnant film which are subject to uniaxial diffusion and reaction in the z direction are... 

As an alternative to film models, McNamara and Amidon [6] included convection, or mass transfer via fluid flow, into the general solid dissolution and reaction modeling scheme. The idea was to recognize that diffusion was not the only process by which mass could be transferred from the solid surface through the boundary layer [7], McNamara and Amidon constructed a set of steady-state convective diffusion continuity equations such as... 

In a fixed-bed catalytic reactor for a fluid-solid reaction, the solid catalyst is present as a bed of relatively small individual particles, randomly oriented and fixed in position. The fluid moves by convective flow through the spaces between the particles. There may also be diffusive flow or transport within the particles, as described in Chapter 8. The relevant kinetics of such reactions are treated in Section 8.5. The fluid may be either a gas or liquid, but we concentrate primarily on catalyzed gas-phase reactions, more common in this situation. We also focus on steady-state operation, thus ignoring any implications of catalyst deactivation with time (Section 8.6). The importance of fixed-bed catalytic reactors can be appreciated from their use in the manufacture of such large-tonnage products as sulfuric acid, ammonia, and methanol (see Figures 1.4,11.5, and 11.6, respectively). 

The rate of diffusion is proper tional to the difference of concentrations between the bulk of the fluid and at the interface. In the steady state, the rates of diffusion and reaction are equal. 

In many situations of practical interest, an appreciable drop in concentration arises between a fluid phase and the external surface of the catalyst because of diffusional resistance. In the steady state, the rate of diffusion to the external surface equals the rate of input to the pore mouth, rd = kga(Cg-Cs) = D(dC/dr)r=R>c=Cs (7.37)... 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

The mechanisms considered above are all composed of steps in which chemical transformation occurs. In many important industrial reactions, chemical rate processes and physical rate processes occur simultaneously. The most important physical rate processes are concerned with heat and mass transfer. The effects of these processes are discussed in detail elsewhere within this book. However, the occurrence of a diffusion process in a reaction mechanism will be mentioned briefly because it can lead to kinetic complexities, particularly when a two-phase system is involved. Consider a reaction scheme in which a reactant A migrates through a non-reacting fluid to reach the interface between two phases. At the interface, where the concentration of A is Caj, species A is consumed in a first-order chemical rate process. In effect, consecutive rate processes are occurring. If a steady state is achieved, then... 

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