Diffusion through a stagnant fluid

When diffusion occurs through a stationary gas, the situation is slightly different. Imagine a surface exposed to a mixture of gases on which gas A is absorbed and gas B is not. Close to the surface, the concentration of A will fall, thereby setting up a concentration gradient under the influence of which A will diffuse towards the surface. 

If Pick s Law applies wherever there is a concentration gradient, it applies equally to both species. Thus, as A diffuses towards the surface, B will diffuse away. Furthermore, since the total pressure is the same 

As B diffuses away from the surface, bulk flow of B towards the surface must take place so that the total pressure remains uniform and, since there is no net motion of B, these two effects must just balance. 

The bulk flow of B is accompanied by a bulk flow of A in proportion to their respective concentrations in the mixture. 

The total rate of transfer of A is given by the sum of the terms for diffusion and bulk flow, that is 

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Diffusion Through a Stagnant Fluid (Spherical Coordinates). 200... 

For the same driving force (concentration gradient), J is the same for both cases, but different. Since i-y is always less than one, we see that equimolar counter-diffusion (126) is slower than diffusion through a stagnant fluid (127). This can be qualitatively understood as follows. Suppose that to get to class, you need to walk down a corridor that s crowded with other students. If everyone else was standing almost still (stagnant fluid), it would be easier to walk around them than if everyone is walking toward you (counter diffusion). 

Fig. 5. Case 3. Sherwood numbers far the transpart of finite size particles through a stagnant fluid to n spherical cellectnr under the action of diffusion and London forces...

The Validity of the Diffusions] Model for the Rate of Cell Deposition through a Stagnant Fluid... 

In considering the transport of a species from a fluid in turbulent flow toward a solid surface, for example, an electrochemically active species to an electrode, Nemst assumed that the transport was governed by molecular diffusion through a stagnant film of fluid of thickness 6. This model, although having questionable physical relevance, is quite useful for correlating effects such as the influence of chemical reaction on mass transfer. A few simple examples of the use of film theoiy to describe mass transfer in the presence of chemical reaction are considered here. 

I. Operating-line derivation. For the case of solute A diffusing through a stagnant gas and then into a stagnant fluid, an overall material balance on component A in Fig. 10.6-6 for a packed absorption tower is... 

In the film theory description of the mass-transfer process occurring between two fluid phases or between a solid and a fluid phase, the complex mass-transfer phenomenon is substituted by the notion of simple molecular diffusion of the species through a stagnant fluid fUm of thickness <5. The actual concentration profiles of species A being transferred from phase 2 to phase 1 are shown in Figures 3.1.6 (a) and (b) in one phase only for a solid-liquid and a gas-iiquid system, respectively. The concentration of A in the liquid phase at the solid-liquid or the gas-Uquid interface is C. Far away from the interface it is reduced to a low value in the liquid phase. In turbulent flow, the curved profile of species A shown would correspond to the time-averaged value (Bird et al, 1960, 2002). According to the... 

External diffusion of reactants. This step depends on the fluid dynamic characteristics of the system. Reactants must first diffuse from the bulk gaseous phase to the outer surface of the carrier through a stagnant thin film of gas. Molecular diffusion rates in the bulk have the activation energy E1 = 2 to 4 kcal/mol and they vary with Tm. 

The diffusion through the stagnant gas film surrounding the particles as well as the diffusion through the pores can play an important role in limiting the overall rate of reaction [1], In the case of heterogeneous reactions, in which a fluid contacts a solid... 

For a solid-catalyzed reaction to take place, a reactant in the fluid phase must first diffuse through the stagnant boundary layer surrounding the catalyst particle. This mode of transport is described (in one spatial dimension) by the Stefan-Max well equations (see Appendix C for details) ... 

Species A. which is present in dilute concentrations, is diffusing at steady state from the bulk fluid through a stagnant film of B of thickness B to the external surface of the catalyst (Figure El l-I. I). The concentration of A at the external boundary- is and at the external catalyst surface is Ca,. with Cas > Q,. Because the thickness of the "hypothetical stagnant film" next to the surface is small with re.spect to the diameter of the panicle (i.e.. 5 d ). we can neglect curvature and represent the diffusion in rectilinear coordinates as shou-n in Figure El l l. 2. 

The definitions for the mass transfer coefficients can be used to theoretically predict them using the diffiisivity, concentrations, length scales, and fluid flow characteristics, thus rendering the two mass transfer approaches equivalent. This can easily be done in the cases of equimolar counterdiffusion (Maz + A bz = 0) and diffusion of A through a stagnant film (Ab = 0) (Hines and Maddox, 1985, p. 140). Also, the theoretical models of film, penetration, surface renewal, and film penetration have been proposed for the estimation of the mass transfer coefficients at a fluid-fluid interface (Hines and Maddox, 1985, pp. 146-151). 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

When fluid is pumped through a cell such as that shown in Fig. 12, transport of dissolved molecules from the cell inlet to the IRE by convection and diffusion is an important issue. The ATR method probes only the volume just above the IRE, which is well within the stagnant boundary layer where diffusion prevails. Figure 13 shows this situation schematically for a diffusion model and a convection-diffusion model (65). The former model assumes that a stagnant boundary layer exists above the IRE, within which mass transport occurs solely by diffusion and that there are no concentration gradients in the convection flow. A more realistic model of the flow-through cell accounts for both convection and diffusion. As a consequence of the relatively narrow gap between the cell walls, the convection leads to a laminar flow profile and consequently to concentration gradients between the cell walls. 

Another classification involves the number of phases in the reaction system. This classification influences the number and importance of mass and energy transfer processes in the design. Consider a stirred mixture of two liquid reactants A and B, and a catalyst consisting of small particles of a solid added to increase the reaction rate. A mass transfer resistance occurs between the bulk liquid and the surface of the catalyst particles. This is because the small particles tend to move with the liquid. Consequently, there is a layer of stagnant fluid that surrounds each particle. This results in reactants A and B transferring through this layer by diffusion in order to reach the catalyst surface. The diffusion resistance gives a difference in concentration between... 

FIGURE 16 Schematic representation of the origins of zone-broadening behavior and mass transfer effects of a polypeptide or protein due to Brownian motion, eddy diffusion, mobile phase mass transfer, stagnant fluid mass transfer, and stationary-phase interaction transfer as the polypeptide or protein migrated through a column packed with porous particles of an interactive HPLC sorbent. 

In catalytic reactions mass transfer from the fluid phase to the active phase inside the porous catalyst particle takes place via transport through a fictitious stagnant fluid film surrounding the particle and via diffusion inside the particle. Heat transport to or from the catalyst takes the same route. These phenomena are summarized in Fig. 8.15. 

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