Transient mass diffusion

For a number of flow situations, the mass-transfer rate can be derived directly from the equation of convective diffusion (see Table VII, Part A). The velocity profile near the electrode is known, and the equation is reduced to a simpler form by appropriate similarity transformations (N6). These well-defined flows, therefore, are being exploited increasingly by electrochemists as tools for the kinetic characterization of electrode reactions. Current distributions at, or below, the limiting current, transient mass transfer, and other aspects of these flows are amenable to analysis. Especially noteworthy are the systematic investigations conducted by Newman (review until 1973 in N7 also N9b, N9c, H6b and references in Table VII), by Daguenet and other French workers (references in Table VII), and by Matsuda (M4a-d). Here we only want to comment on the nature of the velocity profile near the electrode, and on the agreement between theory and mass-transfer experiment. 

Much work has been focused on the significance of dispersion terms in the transient material and energy equations for packed bed reactors. In general, axial diffusion of mass and energy and radial diffusion of mass have been... 

Figure 14.10 Shape of a mass variation-time curve in the case of osmotic cracking (a) end of the diffusion transient, (b) onset of cracking, and (c) percolation threshold of the crack network. 

In addition to the Navier-Stokes equations, the convective diffusion or mass balance equations need to be considered. Filtration is included in the simulation by preventing convection or diffusion of the retained species. The porosity of the membrane is assumed to decrease exponentially with time as a result of fouling. Wai and Fumeaux [1990] modeled the filtration of a 0.2 pm membrane with a central transverse filtrate outlet across the membrane support. They performed transient calculations to predict the flux reduction as a function of time due to fouling. Different membrane or membrane reactor designs can be evaluated by CFD with an ever decreasing amount of computational time. 

Transient Mass Diffusion 796 14-8 Diffusion in a Moving Medium... 

B Perform a transient mass diffusion analysis in large mediums,... 

The surface hardening of a mild steel component by the diffusion of carbon molecules is a transient mass diffusion process. 

Transient mass difliision in a stationary medium is analogous to transient heat transfer provided that the solution is dilute and thus the density of the medium p is constant. In Chapter 4 we presented analytical and graphical solutions for one-dimensional transient heat conduction problems in solids with constant properties, no heat generation, and uniform initial temperature. The analogous one-dimensional transient mass diffusion problems satisfy these requirements ... 

Analogy between the quantities that appear in the formulation and solution of transient heat conduction and transient mass diffusion in a stationary medium... 

FIGURE 14-27 The concentration profile of species A in a semi-inlinite medium during transient mass diffusion and the penetration depth. 

The diffusion coefficients in solids are typically very low (on the order of 10 to 10" mVs), and thus the diffusion process usually affects a thin layer at the surface. A solid can conveniently be treated as a semi-infinite medium during transient mass diffusion regardless of its size and shape when the penetration depth is small relative to the thickness of the solid. When this is not the case, solutions for one dimensional transient mass diffusion through a plane wall, cylinder, and sphere can be obtained from the solution.s of analogous heat conduction problems using the Heisler charts or one term solutions pieseiited in Chapter 4. 

C Consider one-dimeusional mass diffusion of species A through a plane wall. Does Ihe species A coment of the wall change during steady mass diffusion How about during transient mass diffusion ... 

C In transient mass dilliision analysis, can we treat the diffusion of a solid into another solid of finite thickness (such as the diffusion of carbon into an ordinary steel component) as a diffusion process in a semi-infinite medium Explain. 

C. Deslouis, B. Tribollet, M. Duprat, and F. Moran, "Transient Mass Transfer at a Coated Rotating Disk Electrode Diffusion and Electrohydrodynamical Impedances," Journal of The Electrochemical Society, 134 (1987) 2496-2501. 

The basis for the solution of mass diffusion problems, which go beyond the simple case of steady-state and one-dimensional diffusion, sections 1.4.1 and 1.4.2, is the differential equation for the concentration held in a quiescent medium. It is known as the mass diffusion equation. As mass diffusion means the movement of particles, a quiescent medium may only be presumed for special cases which we will discuss first in the following sections. In a similar way to the heat conduction in section 2.1, we will discuss the derivation of the mass diffusion equation in general terms in which the concentration dependence of the material properties and chemical reactions will be considered. This will show that a large number of mass diffusion problems can be described by differential equations and boundary conditions, just like in heat conduction. Therefore, we do not need to solve many new mass diffusion problems, we can merely transfer the results from heat conduction to the analogue mass diffusion problem. This means that mass diffusion problem solutions can be illustrated in a short section. At the end of the section a more detailed discussion of steady-state and transient mass diffusion with chemical reactions is included. 

In section 2.5.3 it was shown that the differential equation for transient mass diffusion is of the same type as the heat conduction equation, a result of which is that many mass diffusion problems can be traced back to the corresponding heat conduction problem. We wish to discuss this in detail for transient diffusion in a semi-infinite solid and in the simple bodies like plates, cylinders and spheres. 

Transient mass diffusion in a semi-infinite solid... 

Transient mass diffusion in bodies of simple geometry with one-dimensional mass flow... 

It was shown in [349] that at high Peclet numbers (in the diffusion boundary layer approximation), by solving the corresponding nonlinear problem on transient mass exchange between drops or bubbles and the flow, one obtains the following expression for the mean Sherwood number ... 

The transient mass transport in spheres plays an important role in many process engineering appheations. At adsorption the adsorptive moves through porosities and is aeeumulated on the internal surface of the spherical adsoibent. At regeneration of such adsorbents, as well as at drying of capillary active solids the opposite process occurs. Mass transport caused by transient diffusion is also existent in fluid particles, as long as no convection occurs in the particle. This applies for small viscous droplets. 

FIGURE 10.10 Evolution of a transient mass transfer in a restricted space from the early stage (thin-layer behavior due to the small diffusion layer) to the restricted diffusion due to the finite space. 

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... 

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

In Fig. 20 we show the MSQ of a system of GM [66] with different mean chain lengths (depending on 7, cf. Eq. (12)) for three values of LO=l, 0.1, 0. 01. Since the individual chains have only transient identity, it is meaningless to discuss their center of mass diffusion. It is evident from Fig. 20 that the MSQ of the segments, g t) = ([x( ) - x(O)j ), follows an intermediate sub-diffusive regime, g(t) oc which is later replaced by conventional diffusion at some characteristic crossover time which grows... 

The transient response of DMFC is inherently slower and consequently the performance is worse than that of the hydrogen fuel cell, since the electrochemical oxidation kinetics of methanol are inherently slower due to intermediates formed during methanol oxidation [3]. Since the methanol solution should penetrate a diffusion layer toward the anode catalyst layer for oxidation, it is inevitable for the DMFC to experience the hi mass transport resistance. The carbon dioxide produced as the result of the oxidation reaction of methanol could also partly block the narrow flow path to be more difScult for the methanol to diflhise toward the catalyst. All these resistances and limitations can alter the cell characteristics and the power output when the cell is operated under variable load conditions. Especially when the DMFC stack is considered, the fluid dynamics inside the fuel cell stack is more complicated and so the transient stack performance could be more dependent of the variable load conditions. 

Transient cavitation is generally due to gaseous or vapor filled cavities, which are believed to be produced at ultrasonic intensity greater than 10 W/cm2. Transient cavitation involves larger variation in the bubble sizes (maximum size reached by the cavity is few hundred times the initial size) over a time scale of few acoustic cycles. The life time of transient bubble is too small for any mass to flow by diffusion of the gas into or out of the bubble however evaporation and condensation of liquid within the cavity can take place freely. Hence, as there is no gas to act as cushion, the collapse is violent. Bubble dynamics analysis can be easily used to understand whether transient cavitation can occur for a particular set of operating conditions. A typical bubble dynamics profile for the case of transient cavitation has been given in Fig. 2.2. By assuming adiabatic collapse of bubble, the maximum temperature and pressure reached after the collapse can be estimated as follows [2]. 

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