Constant flux diffusion problem

Fig. 4.22. Graphical representation of the variation of the concentration c with distance x from the electrode or diffusion sink, (a) The initial condition at f = 0 (b) and (c) the conditions at f, and fg. where tg > f, > f= 0. Note that, at f > 0, (dc/dx)j is a constant, as it should be in the constant-flux diffusion problem.

How the Solution of the Constant-Flux Diffusion Problem Leads to the Solution of Other Problems... 

One approach to this question is to set up the new diffusion problem with the initial and boundary conditions characteristic of the sinusoidally varying flux and to obtain a solution. There is, however, a simpler approach. Using the property of Laplace transforms, one can use the solution (4.65) of the constant-flux diffusion problem to generate solutions for other problems. 

These boundary conditions correspond to the constant surface-flux diffusion problem in Section 5.2.5, in which the surface concentration increased proportionally to t1/2. Therefore, adapting the solution given by Eq. 5.69 to the grainboundary grooving model,... 

The three conditions just listed describe the special features of the constant (unit)-flux diffusion problem. They will now be used to solve Pick s second law. 

In connection with the interphase mass transfer in liquid-liquid and liquid-gas systems, the diffusion equations (and indeed all the equations of change) are valid in both phases. Hence, in principle, diffusion problems in a two-phase system may be solved by solving the diffusion equations in each phase and then choosing the constants of interaction in such a way that the solutions match up at the interface. It is customary to require that the following two conditions be fulfilled at the interface, in a system in which the solute is being transferred from phase I to phase II (1) the flux of mass leaving phase I must equal the flux of mass entering phase II if the diffusion... 

To link the constant-flux problem (of Section 4.2.12) to the constant-current problem discussed here, one can assume that the constant flux arises only from the imposed constant current i. Thus, one considers that the boundary of the diffusion problem is the electrified interface x = 0 at which there is equality of the charge transfer (from electrode to ion) and diffusion fluxes (from solution to electrode), i.e.,... 

Instead of attempting a general discussion of the three conditions characterizing a particular diffusion problem, it is best to treat a typical electrochemical diffusion problem. Consider that in an electrochemical system a constant current is switched on at a time arbitrarily designated t=0 (Fig. 4.17). The current is due to charge-transfer reactions at the electrode-solution interfaces, and these reactions consume a species. Since the concentration of this species at the interface falls below the bulk concentration, a concentration gradient for the species is set up and it diffuses toward the interface. Thus, the externally controlled current sets up a diffusion flux within the solution. 

What is characteristic of one particular electrochemical diffusion process and distinguishes it from all others is the nature of the diffusion flux that is started off at t = 0. Thus, the essential characteristic of the diffusion problem under discussion is the switching on of the constant current, which means that the diffusing species is consumed at a constant rate at the interface and the species diffuses across the interface at a constant rate. In other words, the flux of the diffusing species at the x = 0 boundary of the solution is a constant. 

There is another important diffusion problem, the solution of which can be generated from the concentration response to a constant current (or a flux). Consider that in an eleetroehemieal system there is a plane electrode at the boundary of the eleetrolyte. Now, suppose that with the aid of an electronic pulse generator, an extremely short time eurrent pulse is sent through the system (Fig. 4.28). The current is direeted so as to dissolve the metal of the eleetrode hence, the effect of the pulse is to produee a burst of metal dissolution in whieh a layer of metal ions is piled up at the interface (Fig. 4.29). 

We apply simple effective medium models in an attempt to understand the diffusion process in the complex pore network of a porous SiC sample. There is an analogy between the quantities involved in the electrostatics problem and the steady state diffusion problem for a uniform external diffusion flux impinging on a coated sphere. Kalnin etal. [17] provide the details of such a calculation for the Maxwell Garnett (MG) model [18]. The quantity involved in the averaging is the product of the diffusion constant and the porosity for each component of the composite medium. The effective medium approach does not take into account possible effects due to charges on the molecules and/or pore surfaces, details in the size and shape of the protein molecules, fouling (shown to be negligible in porous SiC), and potentially important features of the microstructure such as bottlenecks. 

At that time chemists became more and more interested in process where the electron transfer at the electrode surface is associated with some sparingly soluble compounds The application of polarography (voltammetry) to the study of suspensions was such case (see also the Sect. 5.5.1). In case of a dropping mercury electrode, two processes happen (1) The attachment of solid particles to the mercury surface due to capillary forces, and (2) the cathodic reduction of these particles. In case of solid particles of ethylen-bis-dithiocarbamate of zinc, a peak is observed at —0.7 V, both in DCP and in ACP. However, in polarography of suspensions, one faces problems the stabilization of particles in the suspension plays an important role, and the signal of the solid particles overlaps with the wave of dissolved compound, and the constant flux of particles to the surface has to be maintained. Unlike the flux of ions, which is controlled by diffusion, the flux of particles is controlled by the stirring of the solution, which has to be very well controlled to achieve reproducible results (Lyalikov, Sister) [208]. 

The diffusion equation is the partial-differential equation that governs the evolution of the concentration field produced by a given flux. With appropriate boundary and initial conditions, the solution to this equation gives the time- and spatial-dependence of the concentration. In this chapter we examine various forms assumed by the diffusion equation when Fick s law is obeyed for the flux. Cases where the diffusivity is constant, a function of concentration, a function of time, or a function of direction are included. In Chapter 5 we discuss mathematical methods of obtaining solutions to the diffusion equation for various boundary-value problems. 

In equation 13, C1 and Cs are the total concentrations in the liquid and solid phases, respectively. This statement of the problem assumes that the convective flux due to the moving boundary (growing surface) is small, the diffusion coefficients are mutual and independent of concentration, the area of the substrate is equal to the area of the solution, the liquid density is constant, and no transport occurs in the solid phase. Further, the conservation equations are uncoupled from the equations for the conservation of energy and momentum. Mass flows resulting from other forces (e.g., thermal diffusion and Marangoni or slider-motion-induced convective flow) are neglected. 

Law Simplified flux equations that arise from Eqs. (5-189) and (5-190) can be used for unidimensional, steady-state problems with binary mixtures. The boundary conditions represent the compositions xAl and xA(l at the left-hand and right-hand sides of a hypothetical layer having thickness Az. The principal restriction of the following equations is that the concentration and diffusivity are assumed to be constant. As written, the flux is positive from left to right, as depicted in Fig. 5-25. 

The analysis can be significantly simplified by reahzing that the rate with which the vorticity diffuses inwards, and hence establishes the fluid motion, is represented by the kinematic viscosity coeflBcient, which is of the order of 10 cmVsec and is at least one order of magnitude greater than the droplet surface regression rate. Hence quasi-steadiness for both the gas and liquid motion, with a stationary droplet surface and constant interfacial heat and mass flux, can be assumed. Once the fluid mechanical aspect of the problem is solved, the transient liquid-phase heat and mass transfer analyses, with a regressing droplet surface, can be performed. 

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