Transient diffusion concentration-dependent

In Example 10.1 the case where the aerosol concentration does not change with time was considered. In many practical situations, however, the aerosol concentration does change with time, possibly as a result of diffusion and subsequent loss of particles to a wall or other surface. In this event, Fick s second law, Eq. 9.2, must be used. Solution of this equation is possible in many cases, depending on the initial and boundary conditions chosen, although the solutions generally take on very complex forms and the actual mechanics involved to find these solutions can be quite tedious. Fortunately, there are several excellent books available which contain large numbers of solutions to the transient diffusion equation (Barrer, 1941 Jost, 1952). Thus, in most cases it is possible to fit initial and boundary conditions of an aerosol problem to one of the published solutions. Several commonly occurring examples follow. 

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. 

Concentration-dependent activity coefficients can be accommodated with relative ease by an added term (e.g., [see Helfferich, 1962a Brooke and Rees, 1968] and variations in diffusivities are easily included in numerical calculations (Helfferich and Petruzzelli, 1985 Hwang and Helfferich, 1986). In both instances, however, a fair amount of additional experimental information is required to establish the dependence on composition. Electro-osmotic solvent transfer and particle-size variations are more difficult to deal with, and no readily manageable models have been developed to date. A subtle difficulty here is that, as a rule, there is not only a variation in equilibrium solvent content with conversion to another ionic form, but that the transient local solvent content is a result of dynamics (electro-osmosis) and so not accessible by thermodynamic considerations (Helfferich, 1962b). Theories based on the Stefan-Maxwell equations or other forms of (hcrniodyiiainics of ir-... 

Figure 54 illustrates the way by which Eq. 16 allows the determination of the complete concentration dependence of the diffusivity from a single transient sorption profile [98]. 

Very often, the transport of the liquid through the rubber is controlled by diffusion with either a constant or concentration-dependent diffusivity. This transport can be performed under stationary conditions when the concentration of the liquid varies with position, and under transient conditions when the liquid concentration varies with position and time. 

In Chapter 7, in-depth studies are made on the resistance of rubbers to liquids, which is expressed by the amount of the liquid absorbed or, much better, by the kinetics of absorption. The process of liquid absorption in the rubber material, as well as of that of desorption of the liquid previously absorbed, are studied. These processes are driven by transient diffusion, which is examined in terms of rubber. As rubber may absorb a large quantity of liquid, a swelling takes place thus, the diffu-sivity is not only concentration-dependent, but also the boundary is moving with the amount of the liquid absorbed. On the other hand, it will be seen that the processes of... 

Refinements of the above volume diffusion concept have been made by a model that includes a contribution of surface-diffusion processes to the dissolution reaction of the more active component at subcritical potentials. By adjustment of different parameters, this model allows for the calculation of current-time transients and concentration-depth profiles of the alloy components [102]. In addition to this, mixed control of the dissolution rate of the more active component by both charge transfer and volume diffusion has been discussed. This case is particularly interesting for short polarization times. The analysis yields, for example, the concentration-depth profile and the surface concentration of the more noble component, c, in dependency on the product ky/(t/D), where is a kinetic factor, t is the polarization time, and D is the interdiffusion coefficient. Moreover, it predicts the occurrence of different time domains in the dissolution current transients [109]. 

To operate the diffusion cell under transient condition, the concentration of one of the solutes is perturbed in one chamber and its concentration in the other chamber is monitored. The time-variation of that concentration will depend on the interplay of various processes occurring inside the pellet. Those processes responsible to the flow through the pellet are reflected in the response, while those processes occurring inside the pellet but not directly contributing to the through flux will be reflected in the response as a secondary level. This will be clear later when we deal with the analysis of the transient diffusion cell. 

The transient operation of the diffusion cell depends on the shape of this input versus time. Usually the following three inputs in concentration are normally used ... 

Equation 4.15 is a second-order partial differential equation. When treating diffusion phenomena with Pick s second law, the typical aim is to solve this equation to yield solutions for the concentration profile of species i as a function of time and space [Ci(x,f)]- By plotting these solutions at a series of times, one can then watch how a diffusion process progresses with time. Solution of Pick s second law requires the specification of a number of boundary and initial conditions. The complexity of the solutions depends on these boundary and initial conditions. Por very complex transient diffusion problems, numerical solution methods based on finite difference/finite element methods and/or Fourier transform methods are commonly implemented. The subsections that follow provide a number of examples of solutions to Pick s second law starting with an extremely simple example and progressing to increasingly more complex situations. The homework exercises provide further opportunities to apply Pick s Second Law to several interesting real world examples. 

This solution consists of two pieces, a position-dependent piece, given by the preexponential term, and a time-dependent piece, given by the exponential. The fact that the position dependence and the time dependence can be separated from one another embodies the concept of self-similarity. This concept came up previously in our discussion of transient finite diffusion in a thin plate (Equation 4.42). Self-similarity is a common and important property of many transient diffusion problems. Self-similarity means that the concenfiation at each point in space along the profile evolves with time in precisely the same way. For the example discussed here, this means that everywhere inside the sphere the concentration of hydrogen increases exponentially in time at a —Dt( — V... 

Thus, the shortest wavelength surface roughness features decay most rapidly while the longer wavelength roughness features decay more slowly. This feature-dependent decay speed is analogous to other transient diffusion processes (recall, e.g., the discussion of transient diffusion in Chapter 4, where we saw that the sharpest features in a concentration profile decay most rapidly). 

Were such an effusion to resolve by diffusion, the process could lead to an increase in the intrasynovial concentration of urate. This would occur because diffusion is dependent on molecular size. Since water is much smaller than urate, it diffuses faster. As water leaves the joint, urate will lag behind and its concentration will transiently rise. 

The measurement of induction times (see second of Eqs. 7.31) or the measurements of Ihe temporal change of concentration profiles for transient diffusion, i.e. diffusion before reaching the steady-state condition or with time-dependent concentration boundary conditions, only provide values for experimental quantities in which both De and R are included. Usually, only the so-called apparent or retarded diffusion coefficient Da = DJR is determined. Due to the t)q)ical ranges of values for 0, /"and R the values for the three diffusion coefficients De, De and Da differ by up to 2 orders of magnitude. Since the terminology is sometimes ambiguous, literature data about diffusion coefficients must be scrutinised very carefully to see which coefficient has actually been determined or used. 

Slow relaxation processes appear to be the dominant factors causing the long times required to reach steady state permeation rates. Transient permeation experiments would yield incorrect diffusion coefficients for membrane materials exhibiting this behavior. Relaxation processes, highly concentration dependent diffusion coefficients and solubility coefficients, therefore, require a more detailed approach to studying transport. This paper describes a preferred method of analyzing diffusion process. 

AES — Auger Electron Spectroscopy DLTS - Deep Level Transient Spectroscopy SEM - Scanning Electron Microscopy SIMS - Secondary Ion Mass Spectrometry D(c) - Concentration Dependent Diffusion Coefficient Maximum Diffusion Coefficient (f) - East Diffusion Component (i) - Interstitial Diffusion Component (s) - Slow Diffusion Component (II) - Parallel to c Direction (JL) - Perpendicular to c Direction... 

Under transient conditions the concentration distribution depends not only on the coordinate but also on time. The relevant functions can be found by considering the linear diffusion occurring along the x-axis in a volume element (iU bounded by the two planes S which are a distance dx apart (Fig. 11.1) it is obvious that dV=S dx. The rate of concentration change dcj/dt in this volume is given by the ratio of -S dJj (the... 

---