Transient diffusion profile

Notes (a) DT2 determined from transient diffusion profiles by Eq. (25) ... 

FIGURE 4.10 (a) The transient infinite diffusion of a rectangular eoneentration profile of thickness 21 can be obtained by subtracting the solutions for two semi-infinite step functions located atx = —I and x = +1, respectively, (b) IllusiTation of the subtraction of the two semi-infinite step functions as they evolve with time, yielding the correct evolution of the transient diffusion profile for the rectangular source. 

The solution for a diffusion couple in which two semi-infinite ternary alloys are bonded initially at a planar interface is worked out in Exercise 6.1 by the same basic method. Because each component has step-function initial conditions, the solution is a sum of error-function solutions (see Section 4.2.2). Such diffusion couples are used widely in experimental studies of ternary diffusion. In Fig. 6.2 the diffusion profiles of Ni and Co are shown for a ternary diffusion couple fabricated by bonding together two Fe-Ni-Co alloys of differing compositions. The Ni, which was initially uniform throughout the couple, develops transient concentration gradients. This example of uphill diffusion results from interactions with the other components in the alloy. Coupling of the concentration profiles during diffusion in this ternary case illustrates the complexities that are present in multicomponent diffusion but absent from the binary case. 

For a case that one stable steady state exists transient temperature profiles calculated agree satisfactorily with the measurements. For a case of three steady states the situation is quite complicated. The model used describes propagation of the fronts however, apparently cannot describe front multiplicity. A detailed calculation of the two-dimensional steady state equations including also the radial dispersion terms indicates that the onedimensional model is a very rough approximation for the diffusion" regime. We expect that dynamic calculations with the one-phase two-dimensional model could explain multiplicity of the fronts. 

The former variables affect the deposition of heat in the solid fuel and its transient temperature-profile, as well as the diffusion of the volatile pyrolysis products and their distribution and mixing with the surrounding atmosphere. The latter factors influence the nature and sequence of the primary and secondary reactions involved, the composition of the flammable volatiles, and, ultimately, the kinetics of the combustion. Consequently, basic study of the combustion of cellulosic materials or fire research has been channeled in these two directions. 

Ion-Exchange Rate and Transient Concentration Profiles The numerically implicit finite difference method was used to solve the set of nonlinear differential equations (8) for a wide range of model parameters such as diffusivities, Dg, D,, and Dy, dissociation constants. Kg, exchanger capacity, ag, and bulk concentration, Cg, of the solution. 

We will consider transient diffusion of a substance in a semi-infinite body B. At time t = 0, substance A is stored in the body at a concentration cAa. The desired concentration profile cA = cA(x,t) satisfies the following differential equation, under the assumption cD = const... 

Fig. 17 Transient concentration profiles in y-direction (i.e., along 8-ring channels) measured by interference microscopy for a adsorption and b desorption of methanol in a large crystal of ferrierite for pressure steps 5 -> 10 and 10 5 mbar. The form of the profiles shows that both surface resistance and internal diffusion (along the 8-ring chan-...

Fig. 45 Comparison of the transient concentration profiles during methanol uptake by the MOF-type crystal as recorded by interference microscopy (symbols) with the corresponding profiles recalculated from the measured diffusivities with surface permeabilities (full line in Fig. 44) which lead to the best fit to the experimental points...

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]. 

Transient spectroelectrochemical studies are possible with modem and fast spectroscopic probes, e.g. based on diode array spectrometers [55]. Compared to steady-state techniques, the design of transient spectroelectrochemical experiments is more challenging and the data analysis has to take into account the development of diffusion- and convection-driven concentration profiles [56]. However, these experiments open up the possibility to study time-dependent processes such as the development of a diffusion profile. For example, it is known, but usually ignored, that changing the redox state of a molecule changes its diffusivity. It has recently been shown [57] that, for the oxidation of AA A A -tetra-methylphenylenediamine (TMPD) in water, ethanol, and acetonitrile (Eq. II.6.1), the diffusion coefficients of the reduced form, TMPD, and the diffusion coefficient of the oxidised form. 

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. 

Transient Interdiffusion in Two Semi-Infinite Bodies The transient diffusion problem illustrated in Figure 4.8, which involves the interdiffusion of two semi-infinite bodies in contact with one another, is closely related to the previous semi-infinite transient diffusion problem. In fact, if you consider just one-half of the problem domain (e.g., consider the evolution of the diffusion profiles for species A for X > 0), diffusion proceeds exactly like the previous semi-infinite diffusion problem. The only difference is that in this case the interfacial concentration of species A is assumed to be pinned at half of its bulk (i.e., pure material A) value. 

FIGURE 4.12 Schematic concept of the linear superposition of a series of thin-film point source solutions to model the transient diffusion of arbitrary concentration profiles. 

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). 

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. 

The migration of the metal was studied by using deep-level transient spectroscopic techniques, or by determining the diffusion profiles of a deep level within depletion regions. The profiles could be accurately described by solutions to Pick s equation. The diffusivity near to room temperature was... 

The dynamics of sulphur uptake in a prereformer is like a fixed-bed absorption as seen in a zinc-oxide bed (refer to Chapter 1). However, in a tubular reformer the pore diffusion restrictions in the sulphur adsorption in a single pellet has a complex influence on the transient sulphur profiles in the reactor and a mathematical model [112] [387] [389] is required to evaluate more exactly the time for fiill saturation and the breakthrough curves of sulphur. 

The analytical solution of the transient diffusion problem of selective anodic dissolution of a binary alloy in the potentiodynamic polarization mode allowed us to obtain the equations for the concentration profiles of an electronegative metal in an alloy, voltammograms as well as modified Randles-Sevcik expressions, taking into accoimt the mixed solid-liquid phase diffusion nature of the kinetic limitations of the process, equilibrium solid phase adsorption of the components, and surface roughness of an electrode. 

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