Finite transient diffusion

Menon and Landau [52] developed a model to describe transient diffusion and migration in stagnant binary electrolytes. Nonuniformity at a partially masked cathode was found to increase during electrolysis as the diffusion resistance develops. The calculations were done using an alternating-direction implicit (ADI) finite difference method. 

TRAFIC (Ref. 6) A core-survey code for calculating the full-core release of metallic fission products. TEIAFIC is a finite-difference solution to the transient diffusion equation for the multihole fuel element geometry with a convective boundary condition at the coolant surface. The temperature and failure distributions required as input are supplied by an automatic interface with the SURVEY/PERFOR code. 

As an alternative to the previous example, we can also solve the problems with inhomogeneous boundary conditions by direct application of the finite integral transform, without the necessity of homogenizing the boundary conditions. To demonstrate this, we consider the following transient diffusion and reaction problem for a catalyst particle of either slab, cylindrical, or spherical shape. The dimensionless mass balance equations in a catalyst particle with a first order... 

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 Finite (Symmetric) Spherical Diffusion So far, we have only examined ID (Cartesian) examples of Fick s second law. Solving Fick s second law in alternative coordinate systems (e.g., for radial, spherical, 2D, or 3D problems) is not really any different. As an example, we examine here the case of transient finite spherical diffusion, which is essentially analogous to the transient finite planar diffusion problem that we just finished discussing. 

Consider the transient finite spherical diffusion problem illustrated in Figure 4.15, which describes the diffusion of H2 into a spherical particle. 

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... 

Pick s second law is a second-order partial differential equation. Solving it in order to predict transient diffusion processes can be fairly straightforward or quite complex, depending on the specific situation. In this chapter, analytical solutions were discussed for a number of cases, including ID transient infinite and semi-infinite diffusion, ID transient finite planar diffusion, and transient spherical finite diffusion as summarized in Table 4.4. In all cases, solution of Pick s second law requires the specification of a number of boundary conditions and initial conditions. 

Importance of mutual diffusion Phase separation, polymer blend processing Relation to interaction parameter Transient diffusion Pick s law of diffusion Damped wave diffusion and relaxation Semi-infinite medium, finite slab Periodic boundary condition... 

Coulometric titration techniques were used to measure chemical diffusion at between 700 and lOOOC. The transient current response to a potentiostatic step was transformed from the time domain to the frequency domain. The equivalent circuit which was used to fit the resultant impedance data contained an element which described the finite-length diffusion of O into the sample. Other elements which were included were the gas-phase capacitance, and the sum of the gas-phase diffusion resistance and that which was associated with the limited surface exchange kinetics of the sample. The chemical diffusion coefficient of the perovskite, Laq gSrq 2C0O3,... 

However, the MF approach is insufficient for particles with 3.3 nm size, if the potential exceeds 0.8 Nshe, that is, in the potential range with rapid kinetics of surface reactions. The MF approach fails for particles with sizes in the range of 1.8 nm. In these cases, it is necessary to account for the finite surface diffusivity of CO and, thus, solve the active site model with the kinetic Monte Carlo simulation approach. Figure 3.11a shows typical results of current transients for particles with mean size of 3.3 nm that are matched closely with the model. Analysis of the data with the kinetic model allows important structural and dynamic parameters of the catalytic system to be extracted and analyzed. 

The derivation of a steady-state solution requires boundary conditions, but no initial condition. Steady-state can be seen as the asymptotic solution (so never mathematically reached at any finite time [43]) of the transient, which -for practical purposes - can be approached in a reasonably short time. For instance, limiting-flux diffusion of a species with diffusion coefficient Di = 10-9 m2 s 1 towards a spherical organism of radius rQ = 1 jxm is practically attained at t r jDi = 1 ms. 

Mathematically, steady-state is never reached within a finite time. For practical purposes, however, one can compute the time necessary to reach steady-state by imposing the condition that a given transient magnitude (concentration or flux) differs from the steady-state value in a reasonably low relative proportion [42], For calculating the proximity to steady-state, the diffusive flux Jm is more convenient than the internalisation flux /u, because of the continuously decreasing behaviour with time of the former. 

In the biomedical literature (e.g. solute = enzyme, drug, etc.), values of kf and kr are often estimated from kinetic experiments that do not distinguish between diffusive transport in the external medium and chemical reaction effects. In that case, reaction kinetics are generally assumed to be rate-limiting with respect to mass transport. This assumption is typically confirmed by comparing the adsorption transient to maximum rates of diffusive flux to the cell surface. Values of kf and kr are then determined from the start of short-term experiments with either no (determination of kf) or a finite concentration (determination of kT) of initial surface bound solute [189]. If the rate constant for the reaction at the cell surface is near or equal to (cf. equation (16)), then... 

Transient is a C-program for solving systems of generally non-linear, parabolic partial differential equations in two variables (that is, space and time), in particular, reaction-diffusion equations within the generalized Crank-Nicolson Finite Difference Method. 

Law et al. [15] determined the diffusion coefficient for benzyl penicillin in thin films of Palacos, Simplex and CMW cements assuming that antibiotic transport can be described by Fick s law using a finite difference approximation to quantify transient non-steady-state behaviour. These investigators found that the diffusion coefficient was increased in the presence of additives and proposed that the finite difference approach could be applied to determine release of antibiotic from preloaded PMMA beads. Dittgen and Stahlkopf [16] showed that incorporation of amino acids of varying solubilities also affected release of chloramphenicol from polymethacrylic... 

Transport by combined migration—diffusion in a finite planar geometry can achieve a true steady state when only two ions are present, as we saw in Sect. 4.2. The same holds true when there are three or more ions present. Under simplifying conditions [see eqn. (89) below], it is possible to predict the steady-state behaviour with arbitrary concentrations of many ions. However, the corresponding transient problem is much more difficult and we shall not attempt to derive the general transient relationship, as we were able to do in deriving eqn. (82) in the two-ion case. 

Since the length scales associated with the thermal lens are on the order of 10 to 1000 times the grating constant, their characteristic time scale interferes with polymer diffusion within the grating. Such thermal lensing has been ignored in many FRS experiments with pulsed laser excitation [27,46] and requires a rather complicated treatment. A detailed discussion of transient heating and finite size effects for the measurement of thermal diffusivities of liquids can be found in Ref. [47]. 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

In order to have theoretical relationships with which experimental data can be compared for analysis it is necessary to obtain solutions to the partial differential equations describing the diffusion-kinetic behaviour of the electrode process. Only a very brief account f the theoretical methods is given here and this is done merely to provide a basis for an appreciation of the problems involved and to point out where detailed treatments can be found. A very lucid introduction to the theoretical methods of dealing with transient electrochemical response has appeared (MacDonald, 1977) which is highly recommended in addition to the classic detailed treatment (Delahay, 1954). Analytical solutions of the partial differential equations are possible only in the most simple cases. In more complex cases either numerical methods are used to solve the equations or they are transformed into finite difference forms and solved by digital simulation. 

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. 

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. 

The origin of the nonlinear density dependences in the diffusion limit arises from the fact that for a single reactive site, the transient terms in diffusion theory become infinitely long lived for d < 2, and a steady-state rate cannot be defined for a single site. This arises from depletion of the equilibrium density for R> Rj, which becomes more severe as the dimensionality is reduced. However, for a finite reactive site density a steady state is eventually produced... 

Some advanced general purpose finite-element codes, well adapted for stress analysis in particular, e.g. ABAQUS or MSC.MARC, have certain capabilities to simulate the stress-assisted diffusion, too. Unfortunately, they still are limited in some rather important aspects. As regards ABAQUS, this allows to perform simulations of the stress-assisted diffusion governed by equation (5) "over" the data of an accomplished solution of a geometrically and physically nonlinear stress-strain analysis, i.e., for the stationary stress field at the end of some preliminary loading trajectory, but not for the case of simultaneous transient loading and hydrogenation. 

In the second chapter we consider steady-state and transient heat conduction and mass diffusion in quiescent media. The fundamental differential equations for the calculation of temperature fields are derived here. We show how analytical and numerical methods are used in the solution of practical cases. Alongside the Laplace transformation and the classical method of separating the variables, we have also presented an extensive discussion of finite difference methods which are very important in practice. Many of the results found for heat conduction can be transferred to the analogous process of mass diffusion. The mathematical solution formulations are the same for both fields. 

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