Transient diffusion spherical

COPAR (Ref. 7) A stand-alone code, as well as a subroutine in the TRAFIC code, which calculates the transient fission product release from failed and intact coated particles with burnup-dependent kernel diffusivities. COPAR is an inf inite - series solution to the transient diffusion equation for a multi-region spherical geometry and arbitary temperature and failure histories. 

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. 

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... 

Models allowing to determine A and No obtain the current transient from the material flux to free , noninteracting, growth centers, considering circular diffusion zones around them, with time-dependent radii r. As shown in Fig. 4, these are two-dimensional projections of three-dimensional fields that define, for a hemispherical nuclei of radius ro, an equivalent area toward which the same amount of matter that diffuses spherically to a three-dimensional nucleus diffuses by planar diffusion. 

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. 

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. 

Microelectrodes with several geometries are reported in the literature, from spherical to disc to line electrodes each geometry has its own critical characteristic dimension and diffusion field in the steady state. The difhisional flux to a spherical microelectrode surface may be regarded as planar at short times, therefore displaying a transient behaviour, but spherical at long times, displaying a steady-state behaviour [28, 34] - If a... 

This expression is the sum of a transient tenu and a steady-state tenu, where r is the radius of the sphere. At short times after the application of the potential step, the transient tenu dominates over the steady-state tenu, and the electrode is analogous to a plane, as the depletion layer is thin compared with the disc radius, and the current varies widi time according to the Cottrell equation. At long times, the transient cunent will decrease to a negligible value, the depletion layer is comparable to the electrode radius, spherical difhision controls the transport of reactant, and the cunent density reaches a steady-state value. At times intenuediate to the limiting conditions of Cottrell behaviour or diffusion control, both transient and steady-state tenus need to be considered and thus the fiill expression must be used. Flowever, many experiments involving microelectrodes are designed such that one of the simpler cunent expressions is valid. 

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. 

Some transient problems tend to a trivial (and useless) steady-state solution without flux and concentration profiles. For instance, concentration profiles due to limiting diffusion towards a plane in an infinite stagnant medium always keep diminishing. Spherical and disc geometries sustain steady-state under semi-infinite diffusion, and this can be practically exploited for small-scale active surfaces. 

COSILAB Combustion Simulation Software is a set of commercial software tools for simulating a variety of laminar flames including unstrained, premixed freely propagating flames, unstrained, premixed burner-stabilized flames, strained premixed flames, strained diffusion flames, strained partially premixed flames cylindrical and spherical symmetrical flames. The code can simulate transient spherically expanding and converging flames, droplets and streams of droplets in flames, sprays, tubular flames, combustion and/or evaporation of single spherical drops of liquid fuel, reactions in plug flow and perfectly stirred reactors, and problems of reactive boundary layers, such as open or enclosed jet flames, or flames in a wall boundary layer. The codes were developed from RUN-1DL, described below, and are now maintained and distributed by SoftPredict. Refer to the website http //www.softpredict.com/cms/ softpredict-home.html for more information. 

In this section, the basic theory required for the analysis and interpretation of adsorption and ion-exchange kinetics in batch systems is presented. For this analysis, we consider the transient adsorption of a single solute from a dilute solution in a constant volume, well-mixed batch system, or equivalently, adsorption of a pure gas. Moreover, uniform spherical particles and isothermal conditions are assumed. Finally, diffusion coefficients are considered to be constant. Heat transfer has not been taken into account in the following analysis, since adsorption and ion exchange are not chemical reactions and occur principally with little evolution or uptake of heat. Furthermore, in environmental applications,... 

SLIDER is a Fortran IV computer program for investigating the diffusion of a single fission-product isotope in a multilayered spherical fuel particle. This code enables one to compute, on the basis of Fick s law of diffusion, the transient and steady-state fission product concentrations and releases in multilayered spherical geometry. 

Because particles of different sizes are distributed throughout the bulk randomly, developing an exact model that couples diffusion to particle size evolution is daunting. However, a mean-field approximation is reasonable because diffusion near a spherical sink (see Section 13.4.2) has a short transient and a steady state characterized by steep concentration gradients near the surface. The particles act as independent sinks in contact with a mean-field as in Fig. 15.2. 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

The most widely used unsteady state method for determining diffusivities in porous solids involves measuring the rate of adsorption or desorption when the sample is subjected to a well defined change in the concentration or pressure of sorbate. The experimental methods differ mainly in the choice of the initial and boundary conditions and the means by which progress towards the new position of equilibrium is followed. The diffusivities are found by matching the experimental transient sorption curve to the solution of Fick s second law. Detailed presentations of the relevant formulae may be found in the literature [1, 2, 12, 15-17]. For spherical particles of radius R, for example, the fractional uptake after a pressure step obeys the relation... 

The characteristic diffusion time for any UME geometry where the transition from semi-infinite linear diffusion (transient) to hemispherical or spherical diffusion (steady state) occurs may be given as... 

The development of ultramicroelectrodes with characteristic physical dimensions below 25 pm has allowed the implementation of faster transients in recent years, as discussed in Section 2.4. For CA and DPSC this means that a smaller step time x can be employed, while there is no advantage to a larger t. Rather, steady-state currents are attained here, owing to the contribution from spherical diffusion for the small electrodes. However, by combination of the use of ultramicroelectrodes and microelectrodes, the useful time window of the techniques is widened considerably. Compared to scanning techniques such as linear sweep voltammetry and cyclic voltammetry, described in the following, the step techniques have the advantage that the responses are independent of heterogeneous kinetics if the potential is properly adjusted. The result is that fewer parameters need to be adjusted for the determination of rate constants. 

A general transient model of diffusion-reaction that uses the effective diffusivity concept described for gas-solid catalytic reactions can be derived here as well, e.g., for a spherical particle ... 

Bartlett et al. presented three basic models to explain the time-dependent transient resistivity response R (t) of PPy sensors to alcohol vapors [22-24]. Two of the models assume a diffusion controlled penetration of the analyte molecules into the polymer film. More specifically, they consider bounded planar as well as bounded spherical diffusion. For bounded diffusion in a plane sheet with thickness / and a diffusion coefficient D they find [23]... 

Long times. At long times, the transient contribution given by the second term of Eq. (6) has decayed to the point at which its contribution to the overall current is negligible. At these long times, the spherical character of the electrode becomes important and the mass transport process is dominated by radial (spherical) diffusion as illustrated in Fig. 4(b). 

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