The Time-Dependent Diffusion Equation

The diffusion of carbon into the steel is described by the time-dependent diffusion equation... 

The time-dependent diffusion equations for Red appropriate to the axisymmetrical geometry, shown in Fig. 10, are identical to Eqs. (9) and (10), given earlier. Although phase 2 is assumed to be semi-infinite in the z-direction, the model can readily be modified for the situation where phase 2 has a finite thickness [61]. 

Because of the assumed dual sorption mechanism present in glassy polymers, the explicit form of the time dependent diffusion equation in these polymers is much more complex than that for rubbery polymers (82-86). As a result exact analytical solutions for this equation can be found only in limiting cases (84,85,87). In all other cases numerical methods must be used to correlate the experimental results with theoretical estimates. Often the numerical procedures require a set of starting values for the parameters of the model. Usually these values are shroud guessed in a range where they are expected to lie for the particular penetrant polymer system. Starting from this set of arbitrary parameters, the numerical procedure adjusts the values until the best fit with the experimental data is obtained. The problem which may arise in such a procedure (88), is that the numerical procedures may lead to excellent fits with the experimental data for quite different starting sets of parameters. Of course the physical interpretation of such a result is difficult. 

In the study of the diffusion of species to and from the electrode surface, we use the notation C.(t,x) to describe the concentration of the /th species as a function of time and distance from the electrode. The time-dependent diffusion equation in its general form is written as ... 

As deposition proceeds, depletion of the metal ion(s) and additive(s) in the electrolyte must be accounted for in the context of the given electrode geometry. A complete description involves solution of the time-dependent diffusion equation ... 

The time-dependent diffusion equations for this case, appropriate axisymmetric cylindrical SECM geometry are (7) ... 

For a large velocity, the dispersion-diffusion coefficient is no longer constant, and is dominated by the dispersivity (Eqn. 17), while for slow fluid velocity the equation reduces to the time-dependant diffusion equation. 

The boundary conditions to solve the time-dependent diffusion equation (Eq. 5.13) are... 

The time dependent diffusion equations may be reduced to time independent equations when exponential flux behavior is assumed. A (-u/v) correction to the absorptlcm cross-sections results. K a prompt critical eigenvalue is known and few-group parameters appropriate tx> the subcritlcal core are used, WANDA will solve for ( /v). 

This model in which the neutron source is taken to be (which implies that neutrons from fission appear at the one velocity of diffusion) is not expected to apply directly to operating reactors however, the techniques to be used later are well illustrated by this formulation of the problem, and the results are useful, if properly adapted, to more general situations. We take then as our neutron-balance relation the time-dependent diffusion equation (5.21) along with the assumed source term... 

This is the MWA transformation mentioned in Sect. i2.3.3.i, but appiied to bands. Inserting into Eq. (i2. i28) gives the time-dependent diffusion equation in conformai coordinates for species k,... 

The Slice simulation package relies on generating exit times from the centre of each overlapping slice and the respective probability of exit at either the upper or lower ends of the slice. In this section the generation of the mean exit time is presented which extends the functionality of Slice for ionic systems. The time dependent diffusion equation for charged species is known to be... 

The basic inhour equation derived from the time-dependent diffusion equation is... 

Before considering some additional examples of the finite difference method for two dimensional boundary value problems, it is useful to consider the possibility of adding another dimension to the problem and this is usually a time dimension. The prototypical example of such an equation is the time dependent diffusion equation ... 

---