Diffusion initial conditions

Figure 9.12 Progress of diffusion with time in an experiment like that shown in Fig. 9.9a, with the initial conditions given by c = Cq for x < 0 and c = 0 for x > 0 (a) c/co versus x and (b) dc/dx versus x.

Example Consider the diffusion equation, with boundary and initial conditions. 

Let us begin by considering the stability of homogeneous solutions and/or initial-conditions i.e. by considering the stability of a simple-diffusive CML when cri(O) = a for all sites i , where a is a fixed point of the local logistic map F(cr) = acr(l—cr). Following Waller and Kapral [kapral84], we first recast equations 8.23 and 8.24... 

First, in composites with high fiber concentrations, there is little matrix in the system that is not near a fiber surface. Inasmuch as polymerization processes are influenced by the diffusion of free radicals from initiators and from reactive sites, and because free radicals can be deactivated when they are intercepted at solid boundaries, the high interfacial area of a prepolymerized composite represents a radically different environment from a conventional bulk polymerization reactor, where solid boundaries are few and very distant from the regions in which most of the polymerization takes place. The polymer molecular weight distribution and cross-link density produced under such diffusion-controlled conditions will differ appreciably from those in bulk polymerizations. 

Specific heat of each species is assumed to be the function of temperature by using JANAF [7]. Transport coefficients for the mixture gas such as viscosity, thermal conductivity, and diffusion coefficient are calculated by using the approximation formula based on the kinetic theory of gas [8]. As for the initial condition, a mixture is quiescent and its temperature and pressure are 300 K and 0.1 MPa, respectively. 

In order to calculate the tip current response, the diffusion equations must be solved subject to the boundary and initial conditions of the system. Prior to the potential step. 

In addition to the assumptions of initial conditions, the validity of U-Pb methodology relies on closed system behavior of U, Pb and intermediate nuclides in the decay chain. Concordance between the two U-series decay chains is most likely to be compromised by Rn loss because Rn is the only gas in the decay chains and has a high diffusivity. Radon-222 in the decay chain has a half life of 3.8 days. This is much longer than the half-life of Rn (3.96 s) in the decay chain. Therefore, partial loss of Rn will give rise to an apparent age younger than the true age, whereas the 207p /235p ... 

For illustration, consider the simplest type of diffusion, described by the partial differential equation (2.5.3), also called linear diffusion. The system will be represented by an infinite tube closed at one end (for x = 0) and initially filled with a solution with concentration c°. Diffusion is produced by very fast removal (e.g. by precipitation or an electrode reaction) of the dissolved substance at the x = 0 plane (the reference plane). The initial concentration c° is retained at large distances from this reference plane (x— < >). The initial condition is thus... 

We consider the process of Brownian diffusion in a potential cp(x). The probability density of a Brownian particle is governed by the FPE (5.72) with delta-function initial condition. The moments of transition time are given by (5.1). 

An easy way to evaluate the initiation rate, Y[ = Iq (1-e " 2,3 e l [pl]) Oj, and a possible deviation from its first-order dependence on the light intensity, is by monitoring the induction period (tj). Under air diffusion-free conditions (laminate), a given number of radicals (N) must be generated by photolysis of the initiator to consume the 02 molecules dissolved in the formulation, before polymerization can start ... 

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. 

Figure 3.5. Sketch of scalar field with slab initial conditions and non-zero diffusivity at 0 < t. 

Starting from these initial conditions, the composition PDF will evolve in a non-trivial manner due to turbulent mixing and molecular diffusion.13 This process is illustrated in Fig. 3.10, where it can be seen that the shape of the composition PDF at early and intermediate times is far from Gaussian.14 As discussed in Chapter 6, one of the principal challenges in transported PDF methods is to develop mixing models that can successfully describe the change in shape of the composition PDF due to molecular diffusion. 

In this case, if the boundary and initial conditions allow it, either ej or c can be used to define the mixture fraction. The number of conserved scalar transport equations that must be solved then reduces to one. In general, depending on the initial conditions, it may be possible to reduce the number of conserved scalar transport equations that must be solved to min(Mi, M2) where M = K - Nr and M2 = number of feed streams - 1. In many practical applications of turbulent reacting flows, M =E and M2 = 1, and one can assume that the molecular-diffusion coefficients are equal thus, only one conserved scalar transport equation (i.e., the mixture fraction) is required to describe the flow. 

Since the molecular diffusivities are used in (5.254), the interval length L(t) and the initial conditions will control the rate of molecular diffusion and, subsequently, the rate of chemical reaction. In order to simulate scalar-gradient amplification due to Kolmogorov-scale mixing (i.e., for 1 < Sc), the interval length is assumed to decrease at a constant rate ... 

This boundary condition does not ensure that the unconditional means will be conserved if the chemical source term is set to zero (or if the flow is non-reacting with non-zero initial conditions Q( 0) 0). Indeed, as shown in the next section, the mean values will only be conserved if the conditional scalar dissipation rate is chosen to be exactly consistent with the mixture-fraction PDF. An alternative boundary condition can be formulated by requiring that the first term on the right-hand side of (5.299) (i.e., the diffusive term) has zero expected value with respect to the mixture-fraction PDF. However, it is not clear how this global condition can be easily implemented in the solution procedure for (5.299). 

This initial condition is rather idealized. In reality, one would expect to see partially premixed zones with f = fst and 7 = 0 which will move towards 7 = 1 along the stoichiometric line. The movement along lines of constant f corresponds to premixed combustion, and occurs at a rate that is controlled by the interaction between molecular diffusion and chemical reactions (i.e., the laminar flame speed). 

With this initial condition, a solution to the (turbulent) diffusion equation 138... 

In the SR model, Cd = 3 is chosen to agree with passive scalar decay from isotropic initial conditions in the absence of turbulent mixing (i.e., pure diffusion). 

As in preceding discussions, we take reductions as an example. Transposition to oxidations just requires a few changes of sign. In the case of a simple A + e —> B reaction, equations (2.30) and (2.31) are obtained from the integration of equations (2.28) and (2.29), with (C )(=0 = C° and (Cg)i=0 = 0 as initial conditions, respectively. In the absence of coupled homogeneous reactions, the gradients of both A and B are constant over the entire diffusion layer (Figure 2.31). Thus, in the case where the potential the surface concentration of A is zero,... 

Because formation ofexcimer E is a diffusion-controlled process, Eqs (4.11)-(4.13) apply to the diffusional rate constant ki for excimer formation. Under the approximation that ki is time-independent, the d-pulse responses, under the initial conditions (at t = 0), [M ] = [M ]o and [E ]o = 0, are... 

The magnitude of Ds is a measure of how fast the molecules diffuse along the particle and therefore sets a time scale for the adsorption process. Two boundary conditions and one initial condition have to be specified in order to obtain a unique solution to Eq. (60). Initially the solid particle is free of adsorbate, which is expressed as ... 

As in Sect. 2.1, Dj is the curvilinear centre-of-mass diffusion constant of the chain, and is given in terms of the monomeric friction constant by the Einstein relation Dj =kT/Nl. L is as before the length of the primitive path, or tube length of the chain, which is Finally, we need the initial condition on p(s,t), which... 

The initial conditions are analogous to those for a diffusion controlled QE mechanism (Sect. 2.4.1). The only difference is that all species involved in the snrface mechanism are immobilized on the electrode surface and characterized by their snrface concentrations, instead of volume concentrations used for diffusion controlled CrE mechanism. In the course of the voltammetric experiment, the following condition holds ... 

CO (20 bar) and styrene at 23 °C under diffusion-controlled conditions. The p-che-lates 10 and 11 were initially formed by 1,2 and 2,1 insertion of styrene in 9, respectively (Figure 7.15, trace a). 

We now instead calculate the drift velocity and diffusivity by directly integrating the traditional formulation of the Langevin equation in terms of random forces, and compare the results to those obtained above by rewriting the Langevin equation as a standard Stratonovich SDE. As in the analysis of the Stratonovich SDE, we calculate the first and second moments of an increment AX (f) = Z (f) — X (0) by integrating Eq. (2.262) from a known initial condition at f = 0. 

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