One Dimensional Diffusion Process

Step 3. Transport within a catalyst pore is usually modeled as a one-dimensional diffusion process. The pore is assumed to be straight and to have length The concentration inside the pore is ai =ai(l,r,z) where I is the position inside the pore measured from the external surface of the catalyst particle. See Figure 10.2. There is no convection inside the pore, and the diameter of the pore is assumed to be so small that there are no concentration gradients in the radial direction. The governing equation is an ODE. 

Equation (13.4) describes the one-dimensional diffusion process. Por a three-dimensional diffusion problem, the diffusion equation is... 

Translational diffusion of a particle can be described by the Green function. For simplicity, we consider here a one-dimensional diffusion process. Let x be the coordinate of the diffusing particle in its path. If no barriers are present in the path, the particle obeys the usual diffusion equation, and the unperturbed Green function associated with this diffusion equation is given by... 

The dyad symmetry of the operator sequence is probably important in providing tight binding to two subunits of the symmetric tetrameric protein.11-13 It is also possible that repressor molecules move along DNA chains in a one-dimensional diffusion process, and that the symmetry of the operator site facilitates recognition by a protein moving from either direction.14 15... 

The transfer of gases at the base of the euphotic zone is by a combination of advection and diffusion processes and in reality is probably dependent on mechanisms that are intermittent rather than constant in time. Here we write the flux as a simple one-dimensional diffusion process dependent on the concentration gradient at the base of the euphotic zone and a parameter that is assumed to be analogous to molecular diffusion, an eddy diffusion coefficient, K ... 

Fick s first law provides a method for calculation of the steady state rate of diffusion when D can be regarded as constant during the diffusion process, and the concentration is a function only of the geometric position inside the polymer. However, concentration is often a function of time as well as of position. We said Equation 14.9 describes a steady state flow, but how does the system reach this steady state The unsteady state flow, or transient state, is described by Fick s second law. For a one-dimensional diffusion process, this can be written as... 

The kinetic data obtained by dynamic TGA designate thermal stabilization effect of nanoclay structure into a polymer matrix, caused in one-dimensional diffusion process of catalytic-charring throughout the thermal degradation of PP nanocomposition. According to provided kinetic analysis, polypropylene nanocomposite demonstrated the transcendent thermal and fireproof behaviour in relation to neat polypropylene. 

Additional discussion of these definitions and relationships between the different reference frames and fluxes are discussed in detail elsewhere (3,4). In principle, any reference frame for analysis may be selected however, a proper choice can reduce the mathematical difficulties. For example, in a one-dimensional diffusion process within fixed boundaries, where ideal mixing of components is a reasonable approximation, selecting the volume average frame of reference is wise because the volume average bulk velocity as defined above is zero, and hence the fluxes viewed from both a fixed reference frame and a reference frame moving... 

As long as St remains sufficiently small, the transition densities for the one dimensional diffusion process with constant drift can be used to calculate the survival probability at every time step St using the equation... 

In the context of this discussion it is important to consider that the dimensionality of the diffusion process is of greatest importance for an evaluation of boundary conditions. Whereas in the case of glycolysis of Saccharomyces carlsbergensis the occurrence of three-dimensional diffusion of molecules of small molecular size is still a valid assumption, in membrane-bound systems two-dimensional or one-dimensional diffusion should be considered, a concept that was discussed early by Bucher, T., Adv. Enzymol. 14, 1 (1953) as well as by Adam, G. and Delbriick, M in Structural Chemistry and Molecular Biology, A. Rich and N. Davidson eds., Freeman, San Francisco (1968). In membrane-bound transport processes, depending on the number of interacting molecules, the rate of... 

Transport Processes and Gauss Theorem One-Dimensional Diffusion/Advection/Reaction Equation Box 22.1 One-Dimensional Diffusion/Advection/Reaction Equation at Steady-State... 

For a random walk, f = 1 because the double sum in Eq. 7.49 is zero and Eq. 7.50 reduces to the form of Eq. 7.47. In principle, f can have a wide range of values corresponding to physical processes relating to specific diffusion mechanisms. This is readily apparent in extreme cases of perfectly correlated one-dimensional diffusion on a lattice via nearest-neighbor jumps. When each jump is identical to its predecessor, Eq. 7.49 shows that the correlation factor f equals NT.6 Another extreme is the case of f = 0, which occurs if each individual jump is exactly opposite the previous jump. However, there are many real diffusion processes that are nearly ideal random walks and have values of f 1, which are described in more detail in Chapter 8. 

In the discussion of separation efficiency thus far, it has been assumed that one-dimensional diffusion is the major contributor to peak broadening in CE, and other factors that contribute to the diffusion process have been ignored. In reality, the total variance of a zone, once corrections have been made for zone velocity and finite detector width, is given by... 

The search for a field dependence of the nuclear relaxation rate in the normal phase of (TMTSF)2C104 has been set out by Caretta et al. [74]. Their results obtained at 200 K (Fig. 13) up to 15 Tesla show a square-root field dependence which is indicative of one-dimensional diffusive spin dynamics. The square root dependence is cut off at low enough field due to processes that do not conserve spin along the stacks [66]. The field dependence is found to become essentially undetectable around 150 K and below, which may indicate either a change of dimensionality in the spin dynamics or that the collisionless regime is reached. 

The exhalation of Rn from material surfaces is controlled by the generation rate of Rn in the material, and the transport by diffusion through the material to the surface. The generation rate is determined by the Rn content of the material, and the emanation fraction. The transport through the material is controlled by the diffusion length through the material. The diffusion process is well described mathematically by one-dimensional diffusion theory, so that knowledge of these parameters will allow accurate calculation of the Rn exhalation rate from the material surface. 

We have already considered steady-state one-dimensional diffusion in the introductory sections 1.4.1 and 1.4.2. Chemical reactions were excluded from these discussions. We now want to consider the effect of chemical reactions, firstly the reactions that occur in a catalytic reactor. These are heterogeneous reactions, which we understand to be reactions at the contact area between a reacting medium and the catalyst. It takes place at the surface, and can therefore be formulated as a boundary condition for a mass transfer problem. In contrast homogeneous reactions take place inside the medium. Inside each volume element, depending on the temperature, composition and pressure, new chemical compounds are generated from those already present. Each volume element can therefore be seen to be a source for the production of material, corresponding to a heat source in heat conduction processes. 

The above butyl-branched alkane was studied by solid-state 13C NMR, alongside its linear analogue C198H398, to establish the solid-state diffusion coefficient.150 Both alkanes were in the once-folded form. The progressive saturation experiments have shown that the longitudinal relaxation of magnetization is consistent with a solid state chain diffusion process. Reptation and one-dimensional diffusion models were demonstrated to satisfactorily represent the data. The addition of the branch to the alkane chain was shown to result in a decrease in the diffusion coefficient, which ranged from 0.0918 nm2 s 1 for the linear chain to 0.016 nm2 s 1 for the branched chain. These diffusion coefficients are consistent with those of polyethylene. 

One-dimensional diffusion through a flat membrane will be treated in the following discussion. The effects of membrane asymmetry will be neglected since the process of permselection occurs in the thin dense layer of effective thickness, Z, at the membrane surface. In such a case, the expression for the local flux of a penetrant at any point in the dense layer can be written as shown in Equation 1 C14) ... 

A prototype of the parabolic equation is the diffusion equation. Diffusion equations are commonly encountered in chemical engineering processes. In the following subsections, the methods of solving one-dimensional diffusion equations are presented. These methods can be easily extended to higherdimensional problems. 

Taking this as a reasonable approximation for PE and PE-n-MMT, the fits with the aid of nonlinear regression were attempted by the model (5), where an one-dimensional diffusion type reaction was used for the first step and the nth-order (Fn) reaction - for the two subsequent steps of the overall thermal degradation process (Figure 6, Table 1). 

The mechanism of nucleation and growth was determined by analysis of deposition current transients as a function of potential. Figure 2 shows a series of current transients for copper deposition on TiN from 50 mM Cu(II) solution for potential steps from the open-circuit potential to deposition potentials in the range from -0.9 V to —1.5 V plotted on a semi-log plot. The nucleation and growth process is characterized by a current peak where the deposition current first increases due to the nucleation of copper clusters and three-dimensional diffusion-controlled growth, and then decreases as the diffusion zones overlap resulting in one-dimensional diffusion-controlled growth to a planar surface [3-... 

Polymorphic transitions of bromovalerylurea from form I to form II and from form III to form I conformed to mechanisms involving one-dimensional diffusion and two-dimensional nuclei growth processes, respectively. Both transitions also exhibited good Arrhenius behavior in the temperature range studied, as shown in Fig. 146.579 Transitions of phenyl-... 

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