Homogeneous diffusion

Great simplification is achieved by introducing the hypothesis of independent reaction times (IRT) that the pairwise reaction times evolve independendy of any other reactions. While the fundamental justification of IRT may not be immediately obvious, one notices its similarity with the molecular pair model of homogeneous diffusion-mediated reactions (Noyes, 1961 Green, 1984). The usefulness of the IRT model depends on the availability of a suitable reaction probability function W(r, a t). For a pair of neutral particles undergoing fully diffusion-con-trolled reactions, Wis given by (a/r) erfc[(r - a)/2(D t)1/2] where If is the mutual diffusion coefficient and erfc is the complement of the error function. 

This is the same result as in Eq. 18-14 with the only exception that diffusivity is replaced by Dipm. The latter is usually smaller than the homogeneous diffusion coefficient, One effect is tortuosity (see Eq. 18-57), but sometimes additional influences are important. They are discussed next. 

Zwanzig s diffusion equation [444], eqn. (211), can be reduced to the stochastic equation used by Clifford et al. [442, 443] [eqn. (183)] to describe the probability that N identical reactant particles exist at time t (see also McQuarrie [502]), Let us consider the case where U — 0, with a static solvent, for a constant homogeneous diffusion coefficient. This is a major simplification of eqn. (211). Now, rather than represent the reaction between two reactants k and j by a boundary condition which requires the... 

Anisothermal Homogeneous Diffusion. Using the reasonable simplifications that the flow of heat is much faster than the transport of matter and that thermal kinetic effects can be neglected, we can dispense with the effect of changing temperature on diffusion within a phase simply by using a reduced time r = tD/R2. Use of a reduced radius p = r/R... 

Figure 6.1 Microdialysis probes based on the concentric assembly of the inlet and outlet tubes provide the highest mechanical strength, as well as homogeneous diffusion paths from the surrounding environment they also cause the lowest tissue damage. The probes differ mainly in the construction of the inlet/outlet lines, as seen when comparing these constructions. The shaft and the capillaries can be either rigid and made of metal (A) or fused silica (fl ) or flexible and made of polyurethane (C). The outer diameter of the membrane on the CMA/11 probe (fl) is 0.24 mm, whereas the membranes of CMA/10 and CMA/20 probes (A and C) are 0.5 mm in diameter.

A convenient way to formulate a dynamical equation for a Levy flight in an external potential is the space-fractional Fokker-Planck equation. Let us quickly review how this is established from the continuous time random walk. We will see below, how that equation also emerges from the alternative Langevin picture with Levy stable noise. Consider a homogeneous diffusion process, obeying relation (16). In the limit k — 0 and u > 0, we have X(k) 1 — CTa fe and /(w) 1 — uz, whence [52-55]... 

For solid homogeneous diffusion (quadratic driving force), it is (6)... 

The breakthrough curve for solid homogeneous diffusion (linear driving force) combined with film mass transfer can be derived ... 

The rigorous description of diffusion and heterogeneous surface-catalyzed chemical reactions in porous catalytic pellets is almost never solved in practice because the partial differential mass balance and the supporting boundary conditions are extremely complex. The approximate solution overlooks intricate details of the pore structure, exploits the symmetry of the macroscopic boundary of one catalytic pellet instead of one of the pores, and invokes the concept of homogeneous diffusion that is not influenced by the orientation of the internal pores. 

Most important, heterogeneous surface-catalyzed chemical reaction rates are written in pseudo-homogeneous (i.e., volumetric) form and they are included in the mass transfer equation instead of the boundary conditions. Details of the porosity and tortuosity of a catalytic pellet are included in the effective diffusion coefficient used to calculate the intrapellet Damkohler number. The parameters (i.e., internal surface area per unit mass of catalyst) and Papp (i.e., apparent pellet density, which includes the internal void volume), whose product has units of inverse length, allow one to express the kinetic rate laws in pseudo-volumetric form, as required by the mass transfer equation. Hence, the mass balance for homogeneous diffusion and multiple pseudo-volumetric chemical reactions in one catalytic pellet is... 

In other words, reactants exist everywhere within the pores of the catalyst when the chemical reaction rate is slow enough relative to intrapellet diffusion, and the intrapellet Damkohler number is less than, or equal to, its critical value. These conditions lead to an effectiveness factor of unity for zerofli-order kinetics. When the intrapellet Damkohler number is greater than Acnticai, the central core of the catalyst is reactant starved because criticai is between 0 and 1, and the effectiveness factor decreases below unity because only the outer shell of the pellet is used to convert reactants to products. In fact, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler number for zeroth-order kinetics exhibits an abrupt change in slope when A = Acriticai- Critical spatial coordinates and critical intrapeUet Damkohler numbers are not required to analyze homogeneous diffusion and chemical reaction problems in catalytic pellets when the reaction order is different from zeroth-order. When the molar density appears explicitly in the rate law for nth-order chemical kinetics (i.e., n > 0), the rate of reaction antomaticaUy becomes extremely small when the reactants vanish. Furthermore, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler nnmber does not exhibit an abrupt change in slope when the rate of reaction is different from zeroth-order. 

The homogeneous diffusion model is slightly more complex in cyUndrical coordinates relative to the model described above in rectangular coordinates. Additional complexity arises because the radial term of the Laplacian operator (V V = V ) accounts for the fact that the surface area across which radial diffusion occurs increases linearly with dimensionless coordinate r/ as one moves radially outward. Basic information for = f(t]) is obtained by integrating the dimensionless mass balance with radial diffusion and chemical reaction ... 

However, the void area fraction is equivalent to the void volume fraction, based on equation (21-76) and the definition of intrapellet porosity Sp at the bottom of p. 555. Effectiveness factor calculations in catalytic pellets require an analysis of one-dimensional pseudo-homogeneous diffusion and chemical reaction in a coordinate system that exploits the symmetry of the macroscopic boundary of a single pellet. For catalysts with rectangular symmetry as described above, one needs an expression for the average diffusional flux of reactants in the thinnest dimension, which corresponds to the x direction. Hence, the quantity of interest at the local level of description is which represents the local... 

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