Diffusion with homogeneous reaction

Notice that equation (1-115) can be written for all the n components, unlike the original Maxwell-Stefan equations, which are valid for only n - 1 components. However, another variable, the total pressure, has been added. Therefore, a bootstrap relation between the fluxes is still needed to complete the problem formulation. 

An explicit expression for the total pressure gradient can be obtained from equation (1-115) by summing with respect to all components  

Equation (1-116) can be substituted into (1-115),written for n - 1 components, to eliminate the total pressure gradient from those equations. The resulting n - 1 equations are then augmented with equation (1-116) and the bootstrap condition to solve for the n fluxes and the total pressure gradient. The resulting boundary-value problem can be solved by modifying the Mathcad program developed in Examples 1.17 and 1.18. 

Your objectives in studying this section are to be able to  

Define the concept of homogeneous chemical reaction, and learn to incorporate it into the continuity equation. 

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Steady-State Diffusion with Homogeneous Chemical Reaction... 

The diffusion of a radioactive component is a relatively easy problem. It is discussed here to illustrate how coupled diffusion and homogeneous reaction can be treated, and to prepare for the more difficult problem of the diffusion of a radiogenic component. The diffusion of a radiogenic component, which is dealt with in Section 3.5.2, is an important geological problem because of its application in geochronology and thermochronology. 

Returning to the survival probability, in Fig. 57, the kinetic theory and diffusion equation [cf. eqn. (132)] predictions are compared. Three values of the activation rate coefficient are used, being 0.5, 1.0 and 2.0 times the Smoluehowski rate coefficient for a purely diffusion-limited homogeneous reaction, 4ttoabD. With a diffusion coefficient of 5x 10 9 m2 s1 and encounter distance of 0.5 nm, significant differences are noted between the kinetic theory and diffusion equation approaches [286]. In all cases, the diffusion equation leads to a faster rate of reaction. In their measurements of the recombination rate of iodine atoms in hydrocarbon solvents, Langhoff et al. [293] have noted that the diffusion equation analysis consistently predicts a faster rate of iodine atom recombination than is actually measured. Thus there is already some experimental support for the value of the kinetic theory approach compared with the diffusion equation analysis. Further developments cannot fail to be exciting. 

Smoluchowski [M. v. Smoluchowski (1917)] treated the problem of diffusion controlled homogeneous reactions in which the reacting particles were initially distributed at random and were non-interacting (except for the collision process). If reaction occurs during the first encounter of the diffusing partners, it is diffusion controlled. If many encounters of the diffusing partner are needed before they eventually react with each other, the process is reaction controlled. If the particles interact already at some distance, one can nevertheless use the concept of diffusion controlled encounters. In this case, one has to carefully define an extended reaction volume as will be outlined later. 

In a system with homogeneous reactions (e.g. reactive absorption), mass and heat transfer is described by the following convective diffusion and convective heat conduction equations (Kenig, 2000) ... 

Steady-state mass diffusion with homogeneous chemical reaction... 

Bieniasz LK (1993) The von Neumann stability of finite-difference algorithms for the electrochemical kinetic simulation of diffusion coupled with homogeneous reactions. J Electroanal Chem 345 13-25... 

FIGURE 5-12 Transmission-line representation of diffusion impedance A. simple diffusion with reflecting boundary B. reflecting boundary diffusion coupled with homogeneous reaction C. simple diffusion with absorbing boundary D. absorbing boundary diffusion coupled with homogeneous reaction... 

FIGURE 5-13 Impedance model for diffusion coupled with homogeneous reaction ... 

The progress of this category of reactions is expected to depend on the composition of the materials within the phase as well as the temperature and pressure of the system. The rate of homogeneous reaction should not be affected by the shape of the container, the surface properties of the solid materials in contact with the phase, and the diffusion characteristics of the fluid. Thus the rate of reaction of component i may be expressed as... 

In theory, one assumes the formation of radicals before the chemical stage begins (see Sect. 2.2.3). These radicals interact with each other to give molecular products, or they may diffuse away to be picked up by a scavenger in a homogeneous reaction to give radical yields. The overlap of the reactive radicals is more on the track of a high-LET particle. Therefore, the molecular yields should increase and the radical yields should decrease with LET. This trend is often observed, and it lends support to the diffusion-kinetic model of radiation-chemical reactions. 

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. 

In addition to this, and in contrast with the homogeneous case discussed in Section 5.2.2, the diffusion of P and Q is therefore not perturbed by any homogeneous reaction. If, furthermore, the P/Q electron transfer at the electrode is fast and thus obeys Nernst s law, the diffusive contribution to the current in equations (5.11) and (5.12) is simply equal to the reversible diffusion-controlled Nernstian response, idif, discussed in Section 1.2. The mutual independence of the diffusive and catalytic contributions to the current, expressed as... 

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,... 

The solution procedure to this equation is the same as described for the temporal isothermal species equations described above. In addition, the associated temperature sensitivity equation can be simply obtained by taking the derivative of Eq. (2.87) with respect to each of the input parameters to the model. The governing equations for similar types of homogeneous reaction systems can be developed for constant volume systems, and stirred and plug flow reactors as described in Chapters 3 and 4 and elsewhere [31-37], The solution to homogeneous systems described by Eq. (2.81) and Eq. (2.87) are often used to study reaction mechanisms in the absence of mass diffusion. These equations (or very similar ones) can approximate the chemical kinetics in flow reactor and shock tube experiments, which are frequently used for developing hydrocarbon combustion reaction mechanisms. 

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