The inner boundary condition

In this section, the various methods which have been developed to treat chemical reaction rates between solutes in solution are discussed, with specific concern for those reactions where the rate of reaction of encounter pairs is of comparable magnitude to the rate of diffusive formation of encounter pairs. Some of the detailed comments on the partially reflecting boundary condition are discussed, the effects of angular variation of the reaction rate and the possibility of using a sink term to represent chemical reaction rather than a boundary condition are presented. Such comments are contrasted with the relatively few instances where experimental data has been obtained for the rate of the concomitant chemical reaction. Recently, attention has been given to a development of aspects of gas-phase reaction rate theory to be applied to reactions in liquids. 

If the rate of reaction of encounter pairs is comparably fast to the rate of formation of encounter pairs, Collins and Kimball [4] suggested that the slowness of the chemical reaction rate could be incorporated into the theory of diffusion-limited reaction rates by modifying the Smoluchowski [3] boundary condition, eqn. (5), to the partially reflecting boundary 

There is sufficient and convincing experimental evidence available already to support the need to consider activation and diffusion processes simultaneously. For instance, in Chap. 2, Sects. 5.2 and 5.4, mention was made of other instances where the reaction rate had been measured and found to be slower than anticipated from the Smoluchowski rate coefficient [eqn. (19)]. Using the Collins and Kimball expression enabled the workers to obtain reasonable estimates of the rate coefficients of encounter pair reactions. There is still some degree of uncertainty that the slower than expected reaction rate might not be attributable to partial 

Wilemski and Fixman [51] have suggested that it is intrinsically more satisfactory to treat chemical reaction by means of a sink term included in the diffusion equation than as a boundary condition imposed on the density distribution. They recommended writing the diffusion equation as 

One further conceptual point in the favour of the inclusion of a noninfinite rate of encounter reaction into the theory of diffusion-limited reactions follows from the behaviour of the time-dependent rate coefficient at short times. The time-dependent Smoluchowski rate coefficient depends on time as r1/2 at short times. Indeed, immediately after formation, the rate of reaction of reactants is essentially infinite. This is because all those reactants within a separation 5R of the encounter 

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The second boundary condition assures total finite existence probability at any time the first boundary condition implies that the recombination is fully diffusion-controlled, which has been found to be true in various liquid hydrocarbons (Allen and Holroyd, 1974). [The inner boundary condition can be suitably modified for partially diffusion-controlled reactions, which, however, does not seem to have been done.]... 

Equation (A4.6) is solved (which has to be done numerically for most values of n) using the inner boundary condition... 

This is the inner boundary condition. It has two serious flaws. The reaction between A and B may not occur at a rate very much faster than the reactants can approach one another. As was discussed in Sect. 3.1, this can lead to an appreciable probability of formation of the species (AB), which can be better described as an encounter pair. This difficulty was neatly handled by Collins and Kimball [4] and is discussed in Sect. 4 and Chap. 8 Sect. 2.4. The other flaw is the specification of one definite distance at which reaction occurs, the encounter distance. Even if the reaction proceeds with similar rates when the separation distance varies by 0.1nm (the largest likely variation of bond distance), this will be a small variation compared with the encounter distance, which is typically >0.5nm. Means to circumvent this difficulty are discussed in Chap. 8 Sect. 2.4 and Chap. 9 Sect. 4. 

When the activation process is comparable with or slower than the rate of approach of reactants to form encounter pairs, it is no longer satisfactory to say that the reactants can not co-exist within a distance R of one another. Because the rate of reaction, /eact, of the activation process is finite, so too is the lifetime (and hence concentration) of encounter pairs non-zero. The inner boundary condition, which describes reaction of A and B together in the diffusion analysis, is unsatisfactory. Collins and Kimball [4] suggested an alternative boundary condition and the remainder of this section analyses their work following Noyes [5]. Firstly, the boundary condition is developed and then included in the diffusion equation analysis to obtain the density distribution. Finally, the rate coefficient is obtained. 

Using Gauss s theorem to reduce the volume integral and applying the inner boundary condition of eqn. (353c) together with eqn. (352) with p = exp (+ G(R, t)... 

Figure 2a Envelope solutions with the outer boundary of log Tefr = 4.2 and the inner boundary conditions that fit to the core at Mr = 5.45 M and r = 0.6 Rq. The ordinate is the mass of the hydrogen-rich envelope (Menv = M— 5.45 M ) given as a function of luminosity L and the mass fraction of helium Y.

When these are substituted into the exact Navier-Stokes equations (183)-(184), and into the inner boundary condition (185b), and terms of the same order in R equated, one finds that the first few terms in the expansion satisfy the relations... 

Both sets of equations are linear. Since it is not expected that the outer expansion is valid near the body, these fields are not required to satisfy the inner boundary conditions on the body. Accordingly, Eqs. (208) and (209) are not unique without the specification of some inner boundary condition. This additional condition is furnished by the matching requirement with the inner expansion. 

The inner boundary condition at the surface of the catalyst fragment. Equation 2.135, states that the diffusional flux at the surface of the fragment (rp = R ) is equal to the rate of reaction R. The outer boimdary condition at the interface of the microparticle (rp = R ic) states that the concentration of monomer in the polymer layer at the interface is in equilibrium with the monomer concentration in the pores of the macroparticle at the position imder consideration. If a partition coefficient (K ) is used between the concentrations of monomer in the pores of the macroparticle and absorbed in the polymer layer surrounding the microparticle, then ... 

The same approach for the PGM phase modeling was adopted in the development of the mathematical model of Dual-Layer ASC monolith catalysts [20]. Such a model, named LSM (Layer + Surface Model), was based on the mathematical model for SCR monolithic converters [12,25,26]. As detailed in the following, the PGM reactivity was directly included in the mentioned SCR converter model by simply modifying the inner boundary condition for the diffiision- -eaction equations within the SCR layer, i.e., the boundary condition at the interface with the PGM phase. It is worth emphasizing that the LSM model reduces to the PGM monolith model when the SCR layer thickness approaches zero. 

We first assessed the impact of internal diffusion limitations in the PGM layer. For this purpose a simulation study was performed progressively increasing the PGM washcoat thickness, from 10 pm up to 71 pm. From the analysis of NH3 concentration profiles a dramatic impact of dififiisional limitations was apparent indeed only the surface of this layer is effectively active due to the extremely high reactions rates of the PGM catalyst. For this reason, we developed a Layer -I- Surface Model (LSM) of dual-layer ASC where we treat the PGM layer as a surface, while we retain the rigorous description of coupled reaction/dififusion in the SCR layer, based on the previous ID -I- ID model of SCR monolithic converters [12,25,26]. Indeed, avoiding the description of diffusion phenomena in the PGM layer enables the direct inclusion of the PGM reactivity in the SCR converter model by simply modifying the inner boundary conditions of the species differential mass balances in the SCR layer, i.e., those now at the interface with the PGM phase. Treating the PGM layer as a surface thus enabled a simple extension of the ID -I- ID SCR converter model to simulate dual-layer catalytic systems too. 

The numerical results presented here are very accurate and well within the usual engineering approximations. Since the basis functions automatically satisfy the outer boundary condition(s), and they are in closed form, the computer time consumption for the satisfaction of the inner boundary condition(s) is very small. Finally, similar basis functions can be obtained for a rectangular region with cavities. However these functions will not be in closed forms. 

Treating partially diffusion controlled reaction involves replacing the inner boundary condition such that pB(a,t) = 0 with a radiation boundary condition [7] of the form... 

Like the neutral case, the inner boundary condition must be replaced from Smoluchowski s condition to pb(a) = actPB- The required inner boundary condition for charged species then takes the form... 

Applying the inner boundary condition p a) = 0) gives B = Similarly using the second boundary condition such that p b) = 1, the coefficient A is obtained as follows... 

The transition density for pa (x, y, t) can be derived using the above procedure, with appropriate changes made to the inner boundary condition or can be derived by using the renewal theorem. In this appendix, the renewal theorem is used as it demonstrates how Pa x, y, t) can be derived without having to solve using the required boundary conditions. Recognising that the transition density Pa x, y, t) can be written using the renewal theorem of a diffusion process in Laplace space as... 

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