One-dimensional reaction-diffusion

When applying a mechanistic model, nearly all of the computational effort resides in step (3).109 In most mechanistic models, step (3) is modeled by one-dimensional reaction-diffusion equations of the form... 

As discussed in Section 4.3, the linear-eddy model solves a one-dimensional reaction-diffusion equation for all length scales. Inertial-range fluid-particle interactions are accounted for by a random rearrangement process. This leads to significant computational inefficiency since step (3) is not the rate-controlling step. Simplifications have thus been introduced to avoid this problem (Baldyga and Bourne 1989). 

We start with the one-dimensional reaction-diffusion equation... 

One-dimensional reaction-diffusion equations (RD-approach) offer a more adequate model than the rj-approach to account for mass transport in the washcoat. The model calculates spatial variations of concentrations and surface reaction rates inside the washcoat. It assumes that the species flux inside the pores is only due to diffusion (Stutz and Pouhkakos, 2008). Therefore, it neglects the convective fluid flow inside the porous layer, because of very low permeabihty assumption (Stutz and Pouhkakos, 2008). Eventually, each gas-phase species leads to one reaction-diffusion equation in the RD-approach, which is written in the transient form, as (Deutschmann, 2011b Deutschmann et al., 2014 Karadeniz et al., 2013)... 

As pointed out in the previous section, the spatially extended open Couette flow reactor [27-33] provides a practical implementation of an effectively one-dimensional reaction-diffusion system with an external concentration gradient imposed from the boundaries. With the specific motivation to provide theoretical and numerical support for the recent experimental observations of sustained dissipative structures in the Couette flow reactor, we will consider the standard reaction-diffusion equation ... 

The ratio vJD can then be used to calculate a chemical reaction rate for a nonconservative solute, S. To do this, the one-dimensional advection-diffusion model is modified to include a chemical reaction term, J. This new equation is called the one-dimensional advection-diffusion-reaction model and has the following form ... 

If the solute imdergoes any chemical changes, a reaction term must be added to Eq. 12.4. In the absence of specific rate law information, diagenetic reactions are generally assumed to be first-order with respect to the solute concentration. Thus, the one-dimensional advection-diffusion equation far a nonconservative solute is given by... 

These solutions to the one-dimensional advection-diffusion model can be used to estimate reaction rate constants Ck) from the pore-water concentrations of S, if and s are known. More sophisticated approaches have been used to define the reaction rate term as the sum of multiple removals and additions whose functionalities are not necessarily first-order. Information on the reaction kinetics is empirically obtained by determining which algorithmic representation of the rate law best fits the vertical depth concentration data. The best-fit rate law can then be used to provide some insight into potential... 

As we saw with the steady-state water-column application of the one-dimensional advection-diffusion-reaction equation (Eq. 4.14), the basic shapes of the vertical concentration profiles can be predicted from the relative rates of the chemical and physical processes. Figure 4.21 provided examples of profiles that exhibit curvatures whose shapes reflected differences in the direction and relative rates of these processes. Some generalized scenarios for sedimentary pore water profiles are presented in Figure 12.7 for the most commonly observed shapes. 

There are two methods to write the diffusion equation for a multispecies component. One is to write the diffusion equation for the conserved component, and then relate the species concentrations by the reaction(s). Using one-dimensional H2O diffusion as an example, the diffusion equation is Equation 3-22a ... 

The catalyst intraparticle reaction-diffusion process of parallel, equilibrium-restrained reactions for the methanation system was studied. The non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components have been established, and solved using an orthogonal collocation method. The simulation values of the effectiveness factors for methanation reaction Ch4 and shift reaction Co2 are fairly in agreement with the experimental values. Ch4 is large, while Co2 is very small. The shift reaction takes place as direct and reverse reaction inside the catalyst pellet because of the interaction of methanation and shift reaction. For parallel, equilibrium-restrained reactions, effectiveness factors are not able to predict the catalyst internal-surface utilization accurately. Therefore, the intraparticle distributions of the temperature, the concentrations of species and so on should be taken into account. 

The experimentally-determined effectiveness factor is determined as the ratio of the experimental macro reaction rate to the intrinsic reaction rate under the same interface (bulk) composition and temperature. Based on the experimental conditions of the macrokinetics, the predicted effectiveness factors of the methanation reaction and the WGSR are obtained by solving the above non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components. Table 1 shows the calculated effectiveness factors and the experimental values. By... 

It is clear in the present system that ET occurs much faster than the diffusive solvation process. To explain exponential and non-exponential kinetics of ET, the idea of Sumi-Marcus two-dimensional reaction coordinate is used. In this treatment, instead of the usual one dimensional reaction coordinate (solvent coordinate), two coordinates are used, i.e., the solvent coordinate and the vibrational nuclear coordinate. A free energy surface is drawn in a two-dimensional plane spanned by the solvent coordinate, X, and the nuclear coordinate, q. 

Figure 16. A double-well potential for reaction IV X) along a one-dimensional reaction coordinate X in the Kramers model, and a reactive diffusive trajectory represented by a zigzag line surmounting a reaction barrier from the reactant to the product well.

Magnesium concentrations as a function of depth (meters below the sea floor) in sediment porewaters from the western flank of the Juan de Fuca Ridge near 48° N in the North Pacific Ocean. The concentration decreases with depth because it is removed from solution by reaction with crustal rocks at the sediment-crustal boundary. The curves are convex upward because of porewater upwelling along the upward-flowing limb of a convection cell. Velocities of the upwelling are determined by using a one-dimensional advection-diffusion model and are indicated by the numbers on the curves. Redrafted from Wheat and MottI (2000). 

We derived in Chap. 4 an evolution equation for slowly varying wavefronts in two-dimensional reaction-diffusion systems. Quite analogously to the dynamics of oscillatory systems with a slowly varying phase pattern, we obtained an asymptotic expansion (4.3.28). If a happened to be small and negative, while the other parameters were of ordinary magnitude and y positive, then the same reasoning advanced in Sect. 7.2 applies, and we get the one-dimensional phase turbulence equation... 

Generally, one has to consider two-dimensional diffusion equations to incorporate the relaxation dynamics from both slow x and fast q coordinates. In the case where the fluctuations of the intramolecular vibrational modes are very fast compared with the solvent relaxation, the adiabatic elimination pro-cedure can lead the two-dimensional diffusive equations to onedimensional ones. Denoting P x,i) and P2 x,t) as the population distributions of the donor and the acceptor states at a given x and time t, respectively, one obtains the one-dimensional coupled diffusion-reaction equations ... 

For the simplest one-dimensional or flat-plate geometry, a simple statement of the material balance for diffusion and catalytic reactions in the pore at steady-state can be made that which diffuses in and does not come out has been converted. The depth of the pore for a flat plate is the half width L, for long, cylindrical pellets is L = dp/2 and for spherical particles L = dp/3. The varying coordinate along the pore length is x ... 

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