Diffusion-reaction problems

Example A reaction diffusion problem is solved with the finite difference method. 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

A recent work has demonstrated that the formulation of reaction-diffusion problems in systems that display slow diffusion within a continuous-time random walk model with a broad waiting time pdf of the form (6) leads to a fractional reaction-diffusion equation that includes a source or sink term in the same additive way as in the Brownian limit [63], With the fractional formulation for single-species slow reaction-diffusion obtained by the authors still being linear, no pattern formation due to Turing instabilities can arise. This is due to the fact that fractional systems of the type (15) are close to Gibbs-Boltzmann thermodynamic equilibrium as shown in the next section. 

Kenig EY, Kholpanov LP. Analysis of formulation and solution of multicomponent reaction-diffusion problems. Theor Found Chem Eng 1992 26 510-521. 

MC is also successful in far from equilibrium processes encountered in the areas of diffusion and reaction. It is precisely this class of non-equilibrium reaction/diffusion problems that is of interest here. Chemical engineering applications of MC include crystal growth (this is probably one of the first areas where physicists applied MC), catalysis, reaction networks, biology, etc. MC simulations provide the stochastic solution to a time-dependent master equation... 

In the following example, we use a simple microkinetic model of CO oxidation on Pt together with the reconstructed porous catalyst to follow the evolution of local concentration profiles within the porous structure. The reaction-diffusion problem of the CO oxidation on the Pt/y-Al203 porous catalyst... 

The reaction-diffusion problem (4) governing the wafers becomes ... 

Figure 56. The internal mass and charge transport as a reaction-diffusion problem.

Finally, the material balance on a slice of the catalyst pellet that is needed to completely specify the reaction/diffusion problem is ... 

The next example is a reaction-diffusion problem in a spherical domain. The reaction rate expression is a nonlinear function of concentration, of a type that is appropriate for the Michaelis-Menten reaction in biological systems. The nondimensional form of the problem is in Eq. (9.25), and it is solved for a = 5, X = 2. 

When the Fickian diffusion model is used many reaction-diffusion problems in porous catalyst pellet can be reduced to a two-point boundary value differential equation of the form of equation B.l. This is not a necessary condition for the application of this simple orthogonal collocation technique. The technique in principle, can be applied to any number of simultaneous two-point boundary value differential equations. 

In order to simplify the model, we assumed an effective diffusMty icould be used to describe diffusion in the catalyst particles. We next presented the general mass and energy balances for the catalyst particle. Next we solved a series of reaction-diffusion problems in a single... 

Consider the isothermal, first-order reaction-diffusion problem in a spherical pellet... 

Chapter 7. Chapter 7 considers the industrially important case of heterogeneous reactions taking place in solid catalyst particles. In the catalyst particle, we must consider the combined reaction-diffusion problem to be able to evaluate, the temperature and the species con-... 

For complex reactions involving many species, we cannot use the simple Thiele modulus and effectiveness factor approach, and must solve numerically the complete reaction-diffusion problem. These problems are challenging because of the steep pellet profiles that are possible. 

Finally, we showed several ways to couple the mass and energy balances over the fluid flowing through a fixed-bed reactor to the balances within the pellet. For simple reaction mechanisms, we were still able to use the effectiveness factor approach to solve the fixed-bed reactor problem. For complex mechanisms, we solved numerically the full problem given in Equations 7.84-7.97, We solved the reaction-diffusion problem in the pellet coupled to the mass and energy balances for the fluid, and we used the Ergun equation to calculate the pressure in the fluid. 

A. Bayliss, D. Gottlieb, B. J. Matkowsky, M. Minkoff, An Adaptive Pseudo-Spectral Method for Reaction Diffusion Problems, J. Comput. Phys. 81 (1989), 421. 

Figure 21 Two-dimensional grid for the simulation of reaction-diffusion problems. The five spatial points used in the discretization of the Lapladan in Eq. [95] are shown.

So far we have found probabilistic solutions to PDEs on the whole space. In fact ltd s formula allows us to represent the solutions to these equations in a bounded domain, e R, with appropriate boundary conditions. For example, let us consider the stationary reaction-diffusion problem... 

A quite general equation of reaction-diffusion problems is ... 

Elnashaie and Elshishini [7] described a sequential numerical algorithm to solve this type of models. However, in this particular case its implementation was not satisfactory. When resorting to Orthogonal Collocation strategies in the solution of reaction-diffusion problem, two type of polynomials can be adopted. When the nature of the problem implies symmetry conditions, symmetric polynomials are the proper choice. An equ ly accurate solution could be obtained resorting to non-syirunetric polynomials, but at the expense of a less efficient approach as the number of equations increase [8]. 

In order not to raise wrong expectations in all of our experiments, the direct numerical integration of the unprepared stiff ODE system was much faster than the integration of the split DAE system, since the splitting and associated transformation cost quite a bit. However, as stated already in the beginning, the purpose of the paper is to derive a reliable dimension reduction tool for application in the context of PDEs. The presentation, however, is much simpler in the ODE context. The eventual effect of the herein advocated methods on the actual numerical solution of challenging reaction-diffusion problems will be shown in a forthcoming paper. 

The first reduction of the two-scale model that we consider is already included in the equations from Table 3.1 and refers to the reaction-diffusion problem inside catalytic particles. The treatment of this question based on the effectiveness factor concept (rf) is widely generalized in the literature, after the seminal works of Damkohler [77], Thiele [78], and Zeldovitch [79]. It may be defined for a reaction j with respect to the conditions prevailing at the pellet surface by Ref. [80] ... 

We note the presence of the reactor scale in the heat transfer boundary condition (3.42b). The formulation in terms of the average solid temperature (7 ) is an approximation, which should be reasonable given that these terms are likely to be subdominant in Equation 3.42b. As in Table 3.2, it is possible to identify the timescales for the different processes in Equations 3.41 and 3.42. The axial conduction term in (3.42b) can be ignored for beds with Rh L. Combination of these scales yields some well-known parameters for the internal reaction-diffusion problem, such as... 

Catalytic monoliths are structured heterogeneous reactors. These two features require the consideration of the reaction-diffusion problem in the catalytic washcoat (the internal region), which can be designed with much more independence from the external domain, when compared to nonstructured reactors. In general, the operating conditions will be such that... 

Mell and Maloy [18] postulated a numerical approach to simulate the steady-state amperometric measurements for an enzyme electrode [18]. Since then, the reaction-diffusion problems describing biochemical processes are often solved numerically [6,7,19]. The analytical solutions are often applied to validation of the corresponding numerical solutions. 

The finite difference technique is a widely used numerical method for solving the reaction-diffusion problems [7]. When applying this approach, the model equations are transformed into a form such that the differentiation can be performed by numerical calculations. There are several difference schemes that can be used for... 

Fenstermacher, P. R., Swinney, H. L., Gollub, J. P. (1979) Dynamical instabilities and the transition to chaotic Taylor vortex flow. J. Fluid Mech. 94, 103 Field, R. J., KOrds, E., Noyes, R. M. (1972) Oscillations in chemical systems. II. Thorough analysis of temporal oscillations in the bromate-cerium-malonic acid system. J. Am. Chem. Soc. 94, 8649 Fife, P. C. (1976a) Singular perturbation and wave front techniques in reaction-diffusion problems. SIAM-AMS Proc. 10, 23... 

Bieniasz LK (2003) High order accurate one-sided finite-difference approximations to gradients at the boundaries, for the simulation of electrochemical reaction-diffusion problems in one-dimensional space geometry. Comput Biol Chem 27 315-325... 

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