Equations of diffusion and reaction

In reacting systems, transfer of matter and heat occurs by bulk flow and diffusion or conduction. Usually transfer in an axial direction is appreciable by bulk flow only. In a rectangular region the various elements of a material balance in one dimensional flow are, 

Similar equations may be developed for other geometries such as spheres and cylinders. To complete the mathematical representation of a problem, initial and boundary conditions are specified. 

Dimensionless variables. Equations may be more convenient to handle in terms of the dimensionless variables, 

A large number of analytical solutions of these equations appear in the literature. Mostly, however, they deal only with first order reactions. All others require solution by numerical or other approximate means. In this book, solutions of two examples are carried along analytically part way in P7.02.06 and P7.02.07. Section 7.4 considers flow through an external film, while Section 7.5 deals with diffusion and reaction in catalyst pores under steady state conditions. 

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The following is an example for the development of a distributed parameter model for the yeast floe in the alcoholic fermentation process. The model takes into consideration the external mass transfer resistances, the mass transfer resistance through the cellular membrane, and the diffusion resistances inside the floe. The two-point boundary-value differential equations for the membrane are manipulated analytically, whereas the nonlinear two-point boundary-value dilferential equations of diffusion and reaction inside... 

Similar considerations concern the irreversible processes of diffusion and reaction in mixtures [5]. A system of M different molecular species is described by the three components of velocity, the mass density, the temperature, and (M — 1) chemical concentrations and is ruled by M + 4 partial differential equations. The M — 1 extra equations govern the mutual diffusions and the possible chemical reactions... 

The study of the intra-phase mass transfer in SCR reactors has been addressed by combining the equations for the external field with the differential equations for diffusion and reaction of NO and N H 3 in the intra-porous region and by adopting the Wakao-Smith random pore model to describe the diffusion of NO and NH3 inside the pores [30, 44]. The solution of the model equations confirmed that steep reactant concentration gradients are present near the external catalyst surface under typical industrial conditions so that the internal catalyst effectiveness factor is low [27]. 

The special form of second-order equation in which the right-hand side is a function only of the dependent variable also turns up in the theory of diffusion and reaction in a slablike particle. Corresponding to equations (123-125) for the sphere, we would have, thanks to the reduction described in Chapter 2 and the example of a first-order nonisothermal reaction given by Eq. (129),... 

While we have contact with this problem, however, we will look at the asymptotics of reactions in a slab of catalyst. In Chapter 3 we used the special form of the equations for diffusion and reaction to get rather general expressions for the Thiele modulus and effectiveness in terms of the center concentration v... 

In the absence of catalysis on the surface, similarity of the concentration and temperature fields is achieved precisely at the ignition limit if the coefficients of diffusion and thermal diffusivity are equal, since in this case both the diffusion gradient and the temperature gradient at the igniting surface are equal to zero, and the equations of diffusion and thermal conductivity with the chemical reaction may be reduced to the form of an identity (see our work on flame propagation [3]). 

This similarity was established in [2] by consideration of the second-order differential equations of diffusion and heat conduction. Under the assumptions made about the coefficient of diffusion and thermal diffusivity, similarity of the fields, and therefore constant enthalpy, in the case of gas combustion occur throughout the space this is the case not only in the steady problem, but in any non-steady problem as well. It is only necessary that there not be any heat loss by radiation or heat transfer to the vessel walls and that there be no additional (other than the chemical reaction) sources of energy. These conditions relate to the combustion of powders and EM as well, and were tacitly accounted for by us when we wrote the equations where the corresponding terms were absent. 

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... 

Derivation of the Differential Equation Describing Diffusion and Reaction 741... 

Dimensionless form of equations describing diffusion and reaction... 

So far we have only considered the irreversible first order reaction A —> B. But we have seen that when adsorption effects are considered even a first order reaction can have more complicated kinetics. For any form of kinetics the equation for diffusion and reaction can be solved for the slab geometry. We shall give the general method and illustrate it by the already familiar case of the first order reaction. 

The following equation governs diffusion and reaction of carbon monoxide in an... 

Closure After completing this chapter, the reader should be able to derive differential equations describing diffusion and reaction, discuss the meaning of the effectiveness factor and its relationship to the Thiele modulus, and identify the regions of mass transfer control and reaction rate control. The reader should be able to apply the Weisz-Prater and Mears criteria to identify gradients and diffusion limitations. These principles should be able to be applied to catalyst particles as well as biomaierial tissue engineering. The reader should be able to apply the overall effectiveness factor to a packed bed reactor to calculate the conversion at the exit of the reactor. The reader should be able to describe the reaction and transport steps in slurry reactors, trickle bed reactors, fluidized-besd reactors, and CVD boat reactors and to make calculations for each reactor. 

In all cases, it is essential to solve the model equations efficiently and accurately. Some techniques are discussed in this book and in the appendices, for the solution of the highly non-linear algebraic, differential and integral equations arising in the modelling of fixed bed catalytic reactors. The most difficult equations to solve are usually the equations for diffusion and reaction in the porous catalyst pellets, especially when diffusional limitations are severe. The orthogonal collocation technique has proved to be very efficient in the solution of this problem in most cases. In cases of extremely steep concentration and temperature profiles inside the pellet, the effective reaction zone method and its more advanced generalization, the spline collocation technique, prove to be very efficient. 

With this more general equation, representation of the effectiveness factor in terms of the Thiele modulus also becomes possible for single-file diffusion. As an example. Fig. 16 shows the result of a computer simulation of diffusion and reaction within a single-file system consisting of A = 100 sites for different occupation numbers in comparison with the dependence... 

We will start by considering the simple case of diffusion and reaction described by equation (7-2). Let us further assume that the catalytic surface is being deactivated by some type of coking mechanism, with the reaction scheme corresponding to (XXIII) of Chapter 3. We will consider two limiting cases of that scheme. 

Numerical Integration of the Mass Transfer Equation with Diffusion and Reaction... 

Our understanding of diffusion and reaction in single-file systems is impaired by the lack of a comprehensive analytical theory. The traditional way of analytically treating the evolution of particle distributions by differential equations is prevented by the correlation of the movement of distant particles. One may respond to this restriction by considering joint probabilities covering the occupancy and further suitable quantities with respect to each individual site. These joint probabilities may be shown to be subject to master equations. 

If the one-equation model of diffusion and reaction in a micropore-macropore system is not valid, one needs to proceed dovm the hierarchy of length scales to develop an analysis of the transport process in both the macropore region and the micropore region. This leads to yet another averaging volume that is illustrated as level III in Figure 1.4. Analysis at this level leads to a micropore effectiveness factor that is discussed by Carberry (1976, Sec. 9.2) and by Froment and Bischoff (1979, Sec. 3.9). 

We consider a two-phase system consisting of a fluid phase and a solid phase as illustrated in Figure 1.7. Here we have identified the fluid phase as the y-phase and the solid phase as the K-phase. The foundations for the analysis of diffusion and reaction in this two-phase system consist of the species continuity equation in the y-phase and the species jump condition at the catalytic surface. The species continuity equation can be expressed as... 

Here we have emphasized the intrinsic nature of our area-averaged transport equation, and this is especially clear with respect to the last term which represents the rate of reaction per unit volume of the fluid phase. In the study of diffusion and reaction in real porous media (Whitaker, 1986a, 1987), it is traditional to work with the rate of reaction per unit volume of the porous medium. Since the ratio of the fluid volume to the volume of the porous medium is the porosity, i.e. 

This result forms the basis for the classic problem of diffusion and reaction in a porous catalyst such as we have illustrated in Figure 1.5. It is extremely important to recognize that the mathematical consequence of equations 1.95 and 1.96 is that the mass average velocity has been set equal to zero thus our substitute for equation 1.38 is given by... 

It is not obvious, but other studies (Ryan etal., 1981) have shown that the reaction source in equations 1.124 and 1.125 makes a negligible contribution to c y. In addition, one can demonstrate (Whitaker, 1999) that the heterogeneous reaction, k CAy, can be neglected for all practical problems of diffusion and reaction in porous catalysts. Furthermore, the non-local diffusion term is negligible for traditional systems, and under these circumstances the boundary value problem for the spatial deviation concentration takes the form... 

Note that in the previous two cases, [/42]b was assumed to be either zero or equal to its interfacial concentration (solubility) [A2]. The situation gets quite complicated mathematically when [/42]b has a finite value between zero and the solubility, that is, 0 < [/42)b < To write the absorption equations for this case, knowledge of the concentration profiles in the films is necessary. Ramachandran and Sharma (1971) assumed that the profiles are linear, based on which an approximate solution was obtained. The assumption of a linear profile is only a first-order approximation, because the simultaneous occurrence of diffusion and reaction can lead to a nonlinear profile in the film. This problem was addressed by Sada et al. (1976) and Chaudhari and Doraiswamy (1974) who assumed certain nonlinear profiles and obtained analytical solutions that gave results comparable to those from numerical solution. 

A front corresponds to a traveling wave solution, which maintains its shape, travels with a constant velocity v, p x, t) = p(x - v t), and joins two steady states of the system. The latter are uniform stationary states, p(x, t) = p, where Ffp) = 0. For the logistic kinetics, the steady states are = 0 and jo2 = 1- While the logistic kinetics has only two steady states, three or more stationary states can exist for a broad class of systems in nonlinear chemistry and population dynamics with Alice effect, but a front can only connect two of them. To determine the propagation direction of the front, we need to evaluate the stability of the stationary states, see Sect. 1.2. The steady state jo is stable if P (fp) < 0 and unstable if F (jo) > 0. Let the initial particle density p x,0) be such that on a certain finite interval, p x,0) is different from 0 and 1, and to the left of this interval p(x,0) = 1, while to the right p x, 0) = 0. In this case, the initial condition is said to have compact support. Kolmogorov et al. [232] showed for Fisher s equation that due to the combined effects of diffusion and reaction, the region of density close to 1 expands to the... 

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