Diffusion system equation

By definition chirality involves a preferred sense of rotation in a three-dimensional space. Therefore, it can only be affected by a modification of the nonscalar fields appearing in the rate equations. For a reaction-diffusion system [equations (1)] these fields are descriptive of a vector irreversible process, namely, the diffusion flux J of constituent k in the medium. According to irreversible thermodynamics, the driving force conjugate to diffusion is... 

As the resulting amplitude equations. Equation (15), have a lower dimension they are simpler to analyze than the original reaction-diffusion systems [Equation (1)] remembering however that they are only valid in some neighbourhood of the point where the reference state linearly looses its stability. Because of this relative simplicity they allow to scrutinize the key nonlinear effects that govern the structure of the bifurcation diagram. In the first place however they allow one to obtain the bifurcated solutions and to discuss their stability. 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Reviews of concentration polarization have been reported (14,38,39). Because solute wall concentration may not be experimentally measurable, models relating solute and solvent fluxes to hydrodynamic parameters are needed for system design. The Navier-Stokes diffusion—convection equation has been numerically solved to calculate wall concentration, and thus the water flux and permeate quaUty (40). 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

In practice, the Peclet number can always be ignored in the diffusion-convection equation. It can also be ignored in the root boundary condition unless C > X/Pc or A, < Pe. Inspection of the table of standard parameter values (Table 2) shows that this is never the case for realistic soil and root conditions. Inspection of Table 2 also reveals that the term relating to nutrient efflux, e, can also be ignored because e < Pe [Pg.343]

The diffusion system. Figure 8.31(B), is a useful and simple apparatus for preparing mixtures of volatile and moderately volatile vapors in a gas stream [388]. The method is based on the constant diffusion of a vapor from a tube of accurately known dimensions, producing a gas phase concentration described by equation (8.12). 

This is Fick s second law of diffusion, the equation that forms the basis for most mathematical models of diffusion processes. The simple form of the equation shown above is applicable only to diffusion in one dimension (x) in systems of rectangular geometry. The mathematical form of the equation becomes more complex when diffusion is allowed to occur in more than one dimension or when the relationship is expressed in cylindrical or spherical coordinate geometries. Since the simple form shown above is itself a second-order partial differential equation, the threat of added complexity is an unpleasant proposition at best. 

The two-state mobility model82 83 was used to describe such system, Equation (23), which states that the phenol self-diffusion coefficient is a weighted average of free and micelle-bound ones ... 

At the mesoscopic level, the reaction-diffusion system is described by a set of partial differential equations,... 

The equations are closed by appealing to the observational result that r = 4 for these diffuse systems, whence... 

An interesting problem arises when we consider solutions or colloidal sols where the diffusing component is much larger in size than the solute molecules. In dilute systems Equation (1.14) would give an adequate value of the Peclet number but not so when the system becomes concentrated, i.e. the system itself becomes a condensed phase. The interactions between the diffusing component slow the motion and, as we shall see in detail in Chapter 3, increase the viscosity. The appropriate dimensionless group should use the system viscosity and not that of the medium and now becomes... 

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. 

We have shown that the microscopic expression for the polymer diffusion coefficient. Equation 2, is the starting point for a discussion of diffusion in a wide range of polymer systems. For the example worked out, polymer diffusion at theta conditions, the resulting expresssion describes the experimental data without adjustable parameters. It should be possible to derive expressions for diffusion... 

Equations were derived to predict response times and the expected steady state current for a pure diffusion system. 

The computer-reconstructed catalyst is represented by a discrete volume phase function in the form of 3D matrix containing information about the phase in each volume element. Another 3D matrix defines the distribution of active catalytic sites. Macroporosity, sizes of supporting articles and the correlation function describing the macropore size distribution are evaluated from the SEM images of porous catalyst (Koci et al., 2006 Kosek et al., 2005). Spatially 3D reaction-diffusion system with low concentrations of reactants and products can be described by mass balances in the form of the following partial differential equations (Koci et al., 2006, 2007a). For gaseous components ... 

Equation (26) is a differential equation with a solution that describes the concentration of a system as a function of time and position. The solution depends on the boundary conditions of the problem as well as on the parameter D. This is the basis for the experimental determination of the diffusion coefficient. Equation (26) is solved for the boundary conditions that apply to a particular experimental arrangement. Then, the concentration of the diffusing substance is measured as a function of time and location in the apparatus. Fitting the experimental data to the theoretical concentration function permits the evaluation of the diffusion coefficient for the system under consideration. 

For a CSTR the stationary-state relationship is given by the solution of an algebraic equation for the reaction-diffusion system we still have a (non-linear) differential equation, albeit ordinary rather than partial as in eqn (9.14). The stationary-state profile can be determined by standard numerical methods once the two parameters D and / have been specified. Figure 9.3 shows two typical profiles for two different values of )(0.1157 and 0.0633) with / = 0.04. In the upper profile, the stationary-state reactant concentration is close to unity across the whole reaction zone, reflecting only low extents of reaction. The profile has a minimum exactly at the centre of the reaction zone p = 0 and is symmetric about this central line. This symmetry with the central minimum is a feature of all the profiles computed for the class A geometries with these symmetric boundary conditions. With the lower diffusion coefficient, D = 0.0633, much greater extents of conversion—in excess of 50 per cent—are possible in the stationary state. 

Equation 6 assumes that the film is in equilibrium with a thin region immediately beneath the monolayer, and that film desorption is limited by diffusion. Before Equation 6 can be used, its validity must be established for each system studied. A theoretical relation developed previously (11) allows one to test whether these conditions hold, and thus whether Equation 6 is applicable. It has been shown, for the conditions... 

We shall now illustrate these aspects through the far-from-equilibrium reaction-diffusion systems, which have been studied in great detail in recent years and which may be taken as a kind of archetype in this context. If chemical reaction and diffusion are the only relevant dissipative processes to be considered, the equations that describe the spatiotemporal variation of the concentrations X (Xt, X2,. . . , A ) are... 

In the linear approximation given by equations (Al), modes selected for amplification grow without bound. The nonlinear reaction-diffusion system... 

The mass transport of the different species in solution is described by the diffusive differential equation system ... 

When a constant potential, E, is applied to the electrode immersed in the solution containing species A and L such that the electron transfer reactions take place, the mass transport supposed by pure diffusion to and from the electrode surface, in the presence of an excess of supporting electrolyte, is described by the following differential diffusive-kinetic equations system ... 

Stute et al. (1992) made numerical calculation to study flow dynamics of the Great Hungarian Plain (GHP) aquifer system with the use of He concentration data. They solved the diffusion-advection equation for He transport for a two-dimensional case. 

Figure 3 summarizes the typical equilibrium data for fluoride sorption on 0.150-0.300 mm surface-tailored zeolite F9. Only the low-concentration range was considered where the Henry s law is applicable. Co-incidentally, fluoride ions exist in natural systems, such as groundwater, at low concentrations. It is observed that the data fit well to the linear isotherm suggesting sorption onto sites with high capacity for fluoride. From Fig. 3, the linear isotherm constant, K ( = 2.337 L/g), was obtained and coupled into the diffusion model, equation (16). 

After the kinetic model for the network is defined, a simulation method needs to be chosen, given the systemic phenomenon of interest. The phenomenon might be spatial. Then it has to be decided whether in addition stochasticity plays a role or not. In the former case the kinetic model should be described with a reaction-diffusion master equation [81], whereas in the latter case partial differential equations should suffice. If the phenomenon does not involve a spatial organization, the dynamics can be simulated either using ordinary differential equations [47] or master equations [82-84]. In the latter case but not in the former, stochasticity is considered of importance. A first-order estimate of the magnitude of stochastic fluctuations can be obtained using the linear noise approximation, given only the ordinary differential equation description of the kinetic model [83-85, 87]. 

The above analysis and Fig. 19-25 provide a theoretical foundation similar to the Thiele-modulus effectiveness factor relationship for fluid-solid systems. However, there are no generalized closed-form expressions of E for the more general case ofa complex reaction network, and its value has to be determined by solving the complete diffusion-reaction equations for known intrinsic mechanism and kinetics, or alternatively estimated experimentally. 

---