Nonlinear diffusion equation

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

It would be of considerable interest to extend the technique just presented to problems involving nonlinear equations because there are many situations in ultracentrifugation where nonideality is a dominant feature. Furthermore, it is known (4, 14) that even for two-component systems with nonideality the theory for estimating the sedimentation constant based on a diffusion-free (c = 0) approximation can lead to systematic error. Therefore, the development of an approximate procedure for nonlinear equations would be useful for further progress in analytical separation methods. 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

The general solutions of the fundamental systems of nonlinear equations [Eq. (2)] will be of the type wherein the state variables are dependent both on time and space, which will manifest in the form of wave propagation. Coupling between several parts of the system will be transmitted through the generalized diffusion coefficient D. If the associated transport process proceeds on a time scale comparable to or slower than the period of the temporal oscillation, macroscopic wave propagation phenomena are to be expected, as, for example, realized with the Belousov-Zhabotinsky... 

Figure 6. (a) The force per unit area between two identical plates with grafted polyelectrolyte brushes, as a function of the separation distance, for N = 1000, a = 1 A, ce = 0.01 M, s2 = 1000 A2, e = 80, <7 = 100 A r = -1.0, w = 0.0, and v = 1.0 A3 calculated in the linear approximation is compared to the numerical result obtained from the nonlinear equations, assumingthatthe counterions diffuse into an infinite reservoir (slow approach), (b) The force per unit area between two identical plates with grafted polyelectrolyte brushes, as a function of the separation distance, for N = 1000, a = 1 A, ce... 

When two electrolyte solutions at different concentrations are separated by an ion--permeable membrane, a potential difference is generally established between the two solutions. This potential difference, known as membrane potential, plays an important role in electrochemical phenomena observed in various biomembrane systems. In the stationary state, the membrane potential arises from both the diffusion potential [1,2] and the membrane boundary potential [3-6]. To calculate the membrane potential, one must simultaneously solve the Nernst-Planck equation and the Poisson equation. Analytic formulas for the membrane potential can be derived only if the electric held within the membrane is assumed to be constant [1,2]. In this chapter, we remove this constant held assumption and numerically solve the above-mentioned nonlinear equations to calculate the membrane potential [7]. 

Figure 2. Values of S(t) at t=(2DR)" for some translational diffusion coefficients. Circles show values by the nonlinear equation for z=0- l, diamonds show values by the linearized equation, triangles show ratios between them.

Mass transfer model that accounted for the mass transfer both inside and outside the emulsion globules, the reaction between the diffusing component and the internal reagent in the globules jointly. A perturbation solution to the resulting nonlinear equations contained the parameters Bi and Da. 

These nonlinear equations define the fluxes u, Q and q implicitly, provided that u h, h) and q(h), are given at a certain reference height h. Here v = / i denotes the kinematic velocity scale, T the in situ temperature, g the gravity acceleration, and Ka the diffusivity. Details on the transfer functions for momentum M and scalar quantities S are given for example in Large (1981). For a recent discussion in the light of nonequilibrium thermodynamics, see Csanady (2001). Because M and 5 are nonlinear, an analytical solution is not known. Instead, parameterizations are commonly used. With the notation... 

Diffusion and mass transfer in multicomponent systems are described by systems of differential equations. These equations are more easily manipulated using matrix notation and concepts from linear algebra. We have chosen to include three appendices that provide the necessary background in matrix theory in order to provide the reader a convenient source of reference material. Appendix A covers linear algebra and matrix computations. Appendix B describes methods for solving systems of differential equations and Appendix C briefly reviews numerical methods for solving systems of linear and nonlinear equations. Other books cover these fields in far more depth than what follows. We have found the book by Amundson (1966) to be particularly useful as it is written with chemical engineering applications in mind. Other books we have consulted are cited at various points in the text. 

Methods for solving mass and heat transfer problems. The convective diffusion equation (3.1.1) is a second-order linear partial differential equation with variable coefficients (in the general case, the fluid velocity depends on the coordinates and time). Exact closed-form solutions of the corresponding problems can be found only in exceptional cases with simple geometry [79,197, 270, 370, 516]. This is especially true of the nonlinear equation (3.1.17). Exact solutions are important for adequate understanding of the physical background of various phenomena and processes. They can serve as test solutions to verify whether the problem is well-posed or to estimate the accuracy of the corresponding numerical, asymptotic, and approximate methods. 

NONLINEAR EQUATION SOLVER PROGRAM Liquid-phase diffusivity of benzene at 55°C (cm /s) ... 

Almost parallel to McKendrick, Hutchinson [215], a well-known ecologist, proposed a time-delayed version for the logistic growth equation, where the nonlinear term was delayed in time. The diffusive Hutchinson equation, also known as the delayed Fisher equation. 

Consequently, if the linearized equations of a reaction-diffusion equation in the neighborhood of a homogeneous steady state are given by Eq. (38), provided Eq. (44) also holds, there must be in addition at least two inhomogeneous steady states with an even number of positive eigenvalues in the original nonlinear equation. 

The effect of laser phase fluctuations on PIER4 has been considered by Agarwal, and detailed line shapes were calculated. In a recent publication we have derived (within the phase diffusion model) equations of motion in the limit of short correlation times. The effect of the stochastic phase fluctuations was shown to be similar to T2 dephasing processes, and a procedure was given for the inclusion of this similarity in many nonlinear processes. In particular, two predictions were made ... 

The nonlinear equations for the adiabatic tubular reactor with diffusion and first-order reaction are given by Clough and Ramirez (1971)... 

Diffusion-Type Mass Transfer Models for Type 1 FacUitation. The state-of-the-art model for Type 1 facilitation is the advancing front model (2,7,8), In this model, the solute is assumed to react instantaneously and irreversibly with the internal reagent at a reaction surface which advances into the globule as the reagent is consumed. A perturbation solution to the resulting nonlinear equations is obtained. In general, the zero-order or pseudo-steady-state solution alone often gives an adequate representation of the diffusion process. 

In the complex systems approach, the microscopic level of interacting neurons should be modeled by coupled differential equations modeling the transmission of nerve impulses by each neuron. The Hodgekin-Huxley equation is an example of a nonlinear diffusion reaction equation with an exact solution of a traveling wave, giving a precise prediction of the speed and shape of the nerve impulse of electric... 

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. 

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