Fractals

Two toy models for surface and interface disaggregation are introduced and some considerations on their relevance for real physico-chemical processes are presented. The models are studied by means of Monte Carlo simulations in 1+1 dimensions and the scaling laws of the interface width w(L, t) are determined. In both cases, the scaling is in agreement with that obtained from the fourth order linear Langevin equations. The result is discussed in relation to another microscopic disaggregation model and to the microscopic growth model of Wolf and Villain.

This article critically discusses the applications of fractal and disordered system theory to chemical engineering problems in order to highlight some promising research directions and the difficulties that may be encountered.

Starting from the analysis of transport and reaction kinetics, the question is addressed, with the aid of some examples, of whether and how engineering research could help in the study of complex phenomenologies on fractals and disordered systems. The effects of thermodynamical nonidealities in transport and adsorption, and the influence of nonlinearities in reaction kinetics are discussed in some detail. Examples of typical engineering problems in which fractal analysis may help towards a better understanding of the physical phenomenologies in the presence of complex porous substrata and fluid mixtures are discussed. The role played by the boundary conditions on transport phenomena involving fractal structures is also analyzed. A critical discussion on the perspectives in the characterization of disordered and fractal porous structures, and in the study of turbulent transport and mixing is also developed.

We present a review of the description of disordered porous media according to the Dual Site-Bond model introduced by Vicente Mayagoitia. The model formulation, percolation properties and applications to problems of practical interest are discussed.

Diffusion on self-similar structures is reviewed within a unified theoretical framework. Much attention is devoted to the asymptotic form of the probability density of random walks on fractals, for which analytical solutions are discussed. New predictions for the structure of percolation clusters at criticality are presented.

The first half of this paper reviews mixing in chaotic flows. The sine-flow is employed as a two dimensional example to demonstrate techniques used for characterizing mixing behavior. Several manifestations of self-similarity are readily apparent. The results of tracer mixing simulations demonstrate a self-similar, iterative development of partially mixed structures in the flow. The spatial distribution of mixing intensities present in the flow is examined via computation of stretching. The probability density function (PDF) of the logarithm of stretching values reveals a Gaussian distribution over the central spectrum of stretching intensities for the globally chaotic case, but contains two peaks for a case with coexisting chaotic and regular regions: a broad Gaussian peak for higher stretching values, corresponding to the chaotic region, and a sharp peak of low stretching values corresponding to the regular regions. The self-similar stretching distributions can be collapsed to a single invariant distribution using an appropriate scaling based on the central limit theorem. The folding processes in the flow are examined through curvature calculations; PDFs for curvature collapse to time invariant self-similar distributions without the need for scaling. Direct computation of the striation thickness distribution (STD) provides the most fundamental (and computationally most expensive) measure of mixing; STDs develop a self-similar form that can be collapsed to an invariant distribution using a simple scaling. The second half of the paper focuses on a real, three-dimensional mixing system: the Kenics static mixer. Two alternate configurations of the Kenics mixer were analyzed: one in which elements have alternating right-handed and left-handed twist (R-L) and a second in which all elements have right-handed twist (R-R). Poincaré sections as well as experiments indicate that the R-L configuration is globally chaotic, while the R-R configuration contains significant segregated, regular regions. Stretching histories of material elements in the two flows were computed, once again revealing self-similar distributions that can be collapsed to an invariant limit using a scaling based on the central limit theorem.

Turbulent intermittency plays a fundamental role in fields ranging from combustion physics and chemical engineering to meteorology. There is a rather general agreement that multifractals are being very successful at quantifying this intermittency. However, we argue that cascade processes are the appropriate and necessary physical models to achieve dynamical modeling of turbulent intermittency. We first review some recent developments and point out new directions which overcome either completely or partially the limitations of current cascade models which are static, discrete in scale, acausal, purely phenomenological and lacking in universal features. We review the debate about universality classes for multifractal processes. Using both turbulent velocity and temperature data, we show that the latter are very well fitted by the (strong) universality, and that the recent (weak, log-Poisson) alternative is untenable for both strong and weak events. Using a continuous, space-time anisotropic framework, we then show how to produce a causal stochastic model of intermittent fields and use it to study the predictability of these fields. Finally, by returning to the origins of the turbulent "shell models" and restoring a large number of degrees of freedom (the Scaling Gyroscope Cascade, SGC models) we partially close the gap between the cascades and the dynamical Navier–Stokes equations. Furthermore, we point out that beyond a close agreement between universal parameters of the different modeling approaches and the empirical estimates in turbulence, there is a rather common structure involving both a "renormalized viscosity" and a "renormalized forcing". We conclude that this gives credence to the possibility of deriving analytical/renormalized models of intermittency built on this structure.

Green function renormalization is applied to obtain exact recursions for integral quantities and integral transforms of concentration fields, involving a summation over a manifold of lattice sites or over the entire lattice. Definition of the Fourier transform of the Green functions, to some extent equivalent to the dynamic structure factor, is extremely useful to obtain quantitative information on the spatial behavior of concentration profiles and eigenfunctions. Some preliminary results on sorption properties of product lattices are presented. The application of anisotropic lattice models to study the volume-to-surface effects in adsorption and reaction is discussed. The properties of the Green function generating function are also analyzed.

Many porous catalysts have a fractal surface, but only rarely do they have a fractal volume, the main exceptions being extremely porous aerogels. It has been suggested that a fractal shape of their volume would be ideal, because it has an infinite area per unit mass that is easily accessible by the reactants. This paper investigates the efficiency of mass fractals by comparing them with nonfractal catalysts. It is found that the specific surface areas of comparable nonfractal catalysts are of the same order of magnitude, if not higher than those of mass fractals. Despite the high effectiveness factor of mass fractals due to the exceptionally easy accessibility of their active sites, production in a nonfractal catalyst is often higher than in a mass fractal, because of the high porosity of the latter. For some strongly diffusion limited reactions, especially in mesoporous catalysts, an added mass fractal macroporosity, with a finite scaling regime, would increase the yields beyond what is possible with a nonfractal catalyst. Nonetheless, when transport through viscous flow in macropores is very rapid the effective reaction rates in classical bimodal catalysts are higher than in fractal catalysts with their high macroporosity.

The hydrodynamic properties of fractal aggregates are investigated numerically by solving the three-dimensional Stokes equations. Aggregates are built with a variety of sizes and fractal dimensions. Their hydraulic radii conductivities and mobilities are evaluated as functions of their volume fraction in the suspension. An influence of the fractal dimension is visible on the hydrodynamic and electrokinetic characteristics of the aggregates. In the dilute limit, our results compare well with the predictions of various mean-field arguments.

A neural-network-based model that has learnt the chaotic hydrodynamics of a fluidized bed reactor is presented. The network is trained on measured electrical capacitance tomography data. A training algorithm is used that does not only minimize the short-term prediction error but also the information needed to synchronize the model with the real system. This forces the model to focus more on learning the longer term dynamics of the system, expressed in the average multi-step-ahead prediction error and dynamic invariants such as correlation entropy and dimension. The availability of the model is an important step towards control of chaos in gas-solid fluidized beds.

The nonlinear dynamic behavior of a Proportional-Integral controlled Continuously Stirred Tank Reactor (CSTR) is analyzed in depth progressing from chaos characterization, through the high codimension bifurcation theory, up to the application of Controlling Chaos techniques.

All these tools can be successfully applied to recognize, to avoid and to use chaos in practical applications, so that the nonlinear dynamic theory turns out to be an indispensable science to constrain dynamic systems to work in the most suitable operative conditions.

This paper presents a condensed exposition of some of the research work on the bifurcation and chaotic behavior of industrially and fundamentally important chemical and biochemical reaction engineering systems. Two cases are presented: a forced industrial fluid catalytic cracking unit and an enzymatic system. The Poincare bifurcation diagrams obtained for both cases reveal their richness in dynamic behavior. Various routes to chaotic behavior have been identified including period doubling, intermittency, crisis.