A mathematical model, which simulates the complicated dynamic behavior experimentally observed during CO oxidation over Pd zeolite catalysts is presented. It describes the coupling of reaction rate oscillations, generated by various parts of the inhomogeneous catalytic layer through the gas phase. It can be shown, that the resulting dynamic behavior depends upon the difference between natural frequencies of local oscillators and the strength of coupling, which is defined mostly by the degree of conversion. Chaotic behavior could be identified under the condition of weak coupling for local oscillators with widely different natural frequencies. In the range of strong coupling the phenomenon of phase death has been obtained. A special type of intermittency chaos (“on–off” chaos) was observed in a small region of parameters under the conditions of strong coupling.
Self-similar solutions of the nonlinear heat equation with a three-dimensional source and density that varies as a power function of the radius are considered in planar, cylindrical, and spherical geometries. The self-similar solutions evolve in a blow-up setting and constitute time-dependent dissipative structures. The eigenfunction spectrum of the self-similar problem is investigated for various values of the model parameters by computational methods involving continuation in a parameter and bifurcation analysis. It is shown that the spectral problem may have a nonunique solution. We establish the number of eigenfunctions and their existence domain in the parameter space. The evolution of the eigenfunctions with changes in the parameter is examined. The stability of the self-similar solutions is shown to depend on the parameter values, the eigenfunction index, and the eigenfunction parity. New structurally stable and metastable self-similar solutions are obtained. The metastable solutions follow the self-similar law almost during the entire blowup time and preserve their complex structure as the temperature is increased by two orders of magnitude.
The article considers self-similar solutions of the nonlinear heat equation with a three-dimensional source that evolve in a blow-up setting. The self-similar problem is a boundary-value problem for a nonlinear equation of elliptical type that has a nonunique solution. We investigate the eigenfunction spectrum of the self-similar problem in two-and three-dimensional space. The problem is solved on a grid by Newton's iteration method. The implementation of Newton's method requires analysis of a linearized equation and construction of initial approximations. The eigenfunctions are continued in a parameter. Structures of various symmetry are obtained. New types of multidimensional structures are observed: these are multiply connected three-dimensional heat localization regions.
The oscillatory behavior during CO oxidation over Pd supported on glass "bers was studied in a recycle reactor. The properties of oscillations as a function of temperature and inlet CO concentration were investigated in detail. The peculiarity of the observed oscillations is their long period up to 6 h. Mathematical model considering oxidation}reduction processes of the Pd has been developed to describe the experimental results. The model accounts for the observed reaction rate dependence on the CO inlet concentration, the region of oscillations and the dependence of the oscillatory behavior upon temperature and CO inlet concentration.
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