Limits Of Instability Research Articles

Stability analysis of steady-state solutions for porous catalytic pellets: Influence of the shape of the pellet

Based on the established method of Berezowski and Burghardt (1993, Chem. Engng Sci. 48, 1517–1534) for analysing the bifurcation to oscillatory solutions, the effect of the pellet shape on the occurrence of oscillatory destabilization is studied. A linear analysis of the dynamics of the system studied led to a relation between the critical values of the Lewis number (i.e. those above which an oscillatory destabilization becomes possible) and the parameters of the system (γ, β ∗ , θ 0). The nonlinear analysis confirmed fully the results of the linear approach. An approximate analysis of the system of differential equations describing the process enabled analytical relations to be determined between the parameters of the system which define the limits of the oscillatory instability (Hopf bifurcation), limits of saddle-type instability and an approximate formula to determine the critical values of the Lewis number. The results of extensive calculations reveal a considerable increase in the critical value of the Lewis number when the pellet shape is changed from the infinite slab to infinite cylinder to sphere. It may therefore be concluded that the infinite slab and sphere represent the limiting dynamic characteristics with respect to the bifurcation to oscillatory solutions. The susceptibility of other shapes to oscillatory destabilization should lie between these two limiting dynamic characteristics corresponding to the sphere and infinite slab.

Stability analysis of steady-state solutions for porous catalytic pellets: influence of the Lewis number

A stability analysis concerning periodic bifurcation of the steady-state solutions has been performed for a model which describes transport phenomena and chemical reaction in a porous catalyst pellet. A linear analysis of the dynamics of the system studied led to a relation between the critical values of the Lewis number Le c (i.e. those above which an oscillatory destabilization becomes possible) and the parameters of the system (γ, β*,θ o). The nonlinear analysis, while confirming the results of the linear analysis, revealed a certain particular characteristic of the system. It has been found that for certain steady states a range of Lewis numbers, Le*< Le< Le c , exists for which both stable and unstable limit cycles may appear. The unstable limit cycle divides the phase plane into two regions with two attractors—the stable steady state and the stable limit cycle. An approximate analysis of the system of differential equations describing the process enabled us to establish analytical relations between the parameters of the system, which determine the limits of oscillatory instability (Hopf bifurcation), the limits of saddle-type instability and an approximate formula to determine the critical value of the Lewis number. The results of calculations concerning the limits of these regions show a perfect agreement with the numerical predictions. The possibility of an exact determination of the limits of the oxcillatory instability region, without resorting to tedious and time-consuming numerical calculations, seems to be extremely useful in modelling the processes taking place in heterogeneous chemical reactors. The values of the Lewis number typical of the actual processes in the reactors have been estimated for varying operating parameters. It may be seen that the feasible values include the critical values of Le obtained in this paper.

One-dimensional model for the Rijke tube

We develop a simple model in which longitudinal, compressible, unsteady heat transfer between heater and gas is computed in the small-Mach-number limit. This calculation is used to determine the transfer function of the heater, which plays an important role in the stability limits of the thermoacoustic instability of the Rijke tube. The transfer function is determined analytically in the limit of small expansion parameter γ, and numerically for γ of order unity. In the case ρμ/cp = constant, an analytical solution can be found.

Boundaries of interchannel flow instability in channels with intermediate collectors

Results are given of an experimental and theoretical determination of the limits of interchannel instability in the parallel channels of reactor cooling systems and steam generators under conditions of unsteady vapor and liquid flow. Experimental results were derived from a system of two geometrically identical channels with vertical coolant flow. Uniform heating was obtained with an alternating current heater and the coolant used was distilled water. The intermediate collector was a tube with a pressure-sampling chamber. The model derived from the experiment is based on a parametric stabilization analysis of the channel configuration.

Upper and Lower Temperature Limits of Instability in Belousov-Zhabotinskii Systems

Upper and Lower Temperature Limits of Instability in Belousov-Zhabotinskii Systems

An experiment on the stability of hypersonic laminar boundary layers

An experimental investigation of the hydrodynamic stability of the laminar hypersonic boundary layer was carried out with the aid of a hot-wire anemometer. The case investigated was that of a flat surface at zero angle of attack and no heat transfer.The streamwise amplitude variation of both natural disturbances and of disturbances artifically excited with a siren mechanism was studied. In both cases it was found that such small fluctuations amplify for certain ranges of frequency and Reynolds numberRθ, and damp for others. The demarcation boundaries for the amplification (instability) zone were found to resemble the corresponding limits of boundary-layer instability at lower speeds. A ‘line of maximum amplification’ of disturbances was also found. The amplification rates and hence the degree of selectivity of the hypersonic layer were found, however, to be considerably lower than those at the lower speeds. The disturbances selected by the layer for maximum amplifications have a wavelength which was estimated to be about twenty times the boundary-layer thickness δ.