Application of Deterministic Chaos Theory to Local Instantaneous Temperature, Pressure, and Heat Transfer Coefficients in a Gas Fluidized Bed

Abstract

The local instantaneous temperature, heat transfer coefficient, and pressure data, gathered around a horizontal tube in a fluidized bed, have been analyzed using the deterministic chaos theory. A stainless steel heat transfer tube, carrying a hot water flow, was placed in a cold bubbling fluidized bed. The tube was instrumented in the circumferential direction with five fast-responding surface thermocouples and a vertical pressure differential sensor. The local temperature and pressure data were measured simultaneously at a frequency of 120 Hz. Additionally, the local instantaneous heat transfer coefficient was evaluated by solving the transient two-dimensional heat conduction equation across the tube wall numerically. The mutual information function (MIF) has been applied to the signals to observe the relationship between points separated in time. MIF was also used to provide the most appropriate time delay constant τ to reconstruct an m-dimensional phase portrait of the one-dimensional time series. The distinct variation of MIF around the tube indicates the variations of solid-surface contact in the circumferential direction. The correlation coefficient was evaluated to calculate the correlation exponentv, which is closely related to the fractal dimension. The correlation exponent is a measure of the strange attractor. The minimum embedding dimension as well as the degrees of freedom of the system were evaluated via the correlation coefficient. Kolmogorov entropies of the signals were approximated by using the correlation coefficient. Kolmogorov entropy considers the inherent multi-dimensional nature of chaotic data. A positive estimation of Kolmogorov entropy is an indication of the chaotic nature of the signal. The Kolmogorov entropies of the temperature data around the tube were found to be between 10 bits/s and 24 bits/s. A comparison between the signals has shown that the local instantaneous heat transfer coefficient exhibits a higher degree of chaos than the local temperature and pressure signals.