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Resonant geometries for circulation pattern macroinstabilities in a stirred tank

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Abstract

Circulation pattern macroinstabilities and precessing vortices are known to exist in stirred tanks, but until recently they have eluded a full quantitative analysis. In this article, frequency analysis of the axial velocity reveals a resonant frequency and geometry for the 45° pitched-blade turbine (PBT) in a flat-bottom cylindrical tank. Unlike other axial impellers, such as the hydrofoils A310 and HE3, the PBT produces a large-scale circulation macroinstability at the scale of the vessel diameter. The macroinstability is superimposed on the turbulent random fluctuations with a timescale that is very long compared to the blade passage frequency and the inertial convective range of the turbulent cascade. In this article it is shown that the circulation pattern macroinstabilities generated by the 45° PBT are coherent and propagate throughout the tank only under very specific conditions (D = T/2 and C/D = 0.50). The normalized frequency of the coherent macroinstability, or Strouhal number, is fMI/N = 0.186. This corresponds to a period of roughly five rotations of the impeller, or 20 individual blade passages. The laser Doppler velocimeter data used in this work are unevenly spaced because they are collected in burst detection mode. This severely complicates spectral or frequency analysis. Standard algorithms such as the fast Fourier transform, autocorrelation methods, and wavelet analysis all require evenly spaced data. The Lomb periodogram was successfully used to extract the low-frequency content of the unevenly spaced data with a higher accuracy than is possible with other methods. The Lomb method also eliminates a short time-averaging step that was required in earlier work. © 2004 American Institute of Chemical Engineers AIChE J, 50: 2986–3005, 2004

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