Largest Lyapunov exponent and Shannon entropy: Two indices to analyze mixing in fluidized beds

The quality of mixing in fluidized beds greatly influences performance in many applications. Assessing quality of mixing involves measuring the mixing rate and evaluating the bed mixedness. Quantifying the bed mixedness is typically done using mixing indices. However, the application of existing mixing indices to fluidized beds can be problematic due to aeration and complications from the particle phase in the Two-Fluid Model (TFM). The objectives of this study are twofold. First, the largest Lyapunov exponent is proposed to quantify mixing in fluidized beds. Its effectiveness is shown on a mono-disperse bed with varying gas velocities. The increase in mixing rate with higher gas velocity is accurately represented by the largest Lyapunov exponent. Second, the Shannon entropy mixing index is adopted to quantify the bed mixedness for TFM results. This index is tested on a bi-disperse bed to predict segregation and evaluate the effect of bed composition and superficial gas velocity on this process. The effect on segregation is reflected in the entropy components: distributional entropy showed minimal variation, whereas conditional entropy was significantly affected. Evaluating bed mixedness at different length scales showed that increasing bin spatial resolution slightly reduced conditional entropy. The results are validated against experimental data. • The largest Lyapunov exponent quantifies chaos and mixing in a fluidized bed. • The Shannon entropy mixing index effectively measures bed mixedness. • It characterizes overall bed distribution and change in bed composition. • Conditional entropy is key for tracking segregation. • The length scale of assessment impacts the observed mixing level in the bed.