Analysis of mass transfer equations during solid-liquid extraction and its application for vanilla extraction kinetics modeling

Rigorous dimensionless analysis and analytical solution of mass transfer equation (punctual and space averaged) governing the solid-liquid extraction dynamic were performed. Analytical solutions result in continuous and analytical maps between solute mass fractions in extract with respect to time which were used both for solute diffusivity estimation and process modeling. In order to validate the models, experimental extraction at equilibrium of 1 cm long vanilla beans pieces were obtained at 0.1, 0.2, 0.3 and 0.4 g vanilla/mL solvent (60% w/w ethanol-water) and 30, 40 and 50 °C; and extraction kinetics were developed at 0.1 g/mL and 30, 40 and 50 °C. Apparent equilibrium constant and solute effective diffusivity estimation were performed by linear regression on experimental results. Apparent equilibrium constant was 0.86, and diffusivities were 1.22, 1.98, and 2.43 × 10−11 m2/s at 30, 40 and 50 °C respectively. Diffusivities and analytical solutions predicted the experimental results at 30 and 50 °C with 0.3, and 0.4 g/mL respectively.