Effect of particle size distribution on mass transfer during solid-fluid extraction and its application to coffee brewing

Solid-fluid extraction (SFE) is one of the most important unit operations in the food and chemical engineering fields. This operation is very often modeled by assuming particles of single particle size (SPS); however, experimental evidence indicates that particle size distribution (PSD) affects mass transfer rate and therefore the extract characteristics at a given time. This study introduces a model based on global mass transfer coefficients to describe the SFE considering the PSD. The model was written in terms of dimensionless variables, arranged in a state-space representation for its analytical solution, and further used to investigate the effect of different PSDs on SFE, including normal, uniform, left and right skewed, and bimodal distributions, each one having different characteristics such as their median, fraction of particles above the mean, standard deviation, among others. Besides, the model was applied to estimate the diffusivity ($D_P$) of total soluble solids during coffee brewing of fine and coarse grinds in a packed bed system using data from literature. Results indicated that, depending on the PSD properties and solids-to-fluid volume ratio (between 0.01 and 1), the use of a SPS underestimated up to 16.5, 30.3, and 48.3% the required time to achieve high extraction yields (EY) of 90, 95, and 99%, respectively, but this trend reversed for low EYs, where time was overestimated in up to 47.5 and 34.2% for EYs of 25 and 50%, respectively. The use of a SPS produced significant underestimation of $D_P$ (about 30%) during the mass transfer analysis of coffee brewing ($p<0.05$), where it was evaluated as 1.3 × 10⁻¹⁰ m²/s when considering the PSD. The proposed model is a valuable tool to improve the understanding of the ongoing mass transfer mechanisms during SFE as well as to increase the knowledge about the processing of products depending on the PSD such as coffee brews.