Mathematical model for scaling up bioprocesses using experiment design combined with buckingham Pi theorem

Scaling up bioprocesses from the experimental to the pilot or industrial scale involves heuristics and scale relationships that are far from the specific phenomena and are usually not connected to the experimental data. In complex systems, the scaling-up methodology must connect the experimental data with the tools of engineering design. In this work, a two-stage gold bioleaching process was used as a case study to develop a mathematical model of bioprocess scaling that combines the design of experiments with dimensional analysis using the Buckingham Pi theorem to formulate a predictive model that allows scaling up bioprocesses. It was found that the C/N, C/K, and T/C ratios are dimensionless factors that can explain the behavior of a system. Using the Pearson Product– Moment bivariate analysis, it was found that the dimensionless factors C/N and C/K were correlated with the leaching potential of the fermented broth at 1060 cm⁻¹. With these results, a non-linear logarithmic model based on dimensionless parameters was proposed to explain the behavior of the system with a correlation coefficient of R² = 0.9889, showing that the optimal conditions to produce fermented broth comprised a C/N ratio close to 50 and a C/K ratio close to 800, which allows predicting the scaling of the bioprocess.