Time-delay mathematical model of lagged lactic acid production using agro-industrial wastes as substrate

The aim of this study was to obtain a feasible mathematical model that may describe the lactic acid production by Lactobacillus casei using residual agro-industrial wastes. The model includes a delay term which characterizes the lagged biomass growth by the acclimation period induced by the considered complex substrate. A novel parametric identification algorithm leads to the characterization of the time delay as well as the kinetic parameters. The application of the estimated parameters in the proposed model defined the behavior of biomass growth, substrate consumption and lactic acid accumulation. A combination of robust exact differentiators (based on the sliding mode algorithm named super twisting) and a time-delay least mean square identification algorithm provides the on-line calculus of the required parameters. The identification algorithm was tested with the experimental data obtained from a bacterial culture system containing all the required nutritional sources to complete the target carbon to nitrogen ratio of 4.13. This value has been characterized as an adequate ratio to promote the efficient production of lactic acid. The proposed model reproduced the experimental data with correlation factor of 0.98 in average for the evaluated experimental conditions. The identified parameters agreed with the reported results in similar studies showing the effectiveness of the identification method as well as the possibility of using non-conventional substrates (milk whey and vegetable residues) to produce lactic acid.