PAT-driven dairy processing: a rheo-Raman-based kinetic model for in-line prediction of milk coagulation dynamics

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Abstract
The rennet-induced coagulation of reconstituted skimmed milk powder was investigated in this work. An analysis of Raman spectra, coupled with rheological measurements and multivariate statistical analysis, was conducted in order to develop a mathematical model for describing the dynamic behavior of milk constituents involved in coagulation. The first principal component from Raman data was used to describe the κ-casein concentration evolution, while the second principal component served to estimate the number of cross-linking sites, of which consumption generates the permanent cross-links, which were assumed to be proportional to the elastic modulus. Three levels of temperature and two levels of rennet concentration were explored. A system of three ordinary differential equations (ODEs) was proposed and analytically integrated to predict the temporal evolution of the first two principal components from Raman spectra and the elastic modulus, which is the parameter of interest in the industrial applications. An activation function was employed to activate/deactivate the generation term of the elastic modulus, as a result of gelation occurring after a delay period. Pseudo-kinetic constant of aggregation and lag time resulted in depending on both temperature and rennet concentration. A test sample was used to validate the prediction capability of the model, which resulted in high prediction performances (R2G′ = 0.9992). Afterwards, the calibrated model was coupled with a Kalman filter algorithm to provide a real-time monitoring procedure for the in-line estimation of the elastic modulus. Graphical abstract

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