Alginate-based hydrogels are extensively used to create bioinks for bioprinting, due to their biocompatibility, low toxicity, low costs, and slight gelling. Modeling of bioprinting process can boost experimental design reducing trial-and-error tests. To this aim, the cross-linking kinetics for the chemical gelation of sodium alginate hydrogels via calcium chloride diffusion is analyzed. Experimental measurements on the absorbed volume of calcium chloride in the hydrogel are obtained at different times. Moreover, a reaction-diffusion model is developed, accounting for the dependence of diffusive properties on the gelation degree. The coupled chemical system is solved using finite element discretizations which include the inhomogeneous evolution of hydrogel state in time and space. Experimental results are fitted within the proposed modeling framework, which is thereby calibrated and validated. Moreover, the importance of accounting for cross-linking-dependent diffusive properties is highlighted, showing that, if a constant diffusivity property is employed, the model does not properly capture the experimental evidence. Since the analyzed mechanisms highly affect the evolution of the front of the solidified gel in the final bioprinted structure, the present study is a step towards the development of reliable computational tools for the in silico optimization of protocols and post-printing treatments for bioprinting applications.