A universal numerical model based on the particle size distribution (PSD) approach has been developed for the simulation of precipitation kinetics in multicomponent alloys during isothermal ageing. Nucleation was implemented utilizing the classical nucleation theory (CNT). Growth and coarsening were modeled by a single growth kinetics equation, which is constructed based on the interfacial diffusion flux balance and the capillarity effect. Only partial off-diagonal terms in the diffusion matrix (diffusion of individual components in the matrix) were taken into account in the calculations to minimize the computational cost while coupling with CALPHAD to extract thermodynamics equilibrium around the interface. A new feature of the model is the incorporation of a more realistic spatial site distribution via a Voronoi construction in the characteristic cell, for the purpose of modifying the diffusion distance. Computational predictions of the precipitate dimensions and the precipitation kinetics were compared with the atom probe tomography (APT) measurements on ternary Ni-Al-Cr alloys isothermally aged at 873 K. It is found that the temporal evolution of the dimensions and composition of the precipitates is well captured, as is the dependence on changes in the alloy composition. The new modification with Voronoi construction demonstrates that the overall precipitation kinetics depends on the density and the spatial site distribution of precipitates. The ability to handle sophisticated alloy chemistries by quantitative equations, the compositional sensitivity of microstructural characteristics emerging from the simulation results, and the ability to visualize the spatial distribution of precipitates make the work very promising for multicomponent alloy design and optimization.