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Accurate simulations of high-pressure transcritical fuel sprays are essential for the design and optimization of next-generation gas turbines, internal combustion engines, and liquid propellant rocket engines. Most important and challenging is the accurate modelling of complex real-gas effects in high-pressure environments, especially the hybrid subcritical-to-supercritical mode of evaporation during the mixing of fuel and oxidizer. In this paper, we present a novel modeling framework for high-fidelity simulations of reacting and non-reacting transcritical fuel sprays. In this method, the high-pressure jet disintegration is modeled using a diffuse interface method with multiphase thermodynamics, which combines multi-component real-fluid kinetic and caloric state equations with vapor-liquid equilibrium calculations in order to compute thermodynamic properties of the mixture at transcritical pressures. All multiphase thermodynamic formulations are presented for general cubic state equations coupled with a rapid phase-equilibrium calculation method. The proposed method represents multiphase turbulent fluid flows at transcritical pressures without relying on any semi-empirical break-up and evaporation models. Combustion source terms are evaluated using a finite-rate chemistry model, including real-gas effects based on the fugacity of the species in the mixture. The adaptive local deconvolution method (ALDM) is used as a physically consistent turbulence model for large-eddy simulation (LES). LES results show a very good agreement with available experimental data for the reacting and non-reacting ECN Spray A at transcritical operating conditions.