We report on the fundamentals of a new method for injection of fuel droplets into a cross-flow within a premixer, to improve its performance. We investigate evaporation history of various-sized n-heptane droplets in high-pressure and - temperature nitrogen gas, using a Lagrangian-Eulerian numerical approach. Deploying the Peng-Robinson equation and Van der Waals mixing rule for phase equilibrium of the n-heptane-nitrogen system, we consider gas solubility at the droplet surface and real gas behavior in high pressures. A number of different-sized droplets are injected both individually and collectively into cross-flow. The results indicate that in this situation, the interaction among droplets is negligible. To demonstrate coherence among droplets, a correlation is provided to describe droplet evaporation history with respect to initial droplet size. In addition, by injecting a continuous radial spray perpendicular to the flow in the premix tube, results reveal that droplets are arranged radially based on their initial sizes in the gas flow.