Two-Phase Stefan Problem for the Modeling of Particle Solidification in a Urea Prilling Tower

Urea production plays a crucial part in the worldwide agricultural economy, providing a primary supply of nitrogen for fertilizers. For storage and transport, urea is synthesized in granular form, and the prilling technology is frequently employed. In this technique, the hot liquid feed passes through an atomizer to produce small droplets, which then fall along the high tower. During the falling process, the liquid droplets gradually become solid because the internal energy is removed by the cooling air, which flows upward from the bottom. Typically, three consecutive thermal phases are analyzed for the solidification process: the liquid droplet cooling, solidification when the surface reaches freezing point, and the solid particle cooling. In this paper, the temperature distribution across the radius of the urea particles was analyzed using a heat transfer equation, which is considered a two-phase Stefan problem. The system of partial differential equations is solved numerically using the finite difference method and the enthalpy method. The temperature of the cooling air at various heights of the tower and the degree of solidification of different particle sizes were estimated and compared with data obtained from the urea factory to assess their reliability. The validation demonstrated a strong correlation between the model estimates and the real plant observations.