Two-phase Stefan problem for the modeling of urea prilling tower

The prilling technique is frequently used to make granular urea and ammonium nitrate. This basic procedure involves spraying a liquid flow from the top of a tower. At the same time, a stream of cooling air collected from the surrounding is fed from the bottom. The generated droplets fall counter-currently and become solid due to the heat removal by the cooling air. The process produces spherical particles with a nearly uniform size. In practice, prilling towers easily suffer operating issues due to incomplete solidification. Because of the poor efficiency of the solidification, a low-quality structure is generated, resulting in productivity and profit losses. Despite of the importance of the process, only a few studies have been conducted on the modeling of a prilling tower. In the study of Wu et al. (2007), a simple shrinking core model was used to design a new prilling tower. The model is based on a lumped technique in which the temperature is uniform over the entire particle. Alamdari et al. (2000) developed a distributed model. The temperature distribution within the particle was described by a heat transfer equation. Rahmanian et al. (2013) also applied this model to a local industrial tower with a rectangular cross-sectional area. Mehrez et al. (2014) also employed simultaneous mass, heat, and momentum transfers between the two phases to simulate the process. However, in these models, the same three sequential thermal intervals for the solidification of urea droplets are considered: cooling of liquid drops, solidification at freezing temperature of the liquid phase, and cooling of complete solid particles. In this approach, the solidification interval is classified as a Stefan one-phase problem, in which the temperature of the liquid phase is assumed to be constant. This assumption is not natural because the temperature distribution within the particle should change gradually with time. Therefore, in this report, the solidification of urea particle is considered as a two-phase Stefan problem, in which the heat fluxes occur in both two phases, liquid and solid. The cooling and solidification are treated as one process from liquid droplets to complete solid particles instead of dividing into three intervals. The model also considers the hydrodynamic of falling particles, mass transfer of the moisture. This velocity is used to estimate the convective heat transfer coefficient. Boundary condition is the convection cooling with air. The variation of the air temperature along the tower is also considered in this study.