Verbal description and development of a mathematical model for cooler of blast furnace slag with different particle sizes in a dry granulation process

A mathematical model based on a verbal problem statement has been developed to describe the hydraulics of a dry granulation unit for liquid blast-furnace slag in vertical coaxial channels blown with air. The model is intended for computer-based optimization of operating parameters in the cooling of hot slag granules of various particle size distributions, as part of the input data preparation for unit design.

In practical applications, vertical annular zones differing in the diameters of the falling and solidifying slag granules may form within the unit. To determine the required aerodynamic drag regime for granules of different sizes during falling, the unit is represented by coaxial vertical annular channels of different cross-sectional areas.

The hydraulic model is based on the well-known empirical Darcy–Weisbach equation. To facilitate computations, the model for optimizing the cooling air flow rates through the annular channels is formulated as a system of nonlinear equations that define the functional relationship between air flow rates and velocities in a parallel-channel configuration.

The aerodynamic drag regime for slag granules is determined based on a quick estimate of their terminal velocity, derived from classical Newtonian mechanics.

The model is exemplified by optimizing the air flow rate in the coaxial channels of the unit for a conventional slag composition containing granules of 1–1.5 mm and 2 mm in diameter.