Theoretical Modeling and Geometric Optimization of Flat Sieves for Grain Mixture Separation

The separation of grain mixtures using flat mechanical sieves is a probabilistic process that depends significantly on the separator’s geometric parameters. This study investigates the relationship between the length-based separation coefficient $\mu (x)$ and the area-based separation coefficient $\mu (xy)$, emphasizing the critical role of the sieve’s shape and working area. Through theoretical modeling, we demonstrate that the separation process follows an exponential decay pattern along the sieve length, while the overall efficiency is determined by the total sieving area. For sieves with equal diagonals, the square-shaped sieve maximizes the working area (at $\beta =\pi /2$) and minimizes grain losses, resulting in optimal separation performance. The area-based coefficient $\mu (xy)$ remains constant under fixed diagonal conditions, whereas the length-based $\mu (x)$ varies with the length $x$, as proven by the derived dependency $\mu (x)_{e}=\mu (xy)_{e}\cdot \sqrt{1+tg^{2}(\alpha _{e})}$. Experimental similarity criteria ($\pi _{1}$,$\pi _{2}$) confirm that grain losses are identical for rectangular and square sieves with equal diagonals; however, the square sieve provides a higher sieving probability per unit area. The study proposes a geometric optimization framework for flat sieves, recommending square configurations with dimensions derived from the equivalence $\mu (x)_{0}\cdot x_{0}=\mu (xy)_{e}\cdot r_{e}$. These results provide a theoretical foundation for designing high-efficiency separators, though experimental validation is suggested for future work.

flat sieves; separation coefficient; grain losses; geometric optimization; probabilistic modeling