Mathematical modeling of vertical pneumatic conveying for solid waste transportation.

The aim of this study is to develop a height-dependent analytical-numerical model describing capsule motion in a vertical pneumatic gravity conveyor designed for the transportation of encapsulated solid household waste in multi-story buildings. The proposed formulation combines a force-balance analytical framework with ANSYS-based numerical implementation to evaluate capsule dynamics under varying mass and fall-height conditions. The governing equations are derived under engineering assumptions including rigid-body axial motion, lumped aerodynamic drag representation, hydraulically smooth pipe walls, and steady annular airflow conditions. Capsule motion is analyzed considering gravitational acceleration, aerodynamic resistance, and pressure losses in the confined annular gap between the capsule and the pipe wall. The model predicts non-linear dependence of capsule velocity and pressure loss on capsule mass and fall height. The results demonstrate a nonlinear relationship between capsule velocity, mass, and fall height under the adopted modeling assumptions, including rigid-body axial motion, quasi-steady annular airflow, and lumped drag representation. For the investigated parameter range, capsule velocity varies from approximately 1 to 13 m/s depending on mass (1-16 kg) and fall height (5-50 m), while pressure losses remain within 0-1.2 × 10⁴ Pa. The obtained relationships enable estimation of airflow requirements necessary to safely decelerate capsules at the base of the system. The proposed framework is intended for preliminary engineering design and parametric analysis. In this study, ANSYS is employed strictly as a numerical implementation environment for the analytically derived governing equations rather than as a full CFD flow solver. The model does not account for capsule rotation, eccentric motion, wall roughness evolution, or transient airflow effects, which define the applicability limits of the proposed framework. These aspects are identified as directions for future experimental validation and refinement.