Mathematical modeling of the reliability of mine ventilation systems using integro-differential equations with additional invariance

Abstract

Modern mining operations are conducted under complex geological conditions characterized by variable gas-dynamic and aerodynamic parameters, placing stringent demands on the reliability and stability of mine ventilation systems. This study presents the results of mathematical modeling based on integro-differential equations describing unsteady airflow distribution and accounting for distributed parameters, including airway geometry, air leakage, and heat and mass transfer. A non-Lie method was used to identify an additional invariance that cannot be detected by classical Lie–Ovsiannikov symmetry analysis. A conserved quantity – the generalized circulation Γ(t) – was identified. It comprises the classical circulation term and an additional term accounting for flow unsteadiness, thereby enabling airflow reversal in diagonal connections to be predicted without iterative recalculation of the entire ventilation network. A numerical algorithm combining finite-difference and finite-element methods was developed. Application of the model to the airflow distribution network of the No. 15 West longwall face identified critical diagonal branches with a high risk of airflow reversal and determined the ranges of the stability indices. The findings provide a theoretical basis for systems designed to provide early warning of emergency ventilation modes and for adaptive ventilation control, with the potential to improve mining safety.