Algorithm For Exact Solutions Of Mesh Equations In Cross' Method For Mine Ventilation Network Analysis

The Hardy Cross iterative method is the most popular technique for solving a ventilation network problem with natural air splitting. This method traditionally solves each mesh equation by an approximation during each iteration. Thus, it is interesting from both a theoretical and a computational viewpoint to be able to solve for an exact solution to the mesh equation. Presented in this paper is an algorithm for computing the exact solution to the mesh equation with a variable fan pressure. The algorithm uses signum functions for air quantities to generate an alternative mesh equation in the form of a quadratic equation that has two solutions defined by the well-known quadratic formula. The condition that the derivatives of the alternative and the original equations must be the same provides a basis for the correct choice between the solutions.