Extension of Newton-Raphson Method with Variable Directions in Ventilation Network Analysis of Underground Excavations

Computer method is one of the ventilation network design methods in underground excavations. Computer method has been designed based on mathematical approximate methods. Methods such as Newton-Raphson method, Hardy Cross method and its modified versions, critical path, linear analysis, non-linear programming and optimization techniques are considered as some mathematical approximate methods. Newton-Raphson method is one of the methods of solving nonlinear equations in numerical calculations. This method is designed based on the definition of derivative and its correction. Using Newton-Raphson method for the analysis of ventilation networks in underground excavations has been common. However, so far, perfect investigation of this method has not been carried out. The purpose of this paper is to investigate Newton-Raphson method and its improvement in the analysis of ventilation networks in underground excavations. The analysis indicates that this method in some models is unable to find the final true answer because instead of convergence, it diverges. Thus, the improvement of this method seems to be necessary. Ventilation design of underground excavations is done based on preparing underground excavation map, identification of branches and injunctions in ventilation network, calculation of mine resistance for any of branches, calculation of air flow quantity for any of branches, calculation of pressure loss for any of branches, calculation of natural ventilation, network adjustment, selection of regulator doors and selection of main fans. Various methods have been presented such a manual and computer methods for the analysis of ventilation networks of underground excavations. The choice of analysis method depends on the purpose of ventilation network analysis.
One of the methods for solving nonlinear equations in numerical calculations is Newton-Raphson method. This method is designed based on the definition of derivative and its correction. In this method, the error amount of the initial guess is calculated for purpose function, and then, the initial guess is corrected. In this method, based on the equation  and injunction rules, supposition flow quantity for any of branches.is supposed. Based on the ventilation network fans and mathematical approximate equations, the error of any ring in the ventilation network is calculated, and then, supposition flow quantity is corrected. Based on the mathematical equations, correction operations of air flow quantity are repeated until the calculation accuracy becomes equal to or smaller than the calculation error. Newton-Raphson method is one of solution methods in ventilation networks analysis. Convergence to the final solution in this method depends on supposition flow quantity, the direction of supposition flow and the arrangement of ring selection. If the direction of the hypothetical flow in the network is in accordance with the actual flow direction, the fastest method for reaching the final solution in the analysis of underground ventilation networks will be the use of Newton-Raphson method. However, if the direction of the hypothetical flow in the network is not in accordance with the actual flow direction, the validation of this method will indicate that this method in some models becomes divergent. Therefore, improvement of Newton-Raphson method is necessary. The improvement of Newton-Raphson method in this paper is made in two stages.  The first stage is the identification of negative flow intensity, and then, the flow direction is reversed. In the second stage, the correction of the equation  is performed, and then, the calculations are repeated.