An Empirical Study Of The Effects Of Mesh Selection Procedures On Efficiency Of Mine Ventilation Analysis Methods

Introduction Since the Hardy Cross numerical algorithm was applied to the solution of mine ventilation networks, numerous developments have taken place in ventilation network analysis procedures. The Newton-Raphson method, the linear theory method and the second-order approximation method are three of the additional approaches that have been applied. The efficiency of a ventilation network analysis procedure depends not only on the method of solving the system of equations derived from ventilation networks but also on the method of deriving the equations through selection of the meshes. This paper investigates the impacts of three mesh selection methods on the efficiencies of four ventilation network procedures. The minimum-resistance spanning tree method is the most popular procedure for selecting meshes. The shortest-path method was suggested by Epp and Fowler (1970) for improvement of the Newton-Raphson method. The minimum-resistance-path method is suggested here as the third method of selecting meshes. The effects of these three methods on the Hardy Cross method, the Newton-Raphson method, the linear theory method and the second-order approximation method are analyzed in this paper. These network analysis methods have been summarized in the paper by Kim and Mutmansky (1991). Mesh selection methods A procedure for mine ventilation network solution based on mesh equations typically consists of two steps: • the mesh selection step and • the network solution step. The mesh selection step establishes a set of independent equations to solve the network. The resulting equations are nonlinear. One method of solving nonlinear equations is to linearize the equations and iteratively solve these linear equations. The four analysis methods previously discussed in this paper use this approach. The linear equations required for solution can be written in matrix form as: P•dQ = dF (1) where P is the pseudo-resistance matrix, Q is the quantity matrix, and F is the Jacobian matrix (Kim and Mutmansky, 1991). The efficiency of the solution method depends on the characteristics of the P matrix, which in turn depend on the method of selecting meshes. To understand the impacts of the mesh selection methods on the efficiency of the ventilation network analysis procedures, the characteristics of the P matrix must be understood first. The diagonal elements of the P matrix, m in number, are the sums of the pseudo-resistances of the branches contained in the m meshes necessary for solution. The off-diagonal elements are the sums of the pseudo-resistances of the branches shared by two meshes. For example, P12 (first row, second column element of the P matrix) is the sum of the pseudo-resistances of the branches common to mesh 1 and mesh 2. When solving the system of equations expressed in Eq. (1), three of the iterative procedures (the Newton-Raphson, the linear theory and the second-order approximation methods) use the direct factorization method (Burden and Faires, 1985) on the P matrix. When the direct factorization method is used, the computational efficiency is enhanced by larger diagonal elements and a sparser P matrix. Generally, the sparsity of the P matrix is more important than the size of the diagonal elements. There is no known mathematical method of predicting the efficiency of these procedures, but heuristic methods can be used to study the efficiency. The Hardy Cross algorithm applies the Gauss-Seidel method of splitting the P matrix and applies the Newton-Raphson procedure. The efficiency of the Hardy Cross method depends on the spectral radius of the matrix[[D-L]-1U], where D is the diagonal, L is the lower triangle and U is the upper triangle of the P matrix for the Newton-Raphson method (Ortega, 1972). The smaller the value of the spectral radius, the faster the convergence. As with the other three methods, larger diagonal elements and a sparser matrix aid in faster convergence. With the Hardy Cross method, however, the size of the diagonal elements is more important than the sparsity of the matrix. Remembering that each of the diagonal elements of the P matrix is the sum of the pseudo-resistances of the branches in a mesh and that each off-diagonal element is the sum of the pseudo-resistances of the branches shared by two different meshes, the following strategies can be used: • Avoid having branches with larger resistance values shared by multiple meshes. This increases the size of the diagonal elements and reduces the chance of off-diagonal elements being large.