Improving the Performance of the Hardy Cross Algorithm for Large Ventilation Models

"The Hardy Cross algorithm offers a reliable method of solving network systems of fluid flow and has become widely used for solving water and ventilation flow networks. A limitation is that computational iterations and time to solve a network rises rapidly with the size of the model and modern detailed ventilation networks have typically grown to thousands of airways. Non-linear matrix solving methods can offer improved performance, however these are more complex and may be unstable if initial estimates are poor. This paper presents improvements that can be applied to the traditional Hardy Cross algorithm to greatly reduce iterations and solving time for large ventilation models. INTRODUCTION The Hardy Cross algorithm (Cross, 1936), was originally developed to solve water flow in city and urban pipe networks. Using principles of Kirchhoff’s electricity current laws, Hardy Cross developed an iterative algorithm for solving non-linear equations associated with network water flow. Traditional network analysis research refers to the various members of system networks as branches, nodes and meshes. This terminology has been changed in this paper to airways, junctions and loops respectively which are terms more familiar to engineers in mine ventilation. When Kirchhoff’s laws are applied to fluid networks, the following assumptions can be made: •The sum of fluid flows into a junction must equal the sum of fluid flows out of the junction.•The sum of pressure losses around any loop through a network system must be equal to zero.An iterative solution can therefore be achieved by initially guessing flow in a network, calculatingthe pressure losses around the network loops (Figure 1), and applying flow corrections using Newton’s method for each loop if the sum of the pressure losses does not equal zero. To solve the system, a loop must be defined for every segment or series of segments between a junction of 3 or more airways.The Hardy Cross algorithm has been used extensively for ventilation network analysis since the introduction of the first electronic computers (Tien, 1997), however numerous other variations and methods have since been developed through the use of matrices to solve the required system of nonlinear equations. Commonly used algorithms such as the Newton Raphson method (Brown, 1969), the Gradient method (Todini & Pilati, 1988), the Linear Theory method (Gupta & Prasad, 2000), and Coupled Network Solver (G. Danko, 2008) have found general use in fluid flow network solvers (Rossman, 2000). Despite poor and sometimes unpredictable convergence performance however, the Hardy Cross algorithm still has unique properties that make it attractive for ventilation models. Some benefits are:"