Strategy for solving flow networks using iterative methods

5 Strategy for solving flow networks using iterative methods 

Invariably the flow network will form part of a larger model, which will define the boundary pressures. These will remain fixed during the process of iteration. The basic strategy is to use initial guesses of the pressures at the intermediate nodes and, in some cases, of the nodal temperatures or specific volumes, in order to calculate the flows and then to see if they balance. The guesses will then be refined successively using an algorithm such as Newton-Raphson until the flows balance to a predetermined tolerance. 

Using the notation of Section 2.10, the implicit flow equations to be solved at each timestep will take the form  

Successively better estimates for z at each time, r, may be calculated using the Newton-Raphson 

A problem for the modeller is that all the normal flow equations, liquid and gas, contain a point of inflexion at flow reversal, when the derivative takes an infinite value. Thus all the normal flow equations possess an inflexion region from which, once there, the algorithm will emerge only slowly. 

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