4. Conclusions
On the looped pipeline network problem, a simultaneous resolution is needed to solve some of its equations, making not possible the direct employment of a modular procedure. The impossibility on use the sequential resolution lies on the required attendance of both Kirchhoff’s laws since, initially satisfying the node mass balance (firstKirchhoff’s law), itis needed a simultaneous convergence of the loop equations obtained from the second Kirchhoff’s law. The proposed method uses modular procedure to identify the problem pressure difference equations, grouping the equations that can be solved sequentially. The identified pressure difference equations replace the traditional loop and pseudo-loop equations and form the nonlinear system to be solved simultaneously. Since the proposed method uses features from both modular and equation-oriented procedures, it was characterized as simultaneous-modular. Furthermore, was proposed the employment of the Newton-Raphson method to solve the problem nonlinear system of equations and, to obtain better initial guesses for the dependent variables, it was proposed a procedure to formulate an objective function for the initial employment of non-deterministic optimization methods. The presented case studies showed that the pressure difference equations are easily identified and, different of the loop equations, are always applicable to the problem. It was also shown that the pressure difference equations are equal to the loop equations formulated by the traditional employment of the second Kirchhoff’s Law, making no difference between problems solved by both procedures. The particularity ofthe lopped pipeline network problem leaded to the development of simulators to solve this specific problem, as the EPANET (2015) software. However, the development of simulators capable to solve problems composed by both looped pipeline and others process equipment’s arenot a simple task. Using the proposed method to identify the pressure difference equations, there is no more need to identifying and choosing the independent loop equations, and was also verified that the equations can be used on problems that aims the obtainment of any design variable, as showed at the case study Industrial installation with lopped pipeline network (item 3.3). Itis understood the described simultaneous-modular procedure makes possible the development of a process simulator capable to solve any process problem, with or without looped pipeline. The main advantage of the procedure is groups all sequential equation to formulate the smallest possible problem system of equations, turn easier and less computationally costly a simultaneous convergence by numerical procedures. The big looped pipeline problem resolution (item 3.2) exemplifies the conclusion, where the whole problem was represented by a system of equation composed by only 30 equations.
