Comparison of Newton Raphson and Hard Darcy methods for gravity main nonlinear water network

Abstract: A water network of 24 pipes depending on mainly gravity and covers an area of 3.78 square kilometers was taken an as a case study to test and compare the analysis. The governing equation of this network are internal flow in pipe equations, which consist of the continuity equation, the modified Bernoulli's equation, and the head loss due to the length of the pipe. The three equations are nonlinear algebraic equations because of the square power of the discharge in the head loss equations, which need to be solved numerically. Hard Darcy method and Newton Raphson method are used to solve the system of nonlinear equations, and to compare the solution.So, twenty four nonlinear equations (nine Bernoulli's equations and fifteen continuity equations) in twenty four unknowns discharges were got by these two method by using MATLAB code. There are not differences in the resulted discharges between Hard Darcy and Newton Raphson methods. Also, it was found that Newton Raphson was faster than Hard Darcy Method when they compared by the number of iteration. The final solution of the discharges have tested by the basic of fluid mechanics that says the summation of head losses inside a loop must be equal zero which can be seen clearly in the plots of the two methods.