Substitution-Newton-Raphson method applied to the modeling of a vapour compression refrigeration system using different representations of the thermodynamic properties of R-134a

Abstract

This paper gives a detailed presentation of the Substitution-Newton-Raphson method, suitable for large sparse non-linear systems. It combines the Successive Substitution method and the Newton-Raphson method in such way as to take the best advantages of both, keeping the convergence features of the Newton-Raphson with the low requirements of memory and time of the Successive Substitution schemes. The large system is solved employing few effective variables, using the greatest possible part of the model equations in substitution fashion to fix the remaining variables, but maintaining the convergence characteristics of the Newton-Raphson. The methodology is exemplified through a simple algebraic system, and applied to a simple thermodynamic, mechanical and heat transfer modeling of a single-stage vapor compression refrigeration system. Three distinct approaches for reproducing the thermodynamic properties of the refrigerant R-134a are compared: the linear interpolation from tabulated data, the use of polynomial fitted curves and the use of functions derived from the Helmholtz free energy

Substitution-Newton-Raphson Method for the Solution of Electric Network Equations

“…When solved by the NR method, this system is implemented in the way given by (4) and (5). As Equation (4) is written as follows:…”

“…On the other hand, literature that deals with the resolution of non-linear equations systems refers to two basic methods: the Successive Substitution (SS) and the Newton-Raphson (NR) [5]. The first one, in general, is simple for programming and does not require much computational memory; however, it exhibits a divergence risk according to the sequence that the equations are listed in the iterative process.…”

“…Thus, this work evaluates the Substitution-NewtonRaphson (SNR) method, described in [5], which combines…”

“…As presented in [5], the SNR method is applicable to a nonlinear algebraic equations system with N variables and N equations. Variables are divided in n effective variables and N-n substitution variables, and, consequently, equations are subdivided in n residual equations and N-n substitution equations.…”

Determination of Wireless Communication Links Optimal Transmission Range Using Improved Bisection Algorithm

“…However, the two methods in their original forms have been associated with some shortcomings. The classical Newton-Raphson method requires some extra mathematical processes that make it difficult to be applied in many cases [10,11,12,13]. On the other hand, the classical bisection method is quite simple; however, when compared to the other methods, the bisection method takes longer time to converge [14,15,16,17,18].…”