Bracketing and convergence in thermal systems simulation

Abstract

This paper addresses computational problems encountered in the simulation of thermal systems, particularly in the bracketing and convergence procedures required to find a value x such that f(x) = x. This is the basic numerical problem in steady-state system simulations or in finding the instantaneous operating condition during transient simulation. Some procedures are described, and approaches to reduce computational time are detailed. These techniques are especially suited to mathematical functions that are single valued, continuous, and confined to the first quadrant (x,f(x) > 0). These are representative features, either inherent or by imposition, of the functions describing thermal systems processes or thermodynamic properties. A procedure for bracketing or finding a convergent interval is proposed and tested on representative example analytical functions. Bracketing of a solution with such a procedure is shown to require up to two iterations in most cases. Several established convergence methods along with a newly modified method and an innovative method are discussed and compared. The importance of working within the bracketed interval is emphasized.