Thermodynamically consistent quasi‐Newton formulae

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Abstract
Newton and quasi‐Newton methods have been used in chemical process design and optimization calculations for quite some time. They continue to be used today, both in the traditional sense and as part of the more recent hybrid method. While Newton‐based fixed‐point methods have been used to solve many different kinds of chemical process design and optimization problems, perhaps the point of single largest application has been that of multicomponent separation problems, especially equilibrium stage distillation.Quite aside from this, classical thermodynamics provides us with certain fundamental mathematical and physical relationships governing the behavior of nonideal solutions, those being the homogeneity of partial molar excess properties and their derivatives and the Gibbs‐Duhem equation.In this work, we draw a connection between the class of Newton‐based fixed‐point methods and classical nonideal thermodynamics. That is, it is shown that, with the exception of Newton's method, none of the conventional Newton‐like methods gives matrix approximations that are thermodynamically consistent. In other words, all existing nonsymmetric and symmetric quasi‐Newton formulae generate matrix approximations that do not satisfy, for either the zerodegree homogeneity or Gibbs‐Duhem equations.In light of this, a new class of quasi‐Newton formulae is presented for use in chemical process problems whose models include chemical and/or phase equilibrium. In particular, several new quasi‐Newton formulae are presented that give Jacobian or Hessian approximations that satisfy the zero‐degree homogeneity and/or Gibbs‐Duhem equations, in addition to the usual secant and perhaps symmetry and sparsity condition. This new class of formulae is called thermodynamically consistent quasi‐Newton formulae. Some numerical results are presented that show that these thermodynamically consistent quasi‐Newton formulae can provide improvements in reliability and computational efficiency when compared to existing Newton‐like methods.