Partial molar excess properties, null spaces, and a new update for the hybrid method of chemical process design

Abstract

AbstractA new thermodynamically consistent quasi‐Newton formula is proposed for building iterative representations of the approximated part of the Jacobian matrix in the hybrid method of chemical process design. This update exploits the fact that the excess enthalpy and activity coefficient functions are homogeneous functions of degree zero in mole numbers which, in turn, gives rise to a natural null space for the approximated part of the Jacobian matrix that is defined by the component molal flow rates. The new formula builds iterative approximated parts that satisfy this null space constraint in addition to the usual overall secant condition and the desired sparsity constraints.The new quasi‐Newton formula is derived using the variational calculus approach first for the nonsparse case. This solution is then extended to the sparse case.The new updating formula, which is applicable to any chemical process design problem that involves nonideal phase equilibria and/or enthalpy of mixing considerations, is compared to the modified Schubert update in the context of the hybrid method. It is also compared to an implementation of Newton's method in which the approximated part of the Jacobian matrix is calculated by finite differences. Using an adiabatic single stage flash problem for a binary mixture with strongly nonideal liquid phase behavior, it is shown that the new updating formula compares favorably with finite differencing and can result in solutions to problems for which the original hybrid method fails.