(331g) A New Phase Equilibrium Calculation Method at Volume, Temperature and Moles Specifications

In this paper, a new method for phase equilibrium calculations at temperature, volume and moles specifications (VTN flash) is proposed. The VTN flash had been for a long time an option in many commercial and research simulators. However, the technique traditionally used, which is robust but computationally expensive was to perform nested PT flashes and update the pressure in an outer loop until the volume specification is honoured. VTN flash calculations received an increased interest in the recent years and several customized minimization procedures were proposed. The problem was formulated either as a constraint minimization of the Helmholtz free energy with respect to mole numbers and volume (which are the natural variables for a pressure-explicit equation of state), or as an unconstrained minimization of the Helmholtz free energy with respect to mole numbers (D.V. Nichita, 2018, New unconstrained minimization methods for robust flash calculations at temperature, volume and moles specifications, Fluid Phase Equilib. 466, 31-47), using so-called “PT-like” iterations (VTN-PT).

In this work, a new formulation is presented, in which an unconstrained minimization of the Helmholtz free energy is performed with respect to mole numbers and pressure. An analysis of the block structure of the Hessian matrix reveals the formal links between the proposed method and PT flash and previous VTN flash methods and suggests how existing codes for PT and VTN flash calculations can be easily modified to incorporate the new method. It is shown that the Hessian in the proposed method has a lower implicitness level, thus Newton iterations are expected to converge slightly slower (in terms of number of iterations), than the VTN-PT method. However, the computational cost of an iteration is considerably smaller in the proposed method, since the equation of state is solved only twice, instead of being repeatedly solved in a volume-balance inner loop. Moreover, the convergence paths of the two methods differ only in the early iteration stages and are essentially identical when the solution is approached. The proposed method is also faster than the traditional nested approach, in which a PT flash is converged in each iteration on pressure.

Highly robust and efficient modified Newton iterations are used, consisting in a modified Cholesky factorization (to ensure descent directions) and a two-stage line search procedure (keeping the iterates within the feasible domain and ensuring a decrease of the objective function at each iteration). Initialization is obtained either from a VTN phase stability testing (D.V. Nichita, 2017, Fast and robust phase stability testing at isothermal-isochoric conditions, Fluid Phase Equilib. 447, 107-124), or from ideal equilibrium constants. A predetermined small number of successive substitution iterations are performed before switching on the second-order method. An extrapolation procedure is also proposed for isochores calculation in the pressure-temperature plane. A two-parameter cubic equation of state was used in this work, but the calculation framework is not model-dependent. The proposed method is tested for a variety of mixtures of various complexities, with a special attention to near-critical conditions, and proved to be fast and robust.