Angelo Lucia

Distillation pinch points and more

Abstract Rising energy costs have spawned renewed interest in improving methodologies for the synthesis, design and/or retrofitting of separation processes. It is well known that energy use in many process industries is dominated by separation tasks—particularly distillation. In this work, the shortest stripping line approach recently proposed by Lucia, Amale, & Taylor (2006) is used to find minimum energy requirements in distillation. The new aspects of this work show that this shortest stripping line approach can find minimum energy requirements for (1) Distillations with feed pinch, saddle pinch, and tangent pinch points. (2) Distillations for which the minimum energy solutions do not correspond to a pinch point. (3) Processes with multiple units (e.g., reactive distillation, extraction/distillation, etc.). Other novel features of this work also shows that the shortest stripping line approach (4) Can be used to identify correct processing targets in multi-unit processes. (5) Encompasses longstanding methods for finding minimum energy requirements including the McCabe-Thiele method and boundary value methods. A back-to-front design approach based on shortest stripping lines is used so that correct processing targets can be identified so that all tasks can be synthesized simultaneously in such a way that the most energy efficient designs are achieved. New problem formulations that take the general form of nonlinear programming (NLP) and mixed integer nonlinear programming (MINLP) problems are given and a novel global optimization algorithm is presented for obtaining energy efficient process designs. A variety of ideal and nonideal distillations, including examples with four or more components, are used to demonstrate the efficacy of the shortest stripping line approach. The examples with more than three components are particularly significant because they clearly illustrate that the proposed approach can be readily used to find minimum energy requirements for distillation problems involving any number of components. Many geometric illustrations are used to highlight the key ideas of the method where appropriate.

Multiphase equilibrium flash calculations

Abstract The determination of the correct number of equilibrium phases and their corresponding compositions at fixed temperature and pressure (TP flash) is studied. The novel aspects of this work center around unique initialization strategies and successive quadratic programming (SQP) enhancements that include the use of (1) only binary tangent plane analyses; (2) the determination of all partially miscible binary pairs and a dominant immiscible pair; (3) novel relative solubility calculations based on component activities and double tangency separation; (4) least squares solutions to compute phase fraction estimates; (5) a variety of algorithmic features that dynamically trap difficulties such as compositions well below machine accuracy, and trivial and collapsed solutions and (6) a posteriori testing of phase and solution stability. The overall algorithmic framework is one based on using a combination of binary tangent plane analyses, bubble point calculations and dimensionless Gibbs free energy minimizations. Binary tangent plane analyses are used to identify all immiscible or partially miscible binary pairs and to avoid dimensionality difficulties associated with locating all stationary points in the tangent plane distance function in the full composition space. The proposed approach consists of solving a sequence of subproblems (i.e. LE, LLE, VLLE,…) until the global minimum dimensionless Gibbs free energy ( G /RT) is found. Maximum information from binary tangent plane analyses and previously solved subproblems are used to generate initial values for the next subproblem. The concept of relative solubilities is introduced and used to initialize phase compositions in all LLE calculations (i.e. phase split or flash). All completely miscible component relative solubilities are calculated using component activities while those for immiscible or partially miscible components are initialized using double tangency separation. Phase fractions are initialized using a least-square solution to the set of component mass balances. All subproblems are formulated in terms of component flows and solved using a full space SQP method using a modified Broyden–Fletcher–Goldfarb–Shanno (BFGS) update of the Lagrangian Hessian matrix. The proposed algorithm was tested within the Aspen Plus process simulator using a variety of physical properties options. Twenty six multicomponent mixtures including some four-phase (VLLLE) emulsion polymerization problems were used to test the proposed algorithm. All problems were easily solved and clearly demonstrate the capabilities of the present multiphase TP flash model.

Global terrain methods

Abstract The task of finding all physically relevant solutions to mathematical models of physical systems remains an important and challenging area of active research in many branches of science and engineering. While there are several useful ‘global’ methods for finding one or more solutions to models for chemical and other processes, it is the solutions that are generally the primary focus. Singular points are often viewed as something to be avoided and no use is made of the natural connectedness that exists between solutions and singular points. This paper describes a completely different, novel and general approach to finding all physically meaningful solutions and singular points to mathematical models of physical systems by intelligently moving up and down the landscape of the least-squares function. Theoretical foundation for this work rests on the fundamental observations that (1) solutions and singular points are smoothly connected when the model functions are smooth; (2) valleys, ridges, ledges, etc. provide a natural characterization of this connectedness; (3) valleys, ridges, etc. can, in turn, be characterized as a collection of constrained extrema over a set of level curves; (4) there is an equivalent characterization of valleys, ridges, etc. as solutions to generalized, constrained eigenvalue–eigenvector problems; and (5) the natural flow of Newton-like vector fields tends to be along these distinct features of the landscape. Differential geometry is used to provide theoretical support for these fundamental observations and related issues. These observations also form the basis for a new family of algorithms for finding all physically meaningful solutions and singular points called ‘global terrain methods’, which consist of a series of downhill, equation-solving computations and uphill, predictor–corrector calculations. Downhill movement to either a singular point or solution is conducted using reliable, norm-reducing (complex domain) trust region methods. Uphill movement, on the other hand, is necessarily to a singular point and uses approximate uphill Newton-like predictor steps combined with intermittent corrector steps. Each corrector step is defined by calculating an extremum in the gradient norm on the current level set for the least-squares function, can be shown to be equivalent to a solution to a generalized, constrained eigenvalue–eigenvector problem and helps ensure that valleys and ridges are tracked as closely as desired. Initial starting points are arbitrary, while starting points for subsequent subproblems defining movement from one stationary point to another are along appropriately determined eigendirections, since valleys and ridges are generalized eigenpathways. Collisions with boundaries of the feasible region and the presence of points at infinity are also addressed and the heuristic termination criterion based on the concept of limited connectedness is presented. A variety of numerical results and geometric illustrations for two-dimensional chemical process models are used to make clear key theoretical concepts, to demonstrate the reliability and efficiency of global terrain methods on small scale problems and to show their potential promise on large scale process models problems.

Chemical process optimization using Newton-like methods

Various interrelated issues that effect the reliability and efficiency of Newton-like methods for chemical process optimization are studied. An algorithm for solving large, sparse quadratic programming (QP) problems that is based on an active set strategy and a symmetric, indefinite factorization is presented. The QP algorithm is fast and reliable. A simple asymmetric trust region method is proposed for improving the reliability of successive QP methods. Ill-defined QP subproblems are avoided by adjusting the size of the trust region in an automatic way. Finally, it is shown that reliable initial values of the unknown variables and multipliers can be generated automatically using generic problem information, short-cut techniques and simulation tools. Many relevant numerical results and illustrations are presented.

Energy targeting and minimum energy distillation column sequences

Abstract The determination of distillation column configurations that consume the least total energy is studied. The novel contributions of the proposed design methodology for finding global minimum energy column sequences presented in this article include: (1) the definition of a total stripping line distance function for any sequence, (2) a robust energy targeting strategy that provides a continuously differentiable description of column sequences, (3) the flexibility to use any phase equilibrium model, (4) the ability to find column sequences that contain non-pinched, minimum energy columns within a sequence, and (5) the ability to include heat integration. The proposed energy targeting approach, which is used in conjunction with the two-level design methodology of Amale & Lucia (2008b) , is shown to be a reliable and effective tool for finding minimum energy distillation column sequences. A number of example problems are presented to show the efficacy of the proposed design methodology.

Fully compositional and thermal reservoir simulation

Abstract Fully compositional and thermal reservoir simulation capabilities are important in oil exploration and production. There are significant resources in existing wells and in heavy oil, oil sands, and deep-water reservoirs. This article has two main goals: (1) to clearly identify chemical engineering sub-problems within reservoir simulation that the PSE community can potentially make contributions to and (2) to describe a new computational framework for fully compositional and thermal reservoir simulation based on a combination of the Automatic Differentiation-General Purpose Research Simulator (AD-GPRS) and the multiphase equilibrium flash library (GFLASH). Numerical results for several chemical engineering sub-problems and reservoir simulations for two EOR applications are presented. Reservoir simulation results clearly show that the Solvent Thermal Resources Innovation Process (STRIP) outperforms conventional steam injection using two important metrics – sweep efficiency and oil recovery.

Solving distillation problems by Newton-like methods

Abstract Literature examples are used to show that failure of direct prediction Newton-like methods is always due to physical inconsistencies that arise during Based on these observations, a generic initialization procedure is presented that makes certain Newton-like methods very reliable. The thermodynamically consistent hybrid method is shown to be more efficient than Newtons method with analytical derivatives and conditions under whic The effect of machine precision on the solution of distillation problems involving pinch points is also illustrated. Finally, multiple solutions for an azeotropic distillation operating in a homogeneous regime are reconfirmed; two regular turning points are located wi Many numerical examples are presented.

A multi-scale framework for multi-phase equilibrium flash

Abstract A general multi-scale framework for multi-component, multi-phase equilibrium flash calculations, which uses information at the molecular and bulk fluid length scales, is described. The multi-scale Gibbs–Helmholtz constrained (GHC) EOS approach of Lucia (2010) is extended to include the use of (1) coarse grained NTP Monte Carlo simulations to gather pure component internal energies of departure, (2) a new linear mixing rule for internal energies of departure for mixtures, (3) a novel expression for partial fugacity coefficients for the GHC EOS, and (4) a novel flash algorithm that uses complex-valued compressibility factors and densities to assist in phase existence determination. Many numerical results for mixtures of CO2–water, NaCl–water, and CO2–NaCl–water are used to show that the GHC EOS flash approach is superior to all other approaches currently available. Many geometric illustrations are presented to elucidate key concepts and many experimental validations are used to substantiate claims of superiority.

A Geometric Terrain Methodology for Global Optimization

Global optimization remains an important area of active research. Many macroscopic and microscopic applications in science and engineering still present formidable challenges to current global optimization techniques. In this work, a completely different, novel and general geometric framework for continuous global optimization is described. The proposed methodology is based on intelligent movement along the valleys and ridges of an appropriate objective function using downhill, local minimization calculations defined in terms of a trust region method and uphill integration of the Newton-like vector field combined with intermittent SQP corrector steps. The novel features of the proposed methodology include new rigorous mathematical definitions of valleys and ridges, the combined use of objective function and gradient surfaces to guide movement, and techniques to assist both exploration and termination. Collisions with boundaries of the feasible region, integral curve bifurcations, and the presence of non-differentiabilities are also discussed. A variety of examples are used to make key concepts clear and to demonstrate the reliability, efficiency and robustness of terrain methods for global optimization.

Multi-scale methods and complex processes: A survey and look ahead

Abstract A comprehensive overview of multi-scale methods is presented with a focus on the ways in which information is communicated between scales. Liquid density computations, which are important in phase equilibrium of carbon storage, are used to illustrate multi-scale process engineering ideas. It is shown that using the Gibbs–Helmholtz equation to constrain the energy parameter in cubic equations of state leads to (1) a natural bridge between bulk and molecular length scales, (2) a new Gibbs–Helmholtz constrained equation of state, and (3) a novel mixing rule, and that the GHC equation can be coupled to NTP molecular simulations to provide more accurate predictions of density for use in phase equilibrium computations and reservoir simulations. Improvements for the next generation of multi-scale tools for analysis, visualization, simulation, and optimization of complex processes including the need to proceed with partial knowledge, multiplicity at various length scales, problems with many scales, and methods for predicting behavior over very long time and length scales are also discussed.