(698a) Dynamic Continuation Strategies for Flowsheet Simulation and Optimization

Dynamic Continuation Strategies for Flowsheet Simulation in Process Design and Optimization

Richard Pattison and Michael Baldea

Department of Chemical Engineering

The University of Texas at Austin, 1 University Station C0400, Austin, TX 78712

email: mbaldea@che.utexas.edu

An important component of process design and optimization is the steady-state simulation of the process flowsheet. From a mathematical perspective, this requires solving the detailed equations describing the material and energy balances of the process units, as well as the correlations defining the physical properties of the components present in the process. The resulting system of algebraic equations (model) is typically highly nonlinear and poses significant solution challenges. Sequential modular (SM) simulation environments solve this system in an iterative manner by “tearing” the recycle streams and solving individual units in sequence until convergence at the flowsheet level is reached. Equation oriented (EO) simulation entails solving all the nonlinear model equations simultaneously. SM simulators are currently predominant in practical industrial applications owing to their superior numerical robustness - they are typically far more likely to obtain a solution for the process model starting from a poor initial guess than their EO counterparts [1-2]. However, EO approaches are advantageous from an optimization perspective due to the simplified calculation of Jacobian and Hessian matrices via automatic differentiation. Thus, in view of accelerating the development and optimization of novel process and energy systems, improving the convergence properties and robustness of EO process simulators is an imperative need.

Newton or Quasi-Newton methods are the most widely used methods for solving nonlinear algebraic equations in process simulators due to the super-linear convergence rate. However, it is recognized that Newton methods are only locally convergent from close initial guesses [3-4]. Several continuation methods have been implemented to address this issue [5], but in all cases, these are specific to individual unit operations, like reactors or distillation columns, and are not suitable for initializing the entire process flowsheet.

To overcome these challenges, we propose a novel pseudo-time continuation approach to solving the steady state process model. Within this context, the nonlinear algebraic system is converted using a set of filters with tunable constants to a set of ordinary differential equations and algebraic equations resulting in a differential algebraic equation (DAE) system [6]. Solving the system consists of i) a simplified consistent initialization and ii) a time stepping routine applied until steady state is reached. We also introduce a set of specific stability constraints that ensure that, at steady state, the solution of the pseudo-dynamic system is identical to that of the original steady-state model.

We present a preliminary implementation of these concepts in a new process simulation environment, featuring the most frequently used unit operations, including reaction, distillation, heat exchange, and compression. Furthermore, we show that the developed pseudo-dynamic models lend themselves naturally to use in process optimization calculations, using our previously-developed time relaxation-based optimization algorithm [7-8].

We present a validation of the proposed concepts using an ethylene-to-ethanol process as a testbed [9]. We show that the proposed solution approach exhibits superior convergence properties compared to commercially available simulators, and demonstrate the use of gradient-based rigorous optimization to improve the energy efficiency of the process.

References

[1] Pantelides, C.C.; Renfro, J.G. The online use of first-principles models in process operations: review current status & future needs. Comput. Chem. Eng., 2012.

[2] Biegler, L.T. Nonlinear programming: concepts, algorithms, and applications to chemical processes. SIAM, 2010.

[3] De P Soares, R. Finding all real solutions of nonlinear systems of equations with discontinuities by a modified affine arithmetic. Comput. Chem. Eng. 2012.

[4] Stuber, M.D.; Kumar, V.; Barton, P.I. Nonsmooth exclusion test for finding all solutions of nonlinear equations. BIT Numerical Mathematics. 2010, 50, 885-917.

[5] Malinen, I.; Tanskanen, J. Homotopy parameter bounding in increasing the robustness of homotopy continuation methods in multiplicity studies. Comput. Chem. Eng. 2010, 34,1761-1774.

[6] Kelley, C.T.; Keyes, D.E. Convergence analysis of pseudo-transient continuation. SIAM Journal on Numerical Analysis, 1998, 35,508–523.

[7] Baldea, M.; Jibb, R.J.; Cano, A.; Ramos, A. Optimal design of plate-fin brazed aluminum heat exchangers, AIChE Annual Meeting, Nashville, TN, 2009

[8] Zanfir, M.; Baldea, M.; Daoutidis, P. Optimizing the Catalyst Distribution for Counter-current Methane Steam Reforming in Plate Reactors, AIChE Journal, 2011, 57, 2518–2528.

[9] Biegler, L.T.; Grossmann, I.E.; Westerberg, A.W. Systematic Methods of Chemical Process Design. Prentice Hall: Upper Saddle River, NJ, 1997.