Terje Hertzberg

Long term dynamic optimization of a catalytic reactor system

Abstract The purpose of this work is to find an optimal operating strategy for a reactor system with a slowly deactivating catalyst. The process studied is Lurgis methanol synthesis, but the methods also applies to other fixed bed reactor systems. The catalyst deactivates irreversibility and must be replaced after some years. The catalyst life time is considered to be known. The problem is formulated as a dynamic optimization problem with time-varying coolant temperature as the free variable. A sequential method is used to solve the dynamic optimization problem.

Dynamic simulation and optimization of a catalytic steam reformer

Abstract The operational performance of a heated fixed-bed reactor has been studied. Previous work indicates that the distribution of heat transfer resistances is important. No generally accepted correlation exists for the wall heat transfer coefficient and the effective radial thermal conductivity, and correlations proposed in the literature have been tested and evaluated. It is found that the outer reactor tube wall temperature is very sensitive to the applied correlation for the wall heat transfer coefficient and none of the evaluated correlations match the real situation perfectly. However, the methane conversion is rather insensitive to the choice of correlation. The effective radial thermal conductivity can be determined from the assumption of equal radial heat- and momentum Peclet number. It is found that using spatially varying physical properties and gas velocities only has a minor effect on the temperature distribution. Thus, the inlet conditions can be used to determine the effective radial thermal conductivity. By applying the correlation by De Wasch and Froment (1972) for the wall heat transfer coefficient, the resulting model is demonstrated for two simulation scenarios: (1) stop in steam supply and (2) stop in gas feed supply (CH 4 , H 2 , CO and CO 2 ). Finally, the optimal methane conversion is obtained for changing feed flow with a limiting value for the outer reactor tube wall temperature applied as a constraint.

Optimization of PSA systems : studies on cyclic steady state convergence

Abstract Two types of trace separation systems are used as case studies to investigate different methods to accelerate cyclic steady state convergence. The Aitken, Muller and the more rigorous Broyden based updating schemes seem to speed up the convergence rate considerably, at least for systems for which the method of direct substitution converges slowly. A damped Newton-based approach is also investigated, but since this method requires an updated Jacobian at each iteration step, the integration and sensitivity calculation time is increased correspondingly. For the two systems considered here, the total calculation time is in fact longer than in the direct substitution case. In the context of optimization, the key factors to success seem to be how the test on the CSS convergence is performed, how the acceleration methods are initiated and how close the initial guess is to the optimal value. Each of the investigated acceleration methods are preoptimized according to these aspects and the Broyden method definitely shows the best result. The Aitken and Muller methods are rather simple and they do not seem reliable enough to be applied during an optimization process.

Multiplicity in reactive distillation of MTBE

This paper presents an explanation of why methyl tert-butyl ether (MTBE) production by reactive distillation may yield multiple solutions. Widely different composition profiles and conversions may, as already reported by Krishna and others, result with identical column specifications, depending on the initial estimates provided and the VLE models being used. A hypothesis yielding a qualitative understanding of this phenomenon has been developed. The inert n-butene plays a key role in the proposed explanation. As the reaction mixture is diluted with n-butene, the activity coefficient of methanol increases substantially and the temperature decreases. This dilution has a profound effect on the equilibrium conversion, enabling MTBE to escape from the reactive zone without decomposition. The minimum boiling point azeotrope between MTBE and methanol plays an important role in the internal transport mechanism for the heavy methanol, in particular when fed below or in the lower part of the reactive section.

Dynamic simulation of chemical engineering systems by the sequential modular approach

Abstract An algorithm for dynamic simulation of chemical engineering systems, using the sequential modular approach, is proposed. The modules are independent simulators, and are integrated over a common time horizon. Interpolation polynomials are used to approximate input variables. These input polynomials are updated before modules are integrated in order to interpolate output from the preceding module(s) and thereby increase coupling and stabilize the computation. Tear-stream variables have to be predicted at future time t n +1 , and various prediction methods are proposed.

The influence of gas-gas interactions on the Langmuir constants for some natural gas hydrates

Abstract Kvamme, B., Lund, A., and Hertzberg, T., 1993. The influence of gas-gas interactions on the Langmuir constants for some natural gas hydrates. Fluid Phase Equilibria 90: 15-44. Hydrates of structures I and II are investigated using Monte Carlo simulations. The empty lattice is assumed to be approximately rigid and unaffected by the guest molecules. The influence of guest-water and guest-guest interactions on the Langmuir constants are estimated using interaction-site models for the guest molecules. It is shown that the guest-guest interactions have a significant effect on the size of the Langmuir constants in the absorption theory of van der Waals and Platteeuw (van der Waals, J.H., 1956. Trans. Faraday Soc., 52: 84; van der Waals, J.H. and Platteeuw J.C., 1959. Adv. Chem. Phys., 2: 1). For reasons of consistency and extrapolation it is recommended that these effects should be included in mean-field terms rather than by empirical corrections.

The sequential linear quadratic programming algorithm for solving dynamic optimization problems - a review.

Abstract The optimum principle for dynamic systems as formulated by Pontryagin in 1962 may be used for development of numerical algorithms to solve dynamic optimization problems. This as opposed to the well known methods which discretize controls (and states) to transform the problem into a NLP framework. An obstacle for its use has been the extensive symbolic manipulations needed to derive the optimality equations for a specific problem, and the difficulty of solving the resulting nonlinear two point boundary value problem. There are methods which make use of the optimality conditions for dynamic systems (Pontryagin Minimum Principle) just as SQP methods use the Kuhn-Tucker conditions. As in SQP, a problem with linear constraints and quadratic objective function is solved iteratively. Such a method is presented in this work. This is closely related to the dynamic optimization method based on a combination of a SQP solver and total discretization of the dynamic system. The dynamic linear-quadratic model has a single analytical optimal control solution, and is thus accurately and effectively solved. Thus, at each iteration, the optimal solution is found for the linear-quadratic approximate model. This gives a search direction which can be used in a iterative scheme to ensure good agreement between the linear-quadratic and the nonlinear model.

Dynamic model of a system with phase- and reaction equilibrium

Abstract The assumption of phase- and chemical reaction equilibrium will introduce index problems when the dynamics of such systems are modeled in a conventional manner. By applying a modeling method which guarantees the resulting model to be semi-explicit index one, a system with phase- and chemical reaction equilibrium is modeled. Both the modeling method and the resulting model are presented. Three methods for initialization of differential algebraic equations are discussed, and one of the methods is recommended for the actual model.

On estimating the error of local thermodynamic models—a general approach

Abstract A general method is developed for predicting the deviation between local and rigorous thermodynamic property models. The method is based on a quadratic error structure. The curvature matrix is updated after each parameter revision. The updating scheme utilizes the observed deviation as a measure for correcting the matrix and thus does not require any additional rigorous model evaluations. A recursive least squares procedure with variable forgetting is adopted for the parameter updating. A local model structure for K-value evaluations is applied in a dynamic simulation model of a batch distillation column. The efficiency and accuracy of the proposed method are tested for two different mixtures: a nearly ideal and a strongly nonideal mixture.

Obtaining sensitivity information in dynamic optimization problems solved by the sequential approach

Abstract An alternative way of obtaining sensitivity information when using the sequential algorithm for solving dynamic optimization problems has been tried out. The results show that the pursued approach has at least the same efficiency as the other available methods, yet is much simpler to use, and demands no symbolic differentiations or other manipulations of the equations. A non linear programming (NLP) solver is used to solve the parameter optimization problem obtained after discretization of the controls. All modern NLP solvers need accurate gradient information in order to function efficiently and reliably. Of the three ways available for obtaining this sensitivity information (sensitivity equations, perturbations and adjoins equations), perturbations are investigated in this work. It has been found that when obtaining sensitivity information through a forward difference, the N perturbed trajectories are nearly identical to the nominal. Therefore all N +1 trajectories are integrated simultaneously in a special purpose DAK solver, reusing the nominal Jacobian in the N perturbed trajectories, and saving the calculation time needed to decide steplengths, number of iterations and other administrative overheads. This speeds up calculation, and at the same time gets a more accurate estimate of the gradient. The method is theoretically as accurate as the sensitivity equation approach, without the need of an recalculated Jacobian at each timestep. This gives beneficial speed properties. The approach has been implemented in a modified version of the general purpose DAK solver DASSL, the new code is called the differential algebraic solver with sensitivity calculations by perturbations or DASSP. The DASSP code was compared to DASSL and DASAC, a code based on sensitivity equations. The codes were tried out on three test cases; two reactor problems, and a phase equilibria problem.