(236e) Selection and Comparison of An Efficient Strategy for the Design and Optimal Operation of a Packed Bed Multitubular Reactor

This paper presents a comparison between two optimization strategies for the design and optimal operation of a packed bed multitubular reactor. The compared strategies are two different approaches to the problem of optimization when the restrictions are differential equations: sequential strategy and a simultaneous strategy. As a case study the catalytic oxidation of o-xylene for the production of phthalic anhydride in the presence of undesirable reactions is studied. A pseudo-homogeneous model with heat exchange and without axial and radial dispersion of mass is used for the reactor. The kinetics for the reactions are those suggested by Froment and Bischoff, 1990 [1], [2]. The system is highly exothermic and the undesirable reactions are favored at high temperatures.

The simultaneous strategy[3] involves the discretization of the system of differential equations in a structured mesh of n points. Thus, the system involving k differential equations is transformed into a system of kn equations. These equations are highly nonlinear and are functions of n conversions, n temperatures along the reactor, n wall temperatures and n molar flows corresponding to each of the n grid points. Therefore, the decision variables of the problem of the new discretized problem are the n values of the independent variables along the n mesh points. The discretized equations are then coupled with the objective function and the whole problem is solved using an optimization method. Thus, the problem is solved as a Non Linear Optimization Problem (NLP),[4],[5].

In the present study 50 and 100 mesh points are used for the discretization of the differential equations. A barrier-adapted method for the discretization of the differential equations was used in which the objective function and the restrictions are joined by a barrier function. This optimization strategy was implemented in Wolfram Mathematica v6.0®.

The sequential strategy decouples the optimization problem. Under this strategy the optimization problem can be thought of as a two-stage flow of information in the computer program. In the first stage, the main step, the optimization method is executed using a stochastic optimization method. In this stage the decision variables are manipulated and then sent as parameters to the second stage. In the latter, an ordinary differential equation solver is implemented. The objective function is evaluated and the sensitivities with respect to the parameters are then obtained. Finally, the parameters are sent back to the first stage of the problem. The process continues until certain stopping conditions are met. This strategy permits to find global optimal values, although it does not guarantee them. Furthermore, it is fairly independent of the initial conditions in the optimization problem and avoids the need for the initialization of a great number of variables. This optimization strategy was implemented in MATLAB R2008a®.

Two different objective functions are studied: 1.) Maximization of the annual profitability and 2.) Maximization of the desired product. The obtain results are then compared for each strategy. In each case, the objective function is submitted to two sets of constraints: The mathematical model of the reactor in the form of a system of ordinary different equations and the inequalities constraints (i.e. maximum allowable operating temperature, minimum inlet temperature, etc).

As results of the study it is presented the optimal objective function values and the computational time for both strategies. Likewise, the advantages and disadvantages associated with each strategy are also presented and a recommendation for the best strategy to use when the problem of the design of the packed bed multitubular reactors are modeled in more complex settings, such as bidimensional models with mass dispersion, is made.

REFERENCES:

[1] Froment, G.F. and K.B. Bischoff, Chemical Reactor Analysis and Design, John Wiley & Sons, New York (1990).

[2] B. Rosendall and B.A. Finlayson, Transport Effects in Packed-Bed Oxidation Reactors, Computers chem. Engng Vol. 19 No. 11 (1995)

[3] C. Llinas, J.M Gomez, A Non Linear Programming Formulation for Optimal Design of a Multitubular Packed Bed Reactor.-Simultaneous Solution. AIChE Annual Meeting & Centennial Celebration. Philadelphia, PA, November 16 -21, 2008

[4] Thomas F. Edgar, David M. Himmelblau, ?Optimization of chemical processes?, McGraw-Hill, Second Edition, (2001)

[5]Arturo M. Cervantes, Andres Wäcther, Reha H. Tütüncü, Lorenz T. Biegler, ?A reduced space interior point strategy for optimization of differential algebraic systems?, Computers and Chemical Engineering 24 (2000)