ODE Solver Algorithm

When using an ordinary differential equation (ODE) solver such as Polymath or  

MATLAB, it is usually easier to leave the mole balances, rate laws, and concen- 

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P6-17g Review the oxidation of formaldehyde to formic acid leactions over a van-dium titanium oxide catalyst [Ind. Eng. Ch m. Res., 28, 387 (1989)] shown in the ODE solver algorithm in the Summary. 

Table E6-6.1. ODE Solver Algorithm for Mulitple Reactions (Continued)...

Summary 733 ODE Solver Algorithm 736 CD-ROM Material 736 Questions and Problems 738... 

The classic methods use an ODE solver in combination with an optimization algorithm and solve the problem sequentially. This solution strategy is referred to as a sequential solution and optimization approach, since for each iteration the optimization variables are set and then the differential equation constraints are integrated. Though straightforward, this approach is generally inefficient because it requires the accurate solution of the model equations at each iteration within the optimization, even when iterates are far from the final optimal solution. 

This function is called numerous times from the Matlab ODE solver. In the example it is the ode45 which is the standard Runge-Kutta algorithm. ode45 requires as parameters the file name of the inner function, ode autocat. m, the vector of initial concentrations, cO, the rate constants, k, and the total amount of time for which the reaction should be modelled (20 time units in the example). The solver returns the vector t at which the concentrations were calculated and the concentrations themselves, the matrix C. Note that due to the adaptive step size control, the concentrations are computed at times t which are not predefined. 

The most efficient algorithm for the solution of the sensitivity differential equations is called the decoupled direct method (ddm), which was first applied in chemical kinetics by Dunker [67, 68]. He drew attention to the fact that equations (4.1) and (4.6) have the same Jacobian, so that a stiff ode solver will use the same step size and order of approximation in the solution of both odes. The ddm method first takes a step for the solution of equation (4.1) and then performs steps for the solution of equation (4.6) for / = 1,. . . , m. The procedure is repeated in the subsequent steps. Since the Jacobian of the equations is the same, it has to be triangularized only once for each time interval. This method is applied in the program SENKIN [69]. 

As the MATLAB software packages with Optimisation Toolbox provides both effective ordinary differential equation (ODE) solvers as well as powerful optimization algorithms, the dynamic simulations reported in this paper are carried out by using the MATLAB Optimisation Toolbox (8). 

An optimal control strategy and algorithm using commercial optimization software packages connected to reliable DAE/ODE solvers are successful for the determination of optimal trajectories with good convergence properties. This implies that under certain conditions, the more complicated optimal control algorithms, such as that based on the well-known Pontrya-gin s maximum principle, could be avoided. 

The multiple reaction algorithm can be applied to parallel reactions, series reactions, complex reactions, and independent reactions. The availability of software packages (ODE solvers) makes it much easier to solve problems using moles or molar flow rates Fj rather than conversion. For liquid systems, concentration is usually the preferred variable used in the mole balance equations. 

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