Differential-algebraic system

Petzold, L. R. A Description of DASSL A Differential-Algebraic System Solver, Sandia National Laboratory Report SAND82-8637 also in Stepleman, R. S. et al., eds. IMACS Trans, on Scientific Computing, vol. 1, pp. 65-68. 

Differential-Algebraic Systems Sometimes models involve ordinary differentia equations subject to some algebraic constraints. For example, the equations governing one equihbrium stage (as in a distillation column) are... 

Differential/Algebraic System Solver", Proc. of the IMACS World Congress, Montreal, August 8-13, 1982... 

Taking these effects into account, internal pore diffusion was modeled on the basis of a wax-filled cylindrical single catalyst pore by using experimental data. The modeling was accomplished by a three-dimensional finite element method as well as by a respective differential-algebraic system. Since the Fischer-Tropsch synthesis is a rather complex reaction, an evaluation of pore diffusion limitations... 

Fig. 4. Schematic of ammonia synthesis reactor. Reprinted with permission from Comp. Chem. Eng., 14, No. 10, 1083-1100, S. Vasantharajan and L. T. Biegler, Simultaneous Optimization of Differential/Algebraic Systems with Error Criterion Adjustment, Copyright 1990, Pergamon Press PLC.

Burrage, K., and Petzold, L. R., On order reduction for Runge-Kutta methods applied to differential/algebraic systems and to stiff systems of ODEs," Lawrence Livermore National Laboratory, UCR-98046 preprint (1988). 

Vasantharajan, S., and Biegler, L. T., Simultaneous parameter optimization of differential-algebraic systems, Computers and Chemical Engineering 14(10), 1083-1100(1990). 

When the boundary-layer approximations are applicable, the characteristics of the steady-state governing equations change from elliptic to parabolic. This is a huge simplification, leading to efficient computational algorithms. After finite-difference or finite-volume discretization, the resulting problem may be solved numerically by the method of lines as a differential-algebraic system. 

T. Maly and L.R. Petzold. Numerical Methods and Software for Sensitivity Analysis of Differential-Algebraic Systems. App. Num. Math., 20 57-79,1996. 

L.R. Petzold, A description of DASSL a differential/algebraic system solver, SAND82-8637, Sandia National Laboratories, 1982. 

Pantelides CC. The consistent initialization of differential-algebraic systems. SIAM J Sci StatComp 1988 9 213-231. 

P. Bilardello, X. Joulia, J.M. Le Lann, H. Delmas, and B. Koehret. A general strategy for parameter estimation in differential-algebraic systems. Computers and Chemical Engineering, 17(5/6) 517-525, 1993. 

Petzold, L. R. DDASSL A differential/algebraic system solver. Albuquerque, NM, Sandia National Laboratories Report No. SAND82-8637 (1982). 

The multi-mode model for a tubular reactor, even in its simplest form (steady state, Pet 1), is an index-infinity differential algebraic system. The local equation of the multi-mode model, which captures the reaction-diffusion phenomena at the local scale, is algebraic in nature, and produces multiple solutions in the presence of autocatalysis, which, in turn, generates multiplicity in the solution of the global evolution equation. We illustrate this feature of the multi-mode models by considering the example of an adiabatic (a = 0) tubular reactor under steady-state operation. We consider the simple case of a non-isothermal first order reaction... 

To solve this problem, the iS-measurement equation (Eq. 2) is recalled in its continuous-instantaneous version (ys(t) = S(t)y, later, a one time derivative is taken by replacing the resulting time-derivative of S (in the right-hand side) by the map fs(f, finally, the P-dynamics is recalled to obtain the following differential-algebraic system... 

With previously published kinetic constants [10,11], presented in Table 1, the model was solved for two different initial conditions. The former one (type I) presumes the existence of a reactants and products profile at t=0, whereas the latter (type II) considers that the reactor is empty at t=0. The resultant differential-algebraic system of equations was solved by backward finite differences formula with variable step, implemented in the DASSL code [12,13], The numerical convergence was assured by increasing the number of finite elements until no further modification in the model simulations was obtained. Hence, the number of elements was gradually increased fi om 20 to 100. Nevertheless, no significant diference was observed, i. e., all product yields were obtained with errors smaller than 10 . ... 

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