12 Chemical Akzo Nobel problem

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1 II Chemical Akzo Nobel problem 12.1 General information This IVP is a sti system of 6 non-linear DAEs of index 1 and has been taken from [Sto98]. The parallel-ivp-algorithm group of CWI contributed this problem to the test set in collaboration with W.J.H. Stortelder. We acknowledge the remarks of Dotsikas Ioannis, which improved the formulation of this problem considerably. The software part of the problem is in the le chemakzo.f available at [MM08] Mathematical description of the problem The problem is of the form with M dy dt The matrix M is of rank 5 and given by = f(y); y(0) = y 0 ; y 0 (0) = y 0 0; M = y 2 IR 6 ; 0 t 180: C A and the function f by f(y) = 0 2r 1 +r 2 r 3 r r 4 r 1 r 2 +r 3 r 2 +r 3 2r 4 r 2 r 3 +r 5 K s y 1 y 4 y r 5 +F in 1 C A ; where the r i and F in are auxiliary variables, given by r 1 = k 1 y 4 1 y ; r 2 = k 2 y 3 y 4 ; r 3 = k 2 K y 1 y 5 ; r 4 = k 3 y 1 y4; 2 r 5 = k 4 y 2 6 y ; F in = kla ( p(co 2) H y 2 ): The values of the parameters k 1, k 2, k 3, k 4, K, kla, p(co 2 ) and H are k 1 = 18:7; k 2 = 0:58; k 3 = 0:09; k 4 = 0:42; K = 34:4; kla = 3:3; K s = 115:83; p(co 2 ) = 0:9; H = 737:

2 II-12-2 DAE - Chemical Akzo Nobel problem The consistent initial vectors are y 0 = 0:444; 0:00123; 0; 0:007; 0; K s y 0;1 y 0;4 T y 0 0 = f(y 0 ): It is clear from the denition of r 1 and r 5 that the function f can not be evaluated for negative values of y 2. In the Fortran subroutine that denes f, we set IERR=-1 if y 2 < 0 to prevent this situation. See page IV-ix of the description of the software part of the test set for more details on IERR Origin of the problem The problem originates from Akzo Nobel Central Research in Arnhem, The Netherlands. It describes a chemical process, in which 2 species, FLB and ZHU, are mixed, while carbon dioxide is continuously added. The resulting species of importance is ZLA. In the interest of commercial competition, the names of the chemical species are ctitious. The reaction equations, as given by Akzo Nobel [CBS93], are given in Figure II The last reaction equation describes an equilibrium 2 FLB CO 2 ZLA + FLB k 1 - FLBT + H2 O k 2 =K - k2 FLBT + ZHU FLB + 2 ZHU + CO 2 k 3 - LB + nitrate FLB:ZHU CO 2 FLB + ZHU - k 4 - ZLA + H2 O FLB:ZHU Figure II.12.1: Reaction scheme for Chemical Akzo Nobel problem. K s = [FLB:ZHU] [FLB ] [ZHU] : The value of K s plays a role in parameter estimation. The other equations describe reactions with velocities given by r 1 = k 1 [FLB ] 4 [CO 2 ] 1 2 ; (II.12.1) r 2 = k 2 [FLBT ] [ZHU]; r 3 = k 2 [FLB ] [ZLA ]; K r 4 = k 3 [FLB ] [ZHU] 2 ; (II.12.2) r 5 = k 4 [FLB:ZHU] 2 [CO 2 ] 1 2 ; (II.12.3) respectively. Here the square brackets `[ ]' denote concentrations. One would expect from the reaction scheme in Figure II.12.1, that reaction velocities r 1, r 4 and r 5 would read r 1 = k 1 [FLB ] 2 [CO 2 ] 1 2 ; r 4 = k 3 [FLB ] [ZHU] 2 [CO 2 ]; r 5 = k 4 [FLB:ZHU] [CO 2 ] 1 2 :

3 II-12-3 However, it turns out that the chemical process under consideration is modeled more appropriately using (II.12.1){(II.12.3). The inow of carbon dioxide per volume unit is denoted by F in, and satises F in = kla ( p(co 2) H [CO 2 ]); where kla is the mass transfer coecient, H is the Henry constant and p(co 2 ) is the partial carbon dioxide pressure. p(co 2 ) is assumed to be independent of [CO 2 ]. The parameters k 1, k 2, k 3, k 4, K, kla, K s, H and p(co 2 ) are given constants. The process is started by mixing mol/liter FLB with mol/liter ZHU. The concentration of carbon dioxide at the beginning is mol/liter. Initially, no other species are present. The simulation is performed on the time interval [0; 180 minutes]. Identifying the concentrations [FLB ], [CO 2 ], [FLBT ], [ZHU], [ZLA ], [FLB:ZHU] with y 1 ; : : : ; y 6, respectively, one easily arrives at the mathematical formulation of the preceding section Numerical solution of the problem Tables II.12.1{II.12.2 and Figures II.12.2{II.12.6 present the reference solution at the end of the integration interval, the run characteristics, the behavior of the solution over the integration interval and the work-precision diagrams, respectively. The reference solution was computed by PSIDE on a Cray C90, using double precision, rtol = atol To get more insight in the exact behavior of the second component, we included a plot of y 2 on [0; 3] in Figure II For the work-precision diagrams, we used: rtol = 10 (8+m=4), m = 0; 1; : : : ; 20; atol = rtol; h0 = rtol for BIMD, GAMD, MEBDFDAE, MEBDFI, RADAU and RADAU5. The failed runs are in Table II.12.3; listed are the Table II.12.1: Reference solution at the end of the integration interval. y 1 0: y 2 0: y 3 0: y 4 0: y 5 0: y 6 0: Table II.12.2: Run characteristics. solver rtol atol h0 mescd scd steps accept #f #Jac #LU CPU BIMD :39 10: DDASSL :04 8: GAMD :89 9: MEBDFI :42 9: PSIDE :41 9: RADAU :71 8: name of the solver that failed, for which values of m this happened, and the reason for failing. Apart from H, which is generally known, all parameters have been estimated by W. Stortelder [Sto95].

4 II-12-4 DAE - Chemical Akzo Nobel problem Table II.12.3: Failed runs. solver m reason PSIDE-1 14,16,17,18,19,20 stepsize too small References [CBS93] CBS-reaction-meeting Koln. Handouts, May Br/ARLO-CRC. [MM08] F. Mazzia and C. Magherini. Test Set for Initial Value Problem Solvers, release 2.4. Department of Mathematics, University of Bari and INdAM, Research Unit of Bari, February Available at testset. [Sto95] [Sto98] W.J.H. Stortelder, Private communication. W.J.H. de Stortelder. Parameter Estimation in Nonlinear Dynamical Systems. PhD thesis, University of Amsterdam, March 12, 1998.

5 II-12-5 Figure II.12.2: Behavior of the solution over the integration interval.

6 II-12-6 DAE - Chemical Akzo Nobel problem Figure II.12.3: Work-precision diagram (scd versus CPU-time).

7 II-12-7 Figure II.12.4: Work-precision diagram (scd versus CPU-time).

8 II-12-8 DAE - Chemical Akzo Nobel problem Figure II.12.5: Work-precision diagram (mescd versus CPU-time).

9 II-12-9 Figure II.12.6: Work-precision diagram (mescd versus CPU-time).

6.2 Mathematical description of the problem. i = m j (y j y i )=r 3 2

6.2 Mathematical description of the problem. i = m j (y j y i )=r 3 2 II-6-1 6 Pleiades problem 6.1 General information The problem consists of a nonsti system of 14 special second order dierential equations rewritten to rst order form, thus providing a nonsti system of