Ind. Eng. Chem. Res. 1992,31, 2587-2593 Deahpande, P. B. Distillation Dynamics and Control; IS& Fkaearch Triangle Park, NC, 1986; pp 359-389. Doherty, M. F.; Perkine, J. D. On the Dynamics of Distillation Pro~ ~ 8 8 8 % .VI: Uniqueneee and Stability of the Steady State in HomogeneousContinuous Distillations. Chem. Eng. Sci. 1982,37, 381-392. Jacobaen, E. W.;Skogmtad, 5. Multiple Steady-States in Ideal Two-Product Distillation. AZChE J. 1991,37,499-511. Kingeley, J. P.; Lucia, A. Simulation and Optimization of ThreePhase Distillation Promsees. Znd. Eng. Chem. Res. 1988,27, 1900-1910. Kovach, J. W.,ID, Seider, W.D. Heterogeneous Azeotropic Distillation: Homotopy-Continuation Methods. Comput. Chem. Eng. 1987,11 (6), 593-605. M a g n w n , T.;Michelsen, M. L.; Fredenslund, Aa. Azeotropic Distillation Using UNIFAC. Znst. Chem. Eng. Symp. Ser. 1979,56, 4-211-4.2119.
2687
Prokopakk, G. J.; Seider, W.D.Feasible Specificationsin Azeotropic Distillation. AZChE J. 1983,29 (11,49-60, Rovaglio, M.; Doherty, M. F. Dynamics of HeterogeneousAzeotzopic Distillation Columns. AIChE J. 1990,36,39-52. Shinekey, F. G. Distillation Control; McGraw-Hill: New York, 1977; pp 93-121,167-194. Sridhar, L. N.;Lucia, A. Analysis and Algorithms for Multistage Separation Processes. Znd. Eng. Chem. Res. 1989,28,793-803. Sridhar, L. N.; Lucia, A. Analysis of Multicomponent, Multistage Separation Procesees: Fixed Temperature and Pressure Profilea. Znd. Eng. Chem. Res. 1990,29,1668-1675. Venkataraman, 5.;Lucia, A. Solving Distillation Problems by Newton-Like Methods. Comput. Chem. Eng. 1988, 12 (l),55-69.
Received for review March 5, 1992 Accepted July 28, 1992
GENERAL RESEARCH Radical Scavenging by Hydroaromatics in the Presence of Oxygen Junichi Kubo Central Technical Research Laboratory, Nippon Oil Company, Chidori-cho 8, Naka-ku, Yokohama, 231 Japan
The radical scavenging abilities of hydroaromatics toward DPPH (N&-diphenyl-N'-picrylhydrazyl) in the presence of oxygen were investigated. Tetralin, octahydrophenanthrene (OHP), synthetic H/D (a multicomponent additive containing various hydroaromatics produced by hydrogenation of decrystallized anthracene oil derived from coal tar), and HHAP (a multicomponent additive produced by hydrogenation of a highly aromatic heavy fraction from petroleum) were examined by heating at 50 O C for 3 h with DPPH in air. It was confirmed by the changes in color of the solution and the ESR spectra that those hydroaromatics have obvious radical scavenging abilities and that OHP, synthetic H/D, and HHAP have higher abilitiea than tetralin. From these experimental results, the additive effects of these hydroaromatics on the deterioration of petroleum products, rubbers, and plastics reported in our previous papers can be attributed to their radical scavenging abilities.
Introduction Pure hydrocarbons without functional groups containing heteroatoms such as 0, N, P, and S have not been applied to date to inhibit the deterioration of hydrocarbon products (except in very exceptional cases). In conventional autoxidation studies, results showing that the addition of hydroaromatics accelerates the absorption of oxygen (Larsen et al., 1942; Robertson and Waters, 1948; Yasutomi and Sakurai, 1976) have been reported. It seems that hydroaromatics, typically represented by tetralin, have been considered to be easily oxidized, and they have never been used as inhibitors to deterioration, despite some exceptional results reported to be effective in limited conditions such as the additive effect of tetralin on the oxidation of cumene (Russell, 1955),tetralin with sulfur compounds on the oxidation of light hydrocarbons (Yamaji, 1960), and tetralin hydroperoxide (Thomas and Tolman, 1962). On the other hand, hydrogen-donating hydroaromatics (abbreviated as hydmaromatics) have been widely used in such pmcessea as coal liquefaction and heavy oil upgrading (Carbon et al., 1958; Fisher et al., 1982; Kubo et al., 1988) serving as radical scavengers to reduce coke formation in a reductive atmosphere. In view of these facts, hydroaromatic-type additives were examined with regard to the deterioration of hydrocarbon products, and it was found that they were obviously effective against the thermal and oxidative deterioration of petroleum products (Kubo,
1991a),rubbers (Kubo, 1991b) and plastics (Kubo, 1991b) and also against radiation degradation of polyolefins (Kubo and Otsuhata, 1992). The inhibiting effects of the hydroaromatic-typeadditives seem to be due to their radical scavenging abilities, as judged from the results of the thermal deterioration tests (Kubo, 1991a). To confirm this, the radical scavenging abilities of tetralin, octahydrophenanthrene, and two multicomponent additives, both of which exhibited noticeable effects on the deterioration of hydrocarbon products, were examined with DPPH (NJV-dipheny1-N'picryhydrazyl). In this paper, the relation of theae results to those from the thermal deterioration tests is discussed.
Experimental Section (Kubo et al., 1991) The radical scavenging abilities of the following substances toward DPPH, which contains relatively stable radicals at room temperature, were examined (1)synthetic H/D (Kubo, 1991a);(2)HHAP (Kubo, 1991b);(3) tetralin; (4) OHP (1,2,3,4,5,6,7,&o&ahydrophenanthrene); (5)naphthalene; (6) phenanthrene; (7)decalin; (8) blank (without addition). Synthetic H/D, which exhibited noticeable effects on the deterioration of petroleum products, rubbers, and plastics, was produced by the hydrogenation of decrystallized anthracene oil obtained from a coal tar fraction.
0808-5885/92/2631-2687$03.00/0(8 1992 American Chemical Society
2688 Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992 Table I. Main Components in Synthetic H/D (250-400 O C Fraction) component fraction (wt 96) 1.38 butyltetralin 0.77 diph eny1 4.78 propyldihydronaphthalene 1.22 acenaphthene 1.64 tetrahydroacenaphthene 3.16 dibellzofuran 3.28 methyldibenzofuran 1.49 fluorene 3.85 methyltetrah ydrofluorene 4.39 phenanthrene and anthracene 3.12 9,10-dihydrophenanthrene 9.41 tetrahydrophenanthrene 19.24 octahydrophenanthrene 2.72 octahydroanthracene 2.01 methylphenanthrene and methylanthracene 1.39 tetrahydrofluoranthene 2.44 Pyrene 1.12 dihydropyrene 1.66 hexahydropyrene
Table 11. Properties of Heavy Hydroaromatics from Petroleum (HHAP) density (15OC, g/cm3) 1.028 2.536 X lo4 (25.36cSt) viscosity (100OC, m2 8-l) carbon residue (wt %) 1.15 pour point ("C) 27.5 h h mint ("(2) 234 elemental analysis (% ) 89.7 C 10.1 H 0.08 S
N H/C (atomic ratio) average molecular weight refractive index distillation ("(2) IBP/5% 10130 50170 90/EP composition saturates aromatics 'H NMR of aromatics H*/H H,/H HR/H
HrlH fa (fraction of aromatic carbons)
-
1.35 398 1.5978 3481374 3851420 4471478 5251584 21.3 78.7 0.16 0.30 0.37 0.17 0.38
Table 111. Chemicals Used in the Experiments chemical maker Wako Junyaku DPPH Tokyo Kasei NJV-diphenyl-N'-picrylhydrazine Kanto Kagaku tstralin Kanto Kagaku naphthalene Kanto Kagaku phenanthrene Kanto Kagaku decalin Kanto Kagaku benzene Tokyo Kaeei 1,2,3,4,5,6,7,8dydrophenanthrene a
The main components contained in the synthetic H/D are shown in Table I. The feed source was changed to a petroleum fraction in order to get a heavier (higher molecular weight) additive having lower volatility. HHAP was produced by the hydrogenation of highly aromatic heavy oil and contains more than 300 components. It is difficult and meaningless to define the individual components contained in HHAF' (the individual components cannot be isolated even by gas chromatography), so the propertiee of the mixtures are shown in Table II. HHAP has partly hydrogenated produds (hydrogenation of aromatic rings 55%) of condensed aromatic rings, its average molecular weight is 398, ita boiling point is above 350 O C , and it has a yellow-brown color and solid/liquid state at room temperature. Tetralin and OHP were used as typical hydrogen-donating hydroaromatica, naphthalene and phenanthrene were used as typical aromatics and, decalin was used as naphthene, for comparison. The testing reagents used in these experiments are summarized in Table 111,and they were used without refinement. First, DPPH in a benzene solution of lo+ mol/L was prepared, and then 20 mL of this solution (containing 2 X lod mol/L DPPH) was taken out. The following amounts of the substanma listed above were added to the solution (benzene was added in the case of blank): (1)2 X lob mol (equal to the moles of DPPH); (2) 2 X la-' mol (10 times the moles of DPPH); (3) 2 X low3mol (100 times the moles of DPPH). Thew samples, placsd in an Erlenmeyer flask (200 mL in inner volume, equipped with a cooler/condenser), were heated at 50 "C,for 3 h in air (in a water bath) and analyzed. The changes in the color of the samples before and after the heating were observed by naked eye and an ultraviolet-visible ray absorptiometer (JASCOUBEST30), and also analyzed by ESR (JEOL JES-FE 3XG). The composition of the samples was analyzed by gas chromatography (SHIMADZU GC-SA, column G-100, 40 m; and SHIMADZU QP-2000, column CBP-1, 25 m) and gas chromatography-maas spectrometry (GC-MS)(JEOL JMS-DX300, E170V).
Results and Discussion The color of the solution of DPPH in the benzene 80lution ia dark purple, and when the hydrazyl radicala disappear, the color changes to yellow-brown. The changes made premium premium premium fiit preium premium premium
Duritv (Wt 96) 70.7 75.6 98.7 100.0 99.0 98.5 99.6 92.2
remarks analyzed by CITI" analyzed by CITI analyzed by CITI analyzed by CITI analyzed by CITI maker's analytical value maker's analyticalvalue analyzed by Nippon Oil Company
Chemicala Inspection & Testing Institute.
Table IV. Changem in Color of the Solution through Heating at 60 OC for 3 h (toward DPPH 2 X lod mol) with Varying Amounts of Additives no. additive 2 x lob mol 2 x lo-' mol 2 x 10-8 mol yellow-brown yellow-brown Unchanged synthetic H/D yellow-brown yellow-brown Unchanged "AP yellow-brown yellow-brown (dark purple remaining) Unchanged tetralin yellow-brown yellow-brown 1,2,3,4,5,6,7,8odahydrophenanthrene Unchanged Unchanged Unchanged naphthalene Unchanged Unchanged Unchanged phenanthrene Unchanged Unchanged UnClWgd blank (without addition) 7 unchanged UnChanged unchanged decaline 8 Unchanged
Ind. Eng.Chem. Res., Vol. 31, No. 11,1992 2589
-
scan
Figurn 1. ESR spectrum of DPPH in benzene solution before heating (DPPH lo-* mol/L, 2
X
lo* mol).
(0 AMP 1 0 0 1 1
Figum 2. ESR spectra of DPPH in benzene solution after heating at 60 "C for 3 h by the addition of 2 X lO-9mol of the following subetancee: (1) synthetic H / D (2) HHAP; (3) tetrahq (4) O W (5) naphthalene: (6) phenanthrene; (7) blank (without addition); (8) decalin.
in color of the solutions after the heating at 50 OC,for 3 h, are shown in Table IV. After the addition of 2 X lo+ mol (equivalent to DPPH) of the additives, changes in color did not occur for any of the additives, but after the addition of 2 X lo-' mol (10times DPPH), clear changes in color were observed for 1-4, although a slight darkpurple color remained in the case of tetralin. However, no changes in color could be found at all after the addition of 5-8. When 2 X mol (100 times DPPH) of the additives was added, the colors of the solutions were completely changed by 1-4, but changes in color were not observed with 5-8. Next, the ESR spectmm for DPPH in benzene solution (DPPHl W 3 mol/L) is illustrated in Figure 1, and clear peaks showing a hydrazyl radical can be observed. The ESR spectra showing the samples after the heating at 50 OC for 3 h with 2 X lop3mol of additives 1-8 are shown in Figure 2. In the casea of the samplea to which additivea 1-4 were added, the signale showing the hydrazyl radicals obviously disappeared, but in the cases of the samples to which 5-8 were added, eip;nale remained elm& unchanged. The ESR spectra magnified 10 times for the samples containing additives 1-4 are shown in Figure 3, and it is observed that no peaks can be found when 1,2, and 4 were added, but peaks can be recognizedwhen 3 was added (the
(3) AMP
100 x 10
AMP
100 x 10
(4) '
?+# - - -
F'igure 3. Maenifiea (10times) ESR spectra of DPPH in benzene solution after heating at M) O C for 3 h by the addition of 2 X 10-8 mol of the following substances: (1) synthetic H / D (2) HbIAP; (3) tatralin; (4) OHP.
shapea of the peaks are different from those of the original ones). The relation between the relative signal intensities and
2590 Ind. Eng. Chem. Res., Vol. 31, No. 11,1992 Table V. Changes of Tetralin and Octahydrophenanthmne through the Reactions with DPPH (SO ' C , 3 h) B
theor formation of aromaticsn A/B X 100 (wt % additive) (%)
A
producta
w additive
(wt % additive)
028
1.50
18.7
1.25
97.6
phenanthrene equiv 1.22 Additives (1O-hol)
Figure 4. Change8 of relative signal intensities hy the addition of the foUowing substenees: ( 0 )deealin; (m) phenanthrene; (A)tetralin: (0)HHAP,(A)OHP;(0)Synthetic HID.
the amount of the additives is illustrated in Figure 4. The vertical axis shows the relative signal intensity compared with the one of DPPH in a benzene solution before the heating. Almost no changes can be observed when decalin and phenanthrene are added, but the signal intensities have obviously decreased in the cases of 1-4 due to the increase in the amount of the additives. These results are quite consistent with the changes in color of the samples. From these results, the following can be concluded 1. The radical scavenging abilities of hydroaromatics (synthetic H/D, HHAP, tetralin, and OHP) are obvious. 2. The radical scavenging ability of tetralin is lower than those of the three hydroaromatics. 3. No radical scavenging abilities of aromatics (naphthalene and phenanthrene) and decalin can be found under these conditions. From these experimental results, the radical scavenging abilities of hydroammatics are quite obvious. The radical scavenging abilities of such hydroaromatics as tetralin, 9,lO-dihydroanthracene, 9,10-dihydrophenanthrene,and 1,4-dihydronaphthalene have already been recognized (Blaude et al., 1954; Hogg et al., 1961),and now it bas been confirmed that hydroaromatics exhibit obvious radical scavenging abilities toward DPPH even in the presence of oxygen. It was already proven that the thermal, oxidative, and radiation deterioration of hydrocarbon products is inhibited by the addition of hydroaromatics (tetralin, OHP, synthetic H/D, and HHAP). The experimental results described in this paper support the conclusion that the inhibiting effects can be attributed to the radical scavenging abilities of hydroaromatics. From Figure 4, it can he observed that the radical scavenging ability of tetralin is lower than that of OHP, synthetic H/D, and HHAP when 2 X lo4 mol of the additives was added. The reaction producta from tetralin and OHP after the reactions with DPPH in this case were investigated (Kubo et al., 1992). The results are summarized in Table V. The analytical values of aromaties by gas chromatography expressed as the weight percent relative to the hydroaromatics added are shown in part A, and the theoretical formation of aromatics calculated from the changes in the ESR spedra assuming all changes in the spectra are due to hydrogen donation from hydroaromatics is shown in part B. In the case of OHP, A/B was near 1.0 (0.976), and this means that almost all the radicals were trapped by hydrogen donated from OHP. However, in the case of
"Theoretical formation of aromatics calculated from the changes of ESR spectra assuming all changes of spectra are due to hydrogen donation from hydroaromatics.
a
m
@
@
350T SOT 350T 350T EAT 12h 72h 7Zh 72h 72h
1
@
addition of 5 m%ohmanthme ~
~
L w l t h o u t addition addition of 5 m9a naphthalene addition of 5 m9a tctralin addition of 5 m9a onahydrophenanthrene
Figure 6. Samples after thermal deterioration tecta (360 OC, 72 h).
tetralin, A/B was only 0.187, that is, the portion of radicals trapped by hydrogen donated from tetralin was only 19% and other radicals are supposed to have disappeared for other reasons. From analytical studies (GC-MS),it was estimated that the other radicals were removed by oxygen-containing compounds from the tetralin. From these data, it can be said that tetralii is more easily oxidized than OHP and that the radical scavenging ability of tetralin is lower than that of OHP under these conditions. The radical scavenging abilities of hydroaromaticscan be considered to have a close relation to the hydmgen donation from hydroaromatica, as judged from the results of the thermal deterioration testa (Kubo, 1991a). That is, the additive effects of tetralin and OHP toward lube base oil (SAE-50)were examined a t 350 "C, 72 h, and 395 O C , 6 h, using a high-pressure vessel (50 mL in inner volume, without agitation). Obvious effects were observed after the addition of tetralin and OHP a t 350 O C , 72 h, when the vapor phase was replaced by nitrogen (purity more than 99.99%), hut effects could not be observed in the changes of color after the addition of naphthalene and phenanthrene (Figure 5). The additive effects remained in the case of the addition of OHP, but the additive effeds disappeared in the case of the addition of tetralin when the test conditions were made more severe (395 O C , 6 h; Figure 6). In the same teats,when the oxygen/nitrogen ratio in the vapor phase was changed, carbonyl almorbenca, acid values, and changes in color were observed. Increases in carbonyl absorbance by IR were restricted by the addition of 5 w t
Ind. Eng. Chem. Res., Vol. 31, No. 11,1992 2591
@ @ a @ @
3%72 39572 39572 39572 395t 6h 6h 6h 6h 6h
111
o'20
Laddition of 5 wt%
r
phmanthmc
L w i t h o u t addition
addition of 5 wt70 naphthalene addition of 5 ut% tefrslin addition of 5 wt% oetahydroDhenanthsns I
Figure 6. Samples after thermal deterioration testa (395 'C, 6 h).
50
0
100
Oxygen in vapor phase ( v o i %)
Figure 8. Changes in acid values as a function of oxygen in the vapor phase: ( 0 )addition of 5 wt % tetralin; (A)addition of 5 wt %
om.
e e i l : 0 . 2 5 m th. fixdceell
t
I 0
0
50
100
Oxygen in vapor phase ( "01%)
Figure 7. Changes in carbonyl absarbance as a function of oxygen in the vapor phase: (0)addition of 5 wt % tetralin;(A) addition of 5wtWOHP.
% of both tetralin and OHP, and the addition of OHP was more effective than that of tetralin (Figure 7). The increases in the acid values were also restricted by the addition of tetralin and OHP, and the addition of OHP was more effective than that of tetralin (Figure 8). The hydrogen donation from tetralin and OHP through these tests is illustrated in Figure 9. It can be seen that the hydrogen donation from OHP is larger than that from tetralin, and the hydrogen donation from hydroaromatics is increased by raising the oxygen partial pressure in the vapor phase. Gas chromatograms showing the formation of naphthalene from tetralin, as well as tetrahydrophenanthrene and phenanthrene from OHP, are shown in Figure 10. It can also be seen from these figures that the hydrogen donation from hydroaromatics is increased by raising the oxygen in the vapor phase. The resulta obtained from the testa with DPPH are very consistent with the ones from the thermal deterioration testa in the following respects: 1. The addition of hydroaromatics is obviously effective in inhibiting deterioration, hut the addition of aromatics is not effective a t all. 2. OHP is more effective than tetralin, and hydrogen donation from OHP is larger than that from tetralin. In conventionalautoxidation,hydroammatics have been considered to be easily oxidized, however, the reaction pathways of hydroaromatics a t lower oxygen partial
,I)
*I
10
40
OI
50
in vSp,r
$0
70
I*
,I
1m
phase IVOl R)
Figure 9. Hydrogenation donation fromtetrdin and OHP by oxygen in the vapor phase: (@) tatralin; (A) OHP.
pressure have not been made clear. From the experimental resulta mentioned above, it can he supposed that the reaction pathways of tetralin are different from thw of OHP at relatively low oxygen partial pressure and that they behave differently depending on the oxygen partial pressure at the reaction sites, as illustrated in Figure 11. It seems that oxidation potentials of tetralin and OHP cannot be estimated from their chemical structures alone, and it can he supposed that diffusion rates of oxygen molecules should be related. The radical scavenging abilities of hydroaromaties have not been utiliid to date except in such processes as coal liquefaction and heavy oil upgrading using a hydrogendonor solvent in a reductive atmosphere. The reason seems to be that hydroaromatics have been considered to be easily oxidized in the presence of oxygen. However, it was found that hydroaromatics exhibit obvious radical scavenging abilities even in the presence of oxygen, and it was also shown by very practical testa that they are prominently effective against thermal, oxidative, and radiation deterioration of hydrocarbon producta. It can he suggested now that the radical scavenging abilities of hydroaromatics can be available in many fields even in the presence of oxygen. The fact that a material having prominent radical scavenging abilities can be produced from petroleum is very important because it suggeata that such effective inhibitors can be produced in large quantities and, accordingly, the cost can be reduced. It should he possible to apply this
2592 Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992
a
lelral
1)
I"
jsphthalene
i , Figure 10. Gas chromatogram of the sample after the deterioration test (A) N2 100 vol %, 350 O C , 48 h, addition of 5 w t % tetralin; (B) O2 100 vol %, 350 O C , 48 h, addition of 5 wt % tetralin; (C) Nz100 vol %, 350 O C , 48 h, addition of 5 wt % OHP;(D) 0%100 vol %, 350 O C , 48 h, addition of 5 w t % OHP. tetra 1in
a'+@
oxygen containing compounds
at the reaction sites
(I)
(PI
&
oxygen containing compounds
(II)
a=(.!L) (I)
I
0,partial pressure
+
at the reaction sites
Figure 11. Illustration of differences of tetralin and octahydrophenanthrene in formation of oxygen-containingcompounds and in inhibiting abilities.
kind of material for such purposes as the prevention of coke and sludge formation in refineries and chemical plants. In addition, hydroaromatics seem to be very stable against high temperature and radiation as observed in the thermal deterioration teats and radiation degradation teats (Kubo, 1992), and they can be expected to be used in conditions where conventional radical scavengers such as hindered phenols and amines are difficult to use. In these experiments, the diffusion of oxygen was not fast enough to be neglected. However, in practical uses such as petroleum products, rubbers, and plastics, the oxygen partial pressure at the reaction sites is relatively low because of the low diffusion rates, and the results obtained in these conditions where the diffusion of oxygen cannot be ignored are qualitatively valid for practical uses. In addition, reaction products from tetralin and OHP suggest that OHP is dehydrogenated to aromatics more easily than tetralin, and the multicomponent additives behave similarly to OHP as shown in Figure 4.
Conclusions The radical scavenging abilities of hydroaromatics were examined by heating at 50 "C, for 3 h in air with DPPH (N,N-diphenyl-N'-picrylhydrazyl),and the following results were obtained
1. The radical scavenging abilities of tetralin, octahydrophenanthrene (OHP), synthetic H/D (a multicomponent-type additive derived from a coal tar fraction and effective against the deterioration of hydrocarbon products), and I-€"(a multicomponent-type additive derived from a heavy fraction of petroleum and effective against the deterioration of hydrocarbon products) were obvious. 2. The radical scavenging abilities of OHP, synthetic H/D, and HHAP were higher than those of tetralin. 3. No radical scavenging abilities of decalin, naphthalene, or phenanthrene were found under these conditione. 4. From the analyses of the reaction products, OHP seems to behave differently from tetralin in the presence of oxygen at relatively low oxygen partial pressure. From these results, the radical scavenging abilities of hydroaromatics were confirmed even in the presence of oxygen, and the inhibiting effects of these hydroaromatics on the deterioration of hydrocarbon products can be attributed to their radical scavenging abilities.
Acknowledgment
I thank Dr. Katsumi Tokumaru, profeseor at Tsukuba University, who gave me valuable suggestions about how to conduct these testa, and I also thank Mesers. Ryuji Miyagawa and Sueji Takahashi, Chemicals Inspection and Testing Institute, who conducted the experiments. Registry No. DPPH,1898-66-4; OHP, 5326-97-3; tetralin, 119-64-2; naphthalene,91-20-3; phenanthrene, 85-01-8; decalin, 91-17-8.
Literature Cited Blaude, E. A.; Brook, A. G.; Linstead, R. P. Hydrogen Transfer. Part 5. Dehydrogenation Reactions with Diphenylpicrylhydrazyl. J. Chem. SOC. 1964,3574-3578. Carlson, C. S.;Langer, A. W.; Stewart,J.; Hill, R. M. Thermal Hydrogenation. Znd. Eng. Chem 1968,50, 1067-1070. Fisher, I. P.; Southrada, F.; Wood,H. J. Gulf Canah Donor Refined Bitumen Heavy Oil Upgrading Process. Prepr. Diu. Pet. Chem. 1982,27,83&848. Hogg, J. S.; Lobmann, D. H.; Russell, K. E. The Reaction of 2,2Diphenyl-1-picrylhydrazylwith 9,lO-Dihydroanthracene and l,4Dihydronaphthalene. Can. J . Chem. 1961,39,1394-1397. Kubo, J. Inhibiting Abilities of Hydroaromatics againat Deterioration of Hydrocarbon Products-1. Inhibition of Thermal Deterioration and Ita Application to Oil Products. Fuel Process. Technol. 1991a, 27, 263-277. Kubo, J. Inhibiting Abilities of Hydroaromatice against Deterioration of Hydrocarbon Products-2. Application to Rubbers and Plastics. Fuel Process. Technol. 199lb, 28, 19-34.
Ind. Eng. Chem. Res. 1992,31,2593-2603 Kubo, J. Inhibiting Effect of Hydroaromatic Additive on Radiation Degradation of Mineral Oil. J. Jpn. Petr. Znst. 1992,35,296-299. Kubo, J.; Otsuhata, K. Inhibition of Radiation Degradation of Polyolefm by Hydroaromatics. Radiat. Phys. Chem. 1992, 39, 261-268.
Kubo, J.; Yamaahita, T.; W y a , IC;Katoh, K.; Satoh, M. Cracking of Heavy Oila by Combination of Hydrogen Donor Solvent and Catalyst. J . Jpn. Pet. Znst. 1988,27,838-&18. Kubo, J.; Miyagawa, R.; TakahaAi, S. Radical Scavenging Abilities of Hydroaromatics. J. Jpn. Pet. Znst. 1991,34, 473-476. Kubo, J.; Miyagawa, Takahaahi, S. Reactione between DPPH and Hydroaromatics in the Presence of Oxygen. J. Jpn. Pet. Znst.
2593
Part 1. Investigation of Autoxidation Producb. J. Chem. SOC. 1948,1574-1590.
Ruseell, G. A. The Competitive Oxidation of Cumene and Tetralin. J. Am. Chem. SOC.1966, 77,4683-4690. Thomas, J. R.;Tolman, C. A. Inhibition of Cumene Oxidation by Tetralin Hydroperoxide. J. Am. Chem. SOC.1962,84,2079-2080. Yamaji, T. Antioxidant Action of Sulfur Compounds. J. Jpn. Pet. Znt. 1960,3,389-393. Yastoumi, S.; Sakurai, T. The Effecte of Aromatic Hydrocarbons on the Oxidation Stability of Squalene (Part 1). Oxidation of Squalene as a Model Compound of Paraffin Portion in Mineral Oil. Bull. Jpn. Pet. Inst. 1976,18, 107-112.
1992,35,111-114.
Received for review March 9, 1992 Revised manuscript received June 9, 1992 Accepted July 22,1992
Lareen, R. G.; Thorpe, R.E.; W i e l d , F.A. Oxidation Characteristics of Pure Hydrocarbons. Znd. Eng. Chem. 1942,34,183-193. Robertson, A; Waters, W. A. Studiea of the Autoxidation of Tetralin.
Solving a Class of Near Index Problems as Perturbations of Index Problems Yonsoo C h u g t and Arthur W. Westerberg* Engineering Design Research Center and Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
This paper extends a numerical algorithm of C h u g and Westerberg for solving nonlinear index problems. The extension permits the solving of a special class of near index problems whose ill-conditioned behavior produces one or more small but nonzero pivots when attempting to solve the Newton equations corresponding to the model equations. Such problems are generally stiff differential-algebraic equation systems, sufficiently stiff that they often cannot be solved by existing stiff ordinary differential equation solvers such as LSODI. We present an algorithm for solving such problems both stably and accurately as a perturbation of the index problem which resulta from setting the small pivot(s) to 0.
Introduction There is an extensive literature on the numerical solution of ordinary differential equations (ODES). Several solution algorithms and computer codes are available even for stiff ODES. When ODESare constrained by algebraic equations, severe problems which have been termed “index problems” can occur when attempting their solution. Gear [?‘I, Pantelides [18], Bachmann et aL [I,21,and Chung and Westerberg [4]have p r o p o d solution algorithm for index problems. There are many parallels between the theory for the numerical solution of DAEs and that for stiff differential equations. We shall show that an index problem can be interpreted as an infinitely stiff system. We shall ale0 introduce what we term near index problems and study their relationship to stiff differential equations. This paper presents a numerical algorithm for solving this special class of problems. We begin with a review of previous work on index problems. Previous Work on Index Problems Several papers have appeared in the past decade which are related to the solution of differential-algebraicequation systems (DAEs). Brenan et al. [3] review more than 200 papers on the solution of differential-algebraicequations. In solving differential-algebraic equations, two major numerical difficulties have been recognized consistent initialization and stable propagation. If systems have these
* Author to whom correspondence should be sent. Current addrew Korea Institute of Science and Technology, Seoul, Korea. 0sS8-5885/92/2631-2593$03.00/0
problems, they have what is now termed an =index problem”. Chung and Westerberg [4]review papers [ I , 6, 7,11, 12, 13, 18, 19, 22, 23, 261 presenting solution methods for index problems. In addition the following papers are relevant. Yip and Sincovec [28] investigate the properties of DAEs. They relate classical theories of matrix pencils to the solvability of DAEs. They also extend the concepts of reachability, controllability, and observability of state variable systems to DAEs. Leimkuhler et al. [16,171 present an approximation method for the consistent initialization of DAEs. They characterizethe consistency requirement with a system of equations. The consistency equations comprise the problem itself, user-controlled specifications,and derivative equations depending on the index of the system. They approximate derivative equations with one-sided finite differences and analyze the numerical solution of the resulting system for certain classes of DAEs. C h u g and Westerberg [4]propose a numerical algorithm for detecting and solving nonlinear index problems. They detect the rank of the Jacobian matrix of the original formulation to find whether the system has an index problem or not. If it has an index problem, they analyze the Jacobian matrix to find the equations responsible for the singularity of the Jacobian matrix which are then symbolicallydifferentiated, similar to the first step in Gear and Petzold [Ill, Gear [q,and Bachmann et al. [I, 21. However, they do not require a tedious symbolic elimination required in these earlier works because they prove a theorem which shows that only the new variablea derived with differentiation have nonunique solutions which vary when arbitrarily setting values for an appropriate set of Q 1992 American
Chemical Society
