Chemical Reaction Engineering as an Intellectual Discipline - ACS

1 Chemical Reaction Engineering as an Intellectual Discipline

Downloaded by ILLINOIS STATE UNIV on December 2, 2012 | http://pubs.acs.org Publication Date: July 28, 1983 | doi: 10.1021/bk-1983-0226.ch001

R. ARIS University of Minnesota, Department of Chemical Engineering and Materials Science, Minneapolis, MN 55455

What, you are entitled to ask, do you mean by an intellectual discipline? Before attempting any formal answer, let me turn to two of the traditional disciplines and try to sketch some of the characteristics that they may have or induce in their adepts. I take my text from the advice given to President Gilman, the first of the Johns Hopkins University, to start with the best classical scholar and the best mathematician that he could find. The latter was none other than the great James Joseph Sylvester, retired from Woolwich since 1870 and spending his time in the enjoyment of the classics, playing chess and versification on the principles of his "Laws of Verse"—a pamphlet by which he set great store. With his appointment to Hopkins at the age of 62 there came the second flowering of his genius and with it his exploration of the fundamental system of invariants and the syzygies of algebraic forms. S y l v e s t e r ' s career had not been an easy one ÇL) . A f t e r h i s s t u d i e s at Cambridge had been i n t e r r u p t e d by i l l n e s s he took h i s degree i n 1837 as Second Wrangler i n the same c l a s s as George Green. To say that he took h i s degree i s not q u i t e accurate, f o r S y l v e s t e r , who says of himself that he was one of the f i r s t h o l d i n g "the f a i t h i n which the founder of C h r i s t i a n i t y was educated" t o compete f o r the mathematical t r i p o s , could not comp l e t e h i s degree without s u b s c r i b i n g t o the 39 A r t i c l e s of the Church of E n g l a n d — a s u b s c r i p t i o n he was u n w i l l i n g t o make. He t h e r e f o r e went to T r i n i t y C o l l e g e , D u b l i n from which he received h i s degrees i n 1841. His Cambridge degree he d i d not r e c e i v e u n t i l 1872, when the r e l i g i o u s b a r r i e r s had at length been removed. I n 1890 he was given an honorary Sc.D. at the same time as Benjamin Jowett, Henry Perry Liddon and other notables. The P u b l i c Orator c o u l d , by then, bracket him w i t h Newton as " S y l v e s t e r noster" i n the accolade: "Nonnulla quae Newtonus noster, quae F r e s n e l i u s , Iacobius, Sturmius, a l i i , imperfecta r e l i q u e r u n t , S y l v e s t e r n o s t e r , aut e l e g a n t i u s e x p l i c a v i t aut argumentis v e r i s comprobavit." A f t e r two years at U n i v e r s i t y C o l l e g e , he crossed the A t l a n t i c to the U n i v e r s i t y of V i r g i n i a but 0097-6156/83/0226-0001$06.00/0 © 1983 American Chemical Society In Chemical Reaction Engineering—Plenary Lectures; Wei, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.


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remained there only a matter of months ( 2 ) . Returning to England he spent some time as an actuary f o r a l i f e insurance company, a drudgery which might have s t i f l e d the mathematical genius of a l e s s robust man. During t h i s p e r i o d , however, h i s f r i e n d s h i p w i t h Cayley matured and he took p r i v a t e p u p i l s , the most i l l u s t r i o u s of whom was Florence N i g h t i n g a l e who went out to Crimea i n 1854, the year of S y l v e s t e r ' s r e l e a s e from the l e g a l world. He returned to academic l i f e as p r o f e s s o r of mathematics at Woolwich w i t h a s a l a r y of £550, government quarters and the r i g h t of a pasturage on Woolwich Common. He continued there u n t i l 1870 when he r e t i r e d over what he regarded as an u n f a i r change i n the r e g u l a t i o n s and w i t h some b i t t e r n e s s over h i s pension. Perhaps i t was t h i s that made him, i n coming to Johns Hopkins, request that h i s s a l a r y , the c o n s i d e r a b l e sum of $5,000, be paid i n gold ( 3 ) . At a l l events the move f o r the second time to America was a great success and he found h i m s e l f able to teach mathematics w i t h great freedom and play a l e a d i n g r o l e i n the i n t e l l e c t u a l l i f e of Baltimore s o c i e t y . H i s colleague i n c l a s s i c s was a l s o of the g r e a t e s t d i s t i n c t i o n . C a l l e d from the same u n i v e r s i t y i n which S y l v e s t e r had had so unhappy an experience years b e f o r e , B a s i l Lanneau G i l d e r s l e e v e was a Southerner born and bred who served w i t h the Confederate Army w h i l e the C i v i l War was i n progress during the summer months i n order (as he says) "to earn the r i g h t to teach Southern youth f o r nine months...by sharing the fortunes of t h e i r f a t h e r s and b r o t h e r s at the f r o n t f o r three." Much l a t e r i n the A t l a n t i c Monthly of January 1892 he s t i l l was "not c e r t a i n t h a t a l l " readers might " a p p r e c i a t e the e n t i r e clearness of conscience w i t h which we of the South went i n t o the war" ( 4 ) . I have not been able to f i n d any reference to the d i r e c t i n t e r c o u r s e of these two men, though I would t h i n k t h a t , w i t h the exception of the q u e s t i o n of s t a t e s r i g h t s , they would have had much i n common. I t was S y l v e s t e r s d e l i g h t to read the c l a s s i c s and G i l d e r s l e e v e would have approved the motto which he gave the American J o u r n a l of Mathematics which was founded at Hopkins i n 1878 and e d i t e d by Sylvester: 1

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G i l d e r s l e e v e does r e f e r to S y l v e s t e r s m e t r i c a l theory of "phonetic syzygy" i n one of the B r i e f Mentions of the American J o u r n a l of P h i l o l o g y , which he founded i n 1880 e a r l y i n h i s time at Hopkins and published from there f o r many y e a r s — a pleasant f o i l to S y l v e s t e r ' s American J o u r n a l of Mathematics. G i l d e r s l e e v e was, of course, a much younger man—45 to S y l v e s t e r ' s 62—when he came to Hopkins b u t , i n c o n t r a s t to S y l v e s t e r ' s b r i e f tenure of 8 y e a r s , he presided over the d e s t i n y of the c l a s s i c s there f o r the next 39. During that time he e s t a b l i s h e d graduate s t u d i e s i n the c l a s s i c a l d i s c i p l i n e s , d i r e c t i n g no l e s s than 67 d o c t o r a l d i s s e r t a t i o n s thus having a profound i n f l u e n c e on c l a s s i c a l s t u d i e s throughout the country. I n 1901 some f o r t y - f i v e of h i s o l d p u p i l s put together a volume of l a r g e l y p h i l o l o g i c a l s t u d i e s i n h i s

In Chemical Reaction Engineering—Plenary Lectures; Wei, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

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Reaction Engineering as an Intellectual Discipline

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honor ( 5 ) . L i k e S y l v e s t e r , he was a p r o l i f i c p u b l i s h e r on a wide range of s u b j e c t s . S y l v e s t e r ' s " C o l l e c t e d Works" run t o 4 quarto volumes, e d i t e d by H. F. Baker i n 1912; G i l d e r s l e e v e s papers were as numerous though not as completely c o l l e c t e d . A volume of h i s s t u d i e s and essays appeared i n 1890 (6) and a s e l e c t i o n of h i s B r i e f Mentions published over the years i n the American J o u r n a l of P h i l o l o g y was e d i t e d w i t h a b i o g r a p h i c a l sketch by h i s successor and devoted p u p i l i n the c h a i r of Greek at Johns Hopkins ( 7 ) . G i l d e r s l e e v e published more books than S y l v e s t e r and h i s L a t i n grammar, r e v i s e d w i t h the cooperation of P r o f e s s o r Lodge, i s s t i l l a standard reference book. Both men r e j o i c e d i n an amplitude of s t y l e that betrays a more l e i s u r e d age than ours. For example, i n a d v e r t i n g t o the then r e c e n t l y published book e n t i t l e d "Value of the C l a s s i c s " (Princeton U n i v e r s i t y P r e s s ) , G i l d e r s l e e v e remarks that "appended to these impressive d e l i v e r a n c e s there i s a formidable array of s t a t i s t i c s drawn up i n r e f u t a t i o n of those other s t a t i s t i c s that have been used only too e f f e c t i v e l y t o s t i r up the pure minds of b e l i e v e r s i n L a t i n and Greek. My missionary days are long overpast and s t a t i s t i c s have l o s t whatever charm they had f o r me even i n the s y n t a c t i c a l l i n e , but the book w i l l strengthen the f e e b l e knees of those who are a f r a i d that they w i l l have to bow thems e l v e s down i n the House of Rimmon." Compare S y l v e s t e r . He published a paper i n the American J o u r n a l of Mathematics i n 1882 e n t i t l e d "A C o n s t r u c t i v e Theory of P a r t i t i o n s , Arranged i n Three A c t s , and I n t e r a c t and an Exodion." Act One i s on p a r t i t i o n s regarded as e n t i t i e s and i s prefaced w i t h the quotation from Twelfth-Night..."seeming parted, but yet a union i n p a r t i t i o n . " In a paper on the three laws of motion i n the world of u n i v e r s a l algebra published i n 1884 he has a charming footnote i n which, a f t e r r e f e r r i n g t o a theorem i n the t e x t , he remarks " I have not had l e i s u r e of mind, being much occupied i n preparing f o r my departure, t o reduce t h i s theorem t o a p o d i c t i c c e r t a i n t y . I s t a t e i t t h e r e f o r e w i t h due reserve." But we must t u r n from the b i o g r a p h i c a l d e t a i l s of these g i a n t s , however f a s c i n a t i n g they may be, and ask about the characters of t h e i r subjects as d i s c i p l i n e s . The very name " c l a s s i c s " i s an acknowledgement of our debt t o the c u l t u r e s of ancient Greece and Rome, f o r i n u s i n g i t we are reminding ours e l v e s that i t i s they who f i r s t recognized the c l a s s e s , o r c a t e g o r i e s , of thought that we, f o r a l l our neglect of t h e i r sources, s t i l l observe today. Whether we l i k e i t or not, the f u r n i t u r e , which we bump i n t o (to borrow a metaphor from J . K. Newman of I l l i n o i s ) i n our gropings through the room of i n t e l l e c t , was b u i l t and put there by the Greeks. I n our everyday and s c i e n t i f i c language we pay hidden t r i b u t e t o the notions of c l a s s i c a l times using words that were c o n s c i o u s l y coined t o r e f l e c t t h e i r thought. Thus "gas" f o r the t h i r d e s t a t e of matter has now been common f o r more than three c e n t u r i e s but goes back t o the a l c h e m i c a l n o t i o n of the i n d w e l l i n g s p i r i t of a t h i n g , which


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Van Helmont (1577-1644) i n h i s "Ortus Medecinae" (1652) r e f e r r e d to as " h a l i t u m i l i u m Gas v o c a v i , non longe a Chao veterum secretum." This was charmingly t r a n s l a t e d i n Chandler's "Van Helmont's O r a t i o n s " (1662) as: " I have c a l l e d that vapour, Gas, being not f a r severed from the Chaos of the A u n t i e n t s . . . a f a r more s u b t i l e or f i n e t h i n g than vapour, mist or d i s t i l l e d O y l i n e s s e s , although as y e t i t be many times t h i c k e r than a i r . But Gas i t s e l f , m a t e r i a l l y taken, i s water as y e t masked w i t h the Ferment of composed Bodies." I f 'gas' i n the p h y s i c a l sense s t i l l bears echoes of the formless darkness over the f a c e of the deep, the c u r r e n t l y popular study of mathematical chaos uses the Greek work i n simple t r a n s l i t e r a t i o n . Yet i t i s pleasant to r e f l e c t that a recent meeting on t h i s s u b j e c t i n Los Alamos (May 1982) could be s u b t i t l e d "ΚΟΣΜΟΣ EN ΧΑΩ" and thus make i r o n i c comment on the world as w e l l as r e c a l l i n g the o r i g i n a l sense of the beauty of order inherent i n the word κόσμοβ. For of i t s o r i g i n a l meaning—adorning or ornament—the only t r a c e l e f t to us i n E n g l i s h i s a s u p e r f i c i a l , almost m e r e t r i c i o u s , o n e — c o s m e t i c . But that i t should a l s o mean order, a 'lucidus ordo', i s n a t u r a l enough and was so t r a n s f e r r e d by Pythagoras to the world as expressing that ordered beauty whose a p p r e c i a t i o n i s the b a s i s f o r a l l s c i e n t i f i c — n o t to say humanistic—endeavour ( 8 ) . 'Mundus' i n L a t i n f o l l o w s a p a r a l l e l course as we see i n P l i n y ' s comment: "Quem κόσμον G r a e c i nomine ornamenti a p p e l l a v e r u n t , eum nos a p e r f e c t a absolutaque e l e g a n t i a mundum": or C i c e r o ' s : "Hunc hac v a r i e t a t e d i s t i n c t u m bene G r a e c i κόσμον, nos lucentum mundum nominamus". To study of the c l a s s i c s i s s u r e l y to submit to a d i s c i p l i n e that r e j o i c e s i n the r e l a t i o n of words and ideas. I t i s not j u s t to l u x u r i a t e i n e t y m o l o g i c a l overtones (a v i c e i n t o which I have j u s t allowed myself to be betrayed) but to l e a r n the r e l a t i o n of word and thought as expressed by the p r e c i s e usage of words and t h e i r p l a c e i n the s y n t a c t i c a l s t r u c t u r e of a sentence. The act of t r a n s l a t i o n from one language i n t o another demands t h i s hard s t r i v i n g f o r p r e c i s i o n , and i s p a r t i c u l a r l y c a l l e d f o r t h by an i n f l e c t e d tongue and one, such as L a t i n , where the thought i s marshalled i n t o p r i n c i p a l and subordinate clauses. By c o n t r a s t E n g l i s h , f o r a l l i t s m e r i t s , has a l a x and i n d i v i d u a l s t r u c t u r e , rambling comfortably on i n coordinated measures. I t i s almost l i k e the c o n t r a s t you can experience by t u r n i n g aside from Watling S t r e e t or the Fosse Way i n t o a country road or lane as you go "to Birmingham by way of Beachy Head." The very f a m i l i a r i t y of our n a t i v e speech and i t s c o l l o q u i a l u s e — c o r r e c t and e f f e c t i v e though t h i s b e — d e n i e s i t the c u t t i n g edge needed to shape our minds. As L i v i n g s t o n e wrote years ago "No doubt there are more important things i n education than the study of grammar; but i t i s not an overstatement to say that not to know Greek i s to be ignorant of the most f l e x i b l e and s u b t l e instrument of expression which the human mind has d e v i s e d , and not to know L a t i n i s to have missed an admirable t r a i n i n g i n p r e c i s e and l o g i c a l thought" (11).



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Of the converse d i s c i p l i n e , that o f t u r n i n g E n g l i s h i n t o L a t i n or Greek i t i s i n t e r e s t i n g t o quote the P h y s i c s master a t Winchester. "The t a s k " , he says, " i n v o l v e s c o n c e n t r a t i o n , c l o s e a t t e n t i o n t o d e t a i l , and considerable l o g i c a l reasoning; there are no short c u t s , no formulae as i n the science problem, the reasoning i n v o l v e d cannot be avoided by mere e f f o r t of memory as i n the w r i t i n g out of a p r o p o s i t i o n i n geometry" (12). I t has been popular among great men from Winston C h u r c h i l l to P e t e r Danckwerts t o deplore the hours spent on L a t i n and e x t o l t h e i r i n s t r u c t i o n i n E n g l i s h , b u t , as I have ventured t o comment e l s e where i n reviewing " I n s i g h t s i n t o Chemical Engineering" (13), t h e i r E n g l i s h may owe more t o the d i s c i p l i n e of the c l a s s i c s than they a f f e c t t o admit. But the study of the c l a s s i c s i s not merely a matter of language however great the general b e n e f i t to our general understanding of language t h i s may a f f o r d . Language i s the entry t o l i t e r a t u r e as l i t e r a t u r e i s the r e f l e c t i o n of the l i f e and thought of i t s time. From t h e i r p l a c e at the root of western c i v i l i z a t i o n , the Greek and L a t i n c u l t u r e s have an uncanny way of a n t i c i p a t i n g and i l l u s t r a t i n g modern p o l i t i c a l and s o c i a l problems. C l a s s i c indeed i s Thucydides' d e s c r i p t i o n of the i n t e r n a l corrosiveness of war and r e v o l u t i o n "Words had t o change t h e i r o r d i n a r y meaning and to take that which was now given them...Frantic v i o l e n c e became the a t t r i b u t e of manliness; cautious p l o t t i n g a j u s t i f i a b l e means of s e l f - d e f e n s e . The advocate of extreme measures was always trustworthy; h i s opponent a man to be suspected" (14). As s o c i o l o g y or s o c i a l psychology, i t s p e n e t r a t i o n should be the envy of the modern s o c i a l s c i e n t i s t . As l i t e r a t u r e , i t s quick a n t i t h e s e s "breathe an almost human a l i v e n e s s " i n t o an otherwise a b s t r a c t a n a l y s i s (15). Archaeology r e v e a l s the c i r c u m s t a n t i a l l i f e as l i t e r a t u r e does the i n t e l l e c t u a l , and i t too has i t s own d i s c i p l i n e s — i t s sherds o f t e n to be handled as d e l i c a t e l y as p a r t i c l e s , i t s a r t i f a c t s as l o v e l y as odes and epodes, and i t s s i t e s as complex and i n t r i c a t e as the p l o t of a tragedy or i n t e r play of a dialogue. T h i s , however, i s not the p l a c e to e x t o l the c l a s s i c s — n o r do they need i t — w h a t I am concerned w i t h i s the f e a t u r e s of i n t e l l e c t u a l l i f e that t h e i r study c u l t i v a t e s . I n the f i r s t p l a c e , there i s the p h i l o l o g i c a l foundation. This necessary occupation w i t h w o r d s — t h e i r p r e c i s i o n of meaning & the exactness of the way they are put t o g e t h e r — i s fundamental t o the c a r e f u l r e c o n s t r u c t i o n of a t e x t and i t s c r i t i c a l e v a l u a t i o n . A t r a i n i n g i n t h i s sometimes leads t o a c e r t a i n cautiousness and care i n q u o t a t i o n . Housman gave h i s i n a u g u r a l as Kennedy P r o f e s s o r of L a t i n a t the U n i v e r s i t y of Cambridge on 9 May 1911, but he never p u b l i s h e d i t as was customary, f o r he was never able to V e r i f y a statement i t contained as t o the t e x t of S h e l l e y ' s Lament of 1821' (16). The manuscript was destroyed a f t e r h i s death but a f u r t h e r copy found among h i s papers was published i n T.L.S. i n 1968, o m i t t i n g , however, the passage of S h e l l e y (and Swinburne's remarks on i t ) that he had been unable to v e r i f y (17). I t was not u n t i l a f t e r



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f u r t h e r work on S h e l l e y manuscripts had j u s t i f i e d the quotation as Housman had used i t , that John Carter published the l e c t u r e as "The Confines of C r i t i c i s m " i n 1969 (18). I mention t h i s example not merely because of t h i s care f o r exactness, which by some might be thought excessive, but because Housman used i t to show the k i n d of confusion that can a r i s e i f e x a c t i t u d e of t h i s order i s not maintained. Swinburne had p r a i s e d an u n m e t r i c a l l i n e i n the commonly p r i n t e d v e r s i o n of the second stanza as "a t h i n g to t h r i l l the v e i n s and draw t e a r s to the eyes of a l l men whose ears were not closed against a l l harmony" by "the melodious e f f e c t of i t s e x q u i s i t e i n e q u a l i t y . " But the l i n e that so t h r i l l e d Swinburne's veins was a m i s p r i n t and the "sovereign sweetness" of the stanza not S h e l l e y ' s c r a f t but some unknown compositor's carelessness. "These, "added Housman s e v e r e l y , " are the performances of the l i t e r a r y mind when, w i t h i t s f a c i l e emotions and i t s i n c a p a c i t y f o r s e l f - e x a m i n a t i o n , i t invades the province of science". I t i s only f a i r to add that having smitten the E n g l i s h tool i t e r a r y carelessness he went on to denounce the German methodisch unimaginativeness; the one breeding the fatuousness of " t a s t e " , the other the m e n t a l i t y of the s l a v e . Against n e i t h e r could he give an i n f a l l i b l e a l e x i k a k o n , though he warned against that " s e r v i l i t y shown towards the l i v i n g " that " i s . . . s o o f t e n found i n company w i t h l a c k of due v e n e r a t i o n towards the dead" (18, p. 44). I f the c l a s s i c s demand exactness they do not r u l e out i m a g i n a t i o n — t h e t e x t u a l c r i t i c , l i t e r a r y c r i t i c , philosopher and a r c h a e o l o g i s t a l l need i t — b u t the d i s c i p l i n e demands that t h i s imagination should be c o n t r o l l e d by p r e c i s i o n of s c h o l a r s h i p r a t h e r than the r a m i f i c a t i o n of fancy. To the n i c e a p p r e c i a t i o n of language the c l a s s i c a l s c h o l a r must add a c e r t a i n copiousness of l e a r n i n g . A very broad reading of l i t e r a t u r e , h i s t o r y and philosophy i s p a r t of the g r a i t h of c l a s s i c a l s c h o l a r s h i p j u s t as a wide experience of d i f f e r e n t aspects of ancient c i v i l i z a t i o n and a keen and d i s c e r n i n g eye f o r i t s a r t i f a c t s i s needed by the a r c h a e o l o g i s t . Indeed the c o n t r a s t that most f o r c i b l y s t r i k e s a s c i e n t i s t i n t a l k i n g to the humanist i s the comparatively immense amount of reading t h a t the l a t t e r must do and the recourse that must be had to primary sources. S i r Andrew Aguecheek's lament i n "Twelfth Night" (provoked you w i l l r e c a l l by S i r Toby's "Pourquoi") "0 that I had but f o l l o w e d the A r t s " (19) i s a r e a l cry of the h e a r t , f o r , save f o r the e x c e p t i o n a l l y g i f t e d , the a c q u i s i t i o n of the v a r i e d and extensive knowledge which i s needed i n ancient s t u d i e s i s an impossible task i f not begun i n childhood. But l e t us t u r n to a d i s c i p l i n e which, as Hardy says, i s "more than any other a r t or science,...a young man's game"— mathematics (20). As an i n t e l l e c t u a l d i s c i p l i n e , i t s l e a d i n g c h a r a c t e r i s t i c s are the power of a b s t r a c t i o n , the passion f o r elegance and the p u r s u i t of r i g o u r . I put them i n that order, f o r , though they share these c h a r a c t e r i s t i c s i n some degree w i t h other d i s c i p l i n e s , the power of a b s t r a c t i o n i n mathematics i s



1.

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1

r a i s e d to a height u n a t t a i n a b l e i n any other s u b j e c t , save perhaps music. As a r e s u l t connections can be found between things which are very f a r from obvious. I t was, as Whitehead pointed out (21), an immensely important step forward i n the h i s t o r y of thought when i t was f i r s t r e a l i z e d that there was a commonality of some s o r t between a p a i r of twins and a brace of pheasants. We take the concept o f number so completely f o r granted that we tend t o f o r g e t what an achievement t h i s a b s t r a c t i o n of concept must have been. Perhaps we can r e c a l l the i n i t i a l con f u s i o n and u l t i m a t e d e l i g h t of r e c o g n i z i n g that the c a r d i n a l number of a c l a s s was indeed the c l a s s of a l l c l a s s e s s i m i l a r t o i t , or at a more lowly l e v e l our l i b e r a t i o n from χ and y as apples and oranges i n our kindergarten s t r u g g l e s w i t h elementary algebra. I t i s of course a long way from the f i v e and country senses of the engineering world t o the t h i n a i r o f the heights that mathematicians seem to breathe so e f f o r t l e s s l y , yet we are f o r tunate to have had as p a r t of our t e c h n i c a l t r a i n i n g a t l e a s t a brush w i t h some o f the s t r u c t u r e s that pervade mathematics. The group, r i n g or f i e l d i n algebra w i t h the notions of equivalence r e l a t i o n s , morphisms, and mappings are p a r t of our mental f u r n i t u r e , not only i n t h e i r i n s t a n t i a t i o n s , but as a b s t r a c t e n t i t i e s forming our powers of apprehension. I t i s thus that we p e r c e i v e t h a t , from a purely s t o i c h e i o m e t r i c point of view, any non-singular transformation of a set of chemical r e a c t i o n s gives an equivalent s e t . A t the same time we are conscious of the f a c t that stoicheiometry i s not everything and that we may have to move beyond i t t o an aspect i n which a p a r t i c u l a r set has a p r e f e r r e d character. Not to have perceived the equivalence i s to be l e f t i n a confusion of c o n c r e t i o n s ; not to have perceived i t s l i m i t a t i o n s i s to have f a i l e d to understand the nature of a b s t r a c t i o n . I suppose the root of these notions goes back to Pythagoras who discerned the pervasive i n f l u e n c e of number i n the order of nature. Of course t h i s was not developed u n t i l much l a t e r when, w i t h the understanding of the n o t i o n of f u n c t i o n , i t became p o s s i b l e t o express p h y s i c a l laws i n mathematical form, but h i s proof (or r a t h e r the proof which goes back to h i s school i n the s i x t h century B.C.) of the i r r a t i o n a l i t y of the square root of two i s c i t e d by G. H. Hardy (20) as opening up the whole realm of r e a l numbers, making that d e f i n i t i v e step beyond the everyday common places of i n t e g e r s and r a t i o n a l e . I f I may, I would l i k e t o advert f o r a moment t o the recent development of non-standard a n a l y s i s and sketch how i n f i n i t e and i n f i n i t e s i m a l numbers can be presented. In this I follow a b e a u t i f u l expository a r t i c l e of Ingleton (22) though, i n my haste s c a r c e l y doing him, or Luxemburg on whom he l e a n s , f u l l j u s t i c e . Consider a l l i n f i n i t e sequences of r e a l numbers Χ (χ^,χ^,···,Χη, ···) and l e t two such e n t i t i e s be equivalent i f they d i f f e r only i n a f i n i t e number of elements i . e . , Χ Ξ X i f χ = x f o r a l l but η η a f i n i t e number of n. From now on we can consider the e n t i t y Χ to be the equivalence c l a s s and representable by any of i t s members j u s t as the r a t i o n a l 1/2 i s the c l a s s (1/2, 2/4, 3/6,..). We 1

T


8


r e t a i n the r e a l numbers χ as the elements (χ,χ,···,χ···)(ί.ε. χ χ f o r a l l η) and can d e f i n e a r i t h m e t i c a l operations such as X±Y = (χ ±Υ ,···,Χ ±Υ ,···) X.Y = < ;|7i»'* » y '' > x

1

1

η

e

η

x

ee

n

=

a n d

n

1/X = (1/x-^, · · · , l / x , · · ·) provided χ ^ 0 — i f i t i s , ignore i t and use 1 i n s t e a d of 1/x s i n c e t h i s can only happen a f i n i t e number of times. These d e f i n i t i o n s a l l o w us to comprise the o r d i n a r y operations of the a r i t h m e t i c of r e a l s w i t h i n t h i s new world of h y p e r r e a l s . I f we use the u s u a l n o t a t i o n f o r the r e a l s we can say that χ ε R, χ Ξ (Χ,Χ,···,χ,···) provides a mapping between R* the space of h y p e r r e a l s X = (χ-,···,χ ,···) and the r u l e f ( X ) = (f(x ),···ί(x )···) i s a n a t u r a l extension of the f u n c t i o n f from R to R*. But now we n o t i c e t h a t we have i n R* a much r i c h e r e n t i t y than we had i n R. We have h y p e r r e a l numbers l i k e Ζ = (1,2,3,···,n,···) which i s i n any sense of the word an i n f i n i t e i n t e g e r . Yet we have not the u n d i f f e r e n t i a t e d world of aleph-zero, i n which, you r e c a l l , ff^ = Ν $ or K = Ν + fÎ f o r any f i n i t e n


1

q

Q

Q

Ν. But here Z+N = (N+l,···,N+n,···) and or N.Z = (Ν,2N, · · · ,nN· · ·) are w e l l - d e f i n e d and g i v e (Z+N) - Ν = Z, (Z+N) - Ζ = N, (N.Z)/N=Z and (N.Z)/Z = N. S i m i l a r l y Z = (1,4,···,n ) o r Z = (1,4,27,···,η ,···) a r e t r u l y " i n f i n i t e " numbers. On the other hand 1/Z = (1,1/2,···,1/n,···) i s an " i n f i n i t e s i m a l " , as are 1/(Z+N), 1/Z e t c . Sums of i n f i n i t e s i m a l s are i n f i n i t e s i m a l s , as are products of i n f i n i t e s i m a l s w i t h f i n i t e h y p e r r e a l s . Products of i n f i n i t e and i n f i n i t e s i m a l numbers can be f i n i t e however as can be q u o t i e n t s of i n f i n i t e s or of i n f i n i t e s i m a l s . Two h y p e r r e a l numbers a r e i n f i n i t e s i m a l l y c l o s e i f t h e i r d i f f e r e n c e i s i n f i n i t e s i m a l . For example (Z+l)/Z and Z/(Z+1) are both i n f i n i t e s i m a l l y c l o s e t o 1 though a l l three are d i s t i n c t h y p e r r e a l numbers. I n f a c t every f i n i t e h y p e r r e a l number i s i n f i n i t e s i m a l l y c l o s e t o a r e a l number and every r e a l number has an i n f i n i t y of i n f i n i t e s i m a l l y c l o s e h y p e r r e a l s . The r e a l number i n f i n i t e s i m a l l y c l o s e t o any h y p e r r e a l i s c a l l e d i t s standard p a r t , i . e . s t X = χ i f (X-x) i s i n f i n i t e s i m a l . This makes the d i s c u s s i o n of convergence r a t h e r simple. A sequence y^ i n R i s a mapping from Ν the space of i n t e g e r s t o R. This can be extended i n the non-standard r e a l s as a mapping from the extension of Ν t o N*, the h y p e r r e a l i n t e g e r s , i n t o R*. Then i s an element of a non-standard sequence i f Υ ε R* and Μ ε Ν*. We say that Y^ converges to Y i f Y = s t Y^ f o r a l l i n f i n i t e i n t e g e r s M. C o n t i n u i t y and d i f f e r e n t i a b i l i t y can be s i m i l a r l y d e f i n e d , f o r example f ( a ) = s t {f(a+h) - f ( a ) } / h f o r a l l i n f i n i t e s i m a l s h. The D i r a c d e l t a f u n c t i o n can be defined pointwise and new approaches t o the theory of p r o b a b i l i t y and s t o c h a s t i c d i f f e r e n t i a l equations a r e opened up. But these are a p p l i c a t i o n s and p a r a d o x i c a l l y I have been d e s c r i b i n g the a b s t r a c t i o n that i s the non-standard world i n very concrete terms. This i s no p l a c e t o go back and t r y t o present i t a b s t r a c t l y , but i t stands as an example of a recent step forward i n the path of a b s t r a c t i o n that began two and a h a l f m i l l e n i a ago (see a l s o (23,24)) . 2

z

2

11

2

V

M

f



1.

ARis


9

In d e s c r i b i n g the c h a r a c t e r i s t i c s o f mathematics I put elegance before r i g o u r , f o r so i t appears t o the mathematically c u l t i v a t e d mind. I t i s not that s l o p p i n e s s of reasoning can be t o l e r a t e d i n the l e a s t , but r a t h e r that r i g o u r i s a d i s c i p l i n e learned w i t h care and p r a c t i c e d w i t h scrupulousness, whereas i t i s the i m a g i n a t i v e leap that reaches out t o a r e s u l t and f e e l s that i t i s r i g h t before the perhaps lengthy c o n f i r m a t i o n i s undertaken that the mathematician c h i e f l y p r i z e s . I n t h i s leap the d r i v i n g f o r c e i s i m a g i n a t i v e rather than l o g i c a l and i t i s the i n s t i n c t f o r beauty which i s the avenue t o t r u t h . Thus e l e g a n c e — a much deeper q u a l i t y than p r e t t i n e s s , though more r e s t r i c t e d by context than beauty i n the l a r g e r s e n s e — i s g r e a t l y valued i n mathematical c i r c l e s . I n i t s h i g h e s t forms, mathematics has "the s i m p l i c i t y and i n e v i t a b l e n e s s " of great a r t and, v a s t l y though he admired him, Hardy's judgment, as a mathematician, o f the mathematician Ramanujan was that h i s work f e l l short of t h i s — " i t would be g r e a t e r i f i t were l e s s strange" (25) was Hardy's way of expressing i t . Yet he paid f u l l t r i b u t e to Ramanujan's "profound and i n v i n c i b l e o r i g i n a l i t y " . Though denying that mathematicians have c o n s c i o u s l y aimed a t beauty r a t h e r than l o g i c , R u s s e l l (26) has w r i t t e n e l o q u e n t l y of the a e s t h e t i c appeal of mathematics i n one of h i s e a r l y essays. "Mathematics, r i g h t l y viewed, possesses not only t r u t h , but supreme b e a u t y — a beauty c o l d and austere, l i k e that of s c u l p t u r e , without appeal t o any p a r t of our weaker n a t u r e , w i t h out the gorgeous trappings of p a i n t i n g o r music, y e t sublimely pure, and capable of a s t e r n p e r f e c t i o n such as only the g r e a t e s t a r t can show. The t r u e s p i r i t o f d e l i g h t , the e x a l t a t i o n , the sense of being more than man, which i s the touchstone of the h i g h e s t e x c e l l e n c e , i s to be found i n mathematics as s u r e l y as i n poetry. What i s best i n mathematics deserves not merely to be l e a r n t as a task, but to be a s s i m i l a t e d as p a r t of d a i l y thought, and brought again and again before the mind w i t h ever-renewed encouragement." But from the realm of the c l a s s i c s , so r i c h and warm w i t h human a s s o c i a t i o n s , and the " c o l d and a u s t e r e " beauty of mathematics we must t u r n t o the banausic domain of chemical r e a c t i o n engineering. I t was, i f I remember r i g h t l y , at one of these meetings that P r o f e s s o r Wei (27) gave us the connection by p o i n t i n g out that one of the o l d e s t t e c h n o l o g i c a l processes i s the fermentation r e a c t i o n , as pure a r e a c t i v e process as we could wish f o r — r i c h and humane w h i l s t i t stayed w i t h i n our domain and took not up w i t h d i s t i l l a t i o n p r o c e s s e s — a n d r i g h t l y respected i n the very t i t l e of our meeting. But important though the business of wine making may be i t i s but one of many r e a c t i o n s which l i e at the cores of the d i f f e r e n t processes o f chemical i n d u s t r y and


10


ENGINEERING

i t i s out of the experience of these that the d i s c i p l i n e of chemical r e a c t i o n engineering has grown. I t i s a recent growth i n a s u b j e c t of no long h i s t o r y f o r I suppose that i t i s only i n the l a s t f i f t y years that i t has emerged as a d i s t i n g u i s h a b l e t r a i n of thought. Damkohler's work i n the 3 0 s , Hougen's and Watson's i n the 40's, Wilhelm's and Amundson's i n the 50's are but p o i n t s of touch-down i n the immense s t r i d e s that have taken us onward. The t a l e of t e x t s has lengthened decade by decade, the wen of papers has grown year by year and I h e s i t a t e to mention any more names for I am sure to leave out people of great s i g n i f i c a n c e . ( I have t r i e d to survey some aspects of the h i s t o r i c a l development e l s e where (28).) Downloaded by ILLINOIS STATE UNIV on December 2, 2012 | http://pubs.acs.org Publication Date: July 28, 1983 | doi: 10.1021/bk-1983-0226.ch001

T

But what has emerged as c h a r a c t e r i s t i c i n t h i s development? I would suggest f i r s t a sense of balance, then a f e e l i n g f o r s t r u c t u r e and f i n a l l y a refinement of concept l e a d i n g to a h i g h degree of awareness. Those are r a t h e r vague terms so l e t my t r y to make them more p r e c i s e and then i l l u s t r a t e some aspects i n greater d e t a i l . Under balance I t h i n k of the balance t h a t needs to be s t r u c k between theory and p r a c t i c e on the one hand and between the general and the p a r t i c u l a r on the other. By s t r u c t u r e I mean the way i n which we have come to see the subject standing on the foundation of thermodynamics, t r a n s p o r t processes and chemical k i n e t i c s . The nature of the c o n s t r a i n t s i m p l i e d by t h i s and the inherent s t r u c t u r e o f , f o r example, s t o i c h e i o m e t r y or mass a c t i o n k i n e t i c s are at i s s u e here. C e r t a i n ideas have been of great importance i n t h i s development and no p a r t of chemical e n g i neering i s r i c h e r i n dimensionless numbers; they range from the four of Damkohler (29) or the modulus of T h i e l e (30) to the Carberry number that B i s c h o f f (31) minted a year or two ago. I t was e a r l y recognized that r e a c t o r s presented p e c u l i a r scale-up problems and the height of r e a c t i o n u n i t never a t t a i n e d the popul a r i t y of the H.T.U. S i m p l i f i e d s t r u c t u r e s p l a y an important r o l e as i n the i n s i g h t gained from Wei and P r a t e r ' s (32) monomolecular systems and the r e s u l t i n g shape i n v a r i a n c e (33). A great refinement of concepts has taken p l a c e over the years as, f o r example, i n the way that we now d i s t i n g u i s h s e n s i t i v i t y , m u l t i p l i c i t y and s t a b i l i t y (34). With the e x p l o r a t i o n of instances of these concepts has come an awareness of the v a r i e t i e s of b e h a v i o r , the comprehensive d e s c r i p t i o n of a l l forms of p o s s i b l e behavior of r e a c t o r s as dynamical systems the pervasiveness of o s c i l l a t i o n and chaos and, w i t h a l l t h i s , an exhanced a b i l i t y to f e e l a f t e r the form of the s o l u t i o n . Chemical r e a c t i o n engineering has been n o t a b l e f o r the balance between experiment and theory that has marked i t from the beginning. I t i s t r u e t h a t , save i n the r a r e i n s t a n c e of a Shinnar at the i n d u s t r i a l - a c a d e m i c i n t e r f a c e or a Schmitz i n the more p u r e l y academic context, t h i s blood and judgment have seldom been "so w e l l commingled" i n one person, but there has always been a p r o p i t i o u s i n t e r c o u r s e between those c h i e f l y concerned w i t h


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p r a c t i c e and those pursuing the e u r i s t i c s of mathematical modelling. The i n g e n u i t y of the p r a c t i c i n g r e a c t i o n a r y i s always r e f r e s h i n g : on the i n d u s t r i a l s c a l e we have the c l a s s i c a l case of the f l u i d i z e d bed as w e l l as the s e r e n d i p i t y of the t r a n s f e r l i n e or the i n t r i c a c y of some of the p o l y m e r i z a t i o n r e a c t o r s ; on the research s c a l e there i s Wicke's creeping zone (35)» Rony s bundle of m i c r o f i l a m e n t s (36), Petersen's s i n g l e p e l l e t r e a c t o r (37), Carberry's spinning basket (38) and the Berty r e a c t o r , the s t r i n g of beads used by Hegedus and Oh (39) and Wei and Degnan's (40) combination of c r o s s - f l o w monoliths or Schmitz' polythene bag. I t i s a p r a c t i c a l i n g e n u i t y matched on the t h e o r e t i c a l s i d e by some of the devices that have been used i n a n a l y s i n g r e a c t o r models. The n o t i o n of an e f f e c t i v e n e s s f a c t o r introduced by T h i e l e , Amundson's e x p l o i t a t i o n o f the phase plane (34) , Gavalas' use of the index theorem (41), the S t e i n e r symmetrization p r i n c i p l e used by Amundson and Luss (42) and the l a t t e r ' s e x p l o i t a t i o n of the formula f o r Gaussian quadrature (43)—perhaps the p r e t t i e s t connection ever made i n the chemical engineering l i t e r a t u r e — a r e t h e o r e t i c a l counterparts, l a r g e and s m a l l , of the c a r e f u l c r a f t of the e x p e r i m e n t a l i s t . So perhaps a l s o the very important i n s i g h t that Danckwerts c o n t r i b u t e d i n h i s f o r m u l a t i o n of the residence time d i s t r i b u t i o n i s a happy f o i l t o h i s h e r o i c ambition t o t r a c e a b l a s t furnace (44). Faced w i t h the enormous v a r i e t y of p r a c t i c a l reactor, chemical engineers are f o r c e d t o t h i n k of the general shape and s t r u c t u r e of r e a c t o r a n a l y s i s i f they are to r e t a i n i t as an i n t e l l e c t u a l d i s c i p l i n e . Wilhelm used t o speak of the "morphology" of the subject years ago i n h i s l e c t u r e s a t P r i n c e t o n and h i s i n f l u e n c e on t h i s s i d e of the A t l a n t i c , together w i t h that of Hougen and Watson, was as formative on the engineering s i d e , as Amundson's was on the t h e o r e t i c a l (45). There are many l e v e l s of s t r u c t u r e from the r a t h e r obvious f a c t that stoicheiometry i s a manifes t a t i o n of l i n e a r algebra t o the profound a n a l y s i s of mass a c t i o n k i n e t i c s that Feinberg (46) has developed so f u l l y from h i s e a r l i e r work w i t h Horn and Jackson. Wei and P r a t e r ' s (32) now c l a s s i c treatment o f f i r s t - o r d e r r e a c t i o n s showed how the s t r u c t u r e of l i n e a r d i f f e r e n t i a l equations leads to both theo r e t i c a l and p r a c t i c a l i n s i g h t s i n t o k i n e t i c systems. More r e c e n t l y Shinnar has shown how fundamental thermodynamic con s t r a i n t s , such as the requirement that the f r e e energy decrease along a r e a c t i o n path, g i v e shape t o the problem of understanding a r e a c t i v e system and i n c o r p o r a t i n g i t i n t o u s e f u l design objectives. The i n t r o d u c t i o n of s i n g u l a r i t y theory by G o l u b i t s k y and K e y f i t z (47) and i t s development by Luss and B a l a k o t a i a h (43,48) i s another s t r u c t u r a l landmark. The l a t t e r show, f o r example, that a l l the p o s s i b l e b i f u r c a t i o n diagrams can be determined f o r c e r t a i n networks of r e a c t i o n s . They use a r e d u c t i v e scheme which allows the system w i t h Ν r e a c t i o n s t o be analysed by l i m i t i n g cases i n which only η r e a c t i o n s proceed at a f i n i t e r a t e and the


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C H E M I C A L R E A C T I O N ENGINEERING

others are e i t h e r instantaneous or g l a c i a l l y slow. I t would be i n a p p r o p r i a t e , even i f i t were n e c e s s a r i l y r e l e v a n t , to say more on t h i s s i n c e a paper at t h i s meeting w i l l b r i n g us up to date on t h e i r t h i n k i n g . But i t r a i s e s two p o i n t s of importance. The f i r s t i s the i n s t i n c t f o r comprehensiveness; the second, the immensity of many dimensions, which I w i l l advert to at the end. As a p r a c t i c a l device f o r making a product the chemical r e a c t o r presents i t s e l f as a matter of design and c o n t r o l and the engineer may be' s a t i s f i e d to o b t a i n economy i n the one and r e l i a b i l i t y i n the other. I t i s not a q u e s t i o n of c o n s i d e r i n g a l l p o s s i b l e r e a c t o r s but of a c h i e v i n g a working r e a c t o r that w i l l be i n t e g r a t e d i n t o a whole p l a n t and come on stream i n time to make a p r o f i t . As an object f o r i n t e l l e c t u a l comprehension, however, the r e a c t o r must be regarded comprehensively, we have the r i g h t and duty to enquire of i t s behavior f o r any, not j u s t the s o - c a l l e d " r e a l i s t i c " , values of the parameters. T h i s i s where such t o o l s as s i n g u l a r i t y theory or that of catastrophes, the q u a l i t a t i v e theory of d i f f e r e n t i a l equations or t o p o l o g i c a l methods are so important, f o r they can, when r i g h t l y used, o f t e n g i v e comprehensive i n f o r m a t i o n . A case i n which the s t r i c t numerical d e f i n i t i o n of regions of parameter space could be replaced by geom e t r i c a l l y d i s t i n c t p o s s i b i l i t i e s arose w i t h the competition of two m i c r o b i o l o g i c a l populations f o r a s i n g l e n u t r i e n t i n a continuous fermentor which Arthur Humphrey and I considered a few years ago (49). We were able to show how the r e l a t i v e d i s p o s i t i o n s of the two growth curves and the p o i n t r e p r e s e n t i n g the d i l u t i o n r a t e and feed c o n c e n t r a t i o n could be encoded i n t o a l a b e l which was a s s o c i a t e d w i t h a phase p o r t r a i t . Thus i t could be s a i d that the dynamical behavior of any such system was t o p o l o g i c a l l y e q u i v a l e n t to any other w i t h the same l a b e l . I t i s tempting to d i g r e s s at t h i s p o i n t and enlarge on the v i r t u e s of chemical r e a c t i o n engineering as a source of m a t e r i a l f o r the c r a f t of f e e l i n g out a s o l u t i o n . T h i s i s as important a technique to l e a r n and teach as i t i s enjoyable i n i t s e x e r c i s e . However i t would take us too f a r a f i e l d and, as I have t r i e d to expound i t elsewhere (see Chap. 2 of 28; 50,51), I w i l l r e s i s t the temptation even though I have a brand new example making p r o p i t i o u s ferment i n my v i t a l s . A f u r t h e r c h a r a c t e r i s t i c of chemical r e a c t i o n engineering i s i t s f a c i l i t y i n r e f i n i n g concepts. I t i s a d e r i v e d c h a r a c t e r i s t i c , stemming from the mathematical component of the s u b j e c t , but i t has been t y p i c a l of i t s development s i n c e the d i s t i n c t i o n between s t a b i l i t y and s e n s i t i v i t y was f i r s t drawn (34). There have been l a p s e s , a l a s , as i n the case of an e x p o s i t o r y paper (52) which, to judge by the demand f o r o f f p r i n t s , proved q u i t e u s e f u l , but which r a t e d a very low Watkins number (53), f o r i t s t i t l e was "On the s t a b i l i t y c r i t e r i a of chemical r e a c t i o n engineering" w h i l e i t s burden was as much the m u l t i p l i c i t y c r i t e r i a as the s t a b i l i t y c o n d i t i o n s . But there has been a steady i n c o r p o r a t i o n of prec i s e l y defined concepts as f o r example i n the burgeoning study of


s



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13

chaos which i s evoking i n t e r e s t from mathematicians, p h y s i c i s t s and chemists as w e l l as engineers. I n the 50's i t seemed l i k e l y that o s c i l l a t i o n s i n r e a c t o r s would be most l i k e l y t o a r i s e when a c o n t r o l system was added t o the r e a c t o r (54) . The n o t i o n of m u l t i p l i c i t y c a r r i e d w i t h i t the p i c t u r e of the a s s o c i a t e d r e g i o n of a t t r a c t i o n , an open set of i n i t i a l s t a t e s which would l e a d t o the same steady s t a t e . Denbigh (55) had shown t h i s i n h i s p i o n e e r i n g work using the n o t i o n of " e q u i f i n a l i t y " adumbrated i n a b i o l o g i c a l context by Burton. But, u s i n g the newly r e v i v e d notions of Hopf b i f u r c a t i o n , Uppal, Ray and Poore (56) were able to show that the u n c o n t r o l l e d non-adiabatic s t i r r e d tank w i t h a s i n g l e exothermic r e a c t i o n — t h e s i m p l e s t non-isothermal s y s t e m — could e x h i b i t a wide v a r i e t y of behavior i n c l u d i n g m u l t i p l e steady s t a t e s and l i m i t c y c l e s w i t h i n l i m i t c y c l e s . T h e i r t a l e of p o s s i b l e modes of behavior was s c a r c e l y t o l d before a group of Russian authors (57) augmented i t . An important r e a l i z a t i o n that has been a t t a i n e d i n recent years i s that the processes of a d s o r p t i o n , desorption and rearrangement on the c a t a l y s t surface may themselves produce m u l t i p l e r e a c t i o n r a t e s o r o s c i l l a t i o n s . Wicke (35) and h i s colleagues found t h i s some years ago and the reviews of Sheintuch and Schmitz (58) and Scheintuch (59) show how wide-spread the phenomenon i s . I t i s dangerous to mention names when I am sure to omit many important ones but those of Y a b l o n s k i i , S l i n k o and t h e i r colleagues i n R u s s i a , Eigenberger and Hugo i n Germany, Kenney i n England and Luss, Takoudis, Schmidt, Ray, Jensen over here s p r i n g to mind i n a d d i t i o n to those already mentioned. The B e l o u s o v - Z h a b o t i n s k i i r e a c t i o n has of course generated a minor i n d u s t r y of experimental and t h e o r e t i c a l s t u d i e s , d i s s i p a t i v e systems and l o c u s - a t o r s which i t would be rash t o rush i n t o here. But even as the n o t i o n of p e r i o d i c s o l u t i o n s t o the n o n l i n e a r d i f f e r e n t i a l equations of chemical r e a c t o r theory was growing i n c r e a s i n g l y f a m i l i a r i n the '60*s and 7 0 s , there was s p r o u t i n g beside i t the new ideas of chaos and non-periodic behavior. This stemmed from a v a s t l y s i m p l i f i e d m e t e o r o l o g i c a l model of three modestly n o n l i n e a r equations s t u d i e d by Lorenz (60) which e x h i b i t e d s o l u t i o n s which were n e i t h e r p e r i o d i c nor asymptotic to p e r i o d i c s o l u t i o n s . The p r o p e r t i e s of such s o l u t i o n s soon a t t r a c t e d a t t e n t i o n of mathematicians and others and there i s a t h r i v i n g i n d u s t r y t r y i n g to put order i n t o chaos a t the moment (61). Already a t the l a s t of these symposia, Pismen (62) i n h i s e x c e l l e n t review of k i n e t i c i n s t a b i l i t i e s was able t o g i v e s e v e r a l references and has h i m s e l f shown how data sampling can c o n t r i b u t e i t s meed of randomness. I t i s c r e d i b l y h e l d that at l e a s t three equations are needed f o r c h a o t i c behavior and i t i s not s u r p r i s i n g that i t has been found i n the s t i r r e d tank when two r e a c t i o n s are t a k i n g p l a c e . R b s s l e r , Varma and K a h l e r t (63) have found i t f o r consecutive r e a c t i o n s , one exo- and the other endothermic; Lynch and others (64) have used p a r a l l e l r e a c t i o n s ; the exothermic sequence T

T

f


14


A->B->C has a l s o been considered by Jorgensen (65) . This l a s t i s a system of three equations f o r u and v, the concentrations of A and B, and w, the temperature, w i t h seven parameters: a, the Damkôhler number f o r A-*B; 3, i t s dimensionless heat of r e a c t i o n ; γ, i t s A r r h e n i u s number; κ, a dimensionless heat t r a n s f e r c o e f f i c i e n t ; v, the r a t i o of a c t i v a t i o n energies of the two r e a c t i o n s ; p, the r a t i o of the heats of r e a c t i o n ; σ, the r a t i o of the Damkohler numbers. They c o n t a i n a c h a r a c t e r i s t i c n o n - l i n e a r i t y E(w)

=

expiyw/(γ+w)}


(which reduces to expw i f γ—*») and are: ύ

=

1 - u{l+aE(w)}

ν w

= =

auE(w) - v { l + a a E ( w ) } -(1+K)W + a3uE(w) + α$ρσ vE (w)

V

V

1

Jorgensen s f i r s t endeavour was to map out the regions of m u l t i p l i c i t y , s t a b i l i t y and i n p a r t i c u l a r the Hopf b i f u r c a t i o n l o c i . Then t a k i n g a l l parameters except κ, the c o o l i n g r a t e parameter, to be constant she v a r i e d t h i s over an i n t e r v a l (5 < κ < 8) i n which there was a s i n g l e unstable steady s t a t e and t h e r e f o r e at l e a s t one o s c i l l a t o r y s o l u t i o n , the p r i n c i p a l o s c i l l a t i o n . There were two s u b i n t e r v a l s , (5.49,5.60) and (6.95,7.29), w i t h i n which two i n t r i c a t e l y complex p a t t e r n s of t r a n s i t i o n s could be seen, the p a t t e r n s being t o p o l o g i c a l m i r r o r images of each other. The g r e a t e r p a r t of each i n t e r v a l was occupied by a c y c l e of approximately twice the p e r i o d of the p r i n c i p a l o s c i l l a t o r y s o l u t i o n , but then as the values of κ moved outward through the l a s t f r a c t i o n s of the s u b i n t e r v a l s , cascades of p e r i o d doublings l e d to chaos, which subsided i n t o regimes of o s c i l l a t i o n s of s i x times the p e r i o d of the p r i n c i p a l . These cascaded i n t o chaos, recovered as p e r i o d 10 o s c i l l a t i o n s , r e l a p s e d and recovered as p e r i o d 8's b e f o r e t h e i r f i n a l widdershins as fandangos f o r c h a o t i c and p e r i o d i c s o l u t i o n s . "Morris a n t i c s " (6_6) indeed and a l l w i t h i n the f i f t h decimal p l a c e ! "So what", says the p r a c t i c a l l y - m i n d e d engineer: " s t u l t i t i a est enim i l l i " (6 7). But n e v e r t h e l e s s i t moves and i t i s not so many years s i n c e Amundson's c r i t e r i a f o r s t a b i l i t y were thought to be i d l e s o p h i s t r i e s . There may indeed be systems f o r which some of these regions of complexity have a s i g n i f i c a n t s i z e or are s i t u a t e d i n an otherwise d e s i r a b l e r e g i o n of parameter space. As a branch of p r a c t i c a l engineering chemical r e a c t o r s may be t o l e r a b l y w e l l understood without a l l the refinements of t h i s i n t r i c a t e behavior. As an o b j e c t of i n t e l l e c t u a l study these apparently e s o t e r i c f e a t u r e s are of great moment; they show that the v e i n , r i c h as i t has proved to be, i s not mined out y e t . The v a l u e of comprehensiveness makes demands t h a t open up a question which I b e l i e v e w i l l be a l l important i n t h i s decade, the question of how to master the daunting "espaces i n f i n i s " (68) of



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a l a r g e number of parameters. We need theorems of great power, such as those of Feinberg i n k i n e t i c s , that w i l l b l o c k out whole subspaces and ensure that no e x o t i c type of behavior can hurt o r devour there. We need ways of keeping i n the important i n t e r f a c e s so that the boundaries of d i f f e r e n t regions do not have t o be determined by t r i a l and e r r o r . I t i s f a i r l y easy t o see how t o keep on the margin of s t a b i l i t y , but how can we tread the edge of chaos? Even t h i s , however, only reduces the d i m e n s i o n a l i t y by one or two and enormous d i f f i c u l t i e s s t i l l remain. P r a c t i c a l i t y can sometimes come t o our a i d by c o n f i n i n g a t t e n t i o n t o subspaces which a r e , i n some sense, o p t i m a l o r , i n some sense, s a f e . But u l t i m a t e l y i t w i l l r e q u i r e some new i n s i g h t t o expand our imaginations, r e v e a l i n g the power of a s u b j e c t apparently so humdrum as chemical r e a c t i o n engineering t o charm "the wide casements opening on the foam of k e e l l e s s seas i n f a i r y lands f o l o r n " (69). Literature Cited 1.

2.

3.

4. 5. 6. 7.

Baker, H. F., E d . ; "The Collected Mathematical Papers of James Joseph Sylvester" (four volumes); Cambridge, 1912 (biographical notice in V o l . 4). There is some confusion over the reason for Sylvester's untimely departure from the University of V i r g i n i a . According to Ε. T. B e l l (Men of Mathematics) he resigned after three months on account of the "refusal of the University authorities to discipline a student who had insulted him." According to Woolf "within three months he was on his way back to England, convinced that he had k i l l e d a student in self defense—a conviction that fortunately proved to be false". Of his experiences in 1841 H. F . Baker writes in the Biographical Notice in Volume 4 of the Collected Mathematical Papers; "Such a considerable change deserved a better fate than b e f e l l ; in Virginia at this time the question of slavery was the subject of bitter contention, and Sylvester had a horror of slavery. The outcome was his almost immediate return; apparently he had intervened vigorously in a quarrel between two of his students." Woolf, H., "An Introduction to J. J. Sylvester"; in "Algebraic Geometry"; Igusa, J-I., E d . ; the Johns Hopkins Centenial Lectures: Supplement to the American Journal of Mathematics. "The Creed of the Old South"; Atlantic Monthly Jan. 1892, 75. "Studies in Honor of Basil L . Gildersleeve"; The Johns Hopkins Press, Baltimore, 1902. Gildersleeve, B. L., "Essays and Studies—Educational and Literary"; Baltimore, 1890. M i l l e r , C. W. E., E d . ; "Selections from the Brief Mention of B. L . Gildersleeve"; Johns Hopkins Press, Baltimore, 1930.


16

8. 9. 10. 11. 12. 13.


14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29.

30. 31. 32. 33.

34. 35.


Plutarch, De Plac. Phil, i, 5. Pliny, Hist. Nat. ii, 3. Cicero, De Universo, 10. Livingstone, R. W., "A Defense of Classical Education"; Macmillan: London, 1916. Ibid. p. 223. Aris, R., Review of "Insights into Chemical EngineeringSelected Papers of P. V. Danckwerts"; Chem. Engng. Sci. 1982, 37, 1123. Thucydides, Peloponesian War III.82-3. Finley, J. Η., "Four Stages of Greek Thought"; p. 73. Gow, A. S. F., "Housman, A. E. A Sketch"; Cambridge University Press: Cambridge, 1936. Times Literary Supplement. 9 May 1968. Housman, A. E., "The Confines of Criticism"; Carter, J., Ed.; Cambridge University Press: Cambridge, 1969. Shakespeare, W., "Twelfth Night"; I, iii, 99-101. Hardy, G. H., "A Mathematician's Apology"; Cambridge University Press: Cambridge, 1940. Whitehead, A. N., "Science and the Modern World"; Macmillan: London, 1925. Ingleton, A. W., Bull. Inst. Math. Applies. 1982, 18, 34. Hoskins, R. F., Bull. Inst. Math. Applies. 1982, 18, 49. Cutland, N. J., Bull. Inst. Math. Applies. 1982, 18, 52. Hardy, G. Η., "Ramanujan—Twelve Lectures Suggested by His Life and Work"; Cambridge University Press: Cambridge, 1940. Russell, B. A. W., "The Study of Mathematics" in "Philosophical Essays"; Longmans Green: London, 1910. Wei, J., This observation, despite its interest and importance, seems to have escaped the public record. Aris, R., "Reflections on Some Trends in Chemical Reaction Engineering"; Chap. 4 of "Chemical Engineering in the University Context"; University of Wisconsin Press: Madison, 1982. Damköhler, D., Einflüsse der Strömung, Diffusion und Wärmeiiberganges auf die Leistung von Reacktions öfen, Z. Electrochem. 1936, 42, 846; 1937, 43, 1,8; 1938, 44, 240. Thiele,E.W., Ind. Eng. Chem. 1939, 31, 916. Bischoff, Κ. Β., Ind. Eng. Chem. Fundamentals 1976, 15, 229. Wei, J. and Prater, C. D., "The Structure and Analysis of Complex Reaction Systems". Advances in Catalysis. Academic Press: New York; Vol. 13. Aris, R., "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts"; Clarendon Press: Oxford; Vol. I, Ch. 5. Amundson, N. R. and Bilous, O., A.I.Ch.E. J. 1955, 1, 513. Wicke, E., "Physical Phenomena in Catalysis and in Gas-Solid Surface Reactions"; Proc. 1st Int. Conf. on Chem. Reac. Engng., Washington, 1970; American Chemical Society: Washington, D.C., 1972.


1.

36. 37. 38. 39.


40.

41.

42. 43. 44. 45. 46.

47. 48. 49. 50. 51. 52. 53.

54. 55. 56. 57. 58. 59. 60.

ARIS


17

Rony, P. R., J. Am. Chem. Soc. 1972, 94, 8247. Petersen, Ε. E. and Hegedus, L. L., Ind. Eng. Chem. Fundamentals 1972, 11, 579. Carberry, J. J., Ind. Eng. Chem. 1964, 56, 39. Oh, Se H., Hegedus, L. L., Baron, K. and Cavendish, J. C., "Carbon Monoxide Oxidation in An Integral Reactor. Transient Response to Concentration Pulses in the Regime of Isothermal Multiplicities"; Proc. ISCRE5 ACS Symposium Series 65, American Chemical Society: Washington, D.C., 1978, p. 461. Wei, J. and Degnan, T. F., "Monolithic Reactor-Heat Exchanger"; Proc. ISCRE5 ACS Symposium Series 65, American Chemical Society: Washington, D.C., 1978, p. 83. Gavalas, G. R., "Nonlinear Differential Equations of Chemially Reacting Systems"; Springer-Verlag: Heidelberg, 1968. Amundson, N. R. and Luss, D., A.I.Ch.E. J. 1967, 13, 759. Luss, D. and Balakotaiah, V. Chem. Engng. Sci. (to appear). Danckwerts, P. V., "Insights into Chemical Engineering (Selected Papers of P. V. Danckwerts)"; Pergamon Press: Oxford, 1982. I have touched on some aspects of this in the Wilhelm Lecture for 1981 and in 24. Feinberg, Μ., Chemical Oscillations, Multiple Equilibria and Reaction Network Structure in "Dynamics and Modelling of Reactive Systems"; Stewart, W. E., Ray, W. H. and Conley, C. C., Eds. Golubitsky, M. and Keyfitz, B. L., SIAM Journal Math. Anal. 1980, 11, 216. Balakotaiah, V. and Luss, D., A.I.Ch.Ε. J. (to appear). Humphrey, A. E. and Aris, R., Biotech, and Bioeng. 1977, 19, 1375. Aris, R., "Mathematical Modelling Techniques"; Pitman: London, 1978. Aris, R., Chem. Eng. Educ. 1976, 10, 114. Aris, R., Chem. Engng. Sci. 1969, 24, 149. The Watkins number is a dimensionless measure of appropriate ness in the closed interval 0 ≤Wa ≤ 1. It was introduced by Re in "a conversation on some aspects of mathematical modelling"; Appl. Math. Modelling 1977, 1, 386. Amundson, N. R. and Aris, R., Chem. Engng. Sci. 1958, 1, 122. Denbigh, Κ. G., Trans. Faraday Soc. 1944, 40, 352. Uppal, Α., Ray, W. H. and Poore, A. B. Chem. Engng. Sci. 1974, 29, 967. Vaganov, D. Α., Samoilenko, N. G. and Abramov, V. G., Chem. Engng. Sci. 1978, 33, 1133. Sheintuch, M. and Schmitz, R. Α., Cat. Rev. Sci. and Eng. 1977, 15, 107. Sheintuch, M., Kinetic Instabilities in Catalytic Oxidation Reactions (a review to appear). Lorenz, Ε. N., J. Atmos. Sci. 1963, 20, 130.



18

61.

62. 63.


64. 65. 66. 67. 68. 69.

ENGINEERING

It would be impossible to give an adequate bibliography of the work on chaos and strange attractors, but the proceedings of a Los Alamos Conference held in May 1982 under the title of "Order in Chaos" will, when they appear, give a good picture of the variety of activity. An introductory sketch of one aspect is given in 53. Pismen, L. Μ., Chem. Engng. Sci. 1980, 35, 1950. Rössler, O., Varma, A. and Kahlert, C. in "Modelling Chemical Reaction Systems"; Ebert, K. and Jaeger, W., Eds., Springer Verlag: Heidelberg, 1981. Lynch, D. T., Rogers, T. D. and Wanke, S. Ε., Math. Modelling 1982, 0, 000. Jorgensen, D. V. and Aris, R. will be in the Jan. 1983 issue of Chem. Engng. Sci. Bridges, R. W., "The Testament of Beauty. III"; Oxford, 1929, 1. 938. Hieronymus, E., Vulgate version of I. Cor. 2, 14. Pascal, B., Pensées. III, 206. Keats, J., Ode to a Nightingale, u. 69-70. I quote Keats' original version so that the reader may have the enjoyment of discussing its variations from the well-known final text.

RECEIVED March 10, 1983


Chemical Reaction Engineering as an Intellectual Discipline - ACS

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