Heuristic Approach to Complex Kinetics - ACS Publications - American

24 Heuristic Approach to Complex Kinetics

Downloaded by UNIV OF MARYLAND COLL PARK on December 16, 2014 | http://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/bk-1978-0065.ch024

J. B. CROPLEY Research and Development Department, Union Carbide Corporation, Chemicals and Plastics Division, P.O. Box 8361, South Charleston, WV 25303

N o n - l i n e a r r e a c t i o n r a t e e x p r e s s i o n s t h a t are adequate f o r r e a c t o r d e s i g n f r e q u e n t l y c o n t a i n more unknown parameters than can be e v a l u a t e d by e i t h e r classical kinetics or n o n - l i n e a r statistical methods. Sucl e x p r e s s i o n s are e n c o u n t e r e d i n both heterogeneous and homogeneous catalysis, and i n b i o c h e m i s t r y . F i f t e e n or more n o n - l i n e a r parameters are not uncommon i n r a t e models f o r complex industrial reactions. The heuristic approach d e s c r i b e d i n this paper utilizes linear statistical methods t o f o r m u l a t e the b a s i c h y p e r b o l i c n o n - l i n e a r model i n a particularly u s e f u l d i m e n s i o n l e s s form. E s s e n t i a l terms are identified and o t h e r s r e j e c t e d at t h i s s t a g e . Reaction stoic h i o m e t r y is combined w i t h the i n h e r e n t m a t h e m a t i c a l characteristics of the d i m e n s i o n l e s s r a t e e x p r e s s i o n to reduce the number o f unknown parameters t o the critical few t h a t must be e v a l u a t e d by n o n - l i n e a r e s t i m a t i o n . Typically, o n l y f o u r o r f i v e parameters remain at t h i s p o i n t , and initial e s t i m a t e s are a v a i l a b l e f o r t h e s e . The approach is e q u a l l y a p p l i c a b l e t o c a s e s where the rate-limiting mechanism is known and where it is not. The heuristic approach is illustrated by an example t h a t utilizes the fictitious but realistic catalytic vapor-phase o x i d a t i o n o f dammitol t o v a l u a l d e hyde and water. Some carbon d i o x i d e i s a l s o produced. I t w i l l be assumed t h a t water i s known t o have no e f f e c t on the r e a c t i o n , but the e f f e c t s of the o t h e r components and temperature are unknown. No knowledge o f the r a t e - l i m i t i n g s t e p i s a v a i l a b l e . The f i r s t s t e p i n the development of the model i s to o b t a i n a s e t of s u i t a b l e e x p e r i m e n t a l d a t a . In t h i ; case s u i t a b l e means t h a t the n e c e s s a r y i n f o r m a t i o n i s a c t u a l l y l a t e n t i n the d a t a , t h a t the ranges of the v a r i a b l e s are r e a l i s t i c , and so on. The experimental reactor s h o u l d be s p e c i f i c a l l y d e s i g n e d t o o b t a i n dat; ©

0-8412-0401-2/78/47-065-292$05.00/0

In Chemical Reaction Engineering—Houston; Weekman, V., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.


24.

293

Heuristic Approach to Complex Kinetics

CROPLEY

f o r t h e k i n d o f r e a c t i o n k i n e t i c s one i s working w i t h . The e x p e r i m e n t a l p l a n s h o u l d be c a r e f u l l y drawn so t h a t the p r i m a r y , i n t e r a c t i v e , and c u r v i l i n e a r e f f e c t s o f the v a r i a b l e s can be o b s e r v e d . U s u a l l y t h i s w i l l mean some s o r t o f s t a t i s t i c a l l y - d e s i g n e d e x p e r i m e n t a l a r r a y . For most vapor-phase c a t a l y t i c r e a c t i o n s , use a g r a d i e n t l e s s r e a c t o r l i k e the back-mixed c a t a l y s t a u t o c l a v e d e v e l o p e d by J . M. B e r t y ( 1 ) . T h i s r e a c t o r p e r m i t s t h e independent v a r i a t i o n o f mass- and s p a c e - v e l o c i t i e s and the d i r e c t o b s e r v a t i o n o f r e a c t i o n r a t e s . F o r t h e experiment a r r a y , I p r e f e r an o r t h o g o n a l central-composite d e s i g n ( 2 ) , (3), which c o n s i s t s o f t h r e e main p a r t s , as shown i n T a b l e I . The f i r s t i s a c o n v e n t i o n a l 16-experiment f r a c t i o n a l f a c t o r i a l de s i g n f o r f i v e v a r i a b l e s a t two l e v e l s . The second com p r i s e s t h r e e i d e n t i c a l e x p e r i m e n t s a t t h e average, o r c e n t e r - p o i n t , c o n d i t i o n s f o r t h e f i r s t 16 e x p e r i m e n t s . The f i n a l p a r t comprises two o u t - l i e r e x p e r i m e n t s f o r each v a r i a b l e . These augment the b a s i c two l e v e l de s i g n t o p r o v i d e an e s t i m a t e o f c u r v a t u r e f o r t h e r e sponse t o each v a r i a b l e . The o v e r a l l e f f e c t o f t h e d e s i g n i s t o s a t u r a t e e f f e c t i v e l y the multi-dimensional v a r i a b l e space. I t i s more e f f e c t i v e than t h e conven t i o n a l one-variable-at-a-time" approach. F o r purposes o f i l l u s t r a t i o n , t h e " o b s e r v e d " r a t e s i n T a b l e I were d e v e l o p e d from an assumed " t r u e " r a t e e x p r e s s i o n by a p p l y i n g random e r r o r as f o l l o w s ; f!

(Rate) , = (Rate)^ 'observed true x

u

v

χ INL R

7

where N i s a random normal number from a s e t w i t h a mean o f 1.0 and a s t a n d a r d d e v i a t i o n o f 0.20. Thus, the " o b s e r v e d " d a t a c o n t a i n 20 p e r c e n t random e r r o r . T h i s e r r o r l e v e l i s n o t i n o r d i n a t e l y h i g h f o r complex s i t u a t i o n s t h a t may i n v o l v e d i f f i c u l t a n a l y t i c a l methods and m u l t i p l e p r o d u c t s . The " t r u e " r a t e e x p r e s s i o n used i n t h i s example i s r

-20000 4.67(10 )e 5ÏÏ0ÏÏ U

r

l+5.52(10" )e 4

R

T

R

T

(P

0

2

)' (P ) '° 5

1

D

5ÏÏUÏÏ

(P )· +7.64(10' )e 5

Q 2

4

R

T

(Ρ )

2

γ

where r i s r e a c t i o n r a t e , g m o l s / v a l u a l d e h y d e / k g c a t a l y s t /hr; Τ i s temperature, °K; R i s t h e gas c o n s t a n t ; Ρ i s p a r t i a l p r e s s u r e , p s i a ; and s u b s c r i p t s 02, D, and ν r e f e r t o oxygen, dammitol, and v a l u a l d e h y d e , r e s p e c t ively. This rate expression implies that the r a t e -


CHEMICAL

294

REACTION

ENGINEERING—HOUSTON

TABLE I A SUMMARY OF SYNTHETIC DATA THE OXIDATION OF DAMMITQL TO VALUALDEHYDE


#

OBSERVED RATE GMOL/KG/HR

TEMPERATURE, °C

PARTIAL PRESSURES, PSIA OXYGEN DAMMI- VALUALC02 TOL DEHYDE

THE BASIC FRACTIONAL FACTORIAL SET 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

4. 07 8. 70 2. 29 5. 69 1. 58 10. 58 1.40 4. 63 1. 25 4. 34 0. 62 1.65 1. 15 3. 66 0. 63 1.44

7 7 7 7 3 3 3 3 7 7 7 7 3 3 2 3

10 10 4 4 10 10 4 4 10 10 4 4 10 10 4 4

3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1

3 1 1 3 1 3 3 1 1 3 3 1 3 1 1 3

100 100 100

5 5 5

7 7 7

2 2 2

2 2 2

91.,7 108. 3 100. 100. 100. 100. 100. 100. 100. 100.

5 5 1,.67 8,.33 5 5 5 5 5 5

2 2 2 2 2 2 0..34 3.,66 2 2

2 2 2 2 2 2 2 2 0.,34 3.,66

105 105 105 105 105 105 105 105 95 95 95 95 95 95 95 95

THE CENTER-POINTS 17 18 19 THE 20 21 22 23 24 25 26 27 28 29

2. 92 3. 16 2. 72 H

OUT-LIERS 1,,66 3.,57 1.,96 4.,11 1.,15 4.,70 4.,12 0..91 2.,95 3..16

!f

7 7 7 7 2 12 7 7 7 7



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295


CROPLEY

l i m i t i n g s t e p i s the s u r f a c e r e a c t i o n between chemisorbed oxygen and unadsorbed dammitol. Two molecules o f v a l u a l d e h y d e chemisorb (perhaps as a dimer) on a s i n g l e c a t a l y s t s i t e , and carbon d i o x i d e does not i n f l u e n c e the r e a c t i o n . H y p e r b o l i c e q u a t i o n s l i k e the above are t y p i c a l l y used t o d e s c r i b e r a t e phenomena i n b o t h heterogeneous and homogeneous c a t a l y s i s , b i o c h e m i s t r y , and i n many homogeneous r e a c t i o n s i n which c h e m i c a l e q u i l i b r i a a r e important. They are more d e s c r i p t i v e o f the r e a c t i o n ' c h e m i s t r y than i s the u s u a l e x p o n e n t i a l r a t e e x p r e s s i o n , and they a v o i d the problems e n c o u n t e r e d i n the e x p o n e n t i a l models i f one of the components i s absent. In t h i s paper, we w i l l use t h e e x p o n e n t i a l e x p r e s s i o n as a s t e p p i n g - s t o n e t o the more r i g o r o u s and u s e f u l h y p e r b o l i c model. The f o l l o w i n g r u l e s com p r i s e the h e u r i s t i c approach: Rule

1:

Develop a m o d i f i e d e x p o n e n t i a l r a t e model of f o l l o w i n g t y p e , u s i n g l i n e a r r e g r e s s i o n methods: l n ( r ) = f [In X., where the Χ . are p r o c e s s Ί

2

(In X . ) ,

X.)]

( I n Χ.)(1η

v a r i a b l e s o f the

the

form:

The s u b s c r i p t cp r e f e r s t o the c e n t e r - p o i n t c o n d i t i o n s of T a b l e I and Ε i s the apparent a c t i v a t i o n energy i n cal/gmol. The e x p o n e n t i a l form of the model d e v e l o p e d from the d a t a i n T a b l e I i s :

r

=

2

.

7

7

e " ^ ^ )

3

2

( ^ )

7

3

( ! v )

(

- -

5

9

L

^

-

l

-

M

)

The " t r u e " and " o b s e r v e d " r a t e s are compared w i t h the r a t e s p r e d i c t e d by t h i s model i n T a b l e I I . The s t a n d a r d d e v i a t i o n of the e r r o r w i t h r e s p e c t t o the "ob s e r v e d " d a t a i s 22 p e r c e n t , o r v e r y l i t t l e more than the random e r r o r i n the d a t a . The model a l s o a c t s as a f i l t e r f o r the random e r r o r , i n the sense t h a t the s t a n d a r d d e v i a t i o n w i t h r e s p e c t t o the " t r u e " v a l u e s i s o n l y 14 p e r c e n t . A l l i n a l l , i t i s not a bad model, but we can do b e t t e r , p a r t i c u l a r l y f o r d e s i g n p u r p o s e s where the c o n c e n t r a t i o n o f v a l u a l d e h y d e at the e n t r a n c e


CHEMICAL REACTION ENGINEERING—HOUSTON

296

TABLE I I A COMPARISON OF RESULTS

"TRUE" RATES 4. 50 12. 34 1. 80 4. 94 3. 11 9. 43 1. 24 3. 77 1. 86 5. 32 0. 75 2. 13 1. 29 4. 11 0. 52 1. 64 3. 00 3. 00 3. 00 1. 44 6. 05 1. 92 3.63 0. 86 5. 14 6. 56 1. 30 3. 00 3. 00


#

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

MEAN MEAN ERROR(a) obs ( > true ( ) s

s

NOTES:

3.47

DATA "OBSERVED" RATES ( c ) 4. 07 8. 70 2. 29 5. 69 1. 58 10. 58 1.40 4. 63 1. 25 4. 34 0. 62 1. 65 1. 15 3. 66 0. 63 1. 44 2. 92 3. 16 2. 72 1. 66 3. 57 1. 96 4. 11 1. 15 4. 70 4. 12 0. 91 2. 95 3. 16 3.13

b

(c)

3.08 0.07 °· ° ·

3.24 0.01 °·

2 2

b

(a) (b)

MODEL PREDICTIONS EXPOHEURISTIC NENTIAL #1 #2 4.24 4. 13 3. 34 10. 28 10.54 10. 68 2. 05 1.85 1. 71 5.31 5. 45 5. 48 2. 76 2. 54 2.91 7. 58 8. 14 7.76 1. 37 1.27 1. 31 4. 10 4. 17 4.05 1. 78 1.83 1. 44 4. 43 4.54 4. 60 0. 88 0.80 0. 74 2. 35 2. 36 2.29 1. 10 1. 19 1.25 3.34 3. 26 3. 50 0.55 0. 56 0. 59 1.74 1. 80 1. 77 2. 93 2.93 2. 77 2. 93 2.93 2. 77 2.93 2. 77 2. 93 1. 35 1. 43 1.43 5.81 5. 49 5. 81 1. 95 1. 80 1.85 3. 62 3.58 3. 26 1. 11 1. 13 0.95 4. 29 4. 10 4.48 6.03 3. 79 5. 89 1.31 1. 06 1. 32 2. 77 2. 93 2.93 2. 93 2. 77 2.93

1

2 6

4

0

Λ

2

3.25 0.02 °°' 2 7

0 8

E r r o r i s d e f i n e d as ( D a t a - P r e d . ) / D a t a . S k and S ^ are standard deviations of the e r r o r as d e f i n e d above, based on t h e " o b s e r v e d " and " t r u e " d a t a , r e s p e c t i v e l y . "Observed" r a t e s were d e r i v e d from " t r u e " r a t e s by imposing a 20 p e r c e n t random normal e r r o r . Models a r e based on "Observed" R a t e s . Q

s

r u e



24.

CROPLEY


297

t o t h e c a t a l y s t bed may be z e r o . We w i l l use some o f the i n f o r m a t i o n i n t h i s model i n t h e development o f t h e h y p e r b o l i c model. I f temperature and t h e c o n c e n t r a t i o n s o f a l l t h e c h e m i c a l s p e c i e s i n t h e study were important i n both the numerator and denominator o f t h e h y p e r b o l i c model, t h e r e would be, f o r η s p e c i e s , ( 4 n + 2 ) unknown p a r a meters i n t h e model. In t h e example study, t h e r e would be 18 parameters. In t h e " t r u e " expression, t h e r e are 10 parameters, because not a l l s p e c i e s a r e impor t a n t i n both numerator and denominator. Even i f t h e " t r u e " mechanism were known, t h e r e would be t o o many parameters t o e s t i m a t e s i m p l y by t o s s i n g t h e d a t a i n t o a n o n - l i n e a r e s t i m a t i o n program. The h e u r i s t i c p r o c e s s i s one o f s c a l i n g t h e p r o b lem down t o a s i z e t h a t i s manageable and o f c h a n g i n g the form o f t h e e q u a t i o n a b i t t o permit good v a l u e s of t h e parameters t o be c a l c u l a t e d . The problem o f f i n d i n g i n i t i a l estimates i s completely e l i m i n a t e d . Rule 2: E l i m i n a t e from c o n s i d e r a t i o n any v a r i a b l e t h a t was not s i g n i f i c a n t i n t h e e x p o n e n t i a l model. In t h e example, t h e p a r t i a l p r e s s u r e o f carbon d i o x i d e i s thus e l i m i n a t e d . Rule 3: E l i m i n a t e t h e temperature terms i n t h e denomina tor. I f they a r e i n d e e d n e c e s s a r y , they can be added later. U s u a l l y a change o f t h r e e o r f o u r k i l o c a l o r i e s i n t h e apparent a c t i v a t i o n energy compensates ade q u a t e l y f o r t h e i r absence. Rule 4: Develop t h e h y p e r b o l i c e q u a t i o n i n terms o f t h e d i m e n s i o n l e s s v a r i a b l e s o f Rule 1. By d o i n g so one b r e a k s t h e i n t e r d e p e n d e n c e o f e x p o n e n t i a l and p r e e x p o n e n t i a l terms, and one o f t h e b i g problems o f non l i n e a r estimation i s minimized. (See Rule 7.) Rule 5: A r b i t r a r i l y a s s i g n exponents t o t h e terms i n t h e denominator. Here one u t i l i z e s h i s knowledge o f t h e c h e m i s t r y and t h e s t o i c h i o m e t r y o f t h e assumed ad sorption e q u i l i b r i a . These may be changed l a t e r , but f o r now use v a l u e s o f 1/2, 1.0, o r 2.0. (The 1/2 i s


CHEMICAL REACTION ENGINEERING—HOUSTON

298

c h a r a c t e r i s t i c f o r c h e m i s o r p t i o n o f d i a t o m i c gases l i k e oxygen, the 2.0 f o r m o l e c u l e s t h a t adsorb as d i m e r s . ) Rule


was

6: Use the v a l u e f o r E, the a c t i v a t i o n energy, t h a t o b t a i n e d i n the e x p o n e n t i a l model.

Rule

7:

Determine the p r e - e x p o n e n t i a l term i n the numer a t o r by o b s e r v i n g t h a t the c o n c e n t r a t i o n and tempera t u r e terms are a l l u n i t y at the c e n t e r - p o i n t l e v e l s . Hence, the h y p e r b o l i c e x p r e s s i o n becomes, f o r the a v e r age r a t e at the c e n t e r p o i n t , merely: r?

A

=

C P

1

+ΣΚ.

or, A

=

r

c

p

(l

+EK ) 1

where A i s the p r e - e x p o n e n t i a l f a c t o r and the are the adsorption e q u i l i b r i u m constants. At t h i s p o i n t , the h y p e r b o l i c model i s ready f o r n o n - l i n e a r e s t i m a t i o n and l o o k s l i k e t h i s f o r our example :

-23295/1

V r

i + K

o2 vy +

e

R

1\

a

b

^ ^(!og) (V) 37

-

T h e r e are f i v e parameters t o e s t i m a t e : a, b, K Q 2 , K ^ , and K . V a l u a l d e h y d e was e l i m i n a t e d from the numerator because i t s exponent i n the e x p o n e n t i a l e q u a t i o n was s t r o n g l y n e g a t i v e , which a l s o suggested the denominator exponent of 2.0. y

Rule

8:

Set the n o n - l i n e a r s e a r c h ranges f o r the numerator exponents between 0 and 2.0. They w i l l seldom be found o u t s i d e t h a t range. Set the s e a r c h ranges f o r the denominator


K's

24.

CROPLEY


299

between 0 and 5. The l o g i c f o r t h i s r u l e i s q u i t e s i m p l e . At t h e c e n t e r - p o i n t c o n d i t i o n s , t h e v a l u e s f o r a l l t h e d i m e n s i o n l e s s v a r i a b l e terms w i l l be u n i t y , and the denominator w i l l be, as noted b e f o r e i n Rule 7: 1

+ΣΚ

£


f

In o r d e r t o have i n f l u e n c e on t h e r a t e , each o f t h e K s must be s i g n i f i c a n t w i t h r e s p e c t t o t h e 1.0, and, a t the same time, not overwhelm i t . Most K s w i l l be found i n t h e 1.0 t o 3.0 range. f

Rule 9: U t i l i z e computerized n o n - l i n e a r e s t i m a t i o n o f the numerator exponents and t h e denominator K - v a l u e s . The f i n a l r e s u l t i n o u r example i s : •23295/1

1 \

s

s

ftl

20.18e r =

The p r e d i c t e d v a l u e s u s i n g t h i s model a r e shown i n T a b l e I I under H e u r i s t i c Model #1. The s t a n d a r d d e v i a t i o n s o f t h e e r r o r f o r t h e " o b s e r v e d " and " t r u e " d a t a a r e about 26 and 12 p e r c e n t , r e s p e c t i v e l y . Thus the h e u r i s t i c model p r e d i c t s t h e " t r u e " d a t a s l i g h t l y b e t t e r than t h e e x p o n e n t i a l model does, but doesn't f i t t h e "observed" d a t a as w e l l . I t i sactually the b e t t e r model, a l t h o u g h t h e comparison seems t o pose a paradox. A c t u a l l y , t h e h e u r i s t i c model i s t h e more c o n s t r a i n e d o f t h e two, because i t has been s t r u c t u r e d to r e f l e c t a p a r t i c u l a r chemistry. It i sless likely t o be i n f l u e n c e d by t h e e r r o r s t r u c t u r e o f t h e d a t a than i s t h e e x p o n e n t i a l model, which i s more g e n e r a l . Rule 10: Examine t h e h e u r i s t i c model f o r any c l u e s o r s u g g e s t i o n s t h a t may improve t h e form o f t h e model. Two p o s s i b i l i t i e s a r e immediately apparent. The numer a t o r exponent f o r t h e oxygen i s v e r y c l o s e t o 0.5, and i t i s t h e r e f o r e a good i d e a t o s i m p l y s e t i t t o 0.5 t o agree w i t h t h e exponent i n t h e denominator. Likewise, the exponent f o r dammitol i n t h e numerator i s n o t f a r removed from 1.0 ( i t s exponent i n t h e denominator) and so we w i l l s e t i t t o 1.0. The s e a r c h v a r i a b l e s have


300

CHEMICAL

REACTION


been narrowed t o t h r e e : the e q u i l i b r i u m constants i n the denominator. A f t e r r e - f i t t i n g , the r e s u l t i s

21.lie


r

2.0

1+1.83 The r e s u l t s o f t h e p r e d i c t i o n s o f t h i s model a r e shown i n T a b l e I I as H e u r i s t i c Model #2. The s t a n d a r d d e v i a t i o n o f t h e e r r o r w i t h r e s p e c t t o t h e "observed" d a t a i s not q u i t e as good as e i t h e r o f t h e o t h e r models, but i t p r e d i c t s t h e " t r u e " v a l u e s markedly b e t ter — the standard d e v i a t i o n f o r the " t r u e " values i s o n l y about 8 p e r c e n t . Once a g a i n we see t h e paradox. The b e s t model o f t h e t h r e e f o r p r e d i c t i o n and d e s i g n purposes a c t u a l l y f i t s t h e d a t a from which i t was developed t h e p o o r e s t . I t i s , t h e r e f o r e , the l e a s t l i k e l y t o be used i n p r a c t i c e . T h i s happens q u i t e f r e q u e n t l y w i t h e r r o r - c o n t a i n i n g d a t a , and i s , i n t h e a u t h o r ' s judgment, a f r e q u e n t cause f o r t h e f a i l u r e o f models t o p r e d i c t d e s i g n e d performance a d e q u a t e l y . The o b v i o u s q u e s t i o n i s how t o t e l l when a model i s adequate f o r d e s i g n and when i t i s not — o r which i s t h e b e s t o f s e v e r a l . One o b v i o u s l y cannot use a model merely because i t conforms t o a mechanism o f some s o r t and doesn't f i t t h e d a t a v e r y w e l l . But j u s t f i t t i n g t h e d a t a w e l l i s o b v i o u s l y not enough. There a r e t h r e e answers t o t h e above q u e s t i o n . The f i r s t i s t o improve t h e a c c u r a c y o f t h e d a t a t o the extent p r a c t i c a b l e . Ten p e r c e n t d a t a i s d i f f i c u l t t o produce, but t h e problems c i t e d here a r e u s u a l l y not s e r i o u s with data of that q u a l i t y . The second answer i s t o t e s t t h e model e x t e n s i v e l y a g a i n s t d a t a t h a t were not used i n i t s development, and p r e f e r a b l y i n a r e a c t o r o f d i f f e r e n t geometry. If a f i x e d - b e d c a t a l y t i c r e a c t o r i s t o be used, then t h e model s h o u l d be t e s t e d i n a t e s t r e a c t o r o f s i m i l a r d e s i g n — a s i n g l e p l a n t - s c a l e tube makes an e x c e l l e n t test reactor. The t h i r d answer i s t o u t i l i z e n o n - k i n e t i c means to d i s c e r n t h e t r u e mechanism o f the r e a c t i o n . I t i s p o s s i b l e t o determine, f o r example, whether a s p e c i e s l i k e dammitol chemisorbs t o an a p p r e c i a b l e e x t e n t on a catalyst. Had such i n f o r m a t i o n been a v a i l a b l e i n the study o f t h e example, dammitol would have been e l i m i n a t e d from t h e denominator and t h e s t a n d a r d d e v i a t i o n o f t h e e r r o r f o r t h e " t r u e " d a t a would have been


24.

CROPLEY


301


o n l y about two p e r c e n t . But the paradox would s t i l l remain — the s t a n d a r d d e v i a t i o n f o r the " o b s e r v e d " d a t a would have been about the same as f o r the o t h e r h e u r i s t i c models, and not as good as t h a t f o r the e x p o n e n t i a l model. But the a b i l i t y o f the h e u r i s t i c models t o p r e d i c t " t r u e " v a l u e s i s t y p i c a l l y b e t t e r than t h a t of the more f l e x i b l e p o w e r - s e r i e s model f o r complex r e a c t i o n s . A Comparison With Modern S t a t i s t i c a l

Methods

There has been s u b s t a n t i a l p r o g r e s s o v e r the past twenty y e a r s i n the use o f modern s t a t i s t i c a l methods t o d e v e l o p n o n - l i n e a r k i n e t i c models. Generally, t h e s e approaches emphasize the placement o f e x p e r i m e n t s t o narrow the c o n f i d e n c e i n t e r v a l s of the e s t i m a t e d parameters and t e c h n i q u e s t o d i s c r i m i n a t e between r i v a l mechanisms. There i s an e x t e n s i v e l i t e r a t u r e on the s u b j e c t , and the r e a d e r i s r e f e r r e d t o one or two e x c e l l e n t r e v i e w a r t i c l e s as s t a r t i n g p o i n t s ( 4 ) , ( 5 ) . In g e n e r a l , t h e s e approaches have not met w i t h w i d e - s p r e a d s u c c e s s i n i n d u s t r y because they are v i r t u a l l y l i m i t e d t o n o n - l i n e a r models o f fewer than f o u r o r f i v e unknown p a r a m e t e r s . As the number o f p a r a meters i s i n c r e a s e d t o d e s c r i b e complex c h e m i s t r y more a d e q u a t e l y , the ranges of the r e a c t i o n c o n d i t i o n s r e q u i r e d f o r the e s t i m a t i o n of unique parameter v a l u e s c o r r e s p o n d i n g l y widen. I t o f t e n happens t h a t the r e q u i r e d ranges are u n r e a l i s t i c because the r a t e l i m i t i n g mechanisms change b e f o r e such ranges can be reached. The h e u r i s t i c approach was d e v e l o p e d t o b l e n d mathematics w i t h a knowledge o f c h e m i s t r y t o accomp l i s h f o r complex systems what n e i t h e r i s a b l e t o alone. Conclusions In t h i s paper I have attempted t o demonstrate a method f o r the development of h y p e r b o l i c r a t e models t h a t are adequate f o r the d e s i g n o f c h e m i c a l r e a c t o r s . The method i s r a p i d and overcomes most of the problems t h a t h i s t o r i c a l l y have hampered the development o f such models f o r complex r e a c t i o n s . I have shown t h a t the q u a l i t y o f f i t o f a model t o e r r o r - c o n t a i n i n g d a t a i s a poor c r i t e r i o n f o r model d i s c r i m i n a t i o n , and t h a t s e v e r a l models may p r e d i c t almost e q u a l l y w e l l . This, of c o u r s e , has been known f o r a l o n g time, but i t has not been w i d e l y r e c o g n i z e d t h a t the model t h a t f i t s the d a t a l e a s t w e l l may be the b e s t model, and t h a t the c o n v e r s e a l s o may be t r u e . In the f i n a l a n a l y s i s


CHEMICAL REACTION

302


a model must be t e s t e d t h o r o u g h l y b e f o r e r e a c t o r design i s f i x e d . Because o f i t s r a p i d i t y and s i m p l i c i t y , t h e h e u r i s t i c approach p e r m i t s t h e i n v e s t i g a t o r more time f o r c r i t i c a l t e s t i n g and m o d i f i c a t i o n .


Literature (1) (2) (3)

(4) (5)

Cited

B e r t y , J., Chem. Eng. P r o g . , (1974), 7 0 ( 5 ) . Box, G.E.P., B i o m e t r i c s , (1954), 10, 16-60. D a v i e s , O.L., "Design and A n a l y s i s o f I n d u s t r i a l Experiments", 534-5, Hafner P u b l i s h i n g Company, New York, 1960. Reilly, P.M., and B l a u , G.E., Can. J . Chem. Eng., (1974) 52, 289-299. Kittrell, J.R., and Mezaki, R., " A p p l i e d K i n e t i c s and Chemical R e a c t i o n E n g i n e e r i n g " , 119-32, American C h e m i c a l S o c i e t y P u b l i c a t i o n s , Washington, D.C., 1967.


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