Engineers (Chemical Industries) by Peter Englezos [Cap 16]

8/22/2019 Engineers (Chemical Industries) by Peter Englezos [Cap 16]

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16Parameter Estimation in ChemicalEngineering Kinetic Models

A n u m b e r of e xam ple s have be e n presented in Chapte rs 4 and 6. The solu-t ions to all t he se p rob le ms are g iv en here except for the two n u m e r i c a l p r o b l e m sthat were solved in C hapt e r 4. In addit ion a n u m b e r of p r o b l e m s have been in -cluded for so lu t ion by the reader .

16.1 A L G E B R A I C E Q U A T I O N M O D E L S16.1.1 Chem ical Kinetics: Catalytic Oxidation of 3-H exanolGallot et al . (1998) s tudied the catalyt ic ox id a t ion o f 3 -h ex an o l wi th h y d r o -gen peroxide. T he data on the effect of the solvent (C H 3O H ) on the partial conver-sion, y, of hydroge n pe rox ide are given in Table 4 .1. The proposed m o d e l is:

y = k , [ l - e x p ( - k 2 t ) ] (16.1)As men t io n ed in Chapte r 4, al though th is is a d y n a m i c e x p e r i m e n t w h e r edata are collected over t ime, we consider i t as a s imp le algebraic equat ion m o d e l

wi th two u n kn o wn parameters . T he data were given fo r two different condi t ions:(i ) with 0.75g an d (ii)with 1.30 g o f m e t h a n o l as so lve n t . A n in i t ia l g u e s s o fk!=1.0 and k2=0.01 was used. The m e t h o d converged in s ix and seven i terat ionsrespect ively wi thou t the need fo r Marquardt ' s modif ica t ion . Actua l ly , if Mar-quard t ' s modif ica t ion is used, the algori thm s low s dow n some w hat . The est imatedparame t e r s are given in Table 16.1 In addi t ion , the model-ca lcu la ted va lues a re285
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c o m p a r e d wi th t he e xpe r ime n t a l data in Ta bl e 16.2. As seen the a g r e e m e n t is verygood in th is case . T h e q u ad r a t i c conve r g e nce of the G a u s s - N e w t o n m e t h o d iss h o w n i n Ta bl e 16.3 w h e r e the reduct ion of the LS objec t ive func t ion i s show nw h e n an ini t ial guess of k,=100 and k2= 10 was used .Table 16.1 Catalytic Oxidation of3-Hexanol: Estimated Parameter

Values and Standard Deviations

M a s s ofC H 3O H0 . 7 5 g1 . 3 0 g

Paramete r V a l u ek,

0.17760.1787

k.0.10550.0726

Standard Devia t ionO kl

0.00950.0129

Ok2

0.01580.0116

Table 16.2 Catalytic Oxidation of 3-Hexanol: Experimental Data andModel CalculationsRe a c t i on

T i m e3613182628

Par t ia l C o n v e r s i o nRun wi th 0.75g m e t h a n o lDa ta

0.0550.0900 . 1 2 00 . 1 5 00 . 1 6 50 . 1 7 5

M o d e l0.0480.0830 . 1 3 30 . 1 5 10 . 1 6 60 . 1 6 8

R un w i t h 1 .30 g m e t h a n o lData0 . 1 4 00.0700.1000 . 1 3 00 . 1 5 00 . 1 6 0

M o d e l0.0350.0630 . 1 0 90 . 1 3 00 . 1 5 10 . 1 5 5

Table 16.3 Catalytic Oxidation of 3-Hexanol: Redu ction of th eLS Objective Function (Data fo r 0.75 g CH 3OH)Iteration

012oJ>456

Object ivefunc t ion59849.159849.132775.6

. 00035242

.00029574

.00029534

.00029534

k,100

1 0 0 . 096.88

0 .1 7 6 90 .17690 .17760 .1776

k210

4.3600 .11400 .11400.10640.10560.1055

Copyright 2001 by Taylor & Francis Group, LLC


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Parameter Estimation in Chemical Engineering Kinetics Models 287

16 .1 .2 Chemical K inetics: Isomerization of Bicycle [2 ,1 ,1 ] HexaneData on the t he rmal i somer iza t ion of bicyclo [2,1,1] he xane were me asure dby Srinivasan and Levi (1963). T he data are given in Table 4.4. T he fo l lowingn o n l in ea r mo de l w as proposed to describe the fraction of or ig ina l mater ia l re -

m a i n i n g (y) as a funct ion o f t i m e (\ ]) and tem perature (x2).y = expl - k i x i exp 1

X2 620 (16.2)Thi s p r o b l em w as d es c r i b ed i n C h a p t e r 4 ( P rob le m 4.3.4). A n in i t ia l g ue ss o fk (0 )=(0 .001, 10000) w as use d a nd conve r g e nce o f t he G a uss - N e wton m e t h o dw i t h o u t th e n eed fo r M a r qua r d t ' s modif ica t ion was ach ieved in f ive i terat ions . T he

reduc t ion in the LS objective func t ion as the iterations proceed is s h o w n in Table1 6 . 4 . In this case the in i t ia l guess w as fairly close the op t imum, k* . As the ini t ialgue ss is further aw ay from k*, the n u m b e r of i terations increases. For example , i fwe use as init ial guess k (0 )=( l , 1000000), conve rge nce is achieved in e igh t itera-t ions. A t th e o p t i m u m , th e fo l l o w i n g parameter values and standard deviat ionsw e r e o b t a i n ed : k, = 0 . 003 7 8 3 8 0 . 000057 a nd k2= 2 7 6 4 3 4 6 1 .

Table 16.4 Isomerization of Bicyclo [2,1,1] Hexane:Reduction of the LS Objective FunctionIteration

012->j45

Objectivefunct ion2.2375

0.2066680.029579

0 .01028170.01028170.0102817

k , x ! 0 20 .1

0.226630.366500.373800.373840.37383

k2100003 8 1 1 1250382767727 6 4 327643

U s i n g th e abo v e p a ramete r est imates the fraction of or ig ina l mater ia l of bicy-clo [2,1,1] h e x a n e was ca lcu la ted and i s show n together with the data in Table16.5. A s seen t he m ode l m a t c h e s th e data wel l .This problem w as also solved wi th M at lab (St ude n t vers ion 5 , T h e M A T HW O R K S I n c . ) U s i n g as ini t ial guess th e va l ue s (k ^ O.OOl , k2= 10000) conve rge nceof the Gauss-Newton me t hod was achieved in 5 i terations to the values(k,=0.003738, k2=27643). As expected, the parameter values that were obtained

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are the s a m e wi th those ob t a in e d w i t h th e Fortran p r o g r a m . In ad d i t i o n t he sa mestandard errors of the paramete r est imates were c o m p u t e d .Table 16.5 Isomerization ofBicyclo [2,1,1] Hexane: Experimental Data and

Model Calculated Values*1

120.60.060.0120.120.60.060.030.015.060.04 5 .190.01 5 0 .60.060.060.030.090.01 5 0 .90.4120.

X2

600600612612612612620620620620620620620620620620620620620620620

y (data).900.949.886.785.791.890.787.877.938.782.827.696.582.795.800.790.883.712.576. 7 1 5.673

y (model).9035.9505.8823.7784.7784.8823.7991.8939.9455.7991.8448.7143.5708.7991.7991.7991.8939.7143.5708.7132.6385

x .60.060.060.060.060.060.030.04 5 . 130.030.045.015.030.090.025.060.160.030.030.060.0

X2

6206206206206206206 3 16 3 16 3 16 3 16 3 1639639639639639639639639639

y (data).802.802.804.794.804.799.764.688.717.802.695.808.655.309.689.437.425.638.659.449

y (model).7991.7991.7991.7991.7991.7991.7835.6930.7835.7835.6935.8097.6556.2818.7034.4292.4298.6556.6556.4298

16.1.3 Catalytic Reduction of Nitric OxideA s another e x a m p l e from che m i ca l kinet ics , we consider the catalytic re -duct ion of nitr ic oxide (NO) b y hydroge n which was s tudied u s i n g a flow reactoroperated different ia l ly at a tmospher ic pressure ( Ay en an d Peters, 1962). T he fo l -l owi ng reaction w as considered to be imp o r tan t

NO + H , H 7O + - N 2Data were taken at 375C, and 400C, and 425C us i ng nitrogen as the dilu-

e n t . The react ion rate in gmol/(min-g-catalyst) and the total NO convers ion wereme asure d at different part ia l pressures for H 2 and NO.Copyright 2001 by Taylor & Francis Group, LLC


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A L a n g m u i r - H i n s h e l w o o d react ion rate m o d e l for the reac t ion be tween ana dso r be d n i t r i c o x i d e m o l e c u l e an d o n e ad jacen t ly a dso r be d h y d r o g e n m o l e c u l e isdescribed b y :

r = ( l + K NO rNO(16.3)

w h e r e r i s the react ion rate in gmol/(min-g-catalyst), ?m is the par t ial pr e s su r e o fh y d r o g e n (atm), P NQ is the par t ial p ressure o f NO (atm), K N O = A 2exp{-E2/KT}atm1 i s the adsorpt ion e q u i l i b r i u m c o n s t an t for NO, K H 2= A.3exp{-E3/RT} atm' isth e adsorpt ion e q u i l i b r i u m constan t fo r H 2 an d k=A }exp{-E }/RT} gmol/(min-g-catalyst) is the fo rward react ion ra te c o n s t an t fo r sur face react ion . T he da ta for theabove problem are given in Table 4.5.T he objective of the es t imat ion procedure is to de te rmine the parameters k,K H2 an d K N O (i f data from on e isotherm are on l y cons idered) or the param ete rs AbA 2, A 3, E I , E 2, E 3 (when al l data are regressed together). T he uni ts of E , , E 2, E 3 arein cal/mol and R is the unive rsal gas cons tant (1.987 cal/mol K).Kittrell e t al. (1965a) cons idered three models for the description of the re-duction of nitr ic oxide . The one g iv en in Chapte r 4 cor responds to a react ion b e-t w e e n one adsorbed mole cu le o f nitr ic oxide and one adsorbed m o l e c u l e of hydro-gen. This w as done on the basis of the shape of the curves pass ing through theplotted data.

In this work, w e first regressed the isothermal data. T he es t imated parame-ters from the t rea tment of the i so thermal data are given in Table 16.6.A n init ialguess of (1^=1.0, k2=1.0 , k3=1.0) was used for a l l i so therms and convergence ofth e Gau ss -Newto n me th o d w i t h o u t the need fo r Marquardt ' s modif ica t ion w asachieved in 13, 16 and 15 iterations for the data at 375,400, and 425 C respec-t ively.Plott ing of I n k j ( j=l ,2 ,3) ve rsus 1/T shows that on l y k ] exhibi t s A r r h e n i u stype of behavior . How e ve r , given the large s tandard deviat ions of the other tw oes t imated paramete rs one canno t draw defini te conc lusions abou t these two pa-rameters .Table 16.6 Catalytic reduction of NO: Estimated Model Parametersby th e Gauss-Newton Method Using Isothermal Data

Te mpe ra t u re(O3754 00425

( k , a k , ) x l 0 45.2 1.25 .5 3.213.5 8.0

k2 a k218.5 3.4

31.5 13.025.9 10.3

k3 a k 313.2 3.4

35.9 14.014.0 8.9

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290 Chapter 16

Kitt rel l e t a l. (1965a) also p e r f o r m e d tw o t y p es o f e s t ima t ion . Firs t th e dataat each isotherm were used separa te ly and subsequent ly a l l data w e re regresseds imu l t a n e ou s ly . The regress ion of the i so thermal data w as also d o n e with l inearleast squares by l inear iz ing the m o d e l equation. In Tables 16.7 and 16.8 the re-ported parameter es t imates are given toge ther with the reported standard error.A y e n an d Peters (1962) have also reported values for the u n k n o w n paramete rs an dthey are given here in Table 16.9.Table 16.7 Catalytic reduction of NO: Estimated Model Parameters byLinear Least Squares Using Isothermal Data

Tempera ture (Q3754 0 04 25

(k, + a k l ) x l 0 44.910.75 . 3 8 . 58 .8 2.3

k2 a k218.8 4.6

38.6 19.648.9 3 1.3

k3 o k314.6 2.9

35.4 11.330.9 + 20.2

Source: Kittre l l et al. (1965a).Table 16.8 Catalytic reduction of NO: Estimated Model Parameters byNonlinear Least Squares Using Isothermal DataT e m p e r a t u r e (C)

37540 04 25

( k , a k l ) x l 0 45. 19 0.95.51 1 . 210.1 3.0

k2 + a k 218.5 + 3.43 1 . 6 + 1 2 . 934.5 15.2

k3 o k313.2 3.4

36.0 13.923.1 + 11.6

Source: Kittre l l et al. (1965a).Table 16.9 Catalytic reduction of NO: Estimated Model Parameters byNonlinear Least Squares

Temp era tu re (C)3754 004 25 J

k , x ! 0 44.947.088.79

k219.0030.4548.55

k314.6420.9630.95Source: A y e n an d Peters (1962).

Table 16.10 Catalytic reduction of NO: Estimated Model Parameters byNonlinear Least Squares Using Nonisothermal DataTempera ture (Q

37540 04 25

( k , o k l ) x l 0 44.92 3.716.58 3.948.63 3 . 9 2

k2 o k215.5 13.426.3 18.342.9 2 3 . 6

k3 o k317.5 11.523.8 12.831.7 14.6

Source: Kittre l l et al. (1965a).



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Parameter Estimation in Chem ical Engineering Kinetics M odels 291

Kitt rel l e t a l. (1965a) also use d al l the data s i m u l t a n e o u s l y t o c o mp u t e t h ep aramete r v a l u es . Th ese p a r a m e t e r v a l u e s are repor ted for each tempera ture andare given in Table 16.10.W ri t in g Ar r he n i us - type e xpr e s s i ons , kj=Aje;tp(-Ej/RT), for the kine t ic con-s tan ts , the mathemat ica l model with s ix u n k n o w n parame t e r s ( A j , A 2, A 3, Eb E 2an d E 3) b e c o m e sEL

f ( x , k ) =A2e R T x 2 + A 3e R Tx,

(16.4)

T he e lemen ts of the ( /x


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292 Chapter 16

E]+2E2+E3S f ^ 2 A , A 2 ? A 3 e RT v . v 2

SEj } RTY3K 1 Y (16.9)E ] + E 2 + E 3A1A2A3e RT X j X 2

of } 2A,A2A3e

R T Y ^E ] + E 2 + 2 E 3

?p" R~~Y?v9E3 I RTY3Jy (16.10)Ei+E2 +E3

A]A2A3e RT X j \ 2R T Y Z

wh ereE 2 _ E ^

Y = l + A 2 e R T x 2 + A 3e R T x , ( 1 6 . 1 1 )T he results were obtained using three different sets of initial guesses whi ch

w e re given b y Kittrell e t al. (1965b). None of them w as good en o u gh to convergeto the global o p t i m u m . In par t icula r the first tw o converged to local opt ima and thethird diverged . The lowest LS objective funct ion w as obta ined with the first in i t ia lg ue ss and i t was 0 . 1 4 6 4 x l O ~ 6 . T h e c o r r es p o n d i n g es t i ma t ed pa r a m e te r va lu e s w e r eA^O.803910.3352, A 2= 1.371 x 10 56.798x 10 4, A 3 = l .768x 10 78.739x 10 6,E , = 9 5 2 0 + 0 . 4 x 10" 5, E 2= 115000.9x 10"7 and E 3= 17,9000.9x 10" 7.In th i s prob lem i t i s very di f f i cu l t to obtain conve r g e nce to the g l o b a l opti-m u m a s t h e c o n d i t i o n n u m b e r of matr ix A a t t he a b o v e l o c a l o p t i m u m is 3 x l 0 1 8 .E v e n if th i s was the g lobal o p t i m u m , a sm a l l change in the data w o u l d resul t inw i d e l y different paramete r es t imates since th is parameter est imat ion problem ap-pears to be fairly i l l - condi t ioned .A t th i s poin t we should always t ry and see w h e t h e r there is anything e lsethat cou l d b e done to reduce the i l l -condi t ion ing of the p roblem. U p o n r e e x a m i n a -t ion of the structure of the model given by E q u a t i o n 16.4 we can readi ly noticethat i t can be rew ri t ten as



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f ( x , k ) =

+ A 2 eR T

x 2 + A 3eRT

xvw h e r eA J ' = A 1 A 2 A 3 (16.13)

andE ^ = E, + E 2 +E 3 (16.14)

The reparamete r ized m o d e l has the same n u m b e r of u n k n o w n parameters(A/, A 2, A 3, E I * , E 2 an d E 3) as the original p ro b lem, how e ve r , it has a s imp le rst ructure . Th is often results in m u c h better converge nce charac te ris tics of the it-erat ive est imat ion a lgo r i th m. In deed , co nv e rgence to the global o p t i m u m w as ob-tained after m a n y i terat ions us ing M a r q u a r d t ' s modif ica t ion . The va lue of Mar-q u a rd t ' s p a ramete r wa s a l wa ys ke p t one o r de r o f m a g n i t u d e greater than thesma l l e s t e i gen v a l u e o f ma t r i x A . A t t h e o p t i m u m , a v a l u e o f zero f o r M a r q u a r d t 'sparame t e r w as use d and conve rge nce w as m a i n ta i ne d .

The LS objective funct ion w as fo u n d to be 0.7604x10" 9. This value is a lmos tth ree orders of m a g n i t u d e smal le r than the one fo u n d earl ier at a local o p t i m u m .The es t imated p a ramete r values were : A,=22.672, A 2=132.4, A 3=585320,, = 1 3 8 9 9 , E 2=2439.6 an d E 3= 1 3 5 0 6 w h e r e p a r a m e t e r s A ! a n d E , w e r e e s t ima t e dback from A ,* an d E , * . W ith th is reparamete r izat ion w e were able to lessen the i l l-condi t ion ing of the problem s ince th e c o n d i t io n n u mb er of mat r ix A w a s n o w5 . 6 x l 0 8 .T he model-ca lcu la ted reaction rates are compare d to the e xpe r i m e n ta l datain Table 16.11 where i t can be seen that th e match is qui te satisfactory. Based onthe six est imated parameter values , th e kinetic cons tants (kb k2 and k3) were com-p u ted at each tem pera ture and they are s h o w n in Tab le 16.12.Having f ou n d th e o p t i m u m , w e re tu rned back to the or ig ina l st ructure of thepr ob l e m an d used an ini t ial guess fairly close to the global o p t i m u m . In this caseth e parameters converged very close to the o p t imu m wh ere the LS objective func-t ion w as 0.774xlO" 9 . T h e condi t ion n u m b e r o f ma t r ix A w a s f o u n d to be 1 . 7 x l 0 l jw h i c h i s about 5 o rde r s o f m a g n i t u d e h i g h e r tha t the one fo r the r e pa r a m e te r i ze dformulat ion calculated at the same poin t . In conc lusion , reparamete r izat ion shouldb e seriously cons idered for hard to converge prob le ms .



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294 Chapter 16

Table 16.11 Catalytic reduction of NO: Experimental Measurementsand Model Calculated Values (with kfrom Table 16.12)P H 2

(atm)P N O

(atm)

M easu red R a t er x l O 5

gmol/fmin-g-catalyst)

Calcu la ted Rater x l O 5

gmol/fmin-g-catalyst)T=375C, Wei g h t of catalys t=2.39g

0.009220.01360.01970.02800.02910.03890.04850.05000.05000.05000.05000.0500

0.05000.05000.05000.05000.05000.05000.05000.009180.01840.02980.03780.0491

1.602.563.273.643.484.464.751.472.483.454.064.75

1.5142.0912.7753.5213.6074.2534.7241.7392.9243.8854.3364.761

T=400C, W e i g h t of catalyst=1.066 g0.006590.01130.02280.03110.04020.05000.05000.05000.05000.05000.0500

0.05000.05000.05000.05000.05000.05000.01000.01530.02700.03610.0432

2.524.215.416.616.868.793.644.776.617.947.82

2.3763.6905.9547.0117.8038.3573.0624.2756.2497.2987.907

T=425C, Weigh t of cata lys t=l .066 g0.004740.01360.02900.04000.05000.05000.05000.0500

0.05000.05000.05000.05000.05000.02690.03020.0387

5.027.2311.3513.0013.919.299.7511.89

3.5507.93811.6812.8213.299.58610.291 1 . 8 1



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Parameter Estimation in Chem ical Engineering Kinetics Models 295

Table 16.12 Catalytic reduction of NO: Estimated Model Parameters byNonlinear Least Squares Using All the Data.Te mpe ra t u re (Q

375400425

k , x l 0 44.656.9410.07

k21 9 . 9 121.3622.80

k316.2924.0534.53

16.2 P R O B L E M S W I T H A L G E B R A I C M O D E L ST h e f o l l o w i n g pa r a m e te r e s t i m a t i o n p r o b l e m s w e r e f o r m u l a t e d f rom re -search pape rs ava i lable in the l i tera ture and are l e f t as exercises .

16.2.1 Catalytic Dehydrogenation of sec-bu tyl AlcoholD at a for the in i t ia l react ion ra te for the cata ly t ic dehydrogenat ion o f sec-

butyl a lcohol to m e th y l e th y l ketone are g iven in T a b l e 16.13 (Thal l er an d Th o do s ,1960; S ha h , 1965). T h e f o l l o w i n g tw o m o d e l s w e r e con s id e re d for the in i t ia l rate:Model A

(16.15)w h e r e

R = k , K A p A ) 2K A p A (16 .16)

M o d e l B

rA, = ( 1 6 . 1 7 )

wh ere X=-0.7, K A is the adsorption eq u i l i b r i u m cons tant fo r sec-butyl alcohol, kHis th e rate coeff icient for the rate of hydrogen desorpt ion controll ing, kR is the ratecoefficient for surface reaction cont ro l l ing , P A is the partial pressure of sec-butylalcohol.



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296 Chapter 16

Table 16 .13 Datafor the Catalytic Dehydrogenation of sec-butyl Alcohol

Te mpe ra t u re(F)600600600600600600600600600600575575575575575

Pressure(atm)

1.07.04 .010.014.65.58.53.0

0.221.01.03.05.07.09.6

Feed Rate(Ib-moles/h)0.013590.013660.013940.013670.013980.013890.013840.013920.013620.013900 .014110 .01 4 000.014010 .013740 .01342

In i t ia l Ra te (r A i)Ib-moles of al-cohol/ (h)(lb-catalyst)0.03920.04160.04160.03260.02470.04150 .03760.04200.02950 .04 1 00.02270.02770.02550.02170.0183

Source: Thal le r an d T h o d o s (1960); Shah (1965).Table 16.14 Parameter E stimates fo r Models A and B

M o d e l

AB

Te mpe ra t u re( F )57 560057 5600

k H x ! 0 2(Ibmoles alcohol/(hr Ib catalyst)7.657.8911.59.50

k R x ! 0 2(Ibmoles alcohol/(hr Ib catalyst)23.581.720.262.8

K A x l 0 2(atm 1)44.453.540.751.5

Source: Shad (1965).

U s i n g the ini t ial rate data given above do the f o l low in g : (a) Determine theparameters , kR , kH an d K A f o r mo de l -A an d m o d e l - B an d thei r 95% confidenceintervals; and (b) U s i n g the parameter es t im ates calculate the ini t ial rate an d com-pare it with the data. Shah (1965) reported the parame t e r est imates given in Table16.14.



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Parameter Estimation in Chemica l Engineering Kinetics Models 29 7

16.2.2 Ox idation of PropyleneT h e f o l l o w i n g data g i ve n in Ta bl e s 16.15, 16.16 an d 16.17 on the ox i da -tion of p ro p y len e o v er b i s m u t h m o l y b d a t e catalyst were obtained at three t e m -pera tures , 350, 375, an d 390C (Watts , 1994).

One m o d e l proposed for the rate of p ro p y len e disappearance , rp, as a funct ionof the oxygen concent ra t ion, C 0, th e propylene concent ra t ion, C p, and the stoichi-ometr ic n u m b e r , n, isk kK a5r,, = - (16.18)

w he re k0 and kp are the rate parameters.

Table 16.15 Data for the Oxidation of Propylene at 350C.C P

3.051 . 3 73 . 1 73.024 . 3 12.783 . 1 12 . 9 62.841 . 4 61 . 3 81 . 4 21 . 4 93 . 0 11 . 3 51 . 5 25 . 9 51 . 4 65.681 . 3 61 . 4 23 . 1 82 . 8 7

C 03.073 . 1 81 . 2 43 . 8 53 . 1 53 . 8 96 . 4 83 . 1 33 . 1 47.937.798 . 0 37.783.038.008.226 . 1 38 . 4 17 . 7 53 . 1 01 . 2 57 . 8 93 . 0 6

n0.6580.4390.4520.6950.6350.6700.7600.6420 .6650.5250.4830.5220.5300.6350.4800.5440.8930 . 5 1 70.9960 . 4 1 60.3670.8350.609

rP2.732 . 8 63.002 . 6 42 . 6 02.732 . 5 62.692.772 . 9 12.872.972 . 9 32 . 7 52.902 . 9 42.382.892 . 4 12 . 8 12.862 . 5 92 . 7 6

Source: Watts (1994).Copyright 2001 by Taylor & Francis Group, LLC


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298 Chapter 16

Table 16.16 Data for the Oxidation ofPropylene at 375CCD2.94

1.353.042.904 .142.692.992.855.461.391.342.731.461.391.331.377.022.897.301.353.152.75

C 02.963.061 . 1 93.703.033.766.233.037.467.671.153.027.657.567.497.752.932.912.967.667.522.93

n1.1600.6800.7401.1701.3901.1901.2901.1302.0300.8040.6301.0800.8640.7720.7770.7451.3101.1601.3600.7411.4401.050

r D2.372.582.242.192.322.312.162.251.932.632.582.162.642.532.642.512.252.272.222.552.142.15

Source: Wat ts (1994).T h e ob ject ive is to d e t e r m i n e t h e p a r ame t e r s an d t h e i r s tandard errors by theG a u s s - N e w t o n m e t h o d for each t emp er a t u r e an d t h en check to see if the p a r a m e t e res t imates obey A r r h e n i u s type behavior .Watts (1994) reported th e fo l lowing p a ramete r est imates at 350C:ko= 1,334 0.081 [(mmol L)as/(g s) ] an d kp =0 .611 0.055 [(Ug s)] . S i m i l a r re-sul ts w e re f ou n d for the data at the other tw o t e mpe ra t u re s .T h e parame t e r va l ue s w e r e then plotted versus the inverse tempera ture andw e re found to fol low an A r r h e n i u s type re la t ionship

E, jk ; = A j exp\ - - -] J RJ = o ,p (16.19)w h e r e A , is the pre-exponent ia l factor and E j the activation e ne rgy . Both num be r sshould b e posi t ive .



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Table 16.17 Data for the Oxidation ofPropylene at 390CCD

2 . 6 22 . 7 93 . 0 23 . 0 71 . 3 61 . 3 11 . 4 22 . 7 26 . 8 67 . 1 31 . 3 27 . 0 92 . 8 81 . 3 37 . 1 41 . 3 75 . 3 91 . 3 12 . 7 42 . 8 95 . 2 9

C 03 . 6 62 . 9 66 . 1 27 . 3 27 . 5 21 . 1 27 . 4 73 . 4 82 . 8 62 . 8 97 . 4 83 . 2 73 . 7 67 . 8 43 . 2 27 . 8 97 . 2 52 . 9 03 . 5 47 . 4 87 . 2 3

n1 . 4 8 01 . 5 1 01 . 8 0 01 . 9 0 00.9900.8050 . 9 9 11 . 5 2 02 . 2 1 02.3000.9362.4301 . 6 4 00.9752.3000.9962.7600.8231 . 5 3 01 . 7 9 02.760

rD1 . 9 52 . 0 01 . 9 21 . 9 62 . 3 62 . 3 32 . 2 61 . 9 32 . 0 62 . 1 02 . 3 62 . 1 61 . 8 52 . 3 82 . 1 02 . 3 91 . 7 62 . 2 81 . 8 41 . 8 31 . 7 5

Source: W atts (1994).S u b s e q u e n t l y , Wat ts p e r f o r m e d a p aramete r es t imat ion b y us ing th e data

f rom all t e mpe ra t u re s s i m ul t a ne ous l y and by e m p l o y i n g th e fo rmula t ion of the rateconstants as in Equation 16.19. The parameter values that they found as w el l astheir standard errors are reported in Table 16.18. It is noted that they found that theresiduals from the fit were w e l l be have d e xcep t for two at 375C. These res idualswere found to accoun t for 40% of the residual sum of squares of devia t ions b e-tween e xpe r i m e n ta l data an d calcu la ted values .Watts (1994) deal t with th e issue of conf idence interval es t imat ion when es -t i m a t i ng parameters in n o n l i n e a r mode ls . He proceeded with the re formula t ion ofEq u a t io n 16.19 because the p re-exponent ia l paramete r es t imates "behaved h i gh l yn o n l i n ea r l y . " T he rate constan ts w e re fo rmula ted as f o l low s

k ; =A ; e x p \ - - (16.20)w i t h



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300 Chapter 16

(16.21)

Table 16.18 Parameter Estimates for the Model for the Oxidation ofPropylene (Noncentered Formulation)Parameter

A0EoA PE p

Estimated Value8 . 9 4

1 0 5 . 61 4 5 . 1

2 8 . 1

Standard Error9 . 7 75.9

209.47 . 8 9

Source: W at t s (1994).Table 16.19 Parameter E stimates for the M odel for the Oxidation ofPropylene (Centered Formulation)

Parameter i ) ) ] 2 + [ / 7 ( f ( j C Q i) ) - M g G i ) ) ] 2 (16.24)

where j = V-l , W n - ^ l . l x w j , coi=0.01 a n d N = 1 0 0 .In th is pr ob l e m you are asked to de termine the u n k n o w n parameters usingth e d o m i n a n t zeros an d poles of the original system as an i n i t i a l guess . LJ opt imi-zation proce dure can be used to obtain the best pa rame t e r est imates .Redo t he p rob le m b u t take N = 1,000, 10,000 and 100,000.Afte r th e parame t e r s have be e n est imated, generate th e N y q u i s t plots for ther ed u c ed m o d e l s and the o r i g i n a l o n e . C o m m e n t o n t h e r e s u l t a t h i gh f requencies .Is N = 1 0 0 a wise choice?R e d o th i s p r o b l e m . How e ve r , th i s t i m e a s s u m e that th e r e duce d model i s afourth o rde r o ne . N a m e l y , it is of the form

g(s) = - ~ , V (16.25)+ k6s + k 7 sHow important is the choice of C 0 ] = 0 . 0 1 ?



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302 Chapter 16

16.3 O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N M O D E L ST he formula t ion for the next three prob lems of the parameter es t imat ionprob lem w as given in Chapte r 6 . These examples w e r e f o rmu la t e d w i t h data fromth e l i terature an d he n ce th e reader is s t rong ly recommended to read th e or ig ina lpapers for a thorough unders tand ing of the re levan t phys ica l an d chemica l phe-

n o m e n a .

16.3.1 A Homogeneous Gas Phase ReactionB e l l m a n e t al. (1967) h a v e cons idered the e s t ima t ion of the two rate con-s tants k] a n d k2 in the B o d en s t e i n -L i n d e r mod e l f o r t he h o m o g e n e o u s g a s p h as ereaction of NO w i t h O 2:

2NO + O 2 < - > 2N O 2T he m o d e l is described by the fo l lowing equa t ion

= k1(a~x)(p-x)2 -k 2 x 2 ; x(0) = 0 (16.26)dtwhere a=126.2, (3=91.9 and x is the concentrat ion of NO 2. The concentrat ion ofN O 2 w as me a su re d e xpe r ime n t a l ly as a func t ion of t ime and the data are given inTable 6 .1B e l lma n et al. (1967) e mploye d th e quas i l inear iza t ion t echn ique a nd ob-tained th e fo l lowing parameter est imates : k | = 0 . 4 5 7 7 x l O ~ 5 an d k2=0.2797xlO~ J .Bodenste in an d L id n e r who ha d obtained the kine t ic data reported sl ight ly differ-en t va lues : k i = 0 . 5 3 x l O ~ 5 an d k2= 0 . 4 1 x l O ~ J . T he la t ter values were obtained b y ac o m b i n a t i o n o f chemica l theory an d t h e data. T h e res idual s u m o f s q u a r es o f d e -viat ions w a s f o u n d to be equa l to 0.210x10" . The cor r e spond i ng va l u e repor ted b yBodenste in an d Lidner who ha d obtained the kinet ic data is 0.555xlO" 2. B e l lma n e tal. (1967) stated that th e difference does n o t ref lect one se t of parameters beingbet ter than th e other.

U s i n g the compu t e r p ro g ra m B a y e s _ O D E l w h i c h is g ive n in A p p e n d i x 2 thef o l low in g parame t e r es t imates were ob ta ined : k1=0.4577xlO" 5 an d k2=0.2796xlO"3u s in g as initial guess k i = 0 . 1 x l O " 5 an d k2=0 . 1x l O " 5 . Co n v ergen c e w as achieved inseven iterations as seen in Table 16.20. T he ca lcu la ted standard d e v ia t ion s for theparameters k, and k2 were 3.3% an d 18.8% respect ive ly . Based on these parametervalues, the concen t ra t ion o f N O 2 w as computed versus t ime an d compared to theexper imen ta l data as s h o w n in Figure 16.1. The overal l f i t i s quite satisfactory.

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Table 16.20 Homogeneous Gas Phase Rea ction: Convergenceof th e Gauss-Newton Method

Iteration012^>j4567

L S ob je c t i ve f u n c t i o n4 0 1 1 . 71739.9774 . 15123.3522.02021.86721.86721.867

k ,O . l x l O '

5

0 . 33 8 25 x l Q - 50 . 3 5 4 1 0 x l O ' 50 . 3 9 7 0 7 x l O '50 . 4 5 5 8 1 X 1 0 ' 50 . 4 5 7 8 6 x l O " 50 . 4 5 7 7 0 x 1 0" 50 . 45771 x l O '

5

k2O . l x l Q -

5

0 . 2 5 8 7 6 x l O '20 . 1 1 6 5 5 X 1 0 '20 . 4 0 7 9 0 x 1 0'30 . 2 6 4 8 2 x 10" 30.280 1 7x lO" 30 . 2 7 9 5 9 x l O '30 .2 7 9 6 2 x 1 0'

3

10 20 30Time 40 50Figure 16.1 Homogeneous Gas Phase Re act ion: Experimental data andmodel calculated values of th e N O 2 concentrat ion.

16.3.2 Pyrolytic Dehydrogenation of Benzene to Diphenyl and TriphenylLet us now consider the pyro ly t ic dehydrogenat ion of benzene to d i p h e n y land t r i phe ny l (Seinfeld and Gavalas , 1970; Ho u gen and Watso n , 1948):

2C 2H 6 4^ C 12H 1 H 2Copyright 2001 by Taylor & Francis Group, LLC


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304 Chapter 16

T h e f o l l o w i n g kine t ic m o d e l h a s b e e n pr opose d

- -=-ri-r2 (16.27a)dt(16.27b)dt 2

w h e r e-x 2(2-2x1-x 2)/3K1 (16.28a)

r2 =k 2 (x,x 2 -(l-x, -2x 2 X2-2x1-x 2 ) /9K2 ) (16.28b)an d w h e r e X ] de no te s Ib-moles of b e n z e n e pe r Ib-mole of p u r e b e n z e n e feed an d x 2de no te s Ib-moles o f d i p h e n y l p e r Ib-mole o f p u r e b e n z e n e feed . The pa r a m e te r s k tand k2 are unknown react ion rate constants w h e r e a s K ] an d K 2 are e q u i l ib r iu mconstants . T he data cons is t of me asure me nt s of x , an d x2 in a flow reactor at e ightvalues of the reciprocal space velocity t and are given in Table 6.2. T he feed to thereactor w as p u re be nze ne . T h e equi l ib r ium cons tants K , an d K 2 were de te rmin edfrom the run at the lowest space veloci ty to be 0.242 an d 0.428, respect ive ly .Seinfeld and Gavalas (1970) emp lo y ed th e quasi l inear izat ion meth o d toest imate the parame t e r s for the kinetic m o d e l that w as proposed b y Ho u gen an dWatson (1948). Seinfeld an d Gavalas (1970) examined the s ignif icance o f h a v i n ga good ini t ial guess . A s we i g h t i ng matrix in the LS objective funct ion they usedthe identity ma t r ix . It w as f ou n d that with an initial guess k!=k2=300 or k ^ k ^ S O Oconve rge nce w as a ch i e ve d in four i terations. The corresponding est imated pa-rameters were 347.4, 403.1. T he quasi l inear iza t ion a lgor i thm diverged when th einit ial guesses were ki =k2= 1 0 0 or k!=k2=100 0. I t is interesting to note that H o u g e nan d Wa tson (1948) repor ted th e v a l u e s o f k , =348 an d k2= 4 0 4 [(Ib-moles),'3'59(ft3)(hr)(atm2)]. Seinfeld and Gavalas pointed out that i t was a coinciden ce that theva l ue s est imated by H oug e n a nd Watso n in a rather "c rude" m a n n e r were veryclose to the v a l u es est imated by the quas i l inea r iza t ion method.S ubse que n t l y , Seinfe ld an d Gavalas examined the ro le o f the weigh t ingmatr ix and the role of u s in g f e we r data points. I t was f ou n d that the est imated val-ues at the global m i n i m u m are not affected apprec iably b y us i ng different weight-ing factors. They proposed th is as a q u i c k test to see whether the global m i n i m u mhas been reached and as a means to move aw ay from a local m i n i m u m of the LSobject ive funct ion. I t was also f ou n d that there is s o m e variation in the es t imates ofk2 as the n u m b e r of data points used in the regression is reduced. The es t imate o fk| was f ou n d to r e m a i n cons tant . The same problem was a lso s tudied b y Kalo-gerakis and L u u s (1983) to demonst ra te th e subs tant ia l e n l a r g e m e n t of the reg ionof conve rge nce for the Gau ss -Newto n m e t h o d th rough the use o f the in fo rmat ionindex discussed in Chapte r 8 (Section 8.7.2.2).

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U s i n g t h e p ro gram prov id e d in A p p e n d i x 2 a n d star t ing with an in i t ia lg u e s s fa r from th e o p t i m u m (k t=k2= 10000), th e G a uss - N e wton m e t h o d conve r g e dw i t h i n n i n e i terat ions . M a r q u a r d t ' s p aramete r w as zero at al l t i m e s . T h e r e duc t i onin the LS object ive funct ion is show n in Table 16.21. A s weight ing mat r ix , th eidentity m atrix was used . The uncertainty in the parameter est imates is qu i t e smal l ,namely , 0 .181% and 0.857% fo r k i and k2 respect ively. The cor responding mat chbetween th e e xpe r ime n t a l data and the model-calculated values is sh o wn in Table16.22.Table 16.21 Benzene Dehydrogenation: Convergence of th e Gauss-Newton Method

I t e ra t ion0123456789

L S o b jec t i v e f u n c t i on0 .275010.273250.258410 .2 1 3 1 7

0 . 6 5 8 4 8 x 1 0 - '0 . 7 5 5 0 9 x 1 0'20 . 23 06 3 x 1 0" '0 . 7 1 8 1 1 x l O - 50 . 70523x l O '50 . 7 0 5 2 3 x l O '5

Standard Devia t ion

k,10000

4971 . 14254.51885.270 3 . 35436.83343.05354.32354.60354.610.642

k210000

9518.41526.11479.5893.62515.53378.92399.624 00 . 244 00 . 233 . 4 3

Table 16.22 Benzene Dehydrogenation: Experimental Data and ModelCalculated ValuesReciprocal Space

Veloci ty x l O 45 . 6 311.321 6 . 9 722.6234.0039.7045.201 6 9 . 7

X j (data)0.8280.7040.6220.5650.4990.4820.4700 .443

X j (model)0.828330.705410.621470.564330.498980.481100.469340.44330

x 2 (data)0.07370.11300.13220.14000.14680.14770 .1 4 7 70.1476

x 2 (mo de l )0.07380.11220.13130.14070.14730.14810.14840.1477

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306 Chapter 16

I f in s t ead of the iden t i ty mat r ix , we use Q,=diag(y](t i)~ 2, y2(tj)~ 2) a s a t imev a r y i n g w ei gh t i n g m a t r i x , w e arr ive a t k*=[355.55, 402.91]T w h i c h is q u i t e closeto k *= [ 3 5 4 .6 1 , 400.23]1 obta ined wi th Q=I.T h i s choice of the w e i g h t i n g matr ixassum es tha t the er ro r in the m easured concentra t ion i s p ropor t ional to the va lue ofthe measured var iable , wh ereas t he cho ice o f Q=I as sume s a cons tant standarderror i n th e m easu rem en t . Th e paramete r es t imates are essential ly the same b e-cause the measurements o f x , and x 2 do no t dif fer b y m o r e than one order of m a g -nitude. In general , the com puted unce rtainty in the parameters is expected to havea h i g he r dependence on the choice of Q . T he standard est imat ion error of k, al-m o s t doub l e d from 0.642 to 1.17, while that of k2 increased mar g i n a l l y from 3.43to 3.62.

io6

10s

k2*/?2

10 -

io3 iob10 I02 I03k,/XHgnre 16.2 Benzene Dehydrogenation: Region of convergence of the Gauss-Newton method (a ) standard Gauss-Newton method (4* orderRunge-Kut ta for integration); (b) G-N method w i th optimal step-sizepolicy (4 lh order Runge-Kut ta fo r integration); (c) G-N method withoptimal step-size policy (DVERK for integration); (d) G-N methodwith optimal step-size policy (DCEAR fo r integration); (e) G-Nmethod with optimal step-size policy and use of information index

(DVERK fo r integration). Test points for G-N method with optimalstep-size policy and u se of information index (DGEAR for integra-tion) where convergence was obtained are denoted by +. [reprintedfrom Industrial Engineering Chemistry Fundamentals with permis-sion from th e American Chemical Society].



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Parameter Estimation in Chemical Engineering Kinetics Models 30 7

T h e reg ion o f c o n v e r g e n c e ca n b e s u b s t a n t i a l l y e n la rg e d b y i m p l e m e n t i n gt he s imple b i se c t ion r u l e fo r s tep-s ize con t ro l o r e ve n be t te r b y i m p l e m e n t i n g th eo p t ima l step-size policy described in Section 8.7.2.1. Fur t he rmore , it is quite im-portant to use a robus t integrat ion r ou t i ne . Ka logerakis and L u u s (1983) fo u n d thatthe use of a s t i f f differential equation solver (like I M S L rou t in e D G E A R w h i c huses Gear ' s method) resul ted in a large expansion of the region of co nv e rgencecompared to I M S L r o u t in e D V E R K (us ing J.H. V e r n e r ' s R u n ge-K u t ta f o rm u las of5* and 6 th order) or to the s imple 4 th order Runge -Kut t a meth o d . A l th o u gh th i sproblem is nonst i f f at the o p t i m u m , this may not be the case when the initial pa-rameter est imates fa r f rom the opt imum and hence , use of a stiff ODE so lver i sge n e r a l l y b e n e f i c i a l . T h e effec ts of the step-size p o l i c y , u se o f a r o b u s t i n t e g r a t i o nrout ine and use of the informat ion i nde x on the region of convergence are show nin F i g u r e 16.2.A s seen in F igure 16.2, the effect of the in fo rmat ion index and the in tegra-t ion routine on the region of is very s ignif icant . It should be noted tha t use ofD G E A R or any other stiff dif ferential equat ion so lver does n o t r eq u i r e extra pro-g r a m m i n g effort since an ana lyt ica l express ion for the Jacobean (9f r/3x)T is alsorequired by the Gauss -Ne w t on me t hod . Ac tu a l ly qui te often th is also resu l ts insavings in co mp u te r t i m e as the parameters change f rom iteration to iteration andth e sys tem O D E s c o u l d become stiff a n d t he i r i n t e g r a t i o n r e q u i r e s ex c es s i v e com-p u t e r effor t b y n o n s t i f f so lve r s (Ka l o ge r ak i s an d E u u s , 1983). In al l the c o m p u t e rp ro grams fo r ODE mo de l s p ro v ided in A p p e n d i x 2, w e have used D I V P A G , t h elatest I M S L integration routine that e m p l o y s Gear ' s m e t h o d .

16.3.3 Catalytic Hydrogenation of 3-Hydroxypropanal (HPA) tol ,3 -Propanediol (PD)T h e h y d r o g e n a t i o n o f 3 - h y d r o x y p r o p a n a l ( H P A ) to 1 , 3 - p r o p an e d i o l (PD)over Ni/SiO 2/A l2O 3 catalyst p o w d e r w as studied by Professor H o f f m a n ' s gro up atthe F r i ed r i ch -Alex an der Univers i ty in E r la g e n , G e rm a n y (Zhu et al., 1997). PD isa potential ly at t rac t ive monomer f o r p o ly mers like p o ly p ro p y len e terephthalate .

Th ey used a batch st irred autoclave. The e xpe r i m e n ta l data w e re kindly providedby Professor Hof f ma n and consist o f m e a s u r e m e n t s of the concentrat ion of HPAan d PD ( C H p A , C P D ) versus t ime at various operat ing t empera tures and pressures.T h e s a m e group also proposed a reac t ion scheme and a math emat ic a l mo de lthat descr ibe th e rates of HPA c o n su mp t io n , PD format ion as we l l as the forma-t ion o f acrole in (Ac). T h e m o d e l i s a s f o l l o ws

(16.29a)

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308 Chapter 16

dC P Ddt (16.29b)

d C A cdt = r, - r, - r_ (16.29c)

w he re Q is the concentrat ion of the catalyst (10 g/L). T he in i t ia l condi t ions in allex p e r imen ts w e re C Hp A (0)=l .34953, C P D (0) = 0 and C A c(0)=0.T he kinetic express ions for the react ion rates are given next,

HK , P

H 'HPA(16.30a)

r - , = - k 2 C p D C H P AK , P 0 .5 (16.30b)

r =

r4 = k 4 C A c C H P A

(16.30c)(16.30d)(16.30e)

In the a b o v e e q u a t ion s , k, ( j=l , 2, 3, -3, 4) are rate cons tants (L/(mol min g)),K I and K 2 are the adsorption equi l ib r ium constants (L/mol) for H 2 and HPA re-spectively; P is the hydrogen pressure (MPa) in the reactor and H is the H e n r y ' slaw c o n s t an t w i t h a va lu e e q u a l to 1379 ( L bar/mol) at 298 K . T h e s ev en p a r a m e -ters (k], k2, k3, k _ 3 , k4, K ] an d K 2) are to be de t e rmine d from the me asure d concen-t ra t ions of HPA and PD versus t ime . In this e xample , w e shal l cons ider only thedata gathered at one isotherm (318 A T ) and three pressures 2.6, 4.0 and 5.15 M P a .T he ex p e r imen ta l data are given in Table 16.23.In this ex amp le th e n u mb er o f measu red var iables is less than the n u m b er o fstate var iab les . Zhu et al . (1997) m i n i m i z e d an unwe i g h te d su m o f squares of de-viations of calculated and e xpe r i m e n ta l concentra t ions of HPA and PD. Th ey usedM arq u a rd f s modi f ica t ion of the Gauss-New ton m e t h o d an d reported the parame t e rest imates sh o wn in Table 16.24.



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Table 16.23 HP A Hydrogenation: Experimental Data C ollected at 318 Kand Pressure 2.6, 4.0 and 5.15 M PaP r e s s u r e( M P a )

2.6

4.0

5 . 1 5

Time( m m )

1 02030405060801001201401601802001 02030405060801001 2 01401601802001 02030405060801001 2 0140160

CHPA( m o l / L )1.373951 . 2 5 8 2 11.187071 . 1 3 2 9 21.035560.9613390.7344360.5645510.3743850.2147990.1009760.03641920.005308921.32951 . 3 1 1 5 71.228281.0870.9945390.8118250.6009620.3863020.2042220.07823040.02777080.003162960.002108641.363241.258821 . 1 7 9 1 80.9721020.8252030.6971090.4214510.2322960.1280950.02898170.00962368

CPD( m o l / L )0.00.01971090.06425760.1363990.2386330.3045990.4923780.7323260.8872541.042841.173061.257691.260320.002628120.05256240.1207360.2413930.3848880.46820.7731930.9908021.149541 . 2 81.291.301.300.002628120.07003940.1843630.3540080.4697770.6073590.8524311.035351 . 1 6 4 1 31.300531 . 3 1 9 7 1



26/37

310 Chapter 16

U s i n g t h e F O R T R A N p r o g r a m g i v e n in A p p e n d i x 2 a n d s tar t ing w i t h th evalues given by Zhu e t al. (1997) as an init ial guess, the LS object ive function w asc o m p u t e d u s i n g th e iden t i ty mat r ix as a w e i g h t i n g m a t r i x . T h e L S ob ject ive func -t ion w as 0.26325 and the cor r e spond i ng c o n d i t i o n n u m b e r o f ma t r ix A w a s0 . 3 4 5 x l 0 1 7 . I t should b e noted that since the parame t e r values appear to differ b yseveral orders of magn i tu de , w e used th e formula t ion with the scaled matr ix Adiscussed in Section 8.1.3. H e nce , the magni tude of the comput e d condi t i on num -ber o f matr ix A is solely due to i l l - condi t ioning of the p roblem .Indeed, the i l l -condi t ion ing of this p ro b lem i s qui te severe . Usin g programB a y e s _ O D E 3 and using as init ial guess th e parame t e r values reported by Zhu et al .(1997) w e were una b l e to converge . At th is po in t it should be emphas ized that atight test of conve rge nce should b e used (NSIG=5 in Equat ion 4 .11) , o therwise i tm ay a ppe a r tha t the a lg o r i t hm h as c o n v erged . In t h i s t e s t M ar q u a r d t ' s pa r a m e te rwas ze ro and no pr ior informat ion was used (i.e., a ze ro was entered into th e com-pute r program for the inverse of the variance of the pr io r paramete r es t ima tes ) .In prob le ms like th i s on e whi ch are very diff icul t to converge , w e should u seM a r q u a r d t ' s modificat ion first to reduce the LS objective funct ion as mu c h a s pos-s ib le . T h e n w e c a n approach c loser to the g lob a l m i n i m u m in a sequential way byletting only one or two parameters to vary a t a t ime. T he es t imated s tandard errorsfo r the pa r a m e te r s p r o v i d e e xce l l e n t in fo rmat ion on w ha t t he n e x t step shou ld b e .For example i f we use as an ini t ial guess k;=10 for all the paramete rs and a con-stant Marquard t ' s pa rame t e r y=10~ 4, the Gauss-Newton i terates lead to k=[2.6866,O. lOSxlO '6 , 0 . 6 7 2 x l ( T , 0 .68xlO" 5, 0.0273, 35.56, 2.57]T . This cor responds to as ignif icant reduction in the LS Objective funct ion f rom 40.654 to 0.30452. Subse-quent ly , us ing th e last est imates as an init ial guess and using a smal ler va lue fo rM a r q u a r d t ' s directional p a ramete r (any value in the range 10"5 to 10"14 yie ldeds imi la r results) we arr ive at k=[2.6866 , 0.236x10" 8, 0.672x10" 3, 0.126xlO" 5,0.0273, 35.56, 2.57]T . The corresponding reduc t ion in the LS objec t ive funct ionw as rather m a r g i n a l from 0.30452 to 0.30447. A ny further at temp t to ge t closer toth e globa l m i n i m u m us i ng M a r q u a r d t ' s modif ica t ion w as unsuccess fu l . T h e valuesfo r Marquardt ' s parameter were varied from a va l ue e qua l to the smal les t e igen-va lue all the w ay up to the third largest e igen v a lu e .

A t th i s p o in t w e swi tched to our sequential approach. First we examine theest imated standard errors in the parame t e r s obtained us i ng M a r q u a r d t ' s modif ica -t ion. These were 22.2, 0 . 3 7 x l 0 8 , 352., 0 . 3 x l 0 7 , 1820, 30.3 an d 14.5 (%) for kbk2, k3, k _ 3 , k4, K I an d K 2 respectively. Since K 2 ,k b K ] an d k3 have the smal les tstandard errors , these are the parameters we should try to o p t imiz e first. Thus ,le t ting o nly K 2 change , the Gauss-Newton meth o d converged in three i tera t ions toK 2=2.5322 and the LS object ive funct ion was r edu c ed from 0.30447 to 0.27807.In these r uns , Marqua rd t ' s parameter was set to zero. N e x t we le t k{ to vary .Gauss-Newton me t hod conve rge d to k[= 2.7397 in three i terations and the LSobjective funct ion w as further reduced from 0.27807 to 0.26938. Next w e opti-m i z e K 2 an d k, together. This step yie lds K 2=2.7119 an d k|=3.0436 with a corre-sp o n d in g L S object ive func t ion o f 0.26753. N e x t w e opt imize three paramete rs

http://dk5985_appendix2.pdf/http://dk5985_appendix2.pdf/


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K 2, k} an d K j . U s i n g as M a r q u a r d t ' s p a r a m e t e r y = 0 . 5 (s ince the sma l l e s t eigen-value was 0.362), th e Gauss -Ne w t on me t hod y i e l d ed the f o l low in g parameter v a l -ues K 2=172.25, k.!=13.354 and K.^4 .5435 wi th a corresponding reduc t ion of theLS objective func t ion from 0.26753 to 0.24357. The corresponding standard errorsof est imate were 9 . 3 , 3.1 and 3.6 (%). Nex t starting from our curren t best p a r a m e -ter values for K 2, k] and Kh w e inc lude another paramete r in our search. Based o nour earl ier calcu la t ions th i s should b e k3 . Opt imiz ing s i m ul t a ne ous l y K 2, kb K ]and k3, w e obta in K 2=191.30, k,=13.502, K,=4.3531 an d k3=.3922x!0 '3 r edu c in gfur ther the LS objective funct ion to 0.21610. A t th i s po i n t we can try al l sevenparameters to see w h e t h e r we can reduce the object ive funct ion an y further . U s i n gMarqua rd t ' s modif ica t ion with j t aking values from 0 al l the way up to 0 . 5 , therewas no further i m p r o v e m e n t in the per formance index . S imi lar ly , there was noi m p r o v e m e n t b y o p t i m i z i n g f ive or s ix pa r a m e te r s . E v e n w h e n four pa r a m e te r sw e re t r ied with k} instead of k3, the re was no further improve me nt . Th us , w e c o n -c l ude that w e have reached the global m i n i m u m . A ll the abo v e steps are s u m m a -rized in Table 16.24. T he fact that we have reached the global m i n i m u m was veri-fied by start ing from wide ly different initial guesses. F or ex amp le , starting withth e paramete r values reported by Zhu e t al. (1997) as init ial guesses, we ar r ive a tth e same va l ue of the LS objective funct ion.At th i s p o in t i s wo r th wh i l e c o mmen t in g o n th e comput e r standard est ima-t ion er rors of the p a r a m e t e r s also s h o w n i n T a b l e 16.24. As seen in the last fou rest imat ion runs we are at the m i n i m u m of the LS object ive funct ion . T he p a r a m e -te r est imates i n th e ru n wh e re we opt imized four on l y paramete rs (K 2, k, , K , & k3)have the smal les t standard error of est imate. Th is is due to the fact that in the c o m -putation of the standard errors, it is a s s u m e d that all other parameters are knownprecisely. In a l l subsequent r u n s b y in t roduc ing addi t ional parameters th e overalluncertainty increases and as a resul t the standard error of all the parameters in -creases t o o .Final ly , in Figure s 16.3a, 16.3b and 16.3c w e present the e xpe r i m e n ta l datai n graphica l form as w el l as the m o d e l ca lcula t ions based on the parame t e r va lue sreported by Zhu et al . (1997) and from th e parameter es t im ates de t e rmine d here ,n a m e l y , k*=[13.502, 0 . 23 6 x lO '8 , 0 .3922xlO'3 , 0 .126xlO' 5 , 0.0273, 4.3531,191.30]T . A s seen, the di f ference be tween th e two m o de l ca lcula t ions is very sm a l land all the gains realized in the LS objective funct ion (from 0.26325 to 0 .21610)produce a sl ightly better match of the HPA and PD transients at 5.15 MPa.Zhu et al. (1997) have a lso reported data at two other t emperatures , 333 Kand 353 K. The determinat ion of the parameters a t these t e mpe ra t u re s is l e f t as anexerc i se fo r the r e a de r ( s e e Sect ion 16.4.3 for detai ls) .



28/37

Co

aomoL

Co

aomoL

C

aomoL

6'T

So -

o

b-a*

So^VD aO a K3 "TTO as

CD

"a^

Oo oo eo oo

CD

Oo oo Oo OO



29/37

Table 16.24 HP A Hydrogenation: Systematic Estimation of Parameter Values Using th e Data C ol lected at 318 K

In i t i a l G ue ssG-N Marquardt ,y = 1 0 '4G - N Ma rq u a rd t ,r = i o - 1 0

O p t i m i z i n gK ,Op t imiz in g

k,Op t imiz in gK 2 & k ,O pt i m i z i n gK 2, k, & K |O p t i m i z i n g

K 2, k , , K , & k,O pt i m i z i n gal l except k 2 , k _ 3O p t i m i z i n gall except k2

Op t imiz in g a l l 7p ar am e te r sParametersrepor ted by Zhue ta l . ( 1 9 97)

k ,10"

2.68662.6866

22.2%#

2.73973.043613.3543.1%13.5028.8%13.5029.5%13.502

21.1%13.502

21.1%6.533

0.045

k.10"

0.108x10-"0.236x10'8

0.37x1 08%*******

0.236x10'8+0.6x1 0 8%3.048xlO' 41.07x10""

k .i10"

0.672x1 0"30.672x10"

352%#

*

*

*

0 . 3 92 2 x10"135%

0 . 3 92 2 x10"226%0.3922x10"

445%0.3922x1 0' 3547%6.233x1 0'6

k - 310 '3

0.68x1 0'50.126x10" s 0 . 3 x l 0 7 %

******

0.126xlO~ 50.6x1 0 6%0 . 1 2 6 x l O ~ 50.4x1 0 7%7.21910~ 4

k4i o - 3

0.02730.0273

+ 1820%*****

0.02731452%0.02732535%0.02732857%

3.902X10'6

K ,10"

35.5635.56

+30.3%***

4.54353.6%4.35319.8%4.353110.9%4.3531

12.7%4.3531

12.7%95.00 1 . 2 8

K 2i o - 3

2.572.57

14.5%2.5322

*2.7119172.259.3%191.308.5%191.30+8.8%191.30

18.0%191.3019.4%3.227

0.033

LS Object.Fu n c t io n40.654

0.304520.304470.278070.269380.267530.243570.216100.216100.216100 . 2 1610

0.26325

cond(

0 . 6 2 x lO . l l x l0 . 1 2 x l0 . 1 2 x l

0 . 3 4 x l: Parameters with a star are assumed to be known and have a cons tant value equal to the one determined in the previous run.



30/37

314 Chapter 16

16.3.4 Gas Hydrate Formation KineticsG as h y d r a t es are n on - s t o ich iome t r i c crysta ls f o r m e d b y t h e e n c losu re o fm o l e c u l e s l ike m e t h a n e , ca rb on dioxide an d h y d r o g e n sulf ide inside cages f o r m e db y hydroge n-bonde d w at e r mo lec u les . There are m o r e than 1 00 c o mp o u n ds

(guests) that can com bi ne wi th water (host) an d form hydrates. Formation of gashydrates is a problem in oil and gas operat ions because it causes plugging of thep ipel ines an d other facilities. On the other ha nd natura l m e t h a n e hydrate exists invast quant i t i es in the ear th 's crust and is regarded as a future energy resource .A m e c h a n i s t i c m o d e l for the kine t ics of g a s hyd ra t e fo rmat ion w as proposedb y E n g le zos e t a l . (1987). T h e m o d e l c o n t a i n s o n e adjus tab le pa r a m e te r fo r eachgas hydrate f o r m i n g substance . The parameters fo r me t hane an d e thane were de-t e rmin e d from e x p e r i m e n t a l data in a semi-ba tch agi ta ted gas - l i q u i d vesse l . D u r i n ga t y p ic a l e x p e r i m e n t in such a ve sse l o n e m o n i t o r s th e rate o f m e t h a n e or e thanegas c o n su mp t io n , the t emperature and the pressure . G as hydrate fo rmat ion is acrystal l ization process but the fact that it occurs from a gas- l iquid system u n derpressure m a k e s it diff icul t to m e a s u r e and mo n i to r in situ the par t icle size an d par-ticle size distr ibution as w el l as the concent ra t ion of the m e t h a n e or ethane in thewater phase .Th e ex p e r imen ts w e re conduc ted at four d ifferent tempera tures fo r each gas.A t each t emp era tu re e xpe r i m e n t s w e re per formed at dif ferent pressures. A total of14 and 1 1 ex p e r imen ts were p er f o rmed for m e t h a n e and e thane respec t ive ly .Based on crysta l l izat ion t he o ry , and the two fi lm theory fo r gas- l iqu id m a s s t rans-fe r En g lez o s et al. (1987) fo rmula ted f ive different ia l equat ions to describe th ekinet ics of hydrate formation in the vesse l and the associate m a s s t ransfer rates.T he g ove r n i ng O D E s are given next .

d f b H D * y a eqd t c w 0 y , s i n h y h cw0; f b ( t 0 ) = f e q (16.31b)

(16.31c)dt "

; H i ( o ) = n ? (16 .31d)Copyright 2001 by Taylor & Francis Group, LLC


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Parameter Estimation in Chemical Engineering Kinetics Models 3 1 5

dt (1 6 .3 1 e)

T he first equat ion gives th e rate of gas c o n su mp t io n as m o l e s of gas (n) ver -su s t i m e . This is the only state variable that is me asure d . T he ini t ial n u m b e r o fmole s , nO is k n o w n . T he in t r ins ic rate constant , K .* is the only u n k n o w n m o d e lparame t e r and i t ente rs the first m o d e l equa t ion th rough the Hatta n u mb er y . Th eHat ta num ber i s g iven by the fo l lowing equat ion

(16 .32)T he other state variables are the fugaci ty of disso lved m e t h a n e in the b u l k of

the l iquid water phase ( f t , ) and the zero, first and s e c o n d m o m e n t of the part iclesize distr ibution (u 0 , ( i j , ^ 2 ) - T he init ial value for the fugaci ty, f j , is eq u a l to theth ree phase equ i l ib r ium fugaci ty f e q . T he in i t ia l n u m b e r o f particles, ji0, o r n u c le iini t ial ly formed w as calculated f rom a m a s s b a lan c e of the a m o u n t of gas con-sum ed at the turbidi ty poin t . The explanat ion of the o ther variables and parametersas w el l as the in i t ia l condi t ions a re described in detail in the reference. T he e qua-t ions are given to i l lustrate the na ture o f th i s p a ramete r est imat ion problem w i t hfive O D E s , o n e kine t ic p a r a m e t e r ( K * ) an d o n l y o n e m e a s u r e d state va r i a b le .

0.130

C H . isotherm at 276 KT O 25 40 55 70 85 10 0

time (mtn)Figure 16.4: Experimental ( j and fitted (-----) curves for the methanehydrate formation at 279 K [reprinted from Chemical EngineeringScience with permission from Elsevier Science].



32/37

316 Chapter 16

0,055

C,H 6 isotherm at 279 K5 15 25 35 45 55 65 75 85

time (min)Figure 16.5: Experimental ( j and fitted (-----) curves for the ethane hy-

drate formation at 279 K [reprinted from Chemical EngineeringScience with permission from Elsevier Science].T he Gauss-Newton m e t h o d with an opt imal step-size pol icy an d M a r -quardt ' s modi f ica t ion to ensure rapid convergence was used to match the calcu-lated gas c o n s u m p t i o n curve with the me asure d one. The five state an d f ive sensi-t ivi ty eq u a t i o n s were integrated u s in g D G E A R (an I M S L r ou t i ne fo r in tegra t ion o fs t i ff d i fferent ia l e q u a t ion s ) . In i t i a l ly , p aramete r es t imat ion w as pe r fo r m e d fo r eachexper iment but i t was found that for each isotherm the pressure dependence w asnot statistically signif icant . C o n s e q u e n t l y , each i so thermal set of data for each gasw as treated s i mu l t an eo u s l y to obtain the op t i m a l parame t e r value. The experi-men ta l data and the cor responding model calculations fo r m ethane and e thane gashydrate fo rmat ion are sh o wn in Figures 16.4 and 16.5.

16.4 P R O B L E M S W I T H ODE MODEL ST he f o l low in g p r o b l e m s w e re fo rmula ted wi th data from the l i tera ture . Al-though the in fo rmat ion pr ov i de d here is suff icient to solve the parame t e r est ima-

tion p ro b l em, th e reader i s s t rongly recommended to see the papers in order tof u l l y c o m p r e h e n d th e re levant p h y s i c a l an d c h em i c a l p h e n o m e n a .



33/37


16.4.1 Toluene HydrogenationC o n s i d e r th e f o l l o w i n g r eac t ion s c h e m e

r, r2A B > C

w h e r e A is t o lu e n e , B is 1 - m e t h y l - c y c l o h e x a n e , C is m e t h y l - c y c l o h e x a n e , r , is theh y d r o gen a t i o n rate ( fo rward r eac t i o n ) an d r _ ] th e dispropor t ionat ion rate (b ac k w ar dreac t ion) . Da ta are a va i l a b le f rom Be lohlav e t a l . (1997) .T h e pr opose d kine t ic m o d e l d es c r i b i n g t h e above sys tem is g i ve n n e x t (Be-loh lav et al . 1997):A =-r,+r., ; C A ( 0 ) = 1 (16.33a)dt

dCBdt

dC cdt

T he rate eq u a t io n s are as fo l lows

= r , - r _ i -r2 ; CB(0) = 0 (16.33b)

= r2 ; Cc(0) = 0 (16.33c)

K |_| Pv A CAr, =-H A A , (16.34a)v rel r _ i _ r _ L v rel r

-i- (16.34b)

+C(16.34c)

w he re C j ( i= A, B, C) are the reactant concentra t ions and K 'e l the relat ive adsorp-tion coeffic ients .T he h y dro gen a t io n of t o l ue ne w as p er f o rmed at a m b i e n t t emperature andpressure in a semi-batch i so thermal stirred reactor with comme rc ia l 5% Ru-ac tcatalyst . Hydrogen w as automat ica l ly added to the system at the same ra te atw h i c h i t was c o n su m ed . Part icle size of the catalyst used and eff iciency of stirring



34/37

3 1 8 Chapter 16

w e r e su f f ic ie n t for ca r ry ing ou t the react ion in the k i n e t i c r e g i m e . U n d e r t h e e x -p e r i m e n t a l c o n d i t i o n s val id i ty o f H e n r y ' s l aw w as a s s u m e d . T h e data a re g i v e nbelow in Table 16.25.Yo u a re asked to use the Gauss -Ne w t on meth o d an d de te r m i ne th e p a ram e-ters kH , kD , k2, K A - T C I , and K c _ r e i as w e l l as their 95% c o n f i d en c e intervals .F or com par i son purpose s , it is no t e d t ha t Belohlav et a l. (1997) rep orted th efo l lowing p a r a m e t e r est imates : kH=0.023 miri1 , kD =0.005 miri1 , k2=0.011 miri1 ,K A .re l=1.9, andK c . r e i= 1 . 8 .Table 16.25 Data for the Hydrogenation of Toluene

i ( m i n )015304560759012018024 03203603804 0 0

CA1.0000.6950.4920.2760.2250 .1630 .1340.0640.0560 . 0 4 10 . 0 3 10.0220.0210.019

C B0.0000.3120.4300.5750.5700.5750.5330.4620.3620 . 2 1 10 . 1 4 60.0800.0700.073

Cc0.0000 . 0 0 10.0800 . 1 5 10 . 1 9 50.2240.3300 . 4 7 10.5800.7470.8220.8980.9090.908

Source: Be lohlav e t a l. (1997).

16.4.2 Methylester H ydrogenationCo n s ide r th e f o l l o wi n g reaction s c h e m e

A -> B -> C Dw h e r e A, B, C and D a re the m e t h y l esters of l ino len ic , l inole ic , ole ic and stearicacids and r , , r2 an d r3 are the hydr og e na t i on rates . T he propose d kinet ic m o d e lde sc r i b i ng the abov e sys tem is g i ve n ne x t (B e l o h l av et al. 1997).



35/37

Parameter Estimation in Chemical Engineering Kinetics Models 3 1 9

d C Adt

dCBdt

dCcdt

dCDdt

= -![ ; CA(0) = 0 . 1 0 1

= r, -r2 ; C B ( 0 ) = 0 . 2 2 1= r2-r3 ; Cc(0) = 0.657

= r3 ; CD(0) = 0 .0208

T he rate eq u a t io n s are as fo l lows

rl =C C C + K D C D

k 2 K . B C BB C B + K C C C + K D C D

r3 =-

( 16 . 3 5 a )

(16.35b)

(16.35c)

(16.35d)

rel r a. vrel rel

(16.36a)

(16.36b)

(16.36c)

Table 16.26 Data for the Hydrogenation of MethylestersT (min)01 01 41 92434691 2 4

CA0.10120.01500.00440.00280.00290.00170.00030.0001

CB0.22100.10640.04880.02420.00150.00050.00040.0002

Cc0.65700.69410.63860.53610.39560.21880.02990.0001

CD0.02080.19770.30580.44440.60550.78080.96800.9982

Source: Belohlav et al. (1997).Copyright 2001 by Taylor & Francis Group, LLC


36/37

320 Chapter 16

w h e r e Q ( i = A , B, C, D) are the r e a c t a n t c o n c en t r a t i o n s an d K le | the re la t ive ad -sorption coeff ic ients .T he e xpe r i m e n t s were pe r fo r m e d in an autoclave at elevated pressure andt emperature . The Ni catalyst D M 2 w a s use d . T he data are given below in Table1 6 . 2 6 ( B e l o h l a v e t a L , 1997).T he object ive i s to de term ine the p a ramete r s kt, k2, k3, K A .re i, K B . r ei , K C- ie i,K D.rei as w e l l as their standard errors . It is noted tha t K A . r d = l . Belohlav e t al.(1997) reported the f o l low in g parame t e r es t imates : ki =1 . 44 miri1 , k2=0.03 min',k3=0 . 0 9 mm1, K B . re l=28.0, Kc.ld=1.8 and K D .rc ,=2.9.

16.4.3 Catalytic Hydrogenat ion of 3 -H ydroxypropanal ( H P A) to1,3-Propanediol (PD) - Nonisothermal Data

L e t u s r ec o n s i d e r t h e h y d r o gen a t i o n o f 3 -h y d r o x y p r o p an a l ( H P A ) to 1,3-propanedio l (PD) over Ni/SiO 2/A l2O 3 catalyst p o w d e r that used as an exampleear l ier . For the same math emat ic a l model of the system you are asked to regresss i mu l t an eo u s l y the data provided in Table 16.23 as we l l as the addi t iona l datagiven here in Table 16.28 for e xpe r i m e n t s pe r fo r m e d at 60C (333 K) and 80C(353 K ). O b v i o u s l y an A r r h e n i u s t ype r e l a t i o n s h i p m u s t b e used in th i s case. Z h uet al. (1997) r epor ted pa r a m e te r s for the a b o v e c o n d i t i o n s an d they are shown inTable 16.28.

Table 16.27 HPA Hydrogenation: Estimated Parameter Valuesfrom the Data Collected at 333 K and 353 K

k,k2K ^

k-3k 4K ,K 2

Paramete r Es t ima tes Standard Er rorT = 3 3 3 K

19.540.102

1. 537x10 0.27x103 . 0 5 8 x l O '4 0 . 7 5 x 1 0 1.788xlO" 3

0.1 2 0 x 1 0 7.515xlO" 50.90x10

120.01.053.0290.013

T = 353 K52.84

0.4011.475

0 . 1 2 x l O " 22 . 0 6 6 x 1 0

0 . 0 3 6 x 1 0 5 . 3 1 5 x 1 0

0. 025x 1 02.624x10

0. 023 x 1 0160.01.732.7670.021

Source: Zhu et al . (1997).Copyright 2001 by Taylor & Francis Group, LLC


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Parameter Estimation in Chem ical Engineering Kinetics Models 321

Table 16.28 HP A Hydrogenation: Experimental Data Collected at 333 Kand 353 KE x p e r i m e n t a lC o n d i t i o n s

5.15 MPa&

3 3 3 K

5 AS MPa&353 K

4.0 MPa&3 3 3 K

Time( m i r i )0.01 020304050600.051 01 52025300.01 02030405060

CHPA( m o l / L )1.349531.08540.6630430.31655410.009822250.005158740.00206351.349530.8735130.447270.1409250.03500760.01308590.005815971.349531 . 1 3 8 7 60.7855210.4484020.1910580.05300740.0199173

CPD

( m o l / L )0.00.1964520.6023650 . 8 5 1 1 1 71.169381.247041.248360.00.3885680.8160320.9670171 . 0 5 1 2 51.082391.120240.00 . 1 5 1 3 8 00.5583440.8671481.125361.291.32

Sowrce: Zhu et al. (1997).

Engineers (Chemical Industries) by Peter Englezos [Cap 16]

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