Constrained and Weighted Least Squares Procedures for Parameter Estimation
Modified, ittrative weighted linear least squares procedures are proposed for parameter estimation in nonlinear kinetic models. Significant constraints are incorporated in two approaches that do not require replicate data. The techniques have been developed and applied in an experimental modeling study of the methanation synthesis reaction. A comparison of these results with those from a nonlinear least squares analysis shows that the parameter estimates from the proposed methods correspond closely to the best nonlinear treatment.
150 Ind.
Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
appro1)ri:itely constminecl a t each step. Consider, for example, the iioiiliiienr niodel
Table I. Various Combinations of Unconstrained Model Forms for Equation 9 Three-l)arariieter combiliatioil
(5) and its liiicnrized form Two-parameter cotiihiiiatioiis 1- 1 1 K.A.X* r i ; ' D + h.3-
Ind. Eng. Chem. Fundom., Vol. 10, No. 1, 1971
151
Table 11. Comparison of Weighted Parameter Estimates for Equation 8 Values used to generate data Recovery methods Hill (1968) (iterative without replicate data) Hill (1968) (iterative with replicate data) This work (iterative T$ ithout replicate data)
ki
KA
KB
5 00
0 10
0 30
7 22
0 08
0 26
5 05
0 10
0 30
5 08
0 10
0 30
aiice 01 = 1. However, in his puraiiieter e5timatioii study, IIill considered the value of LY ai: uiikiion-ii and tried to recover it, both with aiid without replicate data. -1conip:irisoii of results obtaiiied by these methods with t h o v of this study, showi i n Table 11, indicates t h a t the iterative weighted tecliiiique propoqed in thiq study ljrovides excellent parnmcter ertiiiiates when the correct model is einploytd rvitliout requiriiig rel~licate data.
Application to Experimental Data
The succes of the iterative weighted procedure iii the simulation study suggested its use in experimentation. For this purpose, data were obtaiiied from the investigation of Wentzheimer (1969), who studied the rnetliaiiatioii reactioii over a stainless steel catalyst a t four temperatures over a range of reactor pressurei. Various mechanistic schemes which involved a d ~ ~ r p t i o i idesorption, , or surface reaction mtecoiitrolliiig s t e p were postulated (Vallerscharnp, 1969) and Lniig.niuiur-Hiiislicl\~oodmodels developed in the usual way (Hougeii and Katsoli, 1947; Walas, 1959). The paraineters of the 11 modpls in Ta1)le I11 were then estimated by both nonlinear least squares slid weighted linear least squares aiialysi.. -1first test \vas run 011 tlic noiiiterntive riiethod of 1Iezaki. 111 thix xy,+m some of the six ~~araiiieters were always estiin:itctl to I)e iiegitivc. regurdlcs.; of which iiietliaiiatioii niodel \va* xii:il>-zcd. T o reniedy thix, :i secoiid te.;t, was run on t,lie saiiic h t a , using the uiicoii.straiiied iterative weighted procedurc tli:it ~ v n . bo ,qatiqfnctor\- iii the siniulatioii study. Agaiii, negative estimates were found for one or iiiore i)ararii-
Table 111. Kinetic Models for Methanation Reaction h1ODEL
1
MODEL3
MODEL8
152
Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
Table IV. Comparison of CWSA and Nonlinear Least Squares Results at 880°F k,
KH?
Kco,
M o d e l No.
Method“
G-Mole/Cm2-Hr
PSIA-’
PSlA-’
I
CWSA sLS CWLI NLS CWSA SLS CJVS-1 KLS CWSA SLS C\VS:l
2.62 x 0.67 x 6.78 X 6.79 X 2.84 x 2.88 x 2.21 x 2.35 x 5.18 x 5.18 x 5.18 x 5.18 x 4.97 x 4.97 x 4.97 x 4.97 x 3.06 x 3.06 x 6.00 x 6.00 x 3.63 X 3.63 x
2 3 4
5 6
KLS
c:n-s;l SLS c11-S.s
7 8
KLS CWS1 SLS CWSAi
9
IO
SLS
cn-aA
11
913
10-7 lo-’ loe8 lo-* 10-7 10-7 lo-’ lo-‘ 10-6 10-6 10-6 10-8 low6 10-6 10-6 10-j 10-j lo-; lo-: lo-” 10-3
0.0 0.0
KW, PSIA-’
0.002 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0 0 0.0 0.0 0.0 0.0 0 0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.034 0.034 0,400 0,400 0 177 0.177
0.119 0.119 0.052 0,082 0.084 0 . 084
0.0 0.0 0.0 0.0 0.0 0.0
0,002 0,001 0.034 0.024
0.0 0.0 0.0 0.0 0.0 0.0 0.85 0.85
x x 0.85 x 0.85 x
Khf,
PSIA-’
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.93 x 1.93 x 1.93 x 1.93 x
10-4 10-4 10-4
10-3 10-3 10-3 10-3
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 0 0.0
Sb
4.93 4.77 4.85 4.77 5.09 4.78 5.11 4.80 2.68 2.68 2.68 2.68 2.69 2.69 2.69 2.69 1.68 1 68 1.65 1.65 1.65 1.65
X I S . Noiiliiiear least sqiiares aiinlysii. of rqiiare~i g - ~ n o l e ~ , ’ c m ~ - h r ) ~ . I < c , oe-timated ~ t o tie zero for all models. Siiiii
c t ( w i i i each of the 11 inotlcls, aiid n i n rcsiilt the CV’PA \vas :il)ldietl. Typical results of the latter I)roccdurc :ire prcsciitcd i i i ‘Yiilile IV>n-here a comi)aiisoii is t i i d c to tlic l):iraiiietcr cqtiiiiatcb :iii(I z u n i of square v:ilue> derived from n subscyiiciit noiiliiiear lrnzt .squnreq aiialy.:is. The l):iraiiictcr estiniatc~z from tlic CKSX n i i d the iioiiliiicar nnalysis :ire t~ln.nysvery close aiid iii 3 1 i n i i ~cases ~ itleiiticd. 130th inetliods :ilw piuvitle only Iio-itivc estiiiiatc.: of the 1)ar:iiiietcrs. Aiiothcr :idvaiitag.t’ of tlie ClVS.1 is that it eiinblcs oiie t o dcter~iiiiir the rnost inil)ortaiit I)araiiieters iii the ~iiodcl, 1)emiiw tlic I)ositivc 1):iraiileters xliich riiort reduce tlie suni of sqti:ii’t. :ire earliest entered into the liiiearizcd inotitl form. The restilts of t l i i . study show that the two ~ ~ r ore( p o1 .itera’ tive 1i:irametcr estiriiation method.+ (CJVS=1 aiid CIYCA\) Iirovitlc excellent initial paraineter estini:itcb for n iioii1iiic:ir . Scither method requires re1)licatc t1at:i. One di~atlvniitagc of botli is that tlic valuc of CY niii.:t lie k i i o w i i , \vlicwa* it iicctl not be kiiowii iii Hill’> iiietliotl. A secoiitl dizatlvaiitapc i; that the \veiglitrtl procwlure+ iiiziy he ap1)licd only to iioiiliiiear models Ivhich are cal)ablr of liiieariz:itioii. 1Iodcls of tlic coiii1)rtitive tyl)e gciicr:illy f:ill iiito tlii.: category. h i t there :ire iioii1iiie:ir models which do iiot ( h n y c r : i i i d ;\Iez:iki, 1967). Nomenclature
Ki
=
K
=
W
=
:id~orptioii equilibrium coiistaiit for componeiit i. phia-I equilihriuiii coilstant for methaiiatioii reaction, I)sia-’ weighting matrix for weighted linear lea.*t squares :1]q)’onch
Ti7,.I[ = cornl)oiieiits water and metliniie, iehpectively X = matrix of iiidelieiitieiit, v:iriahles in liiieai,ized model fol,lli k = rezictioii rate constant g-niole,q, cm2-hr 3;‘ = cutlo lo reaction rate coiist:iiit I’, = p:iiti:d pi’essiii e of cornponelit i, p i a ~
r li
T,
= = =
reactioii rate piwlictetl by n inotlel, g-mole. ‘cm2-hr average exl~erimeiitallyobsei.ved reactioii rate from rel~licatioiiof run 11, g-mole~,/cm2-hr es1winieiit:illy otlserved reactioii rate for riiii g-iiioles,’ciii*-lir
wz,,,= diirgoiinl elernelit.; of wciglitiiig matrix CY
x ,3
Y
exl)oiieiit of 1)on.ei’ traiisforni:ttioii of reaction rate which Imdtices coiistnnt error variance = expoiieiit of I m v e r traii.:foi’inatioii needed t o 1iiie:irize iioii1iiie:ir iiiotlel = vector of I)aranieter estimutc,s = vcctoi, of trniiqforinetl del)eiidciit v:iii:ilJe =
literature Cited
Box, G . E. P., IIill, IV.*J,, 1)epartrnent of dtatistics, I-iiiversity of \Iri~coii411, Tech. 1:ept. 142 1196X). Hill, W.J., Ph.1). di**er(:ttion, 1Tiiiver.ity of Wiscoii+iti, Madison, \Vi< - ‘ I -1966. . llougen, 0. .\., Watson, K. lI., “ C h e m i d Proces- Priiiciplen,” Part 111, 1). 133, \Vile!., New York, 1917. Ilrinter, \V. C ; . j Rittrell, J. R., “Stati~tival Ilethodology for Chemicd 1:eactioii Alodeliiig.” 111). 4-33, A.1.Ch.I’;. Today
Series, .\nierican Iiistitiite of‘Ch&iiic*:dI’iigitleew, Kew York, 1967.
Kittrell, J. II., “React ion Kiiietici for Chemical Engineers,” p. 149>l~cC;raw-Hill,S e w York, 1059. JVeritzhelmer, W. LV., Ph.1). dksertation, University of Permsylvniiin, Philadelphia, Pa.j 1969.
lIezaki, I:.> Johriwri, I:.
I~IXP:IVI:D for review January 2, 1970 ACCEPT~:D November 30, 1970
R.E.V. w:is the revipient of fellowships from the N.I).E.A., Ford Foundatioii, and E. I. dii Pont de Nemours & Co., Inc. The National Science Foundation and the Univer-ity of PennsylvaniaCornpii ter Center supported the c.oniprita t ional cost s. Ind. Eng. Chem. Fundam., Vol. IO, No. 1 , 1971
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