Boesiger. The net result 1' that a smaller reactor would be specified for a given conversion using the rate equation of thls work.
I.o
Nomenclature 0.81-
CACB = concentrations of KbOCI3 and COCl,, respectively, E k , k'
'"IL
I I
0.2
m n
R T
= = = =
= =
S= s2
=
se2 =
n
a t the reactor conditions, g-niol/l. activation energy of -1rrheniuq equation, kcal/g-mol rate constants in dimensions consistent with concentrations. g-mol/l , and time, see exponent of NbOC13 concentration term exponent of COCl? concentratioii term ideal gas Ian constant reaction rate, g-mol/l. see. sum of squares of the residuals mean sum of squares of residuals mean sum of squares for pure error
GREEKLETTER Figure 5. Joint confidence region for the frequency factor Y and activation energy E at constant rn and n
nieiits. This demonstrates that tlie sequeiitial design of experiments leads to rapid convergence which is an extremely advantageou< attribute, particularly when the experiineiitare difficult, to perform, expensive, and 'or time-consuming. Higher h%OC13 conver>ion* are predicted from t,lie rat,e equation of this work than froiii the rate equation of Boesiger for a given set, of coiidit'ioiir. This difference is rather substantial, being on the order of 25 to 307, for the experinieiit~al c,onditions of this work. This differeiice 1)ossibly results from deficiencies in mixing in the constant-volume reactors of
v =
frequency factor of drrhenius equation in dimensions consistent w t l i concentrations, g-mol/I , and time, see
literature Cited
Roe+zer. Boesiger, I). I).. I)., Steven5on. Stevenson, F. I).. I)., Allef.Trans.. Trans., 1. 1, 1XC59-61 1859-61 11970). (1970). ~.~~ Brothers, J. -4.,'C.S. Brotceri, 1., C.S. Patent 3,126,150 3,128,150 (April 7, f, 1964). l l u n n , \V. E., C.S. Patent 3,009,773 (Sovember 21, 1961). I h n n , IV.E., U.S. Patent 3,107,144 (October l j l 1963). Gloor, \I.. Gloor. \I., Reiland. Reiland, K., H e h . Chz'ni. A d a , 4 4 , 1098-1120 (1961). Graham, It. J.. Graham. J., Stevenwii. Stevenwn, F. I).. J . Chroiizntoar.. Chroiizntoqr., 47. 47, 555-7 (1 1970j) . RECJXVKD RECJ;IVKD for review J u n e 25, 1970 ACCI:PTI~:D ACCI:PTI~.D September 3, 1971 ~
j
,
Rork was performed in Arne, Lahoratory of the U.S. Atomic Energy Commisioii.
Effects of Experimental Error on Parameter Estimation and Convergence of a Sequential Experimental Design Robert J. Graham' and F. Dee Stevenson2 Institute for Atomic Researcfz and Department of Chemical Engineering, Iowa State Vniversity, Ames, Iowa 60010
111 cheinical kinetics, the choice of the conditions for experimentation-e.g., the esperinientnl design-is a two-part problem. First, an adequate model must be found to represent the data, and second, precise estimates of the ]mraniet,ers must be determined. The first problem ha$ been discussed hy 130s and Hill (1967) and Huixter and Reiner (1965), particularly with regard to the problem of di>crirninationbetween various proposed niodelh. Tlii.: article i. concerned with t'lie second part-Le., the estimation of paraiiiet'ers and, in particular, the effect of the size of the experinieiit,al error oii the accuracy and convergence of tlie parameter estimates for a given kinetic model. The general, four-parameter model
1
Present, address, American Oil Co., Whiting, Ind. 46394. To whom correspondence should he addressed.
164 Ind.
Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 2, 1 9 7 2
describing tlie kinetics of the chlorination of niobium oxychloride (SbOC13) wit,h phosgene (COCl,), Graham and St,eveiisoii (19iO)] was used iii this study to facilitate a coingarison with actual esl)erimental work. The experimental design criterion, described by Box and Lucas (1959), was utilized to select the experimental coiiditions which iniiiimize the volume of the joint confidence region of the parameters. The accuracy and coiivergeiice of the parameter estimates a t each stage of this esperimeiit'al design were examined at four leyel.; of t,he rariance, d,of an independent normal error, E . Hypot,hetical experimental dat'a were generated froin the reaction niodel for esperimeiital coiiditioiis .elected by t,he experimental design from the variable space defined by the actual kinetic study of Graha,m and St.evenson.
The effects of experimental error on the values of parameter estimates of a general four-parameter kinetic model were investigated b y conducting a series of hypothetical experiments a t different levels of an imposed independent normal error. The parameters used were obtained from the chlorination of niobium oxychloride with phosgene. The values of the parameters estimated from the hypothetical data varied from the true values according to the magnitude of the experimental error and the particular parameter in the model. The effects of such errors on the convergence of a sequential experimental design were reflected in uniform parameter convergence for small errors and a lack of uniform convergence for large error values. However, predicted values of the dependent variable, obtained from the kinetic model using the final parameter estimates, were within the limits of the imposed experimental error a t all levels investigated.
Experimental
The criterion for the sequential experimental design was suggested by I3ox et al. (1957, 1959, and 1965). The ability of this criterion to lead to precise 1)arameter estimates i i i a small number of experiment.: has been demoiistrat,ed by Hunter et al. (1966, 19671, Kitt'rell et al. (1966), and Graham and Stevenson (1970). The design consists of the sequelice of picking the best set of operat,ing condition. for the next run from previous experiment'al information, conducting the experiment, and reevahiating the parameter estimates from the new data and the data of all the preceding experiments Iteration of this sequelice is carried out until the parameters are coiisidered to be determined within the limit? of the experimental accuracy. To obtain initial parameter estiniates, at, least a-: many initial data points are required a.; t'here are parameters iii the model. After initial experimental rillis have been completed and the parameters estimated, by noiilinear least square.; in this case, the experimental coiiditioiis for the next esperimental run are chosen such that the setting of the independent variables selected maximize 4, defined ab the determinant of the GTGmatrix. As described i i i the references given, G is the matrix of partial derivatives, g u z , of the dependent variable y (a measure of the extent of coilversion) with rehpect, to the ith parameter at the uth set, of experimental cotiditions and the set of parameter estimates cieteriniiied after the S t h experiment-e.g., /911
912
where GT is the transpose of G. The partial derivative3 appear as a result of linearizing the dependent variable. in a Taylor series. Application of Experimental Design
The sequential experimental desigii is influenced by tlie size of experimental error to the extent that the esperimeiital error affects the accuracy of the parameter e.dimates.. The reaction representing the chlorinatioii of KbOCla with COClz-e.g., NbOCla
+ COCI,
NbC'lj
+ CO2
may be modeled by the rate equation:
r
= -v
exp ( - E / ' R T ) C A ~ C B ? '
where Y = frequency factor of hrrhenius equatioii; E = act'ivation energy, kcal 'g-mol; and C,, CB = coiiceiitratioiis of NbOCla and COCh, respectively, g-mol '1.
The best parameter estimate>, hereafter referred to as tlie true value., were determined by iioiiliiiear leait->quare< analysis of actual data by Graham aiid Yteveiimi (1970). The value,. obtaiiied lvere: v = 1.461 x lo6, E = 23.32, vi = 0.829, and n = 0.921 for units i i i kcal, g-mol 1.) "I defined a> the mole ratio of ('O? to the iioiicotirleii-al)le ga.ei from the reactor. Hypothetical xihie? of the (1el:eiident values variable, y, Ivere obtained by adding aii error, t , to !itrue Lvhicli were calculated b y iiunieric:il integratioii of Equation 2 and the .teacly-state flon eciiintioii for ti tribiilnr reactor. J'alues of the iiidepeiideiit iioriiinl error, e . nith ctaiidtir~d deyiatioii u , were generated by a computer adiroutiiie. . l i i initinl >et of esl)eriiiieuts or s e t t i u p of the iiiclepeiitleiit variables were clioGeii m-hicli corre.pontlet1 to tlie aCtii:tl esperimental work. Appropriately iiitrodwiiig error into the calculated re.poiises led to the fir-t >et of pnrnnieter e-tiinate, from tlie five Iiyl)otlietical exi)erinieiit. clio
