Nomenclature
D, K,
= =
Rn
= = = = = = =
8,
=
sn,,
= =
Xk
= = = = = = =
kt
I L,, LIn Jf
XC -If1
Ti
xl‘ Xn
z,.~ Yn 2,
z,.,
decision vector a t the nth unit vapor-liquid equilibrium constant of the j t h component reaction rate constant for the i t h reaction identity matrix performance matrix in the linearized model method matrix involved in Equation 9 composite matrix defined by Equation 14 matrix associated with the lth recycle loop performance matrix of the linearized model in the nth CSTR performance matrix of the linearized model in the nth distillation column t h e j t h element of S, reactor volume vector a t the kth iteration xk associated with the lth recycle loop input state vector to the nth unit the j t h element of z, system-output state vector from the nth unit output state vector from the nth unit the ath element of z,
SUPERSCRIPT IC = iteration number literature Cited
Campbell, D. P., “Process Dynamics,” p 298, Wiley, New York, NY. 1957. --Cavett, R. H., P r o ~Amer. . Petrol. Inst., 43, 111, 57 (1963). Christensen, J. H., Rudd, D. F., AIChE J . , 15 ( l ) ,94 (1969). Hiraizumi, Y., RIori., A,., Nishimura, H., Kaaaku Koaaku. ” . 33 13). 299 (1969). ’ Kesler, M. G., Griffiths, P. R., Proc. Amer. Petrollast., 43, 111,49 (19631. Kl&h,’H. C., PhD thesis, Tulane University, New Orleans, LA (1967). Lee, W., Christensen, J. H., Rudd, D. F., AIChE. J., 12 (6), 1104 (1966). Lee, W., Rudd, D. F., ibid., 1966, p 1184. Maejima, T., MS thesis, M.I.T., Cambridge, LIA (1970). Nagiev, hl. F., Chem. Eng. Progr., 53 (6), 297 (1957). Naphtali, L. Ll., ibid., 60 (9), 70 (1964). Nishimura, H., Hiraizumi, Y., Yagi, S., Kagaku Kogaku, 31 (2), 183 (1967). Orback, O., Crowe, C. M.,AIChE 68th National Meeting, Houston, TX (1971). . , ( 5 ) , 71 Ravicz, A. E., Norman R. L., Chem. Eng. P T O ~ T60 - 7
(1964) \..__
GREEKLETTERS f,.,
= =
6%
the conversion of the j t h reaction a t the nth CSTR parameter vector
SUBSCRIPTS j 1
= =
n seq sim
=
= =
component number loopnumber unit number sequential approach simultaneous approach
Rosen, I?. hf., ibid., 58 (lo), 69 (1962). Rubin, D. I., Chem. Eng. Progr. Symp. Ser., 58 (3), 54 (1962). Russell, D., “Optimization Theory,” p 251, Benjamin, Inc., New York, NY, 1970. Sargent, R. W. H., Westerberg, A. W., Trans. Inst. Chem. Eng., 42 (179), 190 (1964). Shannon, P. T., Johnson, A. I., Crowe, C. &I.,Hoffman, T. W., Hamielec, A. E., Woods, D. R., Chem. Eng. Progr., 62 (6), 49 (1966). RECEIVED for review May 13, 1970 ACCEPTEDJanuary 24, 1972 Work was supported by Chiyoda Chemical Engineering and Construction Co.
Kinetics of Chlorination of Niobium Oxychloride by Phosgene in a Tube-Flow Reactor Application of Sequential Experimental Design Robert J. Graham’ and F. Dee Stevenson2 Institute f o r Atomic Research and Department of Chemical Engineering, Iowa State rniversity, Ames, Iowa 50010
T h e oxygen content of niobium metal prepared by reduction of NbClS is dependent, to a large extent, on the amount of KbOCL contamination in the pentachloride a t the time of reduction. The preparation of high-purity niobium is therefore dependent on the effectiveness of either the separation of NbOCla from NbCls or the conversion of P‘rTbOCla to NbC16. The former approach is fraught with difficulties and is not considered in this work. The chlorination of YbOCla with COCl?, according to the reaction NbOC13
+ COClz = NbC16 + COz
Present address, American Oil Co., Whiting, Ind. 46394.
* To whom correspondence should be addressed.
160 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 2, 1972
has been reported previously by Boesiger and Stevenson (1970), Brothers (1964), and Dunn (1961, 1963), the work of Boesiger being a study of the kinet’ics of the react’ion carried out in a homogeneous gas phase reactor of constant volume. The present work is an extension of Boesiger’s kinetic study to lower ’f\ibOC13concentrations with the experimental conditions determined by the sequential experimental design method. Increased experimental accuracy a t the low T\;bOCl3 concentrations was also an objective of this work, since having an accurate reaction model a t low concentrations is important to reactor design. Three rate equations were considered as possible models of the reaction: (1) the general-order model, (2) the elementary-
The kinetics of the homogeneous gas phase chlorination of niobium oxychloride (NbOC13) to niobium pentachloride (NbClj) by means of phosgene (COC12) was investigated in an isothermal tube-flow reactor. The data were best fitted by a general four-parameter rate expression, although a satisfactory fit of the data was also obtained for an elementary second-order reaction model. The parameters of the rate constante.g., frequency factor and activation energy-were calculated, by nonlinear parameter estimation, to b e 1.461 X 1 O6 and 23.32, respectively, for units of kcal, g-moles, liters, O K , and seconds. The corresponding reaction orders were 0.829 and 0.921 for the NbOC13 and COCIZ concentrations. Data were obtained according to a sequential experimental design to facilitate the estimation of the model parameters with minimal experimentation. It was found that approximately three properly chosen experiments in excess of the number of parameters in the model were sufficient to obtain precise parameter estimates.
;"1
order model, and (3) a transition-intermediate model. These are respectively represented as:
= v
-rA
exp (-E/RT)CaCB
(2)
and
NDENSER
SUBLIMER
REACTOR
COCIZ
where Y , E , m , and n are the usual kinetic parameters; C A and Ce represent concentrations of xbOCl3 and COCl? (g-molll.) ; and T repreqentq temperature in O K . The rate constants k and k' each have two Arrhenius parameters to reflect their temperature dependency. There is no evidence in the literature to indicate that the reaction is other than qimple second order (Model 2 ) . However, coniideratioii of the formation of a transition intermediate, C0Cl2*,according to the following mechanism: kl
COClZ
c,coc12* kz
C0Cl2*
+ NbOC13 kJ_ XbCls + COZ
provides an alternative reaction model (Model 3) which differs from Model 2 a t high concentrations of xbOCl3. The transition-intermediate model is based on the supposition that the rate is controlled by COClz being activated by molecular collision. The data were also correlated by other rate models, but these were discarded for various reasons-e.g., lack-of-fit and/or negatire parameter estimates. Experimental
The experimental equipment consisted of a SbOCla sublimer, a 4-mm 1.d. tubular reactor, and a condenser for the KbC1, product and unreacted NbOC13 (Figure 1). The entire apparatus was made of Pyrex to preclude the occurrence of side reactions between the apparatus and the reactants and products. NbOC13 a as wblimed into an argon carrier gas m hich R as maintained a t a constant temperature and flow rate. The NbOC13 waq then transported through a heated connector tube to the coiled tubular reactor where it was miyed with preheated C0Cl2.The conditioiis of the floa within the reactor were \et and controlled to ensure virtual plug flom conditions. The air-cooled condenser provided for the removal of the NbCh and NbOCh from the noncondensable argon, COC12, and C 0 2 . Analysis of this gas for COz by means of temperature-programmed gas chromatography (Graham and Steven-
Figure 1.
Tube-flow reactor system
son. 1970) facilitated the determination of the extent of conkrsion of sbOC13to NbClS. Parameter estimation was made with a transformed dependent variable to conform to the actually measured CO1 mole fractions. The other measured variables were reactor temperature, residence time (reactor volumeivolumetric flow rate a t temperature and pressure of the reactor), concentrations of NbOC13and COCl? as calculated from measured and calculated flow rates and molar density. The n'bOC13 concentration was calculated by assuming saturation in the sublimer according to the vapor preqwre data of Gloor and Weiland (1961). Results
A11 experimental data were obtained for XbOC13 concentrations between about 2 and 20 mol %, reactor volumes of 70 and 340 cc, react,or temperatures between 338-457OC, and pressures near 1 atm. These data are given in Table I according to the actual order in which they were taken. Xore experimental data were obtained than was previously anticipated. This was due in part to errors in the experimental design program during the initial stages of the design, which caused experimental conditions to be specified which were not particularly helpful in t,he parameter estimation. -inextension of the limit,s of the operating conditions after the initial runs permitted the taking of data at, higher SbOC13 concentrations than was originally planned. Replications were made a t nearly all operating conditions to provide an independent' estimate of the pure or experimental error. The parameter estimates for each of the three proposed models, as calculated by nonlinear estimation of parameters, are summarized along with pertinent statistical values in Table 11. plot of the predicted final ?;bOCls concentration Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 2, 1972
161
Table NbOCla” concn,
COCIP concn,
g-mol/l.
g-mol/l.
Run no.
I. Experimental Data Reactor temp, O c
Space time,” sec
Molar g-mol/l.
Mole fraction
c02c
9 . 1 (1)d 0.000450 0.0164 0.0105 452.8 90.9 0.0177 9 . 2 (1)d 0,000450 0.0165 84.6 0.0172 0.0110 452.7 9 . 3 (1)d 0.000473 0.0164 0,0106 451.5 88.9 0.0157 10.1 0.000507 0.0187 0.0116 0.0082 357.3 488.7 1 0 . 2 (2)d 0.000457 0.0187 357.3 454.4 0.0121 0.0070 1 0 . 3 (2)d 0.000452 0.0187 0.0123 356.6 439.2 0.0065 1 0 . 4 (2)d 0.000453 0.0186 357.0 447.1 0.0122 0.0071 1 0 . 5 (2)d 356.2 451.6 0.000426 0.0187 0.0122 0,0062 1 1 . 4 (3)d 0.001215 0.0192 0.0153 346.2 487.8 0.0123 1 1 . 5 (3)d 0.0192 0.001256 0.0122 346.5 467.6 0.0129 12.1 0.001145 0,0163 456.7 95.4 0.0094 0.0354 1 2 . 2 (4) d 0.0162 O.OO1085 0.0342 457.0 87.1 0.0100 0,0162 1 2 . 3 (4) d 0,001066 82.7 0.0323 0,0101 456.6 12.4 ( 4 ) d 0.001111 0.0163 87.0 0.0337 0.0099 457.1 1 3 . 1 (5)d 0.001364 0.0190 0.0161 516.4 0.0~10 347.2 13.2 (5)d 0.001254 0.0189 0.0149 348.0 488.0 0.0117 1 3 . 3 (5)d 0,0189 347.5 534.5 0.001396 0.0110 0.0163 13.4 0.001575 0.0189 542.3 0.0164 0.0104 346.9 16.2 0,001615 0.0163 98.8 0,0379 0.0067 448.4 16.3 0.0162 84.8 0.001733 0.0360 0.0066 449.6 17.3 0.0163 0.0327 0.002753 69.6 0.0044 447.0 18.1 0.0189 0.0263 352,9 436.9 0.003186 0.0073 0.0192 19.1 0.0200 0.003227 406.3 0.0078 338.4 19.2 0.0192 0.0197 338.7 447.9 0.003469 0.0067 21.2 (6)d 0 0164 0.0331 449.3 58.5 0.001911 0.0091 0.0164 2 1 . 3 (6)d 56.9 0,0306 0.0094 449.9 0.001853 22.1 0.0190 0,0115 0.001208 473.9 0.0118 347.0 22.2 0.0190 0,0095 346.1 421.7 0.0127 0.001080 0.0162 2 3 . 1 (7)d 0.0348 449.7 68.0 0.001928 0.0085 0,0162 23.2 (7) d 66.5 0.0360 0,001947 0.0086 449.5 0.0192 24.1 (8)d 0.0112 339.6 444.0 0.0058 0.002693 0.0191 0.0111 24.2 (8)d 340.5 431.6 0.002654 0.0060 0.0829 0,0168 25.1 (9)d 429.6 382.4 0,0050 0.002572 0.0781 0,0168 25.2 (9)d 429.7 369.6 0.0050 0.002551 0.0177 0.0653 389.6 394.6 0.002542 28.1 0.0079 0.0177 0.0674 398 3 394.3 28.2 (10)d 0.0079 0.002588 0.0174 0.0770 408.1 461.0 29.1 (1l)d 0.0068 0.002635 0.0173 0.0780 408.0 469.2 29.2 (1l)d 0.002725 0,0065 Concentrations evaluated at reactor temperature and pressure. * Space times less than 100 sec obtained with 72-cc reactor. Based on product gas containing only Ar, Con, and C0Cl2. d Replicate runs; items in parentheses denote replicate set numbers.
Table II. Comparison of Kinetic Models Mad&
3
1
m
n Y
X 10+
General order
2 Elementary
Transition intermediate
0.829 0.921 1.461
1.0 1.0 10.21
1. o 1. o 20.32 Y’
E
23.32
s2/s,2 S
sx
104
4.29 0.00210 1.54
23,95 5.55 0.00239 2.11
= 0.079 24.65 E’ = 0.096 4.64 0.00223 1.74
X
from Model 1 vs. NbOCL concentrations calculated from the experimentally measured COz mole fractions is represented in Figure 2. Figure 3 shows these values superimposed on a similar plot constructed from Boesiger’s data and parameter Y = 0.526 X loe, E = 21.21, m = 0.947, and values-e.g., n = 1.013. Joint 95y0 confidence regions for the parameters Y vs. E and m vs. n were computed for the general model and are 162 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 2, 1 9 7 2
represented in Figures 4 and 5, together with similar confidence regions for Boesiger’s data and parameter estimates. Five initial experimental runs (9-13) were used for the initial parameter estimates even though only four were necessary, the fifth being a n attempt to replicate the third. Parameter estimates obtained a t each stage of the experimental design are given in Table I11 in addition to the standard error estimates for each parameter. The experiment number sequence for the experimental design as shown in Table 111, reflects the preferred sequence of experimentation. Actual experiments followed a somewhat different sequence as a result of the error in the experimental design computer program in the initial stages of the work and the extension of the sample space in subsequent experiments. It is apparent that after three runs past the initial set, the values of each parameter converged to near the final estimates which were obtained from all the data. Discussion of Results
From Table 11, it is apparent that there is little statistical difference between the three proposed reaction models. However, the general-order reaction model, Model l , provides the
Table 111. Parameter Estimates at Each Stage of the Sequential Experimental Design for M o d e l 1 Experiment no.
Y
x
E
10-6
n
m
9-13 16.588 ( 1 2 3 9 . 2 % ) a 24.14 ( + 3 . 4 % ) 25 0.524 ( 1 6 5 . 3 % ) 23.29 ( + 1 . 7 % ) . 28 3.702 ( 1 5 0 . 9 % ) 23.68 ( 1 2 . 0 % ) 23 1.117 ( 1 4 3 . 5 % ) 23.05 ( + 1 . 7 % ) 18 1.169 ( 1 4 2 . 3 % ) 23.00 ( 1 1 , 6 % ) 19 1,105 (+45.5%) 22.75 ( 1 1 . 7 ~ o ) For all 16 experiment's 1.461 ( 1 5 0 . 4 % ) 23.32 (1-1.7'%) a Quantities in parentheses are estimates of the standard error of the paramet,er.
0.784 ( + 8 . l % ) 0.734 ( + 7 . 2 % ) 0.869( 1 4 . 9 % ) 0.818 ( 1 4 . 6 % ) 0.822 ( + 4 . 5 7 ~ ) 0.836 (14.8%)
1.388 (+32.7%) 0.887 ( 1 7 . 5 % ) 1.000 (*5.4%) 0.928 (.t5.0%) 0.930 ( 1 4 . 9 % ) 0.936 ( + 5 . 3 % )
0.829 ( + 5 . 5 % )
0.921 (+5.9%)
20
"I 14.0
MEASURED NbOC13 CONC. x IO3
6.01
Figure 2. Predicted vs. measured NbOC13 concentration as determined from Model 1 (general reaction order)
4.0c
z o u h, ,
7
0.0 16
-
0.006 -
20
24
28
~
E, k c a l
Figure 4. Joint confidence region for the reaction orders m and n a t constant v and E
-,
0.004-
-
0.002-
-
0
sz a
W +
err
W
0.000 0
I
0.002
I
0.004
I 0.006
0.008
MEASURED NbOCI3 CONC.
Figure 3. Comparison of predicted vs. measured NbOCI3 concentration with Boesiger's data
best fit of the data, since s2/seZ, the ratio of the residual mean square to the mean square for pure error, and the sum of squares of the residuals, S,are lower than for the other models. The adequacy of Model 1 over the coiicentratioii range of the data obtained is evidenced by lack of scatter in Figure 2. I t is apparent a190 that there are no outliers in the data set. The transition-state model does not differ significantly
from the elementary second-order model. Even a t the highest NbOC13 concentrat'ions, corresponding to about 20 mol %, the denominator term does not' differ from unity by more than 0.06%. This model, therefore, simplifies to the elementary second-order model. Though the general-order reaction model, LIodel 3, gives a slightly better fit of the data, there is no basis for discarding t,he elementary second-order reaction model. Discrimiliation betvieen Model 3 and the other models would require data a t high XbOC1, concentrations which were not attainable n-ith the existing apparatus. The joint confidence regions of this work (Figures 4 and 5 ) are much smaller than reported for Boesiger's dat'a, indicating that more precise parameter estiniat'es were obtained ill thi.; work. The smaller confidence regions are at'tribut'ed to t'he experiment'alapproach which resulted in reduced experimeiital error and the improved design of experiments. It is apparent from Table I11 t,hat the paramet,ers, estimated after the third desigiied experiment (esperimeiit number 23) , are not significantly different from the parameters estimated from all experimental data. ,ilso, the precision of the parameter est'imat,es, as measured by the standard errors, was not appreciably improved by subsequent experiInd. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 2, 1972
163
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). ~.~~ Brotceri, J. -4.,'C.S. 1., C.S. Patent 3,126,150 3,128,150 (April 7, f, 1964). Brothers, 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 , 44, 1098-1120 (1961). J., Stevenwii. Stevenwn, F. I).. J . Chroiizntoar.. Chroiizntoqr., 47. 47, 555-7 Graham, It. J.. Graham. (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>crirnination between 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.
