Parametric Sensitivity in Fitting R. M m k i
J. R. Kittr$ll
The practical considerations important for the estimation of parameters in the nonlinear hyperbolic classes of kinetics are presented. The sums of squares surfaces of several example cases are examined to disclose the characteristics of these swfaces important in nonlinear parameter estimation.
n the determination of a reaction rate model ade-
I quately describing a heterogeneous kinetic system, one may desire to specify the functional form of a model describing his reaction system and to estimate precisely the parameters within this model. Many techniqiies have been developed for determining the preferred functional form of the rate model (74, 75). In this paper, we shall be primarily concerned with the estimation aspects of kinetic modeling rather than these specification aspects. Two broad classifications of nonlinear kinetic models are of such general use as to merit specific consideration of the estimation of the parameters in these models. These are the power function (nth order) models and the hyperbolic models, a particular example of which is shown in Equation 17. A discussion of some of the parameter estimation problems for the power function models has been presented elsewhere (74, 75, 77). Here, we shall be concerned only with the hyperbolic class of models. Many examples of the hyperbolic model class exist in kinetic literature. These can arise from applications of a steady-state hypothesis; typical results of this type of theory include the Lindemann model for gaseous reactions (781, the Michaelis models for enzymatic reactions (71, and the Hougen-\Vatson or LangmuirHinshelwood models for gaseous reactions on solid surfaces (70). Furthermore, many industrial applications of the hyperbolic rate equations have been made by considering the parameters in these equations to be nothing more than empirical constants. This paper will present techniques for obtaining and assessing the reliability of parameter estimates in any of these important applications of the hyperbolic rate models. Study of three examples for solid catalyzed gaseous reactions will indicate several pitfalls in the parameter estimation that can be avoided if their existence is recognized before the experimentation a n d data analyses. T h e parameter estimation problem often can be quite difficult for hyperbolic models because of several factors. These models are frequently nonlinear with respect to the parameters within these models (9, 76). As pointed out by Box (6),the functional form of these models is such that the parameter estimates are usually highly
correlated and poorly estimated. Finally, the number of parameters contained in each model can be too great for the range of the experimental data (22). Here, we will study this problem largely through examinations of the surface of the sums of squares of residual rates (the difference between the experimentally obtained rate and predicted rate) as a function of the parameter values. A surface which has a very sharp defined minimum for a particular set of parameter values, then, will indicate that only one set of parameter values fits the experimental data well. These parameter estimates are thus “well determined.” Similarly, a very large, flat minimum in this surface would indicate that the parameter values can only be “poorly determined.” Estimation of Parameters
T h e absolute minimum of the sums of squares surface will provide a single set of parameter values termed a point estimate. T h e point estimate then, is that set of parameter values which, when used in predicting reaction rates from a given model, minimizes the sum of squares function : N
S(K) =
c
(ru
u=l
-
(11
rJ2
O n some specific models of the hyperbolic class, the rate equation can be rearranged to obtain an equation which is linear in the parameters. Here, linear least squares techniques may be used, but attention must be given in any case to properly weighting the analysis (9, 75). Although the sunis of squares of residual rates should not always be directly minimized since weighting may be required, only these unbveighted residual rates will be considered in this report. There are two reasons for this. Xo evidence exists that different weighting is required for the data of our examples, and the magnitudes of the reaction rates do not change much in all these data so that any reasonable weighting assumption will provide nearly the same paramzter estimates (see, for instance, Example 1). T h e consequences of improper weighting have been presented elsewhere (9, 72). For intrinsically nonlinear rate models, one can find the parameter values at the minimuin of the sunis of squares surface by steepest descent techniques. AlternaVOL. 5 9
NO. 5
M A Y 1967
63
tively, one could utilize the Gaussian or the GaussNewton procedure, which effectively is a n iterative application of linear least squares procedures. T h e parameter estimation technique used in this paper is a combination of the latter two methods. For this reason a n d to aid in the explanation of the next section, the Gaussian procedure will be briefly reviewed. If a general nonlinear reaction rate equation is r = f ( x 1 , X P , . . ., x,; K1, Kz, . . ., K P )or, in matrix form:
r = f ( x : K)
(2)
Now, since the model specification problem has been assumed to be solved, the functional form of f ( x ; K) is known and the N experimental data points already obtained may be written:
+
u = 1, 2, . . ., N y u = f ( x u ;K) eu (3) Denote initial estimates of all the parameters (K1, K z , . . ., K,) as (KlO, KZO, . . ., Kpo). Then, the Taylor series expansion of Equation 3, about the initial estimates, truncated with the linear terms, may be written: yu
=
f ( x u ;KO)
+
(Kt - KtO) + 5 [”‘TitK’]
e,
(4)
K = KO
i=l
Let
Thus, Equation 4 may be written: P
yu - fU0 =
,z + Pi0ZiU0
E,
2=1
(8)
Note that Equation 8 is now linear with respect to the parameters Po. Thus, the estimates bo, of the parameters Po, may be obtained from the least squares solution of the equations ZobO = Y - fo. T h a t is, for a n invertible matrix Zo’Zo,
bo = (Zo’Zo)-’ Zo’(Y - fo)
(9)
where the subscript or superscript zero refers to the evaluation of these matrices at the initial parameter estimates, KO. These estimates, bo, exactly minimize the sum of squares of residuals : P
N
S O W=
C u=l
{yu - f u o -
i=l
+
b’
=
(Zj’Z,
+ AI)-’
Zj’(Y - f’)
(11)
where X is a scalar that at each iteration is set just large enough to ensure a decrease in S(K). Parameter values obtained by nonlinear least squares are not always those that best typify the absolute minimum of the sums of squares surface, because of the characteristics of this surface. Two extreme cases may occur. Particularly for the more complex problems, the sum of squares surface can have several local minima. Here, the nonlinear estimation procedure may converge at one of these minima whizh may either be in a n unacceptable region of the parameter space or the sum of squares of these converged values may be significantly greater than at the absolute minimum. Both situations are undesirable. T h e other case occurs when the nonlinear estimation procedure converges a t a n absolute minimum around which is a very large region of the parameter space having sums of squares nearly as low as the minimum. This very flat minimum can be improved by a better selection of data points (13), reparameterization ( 3 ) , or improved weighting of the least squares procedure (15). Unless the nature of the sum-of-squares surface can be revealed, it may be difficult to obtain parameter estimates precisely and rapidly. Techniques for describing the nature of this surface are illustrated. Confidence Regions of Estimated Parameters
bioztu0i2
(10)
J . R. Kittrell and R. Mezaki were Postdoctoral Fellows in Chemical Engineering at the University of Wisconsin, Madison, W i s . J . R. Kittrell is now a Research Engineer with Chevron Research Co., Richmond, Calif. R. Mezaki is Professor of Chemical Engineering, Yale University, N e w Haven, Conn. T h i s research was supported by the National Science Foundation under Contract N S F - G P 2755. Computer time by the Wisconsin Alumni Research Foundation through the University Research Committee is acknowledged. AUTHORS
64
a n d thus approximately minimize Equation 1, to the extent that the Taylor expansion is valid. T o refine these estimates, bo, so that they more nearly minimize Equation 1, we can adjust the point about which the Taylor expansion is performed through a rewrite of Equation 6 : K.Il = bio KiO. Now, the entire procedure can be repeated, with the revised estimates K t being used in the place of initial estimates Kio for the next iteration. These iterations can then be continued until the corrections, bi’ = K,’ - Ki’--l become small in the j t h iteration. I n this method, of course, good initial estimates, KO, are necessary to ensure convergence. Methods are available for obtaining these estimates for kinetic data (76). T h e nonlinear least squares technique used here is from Marquardt (20) and is a combination of steepest descent and Gaussian procedures. This method calculates the parameter corrections, b’, for the j t h iteration by :
INDUSTRIAL A N D ENGINEERING CHEMISTRY
As implied above, the least squares minimum may not be sufficiently informative to allow much confidence to be placed in the parameter estimates obtained. Consequently, one must frequently turn to procedures which indicate the size of the region within which the true parameter values might be expected to lie. Of course, the one-dimensional analog of this concept is the well known confidence interval. Beale (7) has presented a comprehensive discussion of confidence regions for parameters in nonlinear models. Consequently, only certain aspects of the theory will be presented here. Two easily applied methods of measuring the un-
certainty associated with the parameter estimates are available. The more approximate method is ~bnply to extend the linear theory used in Equation 9, considering each parameter separately. To the extent that the truncated Taylor expansion represents the reaction rate surface in the vicinity of the least squares estimates, confidenceintervals may be calculated for the individual parameter estimates by: var@) =
(z,' z,)-I?
(12)
Here, Var(ft) is the variance-covariance matrix of the parameter estimates at the converged jth iteration, the diagonal elements of which estimate the variances of these parameter estimates. The quantity Z estimates the error variance of the reaction rate data and is simply the residual mean square if the model is adequate. The least squares estimate plus or minus twice the square root of a diagonal element of this matrix, then, will provide an approximate 95% confidence interval for the appropriate parameter estimate. Equation 12 provides an approximate interval estimate for any single parameter, but does not describe the joint region of uncertainty for the parameter etimates consided collectively. A confidence region of the parameter estimates may be approximately calculated by:
To apply this equation, one must calculate the sums of squares at enough sets of parameter values until a locus of all the parameter values possessing a contour level of magnitude S(K) is obtained. Then a plot of the locus h of these parameter values provides a region ~ t h which one can state with lOO(1 - a)% confidence that the true parameter values lie. If the rate equation under consideration were exactly linear with respect to its parameters, such a confidence region would be elliptical. On the other hand, when the rate equation is nonlinear, this region will not be elliptical and, in fact, its deviation from an elliptical shape is a measure of the nonlinearity of the model (3,5). For one, two, or three parameters in the rate equation, of course, graphical representations of the confidence regions are easily presentable. However, for more dimensions some assistance is required for a visualization of the surface. For near-ellipsoidal surfaces, for example, the ratios of the latent roots of the (Z,'Z,) matrix can provide information about the eccentricity of the ellipsoid (3). Marquardt (27) has employed tangent planes of the ellipsoid to describe the joint confidence region by support-plane limits. Repamm.1.riration for Surface Condfflonlng
It is not uncommon for the contours of the sums of squares surface for hyperbolic models fitted to kinetic data to be long and attenuated, as typified by the contoun in the positive (first) quadrant of Figure 1. This type of surface is termed poorly conditioned; an iterative nonlinear least squares routine will generally con-
Figwe 1. Contours of sum of squans surface farasicivds of l?quaIion 17 fa200' C.
verge quite slowly to any point within the region enc l w d by the lowest contour of Figure 1. Frequently, the poorly conditioned surface can be changed into a well conditioned surface, such as Figure 4, by reparameterizing the model. Reparameterization for estimation p'uposes can often be carried out by redefining the independent variables so that the center of their coordinate system is near the center of the experimental design. In kinetics, one type of reparameterization frequently needed is that a d ated with the exponential dependence of parameWs upon temperature. In particular, a parameter entering many of the hyperbolic models is a rate constant such as
k = 7 exp ( - E / R T )
(14) Better estimation can geuerdy be achieved if this parameter is defined as
k
-
7 ' 9
[-E/&
(k - $)]
(15)
y' = y exp (-E/RT) (16) Now, in an estimation eituation, point estimam of parameters E and y' can be obtained more readily than those o f E and y using Equation 14. Compared to the direct use of Quation 14,the choice of initial parameter estimates is less critical and the estimation routine con-
where
verges to the minimum more rapidly. Since no change has been made in the rate equation, however, there is no change in the size of the Eonfidence region of the parameter estimates. Reparameterization has been die cussed in more detail k h u e (3, 4, 8). An example of the use of thin technique follows. ChamchrisHc b h c o s of Sums of S q w m of Resklwl Rates
Although a set of parameter estimates will be obtained by an application of nonlinear least squares if good initial parameter estimates are available (76), V O L 5 9 NO. 5
MAY 1967
65
these estimates are often imprecise. This can result in the numerical maghudes of the kinetic constants being given far more significance than they deserve. Accordingly, it is worthwhile to investigate the relationship between the rare parameters and the surface of the sums of squares of residual rates to gain knowledge of the type of unreliability to be expected. I t should be recalled that the precision of estimated parameter values depends heavily on the nature of this surface. Example 1. Hougen and his coworkers (70) in=restigated the catalytic hydrogenation of mixed isoxtenes over a nickel catalyst. Experimental conditions itilized in this study were temperatures fmm 200" to 125' C. and pressures fmm 1 to 3.5 atm. The final mechanistic model selected by these investigators is representable by the rate equation:
r=
kKaKu!Bpu
(1
+ KBPE + Ku!u + K.%)'
(17)
Blakemore and Hoerl (2) also analyzed these experimental data using Equation 17. By utilizing the 12 isothermal data points reported for 200' C., the rate parameters in Equation 17 were calculated with a nonlinear least squares routine. The estimates thus obtained (Table I) compare favorably with the values reported by both Hougen and Blakemore, indicating that weighting is unimportant for these data and that either unweighted linear or nonlinear least squares analyses suffice (72). However, it is also of interest to examine the sum of squares surface for addi-
,i
0.5
rafa constant of
66
T
II
Equorhn 17
INDUSTRIAL A N D ENGINEERING CHEMISTRY
TABLE 1.
PARAMETER ESTIMATES FOR ISO-OCTEWE HYDROGENATION, ZW"C.
Poronrtcr k, g. mole/& sat+t/ht.
RE,arm-1 Ku, am.-' Ks, atm.-l
0,728 0.322 0.475 0.414
Sum ofquare of residual ram 6.04 X 10-
0.58 0.41
0.56 0.37
6 . 7 X 10+
0.644 0.383 0.580 0.489
0.65 0.38 0.55 0.54
7.07 X 1 0 4 6.21 X 1 0 4
C w e t d to jromda a x p m t i d tmnjmatwa dcpndanc. reaction rote rasiduolr. e Us& rafmtion idu rcridmls. a
6
Ufitg
tional information concerning the reliability of these parameter estimates. The four-dimensional sum of squares surface (four parameters) presents all of the information concerning the parameter estimates for Equation 17 that the data contain. A plot of the s u m of squares surface for two of the four parametem is shown in Figure 1. This illustration utiliizes coordinates of k and RE while Ku = 0.475 atm-' and Ks = 0.414 atm.-* First, this figure presents the long, skewed wntours frequently occurring with such models. Also, it should be noted that two minima exist in this plot, in the first and third quadrants. Accordingly, in the estimation of rate parameters, one may obtain certain negative parameter estimates. In fact, it can be seen from Equation 17 that any two of the parameters k, Ka, or Ku can simultaneouslybe negative without greatly hindering the fit of the equation to the data because of the fact that they are multiplied together in the numerator. These considerations are of importance in the estimation of the parameter values since negative estimates of any adsorption constants are generally used as a justification for rejecting the model under consideration. Therefore, one should attempt to examine the sums of squares surface before rejecting a model on the basis of negative estimates of the adsorption constants. Even then, one should take into account not only the point estimates of the adsorption constants but also the confidence region for these constants. More will be discussed about this procedure later in the report. Hougen and his coworkers (70) also state in the analysis of these data that some estimated constants did not follow the Arrhenius relation very well. Figure 2 presents the estimated forward rate constants, k, obtained by a nonlinear least squares analysis of the data at each temperature level. It can be seen that they do not exactly lie on a straight line in a log constantreciprocal temperature plot. Also shown in this figure, however, are confidence intervals at each of these temperatures, calculated by Equation 12. It can be seen, since each of the parameter values could lie, with about 95% certainty, anywhere along the vertical lines in Figure 2, that not only is an Arrhenius relation not rejected by the data but also the activation energy, or
TABLE II. PARAMETER ESTIMATESFOR ISO-OCTENE HYDROGENATION Estimates (by PGTmtm Erthaks Re,bmmt&dion) N
-987 1.61, 4,425 -11.0
k
Ash N E
3,838
N U S U N 8
4 Sumofaluaresofreaidualrates
TABLE
-8.22 25,780 -52.7 2.744 X 1 0 4
-3,063 5.40 6,722 -10.1 5,538 -9.47 17,950 -31.0 2.378 X lo-'
ill. PARAMETER
ESTIMATES FOR EQUATION 20 Pmmatm Estimate
h, g. mole/(g. mdpt)(aaa)(min.) KA. atm.-' Ks, am.-' Kw, at!=.-'
Sum of q u a m of midual rates
7.02
+
-
+
+
x 10-4
2.91 8.73 X 10-8 6.34 2.3 X lo-*
the slope of the line, is not well determined. Such large confidence intervals are quite typical! Considerable difficulty might be encountered in correlating estimated constants with members of, say, a homologous series of compounds. I t might be expected that the confidence intervals for the constants would be so large that it would not be apparent whether any correlation is present. Techniques are available for taking data in such a way as to minimize such difficulties (73). Let us now estimate the parameters by analyzing the data of all three temperatures simultaneously, through nonlinear least squares. This may be accomplished by writing each parameter of Equation 17 as
k = exp
dehydration of ethyl alcohol to the ether over a catalytic ion exchange resin. Lapidus and Peterson (79) ana1yz.d their integral conversion data using nonlinear least squares to specify an adequate rate model. We differentiated the integral data at 120' C. and 1 atm. pressure to obtain 37 reaction rate points. An analysis of these data indicated that an adequate surface reaction controlling model could be given by Equation 20 and that estimates of the parameter values are those of Table 111. These estimates were obtaiaed by a nonlinear least square^ analysis of these rate data, using KW = 25.2 as indicated independently by Kabel and Johanson: k&A@A' Pdw/Kea) I = (20) (1 KAPA W I I KWPw))'
I
'/,
c I . :
(-%+%)
The eight parameter estimates thus obtained are contained in Table 11. One section ofthe eight-dimensional 95% confidence region is shown in Figure 3. For this poorly conditioned surface, all parameters except AH and AS of Equation 18 were set at the values of Table 11. A similar calculation was performed after Equations 17, 18, and 19 were reparameterized as described in Equation 15. For example, Equation 19 becomes
,4.2 i 1W4\
The parameter estimates thus obtained are also shown in Table 11. The improved sum of squares surface corresponding to Figure 3 is shown in Figure 4. The original equation required 12 iterations to converge while the reparameterized equation required only eight iterations to achieve a comparable converged sum of squares. Example 2. Kabel and Johanson (77) studied the V O L 5 9 NO. 5 M A Y 1 9 6 7
67
Table IV presents the p&ameter estimates and minimum sum of squares for Equation 21. The similarity of these quantities to tho* of Table I11 also.suggesti that Ka should be eliminated. Figure 6 presents the 97.5% confidence region for the three paramete of Equation 21. Note the Characteristic ellipsoidal shape of this region. I t is sufficiently large that a wide range of combinations of parameter values cap adequately fit the experimental data; considerably larger confidence regions have been obsemed (73). Again, procedurks are available for taking data which will minimize the volume of this region (73), as well as reduce the correlation beyeen the alcohol and water adsorption constants apparent from Figure 6. This correlation can a&e by having water present only as a reaction product, which will always increase in stoichiometric proportion to the decrease in alcohol pressure. Example 3. Antither plausible rate model for the alcohol dehydration system bf Example 2 cowponds to a single site surface reaction-ontrolled dehydration: ,
m
k&(ph - PdJw/&J (22) (1 K ~ AKaPn KwPw) The parameter estimates obtained are shown in Table V. Figure 7 presents a two-dimensional section of the complete sums of squares surface (at kl = 4.8 X Id+ gram mole/&. a m . gram catalyst, and Ka = 0), indicating some of the convergence difficulties encountered. For initial estimates in the first quadrant, the arrows indicate the path taken by subsequent iterations of a nonlinear least squares routine. No convergence was obtained. The second quadrant possesses a relatively high minimum, corresponding to a'residual sum of squares of about lo-' (gram mole/gram catalyst-min.)*~ The third quadrant contains an absolute minimum sum of squares, an reporkd in Table V. Since any of the minima can be obtained, considerable caution should be used when such convergence difficulties aFe encountered. Partial print-outs of the sums of squares surfaces and good initial estimates (76) can help remedy the situation. r=
-"'
9:
'-a ---__
8.
e. 7.
6.5
'"
Figwa 6. CotJidancc r@on fa parametms of Epuotion 21
+
+
TABLE IV. PARAMETER ECTIMATES FOR EQUATION 21 PmornCra
It, can be seen from Table 111 that the estimates of the adsorption constant, Ks, are s m d l in magnitude. Since it is desirable to eliminate this parameter from Equation 20 if it is truly zero, *e s u m of squai.es surface was examined,. a typical section of which is presented in Ka space. The dashed line in Figure 5 fbr &e Ki this figureis the 95.0% confidence region. This region is never far from B zero value of the ether adsorption constant; in the four-dimensional space, this is also true: In addition, static adsorption measurements have indicated that Ka may be zero (17). Thus, with & equal to zero, Equation 20 simplifies to:
-
68
INDUSTRIAL A N D ENGINEERING CHEMISTRY
+
EStinOfrr
x
kl, s. mole/(g. cat.lyn)(aaa)(h)
7.04
KA,am.-'
2.89
Kw, am.-L Sum of quara of residual raw
2.3
TABLE V.
10-4
6.30
x 10-9
PARARrTER EITIMATP FOR EQUATION 22
-
estimate of constant Ki aha jth itaation equilibrium constant = least quara estimate of the constant Ki = ova-all thamodynamic
Kj nl
N
P pi 1.
r
R s' S(K)
S&)
So(K)
A&
matrix ofmameta athata Ki = ma& of parameta atimata K;afta thclth iraation = number of indcpcndmt variables = numba of experimental points
numba of param-
= partial prwure of component i, atm.
=predicted reaction rate at uth expaimmt, gram moles/gram catalystaim predicted reaction rate, gram mola/gram catalyst-
-
min.
univwal gaa constant = athated variance of reaction rates (gram molal gram catalyst-min.)' = sum of q u a r a of residuala for a set of paramefa values K, (gram moles/gram catalyst-fin.)' = sum of aquarm of raiduah for a set of parameta values atimated by least squares, (gram moles/
eram catalvst-min.P =s u k of q u a r e of r&duah baaed on a linearized rate equation and for the initial set of parameta values .= enhopy change in activated admrption of s w i m i, cal./gram mol-e K. = enhopv change in reaction;caI./gram moles-' K. = reparamemized entropy change in reaction, cd./
K.
gram mol-O
= abaolute temperature = a w a r e absolute temnaature of B set of data = a particular erpcrimcntd run I
varimbcovariancc matrix for the matrix of parameta valua afta tbelth itaatian KS = gmaalized indcpcndcnt variable = matrix of independent variable = an expaimentaUy obsavcd reaction rate at the uth expaimmtal run, gram moles/gram catalyst-
I=
CONCLUSIONS Several characteristics of sums of squares surfaces of typical Hougen-Watson models have been presented. In particular, the contours of the sums of squares surface, and thus the confidence region of the parameter estimates, are generally ellipsoidal and occasionally quite skewed. Frequently, the sums of squares surface is quite flat in the region of the minimum so that the parameter estimates are imprecise. Finally, minima may exist in several regions of the parameter space. Consequently, when hyperbolic rate equations are employed in the analysis of heterogeneous reaction rate data, one should be cognizant of the above points. Furthermore, the techniques reviewed in this paper can be used to carry out this analysis in a comprehensive fashion. NOMENCLATURE
fu parameta Ki aha jth itaation, obtained from Quation 9 = matrix of parametem b i j aha@ itaation = pmdicted reaction rate at the uth expaimcntal NR using parameter valua aha jth iteration, gram mok/gram catalyst min. = matrix of rare) f.j = activationmergy Fiaha F statirtic p X p matrix ofunity diagonal and ZQO &-diagonal P
--
atimated -tion
tamI
= mthdpy c b w c in activated adrmption of spaia i,
cal./grnm moka mtbalpy change in reaction, cal./gram mola = S n w a r d reaction rate constant of Equation 17, gram moka/(gram catalyst)(bau) h a r d reaction rate conatant of Equation 20, gram moles/(gram catalpt -atm. -min.) = equilibrium admrption conatant for component i,
-
am.-'
min. = ma& of the obasvcd rcaction rare) = partial derivative of rate equation with rapeet to the ith parameter at the uth cxpaimcntal run &a the ~~~
Y 2 d
jth itaation
Zd
= matrix of m t i a l daivatiw of Eauation 7 after ith
Zj'
= transposcofzj
OL
=
;:
7'
= hue Wection of parameta Ki &athejth itaation = matrix of 8i' afta the jth itaation = prc-exponmtial factor for rate constant reparamctakd paaponmtial factw for rate con-
e"
=
7
itaation
-
con6d-e
cc&cient
stant
expcrimcntal~ n agwiated n with the uth Nn = scalar of Epuation 11
x
LITERATURE CITED (1) W e , E. M.L., J. Rq.Sur. Sa,Pnt B 92,41 (2) El-,
(1960).
1. W., HarL A. E., C h . Emg. Pmr. Symp. Sn. 59 (42). 14 (1963).
(3) Bar,0.E. P., Am. N. I:A d . sd. 86 (3), 792 (1960). (4) Box, 0. E. P., f Stad.tia Tsh. Rep,. No. 25, U n i d t y d W r o Madison, Wb., 1 3 . (5) Box, 0. E. P., Hunts, W. G., Tnkmhv. ' 4 (3,301 (1962). (6)E% 0. E. P.,LH. L., Bimmih 46,77 (1959). 0)Cldmd, W. W., W m . B*
[email protected]#d 67, 1M (1963). (8) h a p s , N. R., smih, n., * * ~ p p ~Ri ~-d ~ hd,,i,wilCy,N~ Y ~ L , 1%. (9) HiU, W h I.. FkD. h&3, U O i d t y of Wirondq Madiron, Wia., 1966. (lGfl%n % ou;& ;.$ K. M., " C h d u l ROSD Rindpls, P u t 111," (11) Kabd, R. L., J o h n , L. N., A.I.Ck.R. J . 8,621 (1962). (iz) KitI. R., ~ ~ n t a G., ; W - U O ~c. c., rbd., i i , i o s i (196s). (13) rw., 19.5 (1966). (14) Kit&, I. R., M=aki, R., Ino. E w . clay. 5s (21.28 (19671. (15)Kit&,J.R.,Mezaki,R.,W~uoqC.C.,Bri~Ch.Ey.'11(1),15(1966). (16) KitJ. R., Mezaki, R., Wauolr, C. C., I n . &e. clau.57 (12), 18 (1965). (17) lbid., 58 (5). 50 (1966). (18 Laidla, K. J., "Chcmiul K i m & : ' 2nd rd., p. 144, M&aw.HIU, New 1965. (19) LaNw L.,Petmm,T. I., A.I.Ck.E. J . ll,pp.5,891 (1965). (20) Muquardt, D.W., J. Sr. d(pI. M&. 2.431 (1963). (21) M q & t D. W. u a m Estimation d Nonlinur Paramcfcn," IBM sh.rr L i ' k y P&%%94, Erbibit E. (22) white, R. R. C h d , S. W..A.I.Ch.E. J . 5,354 (1959).
w.
dark,
V O L 5 9 NO. 5 M A Y 1 9 6 7
69
