For both the nonlinear least squares and the weighted linear least squares techniques, the residuals may be examined for goodness of fit. T h e tebst will only be approximate, since the model is nonlinear. T h e four replications provide a n independent estimate of the error variance, and we find that a significant lack of fit a t the 0.10 level corresponds to a residual sum of squares, RSS 2 10.20. We may conclude, therefore, that there is no significant lack of fit for either model. Acknowledgment
T h e authors expre:]s their gratitude to the National Science Foundation, under Grant GK-1055, for financial support, and acknowledge a grant of computer time by the Wisconsin Alumni Research Foundation through the University Research Committee. Nomenclature = partial pressures of hydrogen, n-pentane, and xl, x 2 , x 3
2-met hylbutane, respectively
= adsorption constants
k l , k2, k8
k
7 8 ,
= reaction equilibrium constant = rate constants in single-site
Yd
=
$0, $1, $ 2 ,
43
eo, B1, e2, 0 3
=
=
R
=
Y,
=
Yd w.9
A,
a d (0
RSS C,,
= = = = =
University of Wisconsin Madison, Wis. REIJI MEZAKI Yale University New Haven, Conn. RECEIVED for review August 28, 1967 ACCEPTEDOctober 9, 1967
EDITOR’SNOTE: T h e following comments on this paper have been received from Norman L. Carr, whose experimental data are quoted by the authors.
= reaction rates calculated from single-site and
yo, ) d
R. A. JOHNSON N. A . STANDAL
and dual-site
model, respectively ra, rd
“GAUSHAUS Nonlinear Regression Program,” Computing Center, University of Wisconsin, Madison, Wis., 1963. Hill, W. J., “Statistical Techniques for Model-Building,” Ph.D. thesis, Department of Statistics, University of Wisconsin, 1966. Hougen, 0. A., Watson, K. M., “Chemical Process Principles,’’ Part 111, Wiley, New York, 1947. Hougen, 0. S., Watson, K. M., 2nd. Eng. Chem. 35, 529 (1943). Mathur, G. P., Thodos, G., Chem. Eng. Sci. 21, 1191 (1966).
dual-site model, respectively, with true values of constants response. variables calculated from linearized single site and dual site model, respectively, with true constants constants related to adsorption constants constants related to adsorption constants observed reaction rate transformed observations defined by (x2 xalk)/R transformed observations defined by Y$lz weight functions for linearized single-site and dual-site model, respectively transformation parameters residual sum of squares diagonal elements of variance-covariance matrix of RX
-
literature Cited
Carr, N. L., 2nd. Eng. Chem. 52, 391 (1960). Draper, N. R., Smith, H., “Applied Regression Analysis,” Wiley, New York, 1966.
authors of the subject paper have used some interesting, new estimation methods for model development. These methods appear to be superior to those that were used 10 years ago in the original analysis. If one is primarily searching for catalytic reaction mechanisms, these tools coupled with proper experimentation appear to be valuable. Even these tools may fail in the absence of the basic insight gained by the experimenter. I n the course of my work on pentane isomerization, it was found but not published that the isopentane yield response exhibited a maximum with respect to total pressure, This result is a well-known characteristic of dual-site kinetics models; thisdoes not reflect the nature of the single-site model. I n conclusion, if we pool the old with the new and add insight, the agreement for acceptance of the dual-site model is clear. NORMAN L. CARR Gulf Research €8 Development Go. Pittsburgh, Pa. THE
CORRESP0NDE:NCE ~
N U M E R I C A L DIFFERENTIATION OF EQUALLY SPACED A N D N O T EQUALILY SPACED EXPERIMENTAL DATA SIR: I should like to take exception to the article, “Numerical Differentiation of Equally Spaced and Not Equally Spaced Experimental Data” [IND. ENG. CHEM. FUNDAMESTALS 6, 413 (1967)l. This is in regard to their discussion on round-off error in fitting least squares power functions (page 414 referring to Table I). I feel their interpretation of Table I is very possibly misleading-namely, had the proper scaling been done with single precision as well as double precision arithmetic, they would have found the effect of double precision per se much less dramatic. Proper scaling of data prior to least squares analysis is a major factor which often eliminates the need for double precision arithmetic (Draper and Smith, 1967). I take further exception to their statement, “As B becomes
smaller in Equation 2, the round-off error becomes serious, because the determinant of X’Xapproaches zero.” T h e matrix X’Xis for the independent variables X , X z . . . . X 5 only and reflects the correlation and range or “leverage” in these values. T h e error E relates to Y and in no way affects the behavior of X’X. Donald
T.Winski
International Flavors €8 Fragrances, Inc. Union Beach, N . J. literature Cited
Draper, N. R., Smith, H., “Applied Regression Analysis,” Wiley, New York, 1967. VOL. 7
NO. 1
FEBRUARY 1968
183
