Ind. Eng. Chem. Fundam. 1984, 23, 267-268 Garza, G.; Rosales, M. A. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 168. Haynes. H. W.; Sharma, P. N. AIChE J . 1973, 19, 1043. Kocirik, M.; Zikanova, A. Ind. Eng. Chem. Fundam. 1974, 13, 347. Ma, Y. H.; Evans, D.M. AIChE J . 1966, 14, 956. Neogi, P.; Ruckenstein, E. AIChE J. 1980, 26, 787. Ruckenstein.. E.:. Vaidvanathan. A. S.:. Younoouist.. G. R. Chem. E . Sei. 1971. , 26, 1305. Smith, D. M. Ph.D. Thesis, University of New Mexico, Albuquerque, NM, 1982.
267
Wakao, N.; Smith, J. M. Chem. E . Sci. 1962, 17, 825.
Department of Chemical Engineering Montana State University Bozeman, Montana 5971 7
Douglas M. Smith
I .
Received f o r review May 9,1983 Accepted December 8, 1983
CORRESPONDENCE Comments on “A Maximum-Likelihood Approach to Treating Error Variance Instability in Hougen-Watson Rate Models” Sir: In apaper by Egy (1982), a procedure is described that corrects for nonconstant error variances when fitting linearized forms of Hougen-Watson rate models. The author compared his procedure with what he thought was a technique described by Box and Hill (1974) and concluded that this latter technique was “inappropriate”. The purpose of this correspondence is to point out that the Box-Hill technique was incorrectly stated and improperly applied by the author. Hence, the results attributed to the Box-Hill technique and reported in the Egy paper are not supportable. The problem stems from the fact that the author incorrectly assumes that Box and Hill used the linear model yi(h)= xiB + ui (i = 1,2,..., N observations) where X is a power transformation of the response, yi, say reaction rate, uI’sare independent errors with nonconstant variance V(uJ = u2z:
(Po, ..., PK-J is a vector of coefficients, and xi is the vector of K independent variables for the ith experiment. Here ziis taken as some independent variable (e.g., partial pressure of a reactant), u2 is a constant, and 6 is an unknown power transformation to be estimated from the data. Egy set zi equal to one or another of the independent variables and then estimated 6. For comparison purposes the author claimed that the Box-Hill approach is to set 6 = 0 and estimate A. This is not the Box-Hill model nor the procedure. The correct application of the Box-Hill method is to use the model with coefficients 0 @ =
y I = dxi,o) + ei
where 7 is a linear or nonlinear model function, the e, are
independent errors with nonconstant variance such that
where 4 is an unknown weighting parameter to be estimated from the data, and Y iis the calculated value for the response yi. For the particular example discussed (Carr, 1960), yi was taken by Box and Hill to be the inverse of the reaction rate or r-l, which gives a linear form of the Hougen-Watson rate model under investigation. Contrary to what Egy indicated, the Box-Hill procedure does not change the model form from nonlinear to linear, or vice versa, by transforming y to y(A). Rather, the procedure actually finds the weighting parameter 4 (and not A) and then estimates 0 by an appropriately weighted linear or nonlinear least-squares analysis. In summary, we do not disagree with the author’s suggestion of attempting to stabilize variance based on weighting the data with an independent variable. In some examples this might work very well. However, we think it important to point out that the author has misunderstood the Box-Hill approach and we take issue with his conclusion that the latter technique is “inappropriate”. This can discourage engineers from using a useful method for modeling chemical reaction rates.
Literature Cited Box, G. E. P.; Hill, W. J. Technometrics 1974, 76(3), 385-389 Carr, N. L. Ind. Eng. Chem. 1960, 52(5).391-396. Egy, D. J. Ind. Eng. Chem. Fund 1982, 21, 337-339.
Department of Statistics University of Wisconsin Madison, Wisconsin 53706 Buffalo Research Laboratory Allied Corporation Buffalo, New York 14210
G . E. P. Box D. M. Steinberg W. J . Hill*
Response to Comments on “A Maximum-Likelihood Approach to Treating Error Variance Instability in Hougen-Watson Rate Models” Sir: In their correspondence referencing my 1982 paper, Mssrs. Box, Steinberg, and Hill take exception to some conclusions which I drew regarding a weighting scheme developed by Box and Hill (1974). The Box and Hill scheme to stabilize the error variances in Hougen-Watson
rate models transformed in some fashion for estimation purposes is, I stated, “inappropriate” for reasons which I attempted to demonstrate in my paper and which I shall attempt to defend here. I would like to apologize to Mssrs. Box and I-iill for what
0196-4313/84/1023-0267$01.50/00 1984 American Chemical Society
Ind. Eng. Chem. Fundam. 1904, 23, 268-269
260
was, perhaps, a poor choice of words when I stated their weighting scheme to be “inappropriate”; however, I respectfully suggest that they missed the point of my paper by re-asserting the advisability of employing a weighted least-squares approach where the weights are derived somehow from the fitted response values. I did not intend to generalize the Box and Hill approach necessarily to the assumption of a linear model; rather I wanted to point out that whatever a priori assumption is made regarding the functional form of the model (‘rl’’ in Box, Steinberg, and Hill’s “comments”), the use of a weighting scheme involving the expected response or some estimation thereof will inadvertently affect the functional form of the model being estimated and, hence, the estimates of the model’s coefficients. To illustrate, a model employing the Box and Cox (1964) power transformation on the response variable will not be able to tell the difference between a log-linear model with stable error variance and a linear model with error variances nonconstant, as in the Box, Steinberg, and Hill equation u ( e , ) = (T? 9L2-2@
Box, G. E. P.; Cox, D. R. J. R . Statist. SOC. Ser. 6 1064, 26(2). 211-252. Box, G. E. P.; Hili, W. J. Technometrics 1074, 76(3), 385-389. Egy, D. J.: Ind. Eng. Chem. Fundam. 1082, 21, 337-339.
In other words, a 4 equal to zero could be indicative of an error variance unstable linear model or a log-linear model with constant error variances or some unknown model in
Department of Mathematical Sciences Russell Sage College Troy, New York 12180
between. Equation 17 in my 1982 paper demonstrates that the functional form parameter, A, and the error variance stability parameter, 6 , cannot be isolated from one another when the response variable is used to act in a weighting method to stabilize error variance. In summary, my representation of the Box and Hill scheme to which an objection has been made was intended to demonstrate that deceptive results can be obtained in model estimation which utilizes the response variable in a weighting method to overcome unstable error variance characteristics. By utilizing a nonstochastic variable for that purpose, one can overcome this problem. As a further note, I have found that, in some cases, it is possible to simultaneously solve for functional form and unstable error variance problems by maximizing (likelihood function) eq 13 in my 1982 paper after setting my “z,” variable to one of the regressors in the model being estimated. Literature Cited
Daniel J. Egy
Comments on “Effect of Vapor Efflux from a Spherical Particle on Heat Transfer from a Hot Gas”
Sir: Kalson (1983) has considered the effect of vapor efflux on heat transfer to a cylinder and to a sphere, using the classical film model with constant physical properties. His results differ from the standard expression, which is
*
-=
ho
ho
h0
k -=
NRc~g ___
h -_
(
, +6 R. In RT )(circular cylinder)
ho (1)
in his notation. Equation 1 was derived for a plane boundary by Ackermann (1937) and generalized to spheres by several authors, including Godsave (1953), El-Wakil et al. (19541, and Ranz (1956). A corollary of eq 1for spheres was derived by Longwell (1956). The validity of eq 1 for cylinders, as well as planes and spheres, has been pointed out by Bird et al. (1960). Kalson’s unusual results are due to the neglect of curvature in his expression for the thermal film thickness 6, which he introduces after eq 6. The correct expressions for 6 are obtainable by requiring that the film model reproduce the heat transfer coefficient in the absence of mass transfer _1 -_ -6 (plane) (2a) h, k
k
(->
(sphere)
R,+ 6
(2b)
(2c)
Kalson’s expressions Cocand Cos,for spheres and cylinders, respectively, were improperly based on eq 2a, the relation for a plane. Use of the correct expressions for 6 leads to the known result, eq 1for each geometry. (Nomenclature follows that of Kalson, 1983). Literature Cited Ackermann, G. Forschungsh. 1937, 382, 1-16. Bird, R. B.: Stewart, W. E.: Lightfoot, E. N. “Transport Phenomena”; Wiley: New York, 1960; pp 328-330, 658-668, 674-675, 683-684. El-Wakil, M. M.; Uyehara, 0. A,: Myers, P. S. “A Theoretical Investigation of the Heating-Up Period of Injected Fuel Droplets Vaporizing in Air”: NACA TN 3179, May 1954. Godsave. G. A. E. “Studies of the Combustion of Droplets in a Fuel SprayThe Burning of Single Drops of Fuel”; I n “Fourth Symposium on Combustion”; Williams and Wilkins Co.: Baltimore, 1953: pp 818-830. Kalson, P. A. Ind. Eng. Chem. Fundam. 1083. 22, 355-357. Longwell, J. P. “Combustion of Liquid Fuels”, I n “High Speed Aerodynamics and Jet Propulsion”; Princeton University Press: Princeton, NJ, 1956: Vol. 11.. pp 407-444. Ranz, W. E. Trans. ASME 1958, 76.909-913.
Department of Chemical Engineering University of Wisconsin Madison, Wisconsin 53706
Warren E. Stewart
Comments on “Effect of Vapor Efflux from a Spherical Particle on Heat Transfer from a Hot Gas” Sir: In a recent communication (Kalson, 1983), the problem of reduction of heat transfer due to blowing is addressed in connection with pyrolysis of coal particles. Kalson suggests that the factor used to correct the heat 0196-4313/84/1023-0268$01.50/0
transfer coefficient by James and Mills (1976),and others, was incorrect since a result derived for flat-plate geometry was applied to a spherical geometry. Kalson proceeds to derive correction factors for both spherical and cylindrical C 1984 American
Chemical Society
