Estimating Rate Constants. An Improvement on the Time-Elimination Procedure SIR: For reaction systems in which starting times are not known accurately, Draper et al. (1969) present a useful procedure for determining rate constants. However, they have also presented an unfavorable analysis of the ‘[time-elimination” procedure. The time-elimination procedure, when properly used, offers advantages and is compatible with and complements the “extra parameter method” described b y Draper et al. For illustration, consider Example 1 of that paper. We note that by letting D = kz/kl, Equations 3.7 and 3.8 may be written as
B
=
(A)
[AD- A ]
c = (D+)[$(AD-l)-(A-
(3 ’ 7 ) I)]
0.8)
Equation 3.12 may be written as
A = l - B - C
(3.12)
A value of D may be found which minimizes the objective function defined by Equation 2.6. This requires a simple onedimensional search. Using this value, Equations 3.10 and 3.11 are written as
A
= exp
[$
(t
+
T)]
( 3 .IO)
and
SIR:I n reply to Newberger and Robinson, we should like to make the following points. 1. We said on page 425 (top) of the original article, “NOW, strictly speaking, it is incorrect to apply the least squares technique as stated above. . . since, for the A’s on the right-hand side, we have to substitute observations subject to error. . . This is one, but not the major, drawback of the time elimination procedure.” This remark still applies to Newberger and Robinson’s suggestion. 2. There is no guarantee (in Newberger and Robinson’s example) that a minimization with respect to D followed by a minimization with respect to k~ and T will produce the minimum of the objective function with respect to all three parameters D , kz, and T (or kl, kz, and T, equivalently). This type of difficulty is illustrated by Figure 11.2(c) of Davies (1956). Without such a guarantee, use of the NewbergerRobinson suggestion as given would be extremely risky. 3. The reduction in the dimensionality of the search is less striking when there are more parameters. Suppose there are p parameters. If the Kewberger-Robinson suggested procedure were used, the first search would be in (p-minus) dimensions, the next in 2 dimensions. Since a nonlinear program would have to be used anyway, the saving in time would be slight and, moreover, the adverse effects of 1 and 2 above would be accentuated by larger values of p , which occur frequently in practice in the analyses of complex chemical systems.
.
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Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
(3.11) Using these equations along with 3.12, kt and T may be found which minimize the same objective function, 2.6. Hence, by using the time-elimination procedure, the minimization problem was transformed from a three-dimensional search to a one-dimensional search followed by a two-dimensional search. I n general, the time-elimination procedure, used properly, will allow the original multidimensional problem to be divided into two lower dimensional minimization problems. Using the time-elimination procedure as outlined by Draper et al. can only lead to ratio’s of reaction rates constant, as demonstrated b y Ames (1960, 1962). However, the timeelimination procedure reduces the dimensionality of the parameter estimation problem and as such is a logical starting point whenever concentration profiles of several reacting species are available. literature Cited
Ames, W. F., Znd. Eng. Chem. 52,517 (1960). 1,214 (1962). Ames, W. F., IND.ENQ.CHEM.FUNDAMENTALS Draper, N. R., Kanemasu, H., Mezaki, R., IND.ENQ. CHEY. FUNDAMENTALS 8,423 (1969). A!?. R . Newberger J . D. Robinson American Cyanamid Co. Wayne, N . J. 07470
4. The only possible use we can see for Newberger and Robinson’s suggestion is to obtain one set of initial estimates for the correct over-all minimization required by the “extra parameter” method. The values obtained from this suggestion would be fed back in as the starting values for a minimization with respect to all parameters of the problem. However, since it is usual to try several sets of initial values in order partially to check that a global minimum has been attained, the work involved to get this particular set might not be worth the extra trouble, although there would be no harm in doing it. I n summary then, we believe that the Newberger-Robinson suggestion cannot be relied upon and that its practical value is slight. literature Cited
Davies, 0. L., ed., “Design and Analysis of Industrial Experiments,” 2nd ed., Hafner, New York, 1956.
Norman R. Draper Hiromitsu Kanemasu University of Visconsin Madison, Wis. 68706 Reiji Mezaki New York University Bronx, N . Y . 10668
