IS SOPHISTICATION REALLY NECESSARY? Rutherford Ark No general answer to this question is possible, but it is clear that sophistication cannot be dismissed unsophisticatedly or anyone acquainted with the true meaning of the F word, the answer to this question is immediate. Quibbling, the use of specious fallacy, disingenuous alteration, deceptive modification, and all other forms of adulteration are neither necessary nor desirable. Unfortunately, I used the word in this title before looking it up in the dictionary and was rather discomfited to find that its common technical usage meaning “complexity” or “wordly wisdom” (of a technical, rather than a reprehensible, sort) was a connotation unknown even in the Supplement. [I refer, of course, to the Oxford English Dictionary, of which the Supplement is the thirteenth volume. I a m saved from feeling that the usage is not current, as well as not correct, by the second entry under “sophisticated” in “Le JokeBook de M. Jacques” (American Scholar 1962 32, p. 181).] Yet complexity would hardly have served my turn, for I have in mind to question the necessity for some of the latest and most subtle techniques of analysis, particularly mathematical analysis. Although it would certainly be more profitable, and perhaps even more interesting, to discuss this shift of meaning and its implications-for the morality of an age is reflected in its words and, in a time of wholesale verbicide, the preservation of some degree of definiteness has become an issue of critical moment-I hope I shall be at least understood if I call Pontryagin’s maximum principle or the theory of singular perturbations sophisticated, a t least in comparison with practicing engineers’ rules of thumb. I t becomes important then to ask whether this sort of thing is really necessary. Here I must seem to be the very devil’s advocate, adding another voice to the clamour against these highfalutin notions that the universities are putting out these days. I have no wish to be guilty of such a “trahison des clercs,” but it does seem to be desirable to raise the question, for undue complexity in a model or an appeal to recondite ideas when simple ones would serve as well is indeed sophistication of the worst kind. The canons of mathematical elegance demand that the most 32
INDUSTRIAL AND ENGINEERING CHEMISTRY
economical method be used in any situation, and if some goal is extremely insensitive to the choice of a particular parameter, then building a computer to evaluate this parameter is futile-particularly if a sufficiently close guess can be made. The problem then is to define situations in which detailed and subtle analysis is necessary and to foresee when it is futile. I do not pretend to give a comprehensive answer to this question, but will give some examples of where quite crude methods are adequate and where they come to grief. Approximation to Arrhenius Rate Constant
The ineluctable nonlinearity of chemical kinetics is the rate constant as a function of temperature, k = A exp -E/RT. (The Eyring form, ATn exp -E/RT, might be called a more sophisticated form of it.) This is a function which tends to zero with all its derivatives as T 3 0 and to a finite upper bound A as T 3 03. The greatest value of dk/dT is (4 AR/E)@ a t a temperature of E / 2 R. Often A is large and E of the order of kilocalories, so that the temperature at the point of inflection is high; under normal circumstances we are concerned with a region of temperature where the rate constant is rapidly increasing. There is in fact an old rule of thumb that a reaction popular rate doubles itself every so many degrees-a figure is 10". It is, of course, incorrect (except for a particular temperature dependent on the activation energy), but, because the linear, or even quadratic, ko'(T - TO) l / 2 ko"(T - TO)^ approximation ko is a poor one, the old rule has arisen again in the form of an exponential approximation. The exponential function of temperature having the same value and derivative as k( T ) = A exp -E/RT at T = To is
+
+
careful study of the situation is needed before it can be established that the error induced by this approximation is serioits. There are several reasons for wishing that the approximation were a good one. It is possible occasionally to get exact solutions with the positive exponential (7). O n an analog computer it is much easier to generate than the Arrhenius form and may be more accurate than the straight line sequence which is often used by function generators. Third, it may lead to a reduction in the number of parameters needed to describe a situation. An example of this is Tinkler and Metzner's (77) and Petersen's ( 7 7) treatment of nonisothermal reaction in the catalyst pellet where only one parameter E( -AH)Dec,/RT,2k, is needed in addition to the Thiele modulus. If the exact form of k ( T ) is used, two parameters, of which Petersen's is the product, are needed [cf. Weisz and Hicks (78)1. If it can be shown that the error in the effectiveness factor so obtained is small, this is an important economy; but it is not always easy to do this. Thus, sophistication cannot be dismissed unsophisticatedly. Figure 2, which is a combination of curves from several of Weisz and Hicks's figures, shows two sets of three curves. For these, 0 = c,(-AH)D,/ T,k,, y = E/RT, and, if the exponential approximation were adequate, the curves for which is the same should coincide. Evidently the approximation is adequate at small values of p ~ . The positive exponential approximation is not only inaccurate but totally misleading when used in certain arguments that depend on its shape, for its curvature at high temperatures is the opposite of the Arrhenius function. Take, for example, the stability problem for the adiabatic stirred tank. The equations for the concentrations, c, of a reactant disappearing by first-order reaction and for the temperature, T , are
(3) Now, although k* has the same value and slope as k at T = To,it has a greater curvature and
I n fact, k* is within 5% of k only if T is within To {1 f 2rdl r2 -I- 2r2] where r = (0.0122 RTo/E)1/2. Figure 1 shows the limits of temperature within which less than error is involved in the approximation, Such a figure as this provides some basis for estimating when the approximation may be tolerable numerically, but it must be remembered that it will often be used at an early stage of a calculation and the inaccuracy may be exaggerated or diminished depending on the particular circumstances. Thus, although it may be badly in error at high temperatures, the reaction may be largely over when or where the temperature is high and so the effect of the gross error is diminished. Obviously, a
+
e -dT = dt
TI - T
+ JBk(T)c
(4)
where 8 is the holding time, c and T f the concentration and temperature of the feed, and J the ratio of heat of reaction to heat capacity. Now, because k +- 0 as T -+ 0 and remains finite as T 03, the derivative dT/dt is positive for sufficiently small T and negative when T is sufficiently large. Similarly, dc/dt is positive when c = 0 and negative when c = c p This means that the solution of these differential equations always moves inward over the boundaries of a sufficiently large rectangle in the strip 0 6 c c p From the theorems of PoincarC and Bendixson we can deduce important, though incomplete, information about the possible number of steady states and their nature. If k ( T ) were replaced by k*(T), then dT/dt would be positive for sufficiently large T and the theorem would not apply.
