”
I
Correlation of Rate Data C. C. D e Witt,MICHIGAN STATE COLLEGE, EAST
LANSING, MICH.
Growth curves are applied to the correlation of data from chemical engineering unit operations, which are ordinarily not thought of as exempltfying growth phenomena. The growth equations are so fundamental in many Pelds of exact science that their application to chemical engineering involves a reinterpretation of the significance of both variables and constants. The relation of growth equations used in the explanation of biological data would seem to have little to do with chemical kinetics. However, these equations are almost identical with those concerned with chemical kinetics, radiation, electrical, and many other physical phenomena. The Gomperto. equation is applicable to growth data which show a maximum in the neighborhood of 37 per cent: the simpliPed logistic
0
NE of the outstanding characteristics of engineering literature, and therefore of engineers in general, is the never-ceasing effort to force the results of hardwon knowledge into the form of straight-line relations. “More and better straight lines” is actually the thesis of this paper. But attention will first be directed to some really crooked lines that have heretofore been regarded as straight. I n the present instance it is appropriate to consider first the unit operation, filtration. For some years there has been growing dissatisfaction with the theoretical treatment of filtration. The number of filtration equations proposed is approximately equal to the number of thoughtful investigators who have had the courage to report their data and draw conclusions. Within the last decade the careful work of Ruth, Montillon, and Montonna (16,16)has been shown by Larian (IS)to yield results identical in form with those of Almy and Lewis (2), Sperry (18, 19, & Underwood I), ($1-24, Hinchley, Ure, and Clarke (11), Baker (S),van Gilse, van Ginneken, and Waterman (6, 7, 8). When equally trustworthy experimental data from different investigators in the same field fail to correlate on any reasonable basis, it has been customary to state that some variable related to all the data has been neglected. Re-examination of the records may bring out a real cause for the differences of opinion in these previous data. First, a general property or method of presentation of filtration data should be sought. Ruth, Montillon, and Montonna (16) generalize thus: “Figure 1 illustrates the typical S-shaped curvature of a V2 = 8 plot as secured by Baker and as observed in hundreds of filtrations performed in these laboratories. Although the relative curvatures may vary in degrcc from case to case, they are present without exception in every filtration carried out to complete filling of the frame.” Figure 2 of the same paper shows a log V-log 8 plot of some of Baker’s data. The latter investigator drew the best straight line through the logarithmically plotted data; a curve drawn through the plotted points is convex in form and quite similar to an exponential curve. Plots of other carefully taken filtration data yield similsr diagrams (Figure 1). I\luch imporJune, 1943
equation of Raymond Pearl has a point of inflection at about 50 per cent (eachper cent reckoned on the y-axis). The more complicated forms of the Pearl equation allow the point of inflection of its characteristic S-shaped curve to vary over a wide range. The curve may be symmetrical or asymmetrical. The Gomperts curve is somewhat limited in application but may be generalized to cover more types of data. These equations appear to be applicable to any closed system where afixed number of particles acting upon one another produce changes in themselves or their surroundings. Any process in which an action begins slowly, speeds up. and then dies away may be interpreted mathematically by these equations.
tance has been attached to the variation of the slope of the straight line drawn on such logarithmic plots; actually this line has merely been through the straightest portion of the logarithmic plot of V vs. 8. The data plotted thus do not fall on any one straight line. As reported and plotted in the previously cited instances, most of the points fall on a well defined exponential-type curve. It is easy to see that other published filtration data of similar scope are amenable to the same treatment. Three general observations regarding most of the published filtration data may now be stated: 1. All theoretical treatment of filtration to date is similar in its principal characteristics. No adequate treatment of the theory as such exists. Virtually no filtration data of which the author is aware show the characteristic demanded of a parabolic functionvia., that for equal intervals of time the second differences of the volumes delivered shall be constant. The variation of such second differences in some cases exceeds 1500 per cent even when the initial time interval is discarded. 2. The V2 us. e curves of all filtration data taken over the entire cycle of com letely flling the press frame are always Sshaped curves wit{ varying degrees of curvature. 3. The actual log-log lots of V us. e are not straight lines. However, a best straight pine may be drawn through the data. In many instances the data so plotted fall on what appears to be an exponential-typecurve.
ITH these three general observations as a basis and Wwithout further reference to the filtration operation as such, purely mathematical conclusions can be drawn from the various plotted data curves. Examining item 3 first, if log V vs. log 8 gives an exponential-type curve, then log ( V ) or logf(V) is proportional to CF(’).If this is true, men loglog V [or f ( V ) ]should be proportional to F(f3). That is, the plot of log-log V [orf(V)] vs, F(8) should yield a straight line. This reasoning follows from the definition of a logarithm: The logarithm of a number is the power to which a constant is raised to produce the number. Intrinsically, the logarithm is also a number. Therefore in the present argument log 0 may be replaced by CF@)where F(0) is so defined that the plot of log V us. C”@) actually is a straight line. Then, a s before, log-log V and V = Ke--eF@).
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
695
Any mathematical analysis of filtration data or curves which does not yield finally an S-shaped curve for the V"0 plot is incorrect. Let us subject the previous conclusion to a test. If log-log f ( V ) is proportional to F ( 0 ) , then what law governs the relation betweenf(V) andf(0)? Clearly as before, log f (V ) N C p ( o )and , f (V ) -bCF@) or f (V ) = KbcF(@.This equation reduces easily t o f(V) = Ke-eF(e), Further inquiry into the nature of the latter equation reveals that it was reported by Gompertz (@,who used it to explain the growth and death rates of populations. The differential form of the latter equation is:
(3)
It is evident that the Gompertz equation y = Re-e-s has certain mathematical limitations; in the present instance some of them may be removed by generalizing the equation. To do this (-2) may be replaced by F ( x ) . This substitution will, however, have an important effect on the location of the point of inflection. A priori reasoning indicates that the possibility of a fixed point of inflection for It is clear from the equation V = Ke-eF(@that, as F(0) all filtration curves may not reasonably be expected, welcome becomes negatively infinite, V will approach zero, and as F(0) though such a relation might be. If F ( x ) is developed as a becomes positively infinite, V will approach K . If F(0) = Taylor series, it is necessary that only odd powers of x be retained: otherwise y will become larger than K . Further, the particular form of the equation does not allow any symmetry in the family of curves y = Ke-eF(x), and when F ( z ) is positive, the curve represents a die-away function. This form of the Gompertz function is put into straightline form by plotting log-log ( K l y ) against F ( z ) . "; INTERESTING transformation of the Gompertz function is obtained by assuming that F ( z ) = m/x, where m is a constant. Then A 90 log-log ( K / y ) = m/x. Now if s log-log K/y is set equal to V , Vx = m; this is the equation of a rectangular hyperbola. If this transformation is within the limits of mathematical legality,it follows that any datawhich may be represented by such a hyperbolic function may likewisebe equally well represented by a probability function of the Gompertz type. If a log-log (K/y) us. x plot turns out to be a hyperbola, then a log-log ( K / y ) us. 1/z can be tried. Other valid .03 .05 .1 .2 .3 .5 1.0 2 3 5 10 20 40 Log e substitutions and transformations 1% ill suggest themselves. Figure 1 . Diagrams of Filtration D a t a of Various InEestigators From the form of the derived equaI. Kiesel uhr, S erry (18). tion it is readily seen that the Gom111. 11. Direct Spentqime, yellow, ATIiott Alliqtt (1). (1). IV. Chromium hydroxide, Almy and Lewis (2). pertz equation is nothing more than a particular form of probability function. This implies that any process whose data may be represented by (a - be), then the abscissa of the inflection point of the Ssuch an equation is likely to obey the laws of probashaped curve is 0 = a/b, and the ordinate a t the point of inbility. How is filtration connected with probability? Let flection is V = K/e. The maximum rate of filtration occurs us assume that a unit quantity of a suspension conwhen 36.8 per cent of the total amount of fluid suspension taining sufficient uniformly distributed particles of varihas gone through the plate; i. e., if the suspension is uniform, ous sizes to make up one layer of filter cake is thrown the plate is filled 36.8 per cent full or the thickness of the against a screen or filter cloth. The arrangement of the filter cake is 36.8 per cent of the maximum thickness- if and particles of various sizes will follow some sort of probability only if F ( 0 ) = 0. The use of the Gompertz equation t o exdistribution pattern. Successive layers will follow more or plain filtration theory implies that the relative growth rate of less similar patterns. When the various layers reach the a filter cake decreases as an exponential function of the limiting thickness (filling of the frame), there will be, based fraction of the volume of filtrate already filtered. The Gomon the average size of the particles, a minimum of (n - 1) pertz curve has another interesting property; that is, the distributions of particles in the various layers. In an actual ordinate of any point on the curve is raised to equal powers of filtration there will be many more probabilities of intradisitself in equal intervals of time. tribution due t o settling phenomena in the partially filled plate. Fitting the Gompertz curve, y = KbC6, choose three conI n short, using a given suspension of different or same-size secutive points on the curve YO, yl, y2. particles stirred a t the same rate and pumped into the press
A
696
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 35, No. 6
A
IY
A
A
6
IY
t
B
may, like the Gompertz function, be transformed by substitution into a number of well-known curve functions. For instance, let 1 meF(@= V ; then Vy = K as before. More important, in so far as “Industrial Reaction Rates’’ are concerned, is the fact that this so-called logistic growth function, of use to the biologist, is mathematically identical with one of the more important theoretical equations of chemical kinetics, which is
+
I
L
processes where rates of progress are involved. Such a function might be even more useful than the Gompertz function. I n its simpler form the logistic growth function yields a symmetrical S-shaped curve. I n reality this function seems to have been borrowed by the biologist and bacteriologist from the chemist. However, after mathematical alterations by these scientists, notably Pearl (14),the logistic equation seems to be capable of faithfully representing a large variety of physical and chemical data. The generalized logistic equation
20
/O
encountered so many times in Semenoff’s classical work (17) on the subject. Any mathematical theory concerned with the growth or rate of production of any substance or quantity must of necessity be concerned with the following factors:
0
-3
-2
-I
0
I
FIGURE 5 I
Figures 2 to 5.
Growth Curves for Various Conditions
under a constant pressure head, it seems remarkable that anyone could possibly duplicate in all respects a set of filtration data previously obtained from even the same suspension. How can one guarantee the position of a particle of given size in a stirred suspension a t any particular time? Poiseuille’s law of flow is not denied by such a hypothesis, but such a hypothesis insists that the instantaneous average pore size is governed by the laws of probability. The over-all viewpoint of the filtration operation must, then, be that it is intimately connected with some sort of probability function. At present no claim is made that the suggested form of probability function will fit all filtration data. The idea is put forward with the thought that subsequent workers in this field may evolve a better, more inclusive explanation based on the application of a probability function to the filtration operation. INCE in the present argument a probability function seems to offer a reasonable representation of filtration data, any other function of a similar general nature which offers both a symmetrical as well as an unsymmetrical form will be useful in representing the data from a number of different June, 1943
1. Any dosed system. 2. The system contains a certain amount of energy or matter which may be converted into something else, no matter what or how. The total quantity of the “something else” theoretically obtainable represents the upper limit of product yield, K . 3. Near the lower limit of time, virtually no product is available; the lower limiting asymptote of product yield is zero. 4. The conversion process is one which, owing t o physical or chemical inertia, starts slowly, increases t o a maximum rate, and finally dies away; the product yield asymptotically approaches a limiting value of total yield, K. 5. If the conversion process occurs in cydes, these cycles
should be additive.
While these five qualifications may be connected with chemical reactions, they are not so limited. The derivation of an equation will be attempted which in its final form will answer all of these requirements.
MOLE
coz, C U M U L A T / v E
Figure 6 . Equilibrium Relation between Water Vapor and Carbon Dioxide i n Contact with Hot Carbon ( f r o m Hitchcock and Robinson, 12)
INDUSTRIAL AND ENGINEERING CHEMISTRY
697
Assuming that F ( x ) may be represented by a Taylor series, K
Y
If
= 1
+
2
alzz
u2 = a3 = a4. . .a, = 0,
+ aJza,
, , a p
then
K Y
60
.5
7i;Me
- HOUR3
65
2.0
.
Pearl (14) has already done this for the growth of population, but unfortunately his book is long out of print and unavailable to many. The field of the final equation seems so much broader than the original application that a repetition1 is not out of line with this presentation. Additional fields of useful application will be cited and illustrated. EARL'S derivation slightly modified follows : Consider Pthe curve, b
e
+ c
(4)
l
(9)
w
If m is negative, the curve becomes discontinuous within finite limits. Negative values of K give negative values of y. Therefore we shall limit y and K to positive values. Also, y can never be less than zero or greater than K. The complete curve lies between the x-axis and a parallel line a distance K above it. The following relations hold: If F ( z ) = + m , y = 0 If F ( r ) = - m, y = K I f F ( z ) = 0, y =
Figure 7 . Survival T i m e of Certain G e r m s i n t h e Presence of a Disinfecting A g e n t (IO)
y = - a z :
=
(8)
K l + m
A
The maximum and minimum points occur where dy/dx = 0. But dv/dx = y ( K - y) . F'(x). Therefore we have maximum or minimum points wherever F'(x) = 0 when dy/dx = 0, y = 0, or y - K = 0; therefore the curve passes off to infinity asymptotic to the lines y = 0 and y = K . The points of inflection are determined by the intersections of Equation 8 with the curve, (10)
Dropping all powers of x above the nth, we have two cases to consider: n is odd or even, and a, is positive or negative. When n is even and a, is positive, the curve will be the type
It may be written
where K
= b/c = l/c K a t = -a
m
The rate of change of y with respect t o x is given by
If y is the variable changing with time 2, Equation 6 amounts to the assumption that the time rate of change of y varies directly as y and as ( K - y); constant K is the upper limit of growth-in other words, the value of the growth of y a t infinite time (z = a). Since the rate of growth of y is dependent upon factors that vary with time, we may generalize Equation 5 by using f(x) in place of -a', where f(x) IS some undefined function of time:
where m
=
where F ( z ) 1
constant of integration
= -KJf(z)dz
With the permission of the publishers, Williams & Wilkins Company.
698
0
l
t
3
4
5
6
7
8
7iM€ -M/uur€.s
Figure 8 . Survival T i m e of Eberthella typhi and Escherichia coli in t h e Presence of Chlorine Solutions ( f r o m Clack, 5 )
shown in Figure 2. If a,,is negative, the curve will have the same form, except that it will be asymptotic to line A B a t and x = - OJ and will lie between A B and the both x = x axis. If n, is odd and a, is negative, the curve will have the form shown in Figure 3. Since it is seldom necessary to use more than five constants, we may limit Equation 8 further by stopping a t the third power of (2). This leaves Equation 11.
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 35, No. 6
If a, is positive, the curve of Equation 8 is reversed and becomes asymptotic to line AB (Figure 2 inverted) a t .z = m and to the z axis a t z = - a . Thus, in Equation 11, a3 negative is a case of growth and a3 positive is a case of decay. Equation 11 has several special forms; among ther,. is a case similar to the autocatalytic curve (i. e., no maximum or minimum points and only one point of inflection) except that
+
I n both cases d represents the total growth attained in all the previous cycles. The term d is therefore the lower asymptote of the cycle of growth under consideration, and (d K ) is its upper asymptote. The general picture of such a growth curve of several successive cycles is shown in Figure 5. Whenever the growth within the different cycles is symmetrical or nearly so, there is considerable advantage in using Equation 12 rather than 13. Not only is the labor of fitting the curve less but the values of constant a' will give the rates of growth of the different epochs or cycles. Fitting the curve we have :
+
-
log m - log T/ F(z) log (IC - y) I n Equation 11, F ( z ) = a1z QX* a323
+
+
(14) (15)
Let log m = a ~ , a0
T€MFt%ATTVRE
+ alx + a& + aaza = log
(16)
- 'C,
Figure 9 . Percentage of Ethylene Burned a t Various Temperatures ( f r o m Campbell and Gray, 4 )
it is free from the two restrictive features-the location of the point of inflection in the middle and symmetry of the two limbs of the curve. Asymmetrical or skew curves of this sort can arise only when Equation 8 has no real roots. Any odd value of n may yield this form of curve; the simplest equation that will do it is the one where n = 3 (Figure 4),which is Equation 11. F T E R determining that the growth within any one cycle A may be represented by Equations 9 or 8, the next question is that of treating several consecutive cycles. Since
FigurelO.
Log y-Log x Curves f o r Various Processes
cycles of growth are to be additive, we may use for any single cycle the equation,
June, 1943
INDUSTRIAL AND ENGINEERING CHEMISTRY
699
From these equations, assuming the ordinates are equally spaced on the z axis, the value of K (the limiting ordinate) is obtained from: - Yo)(K -
Y:Y;(K
-
Y2)6(K
Y4)
=
-
YOY%(K ud4(1{ -
ys)*
(22)
I n most cases we know the limiting value of K , for we know the limiting theoretical yield, the total amount of suspension filtered, or the total amount of dry material filtered per plate. To obtain a, let: B,
=
log
(+) K - Y
- log
B~ = log (K+)
B3
=
log
- log (I+)
(?) K - YO
(33) K - Y -
log ?(a) K - Y
Then Equations 17 to 21, inclusive, can be expressed in terms of cc and B:
Figure 6 represents the equilibrium relation between water vapor and carbon dioxide in contact with hot carbon; the data are calculated to the basis of one mole of water vapor, and the individual fractions of carbon dioxide are plotted cumulatively. Figure 7 is the survival-time curve of certain germs (IO) in the presence of a disinfecting agent. Figure 8 presents similar data (6) on Eberthella typhi and Escherichia coli in the presence of chlorine solutions. Figure 9 shows the amount of ethylene burned (4) at various temperatures. K. M.TVatson (26) brought to the writer's attention a current practice at the Universal Oil Products Company of plotting on semiprobability paper the reciprocal of the absolute temperature against the percentage of hydrocarbon distilled by the Engler method. This practice is in line withi the present argument. I n some cases of differential distillation tentatively investigated, curves similar in form to Engler distillation curves are obtained by plotting 1/(y - z) cumulatively against y or z. The characteristic S-shaped curvature is common to the curves for all the processes cited. Khen the variables related by the above curves are plotted on a log y vs. log n. basis, they yield the convex curves noted to a lesser extent in the case of filtration. The growth curves slope away to the left and the die-away curves slope to the right (Figure 10).
T H E application of the Gomperta and logistic curves to a variety of data related to a growth function has been perhaps qualitative. And in many cases empirical equations not based on a probability function creditably represent much of the data cited. Holvever, it should not be long before the methods of statistical analysis and synthesis will be combined with the well known principles of chemical engineering unit operations as well as other significant physicochemical process data. LITERATURE CITED
a2 =
a3
4Bz
- 5B1 - B3 22;
Bs =
+ 3B1 - 3Bz 62:
The solution for the constants of the symmetrical form of growth curve may follow the lines of the preceding discussion. The symmetrical form, when written in the notation used above, becomes :
where K m
-a
= = =
b/c
l/c
= eaa
K a t = al
Assuming three points uniformly spaced along the x axis, as (0, yo), (zl, y J , and (2z1,y2), the values of the constants become :
(1) Alliott, E. A., J . SOC.Chem. Ind., 39,261T, 273-4T (1920). (2) Almy, C., and Lews, W.K., J. IUD. ENG.CHEM.,4, 528 (1912). (3) Baker, F. P., Ibid., 13,610 (1921)_ (4) Campbell, J. R., and Gray, T . , J . SOC.Chem. Ind.,'49, 451T (1930). Clack, C. E., thesis, Mich. State College, 1938. Gilse, J. P. M.van, Ginneken, P. J. H. van, and Waterman, H . I., J. SOC.Chem. Ind., 49, 444T (1930). (7) , Ibid., 49, 483T, 486T (1930). (8) Ibid., 50,41T, 95T (1930). (9) Gompertz, B., Trans. Royal Sac. (London), 1825,513-25. (10) Heathman. L. S.,Pierce, G . O., and Kahler, P., Pub. Hezlth Rept , 51, No. 40,1380 (1936). T?cLns Inut. (11) Hinchley, J. W., Ure, S.G. AX., and Claike, B W., Chem. Engrs. (London), 3, 24 (1926). (12) Hitchcock and Robinson, "Differential Equations in Applied Chemistry", 2nd ed., New York, John TViley & Sons, 1936. (13) Larian, M. G., Trans. Am. Inst. Ckem. Engrs., 35,623-34 (1939). (14) Pearl, Raymond, "Studies in Human Biology", Baltimore, Williams & Wilkins, 1924. (15) Ruth, B. F., Montillon, G. H., and Montonna, R. E., IND. Esc,. CHEIII.,25, 76-82 (1933). (16) Ibid., 25,153-61 (1933). (17) Scmcnoff, N., "Chemical Kinetics and C h a i n Reactions", London, Oxford Univ. Press, 1935. (18) Sperry, D. R., Cham. & M e t . Eno., 17,166 (1917). ESG. CHmf.. 18.276 (1926). 119) Soerrv. D. R.. IND. (20j 1&,"20, 892 (1928). (21) Underwood, A. J. V., J . SOC.Chem. Ind.. 47, 325T (1928). (22) Underwood, A. J. V., Proc. World Eng. Congr., Tokyo, 19Z9, 31, 245 (1931). (23) Ibid., 31,258 (1931). (24) Underwood, A. J. V., Trans. Inst. C h e n . Engrs. (London) 4, 19 (1926). (25) Watson, K. M., private communication. ~
~I
ao = log
7CO
(yo:) K-V
PREEENTED as part of the Symposium on Industrial Reaction Rates held under the auspices of t h e Division of Industrial and Engineering Chemistry, AXERICAN CHEMICAL SOCIETY,at Chicago, Ill.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 35,
No. 6
