OPTIMUM TIME SCHEDULING OF KINETIC EXPERIMENTS KENNETH L. LINDSAY Ethyl Gorp., Baton Rouge, La.
The design of kinetic experiments has been considered as a problem in selecting the best time intervals for measuring the composition of the reacting mixture. An approach based on transformation of the time variable to a dimensionless variable, 0, i s described. This transformation leads to a time schedule for making composition mieasurements which results in a reasonably close approach to equal changes in composition with successive integral changes in 0 throughout the entire time range of the experiment. This minimizes the number of measurements required for evaluating a rate constant to a desired level of accuracy, and permits facile calculation of the instantaneous reaction rate b y methods of numerical differentiation. The method i s extremely versatile and can b e applied to almost any reaction, simple or complex, regardless of whether or not the form of the rate equation i s known in advance.
industrially practicable chemical processes fail to obey Variations in density, volume, gas solubility, temperature, pressure, and similar factors during the progress of the reaction usually invalidate the simplifying assumption of constant environment o n which the classical equations are derived. If these factors vary in a predictable way, it is possible to modify the classical equations by including terms describing their variation into the rate equations. This leads to rate equations which can be extrapolated somewhat more confidently than wholly empirical equations, but which cannot usually be integrated analytically because of their complexity. Under these circumsi ances, if the kineticist wishes to test several possible rate equations, it will usually prove more practical to differentiate the experimental data rather than attempt numerical integration of each rate equation. I n a typical case, he plans to measure a composition variable, x , related to the degree of completion of the reaction, as a function of the time, t , and to calculate the rate, dxldt, a t each of several values o f t . He wishes to make n different measurements of x corresponding to n different values o f t . It is assumed that he can measure t with sufficient accuracy that all of his error arises in his measurement of x, and that this error is independent of the size of x . He is free to select his n values of t arbitrarily. How should he make the selection? The reaction rate, dx,‘dt, decreases more or les, rapidly as the reaction progresses, excrpt for zero order reactions and occasional freaks such as autocatalytic reactions. ,4n efficient experimental design would be one in which all measured values of dxldt were equally precise. Because of the large variation in magnitude of dx/dt over the course of an experiment, it is the proportional error-the ratio of the error in dxldt to a’xldtwhich should be equalized. This objective requires a sampling OST
M the simple rate equations of classical kinetics.
schedule such that the changes in x between one sampling time and the next should be equal throughout the course of the following experiment. If for simplicity, r = dx/dt 2 &/At; and if V ( x ) , V ( t ) ,and V ( r ) are the variances in measuring x , t, and r; then [if V ( x ) is independent of x and V ( t ) = 0, as assumed above]: 1 2 V(r) 2 V ( r ) = - V(Axj = - V ( x ) , and - = - V ( x ) . At At r Ax
I
Hence, equal values of Ax lead to equal values of V ( r ) .
Such a time schedule will be referred to as an optimum time schedule. For an experiment aimed at evaluating a rate constant for a known rate equation, rather than at selecting from a number of possible rate equations, a true optimum time schedule would differ somewhat. and would depend on the form of the rate equation, but the ..equal Y interval” time schedule would be satisfactory for all practical purposes. A problem arises in the calculation of dx,’dt by numerical differentiation for a n experiment designed with equal r intervals. The computation of the derivative is considerably less difficult if the independent variable is equally spaced. It would be possible to calculate dt/dx and invert. but random errors in x would prevent the attainment of absolutely equal x intervals even in a perfectly designed experiment. The approach described below provide. a means of avoiding this computational difficulty and deriving a near-optimum time schedule without advance knowledge of the rate constant or the form of the rate equation for any reaction in which the rate decreases with time. The recommended approach consists of a change in the time variable from t, measured in any convenient units such as minutes, to a dimensionless time variable, 0, which takes on successive values. VOL. 1
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Table 1. -
+/n 2 4 6
8 10
6
7
8
0.4490 0.8421 1 ,0960 1 ,2894 1 ,4477
0.3375 0.6272 0.8081 0.9430 1.0519
0.2693 0,4982 0,6379 0.7406 0.8226
as a Function of n and 4 / n n Y 10 71 0.2236 0.1909 0,1664 0.4126 0.3518 0.3064 0.5260 0,4471 0.3885 0.6085 0.5159 0.4475 0,6740 0.5702 0.4938 CY
72
13
14
0.1474 0.2713 0.3434 0.3949 0,4352
0.1322 0.2434 0,3076 0.3533 0,3890
0.1198 0.2206 0.2786 0.3196 0.3516
Recommended Approach
80
The recommended approach consists of a change in the time variable from t , measured in any convenient units such as minutes, to a dimensionless time variable, e, which takes n - 1 as each of the on successive values of e = 0 , 1, 2, 3 n samples in turn is collected. This transformation automatically provides equal intervals for the independent variable and permits ready calculation of dx/d8 with either a hand calculator or a digital computer, using, for example, the method of Whitaker and Pigford (7). There remains the problem of defining a relationship between t and e having sufficient flexibility to permit a reasonably close approach to equal x intervals without advance knowledge of the mathematical form of the rate equation. For this purpose, it is convenient to introduce two dimensionless parameters, 6 and CY, and a scaling factor, c. These are defined in terms of the initial time interval of the experiment, tl - to. and the total time range of the experiment. trL-l - to. Thus. if t o = 0, t l = CYG. and = $ac, or Q = t n . - l ~ t l . In terms of these parameters. the relationship between t and 8 is defined by
-
z 0
5
60-
a
I
s I-
z 40a W a
20 -
B -
t = c(&
or its equivalent
8 Figure 1.
+ 1 y - l- c
A typical time-scheduled reaction The two dimensionless parameters, $I and CY, are not independent. By substituting t = t l = CYG into Equation 2> their relationship can be shown. This substitution gives
80-
or (CY
+ 1)n-1 = +a + 1
(3)
In order to convert dx/d8 to dxldt, it is necessary to evaluate the multiplier, dO/dt. Equation 1 can be readily differentiated and inverted. giving a
(4)
Successive values of d8/dt can be readily evaluated, using
1 -
=
0
(+a
+i~-l-~
(5b)
In a p p l y i n i these equations to the design of a kinetic experiment the following steps are taken: 0
1
2
3
4
5
6
7
8 Figure 2. Evolution of a time schedule for an isomerization reaction 242
(de/dt),=j-l
I&EC FUNDAMENTALS
The number of measurements, n, and the total time range of the experiment, t,z- are selected. The time, t l ? required for the reaction to progress l / ( n - 1) of the way toward the degree of completion expected a t t,-l
80
-
- 240
5c -
ln
W 2 60-
-180
8 -
-
z 3
-120
gII--
z
a w a
-
-
-60 I
2
Figure 3.
3
4
5
~
1
= ai -
(a,
9
1
0
0
d x l d t . I n addition, near optimization of the time schedule permits more efficient use of each measurement? so that fewer measurements are required for a given precision in estimation of a rate constant. The resultant saving in analytical time frequently overshadows the other advantages of this approach. Two other characteristics of this method of time scale transformation deserve mention. Once a near-optimum combination of 4; a , and n has been found? this combination will be effective despite changes in temperature, dilution, catalyst concentration, or similar factors which affect the total time range of the experiment. It is necessary only to compute a neiv value of the scaling factor corresponding to the new t,-1 and to adjust the time schedule accordingly. This adjustment requires no more than 10 minutes. The other important characteristic of this time-transformation function is that it is open-ended with respect to the number of measurements, n. and the total time range of the experiment. Suppose that a time schedule has been calculated as described above and put into practice. If? near the end of the experiment, one or more additional measurements are desirable because the
+ I)"-' - dff - 1 + 1y-* - i ~~
(n -
8
8 Time scheduling applied to a simple first-order reaction
is estimated, giving @ = t n - l / t l . This is best done from an inspection of data from a previous experiment, and progressive improvements in the equalization of the x intervals result as more data are accumulated. Hints on the selection of 4 in the absence of previous data will be given later. Using the value of @ cielected in the step above and Equation 3, a is calculated. Equation 3 is not particularly easy to solve for a ; hence it is desirable to select, where possible, a suitable a value from Table I. If intermediate values of a are desired, Newton's method of iteration can be applied to Equation 3 , Convergence is rapid and requires no more than two or three iterations in most cases The pertinent iterative solution is a,
7
6
l)(..i
The scaling factor, c, is then calculated from t, - = Gat. The time schedule (value o f t for each value of 8 from 8 = 0 to 8 = n - 1) is calculated from Equation 2. Equations 5a and 5b are used to compute values of d8/dt for each value of 8. After a little practicmc, these steps can be completed on a hand calculator in about half a n hour. The time thus invested will be later repaid severalfold during the computation of
!rd ln
2 60
180
E
1120
I
I
5
6
7
8
-0
I
9
1
0
e Figure 4.
(A
Time scheduling applied to a second-order reaction
+ A -+ products)
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I
1001
"1
60
CASE
m
- k-1 =0.8 kl
I
0
2
4
I
8
I
6
I
I
8
I
10
Figure 5. Time scheduling applied to a second-order reaction ( A B + products)
Figure 6. Time scheduling applied lo a first-order reversible reaction
reaction is proceeding more slowly than anticipated, these 1, can be scheduled quickly by letting e take on values of n, n n 2, etc., instead of stopping a t 8 = n - 1 as originally planned. This was one of the main reasons for the choice of the present function instead of one of several other timetransformation functions which might have been selected.
carry the reaction further toward completion in the allotted time range. The resultant time schedule gave a highly satisfactory approach to linearity in the per cent completion us. 8 plot. Had further experiments been carried out, it seems likely that setting = 18 or 19 would have resulted in a slight additional improvement. How successful is the present time scale transformation function when applied to several simple types of reaction for which true optimum time schedules can be derived by integration? This has been tested, and the results are illustrated in Figures 3 to 6. In each case, appropriate values of 4 were calculated from the corresponding values of t,-1 and t l in the actual optimum time schedule obtained by integration of the differential rate equation. The reactions tested included: a simple first-order reaction (Figure 3) ; a second-order reaction of the type A A +. Products (Figure 4); three different cases of a second-order reaction of the type A B +. Products (Figure 5) ; and three different cases of a first-order reversible reaction (Figure 6). The fit in each case was adequate, illustrating the versatility of the transformation function. The method used for selecting 4 in these examples did not result in the best possible fit, but was used because it was simple, adequate, and consistent with the method recommended for use in designing actual experiments. A slightly better fit could be achieved using a somewhat larger value or 4 in each case. This would result in an S-shaped plot of per cent completion us. 0, similar to Figure 1, with a slight improvement in over-all fit.
+
+
+
Examples
A typical example of time-scheduling in practice is illustrated in Figure 1. The reaction in this case was a gas liquid reaction without solvent; the composition and density of the liquid phase and the solubility of the gas all changed drastically as the reaction progressed. Despite these complexities, the time scale transformation function permitted a close approach to an optimum time schedule. Not a11 attempts a t time-scheduling are as successful as this one, particularly if little previous kinetic data are available. Figure 2 illustrates the evolution of a time schedule for an isomerization reaction. The reaction in this case was also rather complex and was assumed to involve three successive cycles of elimination-addition reactions. The first run was attempted with a time schedule based on a value of 84 for 4. As seen in Figure 2, this schedule was a poor one. Too many points were concentrated during the first 10% of the reaction. The first sample in Run 1 had been collected after 5 minutes. I n Run 2, collection of the first sample was delayed to 20 minutes after the start of the experiment. The total time range of the experiment was kept the same, leading to a value of 21 for 4, and the temperature was increased 10' C. to 244
I&EC FUNDAMENTALS
$J
+
+
Estimating $
T h e study of the simple, integrable reactions has been helpful in suggesting ways of estimating @ before any experimental data are a t hand. For a simple first-order reaction, good fits are obtained when 4 lies somewhere between 2 n and 2.5n. Similarly, for second-order reactions of the A A type, good fits are obtained with values of $ lying between 7 n and 9n. Second-order reactions of the A B type lie between these two cases. When the initial ratio of B to A (&,/ao in Figure 5) is close to unity, the reaction approaches a second-order reA type, and the upper range of $ values action of the A (perhaps 5n to 7n) applies. When one of the two reactants is in excess, however, the reaction approaches a first-order reaction, and $ should lie in the 3n to 5 n range. Further refinement in $ values is easy after the first experimental run is complete.
+
+
+
Nomenclature c = scaling factor, equal to t, - 1/@a:, minutes
i
=
indexing subscript used to designate a particular one of a series of converging iterative solutions of Equation 3 for a
j = a particular value of0 in Equation 5b n = number of measurements of x which it is planned to make in a kinetic experiment, including the measurement a t
t = time; initial time, to, is taken as 0 always, minutes x = any variable, related to the composition, which is meas-
ured as a function of time in order to follow the progress of a reaction
Greeks a: = nonarbitrary dimensionless parameter used in setting up time schedule; obtained from @ and n by solution of
Equation 3 8 = dimensionless time variable which takes on successive values of 0, 1, 2, 3, . . n - 1 as each measurement of x is made in turn @ = arbitrary dimensionless parameter used in setting up time schedule and equal to t, - l / t l Other units can be used for t and c, provided the same units are used for both. If the time is measured in seconds, however, the assumption that the error in the time measurement is negligible may not be valid. I n this case, another approach to time scheduling might be preferable. I n the limiting case in which the error in measuring x is negligible compared with the error in the time measurement, the best schedule would be one involving equal time intervals.
.
Literature Cited (1) Whitaker, S., Pigford, R. L., Znd. Eng. Chem. 52, 185 (1960).
RECEIVED for review Au ust 18, 1961 ACCEPTED fu1y 30, 1962
t = O
MULTIVARIABLE SYSTEMS Anabsis and Feefivorward Control Synthesis R . E. B O L L I N G E R I A N D D . E. L A M B
Department of Chemical Engineering, Uniaersity of Delaware, hrewark, Del.
Methods are developed for analyzing multivariable systems which may include feedforward and feedback control, and for designing feedforward controllers for such systems. These procedures are applicable to systems which can b e described by linear time-invariant models and for which the control system goal is to maintain outputs constant. This approach provides a convenient framework for selecting variables to b e monitored, mcinipulative inputs, and outputs to b e controlled.
characteristic of any control system is the goal which it seeks to attain. A control system to achieve a specific goal can be designed using a variety of mathematical models of the process to be controlled. Possible goals and models are :
A
BASIC
Controller Goals Maximize profit B. Vary outputs in predetermined manner C. Hold outputs constant A.
A.
B.
Process Models Dynamic 1. Nonlinear 2. Linear time-varying 3. Linear time-invariant Steady state
This work is concerned with the design of control systems which adjust process inputs to hold outputs constant. ProcPresent address, Esso Research and Engineering Co., Madison, N. J.
esses to be controlled are represented by linear, time-invariant models, and it is assumed that the process dynamics are known quantitatively. Some of the techniques discussed have their origin in recent work reported in the electrical engineering literature (2, 4 ) . However, several factors make the design of control systems for most electromechanical devices significantly different from that of most process systems. These factors include performance criteria, the nature of input disturbances, and the dynamics of the system to be controlled (7). These differences have prevented process control designers from using a general approach i n multivariable problems. I t is hoped that the treatment presented in this paper, which makes apparent the role of different types of inputs and different sections of the plant transfer matrix, and which separates VOL. 1
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