Q U A N T I T A T I V E R A T E COEFFICIENTS FROM P U L S E D MICROCATALYTIC REACTORS Ethlene Hydrogensation ouer Alumina B L A N T O N , J R . , C H A R L E S H. B Y E R S , ’ A N D R O B E R T P. M E R R I L L Department of Chemical Engineering, Uniuersity af California, Berkeley, Calif. 94720
W I L L I A M A.
The use of puls8edmicrocatalytic reactors could be broadened considerably if quantitative rate coefficients could b e extracted from the conversion data. Until recently it has not been clear how this could b e done except for first-order kinetics. The problem is solved numerically for the case of a concentration pulse which does not broaden in the reactor. The results of calculations for several nonlinear rate equations are presented. Essentially the same amount o f kinetic information can be extracted from the over-all conversion during a pulse run as from a steady-state flow run. There is potentially more kinetic information in the shape of the output pulse. In the hydrogenation o f ethylene over alumina close agreement was found for rate coefficients derived from pulse and flow reactors. HE pulsed-flow microcatalytic reactor system described by TKokes et al. (1955) is noIv in use in many laboratories, primarily as a qualitative analytical tool. This system consists of a reactor containing a small amount of granular catalyst through which is passed a pulse of reactant in a stream of carrier gas. The resulting. conversion is then usually measured gas chromatographically. An immediately apparent advantage of this system is that a great deal of data can be obtained in a very short time compared to the time required using a steadystate system. Another advantage is that the technique allows one to study more exhaustively initial rates, poisoning effects, catalyst deactivation, and other relaxation phenomena which often give insight into the reaction mechanism on the catalyst. These reactors are also a(dvantageous in isotope tracer studies \\.here the cost of the reactant is too high to permit the use of the large quantities needed in a steady-state reactor. The primary disadvantage of the pulsed-flow system has been the clifliculty of obtaining quantitative results due to the variation of reactant concentration with time and position in the reactor. I n the past, quantitative results have been possible only for first-order reactions, since under certain conditions for a firstorder reaction the conversion is independent of the peak shape (Bassett and Hahgood, 1960; Gaziev et al., 1963). Gaziev et al. (1963) attempted to extend the usefulness of this technique to kinetic equations other thafi first-order by solving analytically the case of square-topped and triangular input peaks with no axial dispersion. However, the nondispersed square-topped peak is mathematically identical to the steady-state case and the triangular peak is difficult to obtain experimentally. Based on Gaziev’s work, Schwab and Watson (1965) compared experimentally the pulse and flow techniques, but as the system used, methanol decomposition on silver, obeys first-order kinetics, a real test of their analysis was not provided. Collins and Deans (1968) have investigated pulsed reactors with axial dispersion for the case of a reaction at equilibrium via a radioactive tracer technique, but here also the kinetics are linearized. A more general treatment has been given (Deans et al., 1967), but solutions so far are available only for linear kinetics. Recently Bett and Hall (1967) have shown how the pulsed microcatalytic technique can be used to complement the steadystate f l o i v technique for very strongly adsorbed products.
Present address, Department of Chemical University of Rochester, Rochester, N. Y .
Engineering,
McGee (Bett and Hall, 1968) and Hattori and Murakami (1968) have given special attention to reversible reactions. T h e present work was undertaken to extend the usefulness of the pulsed-flow technique to the more general cases of nonfirst-order kinetics. Numerical solutions were obtained for several rate expressions assuming a Gaussian input pulse. These results were then applied experimentally and the resulting rate coefficients checked by comparing them to those obtained for the same reaction in a steady-state reactor. Numerical Solutions
The continuity equation for an isothermal catalyst bed through which reactants pass in plug flow is:
Equation 1 can be nondimensionalized using the relationships:
T
=
C=
tU/L
CJC,
Y = y/L
(2)
The continuity equation in dimensionless form is then
(3) T h e quantity in brackets is the inverse of the Peclet number based on the particle diameter, which under conditions for chromatographic separation is generally near unity. Usually the ratio of particle diameter to bed length d / L is small and if the concentration profile changes slowly enough, the secondorder term can be neglected and Equation 3 reduces to:
bC b~
bC by
-+-=-R This equation can now be solved for the initial condition:
C = 0
at
T
=
0 for all Y
and the boundary condition :
C = f ( T ) at
Y = 0 for all T
>0
A Gaussian input pulse is relatively easy to produce experimentally by dispersing a narrow input pulse before it enters the reactor. By using a dispersion column with a length much greater than that of the reactor, the pulse can be made broad enough to approach the condition implicit in Equation 4Le., the axial dispersion term is negligible. Furthermore, this VOL. 7
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K.V"Ld'
K
0
Figure 1. Calculated results for simple power-law kinetics of the form ra = kCa 7~
Figure 2. Dependence of conversion on dimensionless rate constant for power-law kinetics
condition may be verified experimentally by observing the pulse before and after passing it through the catalyst bed, since axial diffusion causes dispersion. A finite difference technique was used to obtain solutions to Equation 4 using a Gaussian pulse forf( T ):
c
L-(A) 1 r
= exp
/T\~I
at Y =
o
(7)
where A equals standard deviation of the Gaussian in units of T . Because of stability considerations, it was necessary to use an implicit finite difference scheme, the details of which are given by Blanton (1966). The results of the computer solutions are shown in Figures 1 through 6. Figure 1 shows the results for simple power-law kinetics having the general rate expression
2
I
'0
3
K Figure 3. Dependence of JdX/(l - x ) " on dimensionless rate constant for power-law kinetics A Calculated from second-order triangular pulse solution of Gaziev
Solutions were obtained for n = 1/2, 1, 3/2, and 2 (see Figure 1). For convenience, the results werep lotted as conversion X us. a dimensionless rate constant, K , defined by the equation (9) where V is the void volume of the reactor and Q is the volumetric flow rate of reactant in the reactor-Le., the cross-sectional area of flow multiplied by U , the velocity of the concentration pulse of reactant measured at the actual conditions of the experiment. For the first-order case the curve obeys the relationship
K = In
1 1 -
(10)
which can be shown to be independent of the peak shape for a nonbroadening pulse. Thus it has often been assumed-see, for example, Bassett and Habgood (1960)-that if data plotted as In 1/(1 - x) us. reciprocal flow rate gives a straight line, the reaction is first-order. I n Figure 2, however, the results of the calculations for n = 2 and n = 1/2 are plotted according to Equation 10 along with the first-order case. At low conversions it is clear that all three are straight lines. Although it is not obvious from Figure 2, i t can be shown that substantial deviations from linearity do not occur for the non-first-order reactions for conversions below 40 to 50%. Equation 10 is strictly true only for the case when the axial dispersion is negligible. For quantitative measurements of the first-order rate constant the validity of this assumption should be adequately demonstrated. The dependence of conversion on the shape of the input 612
I&EC FUNDAMENTALS
et ol. ( 1 9 6 3 )
Table I.
Slope Correction Factor for Pulsed Reactors with Power Law Kinetics Reaction Order, Slope Correction R Factor, m
1 y
4{3
3 /2 2
3/4 2/3
pulse is shown in Figure 3. For the square-topped pulse or steady-state case, a plot of K us. J d r / ( l - x)" Lvould give a straight line with a slope of unity. The plotted points for the triangular pulse are the second-order analytical solution of Gaziev et al. (1963) and the curve for the second-order Gaussian pulse is plotted from the numerical calculations in Figure 1. There is little difference between the conversions for a triangular input and a Gaussian input. The triangular input falls slightly below the Gaussian up to a K value of 4, above which it is somewhat greater. The significant point is that both these pulses give conversions which are nearly straight lines. An analogous result is found with the other nth-order cases, so that for conversions up to 50% all orders can be represented, as shown in Figure 3, by the following equation :
where m is given in Table I. For conversions greater than 50y0 the pulsed cases deviate from Equation 11. The use of 11 for establishing the rate
1
Figure 4. Calculated results for first-order Hinshelwood kinetics
Langmuir-
n = q = l
Figure 7. Effect of reaction order on shape of exit pulse for constant conversion
0.2 0 0.05
0.5
0.1
Figure 5. kinetics
1.0
K O = k,C,
5.0
10
50
100
Calculated results for Langmuir-Hinshelwood n = 2,q = 1
I.
o
w
?
Figure 8. Effect of dimensionless rate constant on exit pulse shape for half-order kinetics
-
\
0.8 -
A dimensionless “adsorption” constant was defined for this case by :
06-
K,
Figure 6. kinetics
k,Coq
(13)
and the results are plotted as conversion us. K , at constant values of K , the dimensionless rate constant defined by Equation 9. No simple relationship between the pulsed and steadystate cases like Equation 11 has been found for the LangmuirHinshelwood kinetics, and the parameters must be obtained by fitting a set of conversion data to the curves in Figures 4, 5, and
0.2 -
0.05 0.1
=
O5
lo
Ko:k,CA’t
5.0
IO
50
Calculated results for Langmuir-Hinshelwood n := 312, q = 112
order would require a set of experiments in which the value of C, is varied, since the functional dependence of the conversion on flow rate alone is insensitive to the reaction order (see Figure 2) in the range of conversions where Equation 11 is valid. The results depicted in Figure 1 may be used for all ranges of conversion. Figures 4, 5, and 6 are the results for Langmuir-Hinshelwood kinetics of the general form:
6. Another source of kinetic information is the shape of the exiting concentration pulse. For the first-order case, the output pulse is also Gaussian, but for all other kinetic equations there will be some distortion of the Gaussian, though the pulses are still symmetrical. This is illustrated in Figure 7 for the power-law kinetics. In the coordinates of Figure 7 the Gaussian is a straight line. As expected, the first-order case is straight with unity slope. All other cases give curves which are concave upward. Those with orders less than 1 have initial slopes greater than unity and those with orders greater than 1 have initial slopes less than unity. The magnitude of the initial slopes and the extent of the curvature depend on the conversion. A typical example of this dependence is seen in Figure 8 for the half-order case. All the pulses illustrated in Figure 7 have nearly equal conversions. VOL. 7
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4
3
I THERMOCOUPLE
COLUMN I
0
1
I
1
I
( T/A
/
3
2
I
4
)2
Figure 9. Exit pulse shape as a function of dimensionless coefficients K and K2 for firstorder Langmuir-Hinshelwood kinetics n = q = l
Pulses from the first-order Langrnuir-Hinshelwood rate equation (Equation 12 with n = q = l ) are plotted in Figure 9 for a series of cases near 60% conversion. This rate equation
gives pseudo-orders between 0 and 1, depending on the value of kpC. As expected from the power-law case, the slopes are all initially greater than unity. The curvature of the lines, however, is different than for the power-law case. For the smaller values of K 1 the upper portion of the curve has a concave downward curvature, while all the power-law results are concave upward. Thus, careful analysis of the shapes of the exit pulses can also reveal additional information about the kinetics of the reaction, supplemental to that obtained from the integrated conversion data. A single pulse could indicate whether the order was greater or less than unity and from three or four pulses it is possible to tell whether a Langmuir-Hinshelwood type equation would be more appropriate than a simple power-law expression. Experimental Studies
To test the validity of these results the hydrogenation of ethylene over alumina was investigated in a pulsed reactor. This reaction had previously been found to yield only ethane and to have non-first-order kinetics over a rather large temperature range (Hindin and Weller, 1956; Sinfelt, 1964). The apparatus consisted essentially of a modified version of the system described by Kokes et al. (1955). The primary modification was the addition of thermal conductivity cells in the line before and after the catalyst bed, allowing the reactor input and output concentration profiles to be measured. A dispersion column (0.18-inch inside diameter stainless steel tubing, 10 feet long) \vas also inserted between the sample injection valve and the reactor, to obtain the broad Gaussian input pulse. The column ivas sized to give pulses broad enough so that additional broadening by the catalyst bed was negligible, as determined by comparison of the input with the output pulse. This system is shown in Figure 10. The hydrogen carrier gas, after final cleanup, is passed through the reference sides of the thermal conductivity cells. I t then flows through the sample injection valve, where a pulse of ethylene can be introduced. This pulse is then carried into the disper614
I&EC FUNDAMENTALS
Figure 18. Schematic diagram of modified microcatalytic reactor system
sion column and recorded just prior to entering the reactor. After passing through the reactor, the product pulse is recorded and analyzed by chromatography with a silica gel column. The reactor was a 6-mm. Vycor glass tube (4-mm. inside diameter) enclosed in a vertical resistance-type furnace having an inside diameter of 1.5 inches and a length of 13 inches. The catalyst charge, consisting of 0.500 =t 0.005 gram of alumina, was held in place with glass \vool plugs and occupied a length of about 1.5 inches midway in the furnace. The ends of the furnace were plugged with glass wool to reduce convection currents. The reactor temperature was controlled with a variable transformer and measured with a Chromel-Alumel thermocouple inserted in a 6-mrn. Vycor tube parallel to the reactor tube. The catalyst was made from A41coa's(2-37 alumina, reported to be a high purity &alumina trihydrate. I t consisted of irregularly shaped particles with an average particle size of 100 microns and a void fraction of 0.50. Activation was accomplished by heating in situ in floiving hydrogen (60 cc. per minute) a t 400' C. for 4 hours and then at 600" C. overnight. This procedure is reported to give 7-alumina (Sinfelt, 1964). The resulting BET surface area was found to be 290 sq. meters per gram. Prepurified hydrogen was obtained from the Matheson Co. and further treated by flowing through a Deoxo unit and then over activated alumina to remove water and catalyst poisons. C.P. grade ethylene, also from Matheson, was used directly from the cylinder. Conversions of ethylene to ethane were measured over the temperature range of 100' to 350' C. with the input peak concentration C, varied over two orders of magnitude and the volumetric flow rate, Q, varied by a factor of 3. Over the pressure range studied, 1000 to 1500 mm. of Hg, the reaction was first-order in hydrogen, as reported by Sinfelt (1964). I t was assumed that the partial pressure of the hydrogen carrier gas was equal to the total pressure over the catalyst, since the peak partial pressure of reactant ethylene was generally less than the total pressure by at least an order of magnitude. The resulting data were found to fit the model for both half-order power-law kinetics and the first-order Langmuir-Hinshelwood case. For the half-order case, the rate equation is given by
where k'P,, is equivalent to k in Equation 8. The rate constant, k', is obtained by plotting the group of variables PH2/ QC212 us. conversion and then superimposing this plot on the half-order curve of Figure 1. If the flow rate is varied at constant temperature, the plot of P H J Q C ~ against ~ / ~ conversion will give a single curve, as shown by the data of Figure 11.
ro
-0
-3
Figure 1 1 .
Half-order results for runs made at 2 9 5 " C.
Flow rater, Q.
0 30.3 cc./min.
056.1 cc./min.
1.5
2 IITOK
A94.2 cc./min.
Figure 13. 1.01
O,lOOo. V, 150". A,200°. 0,250'.
0,300'.
Figure 14.
0,350'
I
I
I
C!
Q
(I7)
and
KO = kaCo
I
= k,Cg
Langmuir-Hinshelwood fit for data of Figure 1 1
0.6
X
0.4
0.2
Figure 15.
and the dimensionless coefficients are given by
v k = ___ Vk'pH, __
1
0.8
0 0. I
=
I
I .o
Since the rate constant i s temperature-dependent, a family of curves will result Lvhen the temperature is varied, as shocvn in Figure 12. A rate constant can then be calculated for each temperature and the usual Arrhenius plot constructed to give the activation energy, as c,een in Figure 13. For comparison a curve calculated from the steady-state data published by Sinfelt (1964) is also shown. The agreement is adequate \vhen one considers the sensitivity of alumina catalysts to small changes in pretreatment and environment (Hindin and TVeller, 1956). For first-order Langniuir-Hinshelwood kinetics, the rate equation is :
A'
x 103
Arrhenius plot for half-order kinetics
K, Figure 12. Half-order results with flow rate of 41.2 cc. per minute at injection conditions
2.5
(18)
The values of constants k' and k , are obtained by plotting C, us. conversion and superimposing the plot on the calculated curves of Figure 4. For the identical data used in Figure 11, the re-
0.5
1.0
K a = kC,,
5.0
IO
50
100
Langmuir-Hinshelwood fit for data of Figure 1 2
sulting fitted Langmuir-Hinshelwood plot is shown in Figure 14. T h e data seem to give a better fit than for the half-order use in Figure 15. A rate constant and an adsorption constant can be calculated for each temperature. If a Rideal-Eley mechanism is assumed, coefficient k' will be the product of a rate coefficient and the adsorption coefficient, k, (Bond, 1962). Thus the activation VOL. 7
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615
x N
-
I
I” E F 0 -” N a
1:
0, I L
c \
-e, E
-
0.5 -
-
-
0,
-
c)
x \
r
0.I
I
I
I
I
1.5 I/T Figure 1 6.
I
I
I
2.0 OK
x
I
I
I
I
I
2.5
103
Langmuir-Hinshelwood Arrhenius plot
A
From data of Figure 14. 0 From dato taken on another catalyst sample. D From data of Figure 15
0.3 I.5
Figure 17. plot
I 7
2 .o OK-’
2.5
x
lo3
Langmuir-Hinshelwood van’t Hoff
0 From data of Figure 15. A From data of Figure 14.
0From data taken on anolher catalyst sample energy for the reaction is found by plotting log (k’lk,) against the inverse of the temperature as in Figure 16. The temperature dependence of the adsorption coeflicient is shown in Figure 17. In both figures the data resulting from another catalyst sample are also plotted to indicate the reproducibility of the system. Although the magnitude of the rate constant and the adsorption constant vary as much as a factor of 2, the slopes in Figures 16 and 17 are essentially the same. Discussion and Conclusions
The nearly quantitative agreement between the pulsed reactor rate parameters and the steady-state data of Sinfelt (1964), as shobvn in Figure 13, establishes the validity of the numerical calculations presented here. There are interesting nonlinear rate equations of forms other than those presented which can be solved numerically by the interested investiyator. The important contribution of this work is that i t is possible to achieve in the laboratory conditions consistent with the assumptions inherent in the use of Equation 4. One such assumption, which has not been explicitly stated previously, is that the gasphase partial pressure of reactants must be that which is in steady state with the catalyst surface during the entire pulse. Put another ivay this means that the time constant for adsorption and/or diffusion processes must be much less than for change in gas-phase partial pressure. If this is not the case, the peaks will be broadened by the reactor, but the simple expedient of operating so that no perceptible broadening occurs ensures that the experiment has been carried out under conditions consistent with the use of Equation 4. The importance of establishing this fact experimentally cannot be overemphasized even for first-order kinetics. Another assumption implicit in the use of Equation 4 is that QC, must be greater than the net rate of reaction. There will always be a portion of the leading and trailing edge of the pulse where this condition is not met. Practically, what is required is that QC, be greater than the reaction rate for those portions of pulse which contribute substantially to the over-all conversion. If this condition is not met, the investigator will observe asymmetrical broadening under reaction conditions which Ivould not be present in the absence of the chemical reaction. The simple 616
l&EC FUNDAMENTALS
remedy is to increase the injection volume until this asymmetrical broadening is no longer evident. The same amount of quantitative kinetic information may be obtained from the conversion data in a pulsed reactor as from a flow reactor, if the pulse is broad enough and concentrated enough. Care should be taken to verify these conditions experimentally and there is, of course, no a priori guarantee that for all possible systems a range of operation may be found where both conditions can be met. \\‘hen they can, however, the measurement of input pulse heights along Jvith over-all conversions permits the extraction of nonlinear kinetic rate coeficients from the conversion data. It is possiblej furthermore, from careful analysis of the output pulse to distinguish between poiver-law kinetics and LangmuirHinshelwood kinetics. The ethylene hydrogenation reaction illustrates how this approach might be used. Sinfelt (1964) used a poiver-laiv expression to correlate his data and argued that the change in the apparent activation energy (see Figure 1) resulted from a change in the rate-determining mechanism at the higher temperatures. The use of the Langmuir-Hinshelwood mechanism results in a kinetic rate coeficient whose activation energy does not vary with temperature. All of the variation in the apparent activation energy is seen to occur because the adsorption constant has little or no temperature dependence at low temperatures, but an apparent heat of adsorption of 10 kcal. per mole at high temperatures (see Figure 17). Adsorption constants derived from kinetic experiments represent only sites which are kinetically active. If the number of sites which contribute to the catalytic reaction is increasing with temperature, the apparent temperature-dependence of kinetically active sites will be less than the temperature-dependence of the adsorption on the total surface. I t is believed that the low apparent heat of adsorption shown in Figure 17 for the low temperatures may result from this sort of surface heterogeneity and that at high temperatures all sites are active and so the value of 10 kcal. per mole is a representative average for the entire surface. The analysis of the shapes of the output pulse, in principle, may be used to distinguish betxveen a mechanism like that of
Sinfelt (1964) and the one presented here for the LangmuirHinshelwood kinetics. I n fact, the constants chosen for the pulses shown in Figure 9 are those found empirically in these studies of ethylene hydrogenation at a conversion of 59%. I t is clear that such shapes are easily distinguishable from a set like those of Figure 8, which are similar to those ope would find for a system obeying half-order kinetics. Unfortunately, it was not possible to distingutsh ethylene from ethane in the thermal conductivity cells used as detectors in this study, so no clear choice between the rate expressions is possible for this system. With the use of an olefin-specific detector, however, it would be possible to choose the more likely rate equation. I n general it would not be possible to make such a distinction from d a t a taken in a steady-state reactor. Nomenclature
A C CA C,
Delf d
E,
f( T )
-AH, k , k’
K
k, K, L m n
pH2 Q
= = = = = = =
= = = = = = =
= = = =
arbitrary constant involved in Gaussian exponential dimensionless concentration, CA/C, concentration of component A input peak concentration effective reactant diffusivity catalyst particle diameter effective activation energy input peak shape function effective heat of adsorption rate coefficients dimensionless rate coefficient, VkC,.-’/Q adsorption coeflicient dimensionless coeficient, k a C a ~ over-all length of catalyst bed slope correction factor defined by Equation 11 reaction order partial pressure of hydrogen exponent generally having values of ‘/2 or 1
Q
= volumetric flow rate of reactant over catalyst a t
R
= dimensionless reaction rate,
rA t
= rate of appearance of A due to chemical reaction
T U
= dimensionless time, t U / L = linear velocity of reactant over catalyst; for weakly
reaction conditions = time
V
=
Y
= = =
X
Y
LrA/UCa
adsorbed species it will approach the carrier gas velocity. void volume of reactor chemical conversion distance along reactor dimensionless distance, y / L
Literature Cited
Bassett, D. W., Habgood, H. W., J . Phys. Chem. 64, 769-73 ( 1 9 6 0 ) . Bett, J. A. S., Hall, W. K., Division of Colloid and Surface Chemistry, 154th Meeting, ACS, Chicago, Ill., September 1967. Bett, J. A . S., Hall, W. K., J . Catalysis 10,105 (1968). Blanton, W. A , , Jr., M.S. thesis, University of California, Berkeley, 1966. Bond, G. C., “Catalysis by Metals,” pp. 128-31, Academic Press, New York, 1962. Collins, C. G., Deans, H. W., A.Z.Ch.E. J., in press, 1968. Deans, H. .4., Horns, F. J. M., Klauser, G., “Perturbative Chromatography,” National Meeting, A. I. Ch. E., New York, N. Y., December 1967. Gaziev, G. A,, Filinovskil, V. Yu., Yanovskil, M. I., Kinetica i Kataliz 4, 688-97 (1963). Hattori, H., Murakami, Y., J . Catalysis 10, 114 ( 1 9 6 3 ) . Hindin, S . G., Weller, S. W., J . Phys. Chem. 60, 1501-6 ( 1 9 5 6 ) . Kokes, R. J., Tobin, H., Emmett, P. H., J. A m . Chem. Soc. 17, 5860-2 ( 1 9 5 5 ) . Schwab, G. M., Watson, A . M., J . Catalysis4,570-6 (1965). Sinfelt, J. H., J . Phys. Chem. 68, 232-7 ( 1 9 6 4 ) . RECEIVED for review December 5 , 1967 ACCEPTED June 21, 1968 Work supported by a grant from the National Science Foundation.
THE MAXIMUM PRINCIPLE AND DISCRETE SYSTEMS O K A N GUREL
New York Scienti’jic Center, I B M Gorp., 410 East 62nd St., New York, N . Y .
10021
LEON LAPIDUS
Department of Chemical Engineering, Princeton University, Princeton, N . J . 08540
A steady-state! discrete-staged system is optimized by considering the staged system as evolving in time, each stage variable being an element in the over-all state vector. This introduces a convexing effect and allows the continuous maximum principle to b e used. Two famous examples which have been used to illustrate the defects of the discrete approach are analyzed b y the present method with positive results.
w
in the area of optimal control are aware of the problems encountered in applying the maximum principle to discrete systems-Le., to steady-state optimization of staged systems (Arimoto, 1967; Athans, 1966; Chang, 1967 ; Halkin, 1966; Holtzman, 1966a, b ; Jordan and Polak, 1964; Pearson, 1965) via the discrete maximum principle. As has been elucidated by Horn and Jackson ( 1 9 6 5 ) and Jackson and Horn (1965), the problems have resulted because of an attempt to deduce the nature of stationary values from a consideration of only first-order variations. If the second-order variations are neglected, it is impossible to say that a maximum in the Hamiltonian yields a corresponding minimum index of performance. ORKERS
Stated in another fashion, the necessary conditions for optimality via the discrete maximum principle require certain guarantees for the reachable states of the system. I n general, these reachable sets must be convex or, as recently developed by Holtzman (1966a, b) and Holtzman and Halkin (1966), directionally convex. On the basis of the assumption that the system and the index fully satisfy this assumption of convexity, it is possible to specify the necessary conditions for the discrete maximum principle. The questions raised i n these analyses are not trivial and Halkin (1966) has written: “The reader should realize that [papers in the reference], together with the present paper, are not the first papers concerned with the optimal control systems VOL. 7
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