LIQUID-PHASE OXIDATION OF CYCLOHEXANOL SACHIO ISHIMOTO, TADAHISA SASANO, AND KENJI Teijin Products Development Institute, Iwakuni City, Japan
K A W A M U R A I
The liquid-phase oxidation of cyclohexanol/ in multistage agitated flow reactors under atmospheric pressure and within the temperature range of 100" to 130" C., was studied. Cyclohexanol was oxidized to hydrogen peroxide and cyclohexanone, accompanied by decomposition of hydrogen peroxide and oxidative degradation of the cyclohexanone. The rates of these reactions correlated as follows, for cyclohexanol conversion of less than 15%. (Rate of oxidation of cyclohexanol) = kl(Hz0z)'" kz(HzO#. (Rate of de-
+
composition of
H202)
= kS(Hz02)
+ kd(Hz02)'.
(Rate of degradation of cyclohexanone) = ks
(c=o)2
Operating variables of reactors were optimized b y using dynamic programming to minimize the loss of hydrogen peroxide.
YCLOHEXANONE, hydrogen peroxide, and peroxy comC p o u n d s which are in equilibrium with hydrogen peroxide and cyclohexanone are produced during the oxidation of cyclohexanol. These products are subjected to acidolysis to obtain e-hydroxycaproic acid which can be utilized as a precursor of e-caprolactam. The liquid-phase oxidation of cyclohexanol has been studied by Brown et al. (1955), Kat0 and Mashio (1957), Kunugi et al. (1963), Parlant (1964), and Buxbaum (1967). Brown et al. (1955) reported that violent agitation to ensure close contact of the reactants with oxygen depressed the decomposition of hydrogen peroxide and that addition of calcium carbonate increased the yield and decreased the reaction rate. Kunugi et al. (1963) studied the oxidation of various secondary alcohols, including cyclohexanol, and obtained a rate equation. Parlant (1 964) investigated initial reaction rates in detail. Buxbaum (1967) investigated the reaction mechanism of the liquid-phase oxidation of cyclohexanol. However, the validity of the reaction kinetics which were worked out by these investigators was limited within the range of the low conversion of the reactant (up to 2% of cyclohexanol) and these kinetics did not correlate well with higher conversion reactions, mainly because hydrogen peroxide, one of the reaction products of cyclohexanol oxidation, is susceptible to many further reactions. In the initial stage of oxidation, these reactions (except the first-order decomposition of hydrogen peroxide) are not appreciable. As conversion increases, a higherorder decomposition of hydrogen peroxide and the reaction between hydrogen peroxide and cyclohexanol become significant. Some of the reactions between hydrogen peroxide and other reaction products-e.g., cyclohexanone, peroxy compounds, and acids-might also become appreciable. Thus, many reactions participate in the system one after another, as oxidation proceeds. Some of the reactions which consume hydrogen peroxide produce free radicals and these free radicals afford more branched reactions. I t is difficult and impractical to correlate all reactions in the system. I n the present work, the empirical approach was chosen to correlate rate data, assuming simple reaction paths. The main objective of this work was to simulate the reactions and to optimize the process for a larger scale operation. Multistaged flow reactors are the most suitable for a large scale plant for cyclohexanol oxidation. After preliminary beaker scale batch studies of the reaction, bench scale multistaged flow
Present address, Teijin Hercules Chemical Co., Ltd., Tokuyama City, Japan.
reactors were used to correlate rate data. (It is also possible to study rate data using a batch reactor, but reaction rates are accelerated as oxidation proceeds and it is difficult to obtain accurate differential data a t high conversion of the reactant in a batch system.) From the data obtained from the bench scale apparatus, which covered the temperature range of 100' to 130' C. and cyclohexanol conversion of less than 15%, reaction rates were correlated empirically by simple equations, and operating conditions were optimized using these equations. Experimental
Preliminary Experiments in a Beaker Scale Batch Reactor. Cyclohexanol was oxidized, batchwise, in a borosilicate glass creased flask (300 ml. in capacity). The reactor was agitated by a paddle-bladed impeller at a speed of 1500 r.p.m. and oxygen was fed to the reactor at the rate of 300 ml. per minute (Figure 1). The main reaction of cyclohexanol oxidation is
The rate-determining step of Reaction 1 is the hydrogen atom abstraction from cyclohexanol by the free radical (R .), unless oxygen is insufficiently supplied. Paper-chromatographic analysis showed that about half of the peroxy compound in the system was hydrogen peroxide and half was made up of the addition products of hydrogen peroxide with cyclohexanone (such as 1-hydroxycyclohexyl hydroperoxide). Equilibrium between hydrogen peroxide and organic peroxy compounds might exist (Equation 2), as mentioned by Brown et al. (1955).
~ H z O4Z mC>===O
peroxy compounds
(2)
These peroxy compounds were analyzed by iodometry. For convenience, all peroxy compounds in the system are, hereafter, assumed to be a mixture of hydrogen peroxide and cyclohexanone. Hydrogen peroxide means the hydrogen peroxide in the system, plus hydrogen peroxide which can be liberated from organic peroxy compounds. Cyclohexanone means the cyclohexanone in the system, plus cyclohexanone which can be liberated from organic peroxy compounds. Hydrogen peroxide decomposes rapidly. H202 4 2 H O .
otherwise
HzO
+
l/2
O2
(3)
O n the other hand, cyclohexanone is also susceptible to secondary oxidation.
0-+ -0
0
2
VOL 7
R, acids NO. 3
JULY 1968
(4)
469
rNT
10.0
7VE.NT
Y
R-2
0
FEED
R
Figure 2.
k
+ PRODUCT
Flow diagram of multistage flow reactors
z 0 U
k U
I
I-
z
m
TIME Figure 1.
min
Batch oxidation of cyclohexanol
Temperature 120' C. Oxygen 300 ml. per minute. Cyclohexanol 2 5 0 grams. NazSOl 2.0 grams. 1,l '-Dihydroxy-1,1 '-dicyclohexyl peroxide 2.5 grams
Analysis of the acids produced showed that adipic acid was the main product and that the material balance could be set up well when all of titrated acids were considered to be adipic acid. The rate of conversion of cyclohexanol increases as the concentration of hydrogen peroxide increases. O n the other hand, the decomposition rate of hydrogen peroxide also increases more rapidly with increase of concentration of hydrogen peroxide. Consequently, as the conversion of cyclohexanol increases, the yield of hydrogen peroxide per converted cyclohexanol decreases. Addition of CaC03 or high speed agitation depressed the decomposition of hydrogen peroxide, as reported by Brown et al. (1955). Addition of Na2S04 (Ishimoto et al., 1967) or Na4P207.10Hz0 (Ishimoto and Sasano, 1967) had a similar effect. The initial rate of cyclohexanol consumption was proportional to the square root of the concentration of hydrogen peroxide and apparently to the zero order of cyclohexanol concentration. The decomposition rate of hydrogen peroxide was proportional to the concentration of hydrogen peroxide when the concentration of hydrogen peroxide was low (under 2 mole %). The reaction temperature was preferably kept within 100' to 130' C., since more hydrogen peroxide was lost above 130' C. and the reaction rate was too slow below 100' C. These conclusions coincide with those of Parlant (1964). Materials and Analysis. Cyclohexanol (Toa Gosei K.K.) was used as received, containing 0.3 to 1.0 weight yo cyclohexanone. T h e additive primarily used was Wako special grade precipitated calcium carbonate. The amounts of cyclohexanol and cyclohexanone were determined by vapor phase chromatography (3000-mm. PEG-6000 column ; carrier gas, 80 ml. of helium per minute; column temperature, 140' C.). Hydrogen peroxide (including organic peroxy compounds) was determined by iodometric analysis. Hydrogen peroxide and cyclohexanone are in equilibrium with their addition products containing peroxy oxygen (Equation 2). A platinum (0.5%) on carbon catalyst was added to samples and the samples were warmed to decompose H202 completely 470
l&EC PROCESS D E S I G N AND DEVELOPMENT
Figure 3.
Continuous oxidation reactor
Vessel Bearing
2. 5.
7. Impeller
8. 11.
1. 4.
10. Thermocouple
Cover plate Shaft Gas iniection pipe Drain
3. 6. 9. 12.
Housing Baffle Feed inlet Product exit
Dimensions R-1, R-2. D = H = 1 3 7 mm. d = 55 mm. b = 5 mm. H, = 4 3 mm. R-3. D = H = 1 0 0 mm. d = 40 mm. b = 4 mm. H, = 35 mm.
I
I
151
Fi
12oOc
I
o experimental
-calculaed
'I
CI
I
0I
s-
1.0
Y
K1i2
I
1
c
mI
mY
&ti--
120T
/
-
I?~,,.
1.0
*i
Figure 4. reactors
Reaction rates of cyclohexanol oxidation in multistage flow Upper left. Rate of cyclohexanal consumption Right. Decomposition rate of hydrogen peroxide Lower left. Rate of oxidative degradation of cyclohexanone
and to shift the equilibrium of Equation 2 to the left-hand side. Then the concentrations of cyclohexanol and cyclohexanone were determined. I n the iodometric analysis ethanol (50 d.), glacial acetic acid (2 ml.), and potassium iodide-saturated water (2 ml.) were added to a weighed sample of ca. 2 ml. of the reaction products. The mixture was heated for 5 minutes on a water bath a t 80' to 85' C. and titrated with 0.1N NazSz03. This procedure was justified experimentally by analyzing the mixture of cyclohexanone, hydrogen peroxide, and cyclohexanol. Conversion and yield of hydrogen peroxide were defined as follows: Conversion =
+
cyclohexanone produced (moles) acids produced (moles) cyclohexanol fed (moles)
x
100 (%)
(5)
cm. in effective depth (10 cm. for R-3) equipped with four vertical baffles and a 12-bladed turbine impeller agitator, capable of a maximum of 1800 r.p.m. Oxygen was fed into each reactor through flowmeters and injected through injection rings which had 25 holes of 1-mm. diameter. T h e reactors were constructed of 99.5% pure aluminum to prevent decomposition of hydrogen peroxide and heated with a steam tracer line made of copper tubing. The excess oxygen was vented through a reflux condenser. Overflow liquid e o m the first reactor, R-1, was cascaded to the next reactor, and so forth. The temperature of each reactor could be controlled separately. If not enough oxygen y a s supplied, the yield of hydrogen peroxide decreased. The action rate and the yield were not affected by an oxygen flp3rate above 800 ml. per minute. Also, the yield was not affected by the m o u n t of added CaCOa above a 0.8 weight % addition. Results
(%) ( 6 )
Experimental Data. Cyclohexanol was oxidiacd continuously in the apparatus shown in Figure 2. T h e operating conditions were: additive, CaCOa; 0.8 weight % of cyclohexanol; oxygen feed rate, 1000 ml. per minute; agitation, 1500 r.p.m.; temperature, 105' to 130' C . ; residesce time, 0.77 to 2.0 hours per reactor (Table I). (Many experiments were conducted under other conditions, but they are not described here.)
Moles of reaction products were adopted as a basis in Equations 5 and 6 because the accurate determination of cyclohexanol consumption was more difficult than the accurate determination of products. Experiments in Multistage Flow Reactors. Figure 2 shows a schematic flow diagram of the entire equipment. Cyclohexanol was fed continuously into the first reactor, R-1, by pump P-1 . As shown in Figure 3, reactors R-1 and R-2 had a reaction section 13.7 cm. in i.d. (10 cm. for R-3) and 13.7
Reaction Rate. T h e rate of hydrogen peroxide dec;omposition was proportional to the concentration of hydrogen peroxide only when the concentration of hydrogen peroxide was very low. I n examining the experimental data in Table I and Figure 4 and those of the batch experiment, it was necessary to postulate an additional higher order term of rate of hydrogen peroxide decomposition for the higher hydrogen peroxide
Yield of hydrogen peroxide = hydrogen peroxide produced (moles) cyclohexanone produced (moles) f acids produced (moles)
x
100
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471
Table 1.
Expt. No. 69
T;mp., C.
Feed R-1 R-2
70
101
R-1 R-2 R-3
1.95 1.87
120 120
1.59 1.53
125 120 117
1.11 1.12 0.985
125 120 117
1.23 1.16 1.12
120 116
1.06 0.871
112 105
1.38 1.24
112 105
1.34 1.24
Feed Feed R-1 R-2
N Obsd. 10.0 9.83 9.28 10.0 9.78 8.83
H
A
Calcd. 9.87 9.43 9.81 8.97
10.0
Feed
R-1 R-2 126
120 120
Feed
R-1 R-2 125
1.18 1.13
Feed
R-1 R-2 R-3 116
125 125
Feed R-1 R-2
91
1.05 1.01
Feed R-1 R-2
72
125 125
Feed R-1 R-2
71
Residence Time, Hours
Experimental Data and Calculated Values of Cyclohexanol Oxidation Additive. CaCOa, 0 . 8 weight yo Oxygen. 1000 cc./min. (reactor) Agitation. 1500 r.p.m.
9.72 8.45 10.0 9.82 9.26 9.86 9.79 9.49 9.23 9.86 9.77 9.55 9.05 9.70 9.35 8.71 9.61 9.34 9.14 9.52 9.29 9.06
9.61 8.37 9.85 9.31 9.70 9.48 9.23 9.63 9.29 8.81 9.34 9.02 9.38 9.25 9.26 9.10
’
Obsd. 0.035 0.16 0.66 0.035 0.24 1.04 0.035
Calcd. 0.16 0.58 0.22 0.98
...
1.36 0.035 0.17 1 .oo 0.05
...
0.42 0.69 0.05 0.24 0.58 0.99 0.16 0.52 0.89 0.39
0.40 1.45 0.18 0.69 0.20 0.41 0.65 0.27 0.60 1.03 0.50 0.80
...
0.60 0.73
...
0.69 0.85
0.77 0.45
...
concentration ranges. The rate of cyclohexanol consumption depends upon the concentration of free radical, which in turn is liberated by the decomposition of hydrogen peroxide. Therefore, it was also necessary to postulate an additional higher order term of rate of cyclohexanol consumption in the higher conversion reaction other than initial decomposition, which was proportional to the square root of the hydrogen peroxide concentration. The rate of cyclohexanone decomposition was considered to depend upon the concentrations of cyclohexanone and hydrogen peroxide. However, since the magnitude of cyclohexanone concentration is nearly equal to that of hydrogen peroxide, it is difficult to evaluate the effect of these two concentration terms separately. For convenience, the rate of cyclohexanone consumption was postulated to be represented only in terms of cyclohexanone. From the abovementioned postulation, the following equations were assumed, where exponents CY, p, and y and rate constants were unknown. (Rate of cyclohexanol consumption) = (rate of hydrogen peroxide production) = (rate of cyclohexanone production)
+ k&“ (Rate of hydrogen peroxide decomposition) = k J 2 + k&O =
kiH’”
(7)
(8) (Rate of cyclohexanone degradation) = (rate of acid production) = k5AY
(9)
I n Equations 7, 8, and 9, the concentrations are expressed in moles per 10 moles of cyclohexanol fed. The expression “moles per 10 moles of cyclohexanol fed” is preferred because it is conveniently used in estimating the material balance. 472
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
Obsd. 0 0.164 0.575 0 0.208 0.772 0 0.273 0.817 0 0.177 0.786 0 0.206 0.370 0.645 0 0.218 0.494 0.731 0.288 0.607 0,893 0.374 0.586 0.764 0.461 0.681 0.801
B Cakd.
Obsd. 0 0.005 0.035 0 0.005 0.073 0 0.008 0.227 0 0,004 0.096 0 0,004 0.015 0.042 0 0.006 0.015 0.046 0.0089 0.0334 0.0698 0.0106 0.0251 0.0480 0.0210 0.0467 0.0642
0.128 0.536 0.184 0.798 0,369 0.994 0.148 0.628 0.155 0.365 0.598 0.223 0.542 0.876 0.606 0.842 0.580 0.700 0.687 0.821
Cakd. 0.001
Concersion, Yo Obsd. Cakd.
Hz02 Yield,
% Obsd. Calcd.
0.026
1.4 6.6
1.1 6.7
87
98 93
0.005 0.086
2.1 10.7
1.9 11.0
74
98 77
0.015 0.206
15.5
...
3.9 16.3
64
96 61
0.002 0.037
1.3 10.6
1.5 6.9
74
98 91
0.003 0.012 0.027
... 3.8 6.8
0.3 5.2 7.7
96 95
98 97 95
0.007 0,028 0.071
2.0 5.6 9.9
3.7 5.6 11.9
88 74
98 95 83
0.0218 0.0414 0.0225 0.0302 0.0352 0,0469
Assuming perfect mixing, material balances in the flow reactors are derived asfollowsfrom Equations 7, 8, and 9. Sj(hTj-i - Nj) = kiHj’’2
+kaja
- Nj) - Sf(Hj - Hj-1) = k3Hj f kaHjO Sj(Nj-1 - N j ) - Sj(A, - Aj-i) = k5AjY Sj(Bj - B p i ) = k5AjY
Sj(Nj-1
(10)
(11) (12)
(13)
Exponents and rate constants in Equations 10 to 13 were estimated at several different temperatures by trial and error. Then, rate constants were plotted against 1/T, and this procedure was repeated until exponents and rate constants satisfied Equations 10 to 13 and, at the same time, reaction constants gave linear plots in a k us. 1 / T diagram (Figure 5)-for example, experimental data a t 120’ C. are shown in Figure 4. Ordinates in Figure 4 are the left-hand side of Equations 10,11, and 13, respectively. I n examining Figure 4 (upper left and right) exponents CY and /3 were postulated as CY = 2 and p = 5, and exponent y was directly determined by plots in Figure 4 (lower left). Rate constants kl, kp, ks, and kq were determined by trial and error as 0.22, 0.45, 0.012, and 0.31, respectively. (Curves in Figure 4 show calculated values.) The rate constants determined in this manner are shown in Figure 5 as functions of reciprocal temperatures. Figure 5 shows good linearity of rate constants. Arrhenius equations were obtained from Figure 5, and the rate constants determined were used to simulate the experimental data (Table I). When the calculated concentrations were compared with the experimental data, a minor correction of kl and k2 was necessary. The frequency factors of k l and k p were decreased by lo%, and the term of the initial cyclo-
Rate Constants and Exponents of Equations 11 t o l l 4
Table II.
R = 1.987 cal./mole
+
kl = (3.1 A0)(lOn) exp. (-21850/RT) k p = (1.7)(101*)exp. (-29,80O/RT) ka = (l.l)(1014)exp. (-28,81O/RT) kq = ( 6 . 8 ) ( 1 O 1 8 ) exp. (-34,77O/RT) ks = (4.0)(101a) exp. (-26,82O/RT)
Reaction constants
CY,
p,
y
K.
in Equations
= 2 p = 5
cy
10 to 13
y = 2
L
I
I
2.5 Figure
3
I
I
I
I
2.55 1 / T x lo3
5.
(
l
I
I
2.6 1/'K
I
l
I
I
2.6!
Arrhenius plot of rate constants
hexanone concentration was included in the equation for k l to take into account the slight acceleration effect of cyclohexanone over the initial cyclohexanol consumption. The final results of rate constants and exponents are shown in Table 11. The observed and calculated concentrations using the exponents and rate constants in Table I1 are compared in Table I. An explanation of the apparently large value for /3 may be in order a t this time. The exponent was determined to be larger than is usual in common chemical rate equations-i.e., /3 = 5. This is explained as follows. Hydrogen peroxide produced is susceptible to further reactions. As oxidation proceeds, reactions which consume hydrogen peroxide participate appreciably one after another in the system. These are not fifth-order reactions. However, the total rate of hydrogen peroxide consumption increases rapidly. The value of /3 reflects the situation where the total consumption rate of hydrogen peroxide accelerates rapidly with time-that is, the values of the exponents were not determined by use of stoichiometric chemical reaction mechanisms, they are the results observed in the simulation. Parlant (1964) reported reaction kinetics which were applicable to only the initial stage of the reaction. The results obtained by Parlant correspond to the first term on the righthand side of Equation 7 and the first term on the right-hand side of Equation 8. However, since the additive calcium carbonate
used in these experiments slowed down the reaction rate significantly [Brown et al. (1955) also reported that calcium carbonate decreased the reaction rates], direct comparison of the rate constants (kl and k t ) with those determined by Parlant is invalid. Deviations of calculated concentrations from observed data in Table I are due to fluctuations in feed rate and temperature. Slight contaminations and changes in the activity of the reactor wall also change the rate of hydrogen peroxide decomposition. Brown et al. (1955) and Kunugi et al. (1963) mentioned this fact. Since free radical liberated from hydrogen peroxide accelerates both the side reaction and the main reaction, slight fluctuations in operating conditions have no great effect on conversion-yield relationships. Equations 10 to 13 and Table I1 can be applied to the continuous oxidation of cyclohexanol (containing less than 1 weight % of cyclohexanone) with an additive of C a C 0 3 (0.8 weight %) and with excess oxygen. Minor correction may be needed on rate constants, when the apparatus is scaled up or the amount of additive is changed. However, the principal form of the equations and exponents may be used essentially as presented. Equations 7 to 9 can be applied to both the flow and the batch reactions. Control of the reaction is an additional problem. If the reaction proceeds abnormally for some reason, the normal state is not recovered rapidly by lowering the temperature. If this occurs, dilution by pure cyclohexanol or the products of low conversion reaction is needed. T o determine the conversion quickly, acid concentration is most practical. Optimization of Multistage Continuous Oxidation. T h e main objective of cyclohexanol oxidation is peroxy compounds. The batch process gives a better yield, it is not suitable for large-scale plants, and the reaction rate is low and poor in reproducibility. Therefore, multistage continuous oxidation is preferred. Optimal operating conditions of multistage continuous reactors can be determined by using the empirical simulation mentioned above and dynamic programming (Aris, 1961). Since cyclohexanone is stable in the practical sense, as shown in Table 11, the criterion for optimal production is the maximizing of the yield of hydrogen peroxide for any given conversion of cyclohexanol. The definition of the problem is the maximizing of an objective function L M ( C )as defined by Equation 15 under a constraint (Equation 14). M
Constraint.
cc5=c
j-1
.M.-
Objective function.
L,w(C) =
E Xj
j=1
=
H M
- Ho
(15)
where X 5 = H5 - H 5-1 Equations 10 and 11 are transformation equations which represent the relation between input and output of each stage. (The concentrations of acids and cyclohexanone are not necessary for optimization.) I n these equations, the variables are S5,N,, and Hiif input conditions and H5-1 and temperature T, are given. Therefore, when the conversion, C5namely, N,-is given, S, and H5can be estimated simultaneously by use of Equations 10 and 11. Cj and T5 were chosen as decision or operation variables. By the definition of the problem, the maximized objective functionfAw(C)is :
VOL. 7
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JULY 1 9 6 8
473
I
Toble 111.
C
b6.i Converi$on
Optimal Operating Conditions Estimated by Dynamic Programming
1st Stage
of Cgelohexand, %
Temp., No. %Stages
a
c.
7
1-stage process 2-stage process 3-stage process
121 120 120
10
1-stage process 2-stage process 3-stage process
121 120 120
2nd Stage
%
H
7 4.5 3.5
0.625 0.431 0.341
10 6 5
c.
110 110
0.781
0.555 0.475
3rd Stage
Conv.,
Temp.,
Conu.,
110 110
Temp.,
70
H
O c .
7 5.5
0.658 0.531 0.864 0,735
10 8
Conv.,
70
H
110
7
0.667
110
10
0.885
By the principle of optimality for the discrete deterministic problem,
80 especially when M = 1,
60
CALCULATION (digital computer used). A. Divide a range of conversion-for example, 0 to 15%--into a large number of lattice points. (These arbitrary divisions of the range of convers'lon are called conversion lattice points.) B. R n d an optimal temperature for a one-stage reaction among 20 different values of temperature-Le., 110' to 130' C. in 1' C. steps-for each conversion lattice point. (Equation 18 is used.) C. Then consider a two-stage reaction. In this situation, the principle of optimality holds as for the first reactor. Take any conversion lattice point value for the second reactor conversion (C, Cz). Then, take any arbitrary conversion value first reactor (CI < CI Cz). The optimal temperac' tureforan output of the first reactor have been computed for C1 in B. Calculate the value of each objective function for all temperlitures to be usetl i A the second reactor. Then take the maximum value &r the objective function. Fix the conversion lattice poinf for the second reactor (the value arbitrarily chosen as stated above). Then take any other arbitrary conversion value for the first reactor and repeat this procedure to find the highest value of the objective function among the maximum values for the objective functions of all the posgible conversion values for the first reactor, and determine the optimal decision C1, T I , and T2 for any value of (C, CJ taken. Repeat the procedure for all the lattice points taken (C1 CZ)for the second reactor. D. A three-stage reaction, four-stage reaction, and so forth can be optimized similarly.
d-d"
+
+
E 50112 Y 40
?- 300
Results of the calculations are shown in Table 111. The following ctmclusion was reached. A higher conversion is preferable in the former reactornamely,
CI> CZ> CB,etc. A higher temperature is preferable in the former reactornamely,
'/o
Nomenclature
A
cyclohexanone concentration, cyclohexanone in the system, plus cyclohexanone which can be liberated from organic peroxy compounds, moles per 10 moles of cyclohexanol fed = concentration of acids, moles per 10 moles of cyclohexanol fed =
B
M
C
Cj, total consumption ofcyclohexanol, molesper 10
= j-1
Cj
H
=
k
= =
M N
I n increasing the number of stages, the yield of hydrogen peroxidd is improved, especially for higher conversions (Figure 6).
Xj
Acknkledgment
0
T h e authors are grateful to T. Naruchi for his contribution to the batch experiments of this study and H. Togawa and M. Hasegawa for their chemical analysis. They thank S. Sat0 for his dynamic programming.
1 2
moles of cyclohexanol fed - N , , consumption of cyclohexanol in j t h stage, moles per 10 moles of cyclohexanol fed maximum total return defined by Equation 16 hydrogen peroxide concentration, hydrogen peroxide in the system, plus hydrogen peroxide which can be liberated from organic peroxy compounds, moles per 10 moles of cyclohexanol fed rate constant total number of stages concentration of cyclohexanol, moles per 10 moles of cyclohexanol fed reciprocal residence time, reciprocal hours absolute temperature, ' K. H, - H 3 - l , increase of hydrogen peroxide in each stage, moles per 10 moles of cyclohexanol fed
= N,-,
fM(C) =
T
J E C PROCESS D E S I G N A N D DEVELOPMENT
1
Figure 6. Relation between conversion and yield at optimal operation
S
474
10
5
CONVERSION
+
+
2 staqe
= = = =
SUBSCRIPTS
3
j j
feed R-1 reactor R-2 reactor R-3 reactor = j t h reactor 1 = ( j - 1)th reactor = = = =
-
literature Cited
Ark, R., “Optimal Design of Chemical Reactors,” p. 16, Academic Press, New York, 1961. Brown, N., Hartig, M. J., Roedel, M. J., Anderson, A. W., Schweitzer, C. E., J . A m . Chem. SOC. 77, 1756 (1955). Buxbaum, L. H., Liebigs Ann. Chem. 706,81 (1967). Ishimoto, S.,Sasano, T. (to Teijin, Ltd.), Japan Patent 1967/16294 (Sept. 4, 1967). Ishimoto, S., Togawa, H., Naruchi, T. (to Teijin, Ltd.), Japan Patent 1967/11819 (July 5, 1967).
Kato, S., Mashio, F., Mem. Fac. Ind. Arts, Kyoto Tech. Univ. Sci. Technol. 6, 67 (1957). Kunugi, T., Matsuura, T., Oguni, S.,J. Japan Petrol Inst. 6, 26 (1963). Parlant, C., Rev. Inst. Franc. Petrole Ann. Combust. Liquides 19, No. 7-8, l(1964).
RECEIVED for review November 24, 1967 ACCEPTEDMarch 28, 1968
ASPECTS OF THE FORWARD DYNAMIC PROGRAMMING ALGORITHM JOHN H. SEINFELDIAND LEON LAPIDUS Department of Chemical Engineering, Princeton University, Princeton, N . J .
The computational features of the forward dynamic programming algorithm as applied to a nonautonomous optimal control problem are discussed and a detailed numerical example is presented.
programming may be used for solving multistage Bellman’s principle of optimality leads to a computer algorithm in which the last stage is optimized first (Aris, 1964; Bellman, 1957) (the backward algorithm). However, to begin at the last stage knowledge of the final state or stage number is required. Thus, in certain nonautonomous systems, solution of the optimization or control problem by this customary procedure is extremely timeconsuming because the dynamic programming algorithm must be solved over and over as in a boundary-value problem. A forward dynamic programming algorithm in which the first stage is optimized initially, exists as a dual of, although much less known than, the backward algorithm. This dual technique has been discussed by Dreyfus (1965) and Nemhauser (1966). Bhavnani and Chen (1966) noted the utility of the forward approach in solving certain time-dependent control problems with fixed and free final times. Aris, Nemhauser, and Wilde (1964) have discussed the general technique of working from the fixed to the free end of straight-chain, cyclic, and branching systems using dynamic programming. I n the present paper, the authors discuss the computational features of the forward dynamic programming algorithm as applied to a nonautonomous optimal control problem. To help in this discussion a detailed numerical example is presented; consideration is given to effective means of reducing the computer storage problem normally associated with dynamic programming. Since computation time (Lapidus and Luus, 1967) may often be a major drawback in the use of dynamic programming, it is also pointed out how this time may vary significantly between the use of the forward and backward algorithms. YNAMIC
D optimization and control problems.
Principle of Optimality
Consider a system governed by the differential equations
Present address, Department of Chemical Engineering, California Institute of Technology, Pasadena, Calif.
where x ( t ) is ap-dimensional state vector and u ( t ) anr-dimensional control vector. I n terms of an optimal control problem u (t)EUmust be chosen such that the scalar performance index
+
4 4 0 , t f l = @[X(t,)l
is minimized.
1 t/
J[x(t),u(t)ld
(2)
I n discrete notation Equations 1 and 2 become X,+I
- x,
=
h(x,,u,,nAt)
(3)
and
+ c J(x,,u,) AJ-1
Z[u,,NI
=
‘P.(XN)
n=l
At
(4)
where N is the number of time stages. I n terms of the well known backward dynamic programming algorithm Bellman’s principle of optimality states (Bellman, 1957): “An optimal policy has the property that whatever the initial state and initial decisons are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.” Application of Bellman’s principle to the problem detailed above leads to the recurrence relation
where FN-,(x,) = cumulative value of the performance index Equation 4, by using the optimal policy from stage n to stage N with x, the state resulting from the nth stage. I n this form the explicit algorithm embedded in Equation 5 requires a backward pass starting from stage N (the last stage) and working toward stage 1 (the initial stage). A second pass in the opposite direction then develops the explicit optimal stage information. This is called the backward algorithm. Bhavnani and Chen (1966) have formulated a dual to the above principle of optimality: “An optimal policy has the property that whatever the ensuing state and decisions are, the preceding decisions must constitute an optimal policy with regard to the state existing before the last decision.” VOL. 7
NO. 3 JULY 1 9 6 8
475
