Ind. Eng. Chem. Res. 1987,26, 1663-1667
1663
were aged and water soaked. Substituting half of the EAA with LDPE caused some loss in both strength and percentage elongation with aging and water soaking; however, cost reduction may outweigh these slight property losses. Also, addition of poly01 plasticizers does not greatly affect properties of these starch-EAA-LDPE systems.
properties. All films containing 40% or more biodegradable material develop mold growth and lose at least 50% of their physical properties within a few months of soil contact. The optimum balance in film properties may require breaking up the film by disking into the soil followed by microbial decay of the buried film.
Conclusions The incorporation of urea into a starch-EAA formulation provides one approach to improving preparation and economics of producing the starch-based film. This approach could allow large-scale continuous production by feeding the various components to a low-energy blender and then to an extruder with a mixing screw to produce pellets or directly into a blown-film extruder. A practical formulation of 40% starch, 20% EAA, 15% urea, and 25% LDPE would have a material cost of 35-40 cents/lb, which is near the price of LDPE. Farmers are reporting removal and disposal costs for nondegradable mulch at 100-200 dollars/acre. If 230 lb of LDPE is used per acre, this removal cost is 43-87 cents/lb of f i i . Hence, considerable economic benefit could be realized by mulching crops with a degradable film that need not be removed. No field studies have been conducted on our starchEAA films but we believe that they will need further modification to provide a balance between optimum rates of deterioration and acceptable physical properties for a range of agricultural mulch applications. Preferably, such films should decompose within 12 months into sufficiently small pieces not to interfere with cultivation. A film composed of 40% Amaizo 742D, 20% sucrose, and 40% EAA appears to meet this criterion when in contact with soil. However, such films have poor initial physical
Acknowledgment We thank R. C. Phelps, M. E. Rivers, and others at Rex Plastics, Thomasville, NC, for conducting the pilot-scale extrusion blowing runs and Sara Walz-Salvador of this laboratory for assistance with the film testing. The mention of firm names or trade products does not imply that they are endorsed or recommended by the US.Department of Agriculture over other firms or similar products not mentioned. Registry No. EAA (copolymer), 9010-77-9; PE, 9002-88-4; NH,OH, 1336-21-6; H2NCONH2,57-13-6;starch, 9005-25-8;hydroxyethyl starch, 9005-27-0; glycerol, 56-81-5; glycol glucoside, 5994-13-8.
Literature Cited Otey, F. H.; Westhoff, R. P.; Doane, W. M. Ind. Eng. Chem. Prod. Res. Deu. 1980,19, 592-595. Otey, F. H.; Westhoff, R. P. Ind. Eng. Chem. Prod. Res. Deu. 1984, 23, 284-281. Otey, F. H.; Trimnell, D.; Westhoff, R. P.; Shasha, B. S. J. Agric. Food Chem. 1984,34,1095-1098. Westhoff, R. P.; Kwolek, W. F.: Otev, - . F. H. StarchlStaerke 1979, 31, 163-165. Zobel, H. F. Starch Chemistry and Technology; Academic: Orlando, FL, 1984; pp 285-309. '
Received for review October 6, 1986 Accepted April 27, 1987
Approximate Lumping Applied to the Isomerization of Methylcyclohexenes Maurice Peereboom Department of Inorganic and Physical Chemistry, Delft University of Technology, 2600 GA Delft, The Netherlands
In view of the very fast isomerization of methylenecyclohexane into 1-methylcyclohexene, the kinetics of the double bond shift isomerization of 1-methylcyclohexene (lMCH), 3-methylcyclohexene (3MCH), 4-methylcyclohexene (4MCH), and methylenecyclohexane (MECH) over y-alumina has been studied on the basis of a pseudo-three-component system in which MECH and l M C H are lumped together. Despite the fact that this system of isomers is only approximately lumpable, it appears that the lumping is a very good one. The kinetic parameters obtained in the temperature range 413-523 K are representative of the behavior of the original four-component system. The Wei and Prater technique appears extremely suitable for calculating the pseudo reaction rate constants, indicating a kinetic model in which the reactions proceed by a series mechanism. In practice it is often expedient to represent the kinetics of reactions in a complex reaction system in terms of lumped (hypothetical) components (e.g., Zhu et al., 1985; Frenklach, 1985). For example, many industrial chemical processes involve complex mixtures that are difficult to analyze in detail, which renders it nearly impossible to determine the kinetics of each of the reactions separately. For that reason, the various species are assigned to a limited number of classes where each class is treated as a single entity. This kind of modeling is also suitable for reaction systems in which, as a result of some very fast interconversions, partial equilibrium is achieved. The 0888-5885/87/2626-1663$01.50/0
choice of the lump compositions has then to be such that the evolution of the lump as a function of time is identical with that of the s u m of the individual components of the lump. A general theoretical lumping analysis for monomolecular reaction systems is presented in Wei and Kuo (1969); more recently, an extension to the lumping of bimolecular reaction systems has been developed by Li (1984). The purpose of the present study is to investigate the lumpability of the isomeric methylcyclohexenes and methylenecyclohexaneand to describe the lumped system in terms of a complex firsborder reaction scheme. Relative rate constants for the lumped system calculated by the Wei 0 1987 American Chemical Society
1664 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
and Prater (1962) technique are compared to those obtained from a known rate constant matrix of the original reaction system, which is estimated by an optimization procedure (Marquardt, 1963).
tion of matrix K, which satisfies all the properties of a monomolecular reaction scheme, is then achieved by the choice of E such that
Theory For complex (pseudo)monomolecular reaction systems of first-order reversible reactions between n components, the rate equations are given by Wei and Prater (1962):
and
where a ( t )is the n x l composition vector and K is the nXn rate constant matrix. Matrix element kjj is the first-order rate constant for the reaction from the jth to the ith species. In the process of lumping, individual species of the original reaction system have been combined into a fewer number of pseudocomponents. If each component in the original system is contained in one and only one lumped component, the lumping is called proper (Wei and Kuo, 1969). Lumping is carried out by means of a nXn lumping matrix M of rank li (fi 5 n),transforming a ( t )into ri ( t ) , a fix1 vector. According to Wei and Kuo (1969),a system described by eq 1 is ex,actly lumpable by M if and only if there exists a matrix K such that the kinetic behavior of the lumped system can be represented by -dsi(t) - - -Ki(t) dt A necessary and sufficient condition for exact lumping is
MK = KM
(3)
For the system of the isomers of methylenecyclohexane, the order of the components in the composition vector a ( t ) can be given as (4)
Because of the very rapid interconversions of MECH with lMCH (eq 4), we do not want to deal with these compounds separately but instead lump the two species together. Then si ( t )should be
using
]!E
M=
M is a proper lumping matrix by definition, as each component in the original system is now contained in one and only one lumped component. If a reaction system is exactly lumpable by M, an error matrix E, defined as
E=MK-KM (7) has to be a null matrix. However, from experimental results (Peereboom and Meijering, 1979; Peereboom, 1984) it appears that there is no direct conversion between MECH and 4MCH, i.e., It14 = It41 = 0. In that case it will be clear, by application of eq 3, that exact lumping by a proper lumping matrix is not possible: E is not a null matrix and the system is only approximately lumpable. Following the theory of Kuo and Wei (1969), the genera-
EAMT = 6 K=
(8)
(9)
MKAMTA-~
The diagonal matrix A is defined as A = M A W , in which A is the diagonal matrix with the equilibrium mole fractions ai*(i = 1-4) of the unlumped reaction system as its ith diagonal elements. With K and K known, the error of lumping can now be estimated from the calculated reaction paths. The correct way to obtain the time behavior of the composition vector in the lumped reaction system is to compute ri,(t) = Me-Kta(0)
(10)
in which a (0) is an initial composition vector. In the case of an approximately lumpable system MeWKt f c K t M ,the reaction path calculated by eq 11 does not equal ii,(t)
riw(t)= e-KtMa(0)
(11)
The error of lumping is then given by the error vector ri,(t)
- iiw(t) = (Me-Kt- e-KtM)a(0)
(12)
Experimental Section Materials, reactor, procedures, and analysis used for the investigation of the kinetics of the catalyzed isomerization of the methylcyclohexenesand methylenecyclohexane over y-alumina have already been described in a previous paper (Peereboom and Meijering, 1979). Results and Discussion Determination of the Lumpability of the Reaction System. A first preliminary investigation of the lumpability of the four-component system depends on the experimental determination of the sequences of compositions obtained from the initial compositions. If MECH and lMCH can be lumped, then the reaction paths iil(t) = a l ( t )+ a z ( t ) ,constructed both from a (0) = (1,0,0,0) and from a (0) = (0,1,0,0),have to coincide in the &dimensional composition space. For 453 K, this procedure leads to the representation given in Figure 1. From this figure, the lumping appears to be very promising, however, a more precise estimation of the lumpability can be attained by the method suggested by Ozawa (1973), as worked out below. By use of the compositions along the reaction path obtained from a feed of pure 4MCH, the rate constant matrix at 453 K is calculated by the optimization technique of Marquardt (1963). K =
[
-100.0000 100.0000 O.OOO0 O.oo00
2.3444 -2.3494 0.0050 0.0000
0.0000 0.0352 -0.1065 0.0713
I I
0.0000 0.0000 0.0592 -0.0592
(I3)
From K, the corresponding eigenvector matrix X can be estimated
x = (x ,x ,xz9x5)=
0.0175 0.0046 -0.0009 -0.0175 0.7432 0.1977 -0.0391 0.0175 0.1063 -0.0743 0.1680 0.0000 0.1280 -0.1280 -0.1280 0.0000
[
(14)
The length of the eigenvectors xi(i = 1-3) is adjusted so that the terminus of the vector (x, + x i ) lies on the boundary of the simplex. In lumping four species into three species, the four eigenvectors xiare reduced to three
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1665 100
Table 11. Comparison of Corresponding Relative Rate Constants, Calculated with the Wei-Prater and Wei-Kuo Methods Wei-Prater Wei-Kuo
'50
50
4MCH
3MCH
Figure 1. Course of the reaction path (1,0,0) of the lumped system, derived from the paths starting from (1,0,0,0) and (O,l,O,O), respectively. Table I. Results of the Scalar Products xkTaij(k= 1-3) a 12 =
X IT X ;T
x ST
1
r-0.74821 ,.0175j
1;.1280j r o
=
i
-0.1063
-0.000 02 0.00001 0.013 40
0.004 10 0.035 11 0.000 00
cjgenyectors 4, which are related to the eigenvector matrix X of K. Thus, there exist three nonvanishing eigenvectors satisfying the condition
Mxi =
(i = 0-2)
(15)
and one vanishing eigenvector, y , to which applies
My=O
(16)
Only if the species Ai and Aj are exactly lumpable will the following scalar product result:
xkTai,= OT
(for k = 1-3)
(17)
On the other hand, products of ai'with the vanishing eigenvector should yield non-zero vaiues. In eq 16 and 17, aijis the four-dimensional vector formed from the ith and jth elements of the equilibrium composition vector xo, where all the elements are zeros except the ith, which is xoj, and the jth, which is -xoi. In the case of approximate lumping, eq 17 leads to the combination of i and j that minimizes the product. For the isomerization of the methylcyclohexenes, some of the results are given in Table I. From this data it is clear that, unlike 3- and 4MCH, MECH and lMCH can be lumped satisfactorily. Calculation of the Rat? Constants for the Lumped System. By use of eq 9, K can be calculated if the rate con?.nt matrix K is known; without prior knowledge of K, K has to be determined experimentally. Since the lumped system contains three species, sjx relative isomerization rates have to be calculated. An extremely useful tool for calculating the rate constants in a triangular network is provided by the Wei and Prater technique (1962) which has found regular application now (e.g., Collins et al., 1982; Collins and Scharff, 1983; Christoffel et al., 1980). In this technique some particular binary compositions on the side of the composition triangle have to be esimated, which unlike most of the reaction paths, will lead to the equilibrium composition via straight-line reaction paths. For a three-component system, there are
ff21
R 23
ff,2
0.1388 0.1390
1.7110 1.6834
2.0613 2.0287
two straight-line reaction paths. These paths can be shown to be the eigenvectors of the rate constant matrix of the system. The estimation of one straight-line reaction path together with the known equilibrium composition is already sufficient for the construction of the eigenvector matrix X. This matrix will transform a ?trongly_cu_rv_ed reaction path ri ( t )= (do,dl,d2)into a path b ( t )= (bo,bl,b2), by means of b = X-li (18) The elements 6, are mole fractions of fictitious species Bi, which follow monoqolecular uncpupled kinetics. By correct estimation of X, plots of In bl vs. In b2 should yield straight lines. The slopes of these lices are the ratios of the eigenvaJues corresponding with X. Wei and Prater show that K may then be obtained from
K
= XAX-1
(19) in which A is a diagonal matrix containing the various ratios of eigenvalues. The drawback of this approach is that many experiments have to be performed to locate the straight-line reaction paths. Moreover, it may be difficult to evaluate accurately a slope from the curved reaction path, as the experimental measurements are subjected to errors. In order to avoid experimental iterations, Lombardo and Hall (1971) developed a computer program to find the compositions that yield these straight-line reaction paths. This study also applies a computerized iteration procedure which, however, uses the correlation c_oefficie@ R as a measure of the goodness-of-fit of the In bl vs. In b2 plot (Peereboom, 1981). An accurate determination of the equilibrium composition and a right choice of the strongly curved reaction path are very important in relation to the estimation of the slope of this plot. At 453 K the experimentally measured equilibrium composition of the four isomers is (Peereboom et al., 1982) ro.01621 0.7343 xo = [o.i079] 0.1416
(20)
from which it follows that go=
c:3
(21)
0.1079
The Wei and Prater technique then yields the relative rate constant matrix KWP
[
-0.1432
Kw= 0.1388 0.0044
1 -3.0613 2.0613
]
0.0272 1.7110 -1.7382
(22)
Making use of the same equilibrium composition and applying the equation of Wei and Kco, in which, however, prior knowledge of K is required, K can be estimated as KWK=
0.1390
-3.0287 2.0287
]
1.6834 -1.6834
(23)
From Table 11, in which a comparison of corresponding rate constants is given, it appears that the two data sets are in perfect accordance. Since K depends on a kinetic
1666 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 M E W t 1 MCH 1
x
60 X
60
4 MCH
20
40
60
80
3MCH
Figure 2. Calculated reaction paths ri (X) and ri (-) in the composition triangle resulting from lumping MECH with 1MCH. Table 111. Calculated Relative Rate Constants for the Lumped System at Different Temperatures temp, K 413 433 453 473 493 523
k,,
&I3
k31
0.122 0.134 0.139 0.159 0.173 0.194
0.07 0.07 0.03 0.14 0.03 0.01
0.01 0.01 0.00 0.03 0.00 0.00
&3
1.99 1.87 1.71 2.07 3.14 2.22
&32
2.67 2.52 2.06 2.72 4.11 2.73
model based on the assumption that conversions between lMCH and 4MCH and betwee_nMECH and 3- and 4MCH are excludsd, evaLuation of KWKnecessarily yields zero values for k13 and kS1. With the use of the Wei and Prater technique, however, no restrictions are imposed with regard to possible reactions. Nevertheless, for the rate constants mentioned above, the same numerical results can be expected, considering that the same underlying mechanism is involved. In order to appreciate the data obtained, simulation experiments have been performed on the kinetic model A l F-! Az e AS,for which a triangular network is assumed. With a normally distributed error of 5% in the compositions, it appears that those rate constants, originally chosen as zero, yield values of 0.025 and 0.004, respectively. So within the lattainable accuracy, the experimental values of k13 and k31 calculated by the Wei and Prater technique can be considered as essentially zero, whence it appears that this method offers a very reliable tool to estimate the rate constants of the lumped system. By use of K and KWPrespectively, the reaction paths ri ,(t) and riw(t)are calculated and plotted in a composition triangle (Figure 2). It appears that the lumping error is practically negligible; we may assume, therefore, that the lumped system does indeed reflect the characteristics of the original reaction system, as can be seen from the following ratios of corresponding-rate constants: k34/k43= 0.5919/0.07133 = 0.830 vs. k B / k S 2= 1.711012.@13= 0.830 and k32/k23= 0.0050/0.0352 = 0.142 vs. k z l / k 1 , = 0.139. Values of the rate constant of the lumped system calculated by the Wei and Prater technique at different temperatures are given in Table 111. All these values were calculated from a single curved reaction path, starting from pure 4MCH. At each temperature, calculated reaction paths agree excellently with measured experimental points, whereas application of the optimization technique (Peereboom, 1981) to the same experimental points sometimes shows a less satisfying result (Figure 3). This pheqomenon is probably due to the fact that in estimating K by the optimization procedures, all the experimental reaction
Figure 3. Reaction paths calculated from the relative rate constants obtained by the Wei and Prater technique using only the data from the isomerization of 4MCH (-) and those obtained from a optimization procedure using all the experimental points (- - -).
l o5
I
1I
1.9
I
I
I
1
I
I
2.0
2.1
2.2
2.3
2.1,
2.5
-.+ +*lo3 ( K
-'
)
Figure 4. Arrhenius plots of the-relative rate constants ijjof the lumped system: kzl (01, k23 (X), k32 (A).
paths starting from each of the isomers have been used. However, evidence is given by Silvestri and Prater (1964) that various paths show differences in behavior as a consequence of experimental errors. Because of this, the Wei and Prater method makes use of the path from 4MCHthe reaction path from 3MCH, though strongly curved as well as being more sensitive to experimental errors. Among the tabulated data in the whole range of temperatures, the values of kI3 and k31 indicate that there is no conversion between MECH or lMCH and 4MCH. This reduces the coupled network to essentially a series reaction mechanism as already proposed previously, but for different reasons. Moreover, as far as the remaining rate constants are concerned, it appears that the temperature
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1667 dependency is rather small. Arrhenius plots are given in Figure 4. Estimation of the slopes leads to an apparent energy of activation for the conversion of the lump MECH/lMCH to 3MCH of 7.7 kJ mole1, which accords well with the value of the corresponding reaction from lMCH to 3MCH (7.8 kJ mol-'). As the 3- and 4MCH isomers are much alike, a small difference in the energy of activation for the interconversion of these two components can be expected, which results in a less accurate estimation of that value. However, within the accuracy obtained, the different? in thtapparent energies of activation calculated from k, and k32,1.4 kJ mol-', agrees with the difference in the apparent energies calculated from k34 and k43,0.3 kJ mol-'. The latter results demonstrate once again that, by using the lumped system, the principle features of the original four-component system can be made manifest.
Conclusions 1. It is worthwhile to lump reaction systems, in which rapid reactions between some of the species achieve a partial equilibrium, into a less complex reaction system. As in the case of the isomeric methylcyclohexenes and methylenecyclohexane,the kinetic behavior of the lumped system still provides sufficient information about the original system. 2. The Wei and Prater technique, combined with a computerized iteration procedure to find straight-line reaction paths, is very useful in estimating the rate constants of complex reaction systems. With knowledge of the equilibrium composition, and on the basis of one strongly curved reaction path, a rate constant matrix can be estimated with which reaction paths are calculated that agree excellently with the points measured experimentally. However, the various curved reaction paths that can be considered useful have to be tested beforehand on their sensitivity to experimental errors in the compositions. 3. As the Wei and Prater method does not exclude preliminary conversions between the various species, calculation of rate constants which are zero within the experimental accuracy now supplies additional evidence for the previous assumption (Peereboom, 1984) that there is no direct conversion between MECH and lMCH on the one hand and 4MCH on the other. Nomenclature Scalars ai = ith element of vector a (i = 1,2,...,n) or composition of species A i ai* = ith element of vector a * (=xo) Bi = ith element of vector 1 (i = 1,2,...,6) Ai = ith species in the n-dimensional composition space
6, = ith element of vector b (i = 0,1,...,( n - 1))or composition of fictitious species Bi 8 = space composed of vector b k , = relative reaction rate constant of reaction A, n = number of elements in vector a 6 = number of elements in vector ti t = time
-
A,
Vectors and Matrices a = composition vector in A space a * = equilibrium composition vector in A space
ti = vector of lumped species ti, = defined in (10) ti = defined in (11) al, = defined in (17) 4 = diagonal equilibrium composition matrix A = defined as MAMT b = composition vector in B space E = error matrix, defined in (7) K = kinetic rate constant matrix for the full system K = kinetic rate constant matrix for the lumped system M = lumping matric x, = ith eigenvector of matrix K (i = 1,2..4) 2l = ith eigenvector of matrix K (i = 1,2,..$) X = eigenvector of matrix K x = eigenvector matrix of matrix It y = vanishing eigenvector Greek Symbol A = diagonal eigenvalue matrix of matrix K Superscript = any property related to the lumped system Registry No. MECH, 1192-37-6;lMCH, 591-49-1; 3MCH, 591-48-0;4MCH, 591-47-9. A
Literature Cited Christoffel, E. G.; Surjo, I. T.; Robschlager, K. H. Can. J. Chem. Eng. 1980, 58, 513.
Collins, D. J.; Mulrooney, K. J.;Medina, R. J. J. Catal. 1982, 75, 291. Collins, D. J.; Scharff, R. R. Appl. Catal. 1983, 8, 273. Frenklach, M. Chen. Eng. Sci. 1985,40, 1843. Kuo, J. C. W.; Wei, J. Ind. Eng. Chem. Fundam. 1969,8, 124. Li, G. Chem. Eng. Sci. 1984, 29, 1261. Lombardo, E. A.; Hall, K. W. AIChE J. 1971, 1229. Marquardt, D. W. J. SOC. Ind. Appl. Math. 1963, 11, 2. Ozawa, Y. Ind. Eng. Chem. Fundam. 1973, 12, 191. Peereboom, M. Thesis, Delft University of Technology, 1981, p 141. Peereboom, M. J. Catal. 1984,88, 37. Peereboom, M.; Meijering, J. L. J. Catal. 1979, 59, 45. Peereboom, M.; van de Graaf, B.; Baas, 3. M. A. Red. J.R. Neth. Chem. SOC.1982,101, 336. Silvestri, A. J.; Prater, C. D. J. Phys. Chem. 1964, 68, 3268. Wei, J.; Prater, C. D. Adv. Catal. 1962, 13, 203. Wei, J.; Kuo, J. C. W. Ind. Eng. Chem. Fundam. 1969,8, 115. Zhu, K.; Chen, M.; Yan, N. Int. Chem. Eng. 1985,25, 542.
Received for review October 24, 1986 Accepted April 24, 1987
