Ind. Eng. Chem. Res. 1992,31,45-53 Sandia National Laboratories: Albuquerque, NM, 1989. Koasiakoff,A.; Rice, F. 0. Thermal Decomposition of Hydrocarbons, Resonance Stabilization and Isomerization of Free Radicals. J. Am. Chem. Soc. 1943,65, 590. Lutz, A. E.;Kee, R. J.; Miller, J. A. SENKIN: A Fortran Program for Predicting Homogeneous Gas-Phase Chemical Kinetics with Sensitivity Analysis. Sandia National Laboratories Report SAND87-8248; Sandia National Laboratories: Albuquerque, NM, 1987. McNab, J. G.; Smith, P. V., Jr.; Betts, R. L. The Evolution of Petroleum. Znd. Eng. Chem. 1952,44, 2556. Rice, F. 0.; Herzfeld, K. F. The Thermal Decomposition of Organic Compounds from the Standpoint of Free Radicals. VI. The Mechanism of Some Chain Reactions. J. Am. Chem. SOC.1934, 56,284. Starling, K. E.; Kumar, K. H. Computer Program for Natural Gas Supercompressibility Factor, Custody Transfer Calculations and
46
Process Calculations Using the OU/GRI Correlation. Preprints of the 64th Annual Gas Processors Conuention, Houston, TX; Gas Processors Association: Tulsa, OK, 1985. Stull, D. R.; Westrum, E. F., Jr.; Sinke, G. C. The Chemical Thermodynamics of Organic Compounds,Kreiger: Malabar, FL, 1987. Voge, H. H.; Good, G. M. Thermal Cracking of Higher Paraffins. J. Am. Chem. SOC.1949, 71, 593. Westbrook, C. K.; Warnatz, J.; Pitz, W. J. A Detailed Chemical Kinetics Reaction Mechanism for the Oxidation of Isooctane and n-Heptane Over an Extended Temperature Range and Ita Application to Analysis of Engine Knock. Twenty-second Symposium (International) on Combustion;The Combustion Instituk Pittsburgh, PA, 1988; pp 893-901.
Received for reuiew April 1, 1991 Revised manuscript received August 8, 1991 Accepted August 22,1991
Comparison of Lumped and Molecular Modeling of Hydropyrolysis Dimitris K. Ligurast and David T. Allen* Department of Chemical Engineering, University of California, Los Angeles, California 90024
This work examines three levels of detail in the modeling of hydropyrolysis: traditional lumped kinetic models, molecular modeling with a single molecular structure used to represent each carbon number in each compound class, and molecular modeling with multiple isomers representing each carbon number in each compound class. Detailed comparisons between the three cases are provided in the text. The molecular level modeling with a single structure per carbon number per compound class appears to offer the best compromise between information value and computation simplicity.
Introduction Kinetic modeling of hydrocarbon mixtures containing very many components is a challenging problem that has received considerable attention. Traditionally, the components of the mixture have been grouped together into kinetic lumps, which are then treated as pseudocomponents (Kuo and Wei, 1969; Wei and Kuo, 1969; Bailey, 1972; Hutchinson and Luss, 1970; Luss and Hutchinson, 1971; Liu and Lapidus, 1973; Lee, 1977,1978; Weekman, 1979; Coxson and Bischoff, 1987a,b). The concentrations of the lump are then followed as reactions proceed. One of the problems with this approach is that information about individual components, and thus all details of the product distribution, is lost. Another problem is that the structures present in the lumps evolve as the reactions proceed, changing the reactivity of the lumps. Thus, the rate constants associated with lumped models are frequently functions of feedstock properties and conversion. Such models can be used to organize data, but they are not generally useful as predictive tools. Molecular modeling of hydrocarbon mixtures can overcome many of the disadvantages associated with lumped kinetic modeling; however, molecular modeling requires a detailed knowledge of feedstock structure, rate constants, and reaction pathways that is not often available for heavy hydrocarbon systems. The purpose of this paper is to compare the predictions of molecular level models to the predictions of much simpler lumped models using the hydropyrolysis of heavy hydrocarbons as a case study. In order to highlight the differences in information content provided by these modeling approaches, several levels of model detail will be
* Author to whom correspondence should be addressed. + Present address: Department of Chemical Engineering, University of Delaware, Newark, D E 19716.
Table I. no. 1 2 3 4 5
Compound Classes name no. paraffins 6 olefins 7 cycloparaffins 8 cycloolefins 9 cyclodiolefins 10
name benzenes tetrahydronaphthalenes dihydronaphthalenea naphthalenes diolefins
examined. The starting point will be lumped kinetic modeling based on compound classes. Then, a very simple level of molecular modeling will be examined. In this level of modeling, each kinetic lump is characterized by a distribution of similar molecules. In this work, the molecules in the distribution are assumed to differ only in carbon number. This is clearly oversimplifying, so a final level of modeling will consider the effect of introducing additional variability into the molecular structures. The information potential of the different approaches will then be compared.
Lumped Kinetic Modeling Ten compound classes will be considered in the lumped model of thermal hydropyrolysis. The kinetic lumps, which are actually compound classes, are listed in Table I and include a variety of alipathic, olefinic, and aromatic groups. A more extensive list could be formulated; however, these classes are readily measurable and include most of the molecular classes present in heavy oils. The kinetic model consists of 29 reactions interconnecting 10 compound classes. A detailed description of the model has been given by Parnas and Allen (1988). A schematic diagram of the major reaction pathways is shown in Figure 1. Each compound class occupies a box, and reaction pathways are shown by the connecting arrows. The 29 reactions and their rate constants are given in Table 11. The rate parameters were taken from the work of Dente and Ranzi (1983) and Allen and Gavalas (1983). This
0888-5885/92/2631-0045$03.00/00 1992 American Chemical Society
46 Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992
of heavy hydrocarbons in a much simpler manner than a detailed molecular level model. The most important simplification is the reduction of the large number of components in a hydrocarbon mixture to a number on the order of 10 in the compound class model. The question remains, can the compound class model provide adequate details of the product distribution to be useful? To address this question, we will compare the predictions of the lumped model to predictions of a more detailed molecular level model.
-.
Olefins
7
3.4
-
1.2
c
Paraffins
10
c
8.10 9.11
Cycloolefins
25
16
In our molecular level modeling we will consider two levels of detail. The greatest level of detail would be to consider a large number of components approaching the number (O(104)) of components actually in the oil mixture. A less detailed molecular level model, containing approximately 250 components, will be used as our starting point. For this less detailed model, we have replaced each of the 10 kinetic lumps with a collection of approximately 25
Cycloparoffins
Benzenes
'
17
Cyclodiolefinr
26
22
29
Tetra-Hydro Naphthalenes Nap ht halenes Di-Hydro Naphthalens.
1 9
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 47 Table 111. Rate Eauatione
(PR), which produces an olefin and a smaller paraffin radical. A paraffin radical, PRi, with i carbon atoms, is consumed by ita j3 scission, but can also be created by the j3 scission of any radical of higher cafbon number. The rate of change in concentration of PRi can be expressed as N d -[PRi] = -k![PRi] + cijkf[PRj] d j (1) dt j=i+l where k[ is the rate constant for j3 scission (reaction 1 of Table 11) of a radical with i carbon atoms, N is the upper limit of the carbon number-distribution, and c.. is the probability that a radical PRj will form a radicd with i carbon atoms when it undergoes j3 scission. Development of rate expression is similar for bimolecular reactions of paraffin radicals. In reaction 3 of Table 11, a paraffin radical PRi reacts with an olefin of. carbon number j (OLJ to form a larger paraffp radical P&+jwith a probability eii. Paraffin radical PRi can be consumed by reaction with any olefin j , but it can be produced only by combinations of smaller paraffin radicals and olefins. Thus, for this bimolecular reaction d[PRiI -dt
-
Rates of change in concentration due to any reaction can be developed in a similar manner. The set of equations describing the model of hydropyrolysis are given in Table 111. Numerical integration of these equations requires knowledge of the initial distribution reaction rate constants as a function of carbon number and probability distribution functions describing the product slates for each reaction. For the hydropyrolysis case where a free-radical mechanism is assumed, the total-free-radical concentration and the free-radical distribution need to be estimated as well. Free-Radical Concentrations. Total-free-radical concentration will be estimated using a partial equilibrium approximation. Total-free-radical concentration is given by [RI = (Keq[Rl)1'2
where Kw is the lumped equilibrium constant for the initiation and termination reactions and [R] is the total concentration of molecular species. Termination by disproportionation is neglected. From data reported in the literature (e.g., Allen and Gavalas, 1983; Dente and Ranzi, 19831, the equilibrium constant can be approximated by
Keel = ki/kt N
i-2
j=2
j=l
-kf[PRi] Ceij[OLj] d j + Ck,3[PRj]ej,i-j[OLi-j]d j (2)
(3)
= IO6 exp(40000 cal/RT)
(4)
Although Kw is currently viewed as an adjustable model parameter, this work can accommodate expressions of KW
48 Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992
as functions of concentration or carbon numbers if such data are available. Since hydrogen abstraction reactions are fast relative to the reactions of Table 11, the radicals in each compound class are in equilibrium with each other and with the species making up the class Ri
+ Rj + Ri + Rj
(5)
,
OO
20
IO
Carbon Number
The relative concentrations of any two classes of radicals, i and j , can then be expressed as a function of the equilibrium constants and the class concentrations. While the total radical concentration and the relative radical concentrations of each class will be a function of time, the equilibrium constants will be assumed to be invariant. Estimation of Rate Constants and Product Distributions. For this work, a functional dependence of reaction rate parameters on carbon number will be assumed known. Specifically, it was assumed that the cracking reaction rate constant depends on the square of the carbon number CNi/CNIef
kf' = (CNi/CN,ef)2kFef
(7)
where kq is the rate constant of reaction CY for a molecule with i carbon atoms and k;ef is the rate constant for the reference carbon number for reaction CY shown in Table 11. The form of eq 7 is similar to the equation presented by Voge and Good (1949) for the rate of cracking of normal paraffins, although the latter considers a limited range of carbon numbers. For disproportionation, polymerization, and addition reactions, it was assumed that the rate decreases as the carbon number increases. The expression
k! = (CNref/CNi)k!ef
(8)
was used to estimate the rate constants. Table I1 lists the carbon number dependence used for each reaction in the model. Detailed model compound studies would, obviously, provide more accurate functional forms. As an example of how product distributions were calculated, again consider reaction l of Table 11. Assuming that all carbon centers in the paraffin will form a radical by loss of a hydrogen with equal probability, the ,i3 scission described by reaction 1 will result in a distribution of products approaching a normal distribution with respect to carbon number. Since the distribution functions are discrete, a normal distribution function with a continuous variable cannot be used. Instead, a discrete probability function will be employed: the binomial distribution function (Papoulis, 1965)
O S p l l q = 1 - p
(:)
n! = x!(n- x)!
(104 (lob) (10d
where n is a distribution parameter describing the carbon number range considered, p is a parameter describing the skewing of the distribution, and n! denotes the factorial of n. By setting p equal to 0.5, a discrete distribution is obtained that has an outline similar to the normal distribution. Replacing the discrete variable x by the carbon
O l dins
I
I
I
1
1
IO 20 Carbon Number
Figure 2. Initial hydrocarbon concentration distributions. Olefin and substituted benzene compound classes are present at a molar ratio of 1:2.
number i and n by the carbon number of the reactant j , the probability cij can be estimated as cij
= BO',i)
(11)
Similar distributions can be used for other reactions where radical formation can be assumed equally probable for all carbon centers. Consider now reaction 3 of Table 11, the combination of a paraffin radical with an olefin. It will be more probable for a small radical to combine with an olefin molecule than for a large radical to be added to an olefin molecule. That probability distribution can be approximated by the gamma distribution function (Papoulis, 1965)
10
x s o
where CY and ,i3 are the distribution parameters determining the spread and the sharpness of the distribution and r(a) is the gamma function
r(a)= &myu-le-Y dy
LY
>0
(13)
Replacing the discrete variable x with the carbon number i of the paraffin radical, the probability eij can be estimated: eii = gi (14) where j covers the range of olefin carbon numbers considered in the particular reaction. The probabilities gi are normalized so that their sum from 1 to j equals 1.
Results The lumped and molecular level models were employed for the initial hydrocarbon concentrations shown in Figure 2. Olefin and substituted benzene compound classes are present a t a molar ratio of 1:2. The reactions of this mixture will be considered over a range of hydrogen pressures. We report the results of both the lumped and molecular level models in Figures 3-7. The lumped model
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 49 I
10.5
- Bonrener
--. .
OIet1n r
--- PDloiefinr a r a t t Ins
..... Cycloparaffins
Figure 3. Evolution of the compound class concentration (a), mean carbon number (b), and standard deviation (c), with no hydrogen added at 700 O C .
predicts only the total concentration of each compound class. The molecular model can predict the evolution of all compound class concentration distributions in reaction time. It is cumbersome, however, to follow the complete distributions by sketching their density functions. A simpler approach is to follow the total class concentration, which is the information provided by the lumped model, and the distribution moments, Le., the mean carbon number and its variance or standard deviation. This method of presentation will be used in this work. The results in the case of no hydrogen added to the mixture are shown in Figure 3. The only transformations observed under these initial conditions are the olefin addition reactions to produce larger olefins and diolefins, albeit at very low rates. The substituted benzenes appear
I
1
I
I
I
p
3.41 0
-.-
I
1 (a)
i
I
I
1
2
3
1
1
4 5 Tlmo (8)
1
1
1
1
6
7
8
9
IO
- Benzenrr
-.--..
---
.....
Olotlnr Diolrflnr Poroftlnr Cycloparatf ins
Figure 4. Evolution of the compound class concentration (a), mean carbon number (b), and standard deviation (c), with hydrogen concentration = 0.1 mol/L at 700 O C .
to be relatively inert as indicated by the concentration and the mean carbon number evolutions for their compound class. The mean carbon number for the olefin and diolefin compound classes is predicted to increase, indicating that olefins are combining to give larger olefins and diolefins. That is supported by the decrease in standard deviation of the olefin distribution caused mainly by the consumption of smaller olefins. The diolefin standard deviation increases, which, coupled with the increase in mean carbon number, indicates that a large spectrum of diolefins is produced. Hydrogenation reactions become important when small amounts of hydrogen are added to the mixture, as shown in Figure 4. The net transformation is the hydrogenation of olefins into paraffins, while the substituted benzenes
50 Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 II
IO
I
I
J
1
1
1
1
1
1
-
18 A.
_.
-.-
--..
--.....
Benzenes OlefIns Diolefins Paraffins Cycloparaff ins
Figure 5. Evolution of the compound class concentration (a), mean carbon number (b), and standard deviation (c), with hydrogen concentration = 10.0 mol/L at 600 "C.
are relatively inert. The mean carbon number for olefins is predicted to go through a maximum a t short reaction times and to continuously decrease after that. The initial increase can be attributed to the olefin-olefin addition reactions, as in the previous case, when a significant pool of small olefins exists in the mixture. The decrease in the rates of addition with increasing size, coupled with the simultaneous increase in the cracking rates, causes the mean carbon number of the olefins to decrease at longer reaction times. This deduction is supported by the standard deviation evolution. After an initial sharp decrease, when olefins are consumed by combination reactions, the standard deviation levels off indicating that the hydrogenation reactions take place at similar rates. That works well with the paraffm class behavior, too. The paraffin class mean carbon number closely parallels the be-
- Benzenes
-.-
_-..
---
.....
Olefins Diolefins P a r a f f Ins Cycloparaffins
Figure 6. Evolution of the compound class concentration (a), mean carbon number (b), and standard deviation (c), with hydrogen concentration = 10.0 mol/L at 600 "C. Rate parameters are assumed to be functions of both the carbon number and carbon center distributions.
havior of the olefin class, reflecting the fact that paraffins are the olefin hydrogenation products. Similar observations can be made when the hydrogen concentration is significant as shown in Figure 5. In this case, however, substituted benzenes are quite reactive, being hydrogenated to cycloparaffins through cyclodiolefm and cycloolefin intermediates. Olefins are similarly hydrogenated to paraffh, which, as the final products of the reaction, show a monotonic increase in class concentration. Cycloparaffiin class concentration goes through a maximum since cycloparaffins are produced by benzene hydrogenation and are consumed by ring opening due to hydrocracking. The dominance of the cracking reactions in this case is evident from the decreasing mean carbon number
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 51 For paraffin distributions, the standard deviation goes through a minimum when the olefin class concentration diminishes and cycloparaffin class concentration approaches its maximum. It increases as cycloparaffin decomposition accounta for most of the paraffin reaction at intermediate times and starts to decrease aa cracking reactions of the paraffins themselves become important. The role of hydrogen in suppressing olefin formation and enhancing paraffii cracking as was found by Brooks (1967) is evident in Figures 3-5. Furthermore, increased hydrogen concentration causes aromatics hydrogenation and accelerates the conversion of naphthenes in agreement with the experimental results of Yamada and Amano (1983).There is qualitative agreement between the model predictions and the experimental results of Shabtai et al. (1979)and Bunger (1988). Finally, we note that Figures 3-5 demonstrate that compound class concentration is not sufficient for characterizing the evolution of the kinetic lumps. The momenta of the carbon number distribution seem t~ be a good indicator of lump structure, but there are alternative lump characterizations that could be useful. In the next section we will examine whether carbon type distributions might affect kinetic lump evolution.
Effect of Carbon Center Distributions on Rate Constants The development of the molecular level model has up to this point assumed that rate parameters are functions of carbon number, but are unaffected by the structural characteristics of the components of the compound c k . This has allowed us to model a heavy oil with approxi/------mately 250 molecular species. If, however, the rate parameters are significantly affected by the structural characteristics, i.e., the carbon centers in each class, then we will need a much larger number of species in our molecular model. To assess the importance of variation in carbon centers, we will employ the free-radical mechanism first proposed by Rice (1933)and later modified by Kossiakoff and Rice (1943)also known as the R-K theory or the radical chain theory. The main assumption in this approach is that radical formation is the rate-determining step; i.e., the rate is directly proportional to radical for2.S0 '1 2 3' 4 I 5 I 6 1 7 I 8 I 9 I1 0 mation. l i m o (s) According to the R-K theory, the rate of radical for._ Benzenes mation by hydrogen removal depends on the carbon center -.- Olefins is removed from. The difference in activation the hydrogen --.. Diolefins energy for removing a primary versus removing a second--- Paraffins ..... Cycloparaffins ary hydrogen is approximately 2.0 kcal/mol. The same difference in activation energy is assumed for secondary Figure 7. Evolution of the compound class concentration (a), mean and tertiary hydrogens. Assuming that the relative rate carbon number (b), and standard deviation (c), with hydrogen conof removal of primary hydrogen is Rpm,the rate of removal centration = 10.0 mol/L at 600 O C . Rate parameters and product of a secondary hydrogen will be distributions are assumed to be functions of both carbon number and
---I
'
carbon center distributions.
for all compound classes. The interplay of reactions, however, becomes clearer if the standard deviation is considered. The standard deviation for both benzenes and olefins decreases at the same rate as their class concentration. The value for the olefins, however, goes through a minimum, at the time the pool of initial reactants is exhausted, and then increases to parallel the value for cycloparaffins. That is to be expected since most of the olefiis are produced by cycloparaffin ring opening and to a lesser extent by paraffin cracking. The cycloparaffin distribution standard deviation increases initially as cyclics are produced from benzene hydrogenation, only to go through a maximum when the benzene pool is exhausted and start decreasing later as cracking reaction dominate.
R,,, = ex~(2000/RT)R,,, and that of a tertiary will be Rtd = exp(4000/RT)Rp,,
A t 600 "C,removal of a secondary hydrogen will be 3.17 times faster than removal of a primary hydrogen, while removal of tertiary will be 10.0 times faster. Consider, for example, the case of paraffins. Since the goal is to examine the influence of carbon center distributions, assume that the rate parameters employed up to now were the rate parameters of normal paraffins. A normal paraffin with carbon number k has a primary hydrogen mole fraction of 2 ( 3 ) / ( 2 h + 2) and a secondary hydrogen mole fraction of (2k - 4 ) / ( 2 k + 2). For example,
52 Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992
C6HI4has a 6/14 = 0.43 primary fraction and a 8/14 = 0.57 secondary hydrogen fraction. At 600 "C, its relative reactivity will be (0.43)(1)+ (0.57)(3.17) = 2.23. Assume now that the paraffin compound class has a known carbon center distribution of 0.30 primary, 0.50 secondary, and 0.20 tertiary carbon fractions. The corresponding hydrogen fractions will be 0.43 for primary, 0.48 for secondary, and 0.09 for tertiary, resulting in a relative reactivity of 2.90 at 600 "C. The rate then would be 2.9012.23 = 1.30 times the rate of normal paraffins. Similar estimates can be computed for the other compound classes. For these preliminary calculations, it was assumed that the carbon center distributions were invariant as reactions proceeded. Incorporating the effect of carbon centers in this way had little effect on the class concentration, mean carbon number, and standard deviation. The general picture can be seen using only one of the cases, when a significant amount of hydrogen is added to the mixture, as shown in Figure 6. The compound class evolution c w e s are quite similar to those for the base case, although the paraffin class concentration increases and the cycloparaffin class concentration decreases slightly faster here. The mean carbon number evolutions are remarkably similar except for the olefin class, which shows a small increase at longer reaction times.
Effect of Carbon Center Distributions on Product Distributions In the section describing the estimation of reaction probabilities, it was assumed that radical formation is equally probable for all carbon centers, and thus a pseudonormal product distribution would result. This section examines both the radical formation and product distribution removing the above assumption. In this section, it is assumed that the final radical distribution on a molecule before cracking depends entirely on the carbon center constitution of the molecule and not on where the radical was initially formed. Taking as an example the molecule c -c-c-c
I
-c-c-c-c
with three primary, five secondary and one tertiary carbon centers, the probability that the radical would, at equilibrium, be located on the tertiary position depends only on the relative stability of primary, secondary, and tertiary radicals. For the molecule shown above, the probability of forming the radical on the tertiary position is 0.173 while the probability that the radical would be on a particular secondary carbon would be 0.125. Establishing the possible carbon radical distribution on a molecule allows the cracking probabilities to be considered directly. When more than one possible cracking position exists (cracking occurs on a bond @ to the hydrogen-deficient carbon), the relative rates of cracking are determined by the resonance stabilization of the radical products. The relative rates shown below assume that the energy difference between each radical class is 1.7 kcal. At 700 "C the relative rates would be radical formed methyl ethyl and higher primary Becondary tertiary
re1 rate 1.00 2.41
5.81 13.99
In order to incorporate the above relative rates with the probabilities of radical formation to estimate the product distribution, data on molecular structures is necessary. Since such data are usually unavailable, a random isomer
distribution was assumed. Specifically, if the carbon center distribution consists of only primary, secondary, and tertiary carbons, random positions for the tertiary carbon centers within a molecule was assumed. Positioning the other carbon centers was then straightforward. The random isomer distribution, however, can be viewed as an adjustable model parameter that will depend on the available knowledge of the system. Any such strategy, combined with the probabilities of formation and reaction, will result in a product distribution different than the pseudonormal one employed earlier. Since the overall reaction rate parameters are not affected by this treatment (only the product distribution is affected), little change in compound class concentration evolution was observed. That is shown in Figure 7. The case when a significant amount of hydrogen is added to the mixture is considered again to facilitate the comparison between the lumped and molecular models. The concentration evolution curves are identical with those of the previous case.
Conclusion This paper has examined several different approaches to the kinetic modeling of complex reacting systems. The approaches differ in their level of detail and their information potential. At the lowest level, traditional lumped kinetic models were able to track the compound class concentrations in time, but their information content is minimal. Once the chemical species are grouped together into lumps, information about individual components is lost and only concentrations of the lumps can be followed. Thus, the details of product distribution are lost. At a higher level are models that lump individual chemical species in compound classes, as in traditional lumped models, but each lump is characterized by a s p e c k distribution which accounts for the evolving characteristics of the lumps. The rates of interconversion are functions of the distributions characterizing the lumps. These models are able to not only predict the evolving lump concentration but also, by following the evolution of the distributions, provide useful insights into the changing nature of the lump, e.g., the size transformations and the associated chemistry in the case examined here. No single distribution is sufficient for characterizing the evolution of the reacting mixture. The use of multiple characterization variables can be demonstrated by using both carbon number and functional group distributions. As was evidenced by the second case study, incorporating the effect of functional groups on the rate parameters will mainly affect the overall rates of reaction. Since the net effect tends to be a broadening of the range of applicable rate parameters, the main influence is exerted on the second moment of the product distribution while the compound class concentration and the first moment show minimal changes. Selection of level of detail for kinetic modeling of complex systems greatly depends on the information available on the system and its kinetic behavior. Information gained by going from traditional lumped kinetic models to models incorporating molecular detail is significant and worth the extra effort. Including mixture structural characteristics in the scheme enhances the quality of the predictions, but a significant number of additional assumptions may be needed and the complexity of the scheme will increase. Even so, molecular level models present clear advantages in describing the behavior of a system in great detail and should be preferred whenever possible. One can always simplify a model by lumping when the system is wellknown. The reverse is not possible.
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 53 = standard deviation of distribution
Acknowledgment
u,
This work was supported by the National Science Foundation through Presidential Young Investigator Award CTS 86 57180.
Literature Cited
Nomenclature B(n,x) = binomial distribution function BZj = benzene molecule with j carbon atoms cij = probabilities for reaction 1 in Table I1 C, = component distribution CLDj = cyclodiolefin molecule with j carbon atoms CLOj = cycloolefin molecule with j carbon atoms CLPj = cycloparaffin molecule with j carbon atoms CNi = carbon number i DHNj = dihydronaphthalene molecule with j carbon atoms DIOj = diolefin molecule with j carbon atoms eij = probabilities for reaction 3 in Table I1 f i j = probabilities for reaction IO in Table I1 f, = density function of C, gij = probabilities for reaction 14 in Table I1 H = hydrogen atom Kq = equilibrium constant for the initiation and termination reactions K j = equilibriumconstant between radicals and molecules for compound class j k,, = rate constant for component x in reaction z ki = rate constant for initiation reactions k , = rate constant for termination reactions N = upper limit of carbon number distribution NP, = naphthalene molecule with j carbon atoms OLj = olefin molecule with j carbon atoms P,,, = probability of reaction PRi = paraffin molecule with i carbon atoms PRi = paraffin radical with i carbon atoms qij = probabilities for reaction 11 in Table I1 rij = probabilities for reaction 15 in Table I1 R = molecule R = gas constant R i = molecule in compound class i R = radical Ri = radical in compound class i Rprm= relative rate of removal of primary hydrogen R,,, = relative rate of removal of secondary hydrogen Rtrt = relative rate of removal of tertiary hydrogen Si. = probabilities for reaction 25 in Table I1 &Nj = tetrahydronaphthalene molecule with j carbon atoms vi, = probabilities for reaction 23 in Table I1 wij = probabilities for reaction 26 in Table I1 Greek Symbols a =
gamma distribution function parameter
/3 = gamma distribution function parameter r = gamma function p = mean value or mean of distribution uX2= variance of distribution
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