Ind. Eng. Chem. Res. 1995,34, 3793-3800
3793
Single-Event Rate Parameters for Paraffin Hydrocracking on a Pt/US-Y Zeolite Glenn D. Svoboda, Erik Vynckier, Birgit Debrabandere, and Gilbert F. Froment* Laboratorium voor Petrochemische Techniek, Universiteit Gent, B-9000 Gent, Belgium
A single-event kinetic model is applied to octane hydrocracking on a PtNS-Y zeolite over a wide range of experimental conditions (2’ = 200-260 “C; P = 10-50 bar). The single-event kinetic approach used in this paper draws on the chemical knowledge of the elementary steps occurring on the catalyst surface, retaining the full detail of the reaction network. The hundreds of rate coefficients of the elementary steps in the reaction network are expressed in terms of a practical number of single-event rate coefficients. Rate equations are developed for the paraffins and the olefin and carbenium ion surface intermediates based on this network. Finally, singleevent rate coefficients are estimated for both isomerization and cracking reactions.
Introduction Many catalytic processes encountered in petroleum refining deal with feedstocks consisting of thousands of components. The kinetic modeling of such processes, required for simulation and optimization purposes, is no simple task. To date, most efforts have focused on lumped kinetic models (Nace et al., 1971; Jacob et al., 1976; Ramage et al., 1980). In lumped models, the complex feed is divided into several “lumps” based on compound classes or boiling point range, and a simplified reaction network between the lumps is developed. A major fundamental limitation with lumped models is that the rate coefficients obtained often depend on feed composition or reactor configuration. The single-event kinetic approach used in this paper, on the other hand, draws on the chemical knowledge of the elementary steps occurring on the catalyst surface, retaining the full detail of the reaction network (Baltanas et al., 1989; Froment, 1991). Because the single-event rate coefficients pertain to elementary chemical steps, they are invariant, regardless of feedstock or reactor configuration. For this reason, single-event rate coefficients obtained from model compounds can then be used t o predict conversion and product distribution for other feed compositions. The single-event approach has recently found applications in hydrocracking (Baltanas et al., 1989, Froment, 1991) and catalytic cracking (Feng et al., 1993). In hydrocracking, the single-event model has been applied successfully to a limited network involving the hydroisomerization of n-octane and monomethylheptanesover a PWS-Y zeolite. Single-event parameters for protonation, deprotonation, hydride and methyl shift, and PCP reactions were determined. However, no singleevent parameters for cracking or for isomerization between more highly-branched isomers have been determined to date. Because of its fundamental nature, the single-event kinetic model requires a molecular analysis of the feedstock, which is not feasible for complex feedstocks. Therefore a certain amount of lumping is inevitable for practical applications. For this reason, Vynckier and Froment (1991) developed formulas by which lumped rate coefficients can be calculated through reconstruction of the single-event kinetics. This exercise requires a complete set of single-event rate coefficients, which is not presently available for hydrocracking. This work, although it does not yet provide a complete set of parameters, extends the application of the single-event
model to a network including di- and tribranched isomers and cracking reactions.
Single-Event Kinetic Model Generation of the Network of Elementary Steps. The first task involved in the application of the singleevent kinetic model is the generation of the network of elementary steps involved in the process. Hydrocracking of paraffins on a noble-metal-loaded zeolite is a bifunctional process and thus contains reaction steps on both the metal and acid sites. The noble metal dehydrogenates paraffins into intermediate olefins, which in turn are protonated on Bransted sites. The resulting carbenium ions undergo rearrangements and cracking steps according to well-known chemistry (Brouwer, 1980; Martens and Jacobs, 1990). The rearrangements can be hydride or alkyl shifts, or isomerizations involving protonated cycloalkanes, while cracking occurs through @-scissions. Deprotonation and hydrogenation results in saturated isomers or cracked products. Examples of reactions occurring in paraffin hydrocracking are given in Figure 1. Generating the thousands of products and elementary steps requires a computerized algorithm. To this end, Clymans and Froment (1984) introduced a computer algorithm utilizing Boolean relation matrices for generating such networks. Each carbon atom is represented by a number, and a Ci-Cj bond is indicated by a “TRUE” value (i.e., 1)in row i and columnj (and r o w j and column i) of the matrix. Formation or scission of a bond is implemented by addition or removal of true values in the corresponding positions.
Elementary Steps and Number of Single Events. The networks encountered in paraffin hydrocracking can contain thousands of species and thousands of individual isomerization and cracking steps. Prediction of product distributions requires knowledge of each of the associated rate coefficients. However, in paraffin hydrocracking, the many reactions involved pertain to only a few types of elementary steps: protonation, deprotonation, alkyl shifts, branching reactions involving protonated cycloalkanes, and ,&scission. Trgnsition state theory can be used to show that these elementarystep rate coefficients are multiples of single-event rate coefficients (Baltanas et al., 1989; Vynckier and Froment, 1991):
k
= ne&
0888-5885/95/2634-3793$09.oo/o 1995 American Chemical Society
(1)
3794 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995
pep/ +
11
v
PCP
Y
+
+
it
PCB
^r ' ES
I
It
It
* +it
Figure 1. Typical elementary steps occurring in the hydrocracking of paraffins.
Figure 2. Determination of the number of single events for an elementary step.
where
In eq 1, k is the elementary-step rate coefficient, ne is the number of sjngle events associated with the elementary step, and k is the single-event rate coefficient. In eq 2, 0.4 is the global symmetry number of the reactant, and a T is that of the transition state. In this way, the multitude of elementary rate coefficients is reduced to a limited number of single-event rate coefficients. A simple example can be used to illustrate the need for distinguishing between elementary steps and single events. Consider the 1,2-methyl shift between 2-methyl-3-heptylcarbenium ion and 3-methyl-2-heptylcarbenium ion depicted in Figure 2. The activated complex for a methyl shift is commonly regarded as an alkylbridged cation where the shifting methyl group is bound
to two carbon atoms via a two-electron, three-center bond. Both reactants are secondary carbenium ions, but two methyl groups can shift in the forward reaction, while only one in the reverse. One might suspect the forward reaction to proceed at twice the rate of the reverse reaction. Equations 1and 2 above confirm this presumption. The global symmetry numbers for 2-methyl-3-heptylcarbenium ion, 3-methyl-2-heptylcarbenium ion, and the transition state are 33, 3312, and 33/22, respectively. These global symmetry numbers correspond to four single events in the forward reaction and two single events in the reverse reaction. The forward reaction rate constant is twice as large as the reverse, as expected. The same result is arrived at when a statistical factor approach is employed (Bishop and Laidler, 1965, 1969). For example, in the forward reaction, two methyl groups can shift on either side of the plane, amounting to a statistical factor of 4. Assumptions Concerning the Single-EventRate Parameters. A number of simplifying assumptions are incorporated in the model discussed here (Baltanas et al., 1989) (the names of the various types of reaction are given in Figure 1 and in the Nomenclature Section): (a) Primary carbenium ions and the methyl ion are far less stable than secondary and tertiary ions and are not retained in the network of elementary steps for the case of hydrocracking. (b) The rate coefficient for protonation is independent of the olefin involved but does depend on the type of the resulting carbenium ion (s = secondary; t = tertiary), introducing two single-event rate coefficients, b r ( S ) and KPr(t). (c) The rate coefficients for isomerization of carbenium ions depend only on the types of the reactant and the product, requiring four single-event rate coefficients, kisom(S;s),&som(s;t), kisom(t;S), and hisom(t$), for each type of isomerization (isom = HS; MS, ES, etc.; PCP, PCB, etc.). (d) The cracking rate coefficients depend only on the types of carbenium ions involved, not on the olefin produced, leading to four single-event rate coefficients, kcr(s;s,O), &r(s;t,O), Kcr(t;s,O),and Ldt;t,O). Thermodynamic Constraints and Further Simplifications. While simple assumptions maintain the number of single-event rate coefficients within tractable limits for the protonation, isomerization, and cracking reactions, the same cannot be accomplished for the deprotonation rate coefficients. The single-eyent rate coefficients of deprotonation, &D,(s;~,) and kDe(t;O,), depend both on the type of carbenium ion and the character of the olefin 0 ~ This . condition leaves an overwhelming number of deprotonation coefficients, which can only be reduced through a thermodynamic analysis. Baltanas et al. (1989) were able t o greatly reduce the number of required deprotonation parameters by employing the following two thermodynamic constraints:
k,,(m;O,)
= k ~ e ~ m ; o r ~ ~ ~ ~ o(3) ~ ~ o r ; o ~
hiSom(t;s)
kiSom(s;t)
hDe(t;Or> kpr(t)
gpr(S)
KDe(s;or)
(4)
In eq 3, 0, is any olefin, Or is a selected reference olefin of the same carbon number, and m is either s or t. 0, is an isomer with the double bond preferably in a position that produces both a secondary and a tertiary
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3796 carbenium ion on protonation. Equations 3 and 4 are valid for any carbon number n. This constraint reduces the number of indepsndent single-eyent deprotonation coefficients to two, kDe(s;Or,,) and kDe(t;O,,), for each carbon number. Equation 4, which applies for all the isomerization steps, leads to two more reductions in the number of parameters. First, concerning isomerization, equation 4 reveals that the rate coefficient for (t;s) isomerization is not independent. This constraint leaves only three parameters for each type of isomeGzation. Second, concerning deprotonation, the ratio of kDe(t;Or) to kDe(s;Or)in eq 4 must be independent of the carbon number n since both the isomerization and protonation parameters are assumed independent of the carbon number:
The final number of independent deprotonation parameters is arrived a t through judicious selection of a homologous series of reference olefins (Froment, 1991). The rates of deprotonation of the carbenium ions in the homologous series are assumed equal. This assumption, coupled with eq 5, reduces the number of independent single-event deprotonation Coefficients to two, kDe( S;Or) and kDe(t;or), where Or is a reference olefin in the homologous series. From C5 onward the deprotonation coefficients for the 2-methyl-2-alkenes are assumed to be equal, since they have the same structure. For the olefins up to Cg, the 1-alkenes are taken as reference olefins and the deprotonation coeffkient is assumed to be independent of the chain length. For C5, two references are chosen (2-methyl-2-butene and 1-pentene). They are connected through eq 3.
Rate Equations for Paraffin Hydrocracking The kinetic rate equations for observables and surface intermediates can be formulated in terms of the elementary steps of the carbenium ions on the acid sites (Baltanas et al., 1989). First, a paraffin Pi is physically adsorbed into the pore system of the zeolite: Pi(gas)t P,(ads)
(9) Protonation of olefins on Brmsted sites results in carbenium ions
0,
+ H+ * Rik+
(10)
which in turn undergo isomerization and cracking steps:
(11) (12) After deprotonation of the carbenium ions and hydrogenation of the produced olefins the resulting isoparaffins and cracked paraffins can desorb from the zeolite. The acid catalyzed steps are assumed to be rate determining, making the reaction rates of the paraffins identical to the reaction rates of the corresponding olefins on the acid sites:
RP, = CRO"
(13)
J
as can be derived from the pseudo-steady-state approximation for the olefins. The rates of formation of the olefins on the acid sites can be expressed in terms of carbenium ion concentrations as follows:
where mik is the type (s or t) of carbenium ion Rik+.The concentrations of the carbenium ions are not directly accessible but can be calculated from the pseudo-steadystate approximation for the carbenium ions:
(6)
Hereafter, the (ads) indication is dropped for ease of notation. The concentrations of the physisorbed paraffins are determined by a Langmuir isotherm, as was concluded in previous lumped kinetic studies (Steijns and Froment, 1981; Baltanas et al., 1983): and a balance on the Brernsted sites:
(16)
(7)
The adsorbed paraffin is dehydrogenated on the metal phase of the bifunctional catalyst:
Pi * 0,
+ H,
(8)
The olefin 0, is given a second indexj because more than one olefin can be formed from a single paraffin Pi. The (de)hydrogenation step is assumed t o be in quasiequilibrium; therefore, the concentration of the olefin is calculated from thermodynamic equilibrium:
Equations 15 and 16 form a linear system which can be solved to yield the unknown CR,~+'S and CH+, which in turn can be used to calculate the net rates of formation of the paraffins from eqs 13 and 14. Similar equations to those above have been developed for naphthenes and aromatics (Vynckier and Froment, 1991).
Parameter Estimation The single event kinetic parameters for isomerization and hydrocrackingof paraf€ins were estimated by means
3796 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995
of a data base consisting of hydrocracking experiments over a PWS-Y catalyst with n-octane and 90/10 molar mixtures of n-octane/2-methylheptaneand n-octane/2,5dimethylhexane as feeds. Octane was chosen as the feed molecule because it is the smallest paraffin from which all the required single-event kinetic parameters can be determined. The catalyst was an LZ-Y20 zeolite from Union Carbide loaded with 0.5 wt % Pt. The experiments were performed over a wide range of temperatures (200-260 “C), pressures (10-50 bar), hydrogen to hydrocarbon ratios (30- 1001, and space times (10-420 (g,,t.h)/mol) in a Berty-type reactor with complete mixing of the gas phase. Conversions ranged from 0 to 75%. No deactivation occurred. This was carefully checked by a standard test. The details of the equipment and procedure have been described by Steijns and Froment (19811,but a few modifications were made since. Methane was used as an internal standard, a 50 m (i.d. = 0.25 mm) RSL-150 column with a 0.25 pm poly(dimethylsiloxane) film was used instead of an OV-101 column, and cryogenic cooling permitted the separation of methane from the light cracked products. The reactor effluent consisted primarily of n-octane, monomethyland monoethyl-branched isomers, dimethyl- and methylethyl-branched isomers, and cracked products. Only trace quantities of trimethyl-branched isomers were found. A gas chromatographic analysis permitted a nearly complete molecular specification of the products for these feed molecules. A few overlapping peaks were present. The 3-ethylhexane and 3-methyl-3-ethylpentane peaks fell under the 3-methylheptane peak, the 3,4dimethylhexane peak fell under the 4-methylheptane peak, and the 2-methyl-3-ethylpentanepeak fell under the 2,3-dimethylhexane peak. As the identity of the components in the overlapping peaks was known, their flows were calculated in the parameter estimation using the network of the reactions and the estimated parameter values. The sum of their flow rates was compared with the flow rate corresponding to the “overlapping” peaks, so that no components were neglected in the parameter estimations. In the network generation, only 1,a-hydride and 1,2alkyl shifts were considered, and protonated cycloalkane branching reactions larger than cyclopropane were disregarded. In addition, all alkyl shift reactions were assumed to proceed at the same rate and were not differentiated in the network. The network consisted of 22 paraffins, 75 olefins, and 57 carbenium ions participating in 75 (delhydrogenations, 108 (delprotonations, 66 hydride shifts, 42 alkyl shifts (including methyl, ethyl, propyl, and isopropyl shifts), 106 PCP branching reactions, and 17 ,&scissions. A simultaneous treatment of the data at all temperatures was performed, requiring the determination of 36 Arrhenius parameters for the 17 single-event rate coefficients (2 Pr’s, 2 De’s, 3 HS’s, 3 AS’S,3 PCP’s, and 4 /?-scissions) and the octane Langmuir physisorption constant. Only one physisorption constant was used for all the octane isomers, and the physisorption constants for the cracked products were assumed zero. Thermodynamic data for calculation of equilibrium constants were obtained from TRC tables (Thermodynamics Research Center, 1982) for the paraffins, while Benson’s group contribution method (Benson et al., 1969) was used for the olefins, since experimental data were not available for all the olefins involved in the network. The regression analysis was performed according to Froment (1975). Parameter estimates were obtained by mini-
mizing the objective function:
” ”
n
Here, u is the number of independent responses, n is the number of observations, and d is an element of the inverse of the covariance matrix of the experimental errors of the responses determined from repeated experiments (Van Parijs et al., 1986). Only the diagonal elements of the matrix were used in this work. In order to reduce the correlation between preexponential factors and activation energies, reparametrization was applied according to Kitrell (1970). The number of independent responses totaled 18, which is less than the number of paraffins, 22, due to the overlapping peaks, which were combined and treated as one response. In addition, one response is not independent and could be determined from a carbon balance of the other responses. The n-octane response was selected as the dependent response. Outlet flow rates were calculated using a Newton-Raphson iteration method since a complete product spectrum was not available for a direct calculation of rates from partial pressure data. In this way, the peaks which were not integrated are not neglected. A total of 267 experiments were performed: 199 with n-octane, 35 with n-octane1 2-methylheptane, and 33 with n-octanel2,5-dimethylhexane. Initial minimization of the objective function was achieved using the method of Rosenbrock (Rosenbrock and Storey, 1966), while closer to the minimum a multiresponse Marquardt routine was applied (Marquardt, 1963).
Results and Discussion Initial parameter estimates revealed that the protonation and deprotonation coefficients were strongly correlated with the isomerization and cracking coefficients, indicating that the (de)protonation step was in quasi-equilibrium. In addition, the value calculated for CH+revealed that the site balance in eq 16 could be neglected (i.e., CH+% Ct). These conditions were used in the remaining estimations, requiring some modifications in the rate equations and parameter estimates. The carbenium ion concentrations can now be calculated directly from the thermodynamic equilibrium relationship:
(18) Substituting Ct for CH+and changing to the single-event equilibrium constant yields
With equilibrium in the protonation step and a trivial site balance, the single-event reference protonation equilibrium constant and the single-event isomerization and cracking rate coefficients cannot be determined independently and appe-ar as a composite parameter, for example, Kpr(Or;mik)~isom(mik;mlo). For this reason, where necessary in the rate equations, single-event rate coefficients were replaced by the corresponding composite rate coefficients, and the concentrations of the
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3797 carbenium ions were replaced by the concentrations divided by Kpr(Or;mik). KR(0G;mjk)in eq 19 can be related to KR(Or;mik)using the thermodynamic constraint in eq 3:
Table 1. Composite Arrhenius and van't Hoff Parameters for Octane Hydrocracking on a PWS-Y Zeolite parameter
k*,, mol/(g,,t:h)
parameter
KO,bar-' 1.73 10-4
E*,,kJ/mol
Rpr(OG;mik) = ~ p r ~ O r ; m i k ~ R i s o m ( O (20) ~;Or~ Substituting eq 20 into eq 19 and dividing through by Kpr(Or;mid yield
KL.~
Once the paraffin and olefin concentrations and carbenium ion concentrations divided by kR(Or;mik)are calculated using eqs 7,9, and 21, respectively, the rates of formation of the paraffins can be calculated directly from the following:
52.2
Table 2. Composite Rate Coefficients and Langmuir Physisorption Constant for Octane Hydrocracking on a PWS-Y Zeolite at Several Temperatures 220 "C 2.75 x 3.27 x 4.80 x 8.91 x 4.68 x 6.45 5.71 x
(22) k
Ah?&, kJ/mol
j
where
parameter
KL,E
104 105 10' lo2 10' 103 103
k', mol/(g,,t.h) 240 "C 3.44 x 104 4.08 105 6.98 x 10' 1.29 x lo3 7.92 x 10' 9.23 103 1.06 104
260 "C 4.24 x 5.00 9.86 x 1.81 x 1.29 x 1.29 1.87 x
104 105
10' lo3 lo2 104 104
220 "C
240 "C
260 "C
5.79
3.52
2.23
arametrized Arrhenius estimates are of the form
(26) where k*o,pcp'(s;t)is the reparametrized composite preexponential factor, E*,,pCp(s;t)is the composite activation energy, and T i s the average temperature of the experiments. The composite preexponential factor can be expressed in terms of the reparametrized composite preexponential factor as follows:
Rg= r
n
In eqs 22-24, RgL\Ycr is the net rate of formation of carbenium ion Rzk+ from isomerization and cracking reactions, and REJ is the net rate of formation of olefin 0, from cracking. &som(mzk;m~o) in eq 23 includes only alkyl shifts and PCP branching reactions. Hydride shifts are not needed here since carbenium ions connected by them are in quasi-equilibrium. The protonation and deprotonation single-event rate coefficients are also not needed since they have been incorporated into the composite single-event isomerization and cracking coefficients. Therefore, the number of parameters is greatly reduced, from the original 36 to 22. The remaining parameters include 2 Arrhenius parameters each for 10 single-event parameters (3 AS'S, 3 PCP's, and 4 p-scissions) and for the octane Langmuir physisorption constant. In practice, relative concentrations are used in the estimation because the quantities Csat and Ct are unknown (i.e., C*p, = Cp/Csat, C*R,~+ = CR,,-/CJ. Csat and Ct become incorporated into the parameter estimates. The following is an example of the composite rate coefficient for PCP(s;t): k*pCp(S;t) = CsatC&p,(O,;s>Rpcp(s;t)
(25)
where k*pcp(s;t) is the composite rate coefficient and kpCp(s;t) is the single-event rate coefficient. The rep-
The composite Arrhenius parameters can be expressed in terms of single-event parameters as follows:
where AHpr(Or;s) is the heat of protonation of the reference olefin to form a secondary carbenium ion. The above changes were made to the rate equations, and a regression analysis was performed. Due to the very low tribranched concentrations present, the parameter estimates for the (t;t)reactions could not be determined significantly. These low concentrations are due t o the very fast rate of the P(t;t)reaction associated with 2,2,4trimethylpentane. For these reasons, the AS(t;t)and PCP(t;t) reactions were eliminated from the network, and the P(t;t) reaction was assumed to be instantaneous. These changes were incorporated in the network, and a final regression analysis was performed. All 16 remaining parameter estimates were significant. Table 1 lists the composite Arrhenius parameters obtained
3798 Ind. Eng. Chem. Res., Vol. 34,No. 11, 1995
//
- a 0.020 -
0,016 -
i LL
0.010
O.Oo0
0.006
0.004
0.0011
w'u 0.004
0
0.008
0
0.012
0.016
0
0.024
0.020
0.002
0.004
0.m
0.008
0.010
0.016
0.0030
C
i
0.0024
0.012
0.000
0.0012 0.004
0.0008
4
I /
/
0
0 0
0.0008
0.0012
0.0018
0.0024
0
0.4 00
0.W8
- e
0.004
0.ow
V
f
0
n
0. 16
0.012
/
0.m-
0.004
/
-
LL 0.004
n 0
0.004
O.Oo0
0.012
0.016
0
0.002
0.004
0.006
0.008
F obs
F obs
Figure 3. Observed and calculated responses for several octane isomers and cracked products. Response identification: a, 2-methylheptane; b, 4-methylheptane 3,4-dimethylhexane; c, 2,3-dimethylhexane 2-methyl-3-ethylpentane; d, 2,5-dimethylhexane; e, 2-methylpropane; f, propane. (The reaction conditions are specified in the text.)
+
from these parameter estimates, while Table 2 lists the corresponding composite rate coefficients at several temperatures. The parameters listed in Tables 1 and 2 are in agreement with what is known concerning carbenium ion chemistry. For each type of elementary step reaction at each temperature, the (s;t)reaction was found to be faster than the (s;s) reaction, and type A (AS) rearrangements were found to be considerably faster than type B (PCP). The model also yielded an excellent fit to the experimental data. Parity plots for several of the responses are shown in Figure 3. The fit to experimental data is quite remarkable considering the complexity of the reaction network, the wide experimental conditions employed, and the few parameters involved.
+
Conclusion
A single-event kinetic model was applied t o octane hydrocracking on a pt/LTS-Y zeolite. Experiments with n-octane and 90110 molar mixtures of n-octanel2-methylheptane and n-octanel2,5-dimethylhexane were performed in a Berty-type reactor over a wide variety of experimental conditions, and single-event parameters were estimated. For the catalyst used here, the (de)protonation step was found to be in quasi-equilibrium, requiring some modifications to the rate equations and parameter estimates. Due t o the very low tribranched isomer concentrations present, the (t;t) parameters could not be determined significantly. The remaining parameter estimates, including alkyl shift, PCP, and
Ind. Eng. Chem. Res., Vol. 34,No. 11, 1995 3799 cracking parameters, were found to agree with what is known concerning carbenium ion chemistry. The fit of the single-event model to experimental data was excellent, providing evidence for the validity of the singleevent approach for the conditions used here. This work represents an advancement in the application of the single-event approach to hydrocracking, providing the first parameters for (s;t) isomerization reactions and the first parameters for cracking. Future work will include experiments with tribranched isomers in order to obtain the complete set of single-event parameters necessary for the calculation of lumped rate coefficients for paraffin hydroisomerization and hydrocracking on PWS-Y zeolite. It would be of interest also to check the parameters with other versions of catalysts of this family.
Nomenclature AS = alkyl shift CH+= concentration of free active acid sites, mol/gcat, CQ, = surface concentration of 0 0 , rnol/gcat, C p , = surface concentration of Pi, moYgcat,
Cpt* = relative surface concentration of Pi, dimensionless CR,~= surface concentration of Rik+, moYgCat, C*R,~= relative surface concentration of Rik+, dimensionless CSat.= concentration of total physisorption sites, moYgcat. Ct = concentration of total active acid sites, moYgCat, De = deprotonation DH = dehydrogenation E , = activation energy, kJ/mol E", = composite activation energy, kJ/mol ES = ethyl shift HD = hydrogenation AHp,(O,;m) = heat of protonation of 0, to form a carbenium ion of type m, kJ/mol k' = composite rate coefficient, moY(g,,t;h) k', = composite preexponential factor, moY(gcat:h) k",' = reparametrized composite preexponentialfactor, mol/ (gcat:h) &(ml;mz,O) = single-event cracking rate coefficient of a carbenium ion of type ml to form a carbenium ion of type m2 and an olefin, h-' hD,(m;Oij) = single-event deprotonation rate coefficient of a carbenium ion of type m to form 00,h-' Kisom(ml;mz) = single-event isomerization rate coefficient of a carbenium ion of type ml to form a carbenium ion of type mz, h-l kisom(Rikf;Rlof) = elementary-step isomerization rate coefficient of Rik+ to form Rlof,h-l I?isom(O~;Or) = single-event equilibrium constant of 0 0 to form Or, dimensionless M m ) = single-event protonation rate coefficient of an olefin to form a carbenium ion of type m, gCat./(mol*h) K D H=, ~dehydrogenation equilibrium constant of Pi to form Oij, bar KL,~ = Langmuir physisorption constant of Pi, bar-l KP,(Oij;Rik*) = protonation equilibrium constant of olefin 00to form carbenium ion Rik+,g of catalystlmol &,(Oy ;m) = single-eventprotonation equilibrium constant of olefin Oy to form a carbenium ion of type m, gC,Jmol I?p,(O,;m) = single-event protonation equilibrium constant of the reference olefin to form a carbenium ion of type m, gcdmol m, ml, mik = type of carbenium ion (s or t) MS = methyl shift ne = number of single events 0,j = olefin with indexj formed from paraffin with index i 0, = reference olefin p~~ = gas-phase partial pressure of HP,bar
p , = gas-phase partial pressure of P,, bar P, = paraffin with index i Pr = protonation PCB = branching via pronated cyclobutane PCP = branching via pronated cyclopropane R,k+ = carbenium ion with index k formed from paraffin with index i Ro, = net rate of formation of 0, , mol/(gcat.h) RGF = net rate of formation of 0, from cracking, mol/ (gcat ah) Rpz= net rate of formation of P,, moY(gcat*h) RR,~+ = net rate of formation of R,k+,mol/(gcat*hr) RikRf 'som/Cr - net rate of formation of R,k- from isomerization and cracking, md(gC,t ah) s = secondary S = objective function t = tertiary T = temperature, K, "C = average temperature of experiments, K Yk, = observed value of the ith response during the kth experiment, mom
Greek Letters = calculated value of the ith response during the kth experiment, mom uoV= global symmetry number of 0, UA = global symmetry number of the molecule A B , = cracking via &scission UR,,- = global symmetry number of R,k+ UT = global symmetry number of the transition state a'] = element of the inverse of the covariance matrix of the experimental errors of the responses, h2/mo12 vkl
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Received for review November 15, 1994 Revised manuscript received May 17, 1995 Accepted J u n e 6, 1995@ IE940667S Abstract published in Advance A C S Abstracts, September 15, 1995. @
