Energy & Fuels 1992,6, 845-853
845
Semiempirical Rate Laws for Rice-Herzfeld Pyrolysis of Mixtures: Capturing Chemistry with Reasonable Computational Burden Abhash Nigam, Concetta LaMarca, Dean Fake, and Michael T. Klein' Center for Catalytic Science and Technology, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received June 2,1992. Revised Manuscript Received July 27, 1992
Mechanism-derivedrate laws for kinetically coupled Rice-Herzfeld pyrolysis were used to deduce the form of semiempiricalrate laws (SERLs) that nevertheless represent the mechanistic chemistry. These SERLs strike a balance between the CPU demands of mechanistic models and the lack of chemical significance of purely empirical models. The mechanism-derivedpyrolysis rate laws were phrased in terms of a pure component, initiation, propagation, and termination groups, akin to the kinetic term, the driving force, the adsorption group, and exponent of Langmuir-HinshelwoodHougen-Watson models. Taylor series expansions of the pyrolysis groups provided polynomial representations of each, which combined to form the SERL. Convergence of the Taylor series expansions further provides a relationship between the elementary step kinetic parameters of the mechanism and the parameters of the SERL. The coupled pyrolyses of (1) dibenzyl ether and phenethyl phenyl ether; (2) pentadecylbenzene (PDB); and (3) PDB and tridecylcyclohexanewere well represented by SERLs.
Introduction The production of petroleum-derived or synthetic fuels generallyinvolves the reaction of a complex mixture. The mathematical modeling of such mixtures has historically been approachedfrom two separate paths. At one extreme, purely empiricalmodels defiie the reactants and products by the results of analytical experiments.' The form of such a model is often an elementarymathematicalfunction which can correctly capture experimental trends with a small number of parameters. However, the incorporated parameters have little, if any, chemical significance and are usually very system specific. This limits the ability of the model to extrapolate and predict the behavior of other systems. Additionally, for mixtures with four or more componente, the number of model parameters undermines attempts to estimate them with accuracy and hinders the optimization of the model. On the positive side, however, an empirical model will generally give a very accurate and easy-to-computesummary of the results it regresses. At the other extreme, mechanistic models can capture the behavior of a reaction system over a wide range of conditions. The parameters in these models are rate constants of elementary steps in the mechanism,and their magnitude may often be estimated on an a priori basis with some knowledge of reactant and product structurea2 Unfortunately, as the number of components in the mixture increases, the complexity of the model rapidly becomesunwieldly. For instance, someheavy oil mixtures may contain more than 10 OOO components, each of which can react via a mechanism of 10 or more significant steps. For these cases analytical solution is extremely difficult.
* Corresponding author.
(1) Paspek, S. C.; Klein, M. T. Fuel Sci. Technol. Int. 1991,9(7), 839854. (2) Benson, S. W. ThermochemicalKinetics 2nd ed.; John Wiley and Son: New York, 1976.
0887-0624/92/2506-0845$03.00/0
Moreover, the computational demand for numerical solution can be extreme. These issues limit a purely mechanistic approach and motivate the search for simplifying approximations that do not sacrifice chemical significance. To this end, we have developed semiempirical approximations to analytical, mechanism-derivedrate laws in an effort to reduce the associated mathematical complexity and hence computational demand for kinetic modeling. We use the example of the exact rate law for a binary Rice-Herzfeld pyrolysis mechanism and show how Taylor series expansionsof this exact law can indicate the correct choice of a semiempirical form. This semiempirical rate law (SERL) maintains predictive ability and allows for straightforward parameter optimization, if desired. This work also provides the opportunity to scrutinize RiceHerzfeld pyrolysis and Langmuir-Hinshelwood-HougenWatson catalysis rate laws side by side, in order to emphasizethe influence of embedded assumptions on the complexity of each rate model. This further helps to suggest simple approximationsto the exact Rice-Herzfeld rate law. Homogeneous and Heterogeneous Reaction Cycles: Comparison of RH and LHHW Models While homogeneous and heterogeneous reaction cycles exhibit many conceptual similarities, the complexity of the rate laws that represent reactions of mixtures of many components varies greatly. The two classic examples of heterogeneous and homogeneous kinetics are LangmuirHinshelwood-Hougen-Watson (LHHW) catalysis and Rice-Herzfeld (RH) pyrolysis, respectively. Both formalisms account for the interactions among components that lead to kinetic coupling.3+ That is, the rate of (3) Tsepalov, V. F. Ruas. J. Phys.Chem. 1961,35,533-535. (4) Tsepalov, V. F. Russ. J. Phys. Chem. 1961,36,710-713.
0 1992 American Chemical Society
Nigam et al.
846 Energy & Fuels, Vol. 6,No. 6,1992 A1 + I
A11
Ri1
*
:.x
-
Kd$
K.?,
A I !3 241
A11
Ri1
o1+
RI+I
a +a
A ~ pi c1I
b P I + PI
241 2
p1+
k'
Pi
% ' 11
~ pt I -&? l
}-.-.
Pi
termination
component i in a mixture will in general differ from its pure component rate. This can be traced to the interactions of intermediates, the radicals in the case of pyrolysis and vacant and adsorbed sites in catalysis. The conservation and energetics of the reaction intermediates can be very different in these two reaction models. In the LHHW formalism,the reactive intermediatea(active centers) are constant. in total number. The sites can be vacant or occupied by any one of the n components in the mixture. Interaction between componentscan be indirect and can occur solely through competition for adsorption onto vacant sites. The natural starting point for the analysis of mixtures is the pure component reactions. Figure 1illustrates the LHHW cycle for first-order reaction kinetics. Aria7solved this mechanism using the steady-state approximation to obtain ral(Al) = Ct[Al - (Rl/Kt)l/
+-+-)+ 1
[ (KA)arl
KAl
1
KtkRl
In practice, various limiting cases of this equation are usually applied. For example,in the limit when the surface reaction is the rate-determining step eq 1 reduces to
The generalization to two components, both following a firsborder LHHW schemewith surfacereaction limiting, is rather straightforward rA1(A1'A2)
C&arlKAl[A1 - Rl/Kt] = 1 + KAIA1+ K A ~ + A KRIR1 ~ + KRzR,
(3)
The n-component rate law also has a simple form: Ctke.rlKAl[Al
rAl(Al,A2,...,An)=
n
+ Pj 27i
Pi +
Figure 1. Elementary steps in the reaction of AI via first-order Langmuir-Hinshelwood-Hougen-Watson catalysis and RiceHerzfeld pyrolysis.
- Rl/Kt] n
(4)
However, the analogous rate law for the pyrolysis of mixtures is much more complicated. In RH pyrolysis the reactive intermediates are the radicals formed from each of the components. Each radical (5) Tsepalov, V. F. Russ. J. Phys. Chem. 1961,35,832-833.
(6) Boudart, M.Kinetics of Chemical Processes; Prentice-Hall Inc.: Englewood Cliffs, NJ, 1968. (7) Aria, R. Introduction to the Analysis of Chemical Reactors; Prentice-Hall: Englewood Cliffs, NJ, 1968.
+ Aj 3 Ai + p j
~
j
-
17jl Wl 1 +
products
Figure 2. Cross-propagationand cross-terminationelementary steps in the RiceHerzfeld pyrolysis mechanism for a mixture. Table I. Number of Multiplication, Division, Addition, Subtraction, and Exponent Operations Needed To Evaluate the Rate from a Rate Law Which Is Explicit in Its Parameters.
n 1 2 3 4
LHHW 9 13 17 21
RH
SERLi
#
54
259 1932
1 (i = 1) (5)
9
19 (16) 38 (29) 63 (45)
a LHHW refers to first-order,surfacereaction rate limited catalysis and the parameters are Ctklri, K A ~K, Rand ~ Ki; RH refers to pyrolysis with geometric termination and the parameters are ai, ~ 1 1 yij, , kit kij, and k'ij; the SERL parameters are the K l s .
will in general have a unique identity and therefore reactivity. The radical concentration is related to the relative component concentrations through the pseudosteady-state approximation. In essencethis is the parallel of catalyst site conservation of LHHW, but it is less restrictive. As the relative concentrations of the reactants change, or if the total concentration is varied, the concentration of radicals is subsequently altered. Therefore, the number and identities of the reactive intermediates change. Interaction between componentscan occur in two ways: the radical engendered from one component can react with a different component resulting in chain transfer, or it can react with a radical engendered from another component in a termination step. The mechanism for one component undergoing RH pyrolysis is also presented in Figure 1. The long-chain approximation6is used to simplify the calculation of the overall rate and allows for the consumption of reactant by initiation reactions to be neglected in comparison to that from propagation reactions. For the reasonable approximation of geometric termination, the rate law of eq 5 is obtained.
The two-component rate law is more complex because the addition of a second componentallows kinetic coupling to occur through H-abstraction and cross-termination reactions. Figure 2 summarizesthe additional elementary steps present in a mixture. The rate law deduced from this mechanism,for the case of long chains with geometric termination, is shown as eq 6. Equation 6 illustrates the dramatic increase in complexity of the rate law for two components over that for one-it is certainly more dramatic than the same change in LHHW. While the complexity of LHHW increases in proportion to the number, n,of components in the system, the complexity of the RH law increases faster than 22n, (Complexityhere is related to the number of multiplication, division, addition, subtraction, and exponent operations needed to (8)Gavalas, G. R.
Chem. Eng. Sci. 1966,21, 133-141.
Rice-Herzfeld Pyrolysis of Mixtures
Energy & Fuels, Vol. 6, No.6,1992 847
eq 6 into the Taylor series of eq 8 seta forth the number of terms that need to be included to capture the correct kinetic behavior.
(6a)
A2(kl, +
k k’
For the case of eq 6a, the series terminates after three termswithcoefficientsfpmp(O, ...,0)= O,afpmddAl= l,afpmd aA2 = k d k l l . Continuing, the propagation group of eq 6b and the identical termination groups of eqs 6a and 6b can be expanded about the low AI, A2 limit into the Taylor series of eq 9. Equation 9 may be truncated after the
+AlAz term and entered into eq 7 provide the SERLs of eq 10. The AlAz term is retained to capture any effects of
evaluate the rate when the rate law is written explicitly in terms of its parameters.) This is illustrated in Table I for n = 1-4 components. Equation 6 also shows the potential for the rate of A1 pyrolysis to be enhanced as well as inhibited by the presence of Az, whereas, in the LHHW formalism for the mechanism of Figure 1, the addition of a second component inhibits. At a more conceptual level, the general form of a multicomponent rate law for Rice-Herzfeld pyrolysis shows similarities to that for LHHW catalysis. Both rate expressionscan be written as a combinationof terms. Y ang and Hougeng decomposed LHHW into “kinetic factor”, “driving force”, “adsorption”, and “exponent” terms. Similarly,eq 6 invites decompositioninto pure component, initiation, and, loosely, propagation and termination groups, the latter two being convenient but not mathematically exact. This is illustrated in eq 7.
An intuitive starting point for simplification and formulation of a semiempirical model is to seek low-order polynomials for each of the groups in the above expression. A Taylor series expansion provides such polynomials. By way of example, substitution of the propagation group in (9) Yang,
K.H.; Hougen, 0.A. Chem. Eng. Prog. 1950,46, 146.
binary interactions between components. The A12, A22, and higher order terms have been discarded since their contributions may be partially lumped into the linear and cross terms. In eq 10, R A is ~ the semiempirical rata law, rAi is the exact rate law, and Ki are SERL parameters. The SERLs of eqs 10a and 10b capture rate enhancement due to both an increase in initiation (or total radical concentration) and to chain-transferreactionsas well as inhibition due to the balance of chain transfer and termination, via the relative magnitudes of the propagation and termination groups. The use of any infinite series representation raises the issue of convergence; if the series converge quickly the parameters in eq 10 ( K )will be closely associated with the elementary-step rate-constant ratios found as coefficients in eq 9 (ki/k,). Alternately,if the series is slowly convergent, or worse, divergent, the parameters in eq 10will be strictly empirical. A region of convergence exists about the expansion point, and its size is governed by the relative magnitudes of the elementary-step rate constants. In-
Nigam et al.
848 Energy & Fuels, Vol. 6, No. 6, 1992
spection of eq 9 shows that for the series to converge at aspecified pair of A142 concentrations,the B-scieeion rate constants must be an order of magnitude greater than H-abstraction rate constants if the concentrations are of order 1,and an additional order greater for each order the concentrations are greater than 1. That is, for Yij 1,the condition kijAJk,
