Detailed Kinetic Models in the Context of Reactor ... - ACS Publications

Energy & Fuels 1999, 13, 1135-1144

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Detailed Kinetic Models in the Context of Reactor Analysis: Linking Mechanistic and Process Chemistry Prasanna V. Joshi, Ankush Kumar, Tahmid I. Mizan, and Michael T. Klein*,† Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received February 10, 1999

Detailed kinetic models for the modeling of complex chemistries, including thermal cracking, catalytic reforming, catalytic cracking, and hydroprocessing, offer the compelling advantage chemical significance at the mechanistic level. They carry a considerable burden, however, in terms of species, reactions, and associated rate parameters. This, together with the batch and the plug flow reactor balances, requires solution of a large system of either stiff ordinary differential equations (ODE) or stiff differential algebraic equations (DAE), for both homogeneous and heterogeneous processes. It is often faster numerically to solve a stiff system of ODEs and, thus, it can be useful to convert a system of DAEs to ODEs for numerical solution schemes. For heterogeneous PFR systems, the reactor steady-state balances result in a set of DAEs, and it would therefore be desirable to construct the associated set of ODEs to minimize CPU demand. To this end, we propose that such a transformation can be achieved by making the “flowing surface species” approximation. This involves approximating the overall rate of reaction of surface species, which is identically equal to zero at reactor steady state, by a spatial derivative. We show that this approximation becomes better as the system of equations becomes stiffer, and, hence, is a reverse analogy of the kinetic steady-state approximation in the case of batch systems. To validate the proposition, we analyze various contrived and real examples of mechanistic kinetics for heterogeneous systems.

Introduction The uses of Detailed Kinetic Models (DKM) have evolved considerably over the past two decades. No longer just “off-line” learning models, DKM have become the kinetics kernels of process chemistry models intended for industrial applications. This trend can be expected to continue.1-4 Many DKM represent the chemistry at the mechanistic level. That is, the equations represent the elementary step reactions of both molecular and intermediate (e.g., radicals, carbenium ions) species. This allows a process chemistry model to serve the dual purposes of process design and optimization as well as catalyst property, and, more generally, chemistry optimizations. Mechanistic models are expected to provide the best basis for extrapolation and thus prediction of process chemistry conditions not in the model-tuning database. The technology of thermal cracking for olefin production provides some of the most carefully tested mechanistic models.3-7 The structure of these models is often * Corresponding author. † Present address: Rutgers University, School of Engineering, B204, Piscataway, NJ 08854. Telephone: (732) 445-4453. Fax: (732) 4457067. E-mail: [email protected]. (1) Quann, R. J.; Jaffe, S. B. Chem. Eng. Sci. 1996, 51 (10), 1615. (2) Quann, R. J.; Jaffe, S. B. Ind. Eng. Chem. Res. 1992, 31 (11), 2483. (3) Broadbelt, L. J.; Stark, S. M.; Klein, M. T. Ind. Eng. Chem. Res. 1994, 33 (4), 790-799. (4) Joshi, P. V.; Iyer, S. D.; Klein, M. T. Rev. Process Chem. Eng. 1998, 1 (2), 111-140. (5) Sundaram, K. M.; Froment, G. F. Chem. Eng. Sci. 1977, 32, 601608.

a set of ordinary differential equations (ODE) describing either the PFR holding-time (τ) dependence of the reacting composition (CAi), in the case of constant volume systems, or the reactor-length (z) dependence of the molar flow rate (FAi), in the case of systems with molar expansion. Conceptually, the classic Batch Reactor (BR) to Plug Flow Reactor (PFR) equation transformation of t f τ, for constant volume systems, and even those including molar expansion, nonisothermal, etc., has been made. The stiff ODE are solved forward in time through a “relaxation holding time (τ/PFR)” given appropriate inlet conditions on molecule (CAi ) C0, or FAi ) FAo) and radical concentrations (Cβi ) 0, or Fβi ) 0). It is noteworthy that both the molecules and radicals are in a true plug flow. The situation for heterogeneously catalyzed mechanistic models (e.g., acid cracking via carbenium ion intermediates)4-11 is a bit more subtle. Unlike the homogeneous thermal cracking case, the heterogeneously catalyzed system is truly heterogeneous: the flowing fluid-phase molecules are transformed through the actions of the solid, surface-phase intermediates. Thus, at reactor steady state, the concentration of surface species is governed by a set of algebraic equations. The question thus arises as to whether the t f τ transformation can be made rigorously. What is the difference, for the solid-phase species reactor balances, (6) Sundaram, K. M.; Froment, G. F. Chem. Eng. Sci. 1977, 32, 609617. (7) Fake, D. M.; Nigam, A.; Klein, M. T. Appl. Catal. 1998, 160 (1), 191.

10.1021/ef990021i CCC: $18.00 © 1999 American Chemical Society Published on Web 10/26/1999

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between a steady-state system (PFR) and an unsteadystate system (BR)? Should the “solid-phase” intermediates have apparent flow terms? DKMs, due to their detailed nature, have a large number of species, reactions, and rate parameters associated with the reaction network. Thus, the numerical solutions of DKM often involve a large system of ordinary differential equations (ODE) or differential algebraic equations (DAE). These systems are often stiff for real problems and hence difficult to solve numerically.12 The system of DAEs poses convergence problems for such large nonlinear systems and hence it is often useful to convert the systems of DAEs to ODEs. For heterogeneous PFR systems, reactor steady-state balances result in a set of DAEs which need to be transformed to a set of ODEs for faster solutions. To this end, we propose such a transformation by using a “Flowing Surface Species Approximation” (FSSA). This involves approximating the overall rate of reaction of surface species by a spatial derivative having a flow term, hence transforming the set of DAEs to a set of ODEs. The validity and use of this approximation is addressed in the present paper as follows. First, the classic case of A f B f C is used to introduce the relationship between the stiffness of the system and validity of various simplifying assumptions for the homogeneous system. This “traditional” small model approach is examined and used to define a “correct reference point” for the t f τ transformation for homogeneous systems. This approach is then extended to the heterogeneous system by considering several “simple” kinetic schemes to test the validity and use of the FSSA, and hence expose the subtleties of the t f τ transformation for heterogeneous systems. This is followed by examination of FSSA approximation for the complex heterogeneous process chemistry of catalytic reforming. Background Table 1 summarizes the time-honored chemical engineering approach to the modeling of several ideal reactors. Its six examples illustrate the BR-to-PFR transformation, the CSTR limiting case, and the axial dispersion reactor (ADR) model of intermediate mixing. Moreover, Table 1 exposes the reactor balances as comprising two terms, the LHS, which describes the mixing, and the RHS, which describes the chemical kinetics. The rate laws of Table 1 might be simple power law schemes or more complicated analytical solutions to Rice-Herzfeld (RH) pyrolysis or Langmuir-Hinshelwood-Hougen-Watson (LHHW) catalysis. The essential concept is that, in principle, they represent the (8) Watson, B. A.; Klein, M. T.; Harding, R. H. Ind. Eng. Chem. Res. 1996, 35 (5), 1506-1516. (9) Watson, B. A.; Klein, M. T.; Harding, R. H. Int. J. Chem. Kinet. 1997, 29 (7), 545. (10) Dumesic, J. A.; Rudd, D. F.; Aparicio, L. M.; Rekoske, J. E.; Trevino, A. A. The Microkinetics of Heterogeneous Catalysis, ACS Professional Reference Book; American Chemical Society: Washington, DC, 1993. (11) Boudart, M. Kinetics of Heterogeneous Catalytic Reactions; Princeton University Press: Princeton, NJ, 1984. (12) Brenan, K. E.; Campbell, S. L.; Petzold, L. R. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations; North-Holland: New York, 1989.

Joshi et al. Table 1. Conventional Reactor Material Balances reactor batch PFR (constant volume) PFR (molar expansion) CSTR axial dispersion reactor (with first-order reaction) simplified riser reactor

material balance dCA ) -rA dt dCA ) -rA dτ dFA ) -rA dV (CA - CAo) ) -rA τ 2 dCA 1 d CA ) - kτC dz PeL dz2 dFA ) -(1 - )FcatrA dV

Figure 1. Analytical rate law from mechanism. Homogeneous chemistry (A f B + C).

solution of the kinetics scheme. In practice, an empirical function devoid of obvious mechanistic origins may be used. In either case, reaction intermediates, such as free radicals or surface species, are absent from the explicit rate law because of their mathematical elimination, in principle, as a consequence of kinetics solution. The mechanistic nature of the RHS of Table 1 is developed more fully in Figure 1 and 2, which summarize the procurement of an analytical rate law for RH pyrolysis and LHHW catalysis, respectively. Figure 1 illustrates how the use of the kinetic steady-state approximation (KSSA) leads to an analytical rate law that could serve any of the reactor balance equations of Table 1. Similarly, Figure 2 shows the Aris13 solution of the LHHW reaction of A f B using the KSSA. Note the reduction of the KSSA rate law to various RDS limitations. It is worth emphasizing that the KSSA provides the rate law in terms of observable fluid-phase species only. The intermediates are eliminated as part of the mathematical analysis. The key teaching of Figures 1 and 2 is that the chemical kinetic scheme is conventionally solved before the introduction of its analytical rate law representation into the reactor balance. There would be no flow terms for the radicals β or µ in the reactor balances. The KSSA implicitly represents these concentrations in terms of the concentrations of the molecular species. Likewise, there are no real or imaginary flow terms for the surface species, l, Al, and Bl in reactor balances using rA from Figure 2 for LHHW catalysis. Again, the different (13) Aris, R. Introduction to the Analysis of Chemical Reactors; Prentice-Hall: Englewood Cliffs, NJ, 1965.

Kinetic Models in Reactor Analysis

Energy & Fuels, Vol. 13, No. 6, 1999 1137

Figure 2. Analytical rate law from mechanism. Heterogeneous chemistry (A f B).

kinetic time constants allow the concentration of the intermediates to be represented by the molecular species. The analytical rate laws of Figures 1 and 2, and their incorporation into the reactor balances of Table 1, provide a conventional reference point for the construction of reactor-sensitive DKM where the KSSA is reasonably appropriate. Thus, the treatment of the intermediates in these situations will define “correct” and guide the treatment of intermediates in numerical DKM, which involve too many components for analytical solutions to be reasonably anticipated. In particular, the solution of the chemical kinetics scheme before its introduction as a rate law into the reactor balance will guide the formulation and solution of reactor-sensitive DKM. Development The analysis will focus on homogeneous and heterogeneous systems in turn. The former systems have the pedagogical value that both the molecular and intermediate species are in a true physical flow. The t f τ transformation is indeed quite natural and obvious for these homogeneous systems. This can have the unanticipated effect of encouraging the same t f τ transformation for heterogeneous systems, however, which introduces the paradox of flowing surface intermediates. Homogeneous System. The likely “batch reactor” origins of DKM are illustrated in Figure 3. The mathematical structure of the model is a set of, in general, nonlinear equations for the time dependence of the concentration of molecules (CAi) and intermediates (Cβi). This can be decomposed into two sets of equations, one set for CAi and the other for Cβi. Frequently, as in longchain pyrolysis, for example, the sets of equations will be linear in CAi and Cβi. The convenient numerical solution of this batch DKM is to integrate the set of ODEs forward in time, given initial conditions for CAi ) CAio, and Cβi ) 0 at t ) 0. The equations are generally stiff as a consequence of the high reactivity of the intermediate species β. The

Figure 3. System of ODE for various ideal homogeneous reactors.

solver integrates through the relaxation time (t/BR),11,14 over which β goes from 0 to, essentially, KSSA values. The relationship between the stiffness of the equations and the KSSA is probed in Figure 4 as a plot of the intermediate concentration B in the scheme in eq 1: k1

k2

A 98 B 98 C

(1)

As can be seen from Figure 4, the KSSA approximation becomes better at shorter relaxation times, t/BR () 1/k2),11 as the ratio λ () k2/k1) increases. The stiffness of the system of ODEs is related to the eigenvalues of the system (1 ) k1, 2 ) k2), as shown in eq 2.

stiffness ratio )

|max| k2 ) |min| k1

(2)

Hence, from eqs 1 and 2, and from Figure 4, it is clear (14) Bowen, J. R.; Acrivos, A.; Oppenheim, A. K. Chem. Eng. Sci. 1963, 18, 177.

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Figure 4. Comparison of exact solution of ODE, and KSS approximation for homogeneous A f B f C system as a function of the inverse stiffness ratio (k1/k2).

that the KSSA approximation becomes better, i.e., t/BR and C/B/CAo become smaller, as the system of equations becomes stiffer. It is noteworthy that this result is of generic nature irrespective of the size and complexity of the system. The reactor balance equations shown in Figure 3 can be analytically integrated, for both the BR and PFR, for the simple reaction scheme shown in eq 1. The resulting analytical expressions for the BR and PFR are shown in eq 3 and eq 4, respectively.

CA ) CAo exp(-k1t) CB )

k1CAo

(exp(-k1t) - exp(-k2t))

(k2 - k1)

(3)

CA ) CAo exp(-k1τ) CB )

k1CAo

(exp(-k1τ) - exp(-k2τ))

(k2 - k1)

(4)

where t is the real time, and τ is the space time. As the stiffness ratio λ () k2/k1) increases, i.e., k2 . k1, the term (k2 - k1) f k2, and the terms (exp(-k1t) exp(-k2t)) and (exp(-k1τ) - exp(-k2τ)) tend to exp(k1t), exp(-k1τ), respectively, after the small relaxation time has passed.11 Thus, the concentration profile for CB in eqs 3 and 4 can be written as shown in eqs 5 and 6, respectively.

CB =

k1CAo k1CA CA (exp(-k1t)) ) ) k2 k2 λ

(5)

CB =

k1CAo k1CA CA (exp(-k1τ)) ) ) k2 k2 λ

(6)

Equations 5 and 6 show that as λ increases, the KSSA, i.e., rB ) k1CA - k2CB = 0, becomes a better assumption. Thus, for stiff systems (large λ), the original systems of ODE, for both the BR and the PFR, can be transformed into a system of DAE, and the two solutions are the same in the limit λ f ∞. In general, the solution strategies summarized in Figure 5 can be applied to integrate the steady-state reactor balances both for the gaseous stable species and radical intermediates shown in Figure 3. When analytical solutions for the overall rate are available the reactor/kinetics issue is clear. The only question is whether the KSS (Rβi ) 0 for βi) or rate-determining step approximation (RDSA) is preferred. When numerical solutions are required, it seems reasonable to consider the use of the BR f PFR transformation and solve the equations through the relaxation time τ/PFR. A numerical solution that invokes the KSSA on the intermediates, however, and transforms the system of ODEs for A and β into a set of ODEs for A and a set of algebraic equations for β, may be the closest conceptual mimic of the conventional “correct” scheme. Heterogeneous System. The relationship between stiffness and the KSSA for homogeneous systems has, likely, invited the BR f PFR transformation for constantvolume systems shown in Figure 3. Note that the dβ/dτ arise both from the simple replacement of t in the BR equation by τ, and also because the actual flow for homogeneous systems would produce a dβ/dτ from a “first-principles” shell balance. The awkwardness of this line of reasoning becomes clear when the catalysis is heterogeneous. There is neither flow in nor flow out of the intermediates in the real sense. This invites a closer look at the shell balances for the BR f PFR transformation for a heterogeneous system.



Figure 5. Summary of common conceptual solution strategies for homogeneous systems.

The batch reactor balances for both the gaseous-phase species and the surface intermediates are similar to that of the homogeneous system, and are shown in Figure 3. But for a PFR, the shell balance-derived transient balances for the surface intermediates are different and are shown in eq 7:

dCβi dt

+ QβT

dCβi dϑR

- Rβi ) 0

(7)

Neglecting surface diffusion (QβT ) 0), inflow (Cβio ) 0), and outflow (Cβi ) 0) of the surface species, eq 7 can be transformed, at reactor steady state, to eq 8:

Rβi ) 0

CB )

dFAi ) RAi dϑR (9)

Thus, the reactor steady state (RSS) condition requires the surface species to be at kinetic steady state (KSS), i.e., Rβi ) 0, for the shell balance-derived model. Hence, for a heterogeneous system, KSSA (Rβi ) 0) is an exact result due to the lack of physical flow of surface intermediates. Figures 6 and 7 show the comparison of (1) the approach to the reactor steady state (transient solution) and (2) the KSSA, for the classic A f B f C prototype system shown in eq 1, for λ ) 1, and λ ) 10, respectively. In the analysis of eq 1 for heterogeneous systems, species B is considered as the surface intermediate. Two things can be noted from Figures 6 and 7. First, for the heterogeneous case, KSS is a prerequisite for RSS, and, second, the time required for RSS becomes smaller as the stiffness ratio λ () k2/k1) increases. For a PFR, at RSS, the concentration of surface species B, is given by eq 10 for the simple case shown

k1 CA CA ) k2 λ

(10)

It is to be noted that since B is considered to be a surface species, the overall mass balance satisfies eq 11; species B is not included in the mass balance. At any given time t, the overall integral mass balance is shown in eq 12, thus it includes an accumulation term for B.

CAo,inlet ) CA,outlet + CC,outlet

(11)

inlet ) outlet + accumulation

(8)

Thus the complete set of the governing equations for both the molecular species and surface species for a PFR, at reactor steady state, are shown in eq 9:

Rβi ) 0, where FAi ) QTCAi

in eq 1:

QT

∫0tCAo,inletdt ) QT∫0t(CA,outlet + CC,outlet)dt + ∫0t(∫0VCB(ϑ,t)dϑ)dt

(12)

The numerical solution strategies for solving the steady-state reactor balances are considered in more detail in Figure 8. For the batch reactor system two solutions are possible, the first being the integration of the stiff system of ODEs over the reaction time, and the second being the use of the KSSA approximation to transform the stiff system of ODEs to a system of DAEs. For a PFR, a solution to the set of DAEs given in eq 9 can be obtained either by solving the nonlinear DAE system or solving two simpler systems: (1) ODEs in molecules, and (2) algebraic system in surface species. It is often found, due to the stiffness of the systems in real heterogeneous chemistries, that the convergence of a numerical scheme for a system of nonlinear DAE is slow and hence time-consuming.11 It is thus often advisable to convert the system of DAE to ODE either by differentiating the set of algebraic equations with respect to the independent variable or by approximating the derivatives of the variables constrained by the algebraic equations. The observation about the relationship of stiffness and the usefulness of the KSSA in homogeneous systems suggests a parallel approximation in the case of stiff heterogeneous systems. To this end,

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Figure 6. Approach to steady state for a PFR (heterogeneous system). λ ) k2/k1 ) 1, CAo ) 1 g mol/cc, τ ) V/Q ) 1 s.

Figure 7. Approach to steady state for a PFR (heterogeneous system). λ ) k2/k1 ) 10, CAo ) 1 g mol/cc, τ ) V/Q ) 1 s.

we propose an approximation to convert a system of DAE to a system of ODE, for the PFR. The KSSA, for the homogeneous system, teaches that the concentration profile of the intermediates in a PFR (as a function of the reactor length) is due to the gaseous molecular species profile along the reactor length. Further, if the system of ODEs is stiff enough, it can be converted to a set of DAEs using the KSSA. This

invites a reference frame shift that gives the surface intermediates an apparent flow with the same velocity as that of the gaseous species for a heterogeneous system, to convert the system of DAEs into ODEs. At RSS, due to the equilibration between the gaseousphase molecules and surface species, the residence time (lifetime) of the surface species is a function of the gasphase superficial velocity. Thus, a “flowing surface



Figure 8. Numerical solution strategies for heterogeneous batch reactor (BR), and plug flow reactor (PFR).

species” reactor balance can be used as an approximation to the exact solution shown in eq 9 to transform the system of DAE to ODE for the PFR. The corresponding approximate balance equations are shown in eq 13:

d(QTCβi) ) Rβi dϑR

(13)

where QT is the total gaseous flow in cm3/s. Since the exact solution for the surface species is Rβi ) 0, appropriateness of this flowing surface species approximation (FSSA) is a function of the stiffness of the system, which, in turn, is a function of the rate constants. This approximation is valid if the term on the LHS of eq 13 is very small. The proposition is that the FSSA will become better as the systems of equations get stiffer. This is analogous to the KSSA in the case of a homogeneous system. The system of eq 1 will be considered to develop this proposition further. The intermediate B is considered to be a surface species, which is, governed by the exact steady-state PFR reactor balances shown in eq 9. The overall rate of reaction of B is given in eq 14, and the first derivative with respect to vR, dCB/dvR, is shown in eq 15:

RB ) k1CA - k2CB ) 0 w

d(QTCB) k1 d(QTCA) k1 d(FA) ) ) dϑR k2 dϑR k2 dϑR

(14) (15)

Thus the proposition is that as the stiffness ratio λ () k2/k1) increases, d(QTCB)/dvR f 0 and thus RB f 0. The analytical solution of the FSSA model of eq 13 and that for the ODE for CA are shown in eq 4. It has already been shown in the homogeneous case that the KSSA solution of the system of ODEs becomes better as the equations become stiffer. Similarly, here, eqs 4 and 6 reveal that as the stiffness ratio (λ ) k2/k1) of the system increases, the FSSA, of the system of DAEs, becomes a better approximation. In short, for faster

solution it is efficient to approximate the 0 ) K‚Cβ algebraic problem with d(QTCβ)/dvR ) K‚Cβ because for stiff systems, d(QTCB)/dvR = 0. Thus the spatial derivative d(QTCβ)/dvR satisfies the 0 ) K‚Cβ RSS (or KSS) condition numerically. This result is of generic nature though proven for a simple A f B f C case. The result can be easily seen from Figure 9, where the FSSA solution approaches the exact RSS solution within a trivially small relaxation time (τ/PFR). The volume at which these two solutions become equal decreases as the system gets stiffer, thus validating the proposition. A comparison of Figures 4 and 9 reveals that the KSSA is an inverse of FSSA. For a homogeneous system, the KSSA gets better with increasing stiffness of the system of ODE shown in Figure 3. Similarly, for heterogeneous systems, the FSSA gets better with increasing stiffness of the system of DAE shown in eq 9. These notions were probed by simulation of the simple heterogeneous sequences of Table 2. The batch reactor solution was achieved by integrating the system of ODE and then comparing this “exact” solution with that of the KSSA. For PFR systems, both the FSSA (eq 13) and the “exact” (eq 9) numerical schemes were solved for all cases at RSS. Figures 10 and 11 show the comparison of various solution strategies for the A f B Aris scheme in a BR and PFR, respectively. The corresponding BR and PFR equations are shown in Table 3. It can be clearly seen that for the system under consideration, all the solutions are equivalent after the negligible relaxation period. A stiffness analysis of the nonlinear system shown in Table 3 was done by evaluating the matrix of partial derivatives (Jacobian matrix) at the inlet condition, CA ) Ao, CB ) 0, l ) l0, and CAl ) CBl ) 0. The stiffness ratio, as defined in eq 2, was found to be 1.0E+08 for the values of the rate constants shown in Table 4. The relaxation times, t/BR, for the BR, and τ/PFR, for the PFR

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Figure 9. Comparison of exact solution of DAE, and FSS approximation for heterogeneous A f B f C system as a function of the stiffness ratio λ (k2/k1). Table 2. Sample Heterogeneous Schemes scheme number

overall reaction

detailed mechanism kA

A + l 798 Al kSR

I

ATB

Al 798 Bl kg

Bl 798 B + l kA

A + l 798 Al kSR

II

ATB+C

Al + l 798 Bl + Cl kg

Bl 798 B + l kC

Cl 798 C + l

Figure 11. Comparison of exact solution, and FSSA solution for PFR (A f B heterogeneous system). τR ) relaxation time ) 1.0E-05 s, stiffness ratio ) 1.0E+08.

Figure 10. Comparison of exact solution, and KSSA solution for batch reactor (A f B heterogeneous system). τR ) relaxation time ) 1.0E-05 s, stiffness ratio ) 1.0E+08.

had the same value (2.5E-05 s). Thus, for a constantvolume system, the t f τ transition from a BR f PFR can be easily made, provided the system of equations is stiff, i.e., the stiffness ratio is greater than 10. Also, this

shows that the FSSA approximation is valid for the PFR case. The catalytic reaction scheme for A f B + C system shown in Table 2 was considered next. Bradshaw and Davidson15 report the analytical form of the rate law in the literature. Indeed, the algebraic complexity brought about by the nonlinearity of this two-site problem motivates the use of numerical DKM. The rate constants were taken from the literature15 and are shown in Table 4. The corresponding BR and PFR equations are shown in Table 3, along with the KSSA for the batch reactor case and FSSA for the PFR case. Figure 12 shows the comparison of various solutions strategies for the PFR. (15) Bradshaw, R. W.; Davidson, B. Chem. Eng. Sci. 1969, 24, 15191527.


Energy & Fuels, Vol. 13, No. 6, 1999 1143 Table 3. Balanced Equations for the BR and PFRa

reactor

BR

exact

BR

KSSA

PFR

exact

PFR

ATB

ATB+C

dCA dCB ) rA, ) rB dt dt dCBl dCAl ) r A - rr , ) -rB + rr dt dt

dCA dCB dCC ) -rA, ) rB , ) rC dt dt dt dCBl dCAl ) rA - rr, ) -rB + rr dt dt dCCl ) -rC + rr dt dCA dCB dCC ) -rA, ) rB , ) rC dt dt dt rA - rr ) -rB + rr ) -rC + rr ) 0 dCCl dCAl dCBl w ) ) )0 dt dt dt dFA dFB dFC ) -rA, ) rB, ) rC dϑR dϑR dϑR rA - rr ) -rB + rr ) -rC + rr ) 0

solution

FSSA

dCA dCB ) -rA, ) rB dt dt rA - rr ) -rB + rr ) 0 dCAl dCBl ) )0 w dt dt dFA dFB ) -rA, ) rB dϑR dϑR rA - rr ) -rB + rr ) 0 dFA dFB ) -rA, ) rB dϑR dϑR dCAl rA - rr dCBl -rB + rr ) , ) dϑR QT dϑR QT

dFA dFB dFC ) -rA, ) rB, ) rC dϑR dϑR dϑR dCAl rA - rr dCBl -rB + rr ) , ) dϑR QT dϑR QT dCCl -rC + rr ) dϑR QT

a The KSSA implies R ) 0 in the case of BR, and the FSSA approximates the exact solution (R ) 0) by a derivative in the case of PFR. β β Here, Fi ) QT‚Ci, QT is the total flow rate (cm3/s), vR is the reactor volume (cc). For A T B case, rA ) kA(CACl - CAl/KA), rr ) kr(CAl CBl/Kr), rB ) kB(CBl/KB - CBCl) Clo ) Cl + CAl + CBl, CA ) CAo, Cl ) Clo, CB ) CAl ) CBl ) 0 when t ) 0. For A T B + C case, rA ) kA(CACl - CAl/KA), rr ) kr(CAl - CBlCCl/Kr), rB ) kB(CBl/KB - CBCl), rB ) kC(CCl/KC - CCCl) Clo ) Cl + CAl + CBl + CCl, CA ) CAo, Cl ) Clo, CB ) CC ) CAl ) CBl ) CCl ) 0 when t ) 0. b

Table 4. Rate Constants for Case Study I and IIa case study

I (A T B)

II (A T B + C)

rate constants

kA ) 1.10 kA/KA ) 0.50 kr ) 10.0 kr/Kr ) 2.0 kB ) 10.0 kB/KB ) 1.0

kA )1.18981 kA/KA ) 0.7377 kr ) 6.2806 kr/Kr ) 0.5673 kB ) 10.8275 kB/KB ) 2.0909 kC ) 47.721 kC/KC ) 2.2303

a Rate constants k , k , k , k , are in g mol/(h atm g-cat), and A r B C rate constants ki/Ki are in g mol/(h g-cat).

Evaluating the eigenvalues of the matrix of partial ∂Fi ) showed the stiffness ratio to be derivatives ( ∂CAi 1.0E+05. The relaxation reactor volume (τ/PFR) was 1.0E-04 (normalized units). Thus, from Figure 12, and the relaxation time for the stiff system under consideration, the FSSA for the PFR is indeed valid. Again, the t f τ transition from a BR f PFR can be easily made, for a system with molar expansion, provided the stiffness ratio of the system is greater than 10. Thus, a set of DAEs, for a PFR, can be transformed into a set of ODEs using the FSSA for stiff systems. Generally, due to the low concentration of the active surface sites and their high reactivity, as compared to the fluid-phase species, real heterogeneous systems have a stiffness ratio greater than 1.0E+03. Hence, the FSSA is a valid approximation for real heterogeneous systems, and results in faster solution of the original DAE system. The CPU time saving obtained by converting the DAEs to ODEs is not obvious for small systems considered above, but would be more apparent for large DKM. A mechanistic DKM was developed for the process chemistry of naphtha catalytic reforming to illustrate

Figure 12. Comparison of exact solution, and FSSA solution for PFR (A f B + C, heterogeneous system). τR ) relaxation time ) 1.0E-04 s, stiffness ratio ) 1.0E+05.

the application of these ideas to a real-system DKM. A comparison of the time required for solution by various strategies and the validity of FSSA assumption was sought. Case Study: Mechanistic Modeling of Catalytic Reforming Catalytic reforming is an industrially important process, which is used to boost the octane number of gasoline and for aromatics production. The feedstock for the process includes hydrocarbons within the range C5 e CN e C11. A bifunctional catalyst with a metal and an acid function is generally used. The metal function

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Joshi et al.

Table 5. Size of Catalytic Reforming Models as a Function of Cumulative Carbon Number cumulative no. of no. of CPU for CPU time carbon mechanistic mechanistic exact using the number species reactions solution (s) FSSA (s) 5 6 7 8

40 81 148 358

107 269 484 1124

48 180 1780 8600

4 14 126 568

provides the required hydrogenation/dehydrogenation activity, and the acid function provides the isomerization and cyclization activity. Table 5 shows the representative size of catalytic reforming mechanistic models as a function of cumulative carbon number. It can be seen that the reaction network including the number of species and their reactions is large. For example, for carbon number 6, at RSS, the complete problem contains about 45 molecules, and 35 surface species. Hence, the whole system contains a set of 45 ODEs and 35 algebraic equations coupled to each other. Table 5 shows the comparison of the time required for a simulation using FSSA with that of the exact (system of DAE) solution for a PFR. It can be seen from the figure that the FSSA solution is 10 times faster than the solution of the nonlinear DAE system. The comparison of the final results is shown in Figure 13. The final results agree well, thus validating the use of the FSSA approximation. Conclusions The t f τ transformation from BR to PFR can be easily made for homogeneous systems. For heterogeneous systems, at RSS, KSS is a prerequisite, and hence

Figure 13. Comparison of the results of the FSSA solution with the exact solution. (Feed includes hydrocarbons ranging from C5 e CN e C8).

the t f τ transformation implicitly assumes flow of surface species, requiring a conceptual approximation. For homogeneous systems, the KSSA is useful to obtain analytical rate laws by eliminating the intermediate species, and becomes better as the system of equations gets stiffer. Similarly, for heterogeneous systems, FSSA is useful to convert a set of DAEs to ODES, and becomes better as the system of equations gets stiffer. The conversion of a set of DAEs to ODEs using the FSSA approximation results in considerable time savings without loss of accuracy. EF990021I

Detailed Kinetic Models in the Context of Reactor ... - ACS Publications

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