Ind. Eng. Chem. Res. 1998, 37, 2193-2202
2193
Moment-Based Analysis of Transient Response Catalytic Studies (TAP Experiment) Gregory S. Yablonskii, Sergiy O. Shekhtman, Shaorong Chen, and John T. Gleaves* Department of Chemical Engineering, Washington University, Campus Box 1198, One Brookings Drive, St. Louis, Missouri 63130
A new moment-based theoretical approach for the analysis of TAP (temporal analysis of products) pulse-response data is described. The distinguishing feature of this approach is that the variables of the set of ordinary differential equations that comprise the model are moments instead of concentrations. The moments are functions of the reactor axial coordinate, whereas concentrations are functions of space and time. The theoretical framework for the extraction of kinetic parameters using this approach is developed, and is used to determine kinetic parameters for CO oxidation over platinum powder. A new phenomenon, called “TAP equilibrium” is described. This phenomenon is governed by the interplay between reaction and Knudsen diffusion. 1. Introduction Since its beginnings, especially since the work of Van’t Hoff (1884), chemical kinetics has been concerned with the observation and description of changes in chemical composition in time. Theoretical models of chemical kinetics (“kinetic models”) are sets of ordinary differential equations that represent changes of substance concentrations in time. Generally, the set of equations that comprise a kinetic model are difficult to solve, because most chemical reactions, and especially catalytic reactions, are complex in nature. Reactions usually involve some intermediates, and as a rule, the concentrations of intermediates cannot be detected. In the case of catalytic reactions, there is an additional obstacle caused by the change in catalyst structure during a reaction. The first attempt to simplify the description of complex chemical mechanisms and circumvent the lack of experimental information was made by Bodenstein and Chapmen who proposed the pseudo-steady-state (PSS) hypothesis [see details in the book of Yablonskii et al. (1991)]. According to the PSS hypothesis, the set of ordinary differential equations that comprises the kinetic non-steady-state (non-SS) model can be transformed into a mixed set. The mixed set consists of a subset of ordinary differential equations for the observed substances (e.g., the gaseous concentrations in heterogeneous catalytic reactions) and a subset of algebraic equations for intermediates (e.g., surface adspecies concentrations). Typically, the intermediate concentrations can be found as explicit functions of the observed concentrations, and substituted into the equations for the observed concentrations. As a result, PSS models don't contain the concentrations of unobserved intermediates. The application of the PSS hypothesis simplifies the description of the change of chemical composition in time, but the validity of the hypothesis, especially the parameter domain, and the time domain, must be * Author to whom correspondence should be addressed. Telephone: (314) 935-4367. Fax: (314) 935-7211. E-mail:
[email protected].
verified. Moreover, PSS models are still time-dependent models (i.e., sets of ordinary differential equations), and sometimes it is difficult to solve them even with computers. The invention of the continuous-stirred-tank-reactor (CSTR) by Temkin and Denbig in the 1940-50s was the next important step in the simplification of the description of complex chemical reactions. CSTRs are described by SS kinetic models that consist of sets of algebraic equations, and by definition, don’t contain time. The advantages of the SS approach are the reproducibility of SS data, and the relative simplicity of SS models. The disadvantage of this approach is that SS data often cannot provide the detailed mechanism of a complex reaction or information about fast reactions. The invention of transient response techniques, first proposed by Eigen (1954), provided a method of obtaining information about the detailed mechanism of complex reactions. However, despite progress in this area [e.g., relaxation times analysis, Eigen (1954), and moment analysis, Suzuki and Smith (1972), Raghavan and Doraiswamy (1977), Ramachandran and Smith (1978), Wen and Fan (1975), Dudukovic’ (1986)], there has not been a general tool, especially an analytical tool, that can be used to describe non-SS kinetic data. Approaches for revealing the steps of complex mechanisms, and for finding kinetic parameters on the basis of nonSS kinetic data are mostly based on computer identification methods. Recently, a new transient response technique, denoted a TAP (temporal analysis of products) pulse-response experiment, has been developed that can provide kinetic information about individual reaction steps. First proposed by Gleaves et al. (1988) and later improved by Gleaves et al. (1997), the TAP reactor system combines novel high-speed transient response experiments (TAP pulse-response experiments) with traditional kinetic measurements (SS, TPD, step-transient, etc.) in the same apparatus.
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2194 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998
A TAP experiment is performed by injecting a narrow gas pulse into an evacuated microreactor containing a catalyst sample. Typically, the number of gas molecules in the inlet pulse is much smaller than the total number of active species of the catalyst (or the total amount of gas molecules that can be consumed by the whole catalyst). As a result, the composition of a catalyst does not change significantly during a pulse and may not change significantly after a series of pulses. The observed characteristic in a TAP experiment is the time dependent gas flow F(t), measured in molec/s or mol/s, that escapes from the exit of the microreactor. The exitflow is the experimental transient response to the inlet pulse and can be modeled by a set of partial differential equations. The flow dependencies also have integral characteristics that are the moments of the “flow-time” dependencies. From a practical standpoint the zero-th and first moments are the most interesting. The mathematical model that corresponds to the physical idea of a TAP experiment has a number of unique features. First, a TAP experiment can be considered a simple “on-off” device (“kinetic tap”) connected to a high-vacuum system. As a result, the mathematical model used to describe transport through the reactor has very simple initial and boundary conditions (a more detailed explanation is presented later). Second, TAP models can often be considered “pseudolinear” with respect to certain reactants because the catalyst composition remains constant. Third, the diffusion of individual reactants does not change during a reaction because experiments are performed in the Knudsen flow regime. These three features give a TAP experiment a unique simplicity that is the foundation for the description of non-SS kinetic dependencies. The goal of this paper is to present a new theoretical approach for describing TAP pulse-response data based on experimentally measured integral quantities (moments) of the exit flow. The approach is applied to a special class of models, denoted linear or quasilinear models that represent TAP experiments in which there are (1) an insignificant change in the catalyst composition during a pulse, and (2) a large excess of one surface substance in comparison with the other substances. From a mathematical perspective, the models are comprised of partial differential equations with linear right-hand sides. From a physicochemical point of view, all the kinetic terms in the models are first-order or quasi-first-order expressions.
∂CA ∂2CA b + asSv(1 - b)(-kAa CA(1 - θ) + ) DeA ∂t ∂x2 kAd θ) (1) b
∂CB ∂2CB ) DeB + asSv(1 - b)(-kBa CB(1 - θ) + 2 ∂t ∂x kBd θ) (2) ∂θ ) (kAa CA + kBa CB)(1 - θ) - (kAd + kBd )θ ∂t
(3)
where t is time (s); x is the reactor axial coordinate (cm); CA and CB are concentrations of gases A and B (mol/ cm3), respectively; θ is the surface coverage; DeA and DeB are effective Knudsen diffusivities of gases A and B (cm2/s), respectively; kAa and kBa are adsorption rate constants of A and B (cm3/mol s), respectively; kAd and kBd are desorption rate constant of A and B (s-1), respectively; b is the fractional voidage of the packed bed in the reactor; as is the surface concentration of active sites (mol/cm2); and Sv is the surface area of catalyst per volume of catalyst (cm-1). If the pulse intensity is sufficiently small, then the surface coverage can be considered negligible because of the small concentration of reactant A compared with the amount of the active catalytic substance, and 1 - θ can be assumed to be equal to 1. If the following substitutions are made:
k1 )
asSv(1 - b)kAa asSv(1 - b)kBa ; k-2 ) ; b b k-1 ) kAd ; k2 ) kBd DA )
DeA DeB asSv ; DB ) ; θ* ) (1 - b) θ b b b
(4)
where k1 and k-2 are the apparent adsorption constants for A and B (s-1), respectively, and θ* is the transformed surface concentration, (mol/cm3), then the set of eqs 1-3 can be transformed into:
∂2CA ∂CA - k1CA + k-1θ* ) DA ∂t ∂x2
2. Moment-Based Analysis of Chemical Mechanisms 2.1. Two-Step Catalytic Mechanism. We begin our discussion of the moment-based analysis of TAP data by considering one of the simplest mechanisms for a complex catalytic reaction, the two-step linear mechanism (Temkin-Boudart mechanism). For an isomerization reaction, this mechanism is represented by the following set of steps:
A + Z a AZ AZ a B + Z
Z is an active site that can also be treated as an intermediate, and B is a product. The “diffusion-reaction” equations corresponding to mechanism (I) are the following:
(I)
AaB where A is a reactant, AZ is an adspecies (intermediate),
∂CB ∂2CB - k-2CB + k2θ* ) DB ∂t ∂x2
(5)
∂θ* ) k1CA + k-2CB - (k-1 + k2)θ* ∂t Another special feature of a TAP pulse-response experiment is that there are no gases in the microreactor before a pulse is injected or after a pulse has exited the reactor. In addition, adspecies formed during a pulse exist for a finite time in the reactor, desorb, and then escape into the vacuum system. These unique characteristics lead to special boundary and initial conditions that have been described in detail elsewhere [Gleaves et al. (1988), (1997), Svoboda (1993)]. In the
Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2195
case of integral transformations, the appropriate initial and boundary conditions are represented as follows:
Initial condition:
t ) 0 f CA ) 0; CB ) 0; θ* ) 0
2.1.1.1. Zero-th Moment Equations. Taking into account the TAP initial conditions, the direct integration in time of equations 9 gives the following equations for the zero-th moments:
Boundary conditions: NpA NpB ∂CB ∂CA ) -2 ) -2 δ(t); δ(t) x)0f ∂x DeAAr ∂x DeBAr
d2MA,0(x)
x ) L f CA ) 0; CB ) 0
d2MB,0(x)
2
2
Mn )
∫
+∞ n t F(t,L) 0
dt
(7)
The moments Mn(x) for the flow through any cross section of the reactor at a given axial coordinate are represented by the set of equations:
Mn(x) )
∫
+∞ n t F(t,x) 0
dt
(8)
For example, the zero-th moment, as a function of the reactor axial coordinate, determines the total number of gas molecules that pass through the given cross section. The quantities Mn(x) are related to the nature of a TAP experiment. After the gas mixture has been injected into the reactor, it propagates through the reactor as a result of diffusion and interacts with the catalyst. As the gas molecules emerge from the reactor, the exit flow time dependencies are detected. During a TAP experiment, the flow time dependencies through any cross section of the reactor change as a function of the cross section positions. The change is governed by the diffusion process and the catalytic reaction. From this perspective, a TAP experiment should be described in terms of the evolution of gas flows through reactor cross section with respect to the axial coordinate. The quantities Mn(x) can be used to directly describe this evolution. To obtain equations for the moments, Mn(x), defined by eq 8, equation set 5 was first transformed to a set of equations for flows by taking derivatives with respect to the axial coordinate x:
0)
(10)
The quantity Mθ,0(x) is not measured in a TAP experiment, and can be eliminated by using the third equation of eq set 10. Then, the final equations for the measurable quantities, MA,0(x) and MB,0(x), are represented as:
d2MA,0(x) 2
)
dx
d2MB,0(x) 2
k1k2 1 1 k-1k-2 M (x) M (x) DA k-1 + k2 A,0 DB k-1 + k2 B,0
)-
dx
k1k2 1 M (x) + DA k-1 + k2 A,0 1 k-1k-2 M (x) (11) DB k-1 + k2 B,0
These equations for the zero-th moments are linear, homogeneous, second-order differential equations. Correspondingly, the boundary conditions (6) are changed as follows:
x ) 0 f MA,0 ) NpA, MB,0 ) NpB x)Lf
dMA,0 dMB,0 ) )0 dx dx
(12)
2.1.1.2. n-th Moment Equations. For the general case, the set of equations for the n-th moments (n is more than zero) can be derived using eq set 9. After multiplying the equations by the n-th power of time and integrating in time, the following equations for the n-th moments are obtained:
2 1 ∂FB ∂ FB k-2 F + k2Fθ ) DB ∂t DB B ∂x2
d2MB,n(x)
2
dx
dx2
where flows are given by: FA(t,x) ) -ArDeA ∂CA(t,x)/∂x; FB(t,x) ) -ArDeB ∂CB(t,x)/∂x; and Fθ(t,x) ) -Arb ∂θ*(t,x)/∂x. Using the set of partial differential eqs 9, the set of ordinary differential equations for the moments defined by eq 8 can be obtained by integrating in time.
k-2 M (x) - k2Mθ,0(x) DB B,0
k-2 k1 MA,0(x) + M (x) - (k-1 + k2)Mθ,0(x) DA DB B,0
d2MA,n(x)
(9)
)
dx
2 k1 1 ∂FA ∂ FA ) F + k-1Fθ DA ∂t DA A ∂x2
k1 k-2 ∂Fθ ) FA + F - (k-1 + k2)Fθ ∂t DA DB B
k1 M (x) - k-1Mθ,0(x) DA A,0
dx
(6)
where NpA and NpB are the numbers of moles of gas A and B in one pulse, respectively, and Ar is the cross sectional area of the reactor (cm2). 2.1.1. Moment Equations. The moments Mn of the exit flows F are represented by the set of integral equations:
)
MA,n-1(x) k1 ) M (x) - k-1Mθ,n(x) DA DA A,n
+n
MB,n-1(x) k-2 ) M (x) - k2Mθ,n(x) DB DB B,n
+n
-nMθ,n-1(x) )
k1 k-2 MA,n(x) + M (x) DA DB B,n (k-1 + k2)Mθ,n(x) (13)
As in the previous case, Mθ,n(x) can be eliminated using the third equation of eq set 13. The parameter Mθ,n(x) can be expressed as the linear combination of the moments MA,n(x), MB,n(x), and Mθ,n-1(x):
2196 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998
Mθ,n(x) )
k1
MA,n(x) +
DA(k-1 + k2) k-2
Mθ,n-1(x) MB,n(x) + n (14) (k-1 + k2)
K+/DA, the second equation by the factor K-/DB and subtracting one equation from the other, the following non-trivial equation is obtained:
DB(k-1 + k2)
The parameter Mθ,n-1(x), and all the other unobservable moments associated with the surface concentration θ, can be eliminated by creating a recurrence relationship from eq 14. After the equations for Mθ,n(x), Mθ,n-1(x), Mθ,n-2(x), ..., Mθ,0(x) have been substituted in the eq set 13, the following equations for the n-th moments are obtained:
d2MA,n(x)
)
dx2
∑ i)0
(
MA,n(x) DA k-1 + k2 k-1k-2
1
n-1
k1k2
1
MB,n(x) - n DB k-1 + k2 k1k-1
n!
MA,i(x)
i! (k + k )n-i+1 -1 2
DA
+
MA,n-1(x)
2
dx
)
1
k-1k-2
∑
(
-1
)
k-1k-2
k-1k-2
DA
(k-1 + k2)n-i+1
MB,n(x) -
2
k2k-2 (k-1 + k2)
DB
)
(16)
x ) 0 f MA,n ) MB,n ) 0 dMA,n dMB,n ) )0 dx dx
(17)
2.1.2. Solution of the Moment Equations. 2.1.2.1. Zero-th Moment Equations. The summation of the zero-th moment equations (eq 11) gives the equation: d2/dx2(MA,0(x) + MB,0(x)) ) 0, that corresponds to the mass-balance condition for a TAP transient experiment, taking into account eq set 12
MA,0(x) + MB,0(x) ) NpA + NpB
(
)
K+ K+ W0(x) DA DB
( ) x (x
(18)
Multiplying the first equation of set (11) by the factor
cosh
cosh
(19)
)
K+ K+ (L - x) DA DB
)
K+ K+ L DA DB
Using the mass-balance eq set 13 and the expression for W0(x), the zero-th moments MA,0(x) and MB,0(x) are determined as follows:
MB,0(x) )
Equations 15 and 16 for the n-th moments are linear, inhomogeneous, second-order differential equations. The homogeneous part of the n-th moment equations is the same as that of the zero-th moment equations. The inhomogeneous part is a linear combination of the (n-1)-, (n-2)-, ..., 0-th moments. The transformed boundary conditions (6) for n-th moments are homogeneous and are represented as follows:
x)Lf
)
where W0(x) ) K+MA,0(x)/DA - K-MB,0(x)/DB is the overall process rate; and K+ ) k1k2/(k-1 + k2) and K- ) k-1k-2/(k-1 + k2) are apparent kinetic coefficients of the forward and backward overall-process, respectively. The solution of eq 19 with boundary conditions (eq set 12) is given by:
MA,0(x) )
MB,i(x)
n-i+1
dx
(
-
DA
DB k-1 + k2 k1k2 MB,n-1(x) 1 MA,n(x) - n DA k-1 + k2 DB n-1 k1k2 MA,i(x) n! + i)0 i! (k + k )n-i+1 DA
2
NpA NpB - KW0(x) ) K+ DA DB
(15) d2MB,n(x)
d2W0(x)
NpA + npB W0(x) + 1 + Keq,T K+ K+ DA DB
(NpA + NpB)Keq,T W0(x) 1 + Keq,T K+ K+ DA DB
(20)
where Keq,T ) K+DB/K-DA. The physicochemical definition of this constant is discussed in Section 2.3.2. The zero-th moments of the exit flows, MA,0(L) and MB,0(L), can be represented as linear combinations in terms of the number of molecules in the inlet pulse
MA,0(L) ) µAA,0NpA + µAB,0NpB MB,0(L) ) µBA,0NpA + µBB,0NpB
(21)
where µˆ Rβ,0 is defined as the zero-th moment transition matrix and is given in Chart 1. Each of the elements in this matrix can be measured experimentally. The different elements are equal to the values of the zeroth moments divided by the number of molecules in the corresponding inlet pulse that contains only one type of molecule. The first index R corresponds to the gas (A or B) whose moment was measured, and the second index β corresponds to the gas (A or B) that was the only component in the inlet pulse. The matrix µˆ Rβ,0 connects the zero-th moments of flows entering the reactor (the inlet pulse composition) with those exiting the reactor. In general, the n-th moment transition matrix, µˆ Rβ,n, can be defined in the same way. The application of these matrices is discussed later. 2.1.2.2. Solution of First Moment Equations. The n-th moment eqs 15 and 16 are used to obtain the
Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2197 Chart 1 1 1+ Keq,T + 1
Keq,T K+
cosh L
DA
+
K–
1 1– Keq,T + 1
1 K+
cosh L
DB
+
DA
K– DB
^ µ αβ,0 =
Keq,T
1
1–
Keq,T + 1
cosh L
K+ DA
+
K–
K+ KMA,1(x) M (x) DA DB B,1 dx k1k-1 k-1k-2 MB,0(x) MA,0(x) 1+ (22) 2 DA (k + k ) (k + k )2 DB
(
2
)
-1
d2MB,1(x)
2
)
-1
2
K+ KMA,1(x) + M (x) DA DB B,1 dx k2k-2 MA,0(x) MB,0(x) k1k2 1 + (23) (k-1 + k2)2 DA (k-1 + k2)2 DB 2
)-
(
)
These equations have a structure similar to the equation of the zero-th moment. The equations are not difficult to solve, but the expressions for the first moments, MA,1(x) and MB,1(x), are complicated and are difficult to analyze in terms of kinetic constants. The analysis is more convenient if an appropriate combination of the first moments is used. The summation of eqs 22 and 23 gives a simple equation for such a combination:
k2 + K+ MA,0(x) d2 (MA,1(x) + MB,1(x)) ) 2 k2 DA dx k-1 + K- MB,0(x) (24) k-1 DB Equation 24 does not contain an unknown function on the right-hand side (the zero-th moments are assumed to be known), and can be easily integrated. Using the solution of eq 24, the sum of first moments at the exit, MA,1(L) + MB,1(L), was obtained:
[
NpA + NpB 1 Keq,T + + MA,1(L) + MB,1(L) ) 1 + Keq,T DA DB (Keq,TNpA - NpB) Keq,T k1 1 L2/2 + + 2 k-1DA k k (1 + Keq,T) 2 -1 1 DB 1 -1 1(25) K- DA K+ Kcosh L + DA DB
]
(
(
))
(
(x
1 cosh L
DB
following equations for the first moments:
d2MA,1(x)
1 Keq,T + Keq,T + 1
))
As shown later, this quantity is the most useful to work with from among different combinations of first moments. For example, the expressions for the first moments MA,1(L) and MB,1(L) contain one additional complicated term. Moreover, ex 25 is usually sufficient to determine all the kinetic constants from the zero-th and first moments. 2.1.3. Determination of Kinetic Constants. Using the analytical expression for the zero-th moments, eq set 20, and the sum of the first moments, eq set 25,
K+ DA
+
K– DB
the kinetic constants k1, k2, k-1, and k-2 can be determined analytically. The values of L, DA, DB are assumed to be known. The equations and solutions for the zero-th moments contain only two independent parameters. Using the zero-th moments, only two parameters (e.g., Keq,T and the argument of the cosh term) can be determined. Using the zero-th moment transition matrix µˆ Rβ,0 defined by eq 21, the following expressions were obtained:
Keq,T )
(
x
L
µBA,0 µAB,0
K+ K1 + ) ln + DA DB µAA,0 - µAB,0
)
x
1 - 1 ) σ (26) (µAA,0 - µAB,0)2
To determine all four constants, two additional parameters must be calculated using the first moments. The first moment transition matrix µˆ Rβ,1 was defined in the same way as the zero-th moment transition matrix. The additional parameters can be calculated using µˆ Rβ,1, and all constants can be determined in terms of the zero-th and first moments:
(
)
(
)
2DA µBA,0 k1 ) 2 µAA,1 + µBA,1 + (µ + µAB,1) k-1 µAB,0 BB,1 L DAµBA,0 - 1 (27) DBµAB,0 µAB,0 k-2 2DB ) 2 (µAA,1 + µBA,1) + µBB,1 + µAB,1 k2 µBA,0 L DBµAB,0 - 1 (28) DAµBA,0
[
]
[
2 2 µBA,0 1 1 ) 2(µAA,1 + µBA,1) + 2 + k-1 µBA,0 + µAB,0 σ σ µAB,0 L2 µBA,0 1 (µBB,1 + µAB,1) - 2 + 1 (29) µBA,0 + µAB,0 σ D µAB,0
[
]
(
)
B
]
[
2 µAB,0 2 1 1 ) 2 + (µAA,1 + µBA,1) + 2 k2 σ µBA,0 µBA,0 + µAB,0 σ L2 µAB,0 1 (µBB,1 + µAB,1) - 2 + 1 (30) µBA,0 + µAB,0 σ D µBA,0
]
(
A
)
Equations 27-30 can be used to calculate directly all the kinetic constants for a two-step reversible catalytic
2198 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998
reaction using the experimental values of the zero-th and first moments of the exit flows. 2.2. Adsorption Mechanism in the High-Coverage Domain of One Substance. The LangmuirHinshelwood adsorption mechanism is one of the most commonly used catalytic reaction mechanisms. It can be represented by the following sequence of the steps:
A + Z a AZ
irreversible. Consequently, only A molecules are present in the inlet pulse. One kinetic parameter can be determined from the zero-th moments, and the expression for the sum of the first moments is insufficient to find the remaining constants. To determine all the constants, the first moment for the A molecules is required:
µAA,1 )
Bn + nZ f nBZ
(II)
(x (x
tanh L
AZ + BZ f C + 2Z
Under TAP experimental conditions, this mechanism can be simplified provided (1) a large quantity of reactant B is pre-adsorbed on the catalyst (uniformly); (2) the input pulse does not contain the reactant B; and (3) the molecular concentrations correspond to the conditions [A] + [AZ] , [Z] and [A] + [AZ] , [BZ]. In this case, the total quantity of gaseous species is much less than the total amount of surface sites Z and BZ. The adsorption mechanism can be treated as a special case of a two-step mechanism (I) (i.e., the first step is reversible, and the second step is irreversible):
µAA,1 + µCA,1 - L2/2DC DC - DA L2 1 ) θ k2 µCA,0 DCDA σ2 B
(
)
1 1 ) k-1 k* 2
(x
cosh L
µAA,1 + µCA,1 )
)
k* 1k* 2
) 1 - µCA,0
DA(k-1 + k* 2)
(
(x
cosh L
1 k* 1k* 2
))
+
L2 2DC
DA(k-1 + k* 2)
The existence of product molecules C in the input pulse cannot provide additional information from analysis of the exit-flow moments because the reaction is
+
L2 DAσ2
σµAA,0x1 - µAA,02 2µAA,1 L2 2 σµAA,0x1 - µAA,02 DAσ
(33)
where:
σ)L
x
k* 1k* 2
DA(k-1 + k* 2)
(
) ln
1 + µAA,0
x
1
µAA,02
)
-1
Equations 31-33 can be used to directly calculate the three kinetic constants of the linearized three-step catalytic reaction from the experimental values of the zero-th and first moments of the exit-flows. For the reversible and the irreversible mechanisms, information from the zero-th and first moments is sufficient to determine all the kinetic constants. Using eqs 32 and 33, another useful expression was obtained:
x (
1 ) θB k2
DC(k-1 + k* 2 + k* 1) - DA(k-1 + k* 2) 1DCk*1k*2
2µAA,1
1 k*2
[BZ] ≈ const.; [Z] ≈ const.
1
(31)
2 k* 2µAA,1 2L 1 L2 1 ) + (1 - θB) 2 k1 D σ2 k* 2 A σµAA,0x1 - µAA,02 DAσ (32)
AfC
µAA,0 )
L 3/2 2xDAk* 1k* 2(k-1 + k* 2)
Subsequently, the following expressions for the kinetic constants were obtained:
(III)
The moment equations obtained for mechanism (I) will also be valid for this case provided the constant k-2 is equal to zero. This condition is result of the irreversibility of the second step in mechanism (III). The kinetic constant for the adsorption of A, k*1, contains a factor proportional to the number of active sites [Z], and the kinetic constant of the reaction k*2 contains a factor proportional to the number of adsorbed B molecules [BZ]: k*1 ) k1(1 - θB) and k*2 ) k2θB, where θB is the surface concentration of B molecules. Using the results of the calculations for a two-step mechanism, the following expressions for the zero-th and first moment matrices, µRβ,0 and µRβ,1, for reactant A and product C were obtained:
k* 1k* 2
DA(k-1 + k* 2)
A + Z f AZ AZ + BZ f C + 2Z
2 (k-1 + k* 2) + k-1k* 1
DA(k-1 + k* 2)
cosh L
nA + Bn f nC
) )
k* 1k* 2
)
2µAA,1 L2 L2 k1(1 - θB) k-1 2 DAσ2 σµ DAσ2 AA,0 x1 - µAA,0 (34)
According to this equation, the constant k2 can be determined using the zero-th and first moments of reactant A, surface coverage of the second reactant θB, and the adsorption-desorption equilibrium constant k1/ k-1 of reactant A. For example, according to Gleaves et al. (1988), this equilibrium constant can be calculated from the following equation:
2DeA M1 k1 ) -1 k-1 L2 M0 b
(35)
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Equation 35 corresponds to a TAP adsorption-desorption experiment using a single substance. 2.3. Physicochemical Properties of the Moment Equations. 2.3.1. Analogy between the MomentBased TAP Model and Pseudo-Steady-State Approach. It can be seen from eqs 11 and 15 and 16 that the moment-based TAP models do not contain intermediates or equations that correspond to the intermediates of the complex chemical reactions (surface substances). The elimination of intermediates is the result of the time integration procedure. As already mentioned, a special feature of a TAP experiment is the complete evacuation of the reactor after a pulse is injected. Because of this feature, the standard TAP model [Gleaves et al. (1988), (1997)] based on a set of partial differential equations can be transformed into a mixed set comprised of a subset of ordinary differential equations for the gaseous substances and algebraic subset of equations for the intermediates. In the linear or pseudo-linear cases just described, the algebraic equations are linear and easy to solve with respect to the intermediates. As a result, the moments for the intermediates can be obtained explicitly as a solution of the algebraic subset. The final moment-based equations are the differential equations with right-hand sides that are functions only of the observed variables that are the moments of the gaseous substances (see eqs 15 and 16). In the zero-th moment case, the equations contain only the zero-th moments and are homogeneous. There is an important analogy between the momentbased TAP model and the PSS model that mainly concerns the homogeneous part of the moment-based equations. Because all the moment equations for a given mechanism have the same homogeneous part, the analogy can be illustrated using the zero-th moment equations for a two-step mechanism (I). The PSS equations that correspond to a two-step mechanism (I) can be written as follows:
-
dCA dCB ) ) K+CA - K-CB dt dt
+ - + where K+ ) k+ 1 k2 /∆; K ) k1 k2 /∆; and ∆ ) k1 + k2 + + k1 CA + k2 CB. Comparing the PSS model equations with those of the moment-based TAP model, the following similarities and differences were discovered: (1) The right-hand sides of both models correspond to the overall reaction. The observed variables of the TAP model are the moments of gaseous flows instead of gaseous concentrations in the PSS model. (2) The ratio of the apparent parameters of both models is related to the equilibrium thermodynamic parameter of overall reactions. In the PSS model, this ratio is K+/K- ) Keq. In the moment-based TAP model, the apparent parameters contain the diffusivities. The relation between these parameters and the equilibrium thermodynamic parameter of overall reaction includes the following ratio of diffusivities: DBK+/DAK- ) Keq,T ) (DB/DA)Keq. The physicochemical meaning of parameter Keq,T is discussed in detail in the next section. (3) The apparent parameters in both models are fractions and functions of the parameters of all the steps of the complex chemical reactions. The numerators of the corresponding apparent parameters of both models are the same, whereas the denominators of the parameters are different. The difference is that the denomi-
nators in the moment equations do not include concentration-dependent terms. In the moment-based model, the total apparent parameter does not depend on gaseous characteristics because the catalyst composition changes insignificantly during the TAP experiment (1 - θ ≈ 1). (4) The zero-th moment equations of a moment-based TAP model provide approximately the same amount of information as a PSS model. The first (second and so on) moment equations of a TAP model give additional information. For example, in a previous section it was shown that all the kinetic constants can be calculated using the zero-th and first moments. The totality of the moment equations makes it possible to describe the catalytic system much more precisely than can be accomplished with a PSS model. (5) The most important difference between a momentbased model and a PSS model is that the PSS model is founded on a special assumption concerning pseudo-SS behavior of the intermediates. This assumption is difficult to verify in the general case [Yablonskii et al. (1991)]. In the moment-based TAP model, this special assumption is not needed. The general analysis of the correlation between a moment-based TAP model and PSS model will be published in a future paper. 2.3.2. Equilibrium-Like Relationships for the Zero-th Moments-“TAP Equilibrium”. The lefthand sides of the TAP equations (11) are related to the change of the zero-th moment (i.e., the total number of molecules passing a given cross section) with respect to the axial coordinate. As discussed in Section 2.3.1., the right-hand sides of the TAP zero-th moment equations are analogous to the right-hand side of the PSS model equation. When the right-hand sides of the PSS equations are equal to zero, then traditional equilibrium state conditions are realized. If the right-hand sides of the TAP zero-th moment equations (11) are equal to zero, analogous TAP reactor state conditions are realized, and the zero-th moments do not depend on the axial coordinate (i.e., they are constant in space). The solutions of eq set 11 are then given by MA,0(x) ) const ) NpA and MB,0(x) ) const ) NpB. These relationships can be experimentally observed if the initial pulse composition satisfies the following condition:
NpB/NpA ) MB,0/MA,0 ) DBK+/DAK- ≡ Keq,T (36) Continuing the analogy between the TAP moment equations and PSS equations, the condition in eq 36 defines the “TAP equilibrium state”, and the parameter Keq,T can be termed the “TAP equilibrium constant”. The difference between Keq,T and the traditional equilibrium constant Keq is the factor DB/DA. The quantity W0(x), which has a mathematical structure similar to the total reaction rate, is equal to zero in the TAP equilibrium state. When the condition in eq 36 is not satisfied, W0(x) is nonzero, and the magnitude of W0(x) can be viewed as the displacement of the reaction system from the TAP equilibrium state. The magnitude of W0(x) decreases, and the ratio of the zero-th moments goes toward Keq,T with length. At the reactor exit, the ratio of the zeroth moments approaches Keq,T as the reactor length approaches infinity. In this respect, the TAP equilibrium state can be viewed as the state (defined by the condition in eq 36) that a single-pulse experiment
2200 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998
naturally evolves toward and would reach if the reactor length were infinitely long. The value of W0(x) at the exit, W0(L), is given by: W0(L) ) [K+(NpA/DA) - K-(NpB/ DB)]/[cosh(xK+/DA+(K-/DB)L)], and is nonzero for a finite L. Although the TAP equilibrium state cannot be attained in a single pulse, it can be determined experimentally using a series of pulses. To determine the equilibrium composition, the composition of each inlet pulse, after the first pulse, is made to equal the outlet composition of the previous pulse. In other words, the number of molecules of a particular species in each inlet pulse is set equal to the zero-th moment of this species in the previous pulse. After the pulse with the initial composition, N B , is injected, the zero-th moments of the n-th pulse, M B (n), can be calculated using eq set 20:
M B (n) ) (µˆ Rβ,0)n × N B
(37)
where
( )
N N B ) NpA pB
is the vector of molecule numbers in the inlet pulse;
M B (n) )
( ) M(n) A,0
M(n) B,0
is the vector of exit-flow zero-th moments for the n-th pulse; and µˆ Rβ,0 is the zero-th moment transition matrix defined by eq 21. The n-th power of the matrix µˆ Rβ,0 can be calculated using standard linear algebra procedure. The zero-th (n) moments for the n-th pulse, M(n) A,0 and MB,0, calculated from eq 37 satisfy the following relationships:
W(n) 0 (L) )
K+ (n) K- (n) MA,0 M ) DA DB B,0
KK+ NpA N DA DB pB
(x
cosh L
M(n) B,0
) Keq,T Lim nf∞M(n) A,0
)
n
K+ K+ DA DB (38) (39)
In the limit of large n, the quantity W(n) 0 (L) goes to zero, and the ratio of the zero-th moments exactly corresponds to the TAP equilibrium condition in eq 36; that is, the ratio equals the TAP equilibrium constant. This result indicates that the series of pulses already discussed converges to the TAP equilibrium state, and that the TAP equilibrium constant can be determined experimentally, without modeling the TAP transient response curves. The TAP equilibrium state can be reached for any nonzero-th values of the kinetic constant. In this state, equilibrium-like relationships occur for the zero-th moments (the integral, non-time-dependent quantities); that is, for the total numbers of molecules passing through any cross section of the reactor. For the ratio of flows of different species (or the ratio of first moments), there are, in general, no equilibrium-like relationships at a given moment of time. For example, for
the first moments (and for the corresponding flows) to exhibit equilibrium-like relationships, the kinetic constants must satisfy the following condition:
k2 + K+ k-1 + Kk-1 + k2 + k1 ) 0 or k2DA k-1DB DA k-1 + k2 + k-2 ) 0 (40) DB This condition is not satisfied in the general case. A detailed discussion of the mathematical basis of eq 40, and the TAP equilibrium state will be presented in a future paper. 2.4. Application of Moment-Based Analysis to CO Oxidation over Platinum Powder. To test the moment-based TAP model, the parameters for the oxidation of CO over platinum powder were determined when oxygen coverage is high. In particular, the equilibrium constant for CO adsorption-desorption and the constant for the reaction between adspecies (CO and oxygen) were determined. The results are compared with literature values and those obtained using traditional TAP analysis techniques based on curve fitting the reactant and product responses to a kinetic model. Experiments were conducted using a TAP-2 multifunctional reactor system [Gleaves et al. (1997)]. Gas and catalyst samples were obtained from commercial sources. Gas samples were used without further treatment. All catalyst samples were initially cleaned in situ to remove surface impurities (e.g., carbon, sulfur, and phosphorus) by alternating treatments with oxygen and carbon monoxide at 773 K. After cleaning, the catalyst was cooled down to the desired reaction temperature. TAP vacuum pulse response experiments were performed to investigate CO adsorption-desorption on cleaned platinum powder, and CO2 formation on oxygen pretreated platinum powder. Mass peaks at m/e ) 40, m/e ) 28, and m/e ) 44 were used to monitor the responses of argon, CO, and CO2, respectively. The m/e ) 28 peak includes the CO parent peak and a fragment peak of CO2. The contribution from the CO2 fragment was determined by measuring the ratio of the fragment and parent peaks of a pure CO2 pulse. Pulse intensities of
