Ind. Eng. Chem. Res. 1997, 36, 3149-3153
3149
Symmetrical Cylindrical Model for TAP Pulse Response Experiments and Validity of the One-Dimensional TAP Model Gregory S. Yablonskii,*,† I. Norman Katz,‡ Phungphai Phanawadee,† and John T. Gleaves† Department of Chemical Engineering, Campus Box 1198, and Department of Systems Science and Mathematics, Campus Box 1040, Washington University, St. Louis, Missouri 63130
A two-dimensional symmetrical cylindrical transport model for TAP pulse response experiments is analyzed and compared with the one-dimensional TAP model. It is shown that both models give the same total exit flow for experimentally realistic boundary conditions. The uniformity of the flux distribution at the outlet face of the cylindrical model is also analyzed. It is shown that after time greater than tpeak/60, the flux distribution on the radial coordinate of the outlet face can be considered uniform. 1. Introduction The application of transient response techniques in homogeneous and heterogeneous kinetic studies is becoming increasingly more widespread. Transient experiments have the potential to provide more information than steady-state experiments and are particularly useful in revealing complex multistep processes. A transient response technique that is being increasingly used in heterogeneous kinetic studies is the TAP (temporal analysis of product) reactor system. A simplified schematic diagram of the key parts of a TAP-2 reactor system is shown in Figure 1. The TAP system was developed to investigate multistep catalytic reactions occurring on industrial catalysts (Gleaves et al., 1988). It can be used to perform a variety of transient response experiments (e.g., temperature-programmed experiments, step transient experiments, isotopic switches experiments) but is distinguished by the TAP vacuum pulse response experiment. This experiment is performed by injecting a narrow gas pulse into an evacuated, packed-bed microreactor. The pulse contains a very small number of molecules (10-10 mol) in comparison to conventional transient response experiments, and it travels through the packed bed by Knudsen diffusion. An important feature of this flow regime is that the diffusivities of the individual components of a gas mixture are independent of the gas composition of the mixture. The standard TAP microreactor is a stainless steel tubular (cylindrical) reactor that is packed with catalyst particles or with a combination of catalyst and inert particles. Currently, a one-dimensional (1D) model is used to describe the TAP microreactor (Gleaves et al., 1988; Svoboda et al., 1992; Zou et al., 1993; Creten et al., 1995; Huinink et al., 1996; Gleaves et al., 1997). The goal of this paper is to develop a more realistic transport model of the TAP reactor, i.e., a cylindrical or twodimensional (2D) model, and to justify the validity of the one-dimensional model.
Figure 1. Simplified schematic diagram of the key parts of a TAP-2 reactor system.
radial and axial directions are the same, the Knudsen transport of gas A in the cylindrical reactor is expressed in two-dimensional form as
(
)
∂CA ∂2CA 1 ∂CA ∂2CA ) R2 + + ∂t r ∂r ∂r2 ∂l2
(1)
where CA is the concentration of gas A (mol/cm3), t is time (s), R2 is DeA/b, DeA is the effective Knudsen diffusivity of gas A (cm2/s), b is the fractional voidage in the bed, r is the radial coordinate (cm), and l is the longitudinal coordinate (cm). The boundary conditions are
∂CA (0,r,t) ) 0 ∂l
(2)
∂CA (l,R,t) ) 0 ∂r
(3)
Assuming that the packed bed and the temperature in the bed are uniform and the diffusivities in both
CA(L,r,t) ) 0
(4)
* Author to whom all correspondence should be addressed. E-mail:
[email protected]. † Department of Chemical Engineering. ‡ Department of Systems Science and Mathematics.
Equation 2 stipulates that there is no flux at the reactor entrance when the pulse valve is closed. Equation 3 stipulates that there is no flux through the reactor wall, and eq 4 specifies that the concentration of gas at the
2. Two-Dimensional Model of the TAP-Reactor
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© 1997 American Chemical Society
3150 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
the following expressions for the exit flux and flow (derivation is provided in the Appendix). For initial condition I (eq 5),
(
fluxA,exit(L,r,t) ) ∞
1+
∑ p)2
J0
( ) R1p r R
J02(R1p)
2
2
)
2 t
e-R (R1p /R ) (1D solution)flux (9)
where (1D solution)flux is the solution obtained from the one-dimensional model and is described by (Gleaves et al., 1988; Huinink et al., 1996) Figure 2. Conceptualization of the initial conditions.
reactor exit is zero. The latter condition arises because the reactor outlet is maintained at vacuum conditions. The introduction of a radial coordinate expands the number of physically realistic and interesting initial conditions. These initial conditions correspond to different ways in which a pulse can be introduced into the microreactor. Two types of initial conditions are considered in this study:
(I) pulse through a point at the center of the inlet face of the reactor (Figure 2a) NpA b
CA(l,r,0) ) δ0(l)δ0(r)
(5)
(II) uniform pulse across a portion or full face of the reactor (Figure 2b) NpA CA(l,r,0) ) δ0(l) bπR12
0 e r e R1
R1 < r e R
CA(l,r,0) ) 0
∂CA fluxA,exit ) -DeA (L,r,t) ∂l
(7)
Flow through the centered partial or whole outlet face, FA,exit, is determined by
(∫
FA,exit ) -
0
∫0
R2
∂CA (L,r,t)r dr dθ DeA ∂l
∑
and J0(x) is Bessel’s function of order zero, J1(x) is Bessel’s function of order one, R1p is the pth zero of J1(x), i.e., J1(R1p) ) 0 for p ) 1, 2, 3, etc. (Note that R11 ) 0, and J0(R11) ) 1.) The exit flux at the center of the outlet face is described as
(
fluxA,exit(L,0,t) ) 1 +
)
∞
e-R (R1p /R )t
p)2
J02(R1p)
2
∑
2
2
(1D solution)flux (11)
The outlet flow through the partial outlet face is described as
∫02π∫0R fluxA,exit(L,r,t)r dr dθ ) 2
(6)
where NpA is the number of moles of gas A in the inlet pulse, R is the diameter of the packed-bed reactor, and R1 is the radius of the partial inlet face. A special case of the partial face input is the full face input when R1 is equal to R. Both initial conditions, eqs 5 and 6, correspond to the same inlet pulse intensity, i.e., NpA. Centering the inlet pulse about the axis of the reactor is consistent with experimental practice and also leads to a symmetrical model. For each of the initial conditions (eqs 5 and 6), we obtain the exit flux and flow. Flux at different positions of the outlet face can be determined by
2π
(1D solution)flux ) DeANpAπ ∞ 2 2 2 (-1)n(2n + 1)e-(n+0.5) π [DeA/(bL )]t (10) 2 bL A n)0
)
(8)
where R2 is the radius of the partial outlet face of interest. Application of initial conditions I and II yields
(
)
πR22(1D solution)flux + R1p R2 J 1 ∞ R 2 2 2 2πRR2 e-R (R1p /R )t (1D solution)flux 2 p)2R J (R ) 1p 0 1p (12)
∑
( )
The flow through the whole outlet face (R2 ) R) is given by
∫02π∫0RfluxA,exit(L,r,t)r dr dθ ) πR2(1D solution)flux ) (1D solution)flow (13) For initial condition II (eq 6),
(
fluxA,exit(L,r,t) )
1+
2R
∞
∑ R p)2 1
( )( )
J1
R1p R1p R1 J0 r R R 2
R1pJ0 (R1p)
2
2
2
)
e-R (R1p /R )t ×
(1D solution)flux (14)
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3151
The solution for the exit flux at the center of the outlet face is given by
(
fluxA,exit(L,0,t) )
1+
2R
∞
∑ R p)2 1
( )
R1p R1 J1 R 2
2
2
2
)
(15)
Note that if R1 ) R, then all terms in the summation are zero since J1(R1p) ) 0. In this case, the flux obtained from the two-dimensional model is identical to the flux obtained from the one-dimensional model. The outlet flow through a portion of the outlet face is given by
∫02π∫0R fluxA,exit(L,r,t)r dr dθ )
(
)
πR22(1D solution)flux + R1p R1p R R2 J J 1 1 1 R2 ∞ R R 2 2 2 2 4πR e-R (R1p /R )t 2 2 R1p)2 R1p J0 (R1p) (1D solution)flux (16)
∑
e-R (R1p /R )t
p)2
J02(R1p)
2
The flow through the whole outlet face (R2 ) R) is given by
∫02π∫0RfluxA,exit(L,r,t)r dr dθ ) πR2(1D solution)flux ) (1D solution)flow (17)
tpeak )
2
e 0.01
(18)
L2b L2 ) 2 6DeA 6R
(19)
and set equal to 0.01: 2
2
2
2
2
e-R (R12 /R )t e-R12 (L /R )/[t/(6tpeak)] ) ) 0.01 J02(R12) J02(R12)
(20)
The tpeak in the above equation is the flow peak time for the one-dimensional solution and the cylindrical solution. Substituting R12 ) 3.832, we obtain 2
2
e-2.452(L /R )/(t/tpeak) ) 0.01 0.1662
(21)
This equation shows that the larger the value of L/R, the smaller the value of t/tpeak that is required to satisfy the target value of 0.01. In the typical TAP microreactor, L ) 2.5 cm, R ) 0.2 cm, and L/R ) 12.5. Substituting L/R ) 12.5 into the above equation and solving for t/tpeak, we obtain t/tpeak ) 1/60 or
3. Discussion 3.1. Validity of the One-Dimensional Model. Equations 13 and 17 show that the flow through the whole outlet face of a TAP microreactor is the same for the two-dimensional case with initial conditions I (eq 5) and II (eq 6) and the one-dimensional case. This result does not depend on the length or diameter of the reactor, the diameter of the input face, or the dimensions of the catalyst particles. Consequently, the onedimensional model is valid for describing the total exit flow from a cylindrical TAP microreactor. 3.2. Uniformity of the Flux. Equations 9, 11, 14, and 15 describe the solutions for the flux for a cylindrical model. Comparing these equations with eq 10 (the solution for the one-dimensional flux), it can be seen that the former equations contain additional terms not contained in eq 10 that are functions of the radius, radial coordinate, and time. The flux distribution on the radial coordinate, i.e., the nonuniformity of the flux, is governed by these additional terms. The additional terms in eqs 9, 11, 14, and 15 are made up of exponential functions that decrease with time and approach zero at sufficiently large times. To determine the domain of times in which the additional terms can be neglected, we look for times when the difference between the solution for the one-dimensional model and that for the cylindrical model is no greater than 1%. At the center of the outlet face for initial condition I, the flux is described by eq 11. To satisfy the condition that
2
To determine the conditions that give this result, only the first term (p ) 2) needs to be considered. To obtain a more generalized answer, the first term is rearranged using a relationship from the one-dimensional model (Gleaves et al., 1988),
2
2
( )( )
∞
∑
e-R (R1p /R )t (1D solution)flux
R1pJ0 (R1p)
the solutions for the two models be within 1%, we require
t ) tpeak/60
(22)
This result shows that when the time is equal to or larger than tpeak/60, the flux at the center of the outlet face for initial condition I is within 1% of the flux of the one-dimensional model. Having determined the time when the flux at the center of the cylindrical model and the flux in the onedimensional model are almost equivalent, it is of interest to determine whether the outlet flux is approximately uniform over the entire outlet face at this time. For this purpose, eq 9 can be examined to determine the magnitude of
∞
∑ p)2
J0
( ) R1p r R
J02(R1p)
2
2
2
e-R (R1p /R )t
To evaluate this sum, only the first term need be considered, which can be written as
( )
R12 r 2 2 2 R e-R12 (L /R )/[t/(6tpeak)] 2 J0 (R12)
J0
At fixed t, the numerical value of this term decreases with increasing r, from a maximum at r ) 0 to minimum
3152 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
at r ) R. When t ) tpeak/60, the numerical value of the first term ranges between -0.004 and 0.01; i.e.,
( )
R12 r 2 2 2 R -0.004 e 2 e-R12 (L /R )/[t/(6tpeak)] e 0.01 J0 (R12) J0
This result shows that at times equal to or larger than tpeak/60, the values of the one-dimensional flux and the flux at any position in the outlet face of the cylindrical model with initial condition I are within 1% of each other, and the maximum difference is at the center of the outlet face. Therefore, at times greater than tpeak/ 60, the flux distribution on radial coordinate of the outlet face can be considered uniform. A similar analysis was made for initial condition II, and an analogous result was obtained. The time that gives a uniform flux distribution for initial condition II is even smaller than the time for initial condition I. The larger the radius of the inlet face, the smaller the time required for the flux to become uniform. If the radius of the inlet face is equal to the radius of the reactor, the flux is rigorously the same as the flux calculated from the one-dimensional model. 3.3. Flow through the Partial Outlet Face. Equations 12 and 16 describe the exit flow through a portion of the outlet face. Similar to the flux equations, they contain additional terms that correspond to the nonuniformity of the flux. According to the previous analysis, these additional terms can be neglected after a short period of time, and the flow through the partial outlet face can be described as a simple one-dimensional solution that is proportional to the area of the outlet face.
l ) longitudinal coordinate, cm k ) constant (see eq A2) L ) length of the reactor, cm m ) constant (see eq A6) n ) summation index (eqs 10 and A10) NpA ) number of moles of gas A in the inlet pulse p ) summation index (eq 9) r ) radial coordinate, cm R ) radius of reactor, cm R1 ) radius of the partial inlet face, cm R2 ) radius of the partial outlet face, cm t ) time, s tpeak ) time at which the exit flow is maximum, s T ) time function portion of CA(l,r,t) (see eq A1) w ) constant (see eq A4) X ) radial coordinate function portion of CA(l,r,t) (see eq A1) Y ) longitudinal coordinate function portion of CA(l,r,t) (see eq A1) (1D solution)flux ) solution for exit flux derived from onedimensional model (eq 10) (1D solution)flow ) solution for exit flow derived from onedimensional model Greek Symbols R ) (DeA/b)0.5 R1p ) pth zero ofJ1(x), i.e., J1(R1p) ) 0 for p ) 1, 2, 3, ... δl (0) ) delta function at l ) 0 b ) fractional voidage of the bed θ ) angle in cylindrical coordinate, rad
Appendix Derivations of analytical solutions for the twodimensional model are described. To solve eq 1, the method of separation of variables is used. Let
CA(l,r,t) ) X(l)Y(r)T(t)
4. Conclusions A symmetrical cylindrical or two-dimensional transport model for TAP pulse response experiments has been described and compared to the one-dimensional TAP model. It was shown that both models give the same total exit flow for experimentally realistic boundary conditions. This result does not depend on the length or diameter of the microreactor, the diameter of the input face, or the dimensions of the catalyst particles. From an experimental standpoint, these results indicate that the simpler one-dimensional model is as accurate as the two-dimensional model for describing the transport in a TAP reactor. The uniformity of the flux distribution at the outlet face of the TAP microreactor was also analyzed. It was shown that after times greater than tpeak/60, the flux distribution on radial coordinate of the outlet face can be considered uniform. Nomenclature
(A1)
Substituting eq A1 into eq 1 and rearranging give
T˙ Y′′ Y′ X′′ + + ) -k2, constant ) 2 Y rY X RT
(A2)
Then 2 2
T(t) ) ce-R k t
(A3)
Y′′ Y′ X′′ + + k2 ) ) w2, constant Y rY X
(A4)
Now
Using eqs 2 and 4 gives
π Xn(l) ) dn cos (n + 0.5) l L
(A5)
Y′′ Y′ + ) w2 - k2 ≡ m2 Y rY
(A6)
[
]
Now cm2
A ) cross-sectional area of the reactor, b ) constant (see eq A7) bp ) constants (see eq A9) CA ) concentration of gas A, mol/cm3 Cpn ) constants (see eq A11) dn ) constants (see eq A5) DeA ) effective Knudsen diffusivity of gas A, cm2/s FA,exit ) flow of gas A at the outlet of the reactor, mol/s fluxA,exit ) flux of gas A at the outlet of the reactor, mol/ (cm2 s) J0(x) ) Bessel’s function of order zero J1(x) ) Bessel’s function of order one
Equation A6 is solved by changing the independent variable to mr and using eq 3. The result is
Y(mr) ) bJ0(mr) mr ) R1p
p ) 1, 2, 3, ...
and R1p satisfies the eq J1(R1p) ) 0. So
(A7) (A8)
Yp(r) ) bpJ0
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3153
( ) R1p r R
p ) 1, 2, 3, ...
(A9)
Combining all the constants into a new set of constants, cpn, we obtain 2
π 2 R1p k2 ) w2 + m2 ) (n + 0.5) + 2 n ) 0, 1, 2, ... L R p ) 1, 2, 3, ... (A10)
[
]
and
CA(l,r,t) )
[
Cpn cos (n + ∑ p,n
R1p 2 2 2 2 2 2 r e-R [(n+0.5) (π /L )+(R1p /R )]t (A11) 0.5) l J0 L R The constants Cpn are obtained from the initial conditions and orthogonality properties of the basis functions:
Cpn ) 2π
( ) [ ] ∫∫( [ ])( ( ))
∫0L∫0RrCA(l,r,0)J0
2π
L
R
0
0
R1p π r cos (n + 0.5) l dr dl R L R1p π r dr dl cos2 (n + 0.5) l rJ02 L R (A12)
Consulting Wylie and Barrett (1982, p 603), it can be found that
∫0rrJ02
( )
( )
R1p R1p r2 r dr ) J02 r R 2 R
(A13)
Substituting eq A13 into eq A12 with the upper limit r ) R and rearranging give
Cpn )
4 LR J02(R1p) 2
∫0L∫0RrCA(l,r,0)J0
( ) [ ]
R1p r cos (n + R
π 0.5) l dr dl (A14) L For initial condition I (eq 5),
∫0 ∫0 δ0(r)r dr dθ ) 2π∫0 rδ0(r) dr ) 1 2π
R
R
(A15)
4NpA 2
∫R rJ0 ) 0 1
2
2
bπR1 LR J0 (R1p
( )
R1p r dr R
(A17)
and it is found that
Cpn )
4NpA
( )
J1 R1p
2
πR1RbLR1pJ0 (R1p)
R1 R
for p g 2 (A18)
C1n )
4NpA bπR1
2
2N
R pA r dr ) ∫ LR πR2L 1
2 0
(A19)
b
Substituting eq A11 into eqs 7 and 8, we can determine the expressions for outlet flux and flow described in the main text for each type of initial conditions. Literature Cited Creten, G.; Lafyatis, D. S.; Froment, G. F. Transient Kinetics from the TAP Reactor System: Application to the Oxidation of Propylene to Acrolein. J. Catal. 1995, 154, 151-162. Gleaves, J. T.; Ebner, J. R.; Kuechler, T. C. Temporal Analysis of Products (TAP)sA Unique Catalyst Evaluation System with Submillisecond Time Resolution. Catal. Rev. Sci. Eng. 1988, 30, 49-116. Gleaves, J. T.; Yablonskii, G. S.; Phanawadee, P.; Schuurman, Y. TAP-2 “An Interrogative Kinetics Approach”. Appl. Catal. 1997, in press. Huinink, J. P.; Hoebink, J. H. B. J.; Marin, G. B. Pulse Experiments over Catalyst Beds: A Window of Measurable Reaction Rate Coefficients. Can. J. Chem. Eng. 1996, 74, 580-585. Svoboda, G.; Gleaves, J. T.; Mills, P. L. New Method for Studying the Pyrolysis of VPE/CVD Precusors under Vacuum Conditions. Application to Trimethylantimony and Tetramethyltin. Ind. Eng. Chem. Res. 1992, 21, 19-29. Wylie, C. R.; Barrett, L. C. Advanced Engineering Mathematics, 5th ed.; McGraw-Hill: New York, 1982. Zou, B. S.; Dudukovic, M. P.; Mills, P. L. Modeling of Evacuated Pulse Micro-Reactors. Chem. Eng. Sci. 1993, 48, 2345-2355.
Received for review December 13, 1996 Revised manuscript received February 23, 1997 Accepted March 8, 1997X IE960792Z
and we obtain
Cpn )
Cpn )
and for p ) 1, R11 ) 0, and J0(0) ) 1, so
]( )
π
For initial condition II (eq 6), we have
2NpA πbLR2J02(R1p)
p ) 1, 2, 3, ... (A16)
X Abstract published in Advance ACS Abstracts, June 15, 1997.
