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Thin-Zone TAP Reactor versus Differential PFR: Analysis of Concentration Nonuniformity for Gas-Solid Systems Sergiy O. Shekhtman† and Gregory S. Yablonsky*,‡ CenTACat, School of Chemistry, Queen’s University Belfast, David Keir Building, Stranmillis Road, Belfast BT9 5AG, Northern Ireland, UK, and Department of Chemical Engineering, Washington University in St. Louis, Campus Box 1198, 1 Brookings Drive, St. Louis, Missouri 63130-4899
This paper discusses on different kinetic devices and strategies to ensure concentration uniformity within the catalyst zone. For the differential PFR and thin-zone TAP reactor (TZTR), in which uniformity is achieved by making the active zone thin, the generated nonuniformity was quantified and related to the type of transport. It was shown for the TZTR that diffusion mixes up the nonuniformity created by reaction and allows maintaining small nonuniformity (1520%) in a wide domain of conversions (up to 80%). Contrary to the TZTR, in the differential PFR, the convectional transport does not mix nonuniformity so that its estimate is equal to the conversion, which limits this reactor to the operation at low conversions (e.g., below 20%). This comparison demonstrates a remarkable role of diffusional transport in achieving concentration uniformity in a kinetic experiment. 1. Uniformity In kinetic catalyst characterization, a useful experimental requirement is the uniformity of the reactant concentration profile within the catalytic active zone. This requirement allows relating the observed rate of chemical transformation to a given (spatial average) reactant composition. This ensures approximately the same catalytic performance within the catalyst zone, particularly uniform catalyst composition, which can be related to the reactant composition and reaction rate. Solely such kinetic information obtained under the condition of spatial uniformity provides the basis for unrevealing catalyst composition/structure-activity relationships. Quantitatively, nonuniformity can be characterized by relating the spatial change of the characteristic to its highest value. Particularly, the reactant concentration can be defined as uniform if its spatial difference is small as compared to its maximum value. The corresponding quantity, (Cmax - Cmin)/Cmax, ranges from 0 to 1 and indicates uniformity when it is much smaller than 1. In contemporary heterogeneous catalytic kinetics, there are three types of ideal devices that can ensure uniformity of the concentration profile within the active zone during the kinetic test: (i) CSTR (continuous stirred tank reactor) proposed by Temkin in 19501,2 (see also ref 3). (ii) “Differential PFR” (plug-flow reactor) proposed by Kobayashi and Kobayashi in 1974.4,5 (iii) TZTR (thin-zone TAP reactor) proposed by Shekhtman and Yablonsky in 1999.6-9 It is a special reactor configuration for the transient technique, called Temporal Analysis of Products (TAP), which was created and modified by John Gleaves in 1988 and in 1997, respectively. * To whom correspondence should be addressed. Tel.: (314)935-4367. Fax: (314)935-7211. E-mail:
[email protected]. † Queen’s University Belfast. ‡ Washington University in St. Louis.
For all the devices, the basic reaction equation can be presented as
accumulation - (transportin - transportout) ) reaction rate term (1) Here (especially under steady-state conditions when accumulation term can be omitted), the transport term establishes a “ruler” for measuring the rate of chemical transformation. Particularly, the conversion can be estimated as
X ) (transportin - transportout)/transportin (2) A kind of the transport and corresponding mathematical form of the transport term differs for these devices: for the CSTR and differential PFR, it is convection, while for TZTR, it is diffusional transport. Also, other differences should be distinguished particularly a method of mixing, domain of conditions, level of achieved conversion, etc. 2. CSTR The key idea of the CSTR is to perform perfect mixing in the catalyst zone using the external or internal circulation, turbulent flow itself, or combined methods. Then, the concentration in the reactor is characterized by one spatial average value equal to outlet concentration C. The transport in and out of the reactor is assumed to be purely convectional. The corresponding transport term and reactor equation are given by
transport ) VflowC Vg
dC - (Vflow0C0 - VflowC) ) R(C)Sc dt
(3)
Historically, the CSTR method was proposed as a very simple and straightforward technique for steady-state studies of industrial catalytic reactions under the assumption of no temperature/concentration gradient.1,2
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This device still remains a very important tool for the steady-state kinetic analysis. However, the discussion on the validity of a simple (one-mode) CSTR model is still continuing. According to the recent results obtained by Balakotajah and co-workers,10 the CSTR model needs two modes and two parameters to capture the effect of overall mixing present in the tank as well as in the feed distribution. These rigorous results are obtained for the homogeneous stirred tank reactor, while in the heterogeneous case, the situation would clearly be more complex. Non-steady-state catalytic experiments in a CSTR (first proposed by Bennett11 using Eigen’s (1963) relaxation method12 as a starting point) face serious obstacles, particularly the influence of the “dead volume” inside and/or outside the reactor and macro-scale hydrodynamic nonuniformity.5 Bennett even lists many reactors including ones with external and internal recycle for which he concludes “it is no longer advantageous to do these experiments in an ideal mixed-flow reactor (CSTR)”.5 Thus, to avoid above-mentioned difficulties and ensure uniformity, the CSTR is typically operated under steady-state regime in the domain of not too high conversion, especially for highly exothermic processes. In some steady-state CSTR devices of special design and at special particle size, the higher conversion can be achieved (see ref 13). Then, eq 3 simplified for steadystate and constant flow rate (Vflow) can be presented as
Vflow(C - C0) ) R(C)Sc
(4)
In some cases, the possible nonuniformity can be estimated from the CFD calculations. 3. Differential PFR A PFR is the most commonly used normal pressure technique that can provide a wealth of both steady-state and non-steady-state kinetic information (e.g., using step-response or wave propagation techniques). The transport throughout the reactor is assumed to be purely convectional, and no mixing is applied. Radial concentration gradients in a typical PFR can be neglected. The axial concentration gradient developed by reaction can lead to significant nonuniformity and, therefore, should be taken into account as
transport ) VflowC Vg
∂C ∂ + Lcat (VflowC) ) R(C)Sc ∂t ∂x
(5)
An important modification of the PFR approach is the transient-response experiment in a “differential PFR” proposed by Kobayashi and Kobayashi.4 The idea is to make the catalyst zone thin (or residence time short) so that profile in the reactor can be approximated by the linear function and the reaction rate can be analyzed as a function of spatial average concentration, (Cin + Cout)/2, which explicitly requires uniformity. Consequently, in the differential PFR, the gas concentration is oftentimes “considered to nearly constant concentration” (the quotation is taken from ref 14). Then, reaction eq 5 can be simplified as
∂C Vg - Vflow(Cin - Cout) ) R((Cin + Cout)/2)Sc (6) ∂t
The nonuniformity is developed by the reaction and can be estimated using ratio of the concentration difference (Cin - Cout) to the maximum concentration, which is typically Cin. This ratio is nothing but conversion determined by eq 2. Thus, to maintain reasonable uniformity, the differential PFR should be operated at small conversion (e.g., not more than 20%). 4. TZTR The TAP (temporal analysis of products) technique has been successfully applied in many areas of chemical kinetics and chemical engineering for non-steady-state kinetic characterization.9,15-17 The experimental technique operates under vacuum conditions (10-1-10-2 Pa), where the Knudsen diffusion regime provides welldefined transport. Corresponding transport term and traditional TAP equation in the catalyst zone are given by
∂C ∂x
transport ) -ArD
∂C ∂ ∂C Vg - Vr D ) RSc ∂t ∂x ∂x
( )
(7)
Here, the gas concentration gradient is the driving force for Knudsen diffusion and, therefore, remains unavoidable. The scale of this gradient is primarily determined by the reactor length and the vacuum boundary condition at the exit. The key idea of a TZTR (proposed by Shekhtman and Yablonsky in 1999) is to make the thickness of the catalyst zone very small as compared to the length of the reactor and thus to compensate for the gas concentration gradient.6 Making the catalyst bed thin makes the change of gas concentration in the bed small as compared to the mean value of the concentration. In the general one-pulse TZTR model (rigorously derived in ref 8), a very thin catalyst zone sandwiched between two inert zones is viewed as a reactive boundary between the inert zones. Mathematically, the boundary can be accounted for by choosing appropriate matching conditions for the two inert zones. These conditions were obtained by expanding the inert zone solutions (CI and CII) to the middle of the catalyst zone (LTZ). The first condition establishes the continuity of the gas concentration, and the second condition relates the diffusional flow rate difference to the reaction rate (reactive boundary) as
CI(t)|x)LTZ - CII(t)|x)LTZ ) 0+
Lc X (C (t)| - CII(t)|x)LTZ+Lc/2) + 1 - X 4Lr I x)LTZ-Lc/2 O(Lc3/Lr3) (8) ∂CI + | ∂x x)LTZ
flowI(t)|x)LTZ - flowII(t)|x)LTZ ) -ArD
∂CII | ) ArLcRTZ(t) + ∂x x)LTZ
ArD
Lc X (flowI(t)|x)LTZ-Lc/2 - flowII(t)|x)LTZ+Lc/2) + 1 - X 4Lr O(Lc3/Lr3) (9) RTZ(t) ) 1/2(R(t)|x)LTZ-Lc/2 + R(t)|x)LTZ+Lc/2) + O(Lc2/Lr2)
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∂CI + | ∂x x)LTZ
flowI(t)|x)LTZ - flowII(t)|x)LTZ ) - ArD
∂CII | ) ArLcRTZ(t) (11) ∂x x)LTZ
A rD
It determines the reaction rate as the difference between two diffusional flow rates in the way analogous to CSTR (eq 4) where the reaction rate is determined as the difference between two convectional flow rates. From this perspective, a TZTR can be considered a diffusional CSTR. For example, the conversion, X, in a single-pulse TZTR experiment is governed by the same relationship as a first-order reaction in a CSTR: Figure 1. Nonuniformity vs conversion for the differential PFR and TZTR at different values of geometrical parameter Lr/Lc.
is the spatial average reaction rate characterized as a function of the spatial average gaseous and surface concentrations. The first terms on the right-hand side of these equations determine the first-order approximation of the TZTR equations with respect to small parameter, Lc/ Lr. First-order TZTR equations were solved in general.17 The second terms on the right-hand side of eqs 8 and 9 are proportional to the small parameter, Lc/Lr, squared (one power of Lc is inexplicitly present in X) and represent the second-order approximation. The magnitude of these terms is controlled by the same two factors: Lc/Lr, which is a purely geometric parameter, and transport/reaction term X/(1 - X), which describes the influence of reaction. The nonuniformity in the TZTR also comes from two contributions: (i) Applied concentration difference. It creates the driving force of diffusion: gradient. This contribution is present even when no reaction occurs. (ii) Chemical reaction. It changes the concentration profile. Using eqs 8 and 9 and assuming the gradient in the second inert zone to be approximately constant (this agrees with numerical results in ref 7), both contributions to the nonuniformity in the symmetric TZTR can be estimated as (see Appendix for the detailed derivation):
Lc Cin - Cout X ≈2 + Cin Lr 1 + (1 - X)Lr/Lc
(10)
In the right-hand side, the first term is purely geometrical and the second term describes the influence of the reaction. Figure 1 compares nonuniformity in the differential PFR and TZTR for different values of conversion and geometrical parameter Lr/Lc. In the TZTR, the nonuniformity is always present (nonzero) but can be maintain at a low level (e.g., below 20%) in a much wider domain of conversions. 5. TZTR as a Diffusional CSTR In the first-order approximation, the TZTR (eq 9) can be presented as
X)
kappτcat 1 + kappτcat
(12)
where kapp is an apparent kinetic constant and τcat ) LrLc/2D, a residence time in the catalyst zone. In this case, diffusion is working as an “external pump”. It is important to stress that TZTR equations (eqs 8, 9, or 11) describe the non-steady-state experiment while the CSTR equation (eq 4) corresponds to the steady-state regime. 6. Concluding Remarks Table 1 summarizes a comparison of the discussed kinetic devices. There are two main methods to achieve uniformity: forced mixing and making active zone thin. The first mixing method is realized in the CSTR. This is a very simple and straightforward technique for steady-state kinetic studies. The limitations are not too high conversion for the highly exothermic reactions to maintain an isothermal profile, unreliability of nonsteady-state data, and difficulty in estimation of nonuniformity under operating conditions. The differential PFR and TZTR both exploit the idea to achieve a small difference of concentration within the active zone by making this zone thin, assuming of course that the ratio of the concentration change to the zone thickness, ∆C/∆L (or gradient), is “not too big” (finite). For the differential PFR, keeping finite gradient in a thin zone leads to smaller conversions. In the TZTR, diffusion naturally keeps the gradient finite with a much weaker restriction to the conversion domain. The key difference between these reactors is the type of transport and its role in creating uniformity. In the differential PFR, transport is purely convectional and caused by the small pressure difference. The rate of such transport is proportional to the concentration. The term “transport-in minus transport-out” in eq 2 is determined by the concentration difference created by reaction. As a result, the nonuniformity estimate is equal to the conversion which is the main limitation of the differential PFR. In the TZTR, the transport is purely diffusional and caused by applying the concentration difference between the reactor inlet and the outlet opened to the vacuum. The transport rate is proportional to the concentration gradient. The term “transport-in minus transport-out” in eq 2 is determined by the difference of the concentration gradients created by reaction (eq 9 or 11). As a result, for not too high conversions (below 90%), the nonuniformity estimate is determined by the geometrical parameter (Lr/Lc) and depends on conversion relatively weakly (see Figure 1).
Ind. Eng. Chem. Res., Vol. 44, No. 16, 2005 6521 Table 1. Comparison of the Reactors reactor
CSTR
key idea
forced mixing
regime
steady-state normal conditions reaction
natural cause of nonuniformity driving force of transport transport term influence of transport on the nonuniformity concentration vs length out of the active zone conversion, x, from eq 2 assuming constant flow rate or diffusivity estimation of non-uniformity
Differential PFR
TZTR
making active zone thin (smaller loading) or residence time short (higher flow rate) steady-state and non-steadystate normal conditions reaction
making active zone thin compared to the length of the whole reactor
pressure difference between inlet and outlet; kept small VflowC internal mixing
VflowC no influence
no change with length
no change with length
non-steady-state Knudsen diffusion regime applied concentration difference and reaction concentration difference; outlet concentration is much smaller than the inlet one -ArD(∂C/∂x) diffusion tends to make the uniform profile change with length
(C0 - C)/C0
(Cin - Cout)/Cin
-((∂Cin/∂x) - (∂Cout/∂x))/(∂Cin/∂x)
depends on the model and setup
X
X/(1 + (1 - X)Lr/Lc) + 2Lc/Lr
pressure difference; kept small
influence of kink caused by reaction):
Lc ∂C ∂C CTZ ≈ and ∆Ccat,G ≈ Lc ) 2 CTZ ∂x Lr/2 ∂x Lr The contribution of reaction is associated with a stepwise change of the gradient (kink) in the catalyst zone. In the first-order approximation, this change is determined by eq 9 as
|
|
∂CII Lc ∂CI ∂C ≈R x)LTZ + x)LTZ ) ∂x ∂x ∂x D TZ
∆
This additional gradient causes the concentration drop within the first half of the catalyst zone, ∆Ccat,R (see Figure 2), which can be estimated as Figure 2. TZTR concentration profile with two contributions to nonuniformity in the catalyst zone (marked by dashed lines).
In the TZTR, diffusion mixes up the nonuniformity created by reaction and tends to keep the concentration profile close to the one when no reaction occurs. Consequently, the TZTR allows maintaining a small nonuniformity (15-20%) in a much wider domain of conversions (up to 80%) than the differential PFR. This makes the diffusional transport very suitable for running non-steady-state kinetic experiments under approximately uniform conditions and generates the challenge of adapting the TZTR to the normal pressure domain. Acknowledgment We thank Prof. J. T Gleaves and Dr. J. M. Yoda for many fruitful discussions. Appendix: Derivation of the Estimate of Nonuniformity in the TZTR Figure 2 presents the typical TZ concentration profile (taken from ref 7) and contributions of two terms from eq 10. The geometrical term, ∆Ccat,G, can be estimated using gradient of the solution in inert zone II (no
2 ∂C Lc Lc ) R ∂x 2 2D TZ
∆Ccat,R ≈ ∆
The reaction rate term RTZ and concentration change, ∆Ccat,R, can be related to the thin-zone concentration and conversion using eq 12 and the following approximation:
RTZ ≈ kappCTZ )
X 1 C and ∆Ccat,R ≈ τcat 1 - X TZ Lc X Lc2 1 X CTZ ) C 2D τcat 1 - X Lr 1 - X TZ
Now, the TZTR nonuniformity can be estimated as (see Figure 2)
∆Ccat,G + ∆Ccat,R 2-X ∆C ≈ ) Cin CTZ + ∆Ccat,G/2 + ∆Ccat,R 1 + (1 - X)Lr/Lc This expression combines both contributions. The purely geometrical term can be extracted by setting conversion to zero:
|
∆C 2 ≈ Cin X)0 1 + Lr/Lc At arbitrary conversion, nonuniformity also includes the term describing the influence of reaction:
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∆C 2-X 2 ≈ ) + Cin 1 + (1 - X)Lr/Lc 1 + Lr/Lc
-1 + Lr/Lc X 1 + (1 - X)Lr/Lc 1 + Lr/Lc
In the first order of approximation that is used for solving TZ (eqs 8 and 9), the unit terms can be neglected as compared to Lr/Lc, which leads to eq 10:
-1 + Lr/Lc ∆C 2 X ≈ + ≈ Cin 1 + Lr/Lc 1 + (1 - X)Lr/Lc 1 + Lr/Lc Lc X 2 + Lr 1 + (1 - X)Lr/Lc Here the first term is purely geometrical corresponding to ∆Ccat,G. The second describes the influence of the reaction where the unit term cannot be neglected as compared to (1 - X)Lr/Lc at high conversions. Nomenclature Ar ) reactor cross-section area, cm2 C ) gas concentration, mol/cm3 Cin, Cout ) gas concentration at inlet and outlet of catalyst zone, respectively, mol/cm3 CI(x,t) ) concentrations in the first inert zones, mol/cm3 CII(x,t) ) concentrations in the second inert zones, mol/ cm3 CTZ(t) ) spatial average concentration in the thin zone, mol/ cm3 D ) Knudsen diffusivity, cm2/s kapp ) apparent kinetic constant, 1/s Lc ) length of the catalyst zone, cm Lr ) length of the whole TAP reactor, cm LTZ ) position of thin-zone (the middle of the zone), cm R ) gas reaction rate, mol cm-2 s-1 RTZ ) spatial average reaction rate in the thin zone, mol cm-3 s-1 Sc ) catalyst surface area, cm2 t ) time coordinate, s Vflow ) flow rate, cm3/s Vg ) gas volume, cm3 X ) conversion
x ) space coordinate, cm τcat ) residence time in the catalyst zone, s
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Received for review May 12, 2005 Accepted June 7, 2005 IE050554G
