Ind. Eng. Chem. Res. 2004, 43, 4535-4539
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Enhancement of Heterogeneous Autocatalytic Reactions Accompanied by Mass and Heat Transfer Phenomena Miroslaw Grzesik,*,†,‡ Maria T. Kulawska,†,§ Jerzy Skrzypek,† and Anna Ptaszek‡ Institute of Chemical Engineering, Polish Academy of Sciences, Gliwice, Poland, and Faculty of Food Technology, Academy of Agriculture, Cracow, Poland
The objective of the present work is the evaluation, modeling, and optimization of heterogenic autocatalytic reactions that show the Langmuir-Hinshelwood kinetics with accompanying external diffusion for nonporous catalyst pellets or intraparticle diffusion and external energy transport for porous pellets in a fixed-bed tubular reactor. Both cases studied are closely related to industrial applications. It was proved that there is an opportunity to increase the yield of the process through the controlled change in mass transfer resistance (catalyst pellet size) along the length of the chemical reactor. 1. Introduction Transport phenomena such as external diffusion of gaseous agents, internal diffusion through porous catalyst pellets, and external or internal energy transport often accompany processes in fixed-bed catalytic reactors or processes in bioreactors with immobilized enzyme. They can occur either individually, or catalytic reactions are accompanied by two or more independent transport resistances. Mass transfer phenomena in their vast majority are undesirable since they cause a reduction in overall reaction rate. Because of that, one prefers to eliminate diffusion effects that slow chemical or biochemical reactions. A different situation can occur in case of autocatalytic processes with accompanying internal or external diffusion.1-4 The simplest set of such reactions is A f B, A + B f 2B, where the second reaction is the proper autocatalytic reaction. Reagent B, the product from the first reaction, shows additional catalytic activity independent of the activity of the solid catalyst used in the reactor. The first reaction is a typical process-initiating reaction, catalyzed only by the solid catalyst. Since reagent B in the second reaction simultaneously takes a role of a catalyst, it is advantageous to keep it on the surface or inside the catalyst particle. This kind of an effect can be achieved simply through appropriately controlling the magnitude of either intraparticle or external diffusion resistance.2,3,5 Similar effects can also appear in cases where catalytic reactions show frequently encountered LangmuirHinshelwood kinetics. We are talking about two- or higher-molecular reactions, where changes on surfaces of catalysts between adsorbed gaseous components are limiting steps. For this reaction type, similar to autocatalytic reactions, internal and external diffusion phenomena can, paradoxically, increase the overall rate of a process within a certain ranges of parameters.6,7 The present work extends our previous study. The objective of it is the evaluation, modeling, and optimiza* To whom correspondence should be addressed. Tel.: +48 12 662 4761. E-mail:
[email protected]. † Polish Academy of Sciences. ‡ Academy of Agriculture. § E-mail:
[email protected].
tion of heterogeneous, autocatalytic reactions that show the Langmuir-Hinshelwood kinetics with accompanying external diffusion for nonporous catalyst particle or intraparticle diffusion and external energy transport for porous pellets. A tubular-type reactor with a fixed-bed was used. The cases studied are closely related to industrial applications. Examples of reactions that exhibit autocatalytic behavior include the reaction of methanol over zeolite catalyst to produce hydrocarbons, the catalytic cracking of paraffins on HY zeolite to produce olefins, acidcatalyzed hydrolysis of esters to carboxylic acids and alcohols, the thermal cracking of some polycyclic nalkylarenes, and several enzymatic processes in biotechnology. 2. Mathematical Models We will study the heterogeneous autocatalytic reactions with the following stoichiometric equations: k1
A 98 B
(1) k2
A + (n - 1)B 98 nB
(2)
These are described by a general Langmuir-Hinshelwood kinetic equation:
xAxBn-1
xA
rA ) k1 + k2 (1 + KAxA + KBxB)m1 (1 + KAxA + KBxB)m2 (3) where
xA ) cA/cAo
xB ) cB/cAo
In eq 3, both the reaction rate and the reaction rate constants have a dimension of s-1. Sorption constants are dimensionless. The analysis below will be conducted with the following assumptions: it is a plug flow in the reactor; steadystate conditions are fulfilled; changes in density and
10.1021/ie034247r CCC: $27.50 © 2004 American Chemical Society Published on Web 04/01/2004
4536 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004
viscosity of reaction mixture could be omitted; catalyst particle is symmetrical. 2.1. Reactions Accompanied by External Diffusion. The mathematical model is formulated separately for processes on the surface of the catalyst and for the tubular reactor with a fixed-bed. For the surface of catalyst pellet (m1 ) 1, m2 ) n), the following equation applies:
(
x As r As ) k 2 β + (1 + KAxAs + KBxBs) xAsxBsn-1 (1 + KAxAs + KBxBs)n
)
) kcA(xA - xAs) (4) Figure 1. Overall reaction rate versus mass transfer coefficient for selected values of conversion.
where
xAs ) cAs/cAo
For an adiabatic fixed-bed catalytic reactor, the mass balance equation takes the following form:
xBs ) cBs/cAo ) 1 + xBo - xAs(xBo ) cBo/cAo) β ) k1/k2 For fixed-bed catalytic reactor, the following equation applies:
-dxA/dτ ) rov(xA) ) rAs
xA(0) ) 1
(5)
To find rov ) rAs, the nonlinear eq 4 with respect to xAs must be solved. 2.2. Reactions Accompanied by Intraparticle Diffusion and External Energy Transport. For the surface of the catalyst pellet, the mass balance equation takes the following form:
{ } (
1 d υdxA z ) dz zυ dz
)
f1xA f2xAxBn-1 Φ22 β + (1 + KAxA + KBxB)m1 (1 + KAxA + KBxB)m2 (6) with boundary conditions
(10)
υ + 1 dxA(1) rov(xAr, tr) ) k2o dz Φ22
(11)
tr ) 1 + θ(1 - xAr) θ ) (-∆H)cAo/cpToF
(12)
where
3. Objective Function The optimization problem is formulated as follows: find an optimum distribution of the size of catalyst pellets (bed porosity) in reactor, Rmin < R*(τ) < Rmax, τ ∈ [0, τf], which maximizes the final conversion of A, R(τf)) 1 - xA(τf) (Rr(τf) ) 1 - xAr(τf)). Thus, the objective function may be written as
R(τf) or Rr(τf) f
max
Rmin e R*(τ) e Rmax
τ ∈ [0,τf]
(13)
To solve the problem formulated above, the rule of Denbigh and Horn was used. This rule is the simplified form of Pontriagin maximum principle8 for one state equation.
xA ) xAr(τ) for z ) 1
(7)
dxA/dz ) 0 for z ) 0
(8)
4. Results and Discussion
βpdxA/dz ) Nu(ts - tr) for z ) 1
(9)
4.1. Reactions Accompanied by External Diffusion. The presented model was solved using numerical methods (Newton method, Merson method, golden section method). The results are presented in the graphs. The relation between the overall reaction rate rov and the mass transfer coefficient kcA for various conversion degree R ) 1 - xA and for n ) 2, k2 ) 25, KA ) 2, KB ) 1, β ) 0.01 is shown in Figure 1. It may be observed that the location of the maximum of the curve rov ) rov (kcA), if such a maximum exists, depends on the conversion, and it moves toward higher values of kcA following increase in conversion, i.e., toward lower resistance produced by external diffusion. The fact that the location of the maximum of the function rov ) rov (kcA) is dependent on conversion makes it possible to increase the yield of product in autocatalytic reactions of L-H kinetics. As the value of the mass transfer coefficient may be related to the size of a
where
xA ) cA/cAo xB ) cB/cAo xB ) 1 + xBo - xA ts ) Ts/To βp ) fi ) exp(γi(1 - 1/ts)) 2
(-∆H)DAcAo λhTo
i ) 1, 2 γi ) Ei/(RgTo)
i ) 1, 2
2
Φ2 ) R k2o/DA β ) k1o/k2o Nu )
dxAr/dτ ) -rov(xAr, tr) xAr(0) ) 1
[
( )]
R hR 1 2RvF 0.5 cpµ ) 2 + 1.8 λh 2 µ λh
(
)
1/3
(Ranz’s equation)
Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 4537
As can be observed in Figure 3, a large difference exists between the values of final conversion, R*(τf) obtained for the optimum profile of the R and for the constant values of R, i.e., Rmin and Rmax. A large difference also appears between R*(τf) and value of R(τf) calculated for the optimum constant value of R. It suggests that practical application may be possible. For n ) 2, eq 4 takes the form
(
)
1 + xBo - xAs xAs rAs ) k′2 β′ + ) kcA(xA - xAs) 1 + KxAs 1 + KxAs (14) where Figure 2. Pellet size distribution in a fixed-bed plug-flow catalytic reactor.
KA - KB ω ω ) 1 + KB(1 + xBo)
k′2 ) k2/ω2 β′ ) k1ω/k2 ) βω K )
In such a case, the problem can be solved by an analytical method.3 The formulas for values of xA/ s, r/ov, R/A(τ), and k/cA and for limiting value of β are given below.
xA/ s)
βω + 1 + xBo 2 + KxBo - Kβω + K
) constant
(15)
r/ov ) rAs(xA/ s) ) constant
(16)
R/A(τ) ) 1 - x/A(τ) ) rAs(xA/ s)τ
(17)
(
xA/ s
Figure 3. Conversion profiles in a fixed-bed plug-flow catalytic reactor.
k/cA )
k′2 βω + 1 + KxA/ s
1 + KxA/ s
1 - R/A - xA/ s βlm )
)
1 + xBo - xA/ s
K-1 1 1 + xBo ω K+1
(
)
(18)
(19)
If xBo ) 0, the equation for k/cA takes the form
βω + 1 (1 + K + Kβω + βω) k′2 (2 + 2K)2 / kcA ) βω + 1 1 - R/A 2 + K - Kβω
Figure 4. Overall reaction rate profiles in a fixed-bed plug-flow catalytic reactor.
catalyst pellet employed, such processes may be intensified by a suitable distribution of the pellet size in the various types of chemical reactors. In the Figures 2-4, the selected optimum profile of the pellet size, optimum profile of conversion, and optimum profile of overall reaction rate have been shown. The profiles of conversion and overall reaction rate obtained for both a constant optimum size of pellet and its limiting sizes have also been given. To find the value of R for given kcA, the WakaoFunazkri equation9 was used, as in the work in ref 3.
(20)
and limiting value of β, βlm, is equal to 1/ω. The limiting value of β, βlm, may be interpreted as follows: if β > βlm, then the curve rov ) rov(kcA) is increasing, so the optimum distribution of pellet size along the chemical reactor is trivial,3 i.e., R*(τ) ) Rmin, τ ∈ [0,τk]. 4.2. Reactions Accompanied by Intraparticle Diffusion and External Energy Transport. At first, two-point boundary value problem (eqs 6-9) was solved using Marquardt method. The endothermic reactions were only taken into account. The results of computer calculations are presented in the graphs. In Figure 5 the dependence of effectiveness factor η ) rov/rs on R for the selected values of parameter β and for n ) 2, υ ) 2 (sphere), k2o ) 1, γ1 ) 12, γ2 ) 7, βp ) -0.2 (endothermic reactions), and xA(1) ) 1 has been shown.
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Figure 5. Effectiveness factor of catalyst pellet versus pellet radius for selected values of β.
Figure 6. Overall reaction rate versus pellet radius for selected value of conversion.
One can find that for β < βlm (in this case βlm = 0.15), the effectiveness factor η assumes values much higher than unity in the certain ranges of R. This means that in the cases discussed internal diffusion may considerably increase the rate of the heterogeneous autocatalytic reactions. This behavior is very different from classical isothermal and endothermic catalytic reactions. The relation between the overall rate rov and R for various conversion degree and for n ) 2, υ ) 2, k2o )1, β ) 0.01, γ1 ) 12, γ2 ) 7, βp ) -0.2 (endothermic reactions), and xBo ) 0 is illustrated in Figure 6. It may be observed that the location of the maximum of the curve rov ) rov(R), if such a maximum exists, depends on the conversion degree, and with increasing conversion it moves toward lower values of R, i.e., toward lower resistance produced by intraparticle diffusion. The possibility of η exceeding the values of unity even for endothermic processes and the fact that the location of the maximum of the function rov ) rov (kcA) is dependent on conversion makes it possible to increase the yield of product in heterogeneous autocatalytic reactions by a suitable distribution of the pellet size in the various types of chemical reactors. In Figures 7-9, the selected optimum profile of the pellet size, R*(τ) ∈ [Rmin, Rmax], τ ∈ [0,τf], optimum profile of conversion and related profile of overall rate of process, r/ov(τ), have been shown. The profiles of conversion and overall reaction rate obtained for both a constant optimum size of pellet and for Rmin and Rmax have also been given (k2o ) 3, γ1 ) 7, γ2 ) 12, β ) 0.002).
Figure 7. Pellet size distribution in a fixed-bed plug-flow catalytic reactor.
Figure 8. Conversion profiles in a fixed-bed plug-flow catalytic reactor.
Figure 9. Overall reaction rate profiles in a fixed-bed plug-flow catalytic reactor.
A relatively large difference is observed between the value of final conversion, Rr(τf), obtained for the optimum distribution of the pellet size R*(τ) and for the selected constant values of R. A large difference in values also appears between the of R/r (τf) and Rr(τf) calculated for limiting values of R, Rmin, and Rmax. It is also worthwhile to mention the different behavior of r/ov ) r/ov(τ) for nonporous catalyst pellet which is consented with eq 16 and for a porous catalyst pellet.
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5. Conclusions There is an opportunity to increase the yield of the process through the controlled change in mass transfer resistance (catalyst pellet size) along the length of the chemical reactor. The optimum profile of the characteristic pellet size is a nonincreasing function of space time. The differences between the yield of product in heterogeneous autocatalytic reactions, obtained for an optimum distribution of the pellet size, and the yields corresponding to the constant values of R are large enough to suggest that practical application of the optimum solution proposed may be possible. Nomenclature c ) concentration (mol/m3) D ) effective diffusion coefficient (m2/s) ∆H ) heat of reaction (J/mol) k ) reaction rate constant (1/s) kc ) mass transfer coefficient (1/s) r ) reaction rate (1/s) R ) pellet size (m) t ) dimensionless temperature T ) temperature (K) x ) dimensionless concentration R ) conversion Rh ) heat transfer coefficient (W/(m2‚K)) β ) auxiliary coefficient λh ) effective thermal conductivity (W/(m‚K)) τ ) space time (s) υ ) shape constant (0, infinite slab; 1, cylinder; 2, sphere) Subscripts f ) final value lm ) limiting value o ) initial value ov ) overall rate r ) reactor s ) surface of catalyst pellet
Superscript * ) optimum value
Acknowledgment The work was supported by KBN (Grant PBZ/KBN/ 14/T09/99/01e). Literature Cited (1) Sapre, A. V. Diffusional enhancement of autocatalytic reactions in catalyst particle. AIChE J. 1989, 35, 655. (2) Grzesik, M.; Skrzypek, J. Enhancement of heterogeneous autocatalytic reactions by intraparticle diffusion. Chem. Eng. Sci. 1993, 48, 2463. (3) Grzesik, M.; Skrzypek, J. Enhancement of heterogeneous autocatalytic reactions by external diffusion. Chem. Eng. Sci. 1993, 48, 2469. (4) Neylon, M. K.; Savage, P. E. Analysis of nonisothermal heterogeneous autocatalytic reactions. Chem. Eng. Sci. 1996, 51, 851. (5) Grzesik, M.; Skrzypek, J. Enhancement of nonisothermal autocatalytic reactions by intraparticle diffusion. Stud. Surf. Sci. Catal. 2001, 133, 411. (6) Grzesik, M.; Kopcinska, J. Optimization of pseudoautocatalytic reactions in multiphase reactors. Chem. Ing. Tech. 2001, 73, 668. (7) Grzesik, M.; Skrzypek, J. Optimization of pseudoautocatalytic reactions in heterogeneous reactor. Proc. of ICheaP-5, Florence, Italy, May 20-23, 2001; Vols. 1 and 2, p 163; Pierucci, S., Ed.; AIDIC Servizi S.r.1: Milano, Italy, 2001. (8) Pontriagin, L. S. The mathematical theory of optimal processes; Interscience Publishers: New York, 1962. (9) Wakao, N.; Funazkri, T. Effect of fluid dispersion coefficients on particle-to-fluid mass transfer coefficients in packed beds. Chem. Eng. Sci. 1978, 33, 1375.
Received for review November 13, 2003 Revised manuscript received January 9, 2004 Accepted January 13, 2004 IE034247R
