Ind. Eng. Chem. Res. 2001, 40, 5215-5221
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Influence of the Heat-Flux Profiles on the Operation of Primary Steam Reformers Juliana Pin ˜ a, Noemı´ S. Schbib, Vero´ nica Bucala´ , and Daniel O. Borio* PLAPIQUI (UNS-CONICET), Camino La Carrindanga, Km 7, 8000 Bahı´a Blanca, Argentina
The influence of the heat-flux axial profiles on the main variables of an industrial primary steam reformer (i.e., outlet methane conversion, process gas temperature, tube-skin temperature, and equilibrium approach) has been studied. By means of adjustments of the heat flux along the tube length, two different optimization problems have been proposed: (a) maximum methane conversion for a given maximum allowable tube-skin temperature and (b) minimum tube-skin temperature for a fixed methane conversion, both for constant heat duty. The defined optimization problems have been solved using a one-dimensional heterogeneous model. This mathematical formulation accounts for the strong mass-transfer resistances by an appropriate solution of the material balances within the catalyst. An equivalent annular model has been used to represent the complex geometry of the catalyst particle. The simulation and optimization results indicate that the heat flux strongly influences the reactor performance. Indeed, adequate selections of the heat-flux distribution along the reactor length would increase the production rate and/or extend the tube lifetime significantly. Introduction Steam reforming of hydrocarbons is one of the most important chemical processes; it is used in the manufacture of ammonia, hydrogen, methanol, and many chemicals made from hydrogen and carbon monoxide (synthesis gas). The largest user of the steam-reforming process is the ammonia industry; about 80% of the world ammonia production (around 126 MMtons/year) is based on the steam reforming of hydrocarbons.1,2 Nowadays, natural gas is the most common feedstock for steam reforming, being used in more than 75% of the currently operating units.3,4 The primary reformer is essentially a furnace, constructed of vertically suspended tubes filled with a supported nickel catalyst. Several primary reformer designs are available today; basically they differ in the arrangement of the tubes and the location of the burners in the furnace chamber. These basic designs are classified as top-fired, bottom-fired, side-fired (radiant-wall), and terrace-wall.5 Different firing arrangements result in different tube-wall temperatures and heat-flux profiles.6 In side-fired furnaces, the tubes are disposed in a single row and burners are placed in the furnace walls at four to seven levels. This configuration allows adjustment and control of the heat input along the length of the tube.7 Steam reforming of natural gas (mainly methane) may be represented by the following system of reversible reactions:
CH4 + H2O T CO + 3H2
(1)
CO + H2O T CO2 + H2
(2)
CH4 + 2H2O T CO2 + 4H2
(3)
* To whom correspondence should be addressed. Tel: 54291-4861700, ext. 266. Fax: 54-291-4861600. E-mail: dborio@ criba.edu.ar.
The overall reaction of the system is endothermic. The conversion rate of methane is strongly affected by the diffusion rates through the pores of the catalyst, and therefore reliable diffusion-reaction models are necessary. Very low and variable effectiveness factors have been obtained by many researchers.8-10 Consequently, the implementation of rigorous models is essential to characterize properly the industrial steam reformers operation. Because of the equilibrium restrictions, high temperatures are required in order to improve the production rates. Due to the strongly endothermic nature of the process, a large amount of heat is supplied by fuel burning (commonly natural gas) in the furnace chamber. A key problem in steam reforming is the balance between the heat input through the reformer tubes and the heat consumption in the endothermic reforming reactions.6 As was mentioned previously, the flexibility in the design and operation of the side-fired furnaces allows one to regulate (up to a certain degree) the heat input along the catalyst tubes by controlling the fuelgas flowrate at each row of burners. This action affects directly the maximum tube-skin temperature, which is a relevant variable. Even a slight increase in the maximum tube wall temperature may result in a serious decline of the expected tube lifetime.11 Simultaneously, the axial temperature profile of the process gas stream depends on the heat flux across the tube wall. Therefore, the way in which heat is transferred to the process gas influences the outlet conversion (the methane slip). Typical heat-flux profiles and measured tube-wall temperatures for top- and side-fired reformers have been reported in the open literature;7,12 however, there is a lack of information about the optimization of the process by means of adjustments of the heat flux along the catalyst tubes of industrial primary reformers. The overall goal of this paper is to analyze the influence of the heat-flux profiles’ shapes on the operation of primary reformers and to estimate optimal heatflux distributions based on a rigorous mathematical model of an industrial reformer.
10.1021/ie001065d CCC: $20.00 © 2001 American Chemical Society Published on Web 07/26/2001
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Mathematical Model A one-dimensional heterogeneous mathematical model is used to describe a single reformer tube. The external mass- and heat-transfer resistances are assumed to be neglible.13 Gas Phase.
Mass balances dxCH4 dz dxCO2 dz
) ΩFB(η1r1s + η3r3s)/FCH40 s
dz
)
0
) ΩFB(η2r2 + η3r3 )/FCH4
Energy balance dTg
s
(4)
1 cpFgus
[
3
FB
(-∆Hi)risηi + ∑ i)1
(5)
]
Q(z) πdto Ω
(6)
where
Q(z) ) U(Tg - Tw)
(7)
( )
dti dto 1 1 ) ln + U 2λst dti Ri
(8)
Momentum equation dpt fFgus2 )dz gdp
(9)
Boundary conditions at z ) 0:
0
0
0
0
xCH4 ) xCO2 ) 0; Tg ) Tg ; pt ) pt (10)
The intrinsic kinetic expressions reported by Xu and Froment14 for reactions (1)-(3) are considered. Fresh catalyst conditions are assumed. The tube wall temperature is calculated from a given heat-flux profile through eq 7. The heat-transfer coefficient Ri is estimated from the heat-transfer parameters of the two-dimensional model using the formulas proposed by Froment and Bischoff15 and De Wasch and Froment.16 The friction factor (f) is computed using the equation of Ergun.17 The equivalent particle diameter (dp) is evaluated following the guidelines given by Froment and Bischoff.15 The
Figure 1. Scheme of the real and equivalent catalyst particles.
operating conditions and parameters used in the reactor model are included in Table 1. Catalyst Particle. The selected catalyst particle is the industrial Haldor Topsoe R-67-7H (tablet-shaped with seven holes and convex ends).18 The complex geometry of the real particle is represented by means of an equivalent annular model (Figure 1). The inner radius of the equivalent geometry (rin) is supposed to be equal to the radius of the internal holes of the original catalyst. The outer equivalent radius (req) results from considering that the equivalent particle (of infinite length) has the same external surface per unit volume as the real one. A more rigorous two-dimensional model for the real particle indicated that the proposed equivalent geometry gives a proper description of the composition distribution inside the particle.19 Assuming an isothermal equivalent particle, the continuity equations for CH4 and CO2 become e DCH 4
e DCO 2
(
(
)
dps,CO2 1 d r ) -RTg[r2(ps,j) + r3(ps,j)]Fp r dr dr
parameter
operating condition 520 °C 38.7 bar 38.06 kmol/h 2.796% 0.699% 0.768% 0.157% 0.0039% 22.78% 72.79% 73.09 kW/m2 1990.6 kg of catalyst/m3r 1016.4 kg of catalyst/m3r of catalyst 0.002 585 m 0.001 58 m
(11)
(12)
Boundary conditions at r ) rin:
ps,CH4 ) pCH4, ps,CO2 ) pCO2
(13)
at r ) req:
dps,CH4/dr ) dps,CO2/dr ) 0
(14)
The partial pressures of the remaining components are related to those of CH4 and CO2 by means of algebraic equations obtained from stoichiometry and the mass balances inside the particle. The values of the effective diffusivities are calculated using the expressions given by Xu and Froment.14 The effectiveness factor for reaction i is defined by
Table 1. Operating Conditions and Parameters for the Simulation of an Industrial Steam Reformer Tg0 pt0 Ft0 yH20 yN20 yCO0 yCO20 yAr0 yCH40 yH2O0 Qt FB catalyst particle Fp req rin
)
dps,CH4 1 d r ) RTg[r1(ps,j) + r3(ps,j)]Fp r dr dr
ηi )
∫0Vri(ps,j) dV V ri(ps,js)
i ) 1-3
(15)
Numerical Solution. The differential equations for the gas phase are integrated by means of a GEAR routine. The differential equations for the particle are discretized by means of second-order finite differences, using an adaptive grid of two elements with variable width. A total of 30 and 10 grid points are assigned to the first (near the catalyst surface) and second elements, respectively. For each axial position, the 88 resultant nonlinear algebraic equations are solved through a Quasi-Newton algorithm. Once the partial pressures for all of the components are obtained (ps,j), the effectiveness factors are calculated through eq 15.
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Figure 2. Outlet methane conversion and maximum tube-skin temperature as a function of the slope of a single linear heat-flux profile (A1). Operating conditions are given in Table 1.
Figure 3. Process gas and equilibrium temperatures at the reactor outlet as a function of the slope of a single linear heatflux profile (A1). Operating conditions are given in Table 1.
The model presented above has been checked successfully against data available from a large-scale sidefired primary reformer used in the ammonia industry. Results and Discussion Influence of the Heat-Flux Profiles on the Reactor Performance. To analyze the influence of the heatflux distribution on the main process variables, the following conditions have been specified to define a common basis (Table 1): (a) reactor dimensions, (b) feed conditions, and (c) heat duty (Qt). In addition, a simple linear variation of the heat flux along the tube length has been assumed [Q(z) ) A1z + B1]. The slope (A1) of the heat-flux profile has been varied to study its effect on the reactor operation. For each A1 value, the parameter B1 is univocally defined to satisfy the constraint ∫L0 Q(z) dz/L ) Qt. Figure 2 shows the significant influence of the slope A1 on the outlet methane conversion. A slope of -11.22 kW/m3 (point 1) is the lowest feasible A1 value for a single linear profile because Q(z) becomes zero at the reactor outlet. When slopes more negative than this value are selected, adiabatic conditions are imposed in the last section of the tube to maintain the specified Qt. For high negative values of A1, the methane conversion tends asymptotically to a maximum value of xCH4,eq ) 62.7%, which can be calculated from integral (inputoutput) mass and energy balances if equilibrium conditions at the reactor outlet are assumed. The maximum values of the external tube wall temperature (Tw,max) are also presented in Figure 2. A nonmonotonic behavior is observed for the curve of Tw,max and a minimum value of around 867 °C is found for A1 ) -6.98 kW/m3 (point 2). Reductions in the slopes between A1 ) 0 (constant heat flux) and A1 ) -6.98 kW/m3 lead to two simultaneous positive effects: a conversion increase and a substantial decrease of the maximum tube-skin temperature. In fact, the difference between the Tw,max values corresponding to points 2 and 3 is higher than 60 °C. According to data previously reported,4 this decrease in the tube-skin temperature (from 935 to 867 °C) could extend the tube lifetime by more than 9 years. The influence of the heat-flux distribution on the temperature of the process stream (TgL) and the equi-
Figure 4. Axial heat-flux and methane conversion profiles for the operating points numbered in Figure 2.
librium temperature (TeqL) at the reactor outlet is shown in Figure 3. For each A1 value TeqL is the temperature necessary to attain the same conversion level as that of the reactor (showed in Figure 2), if equilibrium conditions are assumed. The TgL value monotonically decreases as more negative slopes of the heat-flux profile are imposed. This is an important result, because for the same operating conditions the methane conversion shows an opposite behavior (Figure 2). Simultaneously, the difference between the values of TgL and TeqL (usually an industrial measurement of the equilibrium approach) tends to diminish as A1 decreases (Figure 3). Therefore, the improvement observed in the methane conversion as A1 diminishes can be attributed to a better equilibrium approximation or, in other words, to a more efficient utilization of the same total amount of heat. The axial heat-flux profiles corresponding to the four representative operating points indicated in Figure 2 are drawn in Figure 4. Because of the assumption of constant heat duty, the areas underneath the four linear profiles are equal. Point 3 corresponds to a constant heat-flux distribution, point 2 is located at the minimum of the curve for Tw,max, point 4 represents an increasing profile, and point 1 is the decreasing heat-flux distribution that leads to a zero value of Q(z) at the reactor
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Figure 5. Axial process-gas temperature profiles for the operating points numbered in Figure 2.
profile imposed for the operating point 4 is not convenient. In fact, more energy is supplied toward the reactor outlet where less reactants are available. For industrial steam reformers, Dybkjaer7 and Plehiers and Froment12 have reported slightly increasing tube-skin temperature profiles. However, for the selected linear variation of Q(z), the lowest value for Tw,max is reached for a Tw profile showing a slight maximum at around 70% of the tube length (curve 2 of Figure 6). If a more negative heat-flux slope is imposed (e.g., curve 1), the value of Tw,max increases and its location shifts toward the reactor inlet. For all of the studied operating points, very low effectiveness factors are found. This fact indicates a poor use of the total catalyst volume. Optimal Heat-Flux Distribution along the Tube Length. In this section, the heat-flux axial profile is defined as the optimization variable to solve different optimization problems of practical interest. The procedure is based on the previously mentioned rigorous mathematical model of the reactor, which is a subroutine of the optimization program. The optimization algorithm (variable metric projection), suitable to solve constrained nonlinear problems, requires successive reactor simulations which are very time-consuming. For this reason, the heat-flux distributions are idealized as stepwise linear profiles to minimize the number of optimization variables. a. Maximum Methane Conversion for Constant Heat Duty and a Given Maximum Allowable TubeSkin Temperature. For given feed conditions and specified reactor geometry and heat duty (Qt; see Table 1), this optimization problem can be defined as follows:
max xCH4L
(16)
s.t. Tw(z) e Tw,all
(17)
Q(z)
Figure 6. Axial tube-skin temperature profiles for the operating points numbered in Figure 2.
outlet. Figure 4 also includes the methane conversion profiles for the operating points 1-4. An almost linear conversion growth is observed for the case of constant heat-flux distribution (curve 3). For those reactor positions where the heat fluxes are greater than the mean value [Q(z) ) Qt], the slopes of the xCH4 curves (i.e., the observed reaction rates for methane) are higher than that of the reference case (point 3). Figure 5 presents the gas temperature profiles for the studied operations. The operating point 1 shows the highest axial average gas temperature, and this result justifies the highest outlet conversion found for this condition (Figure 2). It is interesting to note that the TgL value decreases as more negative heat-flux slopes are imposed. This result is consistent with the overall energy balance, because the heat duty is constant and the total amount of heat consumed by the chemical reactions increases with the outlet methane conversion. The axial profiles for the tube-skin temperature are shown in Figure 6. When positive or zero slopes are fixed for the heat flux (points 3 and 4), a monotonic increase of Tw is observed; therefore, the critical (maximum) values of the tube-wall temperature are located at the reactor outlet. According to the endothermic nature of the reforming reactions, the increasing axial heat-flux
∫0L Q(z) dz L
) Qt
Q(z) g 0
(18) (19)
If the reactor is subdivided into K sections of equal length, the heat-flux profile in each section follows the linear expression Qk(z) ) Akz + Bk. The number of optimization variables results equal to K in order to satisfy the continuity of the Q(z) profile between each subinterval and the specified constant heat duty Qt. Figure 7 shows the optimal heat-flux distribution obtained when 1-3 linear sections for Q(z) are considered. A maximum value for the tube-skin temperature (Tw,all ) 925 °C) is selected. This is a relatively conservative temperature according to the reported values for industrial reformers.2,3,11 Independent of the number of optimization sections (K), the optimal heat-flux profiles that maximize the methane conversion are decreasing functions. For all of the cases analyzed, the optimal solutions indicate that no heat flux should be supplied at the reactor outlet. For K equal to 2 or 3, the slope of the Q(z) profile (absolute value) diminishes toward the reactor outlet. Figure 7 also includes the optimal methane conversion as well as the values of Tw,max, which are lower than the upper limit Tw,all. The addition of more optimization variables does not improve the
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Figure 7. Optimal axial heat-flux and tube-skin temperature profiles, for maximum outlet methane conversion and K ) 1-3. Qt ) 73.09 KW/m2.
methane conversion significantly. The low sensitivity shown by the objective function can be attributed to the proximity of the optimal xCH4L values to the maximum allowable conversion for the given heat duty (xCH4,eq ) 62.7%). For this case, the slight conversion increase obtained for K ) 2 or 3 does not justify the increase in Tw,max (i.e., the tube lifetime decrease). The Tw axial profiles corresponding to the optimal solutions are also displayed in Figure 7. As more linear sections are considered, the maximum observed in the Tw curves shifts to the reactor inlet, according to the higher heat fluxes supplied in the first tube section. b. Minimum Tube-Skin Temperature for a Fixed Methane Conversion and Constant Heat Duty. When the production rate is fixed in the desired level, the heat-flux profile can be manipulated to reduce the tube-skin temperature, i.e., to maximize the tube lifetime. For this optimization objective and specified feed conditions, reactor dimensions, and heat duty (Table 1), the problem becomes
min Tw,max
(20)
s.t. xCH4L ) xCH4req
(21)
Q(z)
∫0L Q(z) dz L
) Qt
Q(z) g 0
(22) (23)
Figure 8 shows the optimal heat-flux profiles for K ) 2 and different values of the required methane conversion. The solutions have different shapes depending on the specified conversion level. As the desired conversion increases toward the equilibrium value (xCH4,eq ) 62.7%), the optimal heat-flux distribution changes from axial profiles with maxima to monotonically decreasing functions. These results are analogous to those shown in Figure 2; in fact, when low conversions are required for a given heat duty, the heat-flux slope has to be positive (e.g., point 4 of Figure 2). It is interesting to note the behavior of the maximum tube-skin temperature (see the table in Figure 8). As the specified conversion increases from 58.7% to 60.7%, the optimal Tw,max
Figure 8. Optimal axial heat-flux profiles (two linear optimization sections) for minimum Tw,max and different specified conversion levels. Qt ) 73.09 KW/m2.
Figure 9. Optimal axial heat-flux profiles (two linear optimization sections) for minimum Tw,max at conditions of constant conversion (xCH4L ) 61.7%) and different heat duties.
decreases by around 56 °C up to a minimum value of 855.9 °C, which is around 10 °C less than the minimum for Tw,max found for a single linear section (point 2 of Figure 2). For higher outlet conversions (e.g., 61.7%) the maximum tube-skin temperature rises up to 878.7 °C. From the results shown in Figure 8, it can be concluded that for a given heat duty there is a fixed conversion level (in this case around 60.7%) that minimizes the maximum tube-skin temperature. If for production reasons the reactor needs to be operated at lower conversions than 60.7%, the heat duty should be reduced to avoid an undesired increase in the value of Tw,max. The influence of the heat duty is presented in Figure 9, for a constant conversion value of 61.7%. When the heat duty is increased up to 4.5% above the reference case (Qt), the shape of the optimal heat-flux distribution changes from decreasing functions to profiles with maxima. Even though the reference case provides the desired conversion with the lowest heat duty (i.e., low fuel-gas consumption), the corresponding value of Tw,max is not the minimum one. Consequently, for a given xCH4L,
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Acknowledgment J.P. is grateful to the Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas (CONICET) of Argentina for financial support during the work. Nomenclature
Figure 10. Influence of the number of linear sections on the optimal axial heat-flux profiles for minimum Tw,max. xCH4L ) 61.7% and Qt ) 73.09 kW/m2.
there is an optimal value of Qt that leads to minimum Tw,max values and a maximum tube lifetime. For a fixed methane conversion level of 61.7%, Figure 10 shows the optimal solutions obtained for 1-3 linear sections of Q(z). As the number of optimization variables increases, lower values of Tw,max are found. In fact, for K ) 3 the maximum value of Tw(z) is reduced in 10 °C with respect to the simplest case of K ) 1. If the number of linear sections is increased above K ) 3, the value of the objective function would improve gradually and the optimal heat-flux profile would be more precise. The convenience of using a higher number of optimization variables (or other optimization methods) should be evaluated considering the operating feasibility to obtain the desired profile in industry and the magnitude of the achievable improvements in the objective function. Conclusions When the production rate is desired to be as maximum as possible, a decreasing profile appears as the more efficient heat distribution along the tube length. For those operations where the reactor conversion (i.e., the methane slip) has to be maintained in a specified value, the manipulation of the heat-flux profiles may substantially reduce the maximum tube-skin temperature, allowing an important extension of the tube lifetime. Even though the optimal heat-flux profiles may not be exactly reproduced in the industrial practice, the knowledge of their shapes offers interesting information to improve the performance of existing units. The feasibility of practical implementation of axial heat-flux distributions close to the optimal ones appears to be easier to attain in side-fired reformers, by means of an adequate setting of the fuel-gas consumption at the different rows of burners. To make use of the optimization results, if a reliable mathematical model for the radiant chamber is not available, a trial and error procedure should be used to adjust the burners. To attain the desired (optimal) tube-skin temperature profile, accurate measurements of the tube-skin temperatures at different axial positions would be necessary.
Ak ) slope of the linear heat-flux profile at the subinterval k (kW/m2‚m) Bk ) zero ordinate of the linear heat-flux profile at the subinterval k (kW/m2) cp ) heat capacity of the process gas (kJ/kg‚K) dp ) equivalent diameter of the catalyst pellet (momentum equation) (m) dti ) inner diameter of the reactor tube (m) dto ) outer diameter of the reactor tube (m) Dje ) effective diffusivity of component j (m2/s) f ) friction factor Ft ) total molar flow rate (kmol/s) Fj ) molar flow rate of component j (kmol/s) g ) acceleration of gravity (m/s2) h ) height of the catalyst particle (m) ∆Hi ) heat of reaction i, i ) 1-3 (kJ/kmol) K ) number of linear sections of the Q(z) profile L ) reactor length (m) pj ) partial pressure of component j (gas phase) (bar) ps,j ) partial pressure of component j inside the catalyst particle (bar) pt ) total pressure (bar) Q(z) ) heat flux per unit area at axial position z (kW/m2) Qt ) heat duty per unit area (kW/m2) r ) radial coordinate of the catalyst particle (m) ri ) rate of reaction i, i ) 1-3 (kmol/kgcat‚s) ris ) rate of reaction i at the surface of the catalyst, i ) 1-3 (kmol/kgcat‚s) rin ) holes radius of the catalyst particle (m) req ) outer radius of the equivalent particle (m) rout ) outer radius of the original catalyst particle (m) r* ) dimensionless radius of the equivalent particle [(r rin)/(req - rin)] R ) universal gas constant (kJ/kmol‚K) Teq ) temperature at equilibrium conditions (K) Tg ) process gas temperature (K) Tw ) external tube-skin temperature (K) Tw,all ) maximum allowable tube-skin temperature (K) Tw,max ) maximum value of the Tw axial profile (K) us ) superficial velocity (mf3/mr2‚s) U ) overall heat-transfer coefficient (kW/m2‚K) V ) volume of the catalyst particle (m3) xCH4 ) methane conversion [(FCH40 - FCH4)/FCH40] xCH4,eq ) equilibrium conversion of CH4 xCO2 ) conversion of CH4 into CO2 [(FCO2 - FCO20)/FCH40] yj ) molar fraction of component j z ) axial coordinate (m) z* ) dimensionless axial coordinate (m) [z/L] Greek Letters Ri ) convective heat-transfer coefficient (kW/m2‚K) ηi ) effectiveness factor for reaction i, i ) 1-3 λst ) thermal conductivity of the tube metal (kW/m‚K) FB ) bed density (kgcat/mr3) Fg ) gas density (kg/mf3) Fp ) density of the catalyst particle (kgcat/mcat3) Ω ) cross section of the reactor (m2) Subscript k ) at the kth subinterval of Q(z) Superscripts L ) at the reactor outlet
Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5221 req ) required value 0 ) at the reactor inlet s ) at the catalyst particle surface
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(10) Elnashaie, S. S. E. H.; Adris, A. M.; Al-Ubaid, A. S.; Soliman, M. A. On the Nonmonotonic Behaviour of Methane Steam Reforming Kinetics. Chem. Eng. Sci. 1990, 45, 491. (11) Rostrup-Nielsen, J. R. Production of Synthesis Gas. Catal. Today 1993, 18, 305. (12) Plehiers, P. M.; Froment, G. F. Coupled Simulation of Heat Transfer and Reaction in a Steam Reforming Furnace. Chem. Eng. Technol. 1989, 12, 20. (13) Xu, J.; Froment, G. F. Methane Steam Reforming: II. Diffusional Limitations and Reactor Simulation. AIChE J. 1989, 35, 97. (14) Xu, J.; Froment, G. F. Methane Steam Reforming, Methanation and Water-Gas Shift: I. Intrinsic Kinetics. AIChE J. 1989, 35, 88. (15) Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design; John Wiley: New York, 1979. (16) De Wasch, A. P.; Froment, G. F. Heat Transfer in Packed Beds. Chem. Eng. Sci. 1972, 27, 567. (17) Ergun, S. Fluid Flow through Packed Columns. Chem. Eng. Prog. 1952, 48 (2), 89. (18) Topsøe Steam Reforming Catalysts R-67 Series Catalogue. HALDOR TOPSØE A/S: Lynbgy, Denmark, 1998. (19) Pin˜a, J.; Borio, D. O.; Bucala´, V.; Cortinez, V. Analysis of Equivalent Models to evaluate Diffusional Problems in Catalyst Particles of Complex Geometry. In preparation, 2000.
Received for review December 6, 2000 Revised manuscript received May 3, 2001 Accepted May 11, 2001 IE001065D
