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Modeling and Simulation of a Fixed-Bed Pilot-Plant Hydrotreater Jinwen Chen* and Zbigniew Ring National Centre for Upgrading Technology (NCUT), One Oil Patch Drive, Devon, Alberta, Canada T9G 1A8
Tadeusz Dabros Advanced Separation Technologies (AST), One Oil Patch Drive, Devon, Alberta, Canada T9G 1A8
A pseudo-homogeneous two-dimensional reactor model was proposed to describe the dynamic and steady-state behaviors of a fixed-bed pilot-plant hydrotreater used for the hydrotreatment of a partially stabilized light-coker naphtha. The model accounted for not only the hydrogenation reactions and the mass and heat transfer in axial and radial directions in the catalyst bed, but also the heat conduction in the thermowell of the reactor. Steady-state experiments were conducted at various operating conditions. Dynamic experiments were performed by inducing dynamic changes in the hydrogen rate and monitoring the temperature transition in the reactor bed. The simulated temperature profiles in the thermowell were in quite good agreement with those observed in the experiments. Simulation results indicated that the temperature difference between the thermowell and the catalyst bed could be as high as 60 K, which, if ignored, might cause significant errors in the interpretation of the pilot-plant data. Dynamic simulations quantitatively predicted the temperature transition over time when the hydrogen rate was suddenly decreased or increased by 8-15%. 1. Introduction Hydrotreatment is an important process in bitumen and heavy oil upgrading, as well as the subsequent refining of the upgraded streams. Virtually every refinery stream is hydrotreated at one point or another. Each type of stream creates its own scale-up challenges. Hydrotreatment is usually conducted in a fixed-bed catalytic reactor, either in single gas-phase flow or twophase (gas and liquid) trickle flow. Heavier streams usually undergo trickle-flow operation, which gives rise to incomplete catalyst wetting in short experimental reactors operated at commercial space velocities. This tremendously complicates any scale-up considerations. Lighter streams often undergo a single gas phase flow through a catalytic fixed bed that is relatively easier to scale up. However, the extensive heat release associated with hydrotreatment of highly olefinic light streams can complicate reactor scale-up to a significant degree. With the development of computer technology and modern computation methods, modeling and simulation are more commonly applied in the design, performance analysis, optimization, and scale-up of hydrotreatment reactors.1,2 Mathematical models employed in the simulation of fixed-bed reactors have been described in the literature.3-5 Numerous papers have been published on steady-state modeling and simulation of fixed-bed catalytic reactors with either pseudo-homogeneous models or heterogeneous models.6-8 However, studies on dynamic (unsteady-state) modeling and simulation of fixed-bed reactors are reported less frequently in the open literature. The transient behavior of reactors is especially important for reactor control during start-up, during shutdown, or during an upset. van Doesburg and de Jong * Author to whom correspondence should be addressed. Tel: 780 987-8763. Fax: 780 987-5349. E-mail: jichen@ nrcan.gc.ca.
investigated the transient behavior of an adiabatic fixedbed methanator, both experimentally and theoretically.9,10 Pinjala et al. studied the “wrong-way” behavior of fixed-bed reactors when a sudden change in the reactor inlet temperature is imposed.11 Kobyakov and Mikhailov conducted a simulation of the start-up for a fixed catalytic bed reactor.12 Kvamsdal et al. reported a dynamic simulation and optimization of a catalytic steam reformer.13 However, we are unaware of any publications on the steady-state and dynamic simulation of fixed-bed hydrotreaters. This paper reports experimental findings and simulation results on the steady-state and dynamic behavior of a fixed-bed pilot-plant hydrotreater. The supporting experimental work was carried out at the National Centre for Upgrading Technology (NCUT), Devon, Alberta, Canada. The reactor model developed in this study will be used to facilitate the scale-up of these experimental results, to aid the design and optimization of a commercial unit, and to properly design any future experiments of a similar nature. 2. Mathematical Model In this paper, we consider a pseudo-homogeneous twodimensional fixed-bed hydrotreater model with axial and radial dispersion of heat and mass. We refrain from considering local microscale temperature and concentration profiles within catalyst particles because the kinetic parameters that have been used in this model were extracted from experimental data acquired under similar operating conditions and with the same catalyst bed as part of this overall study. Therefore, the intraparticle effects are accounted for in an indirect way, and the temperature and concentration in the catalyst bed essentially represent locally averaged values. We have also included the axial dispersion terms in the model because the heat dispersion in the axial direction was significant as a result of the high temperature gradient
10.1021/ie001083v CCC: $20.00 Published 2001 by the American Chemical Society Published on Web 06/19/2001
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and the relatively high thermal conductivity of the diluted catalyst bed. As discussed by Ring and Chen14 in detail in a separate paper, it is also important to take into account the thermowell situated in the center of the reactor. This thermowell housed an array of thermocouples used for the measurement of the axial temperature profile in the catalyst bed. It should be noted that the axial and radial thermal conduction in the 1-cm-thick stainless steel reactor wall and the axial and radial thermal conduction in the insulation of the reactor furnace have also been taken into account in our model. The short relaxation time for temperature variations in the reactor and at the reactor wall, combined with the long relaxation time in the insulation, results in a nonadiabatic, variable-wall-temperature operation of the reactor. During experiments, the axial temperature profiles in the thermowell of the reactor and at the outer wall of the reactor were monitored at the same time. In separate simulations, it was shown that the radial temperature gradient within the reactor wall was so insignificant (within 1 K) that a flat radial temperature profile could be assumed. Therefore, it is not necessary here to include the heat balance equation for the reactor wall. In addition, the heat balance equation for the insulation of the reactor furnace is not necessarily included because it is beyond the boundary (reactor wall) of the problem. The heat and mass transfer in the reactor are described with the following set of partial differential equations (PDEs).
z ) Z:
∂T ∂2 T 1 ∂ ∂T r ) λw 2 + λw ∂t r ∂r ∂r ∂z t > 0, 0 < r < Rw, 0 < z < Z (1)
(
2
)
2
∂ CA ∂CA ∂ CA 1 ∂CA ∂CA - u0 + ) BDea + BDer 2 ∂t ∂z r ∂r ∂z ∂r2 FBrA t > 0, Rw < r < Ri, 0 < z < Z (2) Heat balance in the catalyst bed cpB
(
)
∂T ∂2 T ∂T ∂2T 1 ∂T + FBrA ) λea 2 - u0Ffcpf + λer 2 + ∂t ∂z r ∂r ∂z ∂r (-∆Hr) t > 0, Rw < r < Ri, 0 < z < Z (3)
The initial conditions are
t ) 0, CA ) CA0, T ) T0 The boundary conditions are
z ) 0:
(∂T∂z ) |
)
well z)0
( ) | ∂CA ∂z
bed
(∂T∂z ) |
hair (T - Troom) λw end1,well (thermowell)
u0 ) (C - CA0) z)0 BDea A (catalyst bed)
bed z)0
)
u0Ffcpf (T - T0) λea
(catalyst bed)
well z)Z
∂CA ∂z
(∂T∂z ) | (∂T∂r ) | (∂T∂r ) |
r ) 0: r ) Rw:
well r)0
(catalyst bed) )0
(catalyst bed)
)0
(thermowell)
)
well r)Rw
( ) | ∂CA ∂r
hair (T - Troom) λw end2,well (thermowell)
)0
bed z)Z
bed z)Z
bed r)Rw
hw (T - TRw,well) λw Rw,bed (thermowell)
)0 (catalyst bed)
( ) |
hw ∂T ) (TRw,bed - TRw,well) ∂r bed r)Rw λer (catalyst bed)
r ) Ri:
( ) | ( ) | ∂CA ∂r
bed r)Ri
)0
(catalyst bed)
hw ∂T ) - (TRi,bed - TRi,wall) ∂r bed r)Ri λer (catalyst bed)
( )
Mass balance in the catalyst bed
)-
( ) |
Heat balance in the thermowell cpwFw
(∂T∂z ) |
It should be noted that the boundary condition
z ) 0:
( ) | ∂CA ∂z
bed z)0
)
u0 (C - CA0) (catalyst bed) BDea A
can be simplified to
z ) 0: CA ) CA0 (catalyst bed) This is true because the axial dispersion of mass is relatively small and the concentration gradient at the reactor inlet is quite flat. The configuration of the catalyst bed reactor is shown in Figure 1. An explanation of all of the symbols in the above equations is given in the Nomenclature part of this paper. The three PDEs in dimensionless form, together with the initial and boundary conditions, were transformed into ordinary differential equations (ODEs) by discretizition in the axial and radial directions using a finite difference method.15,16 The backward finite difference method was used because of the stiffness of the PDEs in the axial direction. The ODEs were then solved with the integration package LSODA.17 3. Experimental Section The experiments were conducted in one of the NCUTs pilot-plant reactors (a detailed description of the unit will be given in a separate paper). The system can be operated at a maximum temperature of 723 K and a maximum pressure of 180 atm. Its feed mass flow rate is 100-2000 g/h, and its gas volumetric flow rate is
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The feed and liquid products were sampled and analyzed to determine the mean molecular weight, bromine number, simulated distillation (SimDis), and sulfur and nitrogen contents. In total, 10 steady-state experiments were conducted under various operating conditions to cover a wide range of reactor inlet temperatures and naphtha and hydrogen flow rates. Two back-to-back 3-h mass balance runs were performed for each condition. About 2.0 kg of liquid product was collected in each run. Data acquired under steady-state conditions were used for kinetics evaluations. The ranges of experimental conditions used are summarized below:
Naphtha rate (g/h) ) 564-760 Hydrogen rate (NL/h) ) 832-1110 Pressure (atm) ) 36 Reactor inlet temperature (K) ) 505-535 Figure 1. Configuration of the catalyst bed reactor.
0-3000 NL/h (normal liters per hour, under normal conditions) in either up-flow or down-flow mode. The reactor tube (2.1 m long with an i.d. of 2.54 cm) was heated by an eight-zone electric furnace that is capable of maintaining either an imposed or an adiabatic temperature profile in the reactor. A 6-mm-o.d. thermowell housing 12 thermocouples was fitted in the center of the reactor tube. Seven of the 12 thermocouples were equally spaced in the axial direction to cover the distance occupied by the catalytic bed and provide a detailed axial temperature profile. The other 5 were located below and above the catalyst bed for reactor controlling and monitoring purposes. Meanwhile, another 12 “skin” thermocouples were installed on the outer wall of the reactor at the same axial positions as those in the thermowell. The thermowell and skin thermocouples provided the axial temperature profiles in the center and at the outer wall of the reactor. The 15-cm-long catalyst bed, packed with a commercial hydrotreating catalyst (Mo-Ni/Al2O3 extrudates, 1.6 mm in diameter and, on average, 4.7 mm in length) was diluted with 0.2-mm glass beads in a 1:1 volume ratio. The dilution procedure was developed on the basis of thorough tests of uniformity of the resulting catalyst bed composition in a separate study. The diluted bed provided a uniform gas flow (close to plug flow) but resulted in a higher thermal conductivity compared to that of the undiluted bed. Above and below the catalyst bed, 0.5-mm glass beads were packed to provide a uniform gas distribution in the reactor cross section and to position the catalyst bed in the middle of the reactor tube. Prior to the hydrotreatment experiments, the fresh catalyst was conditioned using NCUTs standard sulfiding and de-edging techniques. The partially stabilized naphtha (diolefins removed) was introduced into the catalyst bed at a fixed inlet temperature and pressure. The effluent from the reactor was then introduced into a high-pressure phase separator (HPPS) where the light ends and unreacted hydrogen were separated from the hydrotreated naphtha. The liquid from the HPPS went into a stabilizer column to further refine the separation between the gas and liquid products collected in the back-end of the unit. Combined gases from the high-pressure separator and stabilizer were metered and analyzed by an on-line GC analyzer.
During experiments, it was found that the reactor temperature was very sensitive to changes in the hydrogen volumetric flow rate. Therefore, the dynamic experiments were performed starting at a steady state and inducing transients by a sudden increase or decrease in the hydrogen volumetric flow rate. The transition of the temperature in the thermowell was then monitored over time. 4. Results and Discussion 4.1. Kinetics Evaluation and Steady-State Simulation. Information on the kinetics of the hydrogenation reactions is essential to this reactor simulation work. The kinetic parameters were obtained by fitting experimental data with a steady-state model corresponding to that represented by eqs 1-3. Details of this parameter evaluation work are presented elsewhere by Ring and Chen.14 The reaction system considered in the current study involves a large number of olefin hydrogenation, hydrodesulfurization (HDS), hydrodenitrogenation (HDN), and cracking reactions. It was impractical and unnecessary to determine the apparent kinetics for each individual reaction. For simplicity, we assumed that the heat effects related to all of the chemical reactions were associated with a single olefin hydrogenation reaction and that the kinetics of this reaction was determined in terms of the fraction of bromine number reduction from the feed level. This lumped reaction kinetics equation was assumed to have the form
( )
rA ) k0 exp -
Ea C nC 0.85 RgT A H
(4)
The three kinetics parameters in eq 4 (k0, Ea, and n), together with the corresponding lumped heat of reaction (∆Hr), were estimated by fitting the computed olefin conversion and axial temperature profiles in the thermowell with those measured experimentally. Olefin conversion (reduction in bromine number) and axial temperature profiles in the thermowell and at the reactor wall were experimentally measured under various operating conditions. The solution of the reactor model (eqs 1-3) at steady state was obtained in two different ways. One involved running the dynamic model (for a long time) to get the steady-state solution, whereas the other one involved solving the correspond-
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Figure 2. Axial temperature profiles in the thermowell, both experimentally measured and simulated (naphtha rate ) 619.2 g/h, hydrogen rate ) 1110 NL/h, inlet temperature ) 523.3 K).
ing steady-state model. Identical solutions were obtained in the two cases. The resulting kinetics equation was
(
)
-81 000 rA ) 4.128 exp CA1.12 CH0.85 RgT
(5)
The lumped heat of reaction was found to be 101.1 kJ/mol. This value is slightly lower than the average heat of reaction for olefin hydrogenation of around 120 kJ/mol, as reported by Jaffe.18 The possible reason for this difference is that the current reaction system includes some endothermic reactions (e.g., cracking). This hypothesis was supported by on-line GC analysis of the noncondensing gas and by comparing the SimDis results before and after hydrotreatment of the naphtha. It should be mentioned that, in all of the experiments (steady-state and dynamic), the pressure drop across the reactor was negligible, and the average pressure in the catalyst bed was maintained at 36 atm. In Figure 2, the scattered points represent the temperature measured in the thermowell of the reactor at steady state, while the solid line represents the temperature obtained by model fitting. Reasonable agreement is achieved, as is evident in the figure. The operating conditions were a naphtha mass flow rate of 619.2 g/h, a hydrogen volumetric flow rate of 1110 NL/h, and an inlet temperature of 523.3 K. A very sharp axial temperature gradient of about 150 K was observed over the bed length of only 15 cm, indicating the highly exothermic nature of the reaction system. Figures 3 and 4 show the contour plots of olefin conversion and temperature, respectively, in the catalyst bed under the conditions discussed above. Note that the central part of the contour plots represents the thermowell where there is no conversion distribution (white area in Figure 3) but there is a temperature distribution (see Figure 4). It is evident from these plots that, in the region of the catalyst bed extending between axial positions 0 and 12 cm (zone 1), both axial and radial temperature conversion profiles are relatively flat. In the region extending between axial positions 12 and 15 cm (zone 2), these profiles are very steep. Therefore, we can conclude that most of the conversion takes place in zone 2. In zone 1 (see Figure 4), the temperature in the bed is slightly lower than the temperature in the thermowell, whereas in zone 2, the temperature in the bed is much higher than that in the
Figure 3. Conversion contour plot in the reactor (naphtha rate ) 619.2 g/h, hydrogen rate ) 1110 NL/h, inlet temperature ) 523.3 K).
Figure 4. Temperature contour plot in the reactor (naphtha rate ) 619.2 g/h, hydrogen rate ) 1110 NL/h, inlet temperature ) 523.3 K).
themowell. Figure 5 clearly shows the difference between the axial temperature profiles in the thermowell and in the catalyst bed at a dimensionless radial position of r/Ri ) 0.46 where the maximum radial temperature is observed. At the outlet of the catalyst bed, the temperature difference between the themowell and the bed is about 60 K. The reason for this difference is that the thermowell, a stainless steel tube fully filled with thermocouple wires, has a much higher heat conductivity than the catalyst bed. Heat can be much more easily transferred from zone 2 (higher temperature) to zone 1 (lower temperature) along the thermowell than along the catalyst bed. Similarly, in zone 1 the temperature in the bed is lower than the temperature of the reactor wall (1-cm-thick stainless steel), whereas in zone 2, the relationship is reversed.
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Figure 5. Simulated axial temperature profiles in the thermowell and in the catalyst bed at r/Ri ) 0.46 (naphtha rate ) 619.2 g/h, hydrogen rate ) 1110 NL/h, inlet temperature ) 523.3 K).
The back transfer of heat to the inlet of the reactor along the thermowell and reactor wall enhances the heat dispersion in the catalyst bed, leading to a flattened axial temperature profile. This effect is quite pronounced in the pilot-plant reactor because of the relatively small diameter of the catalyst bed. In other words, a much higher axial temperature gradient would be expected for a commercial adiabatic reactor without a thermowell. However, there are several other factors that affect the axial temperature profile in the bed, and all of these factors have to be accounted for in the comparison between pilot-plant and commercial reactors and in the scaling-up of these experimental data. A preliminary scale-up study carried out using the model developed in this work indicated that, in a commercial adiabatic reactor operated under similar conditions, the maximum temperature can become impractically high, raising concerns about the safety of such operations. Therefore, it is impractical to use a single-bed hydrotreater to accommodate this process. A three-bed hydrotreater with interstage quenching would have to be proposed for such a commercial hydrotreater. More detailed analysis and discussion of scale-up challenges for this process is published elsewhere.19 The radial conversion and temperature profiles at the outlet of the catalyst bed are shown in Figure 6. The maximum radial temperature difference is 76 K, as indicated in the figure. Such a large radial temperature gradient was also observed experimentally. As expected, a larger radial temperature gradient led to a larger radial conversion gradient, as shown in the figure. To verify the reactor model, a separate simulation was performed with the kinetic parameters obtained above. A comparison was made between the simulated axial temperature profile in the thermowell of the reactor and the measured one, which was not used in the estimation of the kinetic parameters. Again, reasonable agreement was attained. 4.2. Dynamic Simulation. As mentioned above, the reactor temperature was very sensitive to changes in the hydrogen volumetric flow rate. Therefore, for the purpose of this experimental work, it was practical to induce transients by bringing the system close to a steady state and providing a stepwise increase or decrease in the hydrogen flow rate. The scattered points in Figure 7 show the observed change in temperature
Figure 6. Radial conversion and temperature profiles at the outlet of the catalyst bed (naphtha rate ) 619.2 g/h, hydrogen rate ) 1110 NL/h, inlet temperature ) 523.3 K).
Figure 7. Dynamic simulation: temperature change with time.
(in the thermowell) with time, starting from steady state, over a time period of about 212 min. During this time period, the hydrogen volumetric flow rate was first decreased from 1090 to 1000 NL/h (at 32 min) and then increased from 1000 to 1150 NL/h (at 110 min). It was observed that the temperature change at the inlet of the catalyst bed was relatively small, whereas the temperature change at the outlet of the catalyst bed was quite significant. For example, when the hydrogen volumetric flow rate decreased from 1090 to 1000 NL/h (8.3% decrease), the inlet temperature increased by only 3.5 K, whereas the outlet temperature increased by nearly 40 K over a time period of 78 min. When the hydrogen volumetric flow rate was increased from 1000 to 1150 NL/h (15% increase), the inlet temperature decreased by only 9 K, whereas the outlet temperature decreased by over 60 K during a time period of 102 min. The decrease in the hydrogen volumetric flow rate caused the increase in the catalyst bed temperature in three ways: (1) it decreased the amount of heat carried out of the reactor by the gas, (2) it increased the heat generation rate (reaction rate) by increasing the olefin concentration, and (3) it increased the olefin conversion by increasing the residence time of the reactants in the reactor. Conversely, the increase in the hydrogen volumetric flow rate caused a decrease in catalyst bed
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hydrogen rate was suddenly decreased or increased by 8-15%. Reasonable agreement was achieved between the model predictions and the experimental data. Acknowledgment The authors are indebted to Cecilia Sin and Tom Crothers for their help in conducting the pilot-plant experiments. Partial funding for NCUT has been provided by the Canadian Program for Energy Research and Development (PERD), the Alberta Research Council, and the Alberta Energy Research Institute. Figure 8. Variation with time and axial position of the temperature in the thermowell.
temperature. It should also be noted in Figure 7 that the temperature increase or decrease at the very outlet of the bed (z )15 cm) was less than those at the axial positions of 10 and 12.5 cm. It is likely that the increased reaction rate led to a slight shift of the highreaction-rate zone toward to the inlet of the bed. The solid lines in Figure 7 represent the simulation results under the same conditions and over the same time period. Reasonable agreement is achieved, except for the axial positions of 10 and 15 cm from the inlet of the bed. This indicates that the reactor model developed in this study can be used to simulate not only the steady-state operation but also the thermal dynamic behavior of the hydrotreater. The results of the dynamic simulation might have been affected by inaccuracies in the estimation of some model parameters such as the heat capacity of the catalyst bed (catalyst diluted with glass beads). Further refinement of those parameters is expected to improve the model predictions. However, because the objective of this study required us to carry out accurate steady-state simulations, no further effort was made to rectify the dynamic simulation results at this time. The dynamic model was used here to show the consistency between the dynamic behavior of the reactor and the model predicions. Figure 8 shows the simulated dynamic transition of the temperature in the thermowell under the conditions discussed above. As seen in the figure, the temperature in the first half of the catalyst bed (close to the inlet) does not change much with time, whereas the temperature in the second half of the catalyst bed (close to the outlet) does change significantly with time. This dynamic characteristic can be used in on-line process control in commercial hydrotreaters. 5. Summary A mathematical reactor model has been developed to numerically simulate the dynamic and steady-state behavior of a pilot-plant hydrotreater. The model took the heat transfer in the thermowell of the reactor into account. Steady-state simulation results, under the different operating conditions investigated in this study, were in reasonable agreement with those observed in the pilot-plant experiments. Simulation results indicated that the temperature difference between the thermowell and the bed could be as high as 60 K. In this case, the temperature measured in the thermowell cannot represent the temperature in the bed. Therefore, caution must be exercised in the interpretation of the pilot-plant data. Dynamic simulations quantitatively described the temperature transition with time when
Nomenclature CA ) concentration of the key component, mol/m3 CA0 ) inlet concentration of the key component, mol/m3 CH ) concentration of hydrogen, mol/m3 CH0 ) inlet concentration of hydrogen, mol/m3 cpf ) heat capacity of the fluid, J/(kg K) cpw ) heat capacity of the thermowell (stainless steel), J/(kg K) Dea ) axial dispersion coefficient in the bed, m2/s Der ) radial dispersion coefficient in the bed, m2/s Ea ) activation energy, J/mol ∆Hr ) heat of reaction, J/mol hair ) thermowell-atmosphere heat transfer coefficient, J/(m2 K) hw ) catalyst bed to wall or catalyst bed to thermowell heat transfer coefficient, J/(m2 K) k0 ) frequency factor of reaction rate constant, (m3/ mol)0.85+n [mol/(kgcat s)] n ) reaction order Rg ) gas constant, J/(mol K) Ri ) inner radius of catalyst bed, m Rw ) outer radius of thermowell tube, m r ) radial coordinate, m rA ) reaction rate, mol/(kgcat s) T ) temperature in the bed, K Tend1,well, Tend2,well ) temperatures at the ends of the thermowell, K T0 ) inlet temperature, K Ti,exp ) experimentally measured temperature, K Ti,simu ) simulated temperature, K Troom ) room temperature, K TRi,bed ) bed temperature at the reactor wall, K TRi,wall ) wall temperature, K TRw,bed ) bed temperature contacting outer surface of the thermowell, K TRw,well ) temperature at the outer surface of the thermowell, K t ) time, s u0 ) superficial gas velocity, m/s Z ) height of catalyst bed, m z ) axial coordinate, m B ) bed voidage (space between particles) FB ) density of the bed, kg/m3 Ff ) density of the fluid, kg/m3 Fw ) density of thermowell (stainless steel), kg/m3 λea ) axial effective conductivity, J/(m K) λer ) radial effective conductivity, J/(m K) λw ) thermal conductivity of the thermowell, J/(m K)
Literature Cited (1) Korsten, H.; Hoffmann, U. Three-Phase Reactor Model for Hydrotreating in Pilot Trickle-Bed Reactors. AIChE J. 1996, 42 (5), 1350. (2) Sanchez, M.; Thah, Y.; Dassori, C. G. Multiphase Hydrotreating Reactor Modeling. Presented at the AIChE Spring Meeting, March 19-24, 1995, Houston, TX.
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(3) Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design, 2nd ed.; Wiley: New York, 1990. (4) Schnitzlein, K.; Hofmann, H. An Alternative Model for Catalytic Fixed Bed Reactors. Chem. Eng. Sci. 1987, 42, 2569. (5) Ramachandran, P. A.; Chaudhari, R. V. Three Phase Catalytic Reactors; Gordon and Breach Science Publishers Inc.: London, 1983. (6) Szukiewicz, M.; Kaczmarski, K.; Petrus, R. Modeling of Fixed-Bed Reactor: Two Models of Industrial Reactor for Selective Hydrogenation of Acetylene. Chem. Eng. Sci. 1998, 53, 149. (7) Derkx, O. R.; Dixon, A. G. Effects of Wall Nusselt Number on the Simulation of Catalytic Fixed Bed Reactors. Catal. Today 1997, 35, 435. (8) Papageorgious, J. N.; Froment, G. F. Simulation Models Accounting for Radial Voidage Profiles in Fixed-Bed Reactors. Chem. Eng. Sci. 1995, 50, 3043. (9) van Doesburg, H.; de Jong, W. A. Transient Behaviour of an Adiabatic Fixed Bed Methanator I. Chem Eng. Sci. 1976, 31, 45. (10) van Doesburg, H.; de Jong, W. A. Transient Behaviour of An Adiabatic Fixed Bed Methanator II. Chem Eng. Sci. 1976, 31, 45. (11) Pinjala, V.; Chen, Y. C.; Luss, D. Wrong-way Behavior of Packed-Bed Reactors: II. Impact of Thermal Dispersion. AIChE J. 1988, 34 (10), 1663. (12) Kobyakov A. I.; Mikhailov G. V. Mathematical Simulation of Startup for a Reactor with a Fixed Catalyst Bed. Theor. Found.
Chem. Eng. 1990, 23, 499(English translation of Theoreticheskie Osnovy Khimicheskoi Tekhnologii). (13) Kvamsdal, H. M.; Svendsen, H. F.; Hertzberg, T.; Olsvik, O. Dynamic Simulation and Optimization of a Catalytic Steam Reformer. Chem Eng. Sci. 1999, 54, 2697. (14) Ring, Z.; Chen, J. An Efficient Method for Parameter Estimation and Reactor Simulation with Fortran DLL and Microsoft Excel. Comput. Chem. Eng. 2000, manuscript submitted. (15) James, M. L.; Smith, G. M.; Wolford, J. C. Appled Numerical Methods for Digital Computation; Harper & Row Publishers: New York, 1985. (16) Lapidus, L.; Pinder, G. F. Numerical Solution of Partial Differential Equations in Science and Engineering; John Wiley & Sons Inc.: New York, 1982. (17) Petzold, L.; Hindmarsh, A. LSODAsA Package for Solving Ordinary Differential Equations; Collected in Netlib (www.netlib.org), 1997. (18) Jaffe, S. B. Kinetics of Heat Release in Petroleum Hydrogenation. Ind. Eng. Chem. Process Des. Dev. 1974, 13 (1), 34. (19) Ring, Z.; Chen, J. Scaling-up Study of Light-Coker Naphtha Hydrotreating, manuscript in preparation.
Received for review December 12, 2000 Revised manuscript received April 23, 2001 Accepted May 9, 2001 IE001083V
