Closed-form solution to the problem of reaction with fixed bed adsorption using delay-differential equations

Delay-differential equations (DDEs) can describe many chemical engineering models. However, the formalism of DDEs appears to be underutilized in chemical engineering. We have recast the canonical chemical engineering problem of batch reaction with fixed bed sorption into the form of a delay-differential equation, obtaining a more intuitive model and a simpler closed form solution than those previously reported. Considerable model reduction is possible through the use of DDE formalism when one considers that chemical processes can be partially represented by networks of transportation and state delays. Analytical and numerical methods for solution, as well as controllability and stability theory for systems of DDEs, are nearly as rich and developed as those for ordinary differential equations. Significant progress thus may be possible in areas such as the modeling, synthesis, and control of chemical processes, if the governing equations can be expressed in the form of delay-differential equations.     

Non-isothermal lipase-catalyzed kinetic resolution in a packed bed reactor: Modeling, simulation and miniplant studies

A rigorous model is developed for the exothermic kinetic resolution of 1-methoxy-2-propanol with vinyl acetate catalyzed by immobilized Candida antarctica lipase B in a packed bed reactor. The non-isothermal two-dimensional heterogeneous model takes into account the coupled mass and energy balances, the uneven flow distribution and irreversible ping-pong bi-bi kinetics with competitive substrate inhibition by both enantiomers. This model is based on kinetic parameters, which were estimated in earlier work. The model simulation is validated with experimental results obtained in a fully automated modular miniplant and is shown to be capable of predicting the key parameters needed for process design of a kinetic resolution, the enantiomeric excess and the extent of conversion at a given superficial velocity.     

Flow distribution and velocity measurement in a radial flow fixed bed reactor using electrical resistance tomography

Electrical resistance tomography is a relatively simple and inexpensive technique for imaging electrically conducting systems. It has been applied to visualise the flow pattern and distribution inside a radial flow packed bed of novel design for improving reactor performance with lower pressure drop. The density of information yielded by electrical tomography is suitable for validation of Computational fluid dynamics. Sets of tomographic images representing slices through a packed bed have been obtained for a 8-plane × 16-electrode sensor configuration which produces of the order 10³ conductivity measurements in three-dimensions. Pulse injections of high conductivity tracer, both uniformly in the feed and localised, can be imaged as multiple tomographic images or 3D solid-body images, revealing the internal flow pattern. Differentiation of the motion of the tracer peak conductivity within pixels in the sensing planes and between the planes allows the local flow velocities and directions to be determined. This quantifies the flow pattern for uniformity and radial distributive properties.     

Effect of recycling the unconverted residue on a hydrocracking catalyst operating in an ebullated bed reactor

The effect of recycling the unconverted bottom on catalyst deactivation as a way to improve the hydrocracking conversion of heavy oil was analyzed using the experimental information obtained in a steady-state ebullated bed reactor. The recycle contained different amounts of partially converted (aged) material. Four sets of experiments were performed to demonstrate that after five passes through the reactor, the reactivity of the unconverted material decreased by 15% and its impact on catalyst deactivation increased by 30%. The results indicated that the higher the conversion, the lower is the reactivity and the higher is the catalyst deactivation. The production of an insoluble and refractory to convert material imposes a limit on the recycling benefit.     

Estimation of catalytic activity using an unscented Kalman filtering in condensation reaction

The catalytic activity of cation exchange resins will be continuously reduced with its use time in a condensation reaction for bisphenol A (BPA). For online estimation of the catalytic activity, a catalytic deactivation model is studied for a production plant of BPA, state equation and observation equation are proposed based on the axial temperature distribution of the reactor and the acetone concentration at reactor entrance. A hybrid model of state equation is constructed for improving estimation precision. The unknown parameters in observation equation are calculated with sample data. The unscented Kalman filtering algorithm is then used for on-line estimation of the catalytic activity. The simulation results show that this hybrid model has higher estimation accuracy than the mechanism model and the model is effective for production process of BPA.     

Dynamic modelling of catalytic liquid-phase reactions in fixed beds—Kinetics and catalyst deactivation in the recovery of anthraquinones

Dynamic modelling of catalytic fixed-bed reactors with liquid-phase feed is of crucial importance, since catalyst deactivation often plays a central role in reaction engineering. General dynamic modelling of liquid-phase fixed beds was considered, including complex reaction kinetics and catalyst deactivation. The modelling concept was applied on a catalytic liquid-phase conversion reaction. The model was tested with pilot-plant data and showed a good predictivibility. The model can be used to optimize the production life cycles of fixed beds with catalyst deactivation.     

Study of gas cavity size hysteresis in a packed bed using DEM

When a high velocity gas jet is introduced into a packed bed a cavity is formed. The size of the cavity shows hysteresis on increasing and decreasing gas flow rates. This hysteresis leads to different cavity sizes at same gas flow rate depending on the bed history. The size of cavity affects the gas flow profiles in the packed bed. In this study the cavity size hysteresis phenomenon has been modeled using discrete element method along with turbulent gas flow. A reasonable agreement has been found between computed and experimental results on cavity size hysteresis. The effect of various parameters, such as nozzle height from the bed bottom and packing height, on the cavity size hysteresis has been studied. It is found that inter-particle interaction forces along with gas drag and bed porosity play an important role in describing the cavity size hysteresis. The injection of gas flow allows the particles to go to an unconstrained state than they were previously in, and their ability to remain in that state, even under decreased gas drag force, leads to the phenomenon of cavity size hysteresis.     

Optimization of the active component distribution through the catalyst bed for the case of adiabatic reactor

This article is the next one in the previous series of publications concerning the optimal active component distribution. The aim of the present work is to extend the optimization problem taking into account the energy balance. The variational problem on searching for the optimal active component distribution profile along the catalyst bed was formulated for the case of an adiabatic reactor at an arbitrary form of reaction rate expression. The Euler differential equation was derived and the existence conditions for the solution of variational problem were obtained. A numerical algorithm was suggested and the optimal profiles of the active component distribution were calculated for the first-order reaction. Under isothermal conditions with linear dependence of the reaction rate on reactant concentration at a constant mass transfer coefficient, the uniform distribution is optimal. As opposed to the case of an isothermal reactor with a first-order catalytic reaction, in an adiabatic reactor considerable economy in the active component loading might be achieved due to optimization of its distribution. It was shown that the optimal active component distribution profiles were axially decreasing for the first-order exothermic reactions and increasing for endothermic reactions. The optimal active component distribution profile was calculated for the case of methane combustion on catalytic monoliths.     

Modeling and simulation of an integrated multi-shell fixed bed membrane reactor with well-mixed catalyst pattern for production of styrene and cyclohexane

A rigorous two-dimensional steady state mathematical model based on the dusty gas model is implemented to investigate the performance of a bench-scale integrated multi-shell fixed bed membrane reactor with well-mixed catalyst pattern for simultaneous production of styrene and cyclohexane. Since the styrene producing reaction is equilibrium limited, significant displacement of the thermodynamic equilibrium is achieved by three simultaneous actions of an auxiliary hydrogenation reaction of benzene using a well-mixed catalyst pattern, the membrane and the multi-shell reactor configuration. The simulation results show that the complete conversion of ethylbenzene is possible at relatively low temperature and shorter reactor length. Effective operating regions with optimal conditions are observed and explanations offered. An effective length criterion for the optimal conditions is presented. The effective operating regions are found to be sensitive to changes of catalyst bed composition, feed temperature, feed pressure and shells ratio. It is also found that the multi-shell configuration is superior to the single shell configuration. Although this investigation is restricted to two catalysts and two shells, some of the rich characteristics of this system have been uncovered.     

Dynamic modeling and operability analysis of a dual-membrane fixed bed reactor to produce methanol considering catalyst deactivation

The goal of this research is dynamic operability analysis of dual-membrane reactor considering catalyst deactivation to produce methanol. A dynamic heterogeneous one-dimensional model is developed to predict the performance of this configuration. In this configuration, a conventional reactor has been supported by a Pd/Ag membrane tube for hydrogen permeation and alumina–silica composite membrane tube to remove water vapor from the reaction zone. To verify the accuracy of the considered model, the results of conventional reactor are compared with the plant data. The main advantages of the dual-membrane reactor are: higher catalyst activity and lifetime, higher CO₂ conversion and methanol production.     

Trickle bed reactors: Effect of liquid flow modulation on catalytic activity

The effect of slow ON-OFF liquid flow modulation on the oxidation of aqueous solutions of ethanol using a 0.5% Pd/Al₂O₃ commercial egg-shell catalyst was investigated in a laboratory trickle bed reactor (TBR). In this mode of operation, the catalyst was cyclically exposed to oxidative and reductive environments.The study was carried out under different gas and liquid flow rates, cycle periods and splits. Cycling results have been compared with the steady-state experiments performed at the corresponding average liquid flow rate. Significant improvements over the continuous operation were obtained when the catalyst was exposed to a short surplus of oxygen (to minimize deactivation by overoxidation in the kinetic regime) after a longer time of working in the mass transfer limited regime. According to the results presented here, it is recommended to work with high liquid flow rates and moderate gas flow rates to ensure complete wetting of the catalyst during the ON cycle and to minimize the overoxidation process during the OFF cycle.     

Simulation of catalyst loss from an industrial fluidized bed reactor on the basis of labscale attrition tests

Attrition of catalyst represents a significant economic penalty due to the necessity to make-up the losses on a continual basis. Designing fluidized bed systems to minimize attrition may sacrifice gas-solid contact efficiency that requires pressure to distribute gases in process vessels uniformly. In this paper, the attrition performance of DuPont's vanadium pyrophosphate oxide catalyst (VPP) used in their commercial Circulating Fluidized Bed (CFB) process to convert n-butane to maleic anhydride has been examined. The analysis is based on simulating the commercial conditions in the individual vessels combined with the physical characteristics of the catalyst determined in the laboratory. The comparison of the catalyst loss rate measured in the industrial plant and the simulation results indicates that the measured values are below the values predicted by the simulation. However, the relative change in the catalyst loss rate with operating time is well described. The simulation allows the quantification of the attrition sources in the process. The simulation results revealed that in the present case over 60% of the catalyst loss originates from attrition in the fluidized beds, whereas only 16% originates from attrition in the cyclones. The simulation is an appropriate tool to investigate the influence of parameter changes. Using the simulation the design parameters and operating conditions can be optimized to minimize the catalyst loss rate in the industrial plant.     

A sequential approach to modeling catalytic reactions in packed-bed reactors

A sequential modeling approach is proposed to simulate catalytic reactions in packed-bed reactors. The hydrogenation of alpha-methylstyrene and wet oxidation of phenol are selected as studied cases. The modeling scheme combines a reactor scale axial dispersion model with a pellet scale model. Without involving any fitting parameters, such an approach accounts for the non-linear reaction kinetics expression and different types of pellet-liquid wetting contact. To validate the developed modeling scheme and the parallel approach reported in the literature, the experimental observations for hydrogenation of alpha-methylstyrene to cumene have been employed. The predicted results by both approaches agree reasonably with the experimental data for both gas- and liquid-limited reaction. The proposed sequential approach was also used to simulate the dynamic performance of the reactor and pellets for the catalytic wet oxidation of aqueous phenol over a newly developed but rapidly deactivated catalyst (MnO₂/CeO₂). The simulation results for the catalytic wet oxidation process by both approaches were compared. The simulation describes the time evolution of the catalyst stability at different pellet points along the reactor axis. The performance of trickle beds and packed bubble columns over a range of operating conditions were also investigated, and packed bubble columns were found to achieve higher phenol conversion at the cost of more rapid catalyst deactivation.     

Experimental studies and modeling of a fixed bed reactor for Fischer-Tropsch synthesis using biosyngas

The aim of this work was to study the Fischer-Tropsch (FT) synthesis of a model biosyngas (33% H₂, 17% CO and 50% N₂) in a single tube fixed-bed FT reactor. The FT reactor consisted of a shell and tube with high-pressure boiling water circulating throughout the shell. A spherical unpromoted cobalt catalyst was used with the following reaction conditions: a wall temperature of 473 K, a pressure of 20 bars and a gas hour space velocity (GHSV) of 37 to 180 NmL.g_cat^− 1.h^− 1. The performance of the FT reactor was also validated by developing a 2D pseudo-homogeneous model that includes transport equations and reaction rate equations. Good agreement between the model predictions and experimental results were obtained. This developed model was extended to predict and quantify the influence of the FT kinetics as well as determine the influence of the tube diameter and the wall temperature. The predicted behaviors for CO and H₂ conversion, productivity of hydrocarbons (mainly CH₄ and C₅₊) and fluid temperature along the axis of the reactor have been analyzed.     

Hydrodynamic study of a toroidal fluidized bed reactor

Hydrodynamic behavior of a newly developed toroidal fluidized bed reactor is studied in this work. The reactor has a gas distributor consisting of angled blades in an annular ring at the reactor bottom. The driving force for particles to move over the distributing blades comes from the velocity head of gas jets accelerated upon entering the blade spacing. Relevant hydrodynamic behaviors are measured with various inert materials in a pilot scale 400-mm toroidal fluidized bed reactor. The observed hydrodynamic behavior is found to be essentially predictable at ambient temperature by conventional hydrodynamic models. Fine particle tracking on the reactor wall is clearly observed through oxidation of zinc dross at a bed temperature of around 1120°C, and is simulated on the basis of a simplified mathematical model. Hydrodynamic issues, such as particle flying trajectory and retention time in the reactor, are discussed based on the developed model.     

Catalyst pore plugging effects on hydrocracking reactions in an Ebullated bed reactor operation

This paper analyzes the deactivation effects of NiMo/Al₂O₃ catalyst during the operation in an Ebullated bed reactor for Heavy residue hydrocracking. The spent catalysts were characterized by chemical analysis, ¹³C NMR, ESM, DRX, and by using thermal programmed oxidation and diffusion studies in a shallow bed micro-reactor. The deactivations were performed in a 5 l continuously stirred tank reactor, while the spent catalysts were tested in a 0.05 l micro-reactor. The study focused on determining the properties of the external layer of the catalyst and on evaluating the internal coke and metal deposition. The results indicated that initial deactivation is mainly due to coke depositions, while its impact on mass transfer reaction control depends on temperature. In long-term deactivation, the metal deposition plays a more important role in blocking the internal micro- and meso-structures and in building up the external layer of the pellets.     

Effect of the wall Nusselt number on the simulation of catalytic fixed bed reactors

Literature correlations for the apparent wall heat transfer coefficient (h_w) in fixed bed catalytic reactors are compared. At low to moderate values of the Reynolds number (Re), different correlations can produce estimates of the dimensionless wall Nusselt number (Nu_w = h_wd_p/k_f) that differ by an order of magnitude or more. Some correlations give Nu_w as a function of Re only, others allow for the effects of tube-to-particle diameter ratio and particle and fluid thermal conductivities. The value of Nu_w that is used in a simulation of a fixed bed catalytic reactor can have a strong effect on the predicted behavior. Two examples of fixed bed reactors are simulated and show that the more general correlations for Nu_w are to be preferred.     

Analysis of volume change effects in a fluidized bed catalytic reactor

A mathematical model based on the concept of an improved bubble assemblage model is developed for calculating the conversion of a reaction system involving a volume change in fluidized beds. The influence of volume change on the hydrodynamic behavior of gas in the bed, such as bubble size variation, superficial gas velocity change, and volume fraction occupied by each phase, is also investigated. It is found that increasing stoichiometric coefficient values results in larger bubble size, higher superficial gas velocities, higher crossflow rate between emulsion phase and bubble phase, and greater volume fraction of bubble phase, but smaller volume fraction occupied by the emulsion phase as well as lower conversions.     

Analysis of a two-phase catalytic reaction in a millimeter-scale reactor

The synthesis of a catalyst for a two-phase catalytic reaction, a millimeter-scale reactor experiment, and an analysis model for the prediction of reactor performance are presented in this paper. The catalyst nano-particulate perovskite La_0.8Sr_0.2CoO₃ was prepared by a modified sol-gel method, in which PAA (poly acrylic acid) was added to catalyst precursors. A millimeter-scale reactor experiment with the prepared catalyst was carried. Concentrated hydrogen peroxide was decomposed in the reactor and the characteristics of the reactor were measured in terms of temperature distributions and liquid production rates. The results indicated a flow regime transition, which caused the change of reactor performance. An analysis model for two-phase catalytic reaction based on the lumped flow reactor model and the diagnostic data obtained on the temperature distributions and liquid production rates is proposed. Temperature distributions and heat transfer characteristics of the reactor were predicted by a semi-empirical analysis. In this analysis, the model of the Nusselt number (Nu) was proposed as . This expression of the model reflects the effects of temperature and coordinate location on the heat transfer characteristics of the reactor. From the modeled reactor, characteristics such as the increase of heat transfer in the mid and rear parts of the reactor with the increase of reactant flow rate were obtained. With the obtained results, a tool for the design and analysis of a down-scaled catalytic reaction device was obtained.