An improved theoretical model for hollow-fiber enzyme reactors

An analysis of hollow-fiber enzyme reactors is presented, for first-order kinetics, which leads to explicit expressions for the concentration fields on both the enzyme and substrate sides. These expressions are easily evaluated numerically. Results are presented which show that the substrate-side profiles of bulk concentration versus axial distance can be expressed as functions of a single parameter (an overall mass transfer resistance). Limiting forms of the equations and of system behavior which arise for very low and very high values of the Thiele modulus (corresponding to the reaction-controlled and diffusion-controlled limits) are discussed in some detail. The method of analysis presented is shown to be applicable to other “conjugated boundary value” problems of a similar nature.     

Homogeneous models for porous catalysts and tubular reactors with heterogeneous reactions

One-dimensional models for porous catalysts and laminar flow tubular reactors with first-order reactions at the walls are developed. The method which is used replaces the transport equations containing the local concentrations with equations, similar to those used in the dispersion theory, for the concentrations averaged over the cross section. The main assumption is the approximation of the coefficients in the dispersion equations by those valid for a pulse of concentration introduced at time zero at the mouth of the pore or at the inlet of the tube.The effectiveness factor depends not only on the Thiele modulus but also on the length-to-radius ratio of the catalyst pore. The one-dimensional model for the catalytic tubular reactor provides a good approximation to the area-mean concentration for β (k_sR/D) up to 1·0. The comparison is made with an exact orthogonal expansion solution developed also in the paper. For larger values of β, say β ≈ 100, the discrepancies between the exact solution and the one-dimensional model may be as high as 50 per cent. Therefore, a generalized dispersion solution is presented to obtain more accurate predictions for β > 1·0.     

Development and validation of a two phase CFD model for tubular biodiesel reactors

The use of biodiesel as an alternative to diesel has gained increasing momentum over the past 15 years. To meet this growing demand there is a need to optimise the transesterification reactor at the heart of the biodiesel production system. Assessing the performance of innovative reactors is difficult due to the liquid–liquid reaction mixture that is affected by mass transfer, reaction kinetics and component solubility. This paper presents a Computational Fluid Dynamic model of a tubular reactor developed in ANSYS CFX that can be used to predict the onset of mixing via turbulent flow. In developing the model an analysis of the reaction mixture is provided before the presentation of experimental data, which includes flow visualisation results and temperature dependant viscosity and density data for each phase. The detailed data and model development procedure represents an advancement in the modelling of the two phase transesterification reaction used in biodiesel production.     

Development and analysis of heterogeneous catalytic processes and reactors

A retrospective review and a review of the current state are given for the problem of the development and analysis of heterogeneous catalytic processes and reactors. Advanced approaches to the study of kinetics and the engineering design of catalytic reactions are discussed. The advantages and drawbacks of the main types and structures of industrial catalytic reactors are demonstrated by the examples of present-day catalytic processes and reactors for basic organic and petrochemical synthesis. The operation of several industrial catalytic reactors is analyzed in order to clarify the causes of deterioration of design performance and/or to find ways for the process intensification in operating plants.     

Modeling of chemical reactors—XXXI: The one-phase backflow cell model used for simulation of tubular adiabatic reactors

The backflow cell model is used to simulate steady state operation of a tubular adiabatic reactor. The model proposed embraces different mechanism of axial dispersion of heat and mass. It will be shown that the backflow cell model may be used for approximation of the dispersion model. While there are differences in qualitative behavior of simple cell and dispersion models, the backflow cell model gives results which are in agreement with the dispersion model. The model may be used for simulation of steady-state behavior of tubular homogeneous and heterogeneous reactors.     

A theoretical analysis of the deactivation of an immobilised enzyme in multi-enzyme reactions in fixed-bed and fluid-bed reactors

The effect of time-averaging on the selectivity to an intermediate R (yR ) and the yield to product S (yR ) in a consecutive reaction A→R→S, occurring in a deactivating fixed-bed reactor, has been examined. Time-averaging smooths the selectivity and increases the yield for high values of the deactivation (λ) and reaction (δ) parameters. The kinetics and dynamics of yR and yS in a deactivating steady-state fluid-bed reactor were also examined. Both non-selective and selective deactivation are considered. The practical implications of the development are considered in terms of an example, in which the values chosen for parameters are arbitrary though reasonable with respect to experimental values.     

A heterogeneous one-dimensional model for non-adiabatic fixed bed catalytic reactors

A heterogeneous one dimensional model which takes separately into account the heat transfer through the solid and fluid phases is introduced. The response of this model is compared with that of the heterogeneous two-dimensional model, and a very good agreement, especially at mild conditions, is obtained. Its performance is better than previous one dimensional models and the deviations are explained in tenns of the operative conditions and of the value of dimensionless groups. Conditions at which the different one and two-dimensional models are recommended to be used are presented.     

A two-dimensional model of second order rapid reactions in turbulent tubular reactors

Previous studies of second order rapid reactions in turbulent tubular reactors have been analyzed in terms of a one-dimensional model. Deviations between theory and experiment have been interpreted in terms of diffusion limited reaction rates. In this paper the experimental results are re-analyzed in terms of a two-dimensional model leading to the conclusion that mixing and entrance region effects are responsible for the deviations between the experimental results and one-dimensional theory. A new upper bound for the Damköhler number associated with diffusion controlled reaction rates is suggested.     

Optimization-based strategies for the operation of low-density polyethylene tubular reactors: nonlinear model predictive control

In this work, we present a general nonlinear model predictive control (NMPC) framework for low-density polyethylene (LDPE) tubular reactors. The framework is based on a first-principles dynamic model able to capture complex phenomena arising in these units. We first demonstrate the potential of using NMPC to simultaneously regulate and optimize the process economics in the presence of persistent disturbances such as fouling. We then couple the NMPC controller with a compatible moving horizon estimator (MHE) to provide output feedback. Finally, we discuss computational limitations arising in this framework and make use of recently proposed advanced-step MHE and NMPC strategies to provide nearly instantaneous feedback.     

The bifurcation behavior of tubular reactors

Methods for studying the bifurcation behavior of tubular reactors have been developed. This involves the application of static and Hopf bifurcation theory for PDE's and the very precise determination of steady state profiles. Practical computational methods for carrying out this analysis are discussed in some detail. For the special case of a first order, irreversible reaction in a tubular reactor with axial dispersion, the bifurcation behavior is classified and summarized in parameter space plots. In particular the influence of the Lewis and Peclet numbers is investigated. It is shown that oscillations due to interaction of dispersion and reaction effects should not exist in fixed bed reactors and moreover, should only occur in very short “empty” tubular reactors. The parameter study not only brings together previously published examples of multiple and periodic solutions but also reveals a hitherto undiscovered wealth of bifurcation structures. Sixteen of these structures, which come about by combinations of as many as four bifurcations to multiple steady states and four bifurcations to periodic solutions, are illustrated with numerical examples. Although the analysis is based on the pseudohomogeneous axial dispersion model, it can readily be applied to other reaction diffusion equations such as the general two phase models for fixed bed reactors.     

Dispersion models of unsteady tubular reactors

Axial dispersion in time-variable laminar flow in a tubular reactor is analyzed using an exact procedure for the case of a homogenous first-order reaction. For the first time since the Taylor Dispersion model was originally introduced for the modeling of reactors, its validity is examined over a wide range of the reaction rate parameter by comparison against an exact analysis. It is shown that a constant coefficient dispersion model can be obtained from first principles for large values of time only for initial distribution problems; however, this simple approximate model also is reasonably good for describing concentration distributions for the present inlet distribution problem for slow reactions and for axial locations sufficiently far away from the inlet. For rapid reactions, while the dispersion model is inaccurate in describing axial concentration distributions, it is surprisingly good for predicting the reactor length required for complete conversion. In contrast to the conclusion of a recent article, it will be shown that the dispersion coefficient is independent of the reaction rate constant.     

Backstepping control of chemical tubular reactors

In this paper, a globally stabilizing boundary feedback control law for an arbitrarily fine discretization of a nonlinear PDE model of a chemical tubular reactor is presented. A model that assumes no radial velocity and concentration gradients in the reactor, the temperature gradient described by use of a proper value of the effective radial conductivity, a homogeneous reaction, the properties of the reaction mixture characterized by average values, the mechanism of axial mixing described by a single parameter model, and the kinetics of the first order is considered. Depending on the values of the nondimensional Peclet numbers, Damköhler number, the dimensionless adiabatic temperature rise, and the dimensionless activation energy, the coupled PDE equations for the temperature and concentration can have multiple equilibria that can be either stable or unstable. The objective is to stabilize an unstable steady state of the system using boundary control of temperature and concentration on the inlet side of the reactor. We discretize the original nonlinear PDE model in space using finite difference approximation and get a high order system of coupled nonlinear ODEs. Then, using backstepping design for parabolic PDEs we transform the original coupled system into two uncoupled target systems that are asymptotically stable in l²-norm with appropriate homogeneous boundary conditions. In the real system, the designed control laws would be implemented through small variations of the prescribed inlet temperature and prescribed inlet concentration. The control design is accompanied by a simulation study that shows the feedback control law designed with sensing only on a very coarse grid (using just a few measurements of the temperature and concentration fields) can successfully stabilize the actual system for a variety of different simulation settings (on a fine grid).     

Parametric sensitivity of the ignition process to mass transfer in adiabatic tubular reactors with exothermic heterogeneous reactions

The alternative effects of reaction kinetics, mass, heat and momentum transport on mass conversion by chemical reactions are examined theoretically for a reacot tube with laminar flow. The reaction enthalpy is considered. A heterogeneous reaction between several gaseous components takes place at the inner surface of this reactor tube. Strongly exothermic reactions lead to self-acceleration of the reaction, unless reaction enthalpy is removed through the tube wall. Under certain conditions, there will be a sudden change from mass transfer controlled by the reaction to that controlled by diffusion. This phenomenon is known as ignition of the reaction. The effect of ignition and its sensitivity to reaction enthalpy, thermal conductivity and diffusivity of the fluid as well as activation energy of the first order heterogeneous wall reaction are investigated by a numerical solution of the transport equations. Axial conduction of heat and mass is neglected both in the fluid and in the tube wall. Non-stoichiometric wall reactions of first order, with temperature dependent reaction rates and equilibrium constants, are considered. The results are presented in graphical form, as plots of the local mass flux at the reacting wall as functions of the dimensionless tube length.     

Design of tubular reactors in recycle systems

The article presents an approach to design tubular reactors in recycle systems, based on non-linear analysis. A pseudo-homogeneous plug-flow reactor model is used. It is assumed that the separation unit delivers product and recycle streams with fixed composition. The stand-alone reactor has a unique stable steady state. The coupled reactor–separation–recycle system shows four types of conversion versus plant Damköhler number bifurcation diagrams. A feasible steady state exists only if the reactor volume exceeds a critical value. For isothermal reactor, the steady state is unique and stable. For non-isothermal reactor, one or two steady states are possible. In the second situation the low-conversion state is unstable. In some parameter regions, the unique state is unstable. The design should ensure state unicity and stability, which are favoured by large heat-transfer capacity, low coolant temperature and high reactor-inlet temperature. A case study demonstrates that these phenomena can be easily found in real plants.     

Optimal location of measurements in tubular reactors

The optimal location of temperature measurements along the length of a non-adiabatic tubular reactor in which a first-order exothermic reaction is occu     

Scaleup factors of tubular emulsion polymerization reactors

This paper examines the important dimensionless numbers that control emulsion polymerization in a tubular reactor. It was found that the activation energy of polymerization was of major importance, while the role of monomer diffusion was not very significant. By selecting certain combinations of the dimensionless numbers, changes occurring during scaleup from a small tubular diameter to a larger diameter can be approximated.     

CFD simulation of gas-liquid contacting in tubular reactors

Gas-liquid contacting in tubular reactors was simulated using an Eulerian-Eulerian CFD approach in which accurate interphase momentum closure relations are incorporated, bubble-induced turbulence is accounted for, and population balance equations are used to describe bubble breakage and coalescence. The ability of two breakup kernels (Luo, H., Svendsen, H.F., 1996. Theoretical model for drop and bubble breakup in turbulent dispersions. A.I.Ch.E. Journal 42, 1225-1233; Lehr, F., Millies, M., Mewes, D., 2002. Bubble size distributions and flow fields in bubble columns. A.I.Ch.E. Journal 48, 2426-2443) and three coalescence kernels (Prince, M.J., Blanch, H.W., 1990. Bubble coalescence and breakup in air sparged bubble columns. A.I.Ch.E. Journal 36, 1485-1499; Luo, H., 1993. Coalescence, breakup and liquid recirculation in bubble column reactors. Ph.D. Thesis, Norwegian University of Science and Technology, Trondheim; Lehr, F., Millies, M., Mewes, D., 2002. Bubble size distributions and flow fields in bubble columns. A.I.Ch.E. Journal 48, 2426-2443) to accurately predict several flow parameters in pipe flow was tested.Good agreement between simulation and experimental results (radial profiles of gas holdup, turbulence intensity, and local Sauter bubble diameter) was achieved without the use of empirically derived relationships (such as Drift flux) by adjusting a single parameter which accounts for the deviation in the coalescence behaviour of tap water from that of pure water. The approach adopted in this investigation may thus be applicable to more complex hydrodynamic situations such as those encountered in mechanically agitated tanks and the need for extensive experimental testing may be replaced by single measurement of the effect interfacial properties have on coalescence rates.