Theoretical analysis of inhibition effect on steady states of enzymatic membrane reactor with substrate and product retention for reaction producing weak acid

The aim of this study is to formulate a model of enzymatic membrane reactor (EMR), i.e., a continuous, stirred tank bioreactor with full enzyme recycle, for a reaction producing a weak acid, and to explore the effect of substrate and product inhibition of different mechanisms coupled with transport properties of the membrane on the static behaviour of the system. The inhibition of an enzyme by a substrate leads to the non-monotonicity of reaction rate expression with respect to the substrate concentration. If a product of enzymatic reaction, taking place in the EMR, influences the pH of a reaction mixture this is also the factor causing the non-monotonicity of the substrate and product dependent reaction rate. The character of these dependencies affects substantially the structure of the steady states of the reactor. The bifurcation diagrams, shown in the work, are of different characters depending on the bifurcation parameter. It has been found, that bifurcation diagrams for competitive and uncompetitive inhibition by a substrate differ in the number and position of bifurcation points. Steady states of multiplicity five have been localised in case of uncompetitive inhibition by the substrate at high affinity of the enzyme to this substrate. Retention of reagents, related to transport properties of a membrane, influences significantly the effectiveness of a process. A specially written software in Delphi™ has been used for the calculations.     

Dynamics of penicillin G hydrolysis in an electro‐membrane reactor

Analysis of penicillin G hydrolysis in a membrane reactor with membrane‐entrapped penicillin G acylase is performed using a mathematical model of the reactor system. An electric field imposed to the reactor is considered to enhance transport rates of reaction components and reaction rate. Diffusion, electrophoretic migration and electro‐osmotic flux across the membrane are considered. The analysis focuses on possible effects of the principal operational parameters (electric field intensity, inlet substrate concentration, membrane thickness) on reactor performance. Multiplicities of steady states are frequently encountered. The membrane reactor performance can be easily targeted towards the required reaction regime by applying a constant or periodically varying electric field to the system. The periodic alternation of the polarity of the electric field substantially increases the effectiveness factor of penicillin hydrolysis compared with the steady state operation. Proper adjustments of electric field intensity may also compensate for the decay in enzyme activity. © 2001 Society of Chemical Industry     

Effectiveness factor approximations for multiple steady states in porous catalysts

The effectiveness factor is formally determined by solving a two-point boundary value problem, often numerically. To enhance the computational efficiency in simulations of large-scale reactor systems with porous catalysts, a simple approximation formula for the effectiveness factor is often used. For some reaction rate functions, however, the effectiveness factor as a function of Thiele modulus can show multiple values or sharp changes for a small change in the modulus. In this case, single-valued approximations of the effectiveness factor may give rise to large errors. Based on the two well-known asymptotes of the effectiveness factor for small and large Thiele moduli, we proposed equations for the approximation of the effectiveness factor for up to three multiple steady states and two catalyst geometries of an infinite slab and a sphere. The proposed equations were demonstrated to be useful in estimating the effectiveness factor, particularly for the stable steady states, and also in quickly estimating the Thiele modulus range where multiple effectiveness factors should be searched.     

高效再生催化裂化装置多稳态分析：反应温度开/闭环控制条件对热反馈机制的影响

针对催化裂化反应-再生系统在提升管反应温度开环和闭环控制条件下的输出与输入多稳态问题,分析了烧焦罐式高效再生催化裂化反应-再生系统在两种条件下随着CO助燃剂添加量变化时的多稳态分布。在反应温度开环条件下,因再生温度与反应温度的耦合程度较低,使系统移热曲线呈单调递增,导致了系统出现3个稳态操作点。在反应温度闭环控制条件下,提升管反应器和再生器间热反馈机制发生改变,由于再生剂循环量可以作为额外的自由度对再生温度和反应温度之差进行补偿,再生器和提升管反应器的耦合程度增强,使得系统只会在助燃剂添加量极低时才会出现多个稳态点,而在基准操作条件下只有一个稳态点,规避了系统在提升管反应温度开环时的多个稳态点的问题。     

Modelling and simulation of anaerobic stratified biofilm for methane production and prediction of multiple steady states

The present work deals with modelling of anaerobic metabolism in a stratified biofilm consisting of two distinct layers of microorganisms in a fixed film reactor. The model is applied to steady state conditions and is based on substrate utilization kinetics and mass transport. The model assumes that the substrate is first metabolized by the outer acidogenic layer and the subsequent products are metabolized by the inner layer of methanogens to produce finally methane and carbon dioxide. The model considers slab geometry of the biofilm, Monod kinetics for growth rate of acidogens and the inhibition effect of volatile fatty acids on growth rate of methanogens. The diffusion equations are solved simultaneously using a modified form of the technique of orthogonal collocation on finite elements to obtain concentration profiles and the rate of methane production per unit volume of the biofilm is determined. Computer simulations are carried out to study the effect of various parameters on the rate of methane production and to investigate the existence of multiple steady states. The simulation shows that for a range of parameters values three steady states exist. It is shown that the upper steady state with highest rate of methane production may lead to unsatisfactory reactor performance.     

Analysis of a pH‐Based Potentiometric Biosensor Using the Homotopy Perturbation Method

A mathematical model of steady‐state and non‐steady‐state responses of a pH‐based potentiometric biosensor immobilizing organophosphorus hydrolase was developed. The model is based on non‐stationary diffusion equations containing a nonlinear term related to the Michaelis‐Menten kinetics of an enzymatic reaction. An analytical expression for the substrate concentration was obtained for all values of parameter a (Thiele modulus) using the homotopy perturbation method. From this result, the concentrations of the deprotonation products of an organophosphodiester (P_hH, ZH and AH) were obtained. Our analytical results were compared with available simulation results. A satisfactory agreement with the simulation data is noted.     

Determination of multiple steady states in oxidation of monophenols by tyrosinase with enzymatic-enzymatic-chemical model

A family of an enzymatically catalyzed reaction network is studied, which involves the oxidation of monophenols by tyrosinase with enzymatic-enzymatic-chemical model in an isothermal continuous flow stirred tank reactor (CFSTR). This system consists of 11 coupled non-linear equations and is determined to have the capacity to exhibit computational multiple steady states. A set of rate constants and two corresponding steady states are computed. The phenomena of bistability, hysteresis and bifurcation are discussed. Moreover, the capacity of steady state multiplicity is extended to its family of reaction networks.     

Multiplicity of steady states in an isothermal batch fluidized bed reactor

An investigation of uniqueness and multiplicity of the steady states in an isothermal fluidized bed reactor and continuous stirred tank reactor is presented. Material balance equations for the two phase model of Davidson and Harrison and the modified Partridge and Rowe model, are transformed into a single equation which is of the same form as the material balance equation for an isothermal continuous stirred tank reactor. Exact uniqueness and multiplicity criteria are derived, and numerical examples illustrating the influence of some parameters on the steady state multiplicity are presented.     

Theoretical Analysis of Reaction and Diffusion Processes in a Biofuel Cell Electrode

A mathematical model for reaction diffusion processes in a biofuel cell electrode is discussed. This model is based on reaction diffusion equations containing a non‐linear term related to the rate of the enzyme reaction. Theoretical treatment of a reaction and diffusion processes in a biofuel cell electrode, for the steady and non‐steady state condition is discussed. Approximate analytical expressions for the steady and non‐steady state current density at the electrode surface are calculated by using the new approach to homotopy perturbation method and complex inversion formula. An analytical expression for the steady state current density is compared with numerical results and found to be excellent in agreement. A novel graphical procedure for estimating the Michaelis‐Menten constants and turnover rate solely from the current‐potential curve is suggested. Influence of the controllable parameters such as diffusion of the mediator, Michaelis‐Menten constant for substrate, second‐order rate constant, thickness of the film, turnover rate and initial substrate concentration on the current density are discussed.     

CFD simulation of mixing and reaction: the relevance of the micro-mixing model

The aim of this work is to understand the role of the micro-mixing model in computational fluid dynamics (CFD) simulations of fast reactions. Using CFD, in fact, the reactor is modelled through a computational grid and the governing equations are discretised using numerical methods. However, the mixing phenomena that occur at scales that are smaller than the grid size remain unresolved. This means that the probability density function (PDF) of all scalars is assumed to be a delta function centred at the mean value. In order to take into account micro-mixing effects a model must be added. In this work the finite-mode PDF approach is used to predict the selectivity of a parallel reaction in a Taylor-Couette reactor working in semi-batch conditions. Experimental data are compared with model predictions in order to investigate the relevance of the micro-mixing model. The case of precipitation is also discussed.     

A diffusion‐reaction model for α‐chymotrypsin carrying uniform thermosensitive gel beads

In this study, the kinetic behavior of α‐chymotrypsin‐immobilized, uniform poly(isopropylacrylamide) gel beads was investigated. The kinetic study was performed by using a continuous reactor operated at steady‐state conditions. In the experiments, substrate feed concentration, residence time, and reactor temperature were changed. The results were explained by a diffusion–reaction model developed for steady‐state conditions. The effectiveness factor and Thiele modulus values of the thermosensitive enzyme–gel system were estimated at different temperatures by using an iterative procedure based on fourth order Runge–Kutta algorithm. The results indicated that the overall hydrolysis rate was controlled by the substrate diffusion through the gel matrix. A bending point was detected for the Thiele modulus at the lower critical solution temperature (LCST) of the thermosensitive gel. The effective diffusion coefficient of substrate and effectiveness factor decreased suddenly at LCST. The mass transfer process within the thermosensitive carrier could be described in detail by the proposed model. The results of our numerical procedure were also compared with an analytical approximate solution available in the literature. The consistency between two different model was reasonably good. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 74: 1025–1034, 1999     

Revisiting the maximum conversion of substrate in an enzymatic cstr with micromixing considerations

The balance equations pertaining to an enzyme undergoing first order thermal deactivation and a substrate undergoing enzyme-catalyzed transformation following Michaelis-Menten kinetics in a CSTR are solved for the two limits of micromixing. The maximum conversions of substrate are obtained under the assumption that the space time of the reactor is infinite. The range of the maximum conversion of substrate is wider at intermediate values for the dimensionless Michaelis-Menten constant and becomes narrower as the ratio of the time scales of the enzyme-catalyzed reaction and the enzyme deactivation increases.     

An Approach for Achieving Unstable Convergence for Non‐Isothermal CSTRs

The nonlinearity introduced by the temperature variation in the non‐isothermal continuous stirred tank reactor (CSTR) exhibits concentration multiplicity for certain parameter ranges. The dynamics of reactor temperature under these conditions would be very difficult to observe and maintain. In this work, a novel algorithm is proposed to stabilize the system by designing a cascade of CSTRs that are capable of achieving this inaccessible steady state. Optimization of reactors parameters is performed in an iterative manner to achieve this solution. For a first‐order reaction rate, this method was successful in achieving the inaccessible steady state temperature of 312.5 K using three CSTRs cascades. The suggested algorithm is presented both graphically as well as using computational optimization techniques. The transient simulation studies using the above three CSTRs showed that the unstable steady state is achieved. The newly designed cascade meets the design criteria and achieves the locally unstable steady state temperature to a high degree of accuracy.     

A study on the states of the non isothermal tubular wall reactor

The thermal behaviour of a cooled tube wall reactor is analyzed.The main goal of this work is the characterization of the steady state. Three different steady state solutions, in which the control through the reactor differs from each other, are presented. The possibility of simultaneous existence of these states is analyzed as a function of the parameters of the model.For the zeroth order reaction here treated, it is possible to get isothermal operating conditions. Minimum thermal levels at the wall reactor for a given conversion are discussed and they result closely related to the isothermal conditions.An analytical solution is developed for the so called “highest steady state” (HISS). This expression makes possible to predict the “quenching” conditions of the reactor.     

Spatially patterned states in systems of interacting catalyst particles

Considering an assemblage of identical catalyst particles exposed to uniform surroundings, we show in this theoretical study that interparticle interactions give rise to spatially patterned steady states under some conditions. In such patterned states, the rate of reaction on one particle differs from the rate on all or some of the neighboring particles. Results, which are based on simple geometric configurations and on the assumption that a single nonisothermal reaction occurs, show that spatially patterned states of two types exist. The first occurs only if a single isolated particle has multiple steady states. The second type, not inferable from the behavior of a single isolated particle, is attributable to the instability of a spatially uniform state caused by particle-particle interactions. This type may occur in cases where the steady state of an isolated particle is unique and globally stable. Though the study is not extended here to complex assemblages, the results suggest that these phenomena may occur in fixed bed reactors and lead to insidious nonuniformities in reactor temperature and catalyst activity.     

Molecular weight distribution of polyethylene terephthalate in homogeneous,continuous-flow-stirred tank reactors

Poly(ethylene terephthalate) (PET) formation in homogeneous, continuous-flow-stirred tank reactors (HCSTRs) operating at steady state has been simulated. The feed to the reactor is assumed to consist of the monomer bis-(hydroxyethyl) terephthalate and monofunctional compound (MF₁) cetyl alcohol. The overall polymerization is assumed to consist of the polycondensation, reaction with monofunctional compounds, redistribution, and cyclization reactions. At a given time, the reaction mass consists of polyester molecules (P_n), polyester molecules with an ending of molecules of monofunctional compound (MF_n), and cyclic polymers (C_n). A mass balance for each of these species in the reactor gives rise to a set of algebraic equations to be solved simultaneously. The MWD calculations show that the redistribution reaction plays a major role and cannot be ignored, This result is in contrast lo the observation for semi-batch reactors, for which redistribution becomes important when the cyclization reaction is included. For the same residence times of semi-batch and HCSTRs, the latter gives considerably lower-number average molecular weight, N_av, and polydispersity index, ρ. However, for the same conversions, the ρ for CSTR is higher. The concentration of the monofurctional compound, [MF₁]₀, in the feed and the reactor temperature both influence ρ, but the effect is small within the range studied.     

Low pressure carbonylation of benzyl chloride

For carbonylations, metal carbonyls, particularly cobalt and iron carbonyls, are often used as catalysts. These reactions take place under rather drastic reaction conditions, e.g. 200–300 °C and 60–100 MPa. In some patents it is stated that similar reactions using the same catalysts can also be carried out under rather mild reaction conditions, such as 0–100 °C and 0–2.5 MPa.We studied the conversion of benzyl chloride to phenyl acetic methyl ester in a semi-batch reactor in which one of the reactants was slowly added to the reactor. In this way a non-catalysed parallel reaction could be oppressed for the greater part.The highest conversions and selectivities were obtained when a solution of sodium methylate in methanol was slowly added to a reaction mixture containing benzyl chloride. In this way under optimized conditions a conversion of benzyl chloride of 100% has been found.It also appears that the conversion and selectivity decrease not only at low but also at high CO partial pressures. A possible explanation for this phenomenon will be given.     

The effect of particle size distribution on the olefin polymer reactor dynamics

Mass and energy balances in a reactor have been derived to study the effect of particle size distribution (PSD) for each reaction mechanism on the reactor dynamics. It was observed that the PSD affects both bed height and particle volume. A feasible region for reactor operation has been calculated by using physical constraints. In a nonisothermal polymerization system, the reactor temperature does not change appreciably as catalyst injection rate increases. A unique steady state solution is found in a gas-phase continuous stirred-bed propylene polymerization reactor. The eigenvalues of the system of equations indicate that the steady state is unstable. A comparison with published data allows the observation that the actual reactor dynamics may be readily explained by using only the PSD derived from a simple reaction mechanism.     

Effect of enzyme inhibition on the performance of packed-bed biological reactor - a theoretical study

The performance characteristic of a packed bed reactor has been analyzed by considering diffusional resistance of the biofilm. Model equations were solved by the method of orthogonal collocation for various classical enzyme inhibition kinetics, including partially non-competitive, partially competitive, partially uncompetitive, partially mixed and fully mixed. For all considered modes of inhibition, an increase in the inlet substrate concentration decreases the steady state conversion in the reactor. However, an increase in the Peclet number has been found to improve the conversion. The effects of various other process variables of physical importance were also investigated parametrically.     

Modelling FCC units under steady and unsteady state conditions

A FCC phenomenological‐based model is presented in this study. This model accounts for the behaviour of the riser, the stripper and the regenerator. Both the reactor and the regenerator are incorporated in the model using detailed reaction kinetics. Catalyst regeneration allows for both catalytic and non‐catalyzed combustion of CO to CO₂. Overall, the model is constituted by a set of ordinary differential equations, relatively easy to solve and suitable for steady and unsteady state simulation of FCC units. The proposed model avoids, as much as possible, empirical expressions being suitable for simulation of industrial FCC units and process control studies.