AXIAL DISPERSION MODELS FOR STEADY-STATE, LAMINAR, OPEN-TUBE SYSTEMS

Mass transfer to the wall and a homogeneous first-order chemical reaction are considered separately for laminar Axial dispersion coefficients for steady-state systems computed from exact solutions for the concentration profiles are shown to be strong functions of axial position. For the homogeneous first-order reaction case, the axial dispersion coefficient is a function of the reaction rate constant.     

AXIAL DISPERSION MODELS FOR STEADY-STATE,LAMINAR, OPEN-TUBE SYSTEMS

Mass transfer to the wall and a homogeneous first-order chemical reaction are considered separately for laminar Axial dispersion coefficients for steady-state systems computed from exact solutions for the concentration profiles are shown to be strong functions of axial position. For the homogeneous first-order reaction case, the axial dispersion coefficient is a function of the reaction rate constant.     

Unsteady tubular reactors—time variable flow and inlet conditions

Unsteady flow in a tubular reactor with inlet concentration of reactant that depends on time and position is studied.It is shown that the coefficients which arise in the expression for the axial flux are different from those which occur in the dispersion model when heterogeneous reactions occur. Therefore a new inlet boundary condition expression is derived. This inlet condition is used to solve for the steady state reactant concentration distribution.     

Recycle reactor models for complex electrochemical/chemical reaction systems

The performance of complex electrochemical reaction sequences in recycle plug flow reactors is mathematically modelled. The reactions include successive electron transfers (EE reactions), chemical reaction interposed between successive electron transfers (ECE reactions), simultaneous electron transfers and simultaneous electron transfer and chemical reaction. Both potentiostatic and galvanostatic operations are considered and the effects of important parameters such as mass transport coefficient, recycle ratio and chemical reaction rate in the recycle loop are highlighted. This is done by considering two important electro-organic synthesis reactions, the production ofp-aminophenol and the reduction of oxalic acid to glyoxylic acid.Nomenclature a activity factor - C _ji concentration of species j at reactor inlet - C _j bulk concentration of species j - C _j ^s surface concentration of species j - C _j ^e concentration of component j returned to reactor inlet stream - C _j concentration from reactor outlet - C _je equilibrium concentration - E electrode potential - F Faraday number - i _n partial current density of stepn - i _T total current density - k deactivation rate constant - k _f n forward electrochemical rate constant of stepn - k _b n backward electrochemical rate constant of stepn - k _f forward chemical reaction rate constant - k _r reverse chemical reaction rate constant - k _Lj mass transfer coefficient for species j - k _a forward adsorption rate constant - k _d reverse adsorption rate constant - L reactor length - r recycle ratio - t reaction time - u velocity - Q flow rate - x reactor dimension - _n constant describing potential dependency of reverse reaction rate constant - _n constant describing potential dependency of forward reaction rate constant - _j surface concentration of adsorbed species j - electrode area per unit length - residence time of fluid in recycle loop     

A mathematical model for the dehydrogenation of methylcyclohexane in a packed bed reactor

A model for the dehydrogenation of methylcyclohexane in a tubular reactor over an industrial catalyst Pt-Sn/Al₂O₃ has been established. This model takes into account the axial dispersion at the inlet of the catalytic bed reactor as well as the heat transfer at the wall of the reactor. The heat transfer at the wall is satisfactorily represented by using a heat transfer coefficient correlation for which the parameters are obtained by fitting to the experimental data. The model provides a good representation of the radial and axial temperature profiles in the packed bed and can be also used to calculate the conversion.     

Modelling of catalytic reactors containing core–shell pellets with various morphologies for series reactions

Modelling of series reactions was performed for core–shell catalysts. Mathematical solutions of concentrations inside the pellets were derived from reaction–diffusion equations considering inert-core thickness (ξ_c) for first-order kinetics. Transient behaviours of catalytic reactors containing core–shell pellets were predicted, assuming pseudo-steady state approximation. In a batch reactor, the removal rate of reactants increased with increasing Thiele modulus and decreasing ξ_c in the order of sphere > cylinder > slab. The transient concentration of the intermediate product was maximum and affected by the distribution coefficient, diffusivity ratio, particle shape, and ξ_c. In a continuously stirred tank reactor, the concentration was affected by feed rate and catalyst loading, and conversion could be enhanced by a cascade connection. In a fixed-bed reactor, the concentration increased with increasing ξ_c due to an insufficient catalyst volume. Péclet number and particle shape also affected the concentration, implying that axial dispersion and interfacial area are important design parameters.     

EXACT ANALYSIS OF UNSTEADY CONVECTIVE DIFFUSION IN A SIMPLIFIED CROSS MODEL FLUID

The article presents the effect of non-Newtonian viscosity on the longitudinal dispersion of tracer molecules released in an incompressible viscous non-Newtonian fluid (known as the simplified Cross model fluid) under the action of a constant pressure gradient. The Gill and Sankarasubramanian model is used to solve the unsteady convection diffusion equation for all time periods. An exact expression is obtained for the longitudinal convection coefficient K ₁(η?), which shows the effect of the non-Newtonian parameter η? on the centerline coefficient. It is seen that the value of the K ₁(η?) for η? > 1 is always smaller than the corresponding value for a Newtonian fluid. Also, the longitudinal dispersion coefficient of the solute K ₂(τ, η?) is determined exactly. The results show that the K ₂(τ, η?) asymptotically reaches a stationary state after a certain time. The effect of the η? on the most dominant dispersion coefficient is clearly depicted. Finally, the axial distribution of the average concentration θ _m of the solute over the channel cross section is determined at a fixed instant after the solute injection for several values of the η?. The results for “pure convection” are also reported     

The backflow cell model of isothermal first order flow reactors with axial dispersion

Equations which predict the course of the reversible first order reaction A ? R in backflow cell models of isothermal flow reactors are developed and compared to those of the continuous dispersion model. The conditions of convergence of the concentration A profile of the backflow cell model to that of the continuous dispersion model were investigated for isothermal first order irreversible (A ? R) reactors of finite size. For this case figures are displayed which allow the models to be compared and the effects of axial mixing to be included in reactor design. The effect of backmixing upon the optimum temperature and maximum product yields of R is considered for the reversible reaction in the finite reactor.     

对二甲苯氧化反应器非均匀混合模型

对二甲苯氧化是一个复杂的气-液-固三相反应过程。今根据实验测取的对二甲苯氧化反应动力学关系与数据,建立了工业氧化反应器的非均匀混合模型,用以考察气、液相的混合状况及其对反应的影响。模型采用三区串联加区问返混的结构,区间返混参数由工业反应器实测轴向温度分布拟合确定。计算表明,釜内液相混合接近全混、流,而气相氧浓度存在明显的轴向梯度：溶剂蒸发与液相混合相互竞争造成反应器内有一定的温度梯度：氧浓度梯度和温度梯度共同作用使得CO2．CO量增多,燃烧副反应加剧,这其中氧浓度梯度起主要作用。     

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.     

Ammonia‐functionalized particulate poly(glycidyl methacrylate‐co‐ethylene dimethacrylate)‐based polymers as stationary phase for protein retention

A poly(glycidyl methacrylate‐co‐ethylene dimethacrylate) polymeric matrix was synthesized, amino‐functionalized and packed to be tested for protein retention and separation. Functionalization was carried out by reacting the epoxy groups with 30% ammonia solution to provide amino groups for anion exchange with proteins. Physical characterization of the particulate material showed that the functionalized polymer is macroporous and exhibits a unimodal pore size distribution. The resin presents good thermal stability. Chromatographic characterization using bovine serum albumin (BSA) and α‐lactalbumin as model proteins showed good retention properties for the amino‐functionalized matrix. The values obtained for the equilibrium constant (K) were around 3 for BSA, both in batch reactor and column‐packed operation mode. The K values for α‐lactalbumin were even higher, 8.01 and 4.65 for the batch and column‐packed experiments, respectively. Finally, axial dispersion coefficients were obtained, with a constant value of 0.0141 cm²s^?1 for BSA. Higher values were obtained for α‐lactalbumin, but in this case the axial dispersion coefficient was not constant and its values were dependent on the flows used. © 2001 Society of Chemical Industry     

Removal of carbon disulfide: Experimental and modeling results

A mathematical model describing the rate of carbon disulfide (CS₂) removal has been developed. Kinetic studies were carried out in a fixed-bed reactor under atmospheric pressure and a range of temperatures (30-70 °C). The effects of flow rate, CS₂ inlet concentration, temperature and relative humidity were analyzed. A kinetic model based on axial dispersion, external and internal mass-transfer resistances, as well as effects of S deposition on the inner-face of the catalyst was in agreement with the CS₂ experimental breakthrough curves. The mathematical model can be used for process design and scale-up of similar systems.     

Micromixing effects on a parallel reaction in flow reactors

Using the micromixing concepts of Danckwerts and Zwietering, the Peclet number Pe has been correlated mathematically to the degree of segregation J for the axial dispersion model. The results were applied to compare the micromixing effects on a model, mixed-order parallel reaction system in continuous flow reactors. Axial dispersion model, and Ng and Rippin's two-environment model were used to find the micromixing effects in tubular and stirred tank reactors, respectively. The performance of these reactors, with varying geometries, has been evaluated in terms of overall conversion, selectivity, and yield under identical operating and reaction conditions. The overall conversion increases in a tubular reactor with the increase in J, irrespective of the kinetic orders. However, in a stirred tank reactor, the conversion is found to be micromixing-sensitive, depending on the order of reaction. For m = 1 and n = 2 (case 1), the conversion is fairly insensitive to micromixing effects while it decreases for m = 0.5 and n = 1 (case 2) with increasing J. For the same extent of micromixing, a tubular reactor gives, in both cases, a higher conversion than a stirred tank reactor. The selectivity, in either case, decreases in both reactors with increasing segregation effects. However, in each case, the selectivity of a tubular reactor was fairly close to that of a stirred tank reactor at the same value of J. As far as the yield is concerned, both reactors achieve nearly the same value, without significant micromixing effects.     

Dynamic modelling of heterogeneous catalytic reactions—I. Theoretical considerations

The problem of modelling heterogeneous catalytic reactions by dynamic methods is considered, using a one-dimensional heterogeneous plug-flow model. Assuming constant reactor inlet temperature, uniform initial temperature profile, reactor inlet concentration step changes and absence of part of the reactants, it is shown that the reactor outlet concentration transients may be used to determine individual steps of the reaction scheme and to estimate kinetic parameters. Reaction steps can be determined by comparison of measured transients with classified transients for typical model reactions. Estimation of kinetic parameters is carried out by using the concentration transients and their time derivatives.     

RTD studies in trickle bed reactors packed with porous particles

The residence time distribution (RTD) for liquid phase in a trickle bed reactor (TBR) has been experimentally studied for air-water system. Experiments were performed in a 15.2 cm diameter column using commerical alumina extrudates with D/d_p ratio equal to 75 to eliminate the radial flow differences. The range of liquid and gas flow rates covered was 3.76 < Re_L < 9.3 and 0 < Re_G < 2.92. The axial dispersion model was used to compute axial dispersion coefficient. The effect of liquid and gas flow rates on total liquid holdup and axial dispersion was investigated. The total liquid holdup has been correlated to liquid and gas flow rates.     

Substrate-inhibited enzyme reaction in a tubular reactor with axial dispersion

For a fixed set of physico-chemical parameters, an isothermal substrate-inhibited enzyme reaction in a tubular reactor with axial dispersion can give rise to multiple steady states. Criteria previously developed for non-isothermal reactions are applied to this case to develop analytic conditions which ensure uniqueness of steady states. The criteria compare well with values obtained from exact numerical calculations.     

Reactor engineering models of complex electrochemical reaction schemes IV. Galvanostatic operation of series reactions with solvent decomposition

A mathematical analysis of series electrochemical reactions with solvent decomposition during galvanostatic batch operation is presented. Performance is compared to the equivalent system during potentiostatic operation by using the reduction of nitrotoluene to azoxytoluene as a representative model reaction scheme.Nomenclature C_j bulk concentration of species j - C_A0 initial concentration of A - C_Bmax minimum concentration of B - E electrode potential - F Faraday number - i _p partial current density of Stepp - i _T total current density - k _lj mass transfer coefficient of species j - k _f P electrochemical rate constant of Stepp - n _p number of electrons associated with Stepp - S electrode area - V batch reactor volume - constant describing potential dependency of reaction rate constant     

Dynamic analysis of a trickle bed reactor by moment technique

The performance of a trickle bed reactor is investigated by the moment technique. Residence time distributions of SO₂ tracer in both gas (Helium) and liquid (distilled water) effluents are used to predict zero reduced and first absolute moments and these values are compared with the derived theoretical expressions. Correlations are suggested for gas-liquid mass transfer coefficient, liquid hold up, and extent of axial mixing in liquid phase.True adsorption equilibrium constant of the system is estimated as 0.378 from liquid full bed experiments and contacting efficiency of the trickle bed reactor is found as 0.987.Effect of axial dispersion is not significant on gas-liquid mass transfer coefficient since absorption factor is small, but is found to be quite important on the true estimation of adsorption factor.     

Liquid‐phase axial dispersion of turbulent gas–liquid co‐current flow through screen‐type static mixers

This article discusses the characteristics of turbulent gas–liquid flow through tubular reactors/contactors equipped with screen‐type static mixers from a macromixing perspective. The effect of changing the reactor configuration, and the operating conditions, were investigated by using four different screen geometries of varying mesh numbers. Residence time distribution experiments were conducted in the turbulent regime (4500 < Re < 29,000). Using a deconvolution technique, the RTD function was extracted to quantify the axial/longitudinal liquid‐phase dispersion coefficient. The findings highlight that axial dispersion increases with an increasing flow rate and/or gas‐phase volume fraction. However, regardless of the number and geometry of the mixing elements, reactor configuration, and/or operating conditions, the recorded liquid‐phase axial dispersion coefficients in the presence of screens was lower than that for an empty pipe. Furthermore, the geometry of the screen was found to directly affect the axial dispersion coefficient in the reactor. © 2016 American Institute of Chemical Engineers AIChE J, 63: 1390–1403, 2017     

Modeling of a Buss‐Kneader as a polymerization reactor for acrylates. Part I: Model validation

The Buss‐Kneader is generally known as a compounding device. Although a reasonable number of papers have been published on extruders as polymerization reactors, only little is known about the behavior of the Buss‐Kneader when used as a polymerization reactor. Its good mixing properties in the radial and axial directions make it a suitable reactor for exothermal polymerization reactions. This paper describes experiments with the co‐polymerization of n‐butyl acrylate and hydroxyethyl methacrylate in a Buss‐Kneader. For model calculations the Buss‐Kneader was treated as a plug flow reactor with axial dispersion. Experimental results on axial temperature profile, monomer conversion and molecular weight are compared with model calculations. Model parameters are based on independently measured data on the heat transfer coefficient, axial dispersion and polymerization kinetics.