Fast binary reactions in a heterogeneous catalytic batch reactor

When heterogeneous chemical reaction is sufficiently fast, transport of reactants becomes limiting. In a well stirred, batch reactor, macroscopic concentration gradients can be eliminated as a factor limiting the rate of reaction, leaving only the mesoscopic mass transfer of reactants to the surface of the catalyst as limiting, if the reaction does not occur inside a porous support. Here, a transformation of the governing equations for the time-dependence of bulk and surface concentrations results in second order ODE in time and a single nonlinear constraint with boundary values at the initial and infinite times for two auxiliary variables termed modified Thiele moduli. This system of two equations—one differential, one algebraic—and two unknowns is an exact consequence of the governing equations (three ODEs and three algebraic constraints). The power of this formulation is demonstrated by analytic solutions for irreversible and nearly irreversible theories. These solutions are corroborated by full nonlinear numerical computations of the boundary value problem, for the case when asymmetric mass transfer coefficients admit the possibility that the mode of operation switches from relative surface depletion of one reactant to depletion of the other in a binary reaction. The modified Thiele modulus formulation reveals the time scale for the switch over, as well as giving a reliable prediction for the time scale for 99% conversion based on the switch time identified from the irreversible theory.     

Simulations of mass transfer limited reaction in a moving droplet to study transport limited characteristics

Transport limited heterogeneous reactions with asymmetric transport rates in the non-reacting phase can exhibit an interesting switch in the concentrations of the reactants in the reacting phase from one limiting reactant to the other. This switch, called “cross-over” [Mchedlov-Petrossyan P.O., Khomenko G., Zimmerman W.B., 2003a. Nearly irreversible, fast heterogeneous reactions in premixed flow. Chemical Engineering Science 58, 3005-3023; Mchedlov-Petrossyan P.O., Zimmerman W.B., Khomenko G.A., 2003b. Fast binary reactions in a heterogeneous catalytic batch reactor. Chemical Engineering Science 58, 2691-2703], relates to the optimum design of the tubular reactor as all the reactants in the reacting phase are completely consumed at cross-over. The cross-over phenomenon, which has been studied by a number of researchers using phenomenological modelling, is investigated here by developing a distributed model using level-set simulations, in order to explore the possibility of the existence of cross-over in the frame of reference of a moving droplet. Cross-over occurs for a droplet moving due to buoyancy with asymmetric transfer rates of the reactants in the non-reacting phase and an instantaneous reaction occurring inside the droplet (reacting phase). The cross-over length obtained using the level-set simulation is found to be within 0.7-8% of that obtained using the phenomenological model. Computational experiments are performed by varying the ratios of the initial concentrations of the reactants and the transfer rates of the reactants, in order to obtain the parametric region for the existence of cross-over which is also compared with the theoretical prediction.     

Non-equilibrium effects on fast reactions in a heterogeneous batch reactor

Recent studies by the authors of the heterogeneous catalysis of fast binary reactions have taken a dynamical systems approach, assuming that fast enough reactions are confined to a manifold upon which surface equilibrium holds. This approximation makes substantial simplification possible, for instance in the case of a batch reactor, it allows a naturally sixth order system to be approximated by a two dimensional manifold for the dynamics of two modified Thiele moduli. Nevertheless, a proper assessment of how much faster must the velocity of surface reaction be than the velocity of mass transfer to the catalytic surface before the quasi-equilibrium on the surface holds should be made. In this paper, a theory for the systematic correction to infinitely fast reactions is made for large but finite velocity reactions. It is compared to full numerical solutions to the model equations. Recommendations about the regime of applicability of the quasi-equilibrium approximation are made. In general, the predictions of the quasi-equilibrium theory hold for ratios of mass transfer coefficients to reaction velocity ξ of less than 1/1000, with qualitative agreement in regimes of less than 1/100. The general trend, however, is that the stronger the kinetic asymmetry between the mass transfer coefficients of the reactants, the slower the reaction rate can be and still have the quasi-equilibrium theory hold. A perturbation analysis demonstrates that the quasi-equilibrium theory is a regular limit of the fast non-equilibrium theory. In the irreversible case, a matched asymptotic analysis gives the same prediction for the switch time from effective surface depletion of one reagent to the other as the quasi-equilibrium theory. Furthermore, it gives an estimate of the smoothing out of the transition zone with a temporal width of ξ^1/2. It should be noted that the continual drive for improved catalyst activities inevitably leads to mass transfer limited reactions, and thus this regime is not uncommon.     

Simulation of an extractive electrochemical reaction operated under limiting current conditions in a parallel-plate reactor

An extractive electrochemical reaction, which uses an extractive liquid (β) to separate the adherent product (S) on the electrode surface (e) in order to allow the continuation of the reaction (R) occurring electrochemically in a reacting phase (α), was investigated with the aim of studying the kinetics and reactor design. A kinetic model based on the competitive surface coverage (θ) by the reactant and the product, was developed to describe the performance in a parallel-plate reactor. The electrochemical reaction rate, defined asKmAX _α/e(1-θ)(C _R/α-C _R/e), is equal to the extraction rate, defined asKdAX _β/Seθ(C _S ^β/sat -C _S/β) under steady-state conditions whereX is the dispersion function andA is the specific surface area. Simulation under limiting current conditions reveals that this system is dependent on the volume ratio of the two liquids, the dispersion effect and the reactor geometry and diffusion coefficients. Three dimensionless parametersΦ _k(=K _m/K _d),Φ _x (=X _α/e/X _β/Se) andΦ _v ( =X _β/X _α) were used to describe this extractive electrochemical reaction.     

Parametric sensitivity of a CSTR

It is shown that the generalized sensitivity criterion recently developed in the context of thermal explosions and tubular reactors can be easily applied in the case of a CSTR as well. An illustrative example concerning sensitivity analysis of a single nth order irreversible exothermic reaction in a nonadiabatic CSTR is presented. A generalized region of parametric sensitivity is identified where the reactor temperature is parametrically sensitive simultaneously to all the input parameters. Asymptotic analysis for large heats of reaction is performed to investigate limiting behaviour, which leads to the classical Semenov limit in the case of large activation energies. It is shown that parametric sensitivity can occur even when unique steady states exist for all Damköhler number (Da) values. Furthermore, if operating conditions are chosen so as to avoid completely the possibility of parametric sensitivity for all Da, then the possibility of steady-state multiplicity is automatically avoided.     

Effects of transport and phase equilibrium on fast, nearly irreversible reactive extraction

Integration of reaction and separation can be exploited to drive reversible reactions in the direction of the desired product using multiphase flow contacting. In the case of nearly irreversible, fast reactions, however, the dynamics of the product have little influence on the reactor efficiency in say liquid-liquid reactive extraction. A similar intensification in reaction efficiency to reactive separation can be achieved by exploiting phase equilibrium or asymmetry in mass transfer rates of the reactants. Here, a model for two-layer biphasic flow and homogeneous reaction is proposed for co-current reactive extraction, demonstrating that localization and intensification of reaction occurs in the region between the entrance and crossover. Crossover occurs if the reactant in stoichiometric deficit preferentially populates the reacting phase due to sufficient imbalance in either mass transfer coefficients or phase equilibrium. We develop an infinite Peclet number (convection dominates over bulk diffusion) model that indicates that crossover occurs when

     

Dynamic shear in continuous-flow rotating-disk catalytic reactors with stress-sensitive kinetics based on Curie's theorem in non-equilibrium thermodynamics

Stress-sensitive response is simulated in a modified parallel-disk reactor that implements steady and unidirectional dynamic shear in the creeping flow regime. Reactants chemisorb on the surface of the rotating plate, and catalytic sites are replenished from the bulk fluid via radial and tangential flow accompanied by transverse diffusion in the z-direction toward the active surface. Chemical reaction is enhanced by viscous shear at the interface between the bulk fluid and the rotating plate. This heterogeneous catalytic rotational reactor is modeled via radial convection and axial diffusion with angular symmetry in cylindrical coordinates. The reaction/diffusion boundary condition on the surface of the rotating plate accounts for stress-sensitive reactant consumption via the zr- and zΘ-elements of the velocity gradient tensor at the catalytic surface. Linear transport laws in chemically reactive systems that obey Curie's theorem predict the existence of cross-phenomena between scalar reaction rates and the magnitude of the second-rank velocity gradient tensor, selecting only those elements of ?v experienced by reactants that are chemisorbed on the surface of the rotating plate. Stress sensitivity via the formalism of irreversible thermodynamics introduces a zeroth-order contribution to heterogeneous consumption rates that must be quenched when reactants or active vacant sites are not present on the surface of the rotating plate. Rotating-disk reactor simulations are presented for simple 1st-order, simple 2nd-order, and complex heterogeneous stress-free kinetics, where the latter considers Langmuir-type dissociative adsorption of one of the reactants. Accurate design of rotating-disk reactors must consider stress-sensitivity when the shear-rate-based Damköhler number (i.e., ratio of the stress-dependent zeroth-order consumption rate relative to the rate of reactant diffusion toward the active surface) is greater than its threshold value which increases at higher stress-free Damköhler numbers. Modulated rotation of the catalytically active plate demonstrates that these rotating-disk reactors must operate above a threshold for the stress-sensitive Damköhler number, identified under steady shear conditions, before dynamic shear has a distinguishable effect on reactor performance.     

Behaviour of an elementary oxidation reaction in a semi-batch reactor

The dynamics of a non-isothermal bi-molecular gas-phase reaction in a semi-batch reactor is investigated. It is assumed that one of the reactants flows into a reactor containing the second. A reduced model is obtained by making a ‘pool-chemical’ approximation on the concentration of the reactant initially in the reactor. The region in parameter space in which oscillations are observable in the full transient model is estimated by determining the Hopf bifurcation locus in the reduced system. The contribution of the current work is its comparative study of the behaviour of the full system to that of the pool-chemical model. Although the reaction scheme is symmetric with respect to the reactants the regions of oscillatory behaviour are not identical because the reactants have different heat capacities.     

STEP COVERAGE PREDICTIONS USING COMBINED REACTOR SCALE AND FEATURE SCALE MODELS FOR BLANKET TUNGSTEN LPCVD

A reactor scale model (RSM) for a stagnation point, single wafer reactor for blanket tungsten LPCVD is used to calculate concentrations at the wafer surface. These concentrations and the wafer temperature, which is assumed to be measurable, are needed to determine the local tungsten deposition rate on the wafer and local film conformality (step coverage) in features on patterned wafers. Two feature scale models (FSMs) are used to determine step coverages in infinite trenches which have rectangular initial cross sections and an aspect ratio of five, as a function of reactor operating conditions; 1. a continuum-like diffusion-reaction model (DRM) for simultaneous Knudsen diffusion and heterogeneous surface reactions, and 2. a flux based model which includes ballistic transport of molecules and heterogeneous surface reactions (BTRM).|The RSM establishes “boundary conditions” for the feature scale models, by providing the flux of each species to the local wafer surface. Step coverages predicted using the FSMs with the reactant partial pressures at the wafer surface can be significantly lower than those predicted using reactant partial pressures at the reactor inlet, due to depletion of reactants. The flux based BTRM predicts higher step coverages than the DRM for the same wafer surface conditions.     

A STUDY ON THE PERFORMANCE OF THE CATALYTIC POROUS-WALL THREE-PHASE REACTOR

A theoretical investigation of a catalytic porous-wall reactor in which gaseous and liquid reactants approach each other from opposite sides of the catalyst is undertaken. Equations for the annular liquid-channel are coupled with those for the catalytic wall and solved numerically and analytically using a simplified model. For the model reaction under study, the main design and operation parameters which affect reactor performance are the Thiele modulus, Peclet number, width of the liquid channel and the inlet concentration of the reactant in the liquid phase.

The effect of reactor configuration is peculiar to the cylindrical geometry because the thickness and relative location of the catalytic wall as well as the selection of the liquid and gas channels can influence the reactor performance. Thin-walled catalyst tubes have larger effectiveness factors and as the tube radius approaches that of the reactor, conversion in the reactor increases especially when the liquid is saturated with the gaseous reactant. Concentration of the liquid reactant in the feed has a significant effect if the reactant is depleted at some point inside the catalyst wall. Since the reaction zone width can be adjusted by changing the feed composition, this might have important implications with respect to selectivity.     

Experimental study of mass transfer limited reaction—Part II: Existence of cross-over phenomenon

In Part II, we extend the inverse methodology which is discussed in Part I to various experiments performed in a tubular reactor to infer mass transfer coefficients. Mass transfer coefficients, inferred from the concentrations of the reactants which are extracted from the absorbance of the reaction mixture using multicomponent spectrum analysis, are then used to solve the convection-diffusion-reaction equations for the concentrations of the reactants in the bulk phase and in the dispersed phase, to explore the possibility of cross-over for a mass transfer limited reaction. Experiments are performed incorporating the asymmetry in the transport rates of the premixed reactants, which is the potential reason for the existence of cross-over. The different mass transfer coefficients of the two premixed reactants indeed indicate a switch in the concentration of the reactants in the dispersed phase, which is termed as “cross-over”. The experimental results are further analysed by validating the theoretical criterion proposed by Mchedlov-Petrossyan et al. (2003, Chem. Eng. Sci. 58, 3005-3023 & 2691-2703) to obtain the parametric space for the existence of cross-over, in order to optimize the length of the tubular reactor.     

An H-shaped design for membraneless micro fuel cells

The geometry of micro fuel cell design influences the reactants’ mixing and the depletion at downstream of the channel and thus effects the cell performance. This paper proposes a design for membraneless micro fuel cells with an H-shaped cross-section and a small passage between the anode and cathode channels. The small passage restricts the mixing of the anode and cathode fluids in the main channel. Numerical simulations with electro-chemical reactions have been carried out to investigate the distribution and crossover of the reactants and also the mixing and depletion regions in the system. Results show that optimizing the size of the passage between the anode and cathode channels plays an important role in reducing the mixing of reactants and in increasing fuel utilization. The H-shaped design shows that the mixing region is reduced in size by about 20%, so the H-shaped design has 10 times less fuel crossover than the conventional rectangular design. Moreover, fuel utilization is increased by about 8% with respect to that of the conventional rectangular design. 90° angles between the passage and the anode and cathode channels provide the best layout for this H-shaped design. The aspect ratio 0.083 for the anode and cathode channels exhibits 23% higher fuel utilization than the conventional rectangular design. Moreover, the size of the passage has a significant influence on the boundary layer thickness, the depletion region and the current density. A micro fabrication of the H-shaped design was made and the open circuit voltages were measured. The results are compared with those in the available literature.     

Nonlinear model reconstruction by frequency and amplitude response for a heterogeneous binary reaction in a chemostat

Here we present an analysis of a binary heterogeneous reaction in a chemostat for the particular case of reagents with unequal mass transfer coefficients. For fast irreversible kinetics, there are two potential steady states—the dispersed phase is preferentially populated by one reagent and essentially depleted of the other. The selection of which steady state occurs depends on the operating parameter which is the concentration ratio σ of the slower (B) to the faster (A) transferring reagent in the feed stream. The demarcation between these operating regimes is the critical value of this ratio:

     

Theoretical analysis of reactant dosing concepts to perform parallel-series reactions

The possibility of enhancing selectivities and yields in networks of parallel and series reactions is investigated theoretically. Isothermal tubular reactors are considered where reactants can be introduced at the entrance and also over the wall. The latter way of dosing could be realised, e.g., in a membrane reactor where one or several reactants can be dosed through a porous reactor wall. Besides numerical solutions of the underlying mass balance equations of simplified reactor models, instructive analytical solutions were derived which are valid under certain constraints. Using these solutions an optimisation of the reactor performance could be performed. As objective function the molar fraction of a desired intermediate product at the reactor outlet was maximised. The impact of influencing via the dosing strategy applied the local composition (and thus the local reaction rates) and the component residence time distributions is elucidated.     

Effect of surface diffusion on the selectivity of catalytic reactions

Analysis is made on the effect of surface diffusion on the selectivity of a catalyst for a consecutive reaction A→B→C in a well-mixed stirred tank reactor. The catalyst is composed of an α region despersed on the β region. A→B is assumed to occur on the α region and B→C on the β region. Migration of B from α to B proceeds both by surface diffusion and gas phase transport. Influence of the flow rate through the reactor, the crystallite size of α, and the loading of the catalyst on the selectivity for C in the presence of surface diffusion are discussed. Under otherwise identical conditions, selectivity is increased by surface diffusion. The optimum condition for the production of C is also discussed.     

Modelling and simulation of a direct ethanol fuel cell considering multistep electrochemical reactions, transport processes and mixed potentials

In this work a one-dimensional mathematical model of a direct ethanol fuel cell (DEFC) is presented. The electrochemical oxidation of ethanol in the catalyst layers is described by several reaction steps leading to surface coverage with adsorbed intermediates (CH₃CO, CO, CH₃ and OH) and to the final products acetaldehyde, acetic acid and CO₂. A bifunctional reaction mechanism is assumed for the activation of water on a binary catalyst favouring the further oxidation of adsorbates blocking active catalyst sites. The chemical reactions are highly coupled with the charge and reactant transport. The model accounts for crossover of the reactants through the membrane leading to the phenomenon of cathode and anode mixed potentials due to the parasitic oxidation and reduction of ethanol and oxygen, respectively. Polarisation curves of a DEFC were recorded for various ethanol feed concentrations and were used as reference data for the simulation. Based on one set of model parameters the characteristic of electronic and protonic potential, the relative surface coverage and the parasitic current densities in the catalyst layers were studied.     

Interaction parameters in ternary polystyrene solutions at high temperature

A combination of dilute solution viscometry and Rayleigh light scattering has been used to evaluate experimentally the interaction parameters χ_ij in solutions comprising tetralin (1), polystyrene (2) and 3-methyl cyclohexanol (3). The measurements were carried out at 371.5 K, where binary solutions of the polymer in 3-methyl cyclohexanol are under θ-conditions and tetralin is a thermodynamically good solvent for polystyrene. For polymer-solvent interaction, values of χ₁₂ = 0.40 ± 0.01 and χ₂₃ = 0.50 were obtained. The solvent-solvent interaction parameter χ₁₃ was composition dependent, having limiting values of 0.52 and 0.73 at X₁ = 0 and X₁ = 1, respectively, where X₁ is the mol fraction of liquid 1 in binary mixtures of liquids (1) and (3).     

A novel reverse flow reactor coupling endothermic and exothermic reactions: Part II: Sequential reactor configuration for reversible endothermic reactions

The new reactor concept for highly endothermic reactions at elevated temperatures with possible rapid catalyst deactivation based on the indirect coupling of endothermic and exothermic reactions in reverse flow, developed for irreversible reactions in Part I, has been extended to reversible endothermic reactions for the sequential reactor configuration. In the sequential reactor configuration, the endothermic and exothermic reactants are fed discontinuously and sequentially to the same catalyst bed, which acts as an energy repository delivering energy during the endothermic reaction phase and storing energy during the consecutive exothermic reaction phase. The periodic flow reversals to incorporate recuperative heat exchange result in low temperatures at both reactor ends, while high temperatures prevail in the centre of the reactor. For reversible endothermic reactions, these low exit temperatures can shift the equilibrium back towards the reactants side, causing ‘back-conversion’ at the reactor outlet.The extent of back-conversion is investigated for the propane dehydrogenation/methane combustion reaction system, considering a worst case scenario for the kinetics by assuming that the propylene hydrogenation reaction rate at low temperatures is only limited by mass transfer. It is shown for this reaction system that full equilibrium conversion of the endothermic reactants cannot be combined with recuperative heat exchange, if the reactor is filled entirely with active catalyst. Inactive sections installed at the reactor ends can reduce this back-conversion, but cannot completely prevent it. Furthermore, undesired high temperature peaks can be formed at the transition point between the inactive and active sections, exceeding the maximum allowable temperature (at least for the relatively fast combustion reactions).A new solution is introduced to achieve both full equilibrium conversion and recuperative heat exchange while simultaneously avoiding too high temperatures, even for the worst case scenario of very fast propylene hydrogenation and fuel combustion reaction rates. The proposed solution utilises the movement of the temperature fronts in the sequential reactor configuration and employs less active sections installed at either end of the active catalyst bed and completely inactive sections at the reactor ends, whereas propane combustion is used for energy supply. Finally, it is shown that the plateau temperature can be effectively controlled by simultaneous combustion of propane and methane during the exothermic reaction phase.     

Simulation and application of response surface methodology to a nylon‐6 hydrolytic polymerization in a semibatch reactor

This work presents an experimental design methodology combined with computational simulation to correlate the influence of operational conditions and reactants charge in the numeric average molecular weight (MWN) as well as on monomer conversion (X_CL), for the hydrolytic polymerization of nylon‐6 in a semibatch reactor. It evaluated the reaction temperature, the pressure profile, and the proportion of reactants in the charge. Experimental design was used to screen the most statistically significant variables and to develop a reliable predictive model for each response. The combined use of the models can be applied for process optimization, by establishing MWN and maximum X_CL as objective functions. Responses surface allowed the visualization of the responses behavior when changing the independent variables and therefore to identify the optimal tendencies. This work demonstrates that such methodology can be applied for optimization of complex processes like the hydrolytic polymerization of nylon‐6. This polymerization has many side reactions occurring at the same time, which are sensitive to different profiles of pressure and temperature that are applied. This evaluation is quite interesting as such profiles are necessary to perform the several polymerization steps and have a significant impact on product characteristics and therefore in its applications. © 2012 Wiley Periodicals, Inc. J. Appl. Polym. Sci., 2013     

Manganese oxide dissociation kinetics for the Mn2O3 thermochemical water-splitting cycle. Part 2: CFD model

A detailed CFD model was developed to better understand the kinetics and transport characteristics of Mn₂O₃ dissociation in an aerosol flow reactor (AFR). Radiation was the dominant mode of heat transfer (T=1673–1873 K) and a relation for radiation from the walls and attenuating volume to any volume element in the reactor was developed. Results compared favorably to what was observed experimentally and showed that solid feed concentration into a high temperature AFR has a significant effect on conversion. If the concentration becomes too high, simply increasing the temperature of the reactor wall will not provide enough energy to drive the reaction to a high conversion. Additionally, the model showed that the bulk gas rate does not have that much effect on conversion in the AFR. This is important as a higher gas flow rate provides a faster quench, limiting the possibility of a recombination reaction. Finally, a case study for the diameter of the reactor tube showed that a higher conversion could be achieved by decreasing the size of the reactor tube. Multiple reaction tubes of smaller diameter are likely needed for scale-up.