An improved shooting method for computation of effectiveness factors in porous catalysts

When applied to compute the concentration profile and the effectiveness factor in a porous catalyst, the conventional shooting method requires that the concentration at the catalyst center y(0) must be greater than zero to yield a nontrivial solution. Since, for reactions with reaction orders less than one, y(0) can become zero in the solution when the Thiele modulus is large, the applicability of the conventional method is limited. In this study we propose an improved shooting method which is absent of such limitations and yet retains the merits of the conventional shooting method. The proposed method is shown to be robust, efficient in computation of the effectiveness factors. Not only a single steady state, but also multiple steady states, can easily be identified with the proposed method. A simple rule for stability of a state is also given in the method.     

A robust iterative method of computing effectiveness factors in porous catalysts

A robust, efficient numerical method for computing the effectiveness factor of a heterogeneous reaction in a catalyst is developed in this study. The method is based on shooting at the outer surface of the catalyst and is optimized for an accurate estimation of the concentration gradient at the outer surface. The shooting at the outer surface, however, is inherently unstable for a cylindrical or a spherical catalyst, and it also can become unstable with an increasing Thiele modulus even for a slab catalyst. From an analysis of the governing equation, however, three criteria are developed to make the method work efficiently in the presence of instability. The numerical method is shown to be effective for diverse kinetic expressions, from simple power-law to sophisticated Langmiur-Hinselwood kinetics, and for isothermal or non-isothermal catalysts. The method is also shown to be easily applicable to a more complex case of multiple steady states, estimating all the corresponding effectiveness factors. In this method, a simple rule that determines the stability of each steady state is proposed.     

An approximation method for the effectiveness factor in porous catalysts

An efficient method for computing approximate value of the effectiveness factor is presented. The method is developed for an arbitrary rate expression and for three representative catalyst shapes, namely, an infinite slab, an infinite cylinder and a sphere. In the method, two new asymptotic expansions for small and large Thiele moduli are developed and joined together at an intermediate value of the Thiele modulus, which can be calculated by a simple equation. The approximation is highly accurate for small and large Thiele moduli, and it has small or negligible errors for intermediate values of the Thiele modulus. As demonstrated with examples, the approximation method is fast, straightforward and is a dependable alternative to numerical methods for computing exact values of the effectiveness factor.     

The determination of possibility of multiple steady states for complex catalytic reactions in an isothermal CSTR

A class of complex reaction networks with deficiency from two to six, involving isothermal catalytic reactions in a continuous flow stirred tank reactor (CSTR), is determined to have the capacity to admit multiple positive steady states by implementation of the combination of the Deficiency One Algorithm and the Subnetwork Corollary. A set of rate constants and two corresponding positive steady states, one stable and the other one unstable, are constructed for four examples.     

A novel kind of multiple steady states characteristics in the dividing wall column

In this article, one kind of multiple steady states(MSS) phenomenon was investigated for a dividing wall column(DWC). The four-section model constructed in Aspen Plus was employed to simulate two DWC cases: mixture of n-hexane, n-heptane and n-octane; system of methanol, ethanol and n-propanol. It can be seen that there is a range of vapor split ratio in which multiple solutions of reflux ratio exist for fixed DWC configuration with the same feed and product streams. The width and the curve shapes of the MSS region, and the number of solutions change with the liquid split ratio. This MSS phenomenon was further explained using the component recovery around the prefractionator and the component recycling flow inside the DWC. This MSS phenomenon is helpful for DWC design by knowing the probable existence of multiple solutions in advance.     

Multiple steady states in two-phase reactors under boiling conditions

In this paper, we analyze the nonlinear behavior of two-phase reactors under boiling conditions. First we focus on a simple nth-order reaction of the form A→B, which allows a rigorous analytical treatment. Three necessary conditions for the existence of multiple steady states have been identified: the reactant A has to be the light-boiling component, the difference in boiling point temperatures between the reactant A and the product B has to be sufficiently large, and the order of the reaction has to be less than some physical parameter α. This parameter α can be interpreted as a measure for the phase-equilibrium-driven self-inhibition of the reaction mechanism. Thus, we have found an elegant explanation for the occurrence of multiplicities. Analytical and therefore general quantitative criteria identifying the regions of multiplicity for the model system are presented. Practical relevance of our results is demonstrated by means of two examples, the Monsanto process for the production of acetic acid and the ethylene glycol reactive distillation system.     

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。     

The effect of non‐ideal mixing on the number of steady states and dynamic behaviour for autocatalytical reactions in a CSTR

In this paper, we analyze the concentration multiplicity and dynamic behavior for an autocatalytical reaction, A + R → (n + 1)R + products with an overall rate expression given by – γ_a= kc_a^pc_r^r(p > 0 and r > 0) in a imperfectly mixed (Cholette's model) CSTR. We proved that non‐ideal mixing had an effect on the number of steady states and dynamic behavior for the reaction orders r > 1 and r = 1. However, the above‐mentioned effect does not happen for the reaction order r < 1. Furthermore, a simulated example was used to demonstrate our results.     

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.     

Use of stirred batch reactors for the assessment of adsorption constants in porous solid catalysts with simultaneous diffusion and reaction. Theoretical analysis

A simple, pseudo-equilibrium model was derived for a catalytic system with a first order chemical reaction and simultaneous diffusive and adsorptive processes, in order to assess the corresponding kinetics and Henry law's-type adsorption parameters. Solutions from this model were compared to exact solutions from a more detailed, general model. It was shown that under most of the experimental conditions used in stirred batch reactors and the usual model considerations, it is only possible to assess apparent adsorption parameters. Also, we observed that a stable relationship between the concentrations in the gas and solid phases is reached. The error produced in assuming that the apparent adsorption constant is the real one was calculated to be very important. The value of the apparent adsorption constant depends on various system properties and experimental conditions, such as the Thiele modulus, the amount of catalyst and the contact time. The ratio between the apparent and real adsorption constants was shown to be the transient effectiveness factor at any moment. This ratio reaches a maximum value for the pseudo-equilibrium state, that is always larger than the steady-state effectiveness factor, becoming closer as long as the system's adsorption capacity decreases. The analysis determines the operative conditions to reduce the parametric correlation. Also a criterion for the applicability of usual approximations in the assessment of kinetics and equilibrium adsorption parameters in porous solid catalysts by means of pulse injection methods is established.     

Effectiveness factor of a catalytic sphere in particle assemblage approached with a cell model

The cell model approach was extended to the catalytic reaction over a spherical catalyst pellet and the effects of several parameters on the effectiveness factor were investigated via numerical simulation. The proposed mathematical model consisted of the partial differential equations for gas reactant flow in the external gas phase around a catalyst sphere, mass transport in the gas phase, and two-dimensional diffusion-reaction in the catalytic sphere. Numerical simulation in the orthogonal boundary fitted reference frame demonstrated the decrease in effectiveness factor of a catalyst sphere due to the external mass transfer resistance and inter-particle hydrodynamic interaction. It was observed that the voidage of the particle assemblage shows more recognized influence than the Peclet and Reynolds numbers of particles and the azimuthal diffusion of reactant inside the particle was negligible. On this basis, a simplified set of cell model equations for catalytic reaction in particle assemblages were proposed. An example of optimizing search indicated the present model may be used to detect the most strong influence on catalytic reaction in particle assemblage over the practical ranges of operating and designing parameters to avoid excessive loss in the effectiveness factor.     

Experimental and numerical investigation of supported rhodium catalysts for partial oxidation of methane in exhaust gas diluted reaction mixtures

The partial oxidation of methane/oxygen mixtures with large exhaust gas dilution (46.3 vol% H₂O and 23.1 vol% CO₂) has been investigated experimentally and numerically over Rh/CeO₂-ZrO₂, Rh/ZrO₂ and Rh/α-Al₂O₃ catalysts. Experiments were carried out in a short-contact-time reactor at 5 bar and included exhaust gas analysis, temperature measurements along the reactor, and catalyst characterization. Additional experiments were performed in an optically accessible channel-flow reactor and involved in situ Raman measurements of major gas-phase species concentrations over the catalyst boundary layer and laser-induced fluorescence (LIF) of formaldehyde. A full elliptic two-dimensional numerical code that included elementary hetero-/homogeneous chemical reaction schemes and relevant heat transfer mechanisms in the solid was used in the simulations. The employed heterogeneous reaction mechanism, including only active Rh sites, reproduced the experiments with good accuracy. The ratio of active to geometrical surface area, deduced from hydrogen chemisorption measurements, was the single model parameter needed to account for the effect of different supports. This indicated that water activation occurring on support sites, resulting in inverse OH spillover from the support to the noble metal sites, could be neglected under the present conditions with high water dilution. An evident relationship between noble metal dispersion and catalytic behavior, in terms of methane conversion and synthesis gas yields, could be established. Both measurements and predictions indicated that an increasing Rh dispersion (in the order Rh/α-Al₂O₃, Rh/ZrO₂, and Rh/CeO₂-ZrO₂) resulted in higher methane conversions, lower surface temperatures, and higher synthesis gas yields.     

Perturbation techniques for accelerated convergence of cyclic steady state (CSS) in oxygen VSA simulations

Non-isothermal PSA bulk gas separations are notoriously slow to converge to cyclic steady state (CSS). Most numerical simulators that model the slow transient behaviour of these processes share this slow pace of convergence. This difficulty with numerical simulators has long been recognised, and has hampered the optimisation of oxygen VSA.This paper outlines the application of perturbation techniques to enable more rapid determination of CSS temperature profiles. A number of different techniques are proposed. The simplest approach based on experimental observations involves a two-time-scale decomposition of the adsorbent temperature profile. This decomposition of the temperature term enables the rapid numerical determination of the fast (or adsorptive) component of the temperature, followed by the direct determination of an estimated cyclic steady-state slow time-scale (or convective) temperature. It is demonstrated that this approach is the same as a zeroth-order multiple scale analysis (MSA) approach and a simple application of the Krylov-Bogoliubov (K-B) method of averaging. This study compares the results of this acceleration technique for determining CSS with a full numerical model. The results indicate that the proposed acceleration technique based on perturbation methods provides fast and reasonable estimates of the CSS temperature profile with some limitations. Recognising these limitations of the zeroth-order approach, a first-order MSA is developed. This compares better with full numerical simulations, and could be used to augment and complement existing acceleration techniques. It is shown that a first-order MSA approach can also be used to capture the dynamic temperature response of the oxygen VSA process, in addition to the CSS axial temperature profile.The perturbation techniques presented here are limited to a series expansion of the temperature term in the energy balance, and mass balance is ignored in this analysis. A procedure is outlined where a K-B approach can be used to incorporate a perturbation series expansion in the mass balance. We show that these perturbation techniques not only offer potential for simulation acceleration, but also offer useful insights into the thermal convergence to CSS in oxygen VSA.     

Effectiveness factor for spatial gradient sensing in living cells

We consider the steady-state pattern of messenger molecules produced in the membrane of a cell perceiving and responding to an extracellular gradient of chemoattractant, which directs cell movement towards the chemoattractant source. Specifically, we analyze the undesirable effect of lateral diffusion in blurring the intracellular messenger profile. The concept of an effectiveness factor, akin to the analysis of reactions in porous catalysts, is applied to the spatial gradient sensing problem, with the distinction that slow, not fast, diffusion is required for effective gradient sensing. Analytical effectiveness factor expressions are derived for ideal geometries and then generalized to arbitrary cell shapes. In the case of mouse fibroblasts responding to gradients of platelet-derived growth factor, we conclude that the cell morphology and orientation with respect to the gradient can dictate whether messenger diffusion obliterates gradient sensing or has very little effect. The analysis outlined here allows the effect of intracellular messenger diffusion on spatial gradient sensing to be quantified for individual cells.     

Multiplicity Regions in a Moving‐Bed Reactor: Bifurcation Analysis,Model Extension,and Application for the High‐Temperature Pyrolysis of Methane

This paper reports studies on the bifurcation analysis and the generation of regions of multiplicity in the operation of moving‐bed reactors. The studies were first carried out for a generalized model, which allows the investigation of the effect of various scenarios, such as changes in heat capacity or the number of moles with reaction, as well as the previously unstudied equilibrium and allothermic reactions. The insights gained from this extension were then applied to an actual reaction, namely the high temperature pyrolysis of methane.     

Flow field investigation in a photocatalytic reactor for air treatment (Photo-CREC-air)

The aerodynamic behavior of a photocatalytic reactor for air treatment, Photo-CREC-air, with demonstrated high quantum efficiency performance, is examined using CFX-5.7.1. Photo-CREC-air consists of a venturi section that features low pressure drop and uniform illumination of the photocatalyst, resulting in high oxidation quantum efficiencies. The numerical simulations allowed the identification of several design issues in the original Photo-CREC-air unit, which include extensive boundary layer separation close to the photocatalyst support and regions of flow recirculation that render ca. 77% of the support surface area inactive. The simulations reveal that this issue could be addressed by replacing the wire-mesh basket sidewalls with perforated plates. This modification causes an increase in the pressure drop downstream of the support and achieves significant uniformization of the mass flow and air-photocatalyst contact time distributions.A modified Photo-CREC-air design is also presented and studied using CFX-5.7.1. This modified design is envisaged with the objective of improving UV-irradiation uniformity, an issue that is not completely addressed in the original design due to the shape of the windows and divergent section. CFD simulations reveal that, although the flow field is uniform, mass flow and contact time distributions are not. Nonetheless, this problem is addressed by increasing the pressure drop downstream of the support through the addition of a region modeled as a perforated plate. The simulations reveal that the mass flow and contact time distributions are significantly uniformized once this modification is implemented.     

Kinetic studies in a batch reactor using ion exchange resin catalysts for oxygenates production: Role of mass transfer mechanisms

Oxygenated compounds are usually produced by a reversible liquid phase reaction using ion exchange resins with a bidisperse pore structure as catalyst. Mass transport is mainly controlled by diffusion through the macropores and the mass transfer resistance in the gel microspheres is negligible. Therefore, in this paper a mathematical model of the batch reactor considering diffusion of the species in the external film and then macropore diffusion inside the particle and reaction in the gel microspheres was developed. The numerical solution is implemented through the numerical package PDECOL and detailed explanations of the procedure used are presented. The model was applied to the diethylacetal synthesis using ethanol and acetaldehyde as reactants and Amberlyst 18 as catalyst. The experimental data are fitted with a two-parameter model based on a Langmuir-Hinshelwood rate expression in order to get the true reaction kinetics. The influence of the mass transfer mechanisms is evaluated in terms of the effectiveness factor history during the transient state of the batch reactor. The values of the effectiveness factor calculated at equilibrium with the batch reactor model are compared with those calculated from a steady state infinite bath model.