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.     

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.     

Revisiting the concepts of competition between reaction and diffusion in poisoned catalysts

The competition between diffusion and first‐order irreversible reaction in poisoned catalysts is revisited. Two cases are considered for isothermal slab catalysts: uniform and shell‐progressive (or pore‐mouth) poisoning. Analytical concentration profiles are derived, and the implications on catalyst performance are evaluated in different regimes (chemical‐ or diffusion‐controlled) for different levels of poisoning. It was found that depending on the poisoning mechanism, the activity decay can be more or less pronounced. Being of particular concern is the pore‐mouth poisoning at high Thiele modulus, conditions under which catalyst performance is drastically affected. The reagent concentration profiles allowed the explanation of the phenomena occurring at the particle scale, in particular the effectiveness factors, observed reaction rates, and poisoning factors' dependence on the Thiele modulus and fraction of the poisoned catalyst; it was found that such relationships are dependent on the mechanism of deactivation. © 2012 Canadian Society for Chemical Engineering     

Preparation, testing and modeling of three-dimensionally ordered catalytic layers for electrocatalysis of fuel cell reactions

The arrays of vertically aligned carbon nano-filaments (VACNF) were synthesized by catalytic chemical vapor deposition on TiO_x substrates, obtained via oxidative treatment of polycrystalline Ti and Ti thin films on Si(1 0 0). VACNF were studied using scanning and transmission electron microscopies. The Pt deposition on VACNF was utilized to prepare a set of model catalysts, which were investigated in two fuel cell related reactions: the oxygen reduction (ORR) and the hydrogen oxidation (HOR) reactions. The experimental data were compared with the results of mathematical modeling performed for a fast (quasi)reversible and a slow irreversible electrochemical reaction. The approach made it possible to study electrochemical reactions (HOR, ORR) on nano-materials under well defined mass transport conditions. The influence of the catalytic layer thickness and the Pt coverage on the penetration depth of the reactive species inside the layer and consequently on the performance and on the Pt effectiveness factor were analyzed.     

Dead zones in porous catalysts: Concentration profiles and efficiency factors

This paper examines the conditions under which a dead zone, or a portion of the catalyst devoid of reactant, can form in a porous catalyst in which simultaneous reaction and diffusion are occurring. The condition that allows for the existence of a dead zone is defined by a critical Thiele modulus. When the Thiele modulus – the ratio of chemical reaction to diffusion – is greater than the critical Thiele modulus, a dead zone exists. This dead zone can be mathematically defined by a change of boundary conditions. We examine nth order reactions in isothermal infinite slabs, infinite cylinders, and spheres. In addition, we provide analytical concentration profiles and efficiency factors for zero-order reactions in non-isothermal infinite slabs (in the so-called low beta approximation). We also discuss some common errors and misconceptions associated with this phenomenon.     

好氧颗粒污泥溶解氧传递

假定颗粒污泥为球状,通过实验确定动力学参数,建立氧传质模型并进行了验证。结果表明,利用失活颗粒污泥DO变化,得出颗粒污泥氧扩散系数为0.45×10^-9 m²·s^-1;利用烧杯实验,得出了氧比消耗速率0.10 g O2·（g MLSS）^-1·h^-1,氧半饱和常数为0.65 mg·L^-1。模型求解发现,表面DO越大,DO梯度越大,颗粒污泥表面径向传质能力越强,氧穿透颗粒污泥的距离越远,但是粒径很大时（如半径1.5 mm）,虽然表面DO为4.0 mg·L^-1,但依然没有穿透颗粒污泥,当r/R为0.49～0.65之间时,DO梯度迅速为0;在颗粒污泥半径很小（＜0.5 mm）时,氧完全能穿透整个粒径,同时梯尔模数足够小,内扩散有效因子接近1不变,氧扩散可忽略不计;对于相同的梯尔模数,表面DO高时,内扩散有效因子越大,说明氧扩散的动力越强、受限制的程度越小。     

Overall effectiveness factor of heterogeneous-homogeneous chain reactions with one limited species in a catalytic slurry reactor

The analysis of the overall effectiveness factor for the reaction system involving heterogeneous-homogeneous chain reactions at low concentration of one species is studied. A more general theoretical analysis for estimating the overall effectiveness factor of reaction systems which can be a heterogeneous-homogeneous chain reaction system at low concentration of one species in a slurry reactor is presented, incorporating all the transport parameters. The concepts of overall effectiveness factors for reactions have been extended from the traditional heterogeneous reaction system to the heterogeneous-homogeneous chain reaction system. The effects of reaction mechanism and kinetic models on the overall effectiveness factor are also proposed. Comparisons of the overall effectiveness factor for a slurry reactor with or without considering the reaction mechanism are also obtained.     

Evaluating the effectiveness factor from a 1D approximation fitted at high Thiele modulus: Spanning commercial pellet shapes with linear kinetics

In the present contribution a one-dimensional model (1D model) is employed to account for the effect of pellet shape in the solution of diffusion-catalytic reaction problems. The 1D model free parameter (σ) is fitted so as to match the second term of the series for the effective reaction rate at fast kinetic conditions. The performance of the model was evaluated for a large collection of pellets available commercially, covering almost all the main shapes offered by catalyst manufacturers. For isothermal linear kinetics the effectiveness factors calculated by the 1D model differ at most by 3% from those of the actual 3D pellets.     

Two-parameter model for evaluating effectiveness factor for immobilized enzymes with reversible Michaelis-Menten kinetics

A two-parameter mathematical model was developed to calculate the effectiveness factor for immobilized enzymes in porous spherical particles. The model was resolved for reversible Michaelis-Menten kinetics, including simple Michaelis-Menten and product competitive inhibition kinetics. Since only two dimensionless moduli are involved in the model, the effectiveness factor for the three kinetic equations considered can be estimated by using only one generalized graph.     

表面中毒甲醇成催化剂效率因子

对ＣＯ、ＣＯ２加氢合成甲醇平行反应提出了表面中毒催化剂效率因子的关键组分计算模型，并用正交配置法求解．硫化氢表面中毒深度由失活速率方程求得．计算结果表明。随Ｈ２Ｓ含量增加和使用时间增加，催化剂效率因子减小；随反应温度升高，表面中毒深度增加，又受内扩散影响，催化剂效率因子明显减小．     

New evaluation method for the effectiveness of platinum/carbon electrocatalysts under operating conditions

To facilitate the development of high performance polymer electrolyte fuel cells (PEFCs), we have proposed a new evaluation method for the effectiveness of Pt (Ef_Pt) under actual PEFC operating conditions. We also have examined the dependence of the values of Ef_Pt on experimental parameters, including the ionomer content in the catalyst layer. The Ef_Pt parameter is defined as the ratio of mass activity (MA) in the membrane-electrode assembly (MEA) to the maximum mass activity (MA_max) at high potential, as estimated by use of the channel flow double electrode (CFDE) technique as an ideal case of complete ionic contact with the Pt surface. For the ionomer content in the catalyst layer (Nafion/carbon ratio, N/C), we found a distinct increase of Ef_Pt with increasing N/C, indicating a beneficial effect of the ionic contact and connectivity of the ionic conduction path within the catalyst layer, even though the apparent utilization of platinum (U_Pt) was nearly constant. Typical Ef_Pt values for the mid-range of N/C ratios were found to only reach ca. 12.5% for pure O₂ and T_cell = 65 °C. Furthermore, the Ef_Pt values in air only reached ca. 7.5%. It must be emphasized that there is still much room for the improvement of Ef_Pt.     

Asymptotic effectiveness of a catalyst particle in the form of a hollow cylinder

An explicit formula is presented for the effectiveness factor of a catalyst particle in the form of a hollow cylinder taking into account the reactant diffusion in the fluid phase of the cylindrical core and diffusion and reaction in the outer reactive annulus. The proper normalization to be used in the definition of the generalized Thiele modulus as well as the aspect ratios that lead to the highest effectiveness are also determined. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4020–4024, 2013     

Advances on the development of novel heterogeneous catalysts for transesterification of triglycerides in biodiesel

This paper describes experimental work done towards the search for more profitable and sustainable alternatives regarding biodiesel production, using heterogeneous catalysts instead of the conventional homogenous alkaline catalysts, such as NaOH, KOH or sodium methoxide, for the methanolysis reaction. This experimental work is a first stage on the development and optimization of new solid catalysts, able to produce biodiesel from vegetable oils. The heterogeneous catalytic process has many differences from the currently used in industry homogeneous process. The main advantage is that, it requires lower investment costs, since no need for separation steps of methanol/catalyst, biodiesel/catalyst and glycerine/catalyst. This work resulted in the selection of CaO and CaO modified with Li catalysts, which showed very good catalytic performances with high activity and stability. In fact FAME yields higher than 92% were observed in two consecutive reaction batches without expensive intermediate reactivation procedures. Therefore, those catalysts appear to be suitable for biodiesel production.     

用于生化膜Monod模型的新Thiele模数

A new Thiele＇s modulus, φF, was developed to provide a gradual transition between zero and the first order of kinetics, and to accurately calculate the mass transfer flux and the effectiveness factor for the Monod biofilm. Values of the effectiveness factor, calculated using the new Thiele＇s modulus, were compared with those obtained from numerical solutions and from other published moduli and empirical formulae. The comparison indi- cated that the new Thiele＇s modulus was the best modulus for the Monod biofilm model. In addition, another Thiele＇s modulus, φG, was developed for a Monod biofilm, covered with an external water layer. The overall effectiveness factor can also be calculated by using both moduli φF and φG. The criteria that were proposed for identifica- tion were based on the values of φF and φG, the limiting processes for biomass growth, and substrate conversion. Developed from φF, a new parameter ψ was related uniquely to such features as the depth and shallowness of the generalized substrate concentration profiles inside a Monod biofilm. Criteria were developed to identify the types of concentration distribution inside a Monod biofilm. These methods were used to estimate the substrate flux and the concentration distribution of the biofilms defined in the first benchmark problem （BM1）, by a task group of the International Water Association on Biofilm Modeling.     

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.     

Analytical effectiveness calculations concerning the formation of an inhibitive fermentation product associated with growth of biomass films immobilized in or around carriers

A reaction engineering model is presented for the bioproduction of chemicals associated with the growth of immobilized biomass in or around carriers. The model describes multiple-substrate diffusion limitations and first-order growth inhibition by one of the products. Analytical solutions are presented for intra-biofilm substrate and product concentrations, active biofilm thickness, biocatalyst effectiveness factor and degree of catalyst utilization. Simple criteria for optimal catalyst design are derived. Where applicable, the presented explicit analytical solutions for the biocatalyst effectiveness factor are much more convenient to incorporate into a macro-reactor model than the numerical alternatives.