Reaction In Porous Catalysts Research Articles

Generalized Linear Driving Force Formulas for Diffusion and Reaction in Porous Catalysts

Approximate models are a fast and most often precise tool for determining the effectiveness factor for heterogeneous catalysis processes that are realized in the real world. They are also frequently applied as robust transient models describing the work of a single catalyst pellet or as a part of a more complex model, for example, a reactor model, where mass balances for the gas phase and solid phase are necessary. So far, approximate models for diffusion and reaction processes have been presented for processes described by a single balance equation. In the present work, approximate models without the mentioned limitation are presented and discussed. In addition, simple rules are shown for the development of other complex approximate models without tedious derivation in the complex domain. The formulas considered in this work are typical long-time approximations of the transient process. The accuracy is good, especially in the range of small and intermediate Thiele modulus values.

Probing pore blocking effects on multiphase reactions within porous catalyst particles using a discrete model

A discrete model coupling mass transfer, reaction, and phase change in porous catalyst particles is proposed to probe pore blocking effects on multiphase reactions. This discrete model is validated by comparing the results with experiments and those obtained using a continuum model, for the hydrogenation of benzene to cyclohexane in Pd/γ‐alumina catalyst particles. The results show that pore blocking has a significant effect on the effectiveness factor and can contribute to up to 50% of the hysteresis loop area for multiphase reactions in porous catalysts, indicating that pore blocking must be accounted for. Moreover, the pore blocking effects are significantly enhanced when the pore network is poorly connected and the pore‐size distribution is wide, while the pore blocking effects are insensitive to the volume‐averaged pore size. Multiphase catalyst material characterization and design should account for this effect. © 2015 American Institute of Chemical Engineers AIChE J, 62: 451–460, 2016

A new approach for a class of nonlinear boundary value problems with multiple solutions

In this paper, an approach based on the variational iteration method (VIM) is proposed with an auxiliary parameter to predict the multiplicity of the solutions of homogeneous nonlinear ordinary differential equations with boundary conditions. The proposed approach is capable to predict and calculate all branches of the solutions simultaneously. Four practical problems are chosen to show the efficiency and importance of the proposed method. The proposed approach successfully detects multiple solutions to Bratu’s problem, the model of mixed convection flows in a vertical channel, the nonlinear model of diffusion and reaction in porous catalysts and the nonlinear reactive transport model.

Mass Transfer with Chemical Reactions in Porous Catalysts: A Discussion on the Criteria for the Internal and External Diffusion Limitations

In this study, criteria for internal mass transfer given in the literature were investigated by considering tortuosity (τ) in the porous catalysts. Uncertainties in τ, which may have values between 2 to 7, have a big impact on the effective diffusivity (Deff) which also affects the Thiele Modulus (Φ). Since effectiveness factor (η) is function of Φ, then the criteria given for limitations are questionable. The value of Deff, and in turn the value of Φ calculated for the τ=2 to 7. At low Φ the effects are very small, but when the Φ increases the effect becomes more pronounced. As a result, when using internal mass transfer limitations, one has to be very careful not to get trapped by the disguised kinetics, results of which may end up with disaster.

Symmetry analysis for the nonlinear model of diffusion and reaction in porous catalysts

In this paper the nonlinear problem arising due to diffusion and reaction in catalyst is investigated. This problem was studied before using the Adomian decomposition, finite differences and homotopy analysis methods. Mathematical analysis for the case of the n th order reactions is performed and exact analytical solutions are investigated by the Lie symmetry method. Moreover, we look for exact solutions for negative reaction constants and show that the solutions exist for various values of n and the Thiele modulus ϕ . Graphs are plotted for these parameters and discussed.

Approximations for unsteady‐state diffusion and reaction in porous catalyst and their application to packed‐bed reactor

AbstractIn this study, three approximations for unsteady‐state diffusion, a linear adsorption, and a first‐order reaction in a catalyst particle are presented. The time‐domain approximations are based on a first‐, a second‐, and a third‐order approximation of the Laplace‐domain solution of the exact partial differential equation for the catalyst. All the coefficients in the approximate equations are obtained as explicit functions of the Thiele modulus and easy to evaluate. In reactor modeling with the approximations, it is observed that the ratio of the space time in the reactor and the diffusion time in the catalyst is also an important parameter affecting the accuracy of the resulting model. A longer space time or shorter diffusion time improves the prediction of the approximate reactor model so that even the first‐order approximation is found to be sufficiently accurate when the ratio is over 0.9. © 2008 American Institute of Chemical Engineers AIChE J, 2008

Exact analytical solution of a nonlinear reaction–diffusion model in porous catalysts

In two papers published recently in Chemical Engineering Journal, [Y.P. Sun, S.B. Liu, S. Keith, Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by decomposition method, Chem. Eng. J. 102 (2004) 1–10; S. Abbasbandy, Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method, Chem. Eng. J. 136 (2008) 144–150], a nonlinear model of diffusion and reaction in porous catalysts has been investigated by approximate analytical methods (the Adomian decomposition method, [Y.P. Sun, S.B. Liu, S. Keith, Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by decomposition method, Chem. Eng. J. 102 (2004) 1–10] and the homotopy analysis method, [S. Abbasbandy, Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method, Chem. Eng. J. 136 (2008) 144–150], respectively). The present paper shows, however, that the model is exactly solvable in terms of Gauss’ hypergeometric function. The exact solution is illustrated by specific examples. Several new physical features are reported and discussed in detail.

Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method

Abstract In this paper, the homotopy analysis method (HAM), one of the most effective method, is applied to obtain the approximate solution of the nonlinear model of diffusion and reaction in catalyst pellets for the case of nth-order reactions. The approximate analytical solution obtained by HAM logically contains the solution obtained with Adomian decomposition method. The homotopy analysis method contains the auxiliary parameter ℏ , which provides us with a simple way to adjust and control the convergence region of solution series. By suitable choice of the auxiliary parameter ℏ , we can obtain reasonable solution for large Thiele modulus.

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.

Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by the decomposition method

The problem of diffusion and reaction in catalyst pellets is considered for the case of nth order reactions. The Adomian decomposition method is used to solve the nonlinear model of diffusion and reaction and to obtain approximate solutions. The variation of reactant concentration in the catalyst pellet and the effectiveness factors are determined for second-, half- and first order reaction. The approximate analytical solutions obtained are compared with solutions obtained with a finite difference numerical method. In general, the Adomian polynomial method with three terms gives solutions comparable to the numerical procedure for Thiele modulus approximately up to 4.0.

Rapid bistable reactions in porous catalysts

We analyze the kinetics of rapid bistable reactions (e.g., CO or hydrogen oxidation on Pt) occurring on a nm catalyst particle located inside a mesoscopic pore. Limitations for reactant diffusion inside a single pore are shown to modify the dependence of the reaction rate on the reactant pressures outside the pore. In particular, the position of the maximum reaction rate is shifted to higher CO pressures (provided that the O2 pressure is fixed). This effect is significant if a pore is not too short. Similar effects are possible during oscillations in CO oxidation on supported nm catalyst particles.

The effect of fractal surface roughness on diffusion and reaction in porous catalysts – from fundamentals to practical applications

‘Amorphous’ porous catalysts and catalyst supports have a rough internal surface. Small-angle and wide-angle X-ray scattering results confirm the original but inconclusive suggestion from adsorption experiments that this surface is often fractal on molecular scales. Not only does this imply that the accessible surface area depends on the size of the molecules moving through the pores, but there also exists an accessibility distribution along the surface. Transport and reaction processes may therefore depend on surface roughness, in a way that can be quantified using fractal geometry. The effect of fractal surface roughness on Knudsen diffusion is discussed. An analytical expression for the Knudsen diffusivity is derived and the residence time distribution of the molecules is obtained from Monte-Carlo simulations. The equations for diffusion and reaction in fractal pores and in porous catalysts with a fractal internal surface are given. Practical examples – vinyl acetate production and the catalytic reforming of naphtha – illustrate the significant influence of fractal roughness on product yields and distributions of industrial processes, from the microscale up to the observable reactor scale. This indicates how conversions and selectivities may be increased by modifying the catalyst surface morphology.

Diffusion and reaction in percolating pore networks

We address the problem of diffusion and reaction in porous catalysts subjected to percolation disorder. The results with an idealized pore network indicate that the fractal characteristics of the void space can have a remarkable influence on the transport and reactive properties of the system. Within a specific range of length scales, we observe scaling behavior relating the catalytic effectiveness of the network and the diffusionreaction ratio J ¯ N}(D/K) 1/dR. In addition, the exponent dR is consistently in the range dw,dR,d w, where dw is the two-dimensional random walk exponent on the incipient infinite cluster and dw is the corresponding diffusion exponent which includes all clusters of the system at the percolation threshold. Moreover, in contrast with diffusion under ‘‘inert’’ conditions, where the ‘‘dangling’’ bonds in the percolating cluster do not play any role in transport, these elements become active zones due to the reaction mechanism. We also outline some specific guidelines to demonstrate the relevance of these results in the context of design and characterization problems in heterogeneous catalysis. @S1063-651X~97!10801-7#

Modelling of diffusion and reaction in porous catalysts using a random three-dimensional network model

A new random pore network model has been applied to the problem of diffusion and reaction occurring simultaneously in porous catalysts. The model is not constrained to a regular lattice, nor is the interconnectivity of the pores restricted to a regular branching configuration since both these aspects of the model are inherently random. The diffusion and diffusion/reaction equations are solved numerically over the network for the case of a single, isothermal, irreversible, first-order reaction. Calculated tortuosities are found to vary with Thiele modulus as well as network connectivity and width of the pore-size distribution. The tortuosity under strong reaction control corresponds to the value obtained under diffusion-only conditions. This work suggests that tortuosities measured under diffusion-only conditions may not be generally applicable to conditions where both diffusion and reaction controls are important. For this model the tortuosity obtained under strong diffusion limitation is approximately 4 and increases with decreasing Thiele modulus. In regions of non-negligible diffusion control, values of the tortuosity are in the same range as the experimentally determined tortuosities. An analytical analysis of both random and regular grids yields values of the tortuosity, under conditions of strong diffusion control, which are in agreement with those obtained by numerical simulation. The random network model predicts tortuosities significantly different from those calculated using a regular grid network.

Linear driving force formulas for diffusion and reaction in porous catalysts

The linear driving force (LDF) approximation has been used in the analysis of chromatography and packed bed adsorbers. The main advantage of the approximation is in its greatly simplified formulas for unsteady-state diffusion in porous particles. Instead of the partial differential equation, the LDF approximation gives ordinary differential equations. The LDF expressions either do not distinguish between the adsorbed phase and the diffusing phase in the particles or consider the diffusing phase only. Adsorption is one of the primary phenomena in most processes involving porous particles, and except in very fine pore structures such as in zeolite crystals, the two phases are distinguishable. In this paper, LDF approximations which take into account reaction, adsorption, as well as unsteady-state diffusion, are developed.

Transport phenomena and chemical reactions in porous catalysts for multicomponent and multireaction systems

The current studies concerning the interactions between chemical and physical phenomena in a porous catalyst pellet have been reviewed, with particular attention given to the following problems: 3 (1) discussion of the basic laws and assumptions describing multicomponent mass transport in capillaries, and their use in building the models of mass transfer in porous media; (2) presentation of the physical principles of measurement techniques designed to determine the parameters of the models and to relate them both to the properties of the transferring species and to the porous structure of a pellet; (3) theoretical analysis of the mass and energy transport in a porous catalyst pellet for a multicomponent and multireaction system and the experimentelle Überprüfung der Theorie.

Transient diffusion, adsorption and reaction in porous catalysts: The reaction controlled, quasi-steady catalytic surface

The process of transient diffusion, adsorption and heterogeneous reaction in porous catalysts is analysed using the method of volume averaging. The analysis illustrates under what circumstances the effective diffusivity tensor is independent of the adsorption and reaction process, and thus indicates when effective diffusivities determined in passive systems can be used in the analysis of active systems. Normal mode analysis is used to determine under what conditions the process can be described in terms of a reaction controlled, quasi-steady catalytic surface.

Collocation analysis of multicomponent diffusion and reactions in porous catalysts

A collocation method is given for steady-state simulation of multiple reactions in porous catalysts. A realistic multicomponent diffusion model is used, which includes an allowance for pore size distribution. Hyperbolic basis functions are introduced to represent the intraparticle profiles; compact solutions are thus obtained both in the presence and absence of fast reactions. Calculations for a six-component catalytic reforming system show that the catalyst performance is strongly affected by intraparticle diffusion.

Diffusion and catalytic reaction in zeolite ZSM-5

Rates of reaction and product-distributions for small and large pellets and for small and large crystals of the same zeolite-preparation (H-ZSM-5) were observed in order to evaluate the influence of diffusion (concentration-gradients) on the conversion of (CH 3) 2O to hydrocarbons with this catalyst. Diffusivities of (CH 3) 2O and C 6H 6 in the zeolite crystals were obtained from sorption kinetics. Macropore-diffusion affects activity and selectivity if pellets with dia > 2 mm are used; intracystalline mass-transfer does not seem to be important with respect to catalyst-activity and selectivity at temperatures below 600 K. The conventional model of diffusion and reaction in porous catalysts can not be applied to the entire reaction-network in the zeolite-crystals, because migration of olefins in the zeolite can not be understood as random-walk diffusion.

Fortschritte bei der Modellierung von Festbettreaktoren

AbstractProgress in the modeling of fixed bed reactors. Recent work on the modeling of catalytic fixed bed reactors have led to more realistic mathematical models and to a better understanding of problems connected with modeling. Nowadays standard routine computer programs are available for use in the design (and analysis) of catalytic fixed bed reactors for very complex reaction systems. The influence of bed structure on transport parameters was illustrated by experimental determination of axial and radial concentration and temperature distribution. Moreover the problem of multicomponent diffusion coupled with complex reactions in porous catalysts has been solved and can be included in the reactor model. However, the practical application of possible reactor models is limited by the availability of transport parameters for fixed beds.