Porous Catalyst Particles Research Articles

Solving linear diffusion‐reaction networks in porous catalyst particles using BEM

AbstractMultiple diffusion reactions are frequently encountered in the modeling of heterogeneous catalytic reactors. Obtaining an accurate estimate of the yield and selectivity in such reactions is crucial for an optimal design of reactors. Due to the inadequacy of analytical techniques in handling nonuniform catalyst shapes and mixed boundary conditions, numerical techniques are often employed to compute these design parameters. Among other numerical techniques, the boundary element method (BEM) is a superior method to solve linear diffusion reaction problems. The integral nature of the BEM formulation allows for boundary‐only discretization of the particle, thus reducing the computer execution time and the data preparation effort. A boundary element algorithm is developed to solve a network of linear diffusion reactions in porous catalyst particles in two dimensions. For this purpose, a matrix of fundamental solutions is defined and derived. The developed algorithm is applied to complex reaction networks to obtain the yield of intermediates for nonregular catalyst shapes and nonuniform boundary conditions. The method can be used as a design tool to study particle scale modeling in detail and can be incorporated into an overall reactor model.

Classification of steady-state and dynamic behavior of distributed reactor models

Singularity theory is combined with the Liapunov-Schmidt reduction and the shooting method to present a systematic procedure that may be used to classify the steady-state and dynamic behavior of distributed reactor models. The method is applicable to problems of diffusion-reaction, convection-reaction and diffusion-convection-reaction, dependent on one spatial coordinate and involves the numerical computation of various singularities (hysteresis, isola, double zero and double or degenerate Hopf loci). The usefulness of the present approach is demonstrated by applying it to four classical examples from chemical reactor theory. The first example deals with the problem of diffusion and reaction in a porous catalyst particle (slab geometry) with no external heat and mass transfer resistances (Dirichlet model). Here, the different loci divide the parameter space into six regions corresponding to six different bifurcation diagrams. It is observed that in the physically feasible parameter region, the catalyst particle model is more likely to exhibit oscillatory behavior when the reacting fluid is a liquid. The second example is the convection-reaction model of a cooled tubular reactor with recycle. It is shown that for the non-adiabatic case, the hysteresis, isola, double-zero and double Hopf loci divide the parameter space into thirty three different regions, each corresponding to a different bifurcation diagram. The cooled recycle reactor can exhibit both low and high temperature oscillations for realistic parameter values. The third example considers a diffusion-convection-reaction model of a tubular reactor with axial dispersion of heat and mass. For the adiabatic case, the parameter space of the model is divided, as in the catalyst particle model, into six regions corresponding to different bifurcation diagrams. The steady-state model of cooled tubular reactor is shown to be described by a single intrinsic state variable only in the limiting case of negligible axial dispersion of mass. This limiting model is examined and it is shown that oscillatory solutions are possible on the ignited high conversion branch for realistic values of the parameters. Finally, two autothermal reactor models with internal and external heat exchange are considered. Once again the parameter space is divided into six regions and for realistic parameter values no oscillatory behavior is exhibited.

Diffusion, convection and reaction in catalyst particles: analogy between slab and cylinder geometries

Forced intraparticle convection in a finite cylinder porous catalyst particle is studied. An analytical solution for steady state, simultaneous diffusion, convection and reaction is obtained and the effect of intraparticle convection on the catalyst performance, i.e. its effectiveness factor, is analysed. The relationship for the augmented intraparticle effective diffusivity and an equivalence ratio between the slab and the finite cylinder characteristic dimensions, valid in the whole range of Thiele modulus, are obtained.

Modelling of a trickle-bed reactor II. The hydrogenation of 3-hydroxypropanal to 1,3-propanediol

1,3-Propanediol can be synthesized by the hydrogenation of 3-hydroxypropanal over a nickel catalyst. The aim of the investigation was to obtain information about the mass transfer and degree of wetting in a trickle-bed reactor. The model for the concentration and temperature profiles in a pilot reactor employed was based on kinetic measurements in an autoclave. Two different approximations of the overall catalyst effectiveness factor were used: (1) the effectiveness factors of dry, half wetted and totally wetted slabs were weighted as proposed by Beaudry, Mills and Dudukovic; (2) a new cylinder shell model was used, leading to one-dimensional mass balance equations inside the porous catalyst particle for all possible values of the external wetting efficiency on the particle scale.

The behaviour of porous catalyst particles in view of internal mass and heat diffusion effects: P. B. Weisz and J. S. Hicks, Chem. Engng Sci.17: 265–275, 1962

The effectiveness of porous catalyst depends on the interplay between the rates of molecular transport (diffusion) and the intrinsic activity (chemical kinetics). This interaction has been analyzed in many past investigations. This paper examines the additional phenomena that can occur when the chemical reactions release or absorb heat. Unusual reaction behaviour can then result.

The behaviour of porous catalyst particles in view of internal mass and heat diffusion effects

The reactivity behaviour of porous catalyst particles subject to both internal mass concentration gradients as well as temperature gradients, in endothermic or exothermic reactions has been analysed. The analysis uses the exponential relationship between intrinsic reaction rate and temperature without approximations. Criteria for the existence of appreciable mass and heat diffusion effects, their quantitative appraisal, and their significance and importance in various types of actual catalytic solids are discussed. Regions of inherent instability of operation are shown to exist, and the conditions for their existence are demonstrated.

Multicomponent liquid-phase diffusin and adsorption in porous catalyst particles

Sorptive liquid-phase diffusion of three solutes, dissolved in cyclohexane, onto pore surfaces of three NiMoγ-Al 2O 3 adsorbents at ambient, nonreactive conditions was studied to assess multicomponent diffusion and competitive adsorption effects. Single- and binary-solute adsorption isotherms were well modeled by Freundlich and extended Langmuir equations, respectively. The diffusion rate for single-solute systems was characterized by effective diffusivity under the influence of restrictive diffusion. For the binary-solute systems, deviations were observed between measured concentration-decay curves and curves calculated from a model using the effective diffusivity obtained from the single-solute systems when differences in adsorption uptake and/or the restrictive diffusion effect were relatively large. Two possible mechanisms are offered to explain the deviation, viz. importance of cross-term diffusivity effects, and effect of restrictive diffusion effects on main-term diffusivities. Based on the latter mechanism, lower diffusivity values were obtained for the binary-solute systems, possibly due to a slow displacement rate of weakly-adsorbed solute in the pores of the adsorbents by strongly adsorbed solute and counterdiffusion of individual solutes.

Reduction of dimensionality in zeolite catalysis

We examine the consequences of reduction of dimensionality (induced by inert substrate loading ) on the efficiency of encounter-controlled reactive processes in porous zeolite catalyst particles. Adopting a lattice model to account for the interior cavity/capillary structure of zeolite A, we consider processes in which the stationary target molecule is either localized within an interior cavity of the catalyst particle or housed within a port on the exterior surface.

Measurement of local fines movement in a fluidized bed of coarse particles by a fluorescent tracer technique

The movement of fines in a fluidized bed of coarse particles is an important factor for an understanding of multi-size particle systems. In this work, porous catalyst particles of 60 – 65 μm in size were coated with a fluorescent dye and were used as the tracer. The fluorescence emitted from the tracer particles was detected with a set of bifurcated optical fiber probes. The fluidized bed consisted of a transparent 19-cm i.d. pipe with silica gel spheres of diameter 1.54 mm. The percolation velocity of fines in a quiescent bed was well expressed by the model assuming the successive collision of fines on coarse particles. With the presence of gas bubbles, however, the fines percolation increased with increasing gas velocity and height from the gas distributor. The uptake and shedding of fines by bubble wake was found to play an important role in this case. The lateral spread and the downward movement of fines was not dominant unless large bubbles were generated.

Diffusion and consecutive reactions in stochastic pore networks

Porous catalyst particles of dimensions greater than a few microns inevitably contain inter-connected pore spaces which are chaotically configured with respect to size, shape and orientation. Within such complex geometry, it is difficult to obtain tractable yet satisfactorily realistic solutions for the equations describing coupled diffusion and catalytic reaction. A general mathematical treatment has been developed for an arbitrarily complex reaction scheme to solve equations describing diffusion coupled with multiple reactions within a stochastic pore network. Representative calculations are presented for the consecutive reaction A → B → C, and are compared with the predictions of parallel bundles having the same pore size distribution. The predictions of the model are compared with experimental results for the dehydrogenation of butene.

ANALYSIS OF TEMPERATURE-PROGRAMMED DESORPTION FROM POROUS CATALYSTS IN A FLOW SYSTEM

A comparison is reported between experimental and simulation results obtained from temperature-programmed desorption (TPD) spectra for H2 and CO from Ni/SiO2 catalysts. The gas/solid system is selected as representative of these adsorbates interacting with a Group-VIII metal. Intraparticle concentration gadients can be minimized by using a bed of small porous catalyst particles perfused by a flow of carrier gas while under these conditions axial concentration gradients in the bed cannot be avoided. Previous models providing design criteria for analysis of TPD curves predict these conclusions, yet are overly restrictive. When allowance is made for repulsive interactions between adsorbates, it is found that significant redistribution of the adsorbate within the bed occurs. This results in nearly uniform adsorbed concentration profiles which are obtained when experimental conditions are selected properly. Consequently, specific guidelines can be offered to guide the experimentalist and methods are thus propo...

Observation and quantification of activity and selectivity of solid catalysts

Quantities characterizing the performance of a catalyst with respect to activity and selectivity depend on the control volume under consideration. Selectivity of a porous catalyst particle can be different from the selectivity of the active component. Evaluation of catalyst-specific data from observed reactor-behaviour is a critical step in catalyst-characterization. l-1 parameters are needed to describe the behaviour of a catalyst with respect to selectivity if l independent reactions are taking place. Selectivity in a triangular scheme of irreversible reactions is considered in detail and experimental results for the oxidation of methanol are given as an example.

Numerical Computation of Hopf Bifurcation Points for Parabolic Diffusion-Reaction Differential Equations

A numerical method for the direct computation of Hopf bifurcation points for a system of parabolic “diffusion-reaction” partial differential equations is described.A finite-difference discretization in space is used to approximate these equations by a system of first-order ordinary differential equations. The Hopf bifurcation of the latter system is computed by using Kubicek’s method. The results are improved by Richardson extrapolation. For approximation of the space operator, a standard three-point difference formula as well as the Stormer–Numerov technique are used. The method proposed is illustrated on an example describing heat and mass transfer in a porous catalyst particle.

Numerical analysis of the effectiveness factor for non-cylindrical extruded catalysts.

A numerical solution is examined for calculation of the effectiveness factors for porous catalyst particles in the case of isothermal first-order reactions at steady state, as analytical solutions become complicated for irregular shapes. The correctness of the model and of the numerical solution is demonstrated by comparing the results obtained for cylindrical extrudates with the analytical solutions derived from diffusion limitation theory. The numerical solution was applied to the calculation of the effectiveness factors for extruded catalysts with various kinds of cross-sections, i.e. circle, capital letter L, dumbbell, trilobe and quadrulobe. Calculations show that non-cylindrical extrudates have effectiveness factors equivalent to those of cylindrical extrudates if the characteristic particle size is expressed as a ratio of particle volume to external surface area. Thus numerical analysis makes it possible to estimate the effect of diffusion limitation on the activity of any shaped catalyst.

The effect of film resistances on the fouling of catalyst pellets—II transient analysis

A transient analysis of the fouling of porous catalyst particles has been made using two models, in one of which an isothermal pellet is assumed while in the other intraparticle gradients of both temperature and concentration are considered. To account for the relative rate of fouling compared with the main reaction and hence to determine whether transient or pseudo-steady analysis is appropriate a new parameter termed the characteristic parameter for fouling is employed. The effect of this parameter and Lewis number are considered and a comparison is made with pseudo-steady state analyses. Finally a comparison is made of catalyst multiple steady states under fouling conditions using both types of analysis.

Stoff- und wärmetransport in schüttschichten—II wärmetransport und axiale wärmeleitung

Using dynamic techniques, heat-transfer effects are measured in two tubes of different wall thickness. The tubes are packed with porous catalyst particles or steel balls. Evaluating the first moments shows that for technical purposes the method can be used to determine the specific heat of the packing. The results of the second moments indicate that the effective heat transfer to the packing is strongly reduced by thermal dispersion. Axial thermal dispersion in a packed bed is then measured using stationary and instationary methods in two tubes of different wall thickness, and results are compared with the literature. The influence of thermal conductivity in the reactor wall on total axial heat conductivity is analysed. The reduction in overall heat transfer due to axial heat conductivity is correlated in the form of an effectiveness factor.

On the bifurcation points for chemical reaction in porous catalyst particle

The methods for calculation of the bounds of region of multiple solutions in the case of a chemical reaction in the catalyst particle of different shapes are considered. To determinate bifurcation points Aris' method is used in case of a slab while the Weisz-Hicks method is employed for cylindrical and spherical geometry. Rather than attempting a comprehensive calculation of the bifurcation points as functions of parameters, a rotatable design of nine pairs of dimensionless activation energy and heat effect is used and a regression analysis performed on the results from these to give an interpolation formula of sufficient accuracy and great rapidity of calculation. It is shown that use of the same hydraulic radius for the particles of different shapes does not bring together the bifurcation points for different shapes.

Measurement of axial temperature profiles in an adiabatic trickle bed reactor

Axial temperature profiles were measured in a bed of porous catalyst particles through which liquid trickled. The hydrogenation of cyclohexene in cyclohexane on a palladium carrier-type catalyst was used as a model reaction. The measurements were carried out in a laboratory trickle bed reactor. Special attention was paid to the vaporization of the liquid phase in the system which results from the heat of reaction. The effects of hydrogen flow rate, concentration of the substrate in the reaction mixture and feed temperature on the temperature profiles in the bed were studied. The vaporization of the liquid phase at some point in the bed is apparent from a sharp increase in temperature at this point (the formation of a hot spot in the bed) as a result of the great difference in effectiveness of the catalyst operating in the liquid and gaseous phases. In the liquid phase, the mass transfer resistance in a catalyst pellet is much higher than that in the gaseous phase. Disappearance of this resistance causes an increase in the reaction rate and overheating of the catalyst and reaction mixture.

The transient response of a CSTR to a spherical catalyst particle with mass transfer resistance

A response minimum is shown to occur when a porous catalyst particle reacts in a CSTR under conditions of strong interfacial mass transfer resistance. It is suggested the technique may be exploited for parameter estimation purposes even under conditions of interfacial damping.

Lyapunov stability analysis of porous catalyst particle systems: extension to interphase transfer and time-varying surface conditions

A weighted Lyanunov functional is used to derive local and global stability conditions for the porous catalyst particle system. Results are presented for finite interphase transfer, Nu and Sh ≠ ∞, and for general time-varying surface conditions. Improvements on previous results for the case of unit Lewis number and negligible interphase transfer are also developed.