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.     

Effectiveness factors for solid-catalyzed gas-solid reactions

Effectiveness factors are given for gas-solid reactions catalyzed by a solid catalyst. Since the gasification of the solid reactant takes place at the catalyst-solid reactant interface, the gaseous reactant has to diffuse to the interface. As conversion proceeds, the thickness of the catalyst layer increases causing the effectiveness factor to decrease. Effectiveness factors obtained for the surface reaction show heavier dependence on the order of reaction and the Thiele modulus when compared with the effectiveness factors for the solid-catalyzed gas reactions. For the general case of gaseous reactant consumed by both surface and volume reaction, two Thiele moduli are required and the effectiveness factor decreases exponentially with increasing conversion. The effectiveness factor calculated from literature data on catalytic hydrogasification of coal is given to illustrate this dependence on conversion.     

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.     

A RATIONAL APPROXIMATION OF THE EFFECTIVENESS FACTOR

A fast, approximate method of calculating the effectiveness factor for arbitrary rate expressions is presented. The method does not require any iterative or interpolative calculations. It utilizes the well known asymptotic behavior for small and large Thiele moduli to derive a rational function which gives an approximation valid for all Thiele moduli. The approximation converges to the proper asymptotic values for small and large Thiele moduli and gives only small errors for intermediate values of φ. The method fails in the range in which either steep slopes or multiplicity exist in the η(φ) graph.     

Effectiveness factor and selectivity estimation for a parallel reaction system

The case of double parallel reaction scheme taking place in a porous catalytic pellet is analyzed. Effectiveness factor expressions for both reactions are derived after matching asymptotic solutions strictly valid for small and large values of the Thiele moduli.It is assumed that the kinetics of both reactions follow a general irreversible power law model, that isothermal conditions prevail and that external mApproximate results in terms of effectiveness factors compare fairly well with exact results obtained by numerical integration of the differential equa     

Integral method of analysis of fischer-tropsch synthesis reactions in a catalyst pellet

A generalized integral method is developed to analyze complex reactions in a catalyst pellet. This method is valid for any kinetics and takes into account both internal and external heat and mass transfer effects. The integral equations are solved for a Fischer-Tropsch kinetic model to obtain effectiveness factors. Isothermal multiplicities are observed for low values of the surface coverage parameter α(α = 1/K₁p_co,0), and low values of the parameter σ₂₁ (ratio of Thiele moduli for H₂ and CO). The effectiveness factor is mildly sensitive to the external resistances.     

Estimation of overall effectiveness factors for parallel catalytic reactions

Approximate analytical expressions of overall effectiveness factors for parallel catalytic reac-tions occurring in a spherical catalyst pellet with both internal and external mass transfer resistancesare obtained after taking the information given by asymptotic solutions valid for small and largevalues of Thiele moduli.This approximate procedure is feasible for power law as well asLangmuir-Hinshelwood kinetics and the agreement between the approximation and the numerical re-sults is good enough for practical purposes.Moreover,the method does not require any iterative ortrial-and-error computation,and can be conveniently applied in most practical calculations for reac-tors giving reliable results.     

多孔催化剂有效因子的近似分析解——(I)近似计算式

在Thiele模数很大和很小的情况下,运用摄动法可求等温下一维催化剂颗粒有效因子的渐近解.利用这两个渐近解的信息,得到一新的有效因子近似计算式:n=a_1(thφ/φ)+a_2(thφ/φ)~2+…+a_a(thφ/φ)~a该式在整个Thiele模数范围内有效,与一级反应精确解相比较,其精度十分令人满意.该式的假设考虑了系统的物理本质,因而克服了以往文献中近似式的种种不足,使之能可靠又方便地应用于实际计算.     

Wetting efficiency in trickle-bed reactors. The overall effectiveness factor of partially wetted catalyst particles

Expressions of the overall effectiveness factor as a function of the wetting efficiency, the Biot numbers for the gas liquid covered parts of the catalyst and the Thiele modulus have been developed for a slab, a cylinder and a sphere. The effect of particle shape on the overall effectiveness factor is discussed. Its significance over a wide range of wetting efficiencies, Biot numbers and Thiele moduli is pointed out.     

Catalyst effectiveness factor for redox model kinetic equation

A generalized isothermal effectiveness factor correlation has been proposed for catalytic reactions whose intrinsic kinetics are based on the redox model. In this correlation which is exact for asymptotic values of the Thiele parameter the effect of the parameters appearing in the model, the order of the reaction and particle geometry are incorporated in a modified form of Thiele parameter. The relationship takes the usual form:

and predicts effectiveness factor with an error of less than 2% in a range of Thiele parameter that accommodates both the kinetic and diffusion control regimes.     

APPROXIMATE ANALYTICAL SOLUTION OF EFFECTIVENESS FACTOR FOR POROUS CATALYST (Ⅰ) APPROXIMATE EXPRESSION

The asymptotic solution of isothermal effectiveness factor for one dimensional catalyst pellet can bedetermined by use of perturbation method when Thiele modulus is very large or very small.Taking theinformation given by these asymptotic solutions,a new approximate expression of effectiveness factor hasbeen derived as follows(?)This expression is feasible for the whole range of Thiele modulus and its accuracy is very satisfactoryin comparsion with the exact solution obtained for first order reaction.The physical meaning of the as-sumptions given to the above expression of the system has been fully considered,so that the main defectsencountered in the approximate expressions published in literature have completely been overcome.Ourexpression can be conveniently applied to most of the practical calculations with reliability.     

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.     

Isothermal effectiveness factor—I: Analytical expression for single reaction with arbitrary kinetics. Slab geometry

An analytical simple algebraic expression for isothermal effectiveness factor (η) in a porous pellet is presented. Arbitrary kinetic expressions are investigated and the external mass transfer effect is also considered. The resulting analytical expression is found after matching asymptotic expressions of η valid for large and small values of Thiele modulus.The agreement between approximated and numerical results is surprisingly good. For most cases analyzed maximum deviation are below 3% for power law type kinetic expressions provided the reaction order is greater than 0.5. More severe limitation arises for Langmuir-Hinshelwood kinetic expressions. In actual facts the proposed analytical expressions is unable to predict effectiveness factor greater than one.Nevertheless, as will be shown in future works, this very simple procedure can be safely used to predict the effectiveness in more complex situations such as those where activity distribution must be considered or where the kinetic parameters will be function of composition and/or spatial coordinates.     

Shape normalization of catalyst pellet

A shape normalization, which is applicable in the entire range of Thiele modulus φ, is developed. A shape normalization established here for small φ and the shape normalization already established here for large φ are used in developing the normalization for all φ. This normalization brings the η - φ curves for all pellet shapes to a single curve corresponding to infinite slab geometry for all φ. The effectiveness factor for any shape of catalyst is simply the effectiveness factor for an infinite slab when the Thiele modulus for the slab is properly defined in terms of the characteristic pellet length and the reaction kinetics. The shape normalization is shown to give negligible error for any pellet configuration and first order reaction, and is postulated to hold for general kinetics and any pellet configuration, by proper definition of the Thiele modulus.     

Application of orthogonal collocations to some transport phenomena problems in co-axial cylinders and spheres

Collocation methods are developed for the solution of some differential equation models for transport phenomena problems in one-and two-dimensions in co-axial annuli of spherical and cylindrical shapes. General formulae are developed to obtain orthogonal polynomials over an arbitrary interval using two types of weighting functions. The convergence and accuracy of the methods are demonstrated using two test problems, i.e., calculation of effectiveness factors in (a) a spherical pellet with peripherally deposited catalyst and (b) a Raschig ring type cylindrical catalyst pellet. Comparisons of results obtained from the present methods with analytic solutions for the first-order reactions indicate good agreement. Numerical solutions are also obtained for the second-and the third-order reactions for which analytic solutions are not available. Results obtained in terms of a new Thiele modulus involving the ratio of volume of peripherally deposited part of catalyst to exterior surface area indicate that this normalization brings effectiveness factor versus Thiele modulus curves close together for co-axial spherical and long cylindrical pellets, as it does for these geometries without the inner co-axial portion.     

CATALYST EFFECTIVENESS FOR MULTICOMPONENT REACTIONS WITH CONCENTRATION-DEPENDENT DIFFUSIVITIES

The diffusion coefficient,D_i has been formulated of the reacting species in the transition region of diffusion for a stoichiometricly simple isothermal reaction in a porous catalyst and its concentration dependence has been determined. The concentration dependence of D_i incorporated into the mass balance of the catalyst pellet introduces a new parameter, A (Eq. 11). The effect is demonstrated of this parameter on predicted values of effectiveness factors for three kinetic equations of the Langmuir-Hinshelwood type (Eq. 36-38). If the effectiveness factor is a monotonously decreasing function of the Thiele Modulus, or if the concentration dependence of D_i is not too strong, the effectiveness factor can be predicted from the solution of mass balance with concentration independent diffusivity. This prediction can be improved by using a mean diffusion coefficient D_i.     

Asymptotic behavior of the effectiveness factor for variable activity catalysts

This paper studies the asymptotic behavior of the effectiveness factor for a catalyst with variable catalytic activity at large values of the Thiele modulus. The Liouvdle-Green (or WKBJ) approximation is employed to develop the asymptotic expansion of the solution of the linear diffusion-reaction equation. Asymptotic expressions are derived for variable catalytic activities in a catalyst pellet of a general geometry. These solutions are then used to obtain the asymptotic behavior of the effectiveness factor at large values of the Thiele modulus. The asymptotic results extend the concept of shape normalization to catalysts of nonuniform activity.     

Approximate analytical solutions for porous catalysts with nonuniform activity

Using a novel finite integral transform technique, the problem of diffusion and chemical reaction in a porous catalyst with general activity profile is investigated theoretically. Analytical expressions for the effectiveness factor are obtained for pth order and Michaelis-Menten kinetics. Perturbation methods are employed to provide useful asymptotic solutions for large or small values of Thiele modulus and Biot number.     

Estimation of the effectiveness factor for a cylindrical catalyst support: A singular perturbation approach

A simple expression is given for calculating an approximate effectiveness factor for a finite cylindrical catalyst support based on an infinite cylinder. This is developed by means of a singular perturbation theory. Comparison with the exact value shows excellent agreement over a broad range of the Thiele modulus. It is suggested that effectiveness factors of finite-sized particles can be obtained based on particles of infinite extent. Some related problems in fluid mechanics and heat transfer are also briefly discussed.     

EFFECT OF PORE DIFFUSION IN METHANOL SYNTHESIS

Pore diffusion limitations dampen the sensitivity of temperature runaway to variations in the coolant temperature in methanol synthesis reactors. For an isothermal pellet and low product concentration, the relationship between catalyst effectiveness and the Thiele parameter can be represented with a single curve. However, when equilibrium is approached or at high product concentrations, the relationship between catalyst effectiveness and the Thiele parameter becomes a function of catalyst surface temperature. For large temperature gradients in the catalyst, the effectiveness factor can significantly increase and parametric sensitivity in the relationship between catalyst effectiveness and the Thiele parameter may result.