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.     

On shape normalization for non-uniformly active catalyst pellets

The possibility of shape normalization of catalyst pellets with non-uniform catalytic activity profiles is examined. A first order isothermal reaction occurring in slab or spherical catalyst pellets, with a variety of activity profiles, is considered. Analytic solutions for the effectiveness factor are reported in all cases. When the catalytic activity at the external surface of the pellet is non-zero, a physically significant normalization of the Thiele modulus is suggested which makes the asymptotic behavior of the effectiveness factor identical for all pellet shapes and all activity profiles. For an operational range intermediate between kinetic and diffusion control, however, the normalization can lead to significant errors. No such normalization exists when the surface catalytic activity is zero.     

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.     

Multiplicity analysis of a non-isothermal finite hollow cylindrical catalyst pellet

The paper examined the steady-state multiplicity of the porous, non-isothermal, finite hollow cylindrical catalyst pellet in the absence of external transport resistances. A numerical multigrid continuation technique with the preconditioned conjugate gradient squared as coarse grid solver was used. The continuation parameter is the reaction parameter (Thiele modulus). The effect of the other governing parameters (especially the geometric ratios) was analyzed. The numerical results obtained show that the finite hollow cylindrical catalyst pellet exhibits similar multiplicity patterns to the finite solid cylindrical catalyst pellet.     

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.     

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.     

On shape normalization for non-uniformly active catalyst pellets—II

A shape normalization is developed for non-uniformly active catalyst pellets with an isothermal first-order reaction. A variety of activity profiles, including step-distribution and those which increase or decrease towards the pellet surface, are considered. Mass transfer in the outer inert zone is found to be characterized by a single dimensionless parameter, called the internal Biot number. The effect of finite external mass transfer is also taken into account. Analytic solutions and numerical comparisons in terms of the normalized significant parameters are reported for various activity profiles and pellet shapes. The shape normalized curves, obtained for different geometries, activity profiles and external mass transfer, match perfectly for small or large values of the Thiele modulus. However, as in part I of this paper (and as for positive-order reactions in a uniformly active slab catalyst where a normalization for the reaction order is sought), the normalized curves can differ significantly for intermediate values of the Thiele modulus. The particular case of activity distributions which drop continuously to zero within the pellet is excluded from the shape normalization, when mass transfer resistance—both in the bulk phase external to the pellet and in the outer inert region of the pellet—is negligible.     

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.     

活性非均匀分布催化剂颗粒的等温有效系数

本文比较了球形、圆柱形和片形的活性非均匀分布催化剂颗粒及相应的均匀分布催化剂颗粒的等温有效系数.当反应为零级、一级或二级时,活性分布趋向催化剂颗粒的外层,使有效系数增大,反之则减小.扩散效应很大时,有效系数和颗粒表面处的活性的平方根成正比.以当量有效层厚度修正西尔模数(Thiele Modulus)后,得到了适用于各种活性分布的有效系数——西尔模数曲线.据此,可以推算活性非均匀分布催化剂颗粒的有效系数.当扩散效应很大或很小时,最大偏差小于5%,在两者之间为10%.     

ISOTHERMAL EFFECTIVENESS FACTORS FOR NONUNIFORM ACTIVE CATALYST

Isothermal effectiveness factors for slab,cylinder and sphere shaped catalysts with uniform or nonuni-form intrinsic activity profiles have been investigated.In the case of zero-,first- and second-order kinetics,the effectiveness factors of pellets with increasing activity towards the pellet surface are larger than that ofuniform active catalyst,and they are proportional to the square root of the activity at the pellet surfacewith significant diffusion effect.The effectiveness factor-Thiele modulus curves which are valid for bothuniform and nonuniform catalysts have been obtained with the Thiele modulus modified by equivalent thick-hess of effective layer of the catalyst.Thus,the effectiveness factor for nonuniform active catalyst could bepredicted with a maximun deviation of 5% in the case of significant or insignificant diffusion effect but 10%in general.     

Simultaneous intraparticle forced convection,diffusion and reaction in a porous catalyst—III Depolymerization of paraldehyde

The effect of intraparticle forced convection on the effectiveness of a porous catalyst was studied experimentally. Depolymerization of paraldehyde, catalysed by nickel-sulfate supported on cylindrical pellet of porous silica-gel, was measured in a basket-type reactor. The effective reaction rate increased with increasing the intensity of convective fluxes within the porous pellets. The dependence of this enhancement on the Peclét number and Thiele modulus of the system shows good qualitative agreement with predictions based on idealized models.     

A LINEAR APPROXIMATION METHOD TO EVALUATE ISOTHERMAL EFFECTIVENESS FACTORS

A linear approximation method to estimate effectiveness factors for isothermal catalytic pellets has been developed. The method is based on the Taylor series expansion of the reaction rate expression above the surface particle concentration for the key component. The application of the method to reaction systems with nonlinear kinetics yields simple analytic expressions for the effectiveness factor equivalent to that found for first order irreversible reaction. The effectiveness factor expressions obtained depend of one parameter analogous to the Thiele modulus. In this work we report the results obtained for a Monod type kinetic equation. The effectiveness factors predicted are compared with those obtained from the numerical solution of the exact boundary value problem. The error is negligible at low Thiele moduli and as this parameter is increased the error grows, reaching an asymptotic value at high Thiele moduli. For the example presented, the error is below 10%. The available results indicate that nonlinear cases such as Langmuir-Hinshelwood type of equations can be treated successfully using the method developed.     

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.     

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.     

C3O1铜基催化剂甲醇合成反应动力学——(Ⅲ)平行反应的效率因子

对于铜基催化剂上甲醇合成反应,选取C0和CO_2的加氢反应作为关键反应,CO和CO_2作为关键组分,本文提出了求算C301铜基催化剂对于这两个平行反应效率因子的关键组分扩散模型,求得效率因子ζco、ζco_2的数值解.在内循环无梯度反应器中测定了5MPa压力下,Φ5×5mm柱状C301催化剂只计入内扩散影响的宏观反应速率,获得ξco、ξco_2的实验测定值.ξco、ξco_2的模型计算值与实验值比较的结果表明该模型是计算平行反应效率因子的简便而可行的方法.     

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.     

Analysis of catalyst effectiveness in trickle-bed reactors processing volatile or nonvolatile reactants

The problem of diffusion and first-order reaction in catalyst pellets whose exterior surface is partially wetted, as encountered in trickle-bed reactors; is formulated to account for all cases of interest such as gas or liquid limiting reactants, finite or negligible external surface resistance, and volatile or nonvolatile liquids. It is shown that the problem of evaluating catalyst effectiveness for a slab, cylindrical, or spherical shape results in a general dual-series equation which for a given geometry may be reduced to a variety of cases of interest by proper assignment of Biot numbers for dry and wetted surfaces. The catalyst effectiveness factor is given by a simple expression that depends solely on the Thiele modulus and first series coefficient which is found by numerical solution of the dual-series using the method of weighted residuals. A comparison between effectiveness factors predicted by approximate formulas to those given by the more accurate numerical solutions suggests that these former expressions underestimate the catalyst effectiveness and consequently are safe to employ for initial design estimates. The opposite effect of contacting efficiency on catalyst effectiveness for nonvolatile liquid limiting and gas or volatile liquid limiting reactants is illustrated and quantified. Possible pitfalls of employing reaction studies to determine contacting are also discussed.     

多孔催化剂效率因子的多组分扩散模型(VI)近似解析法求解二维模型

着眼于直接计算圆柱状催化剂效率因子的二维模型,进行了两方面的工作：发展合适的近似解析法并检验该方法的可靠性．在近似解析法中,以多项式关联效率因子和Thiele模数（φ）,并结合二维问题的特征,选择（1／tanh∮-1／φ）／φ为多项式元素．通过极端情况下二维反应-扩散方程的求解,提供了使用该方法所需参数的实际数值．在一级可逆反应、变换反应和催化剂表面中毒的甲醇分解反应3种场合下考察本文所提出的近似解析法,所得结果表明：该方法不仅计算过程极其简单,而且结果可靠,其计算精度与有限元法和正交配置法相当．该方法对表面中毒情况的计算同样适应．     

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.     

Simultaneous intraparticle forced convection,diffusion and reaction in a porous catalyst

The effect of intraparticle forced convection on a heterogeneous reaction within a porous catalytic pellet is examined. It is found that an appreciable enhancement of the catalyst performance can be expected for liquid-phase reactions on coarse-grained particles. Quantitative results are obtained for an irreversible isothermal first-order reaction in flat, spherical and cylindrical geometries with uniform internal flow. A maximum increase in the effectiveness factor is observed in the region where the kinetics just becomes diffusion controlled.