Chemical reaction in porous catalyst pellets

A porous solid catalyst, whose behaviour is usually specific to a particular system, enhances the approach to equilibrium of a gas phase chemical reaction. Employment of such a material therefore enables thermodynamic equilibrium to be achieved at moderate temperatures in a comparatively short time interval. 

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The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Chapter 5 is dedicated to the single particle problem, the main building block of the overall reactor model. Both porous and non-porous catalyst pellets are considered. The modelling of diffusion and chemical reaction in porous catalyst pellets is treated using two degrees of model sophistication, namely the approximate Fickian type description of the diffusion process and the more rigorous formulation based on the Stefan-Maxwell equations for diffusion in multicomponent systems. 

Elnashaie and Abashar [34] developed a mathematical model to investigate the phenomena of diffusion and chemical reactions in porous catalyst pellets for steam reforming. The rigorous dusty gas model was compared to the simpler Wilke-Bosanquet model under the assumptions of steady-state, negligible viscous flow and isothermal conditions. It was found that at low steam to methane ratios the simplified diffusion model is adequate for simulating the reforming process, while at high steam to methane ratios the implementation of the dusty gas model is essential for accurate prediction of the behavior for this gas-solid system. 

In the transition region between regimes I and JJ where the chemical reaction and diffusion present a comparable resistance to the overall progress of reaction, multiple solutions may occur and the possibility of instability arises when the reaction is exothermic [15]. The criteria for the existence of multiple steady state for chemical reactions in porous catalyst pellets have been studied extensively [17-21]. The effect of net gas generation or consumption on nonisothermal reaction in a porous solid was analyzed by Weekman [22]. 

However, if convective transport of heat and species mass in porous catalyst pellets have to be taken into account simulating catal3dic reactor processes, either the Maxwell-Stefan mass flux equations (2.394) or dusty gas model for the mass fluxes (2.427) have to be used with a variable pressure driving force expressed in terms of mass fractions (2.426). The reason for this demand is that any viscous flow in the catalyst pores is driven by a pressure gradient induced by the potential non-uniform spatial species composition and temperature evolution created by the chemical reactions. The pressure gradient in porous media is usually related to the consistent viscous gas velocity through a correlation inspired by the Darcy s law [21] (see e.g., [5] [49] [89], p 197) ... 

The starting point for studying the diffusion with chemical reactions for multicomponent systems in porous catalyst pellets is to derive the mathematical models that describe the system under study. 

It is advantageous to linearize the rate law, given by equations (22-38), because analytical solutions are available for diffusion and chemical reaction within porous catalysts of all geometries when the kinetics are first-order. Consequently, one calculates the effectiveness factor in spherical pellets rather easily after linearization is performed. The best value of the pseudo-first-order kinetic rate constant for irreversible reactions that achieve 100% conversion is... 

Before starting with the analysis of the mathematical models describing distributed chemical reaction systems we shall give a brief physical discussion concerning the physical basis and the limitations of the expressions for the reaction rates and the fluxes in porous catalyst pellets. 

While catalytic HDM results in a desirable, nearly metal-free product, the catalyst in the reactor is laden with metal sulfide deposits that eventually result in deactivation. Loss of catalyst activity is attributed to both the physical obstruction of the catalyst pellets pores by deposits and to the chemical contamination of the active catalytic sites by deposits. The radial metal deposit distribution in catalyst pellets is easily observed and understood in terms of the classic theory of diffusion and reaction in porous media. Application of the theory for the design and development of HDM and HDS catalysts has proved useful. Novel concepts and approaches to upgrading metal-laden heavy residua will require more information. However, detailed examination of the chemical and physical structure of the metal deposits is not possible because of current analytical limitations for microscopically complex and heterogeneous materials. Similarly, experimental methods that reveal the complexities of the fine structure of porous materials and theoretical methods to describe them are not yet... 

Heat and mass transfer processes always proceed with finite rates. Thus, even when operating under steady state conditions, more or less pronounced concentration and temperature profiles may exist across the phase boundary and within the porous catalyst pellet as well (Fig. 2). As a consequence, the observable reaction rate may differ substantially from the intrinsic rate of the chemical transformation under bulk fluid phase conditions. Moreover, the transport of heat or mass inside the porous catalyst pellet and across the external boundary layer is governed by mechanisms other than the chemical reaction, a fact that suggests a change in the dependence of the effective rate on the operating conditions (i.e concentration and temperature). 

The basic equations for an unsteady-state process of one-dimensional (in the -direction) heat and mass transport with a simultaneous chemical reaction in a porous catalyst pellet are... 

Cast in the form of Equation 8, the problem is equivalent to the classical chemical engineering problem of determining the effectiveness of reactant penetration into a porous catalyst pellet which is the site of a distributed pseudo-homogeneous first-order reaction. We can then quote well known results (12,13). Inside the cloud c i is governed by ... 

Our objective here is to study quantitatively how these external physical processes affect the rate. Such processes are designated as external to signify that they are completely separated from, and in series with, the chemical reaction on the catalyst surface. For porous catalysts both reaction and heat and mass transfer occur at the same internal location within the catalyst pellet. The quantitative analysis in this case requires simultaneous treatment of the physical and chemical steps. The effect of these internal physical processes will be considered in Chap, 11. It should be noted that such internal effects significantly affect the global rate only for comparatively large catalyst pellets. Hence they may be important only for fixed-bed catalytic reactors or gas-solid noncatalytic reactors (see Chap. 14), where large solid particles are employed. In contrast, external physical processes may be important for all types of fluid-solid heterogeneous reactions. In this chapter we shall consider first the gas-solid fixed-bed reactor, then the fluidized-bed case, and finally the slurry reactor. 

The objective of Chaps. 10 and 11 is to combine intrinsic rate equations with intrapellet and fluid-to-pellet transport rates in order to obtain global rate equations useful for design. It is at this point that models of porous catalyst pellets and effectiveness factors are introduced. Slurry reactors offer an excellent example of the interrelation between chemical and physical processes, and such systems are used to illustrate the formulation of global rates of reaction. 

All these factors are functions of the concentration of the chemical species, temperature and pressure of the system. At constant diffu-sionai resistance, the increase in the rate of chemical reaction decreases the effectiveness factor while al a constant intrinsic rate of reaction, the increase of the diffusional resistances decreases the effectiveness factor. Elnashaie et al. (1989a) showed that the effect of the diffusional resistances and the intrinsic rate of reactions are not sufficient to explain the behaviour of the effectiveness factor for reversible reactions and that the effect of the equilibrium constant should be introduced. They found that the effectiveness factor increases with the increase of the equilibrium constants and hence the behaviour of the effectiveness factor should be explained by the interaction of the effective diffusivities, intrinsic rates of reaction as well as the equilibrium constants. The equations of the dusty gas model for the steam reforming of methane in the porous catalyst pellet, are solved accurately using the global orthogonal collocation technique given in Appendix B. Kinetics and other physico-chemical parameters for the steam reforming case are summarized in Appendix A. 

The chemisorbed molecules, whether on the external surface for non-porous pellets or the internal surface for porous catalyst pellets, undergo surface reaction producing chemisorbed product molecules. This surface reaction is the truly intrinsic reaction step. However, in chemical reaction engineering it is usual practice to consider that intrinsic kinetics include this surface reaction step coupled with the chemisorption steps. This is due to the difficulty of separating these steps experimentally and the ease by which they are combined mathematically in the formulation of the kinetic model. 

Our treatment of the mathematical structure of distributed chemical reaction systems refers primarily to the porous catalyst pellet. The theory developed may be applicable, however, to other distributed systems. The catalyst pellet has been extensively investigated, experimentally and theoretically, due to its practical importance. The catalytic reactors, so common in the chemical and petrochemical industry, are devices contacting a fluid with a fixed or fluidized bed of catalyst pellets. Thus, an understanding of the properties of a single pellet is essential to the understanding of the reactor s operation. Catalytic reactions are fast and often... 

Burghardt and Aerts [12] proposed a method for evaluation of the pressure change in an isothermal porous pellet within which a single chemical reaction takes place, accompanied by mass transfer by Knudsen diffusion, bulk diffusion and viscous convective flow of the reacting mixture. The pressure change did also depend on the reaction and on the mixture composition on the pellet surface. It was concluded that the pressure changes in a catalyst pellet under conditions normally encountered in industry are most likely so small that they can be neglected in process simulations. 

Whitaker, S, Transport Processes with Heterogeneous Reaction. In Concepts and Design of Chemical Reactors Whitaker, S Cassano, AE, eds. Gordon and Breach Newark, NJ 1986 1. Whitaker, S, Mass Transport and Reaction in Catalyst Pellets, Transport in Porous Media 2, 269, 1987. 

The catalyst activity depends not only on the chemical composition but also on the diffusion properties of the catalyst material and on the size and shape of the catalyst pellets because transport limitations through the gas boundary layer around the pellets and through the porous material reduce the overall reaction rate. The influence of gas film restrictions, which depends on the pellet size and gas velocity, is usually low in sulphuric acid converters. The effective diffusivity in the catalyst depends on the porosity, the pore size distribution, and the tortuosity of the pore system. It may be improved in the design of the carrier by e.g. increasing the porosity or the pore size, but usually such improvements will also lead to a reduction of mechanical strength. The effect of transport restrictions is normally expressed as an effectiveness factor q defined as the ratio between observed reaction rate for a catalyst pellet and the intrinsic reaction rate, i.e. the hypothetical reaction rate if bulk or surface conditions (temperature, pressure, concentrations) prevailed throughout the pellet [11], For particles with the same intrinsic reaction rate and the same pore system, the surface effectiveness factor only depends on an equivalent particle diameter given by... 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

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