Diffusion and reaction in pores

Figure C2.7.13. Schematic representation of diffusion and reaction in pores of HZSM-5 zeolite-catalysed toluene disproportionation the numbers are approximate relative diffusion coefficients in the pores 1131.

A large number of analytical solutions of these equations appear in the literature. Mostly, however, they deal only with first order reactions. All others require solution by numerical or other approximate means. In this book, solutions of two examples are carried along analytically part way in P7.02.06 and P7.02.07. Section 7.4 considers flow through an external film, while Section 7.5 deals with diffusion and reaction in catalyst pores under steady state conditions. 

As mentioned earlier, if the rate of a catalytic reaction is proportional to the surface area, then a catalyst with the highest possible area is most desirable and that is generally achieved by its porous structure. However, the reactants have to diffuse into the pores within the catalyst particle, and as a result a concentration gradient appears between the pore mouth and the interior of the catalyst. Consequently, the concentration at the exterior surface of the catalyst particle does not apply to die whole surface area and the pore diffusion limits the overall rate of reaction. The effectiveness factor tjs is used to account for diffusion and reaction in porous catalysts and is defined as... 

This review will only focus on the modeling efforts in pore diffusion and reaction in single-catalyst pellets which have incorporated pore plugging as a deactivation mechanism. A broad literature exists on the deactivation of catalysts by active site poisoning, and it has been reviewed by Froment and Bischoff (1979). The behavior of catalytic beds undergoing deactivation is... 

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... 

A Rodrigues, R Quinta Ferreira, Effect of intraparticle convection, diffusion and reaction in a large-pore catalyst particle , AlChE Symp Ser, 1988, 84, 80-87... 

Pore networks in 2-D and 3-D are still being developed as computer-aided representations of real porous materials since the idea was first proposed some 40 years ago (2). Subsequently, 2-D networks were applied to porosimetry (5) and low-temperature gas adsorption (4), and 2-D and 3-D models have been compared (5). More recent work has applied 3-D networks to porosimetry (< ), to flow and transport behaviour (7), as well as to diffusion and reaction in catalysis (S). The equivalence of pore networks to a continuum representation for porosity has lately been established (9) and a review of recent developments and applications is available 10). 

In our discussion of surface reactions in Chapter 11 we assumed that each point in the interior of the entire catalyst surface was accessible to the same reactant concentration. However, where the reactants diffuse into the pores within the catalyst pellet, the concentration at the pore mouth will be higher than that inside the pore, and we see that the entire catalytic surface is not accessible to the same concentration. To account for variations in concentration throughout the pellet, we introduce a parameter known as the effectiveness factor. In this chapter we will develop models for diffusion and reaction in two-phase systems, which include catalyst pellets and CVD reactors. The types of reactors discussed in this chapter will include packed beds, bubbling fluidized beds, slurry reactors, and trickle beds. After studying this chapter you will be able to describe diffusion and reaction in two- and three-phase systems, determine when internal pore diffusion limits the overall rate of reaction, describe how to go about eliminating this limitation, and develop models for systems in which both diffusion and reaction play a role (e.g., CVD). 

The earliest studies of diffusion and reaction in catalysts were by Thiele, Damkoehler, and Zeldowitsch." Thiele considered the problem from the standpoint of a single cylindrical pore (see Prob. 11-9). Since the catalytic area per unit length of diffusion path does not change in a straight cylindrical pore whose walls are catalytic, the results are of the form of those for a flat plate [Eqs. (11-55) and (11-56)]. 

Diffusion of reactants to the external surface is the first step in a solid-catalyzed reaction, and this is followed by simultaneous diffusion and reaction in the pores, as discussed in Chapter 4. In developing the solutions for pore diffusion plus reaction, the surface concentrations of reactants and products are assumed to be known, and in many cases these concentrations are essentially the same as in the bulk fluid. However, for fast reactions, the concentration driving force for external mass transfer may become an appreciable fraction of the bulk concentration, and both external and internal diffusion must be allowed for. There may also be temperature differences to consider these will be discussed later. Typical concentration profiles near and in a catalyst particle are depicted in Figure 5.6. As a simplification, a linear concentration gradient is shown in the boundary layer, though the actual concentration profile is generally curved. 

Note that Eq. (7.25) has the same form as Eq. (4.75) for pore diffusion and reaction in a flat slab, but the boundary conditions are different, since the gradient for A is not zero at the edge of the liquid film. 

Let us refine this crude argument by investigating diffusion and reaction in a long narrow cylinder, the walls of which are covered with catalytic material. This reactor can serve as an idealized model of a pore in a catalyst pellet. Let the length of reactor be L, the length coordinate 4 ( = 0 at the reactor entrance), the radius of the reactor p. Reactant A penetrates into the reactor by diffusion only. Its concentration at = 0 is (A)o. Its diffusivity is D. Reaction on the walls proceeds at a rate ... 

Internal Pore Diffusion and Reaction in a Slab-Shaped Catalyst Pellet... 

Diffusion and reaction in a single pore in the catalyst pellet. 

An industrial DMTO fluidized bed catalyst pellet is basically composed of SAPO-34 zeofite particles and catalyst support (or matrix). The pores of zeolite particles and matrix are interconnected as a complex network. The pores inside zeofite particles are typically micropores (less than 2 nm) and the matrix normally has either mesopores (2-50 nm) or macropores (>50 nm), or both (Krishna and Wesselingh, 1997). The bulk diffusion coefficients in the meso- and macropores might be several orders of magnitude larger than surface diffusion coefficients in the micropores. Kortunov et al. (2005) found that the diffusion in macro- and mesopores also plays a crucial part in the transport in catalyst pellets. Therefore, other than a model for SAPO-34 zeofite particles, a modeling approach for diffusion and reaction in MTO catalyst pellets, which are composed of SAPO-34 zeofite particles and catalyst support, is needed. 

Exercise 6.2 Consider the problem of diffusion and reaction in a cylindrical pore (e.g., in a solid catalyst) where component A reacts at the walls of the cylinder according to... 

Solution You should use diffusion coefficients to describe the simultaneous diffusion and reaction in the pores in the catalyst. You should not use mass transfer coefficients because you cannot easily include the effect of reaction (see Sections 16.1 and 17.1). 

The study of the intra-phase mass transfer in SCR reactors has been addressed by combining the equations for the external field with the differential equations for diffusion and reaction of NO and N H 3 in the intra-porous region and by adopting the Wakao-Smith random pore model to describe the diffusion of NO and NH3 inside the pores [30, 44]. The solution of the model equations confirmed that steep reactant concentration gradients are present near the external catalyst surface under typical industrial conditions so that the internal catalyst effectiveness factor is low [27]. 

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