Fogler_ECRE_CDROM.book Page 813 Wednesday, September 17, 2008 5:01 PM Diffusion 12 and Reaction Research is to see what everybody else sees, and to think what nobody else has thought. Albert Szent-Gyorgyi Overview This chapter presents the principles of diffusion and reaction. While the focus is primarily on catalyst pellets, examples illustrating these principles are also drawn from biomaterials engineering and microelectronics. In our discussion of catalytic reactions in Chapter 10, we assumed each point on the interior of catalyst surface was accessible to the same concen- tration. However, we know there are many, many situations where this equal The concentration in accessibility will not be true. For example, when the reactants must diffuse the internal surface inside the catalyst pellet in order to react, we know the concentration at the of the pellet is less pore mouth must be higher than that inside the pore. Consequently, the entire than that of the catalytic surface is not accessible to the same concentration; therefore, the external surface rate of reaction throughout the pellet will vary. To account for variations in reaction rate throughout the pellet, we introduce a parameter known as the effectiveness factor, which is the ratio of the overall reaction rate in the pel- let to the reaction rate at the external surface of the pellet. In this chapter we will develop models for diffusion and reaction in two-phase systems, which include catalyst pellets, tissue generation, and chemical vapor deposition (CVD). The types of reactors discussed in this chapter will include packed beds, bubbling fluidized beds, slurry reactors, trickle bed reactors, and CVD boat reactors. After studying this chapter, you will be able to describe diffu- sion and reaction in two- and three-phase systems, determine when internal diffusion limits the overall rate of reaction, describe how to go about elimi- nating this limitation, and develop models for systems in which both diffu- sion and reaction play a role (e.g., tissue growth, CVD). 813 Fogler_ECRE_CDROM.book Page 814 Wednesday, September 17, 2008 5:01 PM 814 Diffusion and Reaction Chap. 12 In a heterogeneous reaction sequence, mass transfer of reactants first takes place from the bulk fluid to the external surface of the pellet. The reactants then diffuse from the external surface into and through the pores within the pellet, with reaction taking place only on the catalytic surface of the pores. A schematic representation of this two-step diffusion process is shown in Figures 10-6 and 12-1. Figure 12-1 Mass transfer and reaction steps for a catalyst pellet. 12.1 Diffusion and Reaction in Spherical Catalyst Pellets In this section we will develop the internal effectiveness factor for spherical catalyst pellets. The development of models that treat individual pores and pel- lets of different shapes is undertaken in the problems at the end of this chapter. We will first look at the internal mass transfer resistance to either the products or reactants that occurs between the external pellet surface and the interior of the pellet. To illustrate the salient principles of this model, we consider the irreversible isomerization AB ⎯⎯→ that occurs on the surface of the pore walls within the spherical pellet of radius R. 12.1.1 Effective Diffusivity The pores in the pellet are not straight and cylindrical; rather, they are a series of tortuous, interconnecting paths of pore bodies and pore throats with varying cross-sectional areas. It would not be fruitful to describe diffusion within each and every one of the tortuous pathways individually; consequently, we shall define an effective diffusion coefficient so as to describe the average diffusion taking place at any position r in the pellet. We shall consider only radial vari- ations in the concentration; the radial flux WAr will be based on the total area (voids and solid) normal to diffusion transport (i.e., 4␲r2) rather than void area alone. This basis for WAr is made possible by proper definition of the effective diffusivity De. The effective diffusivity accounts for the fact that: 1. Not all of the area normal to the direction of the flux is available (i.e., the area occupied by solids) for the molecules to diffuse. Fogler_ECRE_CDROM.book Page 815 Wednesday, September 17, 2008 5:01 PM Sec. 12.1 Diffusion and Reaction in Spherical Catalyst Pellets 815 2. The paths are tortuous. 3. The pores are of varying cross-sectional areas. An equation that relates De to either the bulk or the Knudsen diffusivity is D φ σ The effective D = --------------------AB p -c (12-1) diffusivity e τ˜ where Actual distance a molecule travels between two points t˜ ϭ tortuosity1 = ---------------------------------------------------------------------------------------------------------------------------------- Shortest distance between those two points Volume of void space φ ϭ pellet porosity = ----------------------------------------------------------------------------- p Total volume ()voids and solids ␴ ϭ c Constriction factor ␴ The constriction factor, c, accounts for the variation in the cross-sectional area that is normal to diffusion.2 It is a function of the ratio of maximum to minimum pore areas (Figure 12-2(a)). When the two areas, A1 ␤ ϭ and A2, are equal, the constriction factor is unity, and when 10, the con- striction factor is approximately 0.5. A B A2 A1 L L area A1 (a) (b) Figure 12-2 (a) Pore constriction; (b) pore tortuosity. Example 12–1 Finding the Tortuosity Calculate the tortuosity for the hypothetical pore of length, L (Figure 12-2(b)), from the definition of t˜ . 1 Some investigators lump constriction and tortuosity into one factor, called the tortuos- ⁄σ ity factor, and set it equal to t˜ c . C. N. Satterfield, Mass Transfer in Heterogeneous Catalysis (Cambridge, Mass.: MIT Press, 1970), pp. 33–47, has an excellent discus- sion on this point. 2 See E. E. Petersen, Chemical Reaction Analysis (Upper Saddle River, N.J.: Prentice Hall, 1965), Chap. 3; C. N. Satterfield and T. K. Sherwood, The Role of Diffusion in Catalysis (Reading, Mass.: Addison-Wesley, 1963), Chap. 1. Fogler_ECRE_CDROM.book Page 816 Wednesday, September 17, 2008 5:01 PM 816 Diffusion and Reaction Chap. 12 Solution Actual distance molecule travels from A to B t˜ = ------------------------------------------------------------------------------------------------------------ Shortest distance between A and B The shortest distance between points A and B is 2 L . The actual distance the mol- ecule travels from A to B is 2L. 2L t˜ ===---------- 2 1.414 2 L Although this value is reasonable for t˜ , values for t˜ ϭ 6 to 10 are not unknown. Typical values of the constriction factor, the tortuosity, and the pellet porosity are, ␴ ϭ φ respectively, c 0.8, t˜ 3.0, and p 0.40. 12.1.2 Derivation of the Differential Equation Describing Diffusion and Reaction We now perform a steady-state mole balance on species A as it enters, leaves, and reacts in a spherical shell of inner radius r and outer radius r r of the pellet (Figure 12-3). Note that even though A is diffusing inward toward the center of the pellet, the convention of our shell balance dictates that the flux be in the direction of increasing r. We choose the flux of A to be positive in the First we will derive direction of increasing r (i.e., the outward direction). Because A is actually the concentration profile of reactant diffusing inward, the flux of A will have some negative value, such as A in the pellet. Ϫ10 mol/m2иs, indicating that the flux is actually in the direction of decreas- ing r. R CAs r + Δr r Figure 12-3 Shell balance on a catalyst pellet. We now proceed to perform our shell balance on A. The area that appears in the balance equation is the total area (voids and solids) normal to the direc- tion of the molar flux: и ␲ 2 Η Rate of A in at r WAr Area WAr 4 r r (12-2) ϭ и ϭ ␲ 2 Η Rate of A out at (r r) WAr Area WAr 4 r rϩ⌬r (12-3) Fogler_ECRE_CDROM.book Page 817 Wednesday, September 17, 2008 5:01 PM Sec. 12.1 Diffusion and Reaction in Spherical Catalyst Pellets 817 Rate of generation Rate of reaction Mass catalyst of A within a ϭ -------------------------------------- - × -------------------------------- × Volume of shell Mass of catalyst Volume shell of thickness Δr ′ 2 ϭ rA × ρ × π Δ c 4 rm r (12-4) Mole balance for where rm is some mean radius between r and r r that is used to approxi- diffusion and mate the volume ⌬V of the shell and ρ is the density of the pellet. reaction inside the c catalyst pellet The mole balance over the shell thickness ⌬r is (In at r) Ϫ (Out at r+Δr) ϩ (Generation within Δr) ϭ 0 Mole balance (12-5) ()× π 2 Ϫ () × π 2 ϩ ()′ ρ ×Δπ 2 ϭ WAr 4 r r W A r 4 r rϩΔr r A c 4 rm r 0 After dividing by (Ϫ4␲⌬r) and taking the limit as ⌬r → 0, we obtain the following differential equation: dW()r2 ---------------------Ar –r′ ρ r2 = 0 (12-6) dr A c Because 1 mol of A reacts under conditions of constant temperature and pressure to form 1 mol of B, we have Equal Molar Counter Diffusion (EMCD) at constant total concentration (Section 11.2.1A), and, therefore, dyA dCA The flux equation W ==–cD -------- –D --------- - (12-7) Ar e dr e dr 3 where CA is the number of moles of A per dm of open pore volume (i.e., vol- ume of gas) as opposed to (mol/vol of gas and solids). In systems where we do not have EMCD in catalyst pores, it may still be possible to use Equation (12-7) if the reactant gases are present in dilute concentrations.