Chemical Reactor Analysis and Design

1. Chemical Reactor Analysis and Design 3th Edition G.F. Froment, K.B. Bischoff†, J. De Wilde Chapter 3 Transport Processes with Reactions Catalyzed by Solids Part two Intraparticle Gradient Effects

2. Introduction • Transport of reactants A, B, ... from the main stream to the catalyst pellet surface. • Transport of reactants in the catalyst pores. • Adsorption of reactants on the catalytic site. • Chemical reaction between adsorbed atoms or molecules. • Desorption of products R, S, .... • Transport of the products in the catalyst pores back to the particle surface. • Transport of products from the particle surface back to the main fluid stream. Steps 1, 3, 4, 5, and 7: strictly consecutive processes Steps 2 and 6: cannot be entirely separated ! Chapter 2: considers steps 3, 4, and 5 Chapter 3: other steps

3. Molecular-, Knudsen- and surface diffusion in pores (mainly encountered in zeolite catalysts) [Adapted from Weisz, 1973]

4. Molecular-, Knudsen- and surface diffusion in pores Molecular diffusion: • Driven by composition gradient • Mixture n components: Stefan-Maxwell: • Molecular diffusivities (binary): • independent of the composition • inversely proportional to the total pressure (gas) • proportional to T3/2 • Momentum transfer: by collisions between atoms • or molecules • Fluxes: expressed per unit external surface of the • catalyst particle with εs: void fraction of the catalyst particle (m3f / m3cat) = fraction particle surface taken by pore mouths (Dupuit)

5. Molecular-, Knudsen- and surface diffusion in pores Knudsen diffusion: • Mean free path of the components >>> pore dimensions • Momentum transfer: mainly collisions with the pore walls • Encountered • at pressures below 5 bar • with pore sizes between 3 and 200 nm • Knudsen diffusion flux of i : independent of the fluxes of • the other components: • Knudsen diffusivity: and: • function of the pore radius • independent of the total pressure • varies with T1/2 (Graham’s law)

6. Molecular-, Knudsen- and surface diffusion in pores Simultaneous Molecular and Knudsen diffusion and flux from viscous or laminar flow: Darcy’s permeability constant Dusty gas model equation (kinetic gas theory) Example: Binary mixture of A and B: Viscous flow term: • generally negligible • except when • > 10 – 20 • (micron-size pores) For equimolar counterdiffusion: Additive resistance relation (Bosanquet formula)

7. Molecular-, Knudsen- and surface diffusion in pores Simultaneous Molecular and Knudsen diffusion and flux from viscous or laminar flow: Diffusion in a multicomponent mixture: Sometimes Stefan-Maxwell replaced by less complicated equivalent binary mixture equation:

8. Molecular-, Knudsen- and surface diffusion in pores Surface diffusion: • Hopping of molecules from one adsorption site to another • Random walk model • where: • : the jump length • : the correlation time for the motion • k : a numerical proportionality factor Vary with temperature according to the van ‘t Hoff exponential law • pre-exponential factor: • energy factor: ~ 1/number of available sites known for structured surfaces like zeolites, but much less for amorphous surfaces • Depends on the surface coverage • More important in micro- than in macroporous material • Driving force: not fluid phase concentration gradient • (Fickian law can not be applied)

9. Diffusion in a catalyst particle A pseudo-continuum model: Effective diffusivities: Catalyst particles: very complicated (3D) pore structure • Model: • Pseudo-continuum • 1D • « Effective » diffusivity Fick type law: Pellet surface: Sphere:

10. Diffusion in a catalyst particle A pseudo-continuum model: in: « Tortuosity » factor: • Tortuous nature of the pores • Eventual pore constrictions • Typical value: 2 - 3

11. Experimental determination of effective diffusivities of a component and of the tortuosity • Pulse response technique: • column packed with catalyst (fixed bed) • ideal plug flow pattern ( dt/dp) • tracer pulse injected in carrier gas flow • pulse response measured (reactor outlet) In Fixed bed Out Tracer pulse Pulse response • Pulse widens: • Dispersion in the bed: • Adsorption on the catalyst surface • Effective diffusion inside the catalyst particle • Three parameters to be estimated: method of moments

12. Experimental determination of effective diffusivities of a component and of the tortuosity • Wicke-Kallenbach cell: • Steady state or transient operation • Single catalyst particle used as membrane • Above membrane: steady flow of carrier gas • Tracer pulse injected into the carrier gas: • Diffuses through the catalyst membrane • Swept in the compartment underneath by a carrier gas • => to detector • Two parameters to be estimated • Determination tortuosity: • Specific catalyst characterization equipment • (mercury porosimetry & nitrogen–sorption and –desorption)

13. Experimental determination of effective diffusivities of a component and of the tortuosity EXAMPLE 3.5.1.2.A Experimental determination of the effective diffusivity of a component and of the catalyst tortuosity by means of the packed column technique • Pt-Sn-y-alumina catalyst (catalytic reforming of naphtha) • Column internal diameter: 10-2 m • Column length: 0.805 m • Particle radius: 0.975 × 10-3 m • Void fraction of the packing: 0.429 m3f / m3r • Catalyst density, ρcat: 1080 kg cat/m3cat Hg-porosimetry, N2-adsorption and -desorption • Hg porosimetry: • Pore volume as a function of the amount of intruded mercury • Pore radius: calculated from Washburn eq. (cylindrical pores) • At 2000 bar: all pores > 3.3 nm filled with Hg • Nitrogen sorption: • Steep increase of the amount adsorbed at pressure where the macropores • are filled by nitrogen through capillary condensation • From Washburn eq.: total volume of adsorbed N2 • The volume of N2 adsorbed until the sharp rise is the meso pore volume

14. Experimental determination of effective diffusivities of a component and of the tortuosity From Van Melkebeke and Froment [1995]

15. Experimental determination of effective diffusivities of a component and of the tortuosity • Cumulative pore volume distribution: • Derived from N2 adsorption curve: Broekhoff-De Boer eq. (cylindrical pores) • Inflection point • => differential pore volume distribution by a peak at mean pore radius • Tracer pulse injected into packed column: • Fitting data: Kubin and Kucera-model • => De, KA, and Dax • => • Remarks: • Performing experiments at various total pressures • => possible to distinguish between and • Measurements possible in the absence or presence • of reactions

16. Diffusion in a catalyst particle Structure and Network models: (in contrast to Pseudo-continuum model) • More realistic representation • More accurate • Structure models: • The random pore model • The parallel cross-linked pore model • Network models: • A Bethe tree model • Network models for disordered pore media • Monte Carlo simulation • Effective Medium Approximation (EMA)

17. Diffusion in a catalyst particle Structure models: The random pore model: • Macro- micro pore model [Wakao and Smith, 1962 & 1964] • Application: pellets manufactured by compression small particles • Void fraction- and pore radius distributions: each replaced by two • averaged values, for the macro for the micro distribution (often a • pore radius of ~100 Å is used as the dividing point between macro • and micro) • Micro-pores particles: randomly positioned in pellet space • Macro-pores of the pellet: interstices • Diffusion flux: three parallel contributions: • Through the macro-pores • Through the micro-pores • Through interconnected macro-micro pores with:

18. Diffusion in a catalyst particle Structure models: The random pore model: Diffusion areas in random pore model. Adapted from Smith [1970].

19. Diffusion in a catalyst particle Structure models: The parallel cross-linked pore model • Pore size and orientation distribution function: • Pellet flux: integrating flux in single pore with orientation l • and accounting for the distribution function: with: : unit vector or direction cosine between l direction and coordinate axes Example: mean binary diffusivity: with: with: the tortuosity tensor

20. Diffusion in a catalyst particle Structure models: The parallel cross-linked pore model Limiting cases: 1) Perfectly communicating pores Cj(z; r, Ω) = Cj(z) (closest to usual types of catalyst particles) with: κ(r) : a reciprocal tortuosity (results from the integration over Ω) Proper diffusivity: weighted with respect to the measured pore size distribution 2) Noncommunicating pores: Pure diffusion at steady state, dNj/dz = 0 or Nj = constant + no assumption on communication of pores

21. Diffusion in a catalyst particle Structure models: The parallel cross-linked pore model Limiting cases: 3) Pore size and orientation effects are uncorrelated • with: • f(r) : the pore size distribution Completely random pore orientations => tortuosity depends only on the vector component cos = 3

22. Diffusion in a catalyst particle Network models: A Bethe tree model: • Higher coordination numbers possible • Pores can have a variable diameter • Main advantage: can yield analytical • solutions for the fluxes • Disadvantage: absence of closed • loops not entirely realistic • branching network of pores: • coordination number of 3 • no closed loops

23. Diffusion in a catalyst particle Network models: Disordered pore media: • Amorphous catalysts: no regular or structured morphology • Sometimes structure modified during its application (pore blockage) • Pore medium description: • Network of channels (preferably 3D) • Size distribution • Disorder to be included: certain • fraction of pores blocked Random number generator (Monte Carlo simulation) Calculations repeated for same over-all blockage probability & average pore size => calculated set of values of De is averaged • Effective Medium Approximation (EMA): construct small size network • => relation between diffusivity & blockage without considering • complete network

24. Diffusion and reaction in a catalyst particle. A continuum model First-Order Reactions. The Concept of Effectiveness Factor: • Reaction and diffusion occur simultaneous: • Process not strictly consecutive • Both phenomena must be considered together Example: first-order reaction, equimolar counterdiffusion, isothermal conditions, and steady-state: slab of thickness L: Species continuity equation A: with boundary conditions: at the surface at the center line Solution:

25. Diffusion and reaction in a catalyst particle. A continuum model First-Order Reactions. The Concept of Effectiveness Factor: with: for a slab of thickness L

26. Diffusion and reaction in a catalyst particle. A continuum model First-Order Reactions. The Concept of Effectiveness Factor: Effectiveness factor: Observed reaction rate: First-order reaction Extension to more practical pellet geometries: cylinders or spheres: e.g., sphere: Aris [1957]:

27. Diffusion and reaction in a catalyst particle. A continuum model First-Order Reactions. The Concept of Effectiveness Factor: Effectiveness factors for slab (P), cylinder (C), and sphere (S) as functions of the Thiele modulus. Dots represent calculations by Amundson and Luss [1967] and Gunn [1965]. From Aris [1965].

28. Diffusion and reaction in a catalyst particle. A continuum model More General Rate Equations. Single rate equation: Analytical solution not possible • Generalized modulus (10 – 15% error) • Numerical solution Depends on Cs ! Coupled multiple reactions: • Numerical solution: • finite difference • orthogonal collocation Catalytic reforming of naphtha on Pt.Sn/alumina. Dimensionless partial pressure profile inside the particle for the reactant nonane. Total pressure: 7 bar, T = 510 °C, molar ratio H2/Hydrocarbons = 5. From Sotelo-Boyas and Froment [2008].

29. Falsification of rate coefficients and activation energy by diffusion limitations Consider nth order reaction: ~ Introduce generalized modulus: observed rate: order (n+1)/2 only correct for 1st order reaction Also: effective diffusion (strong pore diffusion limitations)

30. Falsification of rate coefficients and activation energy by diffusion limitations Weisz and Prater [1954]: or: Languasco, Cunningham, and Calvelo [1972]: EXAMPLE 3.7.A EFFECTIVENESS FACTORS FOR SUCROSE INVERSION IN ION EXCHANGE RESINS First-order reaction: Studied in particles with different size [Gilliland, Bixler, and O’Connell, 1971]

31. Falsification of rate coefficients and activation energy by diffusion limitations EXAMPLE 3.7.A EFFECTIVENESS FACTORS FOR SUCROSE INVERSION IN ION EXCHANGE RESINS a Calculated on the basis of approximate normality of acid resin = 3N. Separately measured: DeA = 2.69 × 10-7 cm2/s. From data at 60 and 70°C. ED = 34 kJ/mol theor.

32. Influence of diffusion limitations on the selectivities of coupled reactions Parallel, independent reactions: [Wheeler, 1951] Absence of diffusion limitations: • Compare: • Diffusional resistance decreases selectivity ! With pore diffusion limitations: First-order reactions & Strong pore diffusion limitations:

33. Influence of diffusion limitations on the selectivities of coupled reactions Consecutive first-order reactions: [Wheeler, 1951] Absence of diffusion limitations: With pore diffusion limitations: Species continuity equations A and R, for slab geometry: Selectivity

34. Influence of diffusion limitations on the selectivities of coupled reactions Consecutive first-order reactions: [Wheeler, 1951] With pore diffusion limitations: Selectivity: • Compare: • Diffusional resistance decreases selectivity ! with: with i = 1, 2 Strong pore diffusion limitation and for DeA = DeR: ~

35. Criteria for the importance of intraparticle diffusion limitations • Determining kinetic parameters from experimental data: • kρs not available yet • Criteria importance pore diffusion not explicitly containing kρs also needed ! 1) Experiments with two different sizes of catalyst: Assume kρs and DeA same for both sizes with: and: = 1 No intraparticle diffusion limitations: Strong intraparticle diffusion limitations: : 2) Weisz-Prater criterion [1954]: First-order reaction: Extendable via generalized modulus

36. Combination of external and internal diffusion limitations Nonisothermal particles • Practical situations: • Internal temperature gradients unlikely • Internal gradients unlikely to cause particle instability