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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

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Introduction 1.Transport of reactants A, B,... from the main stream to the catalyst pellet surface. 2.Transport of reactants in the catalyst pores. 3.Adsorption of reactants on the catalytic site. 4.Chemical reaction between adsorbed atoms or molecules. 5.Desorption of products R, S, Transport of the products in the catalyst pores back to the particle surface. 7.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

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

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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 T 3/2 Momentum transfer: by collisions between atoms or molecules Fluxes: expressed per unit external surface of the catalyst particle with ε s : void fraction of the catalyst particle (m 3 f / m 3 cat ) = fraction particle surface taken by pore mouths (Dupuit)

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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: function of the pore radius independent of the total pressure varies with T 1/2 and: (Graham’s law)

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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) Viscous flow term: generally negligible except when > 10 – 20 (micron-size pores) Example: Binary mixture of A and B: For equimolar counterdiffusion: Additive resistance relation (Bosanquet formula)

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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:

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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)

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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:

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

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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 ( d t /d p ) tracer pulse injected in carrier gas flow pulse response measured (reactor outlet) InOut Tracer pulsePulse response Fixed bed 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

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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)

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Experimental determination of effective diffusivities of a component and of the tortuosity EXAMPLE 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: m Column length: m Particle radius: × m Void fraction of the packing: m 3 f / m 3 r Catalyst density, ρ cat : 1080 kg cat/m 3 cat Hg-porosimetry, N 2 -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 N 2 The volume of N 2 adsorbed until the sharp rise is the meso pore volume

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

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Experimental determination of effective diffusivities of a component and of the tortuosity Cumulative pore volume distribution: Derived from N 2 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 => D e, K A, and D ax => Remarks: Performing experiments at various total pressures => possible to distinguish between and Measurements possible in the absence or presence of reactions

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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: 1. A Bethe tree model 2. Network models for disordered pore media Monte Carlo simulation Effective Medium Approximation (EMA)

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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: 1. Through the macro-pores 2. Through the micro-pores 3. Through interconnected macro-micro pores with:

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

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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: the tortuosity tensor

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Diffusion in a catalyst particle Structure models: The parallel cross-linked pore model Limiting cases: 1) Perfectly communicating poresC j (z; r, Ω) = C j (z) 2) Noncommunicating pores: Pure diffusion at steady state, dN j /dz = 0 or N j = constant + no assumption on communication of pores (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

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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

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Diffusion in a catalyst particle Network models: A Bethe tree model: branching network of pores: coordination number of 3 no closed loops 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

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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 D e is averaged Effective Medium Approximation (EMA): construct small size network => relation between diffusivity & blockage without considering complete network

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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:

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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

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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]:

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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].

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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 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]. Depends on C s !

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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)

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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]

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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. D eA = 2.69 × cm 2 /s. Separately measured: From data at 60 and 70°C. E D = 34 kJ/mol theor.

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

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

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

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

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

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