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**Computer-aided chemical reaction engineering**

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**CACRE Forms of work: Lectures Demonstrations /case studies**

Teachers: Tapio Salmi, Johan Wärnå, Heikki Haario, Paolo Canu, Matias Kangas, Sébastien Leveneur, Fredrik Sandelin, Dmitry Murzin Forms of work: Lectures Demonstrations /case studies Computational exercises Final exam

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**Contents 1 Introduction 2 Stoichiometry and kinetics**

3 Homogeneous reactors 4 Catalytic two-phase reactors 5 Catalytic three-phase reactors 6 Fluid-fluid reactors 7 Reactors with a reactive solid phase 8 Laboratory reactors and parameter estimation 9 CFD Bilagor 1-8

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Approach- procedure 1 Generalized models for chemical reactors (mass and energy balances) 2 Identification of the mathematical structure of the model (NLE, ODE, PDE…) 3 Modularization of the model 4 Selection of numerical strategy and methods 5 Selection of software 6 Model implementation 7 Test simulations 8 Final simulations Bilagor 1-8

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**Chemical process in general**

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**A chemical reactor Reactor Transforms Raw material ® Products**

out In Transforms Raw material ® Products Can be batchwise, semicontinuous or continuous Can be stationary or non-stationary Classification often basen on the number of phases gas, liquid, solid, catalyst The process chemistry determines very much the reactor selection

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**Why is modelling and computation needed**

It cannot be done in this way !

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**Reactor design in nutshell**

Optimization Idea Parameter estimation Mathematical model Experiment Reactor ready

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**Principles of reactor modelling**

Kinetic model Mass and heat transfer model Flow model REACTOR MODEL

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**Ingredients in the model**

Stoichiometry Kinetics and termodynamics Reaction & diffusion Reactor model

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**Stoichiometry and och kinetics**

Desired reactions Non-desired reactions (side reactions) Often multiple reactions Example: Methanol synthesis CO + 2H2 W CH3OH (desired reaction) CO2 + H2 W CO + H2O (side reaction) Parallel reaction with respect to hydrogen Consecutive reaction with respect to CO

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**Stoichiometric matrix**

Reactants - Products +

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**Stoichiometric matrix - example**

CO + 2H2 = CH3OH CO2 + H2 = CO + H2O (1) (2) -1 CO - 2 H2 + 1CH3OH = 0 - 1CO2 -1 H2 + 1CO + 1H2O = 0

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Reaction kinetics Reaction rate R (mol/s m3) gives how many moles of substance is generated /time/volume Important to know the difference between elementary and non-elementary reactions Elementary reaction reflects directly the events, collisions on the molecular level

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**Rate expressions 2A + B = 2C If the reaction is elementary:**

The construction of rate expressions for elementary reactions is straightforward and can be done automatically by a computer; just the stoichiometric matrix is needed as an input

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**Component generation rate ri**

For methanol synthesis reaction rH2 = - 2 R och rCH3OH = +1 R Systems with many reactions and components rH2 = -2R1 - 1R2 rCH3OH = +1 R1 + 0 R2

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Generation rate 2A + B = 2C rA=-2R rB=-R rC=2R

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Homogeneous reactors Only one phase present (incl. a homogeneous catalyst) Tubular reactor (plug flow reactor(PFR)) Tank reactor (CSTR, semibatch reactor, batch reactor)

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**Homogeneous reactors One phase**

Gas or liquid Tank reactors can be continuous (CSTR), semibatch (SBR) och batch reactors (BR) Tubular reactor with plug flow (PFR) or axial dispersion (ADR, ADM)

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CSTR

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Batch reactors

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**Mixing in tank reactors**

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Tubular reactor (PFR)

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Tubular reactor

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**Advanced reactor technology -parallel tube reactors**

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**Cooling systems for reactors**

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Definitions Total molar flow Total molar amount

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**Definitions: mole fraction**

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**Definitions: concentration**

For continuous systems and batch reactors Concentration and mole fraction

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**Definitions: volumetric flow rate, mass flow, density**

Combination gives

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Conversion

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**STIRRED TANK REACTORS WITH COMPLETE BACKMIXING**

STIRRED TANK REACTORS WITH COMPLETE BACKMIXING General mass balance Rearranged to For batch reactor, all flows zero: For batch reactor, all flows zero:

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**With arrays we can write**

The initial condition is CSTR at steady state: dni/dt = 0 With arrays we can write Batch reactor (BR)

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Special cases a) Isothermal liquid phase CSTR τ = V/V0, τ = space time

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**b) Batch reactor, gas and liquid phases**

V= V0

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**Liquid-phase semibatch reactor (SBR) **

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**Gas-phase CSTR at steady state**

Gas-phase CSTR at steady state

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**Tubular reactors: Plug flow (PFR) and axial dispersion model (ADM)**

An infinitesimal volume element ΔV is considered: DADM Denote which imply

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**Plug flow (PFR) and axial dispersion model (ADM)**

General mass balance The accumulation term The volume element ΔV:

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Let Δl 0: where ADM For the plug flow model, the eqn. is reduced to (Adl = dV): PFR At steady state conditions the time derivative of the concentration vanishes: PDF We obtain ADM PFR

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**Liquid phase reactions:**

where w0 = the superficial velocity at the reaction inlet, ADM PFR

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**Gas-phase reactions PFR**

Gas-phase reactions PFR For both isothermal and steady state conditions, dc/dt = 0 and we get: A differential equation is obtained for δ: For isothermal cases dc/dz is 0, and δ is easily obtained by integration (c = c0)

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**The concentrations which are needed in the rate lows are obtained from**

At steady state conditions the natural choice of variable is ni and we get for the plug flow model: The concentrations which are needed in the rate lows are obtained from The initial condition is The volumetric flow rate is updated by the formula

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**Energy balances general considerations**

Energy balances general considerations A general energy balance for a volume element can be written as The use of molar enthalpies: The difference is split to two terms, What is Σ Δhmini? This becomes clear, when the molar heat capacity is introduced: is valid, provided that the pressure effect is neglected. What is Σ HmiΔni? - The mass balance give

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The generation rate and the molar amount being present in the volume element is The term Σ Hmi Δni becomes where the sum Σi υij Hmi is de facto the reaction enthalpy, Δ Hrj.

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We get

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vmi is the molar volume. The energy balance becomes ΔV 0

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Tank reactors Because of the homogeneous contents, the integration over the entire tank volume can be carried out:

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**Heat transfer from/to the reactor**

Steady state For liquid phase reactions the term Heat transfer from/to the reactor

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**S is the total heat transfer area. For cylindrical tubes**

Tubular plug flow reactor Steady state S is the total heat transfer area. For cylindrical tubes where RT, DT and LT denote the radius, the diameter and the length of the tubes. The temperature outside the reactor is

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**Batch reactor Reactor volume is constant, differentiation leads to**

where and

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**Batch reactors are very frequently used for liquid-phase processes, for which ΔUrj ΔHij.**

where the product ρ0 cp is the heat capacity of the reacting liquid.

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**The mathematical structures of homogeneous reactor models**

The mathematical structures of homogeneous reactor models MODEL PROBLEM STRUCTURE Steady state CSTR f (y) = 0 (N)LE Dynamic CSTR dy/dt = f (y) ODE (IVP) dy/dz = f (y) ODE (IVP) Steady state PFR PDE (IVP) Dynamic PFR dy/dt = -A dy/dz + f (y) Batch reactor (BR) dy/dt= f (y) ODE Semi batch reactor (BR) Steady state axial dispersion model (ADM) A d2y/dz2 + Bdy/dz + f (y) = 0 ODE (PVP) Dynamic axial dispersion model (ADM) dy/dt = A d2y/dz2 + B dz/dz + f (y) = PDE (N)LE = (non)linear equations ODE = ordinary differential equations initial value problem PD = partial differential equations, hyperbolic type IVP = initial value problem BVP = boundary value problem

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**Software build-up; tasks**

Software build-up; tasks • input of chemical engineering data (initial concentrations, temperatures, reactor dimensions, kinetic, thermodynamic as well as mass and heat transfer parameters) • input of data for steering of the numerical solution (numerical methods, system step-length selection, convergence criteria e.t.c. • definition of the kinetic and thermodynamic models (reaction rates, calculation of rate and equilibrium constants) • definition of the mass and heat transfer models (e.g. correlations for diffusion and dispersion coefficients, heat and mass transfer coefficients and areas) • definition of mass and energy balances along with equations for pressure drop • definition of partial derivatives needed for the model solution Tasks • numerical solver for the differential and/or algebraic equations involved in the model • output routines, which tell not only the results (e.g. concentration, temperature and pressure profiles) but also give information about the success or failure of the numerical solution process

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**Program flowsheet Simulation programme modules c, T c0, T0 OUTPUT**

INPUT c0, T0 MAIN PROGRAM Program flowsheet SOLVER -mathematical library routine for NLE or ODES MODEL (FCN) -routine for the model eqs, e.g. dy/dt = f(y) CORRE -routine for calculation of correlations for mass/heat transfer RATE -routine for calculation of reaction rates THERMO -routine for calculation of thermodynamics properties MODEL DERIVATIVES (FCNJ) -Jacobian routine J, contains fi/y Simulation programme modules

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**Solution of algebraic equations**

Solution of algebraic equations Newton-Raphson algorithm where k is the iteration index, f(k), is the function vector and J-1 is the inverted Jacobian matrix End criterion One unknown case

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**Solution of ordinary differential equations**

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**and J is the Jacobian matrix**

Runge-Kutta methods where yn and yn-1 give the solution of the differential equation at x and x-Δx. where h=Δx. If the coefficient ail = 0 for l = i the method is explicit; if ail 0 for l = i the method is implicit. where I denotes the identify matrix: and J is the Jacobian matrix :

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**Semi-implicit Runge-Kutta method**

An extension of semi-implicit Runge-Kutta methods is the Rosenbrock-Wanner method (ROW)

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**Linear multistep methods**

Adams-Moulton (AM) and backward difference-methods;

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**Adams-Moulton method:**

Backward difference-method:

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**Temperature and concentration**

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**Temperature dependence of the rate constant**

Jacobus Henricus van’t Hoff and Svante Arrhenius: Transition state theory

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Activation energy

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**Reaction thermodynamics**

Liquid-phase systems Equilibrium constant often determined experimentally Gas-phase system Equilibrium constant can be calculated theoretically, one knows 0.2 0.4 0.6 0.8 1 1.2 2 4 6 8 10

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**Reaction thermodynamics**

Temperature dependence Often approximately Change in internal energy Change in entropy

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**Reaction thermodynamics**

Equilibrium constant Kp integration gives

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**Reaction enthalpies Reaction enthalpy**

At reference temperature T0 (often 298 K) From enthalpies of formations See for example in Reid,Prausnitz, Poling, The Properties of Gases and Liquids Reaction enthalpy from Molar heat capacities exist as temperature (Cpmi) functions

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**CSTR- steady state multiplicity**

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**Catalytic two-phase reactors**

From reaction mechanism to reactor design Tutkimuksessamme syvennymme kinetiikkaan ja katalyysiin jotka pitävät sisällänsä kaiken reaktiomekanismista reaktorisuunnitteluun.

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**Catalyst materials Cu/SiO2**

Elektron microscopy (SEM) reveals catalyst morhpology 20 m 200 m Laboratorio syntetisoi laajan valikoiman katalyytisia materiaaleja joista zeoliitit on yksi pääalue! Cu/SiO2 20 m SAPO-5

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**Isobutene i 10 MR H–FER pores**

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**Fibre catalysts D = 27% dPt= 4 nm D = 40% dPt= 2.5 nm**

m particles 1 mm 4 nm SEM image of the 5 wt.% Pt/SiO2 fiber catalyst TEM image of the 5 wt.% Pt/Al2O3 (Strem Chemicals) catalyst

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**Catalysts in micro- and nanoscale**

5wt.% Pt/SF (Silikafiber) wt.% Pt/Al2O3 (Strem) 5 m 5 m 4 nm 4 nm SEM-bild TEM-bild

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**New products and processes from renewable sources**

Chemicals from biomaterial, particularly wood Catalytic production of biodiesel Tutkimuksessamme syvennymme kinetiikkaan ja katalyysiin jotka pitävät sisällänsä kaiken reaktiomekanismista reaktorisuunnitteluun.

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From wood to food Isomerization of linoleic acid was first time carried out on a heterogeneous catalyst Linoelihapposta jota esiintyy mäntyöljyssä on mahdollista valmistaa terveydelle hyödyllisiä tuotteita heterogeenisten katalyytien avulla

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**Catalytic two-phase reactors**

Heterogeneous catalytic reactor Solid catalyst accelerates reactions Gas or liquid present in the reactor

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**Molecular path to the catalyst**

diffusion to the outer surface of the catalyst particle diffusion through the catalyst pores Molecules come to the active sites on the surface

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**Molecular path to the catalyst**

molecules adsorb on the active sites and react with each other Product molecules desorb and diffuse out

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**Concentration and temperature profiles in catalyst particles**

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Catalytic Reactors Packed bed Fluidized bed

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**Packed bed: traditional design**

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**Gauze-reactor Oxidation av ammonia High temperature, 890°C**

Network of Pt catalyst

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Monolith reactor

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**Multibed reactor Catalyst beds in series (often adiabatic)**

Heat exchangers between the beds

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**Multibed reactor, SO2 to SO3**

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Multitubular reactor

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**Models for packed beds Pseudo-homogeneous model**

concentration and temperature in the catalyst particle on the same level as in the fluid bulk neither concentration- nor temperature gradients in the catalyst particle Diffusion resistance negligible in the catalyst particlen Pore diffusion can be included with the aid of an effectiveness factor

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**Models for packed beds Heterogeneous model**

Separate balance equations for bulk phase and catalyst particles

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Catalyst bulk density

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**One-dimensional pseudo-homogeneous model - stationary**

[in] + [generated i] = [out] + [accumulated]

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**One-dimensional pseudo-homogeneous model**

The diffusion of the molecules through the fluid film around the catalyst particle and their diffusion through the pores Influences the reaction rate The real reaction rate becomes cB = concentration in the bulk phase ρB = mass of catalyst / reactor volume

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**Effectiveness factor Real diffusion flow/Intrinsic kinetics**

=1 if diffusion resistance negligible

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**Diffusion i porous particle**

Catalyst surface

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Diffusion in pores

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**Fick’s law Ni diffusion flow mol/(time surface)**

Dei effective diffusion coefficient

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**Mass balance for a catalyst particle – steady state**

If diffusion coefficient is constant:

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**Form factor – shape factor (a=s+1)**

R characteristic dimension of the particle Ap outer surface of the particle Vp particle volume

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Form factor S=0 slab S=1 cylinder S=2 sphere

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Biot number Relation between the diffusion resistance in the fluid film and the catalyst particle Biot number is usually >>1 for porous particles

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**Thiele modulus: first order reaction**

Reaction rate / diffusion rate

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Effectiveness factor

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**Asymptotic effectiveness factor**

Semianalytical expressions for arbitrary kinetics Good approximation for positive reaction orders Erroneous if the reaktion order is negative, reaction is accelererated with decreasing concentration

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**Heat effects in catalyst particles**

Fourier’s law (heat conduction) Temperature gradient inside the particle typically small Temperature gradient can exist in the fluid film around the particle The film is thin

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**Heat effects in catalyst particle**

Energy and mass balances for catalyst particle are coupled via reaction rate and should be solved numerically Effectivity factor can be >1 for strongly exothermic reactions, rate constant increases with temperature ! Steady state multiplicity possible

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**Steady state multiplicity**

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**Lactose hydrogenation - diffusion**

Concentration profiles Inside the particle -various particle sizes lactose + H2 = lactitol Center Yta

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**Concentration profiles of lactose in the particle**

200 400 600 .5 1 0.2 0.4 0.6 0.8 1.2 1.4 200 400 600 0.5 1 0.2 0.4 0.6 0.8 1.2 1.4 Concentration profiles of lactose in the particle surface surface center x center time x time No deactivation Deactivation

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**Two-dimensional model**

Large heat effect induce a fradial temperature gradient, which leads to concentration gradients Reaction rates vary in radial direction Concentrations gradients created

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**Catalytic hydrogenation**

Hot spot Hydrogenation of av toluene

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Oxidation of o-xylene Dependence of Hot spot on the the coolant and inlet temperature

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**Two-dimensional model: temperature profile**

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**Two-dimensional model**

Mass balance [in plug flow] + [in radial disp.] + [generated] = [out plug flow] + [out radial disp.]

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**Two-dimensional model**

Energy balance

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**Numerical solution Parabolic PDEs converted to ODEs**

Finite difference + (RK, Adams Moulton, Backward difference) Orthogonal collocation + (RK, Adams Moulton, Backward difference)

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**Fluidised bed Fluidisation Solid particles in vertical bed**

Gas blown from the bottom Particles remain stagnant at low gas velocities At higher gas velocities the particles fluidise The bed is expanded and the particles become dispersed in the gas phase Minimum fluidisation velocity

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Fluidised bed With increasing gas velocity a bubble phase is formed – rich in gas Most part of catalyst in the emulsion phase Fluidised bed resembles boiling liquid

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Fluidised bed If the gas velocity is increased even more, the bubbles become equal to the bed diameter (slug flow) Limit velocity for slug flow = ws

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**Fluidised bed and pressure drop**

Fluidisation is recognised by measuring the pressure drop Pressure drop increases monotonically with the gas velocity (Ergun equation) At minimum fluidisation, the increase of pressure drop stops

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**Fluidiserad bed: hydrodynamics**

Bubble phase Emulsion phase Wake Cloud Reactions proceed in all phases !

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**Phase structure in detail**

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**Fluidised bed Matematical model - principles**

Separate balance considerations for each phase Catalyst particles are very small (in micrometer scale) so internal diffusion can be ignored Vigorous turbulence implies that the bed is isothermal

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**Fluidised bed Matematical model**

Plug flow model Unrealistic but gives the maximum limit Backmixing model Sometimes tanks-in-series model is tried

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**Fluidised bed Matematical model**

Kunii-Levenspiel model Realistic description Bubble phase in plug flow Gas flow in the emulsion phase is negligible Cloud and wakephases have the same concentrations

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**Kunii-Levenspiel model**

Transport of a reacting gas from bubble phase to cloud-wake phase and further to emulsion phase Three parts in the volume element

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**Kunii-Levenspiel model**

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**Kunii-Levenspiel model, mass balances**

Bubble phase Cloud-wake phase Emulsion phase

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**Kunii-Levenspiel model structure**

3 * N mass balances (N= number of components) 1 * N ODEs 2*N algebraic equations Numerical solution with DASSL

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**Kunii-Levenspiel model parameters**

Volume fractions Vc/Vb och Ve/Vb Kbc och Kbe from correlations Mean residence time of bubbles

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**Diffusion coefficients**

Depend on molecular structure – in general dependent on concentrations, too Fick’s law gives a simple relation between the diffusion flux and the concentration gradient

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**Effective diffusion coefficients**

Effektive diffusion coefficient in a porous particle Di molekular diffusion coefficient ep porosity < 1 tp tortuosity or labyrinth factor, > 1)

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**Effective diffusion coefficient**

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**Molecular diffusion Intermolecular diffusion Knudsen diffusion**

collisions between molecules Knudsen diffusion Collisions between molecules and pore walls

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**Diffusion coefficient Gas phase**

Fuller-Schettler-Giddings equation T temperature M molar mass v volyme contribution

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**Volyme contributions Diffusion volumes of simple molecules**

He 2.67 CO 18.0 Ne 5.98 CO2 26.9 Ar 16.2 N2O 35.9 Kr 24.5 NH3 20.7 Xe 32.7 H2O 13.1 H SF6 71.3 D Cl2 38.4 N Br2 69.0 O SO2 41.8 Air 19.7 Atomic and Structural Diffusion Volume Increments C 15.9 F 14.7 H 2.31 Cl 21.0 O 6.11 Br 21.9 N 4.54 I 29.8 S 22.9 Aromatic ring -18.3 Heterocyclic ring -18.3

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**Diffusion coefficient Gases**

Knudsen’s diffusion coefficient Sg specific surface area, which can be determined by nitrogen adsorption, BET (Brunauer-Emmett-Teller)-theory

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**Diffusion coefficient Liquids**

Not as well developed theory as for gases General theory for calculation of binary diffusion coefficients in liquids is missing Correlations typically describe a solute in a solvent Correlations exist for neutral molecules and ions

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**Diffusion coefficient Liquids**

Stokes-Einstein equation Molecule radius RA is the bottleneck !

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**Diffusion coefficient Liquid**

Wilke-Chang equation VA solute molar volume at normal boiling point mB solvent viscosity

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**Molar volumes Isobutyric acid 109 Methyl formate 62.8**

Ethyl acetate 106 Diethyl amine 109 Acetonitrile 57.4 Methyl chloride 50.6 Carbon tetrachloride 102 Dichlorodifluoromethane 80.7 Ethyl mercaptan 75.5 Diethyl sulfide 118 Phosgene 69.5 Ammonia 25 Chlorine 45.5 Water Hydrochloric Acid 30.6 Sulfur dioxide 43.8 Methane 37.7 Propane 74.5 Heptane 162 Cyclohexane 117 Ethylene 49.4 Benzene 96.5 Fluorobenzene 102 Bromobenzene 120 Chlorobenzene 115 Iodobenzene 130 Methanol 42.5 n-Propylalcohol 81.8 Dimethyl ether 63.8 Ethyl propyl ether 129 Acetone 77.5 Acetic acid 64.1

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**Atomic increments for estimation of VA**

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**Diffusion coefficient Liquids**

Wilke-Chang equation has been extended to solvent mixtures Association factor f water 2.6 methanol 1.9 ethanol 1.5 non-polar solvents 1.0

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**Viscosity Use experimental data if available Correlation equations**

A, B, C och D from databanks

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