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**BASIC DESIGN EQUATIONS FOR MULTIPHASE REACTORS**

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Starting Reference 1. P. A. Ramachandran and R. V. Chaudhari, Three-Phase Catalytic Reactors, Gordon and Breach Publishers, New York, (1983). 2. Nigam, K.D.P. and Schumpe, A., “Three-phase sparged reactors”, Topics in chemical engineering, 8, , , (1996) 3. Trambouze, P., H. Van Landeghem, J.-P. Wauquier, “Chemical Reactors: Design, Engineering, Operation”, Technip, (2004)

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**Objectives 1. Review microkinetic and macrokinetic processes that**

occur in soluble and solid-catalyzed systems. 2. Review ideal flow patterns for homogeneous systems as a precursor for application to multiphase systems. 3. Derive basic reactor performance equations using ideal flow patterns for the various phases. 4. Introduce non-ideal fluid mixing models. 5. Illustrate concepts through use of case studies.

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**Types of Multiphase Reactions**

Reaction Type Degree of Difficulty • Gas-liquid without catalyst • Gas-liquid with soluble catalyst • Gas-liquid with solid catalyst • Gas-liquid-liquid with soluble or solid catalyst or solid catalyst (two liquid phases) Straightforward Complex

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**Hierarchy of Multiphase Reactor Models**

Model Type Implementation Insight Empirical Ideal Flow Patterns Phenomenological Volume-Averaged Conservation Laws Pointwise Conservation Laws Straightforward Very little Very Difficult or Impossible Significant

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**Macrokinetic Processes in Slurry Reactors**

Hydrodynamics of the multi-phase dispersion - Fluid holdups & holdup distribution - Fluid and particle specific interfacial areas - Bubble size & catalyst size distributions Fluid macromixing - PDF’s of the various phases Fluid micromixing - Bubble coalescence & breakage - Catalyst particle agglomeration & attrition Reactor Model Heat transfer phenomena - Liquid evaporation & condensation - Fluid-to-wall, fluid-to-internal coils, etc. Energy dissipation - Power input from variouis sources (e.g., stirrers, fluid-fluid interactions,…)

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**Macrokinetic Processes in Fixed-Bed Reactors**

Hydrodynamics of the multi-phase flows - Flow regimes & pressure drop - Fluid holdups & holdup distribution - Fluid-fluid & fluid-particle specific interfacial areas - Fluid distribution Fluid macromixing - PDF’s of the various phases Reactor Model Heat transfer phenomena - Liquid evaporation & condensation - Fluid-to-wall, fluid-to-internal coils, etc. Energy dissipation - Pressure drop (e.g., stirrers, fluid-fluid interactions,…)

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**Elements of the Reactor Model**

Micro or Local Analysis Macro or Global Analysis • Gas - liquid mass transfer • Liquid - solid mass transfer • Interparticle and interphase mass transfer • Intraparticle and intraphase diffusion heat transfer • Catalyst particle wetting • Flow patterns for the gas, liquid, and solids • Hydrodynamics of the • Macro distributions of the gas, liquid and solid • Heat exchange • Other types of transport phenomena

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**Reactor Design Variables**

Qout Tout Cout Qin Tin Cin Reactor Product Feed Reactor Process Reaction Flow = f Performance Variables Rates Patterns • Conversion • Flow rates • Kinetics • Macro • Selectivity • Inlet C & T • Transport • Micro • Activity • Heat exchange

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**Ideal Flow Patterns for Single-Phase Systems**

Q (m3/s) Q (m3/s) a. Plug-Flow Q (m3/s) Q (m3/s) b. Backmixed Flow

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**Impulse Tracer Response**

MT t y(t) x(t) t t Q (m3/s) Reactor System Q (m3/s) Fraction of the outflow with a residence time between t and t + dt E(t) is the P.D.F. of the residence time distribution Tracer mass balance requirement:

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**Fluid-Phase Mixing: Single Phase, Plug Flow**

Q (m3/s)

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**Fluid-Phase Mixing: Single Phase, Backmixed**

Q (m3/s) Mi = Mass of tracer injected (kmol)

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**Idealized Mixing Models for Multiphase Reactors**

Model Gas-Phase Liquid Phase Solid-Phase Reactor Type Plug-flow Plug-flow Fixed Trickle-Bed Flooded-Bed Backmixed Backmixed Backmixed Mechanically agitated Plug-Flow Backmixed Backmixed Bubble column Ebullated - bed Gas-Lift & Loop

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**Ideal Flow Patterns in Multiphase Reactors Example: Mechanically Agitated Reactors**

VR = vG + VL + VC 1 = G + L + C or

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**First Absolute Moment of the Tracer Response for Multiphase Systems**

For a single mobile phase in contact with p stagnant phases: For p mobile phases in contact with p - 1 mobile phases: is the partition coefficient of the tracer between phase 1 and j

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**Relating the PDF to Reactor Performance**

“For any system where the covariance of sojourn times is zero (i.e., when the tracer leaves and re-enters the flowing stream at the same spatial position), the PDF of sojourn times in the reaction environment can be obtained from the exit-age PDF for a non-adsorbing tracer that remains confined to the flowing phase external to other phases present in the system.” For a first-order process: Hp(kc) = pdf for the stagnant phase

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**Illustrations of Ideal-Mixing Models for Multiphase Reactors**

Stirred tank Bubble Column Trickle - Bed Flooded - Bed z z G L G L • Plug-flow of gas • Backmixed liquid & catalyst • Batch catalyst • Catalyst is fully wetted • Plug-flow of gas • Plug-flow of liquid • Fixed-bed of catalyst • Catalyst is fully wetted

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**Intrinsic Reaction Rates**

Reaction Scheme: A (g) + vB (l) C (l)

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**Gas Limiting and Plug-Flow of Liquid**

Key Assumptions 1. Gaseous reactant is limiting 2. First-order reaction wrt dissolved gas 3. Constant gas-phase concentration 4. Plug-flow of liquid 5. Isothermal operation 6. Liquid is nonvolatile 7. Catalyst concentration is constant 8. Finite gas-liquid, liquid-solid, and intraparticle gradients z G L

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**Gas Limiting and Plug flow of liquid**

Constant gas phase concentration valid for pure gas at high flow rate Concentration or Axial Height Relative distance from catalyst particle (Net input by convection) (Input by Gas-Liquid Transport) (Loss by Liquid-solid Transport) + - = 0 (1) (2) Dividing by Ar.dz and taking limit dz (3) (4)

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**Gas Limiting and Plug flow of liquid**

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**Gas Limiting and Plug flow of liquid Solving the Model Equations**

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**Concept of Reactor Efficiency**

Rate of rxn in the Entire Reactor with Transport Effects Maximum Possible Rate

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**Conversion of Reactant B (in terms of Reactor Efficiency)**

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**Gas Limiting and Backmixed Liquid**

Key Assumptions Stirred Tank Bubble Column 1. Gaseous reactant is limiting 2. First-order reaction wrt dissolved gas 3. Constant gas-phase concentration 4. Liquid and catalyst are backmixed 5. Isothermal operation 6. Liquid is nonvolatile 7. Catalyst concentration is constant 8. Finite gas-liquid, liquid-solid, and intraparticle gradients z G L

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**Gas Limiting and Backmixed Liquid**

Concentration or Axial Height Relative distance from catalyst particle Concentration of dissolved gas in the liquid bulk is constant [≠f(z)] [=Al,0] Concentration of liquid reactant in the liquid bulk is constant [≠f(z)] [=Bl,0] A in liquid bulk: Analysis is similar to the previous case

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**Gas Limiting and Backmixed Liquid**

A at the catalyst surface: For Reactant B: (Net input by flow) (Rate of rxn of B at the catalyst surface) = (Note: No transport to gas since B is non-volatile)

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**Gas Limiting and Backmixed Liquid Solving the Model Equations**

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**Flow Patterns Concepts for Multiphase Systems**

B A Single phase flow of gas or liquid with exchange between the mobile phase and stagnant phase. Fixed beds, Trickle-beds, packed bubble columns B - Single phase flow of gas or liquid with exchange between a partially backmixed stagnant phase. Semi-batch slurries, fluidized-beds, ebullated beds

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**Flow Patterns Concepts for Multiphase Systems**

C, D - Cocurrent or countercurrent two-phase flow with exchange between the phases and stagnant phase. Trickle-beds, packed or empty bubble columns E - Exchange between two flowing phases, one of which has strong internal recirculation. Empty bubble columns and fluidized beds E C D

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**Axial Dispersion Model (Single Phase)**

Basis: Plug flow with superimposed “diffusional” transport in the direction of flow @ z = L @ z = 0 Let @ = 1 @ = 0

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