BASIC DESIGN EQUATIONS FOR MULTIPHASE REACTORS

2

Starting Reference

1. P. A. Ramachandran and R. V. Chaudhari, Three-PhaseCatalytic Reactors, Gordon and Breach Publishers, New York,(1983).

2. Nigam, K.D.P. and Schumpe, A., “Three-phase spargedreactors”, Topics in chemical engineering, 8, 11-112, 679-739, (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 thatoccur 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 idealflow 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

• Gas-liquid without catalyst

• Gas-liquid with soluble catalyst

• Gas-liquid with solid catalyst

• Gas-liquid-liquid with soluble

or solid catalyst

• Gas-liquid-liquid with soluble

or solid catalyst (two liquid phases)

Straightforward

Complex

Reaction Type Degree of Difficulty

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Hierarchy of Multiphase Reactor Models

Empirical

Ideal Flow Patterns

Phenomenological

Volume-Averaged

Conservation Laws

Pointwise Conservation

Laws

Straightforward

Implementation Insight

Very little

Very Difficult

or Impossible

Significant

Model Type

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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

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,…)

Reactor

Model

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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

Heat transfer phenomena

- Liquid evaporation & condensation

- Fluid-to-wall, fluid-to-internal coils, etc.

Energy dissipation

- Pressure drop

(e.g., stirrers, fluid-fluid interactions,…)

Reactor

Model

Macrokinetic Processes in Fixed-Bed Reactors

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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

• Intraparticle and intraphase

heat transfer

• Catalyst particle wetting

• Flow patterns for the

gas, liquid, and solids

• Hydrodynamics of the

gas, liquid, and solids

• Macro distributions of

the gas, liquid and solid

• Heat exchange

• Other types of transport

phenomena

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Reactor Design Variables

Reactor Process Reaction Flow

= f

Performance Variables Rates Patterns

• Conversion • Flow rates • Kinetics • Macro

• Selectivity • Inlet C & T • Transport • Micro

• Activity • Heat exchange

Feed ReactorQin

Tin

Cin

Product

Qout

Tout

Cout

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Ideal Flow Patterns

for Single-Phase Systems

Q (m3/s) Q (m3/s)

Q (m3/s) Q (m3/s)

a. Plug-Flow

b. Backmixed Flow

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Impulse Tracer Response

Q (m3/s) Q (m3/s)Reactor System

t

x(t) MT t

t

y(t)

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:

oT dt y(t) Q M

Q /M

dt y(t) dt )t(E

T

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Fluid-Phase Mixing: Single Phase, Plug Flow

Q (m3/s)

13

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

1 Plug-flow Plug-flow Fixed Trickle-Bed

Flooded-Bed

2 Backmixed Backmixed Backmixed Mechanically

agitated

3 Plug-Flow Backmixed Backmixed Bubble column

Ebullated - bed

Gas-Lift & Loop

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Ideal Flow Patterns in Multiphase ReactorsExample: Mechanically Agitated Reactors

L

r G L

L

V

Q

( )1

G

r G

G

V

Q

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:

1 =

V1 + K1j Vj

j = 2

p

Q1

For p mobile phases in contact with p - 1 mobile phases:

1 =

V1 + K1j Vj

j = 2

p

Q1 + K1j Qj

j = 2

p

K1j = C j

C1

equil.

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:

0

H -A

pe = X - dt )t(E 1 extt )(k c

0

( -e = dt )t(E ext

t )Q/Wk 1W

Hp(kc) = pdf for the stagnant phase

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Illustrations of Ideal-Mixing Models

for Multiphase Reactors

z

G L• Plug-flow of gas

• Backmixed liquid & catalyst

• Batch catalyst

• Catalyst is fully wetted

z

G L• Plug-flow of gas

• Plug-flow of liquid

• Fixed-bed of catalyst

• Catalyst is fully wetted

Stirred tank

Bubble Column

Trickle - Bed

Flooded - Bed

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Intrinsic Reaction Rates

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

20

z

G L

Gas Limiting and Plug-Flow of Liquid

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

Key Assumptions

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Gas Limiting and Plug flow of liquid

Constant gas phase concentration

valid for pure gas at high flow rate

Conce

ntr

ation o

r Axia

l H

eig

ht

Relative distance from catalyst particle

0dz= AAAadz- kAAAakAQAQ rslpsrl

*

Bldzzllzll

(Net input by convection)

(Input by Gas-Liquid Transport)

(Loss by Liquid-solid Transport)

+ - = 0 (1)

(2)

(3)

(4)

Dividing by Ar.dz and taking limit dz

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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

RRate 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

z

G L

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

Stirred Tank

Bubble Column

Key Assumptions

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Gas Limiting and Backmixed LiquidConce

ntr

ation o

r Axia

l H

eig

ht

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 LiquidA at the catalyst surface:

For Reactant B:

(Note: No transport to gas since B is non-volatile)

(Net input by flow)

(Rate of rxn of B at the catalyst surface)

=

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Gas Limiting and Backmixed LiquidSolving the Model Equations

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Flow Patterns Concepts

for Multiphase Systems

A BA - 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 EC, 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

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Axial Dispersion Model (Single Phase)

Basis: Plug flow with superimposed “diffusional” transport in the

direction of flow

Rdz

Cu

z

CD

t

Cax

2

2

@ z = 0z

CDuCCu ax

00

@ z = L 0

z

C

Let

L

zη

ax

axD

uLPe

u

Lτ

Rτηd

C

η

C

Pet

Cτ

ax

2

21

@ = 0 η

C

PeCC

ax

10

@ = 1 0

η

C