BASIC DESIGN EQUATIONS FOR
MULTIPHASE REACTORS
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, 11-112, 679739, (1996)
3. Trambouze, P., H. Van Landeghem, J.-P. Wauquier,
“Chemical Reactors: Design, Engineering, Operation”,
Technip, (2004)
2
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.
3
Types of Multiphase Reactions
Reaction Type
Degree of Difficulty
• Gas-liquid without catalyst
Straightforward
• 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)
Complex
4
Hierarchy of Multiphase Reactor Models
Model Type
Empirical
Implementation
Insight
Straightforward
Very little
Ideal Flow Patterns
Phenomenological
Volume-Averaged
Conservation Laws
Pointwise Conservation Very Difficult
or Impossible
Laws
Significant
5
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,…)
6
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
Heat transfer phenomena
- Liquid evaporation & condensation
- Fluid-to-wall, fluid-to-internal coils, etc.
Reactor
Model
Energy dissipation
- Pressure drop
(e.g., stirrers, fluid-fluid interactions,…)
7
Elements of the Reactor Model
Micro or Local Analysis
• Gas - liquid mass transfer
Macro or Global Analysis
• Liquid - solid mass transfer
• Flow patterns for the
gas, liquid, and solids
• Interparticle and interphase
mass transfer
• Hydrodynamics of the
gas, liquid, and solids
• Intraparticle and intraphase
diffusion
• Macro distributions of
the gas, liquid and solid
• Intraparticle and intraphase
heat transfer
• Heat exchange
• Catalyst particle wetting
• Other types of transport
phenomena
8
Reactor Design Variables
Feed
Qin
Tin
Cin
Qout
Tout Product
Cout
Reactor
Reactor
Process
Reaction
Flow
Performance
Variables
Rates
• Conversion
• Flow rates
• Kinetics
• Selectivity
• Inlet C & T
• Transport • Micro
• Activity
• Heat exchange
=f
Patterns
• Macro
9
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
10
Impulse Tracer Response
x(t)
MT t
y(t)
t
Q (m3/s)
t
Q (m3/s)
y(t) dt
E(t ) dt 

MT / Q
Reactor System
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:
MT  Q  y(t)dt
o
11
Fluid-Phase Mixing: Single Phase, Plug Flow
Q
(m3/s)
12
Fluid-Phase Mixing: Single Phase, Backmixed
Q
(m3/s)
Mi = Mass of tracer injected (kmol)
13
Idealized Mixing Models for
Multiphase Reactors
Model Gas-Phase
Liquid Phase
Solid-Phase
Reactor Type
Plug-flow
Fixed
Trickle-Bed
Flooded-Bed
1
Plug-flow
2
Backmixed Backmixed
Backmixed
Mechanically
agitated
3
Plug-Flow
Backmixed
Bubble column
Ebullated - bed
Gas-Lift & Loop
Backmixed
14
Ideal Flow Patterns in Multiphase Reactors
Example: Mechanically Agitated Reactors
or
VR = vG + VL + VC
1 = G + L + C
G 
Vr G
QG
L 
Vr (1   G   L )
QL
15
First Absolute Moment of the
Tracer Response for Multiphase Systems
For a single mobile phase in contact with p stagnant phases:
p
V1 +
 K1j Vj
j=2
1 =
Q1
For p mobile phases in contact with p - 1 mobile phases:
p
V1 +
 K1j Vj
j= 2
p
1 =
Q1 +
 K 1j Qj
j= 2
K1j
C j 
=  
C1 equil.
is the partition coefficient of the tracer
between phase 1 and j
16
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:
 - H (k ) t
p
c
1 - XA =  e
Hp(kc) = pdf for the stagnant phase
0

= e
0
Eext ( t ) dt
- (k W W / Q1 ) t
Eext ( t ) dt
17
Illustrations of Ideal-Mixing Models
for Multiphase Reactors
Stirred tank
Trickle - Bed
Bubble Column







z
Flooded - Bed













z






G


L
• Plug-flow of gas
• Backmixed liquid & catalyst
• Batch catalyst
• Catalyst is fully wetted
G
L
• Plug-flow of gas
• Plug-flow of liquid
• Fixed-bed of catalyst
• Catalyst is fully wetted
18
Intrinsic Reaction Rates
Reaction Scheme: A (g) + vB (l)  C (l)
19
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
z
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
G
L
20
Concentration or Axial Height
Gas Limiting and Plug flow of liquid
Constant gas phase concentration 
valid for pure gas at high flow rate
Relative distance from catalyst particle
(Net input by
(Input by Gas- +
convection)
Liquid Transport)
Ql Al z  Ql Al


(Loss by Liquidsolid Transport)
=0
*

k
a
A
 Al Ar dz- ks a p Al  As Ar dz= 0
l B
z dz
(1)
(2)
Dividing by Ar.dz and taking limit dz  
(3)
(4)
21
Gas Limiting and Plug flow of liquid
22
Gas Limiting and Plug flow of liquid
Solving the Model Equations
23
Concept of Reactor Efficiency
R 
Rate of rxn in the Entire Reactor with Transport Effects
Maximum Possible Rate
24
Conversion of Reactant B
(in terms of Reactor Efficiency)
25
Gas Limiting and Backmixed Liquid
Key Assumptions
Stirred Tank
Bubble Column
1. Gaseous reactant is limiting







z
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


G


L
7. Catalyst concentration is constant
8. Finite gas-liquid, liquid-solid,
and intraparticle gradients
26
Concentration or Axial Height
Gas Limiting and Backmixed Liquid
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
27
Gas Limiting and Backmixed Liquid
A at the catalyst surface:
For Reactant B:
(Net input by
(Rate of rxn of B at
=
flow)
the catalyst surface)
(Note: No transport to gas
since B is non-volatile)
28
Gas Limiting and Backmixed Liquid
Solving the Model Equations
29
Flow Patterns Concepts
for Multiphase Systems
A
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
B - Single phase flow of gas or
liquid with exchange between a
partially backmixed stagnant phase.
Semi-batch slurries, fluidized-beds,
ebullated beds
30
Flow Patterns Concepts
for Multiphase Systems
C
D
C, D - Cocurrent or
E
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
31
Axial Dispersion Model (Single Phase)
C
 2C
C
 Dax
u
R
2
t
z
dz
@z=0
Let
C
u0C0  uC  Dax
z
z
η
L
Peax 
uL
Dax
Basis: Plug flow with superimposed
“diffusional” transport in the
direction of flow
@z=L
C
0
z
L
τ
u
C
1  2C C
τ


 τR
2
t
Peax η
dη
@  = 0
1 C
C0  C 
Peax η
@  = 1
C
0
η
32