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Page 1 of 90
Department of Chemical Engineering
Reaction Engineering II
Lecture Notes
V 4.1 11 May 2012
Lecturer: Dr. Clemens Brechtelsbauer
Lecture notes compiled by
Dr. Klaus Hellgardt
with modifications by
Dr. Clemens Brechtelsbauer
based on a course by
Dr. Esat Alpay
Page 2 of 90
Course Aims The course focuses on heterogeneous and multi-phase reactors. Through understanding the underlying physics
of the different reactor types, the student will be equipped to carry out reactor design tasks for conventional
and novel reactors in a systematic way. This is of particular relevance to the 4th year design project.
Course Structure The course consists of the following components:
Fundamentals of transport processes in heterogeneous reactors
Fixed bed catalytic reactors
Fluidised bed reactors
Gas-Liquid and Gas-Liquid-Solid Reactors
Fundamentals of non-catalytic fluid-solid reactions
Learning Outcomes By the end of the course students should be able to:
Identify critical parameters affecting the performance of heterogeneous and multi-phase reactors
Establish and follow a selection process to determine the most appropriate reactor type for a specific
process
Carry out reactor sizing calculations to the level of detail required
Estimate the margin for and level of error in their calculations
Further Reading
1. Gilbert F. Froment, Kenneth B. Bischoff,
Chemical Reactor Analysis and Design, 2nd
Edition,
John Wiley & Sons, 1990
2. H. Scott Fogler,
Elements of Chemical Reaction Engineering, 2nd
Edition,
Prentice-Hall, 1992
3. Octave Levenspiel,
Chemical Reaction Engineering, 3rd
Edition
John Wiley & Sons, 1999
Page 3 of 90
1 Introduction to Heterogeneous and Multiphase Reactors ..................................................................... 4
2 Transport Processes in Heterogeneous Catalysis ................................................................................... 6 2.1 Interfacial Gradient Effects .................................................................................................................. 6
2.1.1 Reactions at Catalyst Surface ....................................................................................................... 6 2.1.2 Attaining Values of km ................................................................................................................ 8 2.1.3 Concentration (Partial Pressure) Differences across the External Film ..................................... 10 2.1.4 Temperature Differences across the External Film .................................................................... 11 2.1.5 Mass Transfer on Metallic Surfaces ........................................................................................... 12
2.2 Intraparticle Gradient Effects ............................................................................................................. 14 2.2.1 Catalyst Internal Structure ......................................................................................................... 14 2.2.2 Pore Diffusion ............................................................................................................................ 14 2.2.3 Reaction and Diffusion within a Catalyst Pellet ........................................................................ 18 2.2.4 Temperature Gradients within a Catalyst Pellet ......................................................................... 27
2.3 Combined Interfacial (External) and Intraparticle (Internal) Resistances ......................................... 30
3 Fixed Bed Catalytic Reactor (FBCR) Design ........................................................................................ 31 3.1 Pseudo-Homogeneous PFR and Axially Dispersed PFR Models ...................................................... 31
3.1.1 PFR Model ................................................................................................................................. 31 3.1.2 Axially Dispersed PFR Model ................................................................................................... 33
3.2 Heterogeneous Models ....................................................................................................................... 36 3.2.1 Use of Effectiveness Factor ....................................................................................................... 36 3.2.2 Use of Intraparticle Diffusion Equations ................................................................................... 37
3.3 2D Models .......................................................................................................................................... 38
4 Fluidised Bed Reactors ........................................................................................................................... 40 4.1 Overview of Fluidisation Principles .................................................................................................. 40 4.2 Overview of Key Applications .......................................................................................................... 49 4.3 Modelling of Fluidised Bed Reactors: Non-Transport ....................................................................... 51
4.3.1 Two-Phase Models ..................................................................................................................... 51 4.3.2 Three-Phase (Hydrodynamic) Models ....................................................................................... 55
4.4 Modelling of Transport (Riser) Reactors ........................................................................................... 57
5 Multiphase Reactors ............................................................................................................................... 59 5.1 Background ........................................................................................................................................ 59 5.2 Review of Two-Film Theory ............................................................................................................. 61 5.3 General Design Models for Multiphase Reactors .............................................................................. 66
5.3.1 Gas & Liquid Phases Completely Mixed ................................................................................... 66 5.3.2 Gas & Liquid Phases in Plug Flow ............................................................................................ 70 5.3.3 Gas Phase in Plug Flow, Liquid Phase Completely Mixed ........................................................ 71 5.3.4 Effective Diffusion Model ......................................................................................................... 71
5.4 Simplifications to Multiphase Design Models ................................................................................... 72 5.4.1 Instantaneous Reactions ............................................................................................................. 72 5.4.2 Very Fast Reactions ................................................................................................................... 72 5.4.3 Slow Reactions ........................................................................................................................... 72 5.4.4 Solid Catalyzed Reactions ......................................................................................................... 73 5.4.5 Resistances in Series Approximation: Gas-Liquid-Solid Reactions .......................................... 73 5.4.6 Resistances in Series Approximation: Gas-Liquid Reactions .................................................... 75
5.5 Factors in Selecting a Gas-Liquid Contactor ..................................................................................... 77
6 Non-Catalytic Fluid-Solid Reactions ..................................................................................................... 78 6.1 Total Particle Dissolution................................................................................................................... 79 6.2 Shrinking Core Model ........................................................................................................................ 81 6.3 Reactor Design ................................................................................................................................... 84
6.3.1 Plug Flow of Solids .................................................................................................................... 85 6.3.2 Mixed Flow of Solids ................................................................................................................. 86
7 Notation .................................................................................................................................................... 87
Reaction Engineering II: Course Overview
Page 4 of 90
1 Introduction to Heterogeneous and Multiphase Reactors
Reaction Engineering I: material and energy balances for ideal PFR, CSTR, and batch
reactors
Pseudo-homogeneous assumption:
Mass transfer & heat transfer resistances between different phases neglected, such that
reactor contents can be treated as a single phase.
Useful for preliminary design or truly homogeneous systems.
Heterogeneous model used when temperature (T) and composition (C) need to be
distinguished between the phases.
Real reactors may involve multiple phases (i.e. multiphase reactors), which will often need to
be considered as heterogeneous.
However, the phrase “multiphase reactors” is usually used for systems involving fluid-fluid
interactions, i.e. gas-liquid and liquid-liquid systems.
For systems involving solids, 2 general cases exist:
(i) Solid as porous catalyst pellet
Solid not consumed in reaction but its physical and chemical nature may change.
E.g.
(1) Pore blocking due to deposits of carbonaceous by-products of reaction, i.e. coking.
(2) Metal particles (the active catalyst) may coalesce at high temperatures, reducing the
overall surface area for reaction and hence the rate constant, i.e. sintering.
(ii) Solids as non-catalyst
E.g.
(1) Dissolution of solid through reaction with a fluid
(2) Burning off of coke in a catalyst pellet for its regeneration
Most practical (and common) utilisation of solid catalysts is in a fixed bed catalytic reactor
(FBCR), i.e. a tubular reactor packed with catalyst, through which the fluid reaction species
flow.
Advantages of FBCR:
- No solids handling
- Little solids attrition
- High surface area through use of porous catalysts
- Plug flow operation can be approached
- No separation of catalyst from reaction products needed
Disadvantages of FBCR:
- Pressure drop
sintering
Page 5 of 90
- Complex (e.g. multitubular) arrangement for reactions requiring high heat-exchange duties
- Large down-time for catalysts which deactivate rapidly
Where disadvantages of FBCR are important, reactors involving the fluidisation of the
catalyst, or the flow (transport) of the catalyst in some way, are employed.
Such operation may enable better heat transfer between the fluid-solid and the fluid and heat-
exchange surface, and provide a means for the continuous removal of catalyse for
regeneration, and feed of fresh catalyst.
Page 6 of 90
2 Transport Processes in Heterogeneous Catalysis
2.1 Interfacial Gradient Effects
2.1.1 Reactions at Catalyst Surface
First-order reaction: S
AS
S
A SSCkr (1)
sm
molr
S
S
AS 2:
sm
mk
S
f
S2
3
:
3:
f
S
Am
molC
S at z = 0
At steady-state:
)( AA
S
A rNrS
(2)
where
)( S
ASAmA CCkNC
(3)
sm
mk
S
f
mC 2
3
:
CA
NA
CSAs
(CA at solid surface)
CAs
(CA in solid)
external-
film
z
active centres
FLUID
l
0
Page 7 of 90
or:
yPC
S
P
S
y
mmm
S
AA
Am
S
AA
Am
kPkCk
PP
Nk
yy
Nk
)(
)(
A
mS
mS
A
S
AAm
S
AS
Ckk
kC
CCkCk
C
C
S
SCS
)(
Substitute this back into (1):
AA Ckr 0 (4)
where Sm kkk
C
111
0
(5)
k0: overall rate constant
Limiting cases:
(i) Sm kkC (rapid mass transfer), then:
Skk ~0 and A
S
AS CC ~
i.e. overall process is reaction rate controlled.
(ii) CmS kk (rapid reaction), then:
Cmkk ~0 and 0~S
ASC
i.e. overall process is diffusion controlled.
Second-order reaction:
Using similar procedure to above, but equation (1) is replaced with:
2)( S
AS
S
A SSCkr (6)
gives
AmA CkrC
))1(( 2/12 (7)
where AS
m
Ck
kC
21
i.e. neither 1st nor 2
nd order concentration dependence.
Limiting cases:
(i) Sm kkC
2~ ASA Ckr
(ii) CmS kk AmA Ckr
C~
Page 8 of 90
For complex reactions, analytical solution is not usually possible.
Mass transfer can thus lead to difficulties in experimentally determining rate coefficients and
orders.
However, we can work under conditions where we have either reaction or diffusion controlled
process. i.e.
- Sm kkC
or
- CmS kk (in this case should reduce T or increase fluid turbulence)
2.1.2 Attaining Values of km
Usually correlations in handbooks define the mass transfer coefficient under conditions of equimolar
counter-diffusion, k0
m
How is k0
m related to km ?
(i) Equimolar counter-diffusion (ECD):
BA NN (8)
dz
dyCDyNN A
ABATA (9)
But 0 BAT NNN
A
S
SA
y
y
AAB
l
A
AABA
dyCDdzN
dz
dyCDN
0
)( S
AAAB
A Syy
l
CDN (10)
But l
CDk AB
my0
(11)
ECDforkkyy mm 0
(Also, l
D
C
kk ABm
m
y
C )
(ii) For reaction in which total moles are not conserved, e.g.:
bBaA
AB Na
bN (12)
equimolar counter-diffusion cannot be used.
Page 9 of 90
Substitute (12) into (9) and rearrange:
A
S
SA
y
yA
AAB
l
A
ya
b
dyCDdzN
)1(10
)( S
AAmA SyyykN
where:
A
y
y
f
m
my
kk
0
(13a)
S
AA
AA
S
AAAA
f
S
S
A
y
y
yyy
1
1ln
)1()1( (13b)
a
abA
)( (13c)
see Reaction Engineering I
Afy is referred to as film factor
for the general reaction:
...... sSrRbBaA
Equations (13a) and (13b) are applicable but
a
basrA
...)(...)(
for 1,0 AfA y and
0
yy mm kk
common method for predicting k0
m is through the use of the jD factor.
jD factor:
3
20
ScG
Mkj mm
D (14)
Mm: average molecular mass (mol
g)
G: mass flux (sm
g
2)
Sc: Schmidt number Df
where : viscosity
f: fluid density
D: molecular diffusivity
k0
m can be taken as 0
ymk or 0
fmk , as long as it is remembered that:
APAPAy fmfmfmm PkyPkykk 0
(AfP is referred to as pressure film factor)
Page 10 of 90
in addition to (14), a second relationship for jD is available from charts or correlations (see e.g.
Froment & Bischoff, Chemical Reactor Analysis & Design, 2nd
edn, p.129)
e.g. for flow in a bed packed with spherical particles and b = 0.37:
jD = 1.66 Re-0.51
for Re < 190
jD = 0.983 Re-0.41
for Re > 190
pGdRe
Thus, given (14) and a suitable correlation, we can solve for k0
m and thus ymk if
Afy is given (see
section 2.1.3)
Note: similar correlations also exist for the heat transfer coefficient, hf, i.e.:
3
2
PrGc
hj
p
f
H (15)
(Re)fjH
where Pr: Prandtl number
pC
Cp: fluid head capacity (e.g. mean value)
: fluid thermal conductivity
2.1.3 Concentration (Partial Pressure) Differences across the External Film
If CA or PA ~ 0 (i.e. yA ~ 0), the mass transfer will be very fast and rA can be expressed as a
function of bulk CA (or PA) directly. E.g.:
AS
S
AA CkrrS (since
S
AA SCC )
Thus it is useful to have some estimate of CA or PA
We’ll use a different definition of rA, i.e. in terms of catalyst mass instead of surface area:
rA’:
skg
mol
cat
such that, for example:
)(' S
AAmmA SCCCakr
where
sm
mk
p
f
mC 2
3
: (as before)
cat
p
mkg
ma
C
2
:
(see equations (16) and (17) in lecture notes)
(Why redefine rA in this way?)
Method:
ymm
AA
ka
ry
'
Therefore, given am and rA’, yA can be calculated if kmy is known.
Page 11 of 90
Af
o
m
myy
kk (13a & 13b)
However, Afy requires
S
ASy which is not yet known!
Thus, use an iterative procedure, starting with an initial guess forS
ASy .
A good initial guess is for A
S
A yyS
~ , i.e. yA = 0, which from equation (13b) gives (using
L’Hôpital’s rule):
AAf yyA
1
Having attained first estimate of yA, and thus S
ASy , the above procedure is repeated until successful
estimates of yA become negligibly different.
Usually, under practical operating conditions, yA is negligible, but T across the external film may
be significant.
2.1.4 Temperature Differences across the External Film
Energy balance at steady-state:
)()(' TTahHr S
SmfrA (17)
where:
Hr: J/mol
hf : J/(ms2
s K)
But AmmA yakry
' (18)
)18(
)17( (i.e. eliminate rA’), and substitute for kmy and hm using jD and jH (eqn 14 & 15) respectively:
Af
A
pm
r
H
DS
Sy
y
cM
H
Scj
jTTT
))(]
Pr[()( 3
2
(19)
(T increases with yA, i.e. when mass transfer resistance are high)
For gases flowing in packed beds, the values of the various groups are such that:
Af
A
pm
r
y
y
cM
HT
)(7.0~ (20)
cp: J/(kg K)
Mm: kg/mol
Hr: J/mol
Page 12 of 90
Furthermore, T is maximum when 0S
ASy
(for irreversible reaction, or mEquilibriuS A
S
A yy for reversible reaction)
Such that:
)1ln( AA
AAf
AA
y
yy
yy
A
Eqn. (20) then gives:
A
AA
pm
r y
cM
HT
)1ln()(7.0~
max
(21)
2.1.5 Mass Transfer on Metallic Surfaces
For packed beds, it has been shown that C variations are small and usually negligible
But mass transfer may be significant when the catalyst is a metallic surface, e.g.:
(1) Catalyst monolith/honeycomb (e.g. catalytic afterburner in automobiles)
(2) Wire gauzes (e.g. oxidation of ammonia)
Advantages of these units:
(1) low P
(2) particulate matter in feed will not clog up bed
(difference between monolith and honeycomb reactors?)
See Fogler p.575, 577 for references on correlations for Cmk for monoliths and wire gauzes.
Page 13 of 90
Monolith supports:
Example
1st order oxidation of CO in a catalytic converter. Calculate:
(1) Exit molar flow rate of CO, nA, given nA0, kS, Cmk , and av (
3
2
r
S
m
m ).
(2) What is the corresponding equation for very rapid reaction kinetics?
Solution:
Assume PFR, constant volumetric flow rate (v0):
(i)
VAAmAA aCCkr
dV
dnSC)( (*)
00
,
)(
v
nC
v
nC
aCCkCk
S
S
SCS
A
AA
A
VAAmAS
Thus, show: Vak
AA enn0
0
where = V/v0
(ii) 0~, S
AmS SCCkk
Equation (*) can be directly integrated to give: VCm ak
AA enn
0
Page 14 of 90
2.2 Intraparticle Gradient Effects
2.2.1 Catalyst Internal Structure
Reaction rate catalyst surface area
Areas range from 10-200 m2/g (most towards higher end); activated carbon ~ 800 m
2/g
(c.f. sand, 0.01 m2/g)
High areas through a highly porous structure, i.e. high surface area to volume ratio.
Pore size not uniform, i.e. a pore size distribution (PSD) exists.
PSD measured by (nitrogen) porosimetry. Suitable for measuring pore sizes in the range of
1-30 nm.
(How does porosimetry work?)
Pore size often classified as:
(1) micropores: dpore < 0.3 nm
(2) mesopores: 0.3nm < dpore < 20 nm
(3) macropores: dpore > 20 nm
Often use a mean pore size in calculations
For some catalysts, can have a bimodal distribution of pore sizes (e.g. zeolite catalysts).
What are zeolites?
What is a bimodal distribution?
What is a bidisperse structure?
For non-zeolitic catalysts, active metal dispersed and supported within a macroporous
support matrix, such as silica or alumina.
Thus unimodal PSD, but can be rather broad.
2.2.2 Pore Diffusion
For gas diffusion through a single cylindrical pore, the ratio of dpore to the mean free path of the
gas () will determine whether or not the pore well affects the diffusion behaviour.
(What is meant by the mean free path of the gas?)
3 different situations:
(i) dpore >>
- molecular diffusion dominates, i.e. Fickian diffusion
- e.g. gases at high pressure; liquids
dpore . . . . . . . .
. . . .
. .
Page 15 of 90
(ii) dpore << , and dmolecule < dpore
- molecular interactions with the pore wall dominate
- diffusion described by Knudsen’s law
- e.g. gases at low P, but not liquids (why?)
(iii) dpore << , and dmolecule ~ dpore
- complex interaction of diffusing molecules with the force-fields of molecules making up the
wall
- referred to as configurational diffusion
- very difficult to predict
- e.g. very large hydrocarbon molecules (e.g. petroleum desulpurisation); pores of very small size
(e.g. diffusion within zeolite crystals, and through biological cell walls)
(see Froment & Bischoff, Chemical Reactor Analysis & Design, p.143)
Dif
fusi
on
coef
fici
ent
(cm
2/s
)
dpore/2 , (nm)
Page 16 of 90
Correlations for Diffusion Coefficients
- correlations for binary molecular diffusion can be attained from Handbooks.
For gases, P
TD
kim
2
3
,
- usually need an effective binary diffusion coefficient, i.e. diffusion coefficient for the key
component through a mixture of the other components, mimD
,.
i.e. dz
dyDCNyN i
m
N
k
kii mi
C
,
1
Nc : number of components
- given the Stefan-Maxwell equations for diffusion, we can calculate mimD
, from actual binary
diffusivity data, kimD
,:
C
C
ki
mi
N
k i
ki
N
ik i
kik
m
m
v
vy
v
vyy
D
D
1
1
)(1
1 ,
,
where v is the stoichiometric coefficient.
- Knudsen diffusion coefficient, Dk, can be calculated from the kinetic theory of gases as:
pore
M
k dM
TD
i
i
2
1
i.e. )(PfDik
But, as P is increased, the transport regime can switch from Knudsen to molecular diffusion.
- as mentioned earlier, configurational (or micropore) diffusion coefficient is difficult to predict
measurement needed.
- for non-zeolite catalysts (and for the binder phase of zeolite catalysts) molecular and Knudsen
diffusion dominate. The pore diffusion coefficient, Dp, will thus be a function of Dm and Dk.
(Dp: pore diffusion coefficient for a single pore)
Treybal (1981) suggests that for:
(1) dpore/ > 20 , i.e. molecular diffusion controlling
Dp = Dm
(2) dpore/ < 0.2 , i.e. Knudsen diffusion controlling
Dp = Dk
For intermediate values, both diffusion types are important. Can use Bosanquet formula to
approximate Dp for this case:
mkp DDD
111
(c.f. resistance in parallel)
Page 17 of 90
- given Dp, how do we calculate an effective diffusion coefficient, De, which account for the
complex pore structure of the catalyst pellet?
An approximation of De (but often adequate for design purposes) is given by:
p
pp
e
DD
(22)
where
p: intraparticle void fraction (3
3
p
p
m
mV
)
p: tortuosity factor
Basis for equation (22):
Compare diffusion in a single pore and diffusion in a porous pellet:
versus
- cross-sectional area available for diffusion = A p lower NA
- tortuous molecule path, and changing pore cross-sectional area due to constrictions. So dz
dC A is
reduced
Thus: dz
dCDN A
p
pp
A
[ Note: p= tortuosity/(constriction factor)
where tortuosity = (actual diffusion path length)/(shortest length) ]
Given De, we can now consider combined diffusion and reaction within a catalyst pellet.
Unlike reaction at a surface, diffusion and reaction take place simultaneously for this case rather than
consecutively.
CA1 CA2 z
dz
dCDN A
pA A linear molecule
path on average
Page 18 of 90
2.2.3 Reaction and Diffusion within a Catalyst Pellet
Consider the concentration profiles within a porous catalyst pellet:
Chemist measures rate under conditions where external and internal mass transfer resistances are
negligible, say rA*.
How?
When mass transfer is important:
CA > CAs
Thus we cannot use bulk concentrations to calculate actual (observed) reaction rate, rA.
We need to relate rA to rA* . This is done through the effectiveness factor, :
position
external
film
Concentration
negligible external film resistance
significant external film resistance
external-
film
CA
CSAs
CAs (catalyst)
rp r
0 (centre/central axis of pellet)
Page 19 of 90
*
A
A
r
r (<1 for isothermal or endothermic reactions)
is useful in design calculations. But for rigorous calculations, particularly for complex reaction
kinetics and non-isothermal operation, it is better to solve the simultaneous equations governing
diffusion and reaction (see section 3).
have previously shown that for a packed bed, external film mass transfer resistances are small.
Thus we can assume the situation depicted by the solid line in the previous graph.
In other words, rA* is the reaction rate which would be measured if all of the pellet “saw” a
concentration of CS
As: S
A
S
AAA SSSrCkr ][*
and
S
A
A
Sr
r
(We will later consider the case when both external and intraparticle gradients are important.)
(Pseudo-) First Order Reaction (A Product)
Consider material balance through an incremental section of a catalyst slab of cross-sectional
area a:
CA= CSAs
CSAs
CAs
rp r
0
NA ==> (mol/m
2p s)
r
r +r
Page 20 of 90
IN – OUT = CONSUMPTION
raraNaNSArArrA
)()(
divided by (ar) and let r 0:
SS AvAA Ckr
dr
dN
where
rAs: mol/(m3
p s)
CAs: mol/m3
f
For no convective flow in the pellet:
dr
dCDN S
A
A
eA
S
S
A Av
A
e Ckdr
CdD
2
2
(for DeA being constant with r)
Integrating this differential equation with the boundary conditions:
r = rp CAs = CS
As (=CA)
r = 0 0dr
dCSA
gives:
A
A
S
S
e
vp
e
v
S
A
A
D
kr
D
kr
C
C
cosh
cosh
(24)
Note:
Ae
vpslab
D
kr (Thiele Modulus), i.e.
Page 21 of 90
We can derive CAs profile for a spherical pellet in a similar way:
i.e. a = 4r2
IN – OUT = CONSUMPTION
rrrNrNrSArA
rrA
222 4)(4)(4
divided by 4r2 and let r 0:
SAvA CkNrdr
d
r)(
1 2
2 (25)
Again, using dr
dCDN S
A
A
eA and the same boundary conditions as for a slab (rp = pellet
radius for this case) gives (after integration):
A
A
S
S
e
vp
e
v
p
S
A
A
D
kr
D
kr
r
r
C
C
sinh
sinh
(26)
Note:
Ae
vpsphere
D
kr
Likewise for a cylindrical pellet:
SAvA CkrNdr
d
r)(
1 (27)
Boundary conditions:
r = rp CAs = CS
As
r = 0 0dr
dCSA
Integrating gives:
A
A
S
S
e
vp
e
v
S
A
A
D
krI
D
krI
C
C
0
0
(28)
I0: Bessel function
Note:
Ae
vpcyl
D
kr
CA/CAs vs r/rp profiles of cylindrical and spherical pellets are similar to slab pellets
rp
Page 22 of 90
Comparing equations (23), (25), and (27), we can write the general pellet mass balance as:
SAA
m
mrNr
dr
d
r)(
1 (29)
where
m = 0 for slab
m = 1 for cylinder
m = 2 for sphere
and dr
dCDN S
A
A
eA
Boundary conditions:
r = rp CAs = CS
As
r = 0 0dr
dCSA
where rp = characteristic half-length of pellet
Given CAs = f(r) for a first-order reaction, we can calculate the average reaction rate (observed
rate):
p
V
A
p
AA dVrV
rr
p
SS 0
1 (30)
Given rA, we are now in a position to calculate the effectiveness factor, .
Effectiveness Factor: First-Order Reactions
Recall:
= (observed reaction rate)/(reaction rate at pellet surface conditions)
S
A
A
Sr
r (31)
For isothermal (and endothermic) reactions, rS
As represents the maximum reaction rate, since
CS
As > CAs, or in other words:
1
)(
AAv
S
Av
S
A rCkCkrSSS
Also, if there is very high resistances within catalyst, there is negligible penetration of reactant
into the pellet such that:
0,0,0 SS AA rC
10
If can be calculated, equation (31) can be used to determine rA, since rS
As is readily calculated,
i.e. S
AA Srr
where Av
S
Av
S
A CkCkrSS
~ for negligible external film mass transfer resistances.
Page 23 of 90
How do we do this?
For a slab:
- Given CAs from equation (24) and rAs = kvCAs, substitute into equation (30) and integrate to
calculate )( AA rrS
- Given rA, use equation (31) to derive an expression for .
Solution:
slab
slab
)tanh( (32)
where
Ae
vpslab
D
kr (33)
Note: (34)
For a sphere (using a similar procedure for slab, but use equation (26) for CAs):
spherespheresphere
1
tanh
13 (35)
( sphere as for slab but rp = pellet radius)
Note:
sphere
sphere
sphere
ei
3~),9..(
1,0
(36)
Similarly, for a cylinder:
cylcyl
cyl
I
I
1
)2(
)2(
0
1 (37)
Note:
(38)
cyl
cyl
cyl
ei
2~),5..(
1,0
slab
slab
slab
ei
1~),3..(
1,0
Page 24 of 90
versus for 1st Order Reaction
1: slab
2: cylinder
3: sphere
versus equations are rather complex for spheres and cylinders.
But we can see from plots that the trends are very similar but a shift along the x-axis.
We can, in fact, redefine Thiele modulus such that for any pellet geometry, versus
approximately coincide, i.e.
Curves for sphere and cylinder coincide with slab curve, such that the relatively simple
expression:
tanh
can be used, where is a redefined (general) Thiele modulus,
This is achieved for:
Ae
v
p
p
D
k
A
V (39)
where:
Vp: particle volume
Ap: external surface area of particle
i.e.
slab: Vp/Ap = rp ( = slab)
sphere: Vp/Ap = rp / 3
cylinder: Vp/Ap = rp / 2
Page 25 of 90
Given , equation (32) is applicable to all pellet geometries. Maximum discrepancy between
curves ~ 15%.
What are the limits of for 0 and ?
Example 3
Reaction rate measured in lab under conditions where intraparticle and external-film resistances
to mass transfer are negligible, i.e.
rAs = 1.5 CS
As (CA = CS
As for this case)
What is the actual (observed) reaction rate in an industrial reactor where intraparticle gradients
are important?
Ans: rA = 1.5 CA
where
tanh (1
st order reaction)
To calculate , we need Vp/Ap, kv (=1.5), and DeA.
What if the reaction is not 1st order?
Effectiveness Factor: General Order Reactions
For general order and reversible reactions, we can further generalise the Thiele modulus as:
2
1
*
2
S
SA
SASSA
SC
CAAe
S
A
p
pdCrD
r
A
V (40)
where C*
As is the equilibrium concentration of the limiting reactant (=0 for an irreversible
reaction).
(See Froment & Bischoff, p.160-162)
Equation (40) accounts for DeA variation with CAs.
Equation (40) assumes diffusional resistances are high such that we are in the ~ 1/ region. If
not, C*As in equation (40) needs to be calculated from:
p
C
C C
CAAe
Aer
dCrD
dCDS
SA
SA A
SASA
SA
* '
*
'2 (+)
Page 26 of 90
Example:
nth
order irreversible reaction of A in a spherical catalyst pellet, for which DeA ~ constant (assume
strong diffusion limitations)
Ans:
S
n
AvA
p
p
p
Ckr
r
A
V
S
3
2
1
0
1
1
1
23
)(
S
SA
SA
S
C
n
Aev
nS
AvpC
nDk
Ckr
2
1
1)(
2
1
3
A
S
e
nS
Avp
D
Cknr (for n > – 1)
Again, CS
As ~ CA for typical packed bed reactors.
Note: for a non-1st order reaction, = f(CA) and will therefore vary with axial position in a
tubular reactor.
Criteria for Intraparticle Diffusional Limitations
If reaction kinetics are known, can be calculated; < 1 indicates diffusional limitations.
However, in experiments where we need to calculate kinetics (e.g. rate constants), we need to
ensure that we are working in a region where diffusional limitations are negligible so as not to
falsify the kinetic data.
Weisz-Prater criterion:
Rearrange equation (39):
ve
p
pkD
V
A
A
2
2
S
Av
S
AA SSCkrr (1
st order reaction)
Eliminate kv and rearrange:
2
2
p
p
S
Ae
A
A
V
CD
r
SA
(41)
(CS
As ~ CA under typical operating conditions)
L.H.S. of (41) is measurable. Then for:
(i) negligible diffusional limitations:
1
1~,1
Page 27 of 90
(ii) considerable diffusional limitations:
1
1~,1
The above method can be generalised to any reaction scheme using the appropriate form of the
Thiele modulus (see eqn.(40)); see also Froment & Bischoff, p.169)
2.2.4 Temperature Gradients within a Catalyst Pellet
We can calculate temperature gradients within a catalyst pellet by considering simultaneously the
intraparticle mass and energy balances:
Eqn.(25) (for a spherical pellet):
S
S
A A
A
e rdr
dCr
dr
dD
r)(
1 2
2
Similarly for energy balance, given an effective thermal conductivity, e:
rA
S
e Hrdr
dTr
dr
d
r S)(
1 2
2 (42)
Exercise: Eliminate rAs between (25) & (42) and integrate twice to obtain:
)()( S
AA
e
erS
SSS SS
A CCDH
TTT
TS is maximum when CAs = 0 for an irreversible reaction (or C*
As for an equilibrium
(reversible) reaction).
i.e.
S
A
e
er
S S
A CDH
T
max (43)
It can actually be shown that eqn.(43) is applicable to all pellet geometries, i.e. same eqn. Is
derived.
For many industrial applications:
1.0max
S
S
S
T
T
i.e. TS is small, but T(external-film) can be large (other way around to CA). Exceptions
include highly exothermic reactions such as some oxidation and hydrogenation reactions.
Effect of TS on is complex, e.g. will influence DeA as well as kv. For highly exothermic
reactions, can exceed 1. (Why?)
Page 28 of 90
Example: for 1st order reaction in a non-isothermal pellet:
Use equations (25) & (42) and S
S
S A
RT
E
A CeAr
0
Put in dimensionless form by defining:
p
S
S
S
S
A
A
r
rr
T
TT
C
CC
S
S ˆ,ˆ,ˆ
Gives:
)ˆ
11(
2
2
2
)ˆ
11(
2
2
2
ˆ)'(ˆ
ˆ
ˆ)'(ˆ
ˆ
T
T
eCrd
Td
eCrd
Cd
where:
Ae
p
D
eAr
0
2
2)'(
S
SRT
E (Arrhenius number)
S
Se
S
Aer
S
S
S
T
CDH
T
TSA
max
Solution is a function of ’ , , and only. (See Weisz & Hicks (1962) diagram)
Liu (1969) shows:
5
'
1~
e
(See Froment and Bischoff, p.184)
Page 29 of 90
(ref. Weisz & Hicks (1962))
= 0 isothermal
< 0 endothermic
> 0 exothermic
’
Page 30 of 90
2.3 Combined Interfacial (External) and Intraparticle (Internal)
Resistances
In solution of intraparticle diffusion equation, C
SAs was assumed known (i.e. = CA) and constant.
When external-film resistances are important, the boundary conditions for the solution of the
intraparticle diffusion equation become:
p
S
ASC
r
A
e
S
AAmpdr
dCDCCkrr )(
0
0dr
dCr SA
(as before)
E.g. for slab pallet with a 1st order reaction, solution of eqn.(23) with the above boundary
conditions gives:
sinhcosh
)cosh(
C
A
S
mp
e
p
A
A
kr
D
r
rC
C
Can then define a global effectiveness factor, G, as:
G = (rate observed) / (rate at bulk fluid concentrations)
][ AA
A
Cr
r
S
which gives:
mG Bi
211
(44)
where
A
C
e
pm
mD
rkBi (Biot number for mass transfer , or Sherwood number)
i.e. For Bim >> 1, G = .
In the region of strong intraparticle diffusional limitations,
1
, :
mG Bi
21
(45)
Finally, for other reaction schemes (non-1st-order_ and pellet geometries, Aris (1965) has shown
that eqn.(45) is again applicable. But is calculated using generalised Thiele modulus (eqn.(40))
for which CAs is replaced by CA.
Page 31 of 90
3 Fixed Bed Catalytic Reactor (FBCR) Design
We will now consider mathematical models describing FBCRs
We will start with simple homogeneous models, then models accounting for interfacial and
intraparticle gradients through the use of:
(i) an effectiveness factor, or
(ii) actual pellet phase mass and energy balances
We will principally consider 1-D models, i.e. no radial gradients in C or T.
We will also consider a single reaction, in which the consumption of specie i is denoted by ri.
(extension of models to multiple reactions will be demonstrated)
The key reactant will be denoted by A
3.1 Pseudo-Homogeneous PFR and Axially Dispersed PFR Models
3.1.1 PFR Model
Simplest FBCR model (from Reaction Engineering I):
bA
A
ibii
i rv
vrr
dV
dn '' (1)
Noting that: ni = u a Ci , dV = a dz:
bA
A
ibii r
v
vruC
dz
d '')( (2)
If u ≠ constant, we need a model for velocity distribution, i.e. an approximation of the
momentum equation.
E.g. Ergun equation:
2
21 uEuEdz
dPi (3)
where:
32
32
2
1
)1(8.1
)1(180
bp
mgb
bp
b
d
ME
dE
(For laminar flow, the term containing E2 can be neglected; for highly turbulent flow, the term
containing E1 can be neglected.)
Page 32 of 90
For a perfect gas:
g
i
iRT
PC (4)
For non-isothermal operation, energy balance needed to describe T-z variation.
From Reaction Engineering I (basis: J/(m3 s)):
0)( ' vrAbp
i
i aQHrCndV
dTi
(5)
where:
Q = U(TC – T) (J/(m2 s))
av = (surface area) / (reactor volume) (m-1
)
[U: overall heat transfer coefficient, J/(m2 s K)
TC: temperature of cooling (heating) fluid, K ]
0)( ' vrAbp
i
i aQHrCCudz
dTi
(6)
Do we need to solve eqn.(1) for all reaction species?
For no separation of reaction species due to, say, different rates of axial dispersion or
intraparticle diffusion (see later notes), we can relate Ci (i ≠ A) to CA from reaction stoichiometry:
(nAo – nA) = mol A reacted
)(00 AA
A
iii nn
v
vnn
i.e. )(00 00 AA
A
i
ii uCCuv
vCuCu (7)
Other way of relating Ci of all components and thus reducing material balance to one equation is
the use of conversion or extent of reaction!
Boundary conditions:
Equations (2), (3), and (6) are 1st order ODEs. Thus each requires 1 boundary condition for the
dependent variable. Usually we specify conditions at the reactor entrance. Thus:
0
0
1 )..1(:00
TT
PP
NiCCz Cii
(8)
Check on degrees of freedom:
Unknowns: NC ∙ C, u, P, T
i.e. NC + 3
Equations: (2), (3), (4), (6), (7) ∙ (NC – 1)
i.e. NC + 3
Therefore, solution possible.
Page 33 of 90
3.1.2 Axially Dispersed PFR Model
Revision from Reaction Engineering I:
Material balance on incremental section:
where DZ = axial dispersion coefficient (m
2/s)
DZ 0 for PFR
DZ ∞ for CSTR
0)()( ' bA
A
iiZi r
v
v
dz
dCD
dz
duC
dz
di
(9)
Similarly for the energy balance, we can have an effective thermal dispersion coefficient in the
axial direction, kZ, to give:
0)()( '
vrAbZp
i
i aQHrz
Tk
zCCu
z
Ti
(10)
Equations (3) & (4) are again applicable.
Eqn.(9) is only applicable if the dispersion coefficients for all species are the same, i.e. DZi = DZ.
(Why?)
Otherwise, we need to solve eqn.(9) for all i.
Boundary conditions:
Now we have 2nd
order ODEs for C & T (equations (9) & (10)). So we need 2 boundary
conditions for each equation. These are chosen at z = 0 and z = L using the Danckwerts boundary
conditions.
Danckwerts boundary conditions account for dispersion in the entrance/exit regions of the bed:
z = 0:
000
Z
iZZii
dz
dCDuCuC (I)
i.e. 0ZiC gives actual boundary condition for Ci at z = 0.
uCi o
z = 0
.
)(dz
dCDuC i
Zii
z z + z
)(dz
dCDuC i
Zii
zar bi '
Page 34 of 90
z = L:
ie
LZ
iZLZi uC
dz
dCDuC
(II)
But iei CC (otherwise physically meaningless!)
0LZ
i
dz
dC
If 0LZ
i
dz
dC, from (II):
iei CC ! which is an inconsistency.
Thus, the only sensible (feasible) solution for (II) is 0LZ
i
dz
dC
In summary, Danckwerts boundary conditions are given by:
0
)(00
0 0
LZ
i
Z
iZiZi
dz
dCLz
dz
dCDCCuz
(11)
Danckwerts boundary conditions can be derived for energy balance in a similar way, i.e.:
0
)(00
0
LZ
Z
ZZi
pi
z
TLz
z
TkTTCCuz
i
(12)
Boundary condition for eqn.(3) is the same as before, i.e. P = P0 at z = 0.
Eqn.(11), for z = 0, again only needs to be specified for (NC – 1) components, since Ci for Ncth
component is given by eqn.(4) if P0 and T0 are specified.
But, for z = L condition, we need to be a bit more careful!
If Pe is held constant, then g e will be known. Thus (NC – 1) equations need to be specified for
LZ
i
dz
dC
.
uCi e
z = L
.
Page 35 of 90
If the outlet flow through a valve (e.g. to control gas velocity):
)( PfuLZ
(valve equation)
This sets LZdz
dP
from eqn.(3),
LZP
, and again (NC – 1) equations need to be specified for
LZ
i
dz
dC
.
Degrees of freedom analysis as for model in section 3.1.1.
When is axial dispersion important?
Young and Finlayson (1973) show that for:
0
'
A
pbA
Z
p
uC
dr
D
du
Axial dispersion can be neglected.
Note: The dispersion term above is in this case defined using dp rather than L.
As a rule of thumb, for flow velocities used in industrial practice, the effect of axial dispersion
(head and mass) on conversion is negligible when:
L > 50 dp
Pe
u |z =L
P|z = L
Page 36 of 90
3.2 Heterogeneous Models
3.2.1 Use of Effectiveness Factor
If can be readily calculated, we can use this in our material and energy balances.
i.e. replace r’A in equations (2) & (6), or (9) & (10) with:
r’A or G r’A
to give actual (observed) reaction rate.
(remember: (G) can be a function of z)
E.g. PFR model:
(i) If G is given, the equations given in section 3.1.1 are applicable. But r’A needs to be replaced
with G r’A.
(ii) If is given, then if we neglect external film mass & heat transfer resistances, the equations
given in section 3.1.1 are applicable. But r’A needs to be replaced with r’A.
Otherwise:
(a) If we allow for external film heat transfer resistance only, the bed energy balance (eqn.(6)) is
now given by:
0)()( ' S
Smfvp
i
i TTahaQCCudz
dTi
(13)
Another variable, TS
S, is now introduced but can be found from pellet phase energy balance:
rbA
S
Smf HrTTah '' )( (14)
(What are the units of a’m?)
(b) If we allow for both external film heat and mass transfer resistance, in addition to equations
(13) & (14), we need to replace material balance with:
0)()( ' SC iimmi CCakuC
dz
d (15)
and:
bAiimm rCCakSC
'' )( (16)
The boundary conditions given in section 3.1.1 are again applicable for this case. Analogous
equations can be written in the presence of mass and thermal dispersion, i.e. equations (14) &
(16).
Page 37 of 90
3.2.2 Use of Intraparticle Diffusion Equations
We solve for intraparticle diffusion equations when cannot be determined readily, e.g. complex
reaction kinetics and non-isothermal operation.
i.e. we solve diffusion equations for mass and heat in a catalyst pellet simultaneously with bed
mass and heat balance. The former were given in sections (2.2.3) & (2.2.4) as (spherical pellet):
bA
A
iie
S
Si rv
v
dr
dCr
dr
d
r
D'2
2)( (17)
rbA
Se Hrdr
dTr
dr
d
r S
'2
2)( (18)
Equations (17) & (18) are solved at each z so as to give, for example, S
iSC and T
SS, so as to be
able to calculate mass and heat transfer rates from bed to pellet at each z.
In the most rigorous form, equations (13) & (15) are applicable for this case (or the analogous
equations in the presence of mass and thermal dispersion), as well as corresponding boundary
conditions.
Now need boundary conditions for (17) & (18). They are 2nd
order ODEs in Si
C and TS. So 2
boundary conditions are needed for each:
r = 0: (symmetry condition)
0
0
0
0
r
S
r
i
dr
dT
dr
dCS
r = rp:
p
p
S
iSC
rr
S
e
S
Sf
rr
i
e
S
iim
dr
dTTTh
dr
dCDCCk
)(
)(
Page 38 of 90
3.3 2D Models
2D models account for radial variations in C & T (& TS) in addition to axial variations.
These usually arise due to a combination of high Hr and non-adiabatic operation through
heating/cooling of the tube (reactor) walls.
Radial velocity gradients will again be associated with mass and thermal dispersions, which can
be characterised by yzD and
yzk , where y denotes the radial dimension in the tube.
Radial dispersion will be much more significant than axial dispersion. Thus in 2D models, the
latter is usually neglected.
How do we derive mass and energy balances for this case?
Need to consider an incremental annulus within the bed, e.g.:
Thus, for example, by neglecting inter-facial and intraparticle gradients we can derive (pseudo-
homogeneous model):
0)1
()( '
2
2
bA
A
iiizi Sy
rv
v
y
C
yy
CDuC
z (19)
0)1
()( '
2
2
vrAbz
i
pi aQHry
T
yy
TkCCu
z
TSyi
(20)
The equations are now 2nd
order with respect to Ci & T in the y-domain. Thus 2 additional
boundary conditions in the y-domain are needed. There are:
y = 0: (centre of tube)
0
0
0
0
y
y
i
y
T
y
C
z
Z
i
Zidz
dCDuC
ZZ
i
Zidz
dCDuC
y
i
Zydy
dCD
yy
i
Zydy
dCD
Page 39 of 90
y = yw: (tube wall; yw = tube radius)
0
wyy
i
y
C (Why?)
)( wyyw
yy
z TTy
Tk
w
w
y
Where w is an effective heat transfer coefficient at the inner surface of the tube, and Tw the wall
temperature.
w is difficult to predict due to variations in gas flow and bed packing in the vicinity of the wall
compared to the rest of the bed.
For a 2D heterogeneous model, researchers have shown that for the radial dispersion of heat, it is
best to distinguish between the solid and fluid phases.
Thus, neglecting external-film mass transfer resistance:
- eqn.(19) is applicable but r’A is replaced by r’A
- eqn.(20) is rewritten as:
0)()1
()( '
2
2
smf
f
z
i
pi TTahy
T
yy
TkCCu
z
Tyi
(21)
- eqn.(14) is rewritten as:
)1
()(2
2''
y
T
yy
TkHrTTah S
zrbA
S
Smf y
(22)
(f & s denote fluid and solid (pellet) phases respectively)
What are the boundary conditions for the above model?)
Area for background reading:
(i) Multiple steady states in exothermic reactions (see Fogler, p. 447-463).
keywords: ignition and extinction temperatures
(ii) Reaction runaway: critical inlet conditions for reaction runaway.
Page 40 of 90
4 Fluidised Bed Reactors
These involve catalyst beds which are not packed rigid but either suspended in fluid (i.e. fluidised
bed reactors), or flowing with the fluid (i.e. transport reactors).
4.1 Overview of Fluidisation Principles
For downward flow in a packed bed, there is no relative movement between particles. P u for
laminar flow, or P u2 for highly turbulent flow.
For upward flow through the bed, P is the same as downward flow at low flow rates. But when
frictional drag on particles becomes equal to their apparent weight (i.e. actual weight less buoyancy),
the particles become rearranged such that they offer less resistance to the flow, and the bed starts to
expand. As u increases, the process continues until the bed has assumed its loosest stable form of
packing. Particles become freely supported in the fluid, and the bed is said to be fluidised.
Minimum fluidisation velocity, umf: the fluid velocity at the point where fluidisation occurs.
At superficial fluid velocities > umf:
(i) Liquid fluidisation: bed continues to expand with u, and maintains a uniform character, bet
agitation (mixing) of particles increases. i.e. particulate fluidisation.
(ii) Gas fluidisation: gas bubble formation within a continuous phase consisting of fluidised
solids The continuous phase referred to as the dense or emulsion phase. i.e. aggregative
fluidisation.
For aggregative fluidisation, even at high inlet flow rates, the flow in the emulsion phase relative to
the particles remains roughly constant. But bubbling may be more vigorous.
For high flow rates and a deep bed, bubbles can coalesce, and even form slugs of gas which occupy
the entire cross-section of the bed.
For our fluidised bed reactors, we will be principally concerned with gas fluidisation.
- Fluidised bed behaves like a fluid:
hydrostatic forces are transmitted (why is this useful?), and solid objects float if their densities <
that of the bed.
There is intimate mixing and rapid heat transfer. Thus it is relatively easy to control T.
- the “type” of fluidisation (see attached figure) depends on particle size and relative density of
the particles (s – g).
Geldart classification can be used to give an indication of the type of fluidisation achievable (see
figure below).
Why fluidisation?
- As mentioned above, we can achieve good control of T (even for highly exothermic reactions)
- Can work with very fine particles, for which ~ 1
Page 41 of 90
- As catalysts improve, rates of reaction increase, i.e. higher values of kv.
But
Ae
vp
D
kr
3 (spherical pellet)
DeA ~ constant. Thus as kv increases, the only way to keep small (and thus close to 1) is to
decrease rp.
Page 42 of 90
(Why not use small particles in a packed bed?)
P vs. uo in a fluidised bed:
where:
u0 = superficial velocity at bed inlet
ut = terminal velocity, i.e. pellets blown out of bed
Note: in fixed bed region:
laminar flow: 01 uE
L
P
)log()log( 0uCP
turbulent flow: 2
02 uEL
P
)log(2)log( 0uCP
slope=1
for laminar flow
Page 43 of 90
Calculation of P across a fluidised bed:
For a fluidised bed:
Total frictional forces on particles = effective (apparent) weight of particles
F1 = F2
gLAAPAP
bb
S
S m
kg
m
m
m
kg
gs
33
3
3
)1)((21
gLPPP gs )1)(()( 12 (1)
If there is an increase in P1, P at that instant will become higher. But then increases (bed
expands), or low-resistance gas by-pass through bubbling, such that P remains the same.
Calculation of umf:
(i) If laminar flow at the point of fluidisation
guEL
Pgsmfmf
mf
mf))(1(1
1
))(1(
E
gu
gsmf
mf
For 32
2
1
)1(180
mfp
mf
dE
:
gdu
gs
mf
pmf
mf
)(
)1(180
123
(2)
(mf ~ 0.4 for a bed of isometric particles)
(ii) If turbulent flow at the point of fluidisation (usually the base for coarse particles)
guEuEL
Pgsmfmfmf
mf
mf))(1(2
21
Thus we can solve for umf explicitly.
However, it is convenient to work in terms of dimensionless groups:
Page 44 of 90
(1) 2
3)(
pgsg dgGa
(Galileo number)
(2)
pmfg
mf
duRe
to give (using the 3 equations above):
2
33Re
75.1Re
)1(180 mf
mf
mf
mf
mfGa
(3)
In reality we expect Darcy’s law and Ergun equation to overestimate Pmf. (Why?)
For laminar flow, investigations have shown that it is more accurate to use a value of 121 rather
than 180 in equation (2).
No data are available for the adjustment of coefficients in turbulent flow. It is best to measure
them experimentally.
Furthermore, the above theory does not account for:
(1) Channelling of the fluid
(2) Electrostatic forces between particles
(3) Agglomeration of particles
(4) Friction between the fluid and vessel walls
Calculation of ut:
When the drag force exerted on a spherical particle by the upflowing gas > the gravity force
(based on apparent density) on the particle, the particle will be blown out of the bed.
i.e. blow-out when:
gvF gspD )(
But pDtgD ACuF )
2
1( 2
where:
CD = drag coefficient
Ap = dp2 / 4 (projected area of particle)
gvCud
F gspDtg
p
D )(8
2
2
2
1
3
)(4
gD
gsp
tC
gdu
(4)
For spherical particles, and Re < 0.4 (
ptg duRe )
Re
24DC
and eqn.(4) reduces to Stokes’ law:
Page 45 of 90
18
)( 2
pgs
t
gdu
(5)
For: 1 < Re < 103, Trambouze et al (1984) give:
99.7ln(Re)
43.6950.5)ln(
DC
For Re > 103, CD = 0.43, and eqn.(4) reduces to Newton’s law:
2
1
)(1.3
g
gsp
t
gdu
(6)
Fluidisation Regimes:
We can now consider fluidisation regimes for Geldart type A or B particles.
(see figures below)
(i) Key points for fluidisation regimes with coarse particles:
(1) Bubbles appear as soon as umf is exceeded
(2) In turbulent regime, bubble life time is short (bubble burst). Bed looks quite uniform. (short-
circuiting of gas through bubbles is less likely)
(3) ut and particle blow-out coincide
(4) In fast fluidisation regime, there is net entrainment of solids
(5) In transport regime, there is solid flow in the direction of gas flow
(6) Carry-over (entrainment) separates particles by size
Page 46 of 90
Page 47 of 90
(ii) Keypoints for fluidisation regimes with fine particles:
(1) Bubbles do not appear as soon as minimum fluidisation is reached; instead there is the
uniform expansion of the bed
(2) The bed is more coherent rather than with particles behaving independently
(3) Turbulent regime sets in well after u0 exceeds ut of an individual particle. Thus we can
operate at higher u0
(4) Carry-over does not separate particles by size (i.e. a more cohesive bed)
Heat and mass transfer in fluidised beds:
- Heat exchange can be through vessel walls or to an internal heat exchanger. For the latter, we
need to be more careful not to adversely effect the flow of fluidising gas, e.g. we can use
“bayonet” tubes. In some cases fluidising gas supplies or removes the reaction heat.
- Most heat transfer correlations assume a pseudo-single-phase system, i.e. do not distinguish
between heat transfer from bubble and emulsion phases.
- For gas fluidisation, heat transfer is also dependent on the geometrical arrangement of the
vessel, and the type of fluidisation.
- Several correlations are available in the literature for heat transfer to wall or internal heat
exchanger. Usually they are in the form:
Nu = f(Re)
e.g. Dow & Jakob correlation for heat transfer to vessel walls: 8.025.0
17.065.0 )1(55.0
gte
pg
ps
g
tdu
C
C
dp
dt
L
dtdhNu
g
S
(i.e. Nu = f(Re0.8
) )
- For heat transfer between gas and particles, Balakrishnan and Pei (1975) give a jH factor: 25.0
2
2
0
)1)((043.0
g
gsp
Hu
gdj
But, in reality the above correlations for heat transfer is difficult to attain because heat transfer
rates between gas and particles is so large!
Kay & Nedderman (1979):
(Good news:)
“Thus for all beds of depth greater than a few centimetres the gas in the particulate [emulsion]
phase may be assumed to be at the same temperature as the particles”
(Bad news:)
“By contrast, the gas in the bubble phase is not in contact with particles and need not be at the
same temperature. Consequently the mean temperature of the gas leaving the bed may differ
Page 48 of 90
from that of the particulates.”
(Is the second statement strictly true?)
Page 49 of 90
4.2 Overview of Key Applications
As mentioned above, reactors involving fluidisation are useful for highly exothermic systems and/or
systems requiring close temperature control, e.g. oxidation reactions.
(When would close T control be needed, even if reaction is mildly exothermic or endothermic?)
In “classical” fluidised bed operation, the catalyst particles are retained in the bed, i.e. there is little
catalyst entrainment.
E.g.:
(1) Oxidation of naphthalene into phthalic anhydride
(2) Ammoxidation of propylene to acrylonitrile
(3) Oxychlorination of ethylene to ethylene dichloride
(4) Coal combustion (can inject limestone for the in situ capture of SO2)
(5) Roasting of ores
Even with classical fluidised beads, the region above the surface of bed still has some solid
concentration. This concentration becomes constant as we move further away from the surface, i.e.
We can apply cyclones above the TDH to recover and return the catalyst.
(Correlations are available to predict TDH, e.g. see Geldart).
In transport reactors, the total amount of catalyst is entrained by gas
E.g.:
(1) Fischer-Tropsch reactions (Sasol): production of hydrocarbons from “synthesis” gas (CO +
H2)
(2) Catalytic cracking, e.g. gasoline and diesel oil production from the cracking of crude oil (e.g.
BP oil, Mobil)
- Such reactions are associated with catalyst deactivation. The transport operation allows flow of
catalyst to a regeneration process (which may involve burning off the coke in a classical fluidised
bed!). The catalyst is then re-circulated.
Constant solids
concentration Transport
disengagement height
(TDH)
Page 50 of 90
Furthermore, transport operation enables very short reaction space-times, which may be needed
to prevent over-cracking, or a reduction in product selectivity.
- In fluid catalytic cracking (FCC) reactors (also referred to as cat crackers) reactions are
endothermic, but the combustion of coke in regeneration stage is used to re-heat the particles, and
thus provide the required reaction heat.
(typically 1-2% (w/w) coke is reduced to 0.4-0.8% (w/w) in regeneration step)
Modern cat crackers use zeolite catalysts, which are highly active for cracking.
(see diagram below for FCC reactor)
- Usually there is some down-flow of catalyst along the wall of the riser, which can upset the
desired product distribution.
The next generation of transport reactors may involve fluidised flow in the downward rather than
upward direction.
Page 51 of 90
4.3 Modelling of Fluidised Bed Reactors: Non-Transport
4.3.1 Two-Phase Models
Early models considered the fluidised bed as single-phase PFR, CSTR or ADPFR. This is
generally poor description of the process. (Why?)
More accurate description is achieved through two-phase model, with interchange between the
phases:
Bubble phase ~ PFR
Emulsion phase ~ well-mixed or ADPFR
Two-phase model requires 6 parameters:
(1) fb : fraction of bed occupied by bubbles (m3
bubbles / m3
bed)
(2) fe : fraction of bed occupied by gas in the emulsion phase (m3
emulsion gas / m3bed)
(3) kI : gas interchange coefficient between bubble and emulsion phases (m3
g / (m3bed s))
(4) DZe : dispersion coefficient in emulsion phase (m2/s)
(Don’t need DZe for well-mixed emulsion phase)
(5) gb : mass of solids in bubble phase (kg / m3
bubbles)
(6) ge : mass of solids in emulsion phase (kg / m3
emulsion)
Thus for isothermal fluidised bed with ADPFR in emulsion phase, the material balances give
(basis: mol / (m3bed s) ):
Bubble phase:
0)( '
Ab
eb
b
Conbased
AbbAAI
A
bb rgfCCkdz
dCuf (7)
(Analysis on units?)
Bubble
phase
u0, CA0
ub, CAb ue, CAe
Emulsion
phase
CAe|out CAb|out
AC
Page 52 of 90
Emulsion phase:
0)1()( '
2
2
Ae
eb
e
e
e
Conbased
AebAAI
A
ze
A
ee rgfCCkdz
CdDf
dz
dCuf (8)
(Analysis on units?)
Also: eb AeeAbbA CufCufCu 0 (9)
Boundary conditions for solution of above:
Bubble phase:
z = 0 CAb = CAo
Emulsion phase:
0
)(00
dz
dCLz
CCudz
dCDz
e
e
e
e
A
AAe
A
z
Simplification of the two-phase model:
If ub >> ue, i.e. ub >> umf, the emulsion phase will be ~ closed (relatively negligible inlet or outlet
flow)
Thus eqn.(8) reduces to: ')1()( AebAAI rgfCCkeb
(10)
(Also neglecting dispersion)
Eqn.(10) assumes a stagnant emulsion phase, but CAe varies with the bed length (z).
E.g.:
1st order reaction with two-phase model in which the emulsion phase ~ closed.
0)( beb
b
AbbAAI
A
bb kCgfCCkdz
dCuf
and
0)1()( eeb AebAAI kCgfCCk
and
be AA CC
where kgfk
k
ebI
I
)1(
(i.e. for kI >> k, CAe ~ CAb;
for kI << k, CAe ~ 0 )
0)( bbb
b
AbbAAI
A
bb kCgfCCkdz
dCuf
0' b
b
A
A
bb Cdz
dCuf
where kgfk bbI )1('
Page 53 of 90
Integrate with boundary conditions z = 0, CAb = CAo:
)'
(
exp
0
b
b
b f
A
A
C
C
(11)
where b
f
bu
L
Note: since there is gas out-flow from the bubble phase only AA CCb
Estimation of parameters appearing in two-phase model:
(i) ub:
rbmfb uuuu )( 0
where ubr is bubble rise velocity.
(Basis for above?)
Werther (1978) gives for a swarm of bubbles:
2
1
)( gdu bbr
where:
db = bubble diameter
= 0.64 for dt < 0.1m
= 1.6 dt0.4
for 0.1m < dt < 1.0m
= 1.6 for dt > 1.0m
Note: db will depend on gas distributer design (see Werther for correlations). For Geldart A
particles, db ~ 10cm.
(ii) fb:
b
mf
bu
uuf
)( 0 (Why?)
(b
bu
uf 0~ for ub >> umf )
(iii) fe:
fbe ff (Why?)
(f = voidage of fluidised bed)
(iv) Lf and f:
Since total volume of solids is constant:
bedpacked
bmfmfff LLL )1()1()1(
e.g. b
f
mf
mf
ff
L
L
1
)1(
)1(
(Why?)
Page 54 of 90
(given fb & mf (~0.4), Lf & f can be calculated)
(v) Dze:
Correlations available in literature, e.g.:
),( 0 bz dufDe
(vi) ue:
mf
mf
e
uu
(Why?)
(vii) gb and ge:
volumebedfluidised
masssolidtotalgfgf ebbb )1(
i.e. f
ebbbLA
Mgfgf
)1(
Note: typically gb ~ 0
(viii) kI:
For two-phase models, kI is often used as a fitting parameter such that the model agrees with
plant data.
Page 55 of 90
4.3.2 Three-Phase (Hydrodynamic) Models
Davidson & Harrison (1963): they worked on gas flow in the vicinity of a rising bubble in a
fluidised bed of fine particles:
(in large particles: “cloudless” bubble)
Thus we can have gas interchange from bubble to cloud, then from cloud to emulsion. i.e.
sequential steps.
i.e.
Again, different mixing regimes in phases can be assumed
Kunii and Levenspiel model (K-L): assumes emulsion phase with no net gas flow (closed).
(Usually achieved for 60 mfu
u)
e.g. K-L model for 1st order reaction.
Material balances:
(i) bubble phase:
0)( bCbb
b
AbbAAI
A
bb kCgfCCkdz
dCuf (12)
(ii) emulsion phase:
eCe AecbAeAI kCgffCCk )'1()( (13)
i.e.
gekCAe: (mol / (m3
emulsion s) )
(1-fb-f’c): (m3
emulsion / m3
bed)
bubble cloud emulsion
CAb CAc CAe
kIb
kIe
Page 56 of 90
(iii) cloud phase (link bubble & emulsion phases):
CeCeCbb AccAAIAAI kCgfCCkCCk ')()( (14)
In eqn.(14):
fc’: m3cloud / m
3bed
gc: kg / m3
cloud
)1(
~
fractionbubble
b
densitybulk
Be
fg
Note: from experimental correlation, Partridge and Rowe give:
)(17.0
17.1'
e
b
bc
u
uff
Using eqns. (13) & (14), we can express CAc in eqn.(12) in terms of CAb (c.f. previous example).
This gives:
b
b
A
A
b Cdz
dCu (15)
where (believe it or not):
b
cbeI
bb
ccI
b
b
f
ffgk
kff
fgk
kfgk
e
b
)1(
1
1
1
1
'
'
( effective rate constant for a three-phase fluidised bed model, i.e. K-L rate constant)
Integration of (15) with boundary conditions z = 0, CAb = CAo:
)(exp
00
bb
A
A
A
A
C
C
C
C
(16)
where b
f
bu
L
Note: the work of past investigators (e.g. K-L, Partridge & Rowe, Davidson & Harrison) has led
to equations for providing predictions for kIb & kIe based on hydrodynamics.
(see Levenspiel, 1999)
3-phase models are found to give better prediction of reactor performance.
Page 57 of 90
4.4 Modelling of Transport (Riser) Reactors
E.g. FCC processes: fast reactions (small needed) and rapid catalyst deactivation.
Velocity of solids ~ velocity of gas, i.e. no “slip” velocity.
Usually employ fine solids, Thus ~ 1
For no catalyst deactivation, riser is very much like a pseudo-homogeneous PFR, but bedpacked
b :
Calculation of :
p
sb
g
mAu
Au
m
m
0
0
3
3
)(
where: p = pellet density (kg / m3
pellet)
p
s
u
Am
0
)/(1
1
(17)
i.e. when SSm /
<< u0 , 1
when SSm /
>> u0 , 0
Thus for no catalyst deactivation:
pA
A rdz
dCu )1('
0 (18)
Unit analysis on R.H.S. of (18):
sm
mol
m
kg
m
m
skg
mol
bpb
p
333
3
. . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
.
. . . .
.
solid Sm
(kg/s) gas u0
A
Page 58 of 90
Catalyst deactivation in FCC is believed to arise from coke deposition and the adsorption of
certain (basic) species present in the feed.
Will result in reduction in the reaction rate(s); reaction rates will decrease with time. Can be
described using deactivation function.
Deactivation function, :
)(]0[
]['
'
tfr
tr
A
A
A (19)
e.g. (1) = 1 - t
(2) = exp(- t)
Then, eqn.(18) becomes:
pAA
A rdz
dCu )1('
0 (20)
“t” in eqn.(19) represents the amount of time the catalyst has spent in the riser,
i.e. 0u
zt (for no slip )
Sometimes is given as a function of the coke concentration on the catalyst pellets. It is
practical to express this concentration in terms of:
( kgcoke / kgcatalyst ) CC
The rate of formation of coke is then:
Cr kgcoke / (kgcat s)
Cr can itself be deactivated as coke is produced!
Thus, material balance for coke deposition:
)1(ˆˆ
pCc
CSr
dz
Cd
A
m (22)
(Units analysis on above?)
(Eqn.(22) has to be solved simultaneously with eqn.(20) for this case)
Finally, the energy balance for an adiabatic riser can be written as:
)1()]()([ '
pCCcAAA
pspg
HrHrdz
dT
A
CmCmSg
(23)
where Cpg & Cps are the specific heat capacity of the gas and solid respectively (kJ/(kg K)) and
gm
the mass flow rate of gas (kg/s).
( gg MRT
PuAm
0
00 )
What are the additional assumptions of the riser model given by eqns. (20), (22) and (23)?
Page 59 of 90
5 Multiphase Reactors
5.1 Background
Two or more phases are needed to carry out the reaction (i.e. gas-liquid; liquid-liquid).
Majority of multiphase reactors involve gas and liquid phases in contact with a solid.
Solid may be:
- Catalyst particles dispersed in the liquid phase, e.g. slurry reactor
- Packing for liquid distribution, e.g. packed bed absorber (CO2 in MEA)
- Packing for liquid distribution and catalyst support, e.g. trickle bed reactor, packed bubble
reactor
- Plates for liquid-gas contact (c.f. distillation column)
Reactors can also be classified in terms of which phase is continuous and which is dispersed.
e.g.:
Liquid phase continuous; gas phase dispersed:
Bubble reactor, slurry reactor, fermentation vessel.
Liquid phase dispersed; gas phase continuous:
tower (also trickle bed and packed bed reactors).
G
. . . . . . . . . .
. . . . . . . . . .
. . . . .
. . . . .
G
L
Page 60 of 90
Liquid phase continuous; gas phase continuous:
Falling film (wetted wall) reactor (When would such operation be useful?)
If the main mass transfer resistance is located in liquid: use dispersed gas phase and continuous
liquid phase.
If the main mass transfer resistance is located in gas: use continuous gas phase and dispersed
liquid phase.
Residence time of reactant and heat transfer considerations will also dictate the type of reactor,
e.g.:
- Plate columns can achieve long contact time between the gas and liquid, but poor temperature
control.
- Stirred tanks (e.g. bubble or slurry reactors) will have large L:G ratio, but yet cope with high G
flowrates and have good temperature control.
Reactors can have co- or counter- current flow of G and L to utilise driving forces for mass and
heat transfer.
Where reactors are principally employed for gas purification, they are referred to as absorbers.
The theories for G-L and L-L systems are similar: the former uses Henry’s law constant to
describe equilibrium distribution of a component in the G & L phases, the latter uses a partition
coefficient. Our analysis will consider G-L systems.
For the mathematical description of multiphase reactors, two situations arise depending on the
relative rates of reaction and mass transfer:
(i) reaction rate >> mass transfer rate:
E.g. fluid-fluid reactions involving a homogeneous (soluble) reaction, or requiring no catalyst. In
this case, mass transfer across, say, the liquid film (see Whitman’s two-film theory) will be
accelerated due to the reaction.
(ii) reaction rate << mass transfer rate:
E.g. Fluid-fluid reactions involving a catalyst (suspended or fixed). In this case, mass transfer
across the liquid film will not be accelerated due to the reaction.
Page 61 of 90
5.2 Review of Two-Film Theory
Two-film theory assumes that stagnant layers (films) exist in both the G and L phases along the
interface.
All resistance to mass transfer are then assumed to be located in the G and L films.
No resistance at interface, such that Henry’s law is satisfied:
ii AA HCP (1)
Thus:
We consider the reactor scheme:
aA(gl) + bB(l) Products (l)
i.e. A dissolved in liquid and reacts with non-volatile B to produce non-volatile product(s).
When reaction rate is relatively small comparing to mass transfer rate (usual assumption for
solid-catalysed reactions):
AyAA NNNL
0
and
)()(iLii AALAAGA CCkPPkN (2)
i.e. negligible reaction in liquid film, and no mass transfer enhancement due to reaction.
When reaction rate is relatively large comparing to mass transfer rate (usual assumption for fluid-
fluid reactions which require no catalyst, or use a homogeneous catalyst):
LyAA NN
0
and
)(0 iLi AALA CCkN
i.e. reaction is in liquid film, and there is mass transfer enhancement due to reaction.
Calculation of NA|o (NA|yL) (see Separation course):
Solution of diffusion equation in liquid film, i.e.
NA|o
G L
(reaction in this
phase only) PA PAi
CAi
CAL
NA = NA|o NA|yL
0 yL
. .
. .
Page 62 of 90
A
A
A rdy
CdD
2
2
Boundary conditions: y = 0, CA = CAi
y = yL, CA = CAL
where: DA = molecular diffusivity of A in liquid film (m2
L/s)
Then:
y
A
AyAdy
dCDN
E.g.:
1st order irreversible reaction:
AA kCr (mol / (m3
L s))
or
ABA CkCr )( with CB in excess
Then:
sinh
)sinh(])1[sinh(L
A
L
A
A
y
yC
y
yC
CLi
where:
= Hatta number
2
1
A
LD
ky (3)
(c.f. Thiele modulus!)
Since L
A
Ly
Dk :
2
1
2
L
A
k
kD (4)
Thus, for this case:
)cosh(sinh0 Li AA
L
A CCk
N
(5a)
and
)cosh(sinh
LiL
AA
L
yA CCk
N (5b)
Note: when 0 (k very small):
)(0 LiL
AALyAA CCkNN
(Why aren’t moles conserved such that LyAA NN
0 for all ?)
Enhancement Factor:
Sometimes it is convenient to work in terms of an enhancement factor to calculate NA:
Page 63 of 90
)(
0
Li AAL
yA
CCk
NE
(6)
i.e. E = (flux with reaction enhancement)/(flux with no reaction enhancement)
Thus, if E is known, we can model 0AN as:
)(0 Li AALA CCEkN
rather than using a complex equation.
Simple expressions can be derived for E for certain reactions:
E.g.:
(i) 1st order irreversible reaction:
Substitute eqn.(5a) into (6):
)cosh
11(
tanh
i
L
A
A
C
CE
For CAL << CAi (often assumed):
tanhE (7)
(c.f.
tanh
1 for slab)
(Why is this similar result not surprising?)
Then: )(0 iALA CEkN where
LAC is not involved
(ii) Instantaneous and irreversible reaction between A and B:
Can show for this case:
)1(0
i
L
i
AA
BB
ALACbD
CaDCkN (8)
Substitution of (8) into (6) and noting that CAL = 0:
G L
PA PAi
CAi
CBL
reaction plane
. . CA=CB=0
Page 64 of 90
i
L
AA
BB
insCbD
CaDE 1 (9)
i.e. Eins ≠ f()
Plots of eqns. (7) & (9) are given by van Krevelen and Hoftijzer (see figure below).
Criteria for “speed” of reaction, and approximations for E:
(1) < 0.2:
slow reaction (mass transfer non limiting)
E ~ 1
(2) 0.2 < < 2:
intermediate reaction
E ~ 1 + 2/3 (for 1
st & 2
nd order reactions)
(3) > 2:
fast reaction
(i) > 5Eins: instantaneous reaction
(ii) < 0.2Eins: E ~ (for 1st & 2
nd order reactions)
Van Krevelen and Hoftijzer plot:
1. Lines below dashed line: instantaneous reaction.
2. E ~ for > 2 and < 0.2Eins
E~1 (for <0.2)
Eins – 1
E = /(tanh) ~
Page 65 of 90
3. Curves between dashed line and E = /(tanh): from numerical simulation.
4. E ~ 1 + 2/3 for 0.2 < < 2
The above expressions for NA|o do not account for gas-phase mass transfer resistance. Since
transport processes are in series, i.e.
Then for CAL ~ 0:
)()(ii ALAAG CEkPPk where
iAC is not involved
ii ALAAG CEkHCPk )(
Hk
Ek
PC
G
L
A
Ai
Hk
Ek
PEkCEkN
G
L
AL
ALA i
0
LG
A
A
Ek
H
k
PN
10
(10)
G
PA PAi
CAi
gas film
. NA|o
L
resistance in series
Page 66 of 90
5.3 General Design Models for Multiphase Reactors
As mentioned above, for the case of fluid-fluid reaction with no catalyst, we need to account for
LuAA NN 0
.
Design models need to account for plug flow or well-mixed flow of the different phases.
Although there will be specific design issues depending on the actual type of reactor (e.g.
flooding, entrainment, weeping, bubble size control, etc… …), the underlying reaction,
absorption and flow phenomena can be described in a relatively simple way.
E.g.: The simplest model of a stirred vessel involving multiple phases: all phases completely
mixed. Thus just like a CSTR.
(What models could be used for: (i) a bubble column, (ii) a spray tower, (iii) a trickle bed
reactor?)
5.3.1 Gas & Liquid Phases Completely Mixed
E.g. Slurry reactor, liquid-liquid reactor in a stirred vessel
(Basis: mol/s, single reaction: key reactant A)
Gas phase material balance:
)1()(0
gvAAA VaNyyGoutin
(11)
av: (m2 gas-liq interface)/(m
3 liquid)
g: (m3 gas)/( m
3 gas+liquid) (1 - g ?)
i.e. R.H.S. of eqn.(11):
s
mol
m
mm
m
m
sm
mol
GL
L
GL
L
i
i
3
3
3
3
2
2
Liquid phase material balance:
Since reaction in liquid film is already accounted for by NA|yL , we need to consider reaction in
the remaining liquid volume, i.e.:
VL’ = liquid volume neglecting liquid film
G (mol/s)
yAin
G
yAout
L (mol/s)
xAin
L
xAout
V (m3)
Page 67 of 90
){
)1()()1(
3
'
3
,
3
'
3
3
3,
3
3
33
LfilmLL
mm
m
gLv
m
mm
gL
mmm
VyaVV
L
L
filmL
Lm
GL
LGL
)1()1(' LvgL yaVV (12)
(yL can be attained from: L
A
Lk
Dy )
Thus, liquid phase material balance can be written as:
)1()(' gvyAAAAL VaNxxLrVLinout
(13)
(rA in mol/(m3 s).)
Overall material balance:
E.g. Given aA + bB P then:
mol A consumed = b
amol B consumed
Thus:
)()(outinoutin BBAA xxL
b
ayyG (14)
Degrees of freedom analysis on well mixed model (eqns. (11)-(14)):
E.g.:
Given: yAin, xAin, xBin, G, L, V, av, g
Unknowns = 6: yAout, xAout, xBout, VL’, NA|o, NA|yL
Equations = 4: i.e. eqns. (11)-(14)
Thus require 2 more equations, i.e. for NA|o & NA|yL; e.g. eqns. (5a) & (5b).
But NA = f(CAi, CAL)
where out
out
L AL
L
A
A xL
LxC
)/( (L: liquid density, mol/m
3)
H
PC i
i
A
A
If gas-film resistance is negligible,
outi AAA yPPP
Otherwise use:
0
)( AAAG NPPki
Note: In a gas purification problem, yAout & yAin may be specified, in which case we can solve for
L or V.
For slow reaction (e.g. with a solid catalyst):
)(0 outinL
AALLyAA xxkNN
Page 68 of 90
For very rapid reaction: reaction completed in liquid film xAout = 0.
0LyAN
and we don’t need eqn.(13)
Can use E to calculate 0AN for this case.
Example: Well-mixed G-L Reaction:
Liquid phase o-xylene oxidation to o-methylbenzoic acid by means of air in a CSTR.
acidoicmethylbenzoxyleneoO
BA
)()(
25.1
Pseudo-1st-order reaction with respect to O2:
rB = 2.4 x 103 CA (kmol/(m
3 hr))
rA = 1.5rB = 3.6 x 103 CA
Data:
P = 13.8 bar
T = 160oC
L = 172 kmol/hr (xAin = 0.0; xBin = 1.0)
G = 245 kmol/hr (yAin = 0.21)
L = 7.1 kmol/m3
DA (O2 through xylene) = 5.2 x 10-6
m2/hr
HA = 126.6 m3 bar/kmol
av(1- g) = 2089 m2
i/m3
L+G ( '
va )
g = 0.336 m3
G/m3
L+G
kL = 1.485 m3
L/(m2
i hr)
Calculate V, xAout, and yAout for a desired conversion of o-xylene of 16%.
Solution:
If 16% o-xylene conversion:
in
outin
B
BB
Lx
xxL )(16.0
hrkmolxxLoutin BB /5.27)( and xBout = 0.84
Also: mk
Dy
L
A
L
66
105.3485.1
102.5
rA = 3.6 x 103 CA = 3.6 x 10
3 (L xAout)
Material balances:
(1) VaNyyG gvAAA outin)1()(
0
(2) VaNLxryaV gvyAAALvgLout
)1()1)(1(
(3) )()(outinoutoutin BBAAA xxL
b
aLxyyG
Thus, given NA|o & NA|yL: 3 unknowns (V, xAout & yAout) & 3 equations
Page 69 of 90
For a 1st-order irreversible reaction, equations (5a) & (5b) are applicable:
)cosh(sinh0 Li AA
L
A CCk
N
)cosh(sinh
LiL
AA
L
yA CCk
N
where L
A
k
kD 2
1
)(
i.e. )2.0(102.9485.1
)102.5106.3( 22
1
63
Therefore sinh ~ , cosh ~ 1
AAALyAA NCCkNNLiL
)(0
Neglecting gas phase mass transfer resistance:
A
A
A
A
AH
yP
H
PC outout
i
)(out
out
AL
A
A
LA xH
yPkN
Thus, neglect av & yL term!
Can now solve 3 material balance equations to give:
V = 6.78 m3
xAout = 3.6 x 10-4
yAout = 4.15 x 10-2
(q.e.d.)
Note:
Stirrer design and speed will influence bubble diameter, db.
db in turn will influence av, i.e.
v
b
g
gv ad
a '6
)1(
(Why?)
kL can be obtained from correlation.
Page 70 of 90
5.3.2 Gas & Liquid Phases in Plug Flow
E.g. packed tower
Consider a differential volume in the tower:
Also: use a’v (m
2i) rather than av (m
2i/m
3L) since correlations are available for a’v for packed
towers.
Gas phase material balance:
'
0 vA
A aNdV
dyG (15)
Liquid phase material balance:
ALvgvyAA ryaaN
dV
dxL
L
)1( '' (16)
Overall material balance:
Can carry out balance either at top of column or at bottom, depending on what inlet/outlet
conditions are specified, i.e.
top: )()()( BBAAAA xxLb
axxLyyG
ininout (17a)
or bottom: )()()(outoutin BBAAAA xxL
b
axxLyyG (17b)
How would the above model, i.e. eqns. (15)-(17) be modified for co-current operation?
How would the above model boundary conditions be simplified for complete reaction in the film?
Why is the choice of packing so important?
What are some of the hydrodynamic issues which we need to be careful about?
NA dV
G, yA
L, xA
xAin yAout
yAin xAout
V
Page 71 of 90
5.3.3 Gas Phase in Plug Flow, Liquid Phase Completely Mixed
E.g. bubble column; slurry reactor (but with bubble flowing straight through)
(c.f. fluidized bed reactor)
Gas phase material balance:
Eqn.(15) is again applicable:
'
0 vA
A aNdV
dyG (15)
Liquid phase material balance:
Since bubble or gas p hase is in plug flow, NA|o & NA|yL will vary with height. Thus the
calculation of the total moles of A transferred from gas to liquid requires an integration.
Therefore:
)('
0
'
inout AAAL
V
vyLA xxLrVdVaN (18)
(c.f. eqn.(13))
Overall material balance:
Eqn.(14) is again applicable:
)()()(outinoutinoutin BBAAAA xxL
b
axxLyyG (14)
5.3.4 Effective Diffusion Model
Similar to plug flow model (of gas, liquid, or gas & liquid phases) but a “dispersion” term is used to
account for some intermediate degree of mixing.
(see Froment & Bischoff, p. 608)
Page 72 of 90
5.4 Simplifications to Multiphase Design Models
5.4.1 Instantaneous Reactions
CAL = 0 (xA = 0)
NA|yL = 0
NA|o = EinskLCAi
E.g.: gas & liquid phases well-mixed; neglect gas-side mass transfer resistances:
Eqn.(11):
)1()( gvALinsAA VaCkEyyGioutin
(H
yP
H
PC outi
i
AA
A
)
GH
VaPkE
y
ygvLins
A
A
out
in)1(
1
Eqn.(13): not needed
Eqn.(14): )()(outinoutin BBAA xxL
b
ayyG
5.4.2 Very Fast Reactions
For reaction that is essentially completed in liquid film:
CAL~ 0
NA|yL = 0
Thus, equations in section 5.4.1 are again applicable, but need to use E rather than Eins.
5.4.3 Slow Reactions
)()(0 iLiL
AAGAALyAAA PPkCCkNNN
Noting that PAi = HCAi and eliminating CAi using kL & kG terms:
LG
AA
A
k
H
k
HCPN L
1 (19)
(c.f. eqn.(10))
E.g. Gas and liquid phases well-mixed
Eqns. (11), (13), & (14) as before but
)1(' gLL VVV
AyAA NNNL
0
Page 73 of 90
Thus:
Eqn.(11):
)1()( gvAAA VaNyyGoutin
Eqn.(13):
)1()()1( gvAAAAg VaNxxLrVinout
Eqn.(14): unchanged
5.4.4 Solid Catalyzed Reactions
For relatively slow reaction, LyAA NN
0
We may need to account for mass transfer resistances associated with catalyst pellets or catalytic
surface.
E.g.
rA G rA
rA rA (if external-film resistances are negligible)
Exercise:
Given and kmc, write down the appropriate form of eqn.(13). What additional equation is needed?
5.4.5 Resistances in Series Approximation: Gas-Liquid-Solid Reactions
(see also Levenspiel (1999))
Consider reaction:
aA(g) + bB(l) P(l)
but 1st order with respect to A & B:
BAA CkCr ' (mol/(kg s))
(also: ''
AB ra
br )
Assume reaction and transport steps are in series:
RA: reaction rate (mol/(m3
reactor s))
G
(A) (ii)
L
(i)
(B)
cat.
(iii)
(iv) .
Page 74 of 90
i.e.
RA )1('
gLA gr (iv)
)( S
AACm SLCCCak (iii)
)('
Li AAvL CCak (ii)
)('
iAAvG PPak (i)
where: S
B
S
AA SSCkCr '
gL: kgcat/m3
L
ac: m2cat surface/m
3reactor
kmc: mL/s
Why is the above description of RA wrong?
When would it be ~ true?
Noting that PAi = HCAi in (i), and eliminating CAi, CAL, S
ASC using (i)-(iv), i.e.
Hak
akiiii
ak
akii
vG
Cm
vL
Cm CC
'')()()(
Gives:
S
A
A
vGvLCm
A S
C
CH
P
HakakakR
''
111
Then substitute for S
ASC using (iv):
AAA PkR (20)
where:
)1()(
1111''
gL
S
BvGvLCmAgkC
H
akakakkSC
(21)
i.e. resistances in series (quantitatively useful)
If we also assume:
CBL >> CAL
i.e. Pure liquid B and slightly soluble A, then:
LS B
S
B CC = constant
Thus Ak = constant, since )( S
BSkC in eqn.(21) is constant
Design applications of Ak is straightforward:
(i) Gas and liquid phase are well-mixed:
)( BBoAAAo Xnb
aVrXn
(XA & XB denote conversions!)
Page 75 of 90
i.e. VXPkVPkXn AAoAAAAAo )1(
)(1
RTkX
XA
A
A
where
gV
V
0
(ii) Gas in plug flow (and any flow of liquid phase (Why?))
A
A
Ao RdV
dXn
)(exp1
RTk
AAX
5.4.6 Resistances in Series Approximation: Gas-Liquid Reactions
Similar method to above, but no solid resistances and need to account for absorption enhancement.
RA
3
3
3
)1(
reactor
L
L
LL
m
m
g
sm
mol
BAv CCk
)('
Li AAvL CCaEk
)('
iAAvG PPak
(Again, why is this wrong? When would it be ok?)
Can then show:
AAA PkR (22)
where for this case:
)1)((
11''
gBvvLvGA L
Ck
H
aEk
H
akk
For CBL in excess, i.e. pure liquid B in feed, RA again is constant if E is constant (or assumed ~
constant)
Note for very fast or instantaneous reaction (complete reaction in liquid film):
vk , and 0~LAC
''
11
vLvGA
aEk
H
akk
and transport steps are in series.
We could have, or course, derived this model from section 5.4.1 (or 5.4.2) but using:
)(0 AAGALinsA PPkCkEN
ii
Page 76 of 90
Example: Well-mixed G-L reaction:
Reconsider o-xylene oxidation to o-methylbenzoic acid by means of air in a CSTR, but this time
using eqn.(21).
acidoicmethylbenzoxyleneoO
BA
)()(
25.1
Pseudo-1st-order reaction with respect to O2:
rB = 2.4 x 103 CA (kmol/(m
3 hr))
rA = 1.5rB = 3.6 x 103 CA
Data:
kG (no gas phase mass transfer resistance)
nBo = 172 kmol/hr
XB (conversion) = 0.16
PA = P yAout = 57,270 Pa
(CBL = L xBout = 5.964 kmol/m3)
H = 126.6 x 105 m
3 Pa/kmol
(kvCBL) = 3.6 x 103 hr
-1
g = 0.336 '
va = 2089 m2/m
3
kL = 1.485 m3/(m
2 hr)
Also, the reaction showed to be slow E = 1
Thus:
1)2.52960.4081( Ak
(liq. film mass transfer resistance & reaction rate resistance on the R.H.S. respectively)
41066.1 Ak
76.6)(
)(0
AA
BB
Pk
b
aXn
V m3
Calculations consistent for this case because:
(i) CBL >> CAL (pure liquid B as feed)
and
(ii) CAL (i.e. xAout) ~ 0, thus L(xAin – xAout) ~ 0
Note: yAout is specified here (calculated using previous (non-approximate) method). If this was not
specified then:
)1(0 AAA XPP
But: BBAA Xnb
aXn
00
8023.016.021.0245
1725.1
0
0
B
A
B
A Xn
n
b
aX
1.57284)8023.01)(21.0(8.13 AP Pa
Page 77 of 90
5.5 Factors in Selecting a Gas-Liquid Contactor
Mass transfer driving forces in towers are higher than in agitators, but for high L/G rations
agitators are better.
Usually for a packed tower: L/G ~ 10 at 1bar.
Liquid droplet vs. gas bubble:
Droplet:
- kG high (the gas flow around droplet is high)
- kL low (liquid is stagnant in droplet)
Bubble:
- kG low (the gas is stagnant in bubble)
- kL high (there is relative motion between liquid film and bulk liquid)
i.e.
(1) Don’t use spray tower if kL is low
(2) Don’t use bubble column if kG is low
(3) For very soluble gases (H small), there is gas-film mass transfer controls, i.e. kG low. Thus
better use a dispersed liquid (droplet) or packet towers. Should avoid bubble contactors.
(4) For gas of low solubility (H high), there is liquid-film mass transfer controls, i.e. kL low.
Should avoid spray contactors.
(see also Levenspiel (1999), for typical characteristics of G-L contactors in terms of: '
va , (1-g ),
capacity)
Page 78 of 90
6 Non-Catalytic Fluid-Solid Reactions
E.g.
- coal gasification/burning
- ore processing
- iron production (blast furnace)
- regeneration of coked catalysts
- activation of catalysts (i.e. reduction or oxidation)
- Si oxidation to SiO2 for fabrication of microelectronic devices
- pharmacokinetic processes
Can classify different reactions into two general types:
(i) Total particle dissolution
i.e. particle is being completely consumed and thus shrinking with time.
E.g. tablet dissolution, coal gasification
(ii) Shrinking core,
i.e. overall particle size remains unchanged, but the reactive components within the particle is
decreasing in concentrations. Thus a shrinking core of reactive material.
E.g. catalyst regeneration activation
Regeneration of coked catalyst pellet
1.0
Fra
ctio
n o
f co
ke
bu
rned
time 0
Page 79 of 90
6.1 Total Particle Dissolution
Consider A diffusing to surface to react with solid B:
At the surface: aA(g/l) + bB(s) P(g/l)
Reactions of this type are usually zero-order in B, and first-order in A: S
AS
S
A Ckr (mol/(m2 s))
At any time:
)( S
AAm
S
ASA CCkCkNC
A
mS
mS
A Ckk
kC
C
C
Thus:
A
S
AA CkrN 0 (1)
where:
Sm kkkC
111
0
(2)
(c.f. eqns. (4) and (5) in section 2)
Effect of dp on kmc:
For flow around a spherical pellet:
3
1
2
1
Re6.02 ScSh (Frössling correlation)
Ag
pg
A
pm
DSc
ud
D
dkSh C
,Re,
i.e. kmc = f(dp) = f(t)
For small particles and/or low u:
Sh ~ 2, i.e. kmc = 2DA/dp
Thus, eqn.(2) becomes:
SpA kdDk
1
/2
11
0
(3)
i.e. equal resistances when
*2p
S
A
p dk
Dd (4)
when *
pp dd : mass transfer controlled
B NA
(mol/(m2 s))
S
Ar .
Page 80 of 90
when *
pp dd : reaction rate controlled
Now consider solid material balance:
–accumulation = consumption (basis: mol/s)
p
S
BBp ArVdt
d )(
(3
2
3
;;6
p
B
Bpp
p
pm
moldA
dV
)
For a = b = 1, S
A
S
B rr :
B
A
B
S
A
p
Ckrd
dt
d
022)(
(5)
Given excess CA (CA = constant), and eqn.(3) for k0, eqn.(5) can be integrated with boundary
conditions:
dp = dpo at t = to
to give:
B
AS
p
pppp tCk
d
dddd
*2
)(1
2
00 (6)
Therefore, for complete particle dissolution (dp = 0):
*21
2
0
0
p
p
p
AS
Bc
d
dd
Ckt
(7)
Note: For agitated systems and/or systems involving complex kinetics, numerical integration of
eqn.(5) may be necessary. Why?
Page 81 of 90
6.2 Shrinking Core Model
Demonstration of shrinking core model through application to catalyst regeneration (i.e.
decoking):
i.e. O2 diffusion through external-film and shell to “core surface”, where a rapid oxidation of the
carbonaceous material occurs.
Diffusion through shell is usually rate-controlling. (Why?)
Although core is shrinking with time, at any instant we can assume the O2 concentration profile
is the shell to be a steady state profile, i.e. quasi-steady state (QSS) assumption.
(Why do we expect QSS to be ~ true?)
(Bischoff (1963, 1965) shows that QSS is true for: 310B
AC
)
Consider diffusion equation for O2 transport through (spherical) shell:
(i.e. eqn.(29) in section 2)
04)(1 22
2 rr
dr
dCDr
dr
d
r
S
AA
eA
For constant DA:
0)( 2 dr
dCr
dr
d A (8)
Boundary conditions for eqn.(8):
r = Ro, CA = CAo
r = R, CA = 0
(What are the assumptions here?)
Note: R = f(t) but will assume constant in establishing CA profile in the shell (QSS assumption).
Integration of eqn.(8) twice, using the above boundary conditions gives:
0
11
11
0
RR
rR
C
C
A
A
(9)
B
NA
(mol/(m2 s))
. shell
core A
(e.g. O2)
R Ro r
Page 82 of 90
i.e.:
Also:
S
A
Rr
A
eRrA rdr
dCDN
A
S
A
Aer
RRR
CDA
2
0
11
0 (10)
Now consider solid (carbon) material balance:
–accumulation = consumption (basis: mol/s)
2
2
0
411
)( 0 R
RRR
CDfV
dt
d Ae
BBcore
A
where:
3
3
4RVcore (with R = f(t)!)
B molarity of coke = density of coke / molar mass of coke
3
cokem
mol
fB = 3
3
pellet
coke
m
m
RR
R
Rf
CD
dt
dR
BB
AeA
0
00
(11)
Boundary conditions for eqn.(11):
R = Ro at t = 0
1.0
R
0A
A
C
C
r R0 0.0
core shell bulk
Page 83 of 90
Integration gives:
3
0
2
0
2
0 2316
0R
R
R
R
CD
Rft
Ae
BB
A
(12)
Thus, time required for complete decoking (R = 0):
06
2
0
Ae
BB
cCD
Rft
A
(13)
E.g. Decoking of fluidised catalytic cracking (FCC) catalyst:
~ 2% (w/w) coke on 2mm diameter pellets:
(BfB): molcoke/m3p
332300
012.0
1140002.0
pcoke
coke
p
p
p
coke
BBm
mol
kg
mol
m
kg
kg
kgf
Regeneration with 5% O2 at ~ 1000K and 1bar; DeA ~ 3 x 10-5
m2/s:
)6.0)(103(6
)101(23005
23
ct
stc 20~
Page 84 of 90
6.3 Reactor Design
Factors controlling design of non-catalytic fluid-solid reactors:
Reaction kinetics for single particles, e.g. eqn.(1) or eqn.(10).
Size distribution of solids
Flow patterns of solids and fluids in reactor (see figure below)
In systems where the kinetics are complex and not well known, or the products of reaction form a
blanketing fluid phase, or large temperature variations exist from position to position:
Analysis difficult design based on experience
(E.g. Blast furnace for producing iron)
However, some real systems can be adequately approximated by idealised systems. Ideal models can
also be used as a starting point (preliminary design) of complex systems.
Flow patterns for fluid-solid reactors:
(ref. Levenspiel (1999))
(What are the flow patterns in the above reactors?)
Page 85 of 90
Consider two idealised systems:
(i) Plug flow of solid; uniform and constant gas composition; particles of unchanging but
different size.
(ii) As for (i) but solids in mixed flow.
6.3.1 Plug Flow of Solids
F[Ri]volumetric flow rate of material of radius ~ Ri fed to the reactor
i.e. Total feed rate, F:
max
0
][
R
RiFF cm
3/s
Mean conversion of solid material, , can then be attained from:
(mean value for fraction of B unconverted) =
iR
ii Rsizeof
fraction
feed
Rsizeof
particlesin
dunconverteB
i.e.
max
0
][
][)1(1
RR
RBBF
FXX i
i (14)
Note: For Ri < '
iR , where '
iR is the radius of the largest particle completely converted in a reactor
of spacetime , then XB = 1.
E.g.
Reconsider the previous example on catalyst decoking of 2mm pellets (R=1000 m) in 20s. What
is BX if the feed contains particles of the following size distribution:
30% ~ 750 m
30% ~ 1000 m
40% ~ 1250 m
and the reactor is operated with a space-time of ~22s?
Solution:
For this case, there will be complete conversion of 750 and 1000 m particles.
)4.0)(1()3.0)(11()3.0)(11()1( ]1250[BB XX
How do we calculate XB[Ri]?
BB
BBBB
RB
fR
fRfR
Xi
3
0
33
0
][
3
43
4
3
4
3
0
33
0
R
RRX B
where R is the radius of the pellets after 22s in the reactor.
Page 86 of 90
Using eqn.(12) we can solve for R at t = 22s:
R ~ 440 m
956.01250
44012503
33
]1250[
BX
982.04.0)956.01(1 BX
What if the given size distribution data is continuous rather than discrete?
What if CA in gas phase is not constant?
6.3.2 Mixed Flow of Solids
E.g. Fluidised bed reactor
XB = f(residence time of solid)
From Reaction Engineering I, exit-age distribution for material in a CSTR is given by:
t
etE
)( (15)
Again, eqn.(14) is applicable, but this time, for a given particle size Ri, there will be a distribution
of XB due to the distribution of residence time given by eqn.(15).
Thus:
max
0
]['
][ )1(1R
R
RBBF
FXX i
i (16)
where '
BX is the mean conversion of particles of size Ri in the bed, i.e.
dttEXXiRBB )]()1[(1
0
][
'
(17)
and XB is again attained from a suitable kinetic model, such as eqn.(12).
Note: in eqn.(17) corresponds to the time for the complete conversion (e.g. regeneration) of a
particle (core) of original radius Ri.
Page 87 of 90
7 Notation
a, b, ... Stoichiometric coefficients -
a Area available for mass flux; cross-sectional area m2
ac Specific surface area of pellet on a reactor volume basis m-1
am Specific surface area of pellet on a mass basis m2 kg
-1
a’m Specific surface area of pellet on a bed-volume basis m
2 m
-3
av Specific surface area of reactor on a reactor volume basis;
specific surface area of gas-liquid interface on a liquid volume
basis
m-1
a’v Specific surface area of gas-liquid interface on a reactor
volume basis
m-1
A Bed cross-sectional area m2
Ap External surface area of pellet (chapter 2); projected particle
area (chapter 4)
m2
Bim Biot number for mass transfer -
C Total fluid phase concentration mol m-3
CA Bulk fluid phase concentration of A mol m-3
CA0 Initial feed / bulk fluid phase concentration of A mol m-3
CAi Liquid phase concentration of A at gas-liquid interface mol m-3
CAs Fluid phase concentration of A within catalyst mol m-3
CS
As Fluid phase concentration of A at catalyst / solid surface mol m-3
CAb Fluid phase concentration of A within bubbles mol m-3
CAc Fluid phase concentration of A within bubble cloud mol m-3
CAe Fluid phase concentration of A within emulsion mol m-3
C Coke concentration kgcoke kgcatalyst-1
CD Drag coefficient -
Cp Fluid heat capacity (molar basis) J mol-1
K-1
Cpg, Cps Fluid / solid heat capacity (mass basis) J kg-1
K-1
dp Particle diameter m
d*
p Particle diameter for equal kinetic and mass transfer
resistances
m
dpore Pore diameter m
dt Tube / vessel diameter m
DA, DB Molecular diffusion coefficient m2 s
-1
DAB, Dm Molecular diffusion coefficient m2 s
-1
De Effective diffusion coefficient m2 s
-1
Dk Knudsen diffusion coefficient m2 s
-1
Dp Pore diffusion coefficient m2 s
-1
Dz Axial dispersion coefficient m2 s
-1
E, Eins Enhancement factor; instantaneous reaction enhancement
factor
-
E Activation energy J mol-1
K-1
E(t) Exit-age distribution -
E1, E2 Ergun equation coefficients -
fb Fraction of bed occupied by bubbles m3
bubbles m-3
bed
fB Volume of B (e.g. coke) per unit pellet volume -
f’c Cloud fraction in the bed m3
cloud m-3
bed
fe Fraction of bed occupied by emulsion gas m3
eg m-3
bed
F Force; volumetric flow rate of solid material (chapter 6) N; m3 s
-1
FD Drag force N
g Acceleration due to gravity m2 s
-1
Page 88 of 90
gb Mass of solids in bubble phase kg m-3
bubble
gc Mass of solids in cloud phase kg m-3
cloud
ge Mass of solids in emulsion phase kg m-3
emulsion
gL Mass of solids per unit volume of liquid kg m-3
liquid
G Mass flux; gas molar flow rate kg m-2
s-1
; mol s-1
Ga Galileo number -
jD, jH j-factors for mass and heat transfer, respectively -
hf Heat transfer coefficient J m2 s
-1 K
-1
H Henry’s law constant Pa m3 mol
-1
I Bessel function -
k Reaction rate constant mol1-n
m3(n-1)
s-1
kG Gas-film mass transfer coefficient mol m-2
s-1
Pa-1
kmc Mass transfer coefficient based on concentration driving force m3
f m-2
s s-1
kmp Mass transfer coefficient based on pressure driving force mol m-2
s Pa-1
s-1
kmy Mass transfer coefficient based on mole fraction driving force mol m-2
s s-1
ko Overall reaction rate constant mol1-n
m3(n-1)
s-1
ko
m Mass transfer coefficient for equimolar counterdiffusion
conditions
ks Rate constant for a reaction rate based on per unit surface area m3
f m-2
s s-1
(for n=1)
kv Rate constant for a reaction rate based on per unit volume s-1
(for n=1)
kz Thermal dispersion coefficient J m-1
K-1
s-1
KI Gas interchange coefficient between bubble and emulsion
phases
m3
g m-3
bed s-1
KIb Gas interchange coefficient between bubble and cloud phases m3
g m-3
bed s-1
KIe Gas interchange coefficient between cloud and emulsion
phases
m3
g m-3
bed s-1
l Film or slab length m
L Reactor length; liquid molar flow rate m; mol s-1
m Total solids mass kg
g Mass flow rate of gas kg s-1
s Mass flow rate of solid kg s-1
Mm Molecular weight g mol-1
n Reaction order -
nA Molar flow rate of A mol s-1
nA0 Feed molar flow rate of A mol s-1
NA Molar flux of A mol m-2
s-1
Nc Number of components -
NT Total molar flux mol m-2
s-1
Nu Nusselt number -
P Total pressure Pa
pA Partial pressure of A Pa
pAi Partial pressure of A at gas-liquid interface Pa
PAs Partial pressure of A within catalyst Pa
PsAs Partial pressure of A at catalyst / solid interface Pa
Pe Peclet number -
Pr Prandtl number -
Q Heat flux J m-2
s-1
r Position coordinate within the pellet m
rA Rate of consumption of A mol m-2
s s-1
; mol m-3
s-1
rAs Rate of consumption of A within catalyst mol m-2
s s-1
rsAs Rate of consumption of A at catalyst / solid surface mol m
-2s s
-1
r’A Rate of consumption of A mol kg
-1s s
-1
Page 89 of 90
c Rate of coke formation kgcoke kgcatalyst-1
s-1
rp Pellet radius m
R Universal gas constant; radius of particle core J mol-1
K-1
; m
RA Effective reaction rate based on resistance in series model mol m-3
s-1
Re, Rep Reynolds number; Re based on particle diameter -
Sc Schmidt number -
Sh Sherwood number -
t Time S
T Temperature K
Tc Coolant temperature K
Ts Temperature within catalyst K
Tss Temperature at catalyst / solid surface K
u Superficial fluid velocity m s-1
ub Bubble velocity m s-1
ubr Bubble velocity under non-flow conditions m s-1
ue Emulsion gas velocity (interstitial) m s-1
u Superficial fluid velocity m s-1
umf Minimum fluidisation velocity (superficial) m s-1
u0 Superficial feed fluid velocity m s-1
ut Terminal velocity m s-1
U Overall heat transfer coefficient J m-2
s-1
v0 Feed volumetric flowrate m3 s
-1
V Reactor volume m3
Vcore Volume of particle core m3
VL Liquid volume m3
VL’ Liquid volume neglecting liquid film m3
Vp Pellet volume m3
V Reactor volume m3
x Liquid phase mole fraction -
X Degree of conversion -
y Length coordinate in liquid film m
yA Mole fraction of A -
yAs Mole fraction of A within catalyst -
ysAs Mole fraction of A at catalyst / solid interface -
yfA Film factor -
yL Depth of liquid film m
z Length coordinate m
Greek
Constant
See equation (13c), chapter 2 -
r Heat of reaction J mol-1
Voidage -
b Bed void fraction -
g Gas fraction per unit volume of liquid and gas -
p Intraparticle void fraction -
Thiele modulus -
Weisz-Prater modulus -
Catalyst deactivation factor based on the reaction of A -
Hatta number -
Intraparticle effectiveness factor -
Page 90 of 90
G Global (intraparticle + external film) effectiveness factor -
Power law coefficient -
Gas viscosity kg m-1
s-1
= Pa s
Fluid thermal conductivity; mean free path W m-1
K-1
; m
b Bulk density kg m-3
bed
B Moles of B (e.g. coke) per unit volume of pellet mol m-3
pellet
f Fluid density kg m-3
fluid
g Gas density kg m-3
gas
p, s Pellet/ solid density kg m-3
pellet
Space / residence time s
p Tortuosity factor -
Subscript
0 Inlet/initial condition
B Bed
b Bubble phase
c Cloud phase; coke; complete
e, eg Emulsion phase; emulsion phase gas
f Fluid phase; fluidised bed
g Gas phase
G, L Gas / liquid phase
i Component identifier; interface
in, out Inlet / outlet streams
mf Minimum fluidisation
p Pellet phase
s Solid; surface
Superscript
s Pellet surface
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