Reaction Engineering II Script V4_1

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


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!


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.


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.


'

0 vA

A aNdV

dyG (15)


ALvgvyAA ryaaN

dV

dxL

L

)1( '' (16)


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)


Eqn.(15) is again applicable:

'

0 vA

A aNdV

dyG (15)


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))


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

Reaction Engineering II Script V4_1

Documents

sa ssas sc

heat transfer

internal heat

chemical reaction

mk kcgt

1st order

fluidised

gas phase