DY INA IN II CS F AINNý EXOTHERMIC REAM, 10 IN

IN A FIXED BED CATALYTIC REA"TOR - -.. --L" -11! f

by

Richard Steven Jordan, B. E. (Hons. )

April, 1977

A thesis submitted for the degree of, Doctor

of Philosophy of the University of London

and for the Diploma of Ij-. iper-', L-al College.

Department of Chemical Engineering

and Chemical Technology,

Imperial College,

London, S. W. 7.

To My Family

ACKN0W1,1 '1 D G. N'r S

I wish to extend my sincere gratitude to Dr. L.

Korshenbaum for his help and guidance, and especially

his constructive criticisms, throughout the entire course

of the research undertaken.

Thank you to the glass blowers for many hours of

patient work, to the technicians of the electronic and

engineering workshops, to Mr. A. Harrup and Mr. W. Mencer,

to the stores personnel, and to Mr. C. Birmingham for

solution of the many computer programming problems encount-

ered. My lab mates deserve a mention for suggesting many

ideas and for maintaining safe and congenial working

conditions.

A special thank you to Mr. J. Maggs of the Chemical

Engineering departmental workshop for his perseverance

and excellent workmanship in constructing many difficult

experimental rigs until the final operational rig was

achieved.

My love to my wife for her patience and forbearance,

and to my whole family, especially my late mother for

encouragement in all my educational endeavours.

Financial support to make this research possible was

provided by the Chemistry Division of the Department of

Scientific and Industrial Research, Petone, New Zealand.

LIST OF CONTEINTS

Abstract 1

1 Introduction 3

2 Literature Review 5

2.1 Kinetic Review 5

2.2 Mathematical Modelling 19

2.3 Reactor Dynamics 26

2.4 Conductivity Correlations 30

2. S Wall Heat Transfer Coefficients 34

2.6 Heat and Mass Transfer Correlations 39

2.7 Effective Radial Diffusivity 43

3 Experimental Method 44

4 Results and Discussion 57

4.1 Experimental Conditions 57

4.2 Product Distribution 60

4.3 Wrong Way Behaviour 63

4.4 Steady State Modelling 75

4.5 Dynamic Modelling 98

5 Conclusions 128

Suggestions for Further Work 132

Bibliography 134

Appendices:

I Reactor Equations 140

II Orthogonal Collocation Method 145

III Application of the Collocation Method 156

IV Collocation Matrices 168

v List of Symbols

vi Heats of Reaction

170

173

VII Experimental Data 174

VIII Temperature Profile of the Reactor Wall 198

Ix Calculation of the Order of Reaction 199

x Normalised Plot of Reaction Models 200

-I. -

AkSTRAC, T

An investigation was Lujcjoj-ta', ýejj to 111odo t1jo Oxperi- ýI .1

mental results obtainod -IL'or a higj, ýlv ý-,,, o)fLjj(. ýj-jjjic 1,0ýjction

carried out in a non-adiabatic) non-isothormal, tubular

catalytic reactor undor both steady state and dynamic

conditions.

The system studied was the partial oxidation of

o-xylei,. e to plithalic anhydride over commercial vanadium

pentoxide catalyst. The inlet o-xylene concentration

in the air stream was varied from 0.31 to 0.73 mole%

(1.26 x10 -4 to 3.0 x 10-4 Kmo 10IM3 at 2S'C) with the

reactor bath temperature ranging from 340 to 355'C. C> Zý

The orthogonal collocation method was extensively

used -to solve the two-dimensional homogeneous and hetero-

geneous steady state models and to facilitate estimation

of the relevant parameters by implementation of a non-

linear regression analysis. An anomalous effect was found 0

over the first quarter of the reactor. Increasing the

o-xylene concentration, under otherwise constant conditions,

resulted in a decrease of the steady state temperature and

hence the rate of heat generation., This implies that the

apparent order of reaction over this region was negative.

Consequently the system could not be consistently modelled

using a normal first ord--r rate of reaction expression.

The simplest feasible model to demonstrate the anomalous

behaviour was derived by assu-, ming that increasing the

-2-

o-xylciie Concentration callsod -reversible deactivation of

the vallactia catillyst. 'tilis rosulted In a ncgzitývo orclor

of roaction at high o-xylciio conccntratýons .

Tho offect of film hoat ind inass transfer was unimpor-

tant, up to 410'C, the maximum temperature modelled. The

same degree of accuracy therefore was achieved for both

the homogeneous and heterogeneous models.

Based on the steady state parameters the reaction

system was solved for the dynamic case of a step change

in the total gas flow rate and/or the o-xylene concen-

tration. The continuum mathematical model of partial

differential equations was solved by application of the

orthogonal collocation method in all three dimensions.

The model demonstrated all the trends of the dynamic

experimental results, including for an increase in the

inlet o-xylene concentration, the initial small amount of

right way behaviour in the axial temperature profile for

the first part of the reactor; this was followed by the

overall wrong way behaviour to the new steady state.

-3-

ch'i ter 1 INT-RODIR"NON

With the ever increasing dcmand for petroleum fuel

and plastics the major source of basic raw material for

the chemical industries of the ijorld is undoubtedly crude

oil and to a lesser extent natural gas. The catalytic

side of the oil refining industry has developed massively

since the first commercial cracking was carried out in

1936. Now heterogeneous catalysis is one of the main

technologies used in the industry with reactions such as

isomerisation, reforming, oxidation, hydrogenation and

alkylation being possible. Such diverse products as

alcohol and rubber can now be produced synthetically.

For a long time, the design of all but isothermal

and the simplest adiabatic tubular reactors was impossible

without scale-up from extensive pilot plant testing of

the process. With the development of simple mixing cell

models, the availability of heat transfer data and

especially the advent of computers, the modelling of

non-isothermal, non-adiabatic catalytic reactors became

feasible. As computer facilities improved, solution of

the relevant partial differential equations by continuum

models and methods has generally superseded the mixing

cell model. With improved integration algorithms progress-

ing through the predictor-corrector, Runge-Kutta, Crank-

Nicholson methods and now the orthogonal collocation

method, it is possible to solve extremely complex reaction

models even under dynamic conditions. The solutions

however., are still very much dependent on the accuracy

of the heat and mass transfer correlations as well as the

-4-

activity of the catalyst under actuaL openitingy conditions.

The aim of this study was to investiggato oxporinion-

tally and theoretically a tubular exothermic catalytic

reactor under both steady state and dynamic conditions.

This was ultimately for use as a computer controlled

process in the pilot plant laboratory of the department

of Chemical Engineering. Mathematical modelling was to

be carried out using the orthogonal collocation method

for integration radially, axially, and in the time direct-

ion, if possible, to determine the suitability of this

algorithm.

The partial oxidation of o-xylene to phthalic anhy-

dride over a vanadia catalyst was chosen, as being repre-

sentative of modern catalysis technology. Other important

considerations were the ease of feeding the reactantsý

only slow deactivation of the catalyst and a large amount

of literature on the reaction at steady state conditions.

-5-

i, i, ui-'ATUl'E RIN EW II zl_lLt ýý r2

Tho modc1ling of a catalytic roactor involves tho

application of data on chemical kinetics, effective

axial and radial diffusivity, vizill heat transfer cocffic-

ients, effective thermal conductivity and heat and mass

transfer coefficients. For the specific case of the

oxidation of o-xylene on vanadia, the catalyst pellets

are only surface coated and intraparticle effects are not

important.

2.1 Kinetic Review

There are two main types of catalyst used for the

partial oxidation of o-xylene to phthalic anhydride.

The German type which has an optimum yield of appTox-

imately 75 mole% at reaction temperatures of 3SO-4500C.

and the American type which has a slightly lower maximum

yield of 70 mole%, however the rate of reaction is much

faster as temperatures of 500-6000C are used. The

catalysts are normally only surface coated with the

support being silica in the first case and alundum or

carborundum in the second. The German type catalyst is

often promoted by the addition of K2 O/K 2 so 4'

The overall rate of reaction most commonly used is

the power law

=kCnm Rr r o,

Cr

-6-

whore norilially MIC'n L11C Oxida Lion i; Cýl I-V i Cd oil t

using air, the OXYSCR concentration, in the most extreme

case for I mole! inlet o-xylene conccntrýition, would

only decrease from 21 to 17%. Therefore the effect of

the oxygen partial pressure has usually boon ignored, so

that n=0 in the above expression.

Mars and Krevelon (3) proposed the rodox model

where the lattice oxygen of the catalyst is included 0 in the reaction steps according to:

aromatic + oxidised k1_

oxidation + reduced compound catalyst products catalyst

reduced + oxygen k2,.

oxidised catalyst catalyst

with the resulting rate of reaction

kCmkC 2 _o,

1r k2cm+ nk 1c 02

where n is the stoichiometric number of oxygen molecules

for the reaction of one aromatic molecule. Mars and van

Krevelen found a constant value for the catalyst oxidation

rate constant k2 for the oxidation of benzene., naphthalene

and anthracene, with m=l.

Shelstad, Downie and Graydon (4) produced the steady

state adsorption model (SSANI), by assuming that only

oxygen is adsorbed onto the catalyst with a negligible 0

amount of desorption. The reaction takes place between.

the adsorbed oxygen and the gas phase aromatic with the

coilse (III cI It I, ii te, ot rcýlc t ioll

R k-

il 0, r 17

rkC+ nkr Cr

where ka is the oxygen adsorpt-Jon rate constant

k is tho roaction rate constant.

Hughes and Adams (S) studying the vapour phase

oxidation of plithalic anhydride postulated yet another

model derived froi-, i the following set of equations:

k PA + VO --!

L- VOPA k_ I

VOPA 2V+ OPA

V+02 fast, VO

with the rate of reaction

R k2 k, Cr

rk2+ k_ 1+k1 Cr

which simplifies to

RACr r1+BC

r

Originally, the model was developed using lattice.

oxygen but it could equally apply to adsorbed oxygen.

As the forms of the rate expression for the redox

model with m=l and the SSAM are identical., discriminat-

ion between the models cannot be done by simple kinetic

studies. The Hughes and Adams form can be distinguished

by varying the oxygen partial pressure, but when this is

-8-

- rate expressions are Constant, "Al 011-00 lie terogencous

idciitical. III the CýKtrcmc Case) those models Call give

all order of reaction foi- the aromatic 1"11-y-ing froin () to

l as the concentration of the aroinatic decreases to zero.

Cameron., Farl-as and Litz (6) found exchange of

isotopic oxygen between the gas phase and the vanadia

catalyst over all the temperature range that they inves-

tigated, 445 to SS40C. The exchange rate was indepen-

dent of the oxygen partial pressure and had an activat-

ion energy of 1.88 x 108 KJ/mole. -They proposed either

the dissociation of oxygen molecules or the loosening

of V-0 bonds was involved. Margolis (7) supported

this, adding that the exchange rate only becomes obser-

vable above 4350C.

Clark and Berets (8) studying the electronic behav-

iour of vanadium pentoxide, state that the presence of

an electron donating agent such as ethylene or o-xylene

prevents the formation of the oxygen barrier -which would

normally form in an atmosphere of oxygen alone. The

exchange of oxygen at the surface during a catalytic

reaction should then be much faster than has previously

been indicated by measurement of oxygen absorption

alone.

Simard and co-workers (9) demonstrated that there

is sufficient lattice oxygen present in the catalyst to

-9-

react with o-xylene at 460'C, oý, cr freshly prepared

vzInadia, in a stream of nitrogcn, restAting in a (Iccl'casc

in the oxidation state of the catalyst. This shows that

chemically bound oxygen can ta-kc place in the reaction.

For oxygen adsorption by slightly reduced catalyst,

Simard found a square root dependence on the oxygen part-

ial pressure. Roiter (10) however, found no evidence of

exchange of isotopic oxygen for the oxidation of naphth-

alene in the temperature range 340-390'C. Boroskov (11)

states that adding promoters to the catalyst can greatly

increase the catalytic activity in relation to the

exchange of molecular oxygen, for example, adding 10

mole% ceasium sulphate, which is a stronger promoter

than potassium sulphate, results in a hundredfold increase

in the rate of exchange.

Out of all this conflicting evidence, as well as the

large variations possible in the formulation of a

promoted catalyst, it is best to assume both adsorbed and

chemically bound oxygen are active species in the partial

oxidation of o-xylene to phthalic anhydride. The general

forms of the models so far produced cannot make this

distinction, therefore this assumption is in order.

Shelstad (4) and Juusola and co-workers (12),

studied the reaction of naphthalene to phthalic anhydride,

and o-xylene to o-tolualdehyde respectively. Both

workers tried Langmuir-Hinshelwood models where both the

aromatic and oxygen molecules are reversibly adsorbed

onto the catalyst.

-]-0-

k 2- cCr

02 2d

For Sholstad's work the Langmuir-11insholivood modol

gave a slightly bettor fit, however the extra complexity

of the i-. qodel was not considered to be practicable and

both groups of workers settled for the SSAM. Juusola

also rejected various other models including the SSAM

with a square root dependence of oxygen partial pressure

caused by the supposed dissociative adsorption of

oxygen by the catalyst.

Juusola modified the SSAM to

Rka Co, k, Cr

rN kC+ZnkC a o2.

I=IIIr

where the summation term is for all the reacting aromatic

species, especially those involved in series reactions.

This idea can conceivably be extended to the bimolecular

Langmuir-Hinshelwood reaction.

k k, k2.. Co. C

N2 (1 +kC+Zkc

01 1=1 1+1 rl

A multitude of reaction schemes exist in the litera-

ture for the reaction of o-xylene over vanadia type cata-

lysts. These are well described by Ellis (13). Only the

simplest schemes will be presented here.

The basic model.. used by many workers, assumes two

parallel reactions, first order with respect to o-xylene

and both having the same activation energy.

-11-

O-Xvlollc plitlialic alillydride

carbon oxides

This, explains the constant selectivity of tlio system over

a wide range of tomporaturcs found by most workers.

A slightly more complex model, a triangular scheme of

first order reactions, was developed by Fropient (14) as

being fairly representative of the gas phase oxidation of

o-xylene over vanadium pentoxide.

o-xylene plithalic anhydride

carbon

I

oxides

Aliev (1S) and Pant and Chanda (16) both postulated

identical schemes involving only parallel reactions.

phthalic anhydride

o-xylene ). maleic anhydride

carbon oxides

Pant and Chanda developed the rate of reaction expression

in the form of a summation of the SSAM derived by Juusola.

3k1

k2+k3)K4c

C+7. S k2c

c tr

+ 10.5 k- C 3r

Where the numbers 3,7.5 and 10.5 are the stoichiometric

amounts of oxygen necessary for the respective reaction.

The reaction expression however, reduces to the normal

SSAM form, as the activation energy Of the three reactions

are the same and there are no series reactions present

in the scheme.

Dixon, Longfield and Emmet (17) reporting the un-

published work of Simard give the following network.

-12-

o-to I ua I do hydo

0-XY10110 plithalic atihydride

maleic anhydride

carbon oxides

Carra and Boltram (18) found no traces of o-tolualde-

hyde and concluded that their data could be represented

by the following scheme with no direct o-xylene combustion.

", plithalic anhydride ""'ýcarbon

oxides o-xylene-

maleic anhydride

The following network was proposed by Herten and

Froment (19).

o-tolualdeh-de carbon oxides y

O-xylýýPne

ph,,, alide---------)-phthalic anhydride

The overall rate of o-xylene disappearance was modelled

equally well by either the power law or the Mars and van Krevelen form with the exponent m equal to 0.6.

A scheme similar to that of Herten and Froment was

devised by Vanhove and Blanchard (20), with no direct

reaction of o-xylene to either phthalide or to phthalic

anhydride.

Boag and co-workers (21) also proposed a network

similar to that of Herten and Froment, but with no further

oxidation of phthalide. They add that as mutual inhibition

effects are known to be important for this system, as demonstrated by Lubarskii and co-workers (22), it is

possible that the product distribution may not be a unique

-13-

function of convol'sion. me resLilts of workers therefore

who have oxidi-sed various intonnediates ii-e not ne. ce-s-sarily

correct especially as regards the rate of re. action. The

results of Boag were derived by a statistical approach

from the reaction of o-xyleno alone, although only a

maximum plithalic anhydride selectivity of 45% was achieved.

Novella and Bennloch (23,24,2S) deduced the model

o-xylene". -/"

o-tolualdehyde plithalic anhydride

"maleic anhydride--->-carbcIL oxides

Further complex schemes for the oxidation of o-

xylene or the reaction of various intermediates have

been developed by Lyubarskii and co-workers (22), Bern-

adini and Ramacchi (26) and Vrbaski and Mattheivs (27,28)

with even more intermediates namely o-methyl benzaldehyde,

o-toluic acid, per-o-toluic acid and benzoic acid.

Many of the reaction schemes show the further

oxidation of phthalic anhydride. Hughes and Adams (5)

and Ioffe and Sherman (29) agreed that phthalic anhydride

underwent direct combustion to -carbon oxides and partial

oxidation to maleic anhydride. Watt (30) showed that

this was important in his kinetic studies on the reaction

of o-xylene using a spinning basket reactor, due to the

long residence times used. Normally however, even in

long packed bed reactors this effect is not important

as contact times are of the order of 1 second.

Ignoring the results where there have been, obvious

-1.4-

mass transfer limitatioiis on the rate of reaction the

activation energy for the o-xylene reaction *Ls in the

range 0.92 x 10 8 to 1.17 x 108 K, J/Kuiole, wi-th most oxygen

adsorption or kinetic activation energies bctwcoll 1.05

x 10 8

and 1.25 x 10 8 KJ/Kmole.

Some workers comment on a reduction in the rate of

reaction at high aromatic concentrations. In most cases

the full catalytic activity was restored by decreasing

the aromatic concentration or by passage of an air stream

over the catalyst. Simard (9) found that the vanadia

catalyst reversibly deactivated at an o-xylene concentrat-

ion of 3.3 mole'o, however he did not investigate the range

between 1.1 and 3.3%. Ross and Calderbank (31) similarly

had a loss of catalytic activity at naphthalene concentra-

tions greater than 1.0 mole% at 3S3'C as did Calderbank

and co-workers (32) at o-xylene concentrations greater 1.6%.

Calderbank (33) using a spinning basket catalytic reactor,

found that the order of the rate of reaction, initially

nearly one with respect to o-xylene, decreased to zero

in the concentration range 0.6 to 1.1% at 400'C and then

became negative at higher concentrations. A good fit for

the data at 400'C can be obtained by a simplified form of

the Langmuir-Hinshelwood expression, where the oxygen

partial pressure is nearly constant and can therefore be

included in the rate constants

kC Rr=

(1 +rkrc r)

where kl= 1.35 x 10 2

1.71 Kmole/(Kg cat). s

-is-

Sholstad (4) and Jut-isola (11-1) at high concontratiolis

of the aromatics they studied, showed that Lhe order of

the reaction decreased to zero, or may even have become

negative, in the second study, when naphthalene and o-

xylone concentrations wore above 1.0% and 1.21, respect-

ively. As mentioned previously however, they rejected

the bimolecular Langmuir-Hinsholwood model which would

explain this effect in favour of the simpler SSAM.

Lyubarskii and co-workers (22) studying the oxidation

of o-xylene to phthalic anhydride and the reaction of

various intermediates, found that the oxidation of maleic

anhydride proceded as a simple first order reaction.

When phthalic anhydride was present however, the reaction

was inhibited by the adsorption of phthalic anhydride on

the catalyst.

kC Rr ma

r1+b Cpa

where b is an adsorption rate constant. For a complete

description of the oxidation of o-tolualdehyde, an

expression similar to the modified form of the SSAM,

proposed by Juusola, was necessary where

kr Ct ol

1 + b1 c to, + b2 c ma + b3 c

pa

The adsorbtion of phthalic and maleic anhydrides however,

was much weaker than that of the o-tolualdehyde so that

these two terms in the denominator could be ignored

resulting in the rate of reaction equation originally

derived by Hughes and Adams (S).

-16-

Roiter and co-workors (34) claimed that the iiaphth-

oquinone concentration, a by-product in the reaction of

naphthalene to phthalic anhydrido, had a retarding, effect

on its own rate of formation from the naphthalene and

proposed the empirical rate of reaction expression;

In 02

c06 (1 +kC nq 2n

They unsuccessfully tried modelling the system with a

series reaction type and a unimolecular Langmuir-Hinshol-

wood type, however they do not report trying a bimolec-

ular Langmuir-Hinshelwood model.

The decrease in the order and the rate of reaction

at high aromatic concentrations is caused either by a

preferential adsorption on the catalyst of the hydro-

carbons present, resulting in the exclusion of the oxygen,

which can be explained by the Langmuir-Hinshelwood model,

or by a reduction in the oxidation state of some or all of

the catalyst present resulting in a decrease in activity.

Conversely Hughes and Adams studying the reaction of

phthalic anhydride found an increase in the rate of react-

ion when the oxygen concentration was less than 10%;

however above this value the rate of reaction was found

to be independent of the oxygen concentration. This was

explained by the vanadia catalyst being reduced to a

lower oxidation state which was more active for the oxid-

ation of phthalic anhydride.

-17-

Little work has I)COII I)LI10lislied on the active ox-

idation state of the výmadin. catýilyst. SlAnard (9) states

that for o-xyleiio rc"ct'o" V205 ýllld V204.34 ý"ro, "c: t've,

ivhilc Volf-'son (35) Ivork-ing on naphthalono oxidation

claims that only V204 is active. Schaefer (36) using

benzene agrees that V204 is active and adds that V2 0 4.34 is slightly active. Farkas (37) claims that this

variation in active catalyst species is due to the differ-

ent nature of the reactions taking place. There is no

ring breaking in the o-xyleno reaction and for th. e

naphthalene and benzene reactions, the rings break in

a dissimilar way.

Ioffe (38) states that the phase composition of the

operating catalyst is completely determined by kinetic

rather than thermodynamic parameters and postulates the

reactions

MO +02 mo 2

k r mo 2+R MO + RO

Reaction of the aromatic can also occur with adsorbed

oxygen on the reduced MO phase giving the overall rate

of reaction expression as the sum of two terms.

Rr=k (1 - ý) e 02

f(C r)+krE

f`(C r)

where ý is the fraction of N102 present

e0 is the fractional surface covering of adsorbed

oxygen

A steady state occurs when the two phases are in kinetic

equilibrium according to:

fl(Crý k2 E2 (Co

2)

-18-

IOffC and LyUbarskii (37) suidied the partial

oxidation of benzene -to malcic mliydride over a vaiiadium

pontoxide catalyst in a recycling difforontiil reactor.

It was found that the presence of malcic anhydric1o retarded

the reaction and that at oxygen partial pressures loss

than O. latm the reaction was second order in oxygen while

at higher concentrations the reaction was zero order.

For the theoretical analysis the following assumpt-

ions were made:

1 The major portion of benzene oxidation was by the

oxygen adsorbed on the surface of the V20 51

2A minor portion of benzene oxidation was by the oxygen

of the V205 lattice.

3 Maleic anhydride is more strongly adsorbed onto the

surface of the V205 than oxygen.

4 The benzene oxidation rate by adsorbed oxygen is much

greater than that by the lattice oxygen.

These assumptions enabled the second term of Ioffe's

original rate of reaction expression to be neglected. By

extending the theory to include the diffusion of oxygen

vacancies through the catalyst the following simplified

equation results.

kf (C )c bz r 02 1+ ka Cma

where

f(C kC2 at low oxygen concentrations 02 1 02

f(C 02

k2 at high oxygen concentrations

-19-

which v., as zippro. ximated hy:

C ct

Rr k' : F(C 02 cbza

ind

in a

This model demonstrates all effects that were

reported, including the retardation of the reaction by the

presence of maleic anhydride, however at no point was a

negative order of reaction found.

2.2 Mathematical Modellin

Many models have been formulated to describe the

characteristics, and facilitate the design, of fixed bed

catalytic reactors. The partial oxidation of o--xylene

to phthalic anhydride in a non-isothermal, non-adiabatic

tubular catalytic reactor at a reaction rate comparable

to industrial reactors, produces a large radial temper-

ature variation. This is due to the high exothermic

heat of reaction and the need for continuous heat transfer

through the wall to avoid excessive bed temperatures

which may decrease or destroy the activity of the catalyst.

Froment (2,40) has shown, for the o-xylene reaction.,

that there is a considerable difference in the theoretical

temperature profiles between calculations carried out

using one and two dimensional single phase models. Conse-

quently one dimensional models have not been used in this

study, as the reactor used for the experiments was of semi- industrial size.

-20-

carborry inýl Vv'cnJc. I (41) conclu(Iccl Lhat axial

dýsporsion of licat and mass iýas iiii1iort; int only Col- shalloll

bods and is negligible foi- bc(I lcnoths groator than SO

pellet diameters. Axial dispersion of both licat and mass

effects have initially been ignored in this study as the I

bed longth was 120 pellet diameters.

In the studies of reverse operation of catalytic

reactors carried out by Berty and co-workers (42) and

Venk-atachalam and co-workers (43) it is obvious that

axial dispersion of heat is important to propagate the.

reaction zone towards the inlet of the bed. The temp-

erature however, in these cases was normally much greater

than 400'C., therefore radiation effects would be far more

important than for the o-xylene reaction, as well as the

conductivity of the packed beds being much greater than

that of the silica catalyst used in these experiments.

In order to understand the literature on the mathemat-

ical modelling, the partial differential equations govern- Z>

ing the system studied will be presented here. For a detail-

ed derivation of the various dimensionless equations pre-

sented and the symbols used, see appendices I and V respec-

tively.

Two types of steady-state two dimensional models

have been used, the simplest being the homogeneous or

single phase case where temperatures and concentrations at

the pellet surface are assumed to be equal to the bulk

gas temperature and concentration. This leads to the

following dimensionless partial differential equations for

energy and mass balances.

-21-

OT D 9T) g(T, C) +

AC LAYC 1+ DC 3zr

0 with boundary conditions

A- Bi(T -T at r=1 ar C

DC 0 at r=1 ar

TC1 at z=I

As the catalyst pellets used were only surface coated with

the active agents, intraparticle effects are eliminated.

The other two dimensional model considered was the

heterogeneous or two phase case where the temperatures

and concentrations at the surface of the pellet are not

equal to the bulk gas temperatures and concentrations,

but are governed by these dimensionless equations:

Energy Balance

g(T P 'C P St ht (T

P- TO

ýTb St (T -T+a (r

3Tb)

3z ht p b) r 3r 3r

Mass Balance

-g (TP, C

I? )

st (C -c) co mt pb

acb st (C -cD (r

ýCb)

az mt p b) ar 3r

with boundary conditions

ýTb - Bi(T -r at r= ar bc

DC b0 at r= 3r

TbCb at z=

-22-

where the subscripts p and 1) re I -Cc, to I)OI-IoL and bulk

conditions respectively. No solid plizisc lioza transfcr has

been directly considered i. e. no intripellot, pcLlct. to

pellet, or pellet to wall, only solid to gas, gas phase

and gas to wall licat transfer. The solid effects however,

are indirectly included in the overall effective thermal

conductivity and what is really an effective wall heat

transfer coefficient. This is a contradiction as the model

is therefore not a complete heterogeneous model. Consid-

ering solid heat transfer separately would mean the

introduction of an extra effective conductivity and wall

heat transfer coefficient. Most literature correlations

howeverl consider all the heat transfer effects to be

lumped into a single overall effective conductivity and

a wall heat transfer coefficient. Even for these param-

eters there is much variation (see chapters 2.4 and 2. S)

so that the addition of two extra parameters cannot be

justified.

The packed bed was assumed to operate at atmospheric

pressure with negligible pressure drop through the bed.

Plug flow has been assumed throughout all the models in

this study. No radial variation in velocity has been

considered as this appreciably complicates the equations.

There is however, partial compensation for this effect, as

it is inherent in the experimentally obtained values of

the effective thermal conductivity and wall heat transfer

coefficient. Neither of these parameters have been

correlated by any workers for radial variation of the

flow rate.

-23-

As axial dispersion lias been neglected for the inodcli-

ing, there are no axial second derivatives. This reduces

the partial difforentiat equations for both models consid-

ered from elliptic to the more readily solveablo parabolic

form.

These partial differential equations have commonly

been solved by finite difference methods, especially

that of Crank-Nicolson,, where first and second partial

derivatives are approximated by a form of the solution

at two and three points respectively, i. e. linear and

parabolic approximations respectively. Therefore at

best, the solution radially and axially, is a series of

interconnected parabolas and straight lines respectively.

Alternatively the orthogonal collocation method can be

used where the whole set of radial points are fitted by

a symmetrical 2Nth order polynomial. The number of

internal collocation points is N, the positions of these

being selected according to the procedures originally

defined by Villadsen and Stewart (44). This is analagous

to Gaussian quadrature, which is exact for a polynomial

of order 2N. For a full derivation of the orthogonal

collocation method and its application, see appendices

II and III respectively. The first and second partial

derivatives are then obtained by multiplying a weighting

vector by the solution vector so that both derivatives are

approximated by the respective derivative of the poly-

nomial solution. This gives rise to more accurate

solutions than the Crank-Nicolson method, meaning fewer

solution points are necessary for comparable accuracy,

-24 -

tllCfCfO 1'0 (1 CCrCýIS 'L R ýl C0111, I) t It C 1* -1; LO I*C, (I LIL I* ellic"ll L's

11"'llell the Collocation Illethod is applied radially, the

radial dorivatives arc roplaced by colistmit tcrins and the

partial difEcrontial oquations arc roduced to tho ordinary

differential form. Normally thesc cquations havo been

integratod by finito difforonce mothods o. g. Rungo-Kutta

(4S) or Euler (46) , but it is possible to apply the oVth0g-

onal collocation method again axially in steps (47) The

solutions axially are approximated by (N+l)th order

polynomials and for the dynamic case, this method can be

applied yet again in the time direction.

For the dynamic model, the dimensionless partial

differential equations are:

Energy Balance 3T

g(TP)C St (T -T -p- p ht p b) ýt

3T b St (T -T b) +3 (r 3Tb)

3z ht p 3r ar

Mass Balance

g(T C pp St (C -C CO mt p b)

3C bIý 3C b 3C b = St

mt (C

p-C b) +Y- -(r -) -C 3z r ar ar at

Where the boundary conditions are the same as for the

heterogeneous case and the temperature and concentration

profiles are defined at time t=0.

A packed bed reactor has two characteristics response

times, one for concentration changes of the order of a

few seconds, or residence times, and the other for the

-2S-

bed temperature changos of the ordor of a few minutes.

Due to the difference in magnitude of the two

responses, the two transients can be scparatod into two

models to simplify the calculations. The first for

concentration changes with pseudo-constant temperature

and the second for the temperature transients with the

concentration going through a series of pseudo-steady

states following the temperature profile. The equations

governing the concentration transient, after a radial

orthogonal collocation substitution, are most easily

solved by the method of characteristics. Temperature

variations along the bed can be included by bending the

characteristic lines as the solution progresses. The

orthogonal collocation method could be used axially and

in the time direction although the computational time

may be excessive. It cannot however accurately handle

a discontinuity in a variable e. g. when the concentration

undergoes a step change.

For the temperature transient, the original three

dimensional concentration partial differential equation is

replaced by the steady state heterogeneous concentration

equation, resulting in all parabolic equations which can

be solved by finite difference methods, or a mixed or

complete orthogonal collocation method. As large time

steps (much greater than the residence time) can be

taken, any concentration discontinuity can be overstepped

and does not affect the orthogonal collocation method.

-26-

2.3 Ronctor Dviiiiiiics

Very little work has been reported for cojiibincd oxper-

imontal and thoeretical investigations of realistic pilot

plant exothermic tubular reactors under dynamic conditions.

Most workers have dealt with reactors that are nearly

adiabatic or have studied the simple reaction of hydrogen

and oxygen over a supported platinum catalyst.

Simulations of packed bed reactors under dynamic

conditions., using a series of continuous stirred tank

reactors, have been developed by Mc Guire and Lapidus (48),

Vanderveen., Luss and Amundson (49) and Elnashaie and

Cresswell (50) where the size of each mixing cell is

approximately that of a catalyst pellet, or the void

between pellets. To obtain a converged steady state

solution it may be necessary to decrease the cell size

until it becomes meaningless so that a continuum model

based on the solution of the ordinary or partial differ-

ential equations describing the reactor, is more approp-

riate. In the model of Mc Guire and Lapidus, where multiple

steady states were possible, as each row of mixing cells

moved from the lower to the higher steady state, a discrete

heat wave passed along the reactor. This effect has never

been found experimentally and shows a spurious result in

using the mixing cell model.

Berty and co-workers (42) used both the discrete

mixing cell and continuum models, the latter being solved by a predictor-corrector method to model their experiments

on the backward migration of the reaction zone in an

2' 7

adiabzitic reactor. Although the fit obtiincd for tho

exporimontal data was poor, the trends wore demonstrated.

Lui and Amundson (SI, 52, S3), using continuum one-

dimensional models, integrated by the

istics, investigated -the stability of

adiabatic packed bed reactors. Heat

resistances were lumped at the pellet

effectiveness factor being applied to

particle reaction. From these models

state situations were demonstrated.

method of character-

adiabatic and non-

and mass transfer

surface., with an

account for intra-

multiple steady

Hansen and Jorgensen (54) obtained good agreement

between their theoretical and experimental results for the

hydrogen/oxygen reaction on non-porous pellets in an adia-

batic reactor although they used a homogeneous one-dimen-

sional model with all the heat transfer effects lumped

into a total Peclet number. The model was solved by

applying the orthogonal collocation method axially to

reduce the partial differential equations to ordinary

differential equations which were then integrated by a 4th

order Runge Kutta method in the time direction. This paper

was a continuation of the earlier theoretical simulations

of Hansen (SS, 56). In these papers however, heterogeneous

one-dimensional adiabatic models were used to describe the

behaviour of packed beds of porous catalyst pellets under

dynamic conditions. Hansen divided the solution of the

dynamic case into two parts. The fast response dynamic

concentration profiles were solved at constant bed temp-

erature by the application of the orthogonal collocation

-28-

me t1iod foi, tlic i, ntci'iia I I) c1 1c t 1)i, oC i1 0- I'o I loýýed by the LY

mctliod of cliaracterýstics zi-Kiallý, aiid a plýcdictor--Col'rcctor

method in the time direction. 'I'llo slolq response d), 11,111lic

temperature proCilos were solved by assuiiiing that the

concentration profiles went through a series of pseudo-

steady states, so that only the hoat balance equations

had to be solved in the time direction., and the concontrat-

ion profile at any time stop was only depondant on the

temperature profile and the inlet concentration at that

time. This was solved as previously by the orthogonal

collocation method and then by the Runge Kutta method

axially and a prcdictor-corrector method in the time direct-

ion. The response time of the fast concentration profiles

was found to be 2-3 seconds and the slow temperature pro-

file 1000-1500 seconds.

Hoiberg, Lyche and Foss (S7) studied the dynamics of

the hydrog'en/oxygen reaction under non-isothermal and non-

adiabatic conditions demonstrating the classical initial

wrong way behaviour for a 20'C decrease in the inlet temp-

erature of 100'C. The frequency response of the reactor

was rigorously studied over a wide range of frequencies.

In all cases, the Crank-Nicolson method was used to solve

the differential equations.

Elnashie and Cresswell (50) and Karanth and Huahes 0 (58) modelled first order reactions for porous catalyst

pellets under dynamic conditions, demonstrating that for

porous pellets the response speed is proportional to the

Lewis number as is the overshoot, The Lewis number is

defined as:

-29-

pCD Lc p

k p

where Cp= spocific licat capacity of catalyst

Dp = diffusivity of reactant inside pellet

kP= pellet conductivity.

Karanth and Hughes investigated the dynamics of the

hydrogenation of toluene over a supported nickel catalyst

in a nearly adiabatic reactor. Consequently they used a

one-dimonsional model and utilised the limiting non-key

component effect to simplify the solution of the two phase

equations. Their model is in poor agreement with their

experimental results, but they do demonstrate the trend.

The transients that were considered were from no reaction

to final steady state which took approximately 50 minutes.

Unfortunately however, the inlet temperature varied over

this interval.

Stewart and Sorenson (59) modelled the oxidation of

o-xylene to phthalic anhydride on porous catalyst pellets,

based on Froment's (14) triangular scheme of first order

reactions under dynamic conditions., using a two-dimensional

heterogeneous model. This was solved by applying the

orthogonal collocation method to both the intraparticle

and radial profiles and finite difference expressions

axially and in the time direction. The collocation points for this model were calculated depending on the value of

various dimensionless groups at the boundary conditions

by formulas given. It is claimed that this allows good

accuracy to be achieved with fewer collocation points. The Newton Raphson form was used to quasi-linearise the

kinetic expressions. No experimental work was carried out.

-30-

2.4 Ef I cc tive Radial Con (I Lictivity Co I, re I a-t. j oils

The correlations in the literature vary greatly dopen-

ding mainly on the nature of the solid packing, whether

metallic or non-motallic, spherical or cylindrical. The

oxidation of o-xylene is unusual in that the reactor dia-

meter is normally 25mm with a pellet to tube diameter ratio

of around 0.2S to facilitate good heat transfer to limit 4: 1

the hot spot temperature. Most of the experimental work

on heat and mass transfer parameters has been carried out

for tube diameter ratios less than 0.2S. In this study a

reactor of diameter 15. S9mm with a diameter ratio of 0.268

was used.

Effective radial conductivity correlations normally

take the form

kk0 Re Pr

where the first term k0 /k is the contribution of the eg stagnant effective conductivity representing the effects

of both fluid and solid conductivities, particle size,

radiant heat transfer and void fraction. The second term

is the fluid flow contribution with the Reynolds number

based on the pellet diameter.

Kunii and Smith (60) derived the following theor-

etical equation for predicting the stagnant conductivities

in packed beds of unconsolidated particles:

k0

ke k9

rs ks

k9

-31-

where radiation botween voids is given by

0.19S2 1'+ 273 rv 2p

(' 100

radiation between surfaces by

h 0.19S2 I(T+273 3 rs 1+CI. -P 100

2(1-c) P

p= emissivity of the solid surface

ý=1, y=ý in most practical situations 3

ý is given by a graph in Kunii and Smith (60)

This equation was also quoted by Yagi and Kunii (59) who

showed that the fit was better than 20% for most experi-

mental data.

The graphs of Yagi and Kunii (61,62) and Hill and

Wilhelm (63), show that the variations of the stagnant

conductivity term ko/k due to radiation effects is neg- eg ligible over the temperature range 340-440'C for non-

metallic particles. Using the equation derived by Kunii

and Smith and assuming a solid emmissivity p=0.75,

recommended by Kunii and Furusawa (64), k0 /kg = 8.0. e

This compares favourably with the literature values in

fig 2.1, of around 10.0 for non-metallic packing. The

work of Gros and Bugarel (65) differs greatly, as they

experimentally obtained a very high conductivity and a

very low heat transfer coefficient. This is due to the

very high degree of correlation between these two param-

eters when they are calculated from the same experimental

data. Ignoring the work of Gros and Bugarel ý (the slope

of the lines) in fig 2.1 varies from 0.061 to 0.136 with

0.10 being an average value.

-32-

Yagi and Kunii (62) 111cl oll)ri(: Il 111ý1 potter (66) both

obtained values of ý for glass spheres for vat-ious cliainctor

ratios (D p

/D t)

(soo fig 2.2) showing that tho fluid flow

contribution docroases for increasing diametor ratio.

Yagi and Kunii show that this effect is more important for

cylinders and rashig rings, for which most of the literat-

ure correlations have been produced, and that the absolute

value of ý for spheres is less than for cylinders and other

shapes. Olbrich and Potter obtained their conductivity

correlations by assuming a very high value of the modified

Biot number (h wdp

/k eý2.12) so their overall effective

conductivity correlations are doubtful; however the effect

of Dp /D t should be reasonably accurate and does compare

favourably with Yagi and Kunii's results.

Extrapolating both Yagi and Kunii's and Olbrich and

Potter's data in fig 2.2 to a particle to tube diameter

ratio of 0.268, gives a value of 0.053 for ý. Assuming

the contribution of the stagnant conductivity term to be

10, the derived final correlation to use for the radial

effective conductivity is

k e 10 + 0.053 Re Pr 9

-33--

40 Gros & Bugarell (silica cyl. )

11.1autz & Johnstone (67)

30

e 20 k,

9

10

1. 2

3

4

5. 6

1 Coberly & Marshall (68) 2 (68) & (69) modified by (70) 3 Campbell & Huntington (71) 4 Yaýgi & Wa'kao (72) 5 Yagi, Kunii & 11. akao (62) 6 Bunnell et al (73)

0! I- I 0 so 160 150 REYNOLDS No.

Fig 2.1 EFFECTIVE RADIAL CONDUCTIVITY CORRELATIONS

1.5

1.0 , ýbrich and

Potter

Yagi and Kunii

0.5

0.0 iI 0.0 0.1 2 0: 3

Dp

Dt

Fig Z. 2 ý vs DIAMETER RATIO

-34-

2. S Wal I- lloýi It- 'r rzi ns fc

r Co cf fl'i ci en ts

Wall heat transfer coefficients are normally expressed

in two ways:

Nu = Nuo +X Re Pr (1)

Nu =b Re c (2)

wliere Nu = li

wD k

Both forms are equally favoured. The second form initially

appears to be the correct form as the Nusselt number should

be expected to be zero at a Reynolds number of zero. The

effective radial conductivity however is not constant

across the whole packed bed but falls off sharply near the

tube wall due to the increase in void fraction and the

presence of a boundary layer. The increase in the void

fraction was reported by Schwartz and Smith (74) and meas-

ured by Kimura et al (75). To allow for this effect the

decrease in effective radial conductivity near the tube

wall is commonly lumped into the wall heat transfer coeff-

0 icient by adding a stagnant Nusselt number Nu The

-radial variation in conductivity decreases with increasing

gas flow rate so correlations of the second form are

normally only valid at a Reynolds number greater than

about 30.

Various correlations for heat transfer coefficients

are shown in fig 2.3; as can be seen the scatter is

similar to that for the effective thermal conductivity

correlations. There is even a large difference between

correlations based on the same data but produced by

different workers. The Nusselt number for cylinders is

greater than for spheres ignoring the results of Gros and

Bugarol montioned previously, mainly due to the cylinders

making better contact with the tube walls than for spheres.

Yagi and Wakao (72) calculated wall heat transfer coef-

ficients from their experimental work with packed beds of

spheres and correlated their data along with that of Felix

(76) and Plautz and Johnstone (67) by:

Nu 0.18 Re 0.80 20 < Re < 2000

Plautz and Johnstone carried out their experiments

in a large 0.020m tube with glass spheres of 0.0013 to

0.0019m at Reynolds numbers between 100 and 2000 correlat-

ing their data by

hw 0.090 G 0.7s

which is equivalent to

Nu = 0.273 Re""'

Hanratty (77) derived the following correlations for

cylinders

Nu 0.95 (Re)o*s 40 < Re < 240 E

which reduc. es to

Nu = 1.36 Re"

and also correlated the work of Plautz (78) and Felix (76)

for glass spheres at Reynolds numbers between 40 and 1,300

by:

Nu 0.12(Re) 0.77

E:

or Nu = 0.243 Reo*'7

j0--

oli s, (lei ived. a Iva 11 licat

transfor corro. tation for sphcucs in a Imcked bcd bziscd on

Thoonos and Kraillers (80) c\pci-i-mcntal iýork oil 111, ISS trýlilsf-cr-

At low Reynolds nuniber the ettliatiol, is 11111111criczIllY identical

to Hanratty's correlation of Plautz and Felix's work however

the form is very different

Nu 0.203 Re i

Pr i+0.2

20 Roo" Pro*'

Yagi and Kunii (81) re-correlated the results of many

workers to the linear form with the stagnant Nusselt tcrm.

Non-metallic spheresý

Yagi and IVal-, ao (72)

Nu =S+O. OS4 Re Pr 100 < Re < SOO

Felix and Plautz and Johnstone

Nu =8+O. OS4 Re Pr 100 < Re < 2000

Non-metallic cylinders

Coberly and Marshall (68)

Nu 20 + 0.069 Re Pr 80 < Re < 2000

The coefficient X of the RePr group was found to be 0.054 in

most cases. Fig 2.3 shows these correlations for the largest

values of the diameter ratio studied. Yagi and Kunii reason-

ed that in cylindrical packed beds at low Reynolds number,

the radial temperature distribution approached the constant

wall temperature leading to inaccurate measurements. They

carried out their experiments in an annular bed packed with

spheres, with steam and water as the heat transfer fluids to

provide an easily measured radial temperature profile. The

constant X in the fluid flow term of equation (1) was found

to be 0.041 and not O. OS4 which was attributed to the

-37-

difference in the packing states of cyli. ndrical and annular

packed bods. Fig 2.4 shows the largo variation of tho

stagnant Nusselt number that was found for different diam-

oter ratios. No reliable data is available at all for dia-

meter ratios greater than 0.15.

Leva and co-workers (82,83,84) produced correlations

for the overall heat transfer coefficient., where the wall

heat transfer coefficient and the effective radial thermal

conductivity are lumped into one parameter. Consequently

this value is lower than the wall heat transfer coefficient

by itself. Leva (85) developed correlations at diameter

ratios greater than 0.3S, and found that the overall heat

transfer coefficient was much larger than expected when

extrapolated from his original correlations at low diameter

ratios. He postulated that this was due to the channelling

caused by the wall effect in such systems.

Due to the large variations of the correlations and

accepting the trend in fig 2.4 for non-metallic spheres,

the best overall correlation for the present study is to

assume the highest reasonable value of the Nusselt number

correlations for spheres, namely the modified form of Plautz

and Johnson's results produced by Yagi and Kunii.

Nu =8+O. OS4 Re Pr

-j8-

30

20

C, NUSSELT-

No. 2

10 -4

cý ýes svý

REYNOLDS No.

Fig 2.3 WALL HEAT TRANSFER COEFFICIENTS

-c ý

(I c rh ank Po go rs ki

oberly & [, I rs h, III rom (81)

flanratty

TS C-V 1111(je Gros &

Bugarel I Plautz & Johnstonee from (81) 2 Thoenes & 'Kramers from (79)

& Plautz and Felix from (77) 3 Yagi & 1%'akao from (81) 4 Yagi & Wakao (72)

I- -1 ýo 10 0 iso 200 ()

20- Data from Yagi xglass spheres and Kunii (81), (annular packed bed)

.0

Nuo 10 glass spheres (cyl. packed beds) celite/cement

clinker

0iaI- 0 0.1 0.2 0.3

Dp

Dt

Fig 2.4 STAGNANT NUSSELT No. vs DIAMETER RATIO

-

2.0 Pl I- t ic IcH ci olls

Most heat and iliass transfer coi-rclations produced have

been of the Chilton-Colburn i facto. t form

id or j 11 a Re -b

where Kh

d Sc PG Pr 3

p

and the paTticle Reynolds number Re GD p

Al

Carberry (86) carried out a theoretical derivation

from boundary layer theory to obtain

1.1 S (Re)- I

E: Jd -'ý E:

or

1-00 3d1,65 Re 0.5 < Re <ý

where E: = 0.488 the void fraction of the reactor.

Carberry and White (87) used this expression for both the

heat and mass transfer coefficients in their reactor model.

Pfeffer and Happel (88) derived analytically, heat and

mass transfer correlations from the energy equation based

on the free surface model for Rc<100

ih id 3.0 Re

Bradshaw and Bennett (89) calculated two mass transfer'

correlations experimentally from air flowing through a bed C,

of naphthalene pellets, the first without allowing for axial

mixing,

-40-

'93

506 100 < 11,10 < looooo

and the second corrected for zi-xial mixing.

id0.606 R(,, -O*'O" 400 < Re < 100000

The difference between the two curves is less than 810 and

can be ignored for high Reynolds numbers. Bradshaw and

Bennett corrected for axial mixing, the experimental work

of Hobson and Thodos (90) and Chu et al. (91), who worked

in the particle Reynolds iiumber ranges 40-350 and 200-1300

respectively. In both cases the difference is much greater,

up to 16%, showing that in the short packed beds used in

these experiments, axial dispersion is important at low

Reynolds numbers and should be considered when selecting

correlations. Only j factor correlations therelore for

experimental work carried out in long packed beds and/or

corrected for axial dispersion have been considered. This

eliminates most of the work carried out on heat transfer

correlations.

The j factor ratio jhljd is commonly accepted to be

close to unity. Values of 1. OS8 and 1.08 were reported.

by Mc Connachie and Thodos (92) and Gainson, Thodos and

Hougen (93) respectively for experimental investigations.

Gupta and Thodos (94) re-correlated the results of 8 workers

for id factors and 4 for ih factors with overall errors of

17% obt. ained aj factor ratio of 1.076 but as in the prev-

ious papers of Thodos mentioned, no corrections were made

for axial dispersion. De Acetis and Thodos (9S) obtained a

j factor ratio of 1.51. Although they did correct their

experimental data for axial dispersion, they did not correct

-41 -

for conduction and radiation effects. As has been shown by

previous workers, and from the trend in Cig 2. S, the heat

transfer j factor is slightly lar, cr than the mass transfer

j factor, so that aj factor ratio of 1.08 would seem app-

ropriate.

Mc Cune and Wilholin (96) , calculating, mass transfer

rates from 2-naphthol pellets into water flowing in a long

packed bed, postulated a correlation for low Reynolds

numbers almost identical to that obtained by Carberry

id1.625 Re-o* I 'o, Ree < 120

id0.687 Re-""" Re > 120

Glaser and Thodos (97) using solid metallic particles 0

heated electrically in a long packed bed derived the heat

transfer j factor correlation: ih

1+ ýýA-2 log 4984.30. < Re < 2700

Ih D Re 0.933

0h

where O. S3S 3h

0 Re 0"0 1.6 h

Re h

Re (1- E: )

Ap= surface area of 1 particle

D= tube 4iameter

ý= sphericity factor, 1 for spheres.

-42-

'1 0

fl

'-4

C; W C3

-j o CD

ui 01-1

-4

'-4

N

., -4

'-4. D '--4

ýoiouA

C3 V-4

cD Q v-I v-I

--43-

Most of the experii"Iciltal work carricd out llýls booll at

a bed porosity of 0. /10 with a diameter ratio less than

0.16. Of the correlations selected only Carberry and

Glaser and Thodos have included bed porosity (is a lpara-

meter. The effect of increasing the bed porosity from

0.40 to 0.488, the value used for this study, is according

to Carberry's correlation to decrease the j factor by 11%.

To estimate the mass transfer j factor therefore, take the

lowest curve in fig 2.5 for the required Reynolds number.

2.7 Effective Radial Diffusivity

When a packed bed catalytic reactor is non-isothermal

and non-adiabatic., all of the parameters reviewed may be

important. The effective radial diffusivity of mass how-

ever., can vary over a wide range (1) with very little effect

on the axial and radial temperature profiles. Normally the

effective radial diffusivity is correlated by the Peclet

number being equal to constant. Froment (2) recommends that

the Peclet number should be between 8 and 11. A value of 8

has been used in this study due to the high catalyst pellet

diameter to tube diameter ratio.

-4,1-

tor 3 EXPERIPTNTAL NFT1101) Ch, lp

All experimental work, (-, \ccpt for a few c; i] ibratioiis

and the light gas analyses, was carried out on the

equipment shown in fig 3.1.

Supply of Reactants

Air supplied from a compressed air cylinder was

metered through a rotameter before entering the pre-

heating serpentine. To keep the catalyst active the

reactant stream should contain 0.01 mole% sulphur. This

was added to the air stream by metering 99% purity

sulphur dioxide, from a gas cylinder, through a water

bubbler.

The reagent grade o-xylene (99% pure) supplied by

BDH was delivered by gravity feed from a reservoir

through a capillary and a Hoke micro-metering valve to

the pre-heating serpentine where it flowed down the tube

wall and evaporated into the air stream. The glass

capillary (approximately 0.3 mm I. D. x 1SO mm) was

necessary to restrict the o-xylene flow in order to

increase the region of useful control of the micro-

metering valve. Fine control was by the micro-metering

valve., while coarse control., over a limited range, was

effected by varying the liquid head in the reservoir.

A burette was connected to the roservoir so that by

closing the reservoir stopcock the o-xylene flow rate

could be measured by timing the decreasing level in the

burette. Due to the large surface area of the reservoir

-45-

Water Mano- meter

Ceramic Fibre Insulation

Fig 3.1 EXPERIMENTAL APPARATUS

-46-

and the vory low flow rate (10mls/hr o-xvlcn-, - into 480 w

I/hr air = 0.41 moloo) the liquid head Jocreased only

very slowly during a run and could be topped tip occasion-

ally to maintain a constant flow rate. Reproducibility

and accuracy during a run wore bettor than 2%.

Various positive displacement metering pumps were

tried to feed the o-xylene, with a hydraulic accumulator

to dampen any flow oscillations, however no smooth

reproducible flow rates were achieved, possibly due to

the flow rate being at the lower operating limit of the

smallest pumps available.

Evaporation of o-xylene into the air stream was not

considered due to the sensitivity of the vapour pressure

to temperature changes, and the difficulty in producing

a step change in the o-xylene concentration while main-

taining a constant air flow rate.

Reactor

The pre-heating serpentine (12.7mm O. D. stainless

steel) with a heated length of 1.20m, connects directly

to the vertical reactor tube (lS. 59mm I. D. stainless

steel) just below a mesh support for the 0.499m of 4.16mm

diameter spherical catalyst pellets supplied by von

Heyden of Munich, Germany. Both tubes were immersed in

the lead/tin solder bath contained in a 0.102m I. D. x

0.610m mild steel tube closed at the bottom and flanged

at the top. This was heated electrically and insulated

by alumina cement and ceramic fibre.

-47-

To determine tho reactor temperature, 10 evenly

spaced the rmo couples were placed axially with the pairs

of wires passing up thrOLIý, rh tho bed and out the sealed

end of the reactor tube (see fig 3.1). An Ilth. thermo-

couple was implanted in a catalyst pellet at a dimension-

less length of 0.222 (0.111m) from the inlet. To drill a

hole in the core of the silica catalyst support for

insertion of the thermocouple, an ultra-sonic drill had

to be used. Unfortunately as much as 20% of the surface

area may have been damaged, and as water had to be used_

in the drilling process, some of the active catalytic

species, especially potassium sulp hate, may have been

removed.

The voltage, relative to the ambient temperature,

across the thermocouples was measured by a Solartron

data logger with paper tape output capable of scanning

eight channels per second. The ambient temperature

correction was determined both from a thermocouple in

melting ice and by a normal glass thermometer which

agreed very well. The thermocouples were Chromel/

Alumel with each 0.10mm wire insulated by threading

through a glass capillary of approximately 0.23mm O. D.

The thermocouples were assembled so that each temperature

measuring weld was on the surface of the bundle which

was held together, at about 3cm intervals by alumina

cement. The thermocouples, although fairly flexible,

were kept axial during the packing by the use of a

centralising jig that was removed as the catalyst was

tapped into place. Unfortunately the first thermocouple

-4 8-

in the I) c (I at a dimonsioii1ess ICI, ý, tII of (). 11 L (O. OS6iii)

was 1-1111-1101vingly damaged and coasod to I'tinction. At

the outlet of the catal), Lic bed the out-sicle diametor of

the thermocouple bundle ivas 1.5111111, or 11, of the cross-

sectional area, but for most of tho longth, the area

occupied by the thermocouples was oven less. This

decreased by 7% the overall packing density from a trial

value of 1.42gm/cm' to 1.33gm/ciiý The reactor however

was now changed from a cylindrical to an annular type

with less than 2 catalyst pellets across the annulus

which may result in channelling of the gas. C,

In initial trials, stainless steel sheathed thermo-

couples were inserted radially into the centre of the

reactor tube however, not even 80/20 gold/nickel solder C>

could withstand leaching by the lead/tin bath.

All the product lines from the reactor were heated

to 180-200'C by Electrothermal heating tape to prevent

blockage by condensation of any products, especially

phthalic anhydride itself. A side stream was directed

for gas chromatographic analysis before the receiver,

which was packed with cotton waste.

Heavy Gas Analysis

The sample was introduced into a flame ionisation

detection chromatograph by a Taylor Servomex air actuated

gas sampling valve, of sample volume 0.679ml, operated

from a 30 psig. supply by a manual flow reversing valve.

The gas flow through the sample valve was observed via

-49-

. ýJio if3.2. _i, gli of wil i-c '- IVIA il the bubbler, the fiiial dc,,; - - 11 1,; ,*i 17,

The sample vilvo was coatainod in the cli roma tog ralfli oven

at 180-190'C Tlio ovon ivýis InallU. 111y C-011trO. I. Iod by a

Regavolt variable voltago transformer. Temperature

stability was + 20C. which is adequate for a flame ion-

isation detector. A sample port was provided for liquid

phase syringe injections. The column (6.3Smm O. D. x 0.8m)

was 10% XE 60 silica gum on 100-120 mesh diatomite CQ

supplied by Pye Unicam. The gas rates ucre nitrogen

carrier gas 66 ml/min., hydrogen 34 ml/min., and combus-

tion air 300 ml/min.

The detector and

had a maximum useable

mole/l (1 ppm). The

kept at 26S and 2400C

manually by a Regavol

amplifier, supplied by Pye Unicam

sensitivity of about 2x 10-'gm

detector and injection port were

respectively, each being controlled

t variable voltage transformer.

Calibrations up to 0.3 mole% were carried out by

subliming phthalic anhydride into a nitrogen stream at

a known constant temperature from which the vapour

pressure could be calculated by the following expression

developed by Crooks and Feetham (98) for the solid phase

up to 130"C.

p= 12 . 249 - 4632. logio

mm T

Saturation of the phthalic anhydride vapour stream was

checked by varying the nitrogen flow rate. The vapour

stream was then admitted through heated lines to the gas

sampling valve and the gas chromatograph for analysis.

For calibrations above 0.3 mole% the calibration was

-so-

Fr om Chromat - ------ gr aph

Teflon Connection to 6.2pua Copper Tube

I.

Water Manometer

To

Hoat Tracing

Vent

IfI

oks r rings

1somm 35

und ss nt

8min Glass stopcock

Fig 3.2. Bubbler on outlet of gas chromatograph

-51-

linearly extrapolated, as this WZIS still- Within tIIC linear

range of the gas chromatograph. The 01-ution t111110-13 "Ire

shown in table 3.1. Unfortunately the maloic anhydride

and o-tolualdohydo peaks could not be separated. At the

exit of the gas sampling valve a septum was installed so

that when the valve was in its normal by-pass position

a sample could be withdrawn for analysis of the light

, gas components in a second gas chromatograph. C,

Light Gas Analysis

The utilisation of a second chromý-Aograph was

necessary to check the overall mass balance and to

determine the CO/CO2 ratio which is vital when calcul-

ating the overall heat of reaction due to the large

difference in the respective heats of formation. The

light gas analysis was carried out by injection into

a programmeable Taylor Servomex gas chromatograph with

a katharometer detector (see fig 3.3). Due to the total

carbon oxides concentration being about 1 mole% or less

the chromatograph was operated at ambient temperature to

facilitate good resolution of the components.

The first column (4.8mm I. D. x 1.5m. SO-80 A. S. T. M.

mesh Poropak T) separates the sample into two groups of

components. The first group to elute, oxygen, nitrogen

and carbon monoxide is directed into the molecular sieve

column (4.8mm I. D. x 3.66m, 44-60 B. S. mesh, 13x molecular

sieve) for further separation. By switching the hydrogen

carrier gas flow the carbon dioxide is eluted directly

into the second Poropak column (4.8mm x 3. OSm, SO-80

-5? -

llydro,, ýIlcji 1,1,0111

carrier gas switching, Unit

Porapak T

hydrogen from carrier gas switching unit

Inolecular siovc

katlillromoter dotoctor

Porapak S

Fig 3.3 GAS CHRO, FOR LIGHT GAS ANALYSIS

Table 3.1 ELUTTON TIMES OF HEAVY COINIPOUNDS

Compound Time (s)

o-xylene 3S

maleic anhydride ss

o-tolualdehyde ss

citraconic anhydride 60

phthaldialdehyde 110

phthalic anhydride 180

phthalide 200

Table 3.2 KATHAROMETER WEIGHTING FACTORS

oxygen 1.000

nitrogen 0.884

carbon monoxide 0.908

carbon dioxide 1.3S8

where

normalized area = area of peak weighting factor

- 53-

A. S. T. M. mesh Pol'oPak- S) , Wilicli retlar(is tile Ca"bOll Jioxi(lc

Ulltil tile (: oj,, jpollCjlt,; ()f tile f j. ýst ,, roLip zire coiiipletely

separated. The final elLition order is oxý, gcn, nitrogen,

carbon monoxide and carbon dioxide. Ideally any heavy com-

ponents adsorbed on the first Poropak column are removed by

back-flushing The column. In practice this was rather

ineffective as after 60 minutes of operation, normally

four or five samples, no more analysis could be carried

out for a further hour while some heavy components

eluted, probably o-xylene or maleic anhydride.

Taylor Servomex flow control and switching units

were -employed to accurately regulate the hydrogen

carrier flow, which is critical when using katharometer

detectors especially if both arms of the bridge circuit

are used as in this case.

Calibrations were performed with 1% certified gas

mixtures of carbon oxide supplied by British Oxygen

Company. Values obtained for the weighting factors

relative to oxygen are shown in Table 3.2.

Immediately after calibration of the flame ion-

isation chromatograph the overall mass balances were

wi thin + 5%;. however in later runs the error was up

to + 20% even though temperatures and gas flow rates were

kept as constant as was possible. The calibration of

the katharometer detector chromatograph did not vary,

with reproducibility of analysis being within 2%, so

that this analysis was assumed to be absolute and the

-54-

total heavy product concentration was JoLermined by

difference. The various heavy component concentrations

were determined from normalising the area of the respect-

ive peaks relative to the total area of the heavy product

peaks. For all gas chromatographic analysis, a Vitraton

400 chart recorder with integrator was used.

O-Dorational Procedure

The vanadia catalyst was activated overnight by

passing 1SO 1/hr of air containing 0.01 mole% sulphur

dioxide through the reactor.

After setting the air rate at the desired value,

the reaction was initiated by gradually opening the

micro--metering valve to increase the o-xylene flow rate

until this reached the required value. As the reactor

approached the desired operating conditions from start-

up the power input to the bath was gradually decreased as

more heat was evolved by the reaction. To obtain

steady state, the power input had to be very carefully

adjusted, while monitoring the hot spot thermocouples, as

these were very sensitive to changes in the bath temp-

erature. For the last half of the experimental runs,

where higher o-xylene concentrations were used, a

cooling tube in the reactor bath through which air from

a 70psig supply was passed, was used as well as the

Regavolt variable voltage transformer to control the bath

temperature. The air cooling tube gave a faster response

and finer control than controlling the bath temperature

by the transformer.

-55-

As stoady state was approachod tho o-xyleiie flow

rate was measured and at stojdy state the gas chroma-

tographic analyses wore conipleted. '['he valve at the

outlet of the receiver being adjusted ivhcn necessary to

keep a small, but constant flow of gas through the sample

valve, observed by the bubbler.

To initiate a step change transient either the o-xylene

flow rate or the air rate -could be changed. The bath and 0 inlet temperature however, could not be changed step-wise,

due to the large thermal mass of the bath. As the

o-xylene flow rate was dependent on the head of liquid

plus the difference between atmospheric pressure and the

gas pressure in the serpentine, changing the air rate

altered the pressure drop across the system, resulting

in a change in the o-xylene flow rate as well. Changing

the o-xylene flow rate, however did not effect the air

rate. As the heat evolved in the reactor changed after

introducing a transient, the power input to the bath had

to be varied by a corresponding amount, so that this

feed forward control could keep the reactor bath at a

constant temperature.

As a transient was initiated the scanning rate of

the data logger, after the first few runs listed, was

increased from scanning all the thermocouples every

minute, to every five seconds. After about four minutes

the scanning rate was returned to normal.

The pressure at the inlet of the system was normally

-50-

kept at about 3Scm of water (O. Spsig) with a carKul

watch being kept on this in case of tube blocRages,

possibly duo to failure of the heat tracing. The gas

flow rate was corrected to 2SOC and I atm by tomporat ure

calculations and a rotameter calibration carried out at

the operating pressure, by water displacement in a large

inverted measuring cylinder.

The system was seldom completely leak proof, however

the leakage rate when the whole system was pressurisod to

55cni of water (0.8psig) was kept below 0.2 1/hr, or less

than 0.05% of the total flow rate. In actual fact the

operating lealkage rate was lowerýthan this as the pressure

of the system, especially after the reactor, was apprec-

iably lower than the test pressure due to the suction

vent line pressure of -5cm of water.

When not in use the reactor was kept at about 2700C

with a small flow of air passing over the catalyst at

all times.

-57-

Chapter 4 IZI: SIJI, TS AND DLSCUSSION

4.1 Experiniciui]. Conditions

Many trial oxporimental runs werc carried out to min-

imise the operating problems of the system. These were

mainly due to non-reproducible results from the flame ion-

isation detector chromatograph, blockage of the bubbler by

the phthalic anhydride in the off gas from the chromatoZgraph,

failure of the heat tracing of the product lines and instab-

ility of the bath temperature at steady state and especially

under dynamic conditions. Elimination of the chromatogra-Ph

problems are mentioned in chapter 3. Heating tape failures

ocurred initially, but seldom in the later experiments,

while stability of the bath temperature at steady state was

achieved through operating experience. For dynamic condit-

ions, where the inlet o-xylene concentration only was varied,

manual feed forward control of the reactor bath heating was

effectively utilised so that the inlet gas temperature to

the reactor normally remained completely constant.

The outside wall temperature of the reactor was meas-

ured by a moveable thermocouple probe. The temperature

profile is shown in appendix VIII. The maximum variation

over the length of the reactor was 30C. This curve was fitted by a fifth order polynomial and included as the

boundary conditions for the partial differential energy

equations.

The temperature reading of the thermocouple probe and

the two thermocouples in the reactor varied by ±O. S'C over

-ss-

a one minute intorval 'I'lli-S WZIS CIUC tO thC VZII'iý11)10 IlLlt. UrO

of the convective circulation oF tho liquid inctal in the

ba th . The measurod centre lino temperaturo of the roactor

variod by up to 10C at steady state, which is very little

considering the sensitivity of the reactor temperature

profiles to variation of the bath temperature. The vari-

ation of the wall temperature, would only be over a small

area at a time and appeared to be completely random, as

would be expected, so that the overall effect of this would

be fairly minor, as found. At very high concentrations and

hence high hot spot temperature it was very difficult to

obtain a stable operating state. In these cases the random

variation of the wall temperature may have been significant

and it would be doubtful if even the best automatic control

could achieve a steady state. The best solution to this

problem would be to have forced convection of the heat

transfer medium. This could be achieved by using a fluid-

ised bed or a molten salt bath with pumping, mechanical

agitation, or the injection of nitrogen or air as is done

in large scale operations. Nevertheless, some variation in

the hot spot can be expected due to parametric sensitivity-.

For the dynamic condition of a step change in the gas

flow rate it was not possible to keep the inlet gas temp-

erature to the reactor constant., by the application of feed

forward control. For an increase in the total gas flow rate

the inlet gas temperature to the reactor increased by 0.2

to O. S'C. This is contrary to normal heat transfer theory

as at the end of a heated tube the outlet gas temperature

should be lower., the higher the flow rate. This can be

explained by the pre-heating serpentine acting as both a

. -59-

heater and a cooler. From appcndi. x VIII it is obvious tha

-the reactor wall toi'LipcratUrO, and hence the býitli t cmpo r Lit Lire

are not constant througliout the inolton metal. As shown in

fig 3.1 of the equipment design, most of the length of the ZI

pre-heating serpentine is in the middle region of the bath.

This is the hottest part and duo to the length of the pre-

heater the gas approaches this temperature. Over the

final part of the pro-heater, the bath temperature decreases,

so consequently the gas is cooled in this section. By

increasing the gas flow rate therefore, less cooling takes

place, resulting in the observed slightly higher inlet

temperature.

The pellet temperature supposedly measured by a therm-

ocouple inserted in the pellet, never registered a temp-

erature higher than the thermocouple positioned in the cen-

tre line. Dynamic response tests were carried out with the

catalyst bed initially at a steady state temperature with

a very small gas flow rate through the bed. The gas rate

was suddenly increased to maximum flow rate, with the temp-

erature response of the thermocouples being recorded at one

second intervals. In all of these tests the response of

both thermocouples was identical and consistant with a

measuring device of very low thermal mass. This shows that

the thermocouple had probably come out of the catalyst pell-

et. In the construction of the thermocouple bundle the

glass capillary sheaths were cemented together approximately

every 3cm and especially just before each thermocouple to

ensure that they were kept as central as possible. The

thermocouple inserted in the catalyst pellet however was

_o0-

only cemented about zlcm from the end to facilitate beliding,

away from the centre line due to the dizimeter 01' the pellet.

If the pellet came oEf during the paclýing process this large

free length of the thermocouple could easily then mean that

the pellet ended up at, or near the wall of the reactor.

Due to the radial temperature profile being approximately

parabolic this would result in the thermocouple, supposedly

inserted in the catalyst pellet, recording a lowor temper-

ature than the centre line thermocouple, as observed exper-

imentally.

Replicate steady state runs were carried out with both

the temperature profiles and the outlet concentrations

agreeing within experimental error.

4.2 Product Distribution

The yield obtained for the production of phthalic

anhydride from o-xylene, as can be seen from the experimen-

tal data in appendix VII, was 76-80% or 106-112 Kg phtha-

lic anhydride per 100 Kg o-xylene, which is at the top end

of the industrial yield range of 100-110 Kg/Kg. The reactor

used was O. Sm in length compared to the industrial units of

3m with a residence time half that of commercial reactors.

The difference in residence times would explain the decreas-

ed amount of over oxidation observed experimentally.

As the compounds being analysed in the flame ionisation

detector chromatograph were not too dissimilar and a quanti-

tative product analysis was not required, a constant weigh-

ting factor of unity was used for all compounds. Normally

-61-

phthalic anhydride comprised around 97o of the heavy hydro-

carbons in the outlet stream so that this assumption is

quite reasonable when calculating the plithalic anhydride

yield. For the products of the low bath temperature runs

however, where up to 34% of the o-xylene feed was left

unreacted, this assumption may be very misleading. The

phthalic anhydride yield for the low bath temperature runs.,

calculated by the remainder method normally used, was about

10% in error compared with a calibration for the gas chrom-

atograph based on the results of other runs. This can be

corrected if a relative sensitivity for o-xylene to phthalic

anhydride of 0.65 is assumed in which case all comparisons

of yield from the two methods are within 3%.

The assumption of a non-unity relative sensitivity for

o-xylene is only important for the low conversion runs 24

and 29 where it can result in an increase in the phthalic

anhydride yield of up to 1S%. The results in appendix VII

for these runs has been corrected accordingly, however no

modelling was carried out based on these data. In the

rest of the experimental runs the maximum fraction of un-

reacted o-xylene was 0.01 so application of the o-xylene

relative selectivity would only change the phthalic anhy-

dride yield by a negligible amount (a maximum of 0.3%).

The major by-product, after the carbon oxides., was

either o-tolualdehyde or maleic anhydride, but the gas

chromatograph could not separate these components. Exper-

imentally the yield of this product increased from 1.5

to 2.0% as the hot spot temperature increased, so that it

-62-

was assumed to be maickc anhydryde as more over oxidation

would be expected to occur at higher reaction temperatures.

For the low bath temperature runs however, the yield was up

to 6%. This implies that this product may have boon o-

tolualdehyde as it is commonly quoted as an intermediate

in the reaction of o-xylone to phthalic anhydride.

No relative sensitivity other than unity was used for

maleic anhydride as normally this was a minor product, and

any small change in this would only make a slight change

to the phthalic anhydride yield. For the low conversion

runs, although the heavy by-product yield was much higher,

a unity relative sensitivity was satisfactory as no detail-

ed calculations were intended for this case.

Occasionally, especially at the start of a run, a

trace of phthalide was detected, but the poor separation

of the phthalic anhydride and phthalide peaks meant that

the former completely overshadowed the latter. Normally

a trace of citraconic anhydride (methyl maleic anhydride)

was detected and possibly pyrocinchonic anhydride (dimeth-

yl maleic anhydride). At normal operating temperatures no

phthaldialdehyde was detected; however at low bath temp-

erature runs, hence at low conversion, a trace was detected.

In all of the runs carried out no traces were found

of o-toluic acid, benzoic acid or benzoquinine. Occasion-

ally a trace component was detected ývith an elution time

of approximately 12 minutes. This was assumed to be the

black deposit that gradually blocked up the product lines

-63-

although they were kept at nearty 200%. Caldcrhanh (99)

claims that this compound is the result of the reaction of

o-xylcne with phthalic anhydride to form a mutti-ring comp-

ound. Due to the limited operating conditions of the gas

chiomatograph it was not possible to conclusively identify

this substance.

The highest yield obtained of 80% was achieved at low

inlet o-xylene concentrations of 0.31 to 0.40 mole%, with

hot spot temperatures less than 390'C. Under these cond-

itions the total carbon oxides yield was 18% with a CO 2 /CO

ratio of approxiamately 2.7S, while the yield of maleic

anhydride was l. S% with the remainder in all cases of about

0.5% unreacted o-xylene. The overall heat of reaction was

approximately -1.61 x 106 KJ/Kmole. As the concentration

was increased to 0.72% the hot spot temperature increased

and the yield of phthalic anhydride decreased to 76%,

while the total carbon oxides increased to 21. S% with a

CO 2/CO ratio very slightly decreased to 2.6. The yield of

maleic anhydride was 2%. Although a lower carbon oxides

ratio was obtained, the higher overall degree of oxidation

of the products resulted in the heat of reaction changing

to -1.70 x 10 KJ/Kmole.

4.3 Wrong Way Behaviour

At all bath temperatures studied, in the o-xylene

concentration range 0.31 to O. S2 mole%, with a constant

air flow rate, an anomolous effect was found. Increasing

the o-xylene concentration resulted, for a-dimensionless

position of 0.222 (0.111m), in an initial small temperature

-64-

rise followed by a decrease to a now lower steady state

temperature, see graphs 4.1 and 4.3. ' This -temperature

decrease, or wrong way behaviour occurs instead of a mono-

tonically rising temperature which would be expected for

a normal first order system, at a position before the hot

spot. The experimental hot spot moved slightly towards

the outlet of the reactor and increased in height, however 4: 1

the movement of the hot spot was in the opposite direction

to the movement that would occur in a similar first order

reaction system. This effect was completely reversible

when the o-xylene concentration was decreased, see graphs

4.2 and 4.4. The second half of the catalytic bed appeared

to behave in all cases as if a simple first order reaction

was tahing place.

To explain the wrong way behaviour physical factors

were first investigated, namely the change in the overall

heat capacity of the reactant gas when the o-xylene concen-

tration was changed. The inlet concentration was around

0.5 mole% and no more than a 30% step change in the concen-

tration was carried out so that the maximum change in any

physical property of the inlet gas would be less than 0.2%,

which is negligible. It is conceivable that the thermo-

couple supposedly positioned at the centre line was in

actual fact near the reactor wall; however if this anom-

olous behaviour occurred at the wall then surely it must

also occur at the centre line of the reactor. The pack-

ing of the reactor was carried out in four steps with the

packing density being within 1% of the final value for each

of the sections, and even if there was a packing irregul-

- Ob -

a

0

0

C-)

3U 1-19 IL

Gas Flo,., j Rate 480.11hr

Inlet Xylone Concentratton 0.313% ttme<O 0.394% tImo*oO

0.0 mIn 11.0 mTn

A 2.0 mIn * 3.0 mTn * 4.0 m1n

0

0

0.00 0.10 0'. 20 0'. 30 0'. 40 0'. 1; 0 C'. 60 0'. 70 0'. 80 0.90 1'-00 LENGTH (DIMENSIONLESS)

Graph 4.1. Experimental axial temperature profiles for an

inlet xylene concentration step increase.

9 RUN 218

Gas Flow Rate 482. IA)c C! Inlet Xylene ConcentratIon

0.389% time<O 0.324% time), O

0.0 mIn 1.0 mIn 2.0 m7n

+ 3.0 mTn x 4.0 mIn

I?

0

. 00 0'. 10 0'. -, o 0v -30 01 . 40 01 . 50 0 . 60 0.70 0.00 o'-so I Do LENGTH (DIMENSIONLESS)

Graph 4.2. Experimental axial temperature profiles for an inlet xylene concentration step decrease.

-00-

. =O -333

-=O -444

L=O . 222

L--O . 222 (pellet)

L=O . 556

L=O . 667

L=O . 778

L=O . 809

OUTLET INLET

.0 TIME (MIN)

Graph 4.3. Experimental transient for an inlet xylene

concentration step increase.

C3

C; RIJN InA

-67-

L=O . 333

L=O . 222

L=O . 222 (pcllet)

L=O . 444

L=O . 558

L=O . 667

L=O . 778

L=O . 889

OUTLET INLET

-0 TIME (MIN)

Graph 4.4. Experimental transient for an inlet xylene concentration step decrease.

C31

C; RUN 21B

- () CS -

arity which is vel-Y dOL[I)tful-, it (:, ""lot Ove t1le

obsorved.

What initially appoars to bo wrong way hchaviour at

a point, can occur when the hot spot pisses through the

point; however in. the experimental work carried out) the

hot spot was always downstream from the dimensionless pos-

ition 0.222., so this explanation is not possible. Initial

wrong way behaviour has been demonstrated for a normal

first order reaction in a catalytic reactor where -the inlet

temperature has been changed, but this does not lead to a

stable steady state in the wrong direction unless multiple

steady states are possible. This means that the only

possible explanation remaining for the wrong way behaviour

is some kinetic effect.

4.3i Reaction Model Discrimination

The variation in selectivity of the major products,

where the phthalic anhydride/carbon oxides ratio decreases

with increasing concentration and hot spot temperature can

be explained by appropriate selection of the activation

energies for any of the reaction networks surveyed in the

literature review, chapter 2.1. The simplest possible

models are the two following reaction schemes of two first

order reactions in series or parallel

o-xylene 1

1- pht'nalic anhydride 2_

carbon oxides k1

o-xylene phthalic anhydride

carbon oxides k2

As t1lo ýjcti_vzjtiojj ojjcrý, y of the sccoad rezictýon is

slightly greater than that of the Cirst, the yield of car-

bon oxides will incycaso as the temperature increases. It

would not be necessary to allow for the small variation in

the CO 2 /CO ratio. Those simple first order reaction models

however, would not explain the wrong way behaviour of the

first part of the axial temperature profile for an inlet

o-xylene concentration incroase. as the heat evolved per

unit length would increase over the first part of the reac-

tor. To obtain wrong way behaviour the heat evolved must

decrease over the first part of the reactor. This effect

can only be achieved if the rate of one or both of the

reactions concerned decreases; this in turn is dependent

on at least one of the reactions having a negative apparent

order of reaction over part of the concentration range.

All of the complex reaction schemes in the literature

survey would be feasible, but only if a negative order of

reaction could occur for some reactions. Using a model

containing two reactions would mean the estimation of twice

as many kinetic parameters, compared with the single react-

ion case from data where there is very little variation

in the outlet concentrations. To simplify the modelling

therefore, it is best to assume a single overall reaction,

which must have a negative order of reaction at high o-

xylene concentrations, with the selectivity and consequent-

ly the heat of reaction depending to a small extent on

experimental results.

-70-

Obviously tho Simple Power I'm expression is [lot a

contender. The models derived by Mars and van Krevelcii (3)

Sholstad (4) and Hughes and Adams (S) all show, for varyiii, (:,,

o-xylone concentrations, a similar variation in the order

of reaction; however this can only decrease from I to zero

and cannot ever become negative. As only one overall reat-

ion is being considered a summation expression as derived

by Juusola is not suitable.

This leaves models that assume some form of catalyst

deactivation (other than the redox type models proviously

mentioned), the bimolecular Langmuir-Hinshelwood model, and

other models containing inhibiting effects due to the prod-

ucts foTmed. Obviously for the Langmuir-Hinshelwood model

at high o-xylene concentrations, the order of reaction

would become negative due to the squared denominator. For

inhibiting models of the following forms:

kC Rrx

r1+kC a pa

k Ca Rr

where a>O and ý>O c pa

the variation in the apparent order of reaction with con-

centration is not obvious at first glance. By differen-

tiating the rate of reaction according to d(ln Rr )/d(ln C X)

(see appendix IX) the order of the reaction can be deter-

mined, and for neither of the forms above is a negative

order of reaction possible.

-L-

'I'llo billioloCLII, 11. - 1110(. 101 is dCriVCd

via tho followin, ý, stol)s:

k o-xyleno W ---

x . --- I- O-Xylclle (ads)

k 00 -- 0 22 (ads)

z::,

kr O-Xylene (ads) + 02(ads) -, products

where kx and R0 are equilibrium adsorption constants.

The rate of reaction is proportional to -the product of

the adsorbed species.

r=krC02 (ads) Cx(ads)

which can be expressed as

k k k C0 Cx

r 0 X 2

+ k C +k C ) 0 02 x x

For reaction at steady state where the oxygen concentrat-

ion is approximately constant the full bimolecular Lang-

muir-Hinshelwood model can be simplified to:

ký C x

k C ) x X

When a dynamic situation occurs it may be necessary to

consider the effect of the adsorption of oxygen by the

catalyst, which would complicate the dynamic case appreci-

ably. This would mean that two extra kinetic parameters

would have to be evaluated in addition to those required

at the steady state. Both parameters could have possible

temperature variation.

-, 17-? _

Tho remain-ing, option is sonic. type of catalyst deact-

ivation model. Considor the folLowing, equations:

K V0+ 2(o-xylono) I 2S 1--

k 0-xYlene +02 --

r- Products

The rate of reaction is proportional to the amount of

V2 OS Present.

RkCC rT V205 X

As the total number of active and inactive sites is assum-

ed to be constant the overall rate of reaction can be

expressed as:

k C r x

- 1 + K 2 C

eq x

The lower oxidation state of the catalyst I may be active,

but in this study it is assumed to be inactive. As can

be seen, the final form of the catalyst deactivation model

is very similar to the simplified Langmuir-Hinshelwood

model; (see appendix X for a normalised plot of these two

models). At high o-xylene concentration both models can

have an order of reaction that is negative. These two

models are the simplest feasible set of reaction steps that

could explain the wrong way behaviour, but are not necess-

arily the only alternative. Of the two models, the catal-

yst deactivation model is preferred as it has fewer kinetic

parameters to be evaluated.

The next requirement of these two models is that for

an increase in o-xylene concentration, the rate of react- ion should initially increase, as observed experimentally

-73-

in thic lirst part ol t', ie rc,,, ctor t)cforc decroaý7. im,

to a

new steady state.

For the Langmuir-Hinsholwood model (see fig 4.1) an

increase in the inlet o-xylone concentration results in an

increase in the concentration of the o-xylono adsorbed on

the catalyst. As a secondary effect the concentration of

adsorbed oxygen must decrease due to the constant total

number of active catalyst sites, but not necessarily by

the same amount that the adsorbed o-xylene concentration

has increased. The overall rate of reaction is the product

of the concentration of the two adsorbed species resulting

in the initial right way behaviour followed by the overall

wrong way behaviour.

The response of the catalyst deactivation model to a

step increase in the o-xylene concentration is simpler as

the only transient effect is the decrease in the concentra-

tion of the catalytically active V205 sites. The product

of o-xylene and the V205 sites is the overall rate of reac-

tion. This shows a similar anomalous behaviour to that of

the Langmuir-Hinshelwood model.

Other possible explanations of the initial right way

behaviour are the effect of the heat of adsorption of the

reactants on the catalyst, or the heat of reaction of the

V205 to a lowcr oxidation stato. Heats of adsorption are

generally less in absolute terms than -2.0 x 105 KJ/Kmole,

which is considerably less than the overall heat of -

-74-

1, all ýjmui 1- 1 iiiis Iic lwood

ý, I od o1

Stop incroaso in xylone concontration

all

U

f--ý V)

lzý

cis

C4 0

u Lr)

C-ital vs t cl ("I cti X- ati oil

Nlodol

Stop incroaso in xyleno concentration

bJ

LO

0

ul 0 Cý4

b, O

Fig 4.1. Two models that display the anomalous

behaviour observed experimentally

Time Time

- S-

I'CZlctioii of -1.0 x 10 6 K, j/K In oIc. This could not resit'Lt in

the observed initial right way behaviour of tho axial temp-

craturc of up to 1.40C. The reduction of v205 is enclotlicr-

mic so that this POSSibility is also eliminatod.

As mentioned earlier the Langinuir-Ifinshelwood model

requires the evaluation of more kinetic parameters than

the catalyst deactivation model so that the latter was pre-

forred provided it could model the observed steady state

behaviour satisfactorily.

For the case of only a single reaction the overall

heat of reaction is obtained by summing the energy contri-

butions for the formation of each product from o-xylene.

The average contribution for each component was phthalic

anhydride 52%, carbon dioxide 39%, carbon monoxide 7%,

with the remaining 2% from the formation of maleic anhy-

dride. Although no distinction could be made between o-

tolualdehyde and maleic anhydride from the gas chromato-

graphic analysis, the difference in assuming the former

instead of the latter for the overall heat of reaction

would be approximately 1% which is unimportant as this is

within the analytical error .

4.4 Steady State Modelling

For an initial investigation of the reaction system

the homogeneous partial differential equations were solved

using both the Crank-Nicolson and the orthogonal collocat- ion methods. For a simple case the Crank-Nicolson method

-76-

needed 30 radial solution points for comparable accuracy

with the orthogonal collocation method using 6 radial

points. If the solution of the temperature and concontrat-

ion profiles is stored until the end of the calculaton

the orthogonal collocation method has the lower storage

requirement. Conversely if the solution is printed out

after every step the Crank-Nicolson method has a much

lower minimum storage requirement. The computational time

necessary for a "one off" calculation by either meth6d is

very similar. When the model has to be solved many times,

as for a non-linear regression analysis, the collocation

method is far superior. The Crank-Nicolson method is very

wasteful as it has to completely restart for each regress-

ion iteration. Conversely, in the orthogonal collocation

method for changes in some of the parameters (e. g. the

kinetic parameters) the inverted solution matrices remain

the same. These matrices need only be recalculated when

other parameters (e. g. effective radial conductivity and

wall heat transfer coefficients) are changed. For changes

in the effective conductivity a large portion of the matrix,

(see equations 3 and 9 of appendix III) for solution of

the energy partial differential equation, has to be changed;

while for a change in the wall heat transfer coefficient

only some of the diagonal terms have to be changed. To

speed up the calculation, a brief investigation was carried

out to see if a small correction could be applied to the

inverse to allow for small changes in the wall heat transfer

coefficient. Although this was not rigorously investigated

none of the simple schemes tried were of sufficient accuracy

to make the operation successful.

-7 /'-

For the matrices doi, ývcd in thc solution of the liomo-

gencous equations, as descrIbed in chapter 2.2, JI. IlVers Loll by

simple Gaussian elimination, with roiv pivoting, iýorkcd very

succossfully. The equations for the hoterogenocus case,

were ill-conditioned and it w as necessary to use Crouts

factorisation mothod to carry out the inversion.

In the homogeneous model six internal collocation

points were used with less than O. I'C variation from the

case of five collocation points, up to a maximum hot spot

temperature of 420"C. For the heterogeneous model, to save

storage and computing time, only two internal collocation

points were used up to a hot spot temperature of 4000C.

Above this temperature, it was necessary to use four coll-

ocation points. The maximum error in all cases was less

than 0.5"C.

4.4i Non-linear Regression Analysis

A selection of steady state axial temperature profiles,

obtained from the initial and final steady states of the

transient runs, were fitted by the method of non-linear

regression of the kinetic and heat transfer parameters.

The objective function was taken as the sum of squared

errors between the experimental and the computed centre

line temperatures at the same axial position. The sum of

squares was minimised by the Powell conjugate direction

method (100).

-78-

The i-filet temperature was that mcisurcd c\perimentaLly

I with the sum 01 squares hoing calculatcd Croill 7 points

betwoon a dimensionless lotigth of 0.2222 -ind 0.889 inclusive.

The last thermocouple at the outlet was not used as the

temperature occassionally tailed off. This end effect was

due to the surface of the load/tin bath being level with

this point resulting in a small amount of heat conduction

up the tube. The point 0.111 was not included in the sum

of squares due to the thermocouple failing as the result

of the packing process.

The squared error between the theoretical and experim-

ental concentration at the outlet of the reactor was not

included in the modelling due to the arbitary nature of

selecting the necessary weighting factor and the need for

an accurate fit for the temperature profiles for the trans-

ient modelling. As all the temperatures in the reactor bed

were of the same order of magnitude, and were all measured

in the same way, no weighting was applied to the individual

temperatures. Although statistically correct, a natural

consequence of this approach is that the hot spot temper-

ature where most of the reaction takes place is given less

weight than is perhaps desirable.

Preliminary trials modelling the axial reactor temper-

ature profile of a single experiment as a first order reac-

tion were reasonable; however this model could not be

consistently applied to groups of data, and more complex

models were investigated. Simultaneous non-linear regress-

ion analysis of all the data was not possible due to

-79-

Jimitations oll comptutc, ýr týiiie and storage. The liiiiiiiiii-, sa-.

tions wore therefore carrýod out for ,,,, roLil),; of data having

similar hot spot temperatures . Tab lo 4. .1 shows the restil ts

of the nonýlincar regression analysis for hoth the lioiiiogcn-

eous and hoterogencous models using tho catalyst deactiva- 00

tion rate of reaction.

k (exp{S - Ea

0RTx r1+KC2

eq x

where Cx is exprossed as a mole fraction of the total molar

concentration (air + hydrocarbons), k0 is the dimensional

3 constant (I Kmole/m . s) and S is the exponential frequency

factor. By differentiating the rate equation it can be

shown that a maximum rate of reaction occurs at a concent-

ration of IIVR7- As the unusual effects previously ment- eq ioned occurred at least down to a concentration stop bet-

ween 0.39 and 0.31%, it was decided to fix this maximum

rate of reaction at an intermediate value of 0.333%, there- 4 fore defining Keq as 9.00 x 10

For the homogeneous model non-linear regression with 0 four variables viz. frequency factor, activation energy,

effective thermal conductivity and wall heat transfer

coefficient proved to be unfeasible due to the high degree

of correlation between the last two parameters. For runs

-I- 19 and 21, W4tjj moderate hot spot temperatures up to 395"C,

the effective conductivity was fixed at 9.30 x 10- 4 KJ/m. s. OC, which is just slightly above the majority of the corre-

lations in chapter 2.4, thus greatly simplifying the search for the minimum sum of squares. The correlation between

-80-

>-

CD

uý

u3 P-1 cý

-0 ce E-

0

:: s 0

4-) 1 4-1 ý11 u >, ýwl

ý-o Q) 4--) 0H ýý4

9 ýý Q) 0

4-)

U) LH Q) 0 ý4 --I-

00 0')

L--

tn

-zf

rn

G)

t1o

00

to

-7t-

CIA

00

t-

ýt

Oo

(7)

t--

r-I \-C)

LO

Lo

-i

1ý

C71

00

LO

Ln

Ln

tn Ln -1

4--) Cfj

Q)

4--)

Cd

4-, ) 1 "I 4J OD Cýs u 00 0

4-J t-- 00 \ýD ýA

r4 1ý Cý X Q) 4--) 0- ri ý'4 r-A r--i r-i --i ltý U) u Lf) 0 CD : ýi: >1 r-i u (/) r- t- \. o 0 :ý oo 00 CD 0 ýs 4--) tn to (D 01 u coo r-i cd 0

P4

LO

0 LH 0)

=1 0 (D t-) 0) cn Ln -1 r-- Ln -zt t-- -t 0-ý ý

, r4 Cý r 0 1; L4 1; 1ý lzý C7 -zzl- I C'-j C'-j Ln \. O r--i v) -t rH -zt C) Ln ý. -i r-I -1

rij 0 +j (1)

H 4J

Q 1-- 00 tn ýt "D r-i r--A r--ý 00 (-) (--I -, I- C--l 0) Cli -I . . . . . . . . . . . . . . > 44 0U LO r-q C. -j ý, o 0 0 co I-- r-4 N-) C--l caý C--j t- H 0

t, o U (I)o I-- oo 00 r-- 0) 0-ý 00 L-- (D) co C) G) 0 4-) rIJ 0 E- t-, t") tn t") tn tn N'ý ýn V') j- tq z: f, t') u

(1)

C-3 r-- -: t ý-i r--q ýýt Ln tn 0) C7) 4-)

r--l r-i cn -1 co 0) r--l --1 -1 T--A V) -r-i 0 u V) . 14 M 4-4

Q) u LH

tn I-- rl, -I- Lr) 01) C'A LO r'- ýJ- LO C) ýt t-- -4 (1) 0 0) 6 r--ý 0) Cý C'-, -1 0 Lr) tn 0 rl- :r CIJ C'A 1ý r-A U

tn rn tn t") Ln L. 0 :: J- V) LO zzj- Lj-) Lf) Lf) : I- cl rH >1 . * ; -4 x00 0 c 0 (D 0 0 (ý (Z ) 0 0 0 0 CD Iu r-, (4-4

V)

0 G-, 0) -4 C; ) -4 rn \r, r- 0 (D ;? - r-i -ý " -i " (-I (-A (--I Cýj " c- j t'O t'O 4-)

0 4-) m (3) 4) 4--) U) aý

ITJ

4-) y)

-H -I 4-3 CTJ

ri LH

U) (f) r-A r-4

H LH

Pý Q)

LO CD

V)

z

11 -01 -

tho frequency fz,. ctor and the activation onorl, 11y wzis ýillov-

iatod by tlic following standard trýmsforjaat ion:

E- , exp (S -RT

Ezl exp(s - -,

if RT

where T* = 623. 'C

and S, =S Ea

R T*

The kinetic parameters directly dote-rmined from the

regression analysis were the activation energy and the

modified frequency factor S

The resulting wall'-heat -transfer coefficient for runs 2

19 and 21 was found to be 0.143 Ki/m . s. OC... which is

slightly greater than that derixod from the correlations

quoted in chapter 2.4, while the activation energy conver-

ged to 1.105 x 10 8 KJ/mole. This is in the middle of the

range of values found by most workers where mass transfer

effects are not limiting. Most of the transient modelling

was expected to be carried out based on the medium temp-

erature experiments including runs 19 and 21, where no high

temperature limiting effects occur. To simplify further

regression analyses therefore, the activation energy was

kept constant at this value for all the runs while the

frequency factor was free to vary to account for any per-

manent catalyst deactivation as the experiments progressed.

As can be seen from table 4.1 and graphs 4. S to 4.8

the overall fit obtained was good with the accuracy of the

homogenoeus model being equal to that of the more complex

0

E-

Ln

.4

lr_l la Cd ý4

u

1ý 9 0u0 r) u0

x

C3

n

I?

LU I_-

C3

I C-11 Co. ody co-GýE DO-09's Do- oylp (3) 3ýi%tJ63JW31

u

000 13

11 . 0.

0uu0

mc

x

-8? -

x

10

0 (1)

0 rz

c3

, Z) - (n U) LLJ

-1 nz "? a C3 -.

LLJ

C3

Cý La J

00,00), 00,0 (13 3c W oy ý- «2 (D) 3ý3f11tJd3dW31

ci

:j -13 -

c2

Z ui 2ý

ci Uj

loe 00,00), CD* C9Z oo* o? e

ii) 3ýniuýý3jwgl

C) x a,

0

Cý 0 ýj 0 C: )

Z

0x03

10 0 :2 fA

+-3 (3

%n ul

C: ) Z

Z ui n

. 1 r2 . c3-

c2

jai 00-08C cc, 0A 00,0PIP 131 38nld83dW31

(D

>I

E- U)

%0

1; liz r_i

CD

- 8'ý-

14

-- C) x 10

... "- C)

70u C) 0 V 13

'i cj C)

0

x C3

C3 1ý

z ul

on C3

C'i LLJ

. C; -j

CD

00,00Y Go - ogE 00-0ýc 00-0i (3) 3ýJ%UýIIJW31

0X r- 0 ýA

0u0

Kg

CY c: 39 X r_ "0 .

c; - 2 >c 0mr. (t) . uu0

az cz (i -&uQ Wo

c; - - ui

ýc2 c; -

x ui

O-Odr Do - CýE DO-DA 00-001F (0) 38niUa3dW31

99

0u00

cr 0xoI

v

x

0

x

cl, cl,

C3 1!

C, tf) LO

ca

LIJ z

C3-..

091022 CI)* 01; 2 oo- ojP

ý0x 10

92 (1 00

Xr 0 '3

- Z: ýýZ -4

0 x

V) :3 0

b, O 0

4-)

: 31

w Cýz

ui

! C3

Cýuj __j

o-I? 00-00, ) OGE or)* OSE 00*0re 13) 38niU838W3i

ý4

0 >4

E-

E-

CIS $-4

u

00

x 0 "1

u u

-84-

0 0

U') LIJ

-i oz "ý C: ) 0-

(f) z LL)

C;

OZ N Lij

Do-ody Co. 06, c 00.04E Oto (31 36nitj8jdw3lco*

x C., ýo - 01

. -. . 11 -0 u

0ua0

12 Ca oxo

0U 1-1

x

C2 (a

ui

z ui

Cý z ui

00' odw 00,09C Do* 09E Co. orip (3) 38nib83dW31

C3 a

Ij 00

q ;3 " C3.

x 0 It r x V) V) C) u

-a u0

- a. LLJ :: D ý C: ý ox

CD z

C3 0 ý;

x z

CDý 1 C3

. 0-

C: ) Z uj -i

00- ody Co. od-ý 00.0 Cý C 00. Orip (3) 38nidýlldW31

r-4 (D Ili 0

>4

V) ;j 0

- :: ý 0 x " -0 4-J

u w 99 0 u 00 u m

- "- 0? >1 r: 0

1) x 0 mE x

0 -

u M

u ol - Cl. uj

a w ct x C4 0 U0 C3 Z

c)

uj Z

C2

C3

x m CD z uj

--j

Q

00-00 , DO-CdE 0 '00 0c 00, Orip

(J) 38nibýjýdw3i -

0

co

Cd

u

00

", ". u-",

0, uu0

Lý. u ej

-8s-

cl 1:

cl U) U)

CD Z

c2

c3 C. LLJ

,; -i

0 0

oü*Oor Co* 09£ on - cýE 00-0; p .(D) 3dn-1 tib3dw31

c-a

-A - 0x 10 In 040

U C) 9 v 41

9

0u0

'a

C3 ý Q. C: $.. mX

9z. U 4)

bo 0 r- 0

0- (n tn ui

c3

(1) Z

c2

v. ui .Z -i

00-06V Do- odc co 1 9c 00, or; (31 98%U83dW31

0 X "I -t 0

0 u

u00 M

- ý. ý ý3 -0 " 0

<

CD 0 11

- 0m

uu ui

2' ::, ý- W- :4 C4 ".

u ,.. LQ u 4)

ui C3

"UJ -C;

00-ove 2ýjniu8Aiw3l

0

x ýI ýn C% ý

,'ýý "0 ýI 0

9 C? 0ý0u00

cr rj

<0x CD -

C-Z xu

0

0

(D b4 0 ý-4

(D

0 Co

-i

c2

0-

CY Lij

,Z -i

co*OOP oo*CCE ( Die colope (3) 3dnidd3dW31

-86-

liotcrogoneous modol.

Regression analysis based on the simplif iod biiiioloc-

ular Langmuir-Ifinshelwood model proved to be of the same

degree of accuracy as that using the siiiiipler catalyst

deactivation model. Accordingly, no further modelling was C,

carried out based on the L angmuir-Hins lie lwood model duo to

the extra complexities in the transient case.

4.4ii Estimation and Effect of Heat and Mass Transfer

Parameters

The effect of the Peclet number for radial mass trans-

fey was briefly investigated over a moderate range. As

widely published, it was found that both the radial and

axial temperature profiles were very insensitive to changes

in this parameter.

The correlation between the effective thermal con-

ductivity and the wall heat transfer coefficient for runs

19 and 21 is shown in table 4.2 and graph 4.9. This could

be modelled within 1% error by an expression summing the

resistances of the heat transfer effects by:

1-1R uh1.91 k

where U=0.0882 KJ /M2 s. 'C and is a constant.

This form is identical to the overall one-dimensional heat

transfer coefficient derived by Beek and Singer (101).

1=1R uhw4ke

-87-

Table 4 .2 RESULTS OF NON-LINFAR RIEGRTSSION OF A

110MOGEINT. 'OUS MODFI, FOR RUN 19 AND 21 DATA

Effective Radial

Conducti vity KJ

e III . S.

6.74 x 10-4

9.30. x 10-4

11.6 x -4 10

19.8 X 10-4

10

1 hw

5

0

I'l'all Heat Transfer

Coefficient

IV

ým2.

S. Ocl

0.190

0.143

0.127

0.108

Activation Energy

E KJ a

[KiTiole

1.156 x. 10" 1.105 X 108

1.083 x 108

1.020 x 108

Expoliontial SUM of Froquency Squares

Factor S

21.431 193.7

20.437 172.2

20.008 163.6

18.806 151.9

Graph 4.9 CORRELATION OF HEAT TRANSFER PARAMETERS

FOR RUNS 19 AND 21

0.5 1.0 1.5 3

- (X 10 ý) k

-88-

Crider and Foss (102)

uh I%T

6.133 k

and also the lumped oiic-dimciisional heat transfer coef-fic-

ients derived in appendix II by application of the orthog-

onal collocation method to the case of one internal colloc-

ation point. The only difference is the value of the con-

stant in the denominator of the conductivity term, which

was found to be significantly lower in -this study. The

correlations of Crider and Foss, and Beek and Singer were

theoretically derived tinder non-reactive conditions. The

experimental difference may be due to the reaction itself

and/or the appreciable variation in the activation energy

necessary to obtain the converged sum of squares on

varying the heat transfer parameters. The high value of

the diameter ratio used may also affect the constant.

To eliminate the correlation of the heat transfer

parameters therefore it may be possible to carry out a

transformation similar to that for the kinetic parameters.

If U and either hw or ke are used as the parameters for

non-linear regression the problem may be overcome.

To model the wall heat transfer coefficient and the

effective radial conductivity for various flow rates, corr-

elations of the following form were used.

Nu = Nuo +ý Rc Pr

k ko

kee+ Re Pr 99

where X=0.054 as suggested in chapter 2. S. The wall heat

- 81 9-

a ii sfcrcoc Ff i ci o. ntfor ru 11 siad21o A- 0. -113 KJ /m 2. C

at a Rcynolds numbor of 112, co-niplotely (Icfiiies the wall

heat transfer correlation as

Nu = 8.23 + 0.054 Ro Pr

This is the same as the corielation originally selectod in

chapter 2.5. The effective radial conductivity value of

9.30 x 10- 4

KJ/Tn. s. OC was just slightly greater than the

correlations surveyed in chapter 2.4. If ý=0.053, as

suggested in chapter 2.4, then at a Reynolds number of 112 41

the stagnant conductivity term is 1S. 3. Increasing ý to 0

0.10 reduces the stagnant conductivity to a more realistic

11.7 which is not too dif-ferent from the suggested value 10 Cý

of 10. Yagi and Kunii (62) in their work on annular packed

beds show that for the diameter ratio (D p

/D t) of 0.268

used in this study, the stagnant Nusselt number increases

to 18, however there is no other work to support this

result. If this were the case then by extrapolating the

correlation derived for the overall heat transfer coej: fic-

ient U., at a Reynolds number of 112, the effective radial

conductivity must be S. Sl x 10-4 KJ/m. s. 'C which would give

the choice of either of the following equations

k 7.39 + O. OS3 Re Pr

k9

ke 3.74 + 0.10 Re Pr

k9

The first correlation with the slope of 0.053 results

in a stagnant conductivity of 7.39 which is less than the

original correlation recommended but very close to the

value of 8.0 calculated by the theoretical equation of

- 90 -

Kunil an(I Smith (60) . The stýI-IIZIIIL Coodlict i'vity L11 the

second correlation is now much -1-ower tkan the range of

experimental data presented in fig 3.1 an(It cannot be con7

sidored feasible. 'I'lic correlations based on a stagnant

Nussolt number and a stagnant offectivo condtictivity ratio

of 18 and 7.39, respectively could be used, and may result

in a lower sum of squares than originally obtained, as

shown by the -trend in table 4.2. The supporting evidence

for this is even more sparse than for the original correl-

ations derived fron runs 19 and 21. Unfortunately there is

very little literature on heat transfer correlations at the

0 P. t ratios occurring in the catal tic oxidation of hiah D /D y

o-xylene. Although changing the heat transfer parameters

(see table 4.2) causes a noticeable change in the kinetic

parameters, the change in the axial temperature profile is

minimal as long as the overall heat transfer correlation

for U is obeyed.

To obtain a reasonable fit at high inlet o-xylene

concentrations and consequently high hot spot temperatures

it was necessary to increase the amount of heat removed

from the system by either increasing the wall heat trans-

fer coefficient or the effective radial conductivity. It

was decided to hold the former constant as this should be

more stable to temperature variations and vary the latter

as this may include some effect of axial dispersion of

heat. The wall heat tr-ýinsfer correlation used throughout

the modelling was that originally derived from the non- Z>

linear regression of runs 19 and 21 with a stagnant Nusselt

number of 8.23. As shown in table 4.1 the stagnant

- 1-

conductivity had to bo I-acrensed from 9.42 to 26.0 as Llio

inlot o-xyleno conconn-zitton increased. "I'llis col-rcspolicts

to an increase, iii the ef. lect. ive coii(Itictiý, ity from 9.30 x 10- 4

to 16.2 x 10 l(J/ili .s. 0C for tho homogencous model. As im-

ilar increase was necessary for the heterogeneous model.

The necessity of increasing the effective radial con-

ductivity as the hot spot temperature increases may be due

to limiting of the reaction by film mass transfer effects,

a decrease in the catalyst activity or the effect of axial

dispersion becoming important. As can be seen from graphs

4.5 to 4.8 the experimental points show that an improved

fit could be obtained if the axial tempera-Lure profile was

slightly stretched, which would occur if axial dispersion

were included. As the inclusion of axial dispersion apprec-

iably complicates the solution of the equations, this has

not been quantitatively studied.

For the non-linear regression analysis carried out

using the heterogeneous model the sum of squares was deter-

mined from the difference between the calculated bulk gas

temperature and the measured temperature, while the reaction

rate was calculated for the temperature and concentration

at the pellet. The pellet heat and mass transfer coeffic-

ients were not free to vary but were fixed from the litera-

ture correlations as suggested in chapter 2.6. Typical axial

temperature profiles for the heterogeneous modelling are

shown in araphs 4. S to 4.8. As the heterogeneous model not

only includes the radial effective conductivity, which is

a parameter in the homogeneous model, but also the heat

- () 1' -

transfer from the pollot to the gas, for an accurate fit it

may be necessary to have a tower raMal conductivity in the

heterogeneous case compared to the homogeneous case. This

was true at all temperatures. At high-or temperatures the

difference was much loss as can be seen by comparing the

stagnant conductivity term in table 4.1.

The effect of varying the mass transfer coofficient is

shown in table 4.3. At low hot spot temperatures (runs 19

and 21) , decreasing the pellet mass transfer coefficient

increases the converged sum of squares though not necess-

arily by a significant amount. At high temperatures however,

the sum of squares is significantly decreased, improving Zý

the fit. increasing the pellet heat transfer coefficient

has a similar effect but to a lesser degree. The lower mass

transfer coefficient results in a lower -value of the effect-

ive radial conductivity for the higher temperature runs.

Slightly less overall variation is therefore necessary in

the effective conductivity than that originally determined

for both the homogeneous and heterogeneous models.

The improved fit at high temperatures by more than

halving the pellet mass transfer coefficient ordinarily

would imply that the original mass transfer coefficient

obtained from the literature was in error by 100%, and that

mass transfer is the controlling resistance at higher temp-

eraturos. If the mass transfer coefficient is halved then

the heat transfer coefficient must also be halved to keep

the ratio j0d equal to 1.08 (see the literature survey

chapter 2.6). Regression analysis of runs 26 and 27 with

-03-

Table 4.3 RESULTS OF VARYINIC, TlllI FILM MASS

TRANSFF, R C01"IFFICIEN'T

KG"0.163 m/s KGý0.0740 m/s

Run No. Effective Sum of Effective Sum of Radial Squares Radial Squares

Conductivity Conductivity KJ k

I I k KJ I ý

OC e M. S. M. S. OC e

19 8.22 x 10,4 88.4 7.44 x 10,4 123.4

30 12.7 X 10-4 66.0 10.0 X 10-4 23.0

26/27 16.7 X 10-4 246.4 11.8 X 10-4 41.2

Note. KGý0.163 m/s was the original value derived from

the correlations for the mass transfer coefficient at a

Reynolds number of 112.

-9 "1 -

I, CF both CooffiCiO111- , LlltCQI J, 11 the OriO, illIL SLI1,11 Of

thcIIC0 1', '1 t U. 1'C C0r 17 C1t0o tl e , quaros , obtained usitig f 1,1

Lransport coofficionts. Therefore mass traiislor cannot be

the limiting effect at the maximum temperature of 410'C C>

reached in the modelled experiments. Most workers on this

system agree that mass transfer limitation occurs around

440'C. Other possible effects as mentioned previously are

the increased importance of the axial dispersion of heat

at high hot spot temperatures or a change in -the oxidation 00

state of flie catalyst. For simplicity in the catalyst

deactivation model no temperature dependence of the catalyst

equilibrium constant has been included by introduction of

an activation energy parameter. If more deactivated catal-

yst were present at higher temperature then this would

explain the slowing of the reaction at higher temperatures

and give the same result as decreasing the mass transfer

coefficient to give mass transfer control. Vijh (103)

states that for the reaction

v0v0+ 10 25 -k-1 9422

the activation energies are 1.56 x 10 8 and 1.80 x 108 KJ/

Kmole for the forward and reverse reactions respectively.

This would favour more V205 at higher temperatures. The

ina ctive species may therefore be V204.34 with the further

formation of inactive V203, at high temperatures, apprec-

iably decreasing the reaction rate. The slowing of the

reaction at high temperatures may also be influenced by

axial heat dispersion as for a hot spot of 420"C the maximum

axial temperature gradient is 400'C/bi.

-

For run 32) where the hot spot tcmpoi-ature reiched

440'C, no rcasonable dogi, co of fit was achieved by cithor

the homogoncous or hoterogoiicous modcls. This may have

been due to film mas-, transfer limitation at the higher

tomperaturo or to the largo degree of toi-. qporary catalyst

deactivation.

Over all the runs carried out there w

permenent deterioration in the activity of

The exponential frequency factor S derived

modelling carried out, varied between 20.3

homogeneous model and 20.2 to 20.3 for the

model.

as no evidence of

the catalyst.

from all the

and 20.5 for the

heterogeneous

Although the axial temperature profiles were fitted

satisfactorily by both the homogeneous and heterogeneous

models, the extent of the reaction did not compare so

favourably. For the homogeneous model 94 to 98% of the

o-xylene reacted while for the heterogeneous model 91 to

96% reacted, compared with the average experimental value

of 99.5%. In graphs 4.5 to 4.8 of the calculated axial

temperature profiles, no consistent difference occurs between the two models. The decreased amount of reaction

taking place in the heterogeneous model is due therefore

to the slightly lower frequency factor and the slightly lower effective radial conductivity, obtained from the non- linear regression analysis.

-06-

Considering Out t thc ou tlct o--xN, Lclic (: oiikQll t rl L imi

; Ilm of sqlmrcý, 3 ! --'oi- the roý,,, rosstoll .3 llot included in the

analysis, the accuracy in naodollin,; is satislýlctory- 0

The usc of an ovorall heat of roaction with only a

single reaction taking place, implies that the heat of roac-

tion is constant along the packed bed. This is very unlik-

ely to be true. Improved accuracy in modelling the extent

of the reaction may be achieved by employing a series reac-

tion model as this would lengthen the hot spot where most

of the reaction occurs. As no intermediate concentrations

along the bed were measured, no improvements on the present

assumption can easily be made, except by drastically in

creasing the complexity of the non-linear regression-analy-

sis to consider a network of reactions. The limited data

available makes such an alternative unfeasible.

4.4iii L 01ý, T Conversion Runs

During start-up of the reactor, an unusual double hot

spot was sometimes noticed. This generally occurred before

the bath temperature was up to its normal operating value,

with a correspondingly low conversion of o-xylenc. A brief

investigation of this phenomenon was carried out in runs

24 and 29, see graphs 4.10 and 4.11. The operating condit-

ions were found to be completely stable and showed the

anomolous behaviour already mentioned. To model a double

hot spot, the reaction must occur by successive oxidation

steps with the possible intermediates being o-tolualdchyde,

c 0

c

v E E. ct ý:; . jý

Ex ::. c- g ccccc

-c-- S. S Ccc O. C ýX tItI Cý

'l . EEEEE

I ID 00000 i

In 00000

0 co ý? C; -4V; ; 1ý 1ý 0 co 8 -* rý 1; 4 ý0qSoo

la 0 .4+x 04 - 0vEoo 804+x

a 0

0*

4ý 9 (D

.H U)

4-J

4-1

(D

r-14

4-J

: 3:

ý-, D*

a C

C) "rl

I) E

k

4-J cz ýA

Q

P4

r-lI

H

-97-

pjjthýjl-jý(J(ý,, plithalic alillydride lild 111ý110ic '11111ydride. 'I'lle

typical Product CUStributioll 65") plltllýlli-c, 11111N, dri, de,

18'0 carbon oxides with a CO 2 ICO ratio of 3.0, S'O maleic

anhydride or o-toluald. chydo, iýith the reiiialudor being un-

reacted o-xylene. A trace of plithaldialdoliyde and plithal-

ide was normally detected. As not only the ratio of CO 2 /CO)

but also the ratio of carbon oxides to p1lithalic anhydride

was slightly greater than for the higher temperature runs

the overall heat of reaction was approxiviately 5% greater.

To obtain a double hot spot using only first order

reactions there must be at least three series reactions.

The heats of reactions may also be important so that the

simplest probable model is

AH. r=-1.117 X106 AHr=-0.432xlO' Afi, =-1.132xlO6KJ/Kmole

O-xylene phthalic maleic carbon anhydride anhydride oxides

As can be seen, the heats of reaction are very amiable

towards a double hot spot occurring. Another model tried

was

AH r =-0.335xlO6 AH, =-0.782xl 06 AHr=-2.69SX106

o-xylene o-tolualdehyde -, pht alic carbon anhydride oxides

The heats of reaction in this case are less likely to

result in a double hump on the axial temperature profile.

The modelling of this system was carried out using a

homogeneous model., two-dimensional in temperature but only

one-dimensional in concentration. For both models, a double hot spot was achieved by non-linear regression of

the 6 hinetic variables with the same heat transfer para-

meters as described previously for runs 19 and 21. The

-98--

m;. iximum bed teml)erziture howovei,, ývas coii-sidel-al)ly Iligiler

tjýjaii the oxporimciital valtic aiid the sccojid hot sjýot ivas Ilot

as convinciiig, as -that achieved expci-imciitally. Due to the

much higher bed temperatures, there was very little o-

xylone left unroacted. The product distribution hoivover

did separate the two models with the first model containing

the maleic anhydride being favoured. 0-tolualdehyde can-

not be completely eliminated as an intermediate as there

may well be four reactions iii a series.

Although two first order exothermic reactions taking

place in a non-adiabatic non-isothermal reactor cannot lead

to a double hot spot, if the orders of the two reactors are

different this phenonenon could well occur. For two reac-

tions takinOr place by a biTiolecular Lanomuir-Hinshelwood

model, with or without a summation term in the denominator,,

as proposed by Juusola (12) for the steady state adsorption

model., a double hot spot may be possible. Use of the

Langmuir-Hinshelwood or catalyst deactivation model would

also explain the wrong way behaviour observed.

4.5 Dynamic Modelling

The dynamic model was solved, based on the values of

the parameters obtained from the steady state heterogeneous

non-linear regression analysis. Between the initial steady

state and the final steady state., the effective overall heat of reaction (see appendix VII), and occasionally the frequency factor, as determined from the sum of squares

minimisation varied, due to the varying selectivity of the

-99-

re a ct ion. As only an overall rate of rcýicti-on Lt-; C(, t,

with no series or parallel reactions to iccount for sciect-

ivity variatIons, , those effects worc included in the trzins-

ient model by linearly varying the frequency factor, and

the heat of reaction over the first four minutes of the

transient. Graphs 4.12 to 4.27 show the experimental axial

temperature profiles during the dynamic state.

The catalyst deactivation reaction

v205+ 2(o-xylone) :; jýý I

must be reversible, as no irreversible effects were found

experimentally. The order of the two reactions was assumed

to be zero with respect to the o-xylene concentration and

first order with respect to either the active or inactive

species. The rates of these reactions were expressed in

terms of the reciprocals of the rate constants, the

effective time constants.

Solution of the full transient model including the

transient concentration profiles proved not to be feasible

as already mentioned in chapter 2.3, due to the large

difference in the time constants of the energy and mass

balanc6 partial differential equations, giving rise to a

"stiff" system. In an attempt to overcome this problem, the

number of internal collocation. points was increased to

give a cubic approximation in the axial and time directions.

This resulted in a marked. improvement in the approximation

of the step change, but no satisfactory convergence could

be achieved. Rather than investigating the solution of

-100-

RUN 19

Gas Flow Rate 477.1/hr

9 Inlet Xylene ConcentratIon 0.397% tlmeýO 0.515% tTmcAO

9 m 0.0 mTn

1 .0 mIn 2.0 mIn

+ 3.0 mIn UJQ x 4.0 mIn

- cr W-

0

0

C; U-00 0'. 10 0'. 20 0'. 30 - 0'. 43 C'. 50 0'. 60 0'. 70 0'. 00 1.00

LENGTH (DIMENSIONLESS)

Graph 4.12. Experimental axial temperature profiles for an

inlet xylene concentration step increase.

0

0

Or L. 1 CL =9

9

0'. 50 0'. 60 6-70 0'. 80 ý. qu 1'. 00 (01MEN31ONLESS)

Graph 4-13. Experimental axial tomperature profiles for an

inlet xylcne concentration step decrease.

RUN 2IR

Gas Flow Rate 480.1/hr

Inlet Xylene Concentratfon

-101-

=O -444

=o -333

L=O . 556

L=O . 222 L=O . 222 (pellet)

L=O . 667

L=Q . 778

L=O . 889

OUTLET

INLET

.0 TIME (MIN)

Graph 4.14. Experimental transient for an inlet xylene

concentration step increase.

0 tL

-102-

L=O . 333

L=O . 444

L=O . 222

L=O . 222 (pellet)

L=0.556

L=O . 667

L=O . 7713

L=0.889

OUTLET INLET

.0 TIME (MIN)

Graph 4.15. Experimental transient for an inlet xylene concentration step decrease.

C) RIJN 21A C;

-103-

P, UN--23fL C! Cos Flow Rate

370.1/hr tlme<O A04.1/hr tTme;, O

9 Inict Xylone Concentratton

A 0.5077 tlmeýO 0.444% tTme;, O

0.0 mIn

e1 .0 mIn A 2.1) mIn

+ 3.0 mIn x 4.0 mtn

cr U j - n- '? I--

0

T. oo a'. 10 0'. 20 0.30 C'. 40 0,. bo 0'. 60 or... 0'. 90 1 00 LENGTH 101MENSIONLESS)

Graph 4.16. Experimental transient for a xylene concentration

step decrease and a gas rate step increase.

RUN 238 C!

Gas Flow Rate 403.1/hr tme<O 370.1/hr tlme>O Inlet Xylene Concentratton 0.444% tIme<O 0.510% tlme>O m 0.0 min

1 .0 min 2.0 m? n

+ 3.0 m7n x 4.0 min

0

a 't-00 0.10 0.20 0.30 0.40 0.50 O. CO 0.70 0.80 0.90 1.00

LENGTH tDIMENSICNLESS)

Graph 4.17. Experimental transient for a xylene concentration

step increase and a gas rate step decrease.

-104-

-=O -333

-=O -444

L=G . 222

L=0.222 (pellet)

L=0.556

L=0.667.

L=O . 778.

L=O . 889

OUTLET I NLET

-0 TIME (MIN)

Graph 4.18. Experimental transient for a xylene concentration

step decrease and a gas rate step increase.

(I UI r)o

-los-

C- 0

L C

I-

TIME (MIN)

=O . 333

. =O -444

-:: 0.222

L=O . 222 pellet)

L=O . 556

L=O . 667

L=0.778

L=O . 889

THLTkTT

.0

Graph 4.19. Experimental transient for a xylene concentration

step increase and a gas rate step decrease.

RMN 230

-106-

C!

C?

tljc? W- : 30 1-0 cr w

0

0

0

0

'b'. oo

-B UNLUX

Gas Flow Rate 437. l1hr tTme<O 484.1/hr tlmc), O In'c', Xylene Concentratton 0.6,07. time<O 0.545% tTme; -O a 0.0 m1n o 1.0 mIn a 2.0 mtn * 3.0 mIn * 4.0 m1n

10 0.20 0.30 0.40 0.50 C-60 0.70 0.80 0.90 LENGTH (0111ENSIONLESS)

Graph 4.20. Experimental transient for a xylene concentration

stop decrease and a gas rate step increase.

/Th\\

WC! cr

C!

-RUN-2-OfL Gas Flow Rate 484.1/hr tlfrnoýO 437.1/hr t1mvO Inlet Xylene Concentratton 0.508% tTme<O 0.610% VmOO

a 0.0 Mtn 0 1.0 Mtn & 2.0 Mtn * 3.0 Mtn * 4.0 Mtn

01 Txo C'. 10 ý. 20 0'. 30 0'. 40 C'. 50 ý-60 0'. 70 0'. 80 1ý-90

LENGTH (DIMENSIONLESS)

Graph 4.21. Experimental transient for a xylene concentration

step increase and a gas rate step decrease.

-107-

iL

LL

c 0

TIME (MIN)

L=G . 333

L=O . 444

L=O . 222

L=O . 222 (pellet) L=O . 556

L=O . 667

L=O . 778

L=O . 889

OUTLET INLET

.0

Graph 4.22. Experimental transient for a xylene concentration

step decrease and a gas rate step increase.

C3 RUN 26A

C3

-108-

L=O . 333

L=O . 444

L=Q . 222

L=O . 222 (pellet)

L=0.556

L=O . 667

L=0 . 778

L=O . 889 OUTLET INLET'

.0 TIME (MIN)

Graph 4.23. Exporimental transient for a xylene concentration

step increase and a gas rate step decrease.

RUN 28FI

-109-

Rmuza Cos Flow Rote

434.1/hr tTme<O 400.1/hr UrnVIO Inlet Xylene Concentratton

ç)c

LLJý

cc , ý=8 LLIý jLM 2-- uj

'b -oo 0-10 ý-20

Graph 4.24. Experimental transient for a xylene concentration

step increase and a gas rate step decrease.

RUN 323

Gcs Flow Rate 400.11hr t? me<O 434.1/hr tlme, -O

9 C,

Inlet Xylene ConcentratTon

v 0.7197. ttme<O 0.626% t1m0,0

a 0.0 mIn a1 .0 m1n

or 2.0 mIn + 3.0 mTn cr

a- C?

x 4.0 mln

C3

9

le 0.20 Cý. So C . 40 0* So MID 0.70 0.00 0.90 1.06 LENGTH (DIMENSIONLESS)

Graph 4.25. Experimental transient for a xylene concentration

step decrease and a gas rate step increase.

0'. 30 0.40 0*. 50 0.50 0'. 70 o'. 80 a'. 90 1'. 00 LENGTH (DIMENSIONLESS)

-110-

cr oý LL

LL F-

TIME (MIN)

L=O . 333

L=O . 444

L=O . 222

L=O . 222 (pellet)

L=O . 556

L=0.667

L=O . 778 L=O . 889 OUTLET INLET

-0

Graph 4.26. Experimental transient for a xylene concentration

step increase and a gas rate step decrease.

RUN 32fl

- 111-

0 RUN 32B C;

C) 0

LLJ

CE Qý LL

LL

TIME (MIN)

L=O . 333

L--O . 444

L=O . 222

L=O . 222 (pellet)

L=O . 556

L=O . 667

L=O . 778

L=O . 889

OUTLET I NLET

-0

Graph 4.27. Experimental transient for a xylene concentration step decrease and a gas rate step increase.

-112-

stiff sots of equations, fiirtlier study was limitcd to tho

temperature transients with tho concentration profiles

going through a series of pscudo-stcady states.

The solution of a normal first order catalytic reaction

taking place in a packed bed under dynamic conditions can

be solved assuming pseudo-steady state concentration prof-

iles in steps of 20 seconds, which easily oversteps the

discontinuity in the concentration and takes approximately

300 seconds of CDC 6400 computer time. In all cases, two

internal radial collocation points with one internal coil-

ocation point in the axial and time directions were used,

except for a brief satisfactory convergence check with

four internal radial points.

Introducing the catalyst deactivation reaction can

drastically increase the computing time, especially if the

time constant for the catalyst deactivation reaction is

decreased to 10 seconds. The pseudo-time constant of the

reactor temperature itself is now very different from that

of the catalyst deactivation reaction so that the system

has almost reverted to the stiff set of equations of the

full transient model. To maintain accuracy, the step size

in the time direction must be decreased. For a final

steady state average accuracy of ±2*C it was necessary to have a maximum step size of 0.80 seconds. This is still

greater than the residence time of 0.2 seconds; however

the time step is now in the range where it is necessary to

consider the full transient model with the dynamic concen- tration profile. This was not done., due to the complexity

-11.3-

of the problem and the computing time necessary. To save

computer time, for some of the runs with a 10 second

catalyst deactivation time constant, only the more inter-

esting first half of the reactor has been modelled. Graphs

4.28.1 30,33, and 35, show the variation of the temperature

and concentration profiles of the model under dynamic con-

ditions. These transients are compared with the experimen-

tal transients in graphs 4.29,31,32,34, and 36.

Up to a dimensionless length of 0.25, the effect of

decreasing the time constant for the catalyst deactivation

reaction is to decrease the initial amount of "right way"

movement of the temperature so that the "wrong way"

movement occurs earlier. For the dimensionless length

between 0.25 and O. SO., decreasing the time constant decrea-

ses the amount of overshoot of the final steady state.

over the last half of the reactor., variation of the time

constant of the catalyst deactivation reaction has very

little effect.

As can be seen from graphs 4.29 and 32, solution of

the catalyst deactivation model under dynamic conditions

when the inlet o-xylene concentration is increased at a

constant gas flow rate, leads to initial right way behav-

lour for the first part of the reactor followed by wrong

way behaviour to a new steady state at a lower temperature.

The reverse concentration change is shown in graphs 4.31,

34, and 36. The excessive amount of wrong way behaviour

occurring in the dynamic modelling at high o-xylene con-

centration (as t--,. -) is due to the large amount of error,

-114 -

Gas Flow Rate 480.1/hr

Inlet Xylene Concentratlon 0.313% tTme<O 0.394% time)-0

0

00

Graph 4.28. Theoretical axial temperature and concentration profiles for a xylene concentration step increase.

-I Is-

11

- 11

L. J- Ck,

re -

I'll - ov--Oo

C3

t_ -0.111

5.00 13-00 TI ME (M IN)

PI IN I ') A

L . 222

-"I experimental theoretical

CL

cr W

15-00 00 5.03 ! 0.00 15-00

C

TI ME IIN

? L=Oz444

- L

1 Cý cr LLJ

Lli

C)

Pl- U-00

C C

C

L =O . 333

5.00 10 -co TI ME (M IN)

L =0 . 556

LLJ

cro C; Lý,

CL

15-00 ,ý . 00 5-C, 0 10 ý co 15-00 TI ME I MIN)

C3

L=0 . 667

- CD -0 UC? L. )C? _Cý

to_ uj

Lli wW cl a: of Of U; ujýý

LLJ tu 0 C) 99

0 -0

00 ". 03 l'o. oo 1,5 co - ý00

5'. 00 l'o-co Is-00 TIME (MIN) TIME (MIN)

C)

w- LzO. 778 to L=C .8 89

- C3 -0

uc? uc?

uj LLJ cr

CD

cr co cr

C; S-Co 1"0.00 1'i, ou . 00 5 . 0.9 "I ___r - l'o. oo 15-00

71 ME IMN171 ME IMIN

Graph 4.29. Theoretical and experimental axial temperatures

for a xylene concentration step increse..

-116-

Gas Flow Rate 482.1/hr

00

Graph 4.30. Theoretical axial temperature and concentration

profiles for a xylene concentration step decrease.

Inlet Xylene Concentratlon 0.389% ttme<O 0.3,247. flmpý, O

. -1 17-

0

-

CL

-0

ui

CY C? a:: W CL

; ýCo

C? C3

- CD UC?

c2 ci cý w0

'b'. o

0

- c2 UC?

L, -j m

CD

C3

I-n-IAI

L=0.333

5.00 10.00 TIME (MIN)

L=0 . 556

5.00 10-00 TIME (MIN)

L=0.778

RUN 21B

L: 0 . 222

1 i. -x-- cx j) c l' 1

aill, the 0 ret iczi 1 cl: '? tr ýý LAJ I CL Y-

CD

TI ME IMIN

9 L=Cz444

0 Cý

I ui

cr Cý of C, L, J Q- z LL,

C, C? LO

is . 00 co S. 30 l'o-oo TIME (MIN)

C? L=O . 667

- CD

UC? - CD (0. LLJ m

tr

cro w,; UJ ý.

_x r_

115.00 -01) 5, -co i'o. oo TIME (MIN)

c2

0 L)c

cro Ir

17. uj

15-00

15-00

15-00

Ca 5'. 00 JO . 00 is-Do ", 0700 5'. 00 1,0-00 1 . 00

TI ME iM IN ITI ME I MINI

Graph 4.31. Theoretical and experimental axial temperatures

for a xylene concentration step decrese.

T INE (MIN)

-118-

RUN 19B

C3 9 C3 9 C3 L =0 . 222

-C3 -0 O exporimental L. JC? U -Ul W. theoretical t, tim LIM CE w 2 Cý UJ . CL-

LJý- CLM

Z Y- uj - L.

C, 9

-00 5.00 10-00 15-00 'b oo 5'. 00 1'0-00 I'S. 00

C3

TIME (MIN) TIME [MIN)

Cý

5.00 . 00

Graph 4.32. Theoretical and experimental axial temperatures

for a xylene concentration stop increase.

TIME (MIN) TIME (MIN)

-119-

Gas Flow Rate 480.1/hr

Inlet Xylene ConcentratTon 0.508% tTme<O 0.392% tlrne->O

4

30

Graph 4,33. Theoretical axial temperature and concentration

profiles for a xylene concentration step decrease.

I t=O

Q 92 t= 38.4s C3

-120-

RUN 2 IA C,

L. . 222

ul cr

cr cl.

CL a_

't . 00 5.00 : 'c co is-Go lo -c01,5-00 TIME IMINI Tl,, 'IE (MIN)

L: 0-133

00

CL ic r

C?

. LOO

lý, Co ; 's cc TIME (MIN) IME (MIN)

9

L

C3 cr

I .- c o 5 . ýO bI- 11.00 TI 4E

L: 0.776 L: 0.889

ýJ9

cr

cl

7, J 1 t, 7 Irt I Milo I Mt Im1

ý, r. "i I ýIi 4.34. Theoretical and experimental axial temperatures

for a xylono concentration step decrease.

_1) I

RUN 30A

0 C!

Gas Flow Rate 477.1/1-ir Iniet X), Iciie Colicentration

0.52,1"0 tiiiie <0 r) A77ý. t-imp >0

00

C)

Graph 4*. '3S.

profiles for Theoretical axial temperature and concentration

a xylene concentration step decrease.

-12 21 -

C) RUN --) 0A

L70. III

ý; o

C? u)

-n

LLI ui w cr

CL C3 Cl

cr C; cr C, U- -

'b -oo E . 09 10.00 15-00 co TI ME IHINI

C)

9 0 L=0-333

L)

LLJCý q- L'i C? , : DO

" :Dc: )

cr X

C C ? ?

4

0

-4- 'b -oo 0 C3

- CD

uc?

ju -

C3 a

xý ujý CL

koo 0 C?

0

CL9 Ckf ,.

I,

,. r-

5.00 10-00 TIME [MIN)

L: O.? 22

experimental theorcti-al

5 -Do 10 -CO TIME I MIN)

0 9 tn L=0.556 r--, L=0.667

00

9

Cc uj

10-00 15-00 10-OC 15-00 TIME (MIN) 71ME (MIN)

C?

L =0 . 778 LzO. 889

C?

ui-

-0 cr

Loo CL

-C" 3 -CO 115-00 TI ME NI

15-00

Graph 4.36. Theoretical and experimental axial temperatures

for a xylene concentration step decrease.

TIME (MIN)

-123-

yesulting froill the stiffiless of the differential equations,

up to 40C in the worst case.

When the air flow rate was decreased, resulting in an

increase in the o-xylene flow rate and hence a greater in-

crease in the o-xylene concentration than that which would

have been expected (see chapter 3), the temperature at the

dimensionless position 0.222 stayed approximately constant.

The hot spot moved very slightly towards the inlet and in-

creased in height, see graph 4.17. The anomolous behaviour

originally found is not obvious (see graph 4.19), over the

first part of the reactor in this case. Due to the dec-

rease in the total gas flow rate, all the heat and mass

transfer parameters would decrease, and the slight forward

movement of the hot spot would balance the expected temp-

erature decrease of the wrong way behaviour. Again this

effect was completely reversible experimentally, see graph

4.16. The model for the flow rate increase (graph 4.37)

shows that the hot spot is actually stationary, which is

in agreement with the experimental results. Graphs 4.37

to 4.39 of the model, show a small initial overshoot of the

final steady state for the first part of the reactor, how-

ever the trends are well demonstrated for the latter part

of the reactor.

Due to the large amount of computer time necessary

to calculate one transient., normally 1700 seconds on a CDC

6400, no hon-linear regression of the dynamic parameters

was carried out, so that the time constants for the con-

version of V205 to a lower non-active oxidation state, and

-12.1-

DO

Graph 4.37. Theoretical profiles for a xylene concentration

step decrease and a gas rate step increase.

Gas Flow Rate 370.1/hr tTme<O 404.1/hr tlme;, O Inlet Xylene Concentratton

-125-

RtJN

W L3 -11 1L :0 . 222

CX 1) CFI III C 11 'L L )I.

LI 10 o 1,0 tca

cr: cr

a. CL

LU

"'o-Go i's -a C) 05-cc 10.00 15-00 I irit I MIN ITI ME ImINI

0 C? 9 L: 0.333 LmOa444

u

cr a

ui K

C? C? L, J 0 Lýj Ln

C?

L15 o ao

ý5'- co lo-oo 1,5-00 ao 5'. 00 Jo-oo I15.00

IME (MIN) TIME (MIN) cD c3 9 c? Lo a L =0 . 556 r-_ L=O . 667

0

C)

CE cr x af C; LLJ Lo UJ Lo

I, I_- I'-- L-

Uj ui C, C?

0 Go 5.00 10-00 O-oc 15-00 T IN. EMINJTI ME MIN

00 9 c?

U)_ L=0.778 tfl- LzO . 889

-0 -0 L. ) C? uc?

M Ln In. Ln.

LAJ I'm

cro crm

LLJ U). Q_m CL

C, C?

-Do 5'. 00 : 'o

-00 15 I- C-0 'b

.co S-00 1,0. (30 15-00 TI ME (M1N)II ME ( MI N)

Graph 4.38. Theoretical and experimental temperatures for

a xylene concentration step decrease and a gas rate step increase.

-I

Li

t.. O. III

5- co 10 -03 T IME [MIN)

L=0 - 333

MIN

C, r-l D

cr

L: 0-2? 2

Iii-

Lu

W

'b'. oo

Co

In Cl

01 C-)

uJ 0!: ; :D

Ujo

r

(U "

exporiniontal theoretical

týA

CD

'. OD

0 0

0 a, 0)

0 CD

In

u

ujc,

cr

LU

M

ý. Oo 10-00 15-00

I f-I E1 111 N

L=Oz444

t k-----X

0 5.00 lb -00 '15

. 00 'b oo 5'. c0 1'0-00 1 . 00 T IIMEIMINTIEIM 11 N

Graph 4.39. Theoretical and experimental temperatures for

a xylene concentration step increase and a gas rate step decrease'..

-127-

the reverse reaction, were approximately determined by a

few trials. The original literature heat capacity of the

bed 0.839 KJ/Kg. OC was used unchanged in the dynamic mod-

Olling.

The time constant derived from the dynamic modelling,

for the deactivation reaction) was estimated to be 10 sec-

(Ther values of the onds (rate constant = 0.1 s-1). For hi,,,,

time constant the overshoot for run 19B, when the concen-

tration was increased from 0.397 to 0.515 mole%, was very

large and resulted in a breakdown of the computing algor-

ithm. With this low value of the time constant, poor

convergence of the temperature profiles to the final steady

state has resulted, due to the stiffness of the partial

differential equations.

For the reverse case, a decrease in the outlet o-

xylene concentration, a time constant of 10 seconds for

the catalyst reactivation reaction results in the disappear-

ance of the initial right way behaviour completely. This

is due to the exponential temperature dependence of the

rate of oxidation of the o-xylene, so that the overall

system is definitely not first order. All the graphs shown

for the reactivation of the catalyst have a time constant

of 100 seconds. Of these, only run 21A (see graph 4.34)

indicates that the time constant is too large. A final

value therefore for the catalyst reactivation time constant

is approximately SO seconds (rate constant = 0.02 s- i

).

12 8-

Chaotor SC0N C-11, t JS 10 NS

The yield in the pilot plant reactor used for the

partial oxidation of o-xylene to plithalic aiihydride was

found to average 78% or 109 Kg plithilic anhydride per -100 Kg o-xylono witli a purity of greater than 97 mo1c, 90 at a

bath temperature of 3SO'C. This is comparable to the yield

obtained commercially. It was shown thall - the amount of

carbon monoxide produced was not negligible and had to be 0 Z,

included in the calculation of the overall heat of reaction.

The axial temperature profiles showed anomolous

behaviour for the first quarter of the length of the reac-

tor where an increase in the inlet o-xylene concentration

resulted eventually in a decrease to the new steady state

temperature, hence there must have been an overall decrease

in the heat evolved per unit reactor volume by the react-

ions taking place. As the possible network of reactions

was modelled by a single overall reaction, the rate of

this reaction must decrease at high o-xylene concentrations.

The physical properties of the system were first eliminated

as a possible cause, leaving some kinetic effect where the

apparent order of reaction must become negative at high

o-xylene concentrations. This eliminates most of-the

reaction models that have been tried for this or similar

systems leaving the bimolecular Langmuir-Hinshelwood and

the previously derived catalyst deactivation model. The

latter model was selected as the simplest feasible model

for both the steady state and dynamic modelling carried

out.

-129-

v205+2 (o-xy. tc, no)

o-xylono +02r -4- products

As the concontration of o-xýrlciio increases , the activo V20S

is deactivated to a lower inactive oxidation statc. The

overall rate of reaction is proportional to the V205 con-

centration resulting in the equation

k C r x

I + K c2 eq x

All the modelling carried out -was based on the catal-

yst deactivation model with an estimated value for the equi- 4. 'ines the order librium constant Keq of 9-00 x 10 This del

of reaction., initially 1.0 at zero o-xylene concentration

to be zero at 0.333 mole%. Above this concentration, the

order of reaction is negative.

The reaction modelling carried out solving the homo-

geneous partial differential equations proved to be of the

same accuracy as the heterogeneous case using literature

values for the heat and mass transfer coefficients between

the gas phase and the catalyst pellets. The parameters

calculated from the non-linear regression analysis were

1.10S X 10 8 KJ/Kmole for the activation energy E a, with an

error of ±4%, 20.4 for the exponential frequency factor S

with an error of ±61o. The following correlation is sugges-

ted for the wall heat transfer coefficient

Nu = 8.23 + 0.054 Re Pr

This is almost identical to the correlation proposed by

- 130-

Yagi and Walao (72) I-or the data of 11.1autz and Millstone

(67) To 111aillt, 11-11 111 accuritc fit over al. 1 the cIntail Llic

effective rnCiial conductivity had to be increasod, as the

inlet o-xylenc concontration increased, so that a corrola--

tion of -the following type is suggested

Kek Oe

-+0.10 Ro Pr k9k9

where the stagnant conductivity term k0 /k -increases

from e9

11.7 to 26.0 as the inlet o-xylene concentration increases

from 0.31 to 0.6 mole%. The necessity for this is to

compensate for either the effect of axial dispersion of heat

or excessive temporary deactivation of. the catalyst at the

higher temperatures encountered with higher inlet o-xylene

concentrations. The lowest value for the stagnant conduct-

ivity term was just slightly above the literature corre-

lations for non-metallic packed beds. The literature

correlations available for the pellet heat and mass trans-

fer correlations are reasonable. At the temperatures

studied, up to 4100C, film heat and mass transfer limitat-

ions do not occur.

For the modelling of the dynamic case, the trends of

the experimental temperature profiles were well demonstrat-

ed by estimating the forward reaction for the deactivation

of the V205 to have a time constant of 10 seconds with 50

seconds as the time constant for the catalyst reactivation

reaction. The overall wrong way behaviour as well as the

small amount of initial right way behaviour of the first

part of the reactor were shown. This behaviour could also

be demonstrated by the slightly more complex bimolecular

-131-

Lailgilluir-Ilins lie livood model.

The orthogonal collocation mothod proved to bo well 0

suited for the solution of the partial differential oquat-

ions for both the horiogeneous -ITid hotorogcneous models,

and especially for carrying out the non-linear regression 0

analysis. The application of the orthogonal collocation

method in all three dimensions for solution of the dynamic

case proved to be satisfactory for normal first order

reaction models. For the inclusion of the catalyst deact-

ivation effect however, the error and the computing time

increased drastically as the time constant for the catalyst

deactivation reaction decreased resultinc, in a "stiff" set

of equations.

At low bath temperature 340-346'C, two hot Spots Were

observed which implies that at least 3 series reactions

were taking place. The simplest most favoured model was

o-xylene phthalic maleic carbon anhydride anhydride oxides

In these runs the average phthalic anhydride yield was 65%.

-I i 2-

SUGGESTIONS FOP, FURTHER WORK

To improve StýlbilitY Of the aXiZIL tCIIIpCraLUI'0 Profile,

especially at hi, (-h inlet concentration-, of o-xzy I Clio , tile

-tion reactor bath should have somo form of forced convoC

imposed upon it. This could be achieved by using a fluid-

ised bed or a salt bath with mechanical agitation, a circ-

ulation pump or injection of air or nitrogen to improve

the heat transfer.

Improvement of the analysis and the sampling of the

concentration at intermediate points along the bed would

result in improved data to model the system as a network

of reactions each with its own heat of reaction and not an

overall heat of reaction as used in -this study. All of

these are being incorporated in a continued large scale

study.

If possible, heat transfer data should be obtained

under non-reactive conditions to produce correlations for

the system as the values in the literature are extremely

scattered. An improvement may be obtained in the non-

linear regression for the effective radial conductivity 4ý

and the wall heat transfer coefficient, from either

separate analysis or with reaction if the correlation

between these two parameters is eliminated by using the

variables U and either hW or ke as defined by the equation

1-1R

uh1.91 k

or an equation of similar form.

I 33-

A study shouLd be undortakon to det--oriiiiiie if ýi distilic-

tion can be made boLween the catalyst dcacLivýifýion model

and the bi-molecular Laiiý, iiiiiii--Ili-r, ýýliolývoo(I iliodel, especially

under dynamic conditions. To solve the pllrtiaL difforential

equations accurately liowever, both these cases may require

the use of a different algorithm to eliminate the stiffness

in the energy equation. This is due to the large difference

in the time constants between the reactor tempera-Lure and

the kinetic constant either for catalyst deactivation or

adsorption/desorption of the reactants.

-134-

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- -1 11 () -

Appendix I mivrmý, m,, vm, Al, MODELS

1. ffoi, ýiogencous Stcidy_ State

Energy balance

Cp PU -DO, R All 'D

Wý 6

g9 ýz rr Dr

with the boundary conditions

ke 30 1--hwa C) at r'= R

3r

0 at z'= 0

Mass balance

uDCR+cD1, DC

Dz' r0r Dr 3r

with the boundary conditions

I ý-C =0 at r'= R

Dr'

C=C at z'= 0

Dimensionless transformations

DT g(T, C) +1D (r ýT

ýz r Dr 3r

ac g(T, C) +Y1ý (r 3C

3z C0r 3r Dr

with the boundary conditions

DT - Bi (T -T at r=1 C

DC 0 at r=I 3r

TC at z=0

where

T 0-0 base c=c Oo-Obase Co

rZ r=-Z RZ

-AH kZ

c (0 Obase) c GR p90 pg

(T. I)

(1.2)

(1.3)

(1.4)

(1.5)

- 14 1-

RZ (T Qf

G

Bi

ED Z

Ru (1 . 6)

2. Hoterogeneous Steady State

Energy balance 0

- All Rh (0 -0bak1 (r, ý) (1.7) pppr ýr 3r'

30 b19 90 b Pg 9azp P_ b)a+ke--, (r -) (1.8)

br ýr Dr,

with the boundary conditions

DO keP

Dr hw (0

P_ 0c at r'= R

p 90 b k-h (0 at r'= R eb 3r' wb bC

Ob 00 at z'= 0

As the solid radial conductivity ke is to be lumped in

an overall gas phase conductivity, the last term in equat-

ion (1.7) can be ignored. Similarly the pellet to wall

heat transfer is lumped in the following overall gas phase

to wall heat transfer boundary condition.

k 90b

h (e b- 0 at r'= R e

Mass balance

RrKG (Cp Cba (1.9)

II 'Cb K (C CD 1'ý (r'

Kb) (I. 10) ýZ' Gpber 3r 3r'

-142-

with the boundary colid it ýO I' S

A 'Cb

0

cbc0

Dimensionless tr

Energy Balance

g(Tp. *C p

3T b 3z

ans f ormati on s

= st ht (T p-

TO

= st ht (T p-T0

Mass Balance

g(T C pp st (C -C CO mt. p b)

3Cb st (C C

3z mt. p b)

with boundary conditions

DTb Bi(T F

3r bC

3C b Dr

Tb Cb

at r'=

at z'. --

ID 9T b r 3r 3y

(r 3Cb)

r 3r Dr

at r=1

at r=1-

at z=I

(1.11)

(1.12)

(I

(1.14)

The modified Stanton numbers for heat and mass transfer

are defined as:

h st ht c P9

pGaZ K

st ýG aZ mt u

-1,13-

vnzini CII Otei. oiý('Ilco I Is MO (I Ol

Energy balance 30

- All rRr

11 (0 - 0 b) a+C 11 --E , (I IS)

p p pP 3t

ýOb U -5- -; - CP P h (0 - 0b) I-ý ý0ý)

a+ ke ,

(r'- z g 9 p p

CP Pg (1.16) 3t g

with boundary conditions as equation (1.8)

Mass balance

3C Rr K G(C Cb a+a6 --P- (1.17)

p at

9C b ' - 3C i D -b

3z, K G(CP C ) b ' ý(rA

) a+6D er 3r ar, A 3C b

DC

with boundary conditions as for equation (I. 10), and the

temperature and concentr ation profiles defined at time t=O.

As the heat capacity of the gas is negligible compared 0

with that of the catalyst, the last term of equation (1.16)

can be ignored. The volume of the boundary layer is negli-

gible therefore the last term in equation (1.17) can be

ignored.

Dimensionless transformations

Energy balance 3T

g(T St ht (T -T b) + --P- (1.19) pPP at

DTb St (T -, T+13 (r

'Tb) (1.20)

3z ht p b) r 3r ar

Mass balance

g(T C) PPSt (C c (1.21) co nit P b)

DC b1Kb ac b 3z

St III t(cPcb+Yr-)-c r Dr Dr 3t

(1.22)

with boundary conditions as for equations (1.12) and (1.14)

and the temperature and concentration profiles defined at

time t=O.

where Cp-p Pp Zt6z

CPg Pg tr uru

-145-

Appcndix H Tlljý Ol, ', 1'110(', ()NAT,

I ji tro (lit c ti on

The orthogonal collocation method is ono of a gonoral

class of approximate methods 1ýnown as the method of weigh-

ted residuals (104,105), which includes the Galerkin .

integral, and moments methods as special cases. In these

methods the unknown solution of a system of differential

equations, with boundary and initial conditions,, is expan-

ded in a trial series whose functional dopendance on

position is chosen, but which includes undetermined func-

tions of time, or a further dimension. These functions are

found by requiring that the trial solution approximates

the differential equation, according to various criteria.

Derivation Of The Orthogonal Collocation ', Tethod

For Radial Profiles

As a specific example solve the dimensionless equation

(1.3).

DT + g(T, C) +ID -(r

DT 0 3z r 3r 3r

where

3T Bi(T -T at r (II. 1a) 3r c

at z=

The temperature and concentration are expanded in a

series Ea 1

(z)Pi(r) of defined functions of radius, Pi(r),

multiplied by unknown functions of z) ai(z). As the

solution for the radial case must be symmetrical about

r=0 expand the trial function T(z, r) in terms of r2.

Therefore define the trial solution for a fixed radial

-1,16--

soction as

N (z, r) E 1. (Z) P. (Tl) N i=o 11

(II . 2)

This trial function is substituted into tho partial differ-

ential equation (II. 1) to form the rosiULial.

R(ai ih z, r) _ ýTý

+ß g(, F, c) ýz

N Z ai(z) Pi(r')) (11.3)

r Dr Dr i=o

If the trial function were the exact the solution., the

residual would be zero. In the various methods of weighted

residuals, the constants ai(z) are derived in such a way

that the residual is forced to be zero in some average sense.

Therefore set the weighted integral of the residual,, fr

wi R(ai, z, r)dr to zero, with the weighting function still

to be selected.

Taking the weighted integral of equation (11.3)

0w j( _ 3T

+ g(T, C))dr fr

ýz N

+a ai(z) W. (r pi(r2) )dr (11.4) fr

Jr 3r 3r

In the collocation method, the weigliting functions are

chosen to be the displaced Dirac delta function

w 6(r -r

which has the properties fr

wiU dr Ulrj

P dr Iý- (-r -L p 2) )

r ýr 3r r ýr 9r i(r Irj

-147-

So that this weighting function forces the solution to be

exact at the specified points rj, the collocation points.

Therefore equation (11.4) reduces to

0 ýTj N + g(T, C)l +aED. - ai(z) (II. S)

3z rj rj i=1 31

where

D. - =13 (r D Pi(r2)) 31 r ýr 3r

Irj

and i= 1ý2 ........ N

j=1,2 ........ (N-1)

i. e. this is not valid at the boundary.

If the boundary condition is of the type T=l at r=l then

it is included in the trial function by specifying the

ai(z) in equation (11.2) so that the trial function fits

the boundary conditions and equations (11.5) and (11.6)

are now valid at the boundary. The boundary condition in

this case however is

0 3T + Bi (T -T at r=1 (11.7)

3r C

Following the same steps as previously

N 0 Bi (T -T c)lr=l +

_iE,

Cji ai(z) (11.8)

where

c--=dP (r 2) 31 dr i

and i= 1ý2 ........ N

j=N only.

For problems involving chemical reactions with an

Arrhenius temperature dependence, the degenerate orthogonal

collocation method is the most feasible., as the other-

- L48-

mothods of wci, (, -Th. tcd Ily I'C(II-I-LI'C CVýIhlýltiOll

of complicated iiitc(,, rals involving c--ýpoticiitlzil functi-olls. I

In the collocation i,, iothod, the cliffercntia-1 equation

is satisfied only at discrete points , called collocation

points. Originally the choice of both the trial function

and the collocation points for boundary value problems was

somewhat arbitrary until Villadson and Stewart (44) defined

the trial functions as orthogonal Polynomials and the coll-

ocation points as the roots of these equations. The coll-

ocation points are also the optimal quadrature points for

the numerical integration of the solution over the same

region. The orthogonal polynomials are defined by:

I fo 111(r2) p, (-r2) pi (r' ) rot- 1 dr 0

where j=1,2 ........

(i-l)

Wl, r') is the weighting factor

Pi(r 2) is a polynomial in r2 of degree i

a the geometric factor =1 for planar geometry

=2 for cylindrical geometry

=3 for spherical geometry

The weighting function W(r)=1 gives rise to the

Legrendre polynomials. P is taken as 1 therefore this 0

completely defines the trial functions

P11-2r2P21- 6r 2+ 6r 4P3......

whereas W(r2)=l-r' defines the Jacobi polynomials

PO =1 Pi =1- 3r2 P2 =1_ 8r2 + lOr4

The collocation points are denoted by rj and are the roots

-149-

of tho Orthogonal Polynomial

1) N (r2) =01,2

Since the functions PI arc known, C ji and D ji

can be

calculated from equations (11.6) and (11.9) respectively.

Therefore equation (11.5) for collocation points &N-1 and

equation (11.8) for the Nth. collocation point at the

boundary can be solved simultaneously by an iterative

method to find the ai(z) which leads to the approximate

solution in equation (11.2).

Computer programmes however are simpler if written in

terms of the solution at the collocation points rj rather

than trying to evaluate the ai(z) and the trial function.

As PN (r') is a polynomial of degree N in r2 equation (11.2)

can be written as N+l - T(z, r) =Z r2 1-2 1=11

or evaluating at the collocation points

N+1 - T(z, rj) r 21-2

d. i=1-3 1

Taking the first derivative and the Laplacian of this

expression

T (z �rj )

3r Irj

19 (r T(z, rj)) r ýr Dr

lrj

N+l d 2i-2 E -r d.

i=ldr

N+l Id E- -(r dr

i=lr dr dr

(11.12)

ri di

These equations can be rewritten in matrix notation as

follows. Note that the (N+l)th. collocation point is r=1

and square matrices have (N+ 1)2 elements.

-1 50-

Qd aT

= Cd Dr

ID(r DT) = Dd

r Dr Dr (11 . 14)

where:

Qj

Solving for d

dr2i-2 dr

I ri

D. - 1d (r dr 2i--2)

ji r dr dr I-rj (I I. lSa, b c)

CQ-1T = AT (11.16)

Iý (r 2-T) DQ-'T = BT (11.17) r Tr ýr

Therefore the derivatives are expressed in terms of the

values of the function at the collocation points. By

substituting equations (11.16) and (11.17) in equation

(II. 1) and (II. 1a) the original partial differential

equation is now reduced to an easily solvable ordinary

differential equation.

dT N+l + g(T, C) +EB. -T0 (11.18) a-zlr

j

with the boundary condition

N+l -EA-T. Bi (T T) at r (II. 18a) i=l N+l 11 N+l w

For a first approximation, the simplest case N=l has

only one internal radial collocation point. If Jacobi

polynomials are used the collocation point r, =l//3-. From

appendix IV substitute the values for matrices A and B

into equations (11.18) and (II. 18a).

dTj g(T, C) [-6 61 -T (11.19)

dz 1

-T N+l_

-I. Si-

- [3 3] -T

IIi [T TI. 19 a)

-T N+l_

Combining equations (11.19) and (II. 19a)

dT, [ ---Lul -ý) (T T+ (T , C) (11.20)

dz Bi +3

U_ Equation (H. 20ý is very similar to tho ono-dimensional

lumped parameter model

dT - Nu , (T -T)+ß (Y(T, C) (11 . 21)

dz w r>

where Nu' 2UZ CP

9 GR

and U= lumped heat transfer coefficient.

The lumped parameter model assumes a constant temperature.

across the reactor. As the simple orthozaonal collocation

case considered has only one internal collocation point,

equation (11.20) refers to a temperature at a particular

radius where the temperature varies parabolically with r.

Equating 6aBi/(Bi+3) from equation (11.20) to the

modified Nusselt number in equation (11.21) and expressing

in terms of heat transfer resistances

1=1R uh3k (II . 22)

for Jacobi weighting and similarly for Logrendre weighting 0 I=I

-iL uh4k (11.23)

Those equations are of the same form as the expressions

for the lumped heat transfer coefficient derived theoret-

ically by Beek and Singer (101), and Crider and Foss (102).

As Legrendre polynomials have a weighting of unity the

-1 S2

ate Of rezIctio I j"

C, X, , III Z, tCII r

eraturc P)y Lis ing Jacobi po I), nomi a Is, the reac- L ion i'atc

expression is evaluated at a temperature zibove the ýývcragc

radial temperature. In low order collocation sohitions,

this approximates more closely the average rate of reaction

due to the exponential effect of the temperature where the

average temperature does not yield the average rate of

reaction. With higher order solutions, the difference

between weighting methods becomes negligible.

Derivation of the Ortlio, ý, Tonal Collocation I'vlothod -

For Axial Profiles

As the solution for the axial case will not be symm-

etric in the region 0, "zl-l expand the trial function in

terms of z

M+l TM(z, r) j a-(r) Pi(z)' (11.24)

1=0 1

substituting the boundary condition at the inlet

M+1 TM(z, r) T(O, r) + Zoai(r) Pi(z) (11.25)

with the orthogonality requirement

fo W(Z) Pj(z) Pi (Z) dz 0

Unlike the previous section there is no general rule for

choosing the weighting factor W(z), therefore select the

simplest case i. e. the Logrendre polynomials.

W(Z) 1P01 PI 1 2z

P2 1 6z + 6z2 P3......

To simplify the computer solution as before solve . equation (11.25) in terms of the solution at the collocat-

ion points.

-153-

1 14+2 T(z, r) Xz L-1 d

I=Ii (1-1 . 26)

Included in the suiiiiiiation term is T(O, r) Of C(jUatiOn

(II. 2S) as this is normally equal to 0 or 1. Evaluating

equation (11.26) at the collocation points

M+2 - T (zj , r) Zz1-Id. (II . 27) 1=131

Taking the derivative and Laplacian of this expression

M+2 d T(zi, r) E z1_11 d. (11.28) ýz

Izi i=1 -d-z zi

2 M+2 d2 i- 1 T(zj, r) zZ2zd. (11.29) TýT Izi 1=1 d-

Izi 1

Note that the first collocation point is at z=O, the

(NI+2)th. collocation point is at z=l and that each square

matrix has (11,1+2 )2 elements.

T= Q*d ýT = C*d ý2 T D*d (H. 30a, b, cý 3z Z2

where

Qý = i-1 Cý. =di- lizi zi

C, zz

zi a., c) j1 -Z,

Solving for d as before

ýT C*Q*-1 T= ET (11.32) 3z D2 T D*Q*- 1T= FT (11.33) 3Z2

The derivatives are now expressed in terms of the values

of the function at the collocation points.

-lS4-

Interpolation

To determine the solution, at a point othor than a

collocation point, equation (II. 10) and equation (11.26)

are used for interpolating in the radial and axial direc-

tions respectively. The interpolation is of order 2N

radially and M+1 axially. For radial interpolation, use

equation (II. 10)

T(z, r)

where d Q- IT

N+l 2 1-2 (11.10)

For centre line conditions expand equation (II. 10)

N+l - T(z, O) r0d, +Er 21-2

d. i=2 1

As r=O T (z, O) = di

where N+l

Z (Q T =IIji

(II . 34)

For axial interpolation use equation (11.26) where

T

Matrices A, B, E and F are listed in appendix IV and

Finlayson (105) and Villadsen(106) for low order collocat-

ion solutions. For higher order solutions the collocation

points are listed in Stroud and Secrest (107). When the

collocation points are known Q, C, D and Q*, C*, D* can be

calculated from equations (II. 15a, b, c) and (II. 31a, b, c)

respectively. Then A, B, E and F can be calculated from

equations (11.16), (11.17), (11,32) and (11.33) respect-7

ively.

-IS5-

o nV or c"ll-C 0

Rigorous proofs of convergence for the orthogonal

collocation method have only been obtained for extremely

simple problems. Finlayson (108), proves the convergence

of the Galerkin method for the sol-Lition of radial diffusion

and reaction in a tubular packed bed reactor with an

Arrhenius type rate of reaction and postulates (105) that

if the Galorkin method converges then the collocation

method converges. Villadsen and Stewart (44) show that

for linear differential equations with constant coefficients

that the orthogonal co'Llocation and Galerkin methods yield

identical results and that the methods are exact for any

even polynomials of degree <2N where N+l is the total

number of collocation points.

Evaluation of Intenrals

To evaluate integrals accurately the quadrature

formula is used.

1 N+l f(x') xa-1 dx =Z wj f (X? ) fo i=l 3

To determine the vector w integrate f(X2) = X21-2

f1x 2i-2 x a-1 dx =1

o 2i 2+a

N+l therefore wj x

21-2

2i 2+a

2

as Qj xj IVQ

and fQ-1

-S(-

App(iidix III AI'1'1,1(', i%'! '[oN OF TliE C01, U)CAT10,114

To solve the rclcýraiiL partial different iýil equations

by the orthogonal collocation mothod, the reactor is divid-

ed into L equal axial collocation steps, with cach stop

having N+1 radial and M+2 axial collocation points, see

fig III. I. Tile axial subdivision of the reactor is necess-

ary to improve accuracy while kee-ping the resulting matrices

to a manageable size. As each axial collocation step is a 0

'reactor with. length Z/L, any terms that include Z, after

the dimensionless transformations, must be divided by L so

that equations (1.5) and (1.6) have to be modified to

GR2 cp

L R2u L 9

lst. collocation

Z=O step

1

2

N

N+l

0 x A

F i!

--4. --- -i -----

73; 73

2nd. collocation step

r=O

----------

Z=z

2 M+l M+2

known temperature and concentration

unknown temperature and concentration unknown boundary temperature and concentration

III. 1 PLAN OF ORTHOGONAL COLLOCATION POTNTS

For the general case using Jacobi polynomials with

N=2 and M=l., the radial collocation points are /(4- rb)110,1

1-(-4+V6)11O, 1 while the axial collocation points are 0,1,1.

'7- -151

For the solution at the LlItOnin co IIII oca ti on po in ts

equations (11.17) and (11.32) for the collocation forms of

the derivatives, are substituted into equation (1.1) with

a modified as above. Similarly for the external collocation

points or boundary conditions, as in the simple case NO,

demonstrated by equations (11.19) and (II. 19a).

T 21

T 31

E 21

E 31 g(T 'C)21 g(T, C)s

E 22

E 32 g(T 'C)22. g(T 'C)3

T 12

T 22

T 32

E23E33

CL Bi

1B 12 B13T

21 T31

B 21

B 22

B 23

T 22

T 32

-T23T3s.

with the boundary conditions

[A

31A32A3 3] T21T3

1- Bi

I (T

2 3- T (T

3 3- T

T 22

T32

T 23

T 33

J

with the notation for T jk, Cjkl and g(T, C) jk only

j= axial collocation position

radial collocation position.

Similarly for the mass balance equations

I1213 11 213 1- [g (T 'C) 21 g (T Q31

C 12 c

22 c

32 E 22 E 32 CO[g(T

'C)22 g(T 'C)32

-E23E9S.

(I I 1.1)

(I I I. la)

11 B12B13c21 cil

E 21 B 22 B 23

c 22

c 32

c 23

c 33

with the boundary conditions

[A

31 A32 A

33] c

21 C31« =

10 01

c 22

c 32

c23c33

(I I 1.2

-158-

e--%

Cd

c, n r- 0

. r-4 4-) as ; zs al 0)

t4

u u F. pq

aa ca ca ca

PCI

C3 d

0

H PQ +

.

N N

N N N - I "'

N . 0 L 0 N

N

N 14 -. - N

N 01 01

- < 0 0 - I

0

+j

u 9 Cd

ýc

0 (4-4

r-4 H E4

U)

- U 0 0 0 0 u u u u

.4 Cd en en C4 a, Cd N Cd Pl N

UUUUUU

0 cq

pq

ro

rQ 21:

CQ o C)

c4

C4

ý' tl- P

-159-

The temperature and concentration at the start of

each step are known so that the solution for the (N+l)x

(M+l) unknown points of each step can be produced by

iteratively solving the matrix equations (111.3) and

(111.4) until the solution has converged. For the first C>

step, the best initial estimate. for the temperature and

concentration is all T=C=l i. e. the initial conditions

of the dimensionless equations. After the first step,

and for each successive step, the best estimate for the

value of T and C at the new collocation points can be

obtained by extrapolating the solution of the previous

step using a modified form of equation (11.26)

M+2 T(z, r) Z di (1 + z) (111.5)

i=I

If M=l the solution for each step is a quadratic

function of axial distance, with the overall solution

consisting of interconnected quadratic arcs. Normally

N=2 to 6 and M=l or 2 so that excessive computer time is

not used in evaluating the inverse of the left hand

sides of equations (111.3) and (11.4), which are of rank

(M+l)x(N+l) for the 1 phase model and twice this for

the 2 phase model. These inverses, although slow to

calculate using Gaussian elimination., or Crouts factor-

isation method, are only evaluated once and are then

stored and re-used until the final solution to the

complete reactor is obtained.

-160-

To solve the -1; tel(-Iy S Ll Lc lie Lerogencolis model , e(JIMt-

ions (I. 11) a nd (I- 12) are t re atcds im iIai, Iy to th 0 110), 10 -

goncous modol

(T p-

TO 21

(T p-

T b) si g (Tp pC p21g

(-rp .cp)31

St ht (T p-

T b) 22 (T

p- T b) s2 g (T

pcp 22 F, (TpjC P) 12

(T p- b) 23

(T p- b) 3s g (T

pcp23g (T

pcp)33 LL

b, iT b2l Tb31E21E31 St ht

(T p-

T b) 21 (T

p- Tb

Tb 12

Tb 22

Tb 32E 22

E32 (T p -T b) 22

(T p-

T b)

:

21

EE L2333

+ C, B

11 B

12 B

13 Tb2

I TbSl

[B

21 B22B2

31 Tb22 Tbs2

Tb2S Tbs3

with the boundary conditions

[A 31A32A3 3]

Tb7.1 Tbs i"

Bi .

[(T b2 3-TC)

(T b 3'3 -T d] I. 6a) T b22 T b32

Tb23T b3s

Similarly for the mass balance equations (1.13) and (1.14)

(C p-

c b)21 (C p-

c b)si g(T pcp )ZI g(T pcp) 32

St mt

(C p-

c b)22 (Cp- cb)32 g(T C g(T C) cpP Z2 pp 32

L

(C p-

c b)23 (Cp- c

b)33 g(T pc P)23 g(TppC p )33

bii C b2. i C b3l E

21 E

31 St

mt

(C P-

C 021 (Cp-C b): i E

22 E32 (C P-

C b) 22

ýCp- C b) 2

C bi2 CbZ2 Cb32

23E 33

+B 11 B

22 B

13 C

b2l C

b3i B21B22B2Cbz2Cb32. CIII. 8)

C b23 Cbs3

with the boundary conditions [A3,

A32A3 31

c b2i

Cbsi' : -,: ý [o

01

Cb22 Cb3z I. 8a)

Cb2i Cb33

-161-

dl-, %

10

r-

ýlo

cii

G)

-0

00oo000

0000000

00000000

922 rn 00ý0

A

Z 2 . = ý- F- f- _Z Z ýQ t- F- E-

m -0 e t4 ýö t4 ýI a

.0 .0 .0.. 0. _ .0 .00.0.0. I-. -. F' HHHHH

0000000000 Jj

U)

000000 0

000000ý00+-Oo

V) (n

2 rn 41

C)00000cncn + t:

41 U)

41

o I cs I V)

0

4J J--

- 162-

F'. cluations (111.7) , (III S) and (. 111 . ")a) I-or the lictero-

0-011COUS IIIIIS's balance Oquatiolls are collibilled ill all alialogous 0

ma iino r to th c one roy 1) a hinc cc clim tio 11 (. 111 . 9) 0

To apply the orthogonal collocation inothod to the dyll-

amic equations, whore the concentration is assumcd to be

pseudo-steady state, the temperature partial derivative with

respect to time, of equation (1.19), is replaced by a matrix

equation derived in an a analogous way to equation (11.32). 1ý

3T p GT

Similarly to a and -y the dimensionless group ý containing

the heat capacity ratio must be modified by dividing by L,

the number of axial collocation steps.

CPP Pp Z Cpgpg t, uL

To allow variation of the time step a scaling factor

Ts must be introduced into equation (1.19), modifying the

dimensionless transformation to .

g(T pC st (T -T+ L- DT

p (III. 10) pp ht p b) Ts 3t

where Ts t t tr

By increasing Ts the size of each collocation step in the

time direction is increased.

Expressing the dynamic equation (III. 10) in the collo-

cation form similarly to the previous equations, with one

internal collocation point in the time direction i. e. three

-163-

collocation points altogether, (i(Icntic. ii to the axial

Collocation oporation) . (T, g (T c g, (TI) pC cr T (T r (T r

. ', )C pj1 1) 1 p) 2j2p zj 3

St ht ")- b) 2jI

b) 2j2 p- b)2j3

[g(T

pcp Iji g(T pcp) 3ja

g(T pcp 3j3 (T

p- T b)sjl (T

p- T b)3j2 (T

p- T b)3j3

+1G22G23T bij, T blj2 T b1p

YS G3 1G32G33

IT

b2j IT

b2 j2T b2 j 3'

-Tb, j IT b3j2 T b, j 3j

where j=2; 3

Transforming equation (1.20) Z>

bil, T bi2l T bi3l E 21

E 31 (T

p- T b)i2l

(TP-T b)i3l

E22. E32 St

ht (T p-

T b)j22 crp-T b)isz

T bi12 T

bi22 T bi32

E23 E33

11 B12BTT a1

31 bi2 I

bi3l"

B2, B22B23Tbi22T bi32

T bi23

T bi33

where i=2.3

with the reactor wall boundary conditions

[Asl A3 2' A3 31

'T

bi2 IT bi3

* 1"

Bi [(T

bi2 S-TC) (T bi33 -Tc)]

(I I I. 12a) Tbi22T bi32

LT bi23 T bi 3 3j

where i=2,3

Combining the two matrix equations from each of equations (III. 11) to (III. 12a) results in the following set of 24

similtaneous equations.

- 164-

oooooooooooo.., . .00; 000000

00000000000 fo 0w0000 0

000000000 000000000

o0000000000000oooooooo

0000000-; ýo 0 a000000000

000000000000000000a 00

00000000 000000000000

W

ooo ooooo0oo00o0a

oooooocooooooo

oo0 00-a0000000000 :0a0aoo

0 C) 000 C) 0000aa00aC, 0

ýo 00o0ooaoo0ooooo 00oo0

00 mm '0 00000000000000 c> c) 0

0 c> 000

-165-

V., r

I-I

-_1

0 bjO Cd P4

0

ý4 P4

rýz 0

4-4

4J rl 0

u,

71 -

z Z 9 " - * - : - - - , ý ý ý - , - - - ...

0 1 :0 0 0 0 0 0 0 0 6

0 0 0 0 0 0 0 0 0 o o o o o -

0 0 0 0 0 -0 0 0 0 0 0 . j! U" " " 0 00

0 0 0 0 C, 0

0 o o 0 0 o o o 0 0

11, 0 0 0 0 o o o o 0 0

ý 0 0 0 0 0 ý. O o o o cj c) o o 0 0 0 Uý 0 0 0 a 0a

a 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0a

0 0 0 : 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 00

-166-

To solve the matrix of Cquations, I-ot, the ti-ý. tnsient

case, the starting point is to solve the steady state oqua-

tions for the heterogeneous modol and store the temperature

and concentration distributions. The variables to be

changed are selected, the loft hand side matrix of equýition

(111.9) is recalculated, unless only the inlot or bath temp-

eratures are changed and inverted. The left hand side

matrix of equation (111.13) is inverted, and the relevant

boundary conditions-imposed.

As the orthogonal collocation method in the axial

direction consists of smoothly connected arcs, the overall

function has a continuous first derivative. This means

that the method does not lend itself easily to considering

a discontinuity in the function as occurs over the space

of one residence time interval immediately the inlet temp-

erature or o-xylene concentration is stepwise changed. To

overcome this., large time steps are used i. e. Ts>1.

When the bath temperature or the gas flow rate is

changed, the initial estimate of the temperature distrib-

f4 rSt Coll- ution along the length of the reactor for the i

ocation step in the time direction is the same as the

steady state temperature. The -concentration however is

pseudo-steady state so only the first axial collocation

step in the time direction is estimated based on the orig-

inal concontration distribution., unless the inlet concen

tration is changed, and then the concentration is extrapol-

ated axially by equation (111.5) as for the steady state

model. For the second and further time steps, the temper-

-167-

ature is bcst estimated by a modified form of equation

(111.5) in which the extrapolation is of order NO

M+2 T (t, z, r) Ed- (1

p i=l 1

If the inlet temperature is changed, to estimate the

temperature over the second time stop near the inlet,

equation (111.14) has to be reduced to the linear form

due to the temperature discontinuity in the time direction.

Once the unknowns for each step have been estimated

equation (111.13) and the heterogeneous steady state mass

balance equation are iteratively solved until they converge.

Note that the inverses of the left hand side matrices of

these equations are constant for all time steps after the

initial steady state.

U, LL 0 -4

I-

0

"0

--168-

"I Ln 1-n 1-4 0

r*4 m IT Ln co 10

14 L4 ý 1 10 tA

-4

"D 00 Ln 0 10 lo - 1-4 t- CD '0 q Ln -4 cn m tn 10 co In " I-

r. 'n ý Lý C; ; ý 1. 1 " -* I x I

41 N r -------

0) 0 Ln 0 10 C) -Izt 00 -f -4 M q: ) 00 -1 t-r) 1-4 C7, 1ý lzt 10 (14 M \4D (, 4 V) 1-4 t- r

'D -4 tn W) t') I I 1 1.1

C'l , -I tj, " ýo 14 r- r- 00 r- co '. 4 'C' "0 00

10 '1. -IT OC) Ln aý rn r- " 0 m 0 ýl u) co u-) ;t C> 17ý Cý oý rý 1ý oý

Lr) r-4 t) te)

crb

oo oo '0 'o Ln 0

1ý li i 1-4 -4 ýr Ln

1 `4

t- tn oo \0 00 t,

CD 00 0 1- t- tn 10 -4 00 0

1-4 C-4 co t) -4 CA I

4J CIO Ln a ýt 10

Ln ý C, r, ýo Ln r- \0 0 \0 'T r- 'o C14 10 rn C% \AD r- 0 r,: cý 1ý Lý 1ý 1ý

M tn -4 cn Ln -4 t) Ln # I

C) Ln Ln I: r r, 10 T oo \0 Ln Ln co t- m t) -4 rl_ 0 LA 10 ,:, r- C) e4 'o c; ) r- L') C% Go rl

Ln 0) al

L

Ch 1.4 Ln CA 0% a, Ln C% r- (: ) - " " (3%

ul , LA I- rý Ln '0 co r-I c7)

r- rn t- CD 10 00 Ln - u r, t) rý o r- m co

t) co r4 10 Go

-c) C; C; 4-1 -4

41

cli Ln 0 u co zr tn 00 CO . 4r rd 0 1- co co 'D r- Q') q: ) C) tw) CD %0 Ln N 0 C) ý - C) \0 f- C7) CO Ln t- u 0 Ln oo r) C)

0

l A

ý. q r4

co 00 pq

x

co

00 00

Ln 00 co co t") cli -4 C4 I I I 1 .1

.e �zr -e , ý 4 C, 4

ý4 C4

-4

10 V4 " 10 1 C% tn t-I 0% I 1ý 1

< ý I

-4 4 t) ý-q 1-4 co X -4 a

., -4

41

'o C4 " %0 ON tn t) 4m

-: 1ý ": I cc 4 4 Cj

tn tn "I 1ý Ca 4

ul 41

., 4 -d' %C 0 Ln qr

C6 10 91) 00 -4

r_ Ir Ln 0 " t, . 14 W) 110 4J 21: cc to co u 0

0 0

S.

-169-

Cylindrical Goomotry

Collocation Points

w= 1

N=4

0.2634992300

0.574464SI43

0.818S294874

0.9646S96062

1

N=S

0.216S873427

0.4 80 3 804 16 9

0.7071067812

0.8770602346

0.9762632447

1

N=6

0.1837532119

0.41IS766111

0.6170011402

0.7869622S64

0.9113751660

0.9829724091

1 l_-r2

0.2389648430

O. S261S87342

0.7639309081

0.9274913130

1

0.1995240765

0.4449869862

0.6617966532

0.8339 4 5006 2

0.94945S0617 1

0.1712204053

0.3848098228

O. S805038245

0.7474433215

0.8770597825

0.9627801781 1

Planar Geometry

Collocation Points

w=1 N 1 0.5 N=50.0469100771

0.23076534SO N 2 0.2113248654

0.7886751.346 0.5

0.7692346SS1. N 3 0.1127016654 0.9530899230

O. S N60.0337652429

0.8872983346 0.1693953068

N 4 0.0694318462 0.3806904070 0.3300094783 0.619309S931 0.6699905218 0.8306046933 0.930S681558 0.9662347571

Note that the first collocation point x1 =0 and the last

collocation point x N+2= 1.

-170-

LIST OF SYýilll("I, S

a External surface aroa of pellet 0111pty Lube, volume M-

Bi Biot numbor (h 11

R/ke)

C Dimensionless o-xylono concentration

C O-xylene concentration Kmo le/M3

C0 Inlet mole fraction o. -xylene

C Inlet o-xylene concentration Kmole/m 3 0

C Mole fraction oxygen 02

CP Heat capacity of gas KJ/Kmole. 0C

CP Heat capacity of catalyst pe Ilet KJ/Kg. "C P

C Mole fraction aromatic reactant r D Diameter M

D Effective diffusivity Ift 2 /s e

g(T, C) Dimensionless rate of reaction

G Superficial Molar flux Kmole /M2. S

h Heat transfer coefficient KJ /M2. S. OC

AH Heat of Reaction KJ/Kmole r ke Effective radial conductivity KJ/m. s. OC

KG Pellet mass transfer coefficient M/S

k9 Conductivity of gas KJ/m. s. 'C

kr Kinetic rate constant Kmole/m'. s L Number of axial collocation steps

Nu Nusselt number (h D /k ) w P 9

Pec Peclet number (D G/cp D ) g e P

Pr Prandtl number (C ji/k ) p 9

Dimensionless radial distance

Radial distance

R Radius of Reactor M

-171-

R Rate of reaction Kmo 1, /, 3, S

Re G/p) Reynolds number (D p

SC Schmidt number (p/p, D ) 9

St ht ModifAd Stanton number for heat

transfer (h aZ/C G) I p St

mt Modified Stanton number for mass transfer (K aZ/u) 9

t Dimensionless time

t Time S,

T Dimensionless temperature

tr Gas residence time in reactor s

U Superficial gas velocity M/S

U Lumped heat transfer coefficient Ki/m'. s. 'C

z Dimensionless axial distance

z Axial distance m

z Reactor length m

a GR 2 Dimensionless group (k e

Z/C P 9 Dimensionless group (-AH /C {O -e 0b ase T pg

2 Y Dimensionless group (cD Z/R u) e 6 Thickness of boundary layer on

catalyst pellet M

6 Void fraction

e Temperature 0C

P9 Gas density Kmole /M3

PP Overall density of packed catalyst Kg/m 3

P Viscosity Kmole/m. s

I Heat capacity ratio (Cp P PP Z/CP tr U) g g p

T Time scale factor

I

-172-

Sub scripts

b Bulk gas

Co o1 allt

0 Inlet

Pellet

Tub

-173-

AppLndix VI 111ýIATS OF RF-ACTION

c f-l + 8 10 (g) 10 1 (0 22 SCO + Sil 0 22 OR) 5

o-xylone All r= --43.78 x -1-0

'J/Kmole X

c8 11 10 (g) + 61, (02) sco + 51120 (g) All = -21.14 x 10 KJ Kmolo

c8 11 10 (g) + 3(0 2) c8 it 403 (g) + 3H 20 ph tha Iic anhydride 5

All r=-

11.17 x 10 KJ/Kmole

c 11 0 843 (g) 2 (0

- +1 2) >- 2 C 41120 3 (g) maleic

anhydride 5 AHY = -4.324 x 10 KJ/Kmole

CH0 42 3(g) + 4'(0 2 2) 4CO +H0 22 (g) AHr = -14.14 x 10 KJ/Kmole

CH0 423 (g) 21 (0 +2 2) > 4CO +H0 2 (,, g 0 All

r 2.821 x 10 KJ/Kiaole

c8H 10 (g) + 02 c8 11 8 O(g) +H20 (g ) o-tolual-

clehyde 5

AHr = -3.354 x 10 K, J/'L\'mole

-174-

r4 r-A 04 ý>, tl-) r-- r- r-- 0 --1 0 --1 o 00 0) 00 r-4 00 C) L. 0 Ln Lo r, r-i -ý X

ct ý4 -A rn -f 00 CC C) (: ) -4 r- (: )ý) Ln Ln (: D CY) (X, ) -7p ý, (: r) 00 r- ; -4

= 01 Q CO 00 CO co C; ) CA) C-1 (D) C) C)o m 0) C) C) 0) (D C) 'a) G) C71 Q) <U -4 rý) V) tr) -: T In -zj- ýJ- tr) tl or) > ý4 oI 0 rz;

10 a) 4-) (L) co "r zJ, " Ln LO LO 00 tl- to U) LO Lf) Ln Lf) Lf) tn 0 r- C) r--4 Q) r--l 0000000000000 CD 00 -zl- Ln %0 tn vvvvvvvvvv

t1r) C14 " tn

0 U0 r- --t C'4 -t C-, ] 00 00 CD ýt 00 zzl- Ln 00 tn tO L. 0 rýf) (71 1-1 1-- -r-I "D 00 r- 00 00 ý, o 1.0 00 r- L-- 1ý0 L-- r'- L-0 L-- 1-- 1-0 "-j " tn 010 " +j ................... - Q) CD Cý C-3 C-j cq C-1 C'4 " cq C-4 C', 3 C-4 C-3 (-, I C-3 t-r) tn tn tn r-q u

0 1: ý

tn \. O \10 r-A rl- LO ýXD Itil :: t llzt -. 11 Ln \. O \ýo Lf) 00 \. O V- 0

f-. i 4 4-j rý Cý r--i r-A r-A r-A r-i -4 r-A "I r--l r-q ý-A r--i \. D ID ý10 t" 0

00 00 4--) : 1- IZI, 00 cn cn 0 tn al 0 \10 00 " co CD \0 00 \-D u0** \ý

.. 1ý

1ý14 141ý 414

1; 1; 4.4 44

L41; Lr)Ln m00 cr, r-i r--i r--4 r-A r-A r--i r--l r--i r--q r--i r-i r-4 r-A r-i -4 r--l r-I r-I 0

E- Z

C) tn tn ýt t-- tn tn Ln ', Do) "" "D C) C: ) r--l -, t- "m(: ) 0 C7) C) 00 tn tn C) te) r-i LO rn LO 00 rl- E-- mt 00 Ln ": I, IZI, Lf) Ln Lf) Itzi- Lf) Ln Ln Lf) Lf) Lr) LO Lo C-4 t) t)

Ln t-- t-) -t ýo 00 00 Ln Ln "D 000 00 r- (M Cý cn 00 0) 00 00 E'ý- r, - V- 1.0 1.0 V) V) t*- 0000 00 rl- il- tý rý r- r- r- rl- r- rý r- r- r- -: I- Lo Ln

Lr) Ln +j -, --A 00 . 00 V) V) C'4 CD " Lf) 00 'Z> C) -4 10 I'D 00 r-4 Q) P4 u

ri rg o 1ý rý rý Cý llý 0ý 0ý 1ý rý rl Q w4: 11 I'll qýll 1-t- 171- 'j- lzzt IZI- ý-4 tn V) V) tn tn V) tn rn t1r) to rn l1r) tn tn tn V)

0 (Z? j U

ýl Q) 0 lz*l 00 0 t-I \ýo C: ) 00 r- -: I- CY) 00 r- 00 +-) ; -4 Ln * tý 1ý

Cd 'f-,

C-1 C; C; 1ý Cý 1ý IZ tý 6

t4, ) tn all " C13 cla (n C4 --- . 00 tý 00 00 m r- r- 0) r- 00 \ýo C) 00 00 C: )

>, . ý, o t-) -t r- U-) Oo " C) ýt (-I L-0 Ln CD rý -ýJ- , zj- C: ) c)) LO %. o '*- u C) Cý -4

r-4 m -

m r--q CD 0) 00 " Lfý t-) tn Ln C: > :r -, zj- r-A r-A "D ý, o " - 4-J 00 N ) tn

.. tn Lr)

.. Lf) tn

.. tn V)

.. -4d- tr)

.. t-n zzl-

.. Ln %j- "-r Ln "D 4- "D

C) u 00 C) 0 00 0 CD CD C) C) o ..

o C) . . - C) C)

.. C) C)

0 o o

z0 1: 4 ;?

C) -4

(31 -I

ýq r--4 C14 - r r C14

-17S-

V- 00 tn I'- r-4 00 00 ('4 U) C) -4 Ln r-, C7) It rn tn ct ýý -q " 00 00 C7) ol (: ) r- r- r-) ýt -: I- \o C) 0 r- r--ý 00 " r) 00 ýo 0) , ý, ý--: -ý r) r, ,0 C) 00 c-, C) c C. ) c-, c-, - C) C) --ý C'ý (7) C:, ) C74 0) C) <U r-I -, t tn lZr 'll IT tn tr) I-T V) týll tl-) V) "z-zr

czT V) tn tn tn tn tn ý4 0 0 r-

exi t CO (D \M CD (D) r1q L. 0 e �0 t ýc CD

�0 al Ln CO ý, 0 00 r--1 U) nt c) m Co r-- 0 Lt) \XD \JD 't vi Q) ...... CD (D 0 CD (D C) -4 -ý -ý cý

r, X ýD

CJ 0 LO -i le Lr) Ln CD e r--4 Ul le Lfi r- r- --4 Ln r- 0) .. 1 e ý- - r-i "0 \m �z Ln Lfi Co cý u- (-A (D r-- oo r-- Co t-- (14 4-. )

pw

lui (1) (311 (3) co 0" t- Cý 0) L-- 1.0 C: ) 0 CD \. O Lt) Co r--i CD 0m Ex, (: D m

42 CD CD c3) t', rn m0(: D CD

ZJ uý4 Lý '; '; '; 4

-4 LA 10 _i

0 0) rl- rl- r-- rl- 00 r-I NO I'D I'D CY) tn 0 CO %C m m r--i r- r- (3) m

u rý 00 C: ) 0 C) L-- I- r-- LO 00 C) C) -t 00 (7) t-) -'T CD r-I ý-l v) '-Zt

.4 1ý 1ý .4 1ý L4 44 1ý L4 1; tý tý 4 L4 ý, 4 ý, 4 .4 ,4 LA %D t- r-I IZI, 1ý0 IZI, tl- IýD I'll I'll, 110 " r--A t-- 0) r--4 0 t-) v)

ý-C) %0 11 In 4 1ý 1ý rý 1ý 1ý -ý 1ý r-ý 14 Cý -ý 1ý -ý 0ý rý rý r-ý

rý rý 0 r- r-- tn V) te) \ 0 tn tn El- 00 r-4 4 0 P4 u 1 t ) ý=, 0

(1) 0)0)

- Cý 1ý (ý Cý 1ý 1ý 1ý Iz Iz Cý C4

IZI :: i- ýt 'll Izil lczl- 'll LO I- lzt -t* 1171* ýzf Lf) Ln Ln Ln V) V) V) in tn tn tn V) V) V) V) t1o tn t-1) t1o tn V) V)

0 C-0)

r-q (1) 0 rl- V) tn 0) \C) Ln Ln r- to C14 0 tn " 00 00 t- \ 41 ý-4 Lf) ý

Cý Iz 1ý C4 Lý Iz 14 14 14 1ý Lý 1ý 14 I'D til Ln (M rn 00 C) t-) V) 00 00 rn 00 co 00 tn rn 00 r- r- C14 cd r--f It "Zi" -It Zt . 72* -r .: I- :: I*

>, ,a C) Ln r- Ln Ln C7) 00 C: ) (7) cq tn CC) zzt- 00 r--4 c) ýt r-- Ln :T r-j uo 'ZI, ýZjl " ', Zl- tn " 0 r--l Ln \ýo \, D t- \--) \ýo C) 00 C-1 v- rý " tn zr r--q "D Ln r- ýo %01-0 LO \. D -t tn tn llz: l- T4, Ln LO LO -zl- Ln Ln 4-J 00 (1) u >-ý 00 C)

6 0 (ý 00

60 CD C; 00 Cý

6 C3 C3

6

:: s 0 \0 r" rý 00 (n m C71 m 0 0 0 P4 : il- C14 C14 C14 V) V) tn

-176-

r

0 tn rn tn

a 00 C) (D U -1 ý4 o

4-J (D C) ý1" Tt U) u 00 t-I v) %ýo

Cý Cý Cý

0 -4 tn tn ýq -H r- Lf) Lf) ý10 4-J cz C-4 (-4 C-4

r--i u C4

V) ýA

10 Q) 00 00 0000

0 >4

+j ? -I LO LO te)

ý 4 ý 4 L L L U

10 ---1 r-4 H r-4 0 ý. 4

ý P4 00 C14 "It

0 Ln r-q r-i 00 u

Ln ýo ko Lr)

" OV) V- ".. "

"

N- N- N- N-

t- 00 CO

Ln Ln Lfi Lr) t41 V) t4) 14)

C! D

0 00 C, 4 rq 00

ý4 4-J ; -4 Ul r'a

tn 0 CD tn

00 r- XU C)

Z0 p4 Z tr) t4ý

00 r-- 0-, \o CI-1 r-i r--i c"i �o rý t- ý, 0

t. ) C14 V)

-177-

ol ý, ),;, ;., ýýý, W, ;, ý, ý, ý,;, A,

CD t- a ýD :r --t ;t0

1ý Cý 1ý cl C, ul c C, cc Lr Ir 'r kr Ir. ir Lr tr 0 kr) Ln

In Iýn )

tmtOOooo -

10 11 It ID Ir ýL Ir cýýmN It 10 11 ýwNNNNw

le 1ý rý 1ý 1ý %%týý. 1: ý 1ý 1ý ýý% 1ý %% 1ý 1% 1ý i 11 wIIý. ý ýL Ir I rý I rl ýýý r- r- r- ýýý Ir, X. 'r J- Lr ý ., LC ýdS 'r rFCý ýr ý Lr r U, Lr ýl ý, I, K. 11 ýýýý ýý ý) ý K, ý . 1) ýl 0) ý) 0) M) W,

W) "ýW. ý :ýý,, ýt NmN %ýý cý cý In Irl Irl Ir. ýr, c Ic IL I IL ;Z1ý

Irl It IL ýL ý ý. .)ý, -1 In 11) 11 ,ý -1 , N) ,ýý ý) ýý

,, ýýý-. K, ý Ir ý ý, Z ýL ZZ 112 j. ý Ic Irl In Ir Zý2--ý, ,,,,, ý, ý,,,,, ý> In ýK-. W) ý, ýKKýý1. KýýKý lý ýýA

ccvc 'D 't

Cc Ic mw

w) Pe)

-ýýý=m L% , ?ý rt. c"c20, -ý- ý- ýccz

ZZC4

ý, ý, ý- ý,,,,,; ý ý- ýk,, «. ',,,

17ýC, 3' ý, aIZ.;, ý 0, -ý ý- 'r --, ý ý- 3ý ý- S. izýIýzr

- 'T -3 71 ý, ý' 3' 7P 71 7Pr. ,T-3, 'r ý J, - T.

----------

Ln -

C! cccrcrccaCaa

zr c. -ý. Ir 19) z Ic ID I zr 0 Cý 0

IC Ir Irl Ir It drr tr it v w, Lr Ln it) in ýýýýw, ý ý, ý) w) n) 11) ) #0 Iq

z0cvV Irl ol e) 0) X) PrI W) W) 11) An

3, J, 0,0,0, w Ir, ý d: ý, ýýpX, C ý', 'r. ýS 41 J) J) d) ý, C, 1r) ýr 0 ý ý, ý 1.1,1, ý 1, . 1, ý, -, W, ,, W, ý ý, e) ý) ý W, ") 0)

N C% N. C, N (7,1 N "o -L 'L 'D N r. :r 4

Ir kr tr r Ar, kc 4p If Ir c 'L ýL Cc 'D

CC CD C QQ CC tr Cl, W) 0

ir Ir r Ir

ýý -7-j ý

a, T c- 0, wý ý) mi 9)

(\ N cý Ný lý ý 01 0' ý a, - ý, lý ý2 n Z, .'& <y jl, , Ir ID = ýn

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ý ;! ýmNý ej Lo IL ý7ý :ý( Ic Iv 31 0 71 N ol Ln cr ly

po .1 zr 4t ýr

a a-, u, r ýn Lr tr a, cccmrn w)

............... TýIýXxýxII. zI. xzmýxI. Ix14mvxZ ýn -) ý) ý) ý) 1) 11) ý -) -ýý ý) ý) ýý- ý) ýýý -) ýýýý -) ýý 1) 11 -) 1ý ýý 11 ný1. ý) ý) ý ý) ý) 'I ý) -1 ý) W) 11) ý W) ý) -1 ý) 0)

-D Ic xz 'n 'n J, U, J, r, mN tr, 'n nn4

T tc xxT -T

T 71 ol TcC, 7 ol 777 0' 0' 71 L, ca 71 7aTz 71 T a, 71 ly a, ol 11 a

I. -T Ulu, 'n Wj

.1 ýn P 'r, ý ýi ý7 ýý I- 1. ný -1 In ý. ý ý. ý) -

N -ý N ev IL ý: 0 !ýý= 10 wý tL wN 'M ýj -r 'D ý C. NNNNr. ! E' ýcý; ý-0xN :ý CY rv ý -, N. ýý0 ýý 0 -4 xcýý 'D 0 CY c <Y ID Cý C. Ct Pý Gý rý 3ý a, rý Cý

Ir Ir xP Ir PpT Ir a, Ln Ir Ir It Ir P J, r Ir rrT, 1: 1 r -P .1d, ýn Ir Ir rrTcJ, p In x di d, ý) nýýý -) ý) ýýý, ) -, ý, -ýI ý) ý, -, ý, ýýIýý1, ý ýl 11) ý, 11 ý) ýý -1) ý) ý) -11 In -n

-ý, ýýýý, ýý ,) ýý ý-ý ý .ý ý) 0ý ý 4i-nj, 2 2ý3ý ý. -: tt%

cý Wý ýý Ir ýý;. 4r. ý ul ý 1ý ýý;; ýýýnýTr, ýTýI, ýrýp, r, ý, ýpýý; ýýIýIýýý177ýa, il .0 :r npr, f, 41 .1 In

-C =

%

Z: .- 11. ý1 .1n rl 4) PCP 4) 6r, P J) D ýa ,DDZCLXCDZýýý-ýýI f- ýýýý CC (r, X IC .0ý

-197-

U If I It aduudUI a- Ir Lr; kr ýr

11 ýý ýýý 71 -7 ýý -I -ý ý: ýýIý: ýý "ý 1ý ý :ýIýý ýý ý: ý Ir "I c rý .............. ..... ................ cý ; 4' ý; Ir I rl siJ, a' ,I, 'r ,au. IIrj-, , ý, ifIýr-LCXE4c

Vd1 11 f j, IIIaa,

- :ýý 'n J) 7,7

ýn P

Lr ......... ... zzzr 777-, -7 ----- 'i

ý. zýa rfrL 'r ýz Ir Ic Ic

zzzxzx ma N C CF CjG L7 97 0' C7777 c :77Tr. aa

-c IcxfccI 'I 'o Lz

If 7 Ir J, Ir C, li im Ir a, ccI

ol ý, ý ý- ýZýýýýrL-ýxý Iz ýIýrýý, ý -- :-ýýýý

li : 71 7 J, C, : : 71 7- 71 0: ý :, =-- a Ll 31 7ý I- 17ý 71 T J, :7 C- a77J, a 7ý 71 zZ 71 71 77C, 71

T, N a' cc c7c cc c .1 cy

x cc 7 a, zIj, 'r. zI o- %zxm r- Ic -z L ýr 'r, r. r* I 1ý cl p12pn4z

Ic IL CL 'L L 'r 'r r ir C% 't C% 'D cN

r. .1nn 0)

2jzj innere.

mit rzrý di29i

1ý P- f, ................

77 Ir 41 J, 'r 'r. r Ir If ad Ir

ý: ý ý; =ý0ý. x 1. ,ýýýcIý= "' - 0- -ý25 -5 zmNNNNN. 1! xc IaZ r-

r Lr rrrxU, 'r Irxr. j- r -r

r J, r kr T, c Ir rrr Ir Lr J, rrx ýr rrcr rl p0 ýýýIý, -1 1, ný1; 1) -1 .11ý-ý, II1, .1ý 01 ýIý, ý, ý, ý ý, ý) Ký 11 0)

uJ2 ......... ..................... ,I-ý-ýý., - Ir ý; ýýI

j-- xrrrr -r r r, rr -r r r. r, P.

Ic c 4. X mP

-198-

Appendix VIII ITMITIPATURF PROFILE OF 'Flik RFIACTOR WALL

3S4

352

Temp .( 'C)

350

348

346

7 ts

Lure

0 10 20 30 40 so 60

Distance from reactor outlet and

surface of load/tin bath (cm) .

- 199-

Appendix IX CALCIWATION OF-THE ORDER Q. ý-_AWACTION

For a reaction system with a constant selectivity, the

phthalic anhydrido concentration can be expressed as

C pa " k(C

O-Cx) where C0 is the inlet o-xylene concentration

k is the selectivity

The rate of reaction containing inhibiting effects

kC Rrx

r1+ ka C pa.

can therefore be expressed as

R=-krCx r1+kak (C

o- CX)

which can be differentiated to determine the order of

reaction

d (ln R r)

1+kakC0

d (ln Cx)1+kak (C 0-

C X)

as 0<k<1 and C0>, Cx>, 0 the order of reaction must

always be greater than or equal to 1, and can never be

negative.

Similarly for the reaction model

Ca Rr =kx r0

pa

the order of reaction is given by

d(ln Rr)ý+a (C 0-

C X)

d(In C X)

C0-Cx

which is always positive for ý and a>0

-200-

0

0

u -11 W4

C)

E- 0

F-i

0

k

Id 0

5-1 0 rj

ITJ

Lo 0

0 00 cq

Cý

4A 0 0 -H

4-)

ul

0

V)

C

0

0 "r

0 U

0 U

0

0

0

0

0