Parametric sensitivity and runaway in fixed-bed catalytic reactors

Ch~micol Enngwering Science. Printed in Great Britain.

Vol.

41.

PARAMETRIC

No.

4. pp.

1063-1071.

IYKh. 0

SENSITIVITY

AND RUNAWAY IN FIXED-BED

Massimo

Morbidelli

and

Arvind

CATALYTIC

1)009-2X39/86 $3.W + 0.00 1986. Pergamon Press Ltd.

REACTORS

Vat-ma

Department of Chemical Engineering University of Notre Dame Notre Dame, IN 46556

ABSTRACT The sensitivity behavior of fixed-bed catalytic reactors is examined using a heterogeneous for the case of n-th order irreversible reactions. one-dimensional plug-flow model, A novel sensftfvfty criterion. based on the occurrence of a maximum in the value of the normalized sensitivity of the particle temperature wfth respect to any of the input parameters of the model, The fntrinsfc nature of the proposed criterion allows the deflnftfon of a f s developed. generalized sensi tfvf ty region of reactor behavior in the relevant parameter space. The boundaries of the sensitfvfty region for various values of the involved physicochemfcal These are found to be in good agreement wf th those previously calcuparameters are reported. based on the occurrence of a positive secondlated using the classfcal geometrical crf terfon. order derivative. before the hot-spot, in the partfcle temperature vs. conversion proffle. The importance of transport confirmed by comparison with

liml tatfons experimental

on

the data

sensitivity available

In

behavior of catalytic the literature.

reactors

is

KEYWORDS Parametric

senstfvity;

runaway

behavior;

fixed-bed

reactors;

catalytic

reactors;

reactor

models.

INTRODUCTION The steady state temperature profile along a tubular catalytic reactor, where an exothermic For certain reaction occurs. is usually characterized by a maximum temperature or hot-spot. the maanftude of the hot-SDOt may become auf te sensftfve to small ooeratlna condl tions. v&iatioGs of the reactor iniet conditfons. as w&l1 as- to any oi the other physfcochemfcal characteristics of the system. Sfnce Bflous and Amundson (1956). this has been referred to “parametrically sensi tlve’ or “runaway” reactor behavior.

as

it fs desirable to know a_ priori the location of the boundary For practical applicatfons. between non-sensitive and sensftfve regions in the reactor oarameter soace. so that the latter can be fmnedfately avoided in the early stages of reactor design. The. calculatfon of such boundaries should be based on some a rforf intrinsic sensitfvity criterion which should provide a rigorous deffnftfon of parametr -+ c sensitivity. without relying on any specf flc consideration about the particular system under examination. Most a priori sensitivity criteria were originally derived in the context of thermal explosion and then transferred to chemical reaction engfneering, based on the striking similarfty theory, of the pseudohomogeneous models governing the behavior of an explosive medium and a tubular reactor. Earlier crfterla were based on some characteristic behavior of the temperatureconversion or temperature-time trajectories (or axial coordinate fn reactor theory). These have recently been compared critically by us (Morbfdel li and Vat-ma, 1985a). The mOst reliable among them is the Adler and Enig (1964) crfterfon. which defines as parametrically sensftfve the situation where the temperature-conversion profile exhibits a reigon with positive secondbefore the hot-spot. This crf terion, order derivative. as well as al 1 the others derfved from based mostly on physical intuition. geometrical arguments, are, however, Only recently crf teri a based directly on the rigorous concept of sensitivity have been proposed. This is defined as the derivative of the maximum temperature wf th respect to some reactor inlet condition or physIn the context of homogeneous systems, Lacey (1983) and Boddington f cochemf cal parameter. and co-workers (1983) defined as criticality the situation where the hot-spot sensitivity with respect to the dfmensionless heat of reaction parameter (or Semenov number in thermal explosion theory) is maximum. However, the choice of this specific input parameter involves some arbitrariness. This was removed by the generalized sensitivity criterion (Morbldellf and 1985b). where criticality was characterized by the maximum of the hot-spot sensitivity to Varma. any of the input parameters. (which. as discussed later, has a clearer physical meaning) sensitivity crf terion.

1063

1064

M. MORBIDELLI

The aim catalytic compare

and

K-l

A. VARMA

of

this work is to develop, along these lines, a generalized sensi tivi reactors, explicitly accounting for their intrinsically heterogeneous its predictions with available experimental data.

SENSITIVITY

CRITERIA

ty

criterion nature.

and

for to

FOR HETEROGENEOUS REACTORS

In order to slmulate fixed-bed catalytic reactors, a heterogeneous one-dimensional plug-flow internally isothermal (c-f., Carberry. 1976; model, where the catalyst particle Is assumed Pereira and co-workers, 1979) is considered. It is known that dispersion phenomena, particularly in the radial direction, may be important in non-adiabatic reactors. However, the adopted model exhibits all features due to inter and intra-particle mass and heat transport phenomena, whose effect on reactor sensitivity is the main object of this work. Using fluid

the and

dimensionless solid phases, dv aY

xP

vP with

IC

v=v

quantities respectively,

defined in the Notation. the mass may be written as fol ?ows: v-l

= Q* _ p (l-xpln = x +ti

0

= v + “p*’ 1

at

(v

E

f(vp)

-v) P

I

and

heat

balances

F

the

(1)

n(xp*vp)

G

(l-x$?

for

(2)

f(vp)

n(xp,vp)

1 Q

(3)

x=0.

(la)

Note that quantities without subscript refer to the fluid, while the subscript p indicates the In this notation, the effectivess factor n is given by the ratio of catalyst particle surface. the overall reaction rate in the particle to that in the absence of all internal transport limitations. Using the normalized Thiele modulus 0. n can be computed from the To? lowing relationship n = [30

coth

a*

f(vp)

(30)

-

11/3$

(41

where = *;

(l-xp)n-l

)

which represents a well-known approximation catalyst particles of any shape, where an Froment and Bischoff, 1979). (C-f.,

(5) to the Irreversible

diffuslon-reactlon n-th order

problem in reaction, with

isothermal w-1, occurs

Rajadhyaksha and co-workers (1975) examined the sensitivity behavior of this model, adopting the pseudohomogeneous van We1 senaere and Froment (1970) cri terion, which is intuitively based on a particular feature of the temperature-reactor length profiles (i.e.. the critical profile I s the one which passes through the maximum of the temperature maxima locus). Results were sensitivity of the fluid temreported only for four limiting regimes, and. in all cases, This approach is not conservative, since in heterogeneous reactors it pera ture was exam1 ned. the catalyst particle temmay happen that, al though the fluid temperature does not runaway, This is due to the non? inear relationship between the two temperatures, and perature does so. it occurs when the system approaches the situation where the derivative of the fluid to the particle temperature vanishes, as in the region of incipient multiplicity. This phenomenon was first recognized by McGreavy and Adder’ley (1975) who termed it local sensitivity, as opposed to sensi tl vi ty, lobe1 which is the one encountered al so in homogeneous reactors, as examined by and colleagues (1975). a jadhyaksha In order perature approach controlled reaction

to simultaneously account for both global and local sensitivity, sensitivity has been exam1 ned by us (Morbide? 11 and Varma. 1986a, makes physical sense, because the particle temperature is really in heterogeneous reactors. since this is the temperature which as well as any change in catalyst activity. rate and selectivity,

the article temI98 %K--TliiS the variable to be actually governs the

the sensitivity behavior of the heterogeneous model (1) to (3). accounting In the present work. is Investigated using two sensitivity for both inter and intrapartic?e transport limitations. defines as runaway the situation based on physical intuition, criteria. The first one (II. where a positive second-order derivative region occurs in the particle temperature vs. conversion before the hot-spot (this is the criterion previously used by Morbidelll and Varma. profile. The second (2) is based on the rigorous concept of normalized sensitivity 1986a. 1986b). to any of the Input model parameters. 4 temperature maximum. vp scvp*; $11 of the particle stv;;

cp) = 2, vp

where the asterisk refers to generic model input parameter sensi ti v< ty . normal ized

5

=

d+ the catalyst (indicating

A*

s(vp*

;

e)

(6)

vP particle a,, 8.

Y,

temperature at the vi, n, Ap or owl.

hot-spot, and s is

+ is the the non-

Parametric

K-l

TECHNIQUES

NUMERICAL

and runaway

FOR EVALUATING

in fixed-bed

CRITICAL

1065

reactors

CONDITIONS

the boundaries of the parametrically sensitive region. it heat of reaction a. wf th parameter , say the dimensionless is parametrically value c+o is such that for c+ac the reactor not.

In order to estimate consider one varying The critical ffxed. while for a
sensitivity

to crlterfon second-order

(1). cri derivative

tfcality in the

is given by the incipient vp-x curve. This occurs

occurrence at the point

is all

convenient the others sensf tive.

of a reglon where

to

WI th

$=$=o

(7)

By coupling Eqs (2). (3) and (7). an algebraic nonlinear system is obtained, whose solution yields the coordinates (x, v, and v 1 of the cri tfcal point. The critical value rzc is then obtained by forcing the solutl% of thi! model Eqs (1) to (3). wf th the IC (la), to go through such a critical point, A convenient numerical procedure for solving this problem is offered by the isocline method, which has been described in detail elsewhere (Morbidelli and Varma, 1986a.

1986b). In order to apply criterion (2). the normalized sensl tivity The local non-normalized sensitivity of the fluid temperature, to the generic input differentiation of Eq. (1) with r espect .w where s(vp;+) and particle surface.

=

g

+

g

s(v;+)

s(v;+)

(1

=

- $2

) + 3

e

= g

reduces

aQ Eg

(8) and conversion as follows

E g1+g2 s(v;+)

at

the

(9)

?lz,

Cs(v,;+I

-

s(v;+)l

(10)

= u

,

(11)

,

(lla)

i s(v;+)

H and

CJ are

at

= +

known

x=0

functions

of

v.

vp,

The objective is to evaluate the particle particular conversion value x*. which in may be represented by the integral form $ where

s(xp;+I

to

IC

where

$

+ H s(v;g)

W with

+

+ $

of temierature Eqs (2) and (3).

aq

L

x-7

P

s(xp;+I

+

P

aD

l-

Eq (8)

s(vp;+)

s(xp. -9) indicate the local sensityvitfes These are obtained by differentiating

shp)

so that

+ g

S(vp*; +I needs to be evaluated. s(v;+I is obtained by direct parameter. + as follows

6(x-x*)

= s(v;;

indicates

+I

the

= (

Dfrac

xp and

temperature turn depends

s(vp;+I delta

all

6(x-x*)

the

input

sensitivity upon the

parameters, at input

2.

the hot-spot, i.e. parameter values.

at a Thi s

dx

(12)

function.

Of course, s(vp*; $1 could be evaluated through s(v;+). as given by the above reported Eq. (8). using Eqs (9) and (10). However, significant savings in numerical effort are obtained by adopting nonlinear sensitivity theory developed by Cacuci (1981). This is based on the introduction of the following adjoint problem - -w

+

HS(v;+)

= 8 = g,(x)

6(x-x*)

with f(v;+)=O at x=1, where the adjoint source term, 8 and to

reduce

Eqs

(13)

and

(12)

to

do=Hp a?

the

following ;

P(O)

(13 1 the

IC

have

been

selected

in

order

form

= I

;

xcc0,x*1

(14 x*

-

g,(x*)

+ si(v;+I

f’(v;+)

+ &;+I

P g dx

J

(15 )

0

where P=~(v;+)/~~(v;$) and ji(v;g)=g2(x*)/p(x*); and S(vp*, +) is then obtained from Eq. (6). Note that the particular choice of the input parameter, 0 In the definition of sensitivity affects only Eq. (15). through the terms si(v;+I = avi/a+ and d. Therefore, Eqs. (1) to (3) and (14) can be integrated once and for all. for a given set of parameter values 2, and then the sensitivity of the hot-spot to each specific parameter, $ may be readily evaluated through different integrals (15). This does not hold true when Eqs. (11) and (12) are used directly. ana since Eqs. (11) and (14) exhibit a highly stiff character in the sensitivity region, the above developed procedure does in fact allow significant savings in numerical effort.

1066

M.

GENERALIZED

CRITERION

MORBIDELLI

and

FOR PARAMETRIC

A.

VARMA

K-l

SENSITIVITY

In Fig. 1 the normal?zed sensitivity with respect to the inlet temperature S(vp*;vil
n=l;vi

-I;

y=ZO;A,

Le =I

-0.004;

@, -0

6

Fig.

Note that fndicating

sensitivjty S(v *;vi) as a func;io; of ” . a and 6; n=l. vi=l. 1'20. ~p=o.oo4. Le=l.

I.

for decreasing progressively

Normalized

a (or 13) values the magnitude of the S(vp*;v?) less and less sensitive reactor behavior.

maximum

decreases,

thus

The behavior of S(vp *;*I for various choices of the independent Input parameter + is analyzed in Flg. 2, which shows a generic cross section of the sensitivity surface. for a fixed 8 value. Each curve represents the normalized sensl tlvTty of the maximum particle temperature to various and ApI as a function of the heat of reaction parameter, a. Input parameters (i.e., *=v?.a. It appears that the critical value u characterlred by the maximum S(vp*;+I, is Identical (up to the first four significant d?$its) for all curves. This finding, which holds true also for all the other input parameters (i.e.. Y, 6,. Le and Owl, allows to deffne a generalized parametric senstlvlty region in the parameter space, where the reactor becomes sensitive simultaneously to small changes of any of the involved physicochemical parameters. It should be noted that the same boundaries of the generalized parametric sens? tlvlty region with respect to a or to any other model parameter--as are also obtained by maximizing S(vp *;I$) previously found for the pseudohomogeneous model (Morbidell? and Varma. 1985b).

0

t3

--

Fig.

2.

Heft

13.5 of Reaclion

Normalized sens
14 Poromeler. S(vp*;+) 1: +v’.

14.5 01 as 2:

a function of +‘a’. 3: $=Ap-

Parametric

K-l

sensitivity

and runaway in fixed-bed

1067

reactors

In Table 1 the critical ac values, as predicted by criteria (11 and (2). are reported as a function of 8. In the same Table f. are also reported the ac values whfch maximize the local sensftlvftles. calculated at the particle temperature hot-spot, of the fluid temperature. v and It appears that for sufficiently large 0 values, the conversion at the particle SUrfaCe. x crf tfcali ty (1 .e, the sensltfvity maximU R’1. occurs simultaneously for all cases, regardless of which particular model output or input parameters are considered in the sensitivity analysis. This strongly supports the intrinsic nature of the proposed criterion, and Sndfcates the existence of a generalized sensitivity region for the reactor. TABLE 1. Sensitivity

Critical Values of the Heat of Reaction Parameter, cat for Criteria; n=l, vi=l, y=20, Le=l, Ap=O.OT and #,=O.

> e+

n

\+

Various

Y

(a)

(bl

(cl

(a)

(b)

(c)

(a)

(b)

(c1

(d)

0.1

2.46

2.30

2.49

1.99

1.98

1.46

3.50

3.38

5.62

3.74

5

6.93

6.71

8.06

6.93

6.83

7.16

7.07

6.90

8.20

6.81

10

8.83

3.72

9.11

8.84

a.77

8.94

8.88

8.78

9.08

8.54

30

13.8

13.8

13.8

13.8

13.8

13.8

13.8

13.8

13.8

13.8

50

17.5

17.5

17.5

17.5

17.5

17.5

17.5

17.5

17.5

17.5

(a) (b) (c? Id)

maximum in the S(vp*;+) vs. a curve; the SIv;gl vs. a curve the Stxp;*) vs. a curve

Oxfam in maximum in second and

third-order

derivatives

criterion vanfshing

Note that these ffndfngs also confirm the correctness (1) (as well as in all crfterla based on some physical conversion or temperature-reactor length trajectories). Moreover. reference to any specffic input parameter. tical a= values for sufficiently large 8 values.

(21 in

the

vp vs.

of the assumption fntuftfon about where sensf tfvf the two proposed

x curve;

crfterfon

{I)

implicft in criterion the temperaturety is defined wf th no criteria predict fden-

Some further insight into the relatfon between criteria (1) and (2) is provfded by the asymptotic analysis for large r values reported by Lacey (1983) in the context of homogeneous exploIt can be shown that both crfterlon (1) and that based on the maximum of the sion theory, non-normaltzed sensitivity with respect to Semenov number approach the result of Semenov ideal as Y-B. Using the same asymptotic system (where reactant consumption is neglected). ac/B=e-ir solution it can also be readily shown that this conclusion holds true also for the generalized criterion (21, independently of the specific choice of +. when applied to pseudohomogeneous reactors &lorbldelli and Varma. 19856). the two crfterfa do not agree any longer, and also the answer of crfFor very small f3 values, However, terfon (2) becomes dependent on the partfcular fnput parameter, 4 under examfnatfon. in this case, due to the relatively small amount of thermal energy available to the system, tt?e and so is the numerical value of sensi tivfty. Thus, therhot-spot magnitude is very lfmf ted, Nevertheless, if a situation of this type mal runaway Is not really a sfgnfffcant phenomenon. the best answer is obtained by directly evaluating the sensf tf vi ty needs to be examined, without relying on any intrinsic sensitfvfty criterion, by using the numerical procevalues, dure reported above. some comments about the connection between parametric sensi tivfty and steady state Finally, multiplicity are in order. in heterogeneous plug-flow models, the fluid phase is described by the initial value ordinary differential equation (11, and, therefore. mu1 tiplicity may arfse only from the nonlinear algebraic equations (2) and (3) describing mass and heat balances in The problem of steady state multiplicity of an internally isothermal the catalyst particle. catalyst partfcle has been studied in detail in the literature (Pereira and co-workers, 1979; Hu and co-workers, 1985). the case where the inlet catalyst particle In reactor senstfvfty studies, When operating conversion branch is the only one of practical interest. and the reactant the catalyst temperature fs usually very hfgh. branch, that the reaction cannot provide the amount of heat necessary to sustain acceleration or runaway. Even though the inlet particle operates fn the low-conversion branch, from low to high-conversion branch) may occur further down the reactor. non can occur only within the parametric sensi tivfty region; i.e. lgni sequent to runaway. This can be readily seen recalling that Ignftlon

operates in the lowin the high-conversion is so rapidly depleted temperature selffgnftion (i.e.. a jump However, thf s phenometfon always occurs subcan only occur at a

M. M~RRTDELL~ and A. VARMA

1068

bifurcation perature situation, particle accounted ignition

K-l

1986a), where the fluid to the particle tempoint (c.f., Morbidelli and Varma. when the system approaches such a However, as discussed above. derivative is zero. even in the case where the fluid temperature is not in the sensitive region, the temperature may run away because of local sensitivity. Since this phenomenon is for in the generalized sensitivity criterion developed here. it can be concluded that along the reactor can only occur within the generalized parametric sensitivity region.

This conclusion is confirmed by the sensitivity values shown in Fig. 2. immediately after attaining its maximum value (which defines the critical Sjv *..+I tivity. i.e.. Q =13.84). undergoes a discontinuity (at ~=13.85). somew Rele along the reactor. occurrence of i&i tion Note in passlng that two u values Is not in general true for al 1 parameter values.

REGIONS

OF GENERALIZED

PARAMETRIC

It

may be seen that a value for sensfwhich indicates the the proximity of the

SENSITIVITY

In Figs. 3 and 4 the regions of generalized parametric sensitivity are shown in the a-A plane, for various values of the reaction order and the Thiele Modulus, respectively. B he sensi tivi ty region is enveloped by the two curves shown. The lower curve represents the critical A as given by the sensitivity ac as a function of the mass transfer resistance parameter. criterion (2). The upper curve represents the minimum 0: valu g’ at which the inlet catalyst particle operates in the high-conversion branch. For larger heat of reaction parameter values, OT the entire reactor operates in the ignited region, which, as discussed above. Is considered to be non-sensitive. For increasing Ap values the locatlon where S(vp*;el Is maximum travels towards the reactor inlet, and when it reaches the inlet particle the two curves merge. 3 and 4 it may be noted that botn inter and intra-particle transport resistances From Figs. decrease the size of the sensitivf ty region. In particular. for sufficiently large values of becomes transport-llmi ted, Ap or Qw the reactor and runaway does not occur. However, the 1 ower boundary of the sensitivity region decreases for increasing Ap values. This can be explained noting that, since the Lewis number, Le is kept constant, increasing A means that both interparticle mass and heat transfer resistances increase. in The increased d a fficulty removing heat produced by the reaction from the catalyst particle has the effect of accelerating the runaway process, which then occurs for lower heat of reaction parameter values, a. This same behavior cannot be expected for increasing internal transport resistances, as In fact may be seen from the results shown in Fig. 4. From Fig. 3 it is also apparent that reactions of higher order are previously observed for homogeneous reactors tor sensi ti vi ty-- a fact 1982).

less likely (Morbidelli

to

exhibit reacand Varma,

it is worth mentioning that the critical values of sensitivity, pc shown in Figs. 3 Finally, These are in very good agreement with those obtained and 4 were obtained using criterion (2). from criterion (11. which, when plotted on the same figures, lead to curves that cannot be This confirms the reliability of the sensitivity boundaries, computed from cridistinguished. terion (1). previously reported for various cases (Morbidelli and Varma. 1982. 1986a. 1986b1. Since criterion (1) ii based per se mostly on physical intui tfon, its reliability ii increased substantially by its agreement wi+Fi criterion (2). which is based firmly on the rigorous concept of normalized senstivi ty. and whfch allows the existence of a generalized sensi tivf ty region as discussed above. COMPARISON

WITH EXPERIMENTAL

DATA

Emig and co-workers (1980) have described the on1 y detailed experlmental sensitivity analysis using a fixed-bed reactor where vinyl acetate synthesis reported so far in the literature. The obtained results are summarized in reactlon occurred on a zinc acetate supported catalyst. in the (R/a) vs Q parameter plane, indicate non-sensitive operating 5. where open circles, Fig. The solid lines represent the while closed circles indicate sensitive conditions. conditions, sensitivity boundaries predicted through the analysis reported above, for various values of the The kinetic parameters have been taken from the mass transfer resistance parameter, Ap. detailed kinetic study of El Sawi and co-workers (1977) (i.e., n=l, y=20), the inlet temthe heat and mass transfer J factors were perature was fixed in all experiments (i.e., vi=lI, internal mass transfer resistances were neglected assumed equal (i.e., Le=l). and, finally, For comparison, the sensitivity boundary given by the pseudohomogeneous model (i.e.. e,=Ol. together with that obtained by with a zero-th order reaction (dash-dot curve) is also shown, Emig and co-workers (1980) using again a pseudohomogeneous model where, however. effective radial transport of heat and mass were explicitly taken into account (dash curve). It can be observed that all pseudohomogeneous criteria predict a similar shape for the senwhichTattens for increasing CL. and which does not seem to agree with the si tf vi ty boundary, Only the introduction of interparticle transport lfmitatlons seems to experimental findings. It is worth stressing the qualitative reproduce the correct shape of the experimental boundary. a quantitative one would require accounting in detail for nature of the reported comparison; the different values of the Reynolds number in each experimental run, as well as for the effect the reported comparison certainly indicates the Nevertheless, of radial thermal gradlents.

Parametric

K-l

sensitivity

I ’ ’““i”

Fig.

3.

in fixed-bed

L

’ ’,““”

-c”=3

Mass

Transfer

Parametrically for var*aus

20 -

and runaway

/

Resistance

0.3

,111,

I

=I

p=20;

y=20

in

1069

II

vt=l;Le

Poromeler.

sensttive regions reaction orders. n.

reactors

A,,

the

(I -Ap

plane

\\M\\

15, IO5 -

n=l;

Le =I

Yi=l

i

p

I 10-a

0 10-d -Fig.

w

y-20

= 20

4.

Mass

Parametrically for various

I

IO

Le =I;

.

.

. . . . .1 10-a

Transfer

I

. I

.,.*I

.

Resistance

20

I

Parameter.

sensitive regions In the Thiele modulus values. ew. I v’=l

. . * ,..

10-I Ap EL -Ap

I

I

I

40

60

60

plane

100

-Q Fig.

5.

Comparfson of the sensi tlvi ty region boundaries predicted by various sensitivity criteria and the experimental data of Emlg and co-workers (1980); experimental data: 0 = non- = sensitive; calculated curves: (_ sensitive. criterion (2) and n-l; (----I: criterion (2). A,-O)and n=O: (---I = Emjg and co-workers (19801. as reported in their Fig. 1.

1070

M.

MORBIDELLI

VARMA

and A.

imoortance of interbarticle transnort limitations catalytic reactors.‘This conclusion is substantiated 1986bI. where the exper
jn

determininq the sensitivity behavior of by a more detailed comparison (Morbldelli magnitude in cases where runaway did not

Note that using the classic local a riorl criterion - +let particle to t e effects by Mears (1971). applied concentration and temperature gradients are negligible aAp/Le However, sitivi ty stresses

(

0.05

for the import of transport , one would conclude that for

limitation interpartfcle

.

the curves reported in Fig. behavior is concerned, over the need for overall reactor

it is Finally. Dorai swamy and ditions under changes in the corresponding correlations, values used in

K-l

5 clearly the entire sensi tivf

show this not to be true as far as the reactor range of operating conditions. This finding ty analysis, as that reported in this work.

worth mentioning that using available semfempirical correlations (c.f., Sharma. 1984) the predicted value cif Ap, for the experimental operating conexamination, ranges between 0.001 and 0.004. Moreover, the above mentioned Reynolds number lead to Ap varlations by a factor of about 1.7. i.e.. to the range investigated in Fig. 5. Considering the moderate accuracy of these particularly in the presence of chemical reaction, the agreement wl th the Ap Fig. 5 is quite satisfactory.

NOTATION in-l

A

P av C =P De 2 F f(V 1 G ’

91*92

H

Le n

Q R SP s(k;o) s(vp*;6)

s(a;o) T U V

VP V

transfer

resistance

parameter,

pbk(Tw)C

particle surface per unit reactor volume, m-l reactant concentration, mol/m3 specifSc heat, J/kg K effective intraparticle diffuslon coefficient, diameter of the reactor tube, m act1 vatf on energy, J/mol right hand side of Eq. (1) expEy(v -1)/v 1 right hgnd siie of Eq. (2) functions difined by Eq. (9) defined by Eq. (111 interpartlcle heat transfer coefficient, J/m2 reaction rate constant, mol (m3/mol In/kg s interparticle mass transfer coefficient, m/s reactor length, m Lewis number, h/ ofcpkg reaction order right hand side of Eq. (3) ideal gas constant, J/K mol particle surface area. m2 normalized

sensitivity

objective

normalized

of

model

output.

sensitivity

sensitivity of model output, i temperature, K overall heat transfer coefficient. dimensionless temperature, T/T, particle volume, m3 mean superficial fluid conversion, (Cl-C)/Ci

X

Greek

mass

velocity,

of to

X to the

model

‘kgav

m2/s

K s

model

particle Input,

J/m2 K s

m/s

Letters (1 a0

a/v

0

heat

Y aH &(x-x*)

dimensionless activation energy. heat of reaction, J/ma1 Dl rat de1 ta function effectiveness factor catalyst bulk density. kg/m3 fluid dens.1 ty, kg/m3

‘b Pf

heat

of

reaction

transfer

parameter. parameter,

(-AH)Cir/TwPfCp 4U/dtpfcppbk(Tw)C E/RT,

in-1

input,

$

temperature 4

hot-spot

sen-

K-l

Parametric

sensitivity

and runaway

in fixed-bed

4

catalyst particle density, kg/m3 source tirm. defined by Eq. (11) normalized Thiele modulus, 4wCf(vp)(l-xp)n-IlI/21n_1

t 9

generic

input

cri tical catalyst reactor

condi ti on particle surface wall

reactor particle ad joint

inlet temperature quantity

Ps

0

normalized

Thiele

modulus,

parameter

of

(V,/S,)[(n+l)p,k(Tw)C the

reactors

1071

/20,11’2

model

Subscripts C

P w Superscrl

i

* .

pts maximum along

the

reactor

REFERENCES The critical conditions in thermal explosion theory with Adler. J.. and J. W. Enig (19641. Comb. and Flame, 5, 97-103. reactant consumption. Bilous. O., and N. R. Amundson (1956) * Chemical reactor stability and sensitivity, II. A.1 .Ch.E. Jl, 2. 117-126. Boddington. T ‘P. Gray, W. Kordylewski, and 5. K. Scott (1983). Thermal explosions with extensive rtZctant consumption: a new criterion for criticality. Proc. R. Sot. Fond. A, 390. 13-30. theory for nonlinear systems, II. Cacuci, 0. G. (1981). Sensitivity Extensions to additional classes of responses, J. Math. Phys.. 2. 2803-2812. Carberry, J. J. (1976). Chemical and Catalytic Reaction Engtneerina. McGraw-Hill. New York, Ooraiswamy, L. K., and M. M. Sharma (19841 Heterogeneous Reactions: Analysis, Examples, and Reactor Design, Vol. 1. John Wiley. New York. A study of the kinetics of vinyl acetate syntheEl-Sawi, M., G Emig. and H. Hofmann (1977). Chem. ingng Jl, 13, 201-211. si 5. H. Hofmann. U. Hoffman”. and U. Fiand (1980). Experimental studies on runaway of Emig. G.. catalytic fixed-bed reactors (vinyl-acetate-synthesis). Chem. Engng SCi.. 35, 249-257. (1979). Chemical Reactor Analysis and Des-n. John Wiley, New Froment. G. F.. and K. 6. Bischoff York. V. Balakotaiah. and 0. Luss (1985). Mu1 ti pl ici ty features of porous catalytic Hu. R.. Influence of reaction order and pellet geometry. Chem. Engng Sci.. 40. 599-610. pellets-II. Critical behavior for homogeneous reacting systems with rarge activation Lacey. A. A. (19831, Int. 31 Engng Sci., 21. 501-515. en&gy. McGreavv. 2.. and C. I. Adderlev (1973). Generalized criteria for parametric sensitivity and tempeFatu+& runaway in catalytic reactors. Chem. Engng Sci.. 28,.577-584. Tests for transport limitations in experimental catalytic reactors. Ind. Mears, 0. E. (1971). Engng Chem. Proc. Des. Oev.. 10. 541-547. sensitivity and runaway in tubular reactors. and A. Varma (lm2) _ Parametric Morbidel li e M., A.1.Ch.E; Jl; 8. 705-713. On parametric sensitivity and runaway criteria of Morbidelli. M.. and A. Vat-ma (1985a). pseudohoAo&neous tubular reactors. Cheb. Engng Sci ., 4& 2165-2168, Morbidelli. M., and A. Varma (1985b). A generalized criterion for parametric sensitivity: Application to thermal explosion theory. Proc. R. Sot. Lond. A, submitted for publication. Morbidelli. M.. and A, Varma (1986a). Parametric sensitivity in fixed-bed catalytic reactors: The role of interparticle transfer resistances. A.1.Ch.E. Jl. in press. sensitivity in fixed-bed catalytic reactors: Morbidelli, M., and A. Varma (1986b). Parametric The role of inter and intraparticle transfer resistances. A.1,Ch.E. Jl. submitted for publication. Pereira, C. J., J. J. Carberry. and A. Varma (1979). Uniqueness criteria for first order catalytic reactions with external transport llmltations. Chem. Engng Sci., 34. 249-255. Rajadhyaksha. R. A,. K. Vasudeva. and L. K. Ooraiswamy 1191s) . c sensl tl VI ty 1 n Parametrl fixeh-bed reactors. Chem. Engng SCi.. 30. 1399-1408. van Welsenaere. R. J.. and G. b. Froment71970). Parametric sensitivity and runaway in fixedbed catalytic react&-s. Chem. Engng Sci., 5, 1503-1516.