parametric sensitivity and runaway in tubular reactors

8/3/2019 Parametric Sensitivity and Runaway in Tubular Reactors

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Parametric Sensitivity and Runaway inTubular Reactors

Para me tr ic sensit ivity of tubular reactors is analyzed to provide cr i t ica l valuesof th e heat of r eact ion and h ea t t r ansfe r pa rame te r s de f in ing runaway and s tab leoperatio ns for all positive-order exoth ermic reactions wi th finite activation ener-gies, and for a ll reactor in k t temperatures. Evaluation of the cr i t ica l values doesnot involve an y tr ia l and error.

MASSIMO MORBlDELLland

ARVIND VARMADepartment of Chemkal Engineering

Unlverslty of Notre D a mNotre Dame, IN 46556SCOPE

The knowledge of temperature and concentra t ion prof i lesa long a nonisothermal tubular reactor plays a fundamental rolein the design, operation a nd control of such reactors. Specifi-ca lly, in exothermic gas-solid reactions, t he tem perat ure ma yexhibit sudden sh arp increases along the reactor length. Wh ensuch a phenomen on occurs, the reactor is sa id to opera te in a runaway condi t ion , wi th the a ssocia ted p re sence of th e so-called hot spot, which can adversely affect reactor perfor-mance .This problem was f irs t s tudied by Bilous and Amuodson(1950), who po in ted ou t th a t in some cases , the tempera tureprof i le is extremely sen sit ive to th e operational and physico-

chemical parameters involved; they termed such a reactorcondit ion para met r ic sensitivity.Although a var ie ty of subsequent papers have been devotedto establishing re la t ionships among the reaction and reactoroperation con ditions governing reactor runaw ay, previous workshave been incomplete and have a lso required tr ia l and errorprocedures to o bta in such re la tionships.Ou r purpose is to provide easily applicable exact criterion ofreactor runaway condit ion for all positive-order exothermicreactions, using the full Arrhenius tempera ture depend ence ofth e reaction ra te , and for a l l va lues of th e inle t temperature .

CONCLUSIONS AND SIGNIFICANCEBased on th e method of isoclines, a necessary a nd su ff ic ientcondit ion for reactor ru naw ay is identif ied. For a l l posit ive-order exoth ermic reactions, using the full Arrhenius temperatured e p e n d e n c e of the reaction ra te , an d for a l l reactor inle t tem-peratures, a simple procedure to ob tain critical values of the hea tof reaction and heat transfer parameters def ining runaway an dnon-runaway (i.e., stable) ope ration s is derived. The procedure

is explic i t , and requires only a single numerical integra tion toobta in th e cr i t ica l values.All other param eters being f ixed, reactor runaw ay is morelikely as the reaction orde r decreases, the reaction activationenergy increases, or as the inle t tem peratu re of the reactionmixture increases.

INTROwCTmNAlmost simultaneously with the appea rance of the work of Bilousand Amundson (1956), Cham br6 (1956) applied t he m ethod ofisoclines to establish qualita tive featu res of the nonlinear d iffer-ential equatio ns that govern steady sta te behavior of t he reactorin the temperature-conversion phase plane. Rarkelew (1959) usedthe same m ethod to integrate these equations for many differentcases, and derived empirical criteria between the heat of reactionand reactor cooling parameters to define runaway condition.Adler and Enig (1964) in studying the mathem atically similarproblem of combustion of solid materials with finite heats of re-action, defined the Occurence of a n inflection point in th e tem-perature-conversion phase pla ne as a criterion for the critical ex-plosion condition. T he relation between the critical values of theparam eters, for a first-order reaction an d large activation en ergywas obtained numerically.

( o r m p n d e n r cwrrrniny lhb p p r liuulcl I* a d d r d 0 A Vrrrrw During Ihc c v u m d0001-lJlI-R2-46860705-(2this wort. M Morbidelli was on h v r ro m IcAfccnnn, di Milano,Milnm. Italy,t he Amencan lnrtilutrd Cherninl Engirrcrs. 1982

AlChE Journal (Vol. 28, No. 5)

Due to the large amoun t of com putation involved in this a pproach, various approximations have been sugg ested by diffe rentauthors (HlaviEek et al., lM9; Dente and Collina, 1964; VanWelsenaere and Froment, 1970) in order to obtain simple explicit(altho ugh approxim ate) expressions of the critical parameters.Some num erical solutions of the problem for different ordersof rea ction a nd f init e values of the activation energy have beenreported by Dente et al. (1966). The criterion derived by theseauth ors is based on the observation that at critical conditions, thetem peratu re versus reactor d istance profile has a n inflexion point,wh ere also the third de rivativ e vanishes. However, this criterion,aspointed o ut by A dler and E nig (1964). is always implied by t heone based on the temperature-concentration phase plane, whilethe conv erse is not true.Apparently unaware of the work of Adler and Enig (1964).Oroskar and S tern (1979) have recently derived a proced ure todet erm ine exact conditions leadin g to runaway reactor operationfor a first-order reaction, using the Frank-Kamenetskii approxi-mation to describe th e temp erature dep enden ce of the reactionrate.

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isocline8,. 32/

- onversion, xFlgure l( a) . Pha- plane behaviorfor an unstable case of a first order reaction

with finite activation energy, y. [O = inflexion point].

Th e present paper provides an easily applicable exact criterionof reactor runaw ay for all positive-order exotherm ic reactions,using th e full Arrhenius depend ence of the reaction rate on tem -perature, and for all reactor inlet temperatures. Based on theme thod of isoclines, a simple strategy is presen ted to evaluate, ina single num erical inte gratio n, th e critical values of t he heat ofreaction an d heat transfer parameters which defin e the separationbetween run away a nd no n-runaway (i.e., "stable") reactor oper-ations. This procedu re does not involve any trial-and-error re quiredby previous authors (Adler and Enig, 1964; Dente et al., 1966;HlaviEek et al., 1969; Oroskar and Stern, 1979) and thereforeconsiderably reduces the amoun t of com putation req uired for theevaluation of the critical param eters.It is worth stressing that this classic parametric sensitivity an drunaway problem pertains to the steady state character of thetubular reactor, an d no t to its transient behavior.BASIC EQUATION

Consider an ideal nonisothermal pseudohom ogeneous (i.e ., aone-phase plug-flow model) tubular reactor operating in the steadystate, in which an n-th order reaction occurs. The relation betweenthe tem perature a nd conversion along the reactor length is givenby th e equation (Aris, 1965):

with th e initial condition: 8 = 8' a t x = 0,where

0 0.2 0.4 0.6 0.8- C o n v e r s i o n , x

Figure l(b ). Phase plane behavior for a case 01 criticality (a" 14.31) fora flrst-order reaction with flnile activation energy, y. At the point indicatedby 0, he locus curve, the operating curve and Ihe isocline defined by 8, =

4.55 are all tangent. Note that @+= 358.9, and is therefore not shown.

and the tem perature of th e cooling medium ( T , ) is assumed con-stant.

DEFINITIONS AND OBSERVATIONSFollowing previous notation (C hambr6,1956; O roskar and Stern,Operating c u r w (6,) is the solution of Eq. 1 .Isocline (8b0) is obtained from Eq. 1 considering d8/dx as a

(3)Each point on this curve, 8 = O~,(x), gives the value of th e slopeof the operating curve at that point, and therefore is very usefulfor graphic al solution of Eq . 1.Locus curue (81) is tP , locus of points where the tang ent to theisocline at that point, B i sO , s equal to the value of th e param eter 8,that d efines the isocline itself ( Fi gu re la) . In other words, this is thelocus of the points where th e operating curv e is tangent to theisocline. T he locus curv e 8 = &(r)s therefore given by the solutionof the system:

1979), we define the following curves:

constant param eter (0,):x = l - ___W - 4 1'"

[Lo- 8,)J

AlChE Journal (Vol. 28 , No. 5 )Page 706 September, 1982


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It is useful for what follows, to n ote some characteristics of these1. The deriv ative of the isoclinecurves:

vanishes for two values of 8 (Figure 1):(6)

Thus, d8/dx 5 0 when y 5 ; so, here isno possibility of runawayfor any values of th e other parameters. W hen y > 4, d8/dx > 0 fo r8- < 8 < 8+ ,a n d < O f o r 8 > 8 + o r 8< 8-;also, d8/dx = OD a t 8 =8-t.2. All the isoclines defined by values of 8, smaller than a specific8: are located in a region of space enveloped by t he isocline definedby 8: itself (F igure l a) . This follows from t he observation tha t

(Y- 2) I?(?- )11/225 = Y

(*) < 0 for all 8 > 0.d8, B = constant3. The locus curve has an asymptote at 8- - from above and8 - + from below, as x - 03 . Recalling that when 8- -, (1+ 8 / ~ ) ~- it follows from E q. 4a tha t 8, - from below;so f rom Eq . 4b, x - ). At x = 0, th e system of Eqs. 4a an d 4bhas either two solutions, at 811 nd 81,, or no solutions for 86 [B-, 8+].Therefore, in the range [B- , 8+], the locus curve is located in th e

positive x region of the 8 - phase plane only for 81, < 8 < el2; freal values an d 81 do not exist, the locus curve isalways restrictedin the negative x region. It is worth no ting that for 0 < 8 < 8- an df o r d > 8+, rom Eq. 4a, each point o n the locus curve is definedby negative values of t he param eter 8,; this means that t he locuscurve travels in a region where all the isoclines are defined bynegative 8,.4. Th e locus cur ve increases monotonical,!y in the ra ng e of 8 >0 whe re the first deriya tive of t he isocline, elsos positive, and th esecond derivative, Or s o is negative. This can be observed fromFigure l a, and readily shown as follows. Recall that the locus curve81 is the locus of all points on th e isoclines where 8is, equals 8, thatdefines th e isocline itself. For a fixed 8,&, is independent of x fromE q . 5. From observation 2 above, for a fixed value of 8 > 0, in -creasing values of ,r enco unte r isoclines with sm aller 8,. Thus inthe region where Bi,, 0, for constant8, the value of x on t he locus curve increases with increasing a. Thisfollows from th e fact that from Eqs. 4a and 4b:

> Ofor 8- < 8 < 8,.(Z)o constantCRITERIA FOR RUNAWAY CONDITION

The problem under consideration is a steady state problem, an dsowe are not concerned w ith transient features at all. The problemsof instability present in w ell-agitated reactors are not present inplug-flow tubular reactors. In the form er, they arise du e to steadystate multiplicity; the plug-flow reactor always has a uniq ue steadystate which is globally stable to transient perturbations. Non-adi-abatic plug-flow tubular reactors, on the other hand, exhibit thephenom enon of parametric sensitivity an d runaway (a type ofinstability, distinct from tem poral ones), which a re examined here.It is worth reiterating that both these features are for the stead y-sta t e model.

Before any conditions on the physicochemical p arameters thatdefine runaway can be derived, one first needs a definition ofrunaway itself. It is generally agre ed that t he occurrence of a "hotAlChE Journal (Vol. 28, NO. 5)

spot" in the tem perature p rofile along with the presence of a pos-itive second deriva tive in the profile of te mp erat ure versus con-version before the hot spot is a runaway condition for the reactor(Adler and Enig, 1964; Van W elsenare and From ent, 1970). Fromthis definition, it follows that a maximum in the temperatureprofile along the reactor length is a hot spot, but runaw ay opera-tion occurs only with those hot spots which are accompanied bya positive second derivative in the temperature vs. conversionprofile before the temperature reaches a maximum at the hot spot.The purpose of this stu dy is to iden tify a relation between the pa-rameters a,p,?,an d 8' that give rise to a positive second derivativein the temperature-conversion profile.For th e case of 8' < 0, it can be shown that the value of 8: a t 8= 0 is always negative. Therefore, runaway condition can onlyoccur in th e region of positive 8. This observation allows us toconsider in the follow ing only th e positive portion of t he 8- phaseplane also for 8' < 0.If

81g(&) 3 a / p (7 )from Eq. 1,Bb < 0 at x = 0 and, because the 8 - profile cannothave a mi nimu m (V arma and Amundson, 1972), the value of 8always decreases; thus, in such cases, runaw ay can nev er occur. Inwhat is to follow, we therefore only have to consider conditionsviolating Eq. 7; i.e., 8'g(oi)< a /p .It can also be easily observed from Eqs. 1, 4a and 4b that the signof 8; at x = 0 an d 8 = 8' > 0 s positive only in the range 81, < 8' 0, du e to runaway, and asa consequence of observation 2, the operating curv e has to remainabove th e isocline. Therefore, at the transition point 8: > 8iS,,andso 8i8, < 0.On the other ha nt , if th e operating curve is tangen t to an isocline,then a t that point 8, = 0. If 8:s, < 0,&, < 8: and therefore theoperating curve will be located in a region above th e isocline and,as a consequence of observation 2, after the tangency point, 8; >0; his is runaway condition by definition, and shows the sufficientpart of the argum ent.CRITICALITY CONDITION

Recalling fro m observations3 that the locus curve for 8 < 8- an d8 > 8+ s defined by negative values of Ox, it follows that th e op-eratin g curve can intersect the locus curve in the range mentionedabove, only with a negative slope and so runaway can not occur.Thu s the operating curve, in order to give rise to runaway condi-tion, mu st intersect th e locus curve at values of 8 in the range [ O - ,8+]. This observation allows us to define th e op erating conditionof the reactor in some particula r cases without solving Eq. 1.Fro m observation 3 we recall that if the locus curve does notinterse ct th e &ax is; i.e., the values of 811 nd do not exist, the locuscurve for 8~ O - , 8+ ] s always restricted in the regio n of ne gativeconversion; therefo re there is again no possibility of runaw ay. Onthe other hand, if 81, an d 81 exist, the following possibilitiesarise:

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i) 0 < 8' < 8 1 ~ :0 < 0 a t x = 0, and the operating curve mayintersect the locus curve in the region [O],,1,I-giving rise to run-aw ay condition-depending on the values of the parameters.ii) 81, < 8' < 81: 8; b 0at x = 0,an d so runaway condition occurssoon after th e rea$tor en trance, as already noted before.iii) 8' > 816 8, < 0 at x = 0, the operating curve (observation3) cannot intersect t he locus curve in the range 8 E [el,,el2]with apositive slope; therefo re no runawa y ca n occur.Fro m t he above analysis, it follows that only fo r the case i ) an dfor 8' < 0 is it not possible, at this stage: to define the operatingcondition of the reactor. In th e following we consider the case:8' < 8rl ( 8 )where for 8' b 0,8: C 0 at x = 0, while for 8' < 0,8; < 0 at 8 = 0as noted before.

Let us define the point (f ,8)on th e locus curve where t he isoclinepassing through this point has an inflexion, a nd furtherm ore(9 )

Th e existence of such a point w ill be shown Lhortly; at th e mo-ment i t is worthwhile to note that both 8 an d 8 are independentof a. Th is is clegr for 8- f rom Eq . 6; fo r 8, the exam ination of theexpression for Bis0 , which in the following is stu$ed in mored etai l(Eq . 16), shows that th e value of 8 at which B i s 0 = 0, i.e., 8, is afunction only of y nd n . Now, assum e that, for a given set of valuesof o,y, and O', it is possible to find a value ac or g, such that theoperating curve passes through the point (2 = Tc ,19 )on the locuscurve.In th is case for all values of a > ac, s a consgq uence of obser-vation 5, the operating curve will reach the v alue8 for x < X c . FromE q. 9 and observation 4, he locus curve increases monotonicallyin the range 8- < 8 < 8, and from observation 3 it follows that

o,,< o for all 8 c [o-,8)

8- < 81, (10)Therefore, recalling that 8' < 811 rom Eq. 8, the operating curvsintersects the locus curve defin ed for ac n the region 81, < 8 < 8.Fro m observation 6, the operating curve must also intersect thelocus curv e defined for all valyes of a > ac n the sam e region,wherein from Eqs. 9 and 10,Ois, < 0; this implies runaway fromthe necessary and su fficient condition discussed above.

Fo r all values of a < ac, rom similar argum ents it follows thctthe ope5ating curve can intersect the locus curve only at 8 > 8,where Oiso > 0. This implies that runaway condition cannot occur.Therefore, acwill be called t he critical value of a for a given setof values of p, y, n and 8'.The relative disposition of the locus and opera ting curves, alongwi th seve ral isoclines, fo r a specific case of critic ality is shown inFigure lb .Note that ye still need to prove th e assumption that there existsthe point (?, 8)defin ed by E q. 9, and the corresponding value ofac.Fo r this purpose, we consider separately th e cases of n = 1an dn # 1.

Case 1: n = 1From Eq. 3, the second d erivati ve of an isocline is given by

which has a unique inflexion point at

an d f is obtained by substituting this in Eqs. 4a and 4b, which alsoprovide the 8, that defines the sp ecific isocline. It is easily observedfrom Eqs. 11and 12 and Figure l a tha t < 0 for all 8- < 8 < 8.Noticing from Eqs. 6 an d 12 that 8- 0 at 8 = 8, this means d8 /dx > 0 fo r O t ( e l , O), whichimplies that d8 /dz < 0 fo r 8 in the same range, and so 8 decreasesas z increases-Thus when Eq . 14 is integrated with the initialcondition 8 = 8 at z = Z, as z increases, a valu e of 8 = 8' is guar-anteed to be reached at some value of z which is by definitiona c .- In the limit y + m, s+ 2 from Eq . 12. This limiting value of8 matches w ith that of Adler an d Enig (1964), an d Oroskar andStern (1979). However, even in this limiting case, they requireda laborious trial and error along with numerical integration of Eq.1 o obtain acwhich is here obtained with a single integration ofEq . 14. The fact that a single integration is sufjicient derives fromthe existence of explicit form ulae for Z a n d 8, which can be pb-tained as shown above by the method of isoclines. The othermethods employed heretofore (Dente et al., 1966, Hlav Gek et al.,1969) do not lend themselves to such a simple techniq ue for eval-uation of ac.

Case 2 n > 0 and # 1.Th e procedure becomes scmewhat m ore complex when n # 1.Since we seek the point (2 , 8) we need to first find the inflexionpoints of a n isocline. From Eq . 3,

where

atld@(6)= anonn =O

and so

since we are interested only in 8 > 0; as noted before, runawaycann ot occur with 8 < 0. Note in passing that 8 > 0 implies a > 8,from Eq . 1, assuring the existence of th e 2 / n root on the rhs of Eq.16 .From Observation 1of Eq. 6 (also Figu re 1)8is, - m as 8-8+ from below; O,,,- m as 8- + from above. Also, since th epoles of t he rhs of E q. 1 9 ar e precisely O*, this equa tion implies 4(8,) < 0. Similarly, @ (8-) > 0 by a consideration of Bis0 as 8+ -from above or below.It now becomes necessary to separa te the subcases n > 1 and 0< n < l .Subcase 1;n > 1 . From Eq. 18,

@(O) > 0 a n d @(0)- as 0 -w it h this , and the fact that q5(8-)> 0, @(O+) < 0, it is assured tha t@(8) 0 fo r at least two positive values of 8:one in the interval 8-< 8 < 8+and the other a t 8 > O+. Furth er, from Descartes rule ofsigns, @(0) as at m ost two positive real roots. Thu s it is clear that

Page 7 0 8 September, 1982 AlChE Journal (Vol. 28, No.5 )


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1bc0

C) r = o ,d=o40 - 40b ) 1 100,d=O30 30-

c

Heat Transfer Parameter, /9Figures 2(a-c). Critical valuesof the heat of reaction parameter (a")s a functionof the heat transfer parameter(8 ) or various values of the reaction order

(n). For each n, the runaway region is above the curve, and the non-runaway (1.e. "stable") region below the curve.

40-30 -B6.50 I

1 1 1t o 10 2 0 3 0 4 0 0 10 20 30 40- eat Transfer Parameter, BFigures 3(a-c). Critical valuesof the heat of reaction parameter(a")sa function of the heat transfer parameter (B ) or different valuesof the inlet temperature

(el).For each 8', the runaway region s above the curve, and the non-runaway (i.e. "stable") region below the curve.

creasing values of 0' reduce the runaway region, as can be ex-pecte d, for all values of n (Figures 3a-c).An interesting feature can be observed in these Figures 3 a c ForO i sufficiently negative, the critical curve aC@) as a minimumat relatively low p values. This means that over a small range of theheat transfer param eter 0, for a fixed heat of reaction param etera, decreasing or increasing results in non -runaw ay operations.This is because for sufficiently low inlet tem peratu res relative tothe wall, a decrease in p reduces th e influenc e of t he hotter wall

- w e r s i m . aFigure 4. Operating curves around criticality for zero Inlet temperature, 8' .1 = inflexion point].

resulting in stability; for highe r /3, of course, the normal situationprevails; i.e., an increase of p causes stability.In Figures 4 , 5 an d 6, the tem peratu re profiles in the 0 - phaseplane in non-runaway, critical and runaway opera ting conditions,are shown for various 8' values. Note that despite the small dif-ferences between the v alues of a, he temperature profiles increasedramatically when runaway occurs (i.e., or N > ac) , hich testifyto the reasonableness of t he definition of ru naw ay. In Fi gu re 5 itis shown that the value a = 13,which for 8' = 0 corresponds to a

Figure 5. Operating curves around criti calit y for positive inlet temperature,8'. 10 = infiexion point].Page 710 September, 1982 AlChE Journal (Vol. 28, No. 5)


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i*0 0.2 0.4 0.6 0.8- onversion ,Figure6. Operating curves around critlcalltylor negative inlet temperature,81. [ = infiexion point].non-runaway condition (Figure 4), for 8 = 1 gives rise to runaw ay.It is appare nt that a n increase of one in th e inlet temp erature 8causes the temperature at th e hot spot to increase by a factor greaterthan four . On the other hand, for 8 = - 1, the value of a = 13corresponds to a non-runaway condition an d as shown in Figure6,a decrease in the inlet temperature by the same amount as before,this time gives rise to a decrease of th e maxim um temp erature byonly about 0.25.In Figures 4 an d 5,expa nded versions of t he 8- portraits arealso shown for conversions near one, to point out th at 8- withzero slope as x - (except for n = 0, as shown in th e Appendix);this fact can also be directly verified by investigating th e tem per-atu re and conversion versus reactor length equations (from whichE q . 1 is derived) linearized abo ut the equilibrium point 8 = 0, x= 1.Th e profile of 8, in a case with sufficiently larg e O i so tha t 8; >0 at x = 0, which implies runaway right at the reactor inlet, is alsoshown in Fig ure 5. It can be observed that despite the relativelylow value of a, he 8 - curve is in this case qualitativ ely similarto those in runaw ay conditions of F igure s 4 an d 5.Finally, it is worth clarifying that the term runa way has beenused in other contexts as well. For example, Kim ura an d Levenspiel(1977) investigated an adiabatic tubular reactor in th e limit of largerecycle, where it behaves similar to an adiabatic stirred-tank re-actor. As is well-known, multiple steady states are possible in sucha case. When steady state multiplicity is present, d epen ding on theperturbations, the reactor can move from a low-conversion stateto the high-conversion state, causing th ereby a temp erature in-crease. They term ed such behavior as runaway.It should be obvious that the present w ork deals with a differentproblem.

ACKNOWLEDGMENTM. Morbidelli thanks the Stazione Sperimentale per i Com-bustibili; Rorsa di Studio L. Miani for financial support duringthe course of this study.

NOTATIONA = preexponential factor in Arrhenius rate equationAlChE Journal (Vol. 28, No. 5)

a{ (i =1,4)CPDEgAHknPRTUYX

z

= coefficients defined by Eq. 18= mean specific heat= diameter of the reactor tube= activation energy= heat of reaction= reaction rate constant= reaction order= total pressure= ideal gas constant= temperature= overall heat transfer coefficient= reactan t conversion= molar fraction= integration variable

= ex p l - ~ / l l+ (O/Y)ll

Greek LettersaP = dimensionless heat of reaction, defined by Eq . 2= dimensionless heat transfer param eter, defined b y Eq .

= dimensionless activation energy, E / R T ,= constant, defined by Eq. 25= dimensionless temperature, [( T - T , ) / T,]y= param eter characteristic of an isocline= roots of Eq. 6= function of 8, defined by Eq . 17= parameter, defined by Eq . 28 c

2Y68O X

O*4\kSubscriptsA = reactant0 = operating curveis0 = isocline1 = locus curveW = wallSuperscriptsi = inlet valueC = critical valueAPPENDIX

Case 1: n = 0From Eq. 1, the opera ting curv e is given byd 8dx a - peg(@

Th e isoclines are horizontal lines, where the value of 8, for each8,, is given by

The function [Bg(B)] as a maximum at 8- and a min imum a t8+ , where 8* are given by Eq. 6. Therefore, the pattern of theisoclines in the 8 - plane is of th e form shown in Figu re 7, wherethe arrows represent increasing values of the param eter Ox definingeach isocline. From Figu re 7 and the definition of an isocline fol-lows that the second derivative of t he o perating curve is positivein the range 8- < 8 < 8+,and negative for 8 < 8- or 8 > 8+ .From the same arguments as in the n # 0 case, Eq. 7, and alsofrom Figure 7, it is apparent that if 8, < 0 at x = 0, i.e., Og(8) 2a /@,he operating curve always decreases and therefore runaw aycondition cannot occur. If dig(@)< a/P, he following pi bi l i t i escan arise:i) 8 < 8-: 8; < 0 at x = 0, runaw ay occurs if th e operating curveintersects the straight line 8 = 8-.ii ) 8- < 8 < 8, : 8: 2 0 at x = 0, runaway occurs soon at thereactor entrance.iii) 8 > 19 , : 8:


8/9

16 8 * 5 , n .0 , d = 9, @=20,Oi=0 Ii$ = 1.95/

12

i$ = 1.95/I 1

12

InIn0c0Inc-.-Eiit

0- 1-1 5 -

I I I I i0.2 0.4 0.6 0.8 I- onversion, xFigure 7. Phase plane behavior for a stable case of a zeroth order reactionwith finite activation energy, y.

the reactor is always non-runaway (i.e., stable). W e consider now,as in th en # 0 ase, only the case i) . It is app aren t that if the isoc-line defined by 8, = 0 is located below the value 8-, the operatingcurve canno t intersect such an iy clin e, and is therefore restrictedin th e region f? < 8-, where exis0


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proximation of ac (except forvery negative 8in th e region of low/3), and that LYC- * or larger valuesof the heat of reaction. Thelatter is due to thefact that for highly exothermic reactions theoperating curve approaches the isocline O r = 0 at relatively lowerconversions, and so criticality is well approximated by insisting thatth e 0, = 0 isocline coincide with 0-. Increasing values of the inlettemperature have the same effect on the operating curve; i.e., theyalso give rise to values of a c closer to a*.

C a s e 2 y = aIn this case

g(0)= exp(-o),and so the operating curve equat ion becomes:

d0 0 exp(-0)- = L y - pd x (1 - )nand the isocline equation is:

Differentiating Eq. A5 :

where it is apparent that for 0 > 1, O iso > 0. This result coincideswith the general case noting from Eq. 6

lim 0- = 1; li m 0 + = + aY -- 7 -m

The locus curve equation is given by

It is readily observed, setting x = 0 in Eqs. A7 and A8, that the valueof 0, at which the locus curve intersects the &axis is given by thesolution of the equation

(A9)Noting that for the rhs of Eq. A9 , R ( 0 ) :R(1) = - m, R ( m ) = a -n, nd R > 0, while for the lhs, L ( 0 ) :L ( 1 ) > 0, L ( m) = 0, L < 0for 0 > 1;it then follows that Eq. A9 has at most one solution (01,)in the range 0 > 1, and that the necessary and sufficient conditionfor the existence of this solution is a > n.This result is consistentwith the general case, noting that lfm,,,

Considering now only the case 0, > 0 at x = 0, which happenswhen Eq. 7 is violated, the following possibilities can arise:

i) a > n, < Oil : 0; n, 2 Oil: O0 0 t x = 0;unaway occurs soon at thereactor entrance.

L (0) E 00 exp (-0) = (a- n 0 / ( 0- ) ] R ( 0 )

= a.

iii) a < n: he locus curve for O > 1 s restricted in th e r d 0 re-gion; runaway cannot occur.As in th e general case, only the first possibility needs to be con-sidered further in the following. The second derivative of t heisocline is obtained by differentiating Eq. A6 :

Oiso vanishes for two values of the temperature, 0 = 1 f&, hereonly the larger one is in the range 0 > 1 for all values of n.Thisresult, which was obtained by Adlgr and Enig (1964) in a differentway, coincides with the solutkn , 0 of Eq. 17 in the limit y + a.The uniqueness of the value 0 in the range 6 c [e-, 0+], alreadyrecognized in the general case, is therefore confirmed for all re-action orders and y - as well.LITERATURE CITEDAdler, J., and J . W . Enig, The Critical Conditions in Thermal ExplosionTheory with Reactant Consumption, Comb. and Flame, 8, 97

(1964).Aris, R. , Introduction to the Analysis of Chemical Rektms, Prentice-Hall,Englewood Cliffs, NJ (1965).Barkelew, C.H ., Stabilityof Chemical Reactors,Chem.Eng.Prog. Symp.Ser., 25 (55),37 (1959).Bilous, 0..and N. R. Amundson, Chemical Reactor Stability and Sensi-tivity.11, AIChE J.,2 , 117 (1956).ChambrC, P. L ., On the Characteristics of a Nonisothermal ChemicalReactor,Chem.Eng.Sci.,5,209 (1956).Dente, M., A. Cappelli,G. Buzzi Ferraris, and A. Collina, La sensitivitkdel regime dei reattori chimici a flusso longitudinale-I, Ing. Chim.Dente, M., and A. Collina, I1 comportamento dei reattori chimici a flussolongitudinale nei riguardi della sensitivitl,Chim.e Industria,46,752

(1964).Hlavkkk, V., M. Marek, and T. M. John, Modellingof ChemicalReactors,XII, Cok Czechoslm. Chem. Comm.,34,3868 (1969).Kimura, S. , and 0. Levenspiel, Temperature Excursions in AdiabaticPacked Bed Reactors, Ind. Eng. Chem. Proc. Des. Den, 16, 145(1977).Oroskar, A. , and S. A. Stern, Stability of Chemical Reactors,AIChE J.,25,903 (1979).Tsotsis, T. T., and R. A. Schmitz, Exact Uniqueness and MultiplicityCriteria for a positive-order Arrehnius Reaction in a Lumped System,Chem.Eng.Sci., 4,135 (1979).Van den Bosch, B., and D. Luss, Uniqueness and Multiplicity Criteria foran n-th order Chemical Reaction, Chem.Eng. ci.. 32,203 (1977).Van Welsenaere, R . J. . and G. F. Froment, Parametric Sensitivity andRunaway in Fixed Bed Catalytic Reactors,Chem.Eng. ci., 25,1503(1970).Varma, A. , and N. R. Amundson, Some Problems Concerning the Non-Adiabatic Tubular Reactor,Can.1.Chem. Eng. , 50,470 (1972).

I t d . , 2 ,s (1966).

Manuscript received February 1 9,1980;revision receivedAugust 6, 1981,andac-cepted August 24 ,1981 .

AlChE Journal (Vol. 28, No. 5 ) September, 1982 Page 713

parametric sensitivity and runaway in tubular reactors

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