Exact criteria for uniqueness and multiplicity of an nth order chemical reaction via a catastrophe theory approach

EXACT CRITERIA FOR UNIQUENESS AND MULTIPLICITY OF AN nth ORDER CHEMICAL REACTION VIA A CATASTROPHE THEORY APPROACH HSUEH-CHIA Department

of Chemical

CHANG

Engineering,

and JOSEPH

Pnnceton

University,

M CAL0 Pnnceton.

NJ 08540, U S A

(Received 22 May 1978, accepted 26 October 1978) Abstract-The development of a simple, generahzed techmque for the exact determmatton of regions of unique and multlple solutions to certain nonlinear equattons vm a catastrophe theory--lmphclt function theorem approach, IS presented The apphcatlon of this technique to the nth order chemlcai reaction m the nonadiabatic and adiabatic CSTR yields exact, exphcrt bounds for all n 2 0 To our knowledge, this IS the first report of exact, exphcrt bounds for these systems, except for R = 0, I for the adiabatic CSTR. and n = 1 for the nonadiabatic CSTR For the nonadiabatic CSTR. these bounds show that the hiaher the reaction order, the smaller the remon m uarameter space for which muitlphclty can occur for all y and iZc (dImensionless act&ion energy and coolant temperature, respectively) This behavior IS slmdar to that reported by Van den Bosch and Luss[ll for the adlabatlc CSTR The zeroth order reaction m the nonadiabatic CSTk exh& more complex behavior and assumes charactenstlcs of both high and low reaction orders insofar as mcreasmg and/or decreasing the uniqueness space, in comparison to all other n > 0 An exact lmphclt bound between regons of umqueness and multrphclty IS also derived for the nth order reaction m a catalyst particle with an mtraparWle concentration gradtent and umform temperature, and is fully demonstrated for the first order reacuon In addition, exphclt cnterla, sufficient for uniqueness and multiphcrty of the catalyst particle steady state, stronger than those of Van den Bosch and Luss, are also developed by combmmg the present technique with bounds suggested by these authors

In the present work, we develop a simple techmque for the determmatlon of exact bounds between multldlmenslonal regons of unique and multiple solunons to certain nonlmear equations via the apphcatlon of the lmphclt function theorem, as msplred by catastrophe theory The method 1s proven m general for multlparameter systems and spectically apphed to the nth order reaction m the nonadiabatic and adlabatlc CSTR and the catalyst partlcle To our knowledge this IS the first report of exact, exphclt bounds for all n 2 0 m the nonadiabatic CSTR, except for n = 1[5]. and m the adiabatic CSTR. except for n = 0,l (e g see[6]) The results for the lumped parameter catalyst particle model are ldentlcal to those for the adiabatic CSTR, and an exact lmpltclt cntenon 1s developed for the catalyst particle model with umform temperature but with an mtrapartlcle concentration gradient For the adiabatic CSTR, and both catalyst particle models, very tight but approximate bounds have been obtamed by Van den Bosch and Luss[6] heremafter referred to as VdB&L) for II = 0. 1 for the lumped parameter models, for which these authors succeeded m obtammg the exact bounds, and compansons are made with their work when possible

INTRODUCTION

nonhnear models of physlco-chemical systems can exhlblt multlphcltles of steady states or umque steady states, dependmg on the system parameters and the Inherent bounds on the system vmables Oftentimes m dealing with such systems It 1s of immediate interest to dehneate a pnon where these reaons of different character he m multldlmenslonal parameter space without solvmg the steady state equation(s) In particular, the exact defimtlon of regions of m&r-modal behavior 1s of direct interest insofar as system stability IS mtunately related to the existence of multiple steady states which can give rise to dscontnnntles m system variables as a result of smooth, contmuous vmatlon of the system parameters It IS exactly this latter type of behavior that IS the prmclple feature of Thorn’s catastrophe theory [ l-1 Catastrophe theory has been the subject of much recent controversy (e g see[2]) mamly due to the apphcatlon of Its relatively simple geometnc models to the analysis of extremely complex behavioral phenomena m the social and blolo@cal sciences, for which the models are not well defined (e g see[3]) In the physlcal sciences and engmeermg, however, models tend to be defined more precisely and are usually mathematically ngorous (d not always exactly correct), and thus results from a catastrophe theory-like analysis are quantitative and as exact as the model The most notable examples of systems exhlbltmg such behavior m chemical engmeermg are chemical reactlons coupled with mass and heat transport m chemical reactors and catalyst particles Indeed, hs has elegantly described the behavior of the CSTR m terms of the sunple cusp catastrophe[4] It IS well

recognized

that

certain

hiATRRMATICAL

D-PMRNT

Consider a curve m the x, a plane The equation descnbmg thus curve can be expressed m either exphclt or Implicit form, i e x = f(a) or f(x, a) = 0, respectively However, an equation given m unphclt form does not necessardy represent a function-a one-to-one mappmg from a to I Suppose rt IS of Interest to find when Nx. a) = 0 1s also a function, or equivalently, the set of pomts on the curve described by f(x, a) = 0 that provides 285

HSUEH-CHIA CHANG and JOSEPH M

286

CurustroPhe set. K If (90) E Rep) IS a point for which there exists a GE R’“’ such that (x0, ao) IS singular. then (a,,) IS a catastrophe pomt, and the set of all such pomts IS the catastrophe set, K K IS simply the projection of 2 onto the parameter space Uniqueness set, U If (a@) E R@’ such that there exists one and only one x0 such that (x0, a”> IS on S, then (a? IS a umqueness point, and U 1s the set of all such points Nd set, N The set of all (a”) E R(“’ for which there are no x0 such that (x0, a”> IS on S IS the nil set, iV In addttlon to these defimtlons some aspects of boundary and related terms need to be spectied Thus,

a one-to-one mapping from u to x The unphclt function theorem deals with this problem locally Given a pomt (x,, aO) such that f(xO. no) = 0, under certam condltlons in the netghborhood of (x0, no), the implicit equation f(x, a) = 0 wdl also be a function These condltlons are in some that f and aflax be nonzero and continuous nelghborhood of (x0, ao) In more general form, the theorem applies to a system of n equations In R f p variables, VIZ fr(x1,

,xn, u1,

,a,) = 0

fi(Xl,

.xn, al.

a,) = 0

f”(X1.

7x,, u1,

a,)=0

CALO

(1)

B(x0, .f) = {xl Ix - x01( e) IS termed an open nerghborhood any set m general. then

This system of equatrons can be solved umquely and ,x,, the mtemal or state variables, m locally for xl, ,u,, the external vanahles or parameters, terms of al, provided that certain ptial dertvatives are continuous and that afl axl

afl ax2

= det [J] # 0,

)

afn ax,

IS the exterior of A, where x IS the complement The boundary of A, 8A, IS then defined as

(2)

gEA

(3)

on 00

=(xianyB(x,e)nA#@ and --

1

g(ao)= x0

(2) f(q(a),

a) = 0, Va on

(3 of A

iIA = Int (A) fl Ext (A)

1

where J IS the Jacobian matnx A formal statement of this theorem follows Implicit function theorem Let (x, a) be a point m the R(n+P>, dlmenslonal where x= space, (u+p) ,x,,) E R’ “‘ . a = (a,, ,a,) E R’*’ Also. let f = (Xl, ,fn) be a vector functton defined on an open set S Cfl? m R’“+p’ with values m R ‘“’ Suppose now that f E A on S If (ti, a~) IS a point in S for which f(xo, a~,) = 0 and for, which det [J] f 0, then there exists a p-dlmenslonal open set, Qo. contammg a0 and one, and only one, function g defined on QO and having values m R’“’ such that (1)

(4)

of A, and Ext (A) = {x[3B(x, e) c 2)

>fl ax,

det af” Jf” axlaxp

of x NOW If A C R@’ IS

Int (A) = {xl3B(x. a) C A} 1s the mtenor

(3)

QO

The detaded proof of the imphclt function theorem can be found m advanced calculus texts (e g see Apostol[fl) Definrtlons In order to proceed with the development, a few defimtlons are required Equrflbnum mumfold, S The set of solutions (~,a) forms the equdibnum, p-dunenslonal mamfold SE R
l

Smgulur set, 2 A pomt (a,, a) E S IS said to be a * smgular pomt If det [J(xo. ao)] = 0 The set of singular L pomts m S forms the singular set, 2. which IS the set of solutions (x, a) of eqns (1) and (2), and a p-l dlmenslonal ma&old m R’“+p’ Ordmury set, 0 A nonsingular pomt (x,,, h) E S 1s said to be an ordinary point, 1 e when detU(xo, PO)] : # 0 0 IS the set of ordmary pomts m S

B(x,c)~)

A#

(6) a],

where Int (A) and Ext (A) are the complements of Int (A) and Ext (A), respecttvely, and @ IS the empty set We now proceed with the followmg necessary proofs Lemma 1 If N = @ for an equdlbnum ma&old, S, defined by eqn (1). then U IS bounded by members of the catastrophe set, K, provided U # Cpand that U IS bounded,le N=@, U#@,aU#@jaUCK Coroihy 1 If N # @ for an eqmhbnum mamfold, S, defined by eqn (1) (I e there exist (PO)E Rep’ where no ~0 E R’“’ exist such that (~0, a~) E s), then N IS bounded by members of the catastrophe set, K, d It IS bounded at all,le 8N, N#QjaNCK Proofs Assume that for a,,E R*’ there are exactly m values of x(x1, , Xmh such that (x1. a) E S, I = m, I e for this value of a there are exactly m 1, solutks to eqn (1) In addltlon assume po 65 K, and the f in eqn (1) are continuous, mfinitely contmuously dfierentlable and det [J(x, a,,)] # 0 for all x such that (x, a~) f S Then by the lmphclt function theorem, m each nelghborhood of (x, , a,,) E S, where L= I, , m. x, can be expressed as a function of a, I e there IS a one-to-one mapping from R@ to R’“’ m the nelghborhood of (x,, a,,) If & IS increased by Sa, there are still m vectors (xl + 6x,, , x,,, + 6x,) such that (x, + Sx,, a0 + Sa) E S In short, u0 65 Kj the number m remams constant m the netghborhood of a0 Equivalently, d there IS a change m the number of XE R’“’ such that (x, a,-,)E S+OEK

lvlultlphclty of an nth order chemical reactlon via a catastrophe theory approach If N = @ and U# a’, m order to amve at the boundary of U a transition must occur at ao~ K, and thus U IS bounded by K and Lemma 1 IS proven If N# a, m order to arrive at the boundary of N, a transition must occur at a,, E K and thus N IS bounded by K and Corollary 1 IS proven At this pomt perhaps two examples would best dlustrate the apphcatlon of these results Example 1 Consider the equthbrmm manifold, S,, shown m Fig 1 and defined by (1 1) The smgular ma&old, condition g(x,

X1. 1s defined by eqn (I 1) and the

ul,aZ)=

3x2-u1

=O

on

SI

f,(x,,

x2,

a,) = xl3 - atxl - x2 = 0

fi(xl, x2, a,) = 3x12- a1 = 0

manifold of S1 of Example 1, I e X1 N2 1s composed of points on u1 < 0 and N2 # @ Kt IS the point UI = 0 and N2 IS bounded by K2 accordmg to Corollary 1 Also, NZ defines the region on a, where there 1s one, and only one, possible value of x for every value of a2 that places This (x. al,n3 on S,, I e for a, CO, (al, az)E UlVa2 represents a necessary and sufficient condltlon for umqueness, and conversely, multlpllclty It should be noted that this cntenon IS not global, but rather IS defined wrth respect to a specd’ic parameter uz (a, m general) In other words, m Example 2 we do not seek the set U,, but rather the set {all(al. a*) E UlVa2E R}, which 1s lust N2. and by Corollary 1, bounded by KZ This 1s an extremely useful result which will be formalized and proven m general m the followmg theorem Theorem 1 Let N,, S,, K,, XI, UI be defined for

(1 2)

S, and X1 as shown m Fig 1 are also known as the cnt1ca.l mamfold and the blfurcatlon locus, respectively, of the cusp catastrophe The catastrophe set, K,, m (a,, 03 space can be obtamed from the simultaneous solution of eqns (1 1) and (1 2), VIZ 27a22 = 4a13, and IS also shown m Fig 1 In this case it IS obvious that N1 = @, and from Lemma 1, the catastrophe set, K, bounds the umqueness set, U,, m the (ar, a*) space shown m Fig 1 Example 2 Let us now consider the set S2 defined by,

(2 1)

It should be lmmedlately obvious that by the transformation xZ+a2 and xI +x, S2 IS simply the smgular

287

fl(x, a) = 0 fn(xra)=O where x = (x1, , xd, a = (a,, S,, K2, X2 be defined for

f=+l(x, aPr a~,

(0

, up-l. a,), and let N2,

fdx,a,,ub

.a,-d=O

f&G qF.Ul.

, u&B-1)= 0

, a,-,)

(10

= det [J(x, aP, al,

, a,-,)1 = 0

Note that aP IS used like a state vanable, thus effectively increasing the number of state vmables to n + 1 However, there are also II + 1 independent equations, so the problem 1s well posed If the prolectlon of a along the a,, axis IS defined as a’, ie a=(al,

Pat . a,-~, ap)-a’=

(al,

, %-11,

(111)

where pO, IS a R*’ to R”‘‘-I’ mapping correspondmg to the proJectIon of a along the a,, axis, then (II) can be rewritten as f&5 a,, a’) = 0 f&c, aP, a’) = 0

, (SP)

(4

fn+,(x, aP, a’) = det [J(x, a,, a’)] = 0 Then ao’E Nt+ (I e

(80’, a,) E UIVU, E R

N2 = b’l(llo’,ap)

(VI

E UIVU, E RI),

provtded

Rg 1 DepIctron of vanables, parameters, and sets m Examples land2

(1) (2) (3) (4)

f 1s contmuous and n$irutely dlfferentlable w r t x aflaa, is continuous N, = @ N2#Q, and 8N2#Q)

HSUEH-CHIA CHANG and JOSEPH M

288

(5) UIf*aand8Ut+@ (6) The complement of the a, axls IS a non-empty {apl(a’, a,) E KI) (7) au, c u, Proof By the defimtlon of form of eqns (I) and (us) (see

From the definition

Substltutmg

(VI)

of a

mto

CASE

a singular set and from the examples 1 and 21,

n11 set,

a, ,a0’) E

I;,V(x, a,) E P

of a catastrophe

aO’E NP~(*‘,

(Vlll)

set,

a,) E K,Va,

E R

(1x)

Since au,, Int (CT,), and Ext (U,) are mutually exclusive sets V a E Rfp), either (a) E GUI, or Int (UA or Ext (U,) Let C,, C,, C3 be the proJections of GUI, Int (U,), Ext (U,), respectively, along the a, axis onto the a’ space, I e

Cl

(x1

G C3

I

c,nG=c,

11 c,nB=c3 III

(Xl)

c,nG=cz=c3

where 6 = {(a’l(a’, 0) E au,} and IS simply the set of projected points from JU, for a, = 0 These three posslblhtles are schematized m Fig 2 If I or III IS true, then the region of Interest. VIZ {a’j(a’, u,) E U,Vu, E R} = Nz = @, which vtolates condltlon (4) Thus the only posslbdlty IS II which leads to E C,, by defimtlon

@a~’ E C, fl

of p,+ (m)

(3, from case II

(JaO’E CI e((8o’, a,) E 8U,,

for some value a, E R

and Lemma I, But by aU,C Kl, Ext (U,)j(%‘, a,) E ICI, for some value a, E R, (a~‘, up) g KI for Vu, E R J(zQ,‘,u,“) E Ext (U,)

for Vu,” E R

Rg 2 Schematization of the three posslbllrtles for the relative locanons of aUt, and LT,

G~IIo’, Substituting

a,?E Int (U,)

U

~U,VCZ,,~E R

(Xl0

(IX lnto XII),

a,,’E Nz+(ao’, a,,? E Int (Ud U 8UlVuPo E R (XIII)

where pm., IS defined by (111) Then, since the complement of the proJectIon of aUs onto the a, axis IS a non-empty set by condltlon (b) (I e &ZJ, IS not parallel to the a, axis), It IS clear that there are only three posslbllities

& E Ext (U,)(sao’

I

proJectIon of aU, onto the set, I e A# a, where A =

(VII),

au’ E Nze(x,

From the defimtlon

CALO

aoE

Smce Int (U,) C UI and by condltlon (7) GUI C U,, then {Int (U,) U au,} C UI, and thus finally (XIII) becomes 80’E N&(ao’,

a,“) f U,Vu,” E R

WV)

QED Thus according to Theorem 1, the cntenon that (a’) must satisfy m order for (a~‘, a,) E U,Vu, IS that ao’E NZ and that conditions (l-7) are satisfied The set of all points that satrsfy these cntena IS Nz By Corollary 1, aNz C &, and thus If KZ can be found, the exact bound between unique and multiple solutions for ail a, for the system of equations defined by (1) m Theorem 1 IS determined In the proof of Theorem 1, x IS unbounded m R’“’ In many physlcal problems, however, x 1s often limited to a region m R’“’ due to physical constramts For example, in chemical reactions the dlmenslonless reactant concentration or conversion 1s bounded m [0, I] If such a constramt is present, addItIonal cons1deratlons are necessary Let E, the cusp pomt (which IS the equivalent of an n-dimensional inflection point), be the correspondmg pomt of KZ on S1 and xc and XI! be the mappings of c and P, onto R’“‘, VIZ E = 1(x, adlao E Kzt (x, ad E

-txE)= {x1(x,a) f 8 {xx,) = bl
SJ

Multtpllcltyof an nth order chemical reaction wa a catastrophe theory approach Usmg these defimtions and Theorem 1, the followmg lemma can be proven Lemma 2 If [xLrxU] are the bounds unposed on x where [xL. xu} = {xix, z xL, and x, IX~,, I = 1, n1, and rf {x,_} n {xX,} = {x~} fl {XX,} = Cp. and {x,1 C [XL, ;,I, and the seven comhtlons of Theorem 1 are satisfied m the bounded region, then Kz IS the exact bound between regions of umque and multiple solutions of x tn the Interval [xL,xU] for all a, {xL} and (xv} are singleton sets contammg only one element xL and xu respectively proof Since {xL} n {xx,}= {xv} n {XI,)= a, the projection of 2, m x space can never cross the bounds XL and xu Thus any extrema mterlor or exterior to [xL, x~] can never enter or escape the interval Therefore, any mtenor extrema can vanish only by coalescing with one another at the cusp pomt lying within the interval Thus Theorem 1 with N, L, and U, redefined accordmgly, 1s region applicable m the completely [XL, x.1 x Rep’ QED If {x~)E[x~,x~~ and {X=)n(XS,)={XU)n{x,,}=~, then extrema interior to (xL, x,), d there are any, can never vanish Thus U1 1s either unbounded or U, = @, thereby vlolatmg condltlon (5) of Theorem 1 If {xc) E [xL, xul and (x=1 n {xx,)or 1~~) n Ix%,} Z @, then the bound at I& becomes one-sided and sufficient only for uniqueness In this case the exact bound must be found by an alternate method

Then by Corollary

289

1, ~Nz C K2, where K2 IS defined by

f~ = Rx,, I% B. x B, xzcr n) = 0

fz=-$$ (XI, Da, B, y, fi, x2=. n) = 0

det[J]=det

[

’

k

1

1-O

1

a*f

a=f

m

(10)

dDiidX,

However,

and thus, m general, for (10) 1s

(tif/aDa)lK2

f(xl,Da,B,

# 0, this lmphes that

Y,B,xz~.

n)=O

~(x,,Da.B.y.B.x==.n)=O

(11)

AF’PLICATIONS

nth Order chemrcal reactron m a nonadtabatrc

CSTR

When a smgle nth order, vreverslble chemical reactlon occurs m a nonadlabatlc CSTR, the reactor steady states are described by,

f =

t1

_“:,,-expcl+

- (Bx, + /3x2,) -Da=0 B) + (lir)(Bx, + Bxz,) (7)

where the notation 1s that of Uppal et al 151,viz Da = k,, epTrCI;’

B = ( - AH)C,qipc,Tf x1 = xzc

= @ =

(CAf

-

r(Tc

CA)lCAf -

Tf)iTf

hAriVpc,

Thus K2 IS alternatively defined by eqn (11) if afidDa#O With specfic reference to eqn (7), the bound for the regron in (B, Y, B, xzC. n ) space where there IS one and only one solution for x1 for all values of Da 1s defined by (11) after the ehmmatlon of Da and x1, provided. (1) f IS contmuous and infinitely dtierentlable with respect to x for x E [O. l] (2) afiaDa# 0, and contmuous (3) there is at least one solution for all (Da, B, 7, B, xzc. n) to eqn (7), 1 e NI = @ (4) the regon m which the solution exists 1s bounded and the bound 1s not parallel to the Da axis Equation (7) satisfies all these requuements In general, f, 3f/3x,, Pf/3x,2 are implicit transcendental functions which make the ehmmatlon of Da and x1 dticult However, consider a function h such that

y = E/RTf h=g,f We now present the apphcatlon of Theorem 1 to eqn (7) Let S,, N,, K,, 2,. U, be defined for eqn (7). viz f(x,,

Da,

B,

y,

B.

xzc,

n)

=

0

+i&

(12)

I

where gl and g, are arbitrary functions m

(x,

a) Then,

(8)

and S2, N2, Kz, Z,, UZ for f(x,, Da, B. x B, xzc. n) = 0 $

(xl, Da, B, Y. P. xzE, n) = 0

(9)

ThUS d f = 0, aflax, = 0, and t3*f/8x1*= 0, then ah/ax, = 0 Therefore, we can choose g, and g2 such that ah/ax, =

0 1s a simpler function to mampulate Admittedly, g, and g2 may introduce addltlonal solutions to (11) However,

290

HSUEH-CHIA

CHANG

and JOSEPH M CALO

from Lemma 1, all the solutions to (11) are not necessarily bounds, and thus addltlons to the set of solutions IS not detrlmental Usually, members of the set which are not bounds are obvious Thus, the simultaneous solution of (11) ~111 yield the exact bound between umque and multiple solutions From eqn (17),

Substltutmg finally, C*(B,

this expresslon

for XT mto eqn (18) yields,

y, B, xZcr n) = aPS + bP2Q + cPQ2 + dQ3 = 0. (21)

where

P = qad - bc and K2 c aN, such that

Q = 2b3 - 6ac, and C* defines

(B, Y. P, x2=. n) E Nz+-(B,

Y, B, xacr n, Da) E UIVJW (22)

-BX,

which

((I+

B)+

+ y(1

- x,)"((l

lmphes

(l/y)tBx,

+ Pxzc))(l -XI)”

B(Bx, +

P&XI

+ p) + (llY)(BXI + Px*&

1 0. =

that

(1-c X,(IZ- l))($+

+

Bxl(l

+/3)(l

-x,)

= 0,

(15)

which IS a statement of Theorem 1, and thus eqn (21) IS the exact bound between regions of umque and multiple solutions of eqn (7) for all values of Da for R > 0. by Lemma 2 since x: E [O, 11 To our knowledge this IS the first report of the exact bound for an nth order reactlon m the nonadlabatrc CSTR The character of the regions separated by the bound represented by eqn (21) will be established by examining the n = 1 case, for which the bound IS known[5] For n = 1, a=0

z = 1 + p + pxz,/r

Rather than proceedmg to determine #flaxI*, we employ the method described by eqns (12) and (13) Here we ludlclously choose g, = 0, and thus where

b=BZyZ+B(l+/I) c=2Bz/y-B(l+/t3) d = I’,

RZ =

(BXI + Pxzc) ((1 + B) + (lly)(Bx, +/be))

exp

x [(I - xd”+‘W + P) +tll~)tBx, Substltutmg

and eqn (21) becomes,

+

Px,cN*l

(16)

C*U% Y. $9 x~cr

1) = 26%’ - 4b5c2+

g, and gz mto eqn (12), h=g,g=lhs

Equation

(23)

ofeqn(lS)=O

I

(17) can be rearranged

8b6d = 0

j-c*+4bd=O

(17)

Substltutmg parameters,

to,

for

6,

C, and

d

-4(1 +p

(24)

m terms

of the system

+Bx*c/r)*

B = - cl+ 8) + (4/rWl + B + pxzJr)’ axI + bx,* + cx, + d = 0,

(18)

where a = (n - l)B*/y* b =

Bzly2+

c = [282/y d=

The derlvatlve

2B*(;

- ‘) + B( 1 + @)]

which 1s the bound between umque and multiple steady states for a first order lrreverslble, exothermlc reactton m the nonadlabatlc CSTR Equation (21) 1s exactly the same bound given by Uppal et al [Sj, viz 4u + P +

+ (n - l)r2 - B( 1 + B)J

(25)

Lhdrl’

B > (1 f P) - (4/y)(l+

B + /3x&)’

for multiphcity for some Da

zz

(26)

of eqn (17) IS simply $=3ax12+2bx,+c=0

I

09)

Comparmg the signs of eqns (25) and (26), the necessary and sufficient condltlons for umqueness and multlphclty for all Da are C*(B,

Thus the entire development has resulted m two simple polynomial eqns (18) and (19) which define the bounds m K+ on the ml set AT2 m R Simultaneous solution of the two equations yields 9ad - bc * x’ =2b*-6ac

(20)

y, p, x2_ n) 2 0,

for umqueness

for all Da (27)

and C*(B,

Equataon

y, p, xzc, n) < 0,

(21) IS plotted

for multlpllcity in Fig

for some Da (28)

3 for y = 20 and xZe = 0

Multiphclty

of an nth order chermcal reactlon via a catastrophe

1 1

I 2

3

Fig 3 The exact bound between regons of unqueness and multlphcity for a nonadlabatlc CSTR for selected values of n 2 0 (-y = 20, X& = 0)

(coolant temperature = feed temperature) for various reactton orders, except for n = 0 for whtch It does not space above each curve apply The entie parameter represents the regton of multiple steady states for some Da, while the curve and the parameter space below tt represent the regon of umque steady states for all Da For the y and xdc cirosen, the curves are practically linear Thts IS conslstent with the observation of Uppal et al [5] that the cntenon for the first order reactlon IS lmear for y + 01and x2= = 0, and here this IS shown for all n > 0 also It may also be observed that the region of parameter space m which multiple steady states can exist, shrmks with mcreasmg n (for xZc = 0) This IS the 35

theory approach

291

same trend as observed by VdB&L for an nth order chemical reaction m an adlabatlc CSTR (for n > 0) Ftgure 4 ts a plot of eqn (21) for II = 2 and y = 20 with x2, as a parameter As can be seen, reductron of the cooiant temperature monotomcaiiy increases the region m parameter space m which multlplrclty can occur, up to a point where the curve IS almost horizontal (e g xZc = - 10) Further reduction m xZf beyond this point continues the decrease of parameter space for uniqueness and limits this space to small p and B (e g ~2~ = -20) Comparmg Figs 4 and 5 for n = 2 and n = 1, respectively, increasing n always decreases the parameter space in (B, f3jtn which muitlphclty can occur for aii x2= The exact bound between regons of uniqueness and multlphclty for all Da for n > 0 and B > 0 IS eqn (21) For B < 0 and II < 0 condltlon (5) of Theorem 1 IS vlolated and eqn (21) IS not the bound The posslblhty of multlplrclty for some Da always exists for n < 0. and umqueness IS always guaranteed for B
(29) Therefore, d (30) then uniqueness of the steady state IS guaranteed for a11 Da For n < 0 the bounds on F m x, are x1

=

0, F(0) = 0

x1= l.F(l)=O

n =2 7=20

30-

25-

0

x)-

o0

0

1

’

’

’

’

’

’

’ 2

’

c

a

3

B

FGg 4

Regions of umqueness and multlphcrty for n = 2 fv = 20) m a nonacbabatuz CSTR with xzc as a Parameter

Fig 5 Regions of umqueness

and multlphclty for n = 1 (y = 20) m a nonadIabatIc CSTR with xZE as a parameter

292

HSUEH-CHIA CHANG

and JOSEPH M

CALO

ail Da IS when one of the extrema occurs at x1 = 1, or eqmvalently, when XI = 1 1n eqn (35), VIZ

and thus (31) and there IS always the posslb1hty of multlphclty for some Da for all values of the other parameters In other words, U, = @ which violates condition (5) of Theorem

Thus there are three distinct regons space where X:, may lie Region

Partial ddTerent1atlon of eqn (29) with respect to XI yields, $=exp

I

1

-

I II III

1

(Bx, + Bc2c) (1+ 8) + (l/y)(Bx, + Bx*c) I (1 -XI)“+‘

1

(I+lu,-x,)(~+r)‘-Bxl(l+B#l-Xl)}

(32)

Condlt1on

1n parameter

Exact Bound (38)

X0*=-1,

ew

&31,01*

ew (37)

X?,CO,

uniqueness guaranteed

The loci in (B, p) space where XT, = 0 and X:, = 1 are, respectively,

For S < 0 (endothermic reaction), for all n,

Bo =

- Y((Y/2)- 1) r((v12) - 1) - xzc

(39)

J31=

B - ~(($2) - 1) r((y/2) - 1) - *zc

(40)

and

and therefore (34) and umqueness of the steady state 1s always guaranteed for all values of the other parameters In this case U1 IS not bounded and violates cond1tlon (5) of Theorem 1 For n = 0, eqn (18) becomes

[

(1 + 8) +;

(BX, + 8**=)]‘-

B(l + /3)X,= 0.

(35)

and the derivative of eqn (35) yields

x;“, = Y(l + /3MY/2) - 1)- B-b B

(36)

Substitution of eqn (36) into eqn (35) yields the bound between regions of uniqueness and mult1phc1ty, VIZ Y2_4Y

B= 4y+4x,,-y*=

-P

(37)

It IS important to note that eqn (37) 1sthe exact bound d and only If X :, E [0, 1) If XT, GZ[0, 11, then #?2 p* becomes only a sufficient condtt1on for uniqueness However, an alternative bound which IS exact for XT, E [0, l] can be found as follows Since eqn (35) 1sa quadratic, there are at most two extrema 1n (Xl, Da) space If xz c 0. the two extrema can never enter [0, 11 for physically reahzable values of the parameters, because eqn (35) can never be zero for x1 = 0 Hence uniqueness 1s guaranteed for all values of the parameters for XT, < 0 On the other hand, 1f x7, > 0, the extrema can enter [0, l] smce no s1mllar restnctions exist at the X1 = 1 bound Thus when x7, > 0, the necessary and sufficient condition for uniqueness for

Both these equations are linear 1n (B, 6) space and divide the space into reaons I, II and III However, due to the resmctlons E > 0, /3> 0, and - y 5 xzc I B (the coolant temperature IS restncted to be less than or equal to the maxmum reactor temperature), the following four cases, schematized 1n Fig 6, anse as a result of all possible values of y and xZc Case A y > 2, Xze > y((y/2) - 1 Equation (40) has a negative slope and intersects the B axis at y((y/2) - 1) Although the space 1s divided into three regions. only two apply smce B > y((y/2) - 1) Also, if y > 4. p* > 0, and mult1pltc1tyoccurs for some Da to the left of /3* and uniqueness IS guaranteed for all Da to the right of @* If y > 4, then fi* < 0 and uniqueness IS guaranteed for the entire accessible (B, p) space The net result IS that eqn (37) 1s the exact bound and 1f y < 4, uniqueness IS always guaranteed Case B y z 2, xlc C y((y/2) - 1) For this case PO X2=) 1s divided into two regons The exact bound 1n recon I IS eqn (38) If /3*> 0 (which occurs for y > 4, xts > y((y/4) - 1) or y < 4, xZe C y((y/4) - l)), multiphclty occurs for some Da to the left of fl* Uniqueness 1s guaranteed for all Da to the nght of p* and d /3* < 0 (which occurs for y > 4, x+, < y((y/4) - 1) or y < 4. XZc> y((y/4) - l)), for all of re@on II Case C y < 2, xlc < y((y/2) - 1) In this case the entire positive (B, /3) space IS accessible and divided into three reqons Reaon II IS divided into two parts by eqn (37) since /3*> &, and agam uniqueness IS guaranteed for all Da to the right of /3*and multiphclty occurs for some Da to the left of p* m regon II Uniqueness IS guaranteed for all Da m region I since eqn (38) becomes complex for these condlt1ons Case D y ( 2, Xfc > ~((~12) - 1) Equation (40) has a

HSUEH-CHIA CHANG

294

comparison to two for II > 0 This occurs only for n = 0 because the denominator of F m eqn (29) (1 e (1 - xl)“) IS unity, which allows F to be finite at the x1 = 1 bound, an even number of extrema IS not and therefore, reqmred Although not clearly dlscermble from Fig 7, one solution exists at very low Da, while at mtermedlate Da there are two solutrons, and for Da greater than that at the extremum, there are no solutions, I e 1-2-O multiphclty This IS quite different from the 1-3-1 pattern observed for all II > 0 Even m region II (I e xTOE [O, 11 where two extrema exist for x, E [0, l] when multlphclty occurs, the pattern IS 1-3-2-O or 1-3-l-0, rather than 1-3-l These multlphclty regions are demarcated by (x,, Da) at the x, = I bound and at the extrema These values are simply found by solvmg eqn (35) for x1 at the extrema, viz x,0( -c ) =

- c 2 iI(cZ- 4B2z21y2) 2B2ty=

(41)

where c and z are defined m (23) and (15) For region i multlphclty only d x1,( -) E [0,1] Solving eqn (7) for Da and substituting for x,X 2).

and

JOSEPH M

nth Order chemrcal

The value of Da at the xi = 1 bound,

Dab =exp t (I

CE.,(B, 7, n) 2 0

1 (42)

-(B + BXk)

1

(43)

Thus for the case presented m Fig 7, one solution exists for 0 < Da < Dab, two solutions for Dab 5 Da < Da(-), and no solutions for Da > Da(-)

m an adiabatac

CL(B,

y, n) c 0

CSTR

mult~phcity

for uniqueness for multlphclty

for all Da

(44)

for some Da

(45)

and

R,=l-p,

The exact bound between

(6)

where y”, y* and y* are the values of y satisfying, respectively, our exact bound, and the multlphclty and uniqueness cnterla of VdB & L, I e y* < y” < y* The results of such compansons for n = 0 03,O 5, 1 5 and 2 0, are presented m Fig 9 In alI cases the abscissa (R = 0) represents our exact bound These curves clearly mdlcate that for all but very small B/y, R, c R* such that the uniqueness cnterlon of VdB & L 1s less conservative (closer to our exact bound) than theu multlphclty cntenon Also, it IS of interest to note that the error

B/Y Fig 8

for all

where C$(B, y. n) IS the exact bound between the regions of uniqueness and multlphclty, and IS simply a special case of eqn (21) obtained by setting fi = 0 (the dlmenslonless heat transfer coefficient) This bound IS presented m Fig 8 for selected values of n 2 0, where n = 0 IS a specsal case Although this figure appears slmdar to Fig 2 of VdB & L, It should be noted that we report only one exact bound, whereas VdB & L actually have two sufficient bounds associated with each value of n, which only appear as a single bound on then figure Of our bound, which IS both necessary and course, sufficient, IS itself bounded by the two sufficient bounds of VdB & L (except for n = 1, 0), but they are all very close Therefore, as a means of comparison we define R*=l-$,

Dab, 1s simply,

+ /3) + (B + /3&)/y

reaction

The cnterla for uniqueness and n > 0 m the adiabatic CSTR are,

-@x1,( * I+ B&c)

Da(~)=X1~(~)exp t (l+p)+(Bx,“(~)+##x*e)/y

CALO

regions of umqueness and multipbcrty for an acbabak ofnz0

CSTR for selected values

Multlphclty of an nth order chemical reactlon via a catastrophe theory approach

n-05 i

295

n-003

I

Rg 9 Residuals between the cnterla of VdB & L and the exact bound as a function of B/y for selected values of ?I>0 represented by R* and R, decreases as n + 1 for both n c 1 and n > 1 Evidently thts occurs as a result of the coefficient of the cubic term (n - 1) m the f’ = 0 equation approaching zero as n + 1, thereby unprovmg their quadratlc approxlmatlon to f’ = 0, which becomes exact m the hmlt of II = 1 Thus the closer n IS to umty, the more rapldly theu two bounds converge on our exact bound Concernmg the zeroth order reactlon m the adlabatlc CSTR, settmg p = 0 m eqns (35) and (36), and solvmg simultaneously yields

nth Order reaction LRa porous catalyst parttcie As IS well known, eqn (7) also defines the steady states of the porous catalyst particle with no mtrapartlcle gradients of mass and heat (lumped model), d we redefine the dlmenslonless parameters as

(47)

Y = 4, as the exact bound which is independent eqn (47) IS the exact bound d equivalently,

as derived by VdB & L for n = 0, and are also presented m Ftg 8 It should be noted that m the reaon B/y < 1, conchtlon (47) IS no longer exact, but IS still sufficient for umqueness

of B However, x:,E 10, 11, or,

Ol$Bll,

=

(- AHNL’oY

(50)

hrTo

/3=0 (48)

wluch IS, of course, dependent on B Thus (or x:, > l), eqn (38) with p = 0 becomes bound for the adlabatlc CSTR, VIZ

B

for B/y -=c1 the correct

2

-B=O Smce B and y are always posltlve, I:, rC0, and eqns (47) and (49) are the only exact bounds These are the same

These definrtlons

f =

(1

result m

_-y*,,-

exp(z++

=o,

(51)

where x1, = (C’, - C)/C. and C, IS the surface concentratlon Thus the exact bound developed for the case of the adlabatlc CSTR IS also applicable to the lumped parameter porous catalyst particle model for n z 0 As pomted out by Luss@], the umqueness crlterlon for the lumped parameter model IS also a sufficient condltlon for

2%

HSUEH-CHIACHANG

of the steady state of the umqueness parameter model (non-neghglbie mtrapartlcie of the catalyst partlcie Thus,

dlstibuted gradrents)

Czd:hO,forn>O

and JOSEPH M

CALO

and, IPX:.

(52)

with

Thus B can yielding

be ehmmated

by equating

Da=a

[2/J - 2u’x*. + uUx&] + u -“ufx,,

f = (B -x2.) XZS

q4’-Sh=O,

where

Sh=k!&

(55)

I

(59) and (60).

,

U”X+,

are sufficient condltlons for umqueness of the dlstnbuted parameter model for the catalyst partlcie Perhaps the most representative model of the porous catalyst partlcie, however, allows for mtrapartlcie concentratlon gradients but assumes a constant mtrapmcie temperature which dtiers from that of the bulk fluld The steady state equation for this model can be rotten as

(60)

u’xzs + U”XX,l

B=[2U-2

= O*

which IS a function of xl=, ‘y, and 402 only Thus for any y and 40*, eqn (61) can be solved for xzs which can then be used to solve for the correspondmg value of B on the bound from either eqn (59) or eqn (60) The exact bounds calculated m this manner with &, as a parameter are presented m Figs 10 and 11 for slab and sphencai geometry, respectlveiy As shown m these two figures, space for equal 40 the slab exhlblts a iarber uniqueness than the sphere In addition, the bound for the slab approaches the hmitmg curve By/(B + y) = 8 much more rapldiy w&h mcreasmg 4. than for the sphere, such that the sufficient condlhon for muitiphclty represented by By&B + y) > 8 IS better for the slab at large 4. Conversely, for small 40 the sufficient condltlon for uruqueness represented by By/(B + y) ~4 IS better for the sphere Thus the exact bounds for ail 4,, are m turn bounded by the curves Byl(B + y) = 4 and 8 These bounds are known (e g , see[8]) and can be simply derived as follows For large 4 for slab. spherlcai and cylmdrlcai geometnes, 2 24’ u = 2(x2:+ y)i

and 7 1s the effectiveness factor, 4’ 1s the square of the Thleie modulus, and B IS gven m (50) Equation (54) can be rewritten as f=Zu-Sh=O, Takmg

IA

for Z and

Z

B Y2 2(x2. + y)’ = -GAB - xts)’ or

ZK” = 0,

&X5, U'X** - u'

-x;.(y2+2B)+x~,(By2--By)-2By2=0

(58)

where Z’ = -B/x% and Z”= 2B/xz, Accordmg to Theorem 1, the sunuitaneous soIution of eqns (57) and (58), by ehmmatmg x 2a, yields the exact bound between repons of unique and multiple solutions in parameter space for ail Sh for any II > 0 and catalyst partlcie geometry, as long as an expresslon for ~4’ IS avadabie Due to the hlghiy transcendental nature of the equations, the soiutton must be Iterative For the case of n = 1, however, 4’ and 7 are Independent of B, and eqns (57) and (58) can be solved exphcltiy for B, and equated, ehmmatmg B, Le B=

(62)

eqn (57) and using the expresstons

u’=_z

the first

f’=Z’u+Zu’=O

f” = Z”u + 22’u’+

Rearrangmg 2’9

(56)

where Z = (B - -G~)/x~~ and u = q# and second derlvatlves of eqn (56),

(61)

(59)

Takmg the denvabve accordmg to Theorem

W)

of eqn (64) and solvmg 1,

for xta,

Br(r - 4) x*s = 2( y2 + 28) Fmaiiy,

substitutmg

eqn (65) mto eqn (64). ByW+y)=8

For small

(W

4 for ail three geometries, ul_ u-

Y2 (x2s

+

YIi

eqn (62) becomes,

(67)

Mult~pk~ty of an nth order chemical reactlon via a catastrophe

theory approach

297

MULTIPLICITY

Fig 10 The exact bound between

reqons of umqueness and multlphctty for the first order reactnon m the slab catalyst particle with a concentration gradlent and urnform temperature, as a function of &

MULTlPLlClTY

B/Y

Fig I I The exact bound between regions of umqueness catalyst particle with a concentratron

and proceedmg

m exactly

and multlphclty for the first order reactlon m the spherIcal gradrent and umform temperature, as a function of &,

1

the same manner,

+Z’=o

ByI(B + y) = 4 Exact

w

cntena for this model of the catalyst mcle can be determmed from eqns (57) and (58) for any n > 0, as long as an expresslon for s& IS avdable However, general exphclt uniqueness and multlphclty cntena, mdependent of geometry (I e T#) and the parameter dot, can also be denved by estabhshmg a pnon bounds on (d 1nTld In&) as described by VdB & L nvldmg eqn (57) by 714~~

(69)

Also, by the cham rule of dlfEerentiatlon,

lmphclt

(70) and eqn (69) becomes, z

C-

idlnrl+, 2dln4

I

-dW2+2’=0 dxzs

(71)

HSUEH-CHIA CHANG and JOSEPWM

298

VdB & L have proposed the followmg bounds on d Inn/d In4 for nth order ureverstble kmettcs for the slab, cyhndncal and spherical geometnes,

CALO

b2 = - 1, a sufficient condttton for multrphctty can be developed m exactly the same manner,

C%B,Y,

4%.a)

-c0,

sufficient

02) Smce eqn (71) represents the exact bound between unique and multrple solutrons for _rZ. for all Sh. then z

Cldlw

2dlnq5

:

l

s0 dx.Ls 1dln42+Z’

(73)

represents the uniqueness parameter space If d inn/d In+ IS asstgned the value of rts upper bound, b,, then (74) IS a sullicrent condrtton for uniqueness state, where 5, = ($r + 1) Thus

of the steady

d In+’ .8& dXZr +z’ =’

(75)

represents a one-srded bound such that only condrtron (73) 1s lmplted Substttutmg for dIn#*/dx2, m eqn (751, stmpldymg, and multtplymg by - 1 (which changes the sense of inequality (73)), &,(Cn - l)&) + J&B

+ 2(n - 1)&r + Gr)

+ x2.(- &y+B + 2+

+ (n - l)&&-+

By* = 0,

Ct(B,

x fl, n) = asps3+ bsPs2Qs + c.P.Q,‘+

where C:, IS grven by eqn (77), except that & IS replaced by &( = ib2 + 1) m all the parameter defimhons m (78) These two suffictent bounds for umqueness and multtphcrty are equrvalent to rI+ and $r+ respectrvely, m VdB 8r L Due to the fact that our technrque does not approximate the resultant cubrc equatron (j’ = 0) wrth a quadratrc, our two sufficrent bounds are shghtly tighter than those of VdB & L However, the inherent conservattsm mtroduced by replacing d lnrJ/dln+ wtth the relattvely loose bounds, 6, and bZ, almost completely controls the values of the resultant bounds such that our bounds and those of VdB &c L are practtcally mdtstmgutshabie for n > 0 Of course, as bt and & are Improved and approach one another, our two bounds wtll always approach the exact bound more raptdly than those of VdB & L. and thus represent an tmprovement For the zeroth order reactton, eqn (54) becomes,

f =-$&“-Sh=O, where 4” (81) yrelds

(81)

= 40’ exp {yx&y + x2)} Drfferentratron of eqn

whtch IS analogous to eqn (71) Therefore, bounds m (72) for d inn/d I@‘. and proceedmg the same fashton, y& = y I 4, y& = 5 2 4,

for umqueness for multtphcrty

usmg the m exactly

for all Sh

(83)

for some Sh

(84)

For sphertcal and slab geometnes n = 1 for &‘<2/3 and 2. respecttvely And thus d d’& < 213 or 2. where 4”lnax = 402 exp {By/(B + y)}, then the exact bound between umqueness and multrphcrty for all Sh becomes

a, = (n - lb5 b, = B + 2(n - 1)&r + &y2

(78)

(85)

d, = By= P, = 9a,d. - b,c, Q, = 2b,‘-6a.c. However, due to the approxtmation of d lnq/d In4 by its upper bound, bl, eqn (77) IS only a one-stded bound and thus suffictent for umqueness (79) for all Sh

c:,ro,

By assrgmng

(82)

’

dsQs3= 0, (77)

where

c, =2yB+(n-1)&y*--&1y2B

for multtplrctty for all Sh

(76)

which IS m exactly the same form as eqn (18) for the CSTR, and can be treated m exactly the same fashion to obtam the bound, VIZ taking the denvatrve, solvmg the result srmultaneously wrth eqn (76) for ~2~. and substrtutmg back mto eqn (76) Thts algebraic mampulatron results In

for all Sh vrz

d Inn/d lne5 the value of Its lower bound,

which 1s the same result as (47) for n = 0 m the adtabattc CSTR Equatton (85) IS the exact bound If xf E [O, B], where x2so

*

-_

Y(Y ----, 2 -

2) for 4’L.

which 1s the value of s%me analysrs as for xzro > B, the exact multrpltctty becomes

c $ (sphere) < 2 (slab)

(86)

xZs at the cusp pornt By the exact n = 0 m the adtabattc CSTR, rf bound between umqueness and eqn (40) for the catalyst partrcle

Multlphclty

of an nth order chemical

reacuon

For &‘* > 2/3 or 2, an lmphclt equation for the exact bound can be derrved by simultaneous solution of the first and second derlvatlves of eqn (81) for x& < B For x& > B, the exact bound 1s f’ = 0 with xzs = B VdB & L have also suggested the improved bounds,

of L. = B$(B + 2-y) However, determmatlon these b1 and blL requues an expression for lnv as a fun&Ion of In& m which case the utdlty of this approach dlmuushes somewhat smce the exact bound can be determmed ImplicItly as presented above for the first order reaction for slab and spherical geometnes

where h

CONCLUSION AND SUMMARY

The techmque developed m the present work has been successfully apphed to the nth order chemical reaction m the nonadiabatic and adiabatic CSTR and the catalyst particle with an mtrapartlcle concentration gradient and umform temperature For the CSTR, the exact, exphclt bound between reaons of umqueness and multiplicity for all parameter values and n 2 0, has been obtamed For the catalyst particle, an exact but lmphcit bound results, wtuch was fully demonstrated for the case of the first order reaction In addition, very strong exphclt suffiiclent bounds for umqueness and multlphcity can also be obtamed by the judicious choice of bounds on d InT)/d In+ a la VdB & L In prmclple, the techmque IS apphcable to any nonhnear equation which satisfies the required condlhons of Theorem 1 However, some of these con&Ions may be ddiicult to venfy a pnon Thus, In general one would assume all the condttlons are met and proceed with the techmque Then from the results, one can often decide whether the uutlal assumptions were mdeed vahd The result IS an exact bound between reaons of umqueness and multlphclty of soluuons to the ollgrnal equation Dependmg on the particular equation the bound may be explicit or Imphclt, but even m the latter case, may be of conslderable utility Acknowfdegements-We are grateful to the Hlggms Fund and the Department of ChenncaI Engmeenng of Pnnceton Umverslty for support of thrs work NOTATION

A aA

a a B B(&,

l) b

C* C* c,, Da 0, E

f

any set m general boundary of A vector parameter or element (in general) constant defined m text dlmenslonless heat of reaction open neighborhood of ~0 bound on d Inqld In&a reactant concentration bound between umque and multiple steady states heat capacity Damkohler nuniber effective dtiuswlty actlvatlon energy functions defined m text

CESVOI 34 No 3-B

via a catastrophe

theory approach

299

function defined m text function defined m text i fun&on defined m text heat transfer coefficient hf mterphase -AH heat of reactlon J Jacoblan matnx K catastrophe set ko rate constant kc interphase mass transfer coefficient N nil set reaction order or dlmenslon of variable space ;f ordinary set P constant and mappmg fun&Ion P parameter space dlmenslon constant set defined m text : element of Q0 R% real number space R* residual fun&on eqmlrbnum mamfold S external surface area of catalyst particle sx Sh Sherwood number T temperature u umqueness set l4 4 V reactor volume volume of catalyst particle V, X vartable vector m general dImensIonless reactant concentration XI chmenslonless coolant temperature XZC dlmenslonless catalyst particle temperature XZS z function defined m text function defined m text Z F

Greek

symbols

defined m eqn (50) dlmenslonless heat transfer coefficient dlmenslonless actlvatlon energy effectiveness factor projectton of certam pomts of au, effective thermal conductwlty (ib + 1) density smgular set reactor residence time empty set Threle modulus subspace [I] (a) Thorn R . Structural Stubdrty and Morphogenesw Ben]amm, Readmg. Mass 1975, (b) Poston T and Stewart I , Catastrophe Theory and Its Appltcatlons PItman. Cambndge 1978, (c) Golubltsky M , SIAM Reurews 1978 20. 352 [2] Kolata G B , Scrence 1977 1%. 287 [3] Zeeman E C , Catastrophe Theory-Selected Papers, 197277 Ad&son-Weslev. Readme. Mass 1977 [4] Ans R , Ann N Y-&ad SG Nov 1977 IS]_ Uppal A, Ray W H and Poore A B , Chem Engng Scr _ 197429. %7 [6] Van den Bosch B and Luss D , Chem Engng Scr 1977 32, 203 [7l Apostal T M, Mathematrcai Analysrs, Addison-Wesley, Readmg, Mass 1957 [8] Luss D, Chem Engng SCI 1971 26, 1713