Sufficient conditions for uniqueness of the steady state solutions in distributed parameter systems

Pergamon Press

Chemical Engmeenng Science, 1968, Vol 23, pp 1249-1255

Pnnted m Great Bntam

Sufficient conditions for uniqueness of the steady state solutions in distributed parameter systems DAN LUSS Umverslty of Houston, Houston. Texas 77004 (Recerved

23 October

1967)

Abstract-A

new techmque of obtammg sufficient condltlons for umqueness of the steady state solutlon m dlstrlbuted parameter systems IS presented The method IS apphed for the cases of an adlabatlc or isothermal reactlon m a porous catalyst and for an adlabatlc tubular reactor with axial dlffuslon The condltlons yield estimates which are much less conservative than those obtamed by the previously pubhshed methods Close agreement was observed when the predlcted bounds were compared with exact numerical solutions _ INTRODUCTION

between a chemical reactlon and mass and heat dlffuslon m dlstrlbuted parameter systems can cause vanous pathologlcal phenomena such as the occurrence of multiple steady states, some of which are highly unstable This has been demonstrated by Welsz and Hlcks[9] for the case of an adiabatic simple reaction occurrmg m porous catalyst pellets, by Roberts and Sattefield[8] for an Isothermal Langmulr Hmshelwood reaction, and by Raymond and Amundson[7] for an adlabatlc tubular reactor It IS important to be able to predict a prlorz under what condltlons this phenomenon may occur Gavalas [ l] using modern topological techmques was the first to obtain sufficient condltlons under which a unique steady state exists for an adiabatic chemical reaction m a porous catalyst pellet A different technique was used by Luss and Amundson[3, 41 who discussed also the occurrence of multiple steady states m adlabatlc tubular reactors and complex Isothermal catalytic reactIons Recently Markus and Amundson [6] developed a new techmque to determine sufficlent condltlons for the existence of a unique steady state for a tubular reactor with axial dlffuslon The above works have shown that for certam kmetlc expressions a umque solution exists for systems of arbitrary size These sufficient condltlons were sometlmes rather conservative

THE COUPLING

The techmque developed here yields much less conservative estimates In this work It IS shown that if multiple steady solutions exist, then a certam Integral equation has to be satisfied For certam kinetic rate expressions this mtegral equation cannot be satisfied, and clearly for these kinetic expressions multiple steady states cannot exist In this way the followmg sufficient condltlons for uniqueness were obtamed (10) for an adlabatlc reaction m a porous pellet, (27) for an adiabatic tubular reaction with axial dlffuslon, and (32) for an Isothermal reactlon m a porous pellet The techmque makes use of certain properties of the steady state solutions which have been proved rigorously m [2, 51 by use of topological techmques However, the derlvatlon given here does not require any famlharlty with either topology or functional analysis The technique 1s developed m Sectlon 1 usmg an adiabatic reaction m a porous catalyst as an example In Sections 2 and 3 the same techmque 1s apphed to an adiabatic tubular reactor and to an Isothermal reactlon m a porous catalyst, respectively I

ADIABATIC REACTION IN A POROUS CATALYST PELLET

For a single chemical reactlon occurrmg m a porous catalyst, with umform temperature and concentration on the surface, one can by lose of

1249

D LUSS

the enthalphy and mass balances express the concentrations of all the reactants as a linear function of the temperature The steady state must satisfy the equation

$j--(r2-$ ‘)+f(T,T,,c,)

(1)

=0

Subject to the boundary condltlon T= T,

r=R

(2)

and it can be proven[l] that the value of T IS bounded from below and above, and that there exists at least one solution The expresslonf( T) depends on the reaction mechanism and will be assumed to have the following propertles (1) f(T)

Fig

1s positive for T, s T < T,,,

T,, IS the adiabatic temperature system reaches equlhbnum 61) f(T,,) = 0 Note that (I) and (11)imply thatf’ (T,,)

where at which the

TEMPERATURE AT CENTER OF CATALYST PELLET TM)

< 0

It can be easily shown that the above assumptions are satisfied by most chemical kinetic expressions and hence do not limit the generality of the results The steady state IS described by a spht boundary value problem and may have more than one solution for particles of certain sizes Smcef( T) IS positive m the range of interest, It can be shown that for each value A (T, < A < T,,) there exists a unique pellet size R for which T(0) = A, where T (0) IS the temperature at the center of the pellet Thus, for a given kmetlc expression a plot of the length of the radius R vs T(0) can have either the shape of case 1 or case 2 m the schematic Fig 1 Clearly, for case 2 there 1s a range of particle sizes for which multiple steady states exist The necessary and sufficient condltlon for the existence of a unique steady state for particles of arbitrary size 1s that there will not be any humps m the curve R vs T(0) Mathematically this 1s expressed by the condltlon that the system described by the dlfferentlal Eq 1 has no blfurcatlon pomt

1 SchematIc of T(0) vs R

Our aim 1s to determine for what types of kinetic expressions f(T) no blfurcatlon pomts can exist It was shown (1, 3) that one such sufficient condition 1s SUP

f’(T)

s 0

(3)

T, s T s T,,

The disadvantage of this condltlon 1s that It IS often rather conservative For example m the case of a first order u-reversible reaction It IS conservative by a factor of about five A stronger condltlon wrll be derived here Assume that for T(0) = A there exists a blfurcatlon pomt with a solution Tb(r) satisfying the equation f

$ c r2ds

>

+f(Tb)

= 0

(4)

Subject to the boundary condltlon Tb = T,

r=&,

(5)

and Tb(0) = A. Then it can be shown (2,5) that the equation

1250

+f'[Tb(r)]V

=0.

(6)

Sufficient condltlons for umqueness of the steady state solutions m dlstrlbuted parameter systems

SubJect to the boundary condltlon v=o

r=Rb

(7)

has a solution which does not vamsh anywhere m the range 0 < r =GR (This IS a result of the fact that at the blfurcatlon point A = 1 1s the first positive elgenvalue of the equation

f$ (r2g >+Af’[Tb(r)]v

= 0

Subject to the boundary condltlon (7) A rigorous proof of this fact using topological techniques I was presented elsewhere[2,5] This result ~111 Ta w be used here without making any further use of topology 1 Fig 2 SchematIc of M for vanous kmetlc rate expresslons By multlplymg Eq (4) by v and (6) by Tb(r)T,,, integrating both equations between 0 to R, steady state exists for particles of arbitrary size and subtracting one from the other, the follow1s that mg equation 1s obtained, dlnf (T) < 1 for all T, s T s T,, (T-T,) dT

I ~Lf[W)l = j-y

(10)

-f’[Te(r)l(T*(r)-T,)}r’vdr

[v$(P%)-

(Tb(r)-Ta)$(r$)] dr

= 0

(8)

Equation (8) can be rewritten as

Clearly, condltlon (10) IS always satisfied for Condltlon (10) 1s stronger than the previously obtamed[3] condltlon (3) To check the estimate obtamed by condltlon (10) consider the case of a first order irreversible reaction Here ~

T = T, or T,,

f(T)

=

%$)(Tad-T) w

Y [

r%dr = 0

(9)

Tad-T, -= TCI

dlnf (T) dT

for various kinetic expressions f(T) 1s given m Fig 2 Clearly, Eq (9) can be satisfied only if M changes Its sign in the interval 0 < r S Rb since f(T)r% does not vanish m this interval 0 < r < R Thus, a sufficient condltlon that no bifurcation point can exist, or that a unique

T

1

(11)

Substltutlon off(T) mto Eq (10) yields the followmg sufficient condltlon for umqueness

A schematic description of the term M = 1- (T-T,)

T-T,

yp”4A

T

Tll

(12)

Accordmg to the numerical results of Welsz and Hlcks[9] multiple solutions occur if P-y c 5 Hence the agreement IS excellent Condltlon (3) yields the followmg condltlons for uniqueness (13) rs G 1 which IS much more conservative (10)

1251

than condltlon

D LUSS

The technique developed m this section can be applied to a wide class of problems In the next two sections It will be used for treating the tubular reactor with axial dlffuslon and the case of a complex Isothermal catalytic reaction 2 ADIABATIC

TUBULAR AXIAL

REACTOR

We will assume that there exists a bifurcation point with a solution Tb (s) Thus, the equation $+

(-~+-&T,cs)])

T) = 0

Pe$+f(

(14)

where Pe =

il.

( 15)

dv Pe -=--_v ds 2

s=o

Pe(T-T,) dT -_= ds

dv -=--v ds

Pe

2

=

(25)

1

-

(T-T,)

dlnLiT)]f(

T)e-Pes/2vds = 0

(26) O

s=l

(17)

If there does not exist any steady state solution which satisfies Eq (26) then no blfurcatlon point exists Hence, a sufficient condltlon for umqueness of the steady state solution IS that

d1n.fV’) (T-T,) -Al

One obtains

forall

d-f

Pe2 p-TW+f*

d2w

(w) = 0

(19)

Note that condltlon T=

where = eepes’lf( T)

(20)

Pe -_=-_W ds 2

dw

Pe

2w

T,sT=sT,,

(27) (27) 1s always satisfied for

TaorTeQ

It has already been shown elsewhere[3] that a sufficient condltlon for umqueness m this case 1s

Subject to the boundary condltlons

ds

s=l

has a solution which does not vamsh for 0 < s < 1 (for a proof see [S]) By use of the same techniques as m Section 1 it can be shown that the bifurcation point solution has to satisfy the equation

(16)

(18)

dw

(24)

0

From

f*(w)

s=o

/‘{(f*(w)-yw)v-(f’(T)-~)vw)ds

Subject to the boundary condltlons g=

(23)

SubJect to the boundary condltlon

WITH

DIFFUSION

It has been shown[7] that for an adlabatlc tubular reactor with axial dlffuslon multiple steady states can occur In the case of one reaction the system IS described by the equation

$-

2,= 0

Sup

f’(T)

SF (28)

s=o

s=l

T, s T s T,,

(21)

(22)

The magnitude of the axial dlffuslon coefficient determines which of the two condltlons (27 or 28) is stronger

1252

Sufficient condltlons for umqueness of the steady state solutlons m dlstrlbuted parameter systems 3

The above condltlon IS stronger than the prevlously obtained result [4]

ISOTHERMAL REACTION IN POROUS CATALYST PELLET

For certain complex Isothermal catalytic reactlon multiple steady states may exist [8] The concentration of the reactant A IS governed by the equation (29) subject to the boundary condltlon c = C” r=R.

(30)

It will be assumed that f(c), the kmetlc expression descrlbmg the consumption of the reactant A, sattsfies the followmg condltlons (1) f(c) IS posltlve for c, G c < c,~, where ceq 1s the Isothermal eqmhbnum concentration of the reactant A. (4 f(G,)

= 0

df =- 0 dc -

for all c eq G c s c,

(33)

Any simple kinetic expression which predicts that the reactlon rate decreases with increased conversion wdl satisfy condltlon (32) or (33) Hence, a unique solution will exist for a catalyst pellet of any arbitrary size However, m case of complex kinetics the reaction rate may increase with conversion, as may be the case d the reaction IS strongly mhlblted by the reactants In these cases multiple steady states may exist for particles of certam sizes Condltlon (32) can be used for a check If this phenomenon ~111occur for any specific kinetic expresslon To clarify the concepts consider the followmg example Example

Note that (I) and (11)imply thatf’ (c,J > 0. Agam, It can be shown[5] that if a bifurcation pomt exists, the equation

A second order Langmulr Hmshelwood reaction @+ VB + Products) occurs m a catalytic pellet The same reaction has been treated before[8], and for convenience of comparison the same notation as m [8] ~111 be used The dlmenslonless reaction rate can be expressed as

0= 0 subJect to the boundary condltlons v=O

r=Rb

r(pA) =

has a solution which does not vamsh for 0 < r < Rb By use of the same technique as m Sectlon 1, it can be shown that the blfurcatlon pomt

or m a dimensionless

r(y)

(et,(r) --cJf’[c~(r)l~

1 3 (c-ca)dln[dfc(c)l

or

(l+Kty)2

(35)

E’-+

(32) 1s always satisfied for

K’ = KpA,*

&!!_ P4,s

c,~ G c G c, (32)

Note that condltlon c = c, or ceq

k’ = kp$,8

(31)

forall

=

where

r2d-

Thus, a sufficient condltlon for umqueness, that no bifurcation pomt can exist. IS

(34)

form i’y(y+E)

solution has to satisfy the equation / 0 RbLf[dr)l-

iPY,(P,+X) (1 +Kp,)2

E 1s related

d

DBP,, _ 1 VSDAPA,~ *

(36)

to the stolchlometrlc excess of B over A in the gas outside the catalyst A and B ~111be chosen such that E will be non-negative

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D LUSS

100 but do not exist for K’ = 10 Thus, the agreement 1s good

Substltutlon of the kmetlc expression (35) mto (32) yields the followmg sufficient condltlon for uniqueness

SUMMARY

(Y--l)(E+2Y-=‘Y) 1 2 sup 04y41 (l+K’Y)Y(Y+E)

(37) -

Clearly, condltlon (37) will be satisfied (assuming 1 + K’y to be posltlve) if K’<

1+$

(38)

or for any value of K’ if E=O

Sufficient condltlons for the umqueness of the steady state solution for a chemical reaction occurring in a porous catalyst pellet or m an adiabatic tubular reactor with axial dlffuslon have been denved The technique 1s very simple and requires a mmlmum amount of simple algebraic computations The condltlons obtamed m this work are stronger than any previously published Companson between the analytically predicted bounds and numerical results reported m the literature show close agreement

(39) Acknowledgment-The

The above sufficient condltlons for uniqueness have already been derived m (4) using condltlon (33) Condltlon (37) will now be used to obtain stronger results Define by ys the value of y for which the expresslon on the nght hand side of the Eq (37) obtains Its highest value. It can be shown that

author IS Indebted to Professor N R Amundson, with whom he had many helpful dlscusslons on this subject NOTATION

A

1 (a) Assume a value of K’ > 2~

K’ 1

a constant concentration of reactant dlffuslon coefficient activation energy variable defined by Eq (36) reaction rate expressed m terms of temperature reaction rate expressed m terms of concentratlon of one reactant kmetlc constant for rate expression (Eq (34)) iP& kinetic constant for rate expression (Eq (34)) KP, .s length of reactor

(b) Solve Eq (40) for ys (c) Solve the mequahty (37) for K’ Iterate

M

I- (T-T,)

y J1-2EK’)+~((1-2EK’)2+3EK’2-6K’) s 3K’

0” AE OTT

(40)

f(c)

for EK’ > + which 1s the range of interest (According to condltlon (38) umqueness 1s always guaranteed d EK’ < 4) To determine for a prescribed value of E the maximum value of K’ which satisfies condltlon (37) the followmg procedure can be used

Using this procedure it was found that for E = 10 a sufficient condltlon for umqueness is K’ < 12 This IS a much stronger result than K s 3 obtamed m [4] using condltlon (38) Numerical integration by Roberts and Satterfield [8] mdlcates that multiple solutions exist for K’ = 1254

lc k*I K

P Pe

dlnf( T) &,-

partial pressure Peclet number radial distance R’ radius of particle R’ gas constant s dlmenslonless length

Sufficient conditions for uniqueness of the steady state solutions m dlstnbuted parameter systems

T u u

W

temperature flow velocity solution of equation -(T-T,)

a

hneanzed

exp r!]

’

AE R’T

v x

stolchlometnc coefficient parameter defined by Eq (36)

differential

Subscrrpts

a

y

dlmenslonless

amblent condltlons adlabatlc blfurcatlon point eq equdrbnum s surface

concentration

ad b

Greek symbols

P

U-a,-TN,

REFERENCES GAVALAS R G ,Chem EngngScl 196621477 VI KRASNOSELSKI M A , Topofogrcal Methods In the Theory of Nonhnear Integral Equations Pergamon Press 1964 N R , Chern Engng Scr 1967 22 253 [31 LUSS D and AMUNDSON [41 LUSS D , Ind Engng Chem Fundls 1967 6 457 LUSS D , Can .I them Engng 1967 6 341 ti; MARKUS L and AMUNDSON N R , Mmorsky Anniversary Volume Drfl Equs J 1968 4 102 N R , Can J them Engng 196442 173 [71 RAYMOND L R and AMUNDSON C N , Ind Engng Chem Fundls 1966 5 3 17 [81 ROBERTS R W and SATTERFIELD [9] WEISZ P B and HICKS J S , Chem Engng Scr 1962 17 265

ill

R&m&- II s’a@t d’une nouvelle technique pour obtemr des condlttons suffisantes pour le caracttre umque de la solution B l’ttat stable dans des systemes de paramttres dlstnbuts La methode est utlhsCe pour les cas d’une rkachon adlabatlque ou lsothermlque dans un catalyseur poreux et pour un rbacteur tubulsure adlabahque ?t mslon axmle Ces condlttons donnent des pr&slons beaucoup moms modbrkes que celles obtenues par des mtthodes dCcntes antkneurement Une grande concordance exlstmt lors de compariusons entre les hrmtes prkdltes et les solutions numknques exactes Zusammenfassung-Es wlrd eme neue Techmk dargelegt, urn ausrelchende Bedmgungen fur &e Emdeutigkelt der Stationare Losung m Systemen mlt vettedten Parametem zu erhalten Die Methode wlrd auf &e Falle emer adlabatlschen oder Isothermen Reaktlon m emem porosen Katalysator sowie auf emen adlabatlschen Rohrreaktor mlt AxlalddTuslon angewendet Die Bedmgungen ergeben Schatzwerte, die bedeutend wemger vorslchtlg smd als die nach fruher veroffenthchten Methoden erhaltenen Em Verglelch mlt genauen zahlenmasslgen LoSUngen ergab gute Uberemstlmmung der vorausgesagten Grenzen

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