On global stability in distributed parameter systems

Pergamon Press

Chemical Engmeenng Saence, 1968, Vol 23, pp 1237-l 248

Pnnted m Great Bntam

On global stability in distributed parameter systems DAN LUSS and JAMES C M LEE Umverslty of Houston, Houston, Texas 77004 (Flrsr receroedl0January 1968, m revrsedform I March 1968) Abstract-A method 1s presented for determmmg fimte stabdlty regons for dlstnbuted parameter systems whose transient behavior IS governed by a smgle parabolic dlfferentlal equation The case of an a&abatlc catalytic reactlon IS &scussed m detad However, the same techmques can be apphed to many other systems m which chemical reacttons and dlffuslon are coupled The method 1s based on the maxlmum pnncrple for parabohc partial drfferentlal equations It 1s shown that fimte regions of stabdlty can be determined unmedlately from the knowledge of the steady state profiles, without havmg to perform any ad&tlonal computations A dIscussIon 1s included of the case m which the transient behavior 1s governed by two coupled partml differential equations

INTRODUCTION STABILITY problems associated with lumped chemical reactors have been extensively mvesttgated m the last decade However, relatively very few studies of drstnbuted parameter systems have been reported. The inherent difficulty m treating these systems is due to the fact that they are described by a system of partial drfferentral equatrons Stabrhty of lumped parameter systems, which are described by a set of ordinary dfferentral equations, can be determined by use of certam classical techniques, which were developed cngmally by Liapunov [l-5] Unfortunately, at present there exists no standard mathematical technique for determmmg the stability of partial differential equations Several studies have been reported on the stability of systems m which a chemrcal reaction is coupled with heat and mass diffusion These systems are described by a second order parabolic partial drfferential equation Amundson [6] developed a technique for determmmg the asymptotic stabihty of the steady states m a tubular reactor with axial diffusion by the numerical solutron of an ordinary differential equation His method was later applied by Schmitz[7] for the case of a flame and by Kuo and Amundson[B] for catalyst pellets We1 [9] suggested the use of Lyapunov’s functronals to determine stability of catalyst pellets He showed

that for certam kinetic expressions the system is globally stable In case of multiple steady states his methods enable the determmatron of asymptotic stability from the solutron of an ordinary differential equation Hartman, Roberts and Satter8eld [ lo] drscussed transient computations for the case of complex isothermal catalytic reaction Recently, a new technique was developed [ 1 l] which enables the determmanon of asymptotic stability m the above systems without having to perform any computations While the asymptotic stability of the above systems can now be easily determined, one is very often interested m the response of the system to large dsturbances In the case of lumped systems this IS usually done by construction of a Lyapunov function Certam techmques for devising these functions have been proposed [2-51 Berger and Lapidus [5] suggested to use the same method for a drstnbutedparameter system However, their results were extremely conservative. In this work a method is developed for obtammg stability regrons for the vanous steady states m drstnbuted parameter systems The case of an adiabatic reaction m a catalyst pellet for which the Lewis number is one will be used as an example However, the method can be applied for any other system whose transient 1s described by a parabolic partial drfferentml equations

1237

D

LUSS

and J

Section 1 contams the formulation of the problem and the various assumptions of the mathematical model In Section 2 the global stability of a unique steady state 1s dscussed. In Section 3 a method 1s developed for obtaining regons of stab&y for the case of multiple steady states In Section 4 a dlscusslon 1s given of the special case where the mltlal behavior of the transient 1s governed by a set of two coupled equations, 1 e , the case where the mltlal enthalpy IS different from the steady state value The analysis 1s based on the apphcatlon of a theorem about the maximum prmclple for partial dfferentlal equations, which 1s discussed m Section 2. The results obtained m this work are stronger than any previously published Moreover, the determmatlon of the stablhty regons does not require any mformatlon but the steady state profiles 1 ADIABATIC

CATALYTIC

REACTION

pcpd$ = kV2T+

T)

(-AH)r(c,

x* E d-b*

T)

LEE

(-AfWca

p =

L2

x = x*/L

$

= v2z-r’(Z,

Leg=

y)

of the

x* E Z”

(3)

T=T,

x*Ex”

(4)

c(x*, 0) = cO(x*)

(5)

T(x*, 0) = T,,(x*)

(6) groups,

= L2r(Zca, yT,)/c,D

x E 8-l

(8)

(9)

where n and I; are the volume and surface of the catalyst expressed by the dlmenslonless coordmates x. The dlscusslon ~111be restncted to the case m which the Lewis number (Le) equals one Thus, by addmg up Eqs (8-9) one can obtam g=V2E

xEZ

(10)

We ~111 define E as the restdual enthalpy boundary condltlons are E=O

XEI:

The

(11)

E(x, 0) = y(x, 0) - l+p(Z(x,

0) - 1) = E,(x) (12)

Hence = 2 A,&,(x)

ePtiT

m=1

(13)

where R,(x) 1s an elgenfunctlon determmed by the shape of the catalyst n and the A, are a function of the lmtlal residual enthalpy -E,(x) It can be shown that all the elgenvalues h, are positive [ 131 If the mltlal residual enthalpy 1s zero all the& vanish and E(x, T) = 0 Thus, Y(x,T)+flZ(X,T)

~'(Z,Y)

XEJ-I

vy+pr’(z,y)

E(x,T)

Le=y?

(7)

Eqs (1,2) can be rewntten as

(2)

c = c,

By use of the followmg dlmenslonless

De

7=-

kTa

(1)

x* E Cl*

where r(c, T) IS the rate of consumption reactant A The boundary condltlons are

M

where

The transient equations descnbmg a chemical reactlon A G B occurmg m a porous catalyst pellet of volume Cn* and boundanes Z * are [ 121: -$=m--r(c,

C

= l+p

(14)

By use of Eq (14) one can reduce the system of the two coupled dlfferentlal equations (8,9) to 1238

On global stability m dlstnbuted parameter systems

g=

vy+F(y)

x E

various

n

multiple solutions exist The condltlons for the existence of multiple steady states have been drscussed m [ 14-161. A proof has been gven m (11) that any steady state for which

where F(Y) =P’((l-(Y-l)lB),Y) subject to the boundary conQtlons y=l

values of y, p and C#I(various particle

(15) sizes) It IS seen that under certain condltlons

(16)

XEZ

(19)

The steady state of the system 1s governed by 1s unstable, while all those for which Vzy+F(y)

= 0

rzz

x E

(17)

subJect to boundary condltlons (16) It is well known [9] that under certain condltlons Eq. (17) has multiple solutions For example m the case of an irreversible first order reaction A + B occurmg in a catalyst pellet F(Y) = $2(1+P-~)

exp

(

Y--l YY

(1 g) >

where +

82=R1+J

Figure 1 shows the value of the dimensionless temperature at the center of the pellet y (0) for

dY(0) > o dL

(20)

are asymptotically stable, where y(0) IS the dlmenslonless temperature at the center of the catalyst pellet. Our am will be to determine which of the asymptotically stable steady states will be obtamed from a given mltlal condltlon In Sections 2 and 3 we ~111 discuss the cases for which Eq (14) 1s valid, hence restricting ourselves to the cases for which the mltlal residual enthalpy IS zero, and the transient behavior will be governed by Eq (15) The cases for which the mltlal residual enthalpy 1s non-zero ~111 be discussed m Section 4 2 GLOBAL STABILITY OF A UNIQUE

12

STEADY STATE

We will consider now the case for which a umque steady state exists. We1[9] has been able to prove global stablhty for the specific case of F’(y) < 0 for 1 s y 6 l+p by use of a Lyapunov functional The same result was obtained m [ 151 by use of the maximum prmclple We will discuss the more complicated case where the kinetic expression satisfies the condltlons. 2

Y=30

._

(I) F’(y) (ii) F(y)

._

Y(O)- DIMENSIONLESS PARTICLE CENTER TEMPERATURE

Fig 1 Dlmenslonless particle center temperarure as a function of 8, y, and $Jfor an u-reversible first order reactlon ma spherical pellet

> 0 > 0

for for

y= 1 1
< yes

where yes 1s the adiabatic eqmhbnum ture of the reaction mixture (111)

1239

F(y)

= 0

and

F’(y)

< 0

for

tempera-

y =yes

D LUSS and J C M LEE

(IV) F(y) < 0

and

F’(Y)

<

0

for

yeq

Y >

7)(x,0) = Y (x,0) -Y,(x)

2 0

(24)

720

(25)

c 0

(26)

73 0

(27)

then and F’ (y ) IS contmuous and fimte for all y 3 1 We will define

Y (x,7)

-Y,,(x)

20

for

since F’ ( y ) is fimte. Slmdarly If rl

(x97)= Y (X,T)-us,(x)

(21) rl(x,O) = Y (x,0) -yss(x)

where the subscnpt ss denotes steady state values By substractmg the steady state Eq (17) from the transient Eqs (15) and use of the mean value theorem one obtams V2~+F’(y*(x.~))1)=$

(22)

where F’(Y*(x,~))

F(Y(x,~)

=

-

F~J,,(x))

Y (x97)-Y,,(x)

then y(x,7)-yJ8(x)

q=o

I

ay(x,T) =a7

By partial ddferentlatlon Eq (15) one obtams

xE2. parabohc

(28)

VPq+F’(y(x,r))q=a$

x E iI

(29)

and

(23)

q=o

I

f=l

+c(x,T)u

with respect to time of

partial

a,(~,~)-$&+5 b,(x,~)z f

q(x,7)

=710(x)

Consider now the followmg dlfferentlal equation i f.l=l

for

Hence, ,vf the mrtlal drsturbance IS one srded it remams so for all 7 > 0. We will define as yr(x;r) the transtent obtamed from the mltlal condltlon y(x,O) = f(x) For example y, (x,7) IS the transient obtamed from y(x,O) = 1 It is proven m Reference [ 181 p 4 1, that for any mltlal condltlon f(x) there exists a umque transient y,( x,7). Also define

and y * (x,7) 1s bounded by y (x,7) and yss (x) The boundary condltlons are

7(x,0) =Y(xm-YY,,(x)

c 0

XEX

Consider the case y(x,O) = 1 Accordmg dlfferentlal Eq (15)

=z

where au, b, and c are real and fimte, a, = ajf and U(X,T) contmuous m n Clearly, Eq. (22) 1s a special case of Eq. (23) We shall state an important theorem known as the maxlmum pnnclple which ~111enable us to establish some important propertles of U. A ngorous proof of the theorem 1s given m Reference [ 171 p 6 Maximum pnncrple theorem If c( x,7) < M where M IS some constant, U(x,O) 2 0 and U(X,T) 3 Ofor all x E 8 then the solutzon of Eq. (23) U (x,7) 3 0 for all r and

ayl (x,0) a7

= q,(x,O)

= F(Y)(,=,

> 0.

Hence it follows from the maxlmum and Eq (29) that ql(x,7)

=

ayl (x,7) a0 a7

720

to the

(31)

pnnclple

(32)

while according to (26-27) YlbS)

s Y,(X)

XEil

Thus, accordmg to the maxlmum pnnclple d

(30)

Slmdarly one can show that 1240

(33)

On global stabdlty m dlstnbuted parameter systems aYl+@(x,T)

a7

<

o

example, Fig 2 shows y1 ( X,T) and yl+s (x,7) for an u-reversible first order reaction m a spherical pellet Since any initial transient will converge to the steady state not later than either y1 (x,7) or Y~+~(x,T), it IS clear that m this case the duration of any transient will not exceed T = 5.5. It should be mentioned that the unique steady state shown m Fig 2 does not satisfy the condltlon F’ (Y) < 0 for all 1 < Y G 1 2 for which WeI has&oven global stab&

(34)

-

while Y1+s(x,7) zy,,(x)

(35)

7ao

From Eqs (32-35) it follows that “both y, (x,7) and Y~+@(x,T) converge umformly and with no oscdlatlons to the unrque steady state.” Consider now an arbitrary lmtlal condltlon 1 G f(x) s I+ p By substractmg the transient equation (15) for y1 (X,7) from that for yI (X,7) one obtams

ai

v”~+~‘(y*(X,T))~=

(36)

where [(x,7) F’(Y*(X,T))

= Y,(x,T) -Y,(%T) =

-F(Yl(X,T))

F(Yf(X,T)) Yf(X,T)

-Yl(X,T)

and y “1s bounded by yf and y1 The boundary con&tlons are 0

5 (x,0) = Yr(X,O) -Y1 (x,0) =f (x) - 1 2 0 5=0 Therefore,

XEI;

(37) (38)

by the maxlmum prmclple ,$(X,7)

7 2 0

(39)

t%d = Yl+dXP)- Yfh)

(40)

3

0

Sumlarly if we define

one can prove that 5(X,7)

From

30

720

(41)

The transient computations were done usmg the computational method of Lm, the detads of which can be found m [ 191. A gnd of 100 pomts was used Identical profiles were obtamed by using a gnd of 200 points It 1s assumed that a steady state 1s achieved when the maxlmum deviation from the steady state at any point 1s smaller than 0 002 It was found that the decay of this very small perturbation depended on the gnd size due to numerical round off errors 3 REGION OF STABILITY

(39) and (41) It follows that-all the transrents for which 1 < y (x,0) < 1 + p are

This proves the global stabdlty of a umque steady state Moreover it enables an easy estimate on the duratron of any transient from the knowledge of y, (x,7) and Y~+~(x,T). For

4 6 6 X - OlMNSlONLESS RADIAL POSITION

1 ranslent temperature protiles for an wreverstble first order chemical reactlon m a spherical pellet p = 0 2, y = 30, +=086

Eqs

bounded from above by yI+p (x,7) and from below by y, (x,7) for all T a&d x E fl. Thus, they converge to the steady state

2

STEADY

FOR MULTIPLE

STATES

We ~111now discuss the case m which multiple steady states exist Clearly, m this case no steady state can be globally stable, since this Implies the existence of a umque steady state We ~111 denote each steady state by a superscnpt The higher the value of y (0) the higher the value of the superscnpt For example, for

1241

D

LUSS

and J

the three steady states shown m Fig 3, yk (x) and yi (x) describe the low and high temperature steady states, respecttvely In [14] a proof was given that m general there exist an odd number of possible steady states Among the 2m + 1 steady states the ones with an even superscript are unstable, while the other are asymptotically stable [ 111. Hence, the low and high temperature steady states are asymptottcally stable Our aim 1s to find for each stable steady state y:Jx) a region bounded from above by the mittal condmons u(x) and from below by I(x) such that (I) Both y,( x,7) and yz(x,r) converge to Y”,(X) (11)

Every y,(x,~) for which U(X)
C

M

LEE

v‘%+

[h+F’(y::(x))]v=

0

u=o

r

x E Q

XEZ;.

(43) (44)

By use of the methods of Section 2 and the maximum prmctple it can be shown that since YZ(X)

< Y&O)

(45)

< Y;;+‘(x)

then y;;(x) G Yl(X,7) S y;;+‘(x)

72 0

(46)

Define rl(x,r) =

Y&7)

(47)

-Y;;(x)

For small value of 77 Its behavror can be descnbed by the followmg linearized equation v%I +F’(y$(x))7j

= &

x 6%n

(48)

rl(x, 0) = El&l

XE8-k

(49)

77(x,7) = 0

x E X.

(50)

Hence r)(x,r) = EIul]e-hlT where shown A1< 0 ISvalid Define

(51)

X1 is the first etgenvalue of (43) It was m (14) that yg (x) is unstable and thus It should be emphasized that Eq (5 1) only for small values of 7) q, (x, 7) = ;

(Y&T

(52)

7))

From Eq (5 1) It follows that 0

4 6 8 2 X-DIMENSIONLESS RADIAL POSITION

to

87)(x, 0) = qdx, 0) = --EIZIJX1 > 0. a?-

3 Regions of stablhty for the case of three steady states for an meverslble first order reactlon In a spherical pellet Fig

According to the maximum prmctple it follows from Eqs (29,30,53) that

we will discuss the method of obtammg these regtons for the case of 2m + 1 steady states Consider the transtent yf (x,7) where yl(x,o) =

l(x)

=

~~281 (x)

+ E~Vl~ <

y:,i+l

where E is an arbitrary small posmve and v1 1s the first etgenfunctton of

(x)

(53)

%(x,7)

=;(Y,(V))

2

0

7 >

0.

(54)

(42)

constant

Thus, tt follows from Eqs (42) and (54) (3 a) y,(x,~) converges fo y$+‘(x) with no oscdlatlons where the mltlal condltlon l(x) IS 1242

On global stabhty m dlstnbuted parameter systems

5(x, T) = y,(x, 7) -yr(x,

glum by Eq. (42). Sumlarly, consider the translent y.(x,~) where YUCX, 0)

=

4X)

=

Y%(X)

-EM

>

Y%’

(x)

7) 2 0

Ta 0

(60)

Slmllarly, one can prove that Y!(X,7)

(55)

where E 1s an arbitrary small positive constant The same methods as before can be used to prove (3 b) y,(x,~) converges to y$;-l(x) wzth no oscdlatlons where the mitral conditrons u(x) 1s grven by Eq (55) The same techniques of Section 2 can be used also to prove (3 c) yl(x, 7) converges to yi, (x) wrth no oscdlations (3 d) yl+p (x,7) converges to yzp+l (x) with no oscillations The above results enable us to bound each stable steady state y&(x) by two functions I(x) and U(X) such that both y1 (x, 7) and yU(x, 7) converge to yis (x) with no osclllatlons We will now prove the followmg (3 e) Any translent yf(x,7) for which l(x) < f(x) < u(x) will converge to y&(x) not slower than either yl(x, 7) or y,(x, T) We will first prove that yr(x, T) s yu(x, 7) Define

-yr(x,

7)

s

0

7 5

0

(61)

Equations (60-61) establish (3.e) Thus, we have developed a method to obtain regions of stability which covers the whole region of interest 1 s y(x) G 1 +p Instead of using u(x) and I(x) defined by Eqs (42, 55) as boundanes of the stability regions, it 1s more convenient to use u*(x) and I*(x) which are defined by U*(x)=y:;(x)-6(x)

c U(X)

x E n

(62)

l*(x)=

3 l(x)

x E n

(63)

y;;(x)+S(x)

where 6(x) 1s an arbitrary small positive function This can be done since as proven m (3 e) yJx, 7) and yU*(x, 7) converge to the same steady state, and y1(x,7) and y,.(x,~) converge to the same steady state One can asslgn 6(x) any arbitrary small value, smce we can always choose a value of E which will satisfy Eqs (62,63) The above results are very strong and enable the determination of the stab&y regions from (56) T(x, 7) = YdX, T) -Yr(X, T). the knowledge of the steady state profiles without having to perform any additional computaBy subtracting the transient Eq (15) for yr(x, T) tions As an example, consider the case described from that for y,(x, 7) one obtains m Fig 3 for which there exists three steady states Figure 4 describes four transients at computed for the followmg mltlal condltlons, v”S+F’(y*(xJ))[=-$ x E n (57) f(x) = 1, y~~(X)-o@OO5, y;8(x)+oOO05, 1+p As predicted the first two transients converge where to the low temperature steady state and the F(Y,(X, 7))F(Yf(X, 7)) others to the high temperature steady state F’(y*(x, 7)) = The figure enables an easy estimate of the duraYJX, 7) -u&G 7) tion of many transients For example any transy*(x,~) IS bounded by yU and yr, and F’(y*) IS ient for which Y’,(X)+ 0 0005 s f(x) s 1 +p will converge to steady state not later than T = finite The boundary condltlons are 1.25 4(x, 0) = u(x) -f(x) > 0 x E n (58) Clearly there are many mitral con&tlons which do not belong to any of the above stablhty XEZ 5(x, 7) = 0 (59) regrons For example, the mltlal condmons

It follows from Eqs (57-59) and the maximum prmaple that

y(x,O)=C 1243

1 < c < y;rs”(x>

(64)

D LUSS and J C M LEE

0

2 X-DIMENSIONLESS

4

6 RADIAL

8

0

t0

2 X-

POSITION

Fig 4 Transient temperature profiles for an lrreverslble first order chermcal reactlon m a spherical pellet Same parameters as m Fig 3

intersects at least two regons of stab&y In such a case the methods and results of this section can still be used for reducing the numencal effort of determining global stablhty as well as for estabhshmg stablhty regons for a new set of mttlal condltlons Consider the case m which multiple steady state solutions exist and the mltlal condltlons are not bounded by any of the stab&y regons prevlously obtamed. In order to determine which steady state will be obtamed, it 1s sufficient to integrate numencally the transient Eq (15) tdl the transient is bounded wlthm one of the stablhty regons This reduces to a large extent the duration of the numerical mtegratlon For example Fig 5 describes a case for which y(x,O) = l-095 and where the system converged to yi8 (x) at T = l-4 However, the transient computation could have been stopped already at 7 = O-55, for which the transient 1s bounded by the stab&y regons for y&(x) (see Fig 3) Slmllarly, Fig 6 describes a case for which y (x, 0) = 1 094 and where yz (x) IS reached at 7 = 1 2 Here the computation could have been stopped at 7 = O-2 when the system 1s bounded by the stab&y reDon of yi8 (x) Now according to (3 e) if two transient yl(x, 7) and yu(x, T) converge to the same steady state, so will any transient yf(x, 7) for which f(x) s Thus, It follows from Fig 5 that f(x) =sU(X)

4 DIMENSIONLESS

6

8

RADIAL

POSlTlON

Fig 5 Transient temperature profiles for an u-reversible first order chemical reaction occunng m a spherical catalyst pellet Initial conltlon Qx, 0) = 1 095

any transient for which y (x, 0) 2 l-095 ~111 converge to y$ (x) Slmllarly, it follows from Fig. 6 that any transient for which 1 < y(x,O) G

0

2 4 X - DIMENSIONLESS

6 RADIAL

6 POSITION

IQ

Fig 6 Transient temperature profiles for an irreversible first order chenucal reactIon occunng m a spherical pellet Imhal con&tion y(x, 0) - 1 094

1 094 will converge to y,‘, (x) In this fashion we have obtamed two new regons of stab&y which are different from the ones shown m Fig. 3 However, m this case the constant mltlal temperature which separates the two regons of stab&y had to be determmed by numerical integration of the transient Eq (15).

1244

On global stablhty m dlstnbuted parameter systems 4

EFFECTS

OF RESIDUAL

ENTHALPY

The results obtamed m Sections 2 and 3 are restncted to cases for whrch the residual enthalpy 1s zero for all T zz 0 We will now discuss the more general case where this assumption is no more valid and for which the transient 1s governed by the two coupled differential Eqs (8,9) According to Eq (13) the residual enthalpy decays exponentially with time and one can always determine a fimte value of T, such that ~E(x,T)~=Iy(x,7)+pz(x,7)-(l+p))

< E 7 2 7, (65)

where E is an arbitrary small constant For example m the case of a spherical pellet E(x, 7) =;

i

eXp (-WZ’tiT)

A,sin(mrx)

m=1 (66)

where A,=2

I

i X&(X) sin (m7rx) dr

Thus, the value of Tc depends on the initial residual enthalpy E,(x) and E If E,(x) = 0 then Tc = 0 Hence, by choosmg a sufficiently small Value Of E, the translents for 7 > T, can be com-

I 0

1

I

2

I

puted by the single partial drfferential Eq. (15) and all the results of the previous sections can be applied for T > T, For most systems To 1s of order one. It was proven m Section 2 that a unique steady state which 1s governed by Eq (15) is always stable to any dtsturbance for which E(x, T) = 0. Now the system of the two coupled dtfferenttal equations (8, 9) always degenerates with time to the single drfferential equation (15) as E(x, T) decays to zero exponentially with time Hence, one can conclude that a unique steady state 1s stable with respect to any arbitrary disturbance, 1.e , the system will always converge to the steady state Durmg the initial penod (T < T,) the behavior of the system depends rather strongly on the uuttal restdual enthalpy, and one may observe oscillations which would not have been possible if E(x, 0) = 0 This is demonstrated m Fig. 7 where the mmal condrtrons are y(x, 0) = 1.1, Z(x, 0) = 0 and thus E(x, 0) = l*l- 1.3 = -0.2 It can be seen that until 7 = 0.2 the temperature decreases for all x and only later the temperature rises till the system converges to the unique steady state If E(x,O) were set to zero the temperature at the center of the particle would have risen monotonously and the

t

I

4

I

6

X-DIMENSIONLESS

I

I

I

8

RADIAL POSlTlOtd

Fig 7 Transient temperature profiles for an lrreverslble first order chemical reactlon occurrmg m a spherical pellet Imtral condltlons y(x, 0) = 1 1,2(x, 0) = 0, E(x, 0) = -0 2

1245

CES-F

I 1

D LUSS and J C M LEE

oscrllation m the temperature would not have occurred In the case of multiple steady state solution the imttal residual enthalpy has an important effect m determmmg which steady state will be obtained from a gtven mttml condttton For example, the transients m Fig 8 descnbe a case for which y(x, 0) = 1.095 and 2(x, 0) = 0, thus E(x, 0) = 1.095 - 1.3 = -0.205 It 1s seen that the system converged to the low temperature

0

2

reaction are coupled The method 1s very general and can be used for any reactton rate expression whose first order denvattve IS finite A proof was gtven that a umque state IS stable with respect to any arbitrary disturbance. It was shown that m case of multtple steady states the asymptotically unstable steady states separate the regtons of stab&y of the two bounding asymptottcally stable steady states Thts important fact enables one to determine fimte

6

4

X- DIMENSIONLESS

lo

8

RADIAL POSITION

Fig 8 Transient temperature profiles for an wreverslble first order chemical reactIon Imtlal con&tlons y(x, 0) = 1 095, 2(x, 0) = 0, E(x, 0) = -0 205

steady state. However, d E(x, 0) were zero the system with the same mttral temperature would have converged with time to the htgh temperature steady state (see Fig 5) A series of transient computattons has mdtcated that an excess of mmal restdual enthalpy tends to shift the system to the high temperature steady state An mmal negative residual enthalpy tends to shift the system to the low temperature steady state CONCLUSION

AND

REMARKS

In thts work we have developed a method for obtammg fimte regions of stab&y for a case of a chemtcal reaction occurring m a porous catalyst pellet The same methods can be apphed for any other system m which diffusion and chemical

regions of stabihty from the knowledge of the steady state profiles The technique was applied for reducing the computattonal effort m treating systems whose uutial temperature dtstnbutron intersects several stab&y regions The effect of the mrtial residual enthalpy was discussed NOTATION

concentration specific heat CP D dtffuston coefficient E dimensionless residual enthalpy AE activation energy heat generation F dimensionless defined by Eq 15 -AH heat of reaction

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C

term,

On global stabdlty m dtstnbuted parameter systems

/:

Is L Le

m 4 r I iz T V x ii X*

;

Y

a constant thermal conductrvity reaction rate constant charactenstrc length Lewis number Dpc,lk a constant ay/ar consumption rate of reactant L2rlcaD radms of pellet temperature an eigenfunctron dlmensronless radial positron r/R TIT, CIC, positron vector X*/L (--LW)Dc,lkT, AEIR T,

a small positive constant ?1 a perturbation from steady state 8 trme A ergenvalue P density 7 De/L2 R2d(iiID) external surface of partrcle n intenor of catalyst pellet E

1

Subscripts a ambrent condrtrons ss steady state condrtrons f the uutml condrtrons of a transrent 0 mmal condmons Superscripts I I-th steady state f(x) mmal temperature

drstnbutron

REFERENCES R,AZChEfll9551513 HI BILOUSO andAMUNDSONN PI BERGER J S and PERLMUTTER D D , A I Ch E.Jf 1964 10 233

WARDEN R B , ARIS R and AMUNDSON N R ,Chem Engng Scr 1964 19 173 LUECKE R H and McGUIRE M L . Ind Enana Chem Fundls 1967 6 432 BERGER A J and LAPIDUS L , Paper presented at New York A I Ch E Meeting 1967 AMUNDSON N R , Can J them Engng 1965 43 49 SCHMITZ R A , Combust Flume 1967 1149 KU0 J G W , and AMUNDSON N R , Chem Engng Scr 1967 22 1185 191WE1 J , Chem Engng Scr 1965 20 729 HARTMAN J S . ROBERTS G W . and SATTERFIELD C N . Ind Ennna Chem Fundls I967 6 80 - _ tf:; LUSS D and AM’UNDSON N R , can J them Engng 1967 22 253 iI21 WEISZ P B and HICKS J S , Chem Engng Sci 1962 17 265 [131 INCE E L . Ordman, DuTerentral Euuatrons McMdlan 1927 1141 GAVALAS G R . dhem Engng SC; 1966 21477 1967 22 253 iI51 LUSS D and AMUNDSON N R , Chem EngngSa [I61 LUSS D , Chem Engng Scr 1968 23 I249 A S , and OLEINIK 0 A, Russ Math Surus 1962 17No 3 1 iI71 IL’LIN A M , KALASHNIKOV 1181 FRIEDMAN A , Parfral Dz_ferentud Equarmns ofParabohc Type Prentice-Hall 1966 r191 LIU S L , Chem Engng Scl 1967 22 871 [31 [41 PI WI r71 [81

R&urn&-II s’agrt dune methode pour dCtermmer des regtons de stabthte hmrtee pour des systemes de paramttres dtstnbues dont le comportement transrtone &pond a une equation dtfferentrelle a parabole simple Le cas d’une reaction catalytique adtabattque fan 1’obJet dune longue dtscussron Les m&mes techniques peuvent nCanmoms s’apphquer a plusreurs autres systemes ou reactrons chmuques et dtffuston se trouvent combmies La methode se fonde sur le prmcrpe maximum pour les equations dtfferentrelles parttelles parabohques II est demontre qu’d est possrble de dCtermmer des regions de stabdlte hmrtee de man&e tmmedtate a partu dune connatssance des profits de f&at fixe, sans qu’d y ait besom d’avou recours a d’autres calculs II y a egalement une dtscusston du cas ob le comportement transrtoue r&pond a deux equations dtfferenttelles parttelles comb&es Zusammenfassung-

Es wud eme Methode fur die Bestrmmung endhcher Stabrhtatsberelche fur vertedte Parametersysteme dargelegt, deren Ubergangsverhalten durch eme emztge parabohsche Differentralgletchung festgelegt wud Der Fall enter adrabatrschen katalyttschen Reaktton wud rm

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LUSS and J C M LEE

emzelnen erortert Dieselben Methoden konnen Jedoch auf eme grosse Anzahl anderer Systeme angewendet werden, m welchen die chemlschen Reaktlonen und DdTuslonserschemungen mltemander gekoppelt smd Die Methode grundet such auf das Maxlmalpnnzlp fur parabohsche partlelle Dlfferentlalglelchungen Es wlrd gezelgt, dass endhche Berelche der Stabdltat unmlttelbar aus der Kenntms der Statlonaren Profile, ohne &e Notwendlgkelt zusatzhcher Berechnungen, bestlmmt werden konnen Der Fall, m welchem das Ubergangsverhalten durch zwel gekoppelte partlelle Dlfferentlalglelchungen bestlmmt wlrd, wlrd erortert

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