Asymptotic stability of a nonisothermal plug flow reactor infinite-dimensional model

Copyright © IFAC Nonlinear Control Systems, Stuttgart, Germany, 2004

ELSEVIER

PUBLICATIONS www.elsevier.com/locate/ifac

ASYMPTOTIC STABILITY OF A NONISOTHERMAL PLUG FLOW REACTOR INFINITE-DIMENSIONAL MODEL Ilyasse Aksikas * Joseph Winkin ** Denis Dochain *

* CESAME, Universite catholique de Louvain 4 Av G. Lemaitre, B-1348 Louvain-la-Neuve, Belgium e-mail: [email protected]@csam.ucl.ac.be ** Department of Mathematics, University of Namur 8 Rempart de la Vierge, B-5000 Namur, Belgium e-mail: [email protected]

Abstract: The asymptotic stability property is studied for a nonisothermal plug flow tubular reactor model, which is described by semi-linear partial differential equations (PDE's) derived from mass and energy balance principles. It is reported that, under sorne condition on the model parameters, any constant temperature equilibrium profile is an asymptotically stable equilibrium of such model. The analysis is based on an asynlptotic stability criterion for a class of infinitedimensional (distributed parameter) semi-linear Banach state space systems and the concept of strictly m-dissipative operator. Copyright © 2004 IFAC Keywords: Thbular reactor, nonisothermal plug flow chemical reactor, partial differential equation, semilinear infinite dimensional system, nonlinear semigroup of contractions, asymptotic stability, strict m-dissipativity.

1. INTRODUCTION

On the other hand stability is one of the most important aspects of system theory. The fundamental theory of stability was established by Lyapunov, which is largely developed for finitedimensional systems. Many results on the asymptotic behavior of nonlinear infinite-dimensional systems are known, for which the dissipativity property plays an important role, see e.g (Brezis, 1973)) (Crandall and Pazy, 1969), (Dafermos and Slemrod, 1973),(Luo et al., 1999). Here we are interested in asymptotic stability criteria for a class of infinite-dimensional semi-linear systems, which can be applied to tubular reactor models.

TUbular reactors cover a large class of processes in chemical and biochemical engineering: see e.g. (Dochain, 1994). They are typically reactors in which the medium is not homogeneous (like fixedbed reactors, packed-bed reactors, fluidized-bed reactors, ... ) and possibly involve different phases (liquid/solid/gas). Such systems are sometimes called " Diffusion-Convection-Reaction" systems since their dynamical model typically includes these three terms. The dynamics of nonisothermal plug flow reactors are described by semi-linear partial differential equations derived from mass and energy balance principles: see e.g. , (Laabissi et al., 2001), (Laabissi et al., 2004), (Winkin et al., 2000).

The objective of this work is basically twofold: (a) to develop a theory of asymptotic stability for a class of semi-linear infinite-dimensional systems, by using tools of infinite-dimensional non-

781

the inlet reactant concentration, respectively. In addition T, <;, and L denote the time and space independent variables, and the length of the reactor, respectively. Finally To and C A,O denote the initial temperature and reactant concentration profiles respectively, such that To(O) = Tin and CA,O(O) = CA,in.

linear system theory (Luo et al., 1999), (b) to use this theory for studying the asymptotic stability of a plug flow reactor model; see (Aksikas et al., 2004). It is shown that, under some condition on the model parameters, any constant temperature equilibrium profile is asymptotically stable. The motivation for choosing such an equilibrium profile is the fact that it minimizes the reactant concentration at the reactor outlet and the energy consumption along the reactor : see (Smets et al., 2002).

Comment 1. In the particular case where To (<;,) = Tin and CA,O(<;') = CA,in for all ~ <;, ~ L,

°

the existence and the invariance properties of the state trajectories of such model on [0,00) have been established in (Laabissi et al., 2001)(see also the references therein), by using a dimensionless variable equivalent model. We shall also use such a model here: see Subsection 2.3 below.

2. NONLINEAR MODEL DESCRIPTION 2.1 Nonlinear PDE Model

The dynamics of tubular reactors are typically described by nonlinear PDE's derived from mass and energy balance principles. Let us consider a nonisothermal plug flow reactor with the following chernical reaction:

2.2 Constant Temperature Equilibrium Profile

In this paper we are interested in equilibrium profiles for the model (1)- (3), of the form

X e = [ T e (.)

CA,e(.)

A~bB,

°

where b > denotes the stoichiometric coefficient of the reaction. If the kinetics of the above reaction are characterized by first order kinetics with respect to the reactant concentration CA (mol/l) and by an Arrhenius-type dependence with respect to the temperature T (K), the dynamics of the process in such a reactor are described (without considering the dynamic of the product concentration since the latter is known if the reactant concentration C A and the temperature T are known) by the following energy and mass balance PDE's:

T e (<;,)

4h ~H where f30 := pCpd' k 1 := pCp ko and with the T

2: 0, by:

The initial conditions are assumed to be given, for ~ <;, ~ L, by

°

T(C 0) = To(<;') ,

CA(<;', 0)

= CA,O()'

(4)

,

in the state-space H := L 2(0, L)2 := L 2(0, L) x L 2 (0, L) where the temperature equilibrium profile is assumed to be constant, i.e

aT -E - v - - k1CAexp(-) - f3o(T - T c ) BC RT (1) aCA -E - v - - - koCAexp(-) a<;, RT

boundary conditions given, for

]

(3)

In the equations above, v, ~H, p, Cp, ko, E, R, h, d, T c , Tin and C A . in hold for the superficial fluid velocity, the heat of reaction, the density, the specific heat, the kinetic constant, the activation energy, the ideal gas constant, the wall heat transfer coefficient, the reactor diameter, the coolant temperat ure, the inlet temperature, and

= Te >

°

a.e. in [0, L] .

(5)

Comment 2. (a) An equilibrium profile of the form (4) must satisfy the boundary conditions (2). Hence, by continuity of the function T e , in view of identity (5), one should have T e (<;,) = Tin for all <;, E [0, L], i.e. the value of the inlet temperature (which can be set by the operator) dictates that of the constant temperature equilibrium profile. (b) In (Smets et al., 2002) temperature equilibrium profiles are studied, that minimize different kinds of performance criteria. The specific temperature equilibrium profile (5) corresponds to the minimtun for a cost criterion which is the sum of the reactant concentration at the reactor outlet and the energy consumption along the reactor. In addition to this interesting optimality property, the temperature equilibritun profile (5) is simple and, in principle, should also lead to a closed-loop steady-state behavior which is close to that of an isothermal reactor model, by using appropriate stabilization techniques around this equilibrium. By integrating the equilibrium ordinary differential equations corresponding to (1), it can easily be shown that, in this case, the reactant concentration equilibrium profile is given for ~ <;, ~ L by

°

(6)

782

where the constant Q

e

Qe

is given by

ko E = -exp(--) > 0.

RTe

v

The equivalent state space description of the model (1 )-(3) is given by the following semi-linear abstract differential equation on the Hilbert space

(7)

H := L 2 (0, 1) x L 2 (0, 1):

B(t) { 0(0)

Then the corresponding coolant temperature equilibriunl profile reads as follows

= =

AoO(t) + IV(O(t)) 00 E D(A o ) n F

+ BoOe

(10)

where A o is the linear (unbounded) operator defined on its domain The latter can be interpreted as an open-loop infinite-dimensional control profile distributed along the whole reactor. The dynanlics of the plug flow reactor model above will be described by means of an infinite dimensional systeIn description derived froIn an equivalent nonlinear PD E dimensionless model. Such an approach is standard in tubular reactor analysis: see e.g. (Laabissi et al., 2001) and the references therein. It is briefly developed in the following subsection.

de

e a.c.,

D(A o) := {O EH:

dz

0(0)

E Hand

=

0}(11)

(a.c. means that 0 is absolutely continuous) by - d. - - {3I

A o 11 := [

dz 0

°

_!!:... _(3 I

]

[e ] II~

(12)

dz

and the nonlinear operator N is defined on

F := {O EH:

Ol,min

~ 0 1 ~ 01,max,

°

~ O2 ~ 1 }

(where the inequalities hold almost everywhere on [0, 1]) by

2.3 Infinite Dimensional System Description

Let us consider two new state variables (h (t) and 02(t) , t ~ 0, which are defined via the following state transformation

fh =

T-Tin T,.

tn

The operator B o E L(H) is the linear bounded operator defined by (14)

(9)

'

In the equations above Oe is the dimensionless coolant temperature given by

Let us consider also dimensionless time t and space z variables defined as follows :

t

= TV L -1, Z =

o _ Te -

(L -1 .

and the parameters Q, (3, 0, and J.L are related to the original parameters as follows: E koL J1 = - - , Q = -exp( -J1)

Comment 3. It is experimentally observed that for all z E [0,1], and for all t ~ 0, T min ~ T(z, t) ~ T max and ~ C(z, t) ~ CA,in or equivalently

°

o1,mtn .',-

T min - Tin Tin

~

01(Z,

< -

and

°

0

Tin Tin

e -

RTin

13 - 13

0L

-

t)

v'

V

8 __ t::..H CA,in pCp Tin .

Observe that the state Oe and the input Oee given by

. = _T_m_ax_-_T_i_n l,max' Tin

(15) ~ O2 ( Z , t) ~ 1 ,

and where the lower bound T min is nonnegati ve and the upper bound T max could possibly be equal to ()(). It turns out that the case T max < +00 is the most interesting one in the stability analysis: see (Aksikas et al., 2004). rvloreover this case is physically feasible: see (Dochain, 1994, Theorem 4.1), which shows that under some assunlptions, the ternperature and the reactant concentration are bounded.

(16) satisfy the equation

AoOe + N(Oe)

+ BoOe,e =

°,

Le. they represent an equilibrium profile for the model (10), that corresponds to the equilibrium (4)-(8) for the initial model.

783

3. ASYNIPT'OTIC STABILITY OF SEl\'lI-LINEAR SYSTEtvlS

(b) By (Luo et al., 1999)[Prop. 2.109, p. 100; Cor. 2.120, pp. 106-107], if A is a m-dissipative operator, then A is the generator of a unique nonlinear contraction semigroup r(t) on D := D(A). Hence for any Xo E D, the corresponding state trajectory t ~ x(t, xo) := r(t)xo E D exists on (0,00) and is differentiable almost everywhere whenever Xo E D(A) such that 1tr(t)xo = Af(t)xo for almost all t in (0,00), see e.g. (Luo et al., 1999)[Prop. 2.98 (iii), p. 93]). Otherwise stated, for any Xo E D(A), x(t, xo) := r(t)xo is the unique solution of the following nonlinear abstract Cauchy problem:

In order to study the asymptotic stability of the constant temperature equilibrium profile for the nonisothernlal plug flow reactor model, some prelinlinary concepts and results are needed, that are related to the theory of nonlinear abstract differential equations on Banach spaces. rVlore specifically this section is devoted to (a) SOIue properties of nonlinear contraction senligroup theory (Brezis, 1973), (Crandall and Pazy, 1969), (Dafermos and Slemrod, 1973), (Luo et al., 1999) and to a paranlount result related to LaSalle's invariance principle (see Theorem 8 below), and (b) the asymptotic stability analysis of a specific class of semi-linear infinite-dimensional dynamical systems. Let us consider a real reflexive Banach space X equipped with the norm 11·11. Let D be a nonempty closed subset of X.

x(t) = Ax(t), { x(O) = xo.

Vxo

The infinitesirnal generator Ar of the nonlinear contraction semigroup r( t) (on D) is defined on its domain

{x

E

D: lim t- 1 [r(t)x - x] exists}

= t--+O+ lin1 t- 1 [f(t)x -

x], for every x

E

D(A r ).

x

t--+oo

x.

:=

{r(t)xo : t ~ O},

=

lim r(tn)xo.

n--+oo

Observe that w( xQ) is r( t )-invariant, i.e. for all

II(x - x') - A(Ax - Ax')II,

t ~ 0, f(t)w(xo) C w(xo). Recall that the distance between a point x E X and a subset n c X is

or equivalently, for all x, x' E D(A), there exists a bounded linear functional f on X such that f(xx') = IIx - x'I1 2 = IIfl1 2 and f(Ax - Ax') ::; O. In addition A is said to be strictly dissipative if the conditions above hold with strict inequalities, for all x, x' E D(A) such that x # x'. (ii) A is said to be (strictly) m-dissipative if it is (strictly) dissipative and the range R(1 -A) = X.

given by

d(x, n)

:= inf

{llx - yll : YEn}.

Theorem 8. Consider the system

x(t) = Arx(t), { x(O) = XQ,

Comment 6. (a) By (Luo et al., 1999)[p. 99], if A is a dissipative operator, then for all A > 0 , (1 - AA)-l is welldefined and non-expansive on R(1 - AA) , i.e. for all y, y' E R(1 - AA),

11(1 - AA)-l y - (1 - AA)-l y 'lI

lim x(t, xo) := liIn r(t)xo =

t--+oo

and the w-limit set w(xo) of XQ is the set of all states in D which are the lirnits at infinity of converging sequences in the orbit through Xo, i.e. more specifically x E w(xo) if and only if there exists a sequence t n ~ 00 such that

Definition 5. Let A be an operator with domain D(A). (i) A is said to be dissipative if, for all x, x' E D(A) and for all A > 0,

Ilx - x'll ::;

D

,(xo)

t-O+

by

Arx

E

In the rest of the paper we shall also need the important result stated in Theorem 8. This result is strongly related to the well-known Lasalle's invariance principle, see e.g. (Luo et al., 1999), (Dafernl0s and Slemrod, 1973). In order to state this result, the following concepts and notations are needed. If r( t) is a nonlinear contraction semigroup on D, for any XQ E D, the orbit ,(xQ) through Xo is defined by

(i) 1'(t+s) = f(t)f(s) for every s, t ~ 0; 1'(0) = I. (ii) IIf(t)x - f(t)x'll ::; IIx - x'll for every t ~ 0 and every x, x' E D. (iii) For every x E D, r(t)x ~ x as t ~ 0+.

=

(17)

Definition 7. Let us consider the systenl (17) and assume that A generates a nonlinear contraction semigroup r(t). Let us consider an equilibrium point x of (17), i.e x E D(A) and Ax = o. x is said to be an asymptotically stable equilibrium point of (17) on D if

Definition 4. A (strongly continuous) nonlinear contraction semigroup on D is a family of operators r(t) : D ~ D, t ~ 0, satisfying:

D(Ar)

t>O

t>O

(18)

where A r is a m-dissipative operator, and let r(t) be the nonlinear contraction semigroup on D = D(A r ), generated by A r . Assume that 0 (without loss of generality ) is an equilibriunl point of (18) and (1 - AA r )-1 is compact for some A > O. Then for any XQ E D, x(t, xo) := f(t)xo

::; lIy - y'll·

784

converges, as t ~ 00, to w(xa) C {z : where r ~ Ilxo 11 , i.e.

IlzlI

=

r},

Now we can state the following result: Theorem 10. Consider a semi-linear system given by (19) and satisfying (AI), (A2) and (A3) with M = 1. Assume that w ~ la. Let r(t) be the nonlinear contraction senligroup on D, generated by A. Assume that No (0) = O. Then for any Xa E D, x(t, xa) := r(t)xa converges, as t ~ 00, to w(xa) C {z: Ilzll = r}, where r ~ Ilxall , i.e.

lim d(x(t, xa), w(xo)) = O.

i-co

If in addition A r is strictly m-dissipative, then x(t, xa) ~ 0 as t ~ 00 i.e. 0 is an asymptotically stable equilibrium point of (18) on D.

Proof : The first part follows from Proposition 3.60(iii), Theoren1 3.61 and Theorem 3.65 in (Luo et al., 1999). For the second part the idea is to prove that the sphere {z : IIzll = r} reduces to the origin under the strict m-dissipativity of Ar. See (Aksikas et al., 2004) for more details.

lim d(x(t, xa), w(xa)) = O.

i-DO

If in addition W > la , then x(t, xa) ~ 0 as t ~ 00 Le. 0 is an asymptotically stable equilibriulll point of (19) on D.

Proof : The idea is to write the operator A as follows: A = Aa + laI + J.Va - laI. By leII1IIla 9 one can prove that A is a m-dissipative (strictly m-dissipative) operator if w ~ la (w > la respectively). On the other hand, by using the n1dissipativity property and using the assumption (A2), one can prove that the operator (I - AA)-l is compact. Then the conclusion follows by Theorem 8. See (Aksikas et al., 2004) for more details.

Now let us consider the following class of selnilinear systerns :

x(t) = Aox(t) + Na(x(t)) { x(O) = Xa E D(A o ) n F '

(19)

where the following assumptions hold : (A 1) the linear operator Aa, defined on its domain D(A a), is the infinitesimal generator of a C asernigroup of bounded linear operators So (t) on a Banach space X such that IISa(t)11 ~ A1 e- wi , for all t ~ 0, for some w > 0 and .!vI ~ 1, i.e the Ca-semigroup So (t) is exponentially stable.

By using a standard argument, viz. the use of an equivalent vector norm on X, see e.g (Pazy, 1983), (Martin, 1976), Theorenl 10 can be extended to the general case, Le. !YI ~ 1. Observe that, if in particular X is a Hilbert space (as in the application considered here, where this extension is needed), it is no longer such a space when equipped with this new norm (however it is still a Banach space).

(A2) there exists a positive constant A such that

(I - ..\Aa)-1 is compact,

(20) Corollary 11. Consider a semi-linear system described by (19) and satisfying conditions (AI), (A2) and (A3). Assume that w > !vIl a. Let f(t) be the nonlinear contraction semigroup on D, generated by the operator A (recall that the operator A and the subset D are given by (21)). Assume that Na(O) = O. Then for any Xa E D,

(A3) JVa is a Lipschitz continuous nonlinear operator defined on a closed subset F of X, with Lipschitz constant la. Let us introduce the following notations:

A := Aa

+ No

x(t, xo)

and D := D(A) = D(A a) n F (21)

:=

r(t)xa

~ 0 as

t

~ 00

i.e. 0 is an asymptotically stable equilibriuln point of (19) on D.

The following preliminary lemrna ((Luo et al., 1999)[Theorem 2.27, p.33], (tvlartin, 1976), (Brezis, 1973)) is needed:

4. PLUG FLOW REACTOR 1\10DEL ST'ABILITY ANALYSIS

Lemma 9. (i) If Aa a linear operator such that Aa generates a linear Co-semigroup of contractions on X, then Aa is m-dissipative. (ii) If 1Va is a Lipschitz operator on D eX, with Lipschitz constant la, then lVa -la I is a dissipative operator. lVloreover, for all l > la, No - II is a strictly dissipative operator. (iii) If Aa is a m-dissipative operator and No is a Lipschitz (strictly) dissipative operator in X, then Aa + No is a (strictly) m-dissipative operator.

In this part we are interested in the study of the asymptotic stability of the equilibrium profile described by equations (15)-(16), for the plug flow reactor model (10)-(14). Recall that this equilibrium is obtained by using the coolant temperature equilibrium profile Tc,e as an open-loop control. Let us consider the state transformation

x(t)

785

:=

O(t) - Be

(22)

laboration agreement between CESArvlE (Universite Catholique de Louvain, Belgium) and LINrvlA (Faculty of Sciences, Universite Chouaib Doukkali, rvlorocco), funded by the State Secretary for Development Cooperation and by the CIUF (Conseil Interuniversitaire de la ComIllunaute Franf;aise, Belgi unl).

Then, with ()c = ()c,e , equation (10) can be rewritten as the following abstract differential equation on the Hilbert space H = £2 (0, 1)2 :

±( t) = Aox(t) + No(x(t)) { x(O) = xa E D(A o) n Fa

(23)

where the operator Aa is given by (11)-(12) and the nonlinear operator No is given on the closed subset Fa := {x EH: x + ()e E F} by

Na(x)

:=

N(x

+ ()e) - N(()e),

REFERENCES

(24)

Aksikas, I., J.J. Winkin and D. Dochain (2004). Asymptotic stability of infinite-dimensional semi-linear systems : application to a nonisothennal reactor, submitted. Brezis, H. (1973). Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Mathematics Studies. North-Holland. Crandall, M.G. and A. Pazy (1969). Semi-groups of nonlinear contractions and dissipative sets. Journal of Functional Analysis 3, 376-418. Dafermos, C.M. and M. Slemrod (1973). Asymptotic behavior of nonlinear contraction semigroups. Journal of Functional Analysis 13, 97-106. Dochain, D. (1994). Contribution to the analysis and control of distributed parameter systems with application to (bio )chemical processes and robotics. These d' Agf(~~gation de l'EnseigneInent Superieur, Universite Catholique de Louvain. Belgium. Laabissi, M., M.E. Achhab, J. Winkin and D. Dochain (2001) . Trajectory analysis of nonisothermal tubular reactor nonlinear models. Systems & Control Letters 42, 169184. Laabissi, M., M.E. Achhab, J. Winkin and D. Dochain (2004). Multiple equilibrium profiles for nonisothermal tubular nonlinear models. Dynamics of Continuous, Discrete and Impulsive Systems, 11, 339-352. Luo, Z., B. Guo and O. Morgiil (1999). Stability and stabilization of infinite dimen.sional systems with applications. Springer Verlag. London. I\t1artin, R.H. (1976). Nonlinear operators and differential equations in Banach spaces. John WHey & Sons. New York. Pazy, A. (1983). Semigroups of linear operators and application to partial differential equations. Applied Mathematical Sciences, 44, Springer Verlag. New York. Smets, 1. Y., D. Dochain and J. F. Van Impe (2002). Optimal temperature control of a steady-state exothermic plug flow reactor. AIehE J 48(2), 279-286. Winkin, J., D. Dochain and P. Ligarius (2000). Dynamical analysis of distributed parameter tubular reactors. A utomatica 36, 349-361.

where the operator N is given by (13). The following theorem is reported: Theorem 12. Consider the system (23)-(24). Assume that the Lipschitz constant la of the operator N a is such that la < f3 . Then the operator A := Aa +Na is the generator of a unique nonlinear contraction semigroup r(t) on Fa . Moreover, for any Xa E Fo ,

x(t, xo)

:=

r(t)xa

~

0 as t

--* 00

Le. 0 is an asymptotically stable equilibrium point of the system (23)-(24) on Fa. Proof : The proof follows from Theorem 10 and is mainly based on the following facts: (a) the operator Aa generates an exponentially stable 0 0 semigroup Sa(t). Moreover IISa(t)11 ::; e-{3t Vt ~ 0, (b) the resolvent operator of Aa is a HilbertSchmidt operator, whence it is compact, (c) the operator N a is a Lipschitz operator on Fa. See (Aksikas et al., 2004).

5. CONCLUDING REMARKS A Lipschitz constant lo of the nonlinear operator No can be computed explicitly in terms of the model parameters which are described under equations (1)-(3). Hence the asymptotic stability condition stated in Theorem 12 can be checked. From a physical point of view, the condition lo < f3 means that the heat transfer coefficient must be large enough in order to dominate a weighted value of the kinetic constant ka exp ( - RT~ax ) . Numerical simulations have been performed for different parameter and initial condition values. These simulations confirm the asymptotic stability result reported here. Acknowledgements: This paper presents research results of the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office. The scientific responsibility rests with its authors. The work of the first named author has been partially carried out within the framework of a col-

786