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Thermodynamic stability of chemical reactors
Chemical Engineering Science, 1973, Vol. 28, pp. 2 195-2203.
Pergamoo Press.
Thermodynamic
Printed in Great Britain
stability of chemical reactors
P. COSTA and C. TREVISSOl Istituto Policattedra di Scienze Chimiche per I’Ingegneria, Universitadi Genva, Genva, Italy (Received
6 June 1972; in revisedform
12 October
1972)
Abstract- Reference is made to the general criteria for thermodynamic stability of stationary states far from equilibrium recently established by Glansdortf and Prigogine, in order to find the global sufficient conditions for the continuous well-stirred tank reactor and for the catalytic particle. When the thermodynamic conditions are obtained in a form which permits their being symmetrically compared with the well-known necessary and sufficient kinetic conditions, it appears that the former are in fact more restrictive, but they may be led back to the kinetic conditions by isolating some particular features of the theory of normal modes.
IN THEIR recent work [ 11 Glansdorff and Prigogine established the thermodynamic stability conditions for stationary states of dissipative structures, even in situations far from equilibrium. These conditions may be applied to a wide variety of physical situations, and in open chemical systems make it possible to predict new spatial structures, some of them of particular interest to biology. We felt it of interest to apply these general criteria in the study of some cases of stability in chemical reactors, that is, of chemical processes in open systems in conditions that are not usually considered in textbooks on classical thermodynamics and irreversible processes [2,3]. For a continuous tank reactor (CSTR) and the catalytic particle in particular, we wanted (1) to confirm and render quantitative the sufficient property of the thermodynamic solution, that is, to see if and to what extent it is more restrictive than the kinetic solution, which is necessary and sutIicient[4]; (2) to show that it is possible to achieve the kinetic conditions using the ‘normal mode’ theory-as is suggested in [l] - but with an independent development which clarifies some features of the theory.
function: for any given physical system of volume V, whether open or closed, this condition assumes the form [ 11
I a*2(gv2) dV L
“T
0.
(1)
In the case of the continuous tank reactor with a perfect mixing, our purpose, as has been said, is to find sufficient conditions of thermodynamic stability, which are consistent with the kinetic conditions for incompressible fluids. As the latter ones usually concern the model of a homogeneous system (that is a point) defined by a unique set of variables pv, pe (or T) for the whole mass, the same supposition will be taken into account in the integration of Eq. (1) and in all successive developments which refer to CSTR. In the absence of velocity fluctuations (6~ = 0) Eq. (1) is simplified to
as2s -= at
I a620&,‘>() v
at
(2)
where the integrand, developed in the indepenpe, p,., has the expression
dent variables GENERAL
FORMULAE
The sufficient conditions for thermodynamic stability, for small perturbations in far from equilibrium systems, are reached by considering the second differential of entropy as a Liapounoff
las2(ps)=*_ 1 Wpe) 2
at
Together 2195
T
at --F”--
a+,
CLY
T
at’
(3)
with Eq. (1) or its simplified form Eq.
P. COSTA
and C. TREVISSOI
(2) we must always consider the local thermodynamic equilibrium equations [ 11 SQZ)
= 62(ps+3pu2)
s 0
+x fj?div [py&) +v8ppTI
(4)
and respectively a2(ps) =
t+(pe)- x
+p,
c 0.
(5)
Y
Since Eqs. (4) and (5) form a fundamental postulate of the thermodynamics of irreversible processes, they also provide the basis for the stability conditions for far from equilibrium states. In Eq. (3) there appear the derivatives in time of the perturbations of internal energy 6(pe) and of the densities of the chemical species 6p,, which must be given explicit form by excess balance equations for energy and mass: if the pressure fluctuations (6P = 0) and the external forces are overlooked, the excess balance equations are deduced as a perturbed form from the well-known conservation equations of energy and species [5]. as (pe) -=-div
[SW+vij(pe)+x
at
Y
with which the thermodynamic stability conditions assume explicit form (2). The simplifying hypothesis contained in Eq. (9), SP = 6v = 0, are consistent with one another only if in every point of the system the specific volume does not depend on the chemical and physical variables for which fluctuations are admitted; if it does, the system cannot be simultaneously isobaric and in a stable hydrodynamic condition (and hence isocoric). In future we shall therefore refer to undilatable fluids and chemical reactions which occur without volume change. THERMODYNAMIC
STABILITY
1 aP(ps) --=--Fidiv 2 at
[sW+va(pe)] +S$[&-div
(7)
P
of which the first is expressed in terms differing slightly from these used in [l], but is more convenient for the developments which follow. Having combined Eqs. (3), (6) and (7) and bearing in mind the molar chemical affinity
where we have introduced advancement l
=-=_CY- CvLl VY
1 awp4 =--Sidiv at
2
[
Y
+ z 6$& P
(10)
the molar degree of PY -
PYO
MYVY*
(11)
-qa(pe),]
+s+i~-(q&-q&,)].
6W+v6(pe)
+ 2 %e,A,)]
(v&)1
Since the variables which appear in Eq. (IO), in agreement with the hypothesis of perfect mixing, do not depend on the space coordinates, Eq. (10) can immediately be integrated to the whole volume V of the reactor: 1 as2s --=-S+[-C&6Ws+qS(pe) 2 at
we obtain
IN A CSTR
(a) Global thermodynamic solution If Eq. (9) is applied to the CSTR, the diffusion fluxes (A, = 0) inside the tank vanish. In the case of a single chemical reaction, Eq. (9) then becomes
8(p,eYA,)]
ash = x v~~M~SO~at div [NW%) +vbl
(9)
Y
(12)
Perturbations in the system may be of two kinds, resulting from perturbations in the environment, 2196
Thermodynamic
stability of chemical reactors
that is, in the feed, or arising spontaneously inside the system as fluctuations of the variables: here we shall consider only the second type. Given therefore 6(pe)f = i%, = 0, Eq. (12) reduces to
It is best explicitly, quired pe bered that
therefore to make Eq. (14) contain first these variables and then the reand E. To this end it must be rememfor the heat flux it is possible to write W,=h,(T,-T)
I am --=-S~[-46W.+4S(pe)] 2 at
+S$m,--q&l
whose differential, stant h,, T,, is
or perturbed
form, for con-
(13) -6W,
which, substituted bility condition
(15)
= h$T.
Next, let us consider the differentials of the affinity and the reaction rate in the variables T, E (for 6P = 0)
SA-
(14Yt
where 8 is the emptying time of the tank. It might be objected that Eq. (14) does not include the stability conditions of irreversible processes which occur on entry to the tank, between the feed and the fluid already present in the reactor. But if, as has been stated, we limit ourselves to considering these fluctuations that may arise in the mass within the tank, where the variables are uniform, we are induced to exclude the mixing of the reactants since it is equivalent to a diffusive process whose driving forces, concentration and temperature gradients, arise immediately above the system in question. In order to reach a direct comparison with the kinetic stability conditions, it is convenient to express Eq. (2) as a quadratic form of the variables pe and E. In order to transform Eq. (14) in this sense, it must also be remembered that normally the chemical affinity and the reaction rate are given as functions of the temperature and the degree of advancement, which at all events are assumed to be subject to fluctuations. tEquation (14) may be reached by introducing the excess balance equations characteristic of the CSTR directly into the integral of Eq. (3). v%$Q V?b_
at
(16)
in Eq. (2), leads to the sta-
= c@w*-qs(pe)
- Vv,M,ih
- q6p,.
S$=gST-s&
0 2
T,P
(17)
; PIE
(18) Finally, if it is borne in mind that
6+=-&T
(19)
in order to pass to the variables pe, E there only remains to extract T from the differential or from the excess balance equation of the internal energy, which, in the hypothesis of undilatable fluid (C, = C,) and a reaction with no volume change (AE = AH), becomes 6(pe) = C,tiT+AH&
(20)
and hence 6T=$-
(6a) (7a)
with aB=-T2
Upon substitution
2197
P
6(pe) +SE. P
(21)
in Eqs. (1 l), 16), (18) and
P. COSTA
and C. TREVISSOI
( 19), it is also
(y+c+-ew,)
SW, =+(pe)
-AH&]
+AHew*
(22)
~ (A Hew, + a,BwT)2
P
da, A S,=&[AHS(pe)-(AH2+a.Cp)Se]
(23)
P
So =
+(pe)
+ (C,w,-AHw~)SE]
(24)
P
1 S,=-&[S(pe)-AHSe]
(25)
(31)
which together form a sujicient condition for thermodynamic stability for small perturbations. On the other hand, the condition of local thermodynamic equilibrium (5), after substitution of Eqs. (8), (1 l), (23) and (25), leads to the quadratic form
P
-&{[“(pe)]“-2AH8(pe)& so that, after combining Eqs. (22)-(25) with Eq. (14) and multiplying by CpT2 > 0, we obtain CI[S(pe)]2+C,S(pe)Se+C3(Se)2
2 0
(26)
where
- Cpea,wT c,
=
f&k -+CP+AH8wT
C, =4AH2+
P +
-=+2Cp+AHew,-cC,ew,
x (C,+AHf%r-CpeW,).
c, a 0;
form (26) is positive
c3 2 0;
CC,
z= c22
(27) semi-
(28)
which leads to the conditions flh u+C,+AHHeoT 4
(SE)~} G 0
2 0
(29)
flh 2 + 2C, + A HtkoT - CpBo, 4 (30)
(32)
which is negative semi-definite if C, 2 0, a 2 0.t But if a must be positive for the local equilibrium, the second members of Eqs. (30) and (31) are also positive, and the set of sufficient conditions for stability (29)-(31) is consistent with the well-known necessary and sufficient conditions of kinetic stability [4] obtained by integrating the two balance equations for mass and energy in transient conditions:
(AH2+a,Cp)
But the quadratic definite if
(AH2+a,C,>
a+cp cfiqz
>
(l-8w,)+AHewr
3 0
(33)
3 0.
(34)
To put it more clearly, a system which satisfies Eqs. (29)-(31) also satisfies Eqs. (33) and (34) and is clearly stable, whereas a system which satisfies Eqs. (33) and (34) and is therefore stable, may not satisfy Eqs. (29)-(31): all this agrees with the merely sufficient condition characteristic of the set of Eqs. (29)-(3 1) as already noted, and the sufficient and necessary condition of the set of Eqs. (33) and (34). Moreover, since the derivative o, is normally negative, if Eq. (29) is satisfied so is certainly Eq. (34); thus with tTo say that for the stability of the local equilibrium it is necessary that (I, > 0 is to extend to any system the results shown in [6] for ideal mixtures.
2198
Thermodynamic stability of chemical reactors
Eqs. (30) and (31) we have obtained symmetrical conditions for Eqs. (33) and (34). How much more restrictive the first two are than the second two depends above all on the value of a,, which varies from case to case according to the well-known De Donder formula
the
conjugate, hold[ 11:
aam at
following
3
0;
inequalities
am2(pz) d 0
must
(36)
where the operator ~3,~ indicates the second differential relative to the pair of normal modes complex conjugates where (37)
(b) Thermodynamic solution and the kinetic theory of the normal modes As was said at the beginning, it is possible to lead Eqs. (29)-(3 1) back to necessary and sufficient stability conditions if certain kinetic concepts typical of the normal modes theory are adopted using special techniques. As is known, it is said that after being perturbed a system evolves according to a normal mode when all the variables evolve in time according to the law
For the ideal continuous tank reactor, in conformity with the simplifying hypothesis already stated and in particular with Eqs. (3) and (5), Eqs. (36) may be made explicit in the following way: 1 a6,2s --=
v
(YJ
6
ia[sol* T at
I
gAa(Sc)*+ T at
2Sm2(ps) =8+[8(pe)]*+i3$(&)*+ +(8+)*G(pe)+(Zi$)*Se
(35) where (Y., is a constant characteristic of the normal mode in question; the number of the possible normal modes corresponds to the number of the independent variables in the system and also to the order of the characteristic kinetic equation which has the oJ’s as roots. The response of a system to an arbitrary perturbation of its independent variables is a combination of all the possible normal modes: hence, the stability of a system after an arbitrary perturbation is the direct consequence of the stability of a system subjected to particular perturbations which induce a normal mode as a response. In kinetic terms, the necessary and sufficient condition for stability states that “the real part of COnStant Of every nOImd mode mUSt be negative”. In thermodynamic terms, this condition is equivalent to imposing that, for every normal mode taken separately together with its complex
at
S 0. (39)
Equations (38) and (39), written for each pair of normal modes complex conjugates accessible to the system, form therefore the necessary and sufficient condition for thermodynamic stability. It is obvious that when considering a normal mode characterized by a real root (Ye, then 6~ = (6~) * and Eqs. (38) and (39) degenerate into (3) and (5) bearing in mind however that the latter are to be referred to a single normal mode. Moreover, if a system characterized by the two independent variables pe, E is constrained in such a way as to exclude any perturbation of either of the two variables, then the response of the system is always a single real normal mode, and Eq. (3) (or Eq. 26, which derives from it) together with Eq. (5) give directly, in these special cases, the necessary and sufficient conditions for stability. If for instance the system is con-
2199
P. COSTA and C. TREVISSOI
strained in such a way as to exclude any perturbation of the variable E(& = 0), from Eq. (26) there immediately results that the necessary and sufficient stability condition is represented only by the first of Eqs. (28), that is, by Eq. (29); this agrees with the kinetic solution since conditions (33) and (34) degenerate into (29) if& = 0 is prescribed. Similarly, if the system is constrained in such a way that it is isothermal (6T = 0), it may be deduced from Eq. (20) that the possible perturbations of the variables pe, E are linked by the relation 6(pe) = AH&
(41)
which, in this case also, coincides with the kinetic condition corresponding to 6T = 0, since a, > 0: l-&0,
3 0.
(42)
As was indicated earlier, the stability condition of the isothermal continuous tank (42) is always verified when, as is usually the case, the reaction rate diminishes with the increase of the degree of advancement (w, < 0); it is well known however that in the case of autocatalytic reactions condition (42) may be violated. There now remains to consider the general case in which the system is not constrained. Let us consider, then, a normal mode in the variables pe, E, characterized by a given value of cu.,: a(pe) =
[~(w)l0f+@,
I% = (se), ems. Differentiating
and, dividing member by member Eqs. (43) and (44) and then (45) and (46)
aG-4
(43)
at
a(~4 -=-=
-aa
&
[~@)I0
at
(47)
@~h.J*
The first of the equalities (47) may also be written
aa(f-4 S(pB)$ = acat
(40)
and hence are no longer independent. Equation (40), substituted in Eqs. (26) and (27), leads to the thermodynamic stability condition a,(1 - ew,> 2 0
(46)
(48)
and we have thus obtained a relation valid for any value of I (and therefore also for t = 0) between perturbations S(pe) and CUE.In particular, Eq. (48) imposes a connection between the initial values [6(pe)],, and (a),, of the perturbations, a connection which is characteristic of the normal mode in question. If now the excess balance equations for mass and energy of the ideal continuous tank (cu. Eqs. 6a and 7a in note) are substituted in Eq. (48), e
Wp4
-=~~W8-i3(pe)
(49)
at
(50) we reach the relation 6(pe)(860-&) which, bearing becomes
= & $tjW,-S(pe) [ in mind Eqs.
I
(5 1)
(22) and (24),
(44)
in time, we have
aabe) -= at
aJIS(pe>loeafi
(45) 2200
-~AH(s~)Z=
0
(52)
Thermodynamic
stability of chemical reactors
from which
ik(&)*[-KK*+(K+K*)AH--Hz--a&,] s 0.
,_W_ &
- 1 9A HwT - 9C,w, 2eoT -!!+/A)
Equation (57), which descends from (39) and is equivalent, for a pair of normal modes, to Eq. (5), is always verified inasmuch as it expresses the postulate of local thermodynamic equilibrium; from the comparison between Eq. (56) and (57) therefore, there results that the necessary and sufficient thermodynamic stability condition reduces to
(53)
may be derived, with eAHwT-ea&,-a
Rh
2
4 >
+4%BAHm 4 Observe also be
(K+K*)ew,+2C,eo,-28AHo,-2C,
(54)
T*
aNp4
SE
[S(pe)lo__- at (6e)o
aa at
-
(55) K+K*
=&-
Rh
eAHq-eCpw,--
4 >
T
Equations (53-55) indicate what the ratio must be between the perturbations 6(pe) and & in order to give rise to a single normal mode. If the perturbations [6(pe)lo and (I%), are arbitrary, in agreement with the role of independent variables proper to pe and E, then the response of the system is a combination of the two possible normal modes; on the other hand, in order that the system should respond according to Eqs. (43) and (44), that is, according to a single normal mode, only one of the variables may be arbitrarily perturbed, while the initial perturbation of the other must satisfy Eq. (55). If we use K, K * to indicate the two constants (53) characteristic of a pair of normal modes complex conjugates and substitute Eqs. (55), together with Eqs. (22)-(25), in the inequalities (3 8) and (39), after a bit of manipulation we reach the necessary and sufficient thermodynamic stability conditions made explicit in the pair of normal modes complex conjugates: 8E(8E)*[(K+K*)eWT+2C,ew,-_eAHOT -2C,][-KK*+(K+K*)AH-AH2-ua,Cp] 20
s 0. (58)
If the discriminant A expressed by Eq. (54) is less than zero, that is, if the two possible normal modes of the system are actually a pair of complex conjugates, then, from Eq. (53),
also that in virtue of Eqs. (47) it must
K-~_
(57)
(59)
and from Eq. (58) it may be deduced that *-t 4
2C, + AH&+-
8C,w, 5 0.
(60)
If instead the two roots of Eq. (52) are real (A 2 0), then K = K* and Eq. (58) reduces to K80T + cpeo, - 8AHOT - c, s 0
(61)
which ensures stability for the two real values of K (53). Upon successively substituting these values (53) of K in (61) we have the two conditions Qh
--!-2C,+8AHWT-eCpW, 4
ah 88
4
+ 2C, + 8A HI+ - eC,o,
2 -dA
(62)
3 d/a
(63)
2 0
(60)
which are equivalent to
(56) 2201
f-lb -+2C,+8AHWT-0CpW( 4
P. COSTA
>
and C. TREVISSOI
2
88 flh + 2C, + 8A HmT- flC,o, Q
3 0.
(64)
Finally, upon substitution of definition (54) of A in Eq. (64), we obtain the well-known stability condition of Van Heerden (Q$+c,)(~--So.)
+~AH~,>
0.
volume change, throughout the volume V of a catalytic particle 6v = 0; consequently Eq. (9) for this system is written
1 aWp4 _ --S+div at
2
+S$Sw+x
(65)
[6W-~6(p,e,A,)] Y [8(pyAy)].
SFdiv
(68)
Y
The necessary and sufficient stability condition for the ideal continuous tank is therefore composed of the combination of Eqs. (65) and (60), or of Eq. (60) alone, according to whether the discriminant (53) is more or less than zero; in this way we have obtained the stability formulae which coincide with the kinetic ones [4]. It should be remembered that the proof that (YJ must be negative in order to have asyntotic stability-which results from the solution of the kinetic method-derives directly from the substitution of Eqs. (43) and (44) in Eqs. (38) and (39). In fact, we obtain
ras,zs = v at
201Jrsm2(ps)
(66)
Upon introducing into Eq. (68) the local expressions of the mass and the heat flux pyAy = - D grad py W=--hgradT
s
AE = AH = x vyMyey
This is also stated in [l]. The development that we have just shown reveals more clearly, through Eq. (47) or Eq. (55), the relation between the properties of the normal modes and the response of the system to constrained or entirely arbitrary initial perturbations. On the other hand, the fact of having obtained stability conditions identical to the kinetic conditions confirms the hypothesis of local thermodynamic equilibrium (39) also as regards the stability of stationary states far from equilibrium.
we
obtain, assuming again that AH is constant,
1 awp4 = G+%T+AHDV%e) at
2
Substituted into Eq. (2), sufficient thermodynamic the catalytic particle; we ient form by multiplying the positive function T2:
I[V
T26 +V2GT +
(72)
Eq. (72) represents a stability condition for reach a more conventhe integrand (72) by
+ AHDV26e)
T%+I-DV~SE
1
dV 2 0
STABlLlTY PARTICLE
OF THE
X (w$T + W& + DV26e) ] dV a 0.
In the hypothesis of constant pressure, of undilatable fluid and reaction taking place without
(73)
or also, bearing in mind Eqs. (17)-( 19), J, [- 62-(h~267-+ AHDv%E) + (AHaT-
THERMODYNAMlC CATALYTIC
(71)
Y
+S+DV%e).
(67)
0.
(70)
with D independent of component y; and bearing in mind the definitions (8) and (11) and
and hence, in order to respect the inequalities (38) and (39), ajar
(6%
u,~E) (741
Since a complete treatment, like that of the continuous tank, would at this point be unneces-
2202
Thermodynamic
stability of chemical reactors
sarily laborious in view of a comparison with the kinetic solutions, we shall only take into consideration some particular perturbations; in other words, we shall impose such constraints on the system as to make it dependent on only one variable instead of on the two original ones (pe, E or T, E). (a) Perturbation with E constant in time. If we impose SE = 0, then Eq. (74) degenerates into
I[ V
-8TXV26T+AHw,(6T)2]
dV 2 0
(75)
which coincides with the kinetic stability condition obtainable from Aris’s equations under the same conditions [7]. (b) Perturbation with T constant in time. In this case ST = 0 and Eq. (74) degenerates into - s v [wC(6e)2+ &DV%~]a, dV 2 0
(76)
which, since a, > 0, coincides again with the corresponding kinetic stability condition.
NOTATION
chemical affinity see Eq. (17) specific heat molar concentration diffusivity in the-pores of the catalytic particle partial derivative perturbation or excess reaction heat with constant volume internal energy per mass unit reaction heat or constant pressure heat exchange coefficient
diffusive flux constants characteristic of the normal modes k coefficient of mass transfer M molecular mass n number of moles volumetric flow z entropy entropy per unit mass ; absolute temperature t time V volume velocity of fluid heat flux w,w: z = s - +r~-W
J K, K+
Greek symbols constant characteristic of a normal mode 2 relative mass flux of the component A discriminant of Eq. (52) degree of advancement ; filling time of tank A thermal conductivity P chemical potential V stoichiometric coefficient P density arbitrary variable of the normal mode heat exchange surface ; 0 reaction rate __
Sz@ixes 0 f c p m y
initial feed external reaction normal mode component
REFERENCES P. and PRIGOGINE I., Thermodynamic Theory of Structure, Stability and Fluctuations”, r11 GLANSDORFF 6,7.Wiley,N.Y. 1971. Thermodynamics. North Holland, Amsterdam 1962. [21 DE GROOT S. R. and MAZUR P., Non-Equilibrium J. and REIK M. G., Handbuch der Physik. Springer, Berlin 1962. [31 MEIXNER p. 173. Prentice Hall, Englewood ClilTs 1965. [41 ARIS R., Introduction to the Analysis ofChemicalReactors, E. N., Transport Phenomena. Wiley, N.Y. 1960. [51 BIRD R. B., STEWART E. W. and LIGHTFOOT J. and DEFAY R., Thermodynamique Chimique. Desoer, Liege 1950. [61 PRIGOGINE t71 ARIS R., Chem. Engng Sci. 1969 24 149.
2203
Chapts.
Pergamoo Press.
Thermodynamic
Printed in Great Britain
stability of chemical reactors
P. COSTA and C. TREVISSOl Istituto Policattedra di Scienze Chimiche per I’Ingegneria, Universitadi Genva, Genva, Italy (Received
6 June 1972; in revisedform
12 October
1972)
Abstract- Reference is made to the general criteria for thermodynamic stability of stationary states far from equilibrium recently established by Glansdortf and Prigogine, in order to find the global sufficient conditions for the continuous well-stirred tank reactor and for the catalytic particle. When the thermodynamic conditions are obtained in a form which permits their being symmetrically compared with the well-known necessary and sufficient kinetic conditions, it appears that the former are in fact more restrictive, but they may be led back to the kinetic conditions by isolating some particular features of the theory of normal modes.
IN THEIR recent work [ 11 Glansdorff and Prigogine established the thermodynamic stability conditions for stationary states of dissipative structures, even in situations far from equilibrium. These conditions may be applied to a wide variety of physical situations, and in open chemical systems make it possible to predict new spatial structures, some of them of particular interest to biology. We felt it of interest to apply these general criteria in the study of some cases of stability in chemical reactors, that is, of chemical processes in open systems in conditions that are not usually considered in textbooks on classical thermodynamics and irreversible processes [2,3]. For a continuous tank reactor (CSTR) and the catalytic particle in particular, we wanted (1) to confirm and render quantitative the sufficient property of the thermodynamic solution, that is, to see if and to what extent it is more restrictive than the kinetic solution, which is necessary and sutIicient[4]; (2) to show that it is possible to achieve the kinetic conditions using the ‘normal mode’ theory-as is suggested in [l] - but with an independent development which clarifies some features of the theory.
function: for any given physical system of volume V, whether open or closed, this condition assumes the form [ 11
I a*2(gv2) dV L
“T
0.
(1)
In the case of the continuous tank reactor with a perfect mixing, our purpose, as has been said, is to find sufficient conditions of thermodynamic stability, which are consistent with the kinetic conditions for incompressible fluids. As the latter ones usually concern the model of a homogeneous system (that is a point) defined by a unique set of variables pv, pe (or T) for the whole mass, the same supposition will be taken into account in the integration of Eq. (1) and in all successive developments which refer to CSTR. In the absence of velocity fluctuations (6~ = 0) Eq. (1) is simplified to
as2s -= at
I a620&,‘>() v
at
(2)
where the integrand, developed in the indepenpe, p,., has the expression
dent variables GENERAL
FORMULAE
The sufficient conditions for thermodynamic stability, for small perturbations in far from equilibrium systems, are reached by considering the second differential of entropy as a Liapounoff
las2(ps)=*_ 1 Wpe) 2
at
Together 2195
T
at --F”--
a+,
CLY
T
at’
(3)
with Eq. (1) or its simplified form Eq.
P. COSTA
and C. TREVISSOI
(2) we must always consider the local thermodynamic equilibrium equations [ 11 SQZ)
= 62(ps+3pu2)
s 0
+x fj?div [py&) +v8ppTI
(4)
and respectively a2(ps) =
t+(pe)- x
+p,
c 0.
(5)
Y
Since Eqs. (4) and (5) form a fundamental postulate of the thermodynamics of irreversible processes, they also provide the basis for the stability conditions for far from equilibrium states. In Eq. (3) there appear the derivatives in time of the perturbations of internal energy 6(pe) and of the densities of the chemical species 6p,, which must be given explicit form by excess balance equations for energy and mass: if the pressure fluctuations (6P = 0) and the external forces are overlooked, the excess balance equations are deduced as a perturbed form from the well-known conservation equations of energy and species [5]. as (pe) -=-div
[SW+vij(pe)+x
at
Y
with which the thermodynamic stability conditions assume explicit form (2). The simplifying hypothesis contained in Eq. (9), SP = 6v = 0, are consistent with one another only if in every point of the system the specific volume does not depend on the chemical and physical variables for which fluctuations are admitted; if it does, the system cannot be simultaneously isobaric and in a stable hydrodynamic condition (and hence isocoric). In future we shall therefore refer to undilatable fluids and chemical reactions which occur without volume change. THERMODYNAMIC
STABILITY
1 aP(ps) --=--Fidiv 2 at
[sW+va(pe)] +S$[&-div
(7)
P
of which the first is expressed in terms differing slightly from these used in [l], but is more convenient for the developments which follow. Having combined Eqs. (3), (6) and (7) and bearing in mind the molar chemical affinity
where we have introduced advancement l
=-=_CY- CvLl VY
1 awp4 =--Sidiv at
2
[
Y
+ z 6$& P
(10)
the molar degree of PY -
PYO
MYVY*
(11)
-qa(pe),]
+s+i~-(q&-q&,)].
6W+v6(pe)
+ 2 %e,A,)]
(v&)1
Since the variables which appear in Eq. (IO), in agreement with the hypothesis of perfect mixing, do not depend on the space coordinates, Eq. (10) can immediately be integrated to the whole volume V of the reactor: 1 as2s --=-S+[-C&6Ws+qS(pe) 2 at
we obtain
IN A CSTR
(a) Global thermodynamic solution If Eq. (9) is applied to the CSTR, the diffusion fluxes (A, = 0) inside the tank vanish. In the case of a single chemical reaction, Eq. (9) then becomes
8(p,eYA,)]
ash = x v~~M~SO~at div [NW%) +vbl
(9)
Y
(12)
Perturbations in the system may be of two kinds, resulting from perturbations in the environment, 2196
Thermodynamic
stability of chemical reactors
that is, in the feed, or arising spontaneously inside the system as fluctuations of the variables: here we shall consider only the second type. Given therefore 6(pe)f = i%, = 0, Eq. (12) reduces to
It is best explicitly, quired pe bered that
therefore to make Eq. (14) contain first these variables and then the reand E. To this end it must be rememfor the heat flux it is possible to write W,=h,(T,-T)
I am --=-S~[-46W.+4S(pe)] 2 at
+S$m,--q&l
whose differential, stant h,, T,, is
or perturbed
form, for con-
(13) -6W,
which, substituted bility condition
(15)
= h$T.
Next, let us consider the differentials of the affinity and the reaction rate in the variables T, E (for 6P = 0)
SA-
(14Yt
where 8 is the emptying time of the tank. It might be objected that Eq. (14) does not include the stability conditions of irreversible processes which occur on entry to the tank, between the feed and the fluid already present in the reactor. But if, as has been stated, we limit ourselves to considering these fluctuations that may arise in the mass within the tank, where the variables are uniform, we are induced to exclude the mixing of the reactants since it is equivalent to a diffusive process whose driving forces, concentration and temperature gradients, arise immediately above the system in question. In order to reach a direct comparison with the kinetic stability conditions, it is convenient to express Eq. (2) as a quadratic form of the variables pe and E. In order to transform Eq. (14) in this sense, it must also be remembered that normally the chemical affinity and the reaction rate are given as functions of the temperature and the degree of advancement, which at all events are assumed to be subject to fluctuations. tEquation (14) may be reached by introducing the excess balance equations characteristic of the CSTR directly into the integral of Eq. (3). v%$Q V?b_
at
(16)
in Eq. (2), leads to the sta-
= c@w*-qs(pe)
- Vv,M,ih
- q6p,.
S$=gST-s&
0 2
T,P
(17)
; PIE
(18) Finally, if it is borne in mind that
6+=-&T
(19)
in order to pass to the variables pe, E there only remains to extract T from the differential or from the excess balance equation of the internal energy, which, in the hypothesis of undilatable fluid (C, = C,) and a reaction with no volume change (AE = AH), becomes 6(pe) = C,tiT+AH&
(20)
and hence 6T=$-
(6a) (7a)
with aB=-T2
Upon substitution
2197
P
6(pe) +SE. P
(21)
in Eqs. (1 l), 16), (18) and
P. COSTA
and C. TREVISSOI
( 19), it is also
(y+c+-ew,)
SW, =+(pe)
-AH&]
+AHew*
(22)
~ (A Hew, + a,BwT)2
P
da, A S,=&[AHS(pe)-(AH2+a.Cp)Se]
(23)
P
So =
+(pe)
+ (C,w,-AHw~)SE]
(24)
P
1 S,=-&[S(pe)-AHSe]
(25)
(31)
which together form a sujicient condition for thermodynamic stability for small perturbations. On the other hand, the condition of local thermodynamic equilibrium (5), after substitution of Eqs. (8), (1 l), (23) and (25), leads to the quadratic form
P
-&{[“(pe)]“-2AH8(pe)& so that, after combining Eqs. (22)-(25) with Eq. (14) and multiplying by CpT2 > 0, we obtain CI[S(pe)]2+C,S(pe)Se+C3(Se)2
2 0
(26)
where
- Cpea,wT c,
=
f&k -+CP+AH8wT
C, =4AH2+
P +
-=+2Cp+AHew,-cC,ew,
x (C,+AHf%r-CpeW,).
c, a 0;
form (26) is positive
c3 2 0;
CC,
z= c22
(27) semi-
(28)
which leads to the conditions flh u+C,+AHHeoT 4
(SE)~} G 0
2 0
(29)
flh 2 + 2C, + A HtkoT - CpBo, 4 (30)
(32)
which is negative semi-definite if C, 2 0, a 2 0.t But if a must be positive for the local equilibrium, the second members of Eqs. (30) and (31) are also positive, and the set of sufficient conditions for stability (29)-(31) is consistent with the well-known necessary and sufficient conditions of kinetic stability [4] obtained by integrating the two balance equations for mass and energy in transient conditions:
(AH2+a,Cp)
But the quadratic definite if
(AH2+a,C,>
a+cp cfiqz
>
(l-8w,)+AHewr
3 0
(33)
3 0.
(34)
To put it more clearly, a system which satisfies Eqs. (29)-(31) also satisfies Eqs. (33) and (34) and is clearly stable, whereas a system which satisfies Eqs. (33) and (34) and is therefore stable, may not satisfy Eqs. (29)-(31): all this agrees with the merely sufficient condition characteristic of the set of Eqs. (29)-(3 1) as already noted, and the sufficient and necessary condition of the set of Eqs. (33) and (34). Moreover, since the derivative o, is normally negative, if Eq. (29) is satisfied so is certainly Eq. (34); thus with tTo say that for the stability of the local equilibrium it is necessary that (I, > 0 is to extend to any system the results shown in [6] for ideal mixtures.
2198
Thermodynamic stability of chemical reactors
Eqs. (30) and (31) we have obtained symmetrical conditions for Eqs. (33) and (34). How much more restrictive the first two are than the second two depends above all on the value of a,, which varies from case to case according to the well-known De Donder formula
the
conjugate, hold[ 11:
aam at
following
3
0;
inequalities
am2(pz) d 0
must
(36)
where the operator ~3,~ indicates the second differential relative to the pair of normal modes complex conjugates where (37)
(b) Thermodynamic solution and the kinetic theory of the normal modes As was said at the beginning, it is possible to lead Eqs. (29)-(3 1) back to necessary and sufficient stability conditions if certain kinetic concepts typical of the normal modes theory are adopted using special techniques. As is known, it is said that after being perturbed a system evolves according to a normal mode when all the variables evolve in time according to the law
For the ideal continuous tank reactor, in conformity with the simplifying hypothesis already stated and in particular with Eqs. (3) and (5), Eqs. (36) may be made explicit in the following way: 1 a6,2s --=
v
(YJ
6
ia[sol* T at
I
gAa(Sc)*+ T at
2Sm2(ps) =8+[8(pe)]*+i3$(&)*+ +(8+)*G(pe)+(Zi$)*Se
(35) where (Y., is a constant characteristic of the normal mode in question; the number of the possible normal modes corresponds to the number of the independent variables in the system and also to the order of the characteristic kinetic equation which has the oJ’s as roots. The response of a system to an arbitrary perturbation of its independent variables is a combination of all the possible normal modes: hence, the stability of a system after an arbitrary perturbation is the direct consequence of the stability of a system subjected to particular perturbations which induce a normal mode as a response. In kinetic terms, the necessary and sufficient condition for stability states that “the real part of COnStant Of every nOImd mode mUSt be negative”. In thermodynamic terms, this condition is equivalent to imposing that, for every normal mode taken separately together with its complex
at
S 0. (39)
Equations (38) and (39), written for each pair of normal modes complex conjugates accessible to the system, form therefore the necessary and sufficient condition for thermodynamic stability. It is obvious that when considering a normal mode characterized by a real root (Ye, then 6~ = (6~) * and Eqs. (38) and (39) degenerate into (3) and (5) bearing in mind however that the latter are to be referred to a single normal mode. Moreover, if a system characterized by the two independent variables pe, E is constrained in such a way as to exclude any perturbation of either of the two variables, then the response of the system is always a single real normal mode, and Eq. (3) (or Eq. 26, which derives from it) together with Eq. (5) give directly, in these special cases, the necessary and sufficient conditions for stability. If for instance the system is con-
2199
P. COSTA and C. TREVISSOI
strained in such a way as to exclude any perturbation of the variable E(& = 0), from Eq. (26) there immediately results that the necessary and sufficient stability condition is represented only by the first of Eqs. (28), that is, by Eq. (29); this agrees with the kinetic solution since conditions (33) and (34) degenerate into (29) if& = 0 is prescribed. Similarly, if the system is constrained in such a way that it is isothermal (6T = 0), it may be deduced from Eq. (20) that the possible perturbations of the variables pe, E are linked by the relation 6(pe) = AH&
(41)
which, in this case also, coincides with the kinetic condition corresponding to 6T = 0, since a, > 0: l-&0,
3 0.
(42)
As was indicated earlier, the stability condition of the isothermal continuous tank (42) is always verified when, as is usually the case, the reaction rate diminishes with the increase of the degree of advancement (w, < 0); it is well known however that in the case of autocatalytic reactions condition (42) may be violated. There now remains to consider the general case in which the system is not constrained. Let us consider, then, a normal mode in the variables pe, E, characterized by a given value of cu.,: a(pe) =
[~(w)l0f+@,
I% = (se), ems. Differentiating
and, dividing member by member Eqs. (43) and (44) and then (45) and (46)
aG-4
(43)
at
a(~4 -=-=
-aa
&
[~@)I0
at
(47)
@~h.J*
The first of the equalities (47) may also be written
aa(f-4 S(pB)$ = acat
(40)
and hence are no longer independent. Equation (40), substituted in Eqs. (26) and (27), leads to the thermodynamic stability condition a,(1 - ew,> 2 0
(46)
(48)
and we have thus obtained a relation valid for any value of I (and therefore also for t = 0) between perturbations S(pe) and CUE.In particular, Eq. (48) imposes a connection between the initial values [6(pe)],, and (a),, of the perturbations, a connection which is characteristic of the normal mode in question. If now the excess balance equations for mass and energy of the ideal continuous tank (cu. Eqs. 6a and 7a in note) are substituted in Eq. (48), e
Wp4
-=~~W8-i3(pe)
(49)
at
(50) we reach the relation 6(pe)(860-&) which, bearing becomes
= & $tjW,-S(pe) [ in mind Eqs.
I
(5 1)
(22) and (24),
(44)
in time, we have
aabe) -= at
aJIS(pe>loeafi
(45) 2200
-~AH(s~)Z=
0
(52)
Thermodynamic
stability of chemical reactors
from which
ik(&)*[-KK*+(K+K*)AH--Hz--a&,] s 0.
,_W_ &
- 1 9A HwT - 9C,w, 2eoT -!!+/A)
Equation (57), which descends from (39) and is equivalent, for a pair of normal modes, to Eq. (5), is always verified inasmuch as it expresses the postulate of local thermodynamic equilibrium; from the comparison between Eq. (56) and (57) therefore, there results that the necessary and sufficient thermodynamic stability condition reduces to
(53)
may be derived, with eAHwT-ea&,-a
Rh
2
4 >
+4%BAHm 4 Observe also be
(K+K*)ew,+2C,eo,-28AHo,-2C,
(54)
T*
aNp4
SE
[S(pe)lo__- at (6e)o
aa at
-
(55) K+K*
=&-
Rh
eAHq-eCpw,--
4 >
T
Equations (53-55) indicate what the ratio must be between the perturbations 6(pe) and & in order to give rise to a single normal mode. If the perturbations [6(pe)lo and (I%), are arbitrary, in agreement with the role of independent variables proper to pe and E, then the response of the system is a combination of the two possible normal modes; on the other hand, in order that the system should respond according to Eqs. (43) and (44), that is, according to a single normal mode, only one of the variables may be arbitrarily perturbed, while the initial perturbation of the other must satisfy Eq. (55). If we use K, K * to indicate the two constants (53) characteristic of a pair of normal modes complex conjugates and substitute Eqs. (55), together with Eqs. (22)-(25), in the inequalities (3 8) and (39), after a bit of manipulation we reach the necessary and sufficient thermodynamic stability conditions made explicit in the pair of normal modes complex conjugates: 8E(8E)*[(K+K*)eWT+2C,ew,-_eAHOT -2C,][-KK*+(K+K*)AH-AH2-ua,Cp] 20
s 0. (58)
If the discriminant A expressed by Eq. (54) is less than zero, that is, if the two possible normal modes of the system are actually a pair of complex conjugates, then, from Eq. (53),
also that in virtue of Eqs. (47) it must
K-~_
(57)
(59)
and from Eq. (58) it may be deduced that *-t 4
2C, + AH&+-
8C,w, 5 0.
(60)
If instead the two roots of Eq. (52) are real (A 2 0), then K = K* and Eq. (58) reduces to K80T + cpeo, - 8AHOT - c, s 0
(61)
which ensures stability for the two real values of K (53). Upon successively substituting these values (53) of K in (61) we have the two conditions Qh
--!-2C,+8AHWT-eCpW, 4
ah 88
4
+ 2C, + 8A HI+ - eC,o,
2 -dA
(62)
3 d/a
(63)
2 0
(60)
which are equivalent to
(56) 2201
f-lb -+2C,+8AHWT-0CpW( 4
P. COSTA
>
and C. TREVISSOI
2
88 flh + 2C, + 8A HmT- flC,o, Q
3 0.
(64)
Finally, upon substitution of definition (54) of A in Eq. (64), we obtain the well-known stability condition of Van Heerden (Q$+c,)(~--So.)
+~AH~,>
0.
volume change, throughout the volume V of a catalytic particle 6v = 0; consequently Eq. (9) for this system is written
1 aWp4 _ --S+div at
2
+S$Sw+x
(65)
[6W-~6(p,e,A,)] Y [8(pyAy)].
SFdiv
(68)
Y
The necessary and sufficient stability condition for the ideal continuous tank is therefore composed of the combination of Eqs. (65) and (60), or of Eq. (60) alone, according to whether the discriminant (53) is more or less than zero; in this way we have obtained the stability formulae which coincide with the kinetic ones [4]. It should be remembered that the proof that (YJ must be negative in order to have asyntotic stability-which results from the solution of the kinetic method-derives directly from the substitution of Eqs. (43) and (44) in Eqs. (38) and (39). In fact, we obtain
ras,zs = v at
201Jrsm2(ps)
(66)
Upon introducing into Eq. (68) the local expressions of the mass and the heat flux pyAy = - D grad py W=--hgradT
s
AE = AH = x vyMyey
This is also stated in [l]. The development that we have just shown reveals more clearly, through Eq. (47) or Eq. (55), the relation between the properties of the normal modes and the response of the system to constrained or entirely arbitrary initial perturbations. On the other hand, the fact of having obtained stability conditions identical to the kinetic conditions confirms the hypothesis of local thermodynamic equilibrium (39) also as regards the stability of stationary states far from equilibrium.
we
obtain, assuming again that AH is constant,
1 awp4 = G+%T+AHDV%e) at
2
Substituted into Eq. (2), sufficient thermodynamic the catalytic particle; we ient form by multiplying the positive function T2:
I[V
T26 +V2GT +
(72)
Eq. (72) represents a stability condition for reach a more conventhe integrand (72) by
+ AHDV26e)
T%+I-DV~SE
1
dV 2 0
STABlLlTY PARTICLE
OF THE
X (w$T + W& + DV26e) ] dV a 0.
In the hypothesis of constant pressure, of undilatable fluid and reaction taking place without
(73)
or also, bearing in mind Eqs. (17)-( 19), J, [- 62-(h~267-+ AHDv%E) + (AHaT-
THERMODYNAMlC CATALYTIC
(71)
Y
+S+DV%e).
(67)
0.
(70)
with D independent of component y; and bearing in mind the definitions (8) and (11) and
and hence, in order to respect the inequalities (38) and (39), ajar
(6%
u,~E) (741
Since a complete treatment, like that of the continuous tank, would at this point be unneces-
2202
Thermodynamic
stability of chemical reactors
sarily laborious in view of a comparison with the kinetic solutions, we shall only take into consideration some particular perturbations; in other words, we shall impose such constraints on the system as to make it dependent on only one variable instead of on the two original ones (pe, E or T, E). (a) Perturbation with E constant in time. If we impose SE = 0, then Eq. (74) degenerates into
I[ V
-8TXV26T+AHw,(6T)2]
dV 2 0
(75)
which coincides with the kinetic stability condition obtainable from Aris’s equations under the same conditions [7]. (b) Perturbation with T constant in time. In this case ST = 0 and Eq. (74) degenerates into - s v [wC(6e)2+ &DV%~]a, dV 2 0
(76)
which, since a, > 0, coincides again with the corresponding kinetic stability condition.
NOTATION
chemical affinity see Eq. (17) specific heat molar concentration diffusivity in the-pores of the catalytic particle partial derivative perturbation or excess reaction heat with constant volume internal energy per mass unit reaction heat or constant pressure heat exchange coefficient
diffusive flux constants characteristic of the normal modes k coefficient of mass transfer M molecular mass n number of moles volumetric flow z entropy entropy per unit mass ; absolute temperature t time V volume velocity of fluid heat flux w,w: z = s - +r~-W
J K, K+
Greek symbols constant characteristic of a normal mode 2 relative mass flux of the component A discriminant of Eq. (52) degree of advancement ; filling time of tank A thermal conductivity P chemical potential V stoichiometric coefficient P density arbitrary variable of the normal mode heat exchange surface ; 0 reaction rate __
Sz@ixes 0 f c p m y
initial feed external reaction normal mode component
REFERENCES P. and PRIGOGINE I., Thermodynamic Theory of Structure, Stability and Fluctuations”, r11 GLANSDORFF 6,7.Wiley,N.Y. 1971. Thermodynamics. North Holland, Amsterdam 1962. [21 DE GROOT S. R. and MAZUR P., Non-Equilibrium J. and REIK M. G., Handbuch der Physik. Springer, Berlin 1962. [31 MEIXNER p. 173. Prentice Hall, Englewood ClilTs 1965. [41 ARIS R., Introduction to the Analysis ofChemicalReactors, E. N., Transport Phenomena. Wiley, N.Y. 1960. [51 BIRD R. B., STEWART E. W. and LIGHTFOOT J. and DEFAY R., Thermodynamique Chimique. Desoer, Liege 1950. [61 PRIGOGINE t71 ARIS R., Chem. Engng Sci. 1969 24 149.
2203
Chapts.