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Micromixing and multiple steady state transitions in a CSTR
Chemical Engincermg Scrence. Vol. 41. No. Printed m Great Britain.
12. p,p_ 3 I I1 -31 17. 19x6
MICROMIXING
0009-2509:86 Pergamo”
AND MULTIPLE STEADY TRANSITIONS IN A CSTR A. PUHL
$3.00 + 0.00 Journals Ltd.
STATE
and G. NICOLIS
Facultk des Sciences Universiti Libre de Bruxelles,Campus Plaine, CP. 231, 1050 Bruxelles,Belgium (Accepted
for
publicution
9 April
1986)
Abstract-Using the micromixing concept introduced by Zwietering we describe the effect of incomplete stirring on consecutive reactions exhibiting bistability transitions in a CSTR. In particular we explain the difference in the behaviour of the system when reactant feed streams are premixed or non-premixed. A perturbation expansion around the fast stirring limit provides analytical results.
concentrations a and /3:
1. INTRODUCTION
Experiments[l+ on bistable chemical reactions in a continuous flow stirred tank reactor (CSTR) have demonstrated some unexpected effects when mixing is incomplete. Two major observations have been made: on the one hand, the decrease of the stirring rate significantly reduces the region of bistability; and on the other hand, this behaviour critically depends on whether the two feedstreams are premixed (PM) or non-premixed (NPM) prior to injection. In particular, under PM-conditions the reduction of the bistability region is much less pronounced than for the NPM mode. These results can be understood qualitatively in terms of the competition between turbulent mixing and inhomogeneities due to reactor feeding. There have been several theoretical approaches to modeling the effects of incomplete stirring on bistability transitions [4-81. However, neither of them has so far explained the differences in the stirring effects when premixed or non-premixed reactant feedstreams are used. The investigation of this problem is the main objective of the present work. Our starting point is the micromixing concept introduced by Zwietering [9] to represent the dilution of the incoming streams of fluid by the surrounding bulk fluid. The stirring efficiency is described by a (micro-)mixing time t, which is related to the turbulent mixing process. This model, discussed in Section 2, is valid for small mixing times, i.e. for the fast stirring limit only. Section 3 is devoted to the analytic treatment of the resulting equations by perturbation theory. In Section 4 the general theory is applied to a model of consecutive reactions involving one autocatalytic step.
2. THE
da -= dt
c(o--a ---+R,(a,B) ttot
dfi -= Ir UC
Bo-B 7 %lf
+
R2
(2.1) (a,
Lo.
Here
CQ,, PO are the inlet concentrations entering the CSTR and RI, R2 are the nonlinear chemical rates
containing characteristic chemical time scales. r&’ is the total inverse residence time, i.e. t,,’ is the sum of the inverse residence times t; ’ and tIj 1 for each input species respectively. The restriction to two variables is adopted only for notational convenience. The analysis can be extended in a straightforward way to multivariable systems. However, it is well known that even two-variable systems of the form (2.1) give rise to a variety of dynamical phenomena such as multiple steady states and periodic oscillations [l&12]. The distinction between the premixed (PM) and the non-premixed (NPM) feeding mode (see Figs 1 and 2)
feed
I
feed II
Fig. 1. SchematicCSTR with reactantspremixed prior to injection (PM mode).
MODEL
Consider an isothermal nonlinear chemical reaction system with two variables operating under CSTRconditions. In the limit of perfect mixing, the system is spatially homogeneous and is described by the following mass-balance equations, written for dimensionless
Fig. 2. Schematic CSTR with reactants non-premixed prior to injection (NPM mode).
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requires different definitions of feeding concentrations. Defining a”, /Soas the inlet concentrations in the PM mode, those in the NPM mode have to be of the form
t1 a, - ta”, _‘
j?;
$_p,
=
tot
l/3
(> &
,
(2.2)
where L is a macroscopic length scale of the turbulent eddies and can be associated with the maximum value of the Kolmogorov energy scale E(k). The energy dissipation rate is proportional to the rate of energy injection by the stirrer [lS]: E - w3/i=
a-a
da -=
~
drl
tm
+ RI (0. b) (2.5)
db -‘‘+R dv tm
v/t,,
LZ
tions, a, 6 are given by (see Appendix):
-=-
tot
if eqs (2.1) are to be representative for both feeding conditions. One way to incorporate inhomogenous perturbations due to incomplete mixing would be to resort to the full set of reactiondiffusion equations including the effect of turbulent diffusion arising from stirring [6, 71. However we consider here an empirical model suggested by Zwietering [9]_ Since we limit ourselves to the range of quasi-ideal mixing, we shall use his “simple model” without considering backmixing. In this case the assumption of isotropic turbulence as a requirement of perfect macromixing is justified. In the model, one considers lumps of fluid injected into the tank. For NPM conditions (see Fig. 2) two separated lumps of fluid enter the tank with initial feeding concentrations ub, & and initial feeding rates 4,o = dJ,I,O= V/t,,, respectively, meaning that a volume +1(11),0dt is injected during a time interval [to, t,+dt]. In the PM mode, only one lump is injected with the feeding rate I#,, = #,,a + &r,c and the initial concentrations tie, & _ The injection process introduces on the one hand inhomogeneities into the tank. On the other hand, the mixing process disperses the entering lumps to a molecular scale by a combined mechanism of turbulent dispersion and molecular diffusion. We shall model the competing processes as a dilution of the entering lumps within the bulk, which itself is assumed to remain homogenous. To describe the dilution process dynamically, we introduce a characteristic small scale time n having the same order of magnitude as the characteristic mixing time t,,,. We may regard q as the “age” of a lump. The parameter t, defines the effectiveness of the mixing process and is related to turbulence theory by the relation [13, 147 tm--
G. NICOLIS
(2.3)
where ;i is a stirrer dimension and w is the stirring frequency. Following Zwietering [9]. the volume growth of an entering lump is dv = +(rl) = 40 exp (q/L) dq and the evolution equations for the lump concentra-
2
(a > b)
where a and fi represent the concentrations of the surrounding bulk material. Although a and /? vary over the reaction time scale, they are to be regarded as constant over the time scale q of the lump evolution. We introduce a new time scale: cp = )llL,
czp =
O(l)1
P-6)
and transform (2.5) into da -=a---a+$R1(a,b)
dv
*=
dv
ch
(2.7)
/T-b+$R,(a,b) ‘h
where we have multiplied RI and R2 by the characteristic chemical time, yielding the dimensionless rates a, and a,. At this stage it is important to note that our considerations require the following hierarchy of characteristic times: rln/r,, 6 1 -= t,(,,,l&z. The first inequality is determined by the range of validity of our model, since we operate in the range of quasi-ideal mixing [9]. The second inequality is necessary to guarantee that a sufficient number of reactions takes place before the molecules leave the tank again. The aggregate volume of all lumps younger than rp, is governed by the relation
49) = L
s,,y
= V+
4(t) d5
tot
[exp (9) - 1]
(2.8)
and is assumed to remain, for all time intervals considered, much smaller than the total tank volume V. This means that lumps will not be allowed to grow ad infinitum. We therefore have to define a “truncation age” 9=p beyond which the lumps will be regarded as belonging to the bulk material a and #I.p is, however, not a critical parameter and simply amplifies or weakens the effect of incomplete mixing without changing qualitatively the results. The average concentrations, which are measured, are the bulk concentrations a and 8. In the framework of the model a and B are regarded as homogenous. We are, therefore, not able to explicitly deal with concentration fluctuations due to turbulence. Nevertheless, a
Micromixing and multiplesteady state transitions and /? incorporate the effect of incomplete mixing through the following corrected bulk material balance equations (see Appendix). For premixed feedstreams (PM) (see Fig. 1) we have: da
exp(14Ca(a, A PL)cto, - tmCexp(P) expti.4 Cb(4 A P) t tot - tmCev (~1~
dt= dB dr=
41
-+ Rl
11 PI 11
following perturbation Ansatz:
(I)= (;:)+g)+ ...
+ R2
a(cp = 0) = aO, b(cp = 0) = #So.
The rates i?, (u, b) and &(a, b) are also expanded around a0 and b,. Considering further for the PM mode the initial conditions
of
and remembering that a and /I do not change over the time scale cp,eqs (2.7) are transformed into a system of linear differential equations: d u0 + a0 = a a77;
(2.10)
d _ dVbo+bo=p
a(a, /I, p) and b(a, /I, p) are, in general, nonlinear functions of a and /L Equations (2.7) together with eqs (2.9) form a set of non-linear equations which can be solved numerically by iteration. A perturbation analysis will be given in the next section. For two separated feedstreams (NPM) (see Fig. 2) the corrected equations for the bulk material yield the form:
tto,
dt= dB
-
t,
(3.2)
d -a’ + u1 = R, (a’, b”) dp d 6’ + b’ = R,(a”, b’) drp
1
exp (~1
dt=
(3.1)
a(cp = 0) = ao, b(cp = 0) = fro
Here a(a, /I, p) and b(a, p, ,u) are the solutions eqs (2.7) at cp = ~1and for the initial conditions
da
3113
+ RI (a, 8)
Cexp(14 - 11
(2.11)
1+&(a,S)
~b,(a,p,p)+~b,,(a,p,~)-P
t tot - t, Cexp (14 - 1I
a, (a, p,p) and b, (a, j?, p) denote the lump concentrations of feed stream I and are evaluated by the differential equations
with the initial values a”(cp=O)=ao b’(rp=O)=So
(2.12)
with the initial conditions a,(cp = 0) = a&, b,(cp = 0) = 0.
(2.13)
Analogously, the lump concentrations a,, (a, p, ,D)and b,,(a, fi, p) of feedstream II are found by solving da 2
drp
= a -a,,
+ ”
db I1 = B-bn+dq
ch
wt (a,,, b,,)
L &(a,,, rch
(2.14) b,,)
The solutions of eqs (3.2) have the general form a”(a,q)=a+(aO--a)exp(--q) b”(BT cp) = B + (PO - PI exp Ca’(a,
B, cp) = exp(-v) x
with the initial conditions
EXPANSION STlRRING
NEAR
THE
41112-J
‘p RI Ca’(a,
(5)
0, b”(P,
511
(3.4)
d<
s0 ew
rp)
’ & Ca”(a, 8, b’(P,
<)I
dC.
IDEAL
LIMIT
Since we are only interested in solutions near the idea1 stirring limit (t,/tch Q l), we shall analytically seek approximate solutions of eqs (2.7) using the CES
(0
(2.15) x
3. PERTURBATION
exp
s0
b’(a,B,q)=exp(--)
a,,(~ = 0) = 0, b,,(cp = 0) = &.
(3.3)
u’(cp = 0) = 0 b’(,p=O)=O.
Inserting the expansion (3.1) into eqs (2.9) we obtain the bulk mass balance equations for PM conditions taking into account the effect of micromixing to first
A.
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PUHL
G. NICOLIS
AND
order in the mixing time: dcc dt=
a,-a+exp(~)~a’(a,B,Il)+O(t:lr~~)
t tOf- tlkc=b
+ RI (a, B)
(14 - 13
(3.5) dp -= dt
/%I - P + exp (p) $
b’ (a, 8. &+0(0r$)
+ Rz (a. P).
t fOt- t, CexpCd - 17
Correspondingly, we find for the first order bulk mass balance equations in NPM mode:
da dt
a,-a+exp(p)+
d+O(G/GJ
1
rh
-=
t tof - tm Cew (~0 - 11
+ RI (a. PI (3.6)
dr
2
t &,f- tm Ccxp (p) - 11
+ RZ (a, 8).
Note that to zeroth order (t,/t,,, --B 0), these relations reduce to eqs (2.1) which describe the perfect mixing limit. 4. APPLICATION
TO A MODEL
SYSTEM
EXHIBITING
BISTABILITY
Our aim is to study the effect of incomplete mixing on isothermal autocatalytic reactions in a CSTR. For this purpose we consider the following coupled reactions: 2A+B-+
(4.1 a)
3A
(4.1b)
C+A-+B
C is supposed to be held constant. In the limit of ideal stirring, the mass balance equations in terms of dimensionless concentrations a and p (normalized over B,) read:
extensively by Gray and Scott [ 10, 1 l] for the case of ideal stirring. For simplicity the residence times of both concentrations a and p are henceforth taken to be equal: t, = t,, = 2t,,,.
tch= 1
da
~-cc
a’j3
t-l
=
tot
a
4
8,-as/t2
=
aI(l+ aa - a,)-
l-8 ttot
(4.2)
=
mode, the system is described by eqs (2.9), (2.7) and the initial conditions
-$+;
-R,(a,&
(4.6)
case of non-ideal stirring. Investigating first the PM
where the reaction rates now assume the explicit form R,(a,&
as/t2
as-a0
We shall now seek approximate solutions for the
-+tch-g ttor
dB -=dt
(4.5)
and the (multiple) steady state solutions of eqs (4.2) are given by
as-a0
-= dt
(4.4)
By adequate choice of units we set
=$-E
_
(4.3) da
a(cp = 0) = aO, b(q = 0) = 1.
The perturbation analysis outlined in Section 3 leads to the following corrected mass balance equations: a0 -
dfl -=
1-B-tmexp(jL)a’(a,8,~)+O(t~)
dt
a + tm exp Wa’
(a, B, LC)+O(ti)
-=
dt
(4.7)
t fot - tm Cev (PIt,ot - t, Cexp (p) -
+ a2P _ CL
11
tz
(4.8)
11
The steady state solutions of these equations can be put into the form The special case t2 - cc would reduce our reaction model to eq. (4. la) only. Such a model had been studied
t-1 tot =
a-au
a:8, - as/t2 - t,B(a,, 8,)+O(tL)
(4.9)
Micromixing and multiplesteady state transitions with
with the abbreviations Pk.
41 = - 3as28, + 2(2a0 - as)asBs + (2 - B&x:
A) = exp (&a’ (ck, B,, p)
-C=wW-11(~~8,-~). Inserting the solution ~‘(a,, eq. (4.10) we obtain P(%Bs)
= pLI(I1 +
(4.10)
+E
use
of
(a,--cd
(1 - B&s2 + 2(ao - %)Q,
*Z =
Bs@O
lcl3
(1
=
-
the
following
a,)’
Bs)(ao
+
-
2(1-
&)(a0
-
a,b,
(4.12)
cQ
cl2 = 1 -exp(--) pLj= 1 -exp(-2p). In order to eliminate /I, we derive from eqs (2.7), (2.9) and (4.2) the exact relation fi.=
l+aO-a,
ar(cp=O)=ab=2a, b,(rp = 0) = 0
(4.14)
Qtt(V = 0) = 0 b,,(ga = 0) = /I& = 2. The corrected mass balance equations are now: exp (p)Co:(a, 8, p) + &(a,
B, dl+O(tf)
t tOt- tm Cew 011- 11
dt -
1 - B - L f- exp (A Co: (a. 8, ~0 + &(a,
G(tf)
For steady state conditions eq. (4.10) is replaced by -I-
tot -
~~B,-~,lt2
a - a0 - t,B(a,,
&)+O(t;)
(4.16)
with &a,,
-
a,)*
-
B,) = 4 exp (&Cc: (a,, /L c1)+ oL(a,, B,, p)3 -Cexp(p)-
11($B,-z)
2@,(2a0
-
4
(4.19)
&I
(2-AbC-Bs(2~0
-G*.
(4. la)
as introduced by Gray and Scott [lo, 11). Bistability is found, if the inlet concentration cl, is much smaller than Do = 1 (e.g. a0 = l/32). In that case PM- and NPM-conditions are nearly identical (see Fig. 3). By decreasing the mixing efficiency, i.e. by increasing the mixing time, we observe a shift of the transition point on the upper branch to smaller inverse residence times. The effect on the lower branch is less pronounced. The addition of the second reaction C+A-+B
(4.1 b)
yields bistability for inlet concentrations a0 = /IO = 1 and t, = 54j5.5. Since a0 and #&, are now of the same order of magnitude, we expect to observe a difference in the behaviour of the system when reactant feedstreams are premixed or non-premixed. The results for the PM mode are depicted in Fig. 4. An increase of the mixing time shifts the transition points of both the upper branch and the lower branch to smaller inverse residence times. The region of bistability is not significantly reduced. Now, for higher values of resi-
+a*fl-E (4.15)
B, Al+
t tot - t, Cexp (PC) - 11
t
=
(2
-
Just as for the PM mode, the exact relation (4.13) holds for the NPM condition and can be used to eliminate @, in eq. (4.16). We shall first discuss the results for the special case t2 -S 00, i.e. for the reaction model
(4.13)
assuming steady state conditions. For the case of the NPM mode, we solve eqs (2.1 l), (2.12) and (2.14) with the initial conditions
da __a0--at-t,+
24
&W0
2A+S+3A
+rl =
-
-
43
P2~2+1zP3~3
made
#JZ= a:/% +
j?,, p) of eqs (3.4) into
(4.11) where we have abbreviations:
dB -= dt
3115
(4.17)
a:(~,, &, CL)and &(a,, B,, p) are evaluated by the perturbation analysis in Section 3. We finally obtain for &a,, Bs):
-a2/3+E. dence time the upper branch represents the unique solution. For such values the reagent approaches the behaviour of a closed system, and for this reason this branch will be referred to as “the thermodynamic branch’. Similarly the lower branch, which in the limit of very small residence times is dominated by the boundary conditions, will be referred to as “the flow branch”. The behaviour is qualitatively completely different for the NPM mode when mixing efficiency is decreased (see Fig. 5). Here the transition point on the flow branch is shifted drastically to higher inverse residence times. Since the transition point on the thermodynamic branch is less affected, we observe a complete destruction of the bistability phenomenon for mixing times larger than 0.04. It is indeed remarkable that we obtain an effect of this order of magnitude even for
3116
A.
PUH~
AND
G. NICOLIS
-
.a00
I
J 0
r-8
_
,600
-
,400
.z a ?
t....,....~....,....,....i 1 .OO
1.50 Total
2.00
Inverse
2.50
Residence
3.00
10-l
2.25 Total
Time
Fig. 3. Calculated steady state concentration a, as a function are of total inverse residence time r,;: . PM- and NPM-modes identical. Dashed, f,,, = 0; 0, f, = l/30, 0, t, = l/15; p = 3; a0 = l/32; t, + co. PM
.,....,.“.,,...
[....’
2.00
1.00
Inverse
_
t....t....,....,....,....j 2.00
2.25 Total
,800
,400
,200
2.50 Inverse
2.75 Residence
3.00
3.25
10-l
Time
Fig. 4. Calculated steady state concentration a, as a function of total inverse residence time t,: in PM mode. Dashed, f,,, = 0; 0, f, = l/45; q ,r,,, = 2/45;A, r,,, = 3/45; /I = 3; a~ = 1; tz = S/55. small deviations from the ideal stirring limit. We finally remark, that we obtained nearly identical results after numerical treatment of the full model equations prior to any perturbation expansion. Thus, first order terms in the perturbation analysis are sufficient as long as tm/tch remains small. 5. CONCLUSIONS We have obtained analytical results on the effect of incomplete mixing for two examples of a bistability transition. We find remarkable differences in the stirring dependence of the bistability curves depending on whether premixed or non-premixed reactant feed streams are used. However, this effect is obtained only when the inlet concentrations of the reacting species are of the same order of magnitude. In
2.75 Residence
3.00
3.25
10-l
Time
Fig. 5. Calculated steady state concentration a, as a function of total inverse residence time f&’ in NPM mode. Dashed, t,,, = 0; 0, r,,, = l/45; 0, t, = 2/45;A, t, = 3/45; /I = 3; a, = 1;
t2 =
54/S.
particular, in contrast to the NPM mode, for the PM mode the analysis does not predict a significant reduction of the which is confirmed
_
.200
2.50
bistability region, a behaviour by experimental observation [3].
The results for the NPM mode, illustrated in Fig. 5, are in qualitative agreement with experimental findings on the bromate-bromide-manganous system [4], where the flow branch (high bromide state) is destabilized while the thermodynamic branch (low bromide state) remains relatively unaffected. On the other hand, there seems to be no universal behaviour valid for all bistable systems: the behaviour depends on the chemical reaction considered. This has already been pointed out by Nicolis and Frisch [6]. Thus, in contrast to the behaviour of the bromatebromide-manganous system, experimental findings on the chlorite-iodide system [l-4] show a destabilisation of the equilibrium branch (low iodide state) relative to the flow branch (high iodide state). We believe that the model presented herein is well suited to explain in a realistic manner the injection effects in a nonideal CSTR. Of special interest is the possibility of describing the injection of different species at different inlet ports. This aspect has so f-ar not been considered in existing theories [4-83. The perturbation approach leads to tractabIe equations and guarantees wide applicability. Thus, the concept of incomplete mixing used in this paper is equally useful for more complex bifurcation phenomena such as isolas and mushrooms [lo, 161 or for chemical oscillations. The latter case has recently been studied experimentally for the BZ reaction [17]. Furthermore, there appears to be no difficulty in extending the analysis to non-isothermal problems, as long as the mechanism of heat exchange between injected lumps and the bulk material is similar to the mechanism of material exchange. It should, however, be noted that the case of diffusional boundary conditions, e.g. that of heat exchange by the walls of the
Micromixing
and multiple steady state transitions
CSTR, cannot be handled in the framework of the model used here. A general statement concerning the characteristic times can be made: as mentioned in Section 2 we are restricted by the hierarchy The effect of incomplete mixing does not only depend on the value of tm/tch. It is equally important that the residence times ttof/tch, for which effects like bistability or oscillations are expected, are not too large. Otherwise inhomogeneities induced by very slow injection will easily be wiped out by turbulent mixing. Admittedly, the theory presented in the present paper fails to furnish a solution of the problem of inhomogenous concentration fluctuations in the CSTR. A more fundamental approach is required. Such an approach could be based on turbulent transport theory, leading to an effective turbulent diffusion coefficient depending on velocity fluctuations in the tank. The turbulent diffusion coefficient should be strongly space dependent due to the fact that the velocity fluctuations vanish at the walls of the CSTR (boundary layer). We, therefore, expect a critical dependence of the results on boundary conditions. Work on this problem is in progress. Acknowledgments-We are indebted to H. Frisch for fruitful discussions. The work of A. P. is supported by the Stiftung Volkswagenwerk. Partial support of the U.S. Department of Energy under contract number AS05-81ER10947 and of the European Community under contract number STI-070-J-C (CD) is also gratefully acknowledged. REFERENCES
[l]
Roux, J. C., DeKepper, Left. 1983 97A 168.
P. and Boissonade, J., Phys.
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Engng Sci. 1984 39 c91 Zwietering, Th. N., Chem. 17651778. Cl01 Gray, P. and Scott, S. K., Chem. Engng Sci. 1983 38 2943. Instabilities Cl11 Gray, P. and Scott, S. K., in Chemical (Edited by Nicolis, G. and Baras, F.). Reidel, Dordrecht 1984. Gray, P. and Scott, S. K., J. phys. Chem. 1985 87 22-32. :::: Corrsin, S., A.1.Ch.E. J. 1964 10 870-877. of Fluid motions. c 143 Brodkey, R. S., The Phenomena Addision-Wesley 1967. Cl51 Bourne, J. R., Kozicki, F., Moergeli, U. and Rys, P., Chem. Engng Sci. 1981 36 1655-1663. Cl61 Ganapathisubramanian, N. and Showalter, K., J. them. Phys. 1984 80 41774184. Cl77 Menzinger, M. and Jankowski, P., J. phys. Chem. 1986 90 1217-1219.
APPENDIX
To derive eqs (2.5) and (2.9) we proceed along the lines indicated by Zwietering [9]. The material balance equations for a lump with concentrations crand b are: dCo41 = ~~$dtl+dJW,Wtl 64.1) dCb41 = $Vd~+NW>Wdq incoming streamfrom
chemical reaction
the bulk d(q) denotes the rate of volume growth of a lump given in eq. (2.4). For the NPM mode 4 stands for 4, or &I, depending on which feedstream we consider. For the PM mode # means (+t + r#~,~), as there is only one feedstream. Inserting
d[a@]
= #da+adr$
d[b@]
= @db+hdr$
(A.2)
into eqs (A. 1) and making use of the relations for the volume growth (2.4) we obtain eqs (2.5). The bulk material baiancecan be set up for the PM mode as foIlows: --&a
+ F,B,(n,8) (A.3)
f/?!!=
-9bs
B dt
+
vB R,
Here V, is the bulk volume [see eq. (2.8)] PI
c31 c41 c51
C61 c71
LX1
Roux, J. C., Saadaoui, H., DeKepper, P. and Boissonade, J., in Fluctuations and Sensitivity in Nonequilibrium Systems (Edited by Horsthemke, W., Kondepudi, D. K.), Vol. 1, pp. 7s-78. Springer Proceedings in Physics 1984. Menzinger, M., Boukalouch, M., DeKepper, P., Boissonade, J., Roux, J. C. and Saadaoui, H., J. phys. Chem. 1985 90 313. Kumpinsky, E. and Epstein, I. R., J. them. Phys. 1985 82 53-57. Boissonade, J., Roux, J. C., Saadaoui, H. and DeKepper, P., in Non-Equilibrium Dynamics in Chemical Systems (Edited by Vidal, C. and Pacault, A.), Vol. 27, pp. 172-177. Springer Series in Synergetics 1985. Nicolis, G. and Frisch, H., Phys. Rev. A. 1985 31 439445. Dewel, G., Borckmans, P. and Walgraef, D., Phys. Rev. A. 1985 31 19831985. Horsthemke, W. and Hannon, L., J. them. Phys. 1984 81 436334368.
V,=
V-u(p)=
[
V l-z(exp(p)-1)
(4
fi)-
1.
(A.4)
Inserting eqs (A.4) and (2.4) into eqs (A.3) we obtain eqs (2.9) which describe the dynamical behaviour of the bulk concentrations u and @ in the PM mode. In the same manner we draw up the bulk material balance for the NPM mode:
Note that @ = +, + c$t, and I$~ = &a + @,,,a. Using eq. (A.4) together with eq. (2.4), we finally obtain from (A.5) eqs (2.11).
12. p,p_ 3 I I1 -31 17. 19x6
MICROMIXING
0009-2509:86 Pergamo”
AND MULTIPLE STEADY TRANSITIONS IN A CSTR A. PUHL
$3.00 + 0.00 Journals Ltd.
STATE
and G. NICOLIS
Facultk des Sciences Universiti Libre de Bruxelles,Campus Plaine, CP. 231, 1050 Bruxelles,Belgium (Accepted
for
publicution
9 April
1986)
Abstract-Using the micromixing concept introduced by Zwietering we describe the effect of incomplete stirring on consecutive reactions exhibiting bistability transitions in a CSTR. In particular we explain the difference in the behaviour of the system when reactant feed streams are premixed or non-premixed. A perturbation expansion around the fast stirring limit provides analytical results.
concentrations a and /3:
1. INTRODUCTION
Experiments[l+ on bistable chemical reactions in a continuous flow stirred tank reactor (CSTR) have demonstrated some unexpected effects when mixing is incomplete. Two major observations have been made: on the one hand, the decrease of the stirring rate significantly reduces the region of bistability; and on the other hand, this behaviour critically depends on whether the two feedstreams are premixed (PM) or non-premixed (NPM) prior to injection. In particular, under PM-conditions the reduction of the bistability region is much less pronounced than for the NPM mode. These results can be understood qualitatively in terms of the competition between turbulent mixing and inhomogeneities due to reactor feeding. There have been several theoretical approaches to modeling the effects of incomplete stirring on bistability transitions [4-81. However, neither of them has so far explained the differences in the stirring effects when premixed or non-premixed reactant feedstreams are used. The investigation of this problem is the main objective of the present work. Our starting point is the micromixing concept introduced by Zwietering [9] to represent the dilution of the incoming streams of fluid by the surrounding bulk fluid. The stirring efficiency is described by a (micro-)mixing time t, which is related to the turbulent mixing process. This model, discussed in Section 2, is valid for small mixing times, i.e. for the fast stirring limit only. Section 3 is devoted to the analytic treatment of the resulting equations by perturbation theory. In Section 4 the general theory is applied to a model of consecutive reactions involving one autocatalytic step.
2. THE
da -= dt
c(o--a ---+R,(a,B) ttot
dfi -= Ir UC
Bo-B 7 %lf
+
R2
(2.1) (a,
Lo.
Here
CQ,, PO are the inlet concentrations entering the CSTR and RI, R2 are the nonlinear chemical rates
containing characteristic chemical time scales. r&’ is the total inverse residence time, i.e. t,,’ is the sum of the inverse residence times t; ’ and tIj 1 for each input species respectively. The restriction to two variables is adopted only for notational convenience. The analysis can be extended in a straightforward way to multivariable systems. However, it is well known that even two-variable systems of the form (2.1) give rise to a variety of dynamical phenomena such as multiple steady states and periodic oscillations [l&12]. The distinction between the premixed (PM) and the non-premixed (NPM) feeding mode (see Figs 1 and 2)
feed
I
feed II
Fig. 1. SchematicCSTR with reactantspremixed prior to injection (PM mode).
MODEL
Consider an isothermal nonlinear chemical reaction system with two variables operating under CSTRconditions. In the limit of perfect mixing, the system is spatially homogeneous and is described by the following mass-balance equations, written for dimensionless
Fig. 2. Schematic CSTR with reactants non-premixed prior to injection (NPM mode).
3111
A.
3112
PUHL
AND
requires different definitions of feeding concentrations. Defining a”, /Soas the inlet concentrations in the PM mode, those in the NPM mode have to be of the form
t1 a, - ta”, _‘
j?;
$_p,
=
tot
l/3
(> &
,
(2.2)
where L is a macroscopic length scale of the turbulent eddies and can be associated with the maximum value of the Kolmogorov energy scale E(k). The energy dissipation rate is proportional to the rate of energy injection by the stirrer [lS]: E - w3/i=
a-a
da -=
~
drl
tm
+ RI (0. b) (2.5)
db -‘‘+R dv tm
v/t,,
LZ
tions, a, 6 are given by (see Appendix):
-=-
tot
if eqs (2.1) are to be representative for both feeding conditions. One way to incorporate inhomogenous perturbations due to incomplete mixing would be to resort to the full set of reactiondiffusion equations including the effect of turbulent diffusion arising from stirring [6, 71. However we consider here an empirical model suggested by Zwietering [9]_ Since we limit ourselves to the range of quasi-ideal mixing, we shall use his “simple model” without considering backmixing. In this case the assumption of isotropic turbulence as a requirement of perfect macromixing is justified. In the model, one considers lumps of fluid injected into the tank. For NPM conditions (see Fig. 2) two separated lumps of fluid enter the tank with initial feeding concentrations ub, & and initial feeding rates 4,o = dJ,I,O= V/t,,, respectively, meaning that a volume +1(11),0dt is injected during a time interval [to, t,+dt]. In the PM mode, only one lump is injected with the feeding rate I#,, = #,,a + &r,c and the initial concentrations tie, & _ The injection process introduces on the one hand inhomogeneities into the tank. On the other hand, the mixing process disperses the entering lumps to a molecular scale by a combined mechanism of turbulent dispersion and molecular diffusion. We shall model the competing processes as a dilution of the entering lumps within the bulk, which itself is assumed to remain homogenous. To describe the dilution process dynamically, we introduce a characteristic small scale time n having the same order of magnitude as the characteristic mixing time t,,,. We may regard q as the “age” of a lump. The parameter t, defines the effectiveness of the mixing process and is related to turbulence theory by the relation [13, 147 tm--
G. NICOLIS
(2.3)
where ;i is a stirrer dimension and w is the stirring frequency. Following Zwietering [9]. the volume growth of an entering lump is dv = +(rl) = 40 exp (q/L) dq and the evolution equations for the lump concentra-
2
(a > b)
where a and fi represent the concentrations of the surrounding bulk material. Although a and /? vary over the reaction time scale, they are to be regarded as constant over the time scale q of the lump evolution. We introduce a new time scale: cp = )llL,
czp =
O(l)1
P-6)
and transform (2.5) into da -=a---a+$R1(a,b)
dv
*=
dv
ch
(2.7)
/T-b+$R,(a,b) ‘h
where we have multiplied RI and R2 by the characteristic chemical time, yielding the dimensionless rates a, and a,. At this stage it is important to note that our considerations require the following hierarchy of characteristic times: rln/r,, 6 1 -= t,(,,,l&z. The first inequality is determined by the range of validity of our model, since we operate in the range of quasi-ideal mixing [9]. The second inequality is necessary to guarantee that a sufficient number of reactions takes place before the molecules leave the tank again. The aggregate volume of all lumps younger than rp, is governed by the relation
49) = L
s,,y
= V+
4(t) d5
tot
[exp (9) - 1]
(2.8)
and is assumed to remain, for all time intervals considered, much smaller than the total tank volume V. This means that lumps will not be allowed to grow ad infinitum. We therefore have to define a “truncation age” 9=p beyond which the lumps will be regarded as belonging to the bulk material a and #I.p is, however, not a critical parameter and simply amplifies or weakens the effect of incomplete mixing without changing qualitatively the results. The average concentrations, which are measured, are the bulk concentrations a and 8. In the framework of the model a and B are regarded as homogenous. We are, therefore, not able to explicitly deal with concentration fluctuations due to turbulence. Nevertheless, a
Micromixing and multiplesteady state transitions and /? incorporate the effect of incomplete mixing through the following corrected bulk material balance equations (see Appendix). For premixed feedstreams (PM) (see Fig. 1) we have: da
exp(14Ca(a, A PL)cto, - tmCexp(P) expti.4 Cb(4 A P) t tot - tmCev (~1~
dt= dB dr=
41
-+ Rl
11 PI 11
following perturbation Ansatz:
(I)= (;:)+g)+ ...
+ R2
a(cp = 0) = aO, b(cp = 0) = #So.
The rates i?, (u, b) and &(a, b) are also expanded around a0 and b,. Considering further for the PM mode the initial conditions
of
and remembering that a and /I do not change over the time scale cp,eqs (2.7) are transformed into a system of linear differential equations: d u0 + a0 = a a77;
(2.10)
d _ dVbo+bo=p
a(a, /I, p) and b(a, /I, p) are, in general, nonlinear functions of a and /L Equations (2.7) together with eqs (2.9) form a set of non-linear equations which can be solved numerically by iteration. A perturbation analysis will be given in the next section. For two separated feedstreams (NPM) (see Fig. 2) the corrected equations for the bulk material yield the form:
tto,
dt= dB
-
t,
(3.2)
d -a’ + u1 = R, (a’, b”) dp d 6’ + b’ = R,(a”, b’) drp
1
exp (~1
dt=
(3.1)
a(cp = 0) = ao, b(cp = 0) = fro
Here a(a, /I, p) and b(a, p, ,u) are the solutions eqs (2.7) at cp = ~1and for the initial conditions
da
3113
+ RI (a, 8)
Cexp(14 - 11
(2.11)
1+&(a,S)
~b,(a,p,p)+~b,,(a,p,~)-P
t tot - t, Cexp (14 - 1I
a, (a, p,p) and b, (a, j?, p) denote the lump concentrations of feed stream I and are evaluated by the differential equations
with the initial values a”(cp=O)=ao b’(rp=O)=So
(2.12)
with the initial conditions a,(cp = 0) = a&, b,(cp = 0) = 0.
(2.13)
Analogously, the lump concentrations a,, (a, p, ,D)and b,,(a, fi, p) of feedstream II are found by solving da 2
drp
= a -a,,
+ ”
db I1 = B-bn+dq
ch
wt (a,,, b,,)
L &(a,,, rch
(2.14) b,,)
The solutions of eqs (3.2) have the general form a”(a,q)=a+(aO--a)exp(--q) b”(BT cp) = B + (PO - PI exp Ca’(a,
B, cp) = exp(-v) x
with the initial conditions
EXPANSION STlRRING
NEAR
THE
41112-J
‘p RI Ca’(a,
(5)
0, b”(P,
511
(3.4)
d<
s0 ew
rp)
’ & Ca”(a, 8, b’(P,
<)I
dC.
IDEAL
LIMIT
Since we are only interested in solutions near the idea1 stirring limit (t,/tch Q l), we shall analytically seek approximate solutions of eqs (2.7) using the CES
(0
(2.15) x
3. PERTURBATION
exp
s0
b’(a,B,q)=exp(--)
a,,(~ = 0) = 0, b,,(cp = 0) = &.
(3.3)
u’(cp = 0) = 0 b’(,p=O)=O.
Inserting the expansion (3.1) into eqs (2.9) we obtain the bulk mass balance equations for PM conditions taking into account the effect of micromixing to first
A.
3114
PUHL
G. NICOLIS
AND
order in the mixing time: dcc dt=
a,-a+exp(~)~a’(a,B,Il)+O(t:lr~~)
t tOf- tlkc=b
+ RI (a, B)
(14 - 13
(3.5) dp -= dt
/%I - P + exp (p) $
b’ (a, 8. &+0(0r$)
+ Rz (a. P).
t fOt- t, CexpCd - 17
Correspondingly, we find for the first order bulk mass balance equations in NPM mode:
da dt
a,-a+exp(p)+
d+O(G/GJ
1
rh
-=
t tof - tm Cew (~0 - 11
+ RI (a. PI (3.6)
dr
2
t &,f- tm Ccxp (p) - 11
+ RZ (a, 8).
Note that to zeroth order (t,/t,,, --B 0), these relations reduce to eqs (2.1) which describe the perfect mixing limit. 4. APPLICATION
TO A MODEL
SYSTEM
EXHIBITING
BISTABILITY
Our aim is to study the effect of incomplete mixing on isothermal autocatalytic reactions in a CSTR. For this purpose we consider the following coupled reactions: 2A+B-+
(4.1 a)
3A
(4.1b)
C+A-+B
C is supposed to be held constant. In the limit of ideal stirring, the mass balance equations in terms of dimensionless concentrations a and p (normalized over B,) read:
extensively by Gray and Scott [ 10, 1 l] for the case of ideal stirring. For simplicity the residence times of both concentrations a and p are henceforth taken to be equal: t, = t,, = 2t,,,.
tch= 1
da
~-cc
a’j3
t-l
=
tot
a
4
8,-as/t2
=
aI(l+ aa - a,)-
l-8 ttot
(4.2)
=
mode, the system is described by eqs (2.9), (2.7) and the initial conditions
-$+;
-R,(a,&
(4.6)
case of non-ideal stirring. Investigating first the PM
where the reaction rates now assume the explicit form R,(a,&
as/t2
as-a0
We shall now seek approximate solutions for the
-+tch-g ttor
dB -=dt
(4.5)
and the (multiple) steady state solutions of eqs (4.2) are given by
as-a0
-= dt
(4.4)
By adequate choice of units we set
=$-E
_
(4.3) da
a(cp = 0) = aO, b(q = 0) = 1.
The perturbation analysis outlined in Section 3 leads to the following corrected mass balance equations: a0 -
dfl -=
1-B-tmexp(jL)a’(a,8,~)+O(t~)
dt
a + tm exp Wa’
(a, B, LC)+O(ti)
-=
dt
(4.7)
t fot - tm Cev (PIt,ot - t, Cexp (p) -
+ a2P _ CL
11
tz
(4.8)
11
The steady state solutions of these equations can be put into the form The special case t2 - cc would reduce our reaction model to eq. (4. la) only. Such a model had been studied
t-1 tot =
a-au
a:8, - as/t2 - t,B(a,, 8,)+O(tL)
(4.9)
Micromixing and multiplesteady state transitions with
with the abbreviations Pk.
41 = - 3as28, + 2(2a0 - as)asBs + (2 - B&x:
A) = exp (&a’ (ck, B,, p)
-C=wW-11(~~8,-~). Inserting the solution ~‘(a,, eq. (4.10) we obtain P(%Bs)
= pLI(I1 +
(4.10)
+E
use
of
(a,--cd
(1 - B&s2 + 2(ao - %)Q,
*Z =
Bs@O
lcl3
(1
=
-
the
following
a,)’
Bs)(ao
+
-
2(1-
&)(a0
-
a,b,
(4.12)
cQ
cl2 = 1 -exp(--) pLj= 1 -exp(-2p). In order to eliminate /I, we derive from eqs (2.7), (2.9) and (4.2) the exact relation fi.=
l+aO-a,
ar(cp=O)=ab=2a, b,(rp = 0) = 0
(4.14)
Qtt(V = 0) = 0 b,,(ga = 0) = /I& = 2. The corrected mass balance equations are now: exp (p)Co:(a, 8, p) + &(a,
B, dl+O(tf)
t tOt- tm Cew 011- 11
dt -
1 - B - L f- exp (A Co: (a. 8, ~0 + &(a,
G(tf)
For steady state conditions eq. (4.10) is replaced by -I-
tot -
~~B,-~,lt2
a - a0 - t,B(a,,
&)+O(t;)
(4.16)
with &a,,
-
a,)*
-
B,) = 4 exp (&Cc: (a,, /L c1)+ oL(a,, B,, p)3 -Cexp(p)-
11($B,-z)
2@,(2a0
-
4
(4.19)
&I
(2-AbC-Bs(2~0
-G*.
(4. la)
as introduced by Gray and Scott [lo, 11). Bistability is found, if the inlet concentration cl, is much smaller than Do = 1 (e.g. a0 = l/32). In that case PM- and NPM-conditions are nearly identical (see Fig. 3). By decreasing the mixing efficiency, i.e. by increasing the mixing time, we observe a shift of the transition point on the upper branch to smaller inverse residence times. The effect on the lower branch is less pronounced. The addition of the second reaction C+A-+B
(4.1 b)
yields bistability for inlet concentrations a0 = /IO = 1 and t, = 54j5.5. Since a0 and #&, are now of the same order of magnitude, we expect to observe a difference in the behaviour of the system when reactant feedstreams are premixed or non-premixed. The results for the PM mode are depicted in Fig. 4. An increase of the mixing time shifts the transition points of both the upper branch and the lower branch to smaller inverse residence times. The region of bistability is not significantly reduced. Now, for higher values of resi-
+a*fl-E (4.15)
B, Al+
t tot - t, Cexp (PC) - 11
t
=
(2
-
Just as for the PM mode, the exact relation (4.13) holds for the NPM condition and can be used to eliminate @, in eq. (4.16). We shall first discuss the results for the special case t2 -S 00, i.e. for the reaction model
(4.13)
assuming steady state conditions. For the case of the NPM mode, we solve eqs (2.1 l), (2.12) and (2.14) with the initial conditions
da __a0--at-t,+
24
&W0
2A+S+3A
+rl =
-
-
43
P2~2+1zP3~3
made
#JZ= a:/% +
j?,, p) of eqs (3.4) into
(4.11) where we have abbreviations:
dB -= dt
3115
(4.17)
a:(~,, &, CL)and &(a,, B,, p) are evaluated by the perturbation analysis in Section 3. We finally obtain for &a,, Bs):
-a2/3+E. dence time the upper branch represents the unique solution. For such values the reagent approaches the behaviour of a closed system, and for this reason this branch will be referred to as “the thermodynamic branch’. Similarly the lower branch, which in the limit of very small residence times is dominated by the boundary conditions, will be referred to as “the flow branch”. The behaviour is qualitatively completely different for the NPM mode when mixing efficiency is decreased (see Fig. 5). Here the transition point on the flow branch is shifted drastically to higher inverse residence times. Since the transition point on the thermodynamic branch is less affected, we observe a complete destruction of the bistability phenomenon for mixing times larger than 0.04. It is indeed remarkable that we obtain an effect of this order of magnitude even for
3116
A.
PUH~
AND
G. NICOLIS
-
.a00
I
J 0
r-8
_
,600
-
,400
.z a ?
t....,....~....,....,....i 1 .OO
1.50 Total
2.00
Inverse
2.50
Residence
3.00
10-l
2.25 Total
Time
Fig. 3. Calculated steady state concentration a, as a function are of total inverse residence time r,;: . PM- and NPM-modes identical. Dashed, f,,, = 0; 0, f, = l/30, 0, t, = l/15; p = 3; a0 = l/32; t, + co. PM
.,....,.“.,,...
[....’
2.00
1.00
Inverse
_
t....t....,....,....,....j 2.00
2.25 Total
,800
,400
,200
2.50 Inverse
2.75 Residence
3.00
3.25
10-l
Time
Fig. 4. Calculated steady state concentration a, as a function of total inverse residence time t,: in PM mode. Dashed, f,,, = 0; 0, f, = l/45; q ,r,,, = 2/45;A, r,,, = 3/45; /I = 3; a~ = 1; tz = S/55. small deviations from the ideal stirring limit. We finally remark, that we obtained nearly identical results after numerical treatment of the full model equations prior to any perturbation expansion. Thus, first order terms in the perturbation analysis are sufficient as long as tm/tch remains small. 5. CONCLUSIONS We have obtained analytical results on the effect of incomplete mixing for two examples of a bistability transition. We find remarkable differences in the stirring dependence of the bistability curves depending on whether premixed or non-premixed reactant feed streams are used. However, this effect is obtained only when the inlet concentrations of the reacting species are of the same order of magnitude. In
2.75 Residence
3.00
3.25
10-l
Time
Fig. 5. Calculated steady state concentration a, as a function of total inverse residence time f&’ in NPM mode. Dashed, t,,, = 0; 0, r,,, = l/45; 0, t, = 2/45;A, t, = 3/45; /I = 3; a, = 1;
t2 =
54/S.
particular, in contrast to the NPM mode, for the PM mode the analysis does not predict a significant reduction of the which is confirmed
_
.200
2.50
bistability region, a behaviour by experimental observation [3].
The results for the NPM mode, illustrated in Fig. 5, are in qualitative agreement with experimental findings on the bromate-bromide-manganous system [4], where the flow branch (high bromide state) is destabilized while the thermodynamic branch (low bromide state) remains relatively unaffected. On the other hand, there seems to be no universal behaviour valid for all bistable systems: the behaviour depends on the chemical reaction considered. This has already been pointed out by Nicolis and Frisch [6]. Thus, in contrast to the behaviour of the bromatebromide-manganous system, experimental findings on the chlorite-iodide system [l-4] show a destabilisation of the equilibrium branch (low iodide state) relative to the flow branch (high iodide state). We believe that the model presented herein is well suited to explain in a realistic manner the injection effects in a nonideal CSTR. Of special interest is the possibility of describing the injection of different species at different inlet ports. This aspect has so f-ar not been considered in existing theories [4-83. The perturbation approach leads to tractabIe equations and guarantees wide applicability. Thus, the concept of incomplete mixing used in this paper is equally useful for more complex bifurcation phenomena such as isolas and mushrooms [lo, 161 or for chemical oscillations. The latter case has recently been studied experimentally for the BZ reaction [17]. Furthermore, there appears to be no difficulty in extending the analysis to non-isothermal problems, as long as the mechanism of heat exchange between injected lumps and the bulk material is similar to the mechanism of material exchange. It should, however, be noted that the case of diffusional boundary conditions, e.g. that of heat exchange by the walls of the
Micromixing
and multiple steady state transitions
CSTR, cannot be handled in the framework of the model used here. A general statement concerning the characteristic times can be made: as mentioned in Section 2 we are restricted by the hierarchy The effect of incomplete mixing does not only depend on the value of tm/tch. It is equally important that the residence times ttof/tch, for which effects like bistability or oscillations are expected, are not too large. Otherwise inhomogeneities induced by very slow injection will easily be wiped out by turbulent mixing. Admittedly, the theory presented in the present paper fails to furnish a solution of the problem of inhomogenous concentration fluctuations in the CSTR. A more fundamental approach is required. Such an approach could be based on turbulent transport theory, leading to an effective turbulent diffusion coefficient depending on velocity fluctuations in the tank. The turbulent diffusion coefficient should be strongly space dependent due to the fact that the velocity fluctuations vanish at the walls of the CSTR (boundary layer). We, therefore, expect a critical dependence of the results on boundary conditions. Work on this problem is in progress. Acknowledgments-We are indebted to H. Frisch for fruitful discussions. The work of A. P. is supported by the Stiftung Volkswagenwerk. Partial support of the U.S. Department of Energy under contract number AS05-81ER10947 and of the European Community under contract number STI-070-J-C (CD) is also gratefully acknowledged. REFERENCES
[l]
Roux, J. C., DeKepper, Left. 1983 97A 168.
P. and Boissonade, J., Phys.
3117
Engng Sci. 1984 39 c91 Zwietering, Th. N., Chem. 17651778. Cl01 Gray, P. and Scott, S. K., Chem. Engng Sci. 1983 38 2943. Instabilities Cl11 Gray, P. and Scott, S. K., in Chemical (Edited by Nicolis, G. and Baras, F.). Reidel, Dordrecht 1984. Gray, P. and Scott, S. K., J. phys. Chem. 1985 87 22-32. :::: Corrsin, S., A.1.Ch.E. J. 1964 10 870-877. of Fluid motions. c 143 Brodkey, R. S., The Phenomena Addision-Wesley 1967. Cl51 Bourne, J. R., Kozicki, F., Moergeli, U. and Rys, P., Chem. Engng Sci. 1981 36 1655-1663. Cl61 Ganapathisubramanian, N. and Showalter, K., J. them. Phys. 1984 80 41774184. Cl77 Menzinger, M. and Jankowski, P., J. phys. Chem. 1986 90 1217-1219.
APPENDIX
To derive eqs (2.5) and (2.9) we proceed along the lines indicated by Zwietering [9]. The material balance equations for a lump with concentrations crand b are: dCo41 = ~~$dtl+dJW,Wtl 64.1) dCb41 = $Vd~+NW>Wdq incoming streamfrom
chemical reaction
the bulk d(q) denotes the rate of volume growth of a lump given in eq. (2.4). For the NPM mode 4 stands for 4, or &I, depending on which feedstream we consider. For the PM mode # means (+t + r#~,~), as there is only one feedstream. Inserting
d[a@]
= #da+adr$
d[b@]
= @db+hdr$
(A.2)
into eqs (A. 1) and making use of the relations for the volume growth (2.4) we obtain eqs (2.5). The bulk material baiancecan be set up for the PM mode as foIlows: --&a
+ F,B,(n,8) (A.3)
f/?!!=
-9bs
B dt
+
vB R,
Here V, is the bulk volume [see eq. (2.8)] PI
c31 c41 c51
C61 c71
LX1
Roux, J. C., Saadaoui, H., DeKepper, P. and Boissonade, J., in Fluctuations and Sensitivity in Nonequilibrium Systems (Edited by Horsthemke, W., Kondepudi, D. K.), Vol. 1, pp. 7s-78. Springer Proceedings in Physics 1984. Menzinger, M., Boukalouch, M., DeKepper, P., Boissonade, J., Roux, J. C. and Saadaoui, H., J. phys. Chem. 1985 90 313. Kumpinsky, E. and Epstein, I. R., J. them. Phys. 1985 82 53-57. Boissonade, J., Roux, J. C., Saadaoui, H. and DeKepper, P., in Non-Equilibrium Dynamics in Chemical Systems (Edited by Vidal, C. and Pacault, A.), Vol. 27, pp. 172-177. Springer Series in Synergetics 1985. Nicolis, G. and Frisch, H., Phys. Rev. A. 1985 31 439445. Dewel, G., Borckmans, P. and Walgraef, D., Phys. Rev. A. 1985 31 19831985. Horsthemke, W. and Hannon, L., J. them. Phys. 1984 81 436334368.
V,=
V-u(p)=
[
V l-z(exp(p)-1)
(4
fi)-
1.
(A.4)
Inserting eqs (A.4) and (2.4) into eqs (A.3) we obtain eqs (2.9) which describe the dynamical behaviour of the bulk concentrations u and @ in the PM mode. In the same manner we draw up the bulk material balance for the NPM mode:
Note that @ = +, + c$t, and I$~ = &a + @,,,a. Using eq. (A.4) together with eq. (2.4), we finally obtain from (A.5) eqs (2.11).