Shorter Communications 511

3. AFTLICATIONS OF TEE SCREENING TFSl’

By differentiating (3) and (4), we obtain:

ai LT&=

_ 1 _ eJwIIz-lh,

z= -2-k(2~~ -3.75) t #O/q')e""'-I'"'. (8)

On addition of (7) and (8), the expression for SUM becomes:

SUM = - 3 -2qk + 15k/4 - (506/q’- 1) eso(“*-“*). (9)

Under the assumed conditions for this system the range of q is 1.75-2.25. To determine the sign pattern of SUM in the region of interest we consider the following:

(a) If 4 is between 1.88 and 2.25 then:

SUM= -L+Mt (10)

where L and M are positive constants. (b) If 71 is between 1.75 and 1.88 then depending on the value of

k we have

or

SUM=PtM[ (11)

SUM= -Q+M[ (12)

where P, M and Q are positive constants. Equations (lo)-(12) show that for a given q, SUM can change

sign at most only once as 5 increases from zero. Hence at most only one limit cycle may exist in the region of interest. Since the singular point is a stable focus, the presence of an unstable limit cycle enclosing the singular is possible. The results of the screening test agree with the conclusions of Luus and Lapidus.

The useful application of the screening test to a real system has been clearly demonstrated. In this case, the test has definitely indicated the absence of multiliiit cycles in the phase plane.

In general the test provides information regarding the maximum number of possible limit cycles in the system. It does not guarantee that all of these exist, only that no more than this number may exist. It provides a necessary but not sufficient condition for the existence of more than one limit cycle.

Thus the test serves as a screening test for all systems which can be described in terms of two varaibles as in (1) and (2).

Where it points to the possible existence of limit cycles we can

then look for them by other methods for example, using the PoincarbBendixson Theorem[rl] and/or the theorem of i [2].

School ofMathen#cs Uniuersity of New So&h Wales Kensington, N.S. W., Australia

ALBERT T. DAOUD

k L M

f?

SUM

t

;

i

NOTATION

a proportionality constant a positive constant a positive constant a positive constant a positive constant

aX+aU ax ay time; dimensionless time a variable a function of x and y dx Z a variable a function of x and y dy dr dimensionless temperature

!!!l dt a function of 11 and f dimensionless concentration d5 dt a function of 9 and 5

REFF#RKK!Es [l] Daoud A. T., A theorem for the possible existence of more

than one limit cycle, to be published. [2] Daoud A. T., Ph.D. Thesis, University of New South Wales

1973. [3] Poincart H., J. De Mafhmatiques, 1881 7(3) 375-422. [4] Bendixson Ivar, Acfa Mathemalica 1901 24 l-88. [5] Balth van der Pol, Phil. Mag. 1927 3 (13) 65-80. [6] Higgins J., Ind. and Engng Chem., 1967 59(5). [7] Aris R. and Amundson N. R., Chem. Engng Sci., 1958 7 121. [8] Aris R. and Amundson N. R., Chem. Engng Sci. 1958 7 132. 191 Luus R. and Lapidus, Chem. Engng Sci. 1966 21, 159-181.

[lo] Heberling P. V., Gaitonde N. Y. and Douglas J. M., A.1Ch.E. J. 1971 17(6) 1506-1508.

[11] Gait0ndeN.Y. andDouglas J.M., A.I.Ch.E.J. 1969 15902.

Chemical Engincedh# Science, 1976, Vol. 31. pp. 511-514. Pergamon Press. Printed in Great Britain

Optimal reactor choice for a reaction in a macrofluid

(Received 10 July 1975; accepted 26 November 1975)

The modern chemical reactor theory emphasizes kinetic optimisa- tion problems: the comparison criteria in terms of reactor volume, conversion and selectivity which allow an optimal choice among different kinds of reactors, are well known and widely treated[l, 81. Nevertheless these criteria are usually developed with reference to a microfluid, that is a fluid with a perfect molecular mixing within its fluidodynamic links.

On the other hand, several studies[2-71 beginning with a well known paper by Danckwerts [2] have pointed out in recent years the usefulness of considering, together with the extreme model of a microfluid, other models of fluid with an imperfect molecular mixing @&tial segregation), and especially the alternative extreme model of the wholly segregated macrofluid, which is detined as a group of elements (particles or points) that cannot exchange matter among themselves. Indeed there are some cases of

practical interest where the perfect molecular mixing hypothesis turns out to be far from physical reality: it will be enough to remember the reactions in a very viscous fluid[4], the reactions in a dispersed phase[5,6], and some enzymatic reactions[7l.

So, among the studies on reactions in partially or wholly segregated fltids, it seems justifiable to extend the kinetic optimisation problems to the case of a macrotluid. Recently the effects of segregation on the design volume of a CSTR and on selectivity have been brielly discussed[5]; now this communica- tion aims to pose the problem of reactor choice with reference to a reaction in a macrotluid (complete segregation) and to underline the cases where the optimal solution of the problem is different from the one concerning a microfluid. ’

The aforesaid extension has naturally to be treated with reference to the particular discontinuous structure of a mac-

512 Shorter Communications

rofluid: as no molecular mixing is allowed for a macrofluid, this always behaves as a group of batch reactors or of ideal plug flow reactors. Therefore, in terms of a mere mathematical model, one can thii of any reactor or any series of reactors where a macrofluid flows as being a system of many ideal plug flow reactors in parallel. In short, a kinetic optimisation problem can be reduced, for a macrofluid, to the optimisation of the above mentioned system of plug flow reactors, i.e. to the search for the particular system of plug flow reactors that will provide a certain result with the minimum total volume or, what is the same thing, with the minimum mean residence time.

To make the following treatment as easy as possible, an isothermal macrofluid will be considered where only one reaction A +. . . . . = Products takes place. Let N be a generic number of plug flow reactors, G the residence time of nth reactor and fn the flow fraction which passes through it.t Moreover a flow fraction f,, is explicitly considered, which does not undergo any reaction and so has a residence time r0 = 0. Now it will be only a question of finding the minimum of the mean residence time

7 = 3 TJ” (b 30; j” ro) (1) n=0

keeping in mind that the mean conversion is a datum

G = “J”C.&“) (2)

and that the fn must fulfill the normalization condition

1 = “to f”. (3)

In eqn (2) CA (7”) is the concentration of the reactant A at the exit from an ideal plug flow reactor (or batch reactor) with a residence time 7.. The function C,(t) can easily be obtained if the reaction rate r,(G) is known:

The extreme conditions of (1) with the constraints (2), (3) are equivalent to the extreme conditions of the function

Now these conditions are

g= j,, +Af&),” =0 n = 1,2 ,....., N (6)

+“+Lc_&)+A~=o n =0,1,2 ,..... ,N (7) n

besides the (2) and (3). From eqns (6), (7) one can deduce that: (a) in a graph C., vs t all

the points CA (T.), 7., with n 5 0 will stay on the straight line (7) and obviously on the CA(f) curve (4) too; (b) because of eqn (6) the straight line (7) with a slope - l/A will be tangent to the curve C,(t) in all the points C.,(G), T. with n P 1 and a f. not zero.

tThe number N is finite, but it can be made very great; thus the ideal system of plug flow reactors considered can approach any distribution of residence times consistent with the definition of a macrofluid. Indeed the number of particles of a macrofluid is finite too.

$The only mixing consistent with the definition of a macrofluid is a macromixing, that is a mechanical mixing of fluid elements with different characteristics. During the mixing each element maintains its initial characteristics, nevertheless it is correct to speak of the mean properties (for instance, mean concentration) of the obtained mixture.

On the basis of these conclusions the minimum problem in question can easily be solved case by case in a graphic way if the function C..,(t) is known. Nevertheless one can already deduce some general considerations. The points CA (7”), T,, which satisfy eqns (6), (7) are always few and, for the niost usual reaction types, are not more than two; hence one can exclude that the optimal solution will consist of reactors with a wide distribution of residence times. On the contrary, the optimal solution will always be constituted by an ideal plug flow reactor or at most by a very small number of plug flow reactors working in parallel. Thus the kinetic optimisation problem for a macrofluid attains a particular meaning when the solution of the same problem for a microfluid foresees the use of reactors diierent from the plug flow, such as the CSTR and the tubular reactor with recycle.

In most real cases the function Ir,(C,)I is monotonically decreasing as CA decreases. Then also CA(t) is monotonically decreasing and has no flex point: there is only one point C.,(T.), r,,, coinciding with the CA, 7, which satisfies eqns (6), (7). So the optimal solution will consist of only one ideal plug flow reactor and it coincides with the one concerning a microfluid. This result could be completely foreseen since the behaviour of a plug flow reactor is the same whatever the nature of the flowing fluid: indeed, in connection with the single residence time of the ideal plug flow reactor, the segregation can only be complete.

But in some cases, as for instance in autocatalytic reactions, the function Ir.,(C.,)I presents a maximum. Then, as is known, the optimal solution for the microfluid foresees the use of continuous stirred reactors too[l, 81. Thus these cases seem worth a more detailed discussion and also being further illustrated by means of a quantitative example.

Let CA be the value of the concentration of the reactant A which the maximum of Jr* 1 corresponds to. In connection with C.‘, the C,(f) function will have a flex point and then it will be possible to draw from the point C,(T,) = C,“, 70 = 0 a straight line tangent to C,(t) in a point Cl, T” different from C,“, 0 (see Fig. 1). If C., is greater than C: the optimal solution for the macrofluid now requires a f0 # 0, that is a plug flow reactor with a residence time T' where only a fraCtiOn

c,“- 6 ‘-fo=cAo_c, (81

of the total feed flows, while the remaining fraction f0 does not react at all (reactor with a zero residence time), but is simply by-passed and mixed with the former.S The mean concentration after mixing will in fact be

(1--jcJc::+f&,“=6

while the optimal mean residence time

(9)

C,“-cA ci dC, - - T=(l-fo)7”=C*o_C. cA0 r*(CA) I

c >c: (10)

will be a linear function of r?., and its graphic representation will coincide with the-said tangent C,“, 0; CL T".

Vice versa if C., is less than Ci the optimal solution for a macrofluid consists again of only one plug flow reactor without any by-pass cfO = 0); the residence time is now immediately given by eqn (4).

With reference to an autocatalytic reactiod A + B = 2B and a reaction rate

r,=-KC&=-KC.,(C-C,).

Figure 1 shows both the graph of the C,(t) function

(11)

c.4 = C,“C e-Kcf

C - CA” + CA0 eeKct (12)

and the locus of optimal residence times, which is formed by the straight line C,“, 0; CX, T" and after, for CA < C;, by the curve C,(t) itself. The dimensionless variables x = CA/C and B = KCt have been used in the graph.

c

t

L - -

513

0

Fig. 1. Optimal residence times for an autocatalytic reaction A + B = 2B and for two values of the initial concentration.

In Fig. 1 the loci of optimal residence times have been reported for two different values of the initial concentration C,“: for greater C,“, and the same 6, the optimal residence times are obviously greater.

The results just exposed become more interesting if they are compared with the ones concerning a microfluid. For microfluids the optimal solution, as is well known, consists, when C., < C.‘,[8], of a CSTR where the reaction goes on up to the value CA of the concentration, and of a plug flow reactor in series. The mean residence time results

T_c:-c*“+ % dCA r* (CL) I- ci r,(G)

CA < CL. (13)

On the other hand, if CA > CA, different solution have been proposed (CSTR, plug flow reactor with recycle[8]) but the one which truly presents a minimum residence time consists, in our opinion, of one CSTR working at maximum Ir., 1 and providing the concentration CA at its exit. Only a fraction of the feed will flow through the tank, while the remaining fraction does not react and is by-passed and mixed with the former. So the optimal residence time

CA - C,” T=r*o ~*A>c: (14)

is a linear function of CA,. The loci of optimal residence times for the microfluid are

plotted too in Fig. 1, again for two different values of initial concentration C,“. As can be seen in the figure, the optimal solutions for the macrofluid and for the microfluid respectively are not only qualitatively (different kinds of reactor), but also qua@atively different. In correspondence to every prefixed value of C, the macrofluid always requires a reaction volume greater than the microfluid does, and these diierences increase as the initial concentration C,” grows to its maximum value C.

In an extreme case, if C,” = C (no component B is fed), the reaction volume for the macrofluid grows to i&mite, whatever CA is, while for the microfluid it is still linite: the only difficulty in the former case is to start the reaction in the continuous tank, and subsequently the reaction will go on because of molecular mixing.

Tii now the question has been developed here in reference to both limit fluid models (macrotluid and microfluid) and limit reactor models (CSTR and plug flow). However this ideal approach suits the spirit of many kinetic optimisation problems, which attempt to point out those ideal solutions that the designer will have to approximate in practice. While no doubts remain on the meaning and the usefulness of the contraposition between the said limit models of reactors[l, 81, on the other hand the contraposition between the limit models of macrofluid and of microfluid needs at least some more discussion. In nature, just as there exists no perfect microfluid, so much the more there exists no perfect macrofluid: molecular exchanges among the particles of a macrofluid may be very little, but never zero. This fact will have a certain weight in choosing the best system of reactors and will even, in some cases, appreciably modify the optimal solution, especially when the reaction, for its start, needs some molecular exchange among particles.

For instance, in the foregoing case of an autocatalytic reaction, if no trace of B is present in the feed (C,” = C), no conversion is attainable for a perfect macrofluid (as was said, the reaction volume becomes infinite). On the contrary, when molecular exchanges, though very little, are not zero, the reaction can go on: now one can suitably use, as a starting device, a CSTR which gives an exit mean concentration of A slightly less than C[4]. Afterwards the optimal solutions concerning a perfect macrofluid with C,” < C will again be approximately valid.

A rigorous and complete discussion of the fluids with an intermediate behaviour between the perfect microfluid and the perfect macrofluid could be developed on the basis of the model of partially segregated fluid described in[3,4]. Yet such a discussion would be too complex and wholly inappropriate here; the aim of this note is only to underline the not negligible effects of the amount of molecular mixing (segregation degree) on kinetic optimisation problems. The contraposition of two limit models of microfluid with a perfect molecular mixing and of macrofluid with no molecular mixing seems to fit this purpose. In this way it has been shown that the optimal solutions of the problem of reactor choice concerning the microfluid and the macrofluid respectively are quite different when the former recommends the use of reactors with a continuous distribution of residence times; such a situation is met, as a rule, when the reaction rate r,(C,) is not

514 Shorter Communications

monotone. Moreover the example of autocatalytic reactions has Greek symbols made evident in quantitative terms the diierences between the A, A’ Lagrange multipliers mean optimal residence times (and between the optimal reaction r mean residence time, set volumes too) corresponding to the limit situation of molecular 7. residence time in the nth plug flow reactor, set mixing; differences which may be very significant, as one can see 0 dimensionless time from Fig. 1.

lnstituto di Sciente e Tecnologie dell ‘Ingegneria Chimica Universitb di Genova

PAOLO- Apex for C,, 72’ M9 a concern@ the initial value

’ concerning the maximum of ]r,] Italia

A, B c

c _A C.4

: K n

N rA

t X

components NOTATION

total volumetric concentration, mol/cm’ volumetric concentration of reactant A, mol/cm3 mean final concentration of A. This is always a datum for

the optimisation problem, mol/cm’ objective function (5), set fraction of the total feed flowing in the nth reactor rate constant, cm3/mol set ordinal for the generic plug flow reactor n =

O,L2 ,....., N total number of plug flow reactors reaction rate, mol/cm3 set time variable, set dimensionless concentration

’ concerning the contact point of .the tangent from C,“, 0

REFERENCES

[l] Aris R., Introduction to the Analysis of Chemical Reactors. Prentice Hall, Englewood Cliffs, New Jersey 1965.

[2] Danckwerts P. V., Chem. Engng Sci. 1958 8 93. [3] Costa P. and Trevissoi C., Chem. Engng Sci. 1972 27 653. [4] Costa P. and Trevissoi C., Chem. Engng Sci. 1972 27 2041. 151 Costa P. and Carberrv J. J.. Chem. Enpnp Sci. 1973 28 2257. i6j Costa P., Maga L. r&d C’anepa B., %sta dei Combustibili

1973 27 58. [7] Dohan L. A. and Weinstein H., Ind. Engrs Chem. Fundl.

1973 12 64. [8] Levenspiel O., Chemical Reaction Engineering. Wiley, New

York 1962.