A note on the qualitative theory of lumped parameter systems

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20 I5 IO !!i \5

I Q I 0.01 I

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Fig. 4. Bffect of wall coating on manometer oscillations both before and after shear flow. Indicates damping reduction property unimpaired by shear flow. Cl, &en tube; 0, tube B after use in shear flow; x, tube B before use in shear flow. by Hand and Wiis[J], of using successively poorer solvents to create a’permanent drag-reducting lay&r. Fmally, we report a further ~ous result. This evolved a 1~25cm bore PVC tube (tube C) which bad a much greater wall-thickness than tubes A and B. When tube C was coated with WSR 301 a reduction in ~ of ~rne~ osc~tio~ was observed (as in tubesA and B). But when tube C was test&d in shear flow, ,the,,friction factor w&s actually increased by the coating (Fig. 5). Lesving the tube to soak in water for 24hr destroyed this effect. Overall, the only firm conclusion to emerge is that an adsorbed layer of PEO, which gives consistent reductions of aping of liquid-column oscillations, yields totally inconsistent results iu shear flow. Certainly our original intention to compare LDA me~~ements of flow sect in coated tubes with that in solutions of PEO has been frustrated by this inconsistency.

Scimce, 1976, Vol. 31, pp. 170-173. Pergamm Press.

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Reynd& number x 10e3

Number of cycles

Chemical En&u&#

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Fig. 5: &iomalous iucre&seof drag in thick-wall tube (tube C). q, cleati tube; 0, coated tubs-first run; x, coated tube-second NE (aPter24hr).

Clearly such P study.must be prece&d by a sys~m~c attack on the problem of preparing consistent h-reducing layers on transparent. materials. School of EngineeringSc&nce Universityof Edinburgh E~nbu~k Scotland

S. AYYASH W. D. &COMB

[l] McComb W. D., Na&~~S!X?. [2] Davies G. A. and Potter A. B., Noun 1966212 66. [3] Hand J. H. and Williams M. C., Chem. Engng Sci 1972UI63. [4] Gyr A. and Mueller A., Chem. Engng Sci. 197429 1057. (51 Wells C. S. (car), V~co~s Drag Reactor Plenum, New York 1%9.

Printed in Great Eitain

A note on the qualitativetheory of lumpedparametersystem!d (Received 22 November 1974;accepted 20 February 1975) Luss [I ] has discussed tbe,qualitative influence of positive definite capacitance matrices, C, on the stabitity of lumped parameter systems described by

ci = f(x).

a single exothermic chemical reaction occuring on the surface of a catalytic wire can be modelled by[2]

(1) (3)

The asymptotic stability of the steady states of this system to small disturbances is determined by the linearized equations ¶j = c-‘&j

(2)

where J is the Jacobian matrix of f evaluated at the steady state. The unique feature of eqn (1) is that a set of time-invariant parameters has been isolated and incorporated into the matrix C. Although these parameters are called “capacitance terms” the results do not depend on this interpretation. Many physical models, however, do naturally contain capacitances. For example, tWork performed under the auspices of the U.S. Atomic Energy Co~ission. Sit is not surprising that the equations and the equations for RC circuits are similar: The mathematical structures underlying th~~~~cs apd network theory are nearIy identicaQ6,S-101. In fact, most processes treated by the methods of c&&al irreversible thermodynamics can be described by equations analogous to circuit ~~t~ns[6].

where B and y are the dimensiontess surface concentration and temperature, respectively, and the capacity term ,a is related to the heat capacity of the wire. Similar equations are encountered in electrical circuit theory[3,4] and network ~~~~cs[~lO]. For example, the equations of motioh of a large class of RC circuits, linearized about some steady state, can be written

,j = -GC-‘v

(4)

where q is a vector of capacitor charges, G is a real nons~ incremental conductance matrix, and the capacitance matrix C is real, symmetric and positive definite. The linearized dynamical equations desc~b~ a set of M candent chemical reactions occur& isothermally among N chemical species can also be written in the form of eqn (4) when certain constitutive assump~ns are madef7$# The matrix G is Sykes for all

Shorter Communications

171

circuits constructed from 2-terminal elements[ll], and for all vectors x, and x1 such that (x1- x,)‘[g(x,) - g(x2)]< 0. For eqn (1) reactions that occur close to equilibrium[7]. In many chemical g(x) = -f(x). When G is symmetric the nonlinear equations of motion (1) can engineering problems reactions are far from equilibrium and G is be written as a gradient flow, i.e. not symmetric. Luss considered systems in which C is a positive detinite , dx diagonal capacitance matrix and P is assumed to be nonsingular. C z = -gradB (8) We shall extend Luss’ investigation in two directions: first by considering the stability of eqn (1) when other constraints are imposed on the matrices of C and J, and second by showing that where the potential P is given by the structure of the equations are such that there can exist critical values of capacitance terms at which bifurcations to periodic P = - ’ x’f(ax) da. (9) solutions occur. Although many of the results to be presented are I0 not new, they complement Luss’ remarks and provide a fuller picture of the effects of capacitance terms on lumped parameter AR solutions of (1) flow downhii on the potential P since system behavior.

(10)

QUALITATIVE BEHAVIOR

In the equations treated by Luss C is a diagonal, positive definite, matrix and a conductance is not explicit. Here we shag assume that C is a positive definite, symmetric, matrix. Further by making the identification GE-J

(5)

we can simultaneously consider the

stability properties of both the linearized systems (2) and (4). To see this, observe that C has a unique real, positive definite, symmetnc square root Cl”. Both C’G and GC’ are similar to C-‘,GC-“’ and hence have the same eigenvahtes. In order to simuhaneously discuss the stability of systems described by eqns (2) and (4) we shag let A= -Cm’G

or -GC-‘.

(6)

Thus we consider systems of the form li =Aq.

(7)

Luss[l] examined the influence of the capacitance on the stability of the steady state when the conductance matrix G is nonaingular. If the conductance matrix is also symmetric, as may occur in the examples cited above, then the capacitance matrix has no influence on the stability of the steady state. The following are then. truell31. (a) The eigenvahtes of A are all real and hence all solutions which tend toward stable steady states or escape from unstable steady states are nonoscillatory. fb) If the matrix G has all of its eigertvahrespositive then A has all‘negative eigenvalues and the ste&.state is locally asymptoticaBy stable. (c) If the matrix G has one or more negative eigenvahtes then A has at least one positive eigenvahre and the steady state is unstable. Thus for example if G is symmetric and positive definite then by (b) the steady state is stable. Contrastingly, if det G < 0 then by (c) the steady state is unstable. Even if G is not symmetric, if it is positive dehnitet then the real parts of all eigenvalues of A are negative and the steady state is stable[l4]. Luss has shown that even when the symmetry of G = -J is relaxed the condition (-l)N det (-G) = det G 0 for-all real y-P 0. $Because Ho& original oaoer is difIIcrihto obtain, the author will make available upon request either the frigid Oentm version or an English translation. OThis condition is necessary for the stability but not the existence of a periodic orbit.

at all points except the steady state where x = 0[3]. If f is strictly monotone increasing then there exists a unique steady state which occurs at the unique minimum of P and all solutions will asymptotically tend toward this stable singular point[lS-17. If P is radially unbounded (i.e. P + m as (x)+ m),then .all solutions of (1) will be bounded and will tend as t +m toward the singular points of P [3,4]. The minima ‘of P will correspond to stable steady states while the saddle points and the maxima will be unstable. BIFVRCATION PiIRNOMENA The ditferential equations (1) differ from the usual state equations i =f(x) in that a set of parameters have been singled out and consolidated into the matrix C. It is of interest to study the qualitative behavior of the solutions to eqn (I) as the parameter values are changed If the solution makes an abplpt qualitative change as the parameter crosses some critical hint, for example, from approaching one attractor to another, then a bifurcation is said ta,_,qc.cur. For an ndiiensional system the $r(n + 1) independent components of C can be functions of one or more parameters. Results on bifurcation are available[l8-201 for the variation of only one parameter, which is the case we shall consider here, but Takens[21] has considered the effects of varying more than one parameter. Assume that the capacitance matrix C is a function of a parameter a ; that is, C = C(o). Let x = x,,(a) be a steady state solution to eqn (1). The stability of this steady state is then determined by the eigenvahtes, A,(a), of the matrix A(a)= -CT’(a)G, which also depend on the parameter a. As the value of a is changed a pair of complex conjugate eigenvalues may cross the imaginary axis and a stable steady state solution may bifurcate to a periodic one. The following theorem stated in a manner suitable for our purposes, ,describes this situation. HOPFBIFURCATION TEEOREM[18,22,231$ Let i = F(x, a) be an nth order [i.e. x = (x,, x2,. . . x,)] system of differential equations for each value of the real parameter a, with F(x, a) depending smoothly on its n + 1 arguments. Suppose x = x,,(a) is a steady state solution {i.e. F[x.,(a), a] = 0) and that the matrix A(a) of the linearization of the system about x,, (a) for a in a neighborhood N(aO)of some critical value a,, has one pair of complex conjugate eigenvalues, &(a) + i&(a), with b(a) # 0, # 0 (“transversaMy”). Then there exists An(ao)= 0,(dL/da)I.-, a one-parameter family of periodic solutions of i = F(x, a) for a in the neighborhood N(ao) with period near (2r/lA1(ao)]). Heuristically, we can understand the origin of these periodic solutions by examining the phase portrait of a two-dimensional system. Suppose that for a > a0 both eigenvahtes have strictly negative real parts. Further suppose that as a decreases a pair of complex conjugate eigenvalues cross the imaginary axis at a = no. The phase portrait near the steady state is shown in Fig. l(a) and (b) for a > (lb and a < a, respectively. Assume that when a crosses a,, the vector field at large distances from x.. (a~)continues to propel trajectories inward as in Fig. l(a),%then a closed orbit results (Fig. l(c)). For higher-dimensional systems as other pairs

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Shorter Communications

(a)

(bl

(cl

Fig. 1. Hopf bifurcation. (a) II > a,,; (b) n < of eigenvalues cross the imaginary axis further bifurcations can occur with tori forming as solutions[22]. Luss[l] has shown that for systems described by eqns (1) and (2) where C is positive definite and J is~ nonsingular and nonsymmetric, changes in the system capacitance can shift pairs of eigenvalues across the imaginary axis. Moreover, when a pair of eigenvalues cross the imaginary axis both eigenvalues must be pu~ly imaginary and non-zero. If J is nonsingular and syminetric, all eigenvalues of A are real and no choice of 6: can move the eigenvalues across the imaginary axis. Thus the stability of the system cannot be affected by changing the capacitance. When J is nonsymmetric variations in C produce the type of eigenvalue movement necessary for bifurcations of periodic solutions. For nonequilibrium thermodynamic processes including chemical reactions occuring cldse to equilibrium, J by Onsager’s theorem is symmetric [6]. Consequently, instabilities and ‘bifurcationsbecome possible only as one moves away from equiIiirium. This, of course, is the regime of interest in most chemical engineering problems. In order to be specific we shall show that bifurcations can occur in the case of si@e exothermic chemical reaction occuring oh the exterior surface: of a catalytic wire. Using the notation of Ervin and Luss[2], under conditions of constant fluid velocity, the diflerential equatiotis describing the transient behavior are

where ,$ = a exp (-y/y). If we let x = [e, y J’ and C(a) = [i

(13)

)1”]then eqns (11~12) can

be written in the form of eqn (1). Viewing a as an adjustable parameter eqns (11) and (12) can also be written as

2 =F(x, a).

ao; (c) periodic solution

for (I C ao.

are A

=$A?

J(

:@A)‘-detA).

If det J > 0 then det AZ 0 and as a is varied three situations arise: (i) For a > a,,, tr A < 0 and all the eigenvalges of A have negative real parts; (ii) when a = a~, tr A = 0 and A has purely imaginary eigenvalues +iA, Ar =

q(det Al,_& # 0

(19)

(W (iiii for a < ao, tr A > 0 and all the eigenvalues of A have a positive real part. Thus as a decreases through a, a pair of complex conjugate eigenvalues cross the imaginary axis. At the critical value for II the transversality condition

&R da

o-q

is obeyed. Here we have used the fact that (dy../da) = 0 and hence from eqn (13) (dff.,/da) = 0. From the Hopf bifurcation theorem we conclude that if det J > 0 there exists a one-parameter family of petiodic solutions to eqns (11x12) which bifurcates from the steady state solution at the critical value of the capacitance C(ao). One can also show that this one-parameter family of solutions is unique up to a phase shift in the neighborhood of x,, and exists on one side of a, only [18]. A steady state of a chemically reacting system is stable only if the “slope condition” (i.e. the increase in the rate df heat removal due to a small perturbation of the steady state exceeds the increase in the heat generation rate) is sati&d[24]. Since bifurcations of the type described by Hopf occur from the stable steady states, fulfillment of the slope condition is necessary for bifurcation to oscillatory behavior. For the example above of a reaction on a catalytic wire the condition det J>O is the slope condition

(21) Linearization of the transient equations gives the m$rix (It is not satisfied by n of the 2n t 1 steady states of eqns (11)-(12)[2].)Expressing the right hand side of inequality (21) in terms of a0 gives

-(l + l),. A(a) = C-‘(a)J =

(15)

(1 t k).. > aGO+ iG), + 1 or

Let aO
then tr A = (1 + f)&,/a - 1). The eigenvalues of A, determined by solving the characteristic equation A*-(trA)A tdetA=O

(17)

k

1.

Hence the predicted bifurcations to periodic solutions occur when a c 1. The analysis in[25] can be utilized to determine the stability of these oscill&ry solutions. Rav. Uu~al and Poorel261have shown that oscillatory solutions do n&i e& for a B 1. s&e it has been argued thai flickering (temperature Ructuations) in catalytic wires occurs when a is

173

Shorter Communications much larger than unity [26,27],the predicted periodic solutions are most likely not related to observed flickering[28]. Acknowledgement-I

would like to thank G. Pimbley for his comments on the manuscript.

Theoretical Division University of California Los Alamos Scientific Laboratory Los Alamos, NM 87545 U.S.A.

a A A a

a0 B C

cc i

det E F f G AHP h

J k^ k, L M N R r ; T. t

tr x Y

ALAN S. PERELSON

NOTATION

pre-exponental factor of rate constant linearized stability matrix C-‘J or -GC-’ area of wire capacity term ndh/ApC,,k,c critical value of a defined bv_ ean _..(15) dA(a)/da capacitance matrix concentration of reactant heat capacity of wire diameter of wire determinant activation energy vector function nonlinear vector function conductance matrix nonlinear vector function heat of reaction heat transfer coefficient Jacobian of f dimensionless rate constant adsorption rate constant number of surface sites per unit surface area of wire number of reactions number of chemical species gas constant reaction rate scaled time surface temperature gas temperature time trace vector of state variables dimensionless surface temperature T/T,

Greek symbols p dimensionless heat of reaction (-AH)k,cL/hT, A dimensionless activation energy EIRT, T small perturbation

B A p 7

dimensionless site concentration eigenvalue of A density of wire dimensionless time adht/ApC,

Subscripts I imaginary part R real part

ss

steady state conditions

Superscript T transpose REFERENCES

[l] Luss D., Chem. Engng Sci. 197429 1832. [2] Ervin M. A. and Luss D., Chem. Engng Sci. 197227 339. [3] Brayton R. K. and Moser J. K., Quart Appl. Math. 196422 1; 22 81. [4] Stern T. E., Aspects of Network and Systems Theory (Edited bv R. E. Kalman and N. DeClaris). , Holt. Rinehart & Winston,*New York (1971). [5] Oster G., Perelson A. and Katchalsky A., Nature 1971234 393; (1972)237 332. [6] Oster G., Perelson A. and Katchalsky A., Quart. Rev. Biophy. 19736 1. [7] Oster G. and Perelson A. S., Arch. Rat. Mech. Anal. 1974 55 230. [S] Perelson A. S. and Oster G., Arch. Rat. Mech. Anal. 1974 57 [9] titer G. and Perelson A., Israel J. Chem. 197311 445. [lo] Oster G. and Perelson A., IEEE Trans. Circuits and Systems 1974 CT-21 709.

[ll] Brayton R. K., Mathematical Aspects of Electrical Network Analysis, Vol. III, SIAM-AMSProceedings. American Mathematical Society. [12] Feinberg M., Arch. Rat. Mech. Anal. 1972 46 1. [13] Oster G. and Desoer C., J. Theoret. Biol. 197132 219. [14] Chua L. 0. and Alexander G. R., IEEE Trans. Circuit Theory 1971CT-18 520. [15] Duffin R. J., Bull. Am. Math. Sot. 194753 %3. [16] Desoer C. A. and Katzenelson J., Bell System Tech.J. 1%544 161. [17] Desoer C. A. and Wu F. F., SIAMI. Appl. Math. 1974 26 315. [18] Hopf E., Acad. Wiss. Leipzig, Math-Phys. K. Ber 194294 3. 1191Chafee N.. J. Diff. Eans. 19684 661. [2Oj Friedrichs’ K. 6 Advanced Ordinary Differential Equations. Gordon and Breach, New York 1965. [21] Takens F., J. Di#. Eqns. 197314 476. [22] Ruelle D. and Takens F., Commun. Math. Phys. 1971 20 167. [23] Kopell N. and Howard L. N., Studies in Appl. Math. 1973 52 291. [24] Bilous D. and Amundson N. R. A.1.Ch.E.J. 1955 1 513.

[25] Uppal A., Ray W. H. and Poore A. B., Chem. Engng Sci. 1974 29 967. [26] Ray W. H., Uppal A. and Poore A. B., Chem. Engng Sci. 1974 29 1330. [27] Edwards W. M., Zuniga-Chaves J. E., Worley R. L. Jr. and Luss D., A.I.Ch.E.J. 1974 20 571. [28] Edwards W. M., Worley F. L. and Luss D., Chem. Engng Sci. 197328 1479.