Dynamic modelling of a heterogeneously catalysed system with stiff Hopf bifurcations

Chmricol ihgineering Science, Vol. 41, No. 2, pp. 317-331, 1986. OWC-2509/s6 S3.00+0.00 Printed in Great Britain. Pergamon Press Ltd.

DYNAMIC MODELLING OF A HETEROGENEOUSLY CATALYSED SYSTEM WITH STIFF HOPF BIFURCATIONS

MOBOLAJI ALUKO+ and HSUEH-CHIA CHANGZ

Department of Chemical Engineering, University of Houston, Houston, TX 77004, U.S.A.

(Received 25 May 1984)

Abstract-We report the results of a detailed dynamic modelling of CO oxidation on a platinum catalytic wire, carried out via an iterative parameter estimation scheme which employs both experimental bifurcation- to-oscillation data as well as the actual oscillatory data. By exploiting the stiffness of the system’s model equations, some algebraic equations that describe the oscillatory boundaries in the feed temperature/partial pressure parameter space are extracted. The resulting model is shown to fit quantitatively these boundaries and the dynamic oscillographs of the wire temperature and simulate, with satisfactory accuracy, the unfitted dynamic data. Following an extension of the independent parameters space of the model beyond the reported experimental one, interesting bifurcation sequences (including multiple steady states) obtained via a computational study are predicted. The utility of both the resulting model in catalytic reactor design and the novel iteration scheme in general parameter estimation work on catalytic models is discussed.

INTRODUCTION

Many heterogeneously catalysed systems (mainly oxidation reactions on noble-metal catalysts) exhibit Hopf bifurcations to sustained oscillations (Slin’ko and Slin’ko, 1978; Rathousky et al., 1980; Turner et al., 1981) and simple bifurcations to steady-state multiplicity (Beusch et al., 1972; Cutlip and Kenney, 1978; Plichta and Schmitz, 1979). These distinctly nonlinear and, in the case of oscillations, transient phenomena have challenged the modelling efforts of many indus- trially important reactors. Even for systems which would not normally be operated in regions where these nonlinear behaviours occur, it has been recommended that, for modelling purposes, use of bifurcation data be made whenever possible since these bifurcations are extremely sensitive to the system parameter values and tend to be uniquely characterized by the key nonlinearities in the system. Such a recommendation has been made by Eckert et al. (1973) and put to use more recently by Rader and Weller (1974) and Beskov (1976) who calculated the reaction order of ammonia oxidation from experimental ignition-extinction information. In this connection, mention can also be made of work in the flame combustion field by investigators such as Kirby and Schmitz (1966), Smith et al. (1971), Schmitz (1976) and Heineman et al. (1979, 1980). Recently, it has been independently suggested by Chang and Chen (1984) and Lyberatos et al. (1984) that controllers be used to destabilize systems that do not bifurcate naturally.

Limited use of these bifurcation data to date,

+Present address: Department of Chemical Engineering, Howard University, Washington, DC 20059, U.S.A.

mo whom correspondence should be addressed.

however, has been in the qualitative discrimination of different candidate models, and it is curious that the same data have not been fully utilized to estimate the parameters of the nonlinear model. This is perhaps due to the lack of a modelling scheme capable of estimating both the oscillatory boundaries and the actual oscillations.

In this paper, we present just such a technique and demonstrate it on a well-known dynamic system-the oxidation of CO on unsupported Pt catalyst. The experimental data employed are exclusively those presented by Turner et al. (1981)and Saleset al. (1982). We show that a nonisothermal Langmuir- Hinshelwood scheme incorporating, as a slow step, reversible oxidation-reduction of the Pt surface, is capable of quantitatively predicting the Hopf bifurcation points in the parameter space, and, more revealingly, simulate with excellent fit the extremely stiff surface-temperature relaxation oscillations in the oscillatory region. This is achieved by using the bifurcation data and some dynamic data of the oscillations in an iterative estimation scheme of the parameters.

EXPERIMENTAL DATA

Utilizing a programmable temperature controller to increase or decrease the gas temperature, Tg, of their Pt wire open-flow reactor, while the partial pressure ratio ap( = Pco/Po,) was held fixed, Turner er al. (1981) obtained upper and lower Ts boundaries of the oscillatory region as a function of ap. In Fig. 1, we reproduce these boundaries as presented in Fig. 3(a) of Sales et al. (1982) (see also Fig. 2 of Turner et al., 1981). Reproducible, spontaneous and sustained oscillations in catalyst surface temperature (measured by ther- mocouples), CO concentration and CO2 production

317

318 M. ALUKO and H.-C. CMNG

700 I ! I

EXPERIMENTAL

0

OO 0

A AA A A

0 0

A *Z

Fig. 1. Experimental oscillation domain boundaries (gales et al., 1982). Reproducible oscillations were found everywhere within the region defined by these two boundaries. The oscillations at point Pare shown in Fig. 2.

rate (both measured by mass spectrometry) were obtained anywhere within the ribbon-shaped region of the boundaries. Abrupt or stiff onsets of finite ampli- tude oscillations were observed close to these boundaries. In the steady-state phase of the parameter estimation procedure to be outlined below, a nonlinear set of algebraic equations is solved at each a,, to obtain predicted boundary Ts values, following which an objective function defined from the resulting residuals is minimized by iterative parameter corrections.

Several oscillographs of surface temperature, CO2 and CO exit partial pressures, and CO2 production rate at various ap and TB values were reported by Turner et al. (1981) and gales et al. (1982), some of which will be sketched or reproduced in the course of

subsequent discussions. The spectrum of single-peak, multi-peak periodic and aperiodic oscillations was presented. Most importantly, all the periodic oscillations are of the relaxation type, many with sharp jumps between two distinct and alternating flat portions. The higher flat portion will be referred to as the “pulse” interval, while the lower flat one will be called the “induction” interval in subsequent discussions. Once again, these observations are evidence of the stiff dynamic character of Turner et al.‘s system.

In the dynamic phase of our subsequent parameter estimation procedure, we will employ points from one cycle of the fairly regular surface temperature relax- ations depicted for ap = 0.031 and Tg = 297°C [see Fig. 3(F) of Turner et al. (1981), reproduced as Fig. 2

0 30 60

TIME (min.) Fig. 2. Characteristic waveform observed at point P of Fig. 1 (Turner et ol., 1981).

here] as “experimental data”, after which theoretically calculated oscillations of the dynamic model will be used to define residuals to be employed in the estimation algorithm.

dnb, -= dt'

2k;S9~P,Jn:)~ -2k’_,(n&)’ -kinEon&

-kbx%,WoT-n;ol) (2)

MODEL EQUATIONS AND PARAMETERS

Sales et al. (1982) have proposed an isothermal model for their system which explains several gross trends exhibited by their experimental data. Yet a fundamental question remains about this assumed isothermality since the observed (m 30°C) differences in temperature between the gas phase and the catalyst surface strongly indicate high exothermicity. Obviously, the catalyst temperature oscillations depicted in Fig. 2 can never be simulated by their model-an important fact which we set out to address in this work. The desirability of an isothermal approxi- mation is, however, understandable when one notes that new and formidable exponential nonlinearities and an increased system dimension are introduced by the nonisothermality. We therefore modified gales et al.‘s model to yield a nonisothermal one here. This nonisothermal model is the same as the one we analysed in earlier publications (Chang and Ah&o, 1984a, b; Aluko and Chang, i 984a, b). In those reports, we merely used reasonable parameter values to locate periodic and aperiodic attractors of the model, and did not attempt to match simulation results with any specific experimental system.

The model is essentially a Langmuir-Hinshelwood reaction mechanism for CO and 0s acting in parallel with redox (or activating-deactivating) steps of the Pt atoms:

CO(g) + x ” cox L

O,(g) + 2x k: 20x (1) I

pC$$= -ha,(T,-T,)+~(k,S~~P~o~~~~o A

+ k-,n~0q~0+2k;S~PoI(n~)Zq~~

+2k’-z(nbl)zqd,~+k;n~onblq,

+ kbxnOl(XOT - nT0j)q,,

+ kL&0n?0pkdl

dXo -~ dr’

= k&e,(XOT - n;O,) -k,&on?O)

where t’ G real time ,

nS = X Y 0 -n~o-f%, (3)

X or = X0 + “So,

& = nEolXO Or = n&,/X0 W)

Y = Xo/Xor. (4b)

Furthermore, surface temperature T, is scaled as follows:

z = (T, - T&/T,. (5)

By their definitions in eqs (4), 8i (0,) are fractional coverages of time-dependent active sites by CO (O,), while y is the undeactivated fraction of all active sites. Substituting these expressions, we have, after some rearrangement, the following dimensionless equations:

se, =~(,,_,~-Gh(z)y(r~+~8,~)~8,L Y

(ha)

COX + OX 3 Products

ox 3 (0)

1

Redox scheme

(0) + COX ‘2 Products

In eqs (l), X represents an active site, COX (OX) represents the adsorbed form of gaseous CO (oxygen), while (0) represents the oxide phase.

By applying certain obvious assumptions and further assuming that simple Langmuir model Arrhenius- form mass action kinetics hold, we then have that the dimensional kinetic equations of the change of the number density of adsorbed species n&o, n&, of undeactivated active sites X, and of the surface temperature T, are given by:

se, = F {Q -r,j -dh(r)y{rm

+Fe,u =gd’ >

(6b)

6i= -~+h(z)y2{f,r~+~~rz+~~rs}

+~y2W{t15r5 +w4r4) = sb (W

j = yzh(z){r5 -pr.+} E gi (66)

where the “dot” denotes differentiation with respect to dimensionless time to scaled with respect to the slow characteristic time for reduction of the Pt atoms,

to = t’/(l/k:L = f/z8 (7)

and

%%o = k,S$OPcon: -k_ n dt’

I b -kSonb,

8, = 1 -ei --82

PC, =r=ha=

thermal characteristic time V

- k:dnE0n?0)

aJOT

‘I, = N,pC,T,

Dynamic modelling of a heterogeneously catalysed system 319

(8)

(9)

(10)

320 M. ALUKO~~~ H.-C. CHANG

k, = k;X,, i=2, -2,3,ox,red

= kp exp ( - E,/R T,)

(11)

= kf=‘exp[y,z/(l +z)]

yi = Ei/RT, i = 1, 2, -2, 3, ox, red (12)

and klef are the (reference) values of ki evaluated at the constant gas temperature Tg, from which values we define the following characteristic times and their ratios:

1 /kyf T’

A’ = l/kffS$oPco = z (13)

D, =““;‘_-& 1 /k’_‘f

(14)

AZ = 1 /k y’ r’

1 /k pS,ol PO, = x (15)

Dz=L!%+ l/k!!{ (16)

l/k% rS= fl===- (17)

70x

l,‘k;ef 7’ y=-=-- (18)

=T =T

7’s = l/k?; (19)

6+L thermal characteristic time

S characteristic time of slow redox step ’ (20)

The dimensionless reaction expressions and rate constants in eq. (6) are:

T1 = $ a(z) 8, -+b(z) 81

r2 = 2Azg(z) 0: -2&c(z) 03

rs = e,t&

r, = e(z) e2 P-5 = + d(z) e1

a(z) = expC(h -n)z/(1+41

b(z) = expC(y- I -_y3Ml +@I

4~) = expC(y-2-7dzlU +dl

d(z) = ewC(k -~3)zlU +z>l e(z) = expC(y, -n)z/(l +z)l WI = expC(y3z/(l +z)l g(z)‘= ewC(y2 -~3)zlU +z>l

w(z) = l/h(z)

‘I1 = ‘IplV’480

tl2 = zlplv. 48

tin = VP/V. 4r

v4 = rl,%,

f15 = ?,%d-

(21)

(22)

(231

The appearance of9 on the right-hand sides of eq. (6) is due to the time dependence of X0 in the definition for Bi in eq. (4). There is a total of 19 dimensionless parameters in eqs (6), viz:

y2, Y-~, y3. yrsd, Y,). (24)

Although dimensionless equations and parameters are convenient for theoretical analyses, in order to retain physical relevance, one can only compare the dimensional values to the observed data. Moreover, certain parameters are directly measurable and they appear in the definitions of some of the above dimensionless parameters. On the other hand, one prefers to work with as many dimensionless parameters or groups of dimensional parameters as possible instead of estimating all physical parameters. Hence, to determine the smallest set of dimensionless and dimensional (physical) parameters that allows us to estimate the measured quantities, we consider the following points.

(i) Explicit notice must be taken of the fact that the experimental data indicate that the gas temperature 7’s and partial pressure ratio otP ( = Pco/Po,) are considered the independent variables whose values can be manipulated externally and are therefore known. For example, the oscillation boundary data were obtained at fixed olP for varying Tg values while oscillatory data are given at specific 0~~ and Tg values as previously stated. Consequently, these variables are not to be estimated, and should be extracted from the definitions of the parameters. With respect to T’s, one rather obvious consequence is that the dimensionless activation energy parameter yi should now be reconsid- ered in its original form of eq. (12). with seven Eis

being estimated rather than yi, The parameter redefinitions necessary for all the other ten parameters besides yi, A, and A, are also straightforward, and can easily be shown to be as follows:

D1 = 0; exp & (ES --El)} s

D2 = D; exp & (Es --E-z) g >

S = kF& 7= exp ( -E&R Tg)

v = & exp 6% lR T,) T 3

P = PO exp {(E, -Ed/R T’ I

qi = $ exp ( - E3/R T,) i = 1, 2, 3 8 0

Vi + (25)

i = 4. 5. s

The dimensionless parameters 04, D; and &’ and the dimensionless parameters k&,, TV, k; and q;(i = 1,s) have obvious definitions and thus represent eleven

more parameters to be estimated in addition to the seven activation energies Ei.


We next consider the redefinitions of AI and AZ_ We note from eqs (13) and (15) that Al and A2 depend on PC0 and Po,, respectively. However, suppose we introduce a “sticking coefficient ratio”, S,,

s, = SEO/S$?z (26)

and using the definition of kT’ from eq. (11) in eqs (13) and (15), it is clear that

AZ = A”, expC(EJ -E,)IRTsl

Ai = S,a,A; expC(& -EI)IRT,I (27)

where

(28)

We have thus extracted ap but have thereby introduced two more parameters to be estimated (S, and A;), bringing the total number to 20.

(ii) In light of overwhelming evidence that the desorption of oxygen is negligible under Turner et al.‘s operating conditions, we can neglect the parameters 05 and EmZ.

From (i) and (ii) above, the parameters to be estimated are the following 18,

P = (S,, A;, D;, $9 El, E-I, Ez. E,, E,,, Ered..

All 18 parameters are needed to fit the oscillograph of Fig. 2 while only the first 13 are needed to estimate the oscillatory boundaries of Fig. 1. This will be clear from the analysis in the next section.

It is important to note here that this new mode1 exhibits a well-known feature in traditional Langmuir-Hinshelwood-Hougen-Watson catalytic reaction modelling, namely a large number of parameters. This gives rise to valid questions concerning the reliability of the parameter estimates with respect to model discrimination (Lapidus and Peterson, 1965). However, as pointed out in the Introduction, we strongly believe that our eventual use here of such parametrically sensitive data as oscillatory and bifurcation-to-oscillation experimental points, rather than the often-used steady-state data, significantly reduces the number of different combinations of acceptable parameter values, and hence increases (though by no means assures) the chances of obtaining a unique set.

PERTURBATION ANALYSIS

If the postulated smallness of the dimensionless parameter 6 is correct (this is subsequently validated from our parameter estimate), then eqs (6) are said to be in a two-time scale singular perturbation form (Chang and Aluko, 1981; Aluko, 1983; Chang and Aluko, 1984b) since 6 multiplies the time derivatives of one set of dynamic variables (i.e. et, t& and z, termed the fast variables) to the exclusion of the other (i.e. y, the slow variable). The applicable asymptotic theory then enables us to approximate the dynamic behaviour of the highly nonlinear system in a drastically simpli-

fied manner. Sales et al. (1982) have exploited this same technique in studying their isothermal model. More general approaches to autonomous systems with two or three time scales can be found in previous works of the present authors, to which readers are referred for details of the perturbation analysis.

If we formally set 6 to zero in eqs (6), then the fast variables are in quasi-steady-state with respect to the slow one. Hence the system trajectories are confined to a slow manifold F defined by the fast equations set to zero:

g’(81, &I Y, 4 = 0 i= 1,2,3 (30)

where the absence of the subscript 6 indicates that S has been set to zero in the original Q$ defined in eqs (6). In most cases, the surfaces of this manifold are stable, and they attract all trajectories in the state-space. However, in cases where I possesses singular points, especially when F overlaps itself around a cusp singularity (Chang and C’alo, 1979), the middle sheet is unstable. If, in addition, the equilibrium points of the overall system, eqs (6). defined by eq. (30) and

8%, ol, Y, 2) = 0, (31)

lie on this middle unstable sheet, then the system exhibits relaxation oscillations in the fast variables as the trajectory alternately leaves one stable surface and lands on another as it reaches the boundaries of the unstable sheet. These sustained relaxation oscillations cannot exist if a stable equilibrium point lies on the stable surface, for then trajectories will be attracted to it instead. Hence, the oscillatory boundaries of Fig. l are defined by the coincidence of the overall system equilibrium point with the boundary of the unstable middle sheet (viz. the singular points of F). The corresponding equations are eqs (30), (31) and the definition of the singular points:

det as?‘9 g27 s3) = o

ate,, ezv 4

as prescribed by the implicit function theorem. Equations (30 j(32) represent five algebraic equa-

tions, and upon eliminating 8i, 02, z and y. one obtains a numerical relationship between Ts and at, on the oscillatory boundaries. Hence, by manipulating the other parameters [only the first l3 of P in eq. (29) occur in eqs (30)-(32)], one can attempt to fit the data points of Fig. 1.

It is interesting to note here that the observed linear relationship between (7” -Ts) and the COZ production rate, RCOl, emphatically reported in the experiments of Turner et al. (198 1) can be easily explained by asymptotic arguments as follows. On the slow manifold F, eq. (30) can be represented as follows:

0 = rl -rs; 0 = T2 --r, (33a)

0 = -z+h(z)y*(tlrrl +42r2+f13rs). Wb)

Substitution of eq. (33a) into eq. (33b) then yields:

-z+h(z)_v*(fll +v2+v3)rs=0. (33c)

322 M. ALUKO and H.-C. CHANG

If we recall the definition of z in eq. (5), then eq. (33~) can be alternatively expressed as follows:

Tc-Ts = k$ exp ( --yJ)h(t)y2r, ‘I’ +” +‘I3 =B k:exp(-Yd’

(334

However, we note that the CO2 turnover number is:

R co2 = k”, exp(-_ydh(z)y’r, (34)

and on applying eqs (25) and (30), one obtains the desired linear dependence:

where R co1 = F,(T, - =,) (35)

FT = k”, ,i, VP. I (36)

Note that the linear relationship of eq. (35) is valid not only for a stable steady-state, but also during the slow- manifold intervals of the oscillations, i.e. the “pulse” and the “induction” flat portions_ The linearity is clearly evident in the experiments, and we will subsequently show that our estimated value of F, com- pares favourably with Turner et al.‘s experimental value (FT x 10 s-l K-l).

ESTIMATION SCHEME

The estimation task at hand is to estimate all 18 model parameters by fitting the bifurcation data of Fig. 1 to eqs (30~(32) and the dynamic data of Fig. 2 to eqs (6). While all 18 model parameters occur in eqs (6), and hence can be estimated from the dynamic data, only 13 of them occur in eqs (30)-(32) and can be estimated from the bifurcation data, the remaining five [more precisely the last five parameters listed in set P of eq. (29)] being found only in terms removed from eqs (6) after the perturbation parameter 6 is set to zero to obtain eqs (3Ok(32). Because the parameter sets of the two independent fits overlap, we develop an iterative estimation procedure to carry out both steady-state optimization and dynamic optimization phases. Since both phases involve estimation of numer- ous parameters from highly nonlinear algebraic or stiff differential equations, our algorithm includes a novel use of steady-state and dynamic behaviour sensitivity coefficients (Lehman and Stark, 1982) which are employed to “freeze” relatively unimportant (insensi- tive) parameters during one phase of the iteration to reduce the complexity of the optimization.

(i) Steady-state optimization At each known “experimental” ap (the partial press-

ure ratio) and current parameter estimates P (maximum dimension = 13; less if some parameters are fixed during this pass of the steady-state phase), the five algebraic equations are solved to obtain T;, the corresponding estimated gas temperature at an oscillation domain boundary. The algebraic equations are solved by Kubicek’s DERPAR routine (Kubicek, 1976), and the iterative corrections to P are done using the IMSL library routine ZXSSQ (IMSL, 1976)

which is an implementation of a finite difference Levenberg-Marquardt algorithm (Levenberg, 1944, Marquardt, 1963) solving nonlinear least-squares problems. A combination of Gauss-Newton and steepest descent methods is used in ZXSSQ, the latter for starting the iterative process and the former due to its rapid convergence characteristics near the minimum. The least-squares objective function to be minimized is

Q,(P) = igl (T’~~T’)2 (37)

where the superscript m denotes measured experimental values and N [ > dim (P)] is the total number of experimental boundary points. Twenty-four points are taken in this study. Note that both boundaries in Fig. 1 are included in the estimation.

(ii) Dynamic optimization The dynamic equations, eqs (6), are integrated via a

GEAR algorithm, Hindmarsh version (Gear, 1971; Hindmarsh, 1974) incorporated into a new algorithm PEFLOQ (Aluko, 1983; Ah&o and Chang, 1984a) developed by the authors for the calculation of both stable and unstable periodic solutions of nonlinear ODES. The integration is carried out at the ap (0.031) and Tg (297°C) values of point P in Fig. 2 and at the current estimates of parameters not frozen by the sensitivity analysis. To ensure that the period of estimated surface temperature oscillations agrees with the measured one, we require the estimate to fit the period as well as minimizing the following objective function,

QatP) = ,gl [ =:@d - TP@i) ’ =gw 1 (38)

where tr are M distinct times (= 25 in this study) within one cycle of the oscillation.

(iii) Sensitivity analysis Before the optimization calculations of any one of

the above two phases, certain parameters are removed from the estimation due to the relative insensitivity of some specified characteristic system behaviours to them. For both the steady-state and the dynamic phases, the sensitivity coefficients &J at a particular value of P are defined as:

(39)

where bi are the numerical values of certain bebaviours at P to be specified later. Following Lehman and Stark (1982), we define finite perturbations in P by the relations:

Ap, = pl(fp - l/fp) (40)

where fp is greater than unity (we choose f = 2 for reasons given in [19]; the denominator of eq. (39) is then equal to 1.5) and Abi are the differences in bi evaluated separately at fpP and (l/f,)P.


For the steady-state phase, Q, [see eq. (37)] is chosen as the only b*. and prior to steady-state optimization, the corresponding 13 sensitivity coefficients are evaluated at the final P value of the preceding dynamic optimization. Only the most sensitive parameters are estimated iteratively in the subsequent steady-state optimization.

With respect to the dynamic optimization phase, we have found Qd to be an inadequate characterization of the Tc oscillation. Instead, we choose the following four characterizing dynamic behaviours:

(i) the total period a; (ii) the pulse period t,;

(iii) the maximum surface temperature z,; and (iv) the minimum surface temperature zmin.

Consequently, there are 4 x 18 = 72 dynamic sensitivity coefficients. These sensitivity coefficients are evaluated at a current P value which comprises those parameters (both estimated and tied) from the preceding steady-state and dynamic optimizations. Again, only the most sensitive parameters are corrected iteratively in the subsequent dynamic opttition. The combination of sensitivity analyses of these characterizing behaviours and the subsequent optimization with respect to Qa are sufficient to produce an excellent fit of the dynamic data.

The sensitivity coefficients prior to a typical dynamic and a typical steady-state phase are tabulated in Tables 1 and 2. The values that are fixed (not es-

timated) during the subsequent optimization are lab& led. The infinity sign indicates disproportionate sensitivity as explained in the table legends and the corresponding parameter must be used in the optimization.

The iterative optimization scheme between the two phases is terminated when all the parameters have converged to within a specified limit or when both Q, and Qd do not improve significantly. We tabulate our final parameter values in Table 3 and demonstrate the theoretical curves in Figs 3 and 4 in comparison to the chosen experimental points. Note that the spikes preceding the pulse period of Fig. 2 are not used as experimental points since they do not seem to repeat in every cycle. These could be noise or initial transients that die out at large time. The simulated periodic solution (which, by definition, is well developed) does not show these spikes. Finally, parameter values recalculated from those quoted in Sales et d’s (1982) isothermal model study are also tabulated in Table 3 for comparison.

VERIFICATION OF ESTIMATED PARAMETERS

We now verify the accuracy of the model and the estimated parameter values by comparing simulated results using parameter values listed in Table 3 with experimental data not used in the estimation. However, before doing so, we first a posteriori justify the perturbation analysis used to approximate the oscillatory boundaries by establishing the smallness of

Table 1. Dynamic sensitivity coefficients

Pulse Max. Min. Parameter Behaviour Total period temp. temp.

No. parameter period f, 2, Gin Comment

1 S, w+ * -0.21 -0.75 2 A: 005 -0.67 -0.60 -0.44

3 w CA 0.44 0.62 0.72

4 If -0.37 -1.2 0 0

5 El -0.46 0.133 -0.36 0.08 6 E-1 co+ 11 T ll

7 & cop -1.02 -0.38 0.42 8 & 006 -2.17 -0.83 -0.83

9 E OX 0.53 0.21 0 1.7

10 E red CO+ * -0.95 -3.42

11 tl: -0.14 0.06 0.59 0.05 12 & -0.08 0.01 0.29 0.04 Fix++

13 SJ -0.01 0.02 0.07 0.11 Fix

14 k& - 14.7 -1.19 0 0.95 15 k”, 0 0 0 0 Fix

16 5T -0.29 -0.05 0.001 1.01 17 rt: 0 0 0 0 Fix

18 v”s - 0.29 - 0.05 0.001 -1.01

tsteady state at 2P7, l/2 PJ . *Und&ned. *Steady state at 2P;. IlSteady state at 1/2P,“. *Steady state at 2PT, 1/2P; (2 = 0). +TThat is, during dynamic optimization.


Table 2. Steady-state sensitivity coefficients Table 3. Final parameter values

Parameter Behaviour Sum of squares Comment No. parameter Ss

1 Sr -2.44 2 A02 -0.65 Fix+ 3 D9 - 0.68 Fix 4 p” - 1.93 5 El 0.633 Fix 6 E-1 00% 7 E2 -1.613 8 E3 5.73 9 E ox 11.2

10 E red 16.8 11 tl; 0.160 Fix 12 t102 0.0076 Fix 13 ‘I; 0.0019 Fix

tTbat is, during steady-state optimization. *Both bifurcation curves are significantly displaced.

Parameter Parameters TSM Final estimated No. and units values values

: 3 4 5 6

z 9

10

::

S, A”, DY Ir” E 1, kcal/mol E_,, kcal/mol EZ , kcal/mol E3, kcal/mol E ox, kcal/mol E rd, kcal/mol rlP.K ~5. K &,K k&,,, min- i k”,, min-’

10.0 1.33 x 10-S 29.0 4.60 X 10-3

0 20

1 10

1 10

13 14 15 16 17 18

1.33 X 103 5.6 x 10”

6. Note that the fitting of the dynamic oscillations does not involve the assumption of a vanishingly small 6, hence the dynamic fit provides us with an estimated value of 6. Using the final parameter values in Table 3, we calculate its value at 297°C gas temperature as 6 = 6.0 x lo-‘, which amply justifies the perturbation analysis. We should also note here that the parameters in Table 3 also yield an F, value [the proportionality constant in eq. (35)] of 400 min- r K- ‘, which, being a factor of only 1.5 lower than the experimental value (600), is considered to be in good agreement.

induction periods of the relaxation oscillations at different gas phase temperatures are depicted_ In Figs 7 and 8, the induction rate (= l/induction period) as a function of 7’s at two values of Pco/Po, are compared in Figs 9, 10 and 11.

In Fig. 5, we compare the simulated and measured surface temperature oscillations at a point different from P of Fig. 2. In Fig. 6, the simulated and measured surface temperature increase during the pulse and

It is evident from Figs 7 and 8 that the predicted induction period is about twice the observed value. This is also clear from the two oscillographs of Fig. 5.

It appears that the surface Pt atoms are reduced (and hence reactivated) much faster than model prediction. Hence, it seems that k&t from our estimation is perhaps too low. This conclusion is amply supported by the fact that of the five parameters which affect only

7oc

T9(K)

5oc

11.2 1.96 x lo-’

16.5 1.10 x lo-” 0.20

19.7 0.755 2.94 2.19

10.1 7.98 x lo6 5.20 Y lo6 1.06 X 106 2.70x 10” 5.61 x 10” 1.55 x 10-Z 7.75 4.50 X 10’

I I I

Fig. 3. Theoretical oscillation boundaries corresponding to the final parameter values.

Dynamic modelling of a heterogeneously catalysed system

01 0 IO 2

TIME (min.)

Fig. 4. Relaxation oscillations for the final parameter values (over two periods).

t

I

360 EXPEFhENTA~

Tg =600 K 1 (a)

G 32012 e

I e 360- S1MULATION

I I

Tg=600 K lb)

0 20 40 TIME (min.)

Fig. 5. Surface temperature oscillations at P CO/PO, = 0.031 and Tg = 600 K. (a) Experimental (Turner er al.. 1981); (b) simulation.

325

the dynamic bebaviour (i.e. parameters 14-18 of Table I), perturbations in k&, provide by far the

only - 1.19); hence much of the total period sensitivity to k”

largest sensitivity of the induction period. [Note that rd can be ascribed to that of the induction period.]

the induction period is defined as the total period (with Unfortunately, at other k&, values, our manually

a kL sensitivity coefficient of - 14.2; see Table 1) attempted fits of Fig. 4 deteriorate considerably. If the dynamic data of Fig. 5(a) were to be included in our

minus the pulse period (with a sensitivity coefficient of estimation scheme, we would expect the agreement to

326 M. ALIJKO and H.-C. CHANG

I I I I I

30- -1.5+1 kcal/mole, w

\-

-0.5 kcal /mole

MODEL

II I7 I8 I9

104/Tg (K-l)

Fig. 6. Arrhenius plots ofexperimental and theoretical surface temperature increase during the pulse (upper branch) and induction (lower branch) periods at a fixed eP = 0.02.

i --

; W

- PC, / PO* =o.oz

0.1 z-

A I I I

16.5 17.5 18.5 IO41 Tq (K-‘1

Fig. 7. Arrhenius plots of experimental and theoretical induction rates corresponding to Fig. 6.

the observed values in Figs &8 to improve con- region in Figs 10 and 11 should not be surprising since siderably. The less satisfactory prediction of the induc- the entire oscillatory boundaries are fitted with good tion interval also explains the disagreement in the accuracy in Fig. 3, especially at large ap values. lower branches of Figs 9-l 1. The upper branches, In summarizing the predictive capability of the representing the pulse interval, are predicted ac- estimated model, the pulse interval of a given catalyst curately. The excellent description of the oscillatory temperature oscillation seems to be well described, but

327 Dynamic modelling of a heterogeneously catalysed system

0.1 b I I I I

52 kcal/mole

I I I I I I8 I9 20

104/T, (K -‘)

Fig. 8. Arrhenius plots of experimental and theoretical induction rates at a fixed Pco/Po, = 0.01. Experimental data are from Sales ef al. (1982).

the induction interval tends to be longer than that of the actual system. Improvement of the parameter values can, of course, be effixted by incorporating additional dynamic data in the estimation. On our part, the limited accuracy with which we can read data from graphs provided by Turner et al. (198 1) (our only present source of data) and the irregularity of most of the oscillatory data presented [for example, see Fig. S(a)] do not warrant such an effort, which would in any case require increased computational capa- bilities. Consequently, it would be too optimistic to require our present model to simulate all dynamic data accurately when only one oscillation (we chose the best characterized) is used to estimate the parameters. That the model performs as well as it does must be attributed to the richness of the information already contained in the bifurcation data of Fig. 1. As noted above, the parameter k& whose inaccurate estimate accounts for the major flaw of the model in predicting the induction interval, can only be obtained from the dynamic optimization and not from the bifurcation data.

PREDICTIONS OF SYSTEM BEHAMOUR

In previous sections, we showed how available experimental data were used both to estimate model parameters and then to verify them. We now seek to determine, via a numerical study using these same estimated parameter values, what new behaviour (if any) may be obtained in independent parameter space outside that employed so far (i.e. 0 < aP < O-04,300 K < Tg < 700 K; see Figs 1 and 3). Such extrapolations,

if eventually confirmed by experiment, would further validate the model.

A summary of the results of this effort is shown in Fig. 12. This represents an extension of the theoretical oscillation boundary curves shown in Fig. 3 (maximum aP = 0.04) to aP = 0.3. It is observed from Fig. 12 that while the bifurcation gas temperature Tg of the lower curve of Fig. 3 continues to increase monotonically as a function of the partial pressure ratio CY~, its upper boundary attains a maximum at point E of Fig. 12, decreases thereafter to intersect the other boundary at point F, exhibits a minimum at point G, after which it also begins to increase monotonically. We thus obtain the interesting behaviour whereby the aP-Ts space shown is divided into three disjoint regions labelled:

(a) region I which , for reasons discussed previously, consists of oscillatory states;

(b) region II, comprised of unique stable steady-states; (c) region III, which, as indicated by the crossing of

the oscillatory boundaries at point F, represents simultaneous intersections of the upper and lower branches of I. Hence, there are two stable steady- states (and one unstable one) in this region.

The prediction of multiple steady-states in region III is confirmed by numerical simulations at representative points. For Ts = 400 K, the upper steady-state ranges from 700 to 900 K for aP from 0.06 to 0.25. This upper steady-state is hence lOO-300°C higher than the lower (quenched) state and the corresponding extinction and ignition phenomena should


o -- (a)

100 ” -2 kcal/mole

kca I /mole

EXPERIMENTAL

t I (b) 1 IOOr

-0.4 kcal/mole

yur 6 fi;;A r;rAA

4 kcat/mole A

MODEL IO I I I I I I I

I7 I8 I9 20 21 104/Tg (K-l)

Fig. 9. Arrhenius plots of reaction rate (as CO1 turnover number) at a fixed P,&Po, = 0.01. (a) Experimental, from Sales et al. (1982); (b) theoretical. In each plot, the upper (lower) branch corresponds

to the pulse (induction) period.

be easily detectable. For example, one could exper-

imentally increase aP while holding the gas tempera-

ture constant at Ts = 600 K, and the system will follow the upper branch until point D where it should extinguish dramatically to the lower steady-state. Upon reversal of the scan direction at this point, the system will remain at the lower branch until point C, where it ignites and jumps to the upper branch. If such a hysteresis is located experimentally, the predictive capability of this model is then established.

As a final check of the perturbation analysis, we compare its prediction of bifurcation points of A, B, C and D for Tg = 600 Kin Fig. 12 to the actual computed values in Table 4. The closeness of the sets of values further reveals the utility of our perturbation analysis.

CONCLUSION

In this work, we have employed bifurcation and oscillatory data to estimate the parameters in a highly nonlinear catalytic model via a novel estimation

method. The nonisothermal model incorporates a surface redox scheme with a Langmuir-Hinshelwood reaction mechanism for CO oxidation. Excellent (good) matching between predicted and fitted oscillatory (bifurcation) data and good agreement with unfitted experimental data provide preliminary vali- dation of the model. The model’s ability to simulate the surface temperature oscillations represents a desirable and significant improvement over gales et al.% (1982) isothermal model. We believe that the richness of information already contained in the bifurcation data enabled the success of this effort, but we also expect that improved parameter estimates can be further obtained by the use of additional dynamic data. It would be gratifying if the bifurcation sequences predicted from Fig. 12 are eventually confirmed experimentally.

Finally, two important aspects of this work are worth highlighting:

(1) We strongly believe that our results indicate that


t

EXPERtMENTAL 200 Tg=523K

(0) 0

0

100 t

0 0 A 1

t 0 A

0 * * *A

0 A*

olno 1 I I I 1 0 0.0 I 0.02

pco’ PO,

Fig. 10. CO1 turnover numbers at a f&d TS 7 523 K. (a) Experimental (Sales et al.. 3982); (b) theoretlcal.

1.5 I I I I t 30

EXPERIMENTAL e

Tg=573K e

4 (a) 30

I 1 t 4 ’ t

MODEL Tg=573 K

$ W

up REGtON OF Fig.3 -1

300 I 0 0.1 a2 0.3

PC0 ‘PO3

Fig. 12. Theoretical oscillation boundaries in 7”-ap space. The maximum alp value is now 0.3 (compare with Fig. 3).

--_---__

Tg(K)

!500

(2)

Fig. 11. CO2 production rates (%) at a fixed gas temperature Acknowledgements-This work was supported by the Tp = 300°C = 573 K. (a) Experimental (Turner et al.. 1981); National Science Foundation Grant No. PE 81-05155 and

(b) theoretical. American Chemical Society PRF Grant No. 12949-G7.

Table 4. Comparison of predicted and computed bifurcation points for Ta = 600 K

tap, T, K)

Point Predicted values Numerical label from Fig. 12 values

A (0.0196, 609) (0.0200.609) B (0.0553,611) (0.0555.610) C (0.0681, 700) (0.0652,693) D (0.2539883) (0.255.880)

the multistage parameter estimation procedure

introduced here, which enables the systematic algorithmic “steering” of initial parameter es: timates to such values that ensure good agreement between experiment and theory over a wide set of operating conditions, is worth testing further in future work for possible improvements and detailed study of convergence characteristics. The method is not confined to parameter estimation in heterogeneous catalytic systems alone, and should find application in other physico-chemical systems in which the use of multi-time-scale analysis is justifiable, and/or in those which also yield oscillatory behaviour. In the wider context ofcatalytic reactor design, our model is clearly approximate since we have neglected changes in gas-phase concentrations and mass-transfer resistance between bulk gas and

catalyst surface. However, the parameter estimates

which we have obtained here, which are best considered as representing the intrinsic kinetics of CO oxidation of a platinum surf&e, may then be employed as initial estimates when a more accurate model is proposed.

330

0, Aiv Di

A;, 0;

bi

Ei

fP

FT

h ki, k;, k; kf ” NA nf PO,, co P

Pi

Es* Qd

4i R

:.. s;

&

=“v =c t’

x0

xOT

Y

z

M. ALUKO and H.C. CHANG

NOTATION

catalyst surface area/volume of catalyst ratios of characteristic time of reaction to adsorption, desorption parameters redefined from Ai, Di[see eq.

(WI behaviour number i chosen for sensitivity analysis activation energy a sensitivity analysis coefficient [see eq.

(4O)l constant of proportionality between CO, reaction rate and (T, - Tg) [see eqs (35) and (36)] heat-transfer coefficient reaction rate constants [see eq. (1 1)] reference ki ( = ki (T,)) Avogadro number surface density of species i partial pressure of O2 and CO parameter number i of set P estimated parameter set [see eq. (29)] objective function for steady-state, dynamic optimizations [see eqs (37) and

(3g)l heats of reaction gas constant dimensionless rate functions sensitivity coefficient initial sticking coefficient of species i ratio of sticking coefficients, SgO/Sgl gas, catalyst surface temperature real time

Superscripts

: adsorption desorption

r reaction S surface

Ah&o, M. E., 1983, Multiple-time-scale analysis of heterogc- nexus catalytic reaction systems, Ph.D. Thesis, University of California at Santa Barbara (available form University Microfilms International, Ann Arbor).

Aluko, M. and Chang, H.-C., 1984a. Multi-scale analysis of exotic dynamics in surface catalyzed reactions- II. Quantitative parameter space analysis of an extended Langmuir-Hinshelwood reaction scheme. Chem. Engng Sci. 39, 51-64.

Aluko, M. and Chang, H.-C., 1984b, PEFLOQ: an algorithm for the bifurcational analvsis of oeriodic solutions of autonomous systems. Corn;. them. kngng 8, 355-365.

Beskov. V. S.. 1976. React. Kinet. Caral. L&t. 4, 351 (in Russian).

Beusch, H.. Fieguth, P. and Wicke, E., 1972. Unstable behaviour of chemical reactions at single particles. Adu. Chem. Ser. 109,615-621.

Chang, H.-C. and Aluko, M., 1981, A quasi-steady-state analysis of the dynamics of two-species heterogeneous catalytic reactions. Chem. Enana Sci. 36. 1611-1622.

Char&H.-C. and Ah&o, M.. 19849 Different paths to chaos in a stiff CO oxidation system. In Frontiers in Chemical Reaction Engineering (Edited by Doraiswamy, L. K. and Mashelkar, R. A.), Proc. Int. Conf. Chem. React. Engng, Poona, India, 1984. Vol. II. DD. 361-373. John Wilev. New York.

__ _.

Chang, H.-C. and Aluko, M., 1984b. Multi-scale analysis of exotic dynamics in surface catalyzed reactions. I. Justification and preliminary model discriminations. C&m. Engng Sci. 39, 36-50.

surface density of undeactivated sites maximum value of X0 undeactivated fraction of maximum active sites X,/XoT dimensionless catalyst surface temperature (T, - T,)/T,

Chang, H.-C. and Calo, J. M., 1979, A priori estimation of chemical relaxation oscillations via a sinsular oerturbation technique. Chem. Engng Commun. 3, 437446.

Ghana H.-C. and Chen. L.-H., 1984. Bifurcation character&tics of nonlinear systems unddr conventional PID control. Chem. Engng Sci. 39, 1127-l 142.

CutliD. M. B. and Kennev, C. N.. 1978. Limit cvcle Dhenom- ena huring catalytic oxidation. reactions eve; a s;pported platinum catalyst, Proc. 5th Int. Symp. Chem. React. Engng, Nouston, A.C.S. Symp. Ser. 65, 475486.

Eckert, E., Hlavacek, V. and Marek, M.. 1973, Catalytic

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Dynamic modelling of a heterogeneously catalysed system with stiff Hopf bifurcations

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