J. Phys. Chem. 1995,99, 12835-12845
12835
Further Measurements on the Oscillatory Cool Flame Oxidation of Acetaldehyde and Comparison with Reaction Mechanism Models Lisa L. Skrumeda and John Ross* Department of Chemistry, Stanford University, Stanford, California 94305 Received: June 8, 1999
Experimental and theoretical studies are reported on oscillatory cool flame oxidation of acetaldehyde by oxygen in a CSTR. Several experimental tests are designed to provide information about the reaction mechanism. The concentrations of acetaldehyde, oxygen, methane, methanol, carbon dioxide, water, hydrogen peroxide, and formaldehyde are measured as a function of time with mass spectrometry. Relative amplitude of oscillation, relative phase of oscillation, and the regulation of several species in response to a step perturbation of either acetaldehyde or oxygen inflow are computed from the measurements of concentration. These quantities are calculated for two models of the acetaldehyde reaction. Measurements suggest the eight cited species are chemical “nonessential” variables. (Their concentrations can be individually fixed while other variables oscillate.) Calculations indicate that the “essential” variables in each model are a short-lived chemical species, several radicals, and the internal gas temperature. The measurements of concentration shift regulation of the eight cited species, in response to step perturbations of acetaldehyde or oxygen inflow, and the calculations made with the two models agree except for the responses of carbon dioxide and water to a step perturbation of acetaldehyde inflow. The concentration shift regulation of nine variables for a step perturbation of the bath temperature is inferred from the literature. Calculations of concentration shift regulation for the first model show agreement for three variables and disagreement for four variables; calculations for the second model show agreement for six variables and disagreement for two variables. Measurements and calculations of the relative phases of oscillation between acetaldehyde, oxygen, methane, methanol, internal temperature, and hydroxyl radical agree. Both models are “category 2” oscillators. The “negative feedback” species is peracetic acid in the first model and methyl hydroperoxide in the second. The “autocatalytic” species in both models are groups of radical intermediates.
I. Introduction Cool flame oxidations are one category of oscillatory reaction.’ The oxidation of acetaldehyde has been studied extensively2-” and serves as an exemplar of complex systems with oscillatory and stationary states. Gray et al.4 have shown the existence of at least five different dynamic states for the acetaldehyde oxidation system in a CSTR, including an oscillatory state and a stationary state connected by a supercritical Hopf bifurcation. Pugh et al.6-7and Tsujimoto et al.Il have studied the effects of periodic perturbations on the oscillatory oxidation of acetaldehyde. The construction of complex reaction mechanisms is a difficult task; models of complex reaction mechanisms are typically built up from mechanisms for similar reactions of related simpler molecules and then are evaluated by comparing temporal series and the range of elicited dynamic behavior. Several models have been p r o p ~ s e d ~ , * - ~that , ’ ~ -recreate ’~ to varying degrees of likeness the oscillatory behavior observed in the acetaldehyde system. In a series of new experiments have been devised to provide information that contributes to deducing a reaction mechanism. Eiswirth et al.24and Chevalier et al.27propose tests to identify the roles of the species in the oscillatory mechanism and the category of the oscillator. They describe how to constant a skeleton mechanism from this information. These new experiments also provide a more extensive basis for comparison of theory with experiment. We present the results of applying some of these tests to the acetaldehyde oxidation system. The measurements of relative amplitude of oscillation @Abstractpublished in Advance ACS Absrructs, August 1, 1995.
presented here show that a set of reactants and intermediates of the experimental system are nonessential species. If a particular variable can be fixed at a constant value while the other variables in the system oscillate, then this variable is a nonessential variable. However, if it is no longer possible for the other variables in the system to oscillate when a particular variable is fixed, then this variable is an essential The measurements are compared with the predictions of two thermokinetic models of oscillatory cool flame acetaldehyde oxidation, labeled PR8 and GGGH.I4 Comparisons are made of the relative phase of oscillation2’ and the concentration shift r e g ~ l a t i o n . ~In~ addition, .~~ the relative phase of oscillation and the concentration shift regulation of the essential species in the two models are used to categorize the oscillator in each model. The relative phase of oscillation and concentration shift regulation relationships among the essential species of each category of oscillator are shown in the Appendix. 11. Experiments 1I.A. Experimental Apparatus. The components of the experimental system are represented in Figure 1 and have been described in part A 210 cm3 (approximately 8 cm in height and 6 cm in diameter) Pyrex CSTR is seated in a double-walled, thermostated oven. Acetaldehyde (Baker) vapor is supplied to the CSTR by a 1L cylinder sitting in a temperature-controlled water bath. Oxygen, nitrogen, or helium is introduced into the CSTR in a separate inflow line. Both fuel lines have flow meters, pressure gauges, and electronic mass flow controllers (MKS Instruments, Type 1259A). Inside the oven, the gases flow through Pyrex tubing of sufficient length to allow the gases to come to the temperature of the oven, Tbath,
0022-365419512099- 12835$09.00/0 0 1995 American Chemical Society
12836 J. Phys. Chem., Vol. 99, No. 34, 1995
---
Diessure
Figure 1. Schematic of the acetaldehyde oxidation apparatus, with the inflows of oxygen and acetaldehyde (ACH) vapor; the mass flow controllers (MFCs), which produce either a constant inflow rate or a step inflow rate; the Pyrex reactor (CSTR) with stirring column; the oven, which controls the bath temperature; the metering valve, which controls the pressure within the reactor; the photomultiplier tube (PMT), which is used to monitor the light emission; the capillary, which samples gases from the interior of the reactor; and the mass spectrometer, which is used to monitor the concentrations of several species.
TABLE 1: Typical Values for the Constraints Used in the Experiments constraint
typical values
inflow rate CH3CHO inflow rate bath temperature, Tbath pressure
2.7-6.3 mmoVmin 3.7-9.0 mmoVmin 533-621 K 80- 160 TOIT
0 2
before they enter the CSTR. A stimng column with several blades and a Pyrex-encased magnetic stir bar are used to maintain the gases in the CSTR in a well-mixed condition. A stir plate is underneath the oven. The outflow line is an approximately 5 mm diameter tube connected to an ice-water trap, in which a small amount of soot typically collects. This is followed by a pressure gauge and a metering valve (Nupro SS-4MG). The metering valve is used to control the range of pressures obtained in the CSTR. Next there is a dry ice-2-propanol (IPA) trap that collects unreacted acetaldehyde and other compounds. The last component of the exit line is a mechanical pump. A 391 pm i.d. polymer-coated glass capillary (Polymicro Technologies) runs through the exit tube and extends a short distance (approximately 5 mm) into the reactor. Gases sampled from the interior of the CSTR via this capillary are diverted to a quadrupole mass spectrometer (Ametek Dycor 100M). Six copper-constantan thermocouple probes (Omega) are positioned inside the oven, near the surface of the reactor, and near the gas inflow lines and are used to indicate Tbath. The response time of the thermocouples is less than 1 s but is inconsequential since the bath temperature is held constant. The bath temperature of the oven is stable to within 0.5 K over several hours. 1I.B. Experimental Procedure. In a typical experiment, the oven is preheated to the desired bath temperature Tbath, the exit orifice size is set with the metering valve, and the stirrer is tumed on. Then the acetaldehyde and oxygen flows are started. After a short time, which is usually between a few minutes and an hour, the reaction settles into one of the dynamic states described previously for the acetaldehyde-oxygen oxidation r e a ~ t i o n .For ~ the extemal constraints typically used in these experiments (Table l), the stable attractor reached is either a stationary state or an oscillatory state. The period of the
Skrumeda and Ross oscillatory states studied in this work ranges from 4.0 to 12.3 s depending on the bath temperature. For extemal constraints that correspond to the oscillatory states, the residence time is between 3.6 and 5.2 s. The bath temperature corresponding to a Hopf bifurcation ranged from around 558 K at the higher reactant inflow rates to around 610 K at the lower reactant inflow rates. The period of oscillation very near these Hopf bifurcations is about 1 s. The primary data used for analysis are the intensities of the species fragment concentrations measured by the mass spectrometer. We choose to follow with the mass spectrometer only those values of mle for which a species fragment intensity is easily resolved from noise and background intensities. In order to measure the noise level and background content of the cool flame mass spectrum, helium is flowed through the heated CSTR at the pressures used in the oxidation experiments, and a mass spectrum is recorded for mle values 1-50. This procedure is repeated with both nitrogen and oxygen. The noise level in the cool flame mass spectrum is taken to be the intensity at mle = 6 (for which there is no ion source); the intensity at mle = 6 varied only slightly among helium, nitrogen, and oxygen. The peaks present in the helium spectrum, with the exception of the peak at mle = 4, make up the background spectrum. The intensities of these peaks are in fairly good agreement with the intensity of the corresponding peaks in the spectra taken with nitrogen and oxygen (with the exception of peaks at mle = 7, 14, 8, and 16 for which there are strong contributions made by nitrogen and oxygen). An average of the intensities from the three spectra obtained with He, Nz, and 0 2 is used for the intensity of the background in the cool flame spectrum. As much as half the amount of acetaldehyde that enters the reactor during an oxidation experiment leaves the reactor through the exit tube. In order to subtract out the contribution made by acetaldehyde to an oxidation spectrum, we assume that during an oxidation experiment the intensity at mle = 43 is due entirely to acetaldehyde. Then after measuring the mass spectrum of acetaldehyde alone, we subtract a proportional amount from each of the peaks in the oxidation spectrum to which acetaldehyde makes a contribution. After correcting the oxidation mass spectra for background and acetaldehyde contribution, species sources of the strong peaks are identified. The concentrations of eight species can be measured as a function of time without much interference from other sources. They are CH3CHO (43), 0 2 (32), COZ(22, 44),H20 (17, 18), CH3OH (31), CHz0 (30), C& (16), and H202 (34). Both CH20 and C2H6 are possible sources at mle = 30; however, on the basis of the work of Bradley and Jones3' and consistent with the conclusions drawn by Gray et al.," we assume that CH20 is the predominant source. Examples of concentrations of oscillatory species are shown in Figure 2. Up to five species are followed simultaneously. The concentrations (arbitrary units) of three sets of simultaneously monitored species are shown for a 60 s span. The period of oscillation in all three sets is 6.9 s. Acetaldehyde (mle = 43) is monitored in each set of five species so that comparisons of phase and amplitude may be made between species from different sets by using acetaldehyde as a reference. II.C. Experimental Results. 1. Relative Amplitude. Relative amplitude of oscillation is defined as half the peak-to-peak amplitude divided by the average amplitude. The relative amplitude of oscillation is calculated for the concentrations of oscillatory species acquired at two combinations of reactant inflow rates. For each combination of inflow rates, two bath temperatures are used, and we obtain four separate data sets on
Oscillatory Cool Flame Oxidation of Acetaldehyde 3.057
TABLE 2: Relative Amplitude of Oscillation (in percent) Measured for Eight Species for Four Data Set@
7
1.
.E
2
8
6
HO ,
5.OE-7 3.6E.7
*
. - . . . . . . . . .. . . ........ . .-. ................ I I .. .. .'. .. . .' -, . , ... . ............. :' .: ... .
1
:.I
* W
*:*
. * e
*.
"
:*
:*
**
**'
.*
.
9
-
:*
* e.
*
.
.
9
10
0
.
.
e
G
2.8E-7
inflow rate (mmoYmin) CH3CHO inflow rate (mmoYmin) bath temperature (K) pressure (Torr) period (s) 0 2
8 .
CH,CHO
J. Phys. Chem., Vol. 99, No. 34, 1995 12837
20
30 Time (sec)
50
40
60
6.3 6.3 9.0 9.0 533 533 142-145 156-159 4.6 4.0
CH3CHO
11 8.0
6.6 4.9 b
co2
12 1.9 4.1 8.5
H20 CH4
9.3
0 2
H202 CH30H CH2O
1.9 3.8 5.7 1.0 5.7
1.o
2.1 3.7 537 86-93 12.3
2.7 3.7 550
86-87 6.9
11
9.5 9.1 6.5 10 11 1.0 17
8.7 7.4 8.4 3.5 8.1 13 1.0 13
a The reactant inflow rates and the bath temperature were varied between data sets. Noisy.
TABLE 3: Relative Phase of Oscillation (in degrees) and Estimated Error of Eight Species for Two Experimental Data Set@ 0 2
inflow rate (mmollmin)
CH3CHO inflow rate (mmoVmin) bath temperature (K) 1.20E-7
..
pressure (Torr) period (s) CH3CHO (ref)
. *
4
0 2
H202 CH30H CH2O
c02 H20 CH4 a
a
.*
2.8E-7
0
2.9E-9
HO ,, 2.3E-9
.; . * . .
.
20
10
I
- - . .. . . . '. * . . . . . . . .
.
,
30 Time (sec)
50
40
60
e,
. n
*,;
~
W
W
W
3.8E-7
0.
. . .... ...... . .. .;. . . . . . . . . ..:. . . . . :. ..,.. . .
(P
8
2.2c.7
s
3
0
1
.. .., .. .. ... . . .. . . . ." . .., . ..: . . . .: 9.
8.
.*
*.
.*
.
0 .
:*
0 .
,
2.8E.7
0
10
20
30 Time (sec)
40
.*
*
*,*
50
60
Figure 2. Plots of typical examples of the concentrations of several species versus time during the oscillatory oxidation of acetaldehyde. The species within each of the three sets were monitored simultaneously under the following constraints: bath temperature, 537 K; acetaldehyde inflow rate, 4.5 mmoVmin; oxygen inflow rate 3.2 mmoYmin; pressure, 86-93 Torr. The three sets were taken sequentially with only a few minutes between sets. First set: C h , H20, CH3CHO. Second set: CH20, CH3OH, 02, CH3CHO. Third set: H202, C02, CH3CHO. The period of oscillation is 6.9 s. the concentration of several species as a function of time. Data from one of these sets is shown in Figure 2. The relative amplitude is computed for several periods and averaged to give
2.7 3.7 537 86-93 12.3
2.7 3.7 550 86-87 6.9
0 3167 1 6 % 10 182 f 3 157 f 6 1 3 0 f 11 5 f 4 156 5
0 18f4 O & 10 111 f 9 158 f 15 153f7 341 f 15 174 f 3
+
The phase of CH3CHO is the reference.
the relative amplitude of a particular species. The relative amplitudes are compiled in Table 2. Relative amplitudes should only be compared within one data set taken at one combination of conditions and not compared between data sets taken at different conditions. In Table 2 the difference in period at one combination of flow rates reflects a difference in bath temperature. (A higher bath temperature yields a shorter period.) For each data set, the relative amplitude of CH3CHO is calculated from three time series since this species is monitored in each group of simultaneously monitored species (see Figure 2). In the three 12.3 s time series, the values of relative amplitude of CH3CHO are 11, 11, and 12; in the three 6.9 s time series (shown in Figure 2), they are 8.8, 8.7, and 8.7. These give an indication of the error associated with the measurements and suggest that the system dynamics are not changing significantly during the course of the experiment. 2 . Relative Phase. Relative phase of oscillation is defined as the average of the phase difference between the maxima of two oscillatory time series and the phase difference between the minima of the two oscillatory time series. The relative phase of oscillation is computed for the four data sets described in the Relative Amplitude section of Experimental Results. The maxima and minima of the acetaldehyde time series are used as references in the relative phase calculations. Relative phase is computed over several periods and averaged to give the relative phase of each species with respect to acetaldehyde. Table 3 shows the relative phase computed for the data sets with 6.9 and 12.3 s periods, while Figure 3 depicts the range of relative phase seem across all four data sets. H20 and COZare not included in the relative phase summary because they show large variation among the four data sets. Included in the summary is a result inferred from Pugh et aL8 They measured
Skrumeda and Ross
12838 J. Phys. Chem., Vol. 99, No. 34, 1995 90’ 135”
45”
180“
-90’
Figure 3. Summary of the results of the experimental relative phase of oscillation calculated with respect to CH3CH0, for six stable species from four experimental data sets. For each species, the range of relative phases observed in the four experiments is indicated by an arc connecting the symbols corresponding to the species. The relative phase of oscillation of the OH radical and the intemal temperature (0)are inferred from experimental data presented in ref 8. Experimental constraints are listed in Table 2.
2.5E-7 5.4E-7
-
A
n
580
620
600
L
Bath temperature (K)
Figure 5. Stationary state concentrations of several species and intemal temperature-bath temperature (AQ measured in an acetaldehyde oxidation CSTR system for a range of bath temperatures. Experimental conditions: pressure, 120 Torr; acetaldehyde and oxygen inflow rates are equal and set so that the residence time is 5 s. [Redrawn from Fig. 4 of ref 4.1 TABLE 4: Summary of Concentration Shift Regulation Results (“+” for Normal Regulation, “-” for Inverse Regulation, “0” for No Change)” constraint perturbed CH3CHO bath 0 2 species tempb inflow inflow monitored
I
CH*o
+ I
3
CH3CHO 0 2
HzOz CH3OH CH20 COZ H2O CHI CH3C03H
co
Tintemal 3.35E-7 1.9E-7
.----------------0
30
60
90
120 150 Time (sec)
180
210
240
Figure 4. Example of a concentration shift regulation experiment performed at a stable stationary state of the acetaldehyde oxidation system. Shown are the responses of the eight species listed in the Experimental Procedure section to a 9% decrease of the inflow rate of CH3CHO. The eight species were followed simultaneously. The inflow rate is perturbed shortly after 120 s. Experimental conditions: bath temperature, 595 K; acetaldehyde inflow rate, 9.0 mmol/min before the shift and 8.2 mmol/min after the shift; oxygen inflow rate, 6.3 mmoY min; pressure, 157 Torr. the relative phase between the OH radical, the intemal reactor temperature Tintemal, and CH3CHO. The OH radical and Tintemal are in phase to within 15”, and they lag CH3CHO by approximately 90”. 3. Concentration Shz3 Regulation (CSR). In these experiments the constraints are chosen so that the oxidation reaction settles into a stable stationary state that is near a supercritical Hopf bifurcation after the reactant inflows are started. A perturbation is applied in the form of a small increase or decrease in either the acetaldehyde or oxygen inflow rate. The reaction then evolves to a new stable stationary state. The change from the initial stationary state to the final stationary state is recorded as follows: for normal regulation, an increased response for a positive perturbation, or a decreased response for a negative perturbation; “-” for inverse regulation, an increased response
“+”
-
+
+-
C
+ +
+ +
+or0 +or0
-or0 + or 0
C
C
C
C
C
C
+
The response of eight species was mesaured for shifts in the oxygen inflow rate and for shifts in the acetaldehyde inflow rate. The regulation of several species for a shift in bath temperature was inferred from Figure 4 of ref. 4. “f or 0” indicates that normal regulation was seen in some experiments while no change was seen in other experiments. The results for perturbation of bath temperature are inferred from experimental data presented in Figure 4 of ref 4. Not available. for a negative perturbation, or a decreased response for a positive perturbation; or “0” for no change. An example of the response of several species to a 9% decrease of the inflow rate of acetaldehyde is shown in Figure 4. The regulation of the concentration of several variables for a change in bath temperature is inferred from data published by Gray et aL4 Redrawn here in Figure 5 , the stationary state concentrations of several species and the excess temperature (equal to the intemal temperature minus the bath temperature) are plotted as a function of bath temperature for an acetaldehyde oxidation system similar to the one used here. Concentrations such as H20 and CO that increase with increasing bath temperature exhibit normal regulation; concentrations such as CH2O and CH30H that decrease with increasing bath temperature exhibit inverse regulation. A summary of all the CSR results is shown in Table 4. Both positive and negative shifts of varying size are made in the acetaldehyde inflow rate during four separate CH3CHO CSR experiments. Different combinations of initial inflow rates and bath temperature are used in the four CH3CHO CSR experiments in order to examine the regulation at several stationary states. For H20 and CH4, two of the CSR experiments show a change in concentrations while two of the CSR experiments do not.
Oscillatory Cool Flame Oxidation of Acetaldehyde
J. Phys. Chem., Vol. 99, No. 34, 1995 12839
TABLE 5: Definitions and/or Typical Values for the Parameters Used in the Calculations with the PR and GGGH Models parameter reactor volume, V CHjCHO inflow rate, ja 0 2 inflow rate, jo bath temperature, Tbath total concentration, M
description experimental constraint experimental constraint experimental constraint experimental constraint fixed; use ideal gas law and relation between and P forjexit =j a + j o varies as ideal gas law empirically determined related to experimental constraint exit orifice heat capacity for main constituents in reactor temDerature-indeDendent integration parameters
reactor pressure, P outflow rate,jexit exit rate, ko heat capacity, C, heat transfer coefficient. k initial values [CHsCHOl,E 0 2 1 [CH3CO3Hl [other species] TmtemaI
11 12
H202
CH3
-+
(J/m3)
1.3 x 10-8P- 2.2
x
(moUs)
jexitlMV ( s - ' )
4.184(6.57 + 0.012Tintemd) (J/(Kmol)) 5x (J/(K3'*s))
0.0 Tbath
-- + + -. - +
(M)
MRTintemd
(moum3) (moVm3)
TABLE 6: The PR ModeP no. reaction 1 CH3CHO + 0 2 CH3CO + HO2 2 HO2 + CH3CHO CH3CO + H202 3 CH3CO3H CH3 OH (C02) 4 CH3 + 0 2 OH + (CH20) 5 OH CH3CHO -.. CH3CO + (H20) 6 CH3CO + 0 2 CH3CO3 7 CH3C03 CH3CHO CH3CO3H + CH3CO CH3CO + (M) -. CH3 + (CO) (M) 8 9 2CH3 (C2H6) 10 2H02 H202 0 2
+
definition or typical value 275 x 10-6(m3) 5.4 -6.6 (mmoUmin) 5.4-6.6 (mmoUmin) 542-620 (K) P(ja + j~)/RTba*mol/m3,3-4 mol/m3
+
20H
+ CH3CHO
+
+ (M)
CH3CO
+
+(Ch)
Inert products are enclosed in parentheses, and (M) represents any third body present in the reactor. The same protocol is used in the CSR experiments where the concentration of 0 2 is perturbed.
(K)
TABLE 7: Rate Constant Data and Enthalpies for the PR Model rate constant k = A exp(E/RTintemal) reaction A no. (s-I or mol m-3 s-I) E/R (K) AH( kJ mol-]) 1 1.0 x 107 14595 . 169 2 1.0 x 106 4000 -346 3 3.2 x 1015 21640 247 4 3.2 x 107 1OOOO -293 5 1.0 x 107 0 -50 6 4.1 x 104 -538 -288 7 5.6 x lo6 3442 -50 8 2.0 x 1010 7550 50 9 2.4 x lo8 0 -360 10 1.0 x 107 500 -178 11 1.2 x 10" 22900 210 12 8.5 x 104 3000 0
source" Westley Westley Westley Westley Wamatz Kondratiev Westley Westley Warnatz Westley Westley Westley
As quoted in ref 8.
III. Reaction Mechanism Models 1II.A. Conditions Imposed on the Model. The conditions imposed on the model for oscillatory cool flames are the same as the ones described by Pugh et al.,* except that there is no quasi-steady-state approximation used for the concentration of the radicals. The values of the parameters used in the modeling are given in Table 5 . The Livermore Solver for Ordinary Differential Equations (LSODE)32is used to integrate the stiff, coupled, first-order, nonlinear differential equations for the behavior of the dynamic variables. The period of the oscillatory states studied here ranges from 1.0 to 6.8 s. For the parameter values used in modeling the oscillatory states, the residence time ranges from 4.7 to 5.3 s. The inflow rates of acetaldehyde and oxygen are equal and constant, except when a perturbation is applied to either one. The intemal temperature, Fntemal, is influenced by three sources of heat loss or gain: heating of the inflowed reactants from Tbath to Tiatema], heat loss through the reactor walls, and enthalpy changes associated with the reactions. 1II.B. €'ugh-Ross Model. The 12 steps of the Pugh-Ross (PR) model8 are shown in Table 6. This model is a modification of a model developed earlier by Halstead et al.,12.'3in which peracetic acid is the degenerate branching intermediate in a chain-thermal ~ c h e m e . ~Rate ~ . ~constants ~ and enthalpies for the reaction steps are given in Table 7. There are a total of 15 species (4 stable molecules, 5 radicals, and 6 inert products). These species and the intemal temperature make 16 dynamic variables for this model. For the flow rates and pressures used in the modeling, the Hopf bifurcation computed for the PR model is 606.3 K, where the period of oscillation is 1.0 s. For
bath temperatures less than this, the dynamic states are oscillatory ones; at bath temperatures above 606.3 K, the dynamic states are stable stationary states. Examples of time series of peracetic acid and the intemal temperature generated with the PR model are shown in Figure 6. In these examples, the bath temperature is 600.0 K and the period is 1.5 s. 1. Relative.Amplitude. The predictions of the PR model for relative amplitude of oscillation of the intemal temperature and the chemical species are calculated for several bath temperatures less than the Hopf bifurcation temperature (606.3 K), as well as at the Hopf bifurcation temperature, and are shown in Table 8. The relative amplitudes of the 16 dynamic variables of the PR model at the four bath temperatures less than the Hopf bifurcation bath temperature are calculated by generating time series at each of the bath temperatures and then finding the maxima, minima, and average values. The relative amplitudes at the Hopf bifurcation are calculated with the SNA programs of S ~ h r e i b e r . For ~ ~ the three lowest bath temperatures the relative amplitudes vary over several orders of magnitude. Near and at the Hopf bifurcation temperature (600.0 and 606.3 K, respectively) the spread in relative amplitudes is over 2 orders of magnitude. In all five cases, the radicals and peracetic acid have the largest relative amplitudes. 2. Relative Phase. The relative phase of oscillation of the dynamic variables with respect to the acetaldehyde waveform is calculated for the five sets of conditions listed in Table 8 for relative amplitude of oscillation. As explained in the Experiments section, acetaldehyde is arbitrarily chosen to have a phase of zero. The ranges of the relative phases are shown in Figure 7. Oxygen is approximately in phase with acetaldehyde.
Skrumeda and Ross
12840 J. Phys. Chem., Vol. 99, No. 34, 1995 0.0057
L
0.0055 645.20
Figure 6. Examples of time series of peracetic acid (upper) and intemal temperature (lower) generated with the PR model. Tbsth = 600.0 K, CH3CHO and 0 2 inflow rates = 6.0 mmoVmin each. TABLE 8: Relative Amplitude of Oscillation Calculated with the PR Model" bath temp (K) period (s)
542.6 6.8
560.0 3.7
580.0 2.1
600.0 1.4
606.3 1.o
CH3CHO
2.3 1.o 32 11 220 1200 560 5200 330 12 12 7.4 11 11 1.o 3.5
2.7 1.o 53 9.2 220 650 370 2700 280 11 11 5.7 9.8 9.5 9.2 4.7
3.3 1.o 90 7.0 140 250 170 600 170 10
3.5 1.o 150 5.5 65 120 80 170 100 9.5 9.5 5.5 9.0 7.5 8.5 4.5
3.7 1.o 220 7.7 150 210 160 320 180 12 12 7.0 12 10 10 9.3
0 2
CHsCO3H HzOz HOz CH3CO CH3 OH CH3CO3 COZ HzO CHI C2H6 CH2O
co
Tmtema~
IO 5.3 10 8.3 9.0 5.0
Acetaldehyde and oxygen inflow rates are 6.0 mmoVmin each. 90'
CH,CHO
A CH,CO,H
CH,CO. CH,CO, 0 CH,.OH
CO,, H20. # C2Hg.C0.
CH,O. CH, -90"
indicated in the figure. At the lower bath temperatures, the relative phase of oscillation of the products varies. 3. Concentration Shift Regulation. The regulation for a small shift in the inflow of one of the chemical species or a small shift in the bath temperature is shown for the PR model in the top half of each cell in Table 9. (The regulation for the GGGH model is shown in the lower half of each table cell.) For species that are not flowed into the reactor, such as CH3C03H and CH20, the perturbation can be applied by adding a small inflow term. The results shown in Table 9 are generated with the SNA programs using conditions that correspond to a Hopf bifurcation. The CSR is also determined from time series generated with the LSODE programs at a bath temperature of 620.0 K and oxygen and acetaldehyde inflow rates of 6 mmoYmin each. The regulation predicted at 620.0 K is the same as shown in Table 9 for 606.3 K. The response of peracetic acid to a shift of any of the variables, except the products, exhibits inverse regulation, which is indicated by the minus sign in the row labeled CH3C03H. This is the only variable that shows inverse selfregulation. The radicals and the intemal temperature show mostly normal regulation in response to shifts in all the variables except the products. The response of any variable to a shift in a product is "no change" except for the response of the product variable being shifted (then there is normal regulation). 1II.C. Gibson-Gray-Griffiths-Hasko Model. The 25 steps of the Gibson-Gray-Griffiths-Hasko (GGGH) model14 are shown in Table 10. Rate constants and enthalpies for the reaction steps are given in Table 11. There are 22 dynamic variables for this model (5 stable species, 12 atoms and radicals, 4 inert products, and the internal temperature). In this model methyl hydroperoxide is the putative branching agent, although the steps of the PR model that contain peracetic acid are present in the GGGH model, albeit with different rate constants for some of these steps. For the flow rates and pressures used in the modeling, the Hopf bifurcation computed for the GGGH model is 610.0 K, where the period of oscillation is 5.7 s. At lower bath temperatures there are oscillatory states, while at higher bath temperatures there are stable stationary states. 1. Relative Amplitude. Relative amplitude of oscillation of the variables of the GGGH model is calculated at the Hopf bifurcation bath temperature of 610.0 K with the SNA programs and is shown in Table 12. The relative amplitudes range over 2 orders of magnitude, which is the same as for the PR model. Peracetic acid, methyl hydroperoxide, and some of the radicals have the largest relative amplitudes. 2. Relative Phase. Relative phase of oscillation of the variables with respect to the acetaldehyde waveform is calculated with the SNA programs and is shown in Figure 8. Oxygen lags acetaldehyde slightly. Both methyl hydroperoxide and peracetic acid lead acetaldehyde. Most of the radicals and the internal temperature are clustered and lag acetaldehyde. The products are antiphase with acetaldehyde. 3. Concentration Shift Regulation. The regulation for a small shift in the inflow of one of the chemical species or the bath temperature is shown for the GGGH model in the lower half of each cell in Table 9. The results shown in Table 9 are generated with the SNA programs using conditions that correspond to a Hopf bifurcation. CH302H is the only variable that shows inverse self-regulation.
Figure 7. Summary of the results of relative phase of oscillation for the PR model. The relative phase was calculated with the data sets described in the Relative Amplitude section for the PR model.
Peracetic acid lags acetaldehyde but leads the radicals and the intemal temperature. Near and at the Hopf bifurcation bath temperature the products are antiphase with acetaldehyde as
IV. Discussion of Experimental and Modeling Results 1V.A. Determination of Essential Species. Prior to categorizing a mechanism according to the scheme of Eiswirth et al.,24the essential species of the mechanism must be identified. The relative amplitude of oscillation test is used to separate the
J. Phys. Chem., Vol. 99, No. 34, 1995 12841
Oscillatory Cool Flame Oxidation of Acetaldehyde
TABLE 9: Concentration Shift Regulation Calculated with the PR Model (Upper Entry in Each Cell) and the GGGH Model (Lower Entry in Each Ce1l)n perturbation of
A
0
P
H
+
-
-
-
+ -
+ +
-
-
-
-
-
+ +
-
-
+
+
+ + + + + + + +-
+ + + + + o + ++
+ + + + + + ++
+
-
+
++ + + + + + + + + + ++ + + + + + +
+ + + + + + + + + + + + + + + + + + +
+ ++ + + + + + + + + + ++ + + + + + +
R1 -
-
-
-
R2
R3
R4
-
+ +
-
-
-
-
_
-
-
-
-
-
-
-
R5 -
-
-
0
0
0
-
0
0
0
-
0 0
0 0
+
o
o
+ + + + 0 + + + + 0 + + + + 0 + + + O O + - + + 0 + - - - 0 + + + + 0 + + -+ + + + + o 0 o - - + + + + o o + - + - + + 0 + - - - + + + - + $+ 0+ - + + + + + + + + + - + + o + + + + + O + + + + + + + o + + + + + + o + + + + + O O + + + + + + o + + + 0 0 + + + + - + + - + + + + + 0 - +- +- +- - + - + 00 + - + + + + + 0 - + + + + + 0 + + + + + O O - + + + + + 0 - + + + + + 0 + + + + + + 0 - + + + + + 0 + + + + + + + + + + 0 + + o- ++
M
W
C
0
0
0
0 0 0 0
+
0 (
0
0 0 0
0 0 0 0
F
E
0
0
0 0
0
o
)
0
o o
0
o
-
B
+
(
D
-
0
- - o 0 0 - - o
J
-
-
-
T
0
+ +-
0
-
0
0
0
0
0
-
-
-
-
o
+
o -0 - 0 o
-
-
-
-
-
-
-
-
-
_
-
+
+
+
+
+
+
+
+
+
+
-
-
+
o
-
-
-
0 0 + + O + + 0 0 + + O + + 0 0 0 0 0 0 + + O + +
o
K
J
G
-
-
0 O 0 O 0 0 0 O
0
o
L
N 0
+
-
-
+
+
+
+
-
-
-
-
+ + -
-
0 -
+
-
+-
-
-
-
-
-
-
-
0
+
o
+
+
+
+
-
+
+
o o o o + - o + + + + + + + o + + o o o o + o o o
o
o
o
0 0 0 0 0 0 0 0 0
0
+
o
+
+
o + o o O + o + o o -
+
o
+ + -
0
-
+
O
+
+
O
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
-
+
+ + + + + + + + o - - - - - - o - + + - - o O
+ +
+ +
+ +
+
+
0 -
+
o + + - - + - O
-
-
+
o + - o + + + + + + + o + - o + + + + + + + 0 0 0 o + + o + + + + + + +
+
Each row represents the responses of the variable listed at the left of the row to individual perturbations of the inflow associated with each variable. Each column represents the individual responses of all the variables to a perturbation of the inflow of the variable listed at the top of the column. (The code for the variable names is shown in the leftmost column.) There are blanks where a particular species is not present in one of the models. indicates normal regulation, “-” indicates inverse regulation, and “0” indicates no change in response to the perturbation. Acetaldehyde and oxygen inflow rates are 6.0 mmoYmin each. The bath temperature is 606.3 K for the PR model and 610.0 K for the GGGH model.
“+”
essential variables from the nonessential ones. For oscillatory states arising from a supercritical Hopf bifurcation, the relative amplitude i f essential- species is large, whereas the relative amplitude of nonessential type a and c species is small. The relative amplitude of a nonessential type b species, usually a product, may be small or large. The relative amplitudes of the eight species that are measured in the experiments vary by only 1 order of magnitude. The relative amplitudes of the variables in the calculations vary by 2 orders of magnitude at the Hopf bifurcation and by more than 2 orders of magnitude away from the Hopf bifurcation. Taking into consideration the variation of the relative amplitudes in the calculations and the fact that the measured relative amplitudes were qot obtained at the Hopf bifurcation, it seems probable that the measured relative amplitudes are all of the same size. In the calculations, the relative amplitudes of the eight measured species are small in comparison to the relative amplitudes of the other species in the models, indicating that these species are nonessential in the models, which is not a surprising result. The relative amplitude results for the chemical variables in the PR model (Table 8) suggest that peracetic acid and the five radicals are essential, since they have large relative amplitudes and are not products, and that the other chemical variables are
nonessential, since they have small relative amplitudes. The calculations show the intemal temperature to have a small relative amplitude. Results of another test, in which one variable at a time is held constant in the calculations, are reported by S k r ~ m e d a . ~ ~ This test follows directly from the definition of nonessential species. If one variable is held constant and the remaining variables continue to oscillate, then the variable being held constant is nonessential. If the remaining variables do not oscillate, then the variable being held constant may be essential. It is also possible that the variable being held constant is nonessential and that oscillations are not seen because suitable values of the constraints (bath temperature and flow rates) are not being used in the calculations. It is reported that each of CH3CH0, 02,HzOz, CO;?,Hz0, C h , C z a rCHz0, and CO may be held constant (one at a time) in the calculations while the other variables continue to oscillate. These species are therefore nonessential in the PR model, in agreement with the results from the relative amplitude test. The intemal temperature is indicated to be essential by this test. Results of a third test of the models are reported by S k r ~ m e d a .This ~ ~ test considers the total change in phase shift across the fundamental (1 :1) entrainment band. The phase shift
Skrumeda and Ross
12842 J. Phys. Chem., Vol. 99, No. 34, 1995
TABLE 12: Relative Amplitude of Oscillation Calculated with the GGGH Model"
TABLE 10: The GGGH Model" no.
reaction
-- -+ -.
+ + + +
+
CH3CHO 0 2 CH3CO H02 CH3CO 0 2 CH3CO3 CH3CO3 CH3CHO CH3C03H CH3CO CH3C03H CH3 OH (COz) CH3CO (M) CH3 CO (M) CH3 0 2 CH30z CH3 0 2 C&O2 2CH3 -.+ (C2H6) CH302 CH302 CH30 CH3O f 0 2 CH3O CH3O CHzO (CHsOH) CH3O2 CH3CHO CH30zH CH3CO CH30zH CH3O OH OH CH3CHO CH3CO (H20) CH30 0 2 CH2O + HO2 OH CH2O HCO (HzO) HCO (M)-H CO (M) HCO 0 2 -CO H02 OH CO-H (C02) 0 H 02-OH H 0 2 (M)-HO2 (M) 0 CH20- HCO OH HO2 CH2O HCO + H202 2H02 H202 + 0 2 H202 (M) 2 0 H (M) OH H202 HO2 (H20)
1 2 3 4 5 6 7
+
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
-
+ + +
---
+
+ +
+
- -. -
+ + +
+ + + + + + +
+
+
+
+
+ + +
A
A
+ +
+
+
+
bath temperature (K) period (s)
610.0 5.7
CH3CHO
1.0 1.2 2.0 2.1 2.9 3.0 3.3 3.3 3.4 3.9 5.0
Tintemal
CH20 0 2
c02 H2O H202 CH3OH
co
CH3CO CH302 a
bath temperature (K) period (s)
610.0 5.7
H02 CH30 CH3CO3 HCO C2H6 OH H CH302H 0 CH3 CH3C03H
7.1 7.6 8.8 11 13 14 21 24 28 31 35
Acetaldehyde and oxygen inflow rates are 6.0 mmollmin each.
+ + +
--
+
+
+ +
Inert products are enclosed in parentheses. (M) represents any third body present in the reactor.
'
CH3C0
TABLE 11: Rate Constant Data and Enthalpies for the GGGH Model"
Figure 8. Results of relative phase of oscillation for the GGGH model.
rate constant k = A exp(E/RTi,,,,,,) reaction no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 a
A
(S-I
or mol m-3 s-I) 3.0 x 2.0 x 1.0 x 2.0 x 1.0 x 1.0 x 1.0 x 2.5 1.0 x 1.0 x 1.0 x 2.0 1.0 x 1.0 x 6.0 1.5 x 1.0 x 3.0 x 2.0 x 5.0 x 5.0 x 1.0 x 1.8 x 3.1 x 3.6 x
lo6 105 107 1014 10'2
106 1014 107 106 106 106 1015
106 106 105 lo8 106 105 108
103 107 106 lo6 10" lo6
EIR (K) 12700 0 5100 20206 0 7578 13904 0 0 0 5196 20447 1203 3000 0 9553 3007 302 8296 -654 2313 4022 0 23630 0
;u
0 OH HOz CH302
The relative phase was calculated with the data set described in the Relative Amplitude section for the GGGH model.
AH (kJ mol-I) 164.0
-288.0 -50.0 247.0 66.9 -115.4 115.4 -371.2 -27.3 -338.6 -13.3 184.7 -136.4 - 102.4 - 132.6 70.4 -125.7 - 104.2 66.9 -197.3 -63.5 - 10.0 -176.8 210.3 -122.6
Taken from ref 14.
is measured between a sinusoidal perturbation of a variable and the response of that variable to the perturbation. For example, a sinusoidal component can be added to the inflow of oxygen. The phase shift is then calculated between the sinusoidal inflow term and the concentration of oxygen. Within the entrainment band, the response, which is the concentration of oxygen, assumes a constant phase shift from the sinusoidal perturbation after a transient period. This phase shift changes as the frequency of the sinusoidal term is varied around the autonomous frequency. The difference in the phase shift between one
edge of the entrainment band and the other is called the total change in phase shift. For sufficiently small perturbations of an essential variable, the total change in phase shift across the fundamental entrainment band approaches 180'. For perturbation of a nonessential variable, which in the acts like a constraint parameter, the total change in phase shift approaches 0". The total change in phase shift of acetaldehyde, oxygen, peracetic acid, and internal temperature was studied at an autonomous frequency of 6.76 s. A sinusoidal term (30% amplitude) was added to the inflows of acetaldehyde and oxygen, resulting in total phase shift changes of 48.8" and 54.0', respectively. A small amplitude sinusoidal inflow term was created for peracetic acid, which produced a total change in phase shift of 154.2'. A sinusoidal term (1% amplitude) added to the term for bath temperature resulted in a 160.9' change in phase shift between the sinusoidal term and the internal temperature variable. The calculation suggests that acetaldehyde and oxygen are nonessential and that peracetic acid is essential, as seen already; further, it suggests that internal temperature is essential. The relative amplitudes of the variables in the GGGH model (Table 12) are more difficult to separate into two groups because there are several intermediate values. Methyl hydroperoxide, peracetic acid, and several of the radicals and atoms (including CH3, H, and 0)have large relative amplitudes and so appear to be the essential variables. In these calculations and internal temperature variable has one of the smallest relative amplitudes. The variables shown to be nonessential in the GGGH model, according to whether oscillations persist when one variable at a time is held constant in the calculation^,^^ are CH3CH0, 0 2 , H202, CH20, CO, 0, CHsOH, COZ,H20, C2H6, CH3CO3, and CH3C03H. The results for 0 and CH3C03H according to this test disagree with the results from the relative amplitude test
J. Phys. Chem., Vol. 99, No. 34, 1995 12843
Oscillatory Cool Flame Oxidation of Acetaldehyde
TABLE 13: Summary of the Concentration Shift Regulation Results for Selected Specie@ constraint perturbed
CH~COSH Hop
02 inflow
CH3CHO inflow Tbath exp PR GGGH exp PR GGGH expb PR GGGH CH3CHO 0 2 H202
CH3OH CHzO
coz
H20 CHI
CH3CO3H
co
Tmterna~
-
+ + + + + +
-
+ + + + + ++ +
+ + + - + + - + + +'+ + - + + - + + ++ + + +
+ + + + + +
- + o + + - + + -
+ + + +
-
-
-
o +
-
-
+ - + + + +
+ +
a Each row shows the responses of the species listed to the left of the row to individual perturbations of oxygen inflow, acetaldehyde inflow, and the bath temperature in the experiments (exp) and in the calculations with the PR model (PR) and the GGGH model (GGGH). There are blanks where a particular species is not present in one of the indicates normal models or is not measured in the experiments. regulation, "-"indicates inverse regulation, and "0"indicates no change in response to the perturbation. This column is inferred from data shown in Figure 4 of ref 4.
CH3C03H
-
H02
t
CH3
t
CH3CO
OH
CHsCO3
CH3
Int.Temp.
t
t
t
t
t
t
t
t
t
t
t
Perturbation Variable C H ~ C ~ ~5 Radicals, H Int. Temp.
Response Variable
5 Radicais, Int. Temp.
Figure 9. Concentration shift regulation of the essential variables of the PR model (upper) and reduced CSR matrix of the essential variables
(lower).
"+"
described above. One possible reason for this discrepancy is that the GGGH model might not meet the prerequisities of the classification scheme, namely, that the model is a simple o ~ c i l l a t o r However, . ~ ~ ~ ~ ~ while a large network may have several unstable currents, usually only one unstable current is dominant, which makes the network equivalent to a simple o ~ c i l l a t o r . ~ ~ Since the measurements of relative phase and CSR are made for nonessential species, these measurements cannot be used to categorize the mechanism; however, the measurements for these species can be compared with the calculations. The calculations of relative phase and CSR for the essential variables in the models may be used to categorize the models. IV.B. Comparison of Measurements with the Models. 1. Relative Phase. For the experiment and both models, 0 2 is approximately in phase with CH3CHO (Figures 3, 7, and 8). The experiment shows that H202 is also approximately in phase with CH3CHO. However, the PR model predicts 135" lead for H202, while the GGGH model predicts an antiphase relationship. In the experiment, CH30H leads CH3CHO by nearly 90" to just over 180". The GGGH model predicts 180" phase difference for CH30H. The measured relative phase of CH20 is 145'-180'. The PR model predicts 180' for CH20, but the GGGH model predicts a 125" lag. The PR model correctly predicts an antiphase relationship for C a . Both models predict a 90" lag for the OH radical and the internal temperature, consistent with the measurements made by Pugh et al.* 2 . Concentration Shift Regulation. The concentration shift regulation results from the experiment, and the two models are summarized for several variables in Table 13. For perturbations of 0 2 and CH2CH0, there is agreement among the experiment and the models except for the responses of C02 and H20 after a perturbation of CH3CHO. Both models predict normal regulation, but the experiment shows inverse regulation. The experimental result appears sensible since adding more CH3CHO, which is already present in excess, may choke the reaction. For perturbation of the bath temperature, the PR model correctly predicts the regulation of three variables and incorrectly predicts the regulation of four species. (The comparison is ambiguous for two species). The GGGH model correctly predicts the regulation of six variables and incorrectly predicts the regulation of two species. (The comparison is ambiguous for one species.)
H
t
t
t
t
t
t
t
t
t
t
t
o
t
+
t
t
t
t
t
t
t
t
+
O H +
t
+
t
-
t
-
. -
+
t
t
t +
t b
Figure 10. Concentration shift regulation of the essential variables of the GGGH model. 1V.C. Categorization of the Two Models. 1. The PR Model. The first step in elucidating the roles of the essential variables is to identify the type Z, or negative feedback, species. The diagonal entry of the CSR matrix for a type Z species is a minus. Peracetic acid, which is the only essential species with a minus along the diagonal, is the type Z species in the PR model. Almost all the other entries of the CSR submatrix for the essential variables (Figure 9, top) are pluses. Barring the few minuses in the rows for the radicals and the internal temperature, this submatrix can be reduced to the 2 x 2 matrix shown in the bottom of Figure 9. Since there are only two essential variable types, they must be types X and Z of category 2. This conclusion is confirmed by an examination of the relative phases (Figure 7). Peracetic acid leads the group of radicals and the internal temperature, indicating species types X and Z, respectively, of category 2. 2. The GGGH Model. The concentration shift regulation of the variables in the GGGH model which are likely to be essential variables are shown in Figure 10. Considering category IB, CH3CO and CH3CO3 may be associated with the type X species since the rows of the CSR matrix for CH3CO and CH3C03 have pluses along the diagonal and minuses in the offdiagonal positions. CH30, CH3, HCO, HO2, H, and 0 may be associated with the type Y species since their rows contain only pluses. There is no apparent type Z species unless it is peracetic acid or methyl hydroperoxide, but neither of their rows matches exactly that of a type Z species. Moreover, CH3CHO and CH3CO3H lead CH3CO and CH3C03 in relative phase, but in
S h m e d a and Ross
12844 J. Phys. Chem., Vol. 99, No. 34, I995
category 1B the type X species leads the type Z species. Therefore, the GGGH model is not a category 1B oscillator. Considering category lC, CH3CO and CH3C03 could be type Y species since their rows have all minuses except along the diagonal. The rows for CH302H and CH3CO3H almost match the rows for the type Y and type Z species, but there are no potential type X species in the GGGH model. Therefore the GGGH model cannot be a category 1C oscillator. With regard to category 2, methyl hydroperoxide is assigned to the type Z species since its row contains only minuses. The rows €or CH@, CH3, HCO, H02, H, 0, and the intemal temperature contain only pluses so they are type Z variables. CH302H leads the six type X species in relative phase, consistent with a category 2 oscillator. The rows of the CSR matrix for CH3C0, CH3C03H, and CH3C03 are composed mostly of minuses and so most closely resemble the row for the type Z species. The rows for OH and CH3O2 do not resemble the rows for either the type X or type 2 species. There are several possible explanations for the rows that do not match the regulation of any of the prototypes. First, the species with anomalous entries in the CSR matrix may not be essential despite their relatively large relative amplitudes. Second, it is possible that concentration shift regulation varies a little for different constraints and that a few atypical entries are to be expected. Third, the GGGH model may not meet the prerequisites of the classification scheme, namely, that the model is a simple o ~ c i l l a t o r . ~It~is. ~not ~ unusual for a large network to have several unstable currents, but generally only one unstable current is dominant, which makes the network equivalent to a simple oscillator.24
Perturb:
X
Y
Z
Response
Perturb:
X
Y
Z
Response Variable
Z
Perturb:
1 ;1 ;I x
Response
X
Variable
Z
z
Figure 11. Concentration shift regulation of essential species types X, Y, and Z for category lB, lC, and 2 oscillators. The perturbed variables are listed along the top; the response variables are listed at the left. The responses are normal regulation (+) and inverse regulation
(-1.
V. Conclusions The results of the experiments and calculations reported here represent one of the first applications of a growing collection of experimental tests and analytical tools developed to provide information about a reaction mechanism. Although it is not possible to formulate a reaction mechanism for the oscillatory cool flame oxidation of acetaldehyde from the results of the tests used here (this is because it tums out that the measured species are nonessential variables), the tests provide another point of comparison between the measurements and the predictions of the models. In this way, limitations are put on any reaction mechanism model. Three tests are employed in this work: relative amplitude of oscillation, relative phase of oscillation, and concentration shift regulation. The relative amplitude of oscillation test suggests that eight stable chemical species are nonessential in the experiment and that one short-lived species and several radicals are essential species in each of the models. There is some discrepancy between the results of the relative amplitude test and another test36reported earlier regarding the nature (essential or nonessential) of two species in the GGGH model. The variable intemal temperature is indicated by the calculations to be essential. The relative phase of oscillation and the concentration shift regulation tests show that the reaction mechanism models of oscillatory cool flame acetaldehyde oxidation are category 2 oscillators. In a category 2 oscillator, the oscillations are derived from interactions between autocatalytic species and negative feedback species. Further measurements on the acetaldehyde oxidation system, in particular of peracetic acid, methyl hydroperoxide, and several radicals, are needed to show which species are the negative feedback species and which are the autocatalytic species.
Figure 12. Relative phase of oscillation for the essential species types X, Y,and Z for category lB, lC, and 2 oscillators. Relative phase increases in a counterclockwise direction. Acknowledgment. We thank Igor Schreiber for helpful conversations and for the use of his SNA computer programs. This work was supported in part by the NIH Molecular Biology Training Program and the Air Force Office of Scientific Research. Appendix The concentration shift regulation matrices for the prototypical essential variables are shown in Figure 11 for category lB, lC, and 2 oscillator^.^^^^^ Each row shows the responses of one variable to step perturbations of the variables listed at the top of the columns. Each column shows the responses of all the variables to step perturbations of the variable at the top of the column. In category lB, for example, the type X species shows inverse regulation in response to a step perturbation in the concentration of the type Y species. The relative phase of oscillation for the prototypical essential species is shown in Figure 12 for category lB, lC, and 2 oscillator^.^^ Relative phase increases in a counterclockwise direction. The order of maxima (or some other feature of the oscillation waveform) is X-Z-Y for a category 1B oscillator, Y-Z-X for a category 1C oscillator, and Z-X for a category 2 oscillator. The relative phases indicated in Figure 12 are not exact. Instead, they show the general relationship between the relative phases of the essential variable types.
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J. Phys. Chem., Vol. 99, No. 34, 1995 12845 (19) Di Maio, F. P.; Lignola, P. G.;Talarico, P. Combust. Sci. Technol. 1993, 91, 119.
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