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Computer Simulation of the Kinetics of Complicated Gas Phase Reactions.

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Volume 19 - Number 5
May 1980
Pages 333-41 0
International Edition in English
Computer Simulation of the Kinetics of Complicated Gas Phase
Reactions
By Klaus H. Ebert, Hanns J. Ederer, and Gunther Isbarn**]
Dedicated to Professor Matthias Seefelder on the occasion of his 60th birthday
Modern digital methods and powerful computers make it possible to simulate the time behavior of chemical reactions. These calculations can be performed on systems containing an almost
unlimited number of elementary reactions. Generally, however, the reaction models used
should contain only those elementary reactions which describe the bulk of the conversion.
Such a reaction model may be obtained by reduction of the complete set of elementary reactions. Another possibility is analysis of the chemical system starting from conditions ensuring a
simple chemistry, which is generally the case at low temperatures and low conversions. The
reaction model may then be extended into the range of the reaction variables (temperature,
time) of interest. Mathematical simulations may be helpful during the development of the reaction model, and sometimes even decisive. These methods were applied to the pyrolysis of ethylbenzene and n-hexane, and to CO oxidation. They yield information on the reaction paths, the
importance of particular elementary reactions, and reaction stability. Furthermore, quantitative data can be obtained concerning the influence of single elementary reactions on the product distribution. The sensitivity matrix shows, e.g., whether the determination of kinetic parameters of an elementary reaction from kinetic data of the overall reaction is possible in principle,
and how high the accuracy of the rate constants should be for simulation of the reaction. Both
results are important for modeling chemical reactions.
1. Introduction
The rapid development of modem computers has markedly advanced the possibilities and the potency of chemical kinetics. Until recently the time behavior of only those chemical reactions could be determined exactly whose rate laws
could be expressed as a system of differential equations
r] Prof. Dr. K . H. Ebert, Dr. H. J. Ederer, DipLChem. G . Isbam
Institut f i r Angewandte Physikalische Chemie and
Sonderforschungshereich123 of the University
Im Neuenheimer Feld 253, D-6900 Heidelberg (Germany)
Angew. Chem. I n f . Ed. Engl. 19. 333-343 (1980)
which can be integrated analytically. This is possible with elementary chemical reactions, which cannot be subdivided,
and with a few simple reaction systems. It is not possible to
obtain exact analytical expressions for the time dependence
of most of the chemical systems of practical interest. Therefore the solution of the simultaneous differential equations
requires other methods.
For reactions involving radicals-and that is the majority
of all homogeneous gas phase reactions-the best known approximation method for solving such systems of differential
equations is the assumption of a steady state"'. This approxi-
0 Verlag Chemie. CmbH, 6940 Weinheim, 1980 0570-0833/80/0505-0333$02.50/0
333
mation implies that the reactive intermediates are consumed
as fast as they are produced, resulting in a zero net formation
rate. This greatly simplifies the mathematical treatment (see
Section 3.2).
Another method of solving complicated systems of differential equations is numerical integration, in which the time
behavior is evaluated by stepwise computation. If there is a
large difference in the rates of production i -d consumption
of the various species, which is always the cL? in the initial
period of radical reactions, very short time intervals have to
be used for the calculations. In classical integration methods
(e.g. Runge-Kutta
this will entail considerable
computational effort. Modern integration methods use variable time intervals, and are available as computer programs.
Short time intervals are chosen in the area of large changes
in the concentration of the species, and longer time intervals
are used if these changes are smaller. Nevertheless, numerical integration of a large number of nonlinear simultaneous
differential equations usually requires an enormous computational effort which can only be mastered with modern high
capacity computers. Modern numerical methods tend to decrease computing time and storage size without reducing the
quality of the results. Results are considered as “exact” if a
reduction of the time intervals does not have any effect on
the results.
Before studying the differences in the results of kinetical
calculations obtained by the “exact” method and by assuming a steady state, it is useful to check how well the laws of
chemical kinetics generally describe the time behavior of a
chemical reaction. The classical theory of the rates of thermal reaction^^^.^] is based upon two fundamental equationsf’].
The time law describes the rate of homogeneous gas phase
reactions in dependence upon the partial pressures:
The first term of the right side contains the components of
first order reactions, and the second term those of second order reactions. The sign indicates the production (+) or the
consumption (-) of the species n, in the reaction considered.
The temperature dependence of the reaction rate constant
can usually be expressed by the Arrhenius equation:
k = A exp( - E J R T )
(2)
The number of elementary reactions participating in a
chemical reaction system can vary within wide limits. Examples of reactions consisting of only few elementary reactions
[*] Both laws are based upon the Boltzmann distribution of molecular energies,
which is satisfied within the usual reaction conditions of most chemical reactions.
Chemical reactions are undergone predominantly by the energy-rich molecules
of the ensemble; therefore, weak interactions, which are responsible for the numerous deviations of thermodynamic properties from “ideality”, are of little or
no influence. In general, eqs. ( 1 ) and (2) are satisfied well. This means that reaction rates can be calculated with high accuracy in a wide range of the reaction
variables if the values of the kinetic constants are known exactly. The accuracy
of the calculations is usually very good relative to the experimental methods
available for determining reaction rates. This is true without limitation for the elementary reactions which cannot be further subdivided and whose behavior is
described by the stoichiometric equation. It should be noted that at low pressure
energy transfer may influence the reaction rates, and that the assumption of the
Boltzmann distribution may loose its validity in very fast reactions-at high temperatures or in reactions with a very low activation energy. Both possibilities
cause deviations from eqs. ( I ) and (2)
334
(e.g. formation of hydrogen halides from the elements) can
be found in any textbook of physical chemistry. Most of the
gas phase reactions of practical interest consist of many-often several hundred-elementary reactions. Of course, they
will all participate to some extent in the reaction, but the
question to be answered is: which reactions determine the kinetic behavior and which exert such a slight influence that
they may be neglected? Reaction mechanisms with a large
number of elementary reactions almost exceed the capabilities of even the most powerful computers. As a rule of
thumb, the computational effort increases with the square of
the number of differential equations.
It is therefore very important that complicated reaction
mechanisms be reduced to a reaction model which describes
the reaction with sufficient accuracy while still being suitable
for kinetic calculations using either the “exact” method or
the steady state assumption.
2. Reaction Models
Reaction models should be made up of elementary reactions because of their general applicability. With a knowledge of the kinetic constants, absolute values of the reaction
rates can be calculated. The number of elementary reactions
increases with the number of atoms present in the reactants.
For the fairly simple decomposition reactions of such small
molecules as butane or pentane there exist more than 500
“possible” elementary reactions, even with the limiting assumption that no products are formed with a molecular
weight exceeding twice that of the parent compound15]. On
addition of other reactants, such as oxygen, water, eic., the
number of elementary reactions is multiplied. However, a
reaction model fulfills its purpose if it describes well enough
the course of the reaction within the interesting range of the
reaction variables (p, T ) , or if it provides answers to specific
questions. It may happen that the production rate of a particular by-product is of more importance than an exact
description of the formation of the major products.
Of course, the lower the number of elementary reactions,
the simpler becomes the mathematical treatment. However,
an upper limit for this number cannot be given in general because the complexity of a reaction model also depends on the
number of the participating species. However, if the number
of elementary reactions approaches 50, a reduction should be
considered.
It is usually not easy to decide from the kinetic constants
whether a particular elementary reaction belongs to the
group of those which are involved in an “important” reaction path within the conversion, and therefore cannot be neglected. Furthermore, a reaction model has to be self-consistent and to contain reactions accounting for the consumption
of all radicals. A useful approach to model reduction is the
combination of certain species or elementary reactions; this
is called lumping. However, the “kinetic constants” of the
lumped reactions rarely remain constant, and the applicability of the corresponding rate expression is consequently limited. Lumping is favored if similar species (e.g. radicals from
unsaturated hydrocarbons) form the same major product
(e.g. soot)’6-91.
In minimizing the number of elementary reactions for a
reaction model the chemist usually adopts a different methAngew. Chem. Int. Ed. Engl. 19, 333-343 (1980)
od. He tries to analyze the reaction mechanism using experimental data, and in so doing he first looks for the initial reactions and then proceeds to formulate those elementary reactions describing the fate of the reactive intermediates. In this
way, the construction of a reaction model is continued by
considering known chemical principles and attempting to
formulate the formation of the products and intermediates
with a minimum set of elementary reactions. Additional investigations about particular elementary
are
sometimes useful. Simultaneous kinetic calculations and
quantitative comparisons with the experimentally determined product distribution may also be helpful in formulating the reaction model.
It is convenient to start the development of a reaction
model at low temperatures and low conversions, where the
number of elementary reactions is small, and to extend it
into the interesting regions of the reaction variables. This
method is then always applicable if the model contains only
few initial reactions, and if these form only few primary intermediates, as in the thermal decomposition of ethylbenzene (see Section 4.3). Furthermore, this method allows a
clear description of reaction behavior and prediction of variations in the reaction behavior caused by variations of the
reaction conditions or by the addition of various reactants.
3. Mathematical Methods
3.1. Numerical Treatment of Rate Equations
For numerical treatment, the reaction model is transformed into a system of differential equations containing one
differential equation for each species taking part in the reaction. In general this is not difficult. If the number of species
is greater than ten, the system should be evaluated by computer. The transformed reaction model consists of linear and
quadratic terms depending on the unimolecular or bimolecular nature of the corresponding elementary reactions. (In radical recombination reactions, third order terms may occur
too.)
To obtain a numerical solution of the differential equation
system the values of the kinetic parameters (activation energy, pre-exponential factor) of each elementary reaction are
needed. The number of elementary reactions whose kinetic
constants are known with satisfactory accuracy is still small;
such data have been compiled in tables”3 “I. If the kinetic
constants are not known, their values should be estimated
from chemical analogy or from thermodynamic considerations[‘‘ “1.
The kinetic constants of reactions of the same type often
vary within close limits. These “band widths” are compiled
Table 1 Types of reactions in the pyrolysis of hydrocarbons and band widths of
the kinetic parameters A [ s I: Is ‘ mol ‘1 (pre-exponential factor) and E., [kJ
mol ‘1 (activation energy); M.M’: stable molecules; R.R’: radicals.
Type no.
Elementary reaction
h o A
E,
-R+R’
M + R +R’+M’
R
-M+R’
R+R’ + M(+M’)
15-17
7-1 1
12--15
7-1 1
10-12
200-360
30- 50
1CK--180
M
R
-
R‘
Angew. Chem. Inr. Ed. Engl. 19, 333-343 (1980)
0
50-
80
in Table 1 for five types of reactions, which represent most of
the elementary reactions occurring in the pyrolysis of hydrocarbons.
For these estimates, it is important to have information
about the sensitivity of the reaction model to variation of the
values of respective reaction constants. More detailed data
on the sensitivity of a particular model and its temperature
dependence can be obtained only by simultaneous computations (cf. Section 5.2). Occasionally in simple systems quantitative data are available on the sensitivity of reaction constants, e.g. of the activation energy of unimolecular initial
reactions. In complicated systems it is mostly very difficult, if
at all possible, to predict the effect of the variation of a particular kinetic constant on a model. For some elementary
reactions it is evident, however, that they do not affect the
reaction model noticeably, even if the rates are varied considerably.
Large differences in the sensitivities appear in all systems
of differential equations in which the time constants differ by
orders of magnitude; such systems are called ‘stiff.
The well-known Runge-Kutta methods[21are not suitable
for the numerical integration of stiff differential systems. Because of the huge difference in the time behavior of the individual elementary reactions the time step size has to be set
extremely small in order to obtain correct results. Errors introduced by excessive step sizes in the initial period of the
reactions are propagated also in regions of higher conversions. The computer time required to cover high conversions
of a reaction is then extremely long.
Fast methods for the solution of stiff differential equation
systems have become an important field o f research in numerical mathematics. All modern “stiff solvers” contain an
efficient self-adjusting optimum step size control, i. e. rapid
variations require small step sizes, slow variations permit
large step sizes[r91.
A further important point is the algorithm
for the numerical stability of a stiff solver. The algorithm has
to prevent the accumulation of computational errors.
The program of Gearizo],which is very often used by chemists, is based on a “multistep predictor-corrector’’ algorithm
which also controls the step size. The program has been very
successful; until now no case of a “wrong” solution of a
chemical problem is known.
Other methods for solving stiff differential equation systems are the implicit Runge-Kutta method of Wanner[Z’]
and
which requires a
the semi-implicit algorithm of DeuflhardZZ]
considerably smaller computer storage size than the program
of Gear.
3.2. The Steady State Assumption
The most important and most frequently applied approximative method in treating differential equation systems in
chemical kinetics is based upon the quasi-steady state assumption (QSSA). In this approximation, the right-hand
sides of the differential equations for the reactive intermediates are set equal to zero in order to transform these differential equations into algebraic equations. As a result, the remaining reduced system of differential equations (for the
reactants and the stable products) is substantially less stiff
than the original system. This may permit either an analytical solution of the differential equation system or the appli-
335
cation of considerably simpler numerical integration methods (e.g. Runge-Kutta). The QSSA does not mean, however,
that the concentrations of all reactive intermediates are set
constant during the reaction. According to the mathematical
treatment with the singular perturbation t h e ~ r y f ~ ~ -which
~’I,
will not be outlined in detail, there exist two types of radicals[281depending mainly on the type of reactions in which
the radical is consumed: type rm, whose concentrations remain approximately constant, and type r,, whose concentrations decrease according to a function similar to that governing the disappearance of the reactant in the course of the
reaction.
The QSSA, as usually applied by chemists, neglects the
time behavior of the initial period of the reaction in which
the concentrations of all radicals increase to a limiting value
corresponding to the “steady-state” valuef*].Generally, the
smaller the steady-state concentrations are, the lower will be
the conversion at which the steady state is attained. Therefore, the QSSA is better fulfilled at low temperatures and
high pressures. The question then arises whether there exists
a quantitative measure for the applicability of this approximation. The differential equations of the radicals can be
transformed into a set of dimensionless differential equations. This procedure and a suitable rearrangement attributes
a dimensionless constant E~ to each dimensionless differential
quotient. The &,-valuesare composed of reaction rate constants and initial concentrations of the reactants, and it can
be shown that they are characteristic values for the applicability of the QSSA. Although theory supplies exact solutions
for calculating the G-values, in most cases more or less rough
estimations are more readily available which give satisfactory information on the applicability of the QSSA.
For thermal decomposition reactions (e. g. the pyrolysis of
ethylbenzene; cf. Table 2) the numerical values of E , for the
two groups of radicals rm and r, are approximated by equations (3) and (4), respectively:
Em
=
&=
rate constant of initial reaction
(rate constant of the reaction (r, + M)) x Mo
(3)
rate constant of initial reaction
rate constant of the reaction of r.
(4)
(Mo= initial concentration of the educt M). In cases in which
there are more than one initiation or consecutive reactions of
the radicals, the sum of the rate constants of these reactions
has to be used. As the number of the reactions involved in
consumption of a certain radical is usually small, calculations of E~ can be easily performed if the corresponding kinetic constants are known. The values E, and E. obtained have
to be multiplied by a value T, which considers that part of
the reactant M reacting via the initial reaction.
Finally, the magnitude of the product E; x is the quantitative measure for the applicability of QSSA[Z8.291.
Comparisons of QSSA calculations with “exact” calculations show
that QSSA is a good approximation if E~ x % s l O - ’ . Rough
estimations can be carried out up to E~ x %= lo-’. Theoretical studies confirm the general existence of these critical valu ~ s [ ~ O ~Another
.
question is whether the E; x % values of all
[*I Quantitative information about the initial period can be obtained, however,
by a suitable transformation of the time within the set of differential equations.
336
radicals participating in the reaction have to be calculated.
For the model of hydrocarbon pyrolysis only the E~ x To-values of the “important” radicals have to be considered; “important” radicals are those formed in the reactions of the
starting materials and those which react with the starting materials.
As an example, the comparison of QSSA data with the
“exact” calculations of the ethylbenzene pyrolysis are shown
in Figure 1. The conversion of the reactant and of three “important” radicals as a function of time at 950 K are shownf281.
The results show that the highest gi x %-values are obtained
for the CH3 radical, and that the deviations are already considerable at &, x %= lo-’. Furthermore, it is shown that disagreement is higher at low pressures.
4. Applications
4.1. Oxidation of Isobutane and Isopentane
The peroxide-initiated oxidation of isobutane and isopentane at relatively low temperatures was chosen as one of the
first examples of the application of modem numerical methods of integration to complicated homogeneous gas phase
reactions. The reaction models contained 20 and 32 elementary reactions, re~pectively[~’].
The product distribution of
both reactions showed good agreement with experimental
data. Another result of the calculations was that the concentrations of some of the radicals remained nearly constant
over a longer period of reaction time, confirming the existence of a steady state under these conditions.
4.2. Pyrolysis of Alkanes
A fundamental study of Allara and Edelson deals with the
thermal decomposition of propane, n-butane, and n-pentanel5].Very extensive reaction models were used which contain almost all the possible elementary reactions (293, 515,
and 586, respectively), assuming that the products have a
molecular weight not greater than twice that of the reactant.
The authors adopted the philosophy “that it is far better to
err on the side having too many reactions than too few”. The
calculations were compared with experimental data only for
the pyrolysis of propane, and then only for the formation of
methane. Nethertheless, interesting information was obtained by the calculations, e. g. on the contributions of particular reaction paths to the conversion, or on the concentration-time relation of selected radicals during the reaction,
and on changes in the reaction behavior due to global variations of the reaction constants.
Froment et al.[321studied a similar problem. A reaction
scheme consisting of 133 elementary reactions was compiled,
and it was assumed that this master set is applicable to the
pyrolysis of saturated and unsaturated hydrocarbons generally. In order to set up a special reaction model, a suitable selection of elementary reactions (50 to 80) was taken from the
master set by “trial and error” methods. The mathematical
treatment of those reaction models was camed out using
Gear’s program. Comparison of the calculated results with
experimental data from a pilot reactor demonstrated that
product distributions of the cracking reactions of ethane,
propane, n-butane, and isobutane as well as of the correAngew. Chem. Inl. Ed. Engl. 19, 333-343 (1980)
005
01
fl.
005
01
t / S
R05
01
t/.
005
01
tlS
O3
i
.,‘lo=
2 3.10‘7
aoi
003
---.......---.-.-.--
................f i........“..........
/
1
...-
tls
-
-...-..........
b 01 - 1
-3
I
I
0.2
I
-I
I
.,,,o.~-~~-s
Fig. 1 . Variation of concentration with time (time-conversion diagram) of ethylbenzene (EB). methyl, a-methylbenzyl, and
hydrogen radicals at 950 K and 3 x lo6 N m - 2 (30 bar) (left row) and at 3 x to’ Nm-’ (0.03 bar) (right row) according to
QSSA (. . . . .) and “exact” calculations (-----).
sponding olefins could be simulated satisfactorily. Distributions of the unstable products and the temperature profile
along the flow reactor were further points of information obtained from the calculations. Simulation of the decomposition reactions of mixtures of the above mentioned hydrocarbons showed that the combination of reaction models by
simple addition of the respective elementary reactions gave
results in good agreement with experimental data. This is to
be expected as long as the elementary reactions are independent and do not influence one another.
Angew. Chem. Int. Ed. Engl. 19. 333-343 (1980)
43. Pyrolysis of Ethylbenzene
The thermal decomposition of ethylbenzene was thoroughly investigated experimentally and comp~tationally[~~1.
In order to establish the reaction model (Table 2) the stable
and unstable species formed at low pressures (lo-* mbar)
and 600-1400 K were determined by mass spectr~metry~~~l.
Figure 2 shows the results, containing only those masses
whose signal area contributed at least 1.5% to the total signal
area. (Because of the different ionization propabilities of the
337
particular species this selection is somewhat arbitrary.) U p to
1100 K the following species were detected (the figures in
parenthesis refer to the possible reaction(s) of formation in
Table 2): methane (3), ethylene (8-1 I), benzene (4),benzyl
(I), toluene (14), styrene (7), a-methylbenzyl (3-9, and bibenzyl (15), as well as undecomposed reactant.
reactions (6, 10, 11). Reactions involving a hydrogen atom
attack on the aromatic ring are well known in the literature[14.15,351. The experimental styrene-benzene ratio (Fig. 3)
was used to adjust the competitive reactions (7) and (S), the
ethylene-styrene ratio (Fig. 4) was used for the adjustment of
the competitive reactions ( 5 ) and (6). With the model of Ta-
Table 2. Reaction model of the pyrolysis of ethylbenzene (computer diagram). a: Kinetic parameters
evaluated by analogy.
Elementary reaction
I
( 1 )
( 2 )
C6H5CH2CH3
C6H5CH2CH3
3
4
5
6
C6H5CH2CH3
C6H5CH2CH3
C6H5CH2CH3
C6H5CH2CH3
(
(
(
(
)
)
)
)
( 7
( 8
( 9
(10
+
*
+ CH3
+ C6H5
+
+ H
+ H
+
-+
*
)
)
)
)
(11 )
(12 )
C6H5CHCH3
C6H5CHCH3
C2H5
C8H11
C6H7
C4H3
+
(13 )
(14 )
CH3 + CH3
H + C6H5CH2
C6H5CH2 + C6H5CH2
*
(15 )
-f
+
+.
+
+.
+.
+
C6H5CH2 + CH3
C6H5 + C2H5
(15.3 /305.4 ) (16)
(16.0 1347.3 )
a
CH4 + C6H5CHCH3
C6H6 + C6H5CHCH3
H2 + C6H5CHCH3
C8H11
( 7.821
(11.0 /
(10.5 /
(11.0 /
C6H5CHCH2 + H
C2H4 + C6H5
C2H4 + H
C2H4 + C6H7
C2H4 + C4H3
C4H2 + H
(14.85/125.5
(14.85/182.0
( 1 3 . 0 /125.5
( 1 3 . 0 /105.0
( 1 3 . 0 /105.0
( 1 3 . 0 /105.0
C2H6
C6H5CH3
C6H5CH2CHZC6H5
(10.34/
29.3
31.4
21.0
46.0
) (13)
)
a
)
a
)
a
(16)
)
a
(16)
)
a
)
a
1
a
0.0 )
I 0.0
(11.9 1 0.0
(11.0
Numerical integrations were performed with the most simple reaction models developed from these results (Table 2)
using Gear’s program. The kinetic constants of the elementary reactions were taken from the literature. Values of constants not known in the literature were estimated from analogous reactions. The calculated results were compared with
experimental data from a laboratory flow reactor using a gas
chromatograph for analysis1331.
The mass balance showed that part of the ethylene is
formed from cleavage reactions of the aromatic ring already
at relatively low temperatures. This was accounted for by
(16)
)
)
a
a
ble 2 the experimental results were in good agreement up to
1150 K. At higher temperatures the formation of benzene
and toluene was markedly higher than the calculated values.
The model could only be adapted by adding a second initial
reaction (2) and the consecutive reaction (9). With these the
reaction model of the pyrolysis of ethylbenzene was increased to 15 elementary reactions.
Figure 5 shows that the model simulates the reaction very
well at relatively short reaction times and within a wide temperature range. As Figure 6 shows, the agreement of the
time-conversion plots is not as good. The simulations of the
. . . . . . . . . . . . . . . * . . . . . .
I
.
. . . . . . . . . . . .
’
. * .
. . . . . . . .
. . . . . . . . . .*
.
6
. . . .
. .
*
..
...
m
338
1
I
I
I
65-C5H,
78- Benzene
m r n ~ m . a m
m
I
....
.
......
.....
1
79- C6H,
- 89-CTH5
m
m....
I
-
LL-Propane
- 50- Butadiyne
I
.
I...
&
. .
I
39-C3H3
.......... m
-
. . . . . .
m
-
,
...........
S
15- Methyl
16-Methane
26-Acetylene
- 28-Ethylene
90- Phenylcarbene
91- Benzyl
- 32-Toluene
- 102- Phenylacetylene
- IOL-Styrene
- 105-a-Methylbenzyl
- 106-Ethylbenzene
- 128-Naphthalene
- 182-Bibenzyl
I
Anyew. Chem Int. Ed Engl 19. 3.11-341 (19Xnl
benzene and ethylene formation agree reasonably well, but
the calculated conversion values of ethylbenzene and styrene
disagree strongly with the experiments. A possible reason is
the existence of reactions of styrene, decompositions, as well
1.00
t
=r
1.90
800
900
1000
1100 K
Fig. 3. Styrene/benzene ratio in dependence of the temperature ( A 0 experimental values, ----- calculated values).
1.80
0
2
4
6
Ills
8
Fig. 6. Comparison of relative molar concentrations as function of time at 950 K
(see Fig. 5).
as reactions with radicals. Because these reactions are very
similar to those of ethylbenzene, the calculations for benzene
and ethylene are not so strongly affected.
4.4. Pyrolysis of n-Hexane
Fig. 4. Ethylene/styrene ratio in dependence of the temperature (see Fig. 3)
Fig. 5. Comparison of relative molar concentrations as function of temperature
( 0 ethylbenzene, A styrene; D ethylene: 0 benzene; ---- calculated curves).
Angew. Chem. Inr Ed. EngI. 19. 333-343 (1980)
In the last few years the pyrolysis of n-hexane has been in401. Hexane is a kind of key
vestigated more
substance. The molecule is large enough to be considered as
a model for higher paraffins, but the product distribution is
not too complicated. In order to describe the reaction at relatively low temperatures and conversions a reaction model
consisting of 38 elementary reactionsI2'l is satisfactory (Table
3). Calculations showed that the reaction rate determined in
a static apparatus by the overall pressure rise can be simulated well enough (Fig. 7). Table 4 shows that experimental
and calculated product distributions also agree well. At higher conversions, however, product reactions have to be considered, the reaction model then extends to about 100 elementary reactions.
The decomposition and isomerization reactions of the
C6H,3 radicals (23-28) are of particular irnportan~e'~'
4H1.
Calculations showed that the ratio n-hexyl/l-methylpentyl
has a decisive influence on the ratio of the products ethylene/propene, and that in the temperature range studied the
rate of the isomerization of n-hexyl ---t 1-methylpentyl is considerably higher than the rate of decomposition of the radicals. Therefore the equilibrium, which is shifted in the direction of I-methylpentyl is established rapidly, and the most
important decomposition reaction is (24). Application of the
p-rule to the decomposition reactions of the C6Hf 3 radicals
indicates a ratio of reactions (23-26) of about 1 : 14: 1.5:9.
This also confirms the importance of (24).
339
Fig. 7. Pressure rise in the pyrolysis of n-hexane (. experimental values; ---- calculated curve).
This reaction is explosive in a particular p T range. The reaction model used consists of 24 elementary reactions, selected
mainly from chemical considerations. Most of the kinetic
constants were taken from the literature (Table 5). For regions showing a nonexplosive behavior, computations were
camed out for a flow reactor assuming a steady state and
taking into account heat flow across the reactor wall. The
concentrations of the radicals, the reaction rates of the elementary reactions, and the rate of heat formation inside the
reactor were calculated. The heat flow was determined as a
function of temperature, pressure, and the geometry of the
reactor. The explosion limit was defined as the temperature
at which the heat formed in the reaction cannot be completely removed.
The calculations indicate that the “critical” temperature is
highly dependent on traces of water. An increase in water
content from 1 to 10 to 100 ppm lowers the critical temperature from 1100 to lo00 to 900 K. Further the calculations
showed that the influence of water occurs essentially via the
reaction chains (4), ( 5 ) and (4), (12), (18) (Table 5).
4.6. Reactions in the Atmosphere
4.5. Oxidation of Carbon Monoxide
Pilling and Noyes have done some work on the oxidation
A number of publications deal with photochemically iniof~interest
are
~~~~
].
of carbon monoxide by oxygen in the presence of ~ a t e r [ ~ ~ , ’ ~ I . tiated reactions in the a t m o ~ p h e r e ~Topics
Table 3. Reaction model of the pyrolysis of n-hexanc (computer diagram). a: Kinetic parameters
evaluated by analogy.
Reaction
Elementary reaction
No.
C6H I4
C6H 14
C6H I4
C6HI4 + H
C6H14 + H
C6H14 + H
C6H14 + CH3
C6Hl4 + CH3
C6H14 + CH3
C6H14 + C2H5
C6H14 + C2H5
C6H14 + C2H5
C6H14 + C3H7
C6H14 + C3H7
C6HI4 + C3H7
C2H5
C3H7
C3H7
C4H9
C4H9
C5Hl I
C5HI 1
I-C6H13
2-C6H 13
3-C6H13
3-C6H I 3
I-C6H13
2-C6H I3
CH3 + CH3
CH3 + C2H5
C2H5 + C2H5
H + H
C3H7 + C3H7
C4H9 + C4H9
C5Hll + CSHII
I-C6H13 + I-C6H13
2-C6H13 + 2-C6H13
3-C6H13 + 3-C6H13
340
*
+
A
*
+
-f
-+
+
*
-+
-+
+
+
*
+
+
-+
-+
-f
-+
-+
-+
*
+
+
+
-+
CH3 + C5Hll
C2H5 + C4H9
C3H7 + C3H7
(17.2/353.4)
(16.6/343.3)
(16.0/340.8)
H2 + I-C6H13
H2 + 2-C6H13
H2 + 3-C6H13
( 9.0/ 42.3)
a
(10.3/ 39.4)
(10.W 39.4)
a
a
CH4 + I-C6H13
CH4 + 2-C6H13
CH4 + 3-C6H13
( 8.7/ 48.1)
( 8.9/ 40.2)
( 8.6/ 40.2)
(13)
C2H6 + I-C6H13
C2H6 + 2-CbH13
C2H6 + 3-C6H13
( 7.8/ 52.8)
( 8.0/ 43.5)
( 7.7/ 43.5)
a
a
C3H8 + I-C6H13
C3H8 + 2-C6H13
C3H8 + 3-C6H13
( 7 . 6 f 52.81
( 7.8/ 43.5)
( 7.5/ 43.5)
a
C2H4 + H
C3H6 + H
C2H4 + CH3
C3H6 + CH3
C2H4 + C2H5
C3H6 + C2H5
C2H4 + C3H7
C2H4 + C4H9
C3H6 + C3H7
C5H10 + CH3
C4H8 + C2H5
( I 3.5/ 170.4) (16)
(13.8/159.1) (16)
(13.6/138.6)
(l2.l/ll3.5)
(13.6/121.4)
(12.5/116.8)
(12.6/120.2)
(12.6/120.2)
(13.5/108.9)
(12.6/120.2)
(14.3/113.5)
2-C6HI 3
I-C6H13
(ll.O/ 57.4)
( l l . l / 69.5)
C2H6
C3H8
C4H10
H2
(10.6/
(10.6/
(10.6/
Tar
Tar
Tar
Tar
Tar
Tar
(ll.O/
0.0)
0.0)
0.0)
0.0)
(10.6/
0.0)
(10.6/ 0.0)
(10.6/ 0.0)
(10.6/ 0.0)
(10.6/ 0.0)
(10.61 0.0)
a
a
(41)
(13)
(13)
a
a
a
(16)
(16)
(16)
(
(
5)
5)
a
(15)
a
a
(
(
5)
5)
a
a
a
a
a
a
a
a
a
a
Angew. Chem. Ini. Ed. Engl. 19. 333-343 (1980)
by “mathematical modeling” is an important tool in chemical kinetics. In mathematics this is called the “inverse problem”. A non-linear least-squares fit in dependence of the
reaction parameters is performed using possibly weighted experimental data and values calculated from the model
( W,=weighting factors):
Table 4. Comparison of the product distribution (16)of the pyrolysis of n-hexane:
700 K, 1 3 x lo4 Nm
(0.13 bar), 1516 conversion.
’
1
Product
exp,
-
H2
CH 4
C2H4
C2H6
C3H6
C3H8
C4H8
C5H10
20.83
22.18
13.76
26.64
1.39
12.95
2.24
calc.
1.42
23.51
23.53
11.18
26.78
1.59
11.45
0.46
L
(5)
Function F should be minimized.
I
Table 5 . Reaction model of CO oxidation (computer diagram). a: Kinetic parameters evaluated by analogy.
~
~~
~
Reaction
lOgloA/E,
Ref.
( 8.35 /251.2
( 8.00 / 0.0
(51)
(52)
(53)
Elementary reaction
No.
*
co2 + 0
C O + O + M
H20 + 0
CO + OH
02 + H
OH + 0
H20 + H
H2 + 0
H2 + OH
OH + OH
O H + H + M
-+
C02
2OH
C02
02+H+M
H 0 2 + OH
H02 + H
H02 + H
H02 + H
H02 + 0
CO + H02
H02 + H02
H202 + OH
H202 + H
H202 + H
H202 + 0
H202 + M
-+
co +
02
*
*
*
*
*
*
*
-+
*
*
*
-+
*
-+
*
-+
*
*
*
*
*
+
M
+H
m + o
02 + H
H2 + OH
OH + H
H20 + H
H20 + 0
H20 + M
H02 + M
H20 + 0 2
H20 + 0
H2 + 0 2
20H
0 2 + OH
CO2 + OH
H202 + 0 2
H20 + H02
H2 + H02
H20 + OH
H02 + OH
20H + M
(10.76 /
( 8.49 /
(11.35 /
(10.40/
(10.92 /
(10.24 /
(10.36 /
( 9.76 /
(11.00 /
( 9.51 /
(10.70 /
(10.70/
(10.40 /
(11.40 /
(10.70/
4.2
2.9
8.0
4.2
( 8.00 / 41.9
(10.00 / 4.2
(10.00 / 7.5
(10.37 / 38.5
(11.50 / 37.7
(l0.70/
5.0
(14.07 /190.5
the influence of particular substances emitted from the
earth’s surface on the ozone layer in the upper atmosphere,
and the formation of smog. One of the most interesting examples was the formation of the Los Angeles smog, which
was studied using a model of 92 elementary
In
this work, the main problem is the formulation of suitable
reaction models. The calculated time dependence of the
reaction systems agreed relatively well with the experimental
data obtained in a smog chamber. The computations are
rather complex because of fluctuations in the radiation density and because of the need to include diffusion processes
and convective transport phenomena in the calculations,
which are difficult to describe exactly.
Though experimental results can be only partly explained
by model computations, important information has been attained on the participation of particular substances in smog
formation, and there is no doubt that these methods will help
to increase our understanding of chemical reactions in the atmosphere and their further consequences.
5. The “Inverse Problem”
5.1. Evaluation of Kinetic Constants
Besides the “simulation” of chemical reactions, the evaluation of kinetic constants of particular elementary reactions
Angew. Chem. Inr. Ed. Engl. 19, 333-343 (1980)
)
)
75.4 )
2.5 )
70.3 )
0.0 )
84.2 )
39.6 )
21.8 )
3.8 )
0.0 )
-4.2 )
4.2 )
)
)
)
)
)
)
)
)
)
)
)
(55)
(54)
(55)
(53)
(53)
(55)
(53)
(56)
(54)
(57)
(57)
(57)
(57)
(57)
(57)
(57)
(54)
(54)
(54)
a
(54)
The usual mathematical methods for the solution of such
optimization problems can be divided into two groups:
1) Direct search methods: For each experimental value a
corresponding value has to be calculated according to a model. The set of optimal parameters is obtained by a systematic
search of certain parameter sets, frequently expressed as lattice points in higher dimensions.
2) Descent methods: These methods require in addition to
the function F also the partial derivatives of F with respect to
the parameters. These derivatives are used to find the steepest descent to the minimum of F.
Bremermann and Milstein tested a number of different
methods from each
A direct search method with
random vectors proved to be most effective and robust. The
direction indicating the variation of the parameter vector is
chosen by a random number generator. The “best” parameter set in this direction is determined by a special iteration
method. This new parameter set is the starting point for the
next random vector, and the calculation is continued until the
new random vectors do not result in further improvements of
the parameter sets. This method was applied to experimental
data from the Calvin photosynthesis cyclef6’].The reaction
model consisted of 22 reactions with 18 species. Optimal
reaction constants were calculated for all reactions. The
model calculation agreed very well with the experimental
time behavior of the 18 species within the course of the reaction.
341
Another method, which is often more effective although
more cumbersome, is the frequently used “trial-and-error’’
approach. The “improvement step”, i. e. the selection of the
new parameter set, is executed “manually” taking into account all the previous results.
In the numerical optimization procedure it is the “stiff
solver” subroutine which demands the greatest proportion of
computer time. Therefore, the greatest potential for improving the method lies in this area. The steady-state assumption
decreases the computational effort considerably. Various parameters or relations between parameters can then often be
evaluated and described by simple linear regression. Thus it
is hardly surprising that nearly all parameter fittings in complex reaction systems assume QSSA conditions.
It may happen that in such a parameter optimization procedure the minimum of the function F can be determined,
but no satisfactory consistency between the experimental and
the calculated values is obtained. This is a clear indication
that the chemistry of the reaction model is incorrect.
rectly. This objection is fundamentally wrong, as can be
clearly demonstrated by an error analysis of the individual
parameters. This is best done by evaluating a sensitivity matrix, which contains the variations of the concentrations of
the different chemical species in dependence of the variations of the parameters of the individual elementary reactions. Such sensitivity matrices also depend on temperature,
initial conditions, and reaction time. Table 6 shows the sensitivity matrix of the model for hexane pyrolysis (Table 3) at
718 K and initial concentrations of 0.0023 mol/l within the
steady-state region.
The columns contain the stable compounds, and the
rows contain the elementary reactions. The elements
of the matrix represents the percentage change in concentration of a particular species if the reaction constant
of the corresponding elementary reaction is increased by a
factor of 2. (In fact, however, the coefficients were calculated
with the assumption of a differential variation of the system.)
Negative values indicate that the concentrations decrease.
The matrix of Table 6 was calculated assuming a steady
state, which remarkably reduced the computational effort. A
comparison shows that in this case the agreement of the
QSSA values with the “exact” values is better than 0.1% under the conditions chosen.
The matrix demonstrates that there are only a few reaction
constants which significantly influence the concentrations of
most of the products. Most rate constants are “insensitive”
and their absolute values have little influence on the product
5.2. Sensitivities of Kinetic Parameters
If for a reaction mechanism one of the optimization methods yields a set of parameter values which fits the experimental results well, an often heard objection is that almost any
conceivable set of data can be simulated by so many parameters (even the shape of an eIephant!), and that ail these sets of
parameters will describe the measurements more or less cor-
Table 6. Sensitivity matrix of the reaction model of the pyrolysis of n-hexane (see Table 3) (computer diagram)
Product
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
26
27
28
29
30
31
32
33
34
35
36
37
38
342
H2
CH4
C2H4
C2H6
C3H6
C3H8
18
24
9
18
23
9
18
23
9
I8
24
9
17
24
9
18
23
9
18
24
9
18
23
9
35
47
18
18
23
9
35
47
18
0
0
0
0
-0
0
0
-0
-0
0
0
3
-3
6
37
-32
9
65
9
3
-3
6
36
-31
6
46
4
-4
9
53
-51
10
73
-0
4
-4
9
52
- 50
10
71
4
-4
9
53
-51
10
73
4
-4
9
53
-51
10
73
-1
-1
-0
-0
0
-0
0
2
-2
4
0
1
0
-0
-0
0
-1
0
0
0
1
-I
-
-2
-6
40
0
-1
-0
-0
-0
12
7
50
31
-0
-0
-0
-0
-2
34
66
-61
-4
4
-4
10
-4
6
-4
I0
1
-1
1
0
-0
-0
0
-0
0
-0
-0
0
-1
2
-1
0
1
-1
-2
-6
-40
0
-1
-0
0
0
-0
-0
-0
0
0
1
-0
-0
-0
-0
-1
0
-0
-0
0
-1
1
0
0
-0
2
-2
0
0
1
0
0
-0
0
-0
0
0
-0
-0
0
-1
3
-2
0
0
1
-1
-2
-6
-40
0
-0
-0
0
-2
-6
-40
0
-1
-0
-2
-6
-40
0
-1
-0
-0
0
0
-0
-0
-0
-0
0
-1
0
0
-0
-0
0
-1
3
-2
0
1
-1
-2
-6
- 40
0
0
-0
-0
0
2
-3
4
5
-3
8
61
27
4
-4
-89
0
-1
-0
0
-0
-0
0
C4H8 C4H10 C5H10
Tar
C6H14
1
-0
-1
12
6
47
29
-0
-0
1
4
1
1
1
-0
-1
23
-3
-22
-0
0
0
1
-1
1
2
0
-0
-0
-0
0
-0
0
-1
0
0
-0
0
0
0
0
1
1
-0
0
0
0
4
-4
9
53
-51
10
73
-0
0
3
-2
4
0
1
0
-0
-0
0
8
-8
18
107
- 102
20
145
0
-0
-0
-8
8
-9
-179
0
-0
-0
0
-1
-97
98
-2
5
-4
0
-0
2
-2
-5
-12
-!
-1
-2
-6
-40
0
-2
-6
-40
0
-1
-5
-12
19
1
-1
-2
-6
- 40
0
0
-2
-1
98
-0
-0
0
-0
0
-0
0
-0
0
0
0
-1
-0
0
-0
-0
0
0
-0
-0
0
-0
-0
0
-80
0
0
0
Angew. Chem. Inr. Ed. Engl. 19, 333-343 (1980)
distribution. From different sensitivity matrices evaluated on
the basis of different experimental parameters a new matrix
can be calculated containing the variances (standard deviations) and covariances of the reaction pararneterd6O1.These
variances depend again on the experimental conditions
(number of measurements, initial conditions, temperature,
etc.) and on the accuracy of the experimental data. Thus, if a
given kinetic parameter is to be evaluated with greater accuracy from experimental data, the experimental conditions
must be such that the variance becomes as small as possible.
This can also be done by selecting special initial conditions,
e.g. by the addition of reactants which increase the sensitivity of the elementary reaction in question. Another possibility
is the selection of another reaction system in which this particular elementary reaction has u priori a high sensitivity.
6. Outlook
Work on the kinetics of complicated chemical reactions
with the aim of solving all the details of a given mechanism
and its dynamics were considered to be more or less hopeless
until recently, because no methods were available for evaluating more general quantitative conclusions from the experimental materials. Today the very opposite seems to apply.
Computational methods for kinetic treatment of even
complicated chemical systems have been developed to a high
standard, and they will surely be further improved. As computer capacity becomes much more readily available in the
future, these methods will become more generally applicable.
The quality of the reaction model will then be the limiting
factor. The quality of models, however, largely depends on
the accuracy with which the kinetic constants of the particular elementary reactions are known. Therefore methods of
experimental determination and evaluation of kinetic constants will become more important.
The application of such methods will initially be focused
upon more special problems such as the effect of additional
substances on reaction rates and product distributions. Within this kind of work the “inverse problem” and sensitivity
analysis play an important part in obtaining quantitative information. The scope of these methods will be enhanced if
they can be extended to the computation of heterogeneous
reactions too. For this type of reaction, absorption, desorption, and diffusion processes have to be included in the reaction scheme. This does not mean that the reaction models become more complicated. On the contrary, in catalytic processes the number of “important” elementary reactions is
comparatively small.
Although the methods outlined above may be viewed with
skepsis, perhaps due to the complicated mathematics involved, they nevertheless supply detailed information about
the mechanism and time dependence of complicated reactions and permit a fast and reliable simulation of the influences of different parameters on chemical reactions. This
is a significant progress in chemical kinetics.
Received: October 31. 1979 [A 315 IE]
German version: Angew. Chem. 92, 331 (1980)
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