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ABSTRACT
A discrete activation-energy distribution derived by analysis
of reaction rate data often contains too many energies to be
conveniently used in most petroleum-basin modeling programs.
This program begins with the complete distribution and gradually
reduces the number of energies to a user-specified number, while
doing least-squares refinement to further optimize both the
energies and the percentage at each energy. A user-specified,
fixed frequency factor is assumed. The original experimental
rate data can be used in the optimizations. For three test
samples having broad energy distributions, seven 1st-order
reactions describe the data quite well. Substantially better
agreement, however, is obtained using 2nd-order reactions. If
experimental rate data are not available for use in the
optimizations, synthetic rate data calculated from the parameters
specified for the complete discrete-E distribution can be used.
INTRODUCTION
Kinetics analysis of reaction rate data for samples of interest
in petroleum geochemistry is frequently done to obtain a discrete
activation-energy distribution. When a spacing of 1 kcal/mol is
used in the analysis, the resulting distribution can contain too
many energies to be conveniently used in most petroleum-basin
modeling programs. For example, the LLNL computer program
KINETICS[1] permits up to 25 energies
and it is not uncommon for the maximum number to be used in
analyzing some samples. Simply using a larger energy spacing to
cover the required range of energies is not a good, universal
method of decreasing the number of energies, because this can result
in calculated rate profiles that are multi-peaked, due to resolution
of individual activation energies. Therefore, a better method was
developed to reach a lower number of energies. The computer program
ENERGIES implements the method.
INPUT DATA
Kinetic parameters (A, E’s, and percent of the reaction at
each E) for the complete discrete-E distribution are required. These
parameters can be automatically read from a previous KINETICS output
file or they can be entered from the keyboard. The frequency factor
is assumed to be constant for the entire analysis. Since the same
activation energy range is used throughout the subsequent analysis in
optimizing a smaller set of E’s, it is important that the
E-range for the initial discrete-E distribution be accurately
constrained by the thermal histories of the experimental reaction rate
data to prevent spurious peaks from occurring in the calculated rate
profiles. A suitable E-range is readily obtained by using discrete-E
parameters from a KINETICS calculation in which the range is
determined by the actual time-temperature constraints of the data.
The user specifies the desired number of E’s in the reduced E-distribution and the reaction order (1 or 2) to be used in the analysis. Normally, the user will probably want to use 1st-order reactions, since they are most consistent with further applications of the results. The option of using 2nd-order reactions is given in this program, however, since such reactions can much more efficiently describe rate profiles having long, high-temperature tails.
Data used in the optimization calculations include up to 51 files containing time, temperature, and either rate or integrated-rate measurements. The specifications for these files are identical to those used in KINETICS. The data need not be at constant heating rates. Numerical methods given by Braun and Burnham[2] for integrating reaction rate data for arbitrary thermal histories are used, with the simple substitution of a discrete distribution for the Gaussian distribution used in that paper.
If experimental rate data are not available for use in the optimizations, synthetic rate data calculated from the parameters specified for the complete discrete-E distribution can be used. With this option, three data files containing time, temperature, and rate are calculated for constant heating rates of 2, 10, and 50 C/min from 250 to 700 C. These data are stored in files SYN-02.DAT, SYN-10.DAT, and SYN-50.DAT in the current working directory. This option should be used only if the experimental rate data are not available.
METHOD
During the course of the analysis, many changes in the E-distribution
are made. These will be systematically described below. For each new
E-distribution, the fractions (hereafter called f’s) for the
quantity associated with each discrete-E are recalculated by means of
linear least squares. The f’s are constrained to sum to one, with
each value greater than or equal to zero and less than or equal to
one. This constrained, linear least squares optimization is based on
minimizing the sum of the squares of the residuals (the differences
between the measured and calculated rates of reaction or, if
desired, the differences between the measured and calculated
integrated rates).
The following paragraphs summarize the scheme used in reducing a broad E-distribution. First, the uniform E-spacing of the complete distribution is gradually increased until {the number of nonzero fractions plus up to two zero fractions} is less than {twice the final desired number of energies}. Up to two zero fractions are used in recognition of the fact that some fractions can become nonzero again during the further gradual change of the E-spacing. A maximum spacing of 3 kcal/mol is allowed. This re-sizing algorithm appears to give a satisfactory initial spread in the E’s, while leaving a large enough number of energies left for further optimization based on other criteria. After making these adjustments in the initial E-distribution, the program discards any E’s having zero fraction.
Next, the number of E’s is further gradually reduced by identifying the energy having the smallest fraction and absorbing it into its neighbors. The f’s are used in calculating how much the neighboring E’s are to be moved towards the energy being absorbed (using a linear tie-line algorithm). After eliminating an energy, the remaining E-distribution is tentatively optimized by simultaneously shifting all E’s in small increments of 50 cal/mol and recalculating the f’s by constrained linear regression until a minimum in the sum of the squares of the reaction rate residuals is found. This absorption and optimization process is continued until the number of remaining energies equals the final desired number. When this process is finished, any E’s having zero fraction are discarded. Thus, the final number of E’s having nonzero f’s may be less than or equal to the desired number.
PROGRAM EXECUTION
The user first selects the command file name. This can either be the
name of an existing command file that was previously made or the name
of a new command file to be created. The user then continues with the
"Setup" command, to either modify the existing command file
or create the new one.
There are four steps in the setup:
Once the above information has been provided, the main calculation can be initiated. A Fortran program ENERSOLV reads the command file and does the optimization calculations in a DOS window. Upon completion of the calculation, the remaining three commands are enabled to permit (1) a comparison of the starting and reduced E-distributions, (2) a more detailed inspection of the results of the calculation, and (3) generation of various plots of the results.
CONCLUSIONS
The computer program ENERGIES can efficiently reduce a broad, discrete-E
distribution to a smaller number of optimized energies. Even for initial
distributions that are very broad, as few as 7 energies are frequently
adequate for calculating reaction rates. Most of the discrepancy is
limited to minor structure in the calculated rates for the
high-temperature tail. This discrepancy is hardly discernible when
comparing integrated rates. Using 2nd-order reactions, instead of
1st-order reactions, greatly improves the accuracy of modeling a broad
E-distribution with an even smaller number of energies.
REFERENCES
and KINETICS: A Computer Program to Analyze Chemical Reaction Data,
Lawrence Livermore National Laboratory, Report UCRL-ID-21588
Rev. 2 (1994).
and Analysis of Chemical Reaction Kinetics Using a Distribution of Activation Energies and Simpler Models, Energy & Fuels 1, 153-161 (1987).
ACKNOWLEDGMENT
An earlier MS-DOS version of this program was originally developed
with the support of Mobil Exploration and Producing Technical Center,
Dallas TX.
DISCLAIMER
The program ENERGIES is provided "as
is" without any additional warranties of any kind, either express
or implied, including any warranty of merchantability or fitness for a
particular purpose. In no event will Braun or the program vendor be
responsible for any damages, including but not limited to lost profits,
lost savings, or other incidental or consequential damages arising out
of the use or inability to use this program.
ENERGIES is available exclusively from Humble Instruments and Services, Inc.
$495.00 per copy.
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Copyright © 1998-2000 Humble Instruments &
Services, Inc.. All rights reserved.
Revised: