A hybird / digital software package for the solution of chemical kinetic parameter identification problems by ALAN M. CARLSON
Electronic Associates, Inc. Princeton, New Jersey

INTRODUCTION The modern hybrid computer offers many significant improvements over first generation hybrid systems These improvements include:
1. The increased speed of digital computers en-

abling programs to be written in hybrid FORTRAN without drastically limiting hybrid solution rates. 2. The development of analog/hybrid software (e.g., hybrid simulation languages and analog set-up programs). The net result of these improvements has been an increase in the SCope and complexity of hybrid applications and a reduction in the effort required to program and debug hybrid problems. Unfortunately, the dev'elopment of hybrid applications software has not kept pace with recent hybrid improvements. Applications software for purposes of this discussion is defined as an integrated set of digital/hybrid programs capable of solving the majority of frequently occurring problems in a specific applications area. Based on this definition, little or no tangible information is currently available on the practicality of developing hybrid software packages although its benefits are obvious. In mid-1968, EAT's Princeton Computation Center initiated a development project to· determine the feasibility of hybrid applications software. The objectives of the project were to select a frequently occurring
733

application area, develop general purpose software for it, and assess the resultant software based on the above definition, computer economics, ease of use, etc. The objectives of this paper are to present and illustrate the use of the software package developed as a result of the above mentioned project. The chemical kinetic data analysis problem, which is often referred to as the chemical model building or parameter identification problem was selected as the applications area. Since the software package, which will be referred to as the kinetic data analysis or KDA package, solves chemical kinetic problems via either all-digital or hybird simulations; the question of simulation economics and accuracy was investigated and will also be discussed. The illustrative problem is the "Monsanto Benchmark Problem" which has been welldocumented2 ,8,6-8 and typifies the chemical kinetic problems the KDA package was designed to solve. This problem requ.ires the determination of twenty-two unknown parameters using thirteen sets of experimental data and a mathematical model requiring the simultaneous solution of seven non-linear differential equations.

Problem analysis
Referring to Figure 1 the kinetic data analysis problem, which occurs during the initial phases of, say, plant design· and economic optimization projects, has three essential, related parts. They are:

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7:)..1

Fall Joint Computer Conference, 1969 except for small problems. The adv2,ntage of direct simulation is that it provides the analyst with a great deal of knowledge about the physical behavior of the system being simulated. 2. Parameter Estimation-A variety of digital computer programs that solve kinetic problems using, for example, statistical techniques, line and non-linear least squares, etc. Spec:jfic illustrations may be found in a recent article by Lapidus and Bard. 5 Unless the analyst is familiar with these programs and is capable of using them without making major modifications, their utilization creates a number of problems. These problems include: A. The mathematical techniques restrict the form of the data or the model:, thereby influencing the design of kinetic experiments' (e.g., batch-isothermal experiments). B. The infrequent use of statistical techniques or lack of a working knowledge of statistics makes it difficult for the user to evaluate program results and equate them to the physical problem. Parameter estimation programs do, however, represent a relatively economical means of solving kinetic problems if they can be used efficiently and without major revisions. 3. Parameter Optimization-This technique uses general purpose optimization algorithms (e.g., gradient search) to automate the above mentioned direct simulation technique. Referring to Figure 2, the optimization variables, A, which are unknown parameters in the kinetic model, are varied so as to minimize an objec:tive function. The objective function, F, is a sealar quantity representing the error between eomputed and experimental results which may be obtained using a variety of mathematical relationships (e.g., sum of squares, integral of the absolute error, etc.). As shown in Figure 2, the best current values of the algorithm variables, AB' are those model parameters resulting in the "best fit" between experimental and computed concentration data, AF, when the algorithm can no longer improve the objective function. 'This technique is: A. Theoretically the most general purpose approach to solving kinetic data analysis problems. It may be used in either a11-

DEFINE ClEM ICAl ANALYSIS
TECHNIQ~S

PERFORM

EXPER I MENTAL DATA

DESIGN lAIORATORY EXPER I ME NTS DEFINE COMPUTATIONAL
TECHNIQ~S

~--~~

EXPERlftENTS

COMPUTER I'ROGMMM I NG .....-._ _ _ _ CIECKOUT MOons

~

DEVELOP MODELS FOR PROPOSED MECHANISMS
SET

PERFORM KINETIC DATA ANALYS IS STUDY

STANDARDS FOR RESULTS

Figure I-Typical kinetic data 2,nalysis How diagram

1. Performing kinetic experiments to obtain the data necessary to determine the model. 2. Proposing one or more mathematical models representing alternative kinetic mechanisms, chemical reactions, etc. 3. Computational analysis of the proposed models by determining values for model parameter (e.g., rate constants) that minimize the discrepancy between computed and experimental results.

The technology required to design and perform kinetic experiments is available and the initial derivation of mathematical models to simulate these experiments is not generally regarded as a diffiult task. However, the applications software required to evaluate these . models is either unavailable, restrictive in a physical sense, or fails to provide the user with an efficient solution to his problem. The project manager responsible for the solution of a kinetic data analysis problem, based on an impromptu survey, is not interested in becoming deeply involved in programming or underwirting extensive program development studies to solve his problem. With the exception of a few industrial organizations, the computational alternati.ves at his disposal are not consistent with his interests. The computational alternatives are:
1. Direct Simulation-The classical analog com-

puter or digital simulation language studylO where the analyst adjusts model parameters in a trial and error fashion. This technique is generally successful; however, it is very time consuming' susceptible to human error, and inefficient

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A. Hybrid/Digital Software Package

735

PROGRAM ALGORITHM EXECUTI VE

6~ J =G

:i

J

G;:J
FORMS

t
ANALOG SET-UP PROGRAM INPUT/OUTPUT PROGRAMS

,
OPTIMIZATION PROGRAMS

~
DATA

"

---

PREPARATION PROCESSOR

I" "
ANALOG COMPUTER

ALL-DIGITAL STUDIES

.,

-

HYBRID STUDI ES

"

01 FFERENTI AL EQUATIONS

•

ALGE BRAI C EQUATIONS

•

Figure 3-KDA program organization Figure 2-Simplified parameter optimization flow diagram

digital or hybrid simulation~ and the mathematical forms of the kinetic models and physical systems that can be investigated are not restricted. B. Not generally used because many organizations do not have access to appropriate software and the development of this software imposes an intolerable financial burden on anyone project. In the past, this technique was not widely used due to high digital production costs. The "Parameter Optimization" technique, requires several hundred simulations of individual experiments per optimization run. The results of the above mentioned survey indicated a significant market existed for general purpose kinetic data analysis applications software if it could produce easily interpretable results, require minimal user participation, and solve kinetic data analysis problems at a reasonable cost using the "Parameter OptimizatioI)." technique. These results were used as guidelines for the software development project.
Software description

software package with a convenient mechanism to add programs, etc. The five programs shown in Figure 3 are an Analog Set-Up Program, a Data Preparation Processor, and three optimization programs. The optimization programs are identical with the exception of the mathematical form and/or computer used to simulate the kinetic model or models. These programs, which have identical executive, optimization, and objective function programs are:
1. A hybrid optimization program using the analog

computer to simulate kinetic models. 2. An all-digital optimization program for kinetic models requiring the solution of one or more ordinary differential equations. 3. An all-digital optimization program for kinetic models requiring the solution of a set of algebraic equations (e.g., continuous stirred-tank reactor experiments.) The Analog Set-Up Program is an interactive program used, for example, to static check analog patch panels prior to executive hybrid production runs. Since programs of this type are generally part of the operating system software for a hybrid computer, a description of this program will not be presented in this paper. Subsequent discussions will also exclude the Program Executive, since its function has, for all practical purposes, already been d·3fined. Therefore, the description of the Kinetic Data Analysis package will be limited to the Data Preparation Processor and the optimization programs. A brief description of hmv the user interacts and communicates with the software package to solve a kinetics problem will be discussed first to clarify later discussions.

The Kinetic Data Analysis package consists of several digital/hybrid processors whose individual functions and interactions are too complex to describe in this paper. However, referring to Figure 3, the current version of these processors may be visualized as five FORTRAN programs under the control of a Program Executive. The Program Executive restores and executes programs requested by the user, provides the

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736

Fall Joint Computer Conference, 1969

TOTAL NUMBER OF CHEMICAL SPECIES, • " ••••••• UNKNOWN ARRHENIUS RATE CONSTANTS, •••••

C1]

C USER )

J:L1]

f
DATA PREPARATION PROCESSOR

EXPERIMENTS OR SETS OF DATA, ••••••••••• ~ AND UNKNOWN MODEL PARAMETERS •••••••• ~

+
INSERT KINETIC MODEL IN KDA SUIROUTINES MODEL

ESTIMATE OF DIGITAL! HYIRID ECONOMII CS ANALOG SCALE FACTORS, S TAT IC AND DYN AMIC CHECK SO LUTION:,

--

CATALYST VARIABLE TRANSFORMATION? •••••• NON-ISOTHERMAL EXPERIMENTS? •••••••••

l::(]
DATA KDA PROGRAMS

HYIRID ID IGITAL OPTIMIZA, TION CHECK SO LUTION

--

DIGITAL SOLUTION OF KINETIC MODEL?

~ ••••• ~

l-----

OPTIMIZATION PROGRAM

EXECUTEA, HE PROGRA FOR ALL- DIGITAL SOLUTIOt

--

DATA SET TEMPERATURE DATA IN DEGREES •••••• [£]

Figure 6-Flowchart for first phase of KDA study

-).-3-P-'J MAXIMUM DATA SET TEMPERATURE. •••••••• 1 2, ¢ ti1
MINIMUM DATA SET TEMPERATURE •••••••••
_I

Figure 4-Typical KDA data form

is "yes" regardless of the user's intention to perform a hybrid simulat:on because, referring to Fig:ure 6" the all-digital optimization program has a built-in. mechanism for obtaining:
1. An analog static check and dynamic check solution. 2. A cost estimate of th~ all-digital solution versus the hybrid solution cost for problems where the most economic alternative is questionable. 3. An accurate estimate for all unknown analog scale factors. 4. An overall dynamic test for hybrid simulations

HYBRID INTERFACE ASSIGNMENTS AND SCALE FACTORS DATA PREPARA T 10 N PROCESSOR DATA TAPE WITH ALL PROCESSED DATA CONCENTRATION WEIGHTING FACTORS DATA TRANSFORMATIONS

>-_ _

~OPTIMIZATION

ALGORITHM

DATA ORGANIZATION DATA SUMMARY DIAGNOSTIC MESSAGES CORRECT 10 NS MATERIAL BALANCE ANALYSIS INCLUDING DERIVATIVES

Figure 5--Data preparation processor flowchart

User interaction ,communication
The user's first contact with the Kinetic Data Analysis package is a set of data forms (sec Figure 4) that request experimental data and other related information in kinetic rather than computer terminology. These forms are transformed into a deck of punched cards and fed to the Data Preparation Processor. Referring to Figure 5, if no errors are detected, the data is processed and the results are printed out and stored on tape. This tape contains all optimization algorithm and kinetic information requi~ed for the execution of the optimization program. To complete the data forms the user is required to provide a "yes" or "no" answer to the question, "AllDigital Solution?" The initia,] answer to this question

which are required to program and debug the analog model for hybrid studies. For all-digital studies, the Kinetic Data Analysis package supplies three partially programmed FORTRAN IV subroutines and a "Block Data" subroutine for kinetic models consisting of either algebraic equations (e.g., stirred-tank reactor) or ordinary differential equations (e.g., batch or flow reactors). The integration package uses a fourth order Rumge-Kutta integration algorithm and a readily implemented mechanism is available to obtain the classical "error versus step size" data to determine the correct and most economical step size for the integration process. The three subroutines require the user to:
1. Store initial values of the variables being integrated in an integration initial condition array. 2. Store computed results in a specified array. 3. Compute intermediate variables and model derivatives or, for example, stage outputs usin~ FORTRAN IV statements.

Items one and two, typically, require two or _three statements and the requirements for item three are a function of the complexity of the kinotic model.

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A Hybrid/Digital Sottware
The "Block Data" subroutine is used to define total number of and names of intermediate and integration variables for control and printout purposes. These four programs (in object form) are incorporated into the Kinetic Data Analysis package to form an executable program which, upon request, will read in the data prepared by the Data Preparation Processor and print out the values of intermediate and dependent variables as a function of the independent variable. For all-digital studies, the user now has an executable optimization program capable of solving his problem. For hybrid studies, this program provides static check, dynamic check and scale factor information. If the user executes one digital solution to his problem (this will be clarified later), the results provide the information required to test the overall accuracy of a hybrid simulation and the running time of the alldigital model to compare hybrid versus digital economics. With the exception of reprocessing the card deck obtained from the data forms and requesting hybrid processing, no digital programming is required for hybrid studies. The Data Preparation Processor, in the hybrid mode, assigns hybrid interface channels to operate in conjunction with preprogrammed hybrid interface programs. Since da.ta transferred to and from the analog model is done in a predefined sequence, the analog logic and interface circuits are also predefined and can be prepatched. Therefore, the additional effort required for hybrid studies is limited to the analog programming required to actually simulate the kinetic model. The Kinetic Data Analysis package has, in effect, organized the hybrid study and, with the aid of the static check, dynamic check, and scale factors determined earlier, made programming and debugging the analog model a relatively simple task. The aforementioned card input analog set-up program limits the time required to set up and check out analog programs to a few minutes. At execution time, the user communicates with the Data Preparation Processor, the optimization programs, and the Program Executive through a set of predefined user oriented commands. These commands can be inputed via cards for batch-unattended runs or a console typewriter. Since the Kinetic Data Analysis package uses a' "space" as a delimiter, commands are entered in "free format." For example, the command "INPUT DATA 8", which is used to read in the data tape from FORTRAN I/O unit 8, may start at any location on a punch card. The above mentioned command list, which contains

Packag~

737

more than fifty individual commands, is too extensive to discuss in detail. The commands can, however, be classified into the six areas of control they make available to the user.
1. Program ControL .. Select I/O devices,

call Kinetic Data Analysis Programs, add to the program library, etc.

2. Kinetic Data Handling.. ............. Control I/O options and computations performed on experimental and computed kinetic data. 3. Optimization Data Handling ................. Control I/O options and computations associated with optimization variabIes. 4. Objective Function ControL ................. Control the mathematical form, weighting and the components or data sets used to compute the objective function (see later discussion) . 5. Optimization Algorithm ControL ... Select the mode (e.g., maximize, minimize) and other options (e.g., iterative, cyclic operation) associated with the optimization algorithm. 6. Model Control and Diagnostic............. Select hybrid diagnostic options (e.g., scan for interface error messages) or digital model control options (e.g., set or reset a one/zero model switch to modify kinetic model). The form of the results obtained by the user during program execution will be discussed later.

Data capacity and classification
The Kinetic Data Analysis package is capable of processing up to fifteen sets of experimental kinetic data (or data sets) which may contain concentration data for a maximum of fifteen chemical species or components. Each data set may contain up to ten values

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738

Fall Joint Computer Conference, 1969 parameters into optimization algorithm variables. 2. The transformation and transfer of these v;a,riabIes to the kinetic model. Both transformations are a function of the aforementioned KDA Case Number and the Arrhenius equation
K = A·EXP (-B/T)
(1)

of an independent or sampling variable (e.g., time for batch reactor, volume for flow reactor, etc.) and fifteen concentration points per sampling variable. These data must be common to all data sets and the sampling variable must be a monoatonic increasing function whose initial value is zero. Howover, equal sampling variable increments are not required. Each data set also contains provision for a catalyst concentration, a temperature, and an alphanumeric user identifier. The purpose and manipulation of the catalyst and temperature data will be discussed later. Up to fifteen unknown reaction rate constants, which are assumed to obey the Arrhenius equation, can be processed. This limit is independent of the thermal state of the system (i.e., isothermal or non-isothermal data sets). In addition, the Kinetic Data Analysis package can process up to fifteen unknown model or individual parameters (e.g., reaction orders; heat transfer coefficients, etc.). The above mentioned limits apply to all-digital studies and hybrid systems w:lOse interface contains a minimum of sixteen analog to digital and digital to analog channels. The Data Preparation Processor catagorizes experimental kinetic data into one of three classes called KDA Case Numbers. They are: Case # 1.. One or more experiments performed under nonisothermal conditions Case # 2.. Two or more experiments performed under isothermal conditions where the difference between the maximum and minimum temperature levels is greater than 5°C or OF. Case # 3.. One or more experiments performed under isothermal conditions where the temperature range is less than or equal to 5°C or OF. This data catagorization is one of the key factors required, for example, to organize optimization algorithm input data and the transfer of rate constants to the kinetic model.

where K = reaction rate constant A,

n=

Arrhenius coefficients

T = absolute temperature

The Kinetic Data Analysis package uses an alternative, but rigorously correct, form of the Arrhenius equation whose derivation is shmvn in Appendix A. This relationship is
(2)

where {3 is defined as
{3 = (l/T R
-

l/T)/O/TL - l/T H )

(3)

In equation 3, TH and '1\ are the maximum Emd minimum experimental data temperatures, respectively and T R is a mid-range reference temperature defined by the equation
(4)

In equation 2, KR denotes the reaction rate constant at TR and KHL is the ratio of the maximum to minimum rate constants (KHL = KH/KrJ. For experimental data catagorized as KDA Case :#: 3, the optimization variables, Ai, are defined as:
(5)

\vhere
1

Optimization variable transformations
. Two tranformations, which play an important part In the data flow between the various KDA processors, are:
1. The tranformation of rate constants and model

= rate constant index, i

=1,2,"', NRC
(6)

NRC = the total number of rate constants. For the two remaining data catagories

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A Hybrid/Digital Software Package and (7)
KDA CASE NUMBER
1 2

739

TABLE I- Items influenced by KDA case number

3

Individual or model parameters specified by the user are sequentially added after the last rate constant variable. For example, the first parameter, PI, is assigned to ;\NRc+I for KDA Case # 3. Referring to Figure 7, optimization variables are transferred to the kinetic model as a function of the KDA Case Number as shO\vn in Table I. For hybrid kinetic models, the rate constants are scaled and transferred to the analog computer in a predefined transfer sequence as shown in Table II. Note that for both digital and hybrid models concentration initial conditions, sampling points, and a ramp sloape (i.e., reciprocal of the last data set sampling point) are also transferre4 to the kinetic model.

TOTAL NUMBER OF OPTIMIZATION VARIABLES

2 • NRC + NPR

2' NRC +NPR

NRC +NPR

FORM ASSIGNED TO OPTIMIZATION VARIABLES REPRESENTING RATE CONSTANTS RATE CONSTANT DATA TRANSFERRED TO DIGITAL MODEL

KR KHL

KR K KHL

A, B KR, KHL

A, •

K

K

RATE CONSTANT DATA TRANSFERRED TO HYBRID MODEL

LOG (K

HL

) K K

LOG (K ) R

TABLE II-Typical transfer sequence* for KDA case

#2
N

=N
N

+1 =0

N =0
F

FRUN

=0

CHANNEL NUMBER

D/A DEMULTIPLEXING
ABC

A/D

COMPUTE MODEL INPUTS - - - F O R Nth DATA SET

2

NCS
SIMULATE Nth ---DATASET

1
Q
...... !::z

NRC

z

...1< ~o::

-

-

-

COMPUTE DATA SET OBJEC TIVE FUNCTION

-~
14 15

zw u o

CAT**
TEMP

* Channel zero used by prepatched KDA circuits.
COMPUTE TOTAL OBJEC T IVE FUNCTION

** Transferred when appl icable •

~

________

~

OUT

Optimization algorithm and objective function options
The current version of the Kinetic Data Analysis package uses a slightly modified version of the P ARTAN algorithm described in detail by Harkins4 • Since a detailed description of the algorithm is available,

Figure 7-Simplified objective function flow diagram

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740 Fall Joint Computer Conference, 1969 this paper will only consider the mathematical form of the objective function. However, it should be noted that this algorithm, which can be classified as an "accelerated gradient" algorithm, was selected because of its proven effectiveness on a number of all-digital and hybrid kinetic studies performed in recent years at EAI Computation Centers. The add-on capability of the software package makes it possible to add other algorithms if the need exists. The mathematical form of the objective function is specified by the user at execution time. Referring to Figure 7, the objective function is based on the "total error" or sum of the individual data set errors. For example, to compute the objective function for a problem consisting of ten components and ten dat.1l sets, ten analog runs or one hundred digital integrations are required. The form of the objective function, its weighting factors, the exclusion of a chemical species or data sets from the objective function, etc., are defined by the user at execution time via the Executive Program. The software package provides integral and polynomial objective function options to the user based on the following definitions: In the above relationship (J denotes a positive sampling variable ratio whose maximum value is unity:

em
8

= SV m
SVM

(10)

The weighting factor is unity if PWI and PW2 are zero. If PWI = 1.0 and PW2 = 0, initial values are weighted, and if PW2 = 1.0 and PWI = 0, final values are weighted. Note that both PWI and PVv2 cannot simultaneously be set to one. The integral option defines individual data set objective functions as

FRUN n =

L i-I J

WGTn,i

1
0

En,m,i d(SV)

(11)

where the integrn.l is computed using a "Trapezoidal Rule" approximation and the weighting factor (WGTn,i) is defined as WG'1\,i = 1 + CWl'Cn,i

+ CW2· (l-C?I,i) (l:~)

En,m,i = COMPi'ICn,m,i - C~,m.iIEXPN COlVIP i = 1.0 or 0 when a chemical species is to be excluded OMIT n = 1.0 or 0 when a data set is to be excluded i = index denoting a chemical species, l~i~.J m = index denoting a sampling point, 1 ~ In ~ lV1 n = index denoting a data set or experiment, 1 ~ n ~ N Cn,m ,i = computed results (unscaled) array C~ ,m, i = experimental results array F = total objective function FRUN n = data set objective functions EXPN = a positive, non-zero constant The polynomial option defines individual data set objective function as FRUN n =

The control constants CWI and CW2 are identicalllll behavior to PWI and PW2. the ,1 values are conc.entration weighting factors computed from experimental data by the Data Preparation Processor. If CWI = 1.0 and C\V2 = 0, large concentrations are weighted, and if CW2 = 1.0 and CWI = 0, small concentrations are ~weighted. This weighting factor is useful when, for example, a component whose range is 0 - 0.05 in a given experiment is more sensitive to an analytical error of, say, ± 0.01 than a component whose range is 0.5 - 1.0. Referring to Figure 7, the total objective function F, is. obtained by summing the individual data set errors, FRUN, modified by Ol\1.ITn (1.0 or 0) to control the inclusion or exclusion of the various data setH.

en

F =

L n=1 N

OlVIITn·FRUNN

(13)

:E

M

WGTn,m

L: i=1 J

En,m,i

(8)

where the weighting factor (WGT n ,m) is

WGT n.m = 1

+ PWI + em
8

PW2 .

em

B

(9)

Note that user commands control the values assigned to 01\11'1\, C01\IP i , EXPN, PWl, PW2, CWI and CW2. In addition to the aforementioned objectivE~ functions, the software package has provision for the user to add a digital subroutine to compute the individual data set errors if the "built-in" options are not applicable. For example, if the data set errors are computed on the analog computer this subroutine can be used to transfer them into the digital computer.

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A Hybrid/Digital Software Package
Optimization results include a table containing the objective function, its fractional contribution to the total objective function, and the average error per data point for each data. set. The total absolute error* or standard error is included in all results to allow the user to compare the relative merits of various objectives functions since their magnitudes depend on their mathema.tical form.

741

preparation and optimization phases of the study, and summarize the numerical results obtained from the study. Simulation accuracy, errors in results, and economics will also be discussed.

Problem description
The illustrative problem contains the two essentia, ingredients to perform a kinetic data analysis study; a proposed kinetic model and experimental data. Referring to Table III, each of the thirteen available data sets contained concentration-time data for seven chemical species (i.e., R. S, T, U, W, X, and Y), the concentration of a non-reactive catalyst, and a temperature. These data were obtained from experiments performed under isothermal conditions over a 133 to 181°C temperature range which included a threefold variation in catalyst concentration, 117 to 368. No two data sets had identical initial concentrations and the number of non-zero sampling variable (i.e., time) p;)ints per data set varied from one to four. The proposed kinetic model, which is shown in Table IV, is based on the following chemical equations:

Temperature and catalyst data
Each of the data sets has associated with it a single temperature which is sufficient for experiments performed under isothermal conditions (i.e., KDA Case #2 and 3.) For non-isothermal situations the data set temperature is the initial or feed temperature; therefore, the requirements of kinetic models which include energy balances (i.e., temperature obtained from the solution of a differential equation) are also satisfied. Studies that require the storage of, say, temperature versus time data are simulated by:
1. Using "Data" statements to include these data

in the subroutines supplied by the user for all digital studies. 2. Using, say, card programmed diode function generators (CPDFG) OD the analog computer for hybrid studies. The CPDFGs work in conjunction with preprogrammed logic that automatically associates each function with the appropriate data set during the simulation. The software package also allows the user to associate a catalyst concentration with each data set. The catalyst concentration, which is transferred to the kinetic model, provides the user with a mechanism for simUlating kinetic models involving a non-reactive or reactive catalyst. For example, when catalyst concentration data is not available in studies involving reactive catalysts, the catalyst concentration is the initial condition for the catalyst material balance equation.
Typical application

KI
Ks

K4

R+S==:;T-~U

K2
R+S-~U

Ks
R+S~W

K9

T

+ S -~X.==;U +
K IO

Ko

K6

S

The following discussion will be devoted to the solution of "Monsanto Benchmark Problem" using the Kinetic Data Anaylsis package on a fuJy expanged EAI 8900 Hybrid Computer. This discuss:oh will include a mathematical description of the problem, illustrate the form of the results obtained during the
... Equation 8 with EXPN and WGT n.m equal to unity.

The model contained eleven unknown rate constants (K I -- K ll) and since this study falls under the KDA Case #2 category, there are a total of twenty-two optimization variables. Each rate constant has one KR and one KHL optimization variable associated with it.

Data preparation processor results
Processing the card deck corresponding to the KDA Data Forms produced the results indicated in Figure 5, which are illustrated by Figures 8 through 11. These

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742

Fall Joint Computer Conference, 1969
TABLE III- Typical data set

IDENTIFIER: RUN TWO

TEMPERATURE IN OEC C CATALYST (UNKNOWN

MINIMUM ) IN UNKNOWN

130,11

MAXIMUM

21111,11

MINIMUM II.117E 113

TEMPERATURE: 146°C TIME HOURS 0.0

CATALYST CONCENTRATION: 117

CONCENTRATION IN MASS FRACTION R S T U W Y X 0.425 0.501 0.018 0.005 0.050

---------------------------------------------------SCALED SCALED DATA SET ---------------------------------------------------RUN 2 RUN RUN ONE TWO 3 11,730" 0.730" 11.911011 0,8U0 0,8Ue 11,7900 11,74110 11.83511' 0,85"" 0,890" 11.6650 0,9B51 11,8651 11,3315 ".3179 ".6440 0.3288 11',6576 11',6522 B,6522 1,6522 0,6522 ",6522 11,6739 0,6141 1,I0B0 3 NO, IDENTlrlER TEMPERATURE CATALYST CONC

SCALED CATALYST-TEMPERATURE DATA

---

-0.002
5

1.0

0.359 0.465 0.051

0.017 0.106

RUN rOUR RUN rIVE RUN RUN RUN SIX 7 8

2.0

0'.315 0.442 0.086 0.033 0.120

---

0.004
0.008

6 7

3.0

0.281

0.424 0.123 0.048 0.116

8 9
111
11

RUN NINE RUN RUN RUN RUN TEN
11

........ -- ........ --_ .. -------- -_ .......... _............... -........ -_ ................. -.... --SCALE
ADC C~ANNEL NUMBER

12 13

12
13

.... ------- .. ---------- ... -_ .... -_ ... ---_ ...... ------------_ .... _.. _... rACTDII COM' " COM' :1 COM" CO"' IJ COM'
~

MAX IMUM VALUE

-

1.11"" 11 II.UII.E 11 '.UIIE I i •• U,.F 11 II.,",E I. •• 51 •• E •• •• U •• E ••

II.Ulle 11 •• UIIE 11 I.UI.E .1 ,.U"F. 11 ..211.E In ' . u l l e 11 '.U •• E 12

-------------------------------------------------_ ..
CAT CONC ANO TEMP XrER ON OAC 14 ANO 15 OURING 'B' PERIOD

Figure 9-Temperature-catalyst interface data transfer

COi'!' ~ CO"' 'r

...... ___ ....... __ ....... _____ .. __ ....... ___ ....... ___ ..... w_ ............ ___ ...... --.... _._.-

ii;c -C~;~;E~ --------;;; i;;~E ---------~;; i~~~- ----. -- --·;C~~E-----~~~.-~;~~;;;NAME VALue DilLY' -_ NU.M8EII ...... ----------- --_ .......... --_ ........... -- .. --- .... --_ ....... -.... --- .. rACTOli ICAIE 1 :".- .. -_... - _------ .--_ .. ..
IIATE CON llATE CON IIATE CON llATE CO'" llATE CON 1

OAe ASSICNMENTS

•. n .. E

12

••• 66.E-11 •• U5IE ••

2

I . . . . . E .1 . . . . . . E 11 ' •• I I , F "

3

'.l25.E •• ',251.E 11
•• UIIE 11

• ,
6 7

•• 1I"E 111
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'UTE CO",· llATE CON FlATE CON IIA TE CON

.,l".E 11

'.25..., ••
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9

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11

FlATE CON 11 UTE CON 11

.

-_.- .. -................ -- ................... -............... -...... -.. -- .. _....... --... --_ ........ -.- --_....... -- -_ ........ ---_ ..
Figure g.-Hybrid interface ,assignments

,,4111'" III

figures omit the first phase of the form processing output. That is, the direct playback of the KDA Data Forms with appropriate error messages when errors are detected.

Figure 8 illustrates the hybrid interface assignments for the eleven reaction rate constants and the seven chemical species involved in the mathematical model, their maximum values, and their scale factors (i.e., reciprocal of maximum value). Figure 9 deta,ils the scaled temperatures and catalyst concentrations that will be transferred to the analog model during the "B" demultiplexing period on D/A-channels 14 and 15. Note that this problem is in the KDA Case #' 2 category whose interface transfer sequence has been illustrated in r-rable II. Referring to Figure 10, the Data Preparation PrQi[~­ essor assigns a number to both the data sets and chemical species involved in the study. These numbers are required by the user to execute commands that manipulate specific chemical components or data sets. For example, to exclude the eleventh data set from the study, the command is "EXCLUDE II" not "EXCLUDE RUN II" where "RUN II" is the data set identifier specified by the U!3er. The lower half of 1~'igure 10 Hlustrates a typieal data set printout containin~ the origi.nl "time" units and sc, .led values (i.e., normalized) of the s.vnpling Vltriable. The normalized values were obtamt:d by:

From the collection of the Computer History Museum (www.computerhistory.org)

A Hybrid/Digital Software Package

743

NAME I KOA NUMBER SUMMARY

COMP NAME COMP COMP COMP COMP COMP COMP COMP R S T U
W
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1

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2 3 4

5
6
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8
9

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PAAll\N OATA

SUMMARY

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MI~
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2

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n

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SAMPLING POINTS NORMALIZED

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n

0.10IIE U

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w
~

•• I8.e 18 I.U'E

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Figure ll-Concentration weighting factors and algorithm input data

CONC SUM

------------ ... _--------_ ... -.... -..... ---- ... _----_ ........ --------------- .. -------------- ... __ .. _----_. •• 999& II • '\IIE 11 1.1I1E 11 1.1I1E 01
Figure l~KDA number assignments and processed data set

COMP Y

. . . . ee II 1.2eU-82 '.4"E-12 . . . . . E-12

TABLE IV-Mathematical model

1. Performing the catalyst transformation shown in Table IV, which was the result of a "yes" answer to the question, "CATALYITC REACTIONS?" (see Figure 4). 2. Dividing all values by the maximum sampling point to form the "normalized" values or s~aled sampling points.

DEFINITION OF TERMS Rl =Kl RS R2 = K2 RS R3 = K3 RS R4 = K4 T RS =KS TS t R6 =K6 US R7 = K7 US RS = KS T R9 = K9 W RlO= K10 X CA T = Catalyst Concentratioh

R11 =K11 Y R12 = Rl + R2 + R3 R =R + RS 13 4 R14 =R6 + R7

= time

a =CAT /(CAT)MIN = Catalyst Ratio
MRT, MSR, MST, etc. = Molecular Weight Ratios; MATERIAL BALANCE EQUATIONS
~ = ~t
~ = ~t

e

=a t

= -R 12 + (MRT) RS + (MRW) R9 : -(MSR) R12 - R14 - RS + (MST) RS + (MSW) R9 + (MSX) R10 + (MSY) Rll = (MTR) Rl - R13 - (MTS) RS ; = (MUR) R2
+ (MUT) R4

~ = £Xt
~ = ~t

d9

rN = rN

adt

~

(MYS) R - R 7 11

+ (MUX) R10

+ (MUY) Rl1 - (MUS) R14

These results also contain concentration and rate summations for each time point to assist the user in . evaluating the consistency of the data based on material balance. The rates, which are not shown in Figure 10, were computed numerically by differentiating a polynomial whose coefficients are determined by a least square fit of the concentration data. Figure 11 illustrates the concentration weighting factors and the input data to the PARTAN Algorith~. Note that the Data Preparation Processor has assigned names, for example, "KROl", to the optimization variable and placed them in a "type three" category. This means they are constrained between an upper and lower limit denoted by "MAXVAL" and "MINVAL". The initial values of the variables are in the "VALUE" column. The results of the preprocessing indicated that the eleventh data set should be excluded from the study because its concentration sums indicated as much as

From the collection of the Computer History Museum (www.computerhistory.org)

744

Fall Joint Computer Conference, 1969

ten percent error. Therefore, optimization results were obtained using twelve, rather than thirteen, data sets.

~UM8ER

__ .. J .... __ .............. __ .... _ .................. __ .............. _ .. __ ............................ _ .. _ ...................... _, __ .... ..

DAU SET I OENT I r I ER
RU~

ABSOLUTE EKHOR

ERAOR rUCT I O~

AvERACE ERROR

ONE TwO 3

•• 791~E-11 '.2296E I I '.4674E •• I.4895E I I ,.4.52E II '.1987E •• '.49UE 18 1.5J3aE Ie

1.2"3E-01 A.5961E-81 1.12I3E 81 A.1271e . , •• ln1E

•• 113IE-'1 •• UUE-I! 1.3339E-'1
~.23J1E-1I1

Optimization results
Figures 12 through 15 illustrato the form of some of the results obtained from the hybrid solution of the problem. Figure 12 illustrates the user commands, which are documented as they are processed, and an optimization summa~y. The summary is updated everytime the algorithm detects an improvement in' the

RUN RUN

RUN FOUR RUN F lyE RUN RUN RUN SI X 7

ee

'.1929E-.l '.UUE-'2 '.2H6E-11 '.1·II3E-.1 •• ,'I27E-'2 '.6'I13E-'2 '.1414£-.1
II. '1192£-'2

1.5158E-01 1.1295E II 1.1383E I I •• 4162E-81 e.3697E-e1 •• 1021E 10 •• 6535E-01

•
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•• 2517E I I

TYPE INPUT OATA ,8 aAU SET NUMBER 2 USER IDENTlrtER '.111IE IJ UNKNOWN RUN TwO TEMPERATURE 146.' C

CAULYST CONCENTRATION liEU rACTOR -'.2421

CATALYST RATIO

'.III1E A1

COMPANy ••• •••• • LOCH ION •••••• • PROJ ENGR ••••••

MONSANTO COMPANY Sf. LOUIS. MO. PAUL PA~ISOT

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PROJ NUMBER •••• PROJ ENCR •••••• CURRENT aAlE ••• TAPE UNIT 9 RESTORE PARTAN PLOT nCLuOE 11 INTEGRAL OBJECTiVE fUNCTION wEICHT LARGE CONCE~TRATIONS ERROR E XPONEN' 1. 0
1006~9

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Figure 13-Typical objective function summary . and detailed data set results

OPTIMIZATION SUMMARY

•"u,

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From the collection of the Computer History Museum (www.computerhistory.org)

746

Fall Joint Computer Conference, 1969 on the error surface with the true minimum, sets of optimization runs were always made starting from four points on the error surface. The four sets of starting values used were the maximum and minimum values of the optimization variables, their arithmetic average values, and the initial or "best guess" values. The problem was solved using the following iterative process: 1. Perform four separate, complete optimization runs using the maximum, minimum, average, and initial values of the optimization variables. 2. Examine the results and determine if the final values of the objective function and optimization variables show good agreement. 3. If the results of step two indicate more runs are required, refine the four sets of starting values based on their results and repeat the first step. This iteration process was repeated three times using the integral form of the objective function with large concentration weighting and an error exponent· equal to unity. Referring to Table V, the values of the objective function for these three iterations are reported in standard error form (i.e., the unweighted sum of the absolute concentration errors). After the third iteration, the mathematical form of the objective function was changed to the standard form to eliminate the effects of the concentration weighting and the results of this iteration indicated that for all practical purposes, the "best fit" had been obtained. The four sets of optimization variables obtained from the fourth iteration showed reasonably good but not perfect agreement. The error introduced into specific reaction rate constants by differences in the final values of the optimization variables were computed using the error form of equation 2; namely,

alpha and beta values pertain to the algorithm perturbations, etc. During the optimization process the optimization summary is the only output available to the user with the exception of a percent improvement indicated on the analog computer digital voltmeter. The percent improvement is relative to the initial or base valuJ of the objective function. After the op-timization process has been completed, the previously mentioned objective function summary is obtained (see Figure 13) which includes a reproducibility error. Referring to Figures 13 and 14, the u,ser may also request a detailed comparison of experimental to computed results and a line printer plot 01' the objective function or any of the optimization variables as a function of the number of improvements. The objective function summary allows the user to determine if, for example, anyone data set is making an excessively large contribution to the objectIve function. The reproducilibity factor, which is typically zero for all-digital studies, is obtained by re-evaluating the objective function under "best fit" condItions after the optimization process has been completed. The percent error between the two objective functions is the percent reproducibility error shown in Figure 13. It reflects the total error introduced into the objective function by the hybrid interface, analog components, etc. As shown in Figure 13, this error wa..: typIcally less than one percent. The objective function plots allow the user to graphically follow the path of the optimization proc~ss. However, plots of specjfic optimization variables versus the number of improvements are more important. They indicate the activity or sensitivity of variables during optimization and allow the user to take appropriate action if, for example, a variable always remained essentially constant. Figure 15 shows a concise final results plot that can be requested via the appropriate user command. This plot, which is obtained on the analog strip chart recorder, consists of a sample variable ramp (() and a set of curves for the computed concentrations. The "blips" on the concentration curves represent the deviation between the curves and experimental data points; therefore, the absence of "blips" represents a near perfect or perfect fit. The pulse prior to each ramp denotes the data set number. The first data set is preceded by a 10 volt pulse, the second by a 20 volt pulse, etc.

~~
K

aK + fI (aKHL)
R

(14)

KR

KHL

where .:lKR and .:lKH L are the most probable errors and and KHL are the average values of the individual optimization variables.9 The results of this analysis are shown in Table VI. Note that the absolute percent error of anyone rate constant is a function of temperature or {3 whose range is ±0.5.
KR

Simulation Accuracy
Comparisons between equivalent hybrid and alldigital optimization runs were made to determine how analog component or digital integration errors in-

Problem solution and results
To avoid the possiblity of confusing a local minimum

From the collection of the Computer History Museum (www.computerhistory.org)

A Hybrid/Digital Software Package
TABLE V-Objective function results

747

CD - ALL-DIGITAL STUDY

STARTiNG LOCATION INITIAL STARTING VALUES' ITERATION 1* ITERATION 2' ITERATION 3' ITERATION 4* 8.28 3.76 3.15 3.06 3.00 MAXIMUM 5'.76 3.15 3.08 3.03 2.98 MINIMUM 25.2 3.59 3.12 3.11 2.99 AVERAGE 5.65 3.65 3.08 3.08 3.01

*

TOTAL COST OF STUDY

L

ICC~~_ _ _ _-:::;:~::""----~

CH

-

HYBRID STUDY

'-T:::O::'T

A:-:-L-:-N7':":U~MB::":'E:-R':":OF:-:O:":P':":TI~M':':IZ~AT~IO~N-R-U-NS-~

NOR

F'igure 16--Typical hybrid-digital economic plot
'Standard error equivalent of weighted integral objective function.

TABLE VI-Rate constant error analysis results
ABSOLUTE PERCENT ERROR i Ki R MINIMUM ERROR
0.32 0.44 0.26 0.30 1.06 1.60 2.05 0.59 0.23 3.22 2.46

i

KHL

K.

I

MAXIMUM ERROR
3.48 0.83 5.43 4.27 1.96 4.48 1.98 3.18 3.49 5.14 4.33 2.06 0.86 2.98 2.43 2.04 3.84 3.04 2.18 1.97 5.79 4.62

1 2 3 4 5 6 7 8 9 10

summaries (see Figure 12) were compared, the final funct.on and optimization var~able results obtained were identical for all practical purpose (L.e., one or two perc..:nt dtfference). This would seem to indicate that the errors associated with experimental data and the mathematical model will have a greater influence on results than the relatively minor errors introduced by digital integration or analog components. It was also concluded that double prec~sion integration accuracy was not worth the additional computation time it required compared to single precision integration. objec~ive Simulation economics

The above discussion indicates there is no technical advantage to be gained ,by using a hybrid rather than an all-digital simulation to so~ve a kinetics problem with the KDA package. Therefore, two questions of interest are: 1. Is there an advantage to using )ne type of computer? 2. How does one determine which computer to use for specific problems'? The answer to the first question ;s there is an economic "break-even" point (see Figure 16) that governs the selection of a hybrid computer over a digital computer or vice versa. This "break-even" point is created when the simulation of the kinetic molel requires the solution of a set of differential equations and the digital cost per optimization run is in excess of the equivalent hybrid cost. A hybrid solution is practical when the hybrid economic advantage during the production phase of a kinetic study offsets and surpasses the deficit encountered during the problem preparation phase. Recalling previous discusions to perform a kinetic study

11

fluenced results. This comparison was based on the standard objective function value obtained after one function evaluation. Using both single and double precision digital integration, a co~parison of objective function values showed good agreement between the digital and hybrid results. Both the hybrid and single precision digital integration results were within approximately ± 1% of the results obtained using double precision integration. These minor differences were traced to errors of less than 0.001 in computed concentration data points. One comparison or equivalent all-digital versus hybrid optjmization runs was made. Although both solutions differed slightly when their optimization

From the collection of the Computer History Museum (www.computerhistory.org)

748

Fall Joint Computer Conference, 1969
An approximate relationship to determine the equivalent time, T D, for a digital optimization is: TD
~

using the Kinetic Data Analysls package the analog programming task is superimposed on the normal preparations required for an all-digitaL study. This creates an obvious hybrid deficit which combines with hybrld cost advantage during the execut.on of the opt.mization program to create an economic "break-even" pomt. The economics associated with the hybrid versus alldig~ta, question should be considered care. udy because sufficient savings can be reaLzed by making the correct decision. For example, a recent hybrid ver;::;us alldigital economic study for a reactor control problem! indicated tpat a large seale hybrid computer had approximately a 20:1 time and 40:1 cost advantage over large scale, third generation digital computers (e.g. $1,200 per hour computation center rate), and a 60:1 hybrid time advantage for the solution of the "lVIonsanto Benchmark Problem" has been reported in the literature. 6 The hybrid cost advantage is directly related to the average computation time required to simulate a data set or experiment. The analog computer, typica.1y requ~res 10-20 milliseconds to simulate one data set, which is independent of problem complexity. The time required for the equivalent digital simul~tion is a function of the speed of the digita computer, the number of equations, their degree of nonlinearity, and t he integration algorithm. The influence of the digital .lltegration algorithm on thiK sitnation is miLor since the' analog; compute can be "spel'ded-up" more rea.dily t han the algorithm. . The answer to the question of how one determines t he answer to the all-digital or hybrid question is very difficult due to lack of information. However, based on information obtained from several hybrid optimization studies performed on EAI 8900 Hybrid Computers, it was possible to