OPERATIONS RESEARCH SYMPOSIUM
operational computer system to estimate physical Anproperties of chemical compounds and mixtures
has been developed by Arthur D. Little, Inc., for the American Institute of Chemical Engineera (7). The system accepts known information, such as molecular structure and themnophysical properties, and generates estimates of desired physical properties together with their corresponding uncertainties. Properties are estimated by applying a succession of programmed entimation equations, or correlations, to known input data, rather than by interpolating or extraplating in tables of stored property values. The system has the capability to utilize limited quantities of information regarding a known substance to arrive at an estimate of a d e s a property. Where there are several alternative methods for estimating a property, the system will automatically select the “best” estimation technique from all those available to it. This is done by c h m i n g that sequence of estimation methods which leads to the least uncertainty in the requested property. At all times, however, the user has complete freedom to constrain the system in numerous ways and thereby to exert control over the estimation procau itself. Particular attention has been devoted to the problem of updating the library of estimation techniques within the system. Since the state of the physical property estimation field is so dynamic, obsolescence can be prevented only by a continual dart to add correlations as they are developed and to delete older, lesa m u r a t e ones. In addition, the scope of the system may be extended by broadening the coverage of physical properties in the estimation system library. To further these ends the procarp of updating thii library has been made relatively straightforward and easy. Every effort was made to standardize and simplify computer programming and systems design, without sacrificing operating efficiency. The entire system, with the exception of two FAP subroutines, has been written in the FORTRAN I1 language for operation under the standard FORTRAN monitor system on an IBM 7090/ 7094 computer with 10 tape drives. The minimum eonfiguration required h a 32K memory machine with six magnetic tape drives and an opexating system which accommodates the FORTRAN I1 chain feature. The A.1.Ch.E. system has been converted to FORTRAN IV 0
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Estimation of Physical Properties Minimum trror P
Analysis RICHARD E. HEITMAN GEORGE H. HARRIS
Accuracy and flexibility characterize a newly developed method which computes physical properties of compounds for which limited experimental data are available
and executed on an IBM 7094 (by Gulf Research & Development Co.) and a UNIVAC 1108 (by E. I. du Pont de Nemours & Co., Inc.). T h e system contains in excess of 40,000 FORTRAN statements, including comment cards. Special consideration was given to designing a modular system that could be readily adapted for use on a variety of types of computing equipment. Restricted versions of the system can be operated on equipment less extensive than an IBM 7090/7094 computer, and certain components of the system, in particular many of the individual estimation methods, have been operated on machines as small as a 16K IBM 1401. Effort is presently being expended to convert the system for use on the IBM System/360 series computer. Why an Estimation System I s Needed
There are several cogent reasons for developing a physical property estimation system:
b Estimation methods are becoming increasingly complex; all but the simplest require a computer for evaluation. b With the proliferation of estimation techniques, it is practically impossible for one to be aware of the best methods, together with their limitations, restrictions, and accuracies. b It is extremely difficult for a scientific or technical worker to consider the large number of alternative schemes usually available to estimate a particular property. Frequently, therefore, he selects a poor scheme without realizing that a more accurate one may be readily available. There is no generally available means to estimate the error associated with a calculated physical property. This error has two sources-uncertainties in the original input data and uncertainties in the basic estimation methods. Capabilities of the System
Estimation methods. The system can estimate physical properties of both pure substances and mixtures (Table I). I n general there are several alternative methods for estimating the same property. The present version of the system contains 122 estimation methods. [An estimation method and its programmed inverses are considered distinct correlations (see section on automatic route generation).] These methods have been carefully selected from the literature to provide the most accurate and generally applicable ones as of February 1965 (3). This thorough investigation led to the publication of a book (4),in which may be found detailed reviews of all estimation methods contained in
TABLE 1. PHYSICAL PROPERTIES IN EST IM A T ION SYSTEM (as of November 1967) Acentric factor Compressibility factor Critical and pseudocritical properties (Pc, V,, To,2,) Diffusion coefficients Enthalpy
Entropy
Fugacity Heat capacity at constant pressure InternaI energy Latent heat of vaporization Pressure Reidel factor Surface tension Temperature Thermal conductivity Vapor pressure Viscosity Volume (density)
the A.1.Ch.E. system. Additional estimation methods are readily incorporated into the system as the need arises. For purposes of physical property estimation, substances may exist in either the liquid or gas phase or in the intermediate two-phase region. I t is not necessary to know in advance to which phase a substance belongs, as the system is able to make this determination from the appropriate thermodynamic variables. The pure component gas phase, being the most well understood, is treated in greatest detail, whereas the liquid phase, still poorly understood, is accommodated to a lesser extent. The solid phase, however, has been dismissed because to date there are no generally applicable equations of state, and because chemical plant designers and operators work mainly with fluids. Of course, there is nothing to prevent these latter estimation methods, once developed, from being added to the system. Properties which may be estimated are conveniently subdivided into three classifications :
I. Pure component constants (e.g., critical constants) 2. Thermodynamic properties (e.g., pressure, temperature, enthalpy) 3. Transport properties (e.g., viscosity, diffusivity, thermal conductivity) VQL 60
NO. 2 F E B R U A R Y 1 9 6 8
51
All these properties are dependent in some way on molecular structure, especially the pure component properties. The system permits user entry of certain structural information, to be described later. Estimation of thermodynamic properties follows from an equation of state once the state of a compound has been specified. The selection of equations of state to be incorporated in the system is extremely important, because so many properties may be derived from them by applying the fundamental laws of thermodynamics. The A.1.Ch.E. system contains three for the gas phase: those of (1) Hirschfelder, Buehler, McGee, and Sutton, ( 2 ) Redlich and Kwong (modified version), and (3) Martin and Hou. There is one for the liquid phase, that of Lydersen, Greenkorn, and Hougen as recently adapted by Yen and Woods. (For a detailed discussion of these methods and others, the reader is referred to (4). It should be borne in mind that improved equations of state may now be available in the literature.) By use of these basic equations, the system can estimate any programmed state-dependent thermodynamic property provided any two of the three variables-pressure, volume, and temperature (P,V , T)-are known. I n theory, of course, for a pure-component, singlephase system, any two thermodynamic properties, such as entropy and heat capacity, determine the state. T o permit the system to accept on input any combination of two independent thermodynamic variables and then estimate any other dependent thermodynamic property would necessitate the derivation and subsequent programming of a multitude of complex relationships. If there were m distinct P-V-T correlations (equations of state) available and n thermodynamic variables ‘2%in addition to P, V, T, then at least m(n2 3 n 3) relationships would have to be programmed to handle the (n 3) (n 2 ) / 2 possible combinations of two independent variables. I n the present system for the gas phase, m is 3 and n is 5 (energy, enthalpy, entropy, fugacity, and heat capacity) ; hence 129 relationshps would need to be derived, of which 23 have been programmed. The 23 programmed equations are of the form P = f ( V , T ) , V = f(P,T), T = f(P,V), and Ci = f(P,V , T ); these are considered to be the most widely useful ones for engineering purposes. Additional relationships may be incorporated in the system as the need arises. Descriptor system. The A.1.Ch.E. system has been designed to take advantage of whatever knowledge the user may possess of the material under study. Naturally, the more information supplied to the system, the better are the estimates obtained. I n some cases, an estimation method is applicable only to particular classes of compounds. I n certain other cases, the estimation error is dependent on the type of compound under investigation. T o the extent that the user is able to specify this qualitative information concerning a compound or the components of a mixture, the system can restrict application of estimation equations. If this information is not provided, the most widely applicable techniques are generally employed.
+
+
52
+
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
Descriptors (numeric codes) permit the specification of qualitative information regarding compounds and mixtures. They consist of two major parts-a phase descriptor and a molecular descriptor. The phase descriptor describes the phase of the compound or mixture and may be omitted for the former (but not for mixtures), since the system can make this determination. The molecular descriptor is an attempt to summarize the molecule’s basic characteristics. Briefly, the molecular descriptor specifies : (1) polarity, (2) composition, and (3) structure. The description of molecular composition and structure is merely categorical and is not meant to be completely definitive. For example, trinitrotoluene would be described as polar, organic, nonhydrocarbon, nonmetallic, aromatic, with three nitro groups. Compound and mixture component descriptors, supplied on input by the user, are decoded internally and used to limit the application of estimation methods. Thus, unless an estimation equation’s applicability descriptor matches the supplied compound descriptor, the method cannot be applied to the substance in question. Furthermore, the compound descriptor may be used internally by an estimation method if the correlation is dependent on some descriptor characteristic such as polarity. Route selection. The user has wide control over the sequence of estimation methods, called a route, leading to the desired properties. Of course, if he so desires, he need not intervene in any way and the system will automatically select that sequence of estimation methods producing the least error in requested properties; usually the system will be operated in this way (this is discussed in the section on minimum error analysis). However, the user is completely free to specify the entire computation route, select estimation methods for inclusion in the route, or reject certain estimation methods from the route. When estimation methods are selected or rejected in advance, it should be recognized that the system may not be free to generate the minimum error route. I n addition, a user may request that the route followed in a prior job (task) be applied to a subsequent one. For example, the exact sequence of estimation methods leading to minimum uncertainty in the heat capacity of benzene at a specified temperature and pressure in task 1 may be applied to m-xylene at a different temperature and pressure in task 2 . However, the heat capacity estimate for m-xylene will not necessarily be the best that the system could produce if the route constraint were removed. hlevertheless, the constraint ensures that heat capacity values for the two compounds are consistent according to method of estimation. Table generation. Tables of property values for the same compound may be generated whenever it is desired to have a task solved repeatedly for a set of input properties, One-dimensional tables of physical properties can be requested over a specified range of pressure, temperature, or temperature/pressure points. For example, by fixing the temperature and specifying a series
of pressure points in ascending order, one can generate estimates of enthalpy as a function of pressure due to a sequence of isothermal gaseous compressions. Adjustment of property estimates. The system has the capability to “adjust” physical property estimates for consistency with prior experimentally determined data. For example, suppose a user has a set of known physical property data at specified temperature/pressure points and wishes to estimate the same property at other temperature/pressure points. Estimates obtained from the system may not be consistent with the experimental data, and it becomes desirable to smooth the estimates. This is easily done by fitting a linear regression model to the experimental and calculated data. First, an estimate zi of the known property zi’ is obtained by the system for each temperature/pressure point ( T i , Pi) in the experimental data set. The coefficients a0 and a1 in the model 2%’
= a0
+ altl
can then be determined by least squares regression analysis. This equation may then be used to smooth subsequent estimates made by the system at temperature/ pressure points other than those in the experimental data set. The root-mean-square deviation between known and estimated values is also calculated. Input processor. The input processor edits the jobs (tasks) submitted, interprets various user control options, and translates input specifications for internal system utilization. Numerous checks for conformity of input data with prescribed standards are performed prior to execution. Certain trivial errors are automatically corrected with an appropriate output message informing the user of action taken. More severe difficulties may require that the task be skipped. Special task specification forms have been designed to simplify entering information into the system. This information, when punched on cards, completely defines each separate task. A group of tasks is called a set and is executed as a unit by the system. Within a set, tasks may refer to data supplied to, or calculated in, other tasks. Thus, input property A (and its associated error) and estimated property B (and its associated error)
AUTHORS .Richard E. Heitman is with Arthur D. Little Ltd. in London, England. George H. Harris is with Arthur D. Little, Inc., in Cambridge, Mass. The authors wish to acknowledge the invaluable contributions of their associates: J . E. Murphy, R. C. Norris, R. C. Reid, and C. Hill. They also atpreciate valued guidance from members of the A.I.Ch.E. Machine Computation Committee and the Physical Properties Subcommittee, especially C. D. Alstad, G . E. Jones, Jr., T. L. Leininger, E. L. Meadows, and B . G . Price. T h e project received the support of the A.Z.Ch.E. Council and the jnancial assistance of more than 30 industrial organizations. T h e estimation program is available from the A.1.Ch.E. at 345 E. 47th St., N e w York, N.Y. 10017.
in task 1 may be input to task 2. Furthermore, a task may copy any part or all of the specifications of preceding tasks within the same set. This facility greatly eases the job of task description. Compounds and mixtures may be assigned identification names and/or numbers for purposes of internal task referencing as well as for output display. Each compound or component within a mixture is normally described via its appropriate descriptor codes to prevent the use of inapplicable estimation methods. A set of property cards is used to input all known physical properties of the compound or mixture under study. The physical units employed as well as the numerical value of each identified property must be given, since a wide selection of English and metric units is available for use in the system. Finally, the error in each input property, if known, should be given as either an absolute quantity or as a percentage of the property itself. If no error is specified, it is assumed to be zero. A second set of property cards identifies the property or properties to be estimated by the system, the units in which the output should be expressed, and whether the estimates are to be adjusted for consistency with known data. The known property data must be supplied to the system as a function of temperature and/or pressure. If tables of requested properties over a series of temperature/pressure points are desired, it is necessary merely to enter the temperature/pressure pairs on specially designed input cards. Alternatively, the user may specify that a series of uniformly spaced temperature/ pressure points be generated by the system, given the base point, step size, and number of increments. Numerous options are available to inhibit the execution of certain system functions and to select from among a variety of types of output. The user may, if he so desires, request that “known” input properties be recalculated whenever possible during the normal course of computation. This means that if the system can estimate an input property with less error than that originally attributed to the property by the user, the estimated value will replace the “known” value in all subsequent computations. Of course, if the input property is initially assigned a zero error, it can never be replaced by a more accurate value. Under normal system operation, input properties and their corresponding errors are not recalculated. Another option permits the user to suppress error propagation through the estimation route. If this is done, estimated properties will not have their associated errors evaluated. Therefore, the minimum error criterion is effectively eliminated and there is no optimum route. I n this case, the first method which estimates a property is selected. Other options control message generation during the course of computation and permit suppression of normal system error checks. There are also two controls for limiting the complexity of estimation routes leading to desired properties. The effect of these latter two controls will be discussed later in the section on automatic route generation. VOL. 6 0
NO. 2
FEBRUARY 1968
53
P H Y S I C A L P R O P E R T Y E S T I M A T I O N S Y S T E M - TASK SET 1 . D . D E M O N S l R A T I O N SET NC. 1. J U L Y TASK 1.0. IA PURE CCMPONENT T A S K A ( W I T H O U T P U T O P T I O N S )
A.1.CH.E.
-
1 1 1965
CONTROL CARDS COMPflUNO/MIXTURE NAUE NUMBER
LA
N-BUTANE
T A S K OESCRlPrCKS CONTROL PHASE MnLECOLAR CHAUACTERi 515 P HC S C G C F G l FGZ I T M 0 L R
100COHP12
I
t
I
I
3
0
0
0
0 0 2
LEVEL
-
VALUC
-
1 20
GU T PUT OPTIONS
LO
PUS
INPUT PROPERTIES
-
-
COMPOUNO/WIXTURE NAME NUVBER
100 100 100 100 100 100 100
HW PC TC VC 10 NO NO P
100
M
100
PSTO T
100
N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE .I-BU T ANE N-BUTANE. N-BUTANE +-BUTANE
100 100 100 100 100 100 190 100 ~
~~
PROPERTY E S T I M 4 T E O CODE F U L L NAME NO.
TO
NRO NRO BROT
OMEG ~~~
Na NA
MOLECULARUEL PRESSURElC TEMPERATURlC VOLUNElC TEMPERATURIB COUNT/OOBRIT COUNT/I)OBRAT PI? E S SURE HARTINHOUSLO PRESSURElSrO TEMPFRATURC TEMPERATURlO COUNT/RIliA.IO COUNT/RIHIYO RONOSROTAT I N ACENTR l C F 4 C T COUNl/ATO* CGUNT/ATOH
0 0 0 0
6 I8
-
COMMENTS
0.10
Lss.ooao
cncu
272.7000
KELV
c.5 0.10
ATM
C.
2.. .
0 0 0
IO. 0.6805
-7.0000
i.eaao
1 2
SET TASK
2. 0.
ATH KFLV KELV
310.8900 298.1500
PIGE PAGE
8 5
0. C.
2. 2. 3.0000 0.2010
~~
I
PCT EKRCQ
0.10 0.10
ATH KFLV
425.1100
0 0
6
-
ccn
58.1200 37.4700
0
UNIT
0.
1.0
4.
0 0
10.
YANTED P R O P E R T I E S COUPOUND/MIXTURE NAME NUMBER N-BUTANE N-BUTANE N-BUTAYE Y-BUTANE N-BUTANE N-MU1 ANE
Figure 7
I
PROPERTY E S T I H A I E D - LEVEL F U L L NAME NO.
S
100
v
100
H
ion 100
F E
100
CP
ENTROPV VOLUME ENrHALPV FUGACITY ENERGY HEATCAPICITY
Output from a sample task-control
Output processor. The output processor produces printed output from the system and, on option, generates punched cards which can in turn be used as input for other physical property estimation tasks. Figures 1-3 show the output from a sample task. This task is included to give the reader a visual appreciation of system output and is not intended to illustrate all features available in the system. Certain informative messages generated during execution of the task have been deleted for the sake of brevity, as has the listing of input cards. T h e following major sections are shown in Figures 1, 2, and 3, and are briefly described below : 1. Control cards--compound/mixture name, number, and descriptors; control and output options as provided on input. property and error information 2. Input properties-all specified by user or assigned by the system together with their units of measurement. 3. Wanted properties-values of requested properties in their specified units with associated percentage errors. The meaning of “level” will be explained later. 4. (Final) route information-the particular sequence of estimation methods leading to desired properties. This will usually be the optimum sequence in the sense of minimum error, unless the user has constrained portions of the route in advance. Methods in level 1 are executed prior to methods in level 2, and so forth. The error ratio is the ratio of the lowest property error obtained to that which was generated by the next best method, if one was available. Error ratios for “wanted” properties are always less than or equal to unity, but this is not necessarily true for other properties. 5. Raw road map-flow-chart of all applicable estimation methods without regard to requested properties. The raw 54
VALUE
-
UNIT
CODE
INDUSTRIAL A N D ENGINEERING CHEMISTRY
-
PCT
-
COHRENTS
ERROR L O G I C ~ L L VUNAVAILABLE
2 2 3 2 3
395?4. 3.8659 0.6681
8.3791 0.4365
F3P
2.
OP
3.
ATH RP BPF
2. 3. 4.
cards, input properties, and wanted properties
road map shows in sequence all properties that may be estimated from the specified input properties. 6 . Alternate values of properties-all intermediate and terminal property values calculated in the normal course of road map evaluation. This section shows the actual sequence of estimations obtained in evaluating all routes leading to desired property values. A property value with a zero key either is a minimum error estimate or else was chosen in advance by the user. O n option, all properties and the final route may be punched on cards for use in subsequent tasks. Timing. The computer running time is dependent upon the complexity and number of tasks in a job as well as on the output options selected by the user. The exact nature of an individual task is dictated by the number and types of estimation method subroutines included in the computation route. These subroutines are organized into 14 chain links which are maintained on the estimation method library tape. Since the present version of the system is highly tapeoriented to provide external program storage, it is not surprising that execution is primarily input-output limited. I n fact, the major portion of computer time is spent in searching the estimation method library tape to locate the next subroutine to be executed in the computation route. A typical computer run on an IBM 7090 requires 90 to 120 sec to execute one task, of which more than two thirds is devoted to tape searching. T h e situation has considerably improved in operating the system on a UNIVAC 1108 computer, owing to the
A.I.W.E.
PHYSICAL
P R O P E R T Y ESTIMATION
TASK
1.0.
-
svsrrM T A S K S E T I.O.OEMONSIRAIION s E r NO. I . JULY LA PURE COWPONENT TASK A I W I T H OUTPUT O P T I O N S )
-
I,
1965
ROUTE I N F O R M A T I O N COMPOUNOIMIXTURE NUMRER NAME N-BUTANE N-BUTANE N-BUTANE N-RUTANE N-BUTANE N-BUTANE N- B U T ANE N- B UT A N E N-BUTANE N-BUTANE N-BUTANE N-BUT AN€ N-BUTANE
PROPERTY E S T I M A T E D - L E V E L CODE F U L L NAME NO.
100 100
ZC CP I
100 100 100 100 100 100 100
HI
100
FP CPO F CP
El ED HO V H
E
100 100
LOO
-
I i I 1
COMPRFSSIHlC HEATCAPAC I l I ENTHALPVlI ENERGY11 ENERGVlD ENTHALPVIO VOLUME ENTHALPY ENERGY FUCACITVPKES HEATCAPACIlO FUGACITV H€ATCPPACIrY
VALUE
-UNITS-
0.2739 24.8795 311.8~73 266.5729
CGMK CCM CGM
I -0.oia96469 1 -0.03032120 2 2 2 2
39524. 8.8659 8.3791 0.9818 0.1477 0.6681 0.4565
2 3 3
F3P BP BP
ATM BPF
PC T ERROR
ESTIM4110N MElHOD CODE F U L L NAME
0.5
LCO
4.
CPIRO HIRD ElRO
3. 3. 4. 4. 2. 3.
3. 2. 15. 2. 4.
-ERRORR A T IO
HDRK VMH HO EO FPMH CPCMH FD
CPC
0 0
0.17
ZClDEFINITIC CPIlRIHANIO Hl/RIHANIO EIIRIHANIO EDlREOLICHK HOlREDLICHK VlMPRTINH HlDEFINITION ElOEFINITION FPlMARTINH CPDlMARTINH FIDEFINITION CPIOEFINITIO
EORK
R E F E R E N C E S COMMENTS KEY MESSAGE
8
0.58 0.09
0
41 48 50
0
67
1.57 0.90 1.00
0
69
0 0 0 0
19 44
o
0.77 7.18 0.50
0
0
1.00
0
S E T PAGE TASV
9 6
PAGE
53 58 61
59 62
RAW ROAD MAP COMPOUNO M I X T U R E NAME NUMBER N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE &BUTANE N-BUTANE N-BUTANE N- B U T A NE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N- B UT A N E N-RUTANF N-BUTANE N-BUTANE N-BU T ANE N-BUTANE N-BU TAME N-BUTANE N-BUTANE N-BUTANE N-RUTANE N-BUT ANE N-BUTANE N-BUTANE N-BUTANE N-BUlANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTINE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANZ N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N- B U1A N E N-BUIANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUIANE N-BUTANE N-BUTANE N-BUTANE N-BU 1ANE N-BUTANE N-BUTANE
-
100 100
LOO
-
PROPERTY E S T I M A T E 0 CODE FULL NAME NO. THE1 ZC VBO
100 100 100 100 100 100 100 100 100 100
vsv
100
HO
LOO
FPSV PV ATOM ZC ALPH V V
LOO 100 100 100
IO0 100 100 100
IO0 100 100 100 100 100 100
HVB
nv R HV CP I
HI E1 SD EO FP
vso V PV PV HV
HO H HSV HSV
LOO
E1
100 100 100 LOO 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100
EO
100
100 100 100 LOO 100
100 100 100 100 100 100 100 100 100 100 100
E FP F FPSV CPO
so
ESV HSV FPSV V V
L V
nn HD
n
HSV HSV ED
EO E EO FP FP F FPSV CPO CP
so so
ESV FPSV CPO
HSO PV
LOO
2
100 100
HO H
THETA/ TRB CONPRESSIBlC VOLUMElBQ VOLUME I S V ENTHALPVIVR ENTHALPVlVB ENTHALPVIV HEATCAPACIII ENTHALPVll ENERtVlI ENTROPVlO ENERGVlD FUGACI TVPRES ENTHALPVIO FUGAC I T V P l S V PRESSURElV ATOH COMPRESSIBlC R I EOE LF AC TOR VOLUME VOLUME VIlLUME /SQ VOLUME PRESSURE I V PRESSURE I V ENTHALPVIV ENTHALPVlO ENTH4LPY ENTHALPVISV ENTHALPVlSV ENERGVlI ENERGYlO ENFRGV FUGACITVPRES FUGACITV FUGACITVPISV HEATCAPACIlD ENTRC1PVlO ENERGVlSV ENTHALPYlSV FIJGAC I T V P I S V VOLUME VOLUME COMPRESSIBIL VOLUME ENTHALPYlO ENTHALPVIO ENTHILPV ENTHALPVISV ENTHALPVISV ENERGVID ENERGVID ENERGY EVERCVIO FUGACITVPRES FUGACITVPRES FUGAC I T V FUGAC I l V P l S V HEATCAPACIID H E A T C A P A C I TV ENTROPVIO ENTROPVlO EYERGVlSV FUGACITVPlSV HEATCAPAC I / O ENIHALPVISO PRESSUKElV CflMPKESSIRIL E N T HA L P V I I) EVTHALPV
EVEL
-
V4LUE
-UNITS-
PCT ERROR
E S T I M A T I O N METHOD CODE F U L L NAME
1
THETO
1
ZCO
1 1
VBQB VSVRK HVBG HVBRPM HVP CPIRO
I 1 1
1 1 1 1 1 1 1
HIRO
2 2
EIRO SDRK EORK FPRK HDRK FPSVRK PVRPM ATCMO ZCL ALPHM VHBMS VMH
2
VSQB
2
VRY. PVA PVFM HVW HONH
1 1 1 2
2
? 2 2 2 2 2
HO HSVHBM HSVMH E10 EDMH EO FPHH FO FPSVHB CPOMH
2 2 2 2 2
SOHH ESVRK HSVRK FPSVMH VHBMS VMH
3
20
3
VRK HOHHMS HDMH
3 3
3 3 3 3 3 3
HO HSVHBM HSVM EOHHMS EDMH
EO ED0
3
3
FPHBMS FPMH FO FPSVHB CPDMH CPC SDHBMS SDMH ESVRK FPSVHH CPDHBM HSPD PVR ZD
3 3 3
3 3 3 3 3 3 3
3 3 4 4
THETlDEFINIT ZC/OEFINITIO VBOIBENSON VSVlREOLICHK HVBlGIACALON HVBlRIEOELPM HVlPITZER C P I/RIHANI 0 HIlRIHANID EIlKIHANIO SOlREDLICHK EOlREOLICHK FPlREDLICHK HOlREDLICHK FPSVIREOLICK PVlRIEDELPM ATCMlOEFlNlT ZCILVDERSEN ALPHlMILLER VlHIRSCHFBMS VIMARTINH VSOIBENSON V l R E O L ICHK PVlANTCINE PVlERPENBECH HVlWATSON HOlMARTINH HlOEFINITION HSVlHIRSCBMS HSVIMARTINH EIIDORRATZ EDlMARTINH ElOEFINITION FPlMARTINH FlDEFINITION FPSVlHIRSBMS CPDlMARTINH SDlMARTINH ESVlREOLICHK HSVIREOLICHK FPSVlMARTINH VlHIRSCHFBMS V l M b R T INH ZlOEFINITION VlREOLICHK HOlHIRSCHBMS HDlMARTINH HlOEFINITION HSVIHIRSCBMS HSVlHARTINH EOlHIRSCHBMS EDIMARTINH ElOEFINITION EO/OEFINITIO FPlHIRSCHBMS FPlHARTINH FlOEFlNITION FPSVlHIRSBMS CPOIHARlINH CP/OEFINITIO SOIHIRSCHBMS SOlMARTINH ESVlREOLICHK FPSVlMAHlINH CPDIHIRSCBMS HSQlDEFINITI Y V I R I EOEL ZIOEFINI~ION
HDHBMS HOlHIHSCHBMS
4
Figure 2. Output from
-ERRORRATIO
HO
a
HIDEFINlTION
REFERENCES CCMNENTS KEY MESSAGE -25
0 -25 -25
-11 -12 -25
0 0 0 -25
0 -12 0 -25 -25
- 12 -11 -25 -11 0 -25 -11 -25
- 25 -25
-11 0 -25 -25
-11 -11 0 0 12
-
-25 0 -25 -25
SET P A G E TASK PAGE
13 7
- 25 -25 -31 -31 -25 -31 -12 -31
-31 -25 -25
-11 -31
-31 -11 -11 -31 0 -25
-31 0 -25 -25 -25 -25 12 -25 -25 -25 -31
-
-11
sample task-route information and raw road map VOL. 6 0
NO. 2
FEBRUARY 1968
55
A.1.CH.E.
SET NO. 1- JUL.Y P H Y S I C A L PROPER T Y E S T I W A T I O N SYSTEM - TASK SET I . 0 - 0 E W O N S T R A T I O N TASK 1.0. 1 A - PURE COWPONENT TASK A I W I r H O U T P l J r O P T I O N S )
1.
1965
R A Y ROAD MAP COMPOUNOIMIXTURE NAME NUMBER N-BUTANE N-BUTANE N-B U T A NE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE M-BUTANE N-BUTANE
-
-
PROPERTY E S T I Y A T E D CODE F U L L NAWE NO.
100 100 100 100 100 100 100 100
HSV EO E EO FP F FPSV CP
100 100 100 100 100 100 100 100
CPO Hsa H E EO F CP
100 100
E
so
nsa
LEVEL
-
VALUE
-UNITS-
PCT ERROR
HSVHBM EDHBWS EO EOC FPHRWS FD FPSVHB CPO SOHBMS CPDHRW
4
ENTHALPYISV ENFRGY I O ENERGY ENERCYlD FUGAC I TYPRES FUGAC I T Y FUGACITYPISV HEATCAPACITY ENTROPY I D HEATCAPACIIO ENTHALPYISQ ENTHALPY ENERGY ENERGYIO FUGACITY HEATCAPAC I TY ENTHALPYISP ENERGY
E S T l R A T I C h WETHDO CODE F U L L NAWE
4 4
4 4 4
4 4 4 4 4
HSPD
5 5 5 5 5 5 6
HO EO EO0 FO CPO HSPO
EO
-ERRORRATIO
R E F E R E N C E S CCWWENTS K E Y WESSAGE
-25
HSVIHIRSCBMS EOIHIRSCHRWS EIDEFlNITION EOIDEFINITIO FPlHIRSCHBWS FIOEFINITION FPSVlHIRSBWS CP/OEFINITIO SDIHIRSCHBWS CPOlHIRSCEMS HSPlOEFINITI HIOEFINITION ElOEFINITlON EOIDEFINITIO FlOEFlNITION CPIOEFINITIO HSQlOEFINITI ElCEFINITION
-31 -31
-11 -31 -31 -25
-11 -25 -31 -25 -31 -31 -31 -31 -31 -25 -31
SET P A G E T A S K PAGE
11
a
A L T E R N A T E V A L U E S OF YANTEO V A R I A B L E S COMPOUNDIMIXTURE NAME NUMBER N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-RUTANE N-BUTANE N-BUTANE
-
100 100 100 100 100 100 100 100 100 100
PROPERTY ESTIMATED CODE F U L L NAME
v v
v
n E F F CP H CP
-
LEVEL
-
VALUE
-UNITS-
YO.
VOLUWE VOLUME VOLUUE ENTHALPY ENERGY FUGAC 1T Y FUGAC I T Y HEATCAPACITY ENTHALPY HEATCAPACITY
39600. 39524. 39644. 8.8659 8.3791 0.6701 0.6681 0.4365 8.9167 0.4295
2 2 2 2 2 2
3 3 4 4
F3P F3P F3P 8P ep ATW ATM RPF EP BPF
PC T ERROH 2. 2.
2. 3. 3. 4. 2. 4. 3. 4.
E S T I M A T I O N WETHOD COOE F U L L NAME
VHBMS VWH VRK HO EO
FO FO CPO HO
CPC
VIHIRSCHFBMS VI4ARTINH VIREOLICHK HlOEFlNITION E/OEFINITIGN FlOEFINITION FICEFINITION CPIOEFINITIO HIOEFINITION CPlOEFINITIO
-ERRORRATIO
0.90
~~
1.00
0.50
1.00
R E F E R E N C E S COMWENTS K E Y MESSAGE
-11 0 -11 0 0 -12 0 0 -11 -11
18 19
26 44 53 59 59 62
44 62
A L T E R N A T E V A L U E S OF P R O P E R T I E S E S T I M A T E D BY L E V E L COMPOUNOIMIXTURE NAME NUMRER N-BUTANE N-BUTANE N-RUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-RUTANE N-BUTANE N-BUTANE N- BUT ANE N-BUTANE N-BUTANE N-BUTANE N-BUT ANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUT ANE N-BUTANE N-BU T AN€ N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE N-BUTANE
-
-
PROPERTY E S T I W A T E D COOE F U L L NAWE NO.
100
zc
100 100 100
HvB HVB CPI
100
HI
100 100
E1 EO FP HD ATOM ZC V V V
100 100 100 100
100 100 100 100
no
100
ti
100
E1 ED E FP F CPD HD ED EO FP F CP CPD ti EO CP
100 100 100
100 100 100
100 100 100 100 100 100
100 100 100
COWPRESSIRIC ENTHALPYlVB ENTHhLPYlVB HEATC A P A C I I 1 ENTHALPYll ENERCYlI ENERGYlD FUGACITYPRES ENTHALPYlD A TOM COMPRESSIRlC VOLUWE VOLUWE VOLUME ENTHALPYlO ENTHALPY ENERGYlI ENERGYlD ENERGY FUGACITYPRES FUGAC I T Y HEATCAPAC I I D EN THALP Y 10 ENERGYlO FNERGYlO FUGAC I T Y P R E S FUGAC I T Y HEATCAPACITY HEATCAPAC I I O ENTHALPY ENERGYlO HEATCAPACITY
LEVEL
-
VALUE
-UNITS-
PC 1
ERHOR 1
I 1 1
I 1 1 1 1 1 2
2 7 2 2 2
2 ? 2 2 2 2 3 3
3 3
3 3 3
4 4 4
0.2739 5475.1285 5362.3139 24.8795 311.8873 286.5729 -0.01896469 0.9847 -0.0303212n 14.0000 0.2760 39600. 39524. 39644. -0.04796217 8.8659 245.7G18 -0.03442576 8.3791 0.9818 0.6701 0.2477 -0.02838270 -0.01484630 -0.01678479 0.9902 0.6681 0.4365 0.04175939 8.9167 -0.01484630 0.4295
CGM CGW CGMK CGM CCN
0.5 4. 2.
CPlRU
NIRO
3.
EIRO EDRC FPRK HDRK ATOWO ZCL
*. C. 3.
EP ATM
2. 2. 2. 19. 3. 6. 25. 3. 2. 4. 15. 3. 70. 65.
3. AIM EVF 8P BPF
2. 4. 12. 3. 73. 4.
ZCIDEFINITIO HVElGIACALON
VHBMS VMH VHK
HOMH HO E10 EDMH ED FPMH FD CPOMH HOHEWS
EOHRWS EO0 FPHBMS
FO CPO CPOHBW HD EO0 CPD
CPIlRlHANIO HIIRIHANID EI/RIHANID EOIREOLICHK FPlREDLICHK HOlREDLICHK ATDMlUEFINIT LCILYOERSEN VlHIRSCHFBWS VIHARTINH VlREDLICblK HOlMLRTINH HlOEFINlTION
0.17
0
8
-11
35 37 41 48 50 67
0.90
-12 0 0 0 0 -12 0 -12 -11 -11 0
l.oo
-11 -11
0.58 0.09 1.57
EIIOORRATZ EDlW4RTINH ElOEFINITION FPlMARTINH F/OEFINITIflN CPOlMARTINH HOlHIRSCHRHS EOlHIHSCHBMS EO/OEFINITIO FPIHIRSCHBWS FlOEFINITION C P l O E F I N 1 T IO CPCIHIRSCBWS HlOEFINITION EOlOEFINITIO CPIOEFINITIO
R E F E R E N C E S COMMENTS K E Y MESSAGE
-ERRORRATIO
MVRRPM HVElRIEDELPM
3.
4.
EP CGW
ZCO HVBG
4.
4.
F3P F3P F3P
E S T I M A T I O N METHOD COOE F U L L NAME
-11 -11 0 0.77
0
-12 7.18
0.50 1.00
0 -12 -11 -11 -11 0 0 -12 -11 -11 -11
18 69 105 6
18 19
t:
44
S E T PAGE TASK P A C E
12 9
49 52 53 58 59 61
42 51 54 57 59 62 75 44 54 62
KEY EXPLANATION CODE
0 -11 -12 -25 -31
MEANING T H I S E V A L U A T I O N Y A S R E Q U I R E D ANI3 H A 0 A M l N l M U W U N C E R T A I N T Y . A R E T T E R E S T I M A T E H A S B E E N WADE T H I S E S T I M A T E IS NOT R E Q U I R E D TO O F l r A l N THE WANTED P R O P E R T I E S NOT N E E D E D FOR T H I S PRORLEM I R E J E C T E O B Y ROAD MAP GENERATOR) NOT E S T I M A T E D BECAUSE NO NEY DATA WAS A V A I L A B L E
Figure 3. Output from a sample task-raw road map (continuation) and alternate ualues of properties elimination of magnetic tape as a predominant storage device and a much faster core storage cycle time. The 1108 system, using a magnetic drum as an auxiliary internal storage device, has reduced task execution time to 45 sec, of Ghich 10-15 sec are for the initial transfer of programs from tape to core. Once the system has been allocated to the drum, additional tasks may be run in 56
INDUSTRIAL A N D ENGINEERING CHEMISTRY
less than 10 sec each. Comparable savings in time may be expected when disk-oriented systems are used. Error
Concept of minimum error route. So far we have not discussed how the system determines the sequence of physical property estimation techniques it will follow,
leading from input data to desired properties, nor how property estimates and associated errors are actually obtained. These processes are of fundamental importance in the operation of the system. As previously mentioned, the system can automatically determine from all possible estimation sequences those which lead to property values possessing minimum error. A sequence of estimation methods which passes through intermediate properties and leads to a final, or desired, property is called a route to that property. Associated with each intermediate property as well as the final property (or properties) is an estimate of the error or uncertainty in the computed value of the property. There are two major sources of error in a n estimated property-the correlation itself is usually imperfect (method error), and the input data used by the correlation are not precise or may be inaccurate. Since the output of one estimation method may serve as input to another, the error in a calculated property will tend to propagate through the route, ultimately affecting the uncertainty in the final property. Of course, errors in the basic input data can be propagated only to the extent that the user is aware of them and provides them on input. The route leading to desired properties having minimum error estimates is called the optimum route. Determination of the optimum route to a set of properties is somewhat analogous to the problem of finding the shortest route between two or more cities linked by a road network. The physical properties may be thought of as cities, the estimation methods as roads, and the errors as distances from the origin cities (input properties) to others. There are important differences, however: An estimation method usually involves a “many-to-one” transformation wherein several inputs are combined to form a single output; by comparison, the road network has only one-to-one transformations. I n a road network the distances between cities are commonly, but not always, independent of direction of travel; in the present route-finding problem, an estimation method and its inverse almost always will have different error values. I n theory, a set of estimation methods (such as a method and its inverse) can lead to a n infinite network owing to the possibility of cyclic transformations within the set-i.e., the outputs of certain correlations in the set become inputs to others within the same set, ad injnitum. Finally, unlike the usual road network problem in which the distances between cities are known a priori, the uncertainties in estimated properties can only be determined by actually propagating errors through the network. For the moment, we shall concern ourselves with the problem of automatic route generation and shall return
later to discuss physical property error estimation and optimum route selection. Automatic route generation. Before physical properties and their corresponding errors can be estimated, a “road network,’’ or master sequence of estimation techniques applicable to the particular problem a t hand must be created. This process, known as “road mapping,” proceeds in each case as a n organized search because it is entirely impractical to predetermine the vast number of routes connecting all possible combinations of known with requested properties. We summarize here the work of Norris ( 2 ) on road mapping. At this point, the concept of a transfer function T becomes quite important; it is a programmed physical property estimation method having one or more property inputs x t but only one property output X O , xo =
To (Xl, x2,
*
I
., Xf, . . ., x,)
If a transfer function has n inputs, it could have n inverses, each a distinct transfer function xa = Tt
( X O , XI,
..
a ,
~ a - 1 , xa+1,
. . .,
Road mapping takes place in three steps. First, all transfer functions inapplicable to the compound in question are eliminated from further consideration. This is accomplished by comparing each transfer function’s applicability descriptor with the compound descriptor. Second, by employing the remaining transfer functions, a raw road map is generated which contains routes to all properties that can be reached. I n the third step, the raw road map is weeded of any routes not leading to requested properties. It is worth emphasizing again that during the route generation process no transfer functions are actually evaluated. Generation of the raw road map is fairly complex and proceeds according to the concept of a “level.” All applicable transfer functions requiring only input data supplied by the user are assigned to level 1, as are their associated output properties. Using these newly created level 1 properties together with level 0 properties (original input data), all applicable transfer functions are assigned to level 2. Continuing in this manner, a transfer function and its associated output property are assigned to level N provided all of its input properties have been assigned to preceding levels and provided at least one input property came from level N 1. The same property or transfer function can be assigned to more than one level. Because different transfer functions can yield the same output property and because transfer functions may have several inverses, properties and transfer functions may be assigned indefinitely to higher levels. This is perfectly reasonable
-
VOL. 6 0
NO. 2
FEBRUARY 1968
57
since improved property estimates may result from this process. Generation of the raw road map may be terminated in three ways, the first natural and the other two through user intervention. Road mapping will automatically terminate when a level is reached beyond which no transfer function can be assigned; however, in no case is the number of levels permitted to exceed 25 because computer storage is limited. The user may, however, limit the number of levels to fewer than 25 by setting an appropriate stopping parameter L, but he risks forfeiting the optimum route to a desired property. Finally, route generation may be simplified and brought to early completion by preventing the repetitive assignment of the same transfer function to higher levels. The user may specify the value of a second stopping parameter R in order to prevent reassignment of the same transfer function to levels beyond LO R, where Lo is the initial level to which it was assigned. R is nominally set at 5 , but the user may specify any number between 1 and 2 5 , inclusive. Small values of R (less than 3) prevent unwieldy road maps, but may eliminate the optimum route. I n the final stage of road mapping, known as weeding, routes not leading to requested properties are eliminated from the raw road map. This process is begun by searching the highest level H in the raw road map for any transfer function whose output is a requested property ; transfer functions not satisfying this condition are eliminated from level H. Having found all such transfer functions, level H - 1 is scanned for any transfer function whose output is an input to a transfer function remaining in level H ; those that do not qualify are eliminated from level H - 1. This process continues backward, level by level, until level 0 is reached, at which point weeding has been completed. The optimum route, if one exists, must be contained in the weeded road map. Before describing how the optimum route is determined, we must now consider the propagation of errors through the competing routes. Error propagation. As previously mentioned, there are two sources of error which enter into physical property estimation: (1) error associated with the estimation method itself (method error) and (2) errors in the values of properties provided as input to the system (input errors). Both types of error tend to propagate through a route in a most complex way. To be meaningful, an error value must have a confidence statement associated with it. Furthermore, since most estimation methods have multiple inputs, it is necessary to use the same confidence statement with each error estimate. The method error for each transfer function in the system has been chosen at the 68.3y0 confidence level-Le., for a normally distributed population of estimates of the true property value, the method error is also the standard deviation. Unfortunately, in practice, transfer function method errors are known with relatively little certainty because few trials are reported in the literature.
Since the method error has in theory been chosen at the 68.3y0 confidence level, so too should the input errors. Although the user may specify input errors either in the same units as the property value or as a percentage thereof, within the system errors are always manipulated as the former. If the user believes a n input property is known without error (for example, it may be a truly independent variable) or if he is unable to assess the error, then no input error need be assigned to this property. An estimation method or transfer function T may be thought of as transforming its n input error distributions into a multivariate output error distribution. Given the error distributions of the n inputs X I , x2, . . ., x,, it is in general impossible to determine analytically the joint error distribution of the output y, where
+
58
INDUSTRIAL A N D ENGINEERING CHEMISTRY
y
= q x l ,
x2,
*
.
.)
x,)
However, if we assume that the transfer function is approximately linear in each input variable x f over the range of uncertainty in x f and that the errors in the x f are uncorrelated, the problem becomes greatly simplified. The total change in y from yo, dy, resulting from a small change dxl in each input variable may be approximated by n
where yo is the base case output of T when x = x O , or x2=xz0, . . , x,=xnO. Under conditions of assumed linearity, the partial derivatives may be replaced by partial difference quotients :
x1=xIo,
I
+ Axf, xi, xkO, . . .) - T ( X ; ,x?,
T(x,O
x,',
. . .)
Ax,
where ATi represents the change in T for a small change Axt in x f from x,O. If we let Axt be the estimate of the standard deviation of the probability distribution of x f , and if errors in the x i are uncorrelated, then an estimator of the variance of y is
i-1
i-1
where suoe is the usual sample variance for yo, or in this case the method error variance. Finally, the propagated error iny is n
Hence, to compute sy we first evaluate the transfer function at the base case for x = xO, perturb each input variable in turn by its standard deviation (maintaining all other variables constant), evaluate the net change in T , and combine all such changes according to the above expression. The values for y and s,, thus obtained, serve
as inputs to the next level of computation, if required, and the error analysis is repeated. It is reasonable to assume that both the errors in the input properties provided by the system user and the method errors of the individual transfer functions are uncorrelated. O n the other hand, correlated errors will exist for computed properties whose routes partially overlap-Le., at some point in their respective routes they have a common input variable or transfer function. I n such cases, the propagated error computed according to the preceding equation will be somewhat incorrect, but the magnitude of the error will be small relative to uncertainties in the method errors and input property errors. Optimum route selection. The optimum route yielding a minimum error estimate of a requested property must also produce minimum error estimates of any intermediate properties included in the route. If this were not true, it would be possible to find a route whose intermediate properties had larger errors, this, in turn, leading to a n improved estimate of the requested property. Such a situation is clearly impossible. Because each property in the optimum route is a minimum error estimate, one need merely perform a sequential evaluation of each property contained in the weeded road map, selecting at each stage that value of a property having least error. The process proceeds level by level, beginning at level 1. Whenever a transfer function assigned to a particular level yields a n improved estimate (smaller error) of a previously evaluated property, the newly estimated property and associated error are selected to replace the former values. However, to minimize needless computation which would result if successive estimates of the same property were only slightly improved, the ratio of the new error to the old one must be less than 0.99. If the new property estimate is not an improved value, it is rejected; the former property value is retained as the minimum error estimate through that level. A transfer function encountered at a certain level N should not necessarily be applied if it has been used previously in the route. The transfer function was reassigned to the higher level during road mapping because one or more of its input properties was reassigned to level N - 1. If one of these input properties received its best estimate so far at level N - 1, then the transfer function will yield an improved estimate of its output property. If, however, none of these input properties received improved estimates at level N - 1, there is no reason to apply the transfer function, and it should be bypassed. The process of evaluating transfer functions and propagating errors through the route terminates when the entire road map has been traversed, or sooner if the repetitive application of transfer functions does not yield improved property estimates. When this occurs, the optimum route (if one exists) will have been determined and all properties, requested as well as intermediate, will have minimum error.
Future Work
There are a number of significant improvements that might be made in the A.1.Ch.E. system and in the field of physical property estimation in general. Recommended improvements have two central objectives: (1) to enhance or maintain the usefulness of the system or (2) to reduce computer running time. Due to the rather involved nature of these recommendations, space permits only a brief reference to each. 1. Incorporation of a chemical structure notation system
2.
3.
4.
5.
6.
7.
would permit the complete, unambiguous description of a molecule on input. This would greatly facilitate the handling of structural correlations within the system. More research is required in estimation method error analysis. Very little information is available concerning the accuracy and limitations of various estimation methods reported in the literature. Continued review of the estimation method literature is absolutely essential to ensure that the best techniques are always available in the system. Since the field of physical property estimation is extremely active, the system will remain viable only through a continuing effort to maintain an up-to-date library of estimation methods. I n this regard, there are a number of physical properties not included in Table I, such as enthalpy, entropy, and free energy of formation, that might be added to the system. Extensive consolidation of transfer function subroutines is possible, which would lead to reduced core storage allocation and significantly shorter running times. Because many thermodynamic properties are derived from a basic set of equations of state, there is at present considerable duplication of coding within those transfer functions arising from the same equation of state. Additional savings will result from centralizing various numerical analysis procedures and common checking routines. Inclusion of a data bank containing pure component constants and structural characteristics of commonly used compounds and mixtures would provide a supplement to any information supplied by the user on input. A capability to analyze the sensitivity of estimated property errors to changes in input property errors and estimation method errors would be valuable. Those points in the computation route contributing most to the uncertainty in the estimated property could be identified, thereby focusing attention on those places where improved input property information and better estimation methods would help most. Removal of certain property constraints would greatly increase the over-all flexibility of the system. It would be desirable to permit tables to be constructed as a function of properties other than temperature and pressure. Another improvement would permit the thermodynamic state of a compound to be defined by any appropriate combination of properties, such properties not being limited solely to temperature, pressure, and volume. Thus, it should be possible to specify both the h,eat capacity and energy and to request the temperature.
BIBLIOGRAPHY (1) Meadows, E. L., Chem. Eng. Progr. 61 (5), 93 (May 1965). (2) Norris, R. C., Ibid.,p. 96. ( 3 ) Reid. R. C.. Ibid. (12). . . 58 (December 1965), - \-,
I
(4) Reid, R . C., Sherwood, T. K., “The Properties of Gases and Liquids,’’ 2nd ed., McGraw-Hill, New York (1966).
VOL. 6 0
NO. 2
FEBRUARY 1968
59
