EV ALUA TION AND MAINTENANCE OF
AN ENTHALPY DATABASE
By
Abhishek Rastogi
Bachelor of Engineering
University of Roorkee
Roorkee, India
June 1993
Submitted to the faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
May 1996
EV ALUATION AND MAINTENANCE OF
AN ENTHALPY DATABASE
Thesis Approved:
Dean of the Graduate College
11
ACKNOWLEDGEMENTS
I wish to express my sincere gratitude to my mentor and adviser, Dr. K. A. M.
Gasem, for his invaluable guidance and advice. I deem it my privilege to have worked
under his supervision. I would like to thank Dr. Jan Wagner for his recommendations
and critical assessment of this work. Dr. Martin S. High, as the third member of my
committee, has reviewed this work and his comments are greatly appreciated.
I would like to gratefully acknowledge the financial support received from the
Gas Processors Association, Tulsa OK.
I am indebted to my parents, and brother, Shlok for their love, encouragement and
support without which this endeavor would not have been possible. And finally, thanks
to all my friends, especially Jairam and Monish, for their understanding and support.
III
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION .............................................................................................. 1
Rationale ...................................................................................................... 2
Research Objectives .................................................. ... ............... ................ 3
II. LITERATURE REVIEW .... .... ........ .................... .... ... ...... ... .... ...... .... .......... .. .... 5
Definition of Enthalpy ............ _.. ................. .............................................. 5
Experimental Determination of Enthalpy.... ....... ....... ..... ....... ........ ........... 7
The Enthalpy Equation .............................. ............................................... 9
Ideal Gas Enthalpy Determination ........................................................... 1 0
Enthalpy Departure Function Estimation ................................................. 12
Overview of Volumetric Equations of State ............................................ 13
Enthalpy Departures from Equations of State ......................................... .16
Excess Enthalpies and Equations of State ................................................ 21
Reference States for Enthalpy Data .......................................................... 22
National Institute of Standards and Technology (NIST) ........................ 22
Thermodynamic Research Center TRC-API 44 .................................... .23
III. GAS PROCESSORS ASSOCIATION ENTHALPY DATABASE .......... .. .. 25
GPA Enthalpy Database Format .............................................................. 25
GP A Enthalpy Database Holdings ........................................................... 26
Source of Enthalpy Data in the Database ... .............................................. 27
Lenoir's Reference Bases for Enthalpy Measurements ............................ 27
GP A Reference States for Enthalpy Data ................................................ .29
Thermodynamic Consistency Checks for GP A Data .............................. .29
Smoothing of GP A Enthalpy Data ........................................................... 31
Conversion of GP A Enthalpy Data to the
Standard GPA Reference State ................................................................. 32
GPA Departure Conversion Techniques .................................................. 35
IV. EQUATION-OF-STATE ENTHALPY
DEPARTURE MODEL EV ALUATIONS ....................................................... 37
The Equation-of-State Model ................................................................... 37
Peng-Robinson Equation-of-State Enthalpy Departure Function ............ 38
Pure Fluid Properties ................................ ... ................... ... ....................... 39
IV
Software Used ... ..... ........... .... .. ... .... ...... .. ...... .... .. .. ..... .. ........ ....... .............. 40
Model Evaluations .... ... ... ... .... ... ...... ..... .. ... .... ... .... .. .. .. .. ..... .. .. ..... ..... ......... . 41
V. ENTHALPY DATA QUALITY ASSURANCE .. .......... .. ...... .. .... ...... ...... ....... 62
Data-Entry Checks .... ... ...... ..... ..... .. ....... ....... .. .. .... .. ... .. ...................... ..... ... 62
EOS-Based Data Screening ..... ............ .. .. ...... .. ........ .. ... .... .. ...................... 63
T est Cases .... .. .... ..... ...... .... .. ... ..... ....... ..... .. .. ................. .... ......................... 64
n-Pentane .. ... ... .......... ....... ... ........... ... ... ...... .. ..... ............. .................... 64
Cyclohexane .. .. ....... .... ............................... ........................................ 69
Benzene ... .. ...... ....... ......................... ................ .................................. 73
Discussion ........................ ... ..................................................................... 77
VI. CONCLUSIONS AND RECOMMENDATIONS ................................ ........... 85
LITERATURE CITED ... ........................... .. ............................... .. ........ ......................... 88
APPENDIX A - A DERIVATION OF PR EOS ENTHALPY DEPARTURE
FUNCTION ......... .......... .............. ......................................... ............... 95
APPENDIX B - LITERATURE REFERENCES IN THE
GPA ENTHALPY DATABASE ................................................. .... 101
APPENDIX C - THE PURE FLUID CRITICAL PROPERTIES USED
FOR THE PENG-ROBINSON ENTHALPY DEPARTURE
FUNCTION MODEL EVALUATION .. .... .. ....................... .. ......... 108
v
LIST OF TABLES
Table Page
1. Experimental Enthalpy Reference States for Systems in GPA RR-6 ................. 33
2. Enthalpy Departure Comparisons ....................... ................................................. 41
3. Peng-Robinson EOS Error Analyses: Pure Fluids ............ ............ ...... ... .. ... ..... ... 44
4. Peng-Robinson EOS Error Analyses: Binary Mixtures ...................................... 47
5. Peng-Robinson EOS Error Analyses: Ternary Mixtures ........ ...................... ... ... 58
6. Peng-Robinson EOS Error Analyses: Multicomponent Mixtures .. .. .............. .... 60
7. Flagged Data Records for n-Pentane ..... ..... ... .. ................................................... 66
8. Flagged Data Records for Cyclohexane ... ... ... ..... .. ........... .... ................. .............. 71
9. Flagged Data Records for Benzene .. ..... ... .... ... .... ....... ... ........... .. ................... ...... 76
10. Summary of Possible Outliers in Enthalpy Data for Pure Fluids ........ .. ............. 79
11. Summary of Possible Outliers in Enthalpy Data for Binary Mixtures ............ ... 80
12. Summary of Possible Outliers in Enthalpy Data for Ternary Mixtures .............. 83
13. Pure Fluid Critical Properties Used in Evaluations ... ............ .. ... ...... ..... ..... .. ...... 109
VI
LIST OF FIGURES
Figure Page
1. A Simple Schematic of a Flow Calorimeter .... .. ............ .. .. ........ ...... ...... .. .. .. .... ... 8
2 Pictorial Representation of Loop Closure for Enthalpy Data
on 50-50 mole % H2S-Ethane .. .. ........ .. .. .. .. ........................ ...... .................. .. ....... 30
3. Liquid-Phase Enthalpy Departure Deviations for n-Pentane ........ .. .. .. .. .. .. ........ .. 67
4. Vapor-Phase Enthalpy Departure Deviations for n-Pentane .... .. .................... .. ... 68
5. Liquid-Phase Enthalpy Departure Deviations for Cyc10hexane ...................... ... 72
6. Vapor-Phase Enthalpy Departure Deviations for Cyc10hexane ................ .......... 73
7. Liquid-Phase Enthalpy Departure Deviations for Benzene .. ................ .......... .... 77
8. Vapor-Phase Enthalpy Departure Deviations for Benzene .............. .... .. .. ........... 78
Vll
NOMENCLATURE
a, b, c, d Correlation constants in Equation (2.7)
A Helmholtz energy
A_Ao Helmholtz energy departure function from ideal gas state
B, C, D Virial coefficients in Equation (2.15)
Cp Specific heat capacity at constant pressure
Cy Specific heat capacity at constant volume
gru: Potential energy difference for a unit mass
G Gibbs free energy
G_Go Gibbs free energy departure from ideal gas state
Lili Enthalpy difference
hE Excess enthalpy
W Ideal gas enthalpy
H_Ho Enthalpy departure function from ideal gas state
M1yap Latent heat of vaporization
rh Mass flow rate of fluid
p Absolute pressure
Q Heat input into the system
R Universal gas constant
S Entropy
S_So Entropy departure function from ideal gas state
T Absolute temperature
l-.U Change in internal energy
U_Uo Internal energy departure from ideal gas state
v Volume
Vlll
Ws Work extracted by the tluid per unit mass
z Component Mole fraction
Z Compressibility factor
RMSE Root-mean-squared error
%AAD Average absolute percent deviation
Greek Symbols
f..
P
co
Subscripts
c
r
Dep
calc
meas
Len
API
Change in property
Density
Acentric factor
Critical number
Reduced number
Departure
Calculated
Measured
Lenoir
American Petroleum Institute
S uperscri pts
o Standard state
19 ideal gas
m mixture
E Excess number
lX
CHAPTER I
INTRODUCTION
In the United States, there are a number of cooperative research organizations
sponsored by industry. In the thermodynamic and physical properties areas, there are
three trade organizations or engineering society groups of interest to the energy sector.
The Gas Processors Association (GPA) conducts research on gases, light hydrocarbons,
and process solvents for the recovery and purification of natural gasoline, liquefied
petroleum gas (LPG) and on substitute gas (l). The American Petroleum Institute (API)
Subcommittee for Technical Data, in its Refining Department, conducts research on
petroleum and synthetic crude refining. The Design Institute for Physical Property Data
(DIPPR), under American Institute of Chemical Engineers (AIChE), conducts research
for the chemical/petrochemical area.
Cooperative research of this kind involves participation and sponsorship by
industrial companIes. Some of the advantages accruing to participating organizations
include the availability of research results in a timely fashion. Data used in process
design are obtained at a fraction of the cost of internal or contract measurement. In
addition, when industry and the participants operate from a common database, plant
design and construction is rendered more efficient. In other cases, cooperative data
supplement internal research and also serve as calibration data for such research.
Rationale
Enthalpies are required for economic evaluation and design of chemical processes.
In the chemical process industry, enthalpy data are used in designing separation units,
reactors, heat exchangers, refrigerators, humidifiers, etc. Thus, accurate enthalpy data are
needed for efficient process design and optimization purposes. Inaccurate
thermodynamic properties such as the enthalpy values can lead to inoperative plants, or
more likely, to over-design and superfluous capital investment.
The Oas Processors Association (OPA) maintains an extensive compilation of
enthalpy departure data, among other kinds of thermodynamic data, of use to the natural
gas processing industry in a "databank." The goal of the OP A Project 921, Enthalpy
Database Development and Maintenance, is to compile, evaluate, and maintain
experimental enthalpy, heat of solution, and isothermal enthalpy departure data for pure
components and mixtures of known composition that address the technical needs of the
gas processing industry. In this context, the database is used primarily to (1) evaluate
enthalpy prediction methods and computer models, (2) develop new or system-specific
correlations, and (3) identify experimental measurements for direct application
(interpolation) in process engineering calculations.
Owing to the practical importance of enthalpy data in the process industry and its
importance in model development efforts, it is essential that such data be accurate. Thus,
the database must be free of data-entry errors, and probable errors in the experimental
measurements must be noted.
2
The current enthalpy data in the GP A database are in the form of enthalpy
departure values, which have been generated from experimental enthalpy data reported in
the literature (24). Over the years, different techniques were employed to determine the
enthalpy departure entries. since the experimental data were based to different reference
states. Consequently, assessing the accuracy of the database departure values by direct
comparison with the original enthalpy records in the literature is not possible.
Research Objectives
The goal of this work was two-fold.
• First, to evaluate the ability of the Peng-Robinson equation of state (EOS) for predicting
the enthalpy departure values of natural gas systems maintained in the GP A databank.
• Second, to use the Peng-Robinson EOS enthalpy departure model to screen the enthalpy
entries and help assess the quality of data in the GP A databank.
The thesis is organized into SlX chapters. Chapter II describes the basic
experimental apparatus for enthalpy measurements. It discusses the formulation of
enthalpy departure functions, ideal gas enthalpy correlations, and the reference states used
for enthalpy determination. It includes a literature review of enthalpy departure function
estimation using different equations of state and also discusses the reasons for choosing
the Peng-Robinson equation of state for purposes of data evaluation.
A description of the GPA enthalpy database format and its holdings is given in
Chapter III. The literature sources of the data in the database and the experimental
3
techniques employed by the investigators are discussed in this chapter. Methodologies
for smoothing of raw data and for data quality checks are also reviewed.
Chapter IV presents the model evaluations for all the enthalpy data in the
database. The tables include the liquid-phase and vapor-phase predictions for the pure
components and binary, ternary, and multi component mixtures.
The issues pertaining to enthalpy data quality assurance are dealt with in Chapter
V. The methodology used for identifying data discrepancies and data meriting further
examination, based on the EOS model evaluations, is demonstrated with the help of three
sample test case systems.
Finally, Chapter VI contains the conclusions of this study and recommendations
for future work.
4
CHAPTER II
LITERATURE REVIEW
In this chapter, background concepts which include the thermodynamic definition
of enthalpy, its representation in terms of intrinsic variables, and the basic procedure for
experimental determination of enthalpy are discussed. A pertinent literature review of the
ideal gas enthalpy determination, enthalpy departure functions from equations of state,
and the reference states used for calorimetric measurements are presented. The reasons
for choosing the Peng-Robinson equation of state for data evaluation purposes are also
discussed.
Defini tion of Enthalpy
For a closed system, i.e., a system which does not exchange mass with its
surroundings, the first law of thermodynamics may be mathematically expressed as:
where
~U=Q-W
U = internal energy
Q = heat input into the system
W = work done by the system
(2.1)
5
In addition to internal energy, another thermodynamic function, known as enthalpy is
commonly used owing to its practical importance. For any system, enthalpy, H, may be
mathematically expressed as:
H=U+pV (2.2)
where
p = absolute pressure
v = volume
The units of H , as seen from Equation (2.2) given above, are those of energy.
Enthalpy, H, is an example of a state junction. Thermodynamically, a state
function is a quantity which does not depend upon the past history of the substance nor
the path employed to reach that condition or state. Internal energy, U, pressure, p and
volume, V are all examples of state functions.
The general first law expression for steady-state fluid flow across any two sections,
is given as (55):
where
!1u 2 • .
mM+m--+mg&=Q-Ws
2
M = enthalpy difference between the sections
Q = heat added to the fluid between the sections per unit time
W = work extracted by the fluid per unit time s
m = mass flow rate of the fluid
6
(2.3)
~U2 = kinetic energy difference of the fluid between the sections per unit mass
2
of fluid
g~ = potential energy difference between the sections per unit mass of fluid
Some of the terms in Equation (2.3) are expressions for energy per unit mass of fluid; in
the SI system of units, energy is expressed in joules or in some multiples of joules. For
most thermodynamic applications, the kinetic and potential energy terms are negligibly
small and may, therefore, be omitted. In such a case, Equation (2.3) reduces to:
Mz=Q-Ws (2.4)
where ~h, Q and Ws are per unit mass of fluid.
Experimental Determination of Enthalpy
Enthalpies are almost always determined experimentally using a flow calorimeter
(55). A simple schematic for a calorimeter is given in Figure 1. The main feature of a
flow calorimeter is an electric heater immersed in a flowing fluid. The flow setup is
designed so that the kinetic and potential energy changes of the fluid between Sections
'1' and '2' of Figure 1 are negligible. This requirement is met by ensuring that the two
sections are at the same elevation, the velocities of flow are small, and that no shaft work
is done between Sections' 1 ' and '2.' Hence Equation (2.4) reduces to
Mz = h, - hI = Q (2.5)
The electric resistance is used to add heat to the flowing fluid, where the rate of
energy input is determined from the resistance of the heater and the current passing
7
00
CONSTANT
TEMPERATURE
BATH
SUPPLY. j
SECTION 1
I
PUMP
I i HEATER
APPLIED
EMF
T2 P2
I
I
I
,.-----I'''/'V ~DISCHARGE
SECTION 2
Figure 1. A Simple Schematic of a Flow Calorimeter
through it. The entire apparatus must be kept well insulated. In practice, there are a
number of details that demand careful attention, but in principle the operation of the
calorimeter apparatus is simple. Measurements of the heat input rate and the rate of fluid
flow allow calculation of tlh between the Sections' l' and '2.' Equation (2.5) reveals
that differences in enthalpy rather than absolute values of enthalpy are determined
experimentally. In reference to Figure 1, the enthalpy difference is between the fluid
entering at Section '1' and exiting at Section '2,' h2 - ht • If enthalpy of the fluid at
Section '1' is arbitrarily taken as zero, then tlh = h2 = Q as indicated by Equation (2.5).
Hence, the choice of a reference state assumes importance when dealing with enthalpy
measurements. Generally, all enthalpies are based to an arbitrarily chosen reference state.
Thus, when making comparisons between different enthalpy values, the reference states
to which the values are based should be the same.
The Enthalpy Equation
According to the phase rule, for a homogeneous substance of constant composition,
fixing the values of two intensive properties establishes its state. The molar or specific
enthalpy of a substance may, therefore, be expressed as a function of two other state
variables. The two state variables are, for convenience, chosen as temperature and
pressure. Therelc>re,
H= H(T,p)
The enthalpy ofa compound can be expressed as a summation of three quantities (53):
9
H(T.p) = [H(T,p)-H(T,pO)]+[H(T.pO)-H(T",p")]+ H(TV,p") (2.6)
where
H( T,p) enthalpy of a pure fluid or a mixture
H(T,p) - H(T,pO) enthalpy departure function
H(T,pV) - H(TV ,pO) ideal gas enthalpy difference
H( TO, pV) = enthalpy at the reference state
H(TO ,pO) is the enthalpy of formation of the compound from the elements at yo and
pO, or the reference state chosen for enthalpy calculations. The quantity
[H(T,pO) - H(TO ,pO)) is the difference in the enthalpy of the compound in the ideal gas
state at the temperature of interest and the reference state of YO. [H(T,p) - H(T,p())] is
called the enthalpy departure function and is the difference in the enthalpy of the
compound at the temperature-pressure condition of interest and the enthalpy of the
compound in the ideal gas state at the same temperature. For notational convenience, the
enthalpy departure function and the ideal gas enthalpy difference will, hereafter, be
denoted as H - H() and H", respectively. Enthalpy departures are also termed as
residual enthalpies, and symbolically denoted as HR .
Ideal Gas Enthalpy Determination
The ideal gas enthalpy function is calculated using an exact relation of type given
below.
10
where HU is the ideal gas enthalpy, c~ the ideal gas heat capacity at constant pressure,
and T the absolute temperature. The choice of the functional form of heat capacity in
most correlations is of a polynomial type (2, 3).
(2.7)
In the United States, and for substances of interest to the energy sector, the parameters
(a, b , c, d , ... ) in most cases are regressed from c~ data of the API Research Project
44 and the Thermodynamic Research Center (TRC) Data Project (2, 3). These equations
are derived using conventional least-squares method, minimizing the sum of the squares
of either the absolute deviations or percentage deviations with respect to reported C;
data. The reference base used was 0 Btullb at 0 oR for the enthalpy computations - the
same as those used for the API Research Project 44 tables (2, 3).
Heat capacity correlations of the polynomial form are, by far, the most popular
means of computing ideal gas enthalpy values. This is because they are reasonably
accurate and afford an easy means of ideal gas enthalpies computation by way of
analytical integration. The accuracy of the equations can be improved by increasing the
number of constants in the correlation.
A drawback of the polynomial form of heat capacity correlations is that even
though greater accuracy in fitting of the individual property may be achieved, it is at the
expense of being thermodynamically inconsistent (6). This is so because actual heat
capacity behavior is not constrained to follow any particular polynomial.
II
Some of the more elaborate choices for heat capacity correlations have the form (4,
5) shown below:
co = a + bexp(-c/T") p
(2.8)
This form is derived from theoretical considerations, but it is not readily amenable to
integration. i.e., a series expansion or a numerical integration procedure is required.
However. the predicted values of C~ are more accurate than those calculated from the
polynomial equation with four constants (4).
More complex equations for calculating the ideal gas heat capacity and enthalpy
have been proposed (6). These equations are more rigorous in form since they are
derived based on statistical mechanical formulae for the heat capacity of an ideal gas (7-
9). Comparison with existing heat capacity correlations shows that these equations are
more accurate for most cases.
Enthalpy Departure Function Estimation
The enthalpy departure function, [H(T,p)-H(T,pO)], is obtained from the
pressure-volume-temperature (pvT) properties of the fluid under study. An equation of
state (EOS) capable of describing the pvT behavior of the fluid offers the most efficient
means for determining enthalpy departure functions.
For a pressure-explicit EOS, the departure function for the Helmholtz energy, A, is
developed first using the appropriate fundamental property relations. Then, all the other
departure functions are readily obtained (53), as shown below:
12
-Jv (pR-T-.-) dV - RTlnV- V VO (2.9)
'"
J[( iP) -!i}v + R In_V_
oc or I· V VO
(2.10)
(A - A") + T (S - SO) + RT (Z -1) (2.11)
(A-A") + T(S-SO) (2.12)
G-G" = (A-A") + RT(Z-l) (2.13)
It is worthwhile to note here that the departure functions ( H - HO) and (U - Uo )
do not depend upon the value of the chosen reference state pO (or VO) while ( A - A ()),
(S-SO), and (G-GO) depend upon po (or V").
For the specific EOS, the right-hand side expressions of the above equations have
to be evaluated. A detailed derivation of the enthalpy departure function using the Peng-
Robinson equation of state is given in Appendix A.
Overview of Volumetric Equations of State
Equations of state play a central role in chemical engineering. Equations of state
(EOS) that represent relations between the pressure, p, molar volume, v, absolute
temperature. T, and compositions are referred to as volumetric EOS. A volumetric EOS
in conjunction with interrelationships provided by classical thermodynamics can be used
for estimating enthalpy departure functions from the ideal gas and for calculating phase
equilibria.
13
Since equations of state are so important in engineering, the literature on the subject
is vast. Literally, hundreds of variations of equations of state exist. However, most of
the accurate volumetric equations of state may be classified according to their origin.
The groups of equations may be classified as the van der Waals family of cubic
equations, the family of extended virial equations, corresponding states equations, and
those equations derived from statistical thermodynamics based on lattice models,
perturbation and integral equation theories, or from fitting computer simulation data.
The van der Waals family of cubic equations of state have been the subject of much
attention and research since the famous cubic equation of van der Waals (vdW) was
proposed in 1873:
RT a
p=---v-
b v 2
(2.14)
where b is the excluded volume and a, the cohesion parameter. While the vdW equation
of state is of historical interest, it is not quantitatively accurate. Other, more accurate
equations of state are those of Redlich and Kwong (RK) (62), Wilson-Redlich-Kwong
(63), Soave-Redlich-Kwong (SRK) (64). Peng-Robinson (PR) (51), Schmidt and Wenzel
(SW) (65), Harmens and Knapp (HK) (66), Patel and Teja (PT) (67), Adachi-Lu-Sugie
(ALS) (68) and recently Trebble and Bishnoi (TB) (69). This list is by no means
exhaustive. but it does represent some of the major milestones along the path of cubic
EOS development.
The \Oirial equation of state is an infinite-power series in inverse molar volume, as
given by Equation (2.15) below. This equation, first proposed by Thiesen (20),
14
represents the volumetric behavior of real fluids as a departure from the ideal gas
equation,
BCD
Z = 1+-+-2 +-3 + ...
v v v
(2.15)
Z is the compressibility factor. The coefficients B, C, D, etc., are called "virial
coefficients"; B is the second virial coefficient, C is the third coefficient, and so on.
From statistical mechanics, these coefficients are related to the forces between the
molecules; i.e., the second virial coefficient represents the interactions between two
molecules, the third virial coefficient retlects the simultaneous interaction among three
molecules, etc.
The Beattie-Bridgeman truncated virial equation (70), a variation of the virial
equation of state, was the first satisfactory equation of state for the quantitative
description of real-gas volumetric behavior. This equation was widely used for the
representation of gaseous pvT behavior until it was replaced by the Benedict-Webb-
Rubin (B\VR) equation (71).
Benedict and coworkers (71) modified the Beattie-Bridgeman equation to yield the
BWR equation:
[
Bo - (A) RT) - (C) RTJ )] Z = 1 +
v
(2.16)
The BWR EOS fits the pvT data of methane, ethane, propane, and n-butane; and helped
calculate density and other derived properties, such as enthalpy, fugacity, vapor pressure,
15
and latent heat of vaporization to high accuracy. The equation was initially applied to
mixtures of these four light components, and in 1951 was extended further to include
eight additional hydrocarbons up to n-heptane. In fact, since the BWR equation is
specialized (fine tuned for lower weight hydrocarbons), it is one of the more accurate
equations of state for these mixtures. For this reason, there are many variations for the
BWR equation; Cooper and GoldFrank (72), Orye (73), Morsy (74), Starling (75),
Nishiumi and Saito (76), Schmidt and Wagner (77), and the AGA natural gas equation by
Starling et al. (78). The AGA natural gas equation is a high precision EOS represented
by two sets of 52 terms, one set for pure gases and the other for mixtures. Each
expression is intended for custody transfer of pure gases or mixtures containing paraffins
(methane through n-decane, i-butane, i-pentane), nitrogen, oxygen, argon, carbon dioxide,
carbon monoxide, hydrogen, helium, and/or hydrogen sulfide.
Enthalpy Departures from Equations of State
A literature overview of enthalpy departure predictions using various equations of
state is presented in this section. The enthalpy departure predictions using BWR EOS
were compared to the Peng-Robinson EOS for a mixture of ethane and methane at 38.61
°C (10). The BWR equation gave superior results upon comparison with experimental
measurements (11). A modified BWR equation was compared in its enthalpy predictive
capability against RK, Beattie-Bridgeman and Lee-Edmister for a limited database of
methane, water and ammonia systems (12). The BWR equation was the most successful.
Enthalpy departure predictions are made for eleven nonpolar fluids using the most
16
general cubic EOS of Kumar and Starling. The equation was compared with the PR
equation and the three-parameter corresponding states Modified BWR (3PCS-MBWR)
equation of state (13). According to those comparisons, the most general cubic EOS is
more accurate than the PR equation and compares quite well with the three-parameter
corresponding states modified BWR (3PCS-MBWR) equation of state.
An evaluation of enthalpy departure prediction methods for nonpolar and polar
fluids was performed by Toledo et al. (14). The SRK, Mathias-SRK, PR, Ploecker-LeeKesler
(PLK), and the GCEOS (Skjold-Jorgensen, 1984) equations of state were tested
against a set of eleven-thousand experimental data points comprising 18 pure
components, 23 binaries, and 5 ternaries. The evaluation shows that the PLK and
GCEOS methods are significantly superior to the SRK and PR methods for single-phase,
nonpolar fluids, particularly in the vapor phase. These models perform similarly for polar
fluids but show larger deviations. The GCEOS model has been found to be a promising
method for both nonpolar and polar fluids. The predictions in the two-phase region by
any of these methods show higher deviations.
Similarly, enthalpy departure predictions were compared for a database comprising
four pure substances and two binary mixtures; the pure substances represented by
cyclohexane, nitrogen, octane and pentane; the binaries were varying compositions of npentane
- cyclohexane and n-pentane - n-octane mixtures (15). The van der Waals
(vdW)-711 EOS was compared to the PR equation. The PR EOS yielded much better
results. The explanation for the differences in performance may be attributed to the
presence of the covolume parameter, b, in the PR equation. Adachi and Sugie (16)
17
conclude that the covolume parameter is the controlling factor in enthalpy calculations.
According to their study, a cubic EOS which gives good pvT predictions is found to be
able to predict enthalpy departures well, and PR EOS is a good choice for this purpose.
A fairly comprehensive study involving the comparative capabilities of eleven
cubic equations of state was carried out for a representative sample of 2640 points for
paraffins ranging from methane to n-decane by Adachi, Sugie, and Lu (17). The PR EOS
gave lower deviations for enthalpy departure values compared to SW, SRK, HK, and ICL
(Ishikawa-Chung-Lu) equations of state. The PR EOS predictions were similar to those
of KS EOS. However, the ALS EOS yielded best enthalpy departure prediction results.
The TB EOS was compared in its enthalpy predictive capabilities to PR EOS using
a database involving methane, carbon dioxide, ammonia, and water (18). The PR EOS
proved to be more accurate than the TB EOS for all three components other than water.
In fact, for carbon dioxide the predicted values between the two equations of state
differed by almost lOO%. The PR EOS gave superior enthalpy predictions when
compared to the Trebble-Bishnoi-Salim, TBS EOS (19) for a database similar to the one
used by Trebble and Bishnoi (18).
It is evident from the above discussion that the modified BWR type equation is a
viable tool for calculating vapor-liquid equilibrium and departure functions for nonpolar
mixtures. However, since the middle 1970s, the SRK and PR equations have dominated
VLE and departure function calculations in the hydrocarbon industry. These equations
are surprisingly good for thermal property calculations, except in the critical region (20).
Albeit, the extended BWR equations are still preferred when volumetric and other
18
thermodynamic information of high accuracy are needed, since the cubic equations of
state do not represent volumetric data well.
Despite its success in correlating both gas and liquid light hydrocarbon mixtures,
the BWR equation poses certain disadvantages. The constants of BWR and its related
equations of state have been tabulated by a number of authors for various compounds
(57). These constants were mostly obtained from fitting experimental data at low to
moderate pressures, thus they cannot be easily extended to high pressure without
verification. Furthermore, these equations cannot be employed if the required pure
component constants are not available.
The equations of state described thus far are of an empirical nature and at best
semiempirical. Empirical equations are often useful; albeit, limited in their application.
Such equations must be applied in the range of temperature, density and composition
where experimental data exist; when empirical equations are extrapolated into regions
where no data are available, poor estimates often result. Further, since the equation-ofstate
parameters have little or no physical significance, it is often difficult to estimate
parameters for other fluids, and especially for mixtures. Theoretically based equations of
state address some of the problems mentioned above. Based on statistical
thermodynamics, these modem equations of state result from lattice models, perturbation
and integral equation theory, or from fitting computer simulation data. Among the
theoretically based equations, the perturbed-hard-chain theory in its various forms has
been in general use (58, 59). However, none of these equations have received the
acceptance of the simpler cubic equations of state, such as those of the Soave and PengRobinson
equations.
19
The simplified perturbed-hard-chain theory (SPHCT) EOS, the modified SPHCT
EOS and the PR EOS were evaluated for the prediction of calorimetric properties (60).
The results indicate that the abilities of the PR EOS, the original SPHCT EOS and the
modified model to predict calorimetric properties are similar to their comparative abilities
to predict volumetric properties. The evaluations were conducted using six pure fluids
covering the two-phase and the single-phase regions. The fluids considered were
methane, ethane, propane, benzene, carbon dioxide and water. These evaluations were
limited in scope and further examination will be required to assess the potential of these
equations.
Daubert (54) used the PR EOS for enthalpy departure predictions and comparisons
against selected enthalpy values in the OP A databank. The databank has since been
continually added to and maintained. The model predicted enthalpy departures very well
for light hydrocarbons and gases. although, the accuracy decreased for the heavier
hydrocarbons - pentane and above.
Similary, in this work it was decided to use the PR EOS enthalpy departure model
for data evaluation purposes and to study its predictive capabilities against the enthalpy
departure values in the OPA databank. The advantages of the PR enthalpy departure
model, which suited our requirements. are several. Specifically, the model is:
• Capable of handling multi phase natural gas systems over a wide range of temperature
and pressure conditions.
• Generalized and applicable to multi component systems with established mixing rules.
• Reasonably accurate with an acceptable speed of computation.
20
Excess Enthalpies and Equations of State
Thus far, enthalpy departure functions, which represent enthalpy changes as a result
of deviation from ideal gas behavior, have been discussed. It is a common occurrence
that when two or more pure compounds mix, 'excess' enthalpy is produced owing to
molecular interaction. Excess enthalpies are commonly denoted as hE, where the
superscript' E' signifies 'excess' property. At the same temperature and pressure, the
enthalpy of a binary "ideal mixture" would be:
(2.17)
where Yl and Y2 are the mole fractions for the pure components' l' and '2,' respectively;
and hi and hz are the molar enthalpies of the two components (55). On mixing, excess
enthalpy, hE is expressed as:
(2.18)
where hm (p, T,y) is the enthalpy of the resulting mixture at the same temperature and
pressure. In an ideal mixture, the excess enthalpy, also known as heat of mixing, is zero.
An ideal mixture is to be expected only when the molecular interaction, mass, and size
are sufficiently similar. Excess enthalpies are usually determined from calorimetry data.
The excess enthalpies of gaseous mixtures are, at times, correlated by means of
equations of state (20). The modified Martin-Hou (MH) EOS (1981) was used to
correlate the excess enthalpy of binary gaseous mixtures at pressures up to 100 atm (22).
Equations of state have also been used to correlate h /: in the liquid region. For instance,
Adachi and Sugie (1988) correlated h I: of the water-acetone system by means of a cubic
21
equation of state; Casielles et al. (1989) predicted hi: of a ternary system from binary
experimental data using the PR equation.
Reference States for Enthalpy Data
It is important to note the reference states employed in reporting enthalpy data. No
uniform reference states are currently used for reporting calorimetric property data.
Different sources employ different reference states. A discussion of the reference states
employed and the calculation paths used by two organizations respected for the integrity
of their compilations is given below. The GP A reference states and enthalpy departure
methods are discussed in Chapter III.
National Institute of Standards and Technology (NIST)
NIST (formerly NBS) uses non-analytic equations of state (25, 26) to derive
thermophysical data of pure fluids for a wide range of pressure and temperature
conditions. The general form of the equation of state is the same for all pure
hydrocarbons, but the density and temperature-dependent functions are substancespecific.
The constants in the equation of state are obtained by fitting the equation
specifically to available pressure-density-temperature (ppT) data for the fluid.
The calorimetric properties determination involves the computation of both the
ideal gas enthalpy difference and the enthalpy departure function. For ideal gas enthalpy
difference. an equation is developed to fit the available data on a given calorimetric
property as a function of temperature, T. The property chosen depends on the
22
availability of consistent data. From the equation for the chosen property, the other ideal
gas functions are evaluated by using the appropriate thermodynamic formulae.
The departure function expressions for change in internal energy, !!. U, change in
specific heat at constant volume, !!.C,., and related quantities are first evaluated along an
isotherm from zero density to the required p - T state. Then, M-l and !!.C p are computed
using the appropriate formulae. For the vapor phase, including saturated vapor, the above
method is used to evaluate the departure function. For the liquid phase, including
saturated liquid, M-l"ap is subtracted from the saturated vapor value and integration is
performed similar to the vapor phase.
The reference state chosen by NIST is U = 0 at the liquid triple point, obtained by
use of an arbitrary value for U (ro ,pO). For instance, according to Goodwin (25), nbutane
is assigned an internal energy U = 22644.306 llmol at 0 K and 0 Pa, to
accommodate a triple point U ( ro ,pO) of zero.
Thermodynamic Research Center-American Petroleum Institute
TRC-API44
According to TRC "Thermodynamic Tables-Hydrocarbons" (27), the ideal gas
enthalpy difference is computed by resorting to statistical mechanics. Statistical
thermodynamics along with spectroscopic data are used for the calculation of specific
heats. Specific heats, in tum, yield enthalpy values and other thermodynamic functions.
The predicted values are checked, when possible, against experimental data and, in some
cases, adjustments are made to achieve better agreement.
23
The departure functions estimation techniques have not been indicated. The
tabulated values are based on experimental measurements, estimation procedures or a
combination of both. The reference state used in computing the ideal gas thermodynamic
functions is a temperature of 0 K and a pressure equal to 1 bar.
The ideal gas thermodynamic properties for the elements used by TRC-API 44 are
similar to those in the JANAF Thermochemical Tables (28). Also, TRC-API 44 has used
symbols, units for thermodynamic properties and atomic masses of elements based on the
International Union of Pure and Applied Chemists (lUPAC) review of 1983 (29).
The literature review revealed that a consensus seems to have emerged for the need
to standardize the reference states employed for thermodynamic functions in general, and
enthalpy values in particular. Since 1975, all tabulated values are given in terms of
International System (SI) of units. Temperature and pressure values of 0 K and 1 bar (or
1 atm) appear to be emerging as de/acto standard reference states for thermodynamic
function computations. However, large volumes of enthalpy data still remain which are
based on different reference states.
24
CHAPTER III
GAS PROCESSORS ASSOCIATION (GPA)
ENTHALPY DATABASE
The GP A Enthalpy database contains enthalpy departure data for pure
components and mixtures of materials that include the following: paraffins to C16; alkyl
naphthenes to C9; aromatics to C\O; and nonhydrocarbons include nitrogen, carbon
monoxide, carbon dioxide, hydrogen sulphide, hydrogen, and water. The systems in the
database and the temperature and pressure conditions at which data are reported are of
particular interest to the natural gas processing industry. In this chapter, the GPA
enthalpy database format and holdings are presented. The sources of experimental data
and the experimental techniques employed by the investigators are discussed. Issues
involving thermodynamic consistency checks, smoothing of enthalpy data, and departure
conversion techniques are also reviewed.
GPA Enthalpy Database Format
The database format includes identification of the components, the mole fraction
of each component for each composition and the temperature. pressure, enthalpy, phase
(liquid, vapor, or two-phase), and the literature reference number for each data point (24).
25
The compound identification numbers follow the method used in the OPSA Engineering
Data Book. The database uses a single set of consistent units for reporting temperature,
pressure and enthalpy departure values. The unit for temperature is ,oF' , the pressure
values are reported in 'psia' , and the enthalpy departures have units of 'Btu/lb.' Each
data record also includes the phase code, departure method, literature reference number,
and a code identifying whether the data point is raw or smoothed.
The phase code specifies whether the point is in the liquid phase, vapor phase, or
the two-phase region. These are denoted by: 1 = liquid; 2 = vapor; 3 = two-phase; 4 =
liquid/two-phase and 5 = vapor/two-phase. Although, five different phase codes in the
database are used to describe all the data records, in reality only three phases are present.
These are: the liquid region (represented by phase codes' l' and '4'), vapor phase (phase
codes '2' and '5') and the liquid-vapor region denoted by phase code '3.' The departure
method is indicated by a letter (A, B, C, D), each of which denotes a procedure used to
process the original enthalpy values to their departure values. The four departure
methods are described in a later section in this chapter. For any data record, 'R' and'S'
are used to indicate raw or smoothed experimental values, respectively.
OP A Enthalpy Database Holdings
The enthalpy database holdings include twenty pure fluids, twenty-nine binary
mixtures, five ternary mixtures, one quaternary mixture and two multicomponent natural
gas systems at various compositions. Tables 3, 4, 5 and 6 in Chapter IV give a complete
listing of the current holdings in the database along with the molar compositions of all .
26
systems and the temperature and pressure ranges for the systems. The tables also include
the number of data points for each of the three phases and the corresponding literature
reference numbers. The listing of the reference numbers and the literature sources are
given in Appendix B.
Source of Enthalpy Data in the Database
All the enthalpy departure data in the database are based on experimental data
collected and compiled over a period extending almost twenty five years (mid 1960's to
1990). A fair amount of the data in the database was acquired through projects sponsored
by the GPA (Projects 661,722,731,741,742,792,811). However, the database also
draws heavily upon the extensive compilations of enthalpy data that were experimentally
obtained by Lenoir et al. (1967-1972) at the University of Southern California, Los
Angeles. The database also includes enthalpy data selected from other literature sources
(Appendix B has a complete listing).
Lenoir's Reference State for Ca10rimeteric Measurements
Lenoir et al. (1967-1972) used a reference state of 75 OF and the liquid-phase
condition while carrying out their experiments. This was accomplished by ensuring that
the hydrocarbon leaving the calorimeter was at 75 OF and in the liquid state. The fluid
flows through the calorimeter apparatus at a constant flow rate and has an inlet
temperature in the range of 150-700 OF. The upper range of the inlet temperature may be
lower or higher depending upon the specific hydrocarbon being used. The pressure was
27
usually varied from 0-1400 psia for all fluids in the flow calorimeter experiments. The
change in enthalpy was measured by the quantity of Freon-II evolved as heat transferred
in the calorimeter from the hydrocarbon fluid to the surrounding Freon-ll, which was
maintained precisely at its boiling point and 75 of.
All of the calorimetric experimental data collected by Lenoir et al. were rebased
from the reference state of 75 of, liquid-phase condition to the API Data Book (1966)
enthalpy datum level of saturated liquid at -200 OF. This was accomplished by adding a
constant value of enthalpy to the experimentally-obtained enthalpy values for each
system. This constant value of enthalpy for each system was determined by subtracting
the latent heat of vaporization from the ideal gas state at 75 of. The enthalpy value at the
ideal gas state at 75 of is taken relative to the API reference state of -200 OF from the API
Data Book (1966). The latent heats of vaporization data were referenced to different
sources for different compounds (30-33). In certain cases, for instance, the cis-2-pentene
calorimetric measurements, instead of using the API Data Book value for ideal gas
enthalpy and the heat of vaporizatIOn data, the rebasing from 75 of to the -200 of state
was carried by graphical integration of heat capacity values from Todd et al. (34). For the
binary mixtures. ideal gas enthalpies from the API Data Book were computed as a
weighted average of the two pure component values. Also, heats of mixing data were
used where applicable. The heats of mixing data, in turn, were referenced to different
sources (35-37).
The original enthalpy measurements, relative to 75 OF and the associated
pressures, are deposited with the American Society of Information Science (ASIS).
Having converted all experimental enthalpy data to the -200 OF basis, the data were
28
plotted on large-scale cartesian coordinate graph paper, and smoothing was performed by
visually drawing an appropriate average curve through the plotted values (38). The final
smoothed enthalpy values were presented in tabular form.
GP A Reference States for Enthalpy Data
Prior to 1974, GPA enthalpy research projects have resulted in the publication of
experimental pure component and mixture enthalpies for systems of interest to the gas
industry (39, 40). However, most of the published data were based on different reference
states. To eliminate this variation in the reported data, the GPA enthalpy data were
converted to two common reference states: the ideal gas state at 0 K, and the elemental
states at 25°C. In 1974, Cochran and Lenoir (GPA Project 733) devised techniques for
data conversion to the above two reference states (23). All the GPA sponsored enthalpy
projects from 1974 on have reported experimental enthalpy values referred to the above
two states (41-46).
Thermodynamic Consistency Checks for GP A Enthalpy Data
GPA reports (39, 40, 41, 47) have presented experimental enthalpy values as
isothermal and isobaric enthalpy differences. This means of compiling data aided in
testing for thermodynamic self consistency. Isothermal enthalpy differences were
measured at two temperatures on each system studied. Measurements at the second
temperature were made to permit an evaluation of the data around closed loops. Since the
enthalpy is a state property, the sum of all changes around a closed loop should add to
29
zero. Experimental measurements will not close perfectly due to experimental
uncertainties, and the lack of closure is a measure of the accuracy of data. For
comparitive purposes, the lack of closure is calculated as percentage error by dividing the
residual difference times 100 by the sum of the absolute value of all the enthalpy changes
around the closed loop. As an example, Figure 2 gives a pictorial representation of loop
closure for experimental enthalpy data on 50 mole % hydrogen - 50 mole % ethane (40).
1000
PRESSURE,
psia
200
-40
+3091
-4277
+7342
100
TEMPERATURE, "F
Figure 2. Pictorial Representation of loop closure for
enthalpy data on 50-50 mole% H2S-Ethane
The isothermal and isobaric data were tested for consistency by making loop
checks between temperature and pressures. From Figure 2 above:
a) At 200 psia, from 100 of to -40 of (MI)pl = - 7342 Btullb-mole
b) At -40 of, from 200 to 1000 psia, (MI)n = + 42 Btullb-mole
c) At 1000 psia, from -40 of to + 1 00 of, (MI)p2 = + 3091 Btullb-mole
d) At 100 of, from 200 to 1000 psia. (MI)T2 = + 4277 Btullb-mole
Summation around the closed loop = + 68 Btullb-mole
30
Out of a total enthalpy change around the loop of 14,752 Btuilb-mole, + 68 Btuilb-mole
represents an error of + 0.5%. Such loop closures were performed on GPA enthalpy data,
wherever possible. Set criterion was employed to accept or reject the enthalpy data
analyzed. Typical values of absolute errors in the loop closure calculations for all the
smoothed enthalpy data (39,40,41,47) were between 0 - 1%. A negative value for the
error % indicated a lack of closure while a positive value represented "excess of closure."
Smoothing of GPA Enthalpy Data
The GPA reports most of its enthalpy measurements on isotherms and isobars (39,
40, 41, 47). These data provide direct isothermal information for checking and evaluating
new or existing equations of state or other correlations for predicting enthalpy.
Isothermal data are best suited for this because an equation of state predicts the deviation
from ideal gas behavior as a function of pressure at the system temperature. Isothermal
data can be compared directly with the equation of state predictions.
Smoothing of experimental data can be carried out using several methods. One
would be to plot the data directly on a grid and to smooth it graphically. Indeed, this was
the method employed by Lenoir et al. (38) to perform smoothing of their data. However.
the drawback to this method is that it is difficult to plot data on a reasonable scale with an
accuracy much better than ± 1 %; thus, this procedure suffers from lack of precision, both
in plotting the experimental data and in reading the smoothed data from the plot.
Another method would be to select an analytical function which approximates the
data; plot deviations between the measured and calculated data; draw a smooth curve
31
. ,
through the deviations; and then correct the calculated data by means of the smoothed
deviation curve. This method has the advantage in that the deviations can be plotted on a
much larger scale. so that plotting and reading errors become insignificant compared to
scatter in the experimental data. The GPA relied on this method due to its inherent
advantages.
The GPA chose a modified Redlich-Kwong (RK) equation-of-state (MARK V
program) as the analytical function for smoothing the individual isotherms. The ideal gas
heat capacity constants used were taken from the API Data Book (1966, 1982).
Deviation plots were prepared showing deviation in Btullb versus pressure (in psia) for
the different systems. The correction factor was then applied to the calculated enthalpies
by use of the following relationship:
H = H +(H - H ) smooth calc, meas calc smoothed
Conversion of GP A Enthalpy Data to the
Standard GP A Reference State
(3.1 )
The GPA research report RR-11 (23) describes the two standard GPA reference
states for enthalpy data. The datum levels chosen were:
a) the ideal gas state at absolute zero, and
b) the elemental states at 25 DC.
All the GPA enthalpy data (39, 40, 41, 47), including both raw and smoothed, had to be
adjusted to the above two reference states. The original GPA data obtained were referred
to different temperature and pressure states. For instance. the reference states used while
32
obtaining experimental enthalpy data for certain systems in RR-6 (39) are presented in
Table 1.
As observed, not only different systems. but the same system with different
compositions are each based to a different reference state. To convert all of these data to
System
Methy 1cyclohexane
Methane
n-Heptane
Methane
n-Heptane
Methane
n-Heptane
Table I
Experimental Enthalpy Reference States
for Systems in GPA-RR 6
Mol. Fraction Phase Temperature (OF) Pressure (psia)
1.0 liquid -100 SO
0.25 liquid -100 200
0.75
0.5 liquid -100 600
0.5
0.95 liquid -100 2500
0.05
the ideal gas state at absolute zero involved rebasing on both the pressure and temperature
axes.
All of the enthalpy data were translated from their original pressure states to the
zero-pressure ideal gas state. GPA RR-ll (39) describes the conversion of enthalpy data
for gases to the zero pressure level. The relation used for rebasing enthalpy data for
pressures up to SO psia to the ideal gas state involved the use of virial coefficients
documented in the literature (48). The relation used was:
33
( dB) H-H =1 B-T- P
o \ dT
(3.2)
where B is the second virial coefficient. Experimental values were not available for all
components. nor for all needed temperature ranges, so an alternative general correlation
for predicting B was needed. The correlation of Pitzer and Curl (49) was selected.
RI;.. [ 0.33 0.1385 0.0121
B = - 0.1445r- - - T2 - 3
Pc R R TR
(3.3)
Equation (3.3) is quite accurate except at very low reduced temperatures, and it can be
reliably used to 50 psia. Having rebased enthalpy data up to 50 psia accurately, the
measured enthalpies were based further on the ideal gas, 0 K basis. The API -44 heat
capacity constants were used for computing ideal gas enthalpies based to the 0 OR state.
These values, in tum, were used to perform the rebasing to the 0 K state.
For data tabulations that use a pressure reference state higher than 50 PSIa,
Starling (50) extrapolated the highest temperature isotherm to zero psia; this extrapolated
zero psia value was then used as the basis for calculating ideal gas enthalpy values at all
other temperatures using ideal gas enthalpies from the API Research Project 44 values.
Once all the data were converted to a single standard reference state, the tabulated
enthalpy departure values in the GPA enthalpy database result as the difference between
the enthalpy value at a particular pressure coordinate for an isotherm and the ideal gas
(zero-pressure) value for the same isotherm.
34
OPA Departure Conversion Techniques
OP A converted its entire database of experimental enthalpy values, related to
different reference states, to departure values in 1989-1992 (GP A Project 822). This
section summarizes the four departure calculation techniques that were employed by the
investigators for enthalpy data conversions (24).
Method A
This method is used to convert Lenoir's (23) data. The departure function H Dep is
defined as:
where
H Dep = H ( T, p ) - H ( T, pO)
= H(T,P)Len +[ H(T, p O)API- H(T,pO)Len]- H(T,pO)API
H (T, P )Len = Lenoir's experimental value
[H (T,pO )API - H (T,pO )Len] = enthalpy difference between Lenoir's
experimental base and the API base of -200 of,
saturated liquid
35
Method B
This method is used for data requiring only the ideal gas enthalpy H ( T, pO) for
determining the departure value, i. e., H (YO , p" ) = O. H (T, pO) is either given by the
author or calculated from the API Technical Data Book ideal gas enthalpies.
Method C
This method is used for trans-decalin, cis-decalin and tetralin. H Dep is defined as:
where
[HT - H75L ] = experimental value
[H:5V - H;5] = pressure effect at 75 of
[H75V - H75L ] = heat of vaporization at 75 of
= ideal gas value at T
= ideal gas value at 75 OF
Method D
This method is same as Method C, except for the way the ideal gas values are calculated.
It is used for only cis-2-pentene.
36
CHAPTER IV
EQUATION-OF-STATE ENTHALPY
DEPARTURE MODEL EVALUATIONS
This chapter includes a description of the Peng-Robinson CPR) equation of state
(EOS), and the enthalpy departure expression derived from it. The reasons for choosing
the PR EOS for this study were discussed in Chapter II. The model evaluations
performed along with the database holdings are presented in tabular form.
The Equation-of-State Model
The PR EOS was used in this study (51):
(4.1)
where
b = 0.07780 R ~/ Pc (4.2)
(4.3)
(4.4)
m = 0.37464 + 1.54226 0; - 0.26992 0;2 (4.5)
The mixing rules employed are:
Q = LLz;zJQij (4.6)
I j
37
b = LLZjZjb'l (4.7)
, 1
where
(4.8)
(4.9)
1
-2 (b +b) 1/ JJ
(4.10)
e'l and D'j are adjustable, empirically-determined binary interaction parameters which
characterize the binary formed by component i and component j .
Peng-Robinson Equation-of-State
Enthalpy Departure Function
From exact thermodynamics, the difference between the enthalpy of a pure
compound fluid and the enthalpy of an ideal gas at the same temperature is given by
<Xl
H - HO = J [p- T(£p/OT)v]dv + pv - RT (4.11)
y
Equation (4 .11) is also obtained by adding Equations (2.9, 2.10 and 2.11) presented in
Chapter II, and substituting for the compressibility factor, Z = pv , in Equation (2.11).
RT
When the PR equation is used to determine the integral of Equation (4. 11), we
obtain,
H-HO = ac [(a-Tda/dT)l2.J2b] In [(v-OA14b)/(v+2.414b)]
+ pv - RT (4.12)
38
with
dajdT =(-aj~) (m~-D5)/ [l+m(l- ~05)] (4.13)
Equations (4.12) and (4.13) are easily extended for multicomponent mixtures when used
in conjunction with Equations (4.9) and (4.10). For a multicomponent system,
with
H - W ~ [( tt.z,Zj YJ - aJ / 2v'2b] In [( v -O.414b )/( v + 2.414b)]
+ pv - RT
r iJ = aij {(-0.5miTn-05/Tci[1+mi(1-T:·5)]
- ( -O.5m j Try-05 )/ Tcj [ 1 + mj ( 1 - T;5)]}
A detailed derivation for the PR departure functions is given in Appendix A.
Pure Fluid Properties
(4.14)
(4.15)
The pure component properties which include the critical temperature ~, critical
pressure Pc' and the acentric factor (j) constitute the input variables for the PR equation
enthalpy departure model. Also, the molecular weights of the substances were needed to
report enthalpy departures on a unit mass basis. The pure fluid values of ~, Pc' and OJ
used are those given by Daubert (24). However, the pure fluid properties for cis-2-
pentene, ethylcyc1ohexane, cis-decalin and trans-decalin, tetralin, and hexadecane were
taken from Reid et al. (53). Appendix C contains a listing of the pure fluid properties used
for the present study.
39
Software Used
The enthalpy departure model was incorporated into the 'GEOS' program; GEOS is
an elaborate thermodynamic software for calculating volumetric, phase equilibrium and
calorimetric properties (52). The software has the capability to handle multiple systems
simultaneously.
The program inputs needed for performing the enthalpy departure predictions and
making comparisons with the experimental enthalpies included the pure fluid critical
properties, temperature, pressure, feed composition, experimental enthalpy departures as
reported in the GPA database, and the option to calculate vapor or liquid enthalpy. The
GP A database phase codes discussed in Chapter III are used for determining liquid or
vapor enthalpy options. In the enthalpy departure predictions carried out, the "raw
ability" of the PR EOS was employed, in that, the mixing rules with no interaction
parameters were used (Cij = 0, Dij = 0).
To validate the accuracy of the GEOS software, enthalpy departures generated by
GEOS were compared to similar predictions by the ASPEN PLUSTM simulator. A test
system involving the ethane-propane mixture (76.3 mol % ethane) was selected from the
GP A database for the purpose. The results are shown in Table 2.
As observed from Table 2, the predicted enthalpy values obtained using GEOS to
those generated by ASPEN PLUS were almost identical. The slight differences, which are
generally -v.;thin 0.1 %, may be attributed to differences in the pure fluid critical properties
employed by ASPEN PLUSTM for purposes of enthalpy departure prediction.
40
Table 2
Enthalpy Departure Comparisons
Pt. Temp. Pressure Exp. Cal. (ASPENTM) Cal. (GEOS) Phase
No. "F pSla BtuJlb BtuJlb BtuJlb
-280.0 250.0 -244.5 -236.93 -237.09 liq
2 -200.0 250.0 -225.2 -22l.19 -22l.32 liq
3 68.0 500.0 -15l.7 -149.46 -149.56 liq
4 80.0 500.0 -146.0 -142.17 -142.27 liq
5 115.0 716.0 -125.3 -118.04 -118.24 liq
6 -40.0 2000.0 -183.0 -183.41 -183.50 liq
7 80.0 250.0 -17.7 -17.71 -17.74 vap
8 240.0 250.0 -9.8 -10.70 -10.72 vap
9 152.0 500.0 -30.1 -30.96 -3l.02 vap
10 251.0 1000.0 -43.3 -47.15 -47.23 vap
Model Evaluations
Tables 3, 4, 5 and 6 give summary reports of error statistics for the PR EOS data
screening results, along with the GP A enthalpy database holdings. The model evaluations
encompassed twenty pure fluids, thirty binary mixtures, five ternary mixtures and three
multicomponent enthalpy systems. In all, around fifteen-thousand data records were
included in the evaluations. For each system considered, the absolute average deviation
(AAD) and the root mean square error (RMSE), both expressed in Btu/lb, is given for the
vapor and liquid phases.
Daubert (24) had used the PR model to predict enthalpy departure functions and
make comparisons with the enthalpy departure values in the GP A databank. However, the
41
mixing rules employed in his evaluations were different from those used for the present
study. Daubert relied on pseudo-critical fluid properties, derived using Kay's mixing rules
for the critical properties (56), for binary and multi component natural gas systems. In this
study, pure fluid properties in conjunction with mixing rules, specified earlier in this
chapter, were used to calculate fluid mixture properties. The model statistics resulting
from this work are similar to Daubert's work for a fairly large number of gas systems in
the GP A databank. Therefore, Kay's pseudo-component mixing rules and the mixing
rules used in this chapter result in comparable predictions.
The results obtained indicate that the accuracy of enthalpy predictions is superior for
lower molecular weight hydrocarbons. For example, ethane (liquid-phase AAD = 2.1
Btullb, vapor-phase AAD = 1.8 Btullb) had lower deviations compared to propane (liquid-phase
AAD = 2.6 Btullb, vapor-phase AAD = 4.1 Btullb).
There is no apparent difference in the ability of the PR model to predict vapor-phase
and liquid-phase enthalpy departure values. For certain systems, the liquid-phase
predictions might be marginally better than the vapor-phase, and for others the opposite
may be true. For example, n-octane displays an AAD of l.7 Btullb and RMSE of 2.4
Btullb in the liquid-phase; and for the vapor-phase it shows an AAD and RMSE of 3.4
Btullb and 4.0 Btullb, respectively. In contrast, benzene has a liquid-phase AAD of 4.0
Btu/lb and a RMSE of 4.7 Btullb; and in its vapor-phase, 1.4 Btu/lb and 2.2 Btu/lb are the
observed AAD and RMSE values.
The evaluations revealed that the PR enthalpy departure model, in general,
performed similarly for the pure fluids, binary fluid systems and other multi component
42
)
•I
systems. The model statistics (RMSE and AAD values) reported in Tables 3, 4, 5 and 6
for the various systems in the liquid and vapor phases support these observations.
Again, there was no difference in the PR model's ability to predict enthalpy
departures based on the temperatures. Depending upon the pure fluid system or mixture
under consideration, the deviations were lower for lower temperature isotherms and, at
other times, the opposite was true.
Also, on examining the point-by-point error analyses for each data set, it was
observed that the predictions, as expected, were not accurate in the critical region. This
was true of almost all the enthalpy systems in the database.
43
. I'I
i I I '
Table 3
Peng-Robinson EOS Error Analyses: Pure Fluids
Sys Components Mole Temp. Press. Expt. No. of Absolute Deviation (Btullb) Departure References
No. Fractions Range Range Phase Points Method
(F) (psi a) AVE RMSE Raw Smooth
Methane 1.0 -250.0 250.0 La 14 1.4 1.8
50.0 2000.0 vb 25 2.5 3.1 578
L-VC 0 NRDd 573
2 Ethane 1.0 -250.0 200.0 L 41 2.1 2.8
-500 3000.0 V 28 18 2.3
L-V 0 NRD 592
+0- +0- 3 Propane 1.0 -250.0 200.0 L 40 4 1 55
400.0 2000.0 V 21 2.6 4.4 564
L-V 0 NRD 590
4 n-Pentane 1.0 95.9 15.2 L 142 2.4 3.4
691.5 1400.0 V 253 1.7 2.4 458
L-V 14 663 585
5 n-Heptane 1.0 361.8 50.0 L 105 7 I 7.6
548.6 100.0 V 52 2.8 3.4 556
L-V 157 665 666
6 n-Octane 1.0 150.8 15.2 L 162 1.7 2.4
600.3 1400.0 V 85 3.4 4.0
L-V 7 A,B 663 586
7 iso-Octane 1.0 188.4 290.0 L 18 1.6 1.8
476.4 1450.0 V 0
L-V 0 B NRD 687
-~
Table 3
Peng-Robinson EOS Error Analyses: Pure Fluids (continued)
Sys Components Mole Temp. Press. Expt. No. of Average Deviation (Btullb) Departure References
No. Fractions Range Range Phase Points Method
IF) (psi a) AVE RMSE Raw Smooth
8 n-Hexadecane 1.0 199.5 25.0 L 82 2.3 4.2
657.0 1400.0 V 21 4.3 4.6
L-V 5 584 583
9 CycIohexane 1.0 117.0 15.4 L 133 2.2 2.8
689.0 1400.0 V 181 2.4 3.1
L-V 8 584 677
V...I 10 Methyl- 1.0 1760 17.0 L 122 6.4 7.2
CycIohexane 464.0 1365.0 V 46 5.5 6.2
L-V 0 592 676
11 Ethyl- 1.0 224.4 290.0 L 21 8.2 9.4
CycIohexane 584.4 1450.0 V 0
L-V 0 B NRD 687
12 cis-2-Pentene 1.0 159.8 20.0 L 69 2.7 3.3
448.7 1400.0 V 210 2.5 3.6
L-V 26 581 678
I3 Benzene l.0 200.0 100.0 L 118 4.0 4.7
696.5 1400.0 V 243 1.4 2.2 584
L-V 14 581 679
14 Toluene 1.0 140.0 17.0 L 107 2.9 3.3
464.0 1365.0 V 38 2.3 2.6
L-V 0 592 676
~
Table 3
Peng-Robinson EOS Error Analyses: Pure Fluids (continued)
Sys. Components Mole Temp. Press. Expt. No. of Average Deviation (Btu/lb) Departure References
No Fractions Range Range Phase Points Method
('F) (psi a) AVE RMSE Raw Smooth
15 Ethyl-Benzene 1.0 350.4 22.0 L 33 1.5 2.1
485.4 80.0 V 12 0.8 0.9
L-V 0 B NRD 687
16 cis-Decalin 1.0 149.0 25.0 L 44 7.1 7.9
595.9 1400.0 V 32 3.2 3.8
L-V 6 580 582
+-- 17 trans-Decalin 1.0 120.1 22.0 L 124 3.2 4.5
0-
644.2 1400.0 V 50 3.9 4.5
L-V 14 580 582
18 Tetralin 1.0 IOU 25.0 L 90 3.3 4.9
677.3 1400.0 V 36 1.4 1.6
L-V 58 580 675
19 H2S 1.0 80.3 145.0 L 0
440.3 4351.0 V 61 1.0 1.4
L-V 0 B NRD 686
20 Nitrogen 1.0 -250.0 200.0 L 7 0.8 1.0
-50.0 3000.0 V 48 1.1 1.2
L-V 0 NRD 587
a 'L' refers to data points represented by phase code = 1 & 4 in the database
b 'V refers to data points represented by phase code = 2 & 5 in the database
c 'L-V refers to data points represented by phase code = 3 in the database
d 'NRD' = No raw data
"-' '''~
Table 4
Peng-Robinson EOS Error Analyses: Binary Mixtures
Sys. Components Mole Temp. Press. Expt. No. of Average Deviation (Btullb) Departure References
No Fractions Range Range Phase Points Method
c<>F> (psia) AVE RMSE Raw Smooth
Methane 0.948 -280.0 14.7 La 167 1.3 1.5
Propane 0.052 230.0 2000.0 v" 252 2.4 2.9
L-V 57 B NRDd 570
2 Methane 0.883 -280.0 250.0 L 191 1.6 2.0
Propane o 117 300.0 2000.0 V 213 3.0 32
L-V 65 B NRD 661
~
-..J 3 Methane 0.72 -280.0 250.0 L 210 2.1 2.6
Propane 0.28 300.0 2000.0 V 175 22 2.5
L-V 84 B NRD 573
4 Methane 0.494 -280.0 250.0 L 252 3.6 4.6
Propane 0.506 300.0 2000.0 V 136 0.9 1.0
L-V 84 B NRD 588
5 Methane 0.234 -280.0 250.0 L 322 3.4 4.5
Propane 0.766 300.0 2000.0 V 103 1.1 1.3
L-V 47 8 NRD 590
6 Methane 0.949 90.0 500.0 L 0
Propane 0.051 200.0 2000.0 V 12 4.8 5.3
L-V 0 8 NRD 591
7 Methane 0.874 90.0 500.0 L 0
Propane 0.126 200.0 2000.0 V 12 3.7 3.9
L-V 0 8 NRD 591
Table 4
Peng-Robinson EOS Error Analyses: Binary Mixtures (continued)
Sys. Components Mole Temp. Press. Expt. No. of Average Deviation (Btullb) Departure References
No. Fractions Range Range Phase Points Method
[1<) (psi a) AVE RMSE Raw Smooth
8 Methane 0.951 -100.0 50.0 L 6 9.1 8.8 667
n-Heptane 0.049 500.0 2000.0 V 79 1.1 1.6 592
L-V 50 NRD 666
9 Methane 0.491 -100.0 50.0 L 15 5.4 6.1 667
n-Heptane 0.509 600.0 2500.0 V 44 2.9 4.7 592
L-V 76 NRD 666
+0- lD
00
Methane 0.249 -lDO.O 50.0 L 90 3.7 11.7 667
n-Heptane 0.751 600.0 2500.0 V 35 3.7 4.1 592
L-V 10 NRD 666
II Ethane 0763 -280.0 250.0 L 60 3.0 3.9
Propane 0.237 251.0 2000.0 V 37 1.5 1.9 592
L-V 17 B NRD 671
12 Ethane 0.498 -280.0 250.0 L 48 3.2 4.7
Propane 0.502 300.0 2000.0 V 28 2 3 4.6 592
L-V 15 B NRD 671
I3 Ethane 0.276 -280.0 500.0 L 26 2.4 2.5
Propane 0.724 300.0 2000.0 V 23 1.9 2.3 592
L-V 10 B NRD 671
14 Propane 0.43 111.0 65.0 L 14 4.3 5.3
iso-Pentane 0.57 358.0 1400.0 V 21 7.2 8.4
L-V 20 A 579 NSDc
Table 4
Peng-Robinson EOS Error Analyses: Binary Mixtures (continued)
Sys. Components Mole Temp. Press. Expt. No. of Average Deviation (Btullb) Departure References
No. Fractions Range Range Phase Points Method m (psia) AVE RMSE Raw Smooth
15 n-Pentane 0.809 75.0 200.0 L 34 2.3 3.0
n-Octane 0.191 605.1 1400.0 V 81 2.9 3.5
L-V 22 A,B 663 663
16 n-Pentane 0.597 75.0 200.0 L 53 3.0 3.4
n-Octane 0.403 605.1 1400.0 V 75 40 4.6
L-V 34 A,B 663 663
~ '" 17 n-Pentane 0.392 75.0 15.2 L 53 2.5 3.1
n-Octane 0.608 601.7 1400.0 V 64 4.4 5.0
L-V 21 A,B 663 663
18 n-Pentane 0.218 75.0 15.2 L 67 1.9 2.4
n-Octane 0.782 605.1 1400.0 V 81 3.0 3.4
L-V 14 A,B 663 663
19 n-Pentane 0.167 198.6 25.0 L 28 3.4 4.0
n-Hexadecane 0.833 618.8 1400.0 V 2 2.9 2.9
L-V 32 A, B 584 583
20 n-Pentane 0.386 148.0 25.0 L 39 3.0 3.5
n-Hexadecane 0.614 619.2 1400.0 V' 16 3.0 3.3
L-V 78 A,B 584 583
21 n-Pentane 0.587 138.6 25.0 L 41 102.8 632.5
n-Hexadecane 0.413 625.5 1400.0 V 16 1.6 1.8
L-V 101 A,8 584 583
Table 4
Peng-Robinson EOS Error Analyses: Binary Mixtures (continued)
Sys. Components Mole Temp. Press. Expt. No. of Average Deviation (Btullb) Departure References
No. Fractions Range Range Phase Points Method m (psia) AVE RMSE Raw Smooth
22 n-Pentane 0.794 117,0 25.0 L 60 2.6 2.4
n-Hexadecane 0.206 625.1 1400.0 V 31 2.4 8.1
L-V 152 A,B 584 583
23 Methane 0.5 -100.0 50.0 L 68 8.0 14.1
Methyl- 0.5 600.0 2500.0 V 36 3.5 4.1 667
Cyc10hexane L-V 3 NRD 592
Vl
0 24 n-Pentane 0.197 122.6 15.4 L 102 2.8 3.2
Cyclohexane 0.803 696.0 1400.0 V 177 2.7 3.7 580
L-V 28 A,B 584 677
25 n-Pentane 0.385 119.9 100.0 L 102 1.6 2.0
Cyc10hexane 0.615 704.7 1400.0 V 176 2.2 2.8 580
L-V 25 A,B 584 677
26 n-Pentane 0.612 127.4 100.0 L 92 2.6 3.1
Cyc10hexane 0.388 695.8 1400.0 V 197 2.1 2.9 580
L-V 23 A,B 584 677
27 n-Pentane 0.793 141.4 25.0 L 85 2.6 3.3
Cyc10hexane 0.207 696.0 1400.0 V 244 2.1 2.9 580
L-V 24 A,B 584 677
28 n-Pentane 0.502 331.3 300.0 L 43 3.6 4.4
cis-2-Pentene 0.498 450.2 1400.0 V 116 1.7 2.4
L-V 12 B,C 581 678
Table 4
Peng-Robinson EOS Error Analyses: Binary Mixtures (continued)
Sys. Components Mole Temp. Press. Expt. No. of Absolute Deviation (Btu/lb) Departure References
No Fractions Range Range Phase Points Method
tF) (psia) AVE RMSE Raw Smooth
29 Methane 0.5 -100.0 50.0 L 40 21.9 24.1
Toluene 0.5 600.0 2500.0 V 33 0.8 1.1 667
L-V 62 B NRD 592
30 Propane 0.252 200.0 200.0 L 9 4.0 4.5
Benzene 0.748 400.0 1000.0 V 2 3.7 4.8
L-V 2 B 564 NSD
l.Ji 31 Propane 0.498 200.0 200.0 L 7 3.3 3.4
Benzene 0.502 400.0 1000.0 V 2 0.9 12
L-V 5 B 564 NSD
32 Propane 0.797 200.0 200.0 L 3 4.6 6.0
Benzene 0.203 400.0 1000.0 V 5 2.3 2.9
L-V 4 B 564 NSD
33 n-Pentane 0.186 152.0 15.2 L 68 2.4 3.5
Benzene 0.814 694.1 1400.0 V 210 2.1 2.7 581
L-V 40 A,B 584 681
34 n-Pentane 0.4 150.0 25.0 L 98 2.0 2.5
Benzene 0.6 695.0 1400.0 V 257 2.9 3.9 581
L-V 48 A,B 584 681
35 n-Pentane 0.594 152.5 25.0 L 69 29 3.5
Benzene 0.406 695.0 1400.0 V 234 2.9 3.8 581
L-V 43 A,B 584 681
"___-
'-~--
Table 4
Peng-Robinson EOS Error Analyses: Binary Mixtures (continued)
Sys. Components Mole Temp. Press. Expt. No. of Absolute Deviation (Btullb) Departure References
No. Fractions Range Range Phase Points Method
(<>r) (psi a) AVE RMSE Raw Smooth
36 n-Pentane 0.801 151.8 25.0 L 60 2.1 2.8
Benzene 0.199 692.9 1400.0 V 234 2.5 3.1 581
L-V 34 A,B 584 681
37 n-Pentane 0322 149.6 20.0 L 85 4.3 4.8
trans-Decalin 0.678 597.8 1400.0 V 45 4.8 5.7
L-V 112 D,B 580 582
Ul 38 n-Pentane 0561 148.9 20.0 L 64 3.5 4.3
N trans-Decalin 0.439 599.3 1400.0 V 41 4.6 5.7
L-V 154 C,B 580 5lS2
39 n-Pentane 0.725 118 9 30.0 L 60 2.8 3.5
trans-Decalin 0.275 598.6 1400.0 V 78 3.1 3.5
L-V 136 C,B 580 582
40 n-Pentane 0.884 100.0 20.0 L 76 2.3 2.6
trans-Decalin 0.116 599.4 1400.0 V 88 1.8 2.5
L-V 134 C,B 580 582
41 n-Pentane 0.197 100.9 25 .0 L 82 4.3 5.7
Tetralin 0.803 696.0 1400.0 V 77 21.8 38.8
L-V 89 C,B 580 675
42 n-Pentane 0.399 148.1 25.0 L 40 4.1 5.2
Tetralin 0.601 676.9 1400.0 V 46 4.8 5.8
L-V 125 C,B 580 675
Table 4
Peng-Robinson EOS Error Analyses: Binary Mixtures (continued)
Sys. Components Mole Temp. Press. Expt. No. of Absolute Deviation (Btullb) Departure References
No Fractions Range Range Phase Points Method
(Op) (psi a) AVE RMSE Raw Smooth
43 n-Pentane 0.588 122.5 25 .0 L 39 5.7 6.2
Tetralin 0.412 636.0 1400.0 V 59 5.3 6.1
L-V 146 C,B 580 675
44 n-Pentane 0.795 120.0 25.0 L 50 4.2 4.4
Tetralin 0.205 639.6 1400.0 V 104 4.0 4.6
L-V 138 C,B 580 675
Vt 45 n-Pentane 0.893 119.9 25.0 L 79 2.4 2.6
w Tetralin 0.107 638.0 1400.0 V 158 3.6 4.9
L-V 108 C,B 580 675
46 Benzene 0.93 367.2 200.0 L 71 4.2 5.1
n-Octane 0.07 596.5 1400.0 V 88 3.5 13.4
L-V 36 A,B 581 679
47 Benzene 0.857 179.2 20.0 L 97 4.8 5.8
n-Octane 0.143 596.6 1400.0 V 103 2.8 3.4
L-V 44 A, B 581 679
48 Benzene 0.771 368.5 200.0 L 56 4.6 5.6
n-Octane 0.229 596.5 1400.0 V 78 2.6 3.5
L-V 23 A,B 581 679
49 Benzene 0.446 158.5 20.0 L 132 4.0 5.0
n-Octane 0.554 598.2 1400.0 V 89 4.2 4.9
L-V 53 A,B 581 679
'~", - l6.~,,".L ~ ... .!~ P ...............
Table 4
Peng-Robinson EOS Error Analyses: Binary Mixtures (continued)
Sys. Components Mole Temp. Press. Expt. No. of Absolute Deviation (BtuJIb) Departure References
No. Fractions Range Range Phase Points Method
(~) (psia) AVE RMSE Raw Smooth
50 Benzene 0.271 157.6 20.0 L 100 1.6 3.4
n-Octane 0.729 598.7 1400.0 V 100 3.8 4.5
L-V 46 A,B 581 679
51 Benzene 0.963 150.2 20.0 L 84 6.2 6.9
n-Hexadecane 0.037 597.1 1400.0 V 124 5.5 6.9
L-V 71 A,B 580 682
Ul 52 Benzene 0.92 150.8 20.0 L 65 8.6 9.6
~ n-Hexadecane 0.08 595.0 1400.0 V 69 6.9 7.9
L-V 103 A,B 580 682
53 Benzene 0.814 150.7 20.0 L 64 10.2 11.4
n-Hexadecane 0.186 595.4 1400.0 V 20 9.6 10. I
L-V 120 A,B 580 682
54 Benzene 0.67 151.8 20.0 L 66 10.9 12.4
n-Hexadecane 0.33 594.0 1400.0 V 9 10.2 10.4
L-V 97 A,B 580 682
55 Benzene 0.419 148.2 20.0 L 60 7.6 9. I
n-Hexadecane 0.581 601.5 1400.0 V 8 6.3 6.4
L-V 62 A,B 580 682
56 Benzene 0.21 I 470.2 400.0 L 59 2.0 2.8
Cyclohexane 0.789 579.2 1400.0 V 152 3.2 4.5
L-V 37 A,B 581 680
....,;.~ .. ..-"-'
Table 4
Peng-Robinson EOS Error Analyses: Binary Mixtures (continued)
Sys. Components Mole Temp. Press. Expt. No. of Absolute Deviation (Btu/lb) Departure References
No Fractions Range Range Phase Points Method
(~ (psi a) AVE RMSE Raw Smooth
57 Benzene 0.334 193 .6 20.0 L 68 1.2 1.5
Cyclohexane 0.666 578.7 1400.0 V 128 3.0 4.5
L-V 37 A,B 581 680
58 Benzene 0.812 447.2 400.0 L 80 1.6 1.8
Cyclohexane 0.188 581.7 1400.0 V 172 3.2 4.4
L-V 57 A,B 581 680
VI 59 Benzene 0.613 444.0 4000 L 71 1.6 2.2
VI Cyclohexane 0.387 577.1 1400.0 V 121 3.3 4.3
L-V 43 A,B 5S1 680
60 Methane 0.566 -280.0 250.0 L 7 2.4 2.2
Nitrogen 0.434 40.0 2000.0 V 369 1.8 2.0 667
L-V 95 B NRD 591
61 H2 0.5 -200.0 500.0 L 0
CO 0.5 -150.0 2500.0 V 66 1.8 2.5
L-V 0 NRD 667
62 H2 0.75 -250.0 750.0 L 0
CO 0.25 -200.0 2500.0 V 70 3.0 4.2
L-V 0 NRD 673
63 H2 0.5 -250.0 20.0 L 0
Methane 0.5 0.0 2500.0 V 82 2.7 4.0
L-V I3 B NRD 673
Table 4
Peng-Robinson EOS Error Analyses: Binary Mixtures (continued)
Sys. Components Mole Temp. Press. Expt. No. of Absolute Deviation (BtulIb) Departure References
No. Fractions Range Range Phase Points Method
C'F) (psi a) AVE RMSE Raw Smooth
64 CO2 0.447 -63.7 72.5 L 0
Nitrogen 0.553 116.3 2900.8 V 190 1.0 1.3
L-V 1 B NRD 688
65 CO2 0.4761 -63.7 72.5 L 0
Methane 0.5239 116.3 7252.0 V 187 5.0 6.2
L-V 0 B NRD 688
Ul 66 CO2 0.5 -50.0 100.0 L 0
0\
Methane 0.5 200.0 2000.0 V 40 5.0 6.2
L-V 8 NRD 667
67 CO2 0.5 -50.0 100.0 L 31 13.1 13.4
n-Pentane 0.5 250.0 2000.0 V 28 4.1 7.1
L-V 14 B NRD 685
68 H2S 0.5 -110.0 1000.0 L 6 72 7.4
Methane 0.5 0.0 2000.0 V 25 3.5 7.0
L-V 17 B NRD 670
69 H2S 0.4927 80.3 145.0 L 0
Methane 0.5073 440.3 5076.4 V 81 114 14.4
L-V 0 B NRD 686
70 H2S 0.5 -120.0 20.0 L 25 5.0 6.3
Ethane 0.5 200.0 2000.0 V 18 1.3 18
L-V 4 B NRD 670
Table 4
Peng-Robinson EOS Error Analyses: Binary Mixtures (continued)
Sys. Components Mole Temp. Press. Expt. No. of Absolute Deviation (Btullb) Departure References
No. Fractions Range Range Phase Points Method
tF) (psia) AVE RMSE Raw Smooth
71 H2S 0.8933 170.3 145.0 L 0
Methyl- 0.1067 440.3 6526.5 V 67 11.2 13.8
Cyclohexane L-V 19 8 NRD 686
72 H2S 0.9184 215.3 145.0 L 0
Toluene 0.0816 440.3 9427.5 V 96 8.0 10.9
L-V 25 8 NRD 686
'J> 73 H2S 0.5141 125.3 145.0 L 0
-...I CO2 0.4859 4403 8702.4 V 79 6.2 80
L-V 5 8 NRD 686
a 'L' refers to data points represented by phase code = 1 & 4 in the database
b 'V' refers to data points represented by phase code = 2 & 5 in the database
c 'L-V' refers to data points represented by phase code = 3 in the database
d 'NRD' = No raw data
e 'NSD' = No smooth data
Table 5
Peng-Robinson EOS Error Analyses: Ternary Mixtures
Sys. Components Mole Temp. Press. Expt. No. of Absolute Deviation (Btullb) Departure References
No Fractions Range Range Phase Points Method
('FJ (psia) AVE RMSE Raw Smooth
Methane 0.366 -240.0 250.0 La 19 5.4 6.5
Ethane 0.311 300.0 2000.0 yb 12 l.7 l.9
Propane 0.323 L-V" 0 NRDd 574
2 Methane 0.3702 -240.0 250.0 L 140 2.4 3.4
Ethane 0.3055 300.0 2000.0 V 24 l.l 1.3
Propane 0.3243 L-V 76 NRD 671
VI 3 H2S 0.333 -110.0 20.0 L 12 24.4 24 .7
00 Methane 0.334 200.0 2000.0 V 27 29 5.1
Ethane 0.333 L-V 9 NRD 670
4 CO2 0.3333 -50.0 100.0 L 5 34 4.4
Methane 0.3333 300.0 2000.0 V 38 2.2 2.6
Ethane 0.3333 L-V 3 NRD 672
5 n-Pentane 0.2 350.2 400.0 L 70 7.8 8.6
Cyclohexane 0.202 600.0 1400.0 V 135 5.9 6.3 581
Benzene 0.598 L-V 81 A,B 665 683
6 n-Pentane 0.333 348.4 20.0 L 76 1 9 2.2
Cyclohexane 0.334 600.0 1400.0 V 139 2.6 3.1 581
Benzene 0.333 L-V 66 A,B 665 683
7 n-Pentane 0.601 351.5 40.0 L 92 2.1 2.6
Cyclohexane 0.199 600.0 1400.0 V 154 2.0 2.4 581
Benzene 0.20 L-V 52 A,B 665 683
Table 5
Peng-Robinson EOS Error Analyses: Ternary Mixtures (continued)
Sys. Components Mole Temp. Press. Expt. No. of Absolute Deviation (Btullb) Departure References
No. Fractions Range Range Phase Points Method
(~ (psia) AVE RMSE Raw Smooth
8 Benzene 0.333 201.3 40.0 L 87 2.7 3.1
n-Octane 0.334 600.0 1400.0 V 25 2.6 3.9
Tetralin 0.333 L-V 50 C,8 581 684
9 Benzene 0.45 201.3 40.0 L 84 3.5 4.2
n-Octane 0.45 600.0 1400.0 V 48 3.1 4.4
Tetra Ii n 0.10 L-V 85 C,8 581 684
V>
\0
a 'L' refers to points represented by phase code = 1 & 4 in the database
b 'V refers to points represented by phase code = 2 & 5 in the database
c 'L-V refers to points represented by phase code = 3 in the database
d 'NRD' = No raw data
Table 6
Peng-Robinson EOS Error Analyses: Multicomponent Mixtures
Sys. Components Mole Temp. Press. Expt. No. of Absolute Deviation (BtuIlb) Departure References
No. Fractions Range Range Phase Points Method
(~ (psia) AVE RMSE Raw Smooth
H2S 0.3037 80.3 145.0 L 0
Methylcyc10hexane 0.0986 440.0 4351.2 V 66 10.1 11.6
Toluene 0.1031 L-V 14
Methane 0.4946 B NRD 686
2 Methane 0.84352 391.8 22.79 L 0
CarbonDioxide 0.04054 2383.35 V 13 8.9 13 .6
Water 0.05509 L-V 0
0"- Nitrogen 0.00018
0 Propylene 0.00009
Carbonyl-Sulphide 000009
H2S 0.02882
2-Methylpropane 0.00009
Ethane 0.02136
Propane 0.OlD02 NRD 674
3 Methane 0.85791 409.8 136.76 L 0
Ethane 0.02841 2110.46 V 8 3.7 5.5
Water 0.05000 L-V 0
Hydrogen 0.00010
Helium 0.00133
Nitrogen 0.01587
CarbonDioxide 0.00285
0\
Sys.
No.
3
Components
Propane
Propylene
2-Metbylpropane
n-Butane
2-Methylbutane
n-Pentane
Table 6
Peng-Robinson EOS Error Analyses: Multicomponent Mixtures (continued)
Mole
Fractions
0.01900
0.00010
0.00922
0.00922
0.00295
0.00304
Temp.
Range
tF)
Press.
Range
(psia)
Expt.
Phase
No. of
Points
Absolute Deviation (Btullb)
AVE RMSE
Departure
Method
References
Raw Smooth
NRD 674
CHAPTER V
ENTHALPY DATA
QUALITY ASSURANCE
The main goal of the model evaluations was to provide an EOS-based screening
tool to audit the current OPA enthalpy database and establish the validity of the current
entries. As described in Chapter III, there was no direct method of ascertaining the
validity of the entries in the database by comparing records with the original sources of
data. This is because the original enthalpy data have been manipulated to generate
enthalpy departure values, which are entered into the OP A database. Thus, transcription
errors and gross outliers are identified based on "higher-than-expected" deviations
between the reported and predicted values for any data set.
Data-Entry Checks
As a first step toward assessing the validity of the records in the database, dataentry
checks were performed, which involved visual inspection of database records for
typographical errors and omissions relating to temperature, pressure, composition, and
phase-code entries. The data-entry errors were detected by comparing database records
with the original sources of data. Following are examples of some of the observations
made while checking the pure fluid enthalpy data.
62
• Incorrect references; e.g., for n-heptane the data points 25-130 have 556 listed
incorrectly as the reference number. Reference number 556 is for the VLE data of nheptane.
• Incorrect and/or extraneous records; e.g., cyclohexane has two different records for
enthalpy departure values at 620 OP and 300 psia, both from the same reference
source. In this case, either one of the references is wrong, or one of the data entries is
extraneous and should be deleted.
• Redundant information; e.g., data points 266 and 267 for cyclohexane are identical.
• Omission of records; e.g., for tetralin the enthalpy departure values corresponding to
the isobar of 20 psia are not included in the database.
• Incorrect entries; e.g., for trans-Decalin the data points 1-24 correspond to the isobar
of 22 psia, and not 20 psia as is incorrectly tabulated in the database.
EOS-Based Data Screening
The PR enthalpy model statistics and the detailed point-by-point output analyses
were used to screen and evaluate the GP A enthalpy data. In all, around fifteen-thousand
data records in the single phase region (i.e., data represented by phase codes' l' and '2',
respectively) were scrutinized for possible outliers using the EOS data screening
procedure. To achieve those objectives, certain criteria were used to help identify data
points showing deviations between the reported and predicted enthalpy departure values
that were larger than expected. The data records noted were:
63
1. Data-entry errors not noted by inspection.
2. Data points exhibiting deviations in calculated enthalpy departure values that are
greater than twice the root-mean-squared error (RMSE) for the entire data set.
Near-critical data points were given special attention.
3. Data points showing an abrupt change in the deviation sign.
4. Data values showing gross systematic errors; these are identified by the
disagreement in the deviations among reported data sets for the same system at
identical or similar conditions.
Test Cases
The following section describes the methodology applied and the analyses used to
identify data records meriting further examination. Three test cases involving an alkane
(n-pentane), a cycloalkane (cyclohexane), and an aromatic (benzene) are presented.
n-Pentane
For the n-pentane system, the point-by-point analysis revealed data entry errors
that were not detected by visual inspection of the data records in the database. The output
file statistics of the liquid-phase enthalpy departure function show that there are two
different isotherms for identical isobars and identical enthalpy departure functions. Also,
both database records are from the same reference source. The two records are 193.8 "F,
200.0 psia and 194.0 "F, 200.0 psia; both of these have the enthalpy departure function
equal to -144.4 Btu/lb. The isotherms of 205.2 OF and 205 .3 "F at 200.0 psia also have an
64
identical enthalpy departure value of -138.7 Btu/lb. It would appear that one of the
records in each case is a typographical error and needs to be deleted from the records.
But on verifying with the original reference source (663 in Appendix B), it becomes clear
that the records do not represent a data-entry error. Indeed, the reference source reports
an identical enthalpy value (not the depanure) for the two different isotherms.
Again, for the liquid enthalpy departure functions, most of the isotherms at the 400
psia isobar show a consistently high deviation, which exceeds the value of twice the
RMSE for the entire data set (6.8 Btu/lb). The 300 - 400 "F isotherms at the 600 psia
isobar also display "higher-than-usual" deviations and have, therefore, been flagged. The
two individual enthalpy records at 405.9 "F, 1000 psia and 361.9 "F, 1400 psia have been
flagged for showing deviations exceeding twice the RMSE value for the entire data.
For the vapor-phase enthalpy predictions, the individual enthalpy records at 600.3
"F, 200.0 psia and 400.0 "F, 500.0 psia have been marked for showing abrupt change in
deviation signs and also for having deviations in excess of those for the neighboring
points. For this data set, two records at 400 "F, 800 psia are reported with two different
enthalpy departure values of -102.0 Btu/lb and -96.5 Btu/lb, respectively. But on further
inspection, it is revealed that one record represents raw experimental data while the other
represents smoothed data.
The data meriting further analysis for n-pentane is given in Table 7. Figures 3 and
4 show deviation plots which represent enthalpy departure deviations against
corresponding temperature values. These plots give a visual representation for the flagged
data records. As shown in the figures, the majority of the flagged records show
65
Table 7
Flagged Data Records for n-Pentane
Data Temp. Press. Enth. Dept Exp. Raw/Smooth Reference Critena for
Record No. (F) (psia) (Btu/lbm) Phase data No. Outliers
28 193.8 200.0 -144.4 R 663 1
29 194.0 200.0 -144.4 R 663
30 205 .2 200.0 -138.7 1 R 663 1
31 205.3 200.0 -138.7 1 R 663 1
76 600.3 200.0 -2.7 2 R 663 2&3
86-96 280.1 400.0 R 663 4
-353.8
120 250.0 500.0 -130.0 1 R 458 4
121 300.0 500.0 -122.0 1 R 458 4
122 350.0 500.0 -113.0 1 R 458 4
127 400.0 500.0 -53.0 2 R 458 2&3
134-142 300.0 600.0 1 R 663 4
-406.0
143 426.0 600.0 -59.7 2 R 663 2
158 400.0 700.0 -100.0 2 R 458 2
164 450.0 900.0 -83 .0 2 R 458 2
177 405.9 1000.0 -103 .6 1 R 663 2
178-187 450.0 1000.0 2 R 663 4
-690.6
189 450.0 1100.0 -92.0 2 R 458 2
198 361.9 1400.0 -118 .0 R 663 2
200-213 402.5 1400.0 2 R 663 4
-689.9
66
Pentane
... Flagged llIta Records in Liquid Pha<;e
,.!:l 10.00
~
CO
£.
£.
t £.
~ £.
£.
~ I~ i~
.~
6
0.00 .......... ---~--=----=-,~~~~""-------------i
0.00 200.00 400.00 600.00
Temperature, of
Figure 3. Liquid-Phase Enthalpy Departure Deviations for n-Pentane
67
- "-
n-Pentane • Flagged Data Records in Vapor Phase •
,..0 10.00 - ...... --~- • OS
uf
§ . -~ ro
•. J I • - • - - - • "---' • - -- -
.~
il)
~
~
~
~ ro 0.00 a >..
0..
- ---- - '- - - - - ~ - - "-
L ~----- _ '---"""" ; F'@ - = . . - ;~--, ,- --'- , - " -, ~ ;::= r- 0 -
~ g r- ~ -= . C""l ,, ~ - ~ ---1 i ~ ~9~~lifi L_':::J§ I ~
C D~ L..J - , , " i="'~ p :.....: , HII ~ I --.; :-:E', I , " _ - = - r- - ..~ '-----' ....J l- ~ =Pl ==I Fi L...c ~"F'I t;;;l ~ ~ ,- ' - ~ ~ _...J t:::::I ,~ " c= ~ ~- 3J ,---; ~ ~
~ - = b
n r '.....J - ...... '-- ro
...c: " '--.J 0 r---
L ~
~
t:: Q
r.I.l - •
-10.00 - I I I I I I
200.00 400.00 600.00 800.00
Temperature, of
Figure 4. Vapor-Phase Enthalpy Departure Deviations for n-Pentane
68
enthalpy departure deviations greater than twice the RMSE. However, most of the
flagged records result as a part of a systematic trend in deviations; an observation not
evidenced by the plots. As discussed later, only a few of these flagged data records
eventually are identified as "possible" outliers.
Cyclohexane
For the liquid-phase enthalpy departure predictions, the data record at the
temperature of 181.8 OF, 15.4 psia was flagged since it showed a deviation of -4.7 Btu/lb
compared to similar isotherms at the 15.4 psia isobar, which have a consistent deviation
of around 2.7 Btu/lb. The raw data record at 471.2 OF, 400.0 psia was highlighted for
showing a deviation of 8.5 BtU/lb, which is in excess of twice the RMSE of 3.0 Btu/lb for
the entire data set. The smoothed data points at 520.0 - 530.0 OF, 588 psia and 540.0 OF,
700 psi a were flagged due to the high deviations between the reported and predicted
departure values (almost 9 Btu/lb).
F or vapor-phase predictions, the raw data record at 199.4 of, 15.4 pSla was
flagged for showing an abrupt change in the deviation sign (deviation = 0.2 Btu/lb) while
similar isotherms (194.5 OF, 196.8 OF, 205.2 OF) at 15.4 psia have deviations equal to -2.5
Btu/lb. Two records at 671.9 OF, 15.4 psia and 686.1 OF, 15.4 psia, respectively were
flagged for displaying deviations which are greater than twice the RMSE for the entire
data set (3.15 Btu/lb). The two raw data records at 497.7 OF, 300 psia and 518.8 OF, 300
psia, both of which are from the same reference source, were marked for reporting
identical enthalpy departure values at -21.2 Btu/lb. Indeed, one of these entries is wrong,
69
for on verifying with the reference source, different values of enthalpies are recorded for
497.7 of, 300 psia and 518.8 of, 300 psia data records. Table 8 presents a listing of the
flagged records for cyclohexane. Figures 5 and 6 depict the deviation plots for
cyclohexane in the liquid and vapor phases, respectively. Similar to the n-Pentane
system, most ofthe flagged records display deviations greater than twice the RMSE.
Benzene
The point-by-point analysis of the system revealed that the raw data in the
database for the enthalpy departures for the liquid-phase showed a consistent deviation of
6-7 Btullb for isobars ranging from 500 - 800 psia. Although, the deviations are high,
when compared to the absolute average deviation of 4 Btullb for the entire data set, the
data points are not flagged. First, because the records represent raw experimental data;
second, the deviations are consistently of the same order. One would assume that the
deviations \vould be lower, if the data set had been smoothed using an equation of state or
some other means. For this raw data set, it was also observed that certain data records
corresponding to the 1000 and 1400 psia isobars, showed deviations of the order of 1.5
Btullb and 0.3 Btullb, respectively. This is contrary to what one would expect, since
deviations between reported and calculated departures are lower for high pressure values.
This may indicate the possibility for systematic errors in the original enthalpy
measurements. Further, in this data set, the data record corresponding to 439 OF, 1000
psia showed an abrupt change in deviation sign. This record has, therefore, been
identified for further examination.
70
Table 8
Flagged Data Records for Cyclohexane
Data Temp. Press. Enth. Dept Exp. Raw/Smooth Reference Criteria for
Record No. (F) (psia) (Btu/lb) Phase data No. Outliers
9 181.8 15.4 -147.2 1 R 584 3
14 199.4 15.4 -1.5 2 R 584 3
21 377.0 15.4 0.1 2 R 584 3
25 671.9 15.4 11.8 2 R 584 2
26 686.1 15.4 8.2 2 R 584 2
35-42 350.4- 2 R 584 4
684.8
66 497.7 300.0 -21.2 2 R 584
67 518.8 300.0 -21.2 2 R 584
76 471.2 400.0 -103.2 S 677 '")
.(..
104 531.1 588.0 -91.3 1 R 584 2
119 561.2 700.0 -76.3 2 R 584 2
135 512.1 1000.0 -113.9 1 R 584 2
144 598.5 1200.0 -91 .1 2 R 584 2
237 500.0 500.0 -103.2 1 S 677 2
249 520.0 588.0 -98.1 1 S 677 2
250 530.0 588.0 -92.3 1 S 677 2
260 530.0 700.0 -97.7 S 677 2
261 540.0 700.0 -95.0 S 677 2
262 560.0 700.0 -78.1 2 S 677 2
298 620.0 1000.0 -75.5 2 S 677 2
299 640.0 1000.0 -67.7 2 S 677 2
319 620.0 1400.0 -90.5 2 S 677 2
71
Cyclohexane
... Flagged Data Records in Liquid Phase
10.00
0.00 --~--------------~-C~-7¥r~&-----~~~------~
-10.00 ........I~--r----r----r----r---~-----r-....I
200.00 400.00 600.00
Temperature, of
Figure 5. Liquid-Phase Enthalpy Departure Deviations for Cyclohexane
72
10.00 ........ ---------------------r •
- -
-10.00 -
•••• , ,
---'
•
•
-20.00 -.,L-I----.-l--"T"'--""T1--""'T
'
--""'T1--.....,.----t
200.00 400.00 600.00 800.00
Temperature, of
Figure 6. Vapor-Phase Enthalpy Departure Deviations for Cydohexane
73
For the \'apor-phase enthalpy departure predictions, the raw enthalpy record
corresponding to the temperature of 553.6 of and a pressure of 750.0 psia was flagged
since the deviation of the predicted and reported value equals +7 Btu/lb. This value is
quite high when compared to the average deviation for the other isotherms at 750.0 psia
(from the same original reference) which show an average deviation of -2 Btu/lb. Some
of the smoothed data points corresponding to the 750 psia isobar were flagged since the
deviations between the reported and predicted values were not consistent. The deviations
ranged in sign and magnitude from +7 Btu/lb to -5 Btu/lb. The 800 psia isotherm for the
smoothed enthalpy data contains possible outliers, since the deviations were about 7
BtuJlb for six consecutive data records and -2.5 BtuJlb for the six records which follow it.
For this isobar, not only are the deviations not consistent, they are fairly high when
compared to the absolute average deviation (AAD) for the entire data set of the vaporphase
enthalpies (1.45 Btu/lb). For similar reasons, both raw and smoothed data records
corresponding to the 1000 psia isobar have been marked as data requiring further
analysis. The possible outlier candidates for the benzene system are listed in Table 9.
The enthalpy departure deviation plots for benzene are given in Figures 7 and 8. The
plots depict deviation trends similar to those given earlier for n-pentane and cyclohexane.
Discussion
The analysis presented above for the three case studies highlights the
methodology adopted for assessing the quality of data in the GPA enthalpy database. A
treatment similar to that for n-pentane. cyclohexane and benzene was employed to
74
identify possible outliers for all the pures, binaries, ternaries and multicomponent systems
in the enthalpy database. It is evident from the discussion presented that each system has
to be treated on an individual basis and examined for possible outliers. Upon closer
inspection of the enthalpy deviations for the above mentioned sample systems, it was
observed that a very large number of records that were flagged for displaying higherthan-
usual deviations (greater than twice the RMSE) are actually a part of a systematic
trend in deviation for the particular data set. Consequently, the deviations exhibited by
these data may be a result of model-lack-of-fit, and therefore these records may not
qualify as outliers. The deviation plots given in Figures 3 - 8 present a graphical
depiction of the possible outliers for the test systems considered based on the set criteria;
however, these plots do not indicate the trends in deviations for a given isotherm or
isobar, and as such, they are inadequate for identifying trends in deviations.
Tables 10, 11 and 12 present a summary of the possible outliers in enthalpy data
of the pure fluids, binaries and the ternaries and multicomponent systems, respectively.
As documented in these tables, the data screening have resulted in the following outlier
ratios: For pure components, OR = 2111625~ for binary systems, OR = 122/5814; and for
ternary systems, OR = 2/276. It is important to note here that the assessments made on
the data quality are based on the relative comparisons generated by the EOS screening
procedure employed for this purpose. Therefore, data records have been marked as
"possible" 0utliers. Consequently, further analysis and examination, which includes
comparisons with predictions against other enthalpy models, is required for a more
accurate assessment for probable errors in the experimental enthalpy measurements.
75
Table 9
Flagged Data Records for Benzene
Data Temp. Press. Enth. Dept Exp. Raw/Smooth Reference Criteria for
Record No. (P) (psia) (Btu/lb) Phase data No. Outliers
22 345.9 100.0 -11.0 2 R 581 2
77 535.9 600.0 -47.6
,..,
- R 584 2
137 553.6 750.0 -94.7 '"1- R 581 2
153 439.4 1000.0 -128.6 1 R 584 3
157 559.1 1000.0 -104.0 2 R 584 2
158 581.1 1000.0 -94.9 2 R 584 2
159 600.7 1000.0 -81.4 2 R 584 2
309 554.0 750.0 -93.6 2 S 679 2
310 556.0 750.0 -88.6 2 S 679 2
312 560.0 750.0 -68.1 2 S 679 2
322 554.0 800.0 -99.4 2 S 679 4
323 556.0 800.0 -97.7 2 S 679 4
324 558.0 800.0 -96.1 2 S 679 4
325 560.0 800.0 -93.7 2 S 679 4
326 565.0 800.0 -87.6 2 S 679 4
327 570.0 800.0 -79.8 2 S 679 4
344 556.0 1000.0 -105.2 2 S 679 2
347 565.0 1000.0 - i 0 1.5 2 S 679 L
348 570.0 1000.0 -99.8 2 S 679 2
354 700.0 1000.0 -38.5 2 S 679 ,....,.
76
10.00
6
5.00
- /\ LJ
~
~j
- .~:::,
6 .~ 6 .~~ :-~
/--[~ . \ 1_\
2:,~
-' - 6 2:, /&1} 61'" 6 6
& I ( \
.'\ L\ -' ~
B: 6
6
6 ,~
.\ ~ 6
6 0,. M
0.00
A
- 1 8errzfre ,
I
Aagged ilia Reards in liquid Im;e
!
~
-5.0 0 I I I I I
200.00 400.00 600.00
Temperature, of
Figure 7. Liquid-Phase Enthalpy Departure Deviations for Benzene
77
8.00 --r~======-=-============---------'
•
-. •• • •
0.00 ~I--------------::~"'---:
-4.00
• •
-8.00 --i---""""T"---"T'""'"---,----r----r----t
200.00 400.00 600.00 800.00
Temperature, of
Figure 8. Vapor-Phase Enthalpy Departure Deviations for Benzene
78
Table 10
Summary of Possible Outliers in Enthalpy Data
for the Pure Fluids
No. System Exp. No. of Outlier Total
Phase outliers Ratio, OR OR
Methane 2 2114 3/39
'"I .:.. \/25
2 Ethane 0 0/4\ \/69
2 \/28
3 Propane \ 0 0/40 \/6\
2 112\
4 n-Pentane \ 0 0/141 1/390
2 1/249
5 n-Octane 1 1 11159 11244
2 0 0/85
6 iso-Octane 1118 1/18
7 n-Hexadecane 1 \/81 III 0 1
2 0 0/20
8 Cyclohexane 3 31128 7/304
2 4 41176
9 cis-2-Pentene 1 0 0/50 4/243
2 4 4/193
10 trans-Decalin 0 0/120 1/153
2 1/33
Total Outlier Ratio for Pure Fluids = 2111625
79
Table 11
Summary of Possible Outliers in Enthalpy Data
for the Binary Mixtures
No. System Mole Exp. No. of Outlier Total
Fraction Phase Outliers Ratio, OR OR
Methane 0.951 1 0 0/6 1/85
n-Heptane 0.049 2 1179
2 Methane 0.491 1 0 0115 3/59
n-Heptane 0.509 2 3 3/44
3 Methane 0.249 4 4/90 41125
n-Heptane 0.751 2 0 0/35
4 Ethane 0.498 0 0/48 2176
Propane 0.502 2. 2 2/28
5 Propane 0.43 0 0/0 2/21
iso-Pentane 0.57 2 2 2/21
6 n-Pentane 0.587 1 1/41 1/57
n-Hexadecane 0.413 2 0 0116
7 n-Pentane 0.794 1 1 1/60 2/91
n-Hexadecane 0.206 2 1 1/31
8 Methane 0.5 1168 11104
Meth-Cylohex 0.5 .,...., 0 0/36
9 n-Pentane 0.197 1 1197 2/267
Cyclohexane 0.803 2 11170
10 n-Pentane 0.385 1 0 0/96 1/263
Cyclohexane 0.615 2 1 1/167
1 1 n-Pentane 0.612 1 1/89 5/279
Cyclohexane 0.388 2 4 4/190
12 n-Pentane 0.793 0 0/82 6/315
Cyc10hexane 0.207 2 6 6/233
13 n-Pentane 0.502 1 0 0/43 1/159
cis-2-Pentene 0.498 2 1/116
80
Table 11
Summary of Possible Outliers in Enthalpy Data
for the Binary Mixtures (continued)
No. System Mole Exp. No. of Outlier Total
fraction Phase outliers Ratio, OR OR
14 Methane 0.5 6 6/40 6/73
Toluene 0.5 2 0 0/33
15 Propane 0.252 0 0/9 1111
Benzene 0.748 2 1/2
16 Propane 0.797 1 113 118
Benzene 0.203 2 0 0/5
17 n-Pentane 0.186 2 2/64 5/266
Benzene 0.814 ,., 3 3/202
18 n-Pentane 0.4 I 1193 5/341
Benzene 0.6 2 4 4/248
19 n-Pentane 0.801 0 0/84 1/278
Benzene 0.199 ,., 1/224 ~
20 n-Pentane 0.725 I 2 2/60 21138
trans-Decal in 0.275 2 0 0/78
21 n-Pentane 0.561 1164 11105
trans-Decal in 0.439 2 0 0/41
22 n-Pentane 0.884 I 1/76 1/164
trans-Decalin 0.116 2 0 0/88
23 n-Pentane 0.197 1 0 0/82 201159
Tetralin 0.803 2 20 20/77
24 n-Pentane 0.795 0 0/50 11154
Tetralin 0.205 2 1/104
25 n-Pentane 0.893 I 1/79 11237
Tetralin 0.107 ,...,. 0 01158
26 Benzene 0.93 I 0 0/71 11159
n-Octane 0.07 2 \/88
81
Table 11
Summary of Possible Outliers in Enthalpy Data
for the Binary t-viixtures (continued)
No. System Mole Exp. No. of Outlier Total
fraction Phase outliers Ratio, OR OR
27 Benzene 0.857 0 0/97 1/132
n-Octane 0.143 2 11103
28 Benzene 0.771 1 1156 11132
n-Octane 0.229 2 0 0176
29 Benzene 0.271 1 15 1511 00 15/200
n-Octane 0.729 2 0 0/100
30 Benzene 0.963 1 1/83 1/207
n-Hexadecane 0.037 2 0 0/124
31 Benzene 0.92 1 1/65 3/134
n-Hexadecane 0.08 2 2 2/69
32 Benzene 0.67 2 2/66 2/75
n-Hexadecane 0.33 2 0 0/9
33 Benzene 0.211 2 2/59 4/211
Cyclohexane 0.789 2 2 21152
34 Benzene 0.334 3 3/67 91195
Cyclohexane 0.666 2 6 6/128
35 Benzene 0.812 1 0 0/80 1/252
Cyclohexane 0.188 2 11172
36 Benzene 0.613 0 0/71 3/192
Cyclohexane 0.87 2 3 3/121
37 CO2 0.5 0 0/31 1/59
n-Pentane 0.5 2 1128
38 H2S 0.5 1 0 0/6 3/31
Methane 0.5 2 3 3/25
Total Outlier Ratio for Binaries = 122/5814
82
Table 12
Summary of Possible Outliers in Enthalpy Data
for the Ternary Mixtures
No. System Mole Exp. No. of Outlier Total
Fraction Phase Outliers Ratio, OR OR
Methane 0.3702 1 1/140 11164
Ethane 0.3055 2 0 0/24
Propane 0.3243
2 Benzene 0.333 0 0/87 11112
n-Octane 0.334 2 \/25
Tetralin 0.333
Total Outlier Ratio for Ternaries = 2/276
83
Beyond the EOS comparative studies conducted here. it would have been useful
to evaluate the thermodynamic consistency of enthalpy data. In this case, however, it was
not possible to devise thermodynamic consistency tests of the Gibbs-Duhem type for
evaluating the enthalpy data quality in the GPA database. This is because experimental
volumetric and entropy data were not available for the systems at the required
temperature and pressure conditions. Also, all GP A enthalpy values analyzed are singlephase
data. To apply the Gibbs-Duhem analysis, vapor-liquid equilibrium (VLE)
measurements are required for successfully implementing the thermodynamic consistency
checks.
84
CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
The Peng-Robinson EOS was used to predict enthalpy departure functions for
natural gas systems in the GPA enthalpy database. The predicted departures were then
compared against the departure values generated from experimental enthalpy values in the
database. The model evaluations revealed that the PR EOS provides reasonably accurate
predictions for the enthalpy departures. Typical average deviations of 2-6 Btullb were
observed for all the systems considered; this was true for both the liquid and vapor phase
predictions.
For the homologous series of alkanes, alkyl naphthenes or aromatics, the lower
molecular weight hydrocarbons, in general, gave superior enthalpy predictions. Moreover,
the PR model did not show any apparent difference in its ability to predict enthalpy
departures in the liquid and vapor phases. Similar results were obtained by Daubert (54)
using pseudo-pure component mixing rules.
The model evaluations were used to help screen enthalpy data entries in the GP A
database. Data points exhibiting "higher-than-usual" deviations were marked as possible
outliers and were identified for further examination and analyses. For all systems
considered, on an average, one to two percent of the total records analyzed each for the
pures, binaries and ternaries were marked as possible outliers. Under the enthalpy data
85
quality assurance procedure, the temperature, pressure, and composition records were
also visually inspected for data-entry errors.
It was not possible to devise thermodynamic consistency checks of the GibbsDuhem
type for evaluating the enthalpy data quality in the GP A database. This is because
experimental volumetric and entropy data were not available for the systems at the
required temperature and pressure conditions. Also, most of the enthalpy data in the
database are in the single-phase region.
For future work, graphical deviation plots should be made to help analyze the
systematic trends in deviation displayed between the EOS departure function predictions
and the experimentally obtained enthalpy departure entries in the GP A database. Also,
other enthalpy prediction models should be evaluated against the experimental enthalpy
departure values in the database. The other models could include a variation of the BWR
EOS and the currently popular cubic EOS amended by volume translation and/or equipped
with different mixing rules. Theoretically based modern equations of state, which include
the SPHCT and its variations, should be tested for their enthalpy departure prediction
capabilities. Such models could be of help in assessing the quality of enthalpy data
identified as displaying systematic trends in deviations.
Techniques should be devised to establish smoothing procedures for the raw
enthalpy data in the database. Equations of state or other means could be employed for
this purpose.
Finally, enthalpy data reported in the literature for natural gas systems after 1990
should be added to the GP A enthalpy database. Such data would serve as an excellent
86
complement to the existing enthalpy entries in the database and would better define the
need for additional experimental measurements.
87
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94
APPENDIX A
A DETAILED DERIVATION OF
PENG-ROBINSON EQUATION OF STATE
ENTHALPY DEPARTURE FUNCTION
95
The expression for the PR EOS enthalpy departure model is derived below. Similar
derivations for the PR EOS are given by Edmister and Lee (61). From exact
thennodynarnics, isothennal definition of enthalpy departure is given as:
The Peng-Robinson equation is given as:
RT a(T)
p = --- ------'----
v - b v( v + b) + b( v - b)
Using the volume integral equation,
j OP\ RT T eb(T)
l~or J.. = v-b - v(v+b) +b(v-b) . ~
(4.11)
(4.1 )
(4.8)
(4.4)
"" tP) RT aCT) RT T az(T)
p - 1 "- iJI' \. = V - b - v( v + b) i- b( v - b) - v - b + v( v + b) + b( v - b) . ~
r[p -T iP "Lv = {T. da(T) - a(T)}' r ~v 2 ;
iJI' \. r dT ( v + b) - 2b
@ const. T:
96
( r-) v da(n 1 v+b 1-~2
H-Ho =fr. . . -a(n}--ln +pv-RT
l dT 2 J2b v + b( 1 + J2) 00
. v + b( 1 - .fi)
hm v_ oo in ( r;::;) = in LO = 0.0
v+bl+~2
H-H o =-1- in[ V+b(l-J2)] . {Td·a (T) -a(T)} +pv-RT
2.fib v + b(l + J2) dT
(AI)
H - H o = 1 in (v -OA14b) . {Td·a -(T-) - a( T) } + pv - RT
2.828b \ v + 2A14b dT
(A2)
(4.8)
= a c . [ 1 + m( 1 _ T,0S)] 2
"ro8( TF) = 2·a [m( 0.5)] iJ [ o.S] c • 1+ 1- T, ·m· or 1- T,
ro(T) -m· rO S 2 "8F = r:. [ 1 + m( ~ _ r,0S)] . a c . [ 1 + m{ 1 - r,0S)]
az( T) -m· r: s ( ) --= ·a,T or T,; . ( I + m( I - ~S)]
(A3)
This is the form for aii , hence for a pure substance, it becomes
(A4)
97
Substituting (A4) in (A2) :
H - HO = 1 In ( v - 0'414b) . { -m· Tr0 5 . a T - a T 1
2.828b lv+2A14b [l+m(l- T,os)] () ()
+ pv- RT (AS)
H - HO = C a [(a - Td-a)] In [(V-OA14b)] + v - RT
2.828b dT (v+2A14b) P
(4.12)
with
(4.13)
Equations (4.12) and (4.13) are used to determine the PR enthalpy departure function for
a pure component.
Mixture Departure Function
F or mixtures,
n n
a = LLziZjaij (4.6)
J j
( C) 0.5 0.5 where aij = 1- if ai af (4.9)
Expanding for a binary,
(A6)
98
(A7)
From (A3)
(A3)
c( ai~ . arz) II oa l~ 1/ oali _____ = a l 2 . __ 1_+ a . 2 . __ 2_
2 or 1 or (A8)
oa ~ ~s 12 -m -O.5mT
__ = _. 0.5r-0S = r
::rr' T°.5 T U.l c c
oaj~ -O.5mj r: s
or I;, (A.9)
Substituting (A9), (A8), (AS) into (A7), one gets
= Z2a a [ -m T-O.5] 1/ 11 II 11 1 cl 1 1 rl + 2z z a I 2a I2a I2aI2(1- C )
T 1 (1- T0.5) 1 2 cl c2 1 2 12 .
cl +m rl
I
99
(A10)
So, in general, for a multicomponent mixture,
(All)
Substituting (A10) into Equation (4.12):
n n
LLZjZ1Yi}T-a
H -Ho = 0.3536 _=i_1..:....1=_I_ ___ . 1n ( V-OA14b) + pV- RT
b v + 2A14b
(4 .14)
where
(4.15)
Equation (4.14) is the required expression for a multicomponent mixture.
,.
100
APPENDIX B
LITERATURE REFERENCES IN THE GPA ENTHALPY DATABASE
101
A listing of the literature reference numbers along with their corresponding
literature sources for all the enthalpy data in the GPA Enthalpy Database is presented in
this appendix.
358. Pitzer, K. S., "The Thermodynamics ofn-Heptane and 2.2,4-Trimethylpentane,
Including Heat Capacities, Heats of Fusion and Vaporization, and Entropies," J.
Am. Chern. Soc. 62, 1224 (1940).
458. Storvic, T. S. and J. M. Smith. "Thermodynamic Properties of Polar Substances:
Enthalpy of Hydrocarbon-Alcohol Systems," 1. Chern. Eng. Data 5,133 (1960).
564. Yarborough, L. and W. C. Edmister, "Calorimetric Determination ofthe
Isothermal pressure effect on the Enthalpy of the Propane-Benzene Systems,"
AIChE Journal 11,492 (1965).
570. Bhirud, V. L. and J. E. Powers, "Thermodynamic Properties of a 5 Mole Percent
Propane in Methane Mixture," Report to the NGPA, August 1969.
573. Dillard. D. D., W. C. Edmister, J. H. Erbar and R. L. Robinson, "Calorimetric
Determination of the Isothermal Effect of Pressure on the Enthalpy of Methane
and Two Methane Propane Mixtures," AIChE Journal, 14,923 (1968).
574. Furtado, A. W., D. L. Katz and J. E. Powers, Paper presented at l59th National
ACS meeting, Houston, Texas, February (1970).
578. Jones, M. L., D. T. Mage, R. C. Faulkner and D. L. Katz. "Measurement of the
Thermodynamic Properties of gases at Low Temperature and High PressureMethane,"
Chern. Eng. Prog. Sym. Series, 59 (44),52-60 (1963).
102
579. Lenoir, J. M., "A Program of Experimental Measurement of Enthalpies of Binary
Hydrocarbon Mixtures above 100 deg. F and in the Critical Region," Proceedings
of API 47,640-52 (1967).
580. Lenoir J. M., K. E. Hayworth and H. G. Hipkin, "Some Measurements and
Predictions of Enthalpy of Hydrocarbon Mixtures," Proceedings of API 50, 212
(1970).
581. Lenoir, J. M., K. E. Hayworth and H. G. Hipkin, "Enthalpy Measurements for
Hydrocarbon Mixtures," Proceedings of API 51,405 (1971).
582. Lenoir, J. M., K. E. Hayworth and H. G. Hipkin, "Enthalpies of Decalin and transDecalin
and n-Pentane Mixtures," J. Chern. Eng. Data, 16, 129 (1971).
583. Lenoir, J. M. and H. G. Hipkin, "Enthalpies of Mixtures ofn-Hexadecane and nPentane,"
J. Chern. Eng. Data, 15,368 (1970).
584. Lenoir, J. M., G. K. Kuravila and H. G. Hipkin, "Measured Enthalpies of Binary
Mixtures of Hydrocarbons with Pentane," Proceedings of API 49,89 (1969).
585. Lenoir, J. M., D. R. Robinson, and H. G. Hipkin, "Flow Calorimeter and
Measurement of the Enthalpy of n-Pentane, " J. Chern. Eng. Data, 15,23 (1970).
586. Lenoir, J. M., D. R. Robinson and H. G. Hipkin, "Enthalpies of Mixtures ofnOctane
and n-Pentane," J. Chern. Eng. Data, 15,26 (1970).
587. Mage, D. T., M. L. Jones Jr., D. L. Katz and J. R. Roebuck, "Experimental
Enthalpies for Nitrogen," Chern. Eng. Prog. Sym. Series, 59 (44), 61 (1963).
103
588. Mather, A. E., "The Direct Determination of the Enthalpy of Fluids Under
Pressure," Ph.D. Thesis, University of Michigan (967).
590. Yesavage, V. F., "The Measurement and Prediction of the Enthalpy of Fluid
Mixtures Under Pressure," Ph.D. Thesis, University of Michigan (1968).
591. Starling, K. E., D. W. Johnson and C. P. Colver, "Evaluation of Eight Enthalpy
Correlations," NGPA Research Report RR-4 (1971).
592. Starling, K. E., "1971-1972 Enthalpy Correlation Evaluation Study," NGPA
Research Report RR-8 (1972).
661. Bhirud, V. L. and 1. E. Powers, "Thermodynamic Properties of a 5 Mole Percent
Propane in methane Mixture," Report to the NGP A, Tulsa, Oklahoma, August
1969, and Manker, E. E., Ph.D. Thesis, University of Michigan (1964).
663. Lenoir, J. M., D. R. Robinson and H. G. Hipkin,"Measurement of the Enthalpy of
Pentane, Octane, and Pentane-Octane Mixtures," Proceedings of API 48, 346-96
(1968).
664. Powers, J. E .. University of Michigan, Ann Arbor, Michigan, Private
Communication (1972).
665. Lenoir, J. M., University of Southern California, Los Angeles, California, Private
Communication (1972).
666. Eakin, B. E., G. M. Wilson and W. E. DeVaney, "Enthalpies of Methane-Seven
Carbon Systems," NGPA Research Report RR-6 (1972).
104
667. Cochran, O. A. and J. M. Lenoir, "OPA Experimental Values Referred to Two
Base Levels," OPA Research Report RR-11 (1974).
670. Eakin, B.