DEVELOPMENT OF A NEW EQUATION OF STATE
FOR WATER AND ITS APPLICATIONS - PARTS I & II
by
PYUNG-HUN ,?1ANG
B.S.,Seoul National University
(1974)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS OF THE
DEGREE OF
MASTER OF SCIENCE
IN MECHANICAL ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1984
Q) Pyung-Hun Chang 1984
The author hereby grants to M.I.T. permission to reproduce and
to distribute copies of this
f-hess document in whole or in part.
Signature of Author........ ...
.....................
.............
Department of Mechanical Engineering
Fyruary, 1984
Certif i:
..*
(A1
0 6...-.....................-......
0* 0&0 9 00
A
.
...........0 00 00
William C. Unkel
Thesis Supervisor
Accepted Dy....
Chairman,
Warren M. Rohsenow
Departmental Commitee on Graduate Studies
MASSACHUSETTS INSTlTTE
OF TECHNOLOGY
JUL 1 7 1984 Archives
LIBRARI[3
preface
The purpose of
of
equation
this
in
state
study
develop
to
was
a
simple
the form of a thermodynamic potential
function to represent the right image of pure water substance,
The intended use
accurate thermodynamic properties.
yielding
power
steam
the
of the equation was to apply to problems in
plant engineering.
The first part of this study covers the development of a
new
equation
state,
of
the
application of
second
the
while
part treats the
engineering
two
equation
of
state
to
like
to
thank
Professor
problem.
The author
of the study.
providing
William
He thanks deeply Professor
guidances
timely
of
completion
Professor
Henry
M.
for providing the motivation and the initial guidance
Paynter
the
would
Derek
this
Rowell
for
The
author
valuable
his
for
essential to
and encouragement,
study.
Unkel
also
thanks
suggestions
and
encouragement.
The author thanks his wife
patience
prayers.
and
love,
his
two
Minja
sons
for
Chai
her
exceptional
and June for their
DEVELOPMENT OF A NEW EQUATION OF STATE
FOR WATER AND ITS APPLICATIONS-PARTS I & II
BY
PYUNG-HUN CHANG
Submitted to the Department of Mechanical Engineering
on February 5, 1984 in partial fulfillment of the
requirements for the Degree of Master of Science in
Mechanical Engineering
ABSTRACT OF PART I
An equation of state for pure water was developed in
the form of a thermodynamic potential function, the Massieu
provide
to
was
The goal
Potential Function (MPF).
computationally simple, but accurate, expression that can be
applied to problems associated with steam turbine cycle
calculations, especially in control systems.
A generalized potential function was formulated after
examining several theoretical equations of state including
the van der Waals(VDW) equation. The potential function for
water was determined from this - generalized function by
evaluating the accuracy as additional terms were added.
Accuracy was defined in terms of agreement of the equation
results with the saturation curve behavior of water.
The resulting potential function (or equation of
state) consists of 4 parameters, has one more term than the
VDW equation, yet gives much better accuracy. The accuracy
of pressure calculated with the present equation of state,
as compared to data in the Keenan-Keyes-Hill-Moore steam
within 1 % deviation in the temperature range
table, is:
from 200 F to 1400 F, with the pressure range from 5 psia to
more than 2000 psia in the superheated region; and within 9
% deviation in the pressure range from 1.5 psia to the
critical point in the saturated region.
The main improvement in the accuracy in the saturated
region might be made by allowing the variation of the
parameter b with temperature, which could reshape the
pressure-volume-temperature relationship in the saturated
region. The accuracy in the superheated region might be
improved by adding a few empirical terms in the form of MPF
to correct the compressibility factor.
Thesis Supervisor: Dr. William C. Unkel
Title: Associate Professor of Mechanical Engineering
CONTENT OF PART I
Page
Preface......................
Abstract.....................
List Of Tables...............
List Of Figures..............
Chapter 1. Introduction......
1.1 Background..........
1.2 Problem Statement...
1.3 Organization........
Chapter 2. Historical Review And
Of The Research
2.1 Historical Review...
2.1.1 Improvements To
2.1.2
2.1.3
Improvements
To
Direction.
Repulsion Te rm.
Attraction T erm
Inclusion Of Ro tational And. &...
Vibrational Term
Inclusion Of Polarity..........
Combination Of Improved Results.....
2.1.4
2.2
2.2.1 LHW And Guggenheim Equations...
2.2.2 The Equation By Whiting et al..
2.3 Requirements And Direction For...
The Research
2.3.1 Requirements...................
2.3.2 Direction......................
Chapter 3. Procedures Of Development.........
Review 6f Thermodynamical Relations.
3.1
3.1.1 Thermodynamical Relations......
3.1.2 Reduced Form Of................
3.1.3
Thermodynamical Relations
The MPF In Reduced Form For....
25
.
Existing State Equations
3.1.4 Assessment Of Equations In.....
Terms Of Saturation Properties
.
.
.
.
.
.
.
26
.
Of.........
.
.
.
.
.
.
.
27
.
3.1.5 Assessment
In Terms
Compressibility Factor
3.2 Procedures Of The Proposed Improvement.
3.2.1 The Formulation Of A...............
Generalized MPF
3.2.2 Procedures Of Selection Of.........
31
.
.
.
.
.
33
.
Parameter
35
35
Chapter 4. Results And Discussions...............
4.1 Results.................................
4.1.1 The Resulting Equation Of State....
'b'...
36
Comparison Of The..................
Compressibility Factor
40
4.1.2 The
4.1.3
35
OTher Parameters
'a'
and
.
4.1.4 Techniques To Improve The Agreement
Of Predicted Compressibility
4.2 Discussions.......... -...... ............ ...............
....
Chapter 5. Conclusions And Recommendations....... ....
5.1 Conclusions.............................
5.2 Recommendations.........................
Bibliography..................................... ..........-.Appendix A.
An Algorithm For The Computation..... ....
Of
Saturation Data
.........
.
.
43
.
44
....
46
.
46
47
--
48
49
.
List Of Tables
Table
Page
1
Comparison Of The Three Equations With
Data
2
Comparison Of Saturation Data: Water, The Present
Result And The van der Waals Equation
37
3
Parameter Values Of The Present Equation Of State
39
4
Comparison Of The Compressibility Factor:
The Present Result
5
Comparison Of
Result
Second Virials:
List Of
Experimental
18
Water And
40
Water And The-Present
43
Figures
Page
Figure
28
1
Comparison Of Saturation Curves: Water, The van der
Waals(VDW) And Longuet-Higgins And Widom's Equations
2
Schematic Diagram Of The
3
Comparison Of Saturation Data: Water, The Present
Result And The VDW Equation - P vs. T
38
4
Comparison Of Saturation Data: Water, The
Result And The VDW Equation - P vs. V
Present
38
5a
Comparison Of Second Virial Coefficients
Scale: Water And The Present Result
In Linear
42
5b
Comparison Of Second Virial Coefficients
Scale: Water And The Present Result
In Log.
42
Second Virial Coefficient
30
DEVELOPMENT OF A NEW EQUATION OF
STATE
FOR WATER AND ITS APPLICATIONS
PART I
DEVELOPMENT OF A NEW STATE EQUATION
CHAPTER 1
INTRODUCTION
1.1
BACKGROUND
simulations
Quantitative calculations in the design and
require
plants
power
steam
for
state
of
and
adjustment
correlations
but
where
fit
where
the
theoretical.
fall
into
the
To
obtain
the
three
class
categories:
of
properties,
empirical,
lookup procedures or
table
Simple
such
Equations of state for
water are required.
fluids can be broadly classified into
semiempirical
the
reliable and computationally economical
estimates of thermodynamic properties..
equations
of
empirical,
while
those
basic form is defined on theoretical grounds
coefficients
are
'fit'
to
the
data
are
termed
semi-empirical equations.
Since extensive experimental measurements of
perties
pro-
are available for water, there exist many different empirical
equations of state with a very large number of
that
thermodynamic
have
adjustable
essentially no physical significance.
parameters
Among these are the
1967 IFC Formulation for Industrial Use and the 1968
IFC
Formulation
for
Scientific
and
General
Use,
both
of
which
are based on the
The former consists of
International Skeleton Table of 1963.[1]
about
150 numerical constants and the latter about 300 constants.
These two
IFC formulations have crucial weakness in that the range of
state
up
split
a
is
has
which
curve
fit
different
of
each
rather than a fit to a thermodynamic potential function like
property
is
not
consistency
thermodynamic
maintained
is
nor
each
in
continuity enforced at their
uses
The equation by Keenan, Keyes, Hill and Moore(KKHM)
boundaries.
single
energy,
free
Helmholtz
sub-region
a
formulation
the
of
each
sub-regions,
Since
sub-formulation.
the
6
into
is
function with 59 constants.[2-j The fundamental
fundamental
function of the KKHM equation
the
is
free
Helmholtz
energy
as
a
of temperature and specific volume, from which all the other
function
All the properties derived
steam properties can be derived.
this
in
way are thermodynamically consistent with each other and continuous in
Nonetheless the larger number of constants
the prescribed region.
in
making
interpolative
example, their accuracies are
larger
for the computational
disadvantageous
still
These totally empirical
efficiency.
accurate
is
equation
KKHM
the
of
equations
state
are
quite
For
of properties.
calculations
typically
of
However,
0.01%.
owing
to
number of constants, more than 90% of the CPU time is known to
be spent in evaluating steam properties, when
equations
used,
are
in
typical
types
these
is
one
state
simulations of transient or steady
state behavior of steam power plant components.[3] This
disadvantage
of
computational
reason to develop theoretical or semi-empirical
expressions.
A few theoretical equations exist for such simple substances such
as argon or methane, although considerable mathematical and conceptual
difficulties delay the rigorous theory of dense fluid.
an
equation
proposed
by
(such
as
strong
polarity)
it difficult to derive a totally theoretical equation of state.
Even semiempirical equations are quite rare.
by
is
Longuet-Higgins and Widom.[4] However, the
complicated nature of the water molecule
makes
Among them
Whiting
and
Prausnitz.[5]
Another
is
One of such equations is
proposed
coworkers.[6] Essentially, these equations are modern
by
Haar
expressions
and
of
well known physical concepts suggested by van der Waals [VDW] over 100
The VDW equation can
years ago.
consist
to
of
two
the first one is repulsion term and the other attraction term.
terms:
The
considered
be
by
equation
et
Whiting
uses
al
and
Carnahan
the
Starling
equation[7] as the repulsion term and the molecular dynamic results of
However,
parameters.
be
adjustable
4
of rather sophisticated theoretical
spite
in
and
coefficients
10
has
Whiting
of
approach
can
The result of
as equivalent translational degree of freedom.
treated
equation
On the other hand, the
ideas, the accuracy is not good.
Haar
vibrations
and
rotations
the
that at high
i.e.
Prigogine[9],
density-dependent
densities, all
the
by
proposed
assumtion
simplfied
uses
In addition, the equation
Alder[81 as the attraction term.
by
al is composed of a reference function with some theoretical
et
This reference function is
basis and a purely empirical function.
an
VDW equation which is modified in part with theoretical basis
updated
quantitatively
behavior
at
the
represent
may
and in part empirically so that it
thermodynamic
temperatures and elsewhere at high
high
densities.
The motivation for this study, under the direction
the
from
came
Paynter,
H.M.
desire
accuracy
could
be
Professor
to find an equation which is
engineering
Thus, in
simpler but more accurate than those available.
calculations,good
of
obtained
reasonable
with
computational expense.
1.2
PROBLEM STATEMENT
The
problem
potential
investigated
function
such
as
in
this
study
is
to
construct
the Massieu potential or Helmholtz free
energy from which all thermodynamic property can be derived.
forementioned
equation.
simple
semi-empirical
or
Like the
equation, the starting point is the VDW
The immediate objective is to improve the VDW
theoretically
a
empirically base terms.
the objective is to find a better
functional
equation
by
More specifically,
structure
as
well
as
parameter
values,
so
as
to represent the right image of pure water
functional
,yielding accurate thermodynamic properties with a simpler
form for efficient calculations.
1.3
ORGANIZATION
In Chapter 2, a detailed historical background for
development
development
of
Thermodynamical
Function
is
of
the
with
relations
reviewed.
Chapter
of
equation
equation
the
briefly
presented here.
equation
This is followed by the direction and criteria for
of water is given.
the
the
4
state.
presented
are
regard
Main
contains
Procedures
to
the
procedures
results
and
in
of
Massieu
Chapter
the
3.
Potential
of the thesis are
discussions,
conclusions and recommendations are presented in Chapter 5.
and
CHAPTER 2
HISTORICAL REVIEW AND DIRECTON OF RESEARCH
In this chapter, various fluid models are introduced and reviewed
so that a rational direction for a derivation of new equation of state
might be found.
In addition, the criteria which appear to be
for the development will be presented.
2.1
HISTORICAL REVIEW
The equation originally proposed by van der Waals[10]
(P+a/v
2
)(v-b)=RT
(2.1)
can be rewritten as
P = RT/(v-b) -a/v
2
(2.2)
crucial
where P is pressure,
T
is
temperature,
v
is
which represents the long range attraction force.
may
Eq.(2.2)
term
The first
in
be interpreted as the intermolecular repulsion term Pr,
the second term as the attraction
the pressure is
term
Pa
In
respectively.
other
composed of
P = Pr + Pa.
-
b
volume,
for the excluded volume by molecules and 'a' is the constant
accounts
words,
specific
(2.3)
The VDW equation gives a moderately good description of P-V-T behavior
in
gas phase where the excluded volume of each molecule does not
the
liquid
only
However it gives
overlap.
phase
the
since
volume
a
qualitative
v
in
the
liquid
The
experimentally to be considerably less than b.
the
VDW
equation
the
in
description
is found
phase
improvements
to
can be either in improved repulsion or in improved
attraction term.
2.1.1
Improvements To Repulsion Term
Several attempts have been made to improve the repulsion term
the
VDW
of
equation by adopting the model of hard-sphere molecules i.e.
molecules where the two body potential function ] is given by
I'(r)= 0 , for r <a
I'(r)= oo,
for r >
(2.4)
where r is radial distance and a
molecule.
[11]
For
such
is the radius of hard core of each
molecules,
Pa
=
0.
Sophisticated
mathematical techniques have been applied to the model, in addition to
a variety of assumptions
concerning
the
properties
of
the
radial
distribution function.[Ill] Various assumptions and techniques are well
reviewed in Reference 11.
include
one
by
Relatively
succesful
Percus-Yevic-Frisch(PYF)
equations
equation
and
of
state
another
by
the
PYF
results
for
These equations are similar with
Carnahan and Starling(CS).
equation being
3
2
P =[(1+y+y )/(1-y) }( RT/v)
(2.5)
and the CS being
3
~ 23
P
P =[(l+y+y -y)/(1-y) }(RT/v)
These
compressibility
factor
which agree well with the theoretical results
hundreds)
or
hard-sphere
of
be
perfect
considered
gas[12},
experimental
data
assemblies
small
of
a computer and
with
molecules
The properties thus obtained
generated the properties of this system.
can
functional
Alder solved the equations of motion
by Alder.
(tens
the
give
equations
where y=b/(4v).
(2.6)
as those of an 'ideal liquid' comparable with the
even
they
though
may
because
directly,
model in Eq.(2.4) is an over-simplified
the
be
not
with
compared
assumption of hard-sphere
description
behavior
of
of
real molecules.
2.1.2
Improvements To The Attraction Term
On the other hand, as an improvement over
term,
the molecular dynamic results of Alder show that the constant 'a'
Pa,
in
attraction
the
the attraction
temperature
and
term
VDW
the
of
depends
actually
equation
on
By using the square well potential as the
density.
two-body potential function, he derived the constant 'a' as
4
M
a = -R(e.q/k)(r.vo)Y
I
M-1
(m.Ap/v~
n-1
)(1/T
)
(2.7)
n=1 m=1
where R is
k
is
the gas constant,e.q
is
characteristic energy per molecule,
the Boltzmann constant and r is number of segments per molecule
where vo is
with v=v/(r.vo),
the close-packed volume.
Inclusion Of Rotational And Vibrational Term
2.1.3
addition
In
to
larger
translation,
molecules
exercise
can
5
Beret and Prausnitz[1 1 expressed
rotational and vibrational motions.
the pressure with three terms, that is
(2.8)
P = Pr + Pa + Prv
where Prv is the term to
Since
Prv,
it
for
account
the
rotation
vibration.
and
difficult to construct a satisfying theory for evaluate
is
Prigogine
has
suggested
a
simplifying
that
assumption
all
density-dependent degree of freedom may be considered to be equivalent
translational degrees of freedom with
(2.9)
Prv = (c-1)Pr
where c is an adjustable parameter for
each
substance.
For
fluids
with larger molecules, c will be larger than 1, while for simple fluid
with no rotation and vibration, c=1.
2.1.4
Inclusion Of Polarity
To account for orientational forces of strongly
such
as
water,
Gmehling
et
polar
al[13] proposed an assumption that the
substance is an equivalent mixture of monomers and dimers at
equilibrium
and
substance
chemical
that the thermodynamic properties can be represented
by the hypothetical mixture.
an equilibrium constant, K
The extent of dimerization is
given
by
2
K = (CD
where
2
(2.10)
M P)
is the true mole fraction,
.
#
is the fugacity coefficient,
P is the total pressure and subscripts M
dimer respectively.
of
D
and
monomer
denote
The Eq.(2.10) is used to find the mole fractions.
composition
by
of
set
Once the fractions are known, a
function
and
constants
mixing rule,
which is
given
are
as
a
described more in
detail-in the appendix of Reference 5.
2.2
COMBINATIONS OF IMPROVED RESULTS
VDW
The more successful improvements over the
imporvements
include
equation
to both the attraction and repulsion terms.
They can be
considered in two classes.
2.2.1
LHW And Guggenheim Equations
Longuet-Higgins and Widom(LKW)[4] combined the repulsion term
given
by the PYF equation, Eq.(2.5) with the VDW expression for Pa in
Eq.(2.2).
Both of the terms have physical
same
the
combined
attraction
term
in
Guggenheim(14)
meanings.
Eq.(2.2)
with an empirical
repulsion term
Pr
=
[1/(1-y) 4
(2.11)
RT/v
which is algebraically simpler than the
almost identical predictions.
PYF
equation
but
leads
The equations are
2
3
[ 2+~
;LHW
a/v
RT/v
]
)/(1-y)
P = [(1+y+y
(.2
(2.12)
P = [1/(I-y) ] RT/v - a/v
(2.13)
;Guggenheim
to
bo).
(or
b
a,
Both of them are have two free parameters
Usually,
excluded volume b is determined either by using the relation
3
b = 2 7rNau /3
a/2 is radius of hard core
where Na is the Avogadro's number and
of each molecule , or by matching a point of interest such as critical
The attraction parameter 'a' is evaluated primarily by
point.-
VDW,
Guggenheim(G)
and
LHW
comparison
is
condition,
t
given
the
in
triple
with
Table
1
data on argon.
experimental
[14],where
c
of
accuracies
the
Guggenheim compared
matching points of interest.
denotes
The
critical
point, Tb the Boyle temperature and vi the
specific liquid volume at the triple point.
Table 1. Comparison of The Three Equations
with Experimental Data of Argon
LHW
Guggenheim
0.375
0.36
0.36
0.29
Tb / Tc
3.375
2.68
2.72
2.73
vi / vc
0.377
0.302
0.303
0.374
Tt / Tc
0.350
0.535
0.528
0.557
VDW
(Pv/RT)c
2.2.2
The Equation By Whiting Et Al
Whiting
Starling
term
Experiments
for
and
vibrational
and
the
adopt
term
Prausnitz
used
the
expression
of
Carnahan
and
repulsion term, Alder's results for the attraction
Prigogine's
to
assumption
represent
water.
for
In
the
the
rotational
form
and
of partition
Q, their ruslt is expressed as
function
Nc
Nc
N
N
Q = (1/N!)(V/A ) (Vf/V).exp[-5 /2kT].(q)int
(2.14)
where N is the number of molecules in total volume at temperature T,
Vf is the free volume, k is the Boltzmann constantA is the de Broglie
D/2 is the potential energy per molecule and where
wavelength, and
c is the parameter to account for the rotation and vibration as
The free volume
introduced in Section 2.1.3.
ln (Vf/V)
where
T
=
1/18
=
2
[3( r/v)
(2.15)
and
n m
5
2
/2kT
2.
(1- r/ v)
4( r/v)]/
-
Y. (Ann-/T V
=
)
(2.16)
n=1 m=1
with T
By
=
that
assuming
water
real
mixture of monomers and dimers,
orientational
forces
due
be represented by a hypothetical
can
and
Whiting
equation of state has in principle
Q
8
adjustable
assumptions, they reduced the number of parameters to
4.
by
using
a
fit
to
parameters
of
However,
mixture
of
The resulting
set
parameters
This
to the strong polar nature of water.
and the details are shown in Appendix A of Reference 5.
data.
for
account
Prausnitz
term is expressed in the form of the partition function
experimental
Eq.(2.7).
in
defined
T/(eq/ck) and all other variables are
simplifying
final
The
are Tm*, Vm*, AHo and ASo, where the superscript * denotes
characteristic values, subscript m represents monomers, and where
and
AHo
ASo are the enthalpy and entropy of dimerization in the saturated
state respectively.
Incidentally, the
parameters
'a'
and
'b'
are
4.5%
and
included in Vm*, and 'Anm' respectively.
For water, the equation gives the accuracy of within:
4.6% deviations of saturated liquid volumes and vapor volumes from the
triple point to just below the
superheated
volume;
0.1%
critical
deviation
point;
2.2%
deviation
of
of saturated pressure, up to a
temperature of about 1400F and a pressure of about 14000psia.
2.3
2.3.1
REQUIREMENTS AND DIRECTIONS FOR THE RESEARCH
Requirements
Before the detailed direction of the research
seems
proper
is
discussed,
it
to specify the requirements for the further development
of equation of state, primarily from the previous review of historical
backgrounds,but also partially from consideration of the intended use
of the equation.
The criteria according to which the results must be judged may be
stated as follows:
1) The properties generated by the resulting equation of state
must be as accurate as possible for its level of complexity.
2) The equation of state must be as simple as possible for the
economy of computation.
3) The equation of state should be able to acccurately
represent water in the two phase region, since the intended
use of the equation is mostly to anyalyze performances of
steam turbine cycles.
4) It is preferred that all terms in the equation have
identifiable physical significance.
5) The properties should be generated to be selfconsistent
throughout the desired interval.
2.3.2
Direction
was shown in
It
term
became
an
Eq.(2.9)
resulting
reduces to Eq.(2.3) with a
it
addtion,
is
term.
repulsion
additional
modified
vibrational
and
that the rotational
Accordingly,
Eq.(2.8)
In
term.
repulsion
that the attraction term a(v,T) in Eq.(2.7)
obvious
constant
makes the equation of state much more complicated than
'a'.
Thus, it appears to be worth attempting:
1) to improve the accuracy of equation of state by modifying the
repulsion term; and
2) to improve the simplicity of the equation by keeping 'a' in
the attraction term constant.
modifying
The improvement on the repulsion term would be made by
terms
repulsion
the
introduced in Section 2.1 with empirical terms (even
though they may not be clearly correlated to microscopic
behavior
of
molecules).
Since
potential
satisfy
function
consistent with each other.
continpus
transition
from
phase to another phase.
the
it
a
thermodynamic
relations,
Maxwell's
Additionally,
one
by
generated
properties
thermodynamic
can
give
they
are
smooth
and
state to another as well as from one
Therefore,
it
is
much
more
desirable
to
construct the equation of state in the form of thermodynamic potential
function such as the Helmholtz free energy or alternatively the
Massieu
the
Potential
direction may
Function(MPF).
be stated
According to these considerations,
as:
--------------------------------------------
To construct an equation of state in the form of thermodynamic
potential by making improvement on the repulsion terms in a
macroscopic way.
----------------------------------------------------------
CHAPTER 3
PROCEDURES OF DEVELOPMENT
In this
Massieu
chapter,
Potential
thermodynamical
relations
regarding
Function(MPF) are briefly reviewed.
expressed in terms of a set of reduced properties.
REVIEW OF THERMODYNAMICAL RELATIONS
the
Then they are
After various
are assessed, the procedure of proposed improvement is given.
3.1
to
MPF
Thermodynamical Relations
3.1.1
The Massieu Potential Function(MPF) J, which has been named after
its originator, is defined as
J = S -
Helmholtz
and
F
where S, U,
U/T
respectively
are
T
(3.la)
- -F/T
entropy,
free energy and absolute temperature.
function J is
internal
energy,
The negative Massieu
then
J
= -
=
J
(3.1b)
F/T
By using the well known relation
P = -(
(3.2)
aF/ aV)T
and Eq.(3.1), where P is pressure and V is total volume, we
obtain for temperature and internal energy the expressions
(3.3)
P = (aJ/ a V)T
7-P =P
(aJ/8P) T
and
U = (aJ/ar
where
T
(3.4)
)
is the inverse of temperature and
respectively.
With
Eq.(3.la),
the
p is the density
remaining
expressions
for
properties can be expressed as
S = U/T -
(3.5)
J
and
(3.6)
G = H-TS = JT + PV
where G and H are te Gibbs free
energy
and
enthalpy
respectively.
The
negative
MPF
then
can
be
to
used
completely
construct the
thermodynamic properties of the material.
3.1.2
Reduced Form Of The Thermodynamical Relations
The negative MPF for the VDW equation can be easily derived as
J/m = R ln b/(v-b) -
,
a/v/T + J,/m
where the first term on the right hand
side
can
be
(3.7)
interpreted
as
term, the second as attraction term and the third term Jr/M
as a part of the PF which is function of temperature only, with m and
repulsion
v, mass and specif ic volume of the material.
If the MPF and corresponding state
equations
are
expressed
in
reduced forms, not only they become more compact, but also comparisons
terms
between different
are
easier.
The
reduced
properties
defined as
(3.8a)
bo/v;
y
2
(bo /a)P;
(3.8b)
(a/R bo)/T;
(3.8c)
J
(J/m)R;
(3.9a)
J'r
(JT /m)R ;
(3.9b)
P
=
and
(3.9c)
JyA J - JT
where Jy is
a part of the reduced MPF excluding J and with bo the
actual volume of one mole of molecules, given as
24
are
bo = Na (
where
7r
/6)a
= b/4
a/2 is radius of hard core, Na the Avogadro's number and b
the
in
the excluded volume
VDW
equation.
The
corresponding
thermodynamical relations for the reduced properties become
P = y (aj/
(3.10)
ay)T
(3.11)
u = (aJ/ar)y
G
=
s
u T
(3,13)
- J
u + P/y
h
3.1.3
(3.12)
J + P/y
(3.13)
The MPF In Reduced Form For Existing State Equations
The MPF's of the VDW, LHW, Guggenheim(G) and CS are
(Jy)
= In (4 y)- ln(1-4y) -
(3.15)
VDW
(Jy)
= in y
-
ln(1-y)
(3/2)(1-y)
yr +
-
yHW
(Jy)
=
in y
(3.16)
-
ln(1-y)
-
(1-y7
YT +
+
-2
+
(1/2)(1-y)
+
(1/3)(1-y)
+
(3.17)
and
25
(Jy)
-1
2(1-y)
+
+
= ln y
CS
(3.18)
+
The similarities and differences
the
relationships
expressions
resulting
The
displayed.
of
are
clearly
f or
the
pressure-volume-temperature become
P
=
VDW
3
P
=
2
(3.19)
2.
(3.20)
- y
[y/(1-4y)]/T
[y(1+y+y )/(1-y) I/T - y
~LHW
4
P
=[y/(1-y)
2
]/T
-y
(3.21)
and
P
-
[y(1+y+y2 -y 3 )/(1y)3 ]/
(3.22)
~-CS
3.1.4
Assessment Of Equations In Terms Of Saturation Properties
One way to assess the accuracy of the formulations is to
the predicted and measured saturation properties.
compare
The thermodynamical
relations in saturation states are given as
Tf
= Tg
(3.23a)
P(vf,Tf) = P(vg,Tg)
(3.23b)
G(vf,Tf) = G(vg,Tg)
(3.23c)
and
where f and g designate saturated liquid and vapor.
For the reduced
variables defined in Eqs.(3.8), the saturated expressions become
.rf
=
g
(3,24a)
P(yf,Tf) = P(yg,rg)
(3.24b)
G(yf,rf) = G(yg,rg)
(3.24c)
and
3 2
Eqs.(3.23a)-(3.23c) with Eqs.(3.12),(3.19),( . 0) and (3.21) yield the
The saturation curves of the VDW, LHW and
saturation properties.
water are compared in Figure 1 in reduced coordinates
subscript
the
where
c
denotes
critical
condition.
coordinates, saturation properties are independent of
'a'
and
'b'.
and
T/Tc
In
these
parameters
the
As can be seen, the equation suggested by LHW improves
there
the agreement considerably, even though
mismatch.
P/Pc
may
This
still
is
be viewed as a positive aspect.
the Eq.(3.16) gives better saturation curve than
significant
That is, since
Eq.(3.15)
by
using
one more term, it may be probable that a still better saturation curve
might be obtained with a relatively simple equation of state.
3.1.5
Assessment In Terms Of Compressibility Factor
From Eqs.(3.8a),(3.8b) and
can
it
(3.8c)
be
shown
that
the
compressibility factor Z is
(3.25)
Z = Pv/RT = P r /y
The corresponding expressions for the compressibility factors of the
VDW, LHW and Guggenheim equations are
= 1/(1-4y)
Z
VDW
Z
LHW
and
z
G
-
y r
(3.26a)
= (1+y+y )/(1-y
-
y r
~
(3.26b)
= 1/(1-y)
-
y
(3.26c)
T
101
103
1. 1
1 .0 1 .1
1.2
1.3
1.1-
1.
1.6
1.'7
FIGURE 1. COMPARISON OF SATURATION CURVES OBTAINED
FROM VDW EQUATION AND LHW MODEL WITH WATER
1.8
1.9
Tc/T
2.0
Each expression may be expanded in power series by using the relation
1/(1-x)
= 1 + x2 + X3+
x4 +..........
ixi < 1
;
The series expansions of the three equations are
2
Z
VDW
(3.27a)
+ 16y +...
1 + (4-
T)y
1 + (4-
T)y +
2
Z
LW
1Oy
and
(3.27b)
+...
2
Z
,
G = 1 + (4- r)y + 10y +...
Since the virial coefficients B, C, D ...
are defined as
3
2
Z - 1 = B/v + C/v
(3.27c)
+...
+ D/v
it can be shown from Eqs.(3.8a),(3.8c) and Eqs.(3.27) that the second
virial coefficients, B of all the three equations are linear functions
of inverse temperature, 1/T as
B = 4bo -
(3.28)
(a/R)/T
are
determined,
as
which
bo,
The second virial coefficient depends on values of 'a' and
described in Section 2.2.1, either by matching a
point of interest or by theoretical calculations.
Measurements
coefficient
of
of
show
properties
water
water
schematically
in
Figure
in
2.
Eq.(3.28) gives poor accuracy, especially in
In
fact,
virial
second
water is a complicated function of 1/T.[2] The second
virial coefficient of the three equations
with
that
Eq.(3.28)
is
compared
The comparison shows that
low
temperature
range.
this aspect has been regarded as an intrinsic deficiency of
the VDW-type equations in predicting the compressibility factor.
Since the proposed improvement in this study
is
to
modify
the
-.
1/T
FIGURE 2. SCHEMATIC DIAGRAM OF THE
SECOND VIRIAL COEFFICIENT
repulsion
term
the attraction term kept constant, the
in
'a'
with
second virial has the same functional form as
intended
Thus,
Eq.(3.28).
the
equation of state will have more or less the same deficiency
in predicting the compressibility factor as the three discussed above.
On the other hand, a significant mismatch in saturation data might be
of
accuracy
corrected without much sacrifice in the
compressibility
factor. -This point will be discussed later in Section 4.1.3.
3.2
3.2.1
PROCEDURES OF THE PROPOSED IMPROVEMENT
The Formulation Of A Generalized MPF
Eqs.(3.15),(3.16),
(3.17)
of
forms
functional
of
Observation
suggested
(3.18)
and
MPF's
various
to
in
Professor
H.M.Paynter a generalized repulsion term in MPF, which is expressed as
i
n
J = ln y - Xln(1-y) +
-Y
Y
Mi/(1-y)
i= 1
For example, this equation reduces to the repulsion term of the
following special equations:
X=1;
'Li=
LHW:
X=1;
42 =3/2;
Guggenheim:
A =1;
P1
CS:
A =0;
/1 =
=
for all i
0
VDW:
i
=
for all other i
0
1; )2 =1/2; 1'3 =1/3; li=O for all other i
2; #2=
1 ;
Defining x=4y, P=16(bo /a)P and
"i
=
0
for all other i
T =4(a/Rbh)/T, then the MPF
of the VDW equation, Eq.(3.15) can be expressed as
Jy = in x -
ln(1-x) -
(3.29a) (3,29a)
x
If it is noticed that the MPF
Jy = in y -
(3.29b)
ln(1-y) - y _
and Eq.(3.29a) yield the identical saturation curves in the normalized
coordinates
P/Pc and Tc /T and that the saturation curve of Eq.(3.16)
equation,
is closer to that of water than that of the VDW
1
'i/(1-y)
in
the
This effect is shown
repulsion term becomes clear.
clearly in Figure 1.
of
effect
If the usual VDW attraction term is
included,
the general MPF becomes
n
Jy = In y - Xln(1-y) +
:
y
-
/li/(1-y)
(3.30)
T
i=1
The corresponding equation of state for pressure and compressibility
factor are
n
P
=
[y +xybl-y)2 + y
Yi. P i/(1-y) ]
-
y
(3.31)
i=1
n
Z
=
1 +Xy/(1-y 3 + y
V
i.
Jli/(1-y
-
y
T
(332)
i=1
The theoretical basis of the MPF given by Eq.(3.30)
to
may
be
open
serious questions, but its simplicity and descriptive ability make
it attractive for further use, provided that it offers some advantages
in trade-off for accuracy and computation time.
Procedures Of Selection Of Parameter
3.2.2
Eq.(3.30)
i+1
has
explicitly
free
shown
The
paramenters.
is to choose values of i and, by use of the saturation data,
approach
to determine the best values of x and
for
results
is a best
of
Comparison
i's.
the
the
different values of i is then used to determine if there
value
i.
of
detailed
The
procedures
may
adopted
be
summarized as follows:
1.
Determine number of terms by examining the results for several
values of i in Eq.(3.30).
2.
Change step by step values of
A and I1i's.
3. Obtain the critical point which is uniquely determined by the
above set of parameters by solving the following equations
simultaneously for yc and
(P/
ay),
=0;
Tc, and Pc accordingly,
(9P/ ay )
=0
where c designates the critical point.
4. Determine saturation curves by solving the following two
equations for several
T_'s
betweenr
=
Tc
and
r
=2
Tc,
(3.33)
P(yf,r) = P(yg,T); G(yf,r) = G(yg,r)
where f, g designates saturated liquid and vapor respectively.
Thus, obtain yf, yg and P for each
T.
5. Compare these sets of saturation properties with those of
water for several r using the quantitative measure I, as
n
I = X
i=1
2
[ 1 -
(P/Pc)m /
(P/Pc)wI
(3.34)
which represents deviations between the two, where m and w
represent fluid model and water respectively.
6. Repeat the steps 2 to 6 until I becomes minimum.
For the procedures, an algorithm was implemented on the
computer
to solve the nonlinear equations
P(yfrf) = P(yg,rf)
G(yfTf) = G(yg,rf).
The program
is
emphasized
that
listed
the
in
Appendix
above
values of the parameters a and
A.
procedures
b,
since
In
addition,
it
may
be
were done independent of the
in
saturation properties are independent of these.
the
reduced
form
the
CHAPTER 4
RESULTS AND DISCUSSIONS
4.1
RESULTS
4.1.1
The Resulting Equation Of State
After executions of the procedures described in Section
3.2,
it
was found that neither increasing n in Eq.(3.30) to more than two, nor
chosing
least
X other than unity
the
improve
in terms of saturation properties.
significantly,
accuracy
at
Thus any serious compromise
between accuracy and simplicity is not required if the values of i=1,
X=1
and
sti=9
are
selected.
The MPF and the equation of state for
pressure become as follows:
Jy = in y - ln(1-y) +
2
P =
9
/(1-y)
yr
-
2
2
[y/(1-y) + 9y /(1-y)
(4.1)
/r
-
y
The corresponding saturation data are shown in Table
(4.2)
2
and
compared
those of VDW equation and water obtained from KKHM steam tables.
with
The present results are presented and compared with VDW results of the
properties
saturation
and
water graphically in Figures 3 and 4.
vs.
behavior
temperature
than
better
the
VDW
pressure
the
represents
can be seen, the present results accurately
As
the two phase region and are somewhat
in
pressure-volume
the
predicting
in
equation
relationship in the saturated states.
Since the equation of state of P generated by the general form of
in
MPF
reduces
Eq.(3.31)
of ideal gas as density P-(or y)
that
to
become smaller, as should be the case, the behavior in this region
is
discussion
of
shown
not
in
great
detail
here.
The
detailed
compressibility and the second virial coefficient is given in
Section
4.1.3.
4.1.2
The Other Parameters 'a' And 'b'
equation
In principle, the excluded volume b in the VDW
calculated
from
intermolecular
motions.
However,
may
the
be
large
uncertainties in the knowledge of intermolecular motions of water make
it
more
accurate
to
obtain b by fitting the P,V,T data at a proper
Then the other constants a and
point. of interest.
simultaneously from Eqs.(3.8a) and (3.8b).
determined only by one point of fitting.
are
b = 4bo
bo;
v y
2
a = bo (P/P)
R
=
T(a/bo)/T
and
are
determined
It may be noted that since
the three parameters a,b and R are interrelated to
are
R
each
other,
they
The resulting relations
TABLE 2.
COMPARISONS OF SATURATION DATA
WATER
Tc/T
P/Pc
vf /vc
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
1.000
0.481
0.235
0.115
5.572
2.679
1.272
5.966
2.809
1.263
1.000
0.468
0.410
0.380
0.362
0.349
0.340
0.333
0.328*
0.324
Tc = 1165.44R,
-2
x 10 -2
x 102
x 10 3
x 10 -3
x 10
x 10
vg/ve
1.000
5.297
12.62
25.02
50.53
101.4
204.2
415.0
852.6
1775.
Pc = 3204 psia, vc=0.05053 ft/lb
PRESENT RESULT
Tc/T
P/Pc
vf/vc
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
1.000
0.512
0.256
0.123
5.934
2.731
1.217
5.262
2.029
9.203
-3
1.000
0.656
0.530
0.463
0.420
0.390
0.367
0.349
0.334
0.322
Pc = 2.0145 x 10
,
x
x
x
x
x
x
-- 2
10 -2
10 -2
10
10 -3
10
10
Ic = 18.4 97,
vg/vc
1.000
5.049
10.21
22.24
45.56
94.74
202.1
443.0
1000.
2325.
yc = 0.1062
VAN DER WAALS EQUATION
P/PC
Tc/T
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
1.000
0.675
0.508
0.317
0.222
0.156
-2
0.110
2
7.748 x 10
5.492 x 10 2
3.896 x 10 -2
Pc = 3.7037 x 107
*
vf/vc
subscript
=
3.375,
1.000
0.615
0.540
0.499
0.473
0.454
0.440
0.429
0.420
0.413
vg/vc
1.000
2.227
3.440
5.045
7.100
9.874
13.58
18.53
25.18
34.11
xc = 0.3333
'c' .denotes the critical point;
S= (a/Rbo)/T; y
=
bo/v;
x=4y; - =4T
101
102
v v v v
PRESENT RESULTS
1. 1 1.2
1.0
FIGURE 3.
1.3
1.4
1.5
1.6
1.7
1.8
COMPARISON OF CALCULATED SATURATION DATA
WITH PROPERTIES OF WATER, PRESENT RESULTS
AND THE RESULTS.OBTAINED FROM VDW EQUATION.
1.9 2.0
Tc /1
100
101
102
0310 0
10~1
101
102
10 3
v
FIGURE
4. COMPARISON OF CALCULATED SATURATION DOME
WITH PROPERTIES OF WATER, PRESENT RESULTS
AND THE RESULTS OBTAINED FROM VDW EQUATION
/Vc
where P,v,T were set properties of water at a certain state while
and
r
were
set
reduced
values in the MPF.
points could be a state in superheated
One reasonable fitting
region
where
a point on the saturation curve where the
by
generated
curve
of
water
and
that
For the former case, when
present equation cross.
the
thermodynamical
Another would be
of water approaches that of the ideal gas.
behavior
P,y
T=900 F and for the latter when Tc/T=1.56, the constants
a,bo
and
R
obtained are shown in Table 3.
Incidentally, the constant
R
well
may
be
obtained
from
the
relation:
R = R / M
where R
is
However,
the
universal
gas
constant
and
M
molecular
weight.
for the sake of consistency, it was determined by Eq.(3.8c).
It can be observed in Table 3, however, that the two R's are not
different,
at least for the fitting points chosen either in saturated
vapor region or in the superheated region.
constant
much
As can be
predicted,
the
R gives better agreement when evaluated at a superheated (or
gaseous) state than a saturated state.
Table 3.
4
2
a (ft -lbf/lb )
4
2
g/gm )
(cm 3
bo(ft /lb)
The Values Of Constants Obtained
Superheated
Saturated
State Fitted
State Fitted
4
1.4964 x 10
7
4
1.4184 x 10
-3
-3
8.0848 x 10
7.8267 x 10
(cm /gm)
0.5044
0.4883
R (Btu/R-lb)
0.1103
0.1069
4.622
4.476
3
(J
/K-gm)
7
1.7864 x 10
1.8854 x 10
R = R / M = 4.615 (J / K-gm)
R = Universal gas constant; M = Molecular weight
Comparison Of The Compressibility Factor
4.1.3
The compressibility factor Z of the fluid model of
calculated
by
using
Eq.(4.1)
was
Eq.(3.32) and compared in Table 4 to the values
for water as determined from KKHM steam table data.
Table 4. Comparison Of Compressibility Factor
Tr=T/Tc, yr=y/yc, y=b/4v
subscript c designates critical value
yr
0.8
0.012
0.9760
0.9833
0.065
0.8366
0.9103
0.010
0.9852
0.9910
0.051
0.9266
0.9545
0.104
0.8528
0.9085
0.423
0.4994
0.6593
0.008
0.9930
0.9952
0.085
0.9375
0.9527
0.366
0.7627
0.8200
0.816
0.6000
0.6940
1.0
1.2
To
Zmodel
Tr
better
display
Zwater
the
discrepancies,
the
second
virial
coefficient
B
is derived for the present model.
From Eq.(3.32), the
compressibility Z is expressed as
Z = 1/(1-y) +
9
2
y/(1-y) -
y
(4.3)
T
As a power series, it is expressed as
Z
=
1 + (10
-
+
(4.4)
19y +...
the second virial B becomes
From Eqs.(3.8a) and (3.8c),
B = 10bo -
-)y
a/R/T
(4.5)
By using the two sets of a,bo and R previously obtained, the
corresponding second virial coefficients Bi and B2 obtained
as
3
BI = 4.960 -
5.94617 (cm/g)
(4. 6a)
3
B2 = 4.700 - 5.81837 (cm/g)
(4.6b)
where B1 is for superheated water and B2 for saturated water, with
I A 1000/T K
The empirically derived expression of the second virial equation
for water suggested, in Reference 2, by Keyes is
B = 2.0624 -
- = 0.1008T /
where
The three equations
Figures
5a
and 5b.
2.81204T
10
(1 + 0.03497
Eqs.(4.6a),
(4.6b)
(cm /g)
(4.7)
)
and
(4.7)
are
Also values are tabulated in Table 5.
plotted
shows a linear plot of B vs.
in
Figure 5a
c, while 5b shows a logarithmic plot.
0
-10 -20 -30 -40 -
o2
- 50
-
C
-60 -
a
z
o
-'70
v9
-0
-80 -
-100
0.5
1.0
1.5
2.0
2.5
4.0
3.0
TAU(1000/T K)
FIGURE 5a.
COMPARISON OF THE SECOND VIRIAL COEFFICIENTS
OF THE PRESENT RESULTS WITH WATER(LINEAR SCALE)
102
C)
'K
I
A
-
L-
C)
02
101
-J
WATER
a
z
100
------*
0
* *
Bl(evaluated at the
superheated state)
*
B2(evaluated
at the
saturated state)
(9
w
(:)
10n'
I
O.5
I
1.5
I
I
2.0
3.0
TAUII100
FIGURE 36.
K)
COMPARISON OF THE SECOND VIRIAL COEFFICIENTS
OF THE PRESENT RESULTS WITH WATER
(LOGARITHMIC SCALE)
0-.
Table 5.
3
OF SECOND VIRIAL B(cm 7g)
COMPARISON
(B)water
B1
1.00
-1.21
-0.986
-1.12
1.40
-3.54
-3.36
-3.45
1.80
-7.18
-5.74
-5.77
2.20
-13.0
-8.12
-8.10
2.60
-22.1
-10.5
-10.4
3.00
-36.3
-12.9
-12.8
3.40
-58.0
-15.3
-15.1
(1000/T K)
B2
The examination of the equation for the second virial-coefficient
from the fluid model shows that the two selection techniques
obtained
As
has
been
mentioned previously, the equation of state constructed
deficiency
has an inherent
temperature.
its linear
factor.
compressibility
for a and bo do not significantly affect the
It
in
the
compressibility
at
factor
low
is not so surprising that the mere constant 'a' and
representation,
Eq.(4.5)
have
such
a
deficiency
when
compared to the empirical expression of Keyes.
4.1.4
Techniques
To
Improve
The
Agreement
Of
Predicted
Compressibility
One way to improve the compressibility factor would be to
select
a, b and R so that the deviation from measured data would be minimised
for a certain temperature range.
For instance, the linear regression
of the second virial between r =0.8 and 3.0 gives
B =
17.32 -
15.98 T (g/cm )
which is substantially different to the results
as
points
parameter
represented
considerably different values
undesirable
new
superheated
states.
additional
of
terms
with
by
Eqs.(4.6).
bo
and
at
both
inaccuracies
Thus
a,
the
improvements
a
obtained
R,
be
cause
and
states
to
form.
requires
will
saturation
should
fitting
This
which
functional
different
by
incorporate
For example,
Whiting et al[5] modified by adding an empirical term in the
form
of
the Helmholtz free energy which is
n
Fsv = (d/Tr ) exp (-Akp)
,
(4.8)
where Tr=T/Tc, d, n are constants which are determined from the
second-virial-coefficient data, and A
for a
better
Eq.(4.8)
may
third
be
is a positive constant
The
coefficient.
virial
incorporated
correction
for
the
future
in
in Eq.(4.1) provided that it does not
affect the computed saturation curve significantly.
attemptng
term
It may
be
worth
of the equation of state in
improvements
Eq.(4.1).
4.2
DISCUSSIONS
The new equation may
be
evaluated
in
terms
of
the
criteria
adopted, which were presented in Chapter 2.
Accuracy:
In the saturated region, for most of states from the critical point to
about 1.5 atm., the agreement between calculated pressure and the KKHM
steam table data
agreements
are
is
within
within
1
9%.
%,
with
In
the
superheated
region,
the
a temperature range from 200F to
1400F, a pressure range of 5 psia to more than 2000 psia.
Simplicity:
The equation has of four free parameters, which are a,bo,R and
and has only one more term than the VDW equation.
In addition, it
is easy to convert from one set of independent variables to another.
Saturation curve:
The present equation of state gives two phase region which is
similar
to water.
Physical Significance:
The theoretical basis of Eq.(3.30)
meaning
and
in
particular
of the term, 9/(1-y) are open to questions.
the
physical
It is difficult,
so far, to relate them to the knowledge of intermolecular motion.
45
CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS
5.1
CONCLUSIONS
A new equation
Function
Potential
state
of
for
been
has
water
in
the
The
derived.
of
form
generates
equation
As
relatively accurate-properties, considering its simplicity.
in
Massieu
Figures 3 , 4 and Table 2, it gives much more accurate values than
the VDW equation even though it
While it
has only one more term.
response
be useful when employed either for the analysis of transient
a
steam
turbine cycle, where the Runge-Kutta method or a related
method of solving nonlinear differential equations require
equation
of
derivatives.
for
may
adequate for calculations which require high accuracy, it may
be
not
in
shown
the
for
state
the
evaluation
Further it may be employed as
further
refinements.
form this equation of
state
various
of
a
simplified
steam property
reference
function[61
Finally, the simple closed functional
allows
explicit
evaluation
of
formal
derivatives, and this may be an advantage over the empirical equations
with complicated functional form or tabulated data.
5.2
RECOMMENDATIONS
In
assumed
the previous development of equation of
the
that
al[6] showed
that
b
b is constant.
volume
excluded
depends
on
state,
temperature.
it
has
However, Haar et
They
obtained
of the parameter b by fitting to the steam table data.
expression
been
the
It
is given as
-5
-3
b(T) = b' + c ln T + d T + e T
where T = T/647.073 K and b',c,d
values.
Note
that
since
the
e
and
are
determined
empirically
saturated curve was obtained without
regard to b, varying b with T does not change the basic shape
curve.
of
the
It would be worth attempting to examine whether and how much a
variable b(T) can improve P-V-T relationship.
Another
representation
improvement
to
might
be
to
find
an
approximate
Alder's attraction term as represented in Eq.(2.7)
by some mathematical means like least square fittings.
As suggested in Section 4.1.4, an additional term might
to improve the accuracy of the compressibility factor.
be
used
Bibliography
1. J.Straub, "Equilibrium Properties of Water Substance 1974-1979, Tasks and
Activities of Working Group I," Water and Steam, Proc. of 9th Int. Conf.
on Properties of Steam, edited by J.Straub and K.Scheffler.33
2. J.Keenan, F.Keyes, P.Hill and J.Moore, Steam Tables. John Wiley & Sons:
N.Y.:1969
3. J.Campbell and R.Jenner, "Computational Steam/Water Property Routines,"
" Water and Steam, Proc. of 9th Int. Conf. on Properties of Steam,
edited by J.Straub and K.Scheffler.142
4. H.Longuet-Higgins and B.Widom, "A Rigid Sphere Model for the Melting of
Argon," Molecular Physics, 8: 549(1964)
5. W.Whiting and J.Prausnitz, "A New Equation of State for Fluid Water Based
on Hard-sphere Peturbation Theory and Dimerization Equilibria,"" Water and
Steam, Proc. of 9th Int. Conf. on Properties of Steam, edited by J.Straub
and K.Scheffler.83
6. L.Haar, J.Gallagher and G.Kell, "Thermodynamic Properties for Fluid Water,"
" Water and Steam, Proc. of 9th Int. Conf. on Properties of Steam, edited
by J.Straub and K.Scheffler.69
7. N.Carnahan and K.Starling, "Intermolecular Repulsion and The Equation of
State for Fluids," Journal of AIchE, 18: 1184(1969)
8. B.Alder, D.Young and M.Mark, "Studies in Molecular Dynamics,"
Journal of Chemical Physics, 56: 3013(1972)
9. I.Prigogine, Molecular Theory of Solutions
North Holland: Amsterdam, 1957
10.J.van der Waals,"Over de continueteit van den gas,"
(Dissertation, Leiden), 1873
11.H.Temperley, J.Rowlinson and G.Rushbrooke, Physics of Simple Liquids
John Wiley & Sons Inc.:N.Y., 1968
12.B.Alder and T.Wainwright, " Studies in Molecular Dynamics II," Journal
of Chemical Physics, 33: 1442(1960)
13.J.Gmehling, D.Liu and J.Prausnitz, "High Pressure Vagor-liquid Equilbria
for Mixtures Containing One or More Polar Components
14.E.Guggenheim,"Variations on van der Waals' Equation of State for High
Densities," Molecular Physics, 9(1965)43, 199
15.S.Beret and J.Prausnitz," Perturbed Hard-chain Theory. An Equation of
State for Fluids Containing Small or Larger Molecules," Journal of
AIchE, 18: 1125(1975)
Appendix A
In saturated states, the relations in Eq.(3.21) hold. If Eq.(3.31) is
combined with Eq.(3.24b), then
2
2
2
2
2
2
yf -yg + A ( yf /qf - yg /qg ) + tI(yf /qf - yg /qg ) +
+ 2.L2 yf2 /qf - y2 /qg ) +
r(yf
~~
-
= 0
where q = 1-y.
=
-
yg)
(A-i)
Then
yf
(yf2 /qf
yg +
-
yg 2 /qg ) +
-
(yf 2 /q2- yg
+
q)
(A-2)
...
yg 2
Yg
2
yf
Besides, From Eq.(3.12) and Eq.(3.21), we can derive
Jf - Jg + ' (Pf/yf - Pg/yg) = 0
(A-3)
If we combine Eq.(A-3) with Eqs.(3.30) and (3.31), then
ln
yf/yg
-
Then
-
kln qf/qg + pL(i/qf
-
r (yf
-
yg)
+
I/qg)
z (Pf/yf -
+
Pg/yg)
p
2
(i/qf
-
i/qg
+
= 0
(A-4)
2
2
A
ln (yg/yf)(qf/qg) =I'l(1/qf - 1/qg) + lt2((L/qf -i/qg )+
+.......-
T
(yf-yg)
..
(A-5)
If the right-hand-side of Eq.(A-5) is set X, then Eq.(A-5) becomes
yg = yf (qf/qg)
.exp(X)
(A-6)
Both Eqs.(A-6) and (A-2) constitute the algorithm, which are now
_i
yfi -
ygi +
yf
yg i+1= yfi (qf
where i =1,2,..
.
x (yfi/qfi- ygi/qgi +..
- yg
/qgi)
Incidentally, yg
.exp(X
can be determined empirically, by
yg 1= e qf 2
where e is required to adjust.
The usual values range from 0.1 to o.3
(A-7)
DEVELOPMENT OF A NEW EQUATION OF STATE
FOR WATER AND ITS APPLICATIONS
PART II
APPLICATIONS
ABSTRACT OF PART II
was
substance
water
A new equation of state for pure
applied
to two problems associated with steam turbine cycle
calculations,
in
equation
using
of
the
advantages
evaluate
situations.
engineering
practical
to
the
evaluation was made
speed,
computional
in
primarily
accuracy
set
another
with
comparison
in
of
terms
the
The
and
of
a
This other set of equations
equations of state.
of
of
equations
standard
development
more
follows
engineering calculations and, has
determined
for
coefficients
dependent properties.
function
potential
each
In
empirically
of
combination
energy,
equation
the
for
and
thermodynamic
it was
P-V-T behavior,
the
internal
of
independent
the
to
addition
describing
necessary to determine the
portion
This
entropy.
36
state
enthalpy
of
state
and
was
expression
der
Waals
constructed by incorporating the van
and making a curve-fit to the Keenan- keyes-Hill-Moore(KKHM)
steam table
data.
The first
problem was
line
expansion
problem, the equations of
on
the
program,
to determine
steam
partial
of
were
used
state
in conjunction with an
for evaluating
the
thermodynamic states
performance
loads.
in
a
empirical method by
of
In this
computer
Spencer
turbines.
steam
The
two
sets of
speed
with
the
computational
and
and
compared.
were
measured
of
state
equations
the
transient
simulate
was
to
problem
The
second
of a heat exchanger typical of
part
a
small
of
behavior
and
state
of
equations
The
cycle
systems.
steam
model the dynamic
applied
to
of
were
laws
conservation
accuracy
characteristics of the component.
The
transient
response,
was simulated using the
load
changes,
to
subjected
when
11/780.
on
a
VAX
implemented
DYSYS
as
program package
and computational speed were made
of
accuracy
Comparisons
for
this problem, again using the two
The results
show the
advantage
sets
of
of equations.
the
equations
of
Compared to the more typical
state developed in this study.
and
approach, the accuracy is equivalent or slightly better
the
computational
time is reduced by a factor from 2 to 6,
depending on the particular problem.
Further computational advantages might be
achieved
by
algebraic
manipulation
of the simple equations of state in
the present study.
Considering
that
their
simple
closed
forms
allow
formal
derivatives,
that
most
terms
are
physically identifiable,
the
equations
might
find
their
usefulness
in applications where functional dependences are
needed and where the empirical
equations
with
complicated
expression can not be used.
The accuracy of the equation of
state for the internal energy would be
improved
by
adding
terms similar to those of the KKHM formulation, which should
give more accurate enthalpy and entropy.
CONTENT OF PART II
Page
List
Of Tables..........
List
Of
.
..
.....
Abstract...-...............
Figures.........
55
chapter 1.
Introduction.
Chapter 2.
New Equations Of
Inter nal
State For The
Energy, Entha lpy And Entropy........
Chapter 3.
The Determina tion Of
The Expansion Line End Points.......
3.1
Introduction...
3.2 The Methods
Of
Estimation Of Non.Rehea
Performance
Of
Steam Turbines.........
t
64
3.3 The Determinati on Of The ELEP's.......
3.4
The Procedures
For The
Computation ....
3.5 An Illustrative Example
Applicati on To The
Chapter 4. The
OF
A
Heat-
t
hanger.......
.
-
-. --.-.
-
The Procedures Of Modell
And
Selected
tem
Chapter
The Results..............
5.
Conclusions...........
Bibliography...........
73
Simulation
-
-
-
-
--
0
--
-
-
-
-
--
-
-
-
-
-
--
-
-
-
--
0
77
-
-
--
-
77
78
-.
4.3
69
-........ .....
Introduction............
Simulation Of A
66
-
-.
.
-
-
-
--
84
--***.....88
List Of Tables
Table
1
Page
Comparison Of Property Values
The Present Equation Of State
Steam Table And
61
2
The Efficiencies Calculated by Spencer's
3
Comparison Of Expansion Line End Point's Obtained
With Three Methods : Steam Table, The Present
Equation And The MP Equation
76
Properties And Parameters Of The Selected System
79
4
Method
73
List Of Figures
Page
Figure
Expansion Line With Load Changes
1
Turbine
2
The Flowchart Describing The Determination Of The
Expansion Line End Points Of Turbines
3
Schematic Diagram Of The Selected
4
Transient Response Of Pressure
The Ramp Input
5
6
7
System
In The
Transient Response Of Temperature
The Ramp Input
67
72
79
system With
85
In The system With
85
Transient Response Of Pressure In The system With
The Ramp Input With Exaggerated Control Volume
86
Transient Response Of Pressure In The system With
The Ramp Input With Exaggerated Control Volume
87
CHAPTER 1
INTRODUCTION
The equation of state
in
of
form
the
(MPF) has been derived in Part I.
Function
accuracy,
criteria:
4.2
Section
state was discussed in
The resulting equation of
Part
of
Potential
Massieu
the
I,
terms
in
of
the
simplicity, .identification of saturation curve
and physical meaning of the equation.
In this part, the equation of state is applied to
problems
associated
with
steam
turbine
cycle
calculations.
situations,
above-mentioned criteria.
the
the
equation
of
state
according
be
The secondary reason would
The
practical
the
primary reason for the application was to evaluate, in
engineering
practical
two
to
to
the
examine
feasibility of obtaining a useful scheme of computation which may
be immediately used.
The two problems
for
the
application
equation of state are:
1) the determination of states on the expansion line
including the expansion line end point(ELEP) of
steam turbines at partial loads; and
of
the
the simulation of transient behavior of a heat
2)
exchanger in a steam turbine cycle with the load
changing.
The problems are chosen, because both
state
of
require
them
At the same time, the
the evaluation of steam properties.
for
in
problems are more frequently encountered ones
of
equation
have
and
practice
their own importances in application engineering.
For the purpose of comparison, another set of equations of
was
state
selected and used in parallel with the equation,of state obtained
in this study.
Michael
by
The set of equation of state was made
at
M.I.T.
equations
of
state
have
independent
and
and
Packer
co-workers.[I}
separate
correlations
dependent
variables.
each
for
This
combination
of
Each correlation consists of 36 coefficients in
the superheated region and 9 coefficients in saturated
states,
which
were determined by fitting to the Keenan-Keyes-Hill- Moore(KKHM) steam
This set of equations can be taken
table data.
Thus,
time.
programming
accuracy,
of
their view a good compromise
typical
that
of
engineering works and representing in
for
people
many
by
prepared
as
computational
and
speed
it appears to be reasonable to select such
equations for the purpose of comparisons.
For
evaluation
the
were
times
computation
which
the
efficiency,
computational
the
measured in the two problems for each set of
The measuremets of CPU time were made by using
equations of state.
subroutine
of
a
is available in the Joint Computer Facility(JCF) at
M.I.T.
Since the problems require
the
equations
of
states
for
such
which
the
equation of state so far obtained in this study does not include,
the
properties
as
of
equations
internal
energy,
enthalpy
and
entropy,
state for such properties are derived in Chapter 2.
Chapter 3, the application to the determination of ELEP is
Chapter
4
exchanger.
part.
contains
the
simulation
In
presented.
of transient behavior of a heat
Chapter 5 includes the discussions and conclusions of
the
CHAPTER 2
NEW EQUATIONS FOR THE INTERNAL ENERGY, ENTHALY AND ENTROPY.
The equation of state was derived in the
form
part
previous
for pressure as a function of reduced temperature and reduced
density
The Eqs.(4.1) and (4.2) of Part I are again
Jy
ln y -
P
[y/(1-y)
ln(1-y) + 9/(l-y) -
+ 9y2 /(1-y)
The MPF J has one more term
temperature
the
The MPF yields the equation of state
of the MPF Jy in Eq.(4.1).
y in Eq.(4.2).
in
only.
The
2
T
-
Eq.(3.9)
Jin
The relation may be expressed as
(3J/ar)y
=
or
=
(aJy/ar)y
(4.1)
y2
(4.2)
which
is
a
function
of
Jr accounts for the main portion of the
term
internal energy and enthalpy as displayed in
u
yr
+
(aJr/ar)y
Eqs.(3.10)
and
(3.11).
u
= (aJT /87)y
y
-
(1)
and
h = (aJT /a )y + P/y - y
(2)
Since Jr is a function of temperature or
of
selection
Thus,
already made.
affect
Jr can
the
according
separately,
the
to
the
broader
required
instance, the Helmholtz free
and
independently
determined
required accuracy and to the range of
accuracy
and
range is, the more terms are needed.
For
energy
fo(T)
the
in
KKHM
formulation
8 terms, 6 of which are powers of l/T, one of which is a
of
consists
be
the
P-V-T relationship which was
the
It is evident that the more the required
applications.
the
not
JT does
equivalently T only,
term of ln T, the remaining one is a term of T(ln T).
It is expressed
as
6
'o(T)
i
Ci/
=2
T
+ C 7 ln T + C& (ln T)/r
(3)
i=1
where
of
T
=
state
1000 R/ T andfo(T) is equivalent to J of the MPF formulation
the
On
equations.
other
hand,
the
van der Waals(VDW)
representation of Jrhas only one term with
J
= c ln
(4)
7
where c is a reduced specific heat
representation
at
constant
volume.
the internal energy from Eqs.(3.l1)
yields
The
VDW's
and (3.15)
of Part I as
u
c /
-y
(5)
Considering that the internal energy for the ideal gas is given in
the reduced variables as
u
(6)
c /
=
the
has
which
Eq.(6),
qualitatively
accurate
in
representing
dense gases such as argon or carbon
Eq.(4)
is
used
of
a correctional term representing the reduced
This form for -the internal energy has
density y.
modification
possible
the Eq.(5) may be regarded as the simplest
known
been
to
be
the internal energy of some
the
If
dioxide.[2]
J. in
same
for the representation of internal energy in Eq.(1),
the resulting equation of state for internal energy
-is
of
the
same
form as Eq.(5).
Although the pure water substance
is
different
argon
from
or
carbon dioxide, the Eq.(5) appeared to be a reasonable base upon which
further improvement might be made.
specific
heat
c
would
the
One way to determine
reduced
be to make a least-square fitting to several
region
steam table data of internal energy at superheated
influence of density y becomes negligible.
where
the
A tentative value of c was
determined by fitting ten states between temperatures of 700F and 900F
with a pressure of 10 psia. The resulting value is c = 3.4362. Since
'c' denotes a reduced heat capacity, it is dimensionless. The value c
and
its
corresponding
of state for internal energy Eq.(5)
equation
and
were evaluated by generating enthalpy -and entropy
with
the
steam
table data.
+ JT
= ln y - ln(l-y) +9/(1-y) - yr+ c ln r
2
T
-y
h = u + P /y
and
2
2
P = [y/(l-y) + 9y /(l-y) ]/I
U CI
comparing
The resulting expressions of properties
from Eqs.(3.8), (3.9) of Part I and Eq.(5) are
J = Jy
by
-
y
S =U
- J
The relationships between the reduced and original properties
are again
J/m
J R + constant
) P
P
=
(a/bo
T
=
(a/Rbo)/
v
=
bo / y
s
=
R.s
h
=
(a/bo)h
and
u
(a/bo)u
where a, bo and R were determined in Section 4.1.2 of Part I,
and had the values of
a
=
bo =
R
=
4
4
-3
3
1.4964 x 10 (ft -lbf/lb
2
)
8.0848 x 10 (ft /lb)
0.1103
(Btu/R-lb)
The properties generated by the above equations are listed and
compared with the steam table data in Table 1.
The ranges for
1% accuracy for properties are:
P = 5 psia to 2000 psia
T = 200F to 1400F
for superheated vapor and
T = 10OF to 400F for the saturated vapor.
Considering the accuracy that could be achieved with
the
selected
c
Table 1.
Comparison of Properties : The Present
Equations of States with Steam Table
T = 300 F
(*)
KKRM p
h
s
PR
P
h
s
10.0
1193.7
1.8592
20.0
1191.5
1.7805
50.0
1184.4
1.6722
(67.0)
1180.2)
1.6356)
(*) KKHM: Steam Table
Data
PR
Present
Result
10.0
1187.9
1.8527
20.1
1187.1
1.7752
50.8
1184.9
1.6714
(68.4
(1183.7)
1.6375)
Units: P
bpsia)
h(tu/lb)
50.0
1284.0
1.7887
100.0
1279.1
1.7085
200.0
1268.8
1.6239
500.0
1231.5
1.4923
(680.0)
(1202.5)
(1.4335)
518.9
1256.5
1.5115
(722.0)
(1242.5)
1.4648
T = 500 F
KKHM p
h
PR
s
1.9690
20.0
1286.6
1.8919
P
h
10.0
1285.9
1.9674
20.0
1285.4
1.8905
50.2
1283.8
1.7882
100.6
1281.1
1.7096
202.4
1275.5'
1.6286
10.0
1384.0
2.0601
20.0
1383.5
1.9834
50.0
1381.9
1.8814
100.0
1379.2
1.8033
200.0
1373.8
1.7234
500.0
1356.7
1.6112
1000.0
1324.6
1.5135
10.0
1383.9
2.0601
20.0
1383.5
1.9834
50.1
1382.3
1.8816
100.2
1380.2
1.8038
200.5
1376.0
1.7248
502.5
1362.9
1.6155
1011.5
1338.5
1.5233
100.0
1480.5
1.8838
200.0
1477.1
1.8055
500.0
1466.5
1.6987
1000.0
1448.1
1.6120
100.1
1479.0
1.8925
200.2
1475.7
1.8042
500.4
1465.6
1.6976
1001.6
1448.1
1.6114
10.0
1287.7
s
T = 700 F
KKHM p
h
s
PR
P
h
s
T = 900 F
KKHM
5
s
PR
P
h
s
P
h
s
T =
KKHM
PR
P
h
s
10.0
1483.5
2.1393
20.0
1483.2
2.0627
50.0
1482.2
1.9611
2000
1407.6
1.5126
3000
1361.7
1.4414
4000
1315.2
1.3850
10.0
1481.9
2.1381
20.0
1483.2
2.0627
50.0
1480.6
1.9598
2010.8
1409.3
1.5129
3037.0
1364.6
1.4415
4100.0
1315.2
1.3769
10.0
1586.6
2.2099
20.0
1586.3
2.1334
50.0
1585.6
2.0321
100.0
1584.5
1.9551
500.0
1575.1
1.7731
1000.0
1562.9
1.6908
2000.0
1534.6
1.5950
10.0
1580.0
2.2053
20.0
1580.0
2.1286
50.0
1579.0
2.0272
100.1
1577.5
1.9500
500.5
1566.7
1.7670
1001.9
1553.0
1.6835
2113.0
1521.6
1.5850
10OF
and its corresponding equations of state, the value of c appears to be
a good selection
especially
for
for
the
the
intended
dynamic
applications
problem
which
stringent accuracy than static problems.
value
mentioned
usually
requires
less
of
Since the selection
the
c was made rather arbitrarily except for applying the least
of
square fitting technique in a certain range of pressure,
considered
to
be
the
optimal choice.
it
can
not
A better value of c might be
terms
Addition of
expected if a more systematic study could be made.
similar
before,
to those of the KKHM formulation would also improve accuracy.
The above-quoted accuracy was calculated as
,
% accuracy = 100 x Predicted Properties - Tabulated data
Tabulated data
However, it should be noted that the accuracy of
calculated
in
this
way for the enthalpy and entropy may be less meaningful in engineering
applications.
changes
In
many
engineering
situations,
what
in the enthalpy and entropy in a given process.
matters
are
A better way
of evaluating the accuracy would be
% accuracy = 100 x
A(Predicted data) - A(Tabulated data)
A(Tabulated data)
where A represents the change of a property in a process.
The
accuracy evaluated in the latter way should give worse values.
two problems, the accuracy was evaluated in the latter way.
In the
CHAPTER 3
THE DETERMINATION OF THE EXPANSION LINE END POINTS
3.1
INTRODUCTION
One of the more frequently
encountered
problems
in
steam
the
power plant engineering is the determination of the expansion line end
point(ELEP) of the turbine, or
expansion line.
more
generally,
the
states
The reasons are that:
1) the ELEP of higher pressure sections gives the inlet
conditions of reheaters, while that of lower pressure
section gives the inlet conditions of condensers; and
2) the ELEP and the states of stages in the turbine are
used for the design of extraction lines and they give
on
the
necessary data for the prevention of reverse flows
from the feedwater heaters.
Therefore, it is important to accurately determine the ELEP's and
the
states on the expansion line for the calculations of:
.1) the heat balance(or the static performance) of a
given steam turbine cycle at both design and partial
loads; and
2) the dynamic response of heat exchangers such as
reheaters and condensers.
states
on
expansion line, the proper equation of the state is required.
In
To accurately and quickly determine the ELEP's or the
the
addition
to
reviewed
to
of
state,
a
Therefore, the methods
of
the
for
of
estimation
reliable
efficiency)
performance(or
non-extraction
indispensible.
equation
the
the
turbine
is
will
be
estimation
extent first, and the use of the equation of state
some
will be discussed later.
3.2
THE METHODS OF ESTIMATION
OF
OF
PERFORMANCE
NON-EXTRACTION
STEAM TURBINES
A.
The Stage by Stage Method
The overall
performance
non-extraction
of
a
can
turbine
be
determined by calculating the performance of each individual stage and
then by combining them.
However,
the
stage
by
stage
calculation
method is not, in general, practical for predicting the performance of
new designs in the stage of proposition, since the detailed
of
the
particular
knowledge
turbine design is not available at that time.
addition, the method is usually
long
and
tedious.
Therefore,
method is not preferred except for a particularly unusual case.
In
the
The Empirical Methods
B.
On the other hand, the overall performance of a
of a turbine is closely related to the type of
performance
the
that
experiences
from
More specifically speaking, it was observed
tests.
rating
design and that the type of design is related, in turn, to the
of
turbine
the
Therefore, it was
well as the steam conditions.
as
be
methods which are based on observations and
empirical
by
calculated
may
turbine
observed that the overall performance of the turbine is predictable to
a
if
degree,
certain
rating, the type of design and the steam
the
conditions are given.
method
Warren and Knowlton used these facts to present a general
for
overall
the
of
calculation
the
performance
of
nonreheat
of
turbine-generator units.{3] Elston and Knowlton ex~tended the area
unit.[ ]
reheat
including
by
application
4
Bartlett[5] improved the method by extending the
part
performance
load
and
for
the
prediction
is
which
a
essentially
version of the Hegetschweiler's method.
updated
of
results
experimental
In
method
the
addition,
a
Test
Power
on
based
Code
No.6
is the more useful for the
it
since
computer,
of
performance
with
mathematical
expressions
for computer applicatons.
analysis
is
24 performance tests in power
plants, which were done according to the ASME
requirements.
and
modified
That is, the area of
application of Spencer's method is same, but the method
the
Spencer and
of nonreheat and reheat turbine.
Cotton[6] presented a method
updated
Hegetschweiler
provides
Spencer's method
with
an
Spencer's method predicts the efficiencies of a given turbine
at
predicts the overall performance
of
modern
steam
turbine
accuracy of less than 0.5% deviations.
each
section
both
at
the
design
flow
and
partial flows, if the
following informations given:
1) volume flow;
2) pressure ratio of steam at a point or a stage of
interest to the throttle pressure;
3) initial steam conditions at the governing stage; and
65
4) governing stage design.
The 'section' in this context means a group of similar stages.
It
is
well known that modern large turbines frequently are composed of three
sections:
section.
the high pressure, the intermediate and
the
low
pressure
The detailed formula and the procedures for the calculations
are well described in Reference 6.
3.3
THE DETERMINATION OF THE ELEP'S
As mentioned before,
the
are
estimated
efficiencies
used
to
determine the pressure and temperature of steam(or states of steam) at
However,
various stages and the expansion line end points(ELEP).
it
is not so simple to determine the states, because the pressures at the
Thus the states
can
be
flow
The phenomena are shown
rate of the steam(or equivalent load) varies.
in Figure 1.
the
as
simultanously,
stages, in addition to efficiency, vary
obtained
by
considering
the
variations of both the pressure and the efficiency.
Since the
depends
efficiency
on
the
flow
rate,
may
it
be
expressed as
(7)
E = E(w)
where E is the efficiency of sections of the turbine and w is the mass
On the other hand, the efficiency E is defined as
flow rate.
E = (hl -
where hl is the inlet
enthalpy
of
the isentropic
h2)/(hl -
steam
(8)
h2s)
enthalpy
at
the
section,
h2
is
the
a state on the expansion line including the ELEP, h2s is
enthalpy
at
the
pressure
of
the
state
P2.
The
s
T
~'
W decreasing
PRESSURE
ELEP
W decreasing
FIGURE 1. TURBINE EXPANSION LINE
WITH CHANGES OF FLOW
isentropic
enthalpy h2s is determined by the constitutive relation or
the equation of state
h2s = h2s(sl,P2)
where si is the inlet entropy.
(9)
Then the enthalpy of the state on
the
expansion line including the ELEP is given as
h2
=
Meanwhile, the flow
hl - E(w)(hl - h2s)
rate
at
the
(10)
relationship
another
gives
load
between the pressure P2 and the specific volume v2 as
w = ClP2/v2
(11)
or alternatively, gives the relationship between the pressure and
the
temperature as
w = C2 P2/FT2
(12)
where Cl and C2 are the constants determined by the pressure ratio
the
pressure
at the state to that of the state at the next stage, as
well as the adiabatic exponent k.
The two relationships (11) and (12)
hold for the perfect gas, when a pressure ratio is given.
the superheated steam is not a perfect
valid
provided
that
the
gas,
the
ELEP
are
determined
so
that
properties might satisfy Eqs.(10) and (11) or
and
(12)
two
Even though
equations
are
the values Cl and C2 are correct for the region
under considerations.[7] Therefore the states on
including
of
simultaneously.
Thus
68
the
the
expansion
their
thermodynamic
alternatively
determination
line
of
Eqs.(10)
such states
require iterative calculations of the two equations (10) and
well
quickly
proper
a
mentioned before,
than
important
but
the accuracy
of
the
valuable
not
the
Among the criteria
of
state
could
computation
the
is
computation
the
more
becomes
states
the ELEP are less than about 30i
be
to
30
to
be
a few ELEP's,
turbine
stages.
The
at most in the order of magnitude of 0.1
seconds in a modern minicomputer.
for
that
That is because
several extraction points and about 15
time
and
the equation of state can be easily programmed
that
including
computation
equation
accurately
other factors in the problem of determining the ELEP.
and implemented to any computer.
determined
ELEP
equation of state is required.
The simplicity is also
economical,
the
as
To determine
as the estimation of properties at each iteration.
the states on the expansion line including
(11)
At the same time, it
is
necessary
scheme to be able to recognize in which phase a
state lies, or equivalently where the saturation line is located.
3.4
THE PROCEDURES FOR THE COMPUTATION
By utilizing the relationships and discussions introduced in
previous
the
sections, a scheme for the determination of the state on the
expansion line including the ELEP can be set up.
In the
scheme,
input data are assumed to be:
1) The inlet steam conditions and the reheat pressure
of the turbine;
2) The type and rating of the turbine; and
3) The pressures at states of interest at the design
flow, including the pressures at the ELEP's of each
section.
The detailed procedures adopted may be summarized as follows.
1. Determine efficiencies of sections of the turbine
the
at the design flow(or the rated load) Eo's by using
Spencer's method, where the subscript o denotes
the conditions at the design flow.
2. Determine the inlet steam conditions of the section
of interest.
If the section is the high pressure
.section, the inlet steam conditions given as the
input data are the inlet conditions of that section.
If the section is other than high pressure section,
the steam conditions can be determined by usintg the
equation of state and Eq.(10) successively.
3.
Determine the isentropic enthalpy of the state ofinterest on the expansion line including the ELEP,
by using the equation of state in Eq.(9) as
h2so =
where
f
(sl,P2o)
denotes the equation of state,
subscript 1 the inlet condition of the section,
subscript 2 the state of interest in that section,
while s denotes the isentropic path.
4. Determine the enthalpy of the state at the design
flow by using
Eq.(lO)
h2o = hl
-
as
Eo(hl -
h2so)
5. Determine other properties at state 2, by using
the equation of state as
T2o, S2o, v2o =
f (h2o,P2o)
6. Determine the efficiency E(w) at the partial flow
by using Spencer's method.
7. Guess a tentative pressure at state 2, P2
8. Determine the isentropic enthalpy h2s which
corresponds to P2 and sl, by the equation of
state as
h2s
Y (P2,sl)
9. Determine the enthalpy at state 2, by using the
Eq.(10) as
h2 = hl - E(w)(hl -
h2s)
10. Since P2 and h2 are known, other properties can
be determined by using the equation of state as
T2, s2, v2 =
4,(h2, P2)
11. By using the equation (12), update P2 such as
P2 = (w/wo)V/T2/T2o P2o
12. Repeat procedures 7 to 10 until a satisfactory
convergence is made.
The procedures are described in the flow chart in Figure 2
TIlE PRODEDURES OF DETERMINING ELEP
h
T2
NO
P2
S2
=
-CHECK
=
(W
#'(h2,P2
/
Wo)
)
VT2 /T2o P20
CONVERGENCE
YES
1 X
I Is
1,2;
states
rated load
' s ' : isentropic
* subscripts 1,
2:
FIGURE 2. TIIE FLOWCHART OF PROCEDURES DETERMINING ELEP'S
3.5
AN ILLUSTRATIVE EXAMPLE AND THE RESULTS
As an example of the application of the equation of state to
determination
steam
of
ELEP,
a relatively large turbine and corresponding
given.
are
conditions
the
The
given
conditions
and
basic
assumptions are listed as follows.
(A) Turbine nameplate rating: 250,000KW
(B) Turbine Type: cross compound double-flow,
3600/1800 rpm,
43 inch last-stage backet, 247.6
sq. ft. total last-stage annulus
area.
(C) Steam Condition: Initial 2400psia, 100OF; exhaust
pressure 1.Oin-Hg
(D) Estimated design throttle flow: 1,559,000 lb/hr
(E) Point of interest in expansion line and its pressure
at design flow: ELEP of high pressure section with
480psia at design flow
By using Spencer's method, the efficiencies of high
pressure
at the design flow and partial flows are given in Table 2
Table 2
w
The efficiency calculated
efficiency
1.0
0.8495
0.9
0.8446
0.8
0.8305
0.7
0.8087
0.6
0.7808
0.5
0.7483
0.4
0.7128
0.3
0.6750
section
0.6392
0.2
w = perunit mass flow rate.
The calculation of efficiencies was
Reference
given in
flows
design
the
line
of
pressure
the
at
and
flow
at
then
one
the
is
The
isentropic
ELEP, the other is the ELEP as
In the former state, the enthalpy and
illustrated in Figure 1.
four
explained before.
procedures
the
to
according
properties are evaluated at two points:
expansion
formula
various
using
6.
The ELEP are obtained first at
partial
by
made
other
properties are evaluated for given entropy and pressusre, while in the
are
latter state the temperature and other properties
The
pressure.
and
enthalpy
given
evaluated
for
criteria for the convergence of
iteration at partial loads were less than 0.1%.
Three methods were used to evaluate the steam properties:
1. the steam table by Keenan-Keyes-Hill-Moore,
designated KKHM.
2. the equations of state obtained in this study,
designated PR.
3. the equations of state by Michael Packer et al,
designated MP.
The evaluation by looking up
provide
reference
a
the
steam
was
table
made
mainly
to
for the comparison of accuracy of the two
data
sets of equations of state:the equations obtained in
this
study
and
the MP equations.
The computation time is not so important in this problem, because
the
amount
total
of
time
CPU
required
for
determining
all the
necessary states in the expansion line of the larger turbine including
the
ELEP's
would be at most in the order of magnitude of 0.1 seconds
in a modern minicomputer.
compare
the
However,
computational
the
CPU
efficiencies,
time
by
was
using
measured
the
to
FORTRAN
subroutine 'JPI.FOR' in the Joint Computer Facility
M.I.T.
More
of the same program were made to get a reliable
repetitions
50
than
at
estimation of CPU time for
each
set
of
equations
of
state.
The
for the two sets of equations were done at the same time
measurements
when the computation load
of
the
system
VAX-l1/780
is
relatively
small.
At. partial
flows
expansion
isentropic
of
40%
and
20%
of
the
design
specifically,
More
for
a
given
pressure;
the
subprogram
the
present
computing
the
SGSAT(P).[1] Then the phase can be recognized by
comparing the entropy of steam before the expansion (sl in
with
belongs
this study can compute the entropy by making use of the
in
algotithm in Appendix A of Part I, the MP equations by
function
state
these two sets of equations, can compute the
entropy of saturated vapor Sg
equations
a
the
Both the
lines fell into the two phase region.
sets of equations are able to recognize which phase
to.
flow,
Figure
1)
The criterion in Section 2.3 of Part I which requires
Sg.
the capability to give two phase
region
can
be
justified
in
this
example.
The resulting ELEP's and
are
equations
compared
The equations of
given.
accurate
pressure,
the
more
accuracies
of
using
the
the MP equations.
two
sets
of
Table 3, the measured CPU times are also
state
obtained
in
this
important
more
accurate
study
properties
are
temperature.
pressure
practice, the present equations are more useful.
when
the
give
more
enthalpy and entropy than the MP equations, while
the latter yields somewhat
that
in
the
Considering
and enthalpy in
The average CPU time
present equations is 6 times shorter than that using
Table 3 The Expansion Line End Point Obtained
with Three Methods of Generating steam
Property Values and the cpu Time
The ELEP's Obtained with Three Formula
P (psia)
S (Btu/lb.R)
h (Btu/lb)
T (F)
1.0
480.
480.(0.0)
480.(0.0)
1.5609
1.5612(1.05)
1.5711(36.0)
1296.4
1298.2(1.1)
1308.3(7.2)
KKHM
594.1
573.9(3.4) PR
605.7(1.8) MP
0.8
376.2
376.2(0.0)
366.6(2.5)
1.5701
1.5706(1.33)
1.5773(19.2)
1280.4
1280.9(0.3)
1285.4(2.8)
KKHM
551.8
532.5(3.5) PR
554.4(0.5) MP
0.6
277.1
277.6(0.2)
261.7(5.6)
1.5903
1.5911(1.39)
1.5924(3.64)
1269.3
1270.3(0.5)
1265.7(1.9)
KKHM
515.6
517.2(0.3) PR
505.2(2.0) MP
0.4
181.4
181.7(0.2)
168.0(7.4)
1.6227
1.6237(1.11)
1.6289(6.88)
1260.3
1263.0(1.3)
1256.2(2.0)
KKHM
480.9
481.7(0.3) PR
470.6(2.1) MP
0.2
88.6
89.0(0.5)
79.5(10.)
1.6830
1.6841(0.73)
1.6945(7.65)
1244.6
1247.6(1.4)
1243.5(0.5)
KKHM
429.9
421.0(2.1) PR
424.8(1.2) MP
perunit mass flow rate
*The values in the parenthesis denote the % error from the KKHM steam
table data
* % error of P, T = 100 x (predicted - tabulated)/tabulated value
* % error of h, s = 100 x [A(predicted) -A(tabulated)]/A(tabulated)
where A represents the variation of properties in a process
* The three formula are denoted as:
KKHM: KKHM steam Table data, PR: The Present Equations of state
MP: The Equations of state by Michael et al.
* W
The Average cpu Time Required for Each
Calculations with The Two Sets of Equations
The Present Equations of state : 1.88 msec.
: 10.86 msec.
The MP Equations
CHAPTER 4
THE APPLICATION TO THE SIMULATION OF A HEAT EXCHANGER
4.1
INTRODUCTION
One of important problems in the power plant engineering
simulation
of
transient
of
response
heat
exchangers.
important for the study of dynamic characteristics of
a
is
the
This
power
is
plant
when subjected to normal and abnormal disturbances.
The dynamic model of a heat exchanger requires a set of
laws
such
as
the
conservation
of
mass, momentum and energy.
dynamic model also requires a set of equations of
unknown
thermodynamic
properties
physical
state
to
The
evaluate
with known properties at each time
step.
Among the criteria, the simplicity of the equation
of
state
is
primarily for the economical reason.
desired
A typical simulation of
transient response of a section of heat exchanger would
1000 timesteps.
than
An average dynamic model of a realistic problem
might have more than 150 sections
two
Since
or
three
more
require
with
properties
steam
2
iterations
be
must
per
timestep.
at each
evaluated
iteration, 1/2 to 1 million steam property evaluations are required in
case.
this
economical computation is crucial in this problem.
Thus
The accuracy is also important
In this section, a part of heat
is
exchanger
as
selected
the
system and the simple dynamic model is made under var1ous assumptions.
The transient response is simulated
equations.
The
model
this
by
first
the
state obtained in this study and second the MP
of
equations
present
with
computation
time
and
accuracy
of
are
properties
compared to evaluate the equation of state.
4.2
THE PROCEDURES OF MODELLING AND SIMULATION OF A SELECTED SYSTEM
System
A section of a heat
as
selected
the
exchanger
system
in
a
steam
turbine
cycle
It is
(the control volume) to be analyzed.
assumed that the section was so chosen that it can be regarded
lumped
parameter
system'.
various
existing
techniques
the
and
'distributed
criteria
However, since the main purpose of this study is to
equations
of
state
as
'a
For the selection of sufficient number of
'lumped sections' to adequately describe
system',
was
can
examine
parameter
used.
be
how
the
are applied to practical engineering situations,
the lumping techniques and criteria are
not
treated
A
here.
review of these techniques and criteria appears in Reference 8.
good
Figure 3 shows a schematic diagram of the selected system,
which
is 'a representative tube' instead of bank of tubes.
NI
P1 ,h1,TI
P2,h2,T2
FIGURE
3.
A SCHEMATIC
DLAGRAM OF THE SYSTEM
Table 4
Properties and Parameters of System
Property
Section 1
Inside
P(psia)
50
48
46
T(F)
500
600
600
h(Btu/lb)
1284.0
1332.9
1332.9
s(Btu/lb/R)
1.7887
1.8417
1.8464
u(Btu/lb)
1179.4
1216.9
1216.9
v(ft /lb)
11.305
13.055
13.627
with
W
Section 2
= 1000lb/s
Vcv = 4000 ft
Qin = 4.89 x 10 Btu/s
Some of data and thermodynamic properties necessary
to
describe
the
system
an equilibrium state correspond to a load point are listed
at
wall
tube
in Table 4, where 'Qin' is the heat transfer rate from the
to the system, 'Vcv' designates the control volume and 'W' is the mass
The size and capacity of the system roughly correspond
flow rate.
of
those
a
(about
cycle
turbine
of
section
second reheater of modern larger steam
the
500
to
MW),
1000
to
are
but
a
precise
and
energy
not
representation of a practical piece of equipment.
Modelling
By
the
using
conservation
as
laws
physical
well
of
mass,
momentum
as the equations of state and the constitutive
relationship imposed by the turbine control valve, the system
modelled.
can
be
The governing equations are given in the section below.
Continuity:
Vcv dp/dt = WI
-
W2
where subscripts 1 and 2 denote cross-sections 1 and 2,
(13)
respectively.
Momentum:
2
P1 -
P -
IP
(14)
P2 = K2 W2 2p
(15)
P = KI W
where coeffiecients K1 and K2, determined from the given data in Table
4, are
Energy:
KI = 1.769 x 10
(ft psi sec 2 /lbm)
K2 = 1.532 x 10
(ft psi sec 2 /lbm)
Vcv d( p u)/dt = W1 hl - W2 h2 + Qin (16)
If the enthalpy and temperature inside the tube are assumed to be same
as outlet properties h2 and T2, then Eq.(16) becomes
Vcv d( pu) dt = Wi hl - W2 h + Qin
(17)
Equations of state:
Since the density and the product of density and internal
computed
and
each time step, the density and the internal
at
known
energy can be selected
are
energy
the
as
Then
variables.
independent
other
properties are obtained by
P, h,
where
p
( p ,u)
T = f
(18)
represents the equation of state.
Constitutive relationship of control valve:
Since the steam generated from heat exchangers such as
or
reheater
passes
the
turbine
a
superheater
control valve, it is governed by a
certain law imposed by control valve which is
(19)
Wc = C Pc/Tc
where C is a constant determined by the valve area and other
the
subscript
'c'
denotes
the
control
valve.
If there are other
sections from the selected section to the turbine control
pressure
drop
are
summed
factors;
valve,
the
from P to Pc to yield another equation by
summing Eq.(15) and the same equations in the subsequent sections.
If
it is assumed,for the sake of simplicity, that the selected section is
the last section before the control valve, then the Eq.(19) becomes
(20)
W2 = C P2/fT2
where C ,determined from the given data, is
1/2
C = 707.78 (lb R /psia sec)
Eqs.(15.) and (20) give the relationship
2
W2 =/PT /2 [ (1+4P/k2#3 PT) -1]
where
(21)
B = 1/(C K2)
W2 is determined
thermodynamic
iteratively
properites.
so
that
Additional
it
may
be
assumptions
compatible
which
with
were made
implicitly are:
(1) The properties inside the tube are well mixed
and uniform.
(2) The effect of gravity are negligible and the
flow is one directional.
(3) The heat transfer rate Qin is controllable.
Simulation
One of the inputs or
disturbances
the
to
system
can
be
the
incoming mass flow rate which represents changes or variations of load
in the cycle. The inlet mass flow rate was chosen as a ramp input
which increases 10 % in 5 minutes;
i.e.
Wl = Wo + a t
2
where Wo = 1000 lb/sec and a = 0.3333 lb/sec.
The simulation of the model of the system was made by
program
DYSYS
(DYnamic
Facility(J.C.F.) at M.I.T.
SYstem
using
the
Simulator) within the Joint Computer
The program solves the
time
response
of
systems described by a set of ordinary differential equations with the
EQSIM
was
four
computed
to describe the system with governing equations (14)
used
through (21).
called
The user supplied subroutine
fourth order Runge-Kutta method.
At
each
time
were
equations
governing
the
step,
to ascertain adequate convergence of W2 (mostly
times
within 0.1 %).
in
properties
Eq.(18).
correlations in Eq.(18).
the
pressure
to
the
evaluate
equations do not have the
Meanwhile, the equations
predict
study
present
MP
the
However
used
were
The two sets of equations of state
and
of
temperature
the
in
state
P,T
with
independent variables p, u with an accuracy of better than 0.5% inside
the range of computation, which is less than the upper limit of
error(1%) in the MP equations.[l] Thus the equations of state in
study
were
substituted
for
the
missing
relationships.
this
More
specifically, at the inlet section 1, the missing relationship
TI
=
*
(pi,ul)
Then the enthalpy
was substituted.
was
evaluated
existing
the
by
correlation in the MP equations,
hl
=
i
(P1,Tl)
while, inside the tube, the missing relationship
P
=
f(P ,u)
was substituted and then the enthalpy h was calculated by
u + P/p
h
Temperature only could be obtained by the existing correlation
equations
T
=
(P,h)
in
MP
The time step was kept less than 0.5 seconds.
in
diverged
the
simulation,
owing
Otherwise, the solution
the
to
governing
nonlinear
equations.
THE RESULTS
4.3
The transient response of the system when subjected to
the
ramp
Figure 4 shows the
input was computed and plotted in Figures 4 and 5.
transient response of pressure inside the tube, and Figure 5 shows the
response of temperature and the ramp input, which is the incoming mass
flow rate.
are
state
In the figures, simulations using each set of equations of
plotted
together
to
compare
force stronger
To
them.
transient response, a simulation ,with the control volume
exaggerated
times, was made by using the equations of state in this study and
100
The Figures 6 and 7
the results are plotted in Figures 6 and 7.
show
that the system behaves like a typical linear second order system with
a damping ratio of about 0.5.
The results may be discussed
according
to the two criteria;the accuracy and the computational efficiency.
The Accuracy:
It is hard to examine the absolute accuracy
because
there
of
properties,
the
is no proper reference data available for comparisons.
equations
Instead, the relative discrepancies between the two sets of
of
state were calculated by comparing the intermediate states between
the initial state and final state.
than
0.1% and no more than 0.5%
was checked at the final
comparing
state
.
(or
The
agreements
mostly
are
less
Also the thermodynamic consistency
state)
equilibrium
another
by
the resulting values of enthalpy in the simulation with the
value obtained by letting d(properties)/dt
=0.
The
agreements
are
better than 0.5% in both cases.
The Computational Efficiency:
Since the CPU-time measuring subroutine
(JPI)
was
not
working
57
56
55
C4
53
52
51
50
49
48
47
0
50
100
150
200
250
300
TIMECSEC)
COMPARISON
FIGURE 4.
INSIDE THE SYSTEM
)
OF TRANSIENT
RESPONSES
OF PRESSURE
1250
1200
1150
1100
1050
1000
0
50
100
150
200
250
300
TIMEC3ECJ
FIGURE 5. COMPARISON OF TRANSIENT RESPONSES OF TEMPERATURE
INSIDE THE SYSTEM
57
5
55
54
53
52
51
50
49
0
50
100
150
250
200
300
TIME (SECONDS)
FIGURE 6 TRANSIENT RESPONSE OF PRESSURE WITH EXAGGERATED
CONTROL VOLUME
u) 12501
Li
U-)
N
m
LLi
0
U-
1150-
U-)
INCOMING MASS FLOW RATE
06
1100-
HLiJ
1050
AT OUTLET
Li
H-
1000
50
100
150
200
250
300
TIME (SECONDS)
FIGURE
7
TRANSIENT
RESPONSE
EXAGGERATED CONTROL VOLUME
OF
TEMPERATURE
WITH
with
the
program
DYSYS,
the
subroutine EQSIM which calculates the
properties at each time step was separated into a DO-loop of a simple
main program and repeated 200 times. The measured CPU times are 36
this
msec with the present equations of state obtained in
70msec.
with
the
MP
equations.
Therefore,
efficiency with the former is about twice higher
equations.
87
the
than
study
and
computational
that
with
MP
CHAPTER 5
CONCLUSIONS
In this study,
problems
the
of
steam
turbine
with
associated
state
equations
were
cycle
applied
to
calculations.
purpose for the applications is to evaluate the equations of state
computational speed and identification of the saturation states.
of
state
The
in
the accuracy, the simplicity as represented by
terms of the criteria:
equations
the
obtained
comparing with a typical
set
in
of
The
the present study were assessed by
empirical
equations
developed
by
achieved
by
Packer et al.[1]
In the problem determining the ELEP's, the accuracy
the
present
equations
of
state is somewhat better than that by the
fully empirical equations of Packer.
considering
This is encouraging,
especially
that the equations of state have only 5 coefficients;
a,
bo, R, y and c, while the empirical equations have 36 coefficients for
The computational efficiencies were compared by
each correlation.
measuring the CPU time of the computer.
calculations
alone
The CPU time for the property
as required with the present equations is about 6
times less than with the method of Packer.
Both
equations
have
the
capability
to
recognize
the
saturation lines.
This capability was
phase
two
the
required when the isentropic expansion line fell into
region.
that
The results of measurements indicate
investigate
the
simulation
the
with
the
state requires about half of the CPU time than
of
present, equations
with the empirical equations.
as
to
used
transient calculations of steam system performances.
for
performance
was
problem
A simple heat exchanger
Although the result is not as
dramatic
The comparison of the accuracy
expected, it is still significant.
with each other indicates that the results
are
equigalent.
Further
advantages might be achieved by algebraic manipulations
computational
of the present equations of state.
The results clearly show the advantages of employing the
equation
of
property:
time
is
state rather than developing a set of curve-fits to each
accuracy is slightly better or equivalent
reduced
by
a
of
and
computation
factor of 2 to 6 depending on the particular
problem being invetigated.
equations
present
It
may
be
concluded
that
the
present
state can be utilized effectively in certain classes of
problems as demonstrated in this study,
owing
to
the
accuracy
and
simplicity as well as the intrinsic ability to recognize the two phase
region.
for
Thus it may be also concluded that the purpose
developing
present
the
equations
state
of
and
criteria
can be considered
relevant in the practical engineering situations.
Considering that the present equations of state yield
accurate
properties
with
a
relatively
simple functional form, it would not be
hard to expect that the further improvements may be made on the
basis
The improvements could be made by adding
of these equations of state.
a few terms to the equation representing the internal energy such as
u
=
cl/T
+ c2/r
2
+
...
-
y
which should give more accurate internal energy, enthalpy and entropy
than the equation
'
U = c/r -
y.
BIBLIOGRAPHY
1. M.Packer,"Thermodynamic
Properties of Pure and Saline(Geothermal)
Water,"
Lab. for Manuf. and Production, MIT
2. Hirschfelder,Molecular Theory of Gases and Liquids. John Wiley & Sons Inc.
3. G.Warren and P.Knowlton,Jr.,"Relative Engine Efficiencies From Large Modern
Steam Turbine-Generator Units," Trans. ASME: vol.
6
3,pl 2 5,19 4 1
4. C.Elston and P.Knowlton,Jr.,"Comparative Efficiencies of Central Station
Reheat and Non-reheat Steam Turbine-Generator Units," Trans. ASME, vol.74,
p 13 8 9 , 1952
5. H.Hegetschweiler and R.Bartlett,"Predicting Performance of Steam TurbineGenerator," Trans. ASME, vol. 79, p10 8 5, 1957
6. R.Spencer and K.Cotton,"A Method for Predicting the Performance of Steam
Turbine-Generator..," Trans. ASME Paper No 62-WA-209, 1962
7. J.Salisbury, Steam Turbines and Their Cycles. Robert E. Krieger Publishing
Co., Huntington, N.Y.
8. G.Masada,"Modeling and Control of Power Plant Boiler-Turbine-Generator
System," Ph.D. Thesis, MIT 1979