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The approximation of the critical isotherm with broken rational function
Fluid Phase Equilibria, 69 (1991) l-14 Elsevier Science Publishers B.V., Amsterdam
1
The approximation of the critical isotherm with broken rational function Volker Peters’) Ixwtitut fiir Thermodynamik,
UniversitSt Hannover, Germany
(Received
in final
ZIarch 27,
1991; accepted
form September 8,
1991)
Abstract Equations of state in the form of broken rational functions of the molecular or specific volume are called algebraic equations of state. They can also be seen as Pad6 approximations of the vi&l equation of state. By adapting the general algebraic equation of state from 3rd to 12th degree to the critical isotherms of ten different substances, it turns out that algebraic equations of state with an uneven degree are superior to those with an even degree. Furthermore, it becomes evident that algebraic equations of state of the 7th degree are especially suitable.
Introduction For the estimation and analysis of power and chemical engineering processes, the thermodyn~c
features of the working fluids are needed. These state variables can
be taken from measured values or from equations of state. The calculation of thermodynamic state variables using equations of state is obviously more convenient because only a relatively small number of measured values is needed. Moreover, equations of state can be extrapolated in regions where no measured values are available. For the repeated calculation of state variables the structure of the equation is an important factor in determining the calculation time. It is desirable to calculate the function with the smallest possible number of basic operations. High powers and exponential functions prolong the time of calculation. For these reasons, the usefulness of broken rational functions to describe the equation of state shall be investigated.
Virial equation of state The virial state equation was first given by Kamerlingh-Onnes in 1901. He wrote
%) Now at
BayerAC&Leverkusen.
0378-3812/91/$03.50
All rights reserved 1991 Elsevier Science Publishers B.V.
2 the p-v-T equation of state in the following way :
p=A’+
;+g+...
He named the temperature-dependent
.
(1)
coefficents A’, B’, C’ . . .
‘virial’ coefficients.
The first one is A* = RT. The virial equation is generally written in the form of the real gas factor
z=g=l+B(T)e+C(T)e’+...
.
(2)
B (T) is called the second, C (T) the third, virial coefficient, etc. The virial coefficents are defined through the MC Laurin expansion
The virial co¢s
indicate the deviation of the real fluid from the ideal gas state.
Although the virial equation was solely established empirically to demonstrate the thermal behaviour of gases, it could later, with the help of statistical thermodynamics, be justified physically (see Mason and Spurling 1969).
The algebraic equation of state The term ‘algebraic equation of state’ was introduced by Baehr (1953). It describes the p-v-T equation of state, which can be written in the form vN + cN-~v~-’
+ . . . + cgl* + qv + GJ = 0
(5)
an algebraic equation in v. The coefficents GJ to cN_1 are functions of pressure and temperature (c+ = f&T)).
Ap ar t f rom this form (equation (5)), algebraic equations
of state can also be described aa broken rational functions: P=
polynom (v)~-’ polynom (v)~
’
(6)
Algebraic state equations can also be seen aa Padk approximations of the virial equations of state. The virial equation of state
P(e,T)=RTe+B(T)e2+C(T)e3+...
(7)
3 gives, with few virial coefficients, exact results only with sufficiently large specific volumes. If it is to be used for all states, numerous virial coefficients must be included. An exact description of the thermal equation of state is only possible through a power series with infinite sectors: zJ(e,T)
= RTe+eci(T)e’
.
i=2
63)
This power series expansion of the thermal equation of state can be approached through the [L/M] Pad6 approximation
RTe + p(e,T) = 1+
ig2 A CT)e’
fl
Bi
+ 0
(eL+M+l)
(9)
(T) ei
(Baker and Graves-Morris 1981). It is assumed that the Pad6 approximation converges for all states. The remaining part 0 (eL+M+l) is regarded as negligibly small. A virial expansion, with (L + M) virial coefficients, can be transformed into a rational function:
p(e,T)
=
RTe+A2(T)e2+...+AL(T)eL
(10)
1+&(T)e+B2(T)e2t...tBhf(T)eM *
If the density e is replaced by the specific volume w, this results in RTuN-’ $ A2 (T) vN-’ t . . . t AL (2”)vN-L J’(v7T) = +’ + & (T) yN-l + B2 (T) G-2 $ . . . + BM (T) vN-M
(11)
with
N=
L
for
L 1 M
M
for
M 2 L
(12)
’
This function also includes the virial expansion up to the N-th virial coefficient which applies if M is zero and L is N.
Through the preceding approximation the degree
of the equation of state, which results if the equation is transformed to the specific volume, is lowered from M + L with the virial expansion to the degree N with the broken rational function. From a specific degree (N) of the function arises the greatest possible number of coefficients with equivalent values of L and M. The general form of the algebraic equation of state of the N-th degree therefore is:
RTvN-’ + A2 (T) vN-2 t . .,t AN (2') B2 (T) vN-2 t . ..t&(T)
p(u7 T,= yN t B1(T) G-1 +
’
(13)
The critical isotherm is examined to determine which broken rational function is suitable to describe the thermal equation of state.
This is the isotherm of the fluid
state field surface most difficult to describe.
Degree of the equation of state To examine the critical isotherm it is desirable to use the form which arises when using the reduced variables :
C@N-1 + Az (29)pN-* + . . . + AN (19)
?r= ‘PN+B1(~)(PN-1+&(d)(PN-2+...+BN(fi)
(14)
’
where A = p/p, is the reduced pressure, 9 = T/T= the reduced temperature, cp = v/v, the reduced volume and C = (RT,) / (pcvc) th e reduced gas constant.
Through non-
dimensionalising, the different substances can be compared more easily. From stability considerations it follows that at the critical point, there must be at least a three-point contact between the critical isotherm and the critical isobar. Shifting the origin of the coordinate system in the direction of the critical pressure yields a triple zero position at the critical volume
A= 1_
(‘p- 1)3’ (pN-3 + C4(pN-4 + c5vN-5+ . . . + CAT) pN + BlpNT1 + BzpNs2+ . . . + BN
Such a description of the critical isotherm was first given by Baehr (1953).
(15)
As the
equation (14) for infinite volumes should approach the equation of state of the ideal gas, the coefficients C4 und Bl are not independent of each other, but must fulfill the condition C=Br+3-C4
Determination
.
(16)
of the parameters
To determine the parameters of the reduced equation of the critical isotherm, equation (15), for various pure fluids, the direct method of Ahrendts and Baehr (1979) was used which is a simplification of the Maximum-Likelihood method. With this method the parameters are determined in the way that the sum of squares X2 = 2 j=l
with
(wj,=r - r(qj, fi,, Ci, Bi))” (+)j
5 is minimized. Weighting the individual terms through the variances, the variance of measuring the temperature is also taken into account. The value-pairs (cpj, zj,eq) were generated through the fundamental equations, which represent the state variables usually within the measurement uncertainty. This process was chosen, since few measurements of properties on critical isotherms are available. Table 1 lists the pure substances and the authors of each chosen fundamental equation. Table 1. Fundamentalequations for pure substances.
Ne
-
Katti
Katti et al. (1986)
Ar
-
Stewart
Stewart
02
-
Schmidt
Schmidt and Wagner (1985)
N2
-
Jacobsen
Jacobsen et al. (1986)
CH4
-
Setzmann
Setzmann
C2H4
-
Jahangiri
Jahangiri et al. (1986)
et al. (1989)
(1989)
R12
-
Marx
Marx (1989)
Fu2
-
Kohlen
Kohlen (1987)
NH3
-
Ahrendts
Ahrendts and Baehr (1979)
H2O
-
Saul
Saul (1988)
Hz0
-
Rosner
Rosner (1986)
J
100 value-pairs for the region between the critical point and the maximum pressure given for each of the fundamental equations as well as 100 value-pairs for the region between the critical point and the maximum specific volume, to which the fundamental equation was adapted, were chosen (see table 2). The positions
Qj
=
1/6j were, selected
at intervals according to density, and distributed between the regions. This division is suitable, since the shape of the critical isotherm is very flat in the critical region. In figure 1 the logarithm of the reduced pressure ?r is depicted as a function of the reduced density 6. The critical isotherm is steepest for inert gases and most nearly flat for water. Thus it can be checked whether the model equations are also suitable to describe the critical region.
An exclusion of the critical region, as proposed by
Schreiner (1986) was not made. The variances CT:,a: and uj, for the chosen fundamental equations, cannot be found
6 Table 2. Limits of the critical isotherm. Substance
PC
T,
VC
bar
K
m3/kg
-
-
-
-
Ne
26,80
44,49
0,002075
14,4
2,42
14,3
0,07
Ar
48,60
150,66
0,001884
15,4
2,46
20,O
0,05
6 n&(12
=maz
Q%naz
&nc
02
50,43
154,58
0,002293
16,3
2,51
25,0
0,04
NZ
33,98
126,19
0,003194
16,3
2,45
25,0
0,04
CH4
45,96
190,53
0,006171
16,2
2,45
25,0
0,04
CzH4
50,37
282,32
0,004669
16,l
2,43
25,0
0,04
CFsCls
41,21
384,95
0,001770
16,l
2,40
25,0
0,04
CHClFs
49,88
369,30
0,001949
10,4
2,35
25,0
0,04
NH3
113,33
405,40
0,004439
18,l
2,76
25,0
0,04
Hz0
220,64
647,14
0,003106
18,2
2,85
25,0
0,04
lo5-
0.0
0.5
1.5
1.0
2.0
2.5
3.0
6Figure 1. The critical isotherm of pure substances.
in literature. Therefore they were calculated from estimated measurement errors with the help of the Jo-rule to UT =
5 . (200 Pa/p, + x/104)
(19)
7
08
1
=
3
a 0.01
K/TC
(20)
(21) This rough estimate
is to be preferred
In order to compare therm,
the standard
the different
to arbitrary
( e.g. (o;r)j = 1. ) evaluation.
forms of the model equation
for the critical
iso-
deviation (22)
o= was used as an evaluation and 1 is the number Minimizing
of parameters
of value-pairs
(here m = 200)
(here 1= 2N - 4). leads to a non-linear
of Den&s,
equation
system
which, was
Gag and Welscfs (1979). This method
is, similarly
by Ahrendts and Baehr (1979), a Marquardt-method
to the one proposed Fletcher,
criterion.
the sum of squares
solved with the method
(mx:I) d m is the number
for which, the Newton-
or the gradient
procedure
modified
is chosen according
The Dennis, Gay and Welsch method
results
of an iteration.
robust
in use than the one of Ahrendts and Baehr. Zero was used as the starting
for all adaptable
coefficients,
i.e. the iteration
was started
turned
by
to the
out to be more value
with the ideal gas law.
Performance of the equation for the critical isotherm The general algebraic was taken for equations method
previously
form of the equation
from 3rd to 12th degree and was adapted
of the evaluated
are used for all substances.
equations
give a lower standard
of even degree.
From this results
evident
deviation
deviations
The curve Uneven
than equations
equations
of state of
(15) exceeds the value of 9, convergence
equation
of an uneven
that algebraic
in
of the model
degree.
on the critical isotherm
from the steadily system
8, the curves of even and uneven
the model equations
the standard
is
to those with an even degree.
If the degree N in the equation
exceeds
to the
The same procedures
those with an uneven
the hypothesis
uneven degree are to be preferred
to solve the nonlinear
deviation.
lies below that of the ‘even model equations’.
model equations
This becomes
according
(15))
for oxygen are given as an example
curves can be drawn through
of the ‘uneven model equations’
quired
standard
Only the results
with an even degree and through
will arise.
(equation
. In figure 2 the degree of each of the model equations
described.
shown over the logarithm
figure 3. Separate
for the critical isotherm
increasing
number
of the sum of squares.
model equations
approach
degree lose their superiority.
problems
of iterations
re-
If the degree each other,
and
From the 7th degree
8
13
I
I
I
I
I
I
I
r .
12:
#
11 F
= Y X
10 9 t N
- Katti - stewart - Schmidt 02 N - Jacobsen Ck, - Setzmonn :?I-$ 1 J;ah;giri
No Ar
m
R22
X + 0
a
-
Kohlen
A
7 6 5 4 3 2 0.02 0.050.1
0.2
0.5
1
standard Figure
2
5
deviation
10
20
50 100200
500
-
u
2. Standard deviation u of the critical isotherm for different degrees N of the algebraic equation of state.
onwards, the model equations describe the critical isotherm within the given variances. The calculated standard deviations are smaller than 1. For these reasons, algebraic equations of state of the 7th degree were chosen for the following examination.
Zeros and poles of the function In order to examine the equation of 7th degree for the critical isotherm more precisely, numerator and denominator were reduced to linear factors.
All substances in-
vestigated resulted in the same form of the reduced equation broken down to linear factors for the critical isotherm which is (9 - II3 [b r
-
l
=
-
(9
-
A)
[(P
-
P2)’
- 7d2 + 74 +
A]
[b
-
m2
[(lp +
73)2
a,z]
+
74
[(yJ -
(23) ps)2
+
B:]
1211 lot
9a-
N
76543to2
0.050.1 I I
0.2 I
2I 5I 0.5 I 1I standard deviation u
10 I
20 I
50 I 100200 I I
500
Figure 3. Standard deviation of the critical isotherm of oxygen for different degrees N.
with 71 = f (g - 3 - 273) + $31
+ pz + /A + ps
.
(24)
The equation (24) corresponds to the theorem, first proposed by Baehr (1953), that the reduced gas constant Q is equivalent to the sum of zero positions of (z - 1) minus the sum of all pole positions :
. I = 3 + 2~~ + 27s _ (a
+ 2~s + 2P4 + 2Ps)
(25)
Two pairs of conjugate complex zero positions appear in the numerator of equation (23). The denominator has a real pole and three conjugate complex pole pairs. Model equations of 3rd and 5th degree have a similar partition of numerator and denominator in linear factors.
With model equations of even degree only conjugate complex pole
pairs were found. In figure 4 poles and zero positions of the 7th degree equation are shown. The distribution of poles seems similar for different substances but an explanation cannot be given. However, it becomes clear that the real pole is that region which van der
10 poles
0.6
.-t
zero positions
t
-0.6 1X
R22
- Kohlen
1;;; -0.6
;y[ -0.3
0.0
0.3
real
parf
0.6
0.9
1.2
-
‘0.0
,
,
0.3
0.6
real
part
, 1 0.9
1.2
-
Figure 4. Zero and pole positions of the equation 7th degree (23) for the critical isotherm.
Waals already proposed for the co-volume (/31w 0,25).
One of the conjugate complex
pole pairs lies in a physically unimportant region. The real part of the pole pair is smaller than the only real co-volume, with pressure rising towards infinity. Many cubic state equations in this area either have a real double pole or two different real poles. A further conjugate complex pole pair has a real part around the medium liquid volume (6 M 0,7, see figure 5). The real part of the third conjugate complex pole pair concides approximately with the critical volume (see figure 5). The real pole appearing in model equations of an uneven degree corresponds also to the co-volume of the molecules which can be derived from molecular theory. Taking into account the co-volume, this supports the hypothesis of the superiority of broken rational equations of state. To clarify the explanations, zero positions and poles of the 7th degree equation of the critical isotherm of oxygen are shown in figure 5. The interaction between the
11
0.6
I
I
I
0.4 _ 3rd pole pair I t z.
1 th pole
s 2ndXpoe pair
0.2 -
reel
triple
X
-0.2 -
0
-
2nd
-0.6 -0.2
_
0.0
0 zero
zero
-
x pair
I
I
I
I
0.2
0.4
0.6
0.8
real
m ”
position
1th zero 0pair
X -0.4
pair
pole
” /\
0.0
E .-
I
X
X
P Z ‘$
I
part
I 1.0
1.2
-
Figure 5. Zeros (0 ) and poles ( X) of the equation 7th degree of the critical isotherm of
oxygen.
poles and the zero points can clearly be seen. The triple zero position at the critical point and the first conjugate complex zero position pair facilitate the flat shape of the critical isotherm in the critical region (see figure 1). The real pole and the second pole pair describe the steep increase of the isotherm in the co-volume of the molecules. The second pair of zeros and the first pole pair indicate the transitive area between the flat shape of the isotherm at the critical point and the steep in crease at the co-volume. The third pole pair lies outside the physically important region, defined by the balanced relation between conjugate complex pole pairs and zero pairs.
Summary and Conclusions The approximation of the equation of state for the critical isotherm with broken rational functions of the specific or molecular volume was investigated. By adapting the general algebraic equation of state from 3rd to 12th degree to the critical isotherms of ten different substances the most suitable form will be found. The property values needed for adaptation were found by calculating them from fundamen-
12 tal equations, since few measurements of the critical isotherm are available. Algebraic equations of state with an uneven degree are superior to those with an even degree. For further examination, the algebraic equation of state of the 7th degree was chosen, as it yields the pressures up to the critical isotherm within the prescribed variances. Futhermore, the distribution of zero positions and poles of the equation of state of 7th degree has been depicted (see Peters 1990). F rom these considerations an equation of 7th degree was developed, which is able to represent the one and two phase regions of pure substances and of mixtures with good precision. Acknowledgements : The author wishes to thank Prof. Dr.-Ing. Dr.-Ing. E.h. H.D. Baehr for suggesting this work.
List of Symbols
A,B,C a,b,c
coefficient coefficient
B
second virial coefficient
c
thrid virial coefficient
1
number of parameters
vn
number of data
N
degree of function
P
pressure
R
gas constant
T
temperature
V
specific volume
Z
compressibility factor (z = 5)
a7A-Y
coefficients
6
P reduced density (6 = -= PC
reduced gas constant
(6 =
reduced temperature (19 = $) reduced pressure ( A = E ) ’ density standard deviation variance reduced specific volume ( cp= i
)
13
weighted
X2
sum of squares
subscript
c
critical point value
subscript
ezp
experimental
subscript
i, j
indices of terms
value
References Ahrendts,
J.
and H.D.
zur Aufstellung
Baehr
: Die Anwendung
thermodynamischer
nichtlinearer
Zustandsgleichungen.
Regressionsverfahren
Forsch.
im Ing.-Wesen,
45
(1979) 2, 51-56 . Ahrendts,
J. and H.D.
Forschungsheft Baehr
: Die thermischen
Eigenschaften
von Ammoniak.
VDI-
596, 1979.
H. D.
chung.
Baehr
: Der kritische
Zustand
und seine Darstellung
Abh. d. Akad. d. Wissensch.
u. d. Literatur
durch die Zustandsglei-
Mainz.
Mathe.-naturw.
Klasse,
1953, S. 235-333. Baker,
G.A,Jr.
and P. Graves-Morris
Volume 13. Part II : Extensions matics
and its Applications,
Dennis,
J.E.
regression. Dennis,
J.E.,
gorithm.
and R.E.
Commun. .M.
Technical
: PadC approximants.
and applications,
Addison-Wesley
Welsch
Comput.
Gay and R.E.
summary
Welsch
report
I : Basic theory,
Volume 14. Encyclopedia
Publishing
: Techniques
Statist.-Simul.
Part
Company
for nonlinear
of Mathe-
1981.
least squares
and robust
B’7 (1978) 4, 345-359.
: An adaptive
nonlinear
# 2010, Mathematics
Research
least-squares
al-
Center-Madison
1979. Jacobsen,
equation
R.
, R.B. Stewart, M. Jahangiri
for thermodynamic
property
ring, 31 (1986) 1161-1168, Plenum, Jacobsen,
R.
from freezing
and S.G. Penoncello
correlations.
Advances
: A new fundamental in Cryogenic
Enginee-
New York.
, R.B. Stewart and M. Jahangiri line to 2000 K at pressures
: Thermodynamic
to 1000 MPa.
properties
Journal
of phys.
of nitrogen them.
ref.
Data 15 (1986) 2, 735-794. Jahangiri,
perties phys.
M., R. Jacobsen,
of ethylene them.
R.B.
from freezing
Stewart
and R.D. McCarty
line to 450 K at pressures
: Thermodynamic
to 260 MPa.
Journal
proof
ref. Data 15 (1986) 2, 593-645.
Kamerlingh-Onnes,
H. : Expression
means of series. Comm.
Nr. 71, Phys.
of the equation
of state of gases and liquids by
Lab., Universit5t
Leiden 1901.
14
Katti, R., R. T. Jacobsen,
R.B.
of neon for temperatures Advances
in Cryogenic
Kohlen,
R.
lationen.
Stewart
and M. Jahangiri
from the triple point Engineering,
Universit%t
Bochum
Mason, E.A.
to ‘100 K at pressures
31 (1986) 1189-1197, Plenum,
: Das fluide Zustandsgebiet
Fortschritt-Berichte
: Thermodynamic
von R22 Messungen;
VDI, Reihe 19, Nr.
properties to 700 MPa.
New York.
Berechnungen;
Korre-
14, 1987 und Dissertation,
Ruhr-
1987.
and T.H. Spurling
: The virial equati0.n of state.
Pergamon
Press Oxford
1969 . Marz,
V.
: Fundamentalgleichung
verhaltens
der Substanzen
Dichlordifluormethan
22), Trichlormonofluormethan Ruhr-Universitat Peters,
V.
rationale
Bochum
tion, Universit%t N.
Hannover
bis 10 000 bar erstellt Miinchen
der thermischen
Zustandsgleichung
durch gebrochene
durch multiple
fiir Wasser im Bereich vom Tripelpunkt
Regressionsanalyse.
Dissertation,
Universitat
1986. fiir den fluiden Zustandsbereich
Reihe 3, Nr. 149, 1988 und Dissertation, Schmidt
R.
Sauerstoff. Schmidt,
Ruhr-Universit;it
R. and W. Wagner
Schreiner,
K.
to Oxygen.
: Beschreibung
kubischen
Ruhr-Universitat
Bochum
U.
des thermischen
Universit5t
R.B
und ihre Anwendung
Verhaltens
reiner Fluide mit druckexVDI, Reihe 3, Nr. 125,
1986.
fiir Methan
fiir den Zustandsbereich
bis 700 K bei Driicken bis 1000 MPa. Fortschritt-Berichte
and R. Jacobsen
auf
19 (1985) 175-200.
Fortschritt-Berichte
Kaiserslautern
Reihe 3, Nr. 192, 1989 und Dissertation,
to 1200 K with pressures
1988.
1983.
Fluid Phase Equilibria
: Eine Fundamentalgleichung
der Schmelzdruckkurve
Bochum
VDI,
: A new form of the equation of state for pure substances
Zustandsgleichungen.
1986 und Dissertation, Setzmann,
von Wssser bis
von 1273 K. Fortschritt-Berichte
: Eine neue Form einer Fundamentalgleichung Dissertation,
and its application
639-798.
(R 113). Dissertation,
1990. Zustandsgleichung
Saul, H. : Eine Fundamentalgleichung
Stewart,
(R
VDI, Reihe 3, Nr. 224, 1990 und Disserta-
zu Driicken von 25 000 MPa und Temperaturen
pliziten
Zustands-
(R 12), Monochlordifluormethan
(R 11) und Trichlortrifluorethan
Fortschritt-Berichte
: Thermische
des thermodynamischen
1989.
: Die Approximation
Funktionen.
Rosner,
zur Beschreibung
Ruhr-Universitgt
: Thermodynamic
properties
Bochum
von VDI,
1989.
of argon from triple point
to 1000 MPa. J ournal of phys. them.
ref. Data 18 (1989) 2,
1
The approximation of the critical isotherm with broken rational function Volker Peters’) Ixwtitut fiir Thermodynamik,
UniversitSt Hannover, Germany
(Received
in final
ZIarch 27,
1991; accepted
form September 8,
1991)
Abstract Equations of state in the form of broken rational functions of the molecular or specific volume are called algebraic equations of state. They can also be seen as Pad6 approximations of the vi&l equation of state. By adapting the general algebraic equation of state from 3rd to 12th degree to the critical isotherms of ten different substances, it turns out that algebraic equations of state with an uneven degree are superior to those with an even degree. Furthermore, it becomes evident that algebraic equations of state of the 7th degree are especially suitable.
Introduction For the estimation and analysis of power and chemical engineering processes, the thermodyn~c
features of the working fluids are needed. These state variables can
be taken from measured values or from equations of state. The calculation of thermodynamic state variables using equations of state is obviously more convenient because only a relatively small number of measured values is needed. Moreover, equations of state can be extrapolated in regions where no measured values are available. For the repeated calculation of state variables the structure of the equation is an important factor in determining the calculation time. It is desirable to calculate the function with the smallest possible number of basic operations. High powers and exponential functions prolong the time of calculation. For these reasons, the usefulness of broken rational functions to describe the equation of state shall be investigated.
Virial equation of state The virial state equation was first given by Kamerlingh-Onnes in 1901. He wrote
%) Now at
BayerAC&Leverkusen.
0378-3812/91/$03.50
All rights reserved 1991 Elsevier Science Publishers B.V.
2 the p-v-T equation of state in the following way :
p=A’+
;+g+...
He named the temperature-dependent
.
(1)
coefficents A’, B’, C’ . . .
‘virial’ coefficients.
The first one is A* = RT. The virial equation is generally written in the form of the real gas factor
z=g=l+B(T)e+C(T)e’+...
.
(2)
B (T) is called the second, C (T) the third, virial coefficient, etc. The virial coefficents are defined through the MC Laurin expansion
The virial co¢s
indicate the deviation of the real fluid from the ideal gas state.
Although the virial equation was solely established empirically to demonstrate the thermal behaviour of gases, it could later, with the help of statistical thermodynamics, be justified physically (see Mason and Spurling 1969).
The algebraic equation of state The term ‘algebraic equation of state’ was introduced by Baehr (1953). It describes the p-v-T equation of state, which can be written in the form vN + cN-~v~-’
+ . . . + cgl* + qv + GJ = 0
(5)
an algebraic equation in v. The coefficents GJ to cN_1 are functions of pressure and temperature (c+ = f&T)).
Ap ar t f rom this form (equation (5)), algebraic equations
of state can also be described aa broken rational functions: P=
polynom (v)~-’ polynom (v)~
’
(6)
Algebraic state equations can also be seen aa Padk approximations of the virial equations of state. The virial equation of state
P(e,T)=RTe+B(T)e2+C(T)e3+...
(7)
3 gives, with few virial coefficients, exact results only with sufficiently large specific volumes. If it is to be used for all states, numerous virial coefficients must be included. An exact description of the thermal equation of state is only possible through a power series with infinite sectors: zJ(e,T)
= RTe+eci(T)e’
.
i=2
63)
This power series expansion of the thermal equation of state can be approached through the [L/M] Pad6 approximation
RTe + p(e,T) = 1+
ig2 A CT)e’
fl
Bi
+ 0
(eL+M+l)
(9)
(T) ei
(Baker and Graves-Morris 1981). It is assumed that the Pad6 approximation converges for all states. The remaining part 0 (eL+M+l) is regarded as negligibly small. A virial expansion, with (L + M) virial coefficients, can be transformed into a rational function:
p(e,T)
=
RTe+A2(T)e2+...+AL(T)eL
(10)
1+&(T)e+B2(T)e2t...tBhf(T)eM *
If the density e is replaced by the specific volume w, this results in RTuN-’ $ A2 (T) vN-’ t . . . t AL (2”)vN-L J’(v7T) = +’ + & (T) yN-l + B2 (T) G-2 $ . . . + BM (T) vN-M
(11)
with
N=
L
for
L 1 M
M
for
M 2 L
(12)
’
This function also includes the virial expansion up to the N-th virial coefficient which applies if M is zero and L is N.
Through the preceding approximation the degree
of the equation of state, which results if the equation is transformed to the specific volume, is lowered from M + L with the virial expansion to the degree N with the broken rational function. From a specific degree (N) of the function arises the greatest possible number of coefficients with equivalent values of L and M. The general form of the algebraic equation of state of the N-th degree therefore is:
RTvN-’ + A2 (T) vN-2 t . .,t AN (2') B2 (T) vN-2 t . ..t&(T)
p(u7 T,= yN t B1(T) G-1 +
’
(13)
The critical isotherm is examined to determine which broken rational function is suitable to describe the thermal equation of state.
This is the isotherm of the fluid
state field surface most difficult to describe.
Degree of the equation of state To examine the critical isotherm it is desirable to use the form which arises when using the reduced variables :
C@N-1 + Az (29)pN-* + . . . + AN (19)
?r= ‘PN+B1(~)(PN-1+&(d)(PN-2+...+BN(fi)
(14)
’
where A = p/p, is the reduced pressure, 9 = T/T= the reduced temperature, cp = v/v, the reduced volume and C = (RT,) / (pcvc) th e reduced gas constant.
Through non-
dimensionalising, the different substances can be compared more easily. From stability considerations it follows that at the critical point, there must be at least a three-point contact between the critical isotherm and the critical isobar. Shifting the origin of the coordinate system in the direction of the critical pressure yields a triple zero position at the critical volume
A= 1_
(‘p- 1)3’ (pN-3 + C4(pN-4 + c5vN-5+ . . . + CAT) pN + BlpNT1 + BzpNs2+ . . . + BN
Such a description of the critical isotherm was first given by Baehr (1953).
(15)
As the
equation (14) for infinite volumes should approach the equation of state of the ideal gas, the coefficients C4 und Bl are not independent of each other, but must fulfill the condition C=Br+3-C4
Determination
.
(16)
of the parameters
To determine the parameters of the reduced equation of the critical isotherm, equation (15), for various pure fluids, the direct method of Ahrendts and Baehr (1979) was used which is a simplification of the Maximum-Likelihood method. With this method the parameters are determined in the way that the sum of squares X2 = 2 j=l
with
(wj,=r - r(qj, fi,, Ci, Bi))” (+)j
5 is minimized. Weighting the individual terms through the variances, the variance of measuring the temperature is also taken into account. The value-pairs (cpj, zj,eq) were generated through the fundamental equations, which represent the state variables usually within the measurement uncertainty. This process was chosen, since few measurements of properties on critical isotherms are available. Table 1 lists the pure substances and the authors of each chosen fundamental equation. Table 1. Fundamentalequations for pure substances.
Ne
-
Katti
Katti et al. (1986)
Ar
-
Stewart
Stewart
02
-
Schmidt
Schmidt and Wagner (1985)
N2
-
Jacobsen
Jacobsen et al. (1986)
CH4
-
Setzmann
Setzmann
C2H4
-
Jahangiri
Jahangiri et al. (1986)
et al. (1989)
(1989)
R12
-
Marx
Marx (1989)
Fu2
-
Kohlen
Kohlen (1987)
NH3
-
Ahrendts
Ahrendts and Baehr (1979)
H2O
-
Saul
Saul (1988)
Hz0
-
Rosner
Rosner (1986)
J
100 value-pairs for the region between the critical point and the maximum pressure given for each of the fundamental equations as well as 100 value-pairs for the region between the critical point and the maximum specific volume, to which the fundamental equation was adapted, were chosen (see table 2). The positions
Qj
=
1/6j were, selected
at intervals according to density, and distributed between the regions. This division is suitable, since the shape of the critical isotherm is very flat in the critical region. In figure 1 the logarithm of the reduced pressure ?r is depicted as a function of the reduced density 6. The critical isotherm is steepest for inert gases and most nearly flat for water. Thus it can be checked whether the model equations are also suitable to describe the critical region.
An exclusion of the critical region, as proposed by
Schreiner (1986) was not made. The variances CT:,a: and uj, for the chosen fundamental equations, cannot be found
6 Table 2. Limits of the critical isotherm. Substance
PC
T,
VC
bar
K
m3/kg
-
-
-
-
Ne
26,80
44,49
0,002075
14,4
2,42
14,3
0,07
Ar
48,60
150,66
0,001884
15,4
2,46
20,O
0,05
6 n&(12
=maz
Q%naz
&nc
02
50,43
154,58
0,002293
16,3
2,51
25,0
0,04
NZ
33,98
126,19
0,003194
16,3
2,45
25,0
0,04
CH4
45,96
190,53
0,006171
16,2
2,45
25,0
0,04
CzH4
50,37
282,32
0,004669
16,l
2,43
25,0
0,04
CFsCls
41,21
384,95
0,001770
16,l
2,40
25,0
0,04
CHClFs
49,88
369,30
0,001949
10,4
2,35
25,0
0,04
NH3
113,33
405,40
0,004439
18,l
2,76
25,0
0,04
Hz0
220,64
647,14
0,003106
18,2
2,85
25,0
0,04
lo5-
0.0
0.5
1.5
1.0
2.0
2.5
3.0
6Figure 1. The critical isotherm of pure substances.
in literature. Therefore they were calculated from estimated measurement errors with the help of the Jo-rule to UT =
5 . (200 Pa/p, + x/104)
(19)
7
08
1
=
3
a 0.01
K/TC
(20)
(21) This rough estimate
is to be preferred
In order to compare therm,
the standard
the different
to arbitrary
( e.g. (o;r)j = 1. ) evaluation.
forms of the model equation
for the critical
iso-
deviation (22)
o= was used as an evaluation and 1 is the number Minimizing
of parameters
of value-pairs
(here m = 200)
(here 1= 2N - 4). leads to a non-linear
of Den&s,
equation
system
which, was
Gag and Welscfs (1979). This method
is, similarly
by Ahrendts and Baehr (1979), a Marquardt-method
to the one proposed Fletcher,
criterion.
the sum of squares
solved with the method
(mx:I) d m is the number
for which, the Newton-
or the gradient
procedure
modified
is chosen according
The Dennis, Gay and Welsch method
results
of an iteration.
robust
in use than the one of Ahrendts and Baehr. Zero was used as the starting
for all adaptable
coefficients,
i.e. the iteration
was started
turned
by
to the
out to be more value
with the ideal gas law.
Performance of the equation for the critical isotherm The general algebraic was taken for equations method
previously
form of the equation
from 3rd to 12th degree and was adapted
of the evaluated
are used for all substances.
equations
give a lower standard
of even degree.
From this results
evident
deviation
deviations
The curve Uneven
than equations
equations
of state of
(15) exceeds the value of 9, convergence
equation
of an uneven
that algebraic
in
of the model
degree.
on the critical isotherm
from the steadily system
8, the curves of even and uneven
the model equations
the standard
is
to those with an even degree.
If the degree N in the equation
exceeds
to the
The same procedures
those with an uneven
the hypothesis
uneven degree are to be preferred
to solve the nonlinear
deviation.
lies below that of the ‘even model equations’.
model equations
This becomes
according
(15))
for oxygen are given as an example
curves can be drawn through
of the ‘uneven model equations’
quired
standard
Only the results
with an even degree and through
will arise.
(equation
. In figure 2 the degree of each of the model equations
described.
shown over the logarithm
figure 3. Separate
for the critical isotherm
increasing
number
of the sum of squares.
model equations
approach
degree lose their superiority.
problems
of iterations
re-
If the degree each other,
and
From the 7th degree
8
13
I
I
I
I
I
I
I
r .
12:
#
11 F
= Y X
10 9 t N
- Katti - stewart - Schmidt 02 N - Jacobsen Ck, - Setzmonn :?I-$ 1 J;ah;giri
No Ar
m
R22
X + 0
a
-
Kohlen
A
7 6 5 4 3 2 0.02 0.050.1
0.2
0.5
1
standard Figure
2
5
deviation
10
20
50 100200
500
-
u
2. Standard deviation u of the critical isotherm for different degrees N of the algebraic equation of state.
onwards, the model equations describe the critical isotherm within the given variances. The calculated standard deviations are smaller than 1. For these reasons, algebraic equations of state of the 7th degree were chosen for the following examination.
Zeros and poles of the function In order to examine the equation of 7th degree for the critical isotherm more precisely, numerator and denominator were reduced to linear factors.
All substances in-
vestigated resulted in the same form of the reduced equation broken down to linear factors for the critical isotherm which is (9 - II3 [b r
-
l
=
-
(9
-
A)
[(P
-
P2)’
- 7d2 + 74 +
A]
[b
-
m2
[(lp +
73)2
a,z]
+
74
[(yJ -
(23) ps)2
+
B:]
1211 lot
9a-
N
76543to2
0.050.1 I I
0.2 I
2I 5I 0.5 I 1I standard deviation u
10 I
20 I
50 I 100200 I I
500
Figure 3. Standard deviation of the critical isotherm of oxygen for different degrees N.
with 71 = f (g - 3 - 273) + $31
+ pz + /A + ps
.
(24)
The equation (24) corresponds to the theorem, first proposed by Baehr (1953), that the reduced gas constant Q is equivalent to the sum of zero positions of (z - 1) minus the sum of all pole positions :
. I = 3 + 2~~ + 27s _ (a
+ 2~s + 2P4 + 2Ps)
(25)
Two pairs of conjugate complex zero positions appear in the numerator of equation (23). The denominator has a real pole and three conjugate complex pole pairs. Model equations of 3rd and 5th degree have a similar partition of numerator and denominator in linear factors.
With model equations of even degree only conjugate complex pole
pairs were found. In figure 4 poles and zero positions of the 7th degree equation are shown. The distribution of poles seems similar for different substances but an explanation cannot be given. However, it becomes clear that the real pole is that region which van der
10 poles
0.6
.-t
zero positions
t
-0.6 1X
R22
- Kohlen
1;;; -0.6
;y[ -0.3
0.0
0.3
real
parf
0.6
0.9
1.2
-
‘0.0
,
,
0.3
0.6
real
part
, 1 0.9
1.2
-
Figure 4. Zero and pole positions of the equation 7th degree (23) for the critical isotherm.
Waals already proposed for the co-volume (/31w 0,25).
One of the conjugate complex
pole pairs lies in a physically unimportant region. The real part of the pole pair is smaller than the only real co-volume, with pressure rising towards infinity. Many cubic state equations in this area either have a real double pole or two different real poles. A further conjugate complex pole pair has a real part around the medium liquid volume (6 M 0,7, see figure 5). The real part of the third conjugate complex pole pair concides approximately with the critical volume (see figure 5). The real pole appearing in model equations of an uneven degree corresponds also to the co-volume of the molecules which can be derived from molecular theory. Taking into account the co-volume, this supports the hypothesis of the superiority of broken rational equations of state. To clarify the explanations, zero positions and poles of the 7th degree equation of the critical isotherm of oxygen are shown in figure 5. The interaction between the
11
0.6
I
I
I
0.4 _ 3rd pole pair I t z.
1 th pole
s 2ndXpoe pair
0.2 -
reel
triple
X
-0.2 -
0
-
2nd
-0.6 -0.2
_
0.0
0 zero
zero
-
x pair
I
I
I
I
0.2
0.4
0.6
0.8
real
m ”
position
1th zero 0pair
X -0.4
pair
pole
” /\
0.0
E .-
I
X
X
P Z ‘$
I
part
I 1.0
1.2
-
Figure 5. Zeros (0 ) and poles ( X) of the equation 7th degree of the critical isotherm of
oxygen.
poles and the zero points can clearly be seen. The triple zero position at the critical point and the first conjugate complex zero position pair facilitate the flat shape of the critical isotherm in the critical region (see figure 1). The real pole and the second pole pair describe the steep increase of the isotherm in the co-volume of the molecules. The second pair of zeros and the first pole pair indicate the transitive area between the flat shape of the isotherm at the critical point and the steep in crease at the co-volume. The third pole pair lies outside the physically important region, defined by the balanced relation between conjugate complex pole pairs and zero pairs.
Summary and Conclusions The approximation of the equation of state for the critical isotherm with broken rational functions of the specific or molecular volume was investigated. By adapting the general algebraic equation of state from 3rd to 12th degree to the critical isotherms of ten different substances the most suitable form will be found. The property values needed for adaptation were found by calculating them from fundamen-
12 tal equations, since few measurements of the critical isotherm are available. Algebraic equations of state with an uneven degree are superior to those with an even degree. For further examination, the algebraic equation of state of the 7th degree was chosen, as it yields the pressures up to the critical isotherm within the prescribed variances. Futhermore, the distribution of zero positions and poles of the equation of state of 7th degree has been depicted (see Peters 1990). F rom these considerations an equation of 7th degree was developed, which is able to represent the one and two phase regions of pure substances and of mixtures with good precision. Acknowledgements : The author wishes to thank Prof. Dr.-Ing. Dr.-Ing. E.h. H.D. Baehr for suggesting this work.
List of Symbols
A,B,C a,b,c
coefficient coefficient
B
second virial coefficient
c
thrid virial coefficient
1
number of parameters
vn
number of data
N
degree of function
P
pressure
R
gas constant
T
temperature
V
specific volume
Z
compressibility factor (z = 5)
a7A-Y
coefficients
6
P reduced density (6 = -= PC
reduced gas constant
(6 =
reduced temperature (19 = $) reduced pressure ( A = E ) ’ density standard deviation variance reduced specific volume ( cp= i
)
13
weighted
X2
sum of squares
subscript
c
critical point value
subscript
ezp
experimental
subscript
i, j
indices of terms
value
References Ahrendts,
J.
and H.D.
zur Aufstellung
Baehr
: Die Anwendung
thermodynamischer
nichtlinearer
Zustandsgleichungen.
Regressionsverfahren
Forsch.
im Ing.-Wesen,
45
(1979) 2, 51-56 . Ahrendts,
J. and H.D.
Forschungsheft Baehr
: Die thermischen
Eigenschaften
von Ammoniak.
VDI-
596, 1979.
H. D.
chung.
Baehr
: Der kritische
Zustand
und seine Darstellung
Abh. d. Akad. d. Wissensch.
u. d. Literatur
durch die Zustandsglei-
Mainz.
Mathe.-naturw.
Klasse,
1953, S. 235-333. Baker,
G.A,Jr.
and P. Graves-Morris
Volume 13. Part II : Extensions matics
and its Applications,
Dennis,
J.E.
regression. Dennis,
J.E.,
gorithm.
and R.E.
Commun. .M.
Technical
: PadC approximants.
and applications,
Addison-Wesley
Welsch
Comput.
Gay and R.E.
summary
Welsch
report
I : Basic theory,
Volume 14. Encyclopedia
Publishing
: Techniques
Statist.-Simul.
Part
Company
for nonlinear
of Mathe-
1981.
least squares
and robust
B’7 (1978) 4, 345-359.
: An adaptive
nonlinear
# 2010, Mathematics
Research
least-squares
al-
Center-Madison
1979. Jacobsen,
equation
R.
, R.B. Stewart, M. Jahangiri
for thermodynamic
property
ring, 31 (1986) 1161-1168, Plenum, Jacobsen,
R.
from freezing
and S.G. Penoncello
correlations.
Advances
: A new fundamental in Cryogenic
Enginee-
New York.
, R.B. Stewart and M. Jahangiri line to 2000 K at pressures
: Thermodynamic
to 1000 MPa.
properties
Journal
of phys.
of nitrogen them.
ref.
Data 15 (1986) 2, 735-794. Jahangiri,
perties phys.
M., R. Jacobsen,
of ethylene them.
R.B.
from freezing
Stewart
and R.D. McCarty
line to 450 K at pressures
: Thermodynamic
to 260 MPa.
Journal
proof
ref. Data 15 (1986) 2, 593-645.
Kamerlingh-Onnes,
H. : Expression
means of series. Comm.
Nr. 71, Phys.
of the equation
of state of gases and liquids by
Lab., Universit5t
Leiden 1901.
14
Katti, R., R. T. Jacobsen,
R.B.
of neon for temperatures Advances
in Cryogenic
Kohlen,
R.
lationen.
Stewart
and M. Jahangiri
from the triple point Engineering,
Universit%t
Bochum
Mason, E.A.
to ‘100 K at pressures
31 (1986) 1189-1197, Plenum,
: Das fluide Zustandsgebiet
Fortschritt-Berichte
: Thermodynamic
von R22 Messungen;
VDI, Reihe 19, Nr.
properties to 700 MPa.
New York.
Berechnungen;
Korre-
14, 1987 und Dissertation,
Ruhr-
1987.
and T.H. Spurling
: The virial equati0.n of state.
Pergamon
Press Oxford
1969 . Marz,
V.
: Fundamentalgleichung
verhaltens
der Substanzen
Dichlordifluormethan
22), Trichlormonofluormethan Ruhr-Universitat Peters,
V.
rationale
Bochum
tion, Universit%t N.
Hannover
bis 10 000 bar erstellt Miinchen
der thermischen
Zustandsgleichung
durch gebrochene
durch multiple
fiir Wasser im Bereich vom Tripelpunkt
Regressionsanalyse.
Dissertation,
Universitat
1986. fiir den fluiden Zustandsbereich
Reihe 3, Nr. 149, 1988 und Dissertation, Schmidt
R.
Sauerstoff. Schmidt,
Ruhr-Universit;it
R. and W. Wagner
Schreiner,
K.
to Oxygen.
: Beschreibung
kubischen
Ruhr-Universitat
Bochum
U.
des thermischen
Universit5t
R.B
und ihre Anwendung
Verhaltens
reiner Fluide mit druckexVDI, Reihe 3, Nr. 125,
1986.
fiir Methan
fiir den Zustandsbereich
bis 700 K bei Driicken bis 1000 MPa. Fortschritt-Berichte
and R. Jacobsen
auf
19 (1985) 175-200.
Fortschritt-Berichte
Kaiserslautern
Reihe 3, Nr. 192, 1989 und Dissertation,
to 1200 K with pressures
1988.
1983.
Fluid Phase Equilibria
: Eine Fundamentalgleichung
der Schmelzdruckkurve
Bochum
VDI,
: A new form of the equation of state for pure substances
Zustandsgleichungen.
1986 und Dissertation, Setzmann,
von Wssser bis
von 1273 K. Fortschritt-Berichte
: Eine neue Form einer Fundamentalgleichung Dissertation,
and its application
639-798.
(R 113). Dissertation,
1990. Zustandsgleichung
Saul, H. : Eine Fundamentalgleichung
Stewart,
(R
VDI, Reihe 3, Nr. 224, 1990 und Disserta-
zu Driicken von 25 000 MPa und Temperaturen
pliziten
Zustands-
(R 12), Monochlordifluormethan
(R 11) und Trichlortrifluorethan
Fortschritt-Berichte
: Thermische
des thermodynamischen
1989.
: Die Approximation
Funktionen.
Rosner,
zur Beschreibung
Ruhr-Universitgt
: Thermodynamic
properties
Bochum
von VDI,
1989.
of argon from triple point
to 1000 MPa. J ournal of phys. them.
ref. Data 18 (1989) 2,