Rep rinted from May 1986, Vol. 108, Journal of Heat T ransfer
Use of a Cubic Equation to Predict
Surface Tension and Spinodal
Limits
P. O. 8i ney1
Wei-guo Oong 2
J. H. Lienhard
Fellow ASME
Heat Transfer and Phase-Change Laboratory, Mechanical Engineering Department , University 01 Houston , Houston, TX 77004 The very general Shamsundar- M urali cubic equation is used to interpolate p-v-T
data into the metastable and unstable regions. This y ields a spinodal line that closely
matches the homogeneous nucleation limit predicted by an improved kinetic theory.
Only the pressure, the saturated liquid and vapor volumes, and the liquid com­
pressibility at saturation, as well as one compressed liquid data point. are needed to
use the cubic equation fo r the interpolation process. The equation also yields an ac­
curate prediction of the temperature dependence of surf ace tension when it is
substituted in van der Waals'surface tension fo rmula. Thus, by capitalizing on the
inherent relation among the p-v- T equation, the spinodal prediction, and the sur­
face tension - all three - it is possible to obtain each with high accuracy and minimal
experimental data.
p
Introduction
Surface tension is intimately rel ated to the metastable and
unstable fluid states, and to the p-v- T equation that describes
these states , The aim of this study is to use this interrelation to
assist in the development of means for estim ating and predict­
ing both p-v- T and surface te nsion d ata , W e therefore begin
by reviewing the character of this relationship .
Pc
Surface Tension and the Equation of Stale. It was shown
in 1894 by van der W aals [1] that the temperature dependence
of surface tension could be predicted precisely by the
expression
~=f(Tr)=
r
f'g
---k-[Pr,sa,(Vr-V rj )PrdVrJ 1/2 dvr (I)
ao
Vrj Vr
Vrj
where a is the surface tension ; Tn Pro and Vr are the reduced 3
temperature, pressure, and volume: and where g and f denote
saturated vapor and liquid values. T he constant ao is a
reference value of the surface tension which van der Wa als
showed how to evaluate in terms of molecula r properties. (N o
one to date has managed to make accurate evaluations of ao.)
Little has been done with eq uatio n (I) because its use re­
quires a full knowledge of p-v- T information throughout the
metastable and unstable fluid regimes (see Fig. I). van der
Waals used his own famous equation of state in equation (1)
and - without the aid of a computer - succeeded in making an
approximation valid only near the critical point:
16
·
I1m
a=--a (1- T
Tr-I
V6
0
)3 / 2 r
(2)
Recen t studies (see, e.g., [2]) of the variation of a with Tr near
the critical point suggest that
lim acx(1 _ T
Tr-l
)1.280r 1.29 r
(3)
gives a more plausible temperature dependence for real flui ds
than equation (2) . It is also kno wn that a form of equation (2)
with an exponent of 1119, or 1. 22, represents a wide variety of
fluid s pretty well at lower temperatures.
O ~~~----------------------~
Fig. 1
On Locating the Spinodal Line. A knowledge of the loca­
tion of the liquid and /or vapor spinodal lines can be par­
ticularly helpful in the process of develo ping the p-v- T equa­
tio n of sta te that is needed to comp lete t he integrat io n o f equa­
tion (1). In 1981 , Lienhard and Karimi [3] pro vi ded m olecular
arguments that ha wed tha t the liqu id spinodal limit 4 co ul d be
predicted quite accu ra tely b y h omogeneo us nucleation theo ry.
T hey also ho wed that this was n ot true fo r the vap or spinodal
limit {3. 4] . Vapor spinodal li nes lie nowhere ncar the limit of
homogeneo us nucleation for vapo rs.
In the course of their wor k, Lienhard and K rim i used the
conventiona l ho mogeneous nucleation expression
I PreSent address: Mec hanical Engineering Department, Prairie View A . and
M. University , Prairie View, TX 77445 .
20 n leave from the Thermal Power Engineering Research In slitute, Xi-an,
People's R epublic of China .
3 A "reduced" properly is one divided by its thermodynamic critical value .
Contributed by the Heat Tran sfer Division and presen ted at the 23rd National
Heat Transfer Conference , Denver, CO, Augu st 1985 . Manuscript received by
the Heat Transfer Division No vember 8, 1984.
Journal of Heat Transfer Typical real,lIuid Isotherms
. nu leation events
-Gb
=e molec ule co llisions
}
(4)
4The spinodaiiimit is the locus of points at which (ap/au)T is zero (see Fig.
1).
MAY 1986, Vo l. 108/405
where 5
j=pro bability of nucleati ng a bubble in a given collision (5)
a nd
G b= Wkcn / (kTor k Tc)
(6)
and (see e.g. [5])
(7)
Notice that in equation (6) we suggest that the Gibbs
number Gb sho uld be a ratio of the critical work required to
form a nucleus to either kT or kTc' T he conventional nu clea­
tion theory is based on the average kinetic energy of the sur­
rounding molecules, which is on the order of kT. However, it
was noted in [3] that the energy requi red to separate molecules
from one another is on the order of k T c. T his seemed to be a n
equally plausible candidate for the characteristic energy of the
system.
When equations (4), (6), and (7) are combined, we obtain
the following expression fo r the homogeneous nucleation
pressure P corresponding with a given temperature Tsp
- In j = 167ru 3
2
2
3(kT or k Tc )[P sat (Tsp) - p ] (I - vll v g)
(8)
where vI and Vg are to be evaluated at Tsp- Two issues re­
mained: identifying the value of j that will give the spinodal
limit, and deciding whether to use kT or k T c'
Lienhard and Karimi next curve-fit cubic equations to the
well-doc umented stable equilibrium states of water, constrain­
ing them to satisfy the "Gibbs-Maxwell " requirement that
r
~
J vdp=O I
(9)
which stipula tes that the two regio ns between an isotherm and
a horizontal li ne connecting j an d g must be equal to one
another in area (e.g., Area A in Fig. 1 equals area B). They
chose the H impan form (8) of cu bic equatio n
A
a
Pr=---
(10)
vr-b
(vr-c) (vr +d)
and evaluated all five constants using least squares fit. They
re-evaluated the constants for each of many isotherms.
There were two weaknesses in this curve-fit procedure. The
first is tha t equation ( 10) was not forced to fit the ideal gas law
precisely at high temperatures. T he second is that the H impan
form turns out to be slightly rest rictive. We remedy these
featu res subsequen tly.
Equ at ion (10) with the five stat istically fitted constants gave
interpolated liquid spinodal press ures that showed some
numerical data scatter. T hese pressures corresponded very
5Actually . it is more common (see Skripov [5. 6] and Avedisian [7]) to use J
instead of j. J is equal toj multiplied be{ the rate of molecular co llisions per cubi c
centimeter. For water, J is about 10 3 times j in these units.
closely to the homogeneous nucleation pressures given by
equation (8) with j= 10- 5 and with kTc used as the
characteristic energy. -' he interpolations did not come close to
equation (8) when the conventional energy kT was used. (The
minimal valu~ ofj= 10- 5 was not purely empirical. M olecular
arguments in [3] fixed it within an order of magnitude or so .)
It is important to note th at when these arguments are ap­
plied to the vapor spinodal [3J they show that the limit of
homogeneo us nucleation o f dro plets is far from the vapor
spinodal limit.
A New Form of Cubic i'A:Juation for Fitting p-v- T
Data. Shamsundar and M ura li [9, 10] have recently made ef­
fective use of the follow ing general form of cubic equation in
fittin g indi vidual isotherms
~=
1- (V-VI) (v-v rn ) (v- vg )
(11)
(v+b) (v+c) (v+d)
The quantities P ,al' Vj ' Vrn , vg , b, c, and d all vary with
temperature. This form has the advantage that it automati­
cally satis fies critical point criteria, but it need not be tied to
them. It is also pre factored to simplify fitting the constants.
Three of th e coefficients in equation (II) are the known
temperature-dependent properties: P sal , VI' and vg • Thus the
most straightforward use of the equation is one in which it is
fit to one isotherm at a time.
Murali simplified equation (II) by setting c = d and thereby
red ucing the number of unkno wn coefficients to three. These
three coefficients were determined by imposing the following
three conditions on the equation: the ideal gas limit at low
pressures, the Gi bbs-Maxwell condition, and the measured
isotherm al compressibility of saturated liquid . He thus
evaluated the coefficients of the equation directly (rather than
statistically) using very few data.
The condition under which equation (11) reduces to the
ideal gas law, by the way, is
.0 ' at
RT
--=b+c+d+vl+vm+vg
(12)
P sat
This kind of temperature-by-temperature application of
equation (11) yielded far higher accuracies [10) than any ex­
isting cubic equation, particularly in the liquid range. The
equation also performs extremely well in the stable
superheated vapor range. T he key to this success is, of course ,
the fact that the coefficients do not have to obey any predeter­
mined dependence on temperature .
An nJustrative Application of tbe Preceding Ideas. Infor­
mation of the kind \ e ha ve been describing can be used to ex­
pand existi ng knowledge. Kari mi , for example, developed a
new fu ndamental equation [11 ] for water , based in part on his
interpolations. W hen this fundamental equation was used to
generate p-v-T data fo r use in equ ation (I), the resulting
values of u/uo lay within 4.3 percent of measured values over
all but the lowest range of saturation temperatures. This was
- - - - Nomenclature
A, a, b, c, d Gb
j
J
k
p
T
undetermined constants in the various
cubic equations
Gibbs number (see equation (6»
nucleation pro bability, equation (5)
j expressed as a rate per unit volume
Boltzmann 's con stant
pressure
temperature
speci fic volu me, saturated liquid volume,
saturated vapor volume
v at the root of a cubic p-V- T equation be­
tween vf and Vg
406/ Vol. 108, MAY 1986
u, Uo
w
= surface tension, undetermined reference
=
value of u
the P itzer acentric factor,
- 1 -IOgIOPr.sat (Tr = 0.7)
Superscripts and Subscripts
c
j, g
r
sat
sp
a property at the critical point
saturated liquid or vapor properties
a "reduced" propert y (Xr=XIX c)
a property at a saturation condition
a property at a spinodal point
Transactions of the ASME
c
2
0
O r-------------------------~~~
Ol <lJ
'-
0
-c
::J
0­
-c
-2
-1
-2
2
Q)
(I)
T,= 0 . 5
(I)
....<lJ
T, =0.7 T,= 0 .8
T, = 0.9
r-----~~--~~~~------,_--~O
c.
E
I but ane I
-1
0
0
c
2
-2
<lJ
T, = 0.53
~
<lJ
....
c.
c:
...0
...
T, =0.7 T,=0 . 8
T,= 0.9
eq u atio n
C/)
C/)
\ ".. ....
-6
~
a.
~
~
u
:J
l:l
/'"
-8
/
V
-10
0
./
",
equation (8)
with kT replaced
by kTc
predicted conve nti on al
" " " ' - - - - homogene o us nucleation
Ipentane l
-1
theory. equation (8)
-1 2
-2
2
T,= 0.6
-
T, = 0 . 5 6 T, =0.7
...
Ql
Iheptanel
T, =0 . 8
- 14L-----~--~----~-----L----~--~
T,=0.9
0.4
c
<lJ
0
Cl.
predi c ted by
cubic p- v-T
a:
'-
Ql
-4
~
"
<1>
'­
(I)
(I)
cl
0.3
0 .4
OJ
0.8
0 .9
1.0
Fig. 3 Comparison of the spinodal limit 01 water as predicted by
homogeneous nuclealion theory and by the general cubIc equation
0.46
Reduced specific volume, vr
Fig. 2 Typical values of the error in pressure as predicted by cubic
p-v- T equallon curve·fltted to four data points
the fir st and only such use of the van del' Waals surface ten­
sion theory to verify p -v- T information , to o ur k no wledge,
a nd it d isplays the po tential interaction of (J and p -v- T
information.
Present Objectives. The a vai la bility of Sh ams undar and
M urali's new met ho d fo r interpolati ng iso therms g ives means
for su bstantia ll y impro ing upon the work in (3] and for re­
o pening tbe t wo q ues tio ns: (1) "Is k Tc a better characteristic
nergy tha n k T?" (2) "What minimu m value of j gives t he
spinodal limi t when it is used in eq uation (8)?"
We therefore address these matters usi ng van der Waals'
sur face tension prediction as a hitherto little-used vali dity
check.
Isothermal Curve Fits
We altered two o f M urali ' s assumptions: We did not use the
ass umptio n that e = d. This increased the number o f unknown
coefficiems by one, b ut stood to improve th accuracy o f the
resulting eq uation. The seco nd a lteration dealt with the
restricti ve form of the den o minator of equalion (L1 ) (v + b)
• (v + c ) (v + d) . In t his fo rm the equation is restrictive if b, e,
a nd d are to be real. T hi implica tion was relaxed by writ ing
the denominator in the form (v + a) ( v 2 +/ v +g) where / = b + c
and g = be (9J . This form allows b and e to be com plex without
USi ng complex numbers explicitly.
T o complete the cu rve fit we needed one m ore condition
than M urali d id since we chose not to set c = d. F or this we
selected a high-p ressure liq uid point. We ide n tified the
pressure that m ade the isotherm best fi t the available data for
stable liq uid and vap or stales by trying several p ressures until
we fou nd t he o ne that wo rk ed bes t. For water this press ure
proved to be about 800 bars, o r a reduced pressure o f about
3.7, although substan tially higher or lower values worked
a lmost as well. Data from the IFC Formulation for Scientific
Use (12) were used to d o t his .
Journal of Heat Transfer
0 .6
Reduced temper atu r e, T,
-1
~-L_ _L - - L_ _~~_ _~~_ _~~L-~ - 2
0.26
0.5
When the four conditions were applied to t he cubic equa­
(ion (1 J) they yielded four nonlinear equations which we
solved using t he met hod of su ccessive su bstitutions and linear
Interpolation .
We made sets of isot hermal curve fits for water and for
several straight-chai n hydrocarbons. For the straigh t-chain
hydrocar bons, we used the da ta o f Starli ng [13] as curve-fitt ed
by Reynolds [14], where we could . (Ho wever , in a few cases
the St arling data for liquids disagreed wit h API data [15] or
the data of Vargafti k [16] .) These were the o nly substances fo r
which we co ul d readily ob tain reliable p - v- T and surface t n­
sio n data [15 , 16, 17J over broad ranges o f temperature. In all
cases , the high-press ure p-u- T data point used in the curve f it
was [he value at Pr = 5.
We a l 0 10 ked at am monia, argon, benzene, carbo n di­
oxide, metha ne, hydrogen, oxygen , and nitrogen for which we
did not have complete data over la rge ranges. In these cases
curve fi ts were only made a t one tem perature each . The data
so urces for these cases were [16, J7, 18] .
The fl uids co nsidered here res o lve roughly into two
clas ifi cations: those for which we a re confident o f the ac­
curacy (wa ter . oxygen, hydrogen, nitrogen, butane, heptane,
pentane, and p ro pane) and those for which we found some
level of unresolvable d isagreement in the relevam properties
(ammo nia, argon, benzene, carbon d iOXide, ethane, hexane ,
methane, and oc tane) . In t he subsequent discu ssio ns we take
care to base our conclusions only on the results ob tained in the
for mer fluids .
F igure 2 show the resulting err ors o f t he cubic interpola­
tions for several typical fluids . These plot~ of error in the
p red ict d pressure, at selected values o f Tr in the liquid range,
reflect a very severe test of the curve fit s. They consistently
show err rs su bstan tiall y less than one percent, at reduced
temp ratures below 0.9.
Liquid Spinodal Limits
Figure 3 co mpares the spi nodal limit of water as predicted
by our cu bic equatio n with t he pred icted spinodal limit based
o n equation (8) . The NBS surface tension recom mendation
[19] was used in this calculation . (The value o f j used in Fig . 3
MAY 1986, Vol . 108/407
0.5
0. 6
0.7
- - '-/1 '7 iT
0 .8
0 .9
1.0
0.8
-~ r7
~
-2
c..~
•
0
~
0 .9
1.0
1.0
0 .8
0 .9
1. 0
J
J
f
/
I.
0 .9
..,
/ /
..
/i
- 1
-2
./ /
/
,,'
~. /
., '
j
/'
-4
; ::r ,it' ' (,
ii/ /' ;/:: ~"""'". '"
-4
.-
!
~
~
a:
,'
.,
,
cf..
~f'
I
-1 0
- 12
I'
_
r
- ,5 -
f-' 3,
t
I
uw.H"",UI
I
:::'~:C: ;" 'M
. , .1,
2- .... 1 /
•
"
"
0 .7
0. 5
.II"'~-."
1
' 1f
0. 6
:;0,
.:
-1.A
0 .5
I
..
0.6
0 .7
'
0 .6
~'_:~':'::'Ir '
,1-1-'-' --'---'--0 .7
0.8
0 .6
0.7
0.8
-5
"
j=3(10)
-8
-
~
........:
___ -- ~ -
-
.-
.-
.
2
12
1 -­
C
818
H
- . -.-
•
O2
II CO 2
•
N2
[J
CsHs
... H2
o
NH3
o
.;. Ar
CH 4
'---'
0 .9
1.0
°
0 .5
0 .4
Redu ced tem perat ure , Tr
Fig, 4 Comparison 01 the spinodal limit 01 several lIulds as predicted
by homogeneous nucleation theory and by the general cubic equation
was 2 x 10- 5 . We return to the question o f specifyin g) below.)
The agreement is very good when kTc is used in pl ac~ of k T! n
the equation, except at such low temperatures and hIgh hqwd
tensile stresses tha t both theories are being pushed to the edge
o f their limits of applicability . The choice between k Tand k Tc
makes little difference in the regio n o f positive pre sure, and
this is the only region in which nucleation exper iments have
ever been made for large). At lower pressures the two diverge
very strongly .
Our suggestion that k Tbe replaced with kTc is largel y based
on this kind o f extrapolation . This kind o f demonst ra tion was
made less conclusi ely (with the less flexible Himpan equa­
tion) in [3] . The present evidence is very c.ompe.lIing. indeed .
Several other such comparisons are given In FIg. 4 fo r
butane, heptane, hexane, and propane. (Since urface tensi?n
data were not available for butane and propane over tile en ti re
rang of temperature, the missing values had to be filled in
with the help of equation (1 ) in these cases.) In each case, we
have used a limiting value of j that best fits the extrapolation.
These j's do not all match the value of 2 x 10 - 5 used fo r waler.
The four flu ids selected fo r display in Fig. 4 were chosen
because they embrace a wide range of j values. By the same
token, the four flu ids shown in Fig , 2 were selected because
they typified the error o f th many flu ids thal have been fitted .
The results of an inverse kind o f calculation are show n in
Fig . 5. Val ues of -J -In(j), whic h is in versely proportio nal to
the pressure difference between saturation an d the liquid
spinodal line, were calculated at each point using equation (8) ,
with kTc. and the pressure difference predicted by equation
(1).
Figure 5 strongly suggests that a "best value" of) for the
spinodal limit - if one truly exists - is o ne slight ly in excess of
10- 5 , in preference [ 0 10 - 5 which was prev iously suggested
(3). It is clear that these ) limi ts are fairly sen 'itive to the ac­
curacy of the data upon which they are based. Thus, in choos­
ing the appropriate limiting value, one must be guided strong­
ly by water and the other very weLl- docwnented fluids.
One must also consider whelher or not these ) al ues were
obtained in regimes in which the cubic equation is tru ly ve ry
accurate. Figure 2 makes it clear that the general cubic equa­
tion interpolations begin to lose precisi on at very high
temperatures - typically before T, == 0.9. It also becomes ap­
parent in the subsequent section that, although it interpolates
stable properties very accurately at low temperatures. equa­
tion (11) probably fails to represent metastable and unsta ble
properties with very high accuracy at low temperatures. ThL is
evident in its failure to predict the temperature dependence of
surface tension wi th high accuracy below T, == 0.5.
We accordingly restrict the plots in Fig. 5 to the range
0.5 < T, < 0.85, or to a smaller range in which reliable data are
4081 Vol. 108, MAY 1986
I .-.;.
_
'
\
-_ ,'45
_IIU'"''
•
I
4
-6
-1 0
~
-5
= 10
1. 0
Reduced
temperature, Tr
Fig. 5 Comparison 01 the values of v'=TnUl for which equations (8)
and (11) agree exaclly
bO
b
7
c·
0
'a;
c:
Q)
6
~ lAPS co rrel a ti on of data
predicted u s ing cubic
Q)
()
-...
ca
e quation of state and
equation (1)
5
~
-
4
c:
3
(/)
0
Q)
C,)
Q)
j
'C
C
Q) Cl
Q)
2
'C
...
Q)
::I
ca
1
L. Q) Cl
E
0,)
~
0
0.4
0.5
0 .6
0.7
O.B
0.9
Reduced t empera ture. Tr
Fig. 6 Predicted and measured temperature dependence of the sur·
face tension 01 water
available. Furthermo re, we suggest that the middle
temperature range in Fig. 5 is the most reliable. We have
averaged the ordinate values of the more relia ble data in Fig.
5, giving wa ter double weight, to obtain the recommendation
that
) ,pinodal
== 3 X 10- )
This is just a little higher than the values o f (l or 2) x 10 - 5 ,
used previously. However this must be accompanied by the
warning that we might eventually have to admit some
depen de :~ce o f the limiting ) on T, and the flu id. (Of course
3 X 10- 5 is an approxim ation that o ne would only wa nt to use
if better information about j were u navailable.)
No tice, too , that replacing k Twith k Tc was a pretty revolu­
tionary suggestion. The modi fica tion o f ) by even so much as
an order of magnitude, on the other hand, is far less im portant
because most calculations based on) are very insensitive to its
value.
Prediction of Surface Tension
T he acid test of any p -u- Tequation that purports to predict
Transactions of the ASME
10
O
10
8
b
~
"­
b
u
Q)
4
0
u
2
...
Q)
Q
.
/------­
~
Q)
-4
c:
-6
Karimi & Lienhard's
equation
-8
Uncertainty of
experimental data
-2
Q)
a.
6
CIJ
....U
4
"0
2
...0.
'\
\...,\
Cubic equation
-10
0.4
\
\
...
......
-4
....I:
-6
-2
0
I
Q)
<Il
0.7
0.6
0.5
0.8
0.9
...u
<Il 1.0
temperature, Tr
Reduced
Cl. Fig. 7 Accuracy of the predicted temperature dependence of the sur·
face tension 01 water, based on the present cubic Interpolations of
p-v- T data, and upon Karimi's fundamental equation
7
I-~
-c:
"bO
5
b
4
C
3
"-
0
'c:"
6
5
b·
n
5
0
7
2
Heptane
'"'"
"
"
.~
'"
"E"
6
u
...
:J
...
4
Q)
a.
E
Q)
I­
3
0.2
0
-0 .3
4
Octane
2
1
1
0. 6
0.7
0 .8
0. 9
0
1 .0
Reduced temperature, Tr
Fig. 8 Predicted and measured temperature dependence of the sur·
face tensIon of three fluIds
metastable and u ns table properties is whether or no t it cor­
rectly predicts the temperatu re dependence o f sur face tension
when it is used in van der Waals' equatio n (I ) . We have sub­
jected our cubic equations for water to this test at each
temperature, and the resu lts are shown in Fig. 6.
Figure 6 ma kes it quite clear that, except at the very lowest
temperatures, this prediction has been extraordinarily suc­
cessful. Nevertheless, it is this evidence th at suggests that,
while the cubic fi ts the low-temperature stable points with
great accuracy, it is probably less accurate than we wo uld wish
in the metastable-unstable range . Of course, this observation
is based on water - the only substance for which we had full
data below Tr = 0.5, but one which is also known for its
strange behavior at low temperatures.
• ",0
•
Cs H'2
•
C3H,!I
.. C .. H,O
0
C 6H 1 4
£\. CeHe
0
co,
¢. NH:J
~N,
..,. C2He
.
+ H,
o
C H4
-9-
A,
-0 . 2
- 0 .1
0,
0
0.1
0.2
0.3
0.4
0.5
0. 6
Pitzer acentric factor, w
3
I
2
0
0.5
0.4
0
Q)
~
0 .6
0
5
0. 8
~
Q)
Q
Open symbols d e n o te oth er fluid s
Le ast squares Ii! through solid
points
1. 0
"
"" 4
Q)
Q)
o
1 .4
"0
Q)
c:
Solid symbols denote the fluids tor
whi ch abundant consistent data are
a w. lI able
1.2
3
"'C
'"
;;;'"
3
c:
1.0
C
0
0
•
1.6
d'
2
~
4
-
0.9
0 .8
1.8
3
...'"
'"
:J
0.7
0.6
~
Q)
0
- 10
0.5
6
4
7
-8
Re d uced t e mp e rat ur e , Tr
>-"
Q)
Hexane
Ag. 9 Accuracy ot the predicted temperature dependence of the sur·
face tension of three lIulds, based upon the present cubic Interpola·
tlons of P-'f- T data
7
- - Predicted by
cubic equati on
and equation ( 1)
6
0
I:
Q)
...
b
Q)
...
0
......
0
8
"0
0
c:
bO
.....
6
Fig. 10 Corresponding states correlallon of the surface tension lead
constant
Figure 7 shows the error in the predicted temperature
dependence of surface tension, for water. The prediction is
very nearly wit hin the reported accuracy of the NBS surface
tension data [19) for Tr ~ 0.5 . T he accuracy of Karimi's
prediction is also included for comparison.
Figure 8 includes a comparable set of curves for heptane,
hexane, and octane, the only fluids besides water for whi ch
convincing surface tensio n data were available over a wide
range of temperature [161 . These curves again how that the
cubic interp olations give very good predictions of the
temperature dependence o r surface tension hen they are
substituted in equation (1 ). Figure 9 shows t he percent error in
a/ ao for these fluids . O nce again the results are very accurate
for T r ::;0.85.
While equation (I) only predicts a/ ao , we would like to be
ab le to predict ao as well . T o make the comparisons in Figs. 6
and 8, it was necessary to calculate the average value o f ao for
each substance, based on the surface tension data. W e also
computed some val ues of ao at single points for fluids for
which reliable p-u-Tand a data were not available over ranges
of tem perature.
In 1955, Brock and Bird [20] showed that the appropriate
corresponding states nondimensionalization of a was
Journal of Heat Transfer
MAY 1986, Vol. 108 /409
-­
-­
---­
ao / pc 2/ 3(kTJ 1/3 . Figure 10 presents the correlation of our ao
values as a function o f the P itzer factor w using this non­
dimensionalization . 6 T he points based upon the data in which
we have high confidence are presented as solid symbols. They
define the following correlation
ao/p//3(kTJ I / 3 = 1.08 -0.65w
(13)
with a correlation coeff icient of 0.995. The remaining data are
somewhat more widely scattered, but they do not significantly
alter the correlation.
Others, startulg with Hakim et al. [23], have formed cor­
responding states correlations for a that include expressions
for ao. These can be very useful, but they a re normally based
on assumed forms of the temperat ure dependence of a/ao ,
that differ from that given by van def W aals' integral. Yet,
even though lhese ao expressions might also be linear in w (as is
true in [23)), they do not and should not match eq uat ion (13).
Equation (13) gives lhe lead constant specifically for the van
der Waals integral.
One can thus predict surface tension with acceptable ac­
curacy for many applications, using p-u- T data alone, with
the help of equations (I) and (13 ). As a matter of academic in­
terest, we can predict a dimensionless (]o fo r the van der W aals
equations (for which w = - 0.302) . Th value is 1.276.
Conclusions
It a ppears possi bl to interpolate p-v- T data with great
accuracy and a mi nimu m of experimental data, using equation
(1J). The accuracy o f such predictions has proven to be best
(for the 16 fluids st udied) in the ra nge 0. 5<Tr<0.85.
2 The limiting value of j for which the homogeneous
nucleation theory, eq uatio n (8) , gi ves the spin odal lim it , is on
the order of 3 X 10 - 5 . However, il might ultimate ly show some
variat ion from flu id to fl uid, or fr om one saturation condition
to another.
3 Furt her compelling support is provided for the idea (sug­
gested in (3]) that k Tc should be used in equation (8) in place
of k T.
4 Equation (I) provides convincing support for th e present
predictions of metastable and llllsta ble p -u- T data.
5 T he lead constant ao in equation (I ) is given by equation
(13).
6 T he Shamsundar- Murali cubic equation has on ly been
used for indi vidual isotherms here. We strongly recommend
that the problem of developi ng gene ral corresponding states
correlations to represent the temperature dependence of its
coefficients be undertaken in the future.
Acknowledgments
We are particularly grateful to Prof. N. Shamsundar for his
6The Pitzer acentric factor [21], w'" - I - log I0 Pr.sat (Tr : 0.7), is probably
the best choice of molec ular parameter available for correct ing Corresponding
Stales correlations. See, e.g., [221 for a further discussion of recent applications.
410 1Vol. 108, MAY 1986
very helpful counsel in all aspects of this wor k . This work has
received suppo rt from the Electric Po wer Research Institute
with Dr. G. Srikanti ah as Project Manager, and th e University
of HOll ton E nergy Laboratory.
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