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Full text: Numerical implementation and oceanographic application of the thermodynamic potentials of liquid water, water vapour, ice, seawater and humid air : Part 1: background and equations

R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1 
661 
www.ocean-sci.net/6/633/2010/ 
Ocean Sci., 6, 633-677, 2010 
be unstable if thermodynamic stability criteria such as posi 
tive compressibility are violated. (Independent of any simul 
taneous existence of other phases, a negative compressibil 
ity would amplify any pressure-density fluctuation, causing 
the fluid to collapse.) The boundary between metastable and 
unstable existence is regarded as the spinodal line beyond 
which the phase can no longer stably and homogeneously 
exist. 
Since it is practically impossible to measure thermody 
namic properties on or beyond the spinodal, its location 
in the phase diagram is not exactly known from empirical 
equations of state. Even though the IAPWS-95 formulation 
extrapolates well into the metastable regions (Wagner and 
Pilli), 2002; Feistel et al., 2008), with increasing distance 
from the saturation line the values computed from the 
Helmholtz function outside its validity range will become 
unreliable. This turns out not to be a major issue for de 
termination of the seawater Gibbs function since we require 
only relatively small excursions into the metastable regions 
to deal with the effects of shifted phase transition boundaries 
in the presence of sea salt. 
Panels a and b of Fig. Al indicate the initializations used in 
the library for our iterative solutions for the liquid and vapour 
phases, respectively. Figure A2 provides additional infor 
mation regarding the industrial formulation IF-97 (IAPWS, 
2007; Wagner and Kretzschmar, 2008) referred to in Fig. Al. 
Starting in either fluid state at a point near CP located 
along the saturation line joining TP and CP, we may circum 
scribe the critical point along a closed T—P path. Along 
any such curve, the properties change only gradually; noth 
ing like a transition point between liquid and vapour is en 
countered. Since, for numerical purposes, we distinguish be 
tween the Gibbs functions of liquid, Eq. (Al), and vapour, 
Eq. (A2), we need to specify such a transition point for tech 
nical rather than for physical reasons. Here we define the 
Gibbs functions Eqs. (Al) and (A2) to be different at sub- 
critical conditions (T<Tq and P<Pq) and to be identical at 
supercritical conditions (T>Tc or P>Pc)■ The critical tem 
perature of water is 7c=647.096K and the critical density 
is pc=322kgm -3 (IAPWS, 2009a); the critical pressure fol 
lows from Eq. (4.1) to be Pc=22.064MPa. In order to cover 
metastable states of liquid water as required in the regions of 
vapour-pressure lowering or freezing-point lowering caused 
by the presence of dissolved sea salt, the Gibbs function for 
liquid water is also available for T and P in regions extend 
ing somewhat beyond the saturation curve and beyond the 
melting curve. 
According to our numerical definition of liquid, vapour 
and fluid states, the initial values required for the iteration of 
Eq. (A3) can be chosen identically for the fluid density in the 
supercritical region (T>Tc or P>Pc), as shown in Fig. Al, 
but must be different for liquid and vapour in the subcriticai 
quarter (T<Tq and P<Pq). In the subcriticai region, sepa 
rate Gibbs functions are available from the industrial formu 
lation IF-97 (IAPWS, 2007; Wagner and Kretzschmar, 2008) 
Fig. A2. Thermodynamic relations available from the Industrial 
Formulation IF-97 (IAPWS, 2007; Wagner and Kretzschmar, 2008) 
defined in different temperature-pressure regions, derived with re 
duced accuracy from the IAPWS-95 Helmholtz function for differ 
ent independent variables. Here we make use of the Gibbs func 
tions “g(p, T)" available in region 1 (liquid/fluid) and in region 2 
(vapour/fluid) as shown in Fig. Al. In region 3, separate equations 
for the specific volume are available for various subregions which 
are not used here. Region 4 is the saturation curve. Graphics repro 
duced from IAPWS (2007), with permission of IAPWS. 
in regions 1 and 2 (Fig. A2) as defined therein which pro 
vide excellent starting values for the liquid and the vapour 
state. These Gibbs functions can also be used for the fluid, 
region 1 below 623.15 K and 100 MPa, and region 2 between 
273.15 K and 1073.15 K, and below 16.529 MPa. In the sub 
limation region and in the supercritical region, the ideal-gas 
density, p=P/(RifjT), provides a sufficient starting estimate 
below 273.15 K and above 650 K, and a constant value of 
p=1000kgm -3 can be used below 650K, Fig. Al. These 
latter choices are sufficient to ensure numerical convergence 
but do not necessarily optimize the speed of the code. Addi 
tional considerations apply to the immediate neighbourhood 
of the critical point as discussed below. Note that all of these 
rules are built into the library routines discussed in Part 2 
(Wright et al., 2010a) so that the user can make use of the 
routines without dealing with (or even being fully aware of) 
the details. 
The critical region is defined here as the T — P rectangle 
623.15-650K and 16.529-35 MPa, Fig. Al. The coefficients 
of an auxiliary cubic polynomial equation of state 
have been determined by regression to IAPWS-95 data 
points in the stable liquid, vapour and fluid region with an 
r.m.s. deviation of 1% or less for each phase; the resulting 
coefficients are given in Table Al. The cubic polynomial 
used for Eq. (A4) permits analytical inversion to determine 
p (T. p) for both the liquid and the vapour branches. The 
critical point of Eq. (A4) was chosen to be identical with the
	        
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