S0001433809060139

ISSN 0001-4338, Izvestiya, Atmospheric and Oceanic Physics, 2009, Vol. 45, No. 6, pp. 799–804. © Pleiades Publishing, Ltd., 2009.
Original Russian Text © N.P. Romanov, 2009, published in Izvestiya AN. Fizika Atmosfery i Okeana, 2009, Vol. 45, No. 6, pp. 854–860.
A New formula for Saturated Water Steam Pressure
within the Temperature Range –25 to 220°C
N. P. Romanov
Typhoon Research and Development Enterprise, ul. Pobedy 4, Obninsk, Kaluga oblast, 249038 Russia
e-mail: [email protected]
Received June 25, 2008
Abstract—Instead of approximation formula ln(E(t)/E(0)) = [(a – bt)t/(c + T)] commonly used at present for representing dependence of pressure of saturated streams of liquid water E upon temperature we suggested new approximation formula of greater accuracy in the form ln(E(t)/E(0)) = [(A – Bt + Ct2) t/T], where t and T are temperature in
°C and K respectively. For this formula with parameters A = 19.846, B = 8.97 × 10–3, C = 1.248 × 10–5 and E(0) =
6.1121 GPa with ITS-90 temperature scale and for temperature range from 0°C to 110°C relative difference of approximation applying six parameter formula by W. Wagner and A. Pruβ 2002, developed for positive temperatures, is less
than 0.005%, that is approximately 15 times less than accuracy obtained with the firs formula. Increase of temperature
range results in relative difference increasing, but for even temperature range from 0°C to 220°C it does not higher
than 0.1%. For negative temperatures relative difference between our formula and a formula of D. M. Murphy and
T. Koop, 2005, is less than 0.1% for temperatures higher than –25°C. This paper also presents values of coefficients
for approximation of Goff and Grach formula recommended by IMO. The procedure of finding dew point Td for
known water steam pressure en based on our formula adds up to solving an algebraic equation of a third degree, which
coefficients are presented in this paper. For simplifying this procedure this paper also includes approximation ratio apAT 0
plying a coefficient A noted above, in the form Td(en) = ------------ + 0.0866ε2 + 0.0116ε10/3, where ε = ln(en/E(T0)). Error
A–ε
of dew point recovery in this ratio is less than 0.005 K within the range from 0 to 50°C.
DOI: 10.1134/S0001433809060139
1. INTRODUCTION
Meteorology and other fields of science and technology require knowledge of the dependence of the
pressure of saturated water steams on the temperature.
Currently, Goff and Grach formulas—deduced in
1946 and adjusted in 1957—are accepted in official
IMO documents for calculating this dependence. For
liquid water these formulas were accepted as official
in the IMO protocol [1] in 1975 in the following form:
T
T
log E ( GPa ) = 10.79574 ⎛ 1 – -----1⎞ – 5.02800 log ⎛ -----⎞
⎝
⎠
⎝
T 1⎠
T
(1)
( – 8.2969 ( T /T 1 – 1 ) )
–4
+ 1.50475 × 10 ( 1 – 10
)
+ 0.42873 × 10 ( 10
–3
( 4.76955 ( 1 – T 1 /T ) )
– 1 ) + 0.78614,
where E is the pressure of saturated water steams, í is
temperature of water in Kelvin degrees, and T1 =
273.16 K is the water’s triple point temperature (for
which solid, liquid, and gas phases are in balance).
Formula (1) includes the value E(T1) = 6.1114 GPa that
was known that time. Later on in this paper, we will also
apply the temperature of ice melting í0 = 273.15 K and
temperature in Celsius degrees t, defined as t = T – í0.
Using new experimental data, the value E(T1) =
(6.11675 ± 0.0001) GPa measured in paper [2] introduces the following formula for saturated steam pressure in a temperature range of from 0 to 100°ë in paper
[3], the error of which was estimated as several thousandths of a percent.
ln E ( Pa ) = – 2991.2729T – 6017.0128T
+ 18.87643854 – 0.02835472T + 1.7838301
–2
× 10 T – 8.4150417 × 10
–5
2
× 10
– 13
– 10
–1
3
(2)
T + 4.4412543
4
T + 2.858487 ln T .
The next paper [4] includes formula E(T) for a temperature range 0–200°C. This formula already uses a temperature scale which was officially introduced in 1990
and has the abbreviation ITS-90. A description of this
scale that most fully agrees with the thermodynamic
temperature scale is presented in papers [5, 6]. Here we
will note that, in the ITS-90 scale temperature, a
defined value remains in the triple point equal to
273.16 K. However, the temperature of boiling water
at a normal pressure is 99.974°ë instead of 100°C. For
now, a subscript of 90 for the temperature or the sign
799
800
ROMANOV
/90 after the degrees will indicate the use of ITS-90.
Taking this indication into account, the formula presented in [4] is written in the following form:
– 5.8002206 × 10
ln E ( Pa ) = ------------------------------------------- + 1.3914993
T 90
3
– 4.8640239 × 10 T 90 + 4.1764768 × 10 T 90
–2
–5
2
(3)
– 1.4452093 × 10 T 90 + 6.54596 ln T 90 .
–8
3
For a wide range of temperatures from 273.16 to
647 K, papers [7, 8] include the following formula:
T 90 ⎛ E ( GPa )⎞
------ ln ------------------- = – 7.85951783ϑ
T c ⎝ E(T c) ⎠
+ 1.84408259ϑ
+ 22.6807411ϑ
3.5
1.5
– 11.7866497ϑ
3
4
(4)
7.5
– 15.9618719ϑ + 1.80122502ϑ ,
T 90
-, T = 647.096 K/90 is the temperawhere ϑ = 1 – -----Tc c
ture in the critical point and E(Tc) = 220640 GPa is the
pressure of saturated streams in the critical point.
Let us note that, for the positive temperatures for
which we determine the range of their application, the
difference between these ratios is less than tenths of a
percentage, which we will demonstrate below. The situation with negative temperatures is more complicated, which is evident from the overview of papers on
the pressure of saturated streams of supercooled liquid
water [9]. In [9] it was emphasized that the complexity
of the behavior of saturated stream pressure above
supercooled water is defined by a maximum heat
capacity value at T = 235 K; therefore, it is claimed
that a simple dependence of the saturated steam pressure above the supercooled water should not be
expected. The formula developed by the authors of [9]
for the temperature range of –10 to 59°ë contains nine
parameters and has the following form:
ln E ( Pa ) = 54.842763 – 6763.22T 90 – 4.21 ln T 90
–1
+ 0.000367 + tanh { 0.0415 ( T 90 – 218.8 ) } ( 53.878 (5)
– 1331.22T 90 – 9.44523 ln T 90 + 0.014025T 90 ).
–1
An overview of the ratios presented above (1)–(5),
which we will call initial, shows that all of them have
quite a complex structure; moreover, depending on the
author, one structure dramatically differs from
another. The results of calculations using these formulas are obtained by subtracting big numbers; thus, for
reaching an acceptable accuracy, temperature expansion coefficients for various degrees are set with a lot
of significant digits, requiring doubled accuracy when
calculating using these formulas. It is highly possible
that typographic and other mistakes will arise, and a
list of these errors for formula (1) in various IMO edi-
tions is presented in [9]. Editing mistakes appear in
other official editions, in particular in the reference
book [10, p. 75], where in the second and third member of formula (1), instead of the í/í1 ratio, í1/í was
typed; in the forth member, instead of multiplier
4.76955, multiplier 4.79955 was written. Finally, the
form of expressions makes it impossible to predetermine the influence of separate members or compare
the expressions with each other. The considerations
presented above stimulate the need to develop simpler
and more universal ratios, even with some accuracy
loss.
One large overview [11] includes a little less than
100 approximation formulas of various complexities
and numerically compares them with formula (1). The
approximation formulas presented in reference books
[10, 12] are also included in an examination list in
[11], the conclusions made in this paper are also
applied to them. Conclusions in [11] prove that a lot
of complex expressions are often less accurate than
simple ones. A three parameter ratio that also includes
the possibility of building an analytical expression for
the reverse function is recognized as the most effective of the simple expressions
(a – b(T – T 0))(T – T 0)
- .
E ( T ) = E ( T 0 ) exp -------------------------------------------------------c + (T – T0)
(6)
The value E(T0) is defined by experiment and is not a
parameter. Potentials for using expression (6) to
describe E(T) are almost entirely studied in [13], where
the least-square method was applied to calculate the
parameters that will be used as the initial data in formula (2); this formula was also suggested to be used
for negative temperatures. For two-parameter ratio (6), we
recommend using the set of values E(T0) = 6.1121 GPa,
a = 17.502, b = 0 , c = 240.97. In this case the error of
approximation of ratio (3) reaches 0.2% within the
temperature range of from –20 to 50°ë. Further, [13]
suggests using a three-parameter formula with the
parameters E(T0) = 6.1121 GPa, a = 18.678, b = 1/234.5,
c = 257.14. In this case the error of approximation
within the range of –25 to 105°ë is in the range of
0.1%. Further refining the approximation accuracy
using ratio (6) is achieved only by limiting the temperature range with the respective adjustment of parameters or inserting a new parameter. In this paper we show
that using another approximation relation makes it
possible to reach a considerably less approximation
error of initial ratios even with three parameters.
2. A NEW APPROXIMATION
FORM OF THE DEPENDENCE
OF SATURATED WATER STEAM PRESSURE
UPON TEMPERATURE
The nature of the dependence of the ratio between
the values of the pressure of saturated steams in two
IZVESTIYA, ATMOSPHERIC AND OCEANIC PHYSICS
Vol. 45
No. 6
2009