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