How To Find Vapor Pressure Of Water

1. Introduction

Atmospheric air consists of a number of gaseous components (due east.g., nitrogen, oxygen, carbon dioxide, inert gas, and water vapor). Dry air exists when all water vapor has been removed from atmospheric air. The composition of dry out air is relatively unvarying. Moist air is a mixture of dry air and water vapor. The amount of water vapor in moist air changes from zero to a maximum that relies on the temperature and pressure of atmospheric air. The latter condition is called saturation, a state of neutral equilibrium between moist air and the condensed water phase (ASHRAE 2013, affiliate 8). The saturation water vapor force per unit area, which is a function of air temperature, provides a basis for determining other thermodynamic properties of moist air (humidity ratio, specific enthalpy, specific entropy, specific book, etc.).

Goff and Gratch developed an accurate formula for calculating the saturation vapor pressure level (Goff and Gratch 1945), and modified this formula later (Goff 1957). In 1966 the Globe Meteorological Organization (WMO) recommended its utilize, and the WMO meteorological tables were based on these formulas (Letestu 1966). Wexler and Hyland obtained new laboratory data on the saturation vapor pressure level of water and derived another lengthy formula. Subsequently, this formula and the computed values were incorporated in the ASHRAE Handbook—Fundamentals (Wexler 1976; Hyland and Wexler 1983; ASHRAE is derived from the American Gild of Heating, Refrigerating, and Air-Conditioning Engineers). Wagner and Pruss developed a new formulation for the thermodynamic properties of ordinary water substance for general and scientific use, and afterward this formulation was adopted past the International Clan for the Backdrop of Water and Steam (IAPWS) (Wagner and Pruss 1993, 2002; IAPWS 2016). In 2011, Wagner et al. adult a new equation, which has considerably less dubiety, for calculating the saturation vapor pressure of ice (Wagner et al. 2011). The IAPWS adopted information technology shortly thereafter (IAPWS 2011).

The values of saturation vapor pressure can be obtained by looking them upward in the reference tables. When large-scale computations are needed or temperatures with decimal points are reported, it is routine to calculate the saturation vapor pressure by using an accurate formula.

The Goff–Gratch, Hyland–Wexler, and Wagner–Pruss formulas are ho-hum and inconvenient for apply in calculating the saturation vapor pressure. In that location has been much research on formulating the saturation vapor pressure. Bosen gave a formula for computing the saturation vapor pressure of water with reasonable accuracy in the temperature range from −51.i° to 54.iv°C (Bosen 1960). Langlois developed a fractional formula for calculating the saturation vapor pressure of h2o and compared the results with the observed values in Byers (1959) and the formula in Berry et al. (1945). The per centum error vs the observations was found to be less than 0.5% over the temperature range of 0°–35°C (Langlois 1967). Richards derived a iv-degree polynomial for the saturation vapor pressure of water within the temperature range from −50° to 140°C (Richards 1971). Tabata suggested a quadratic formula that yielded the saturation vapor pressure with a mean percentage fault of 0.17% within the temperature range from 0° to 35°C (Tabata 1973). Lowe provided a sixth-degree polynomial formula for computing the saturation vapor force per unit area of h2o from −50° to 50°C and evaluated it against the Goff's equation. The results showed that the accurateness was very loftier (Lowe 1977). Rasmussen presented a five-degree polynomial and a sixth-caste polynomial for calculating the saturation vapor pressure over water (from −50° to 50°C) and over ice (from −fifty° to 0°C) and another four-caste polynomial for the saturation vapor pressure over ice. The maximum relative error was very low (Rasmussen 1978). Gueymard used a least squares method to obtain three formulas for the saturation vapor pressure over water and compared the computed results with the new ASHRAE reference information. Information technology was shown that these three formulas could calculate the saturation vapor force per unit area with smashing accuracy (Gueymard 1993). Sanjari proposed a nonlinear equation for calculating the saturation vapor pressure of 75 pure substances by using multiple regression analysis, and comparison of the computed results and the information in the literature showed that the hateful relative deviation was about 0.1% for water (Sanjari 2013). The temperature range was non given in Sanjari (2013).

The to a higher place-mentioned formulas are nonetheless complex. Some researchers made various attempts to found elementary equations fitted to the reference data by means of regression assay. Tetens proposed two short formulas, one of which was of the Magnus grade, for calculating the saturation vapor pressure of water and of ice (Tetens 1930). The vapor pressure values given by this formula were acceptable for most meteorological purposes (Murray 1967), but the relative error provided by the Tetens formula increased with decreasing temperatures (Riegel 1974; Xu et al. 2012). In particular, at temperatures beneath −25°C the errors were above i% (Murray 1967). Buck used a minimax plumbing equipment procedure over the temperature range of near interest in meteorology and derived ii equations for computing the saturation vapor pressure of water (from −20° to 50°C) and of ice (from −50° to 0°C), respectively (Buck 1981). Bolton introduced an empirical formula for the saturation vapor pressure of water and noted that the error was 0.iv% over the temperature range from −35° to 35°C (Bolton 1980). Alduchov and Eskridge made a good use of an iterative process and least squares method to optimize the coefficients of the Magnus formula and recommended 2 equations for calculating the saturation vapor pressure over water (from −xl° to l°C) and over ice (from −80° to 0°C), respectively. These two equations yielded maximum relative mistake of less than 0.384% and 0.213%, respectively (Alduchov and Eskridge 1996). Leckner calculated the saturation vapor pressure of water using a simple exponential function (Leckner 1978). Stephens estimated the saturation vapor pressure of h2o by using a simple formula that was based on the Clausius–Clapeyron equation (Stephens 1990). Koutsoyiannis recently derived an equation for computing saturation vapor pressure of water on the basis of the Clausius–Clapeyron equation and indicated that the relative difference from the reference datasets was pocket-sized from −40° to fifty°C (Koutsoyiannis 2012).

In nature, subzero temperatures occur in high-breadth common cold regions. The temperatures in some areas have dropped to tape levels considering of a high frequency of cold weather (Turner et al. 2016). It has been predicted that about 20% of Earth's land surface will feel many rut waves by 2040 (Coumou and Robinson 2013). These changes will require a saturation vapor pressure level conception to be very accurate over a wide temperature range (from −100° to 100°C).

For a vapor pressure formula, accuracy range is divers as the temperature range in which the relative fault (RE) is less than 0.one% (Gueymard 1993). Although those formulas are simple, the computed results are not very accurate within a broad temperature range (from −100° to 100°C). In improver, information technology is desirable to use a very accurate formula when extreme accuracy is required; for example, accurate calculation of rainfall, precise prediction of the rising surface temperature of Earth as a result of global warming, and authentic calculation of evapotranspiration in agriculture. None of those elementary formulas is able to attain this goal. The purpose of this report was to develop a elementary and very accurate formulation for calculating the saturation vapor pressure level of water and of water ice. The computed results are extremely accurate for a wide temperature range from −100° to 100°C.

2. Methods

a. Currently existing equations

There are four reference datasets for saturation vapor pressure in the literature. The IAPWS formulation was adult on the ground of experimental data that are of high quality and accept been converted to the international temperature calibration (ITS-90), and it is able to represent the experimental data to within the experimental uncertainty. Therefore, the IAPWS reference dataset was chosen in this study.

Wagner and Pruss proposed a circuitous equation for the saturation vapor pressure of h2o (Wagner and Pruss 1993, 2002). Later this equation formed the basis of the IAPWS formulation. The differences between results from this formula and the results from the IAPWS conception are extremely small. This formula is as follows:

where P _s is the saturation h2o vapor pressure (Pa), T _c is the critical indicate temperature (647.096 K), and T is the temperature (K).

The equation for calculating the saturation vapor force per unit area of ice, developed past Wagner et al. (2011) and adopted by the IAPWS is given by

where T _t is the triple indicate temperature of water (273.16 Yard).

We here give the 2 most commonly used uncomplicated formulas for the saturation vapor pressure. The showtime is the improved Magnus formula with respect to water:

The 2nd is the improved Magnus formula with respect to water ice:

In both equations, t is the temperature (°C).

b. Developing a new formulation

In accordance with thermodynamic theory, the relationship betwixt the saturation vapor pressure level and the temperature at the equilibrium of two phases of water, because of entropy maximization, is governed by the Clausius–Clapeyron equation (Koutsoyiannis 2014; Ma et al. 2015):

where P _s is the saturation vapor pressure (Pa), T is the temperature of water or water ice (K), 50 is the latent heat of vaporization (J kg⁻¹), and V is the specific volume of h2o vapor (k³ kg⁻ⁱ). The specific volume of water vapor is determined by using the ideal gas law (Çengel and Boles 2011):

where R is the gas constant of water vapor (461.5 J kmol⁻¹ K⁻ⁱ). Submitting Eq. (half dozen) into Eq. (5), after rearranging, gives

The latent rut of vaporization of h2o is a linear function of the temperature (Henderson-Sellers 1984):

where m is a coefficient equal to 3 151 378 and n is a coefficient equal to 2386. Substituting 50 from Eq. (eight) to Eq. (7) gives

Integrating this equation yields

where C is the constant. To obtain this abiding, suppose that the saturation vapor pressure level of water is P ₀ at a known temperature T ₀. The step of putting P ₀ and T ₀ into Eq. (10) and solving it for C yields

The next step of substituting Eq. (11) for C in Eq. (ten) and rewriting it gives

Rewriting Eq. (12) yields

In essence, this equation takes the following class:

where a, b, and c are the coefficients. After conversion of the unit of temperature to degrees Celsius, the in a higher place equation becomes

According to regression analysis, more regression coefficients bring more degrees of freedom and a more accurate fit of the regression equation to the information. If the constant term 273.xv in the denominator of Eq. (20) is viewed to be 1 regression coefficient, the RE is not fairly depression. Therefore, the constant term 273.fifteen is considered to exist two separate regression coefficients. It so follows that

where d ₁ and d _two are coefficients.

c. New formulas for saturation vapor force per unit area of water and water ice

The higher up equation was adopted to produce an accurate fit to the saturation vapor pressure of water in the IAPWS reference dataset. The five coefficients were obtained by using the least squares method, which minimizes the sum of the squared differences between the predicted values and the reference values. The regression equation originally carried as many decimal digits as the capability of the computer permitted. To achieve the least number of significant figures without an appreciable increase of the error, five coefficients take been truncated. The final formula for the saturation vapor pressure of water is given by

In a similar style, Eq. (16) was used to generate an accurate fit to the saturation vapor pressure of ice from the IAPWS reference values. There exist considerable combinations of coefficients d ₁ and d _two that yield an RE of 0.004%–0.007%. The following formula, which provides minimum significant digits with a small sacrifice in accurateness, was obtained for the saturation vapor pressure level of ice:

3. Results

For comparisons, the saturation vapor pressure values of water from IAPWS were viewed to exist the reference values. The saturation water vapor pressure values were calculated past Eq. (17) for temperatures ranging from 0° to 100°C at an interval of i°C and compared with the reference values. Table 1 lists the saturation water vapor pressure level values from this new formula and the improved Magnus formula at an interval of twenty°C. The saturation water vapor pressures from Eq. (17) are virtually identical to the reference values for this wide range of temperatures (0°–100°C).

Table 1.

Saturation vapor pressure over the temperature range from 0° to 100°C at an interval of twenty°C.

The saturation vapor pressure values of water ice were calculated by Eq. (18) for temperatures ranging from −100° to 0°C at an interval of one°C. As exhibited in Table 2, the computed values from Eq. (18) are compared with the reference values. Again, the saturation vapor pressures of ice from Eq. (18) are very close to the reference values within a wide temperature range from −100° to 0°C.

Table 2.

Saturation vapor pressure level over the temperature range from −100°C to 0°C at an interval of 20°C.

iv. Discussion

It is critical to give the calculation mistake for a simple formula. To assess the accuracy of the new formula quantitatively, mean relative error (MRE; %) and maximum relative error (MAXRE; %) are selected every bit error criteria; they are respectively determined by

where Northward is the number of data points, P _si is the calculated saturation vapor pressure (Pa), and the P _ssi are the reference values (Pa).

The MREs from Eq. (17) and the improved Magnus formula [Eq. (3)] are shown in Table 3. The MAXREs are likewise included, as well equally the respective temperatures. The MRE of Eq. (17) is only 0.001%, much lower than that of the improved Magnus formula. In other words, if the reference saturation vapor pressure is 100 000 Pa at a given temperature, the absolute mistake resulting from Eq. (17) is i Pa. Furthermore, this new formula yields an MRE of 0.0005% for the moderate thermal environment (x°–40°C). It is noteworthy that the MAXRE of Eq. (17) is 0.0057%, lower than the MREs of the improved Magnus formula. Figure i shows a vivid picture of accurateness comparison. It appears every bit if the Eq. (17) line and the horizontal axis are indistinguishable from each other because of the extremely small MRE and MAXRE values. Therefore, the superiority of Eq. (17) over Eq. (3) can be readily identified from Tables i and 3 as well equally Fig. 1.

Tabular array 3.

MAXRE, MRE, and accuracy range. Annotation that the MRE is calculated on the basis of 101 data points from 0° to 100°C, with an interval of ane°C.

Equally exhibited in Table 4, the MRE for saturation vapor pressure level of water ice from Eq. (18) is only 0.006%, much lower than that of the improved Magnus formula. The MAXRE of Eq. (18) is 0.023%. Again, the MAXRE from Eq. (18) is lower than the MRE of the improved Magnus formula. Thus, the superiority of Eq. (18) is vividly brought to lite from Fig. 2.

Table 4.

MAXRE, MRE, and accuracy range from four formulas. Note that the MRE is calculated on the ground of 101 information points from −100° to 0°C, with an interval of 1°C.

As mentioned hereinbefore, the accurateness of those elementary formulas is express to narrow temperature ranges. The accuracy ranges for the new formula and the improved Magnus formula are displayed in Tables 2 and 4. The new Eqs. (17) and (18) are able to calculate the saturation vapor pressure very accurately over a broad range of temperatures. The improved Magnus formula yields RE exceeding 0.1% at temperatures above 53°C.

In summary, this new formula is superior to the improved Magnus formula because of the following aspects:

1. It is much more authentic than the improved Magnus formula. The great accuracy can be ascribed to the fact that this new formula was derived from the Clausius–Clapeyron equation, which dictates the equilibrium between 2 phases of matter, and to the fact that more coefficients lead to a better fit to the reference dataset.

2. The great accurateness is valid throughout a wide temperature range from −100° to 100°C.

3. It is simple and easy to think, especially for the formula with respect to water ice. If the same digit of a coefficient is counted 1 time, Eq. (eighteen) has a total of 12 digits. Equations (17) and (eighteen) share very like coefficients (34.494 and 43.494).

Moist air does not, in strict terms, satisfy the platonic gas law. The saturation vapor pressure over h2o or ice should be multiplied by an enhancement cistron to obtain the saturation vapor pressure for moist air. The enhancement cistron, which is a weak function of temperature and pressure, is defined as the ratio of the saturation vapor pressure for moist air to that of pure h2o vapor over a plane of h2o (Cadet 1981). The temperature effect is negligible. Thus, the enhancement factor is determined by (Alduchov and Eskridge 1996)

where f is the enhancement cistron and P is the atmospheric pressure (Pa).

5. Conclusions

Computation of the saturation vapor pressure at a given temperature is ofttimes required for some applications in a broad multifariousness of disciplines. There are a number of formulas bachelor for this purpose. Some formulas are too complex and computationally inefficient. Few people use them. Others are non very accurate, either at low temperatures or at high temperatures. A new formula has been obtained, past integration of the Clausius–Clapeyron equation, for calculating the saturation vapor pressure. Equally compared with the reference data adopted by the IAPWS, this new formula yields mean relative errors of 0.001% and 0.006% for the saturation vapor force per unit area of water and of ice, respectively, within a wide range of temperatures from −100° to 100°C, much lower than the MREs of the improved Magnus formula. Furthermore, the great accuracy of this formula is valid throughout the whole temperature range.

Because this new formula provides significant advantages over the improved Magnus formula in that it is simple and very accurate over a wide temperature range, it can exist used to calculate the saturation vapor pressure for some applications in many disciplines. Despite the fact that the new formula has one more operation than the Magnus formula, it is desirable to select the new formula because of significant error reduction over a wide temperature range. Therefore, this new formula may exist a viable substitute for other uncomplicated and complex formulas because of its first-class performance.

Acknowledgments

This piece of work was financially supported by the Hubei Key Laboratory for Digital Textile Technology.

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How To Find Vapor Pressure Of Water,

Source: https://journals.ametsoc.org/view/journals/apme/57/6/jamc-d-17-0334.1.xml

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