Editorial Type:
Article
Article Type:
Research Article

Direct Calculation of Thermodynamic Wet-Bulb Temperature as a Function of Pressure and Elevation

Sayed-Hossein Sadeghi Washington State University, Prosser, Washington

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Troy R. Peters Washington State University, Prosser, Washington

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Douglas R. Cobos Decagon Devices, Pullman, Washington

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Henry W. Loescher National Ecological Observatory Network, and Institute of Arctic and Alpine Research, University of Colorado, Boulder, Colorado

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Colin S. Campbell Decagon Devices, Pullman, Washington

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Print Publication:
01 Aug 2013
DOI:
https://doi.org/10.1175/JTECH-D-12-00191.1
Page(s):
1757–1765
Full access

Abstract

A simple analytical method was developed for directly calculating the thermodynamic wet-bulb temperature from air temperature and the vapor pressure (or relative humidity) at elevations up to 4500 m above MSL was developed. This methodology was based on the fact that the wet-bulb temperature can be closely approximated by a second-order polynomial in both the positive and negative ranges in ambient air temperature. The method in this study builds upon this understanding and provides results for the negative range of air temperatures (−17° to 0°C), so that the maximum observed error in this area is equal to or smaller than −0.17°C. For temperatures ≥0°C, wet-bulb temperature accuracy was ±0.65°C, and larger errors corresponded to very high temperatures (Ta ≥ 39°C) and/or very high or low relative humidities (5% < RH < 10% or RH > 98%). The mean absolute error and the root-mean-square error were 0.15° and 0.2°C, respectively.

Corresponding author address: Troy Peters, Washington State University, 24106 N. Bunn Rd., Prosser, WA 99350. E-mail: troy_peters@wsu.edu

Abstract

A simple analytical method was developed for directly calculating the thermodynamic wet-bulb temperature from air temperature and the vapor pressure (or relative humidity) at elevations up to 4500 m above MSL was developed. This methodology was based on the fact that the wet-bulb temperature can be closely approximated by a second-order polynomial in both the positive and negative ranges in ambient air temperature. The method in this study builds upon this understanding and provides results for the negative range of air temperatures (−17° to 0°C), so that the maximum observed error in this area is equal to or smaller than −0.17°C. For temperatures ≥0°C, wet-bulb temperature accuracy was ±0.65°C, and larger errors corresponded to very high temperatures (Ta ≥ 39°C) and/or very high or low relative humidities (5% < RH < 10% or RH > 98%). The mean absolute error and the root-mean-square error were 0.15° and 0.2°C, respectively.

Corresponding author address: Troy Peters, Washington State University, 24106 N. Bunn Rd., Prosser, WA 99350. E-mail: troy_peters@wsu.edu
Keywords: Humidity; Temperature; Water vapor

1. Introduction

First principles dictate that for any given ambient air mass, the difference between aspirated (well coupled) air temperature that includes ambient water vapor (dry-bulb temperature Ta) and the temperature of the same air mass (wet-bulb temperature Tw) at saturation provides a direct measurement of the amount of water vapor that air mass contains. This estimate can be determined as both relative and absolute quantities (Loescher et al. 2009). In other words, Tw is the temperature that a volume of air would have if cooled adiabatically to saturation at a constant pressure where all the heat energy came from the measured volume of air (Monteith 1965). The importance of this first principle is better realized when the Ta and Tw measured at both a surface level (boundary condition) and at different heights, that is, reference levels, can be used to estimate the evapotranspiration vertically through these two levels, for example, through a leaf or a canopy surface (Slatyer and McIlroy 1961; Alves et al. 2000; Balogun et al. 2002a,b).

Wet-bulb temperature is a basic hydrostatic, physical quantity that can be used to estimate basic physical weather parameters (Stull 2011). Some applied applications of Tw include linking surface and boundary layer flows (Wai and Smith 1998) and interpreting surface scalar fluxes using physical properties (Loescher et al. 2006), while practical applications may be determining the efficiency of industrial coolers (Gan and Riffat 1999), managing hydrological resources (Dunin and Greenwood 1986), and identifying saturated adiabats on thermodynamic isopleths (Stull 2000).

Another commonplace and important application of Tw is in agricultural research and management, and its determination has important implications on agronomic economies. For instance, Tw is used to determine (i) the amount of time and energy required to dry grain to a stable storage moisture content (Schmidt and Waite 1962), (ii) frost protection for fruit crops (e.g., G. Hoogenboom 2012, personal communication), and (iii) minimum temperature forecasts on relatively clear nights (Angström 1920, abstracted in Smith 1920).

The determination of the theoretical thermodynamic wet-bulb temperature can be solved by the following equation (ASABE 2006):
e1
where ea is the ambient vapor pressure (kPa), Ta is the dry-bulb temperature (K), es(Tw) is the saturated vapor pressure at Tw (kPa), and B is the thermodynamic psychometric constant given by
e2
In Eq. (2), m, n, and q are empirical coefficients; Pa is the atmospheric pressure head (kPa); and hfg is the latent heat of vaporization of water at Tw (J kg−1) defined by Brooker (1967) and where
e3a
Both r and s are empirically determined coefficients. Equation (3a) is valid when 0° ≤ Tw ≤ 65.6°C. For negative values of Tw (−17.8° ≤ Tw ≤ 0°C), hfg in Eq. (2) must be replaced by the latent heat of sublimation (hig) at Tw (Brooker 1967), such that
e3b
where both r and s are empirically determined coefficients. Also, Pa can be determined as (Campbell and Norman 1998)
e3c
with H being the elevation (m).
If we assume that the thermodynamic wet-bulb temperature [from Eq. (1)] and the wet-bulb temperature (empirical, measured by a thermometer) are approximately the same, then Eq. (1) takes the following form:
e4
where γ is the psychometric constant (°C−1) defined by γ = Cp/Lυ ≈ 0.4[gwater (kgair)−1 K−1] (where Cp is the specific heat of moist air at constant pressure and Lυ is the latent heat of vaporization). The historical approach of using B in Eq. (1) converts the temperature depression into vapor pressure deficit through the estimation of empirical coefficients and empirical quantification of the latent energy. Substituting γPa for B in Eq. (4) takes the same form of unit conversion but removes the uncertainty from the empirical coefficients.

Because Eq. (1) [or Eq. (4)] has no direct solution for Tw (or ), typically a trial-and-error process is carried out to find an accurate estimation. Using computers, this requires considerable amounts of CPU time to analyze large weather databases. Alternatively, one can employ a graphical solution by through the use psychrometric diagrams (ASHRAE 1997). This latter method, however, has large associated uncertainties because of human errors (Brooker 1967) and must use a priori, predetermined curve fits for each value of the atmospheric pressure.

Noniterative and analytical approaches have been suggested to calculate Tw. Sreekanth et al. (1998) used artificial neural networks that require relative humidity (RH) and the dewpoint temperature (Td) estimates as input variables. Chappell et al. (1974) calculated Tw by estimating the amount of dry air needed to dry a given mass of moist air through an isobaric and adiabatic procedure, and they reported an accuracy of ±1°C across ambient ranges of atmospheric temperature and pressure when wet-bulb depressions were <30°C. Chau (1980) presented empirical equations to find the Tw by dividing the psychrometric chart into seven arbitrary ranges that span Ta values of −32° to 260°C and Td values of −32° to 40°C, but found this technique only to be accurate (and valid) at sea level.

The crux of all of the above-mentioned solutions requires known Td temperatures to derive Tw. Alternatively, some studies have combined estimates of Ta and RH with either a third-order polynomial equation for es() [as in the case with Tejeda-Martínez (1994)] or with gene-expression programming (i.e., Stull 2000) to derive the wet-bulb temperature. Both studies found this methodology was not valid for ambient conditions with low values of Ta (i.e., <10°C), and/or with low values of RH (i.e., Stull 2000). The Stull (2000) methodology was also only valid at sea level.

Additional uncertainties are the result in the choice of which value of the psychrometric constant is used, particularly when (i) a constant value is chosen [e.g., 6.53 × 10−4 °C−1 (Tejeda-Martínez 1994), 6.67 × 10−4 (Schurer 1981), and 5.68 × 10−4 °C−1 to 6.42 × 10−4 °C−1 when Tw ≤ 0° and 0° < Tw < 30°C (Simões-Moreira 1999)] and (ii) theoretical estimations of γ outlined in Eq. (1) are being questioned empirically (Simões-Moreira 1999; Loescher et al. 2009). As such, these studies have empirically shown that γ is independent of Ta and Pa (Simões-Moreira 1999; Loescher et al. 2009), and they have demonstrated its value is strongly dependent on Tw (Simôes-Moreira 1999) and the wet-bulb depression (Loescher et al. 2009).

The objective of this study is to derive a direct solution for calculating the Tw at any desired elevation while maintaining the high levels of accuracy needed for most applications. This will be achieved by first finding a direct solution for calculating from Eq. (4) and then exploring its mathematical derivations to enhance the accuracy in Tw estimates.

2. Methodology

Mathematical derivations

In this study, we are interested in the range of Ta used for most basic and applied research (−17°C ≤ Ta ≤ 40°C) and assumed the Simões-Moreira's (1999) values for γ are accurate. The saturated vapor pressure in Eq. (4) is estimated by the Magnus equation (Murray 1967):
e5a
e5b
where a, b, and c are empirical coefficients that depend on the interval by which temperature is being measured (Table 1). Buck (1981) reported that Eq. (5a) led to a maximum 5%–6% deviation from “truth” and resulted in a positive divergence (when conditions were 0° ≤ Ta ≤ 50°C) and negative divergence (when conditions were −40° ≤ Ta ≤ 0°C), when compared with the Wexler (1976) derivation of es(Ta). Similarly, es() can be estimated by Eq. (5b). Substitution of Pa from Eq. (3c) and es() from Eq. (5b) into Eq. (4) results in
e6
Table 1.

Coefficients for calculating saturated vapor pressure of pure water as a function of temperature after Buck (1981).

Table 1.
To find a solution to estimate [Eq. (6)], we numerically applied different combinations of , Ta, ea, and H. In each combination, Ta, H, and ea were held constant and only was altered. We used two discrete ranges, that is, −30° ≤ < 0°C and 0° ≤ ≤ 40°C, and in preliminary analyses, Eq. (6) closely followed the shape of a second-order polynomial (Fig. 1). Not surprisingly, this relationship was more curvilinear for positive Ta values. Note that the intersection of each curve with the horizontal axis represents . Based on this, an equivalent form for Eq. (6) can heuristically be considered as
e7
where λ, ϕ, and ψ are empirical coefficients, and the advantage of using Eq. (7) can be rearranged to explicitly calculate as
e8a
where
e8b
It is worth noting that Eq. (7) is inherently an ascendant function (the derivative at any arbitrary point is always positive), making an acceptable value of always the larger root of the second-order polynomial (i.e., Figs. 1a–h) and therefore, the positive sign should always be inserted before in Eq. (8a).
Fig. 1.
Fig. 1.

Variation of F() in Eq. (6) with (top) negative range of the wet bulb where for line a, H = 1700 m, ea = 0.13 kPa, and Ta = −15°C; line b, H = 0 m, ea = 0.08 kPa, and Ta = −11°C; line c, H = 660 m, ea = 0.23 kPa, and Ta = −8°C; and line d, H = 3000 m, ea = 0.3 kPa, and Ta = −4°C; and (bottom) positive range of the wet bulb where line e, H = 4500 m, ea = 0.6 kPa, and Ta = 3°C; line f, H = 1300 m, ea = 1 kPa, and Ta = 13°C; line g, H = 100 m, ea = 3 kPa, and Ta = 9°C; and line h, H = 700 m, ea = 2.4 kPa, and Ta = 40°C.

Citation: Journal of Atmospheric and Oceanic Technology 30, 8; 10.1175/JTECH-D-12-00191.1

Because the solutions for λ, ϕ, and ψ are infinite given any suite of environmental conditions [Eq. (7)], we evaluated them at three different logical conditions of . For the positive region (0° ≤ ≤ 40°C), these fixed conditions were = 0, = Ta/2, and = Ta. This is a logical selection because is always ≤Ta, and in other words, the ambient Ta is the last value to be substituted into Eq. (7). For negative values (−17° ≤ < 0°C), the selection of = Ta could not be addressed mathematically because values of Tw are always <Ta and do not necessarily cover the whole negative range of Eq. (7). Applying the logic from Stull (2000, his Fig. 1), the wet-bulb depressions (Ta − ) did not exceed 6°C when was negative and when the RH was >20%. This similarity of Ta and provides the rationale to use the same set of fixed conditions when is negative as when is positive. Thus, the mathematical derivations yielded
e9a
e9b
e9c
where the function ξ is given by
e10
To determine the dependency of Ta on λ and ξ, a series of numerical operations were again carried out. For this purpose, the values of Ta were varied from −17° to 40°C by an increment of 0.1°C (Fig. 2). For both parameters, the statistical relationships were nonlinear, as presented below and described in the Fig. 2 caption. The proposed analytical solution therefore includes the derivation of λ and ξ from these statistical relationships, calculating ψ and ϕ by Eqs. (9a) and (9c), and finally by Eq. (8a), where
e11
e12
Fig. 2.
Fig. 2.

Relationship between λ (black line) and ζ (dotted line) as a function of temperature when −17° ≤ Ta ≤ 40°C, λ = 0.0014 exp(0.027Ta) with R2 = 0.9998, and ζ = −[3 × 10−07(Ta)3] − [1 × 10−05 (Ta)2] + [2 × 10−05 (Ta)] + 0.0444 with R2 = 0.9999. Note that Ta in the denominator of the λ function has the exponent 2 [Eq. (9b)], whereas its exponent value in ζ is 1 [Eq. (10)], making the shapes of the two functions quite different. Hence, we choose different regression equations.

Citation: Journal of Atmospheric and Oceanic Technology 30, 8; 10.1175/JTECH-D-12-00191.1

3. Results

Validation

The accuracy of the analytical solution derived here for calculating was tested by comparing it with the solution found in Eq. (1). For this purpose, a series of numerical, iterative computations were carried out to calculate a highly accurate and theoretical solution for Tw at various combinations of the input data as follows: (i) Ta between −17° and 40°C (by an equal increment of 1°C) and es(Ta) between 0.01 and 7.2 kPa (by an equal increment of 0.05 kPa), and (ii) elevation from 0 (Pa at sea level = 101.325 kPa) to 4500 m (Pa = 58.5 kPa) in 500-m increments. After excluding results leading to a RH > 100% and RH < 5%, 29 515 remaining combinations were analyzed. For the purpose of a direct comparison, the analytically approximated values of were also calculated and plotted against Tw obtained by the numerical analysis (Fig. 3). We found a good agreement between Tw and , such that the absolute maximum difference between the numerical and analytical solutions was approximately |1.3|°C. However, in order to further improve the accuracy, a regression of Tw with was also derived for both the positive and negative ranges of (see the regression equations in Fig. 3 caption). Using this modification, the maximum difference between and Tw decreased to less than |0.65|°C.

Fig. 3.
Fig. 3.

The quantity versus Tw when (a) < 0°C, Tw = 1.011() − 0.0419 with R2 = 0.9999 and (b) ≥ 0°C, Tw = 1.0301() − 0.213 with R2 = 0.9995.

Citation: Journal of Atmospheric and Oceanic Technology 30, 8; 10.1175/JTECH-D-12-00191.1

Prior to the use of the regression equations in Fig. 3, the sign of has to be determined; as such, when Ta < 0°C, will always be negative. In support of these analyses, the m, n, and q coefficients in Eq. (2), and r and s in Eq. (3a), and t and u in Eq. (3b) were statistically estimated and were 100.9254 (for m), 0.155 77 (for n), 0.621 94 (for q), 2 502 535.259 (for r), 2385.764 24 (for s), 2 839 683.144 (for t), and 212.563 84 (for u). Nevertheless, when the Ta is positive (Ta ≥ 0°C), the positive sign of is not guaranteed. For example, assume that the Ta is 2°C, the elevation is H = 3500 m, and ea is 0.16 kPa (RH ≈ 23%); the calculated by the iterative solution (Eq. 4) is −4.52°C (Ta ≥ 0 and < 0). However, in the same example, if Ta is 14°C, then would be 1.6°C (Ta ≥ 0 and > 0). To find out whether is positive or negative, we define the F(T) function from Eq. (4) as
e13
Notice again that the root of this function represents the [F() = 0]. Considering the graphical derivation of Figs. 1e–h, can only be positive if the root of F(T) falls between F(0) and F(Ta) or in other words, F(0) × F(Ta) < 0. Substituting these values into Eq. (13) yields
e14a
e14b
Because RH < 1 and ea < es(Ta), F(Ta) from Eq. (14b) will always be positive. This means that will be positive only when F(0) < 0 [since F(0) × F(Ta) is negative], or
e15
We further examined the differences between Tw and the modified (Fig. 3) as not only a function of Ta but also differences in RH and elevation as well (Fig. 4). Interestingly, when Ta < 0°C, the maximum difference was only |0.17|°C. However, when Ta ≥ 0°C, the difference ranged between ±0.65°C with the largest observed differences (≥|0.5|°C) occurring when conditions were (i) Ta ≥ 39°C and 31% < RH < 65%, (ii) Ta ≥ 35°C, 5% < RH < 8%, and when (iii) 27° < Ta < 34°C and RH > 98%. This means that the actual temperature and/or the relative humidity values have to be very high to cause a difference larger than |0.5|°C between the theoretical and the analytical wet-bulb temperature. In general terms, when the Ta increased, the difference between our two compared approaches also increased slightly. The effects of H on − Tw were relatively small and its general pattern across the range of Ta and RH remained similar (Fig. 4).
Fig. 4.
Fig. 4.

Difference between Tw Eq. (1) and as a function of Ta and RH at elevations ranging from sea level to 4500 m. The y axis is % RH.

Citation: Journal of Atmospheric and Oceanic Technology 30, 8; 10.1175/JTECH-D-12-00191.1

The variation of − Tw with H, RH, and Ta was also evaluated (Fig. 5). We found that (i) an elevation change from the sea level to H = 2000 m does not significantly affect the difference of − Tw, although a slight increase was observable (cf. Fig. 5a). For H > 3000 m, differences increase at a higher rate when compared to those sea level conditions. These results indicate that larger differences in − Tw are expected when H > 3000 m, and (ii) as shown in Fig. 5b, four distinct behaviors are exhibited for the − Tw as RH increases. The differences decrease as RH increases from 5% to 20%, increases between 20% < RH < 50%, decreases between 50% < RH < 85%, and finally remain approximately constant as RH varies between 85% and 100%. In addition, the largest differences were apparent when RH is near to 5% or about 50%, and (iii) Fig. 5c demonstrates that the proposed analytical method overestimates Tw (relative to ) when 21°C ≤ Ta ≤ 25°C regardless of the value of the RH and H. When Ta > 8°C, the range of − Tw increased as Ta increases, with the rate being considerably higher when Ta ≥ 26°C. The two distinct sections in the range of − Tw when 0°C < Ta ≤ 8°C result from the fact that positive dry-bulb temperatures can provide negative Tw values (i.e., the upper section). Finally, when Ta < 0°C, the − Tw differences decrease as Ta decreases and both maximum and minimum values of − Tw became smaller (Fig. 5c).

Fig. 5.
Fig. 5.

Variation of − Tw versus (top) H, (middle) RH, and (bottom) Ta.

Citation: Journal of Atmospheric and Oceanic Technology 30, 8; 10.1175/JTECH-D-12-00191.1

The mean absolute error (MAE) and the root-mean-squared errors (RMSE) for the proposed analytical solution were less than 0.15° and 0.2°C, respectively. When Ta was positive, MAE and RMSE were 0.15° and 0.21°C, respectively, but were 0.07° and 0.08°C, respectively, when Ta was <0°C.

4. Conclusions

The wet-bulb temperature is an important psychrometric parameter with implications in environmental, meteorological, and agricultural basic research and applied applications. Historically, the theoretical equation for the Tw has no direct solution, and to find and accurate estimation typically relied upon a trial and error process that requires significant CPU time. On the contrary, the methodology here described, based on second-order polynomial fit, is computationally fast and accurate. In this study, an easy-to-use and accurate analytical solution for calculating the thermodynamic wet-bulb temperature for elevations up to 4500 m above MSL is presented. The reason for this upper bound is because the uncertainty in the psychrometric constant increases to >30% above 4500 m above MSL (Simões-Moreira 1999). Moreover, since the equation for calculating the atmospheric pressure head was used in both methods, we assumed it would not be a significant source of uncertainty and not change the difference between the theoretical and the analytical solutions of Tw. It was found that the wet-bulb temperature for both positive and negative ranges of the air temperature can be simulated by a second-order equation. The suggested technique seems to converge with the other known approach when −17° ≤ Ta < 0°C, so that the maximum difference in predicting Tw did not exceed |0.17|°C. When Ta ≥ 0°C, − Tw were within ±0.65°C. The larger absolute differences between the observed comparison Tw calculated with Ta in the positive range contrasted with the negative range was likely due to two major reasons: (i) the positive values spanned a larger range and (ii) the relationship between wet-bulb temperature and Ta is more curvilinear when Ta ≥ 0°C. Therefore, the comparative use of the three environmental conditions leads to larger uncertainties in estimating the second-order polynomial coefficients (i.e., λ, ϕ, and ψ). This study also proved again that values of 5.68 × 10−4 to 6.42 × 10−4 when Tw < 0° and 0° ≤ Tw ≤ 30°C converged with other estimates of the thermodynamic psychrometric constant. The proposed method can provide more detailed understanding of hydrostatic properties of air containing water vapor, particularly in the estimation of the wet-bulb temperature without the use of alternative and computationally cumbersome iterative approaches.

Acknowledgments

The authors wish to thank Prof. R. Stull for his response to our inquiry, and also Dr. Th. Bellinger for suggesting useful references that greatly improved the quality of this work. HWL wishes to thank the National Science Foundation under the Grant EF-102980. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors are thankful for the thoughtful comments from three anonymous reviewers.

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Journal of Atmospheric and Oceanic Technology

  • Alves, I., Fontes J. C. , and Pereira L. S. , 2000: Evapotranspiration estimation from infrared surface temperature. II: The surface temperature as a wet bulb temperature. Trans. ASAE, 43, 591–598.

    • Search Google Scholar
    • Export Citation

  • Angström, A., 1920: Studies of the frost problem I. Geogr. Ann.,2, 2–32.

  • ASABE, 2006: ASABE standards: Psychrometric data. American Society of Agriculture and Biological Engineers ASAE D271.2 APR1979 (R2005), 27 pp. [Available online at http://elibrary.asabe.org/standards.asp.]

  • ASHRAE, 1997: 1997 ASHRAE Handbook: Fundamentals. SI ed. ASHRAE, 1426 pp.

  • Balogun, A. A., Jegede O. O. , Foken T. , and Olaleye J. O. , 2002a: Comparison of two Bowen-ratio methods for the estimation of sensible and latent heat fluxes at Ile-Ife, Nigeria. J. Afr. Meteor. Soc., 5 (2), 63–69.

    • Search Google Scholar
    • Export Citation

  • Balogun, A. A., Jegede O. O. , Foken T. , and Olaleye J. O. , 2002b: Estimation of sensible and latent heat fluxes over bare soil using bowen ratio energy balance method at a humid tropical site. J. Afr. Meteor. Soc., 5 (1), 63–71.

    • Search Google Scholar
    • Export Citation

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Variation of F() in Eq. (6) with (top) negative range of the wet bulb where for line a, H = 1700 m, ea = 0.13 kPa, and Ta = −15°C; line b, H = 0 m, ea = 0.08 kPa, and Ta = −11°C; line c, H = 660 m, ea = 0.23 kPa, and Ta = −8°C; and line d, H = 3000 m, ea = 0.3 kPa, and Ta = −4°C; and (bottom) positive range of the wet bulb where line e, H = 4500 m, ea = 0.6 kPa, and Ta = 3°C; line f, H = 1300 m, ea = 1 kPa, and Ta = 13°C; line g, H = 100 m, ea = 3 kPa, and Ta = 9°C; and line h, H = 700 m, ea = 2.4 kPa, and Ta = 40°C.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Relationship between λ (black line) and ζ (dotted line) as a function of temperature when −17° ≤ Ta ≤ 40°C, λ = 0.0014 exp(0.027Ta) with R2 = 0.9998, and ζ = −[3 × 10−07(Ta)3] − [1 × 10−05 (Ta)2] + [2 × 10−05 (Ta)] + 0.0444 with R2 = 0.9999. Note that Ta in the denominator of the λ function has the exponent 2 [Eq. (9b)], whereas its exponent value in ζ is 1 [Eq. (10)], making the shapes of the two functions quite different. Hence, we choose different regression equations.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    The quantity versus Tw when (a) < 0°C, Tw = 1.011() − 0.0419 with R2 = 0.9999 and (b) ≥ 0°C, Tw = 1.0301() − 0.213 with R2 = 0.9995.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Difference between Tw Eq. (1) and as a function of Ta and RH at elevations ranging from sea level to 4500 m. The y axis is % RH.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Variation of − Tw versus (top) H, (middle) RH, and (bottom) Ta.

Fig. 1.

Variation of F() in Eq. (6) with (top) negative range of the wet bulb where for line a, H = 1700 m, ea = 0.13 kPa, and Ta = −15°C; line b, H = 0 m, ea = 0.08 kPa, and Ta = −11°C; line c, H = 660 m, ea = 0.23 kPa, and Ta = −8°C; and line d, H = 3000 m, ea = 0.3 kPa, and Ta = −4°C; and (bottom) positive range of the wet bulb where line e, H = 4500 m, ea = 0.6 kPa, and Ta = 3°C; line f, H = 1300 m, ea = 1 kPa, and Ta = 13°C; line g, H = 100 m, ea = 3 kPa, and Ta = 9°C; and line h, H = 700 m, ea = 2.4 kPa, and Ta = 40°C.
Fig. 2.

Relationship between λ (black line) and ζ (dotted line) as a function of temperature when −17° ≤ Ta ≤ 40°C, λ = 0.0014 exp(0.027Ta) with R2 = 0.9998, and ζ = −[3 × 10−07(Ta)3] − [1 × 10−05 (Ta)2] + [2 × 10−05 (Ta)] + 0.0444 with R2 = 0.9999. Note that Ta in the denominator of the λ function has the exponent 2 [Eq. (9b)], whereas its exponent value in ζ is 1 [Eq. (10)], making the shapes of the two functions quite different. Hence, we choose different regression equations.
Fig. 3.

The quantity versus Tw when (a) < 0°C, Tw = 1.011() − 0.0419 with R2 = 0.9999 and (b) ≥ 0°C, Tw = 1.0301() − 0.213 with R2 = 0.9995.
Fig. 4.

Difference between Tw Eq. (1) and as a function of Ta and RH at elevations ranging from sea level to 4500 m. The y axis is % RH.
Fig. 5.

Variation of − Tw versus (top) H, (middle) RH, and (bottom) Ta.
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