AUGUST 2013 S A D E G H I E T A L . 1757 Direct Calculation of Thermodynamic Wet-Bulb Temperature as a Function of Pressure and Elevation SAYED-HOSSEIN SADEGHI AND TROY R. PETERS Washington State University, Prosser, Washington DOUGLAS R. COBOS Decagon Devices, Pullman, Washington HENRY W. LOESCHER National Ecological Observatory Network, and Institute of Arctic and Alpine Research, University of Colorado, Boulder, Colorado COLIN S. CAMPBELL Decagon Devices, Pullman, Washington (Manuscript received 6 September 2012, in final form 29 January 2013) ABSTRACT A simple analytical method was developed for directly calculating the thermodynamic wet-bulb temper- ature 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 tem- perature. The method in this study builds upon this understanding and provides results for the negative range of air temperatures (2178 to 08C), so that the maximum observed error in this area is equal to or smaller than 20.178C. For temperatures $08C, wet-bulb temperature accuracy was 60.658C, and larger errors corre- sponded to very high temperatures (Ta $ 398C) 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 0.158 and 0.28C, respectively. 1. Introduction from the measured volume of air (Monteith 1965). The importance of this first principle is better realized when First principles dictate that for any given ambient air the T and T measured at both a surface level (boundary mass, the difference between aspirated (well coupled) a w condition) and at different heights, that is, reference air temperature that includes ambient water vapor (dry- levels, can be used to estimate the evapotranspiration bulb temperature T ) and the temperature of the same a vertically through these two levels, for example, through air mass (wet-bulb temperature T ) at saturation pro- w a leaf or a canopy surface (Slatyer and McIlroy 1961; vides a direct measurement of the amount of water vapor Alves et al. 2000; Balogun et al. 2002a,b). that air mass contains. This estimate can be determined Wet-bulb temperature is a basic hydrostatic, physical as both relative and absolute quantities (Loescher et al. quantity that can be used to estimate basic physical 2009). In other words, T is the temperature that a vol- w weather parameters (Stull 2011). Some applied appli- ume of air would have if cooled adiabatically to satura- cations of T include linking surface and boundary layer tion at a constant pressure where all the heat energy came w flows (Wai and Smith 1998) and interpreting surface scalar fluxes using physical properties (Loescher et al. Corresponding author address: Troy Peters, Washington State 2006), while practical applications may be determining University, 24106 N. Bunn Rd., Prosser, WA 99350. the efficiency of industrial coolers (Gan and Riffat 1999), E-mail: [email protected] managing hydrological resources (Dunin and Greenwood DOI: 10.1175/JTECH-D-12-00191.1 Ó 2013 American Meteorological Society Unauthenticated | Downloaded 09/23/21 02:57 PM UTC 1758 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 30 1986), and identifying saturated adiabats on thermody- 2 1 2 5 ea es(Tw*) gPa(Ta Tw*) 0, (4) namic isopleths (Stull 2000). 2 Another commonplace and important application of where g is the psychometric constant (8C 1) defined by 21 21 Tw is in agricultural research and management, and its g 5 Cp/Ly ’ 0.4[gwater (kgair) K ] (where Cp is the determination has important implications on agronomic specific heat of moist air at constant pressure and Ly is economies. For instance, Tw is used to determine (i) the latent heat of vaporization). The historical approach the amount of time and energy required to dry grain of using B in Eq. (1) converts the temperature depression to a stable storage moisture content (Schmidt and into vapor pressure deficit through the estimation of Waite 1962), (ii) frost protection for fruit crops (e.g., empirical coefficients and empirical quantification of the G. Hoogenboom 2012, personal communication), and latent energy. Substituting gPa for B in Eq. (4) takes the (iii) minimum temperature forecasts on relatively clear same form of unit conversion but removes the uncer- nights (Angstrom€ 1920, abstracted in Smith 1920). tainty from the empirical coefficients. The determination of the theoretical thermodynamic Because Eq. (1) [or Eq. (4)] has no direct solution for wet-bulb temperature can be solved by the following Tw (or Tw*), typically a trial-and-error process is carried equation (ASABE 2006): out to find an accurate estimation. Using computers, this requires considerable amounts of CPU time to analyze 2 5 2 es(Tw) ea B(Tw Ta), (1) large weather databases. Alternatively, one can employ a graphical solution by through the use psychrometric where e is the ambient vapor pressure (kPa), T is the a a diagrams (ASHRAE 1997). This latter method, however, dry-bulb temperature (K), e (T ) is the saturated vapor s w has large associated uncertainties because of human er- pressure at T (kPa), and B is the thermodynamic psy- w rors (Brooker 1967) and must use a priori, predetermined chometric constant given by curve fits for each value of the atmospheric pressure. e Noniterative and analytical approaches have been m[e (T ) 2 P ] 1 1 n a s w a suggested to calculate Tw. Sreekanth et al. (1998) used Pa B 5 . (2) artificial neural networks that require relative humidity qhfg (RH) and the dewpoint temperature (Td) estimates as input variables. Chappell et al. (1974) calculated Tw by In Eq. (2), m, n, and q are empirical coefficients; Pa is the estimating the amount of dry air needed to dry a given atmospheric pressure head (kPa); and hfg is the latent 2 heat of vaporization of water at T (J kg 1) defined by mass of moist air through an isobaric and adiabatic pro- w 6 8 Brooker (1967) and where cedure, and they reported an accuracy of 1 C across ambient ranges of atmospheric temperature and pressure 5 2 2 : when wet-bulb depressions were ,308C. Chau (1980) hfg r s(Tw 273 16). (3a) presented empirical equations to find the Tw by dividing Both r and s are empirically determined coefficients. the psychrometric chart into seven arbitrary ranges that 2 8 8 2 8 Equation (3a) is valid when 08 # Tw # 65.68C. For neg- span Ta values of 32 to 260 CandTd values of 32 to 8 ative values of Tw (217.88 # Tw # 08C), hfg in Eq. (2) must 40 C, but found this technique only to be accurate (and be replaced by the latent heat of sublimation (hig)atTw valid) at sea level. (Brooker 1967), such that The crux of all of the above-mentioned solutions re- quires known Td temperatures to derive Tw. Alterna- 5 2 2 : hig t u(Tw 255 38), (3b) tively, some studies have combined estimates of Ta and RH with either a third-order polynomial equation for where both r and s are empirically determined coef- es(Tw*) [as in the case with Tejeda-Martı´nez (1994)] or ficients. Also, Pa can be determined as (Campbell and with gene-expression programming (i.e., Stull 2000) to Norman 1998) derive the wet-bulb temperature. Both studies found this methodology was not valid for ambient conditions 2H with low values of T (i.e., ,108C), and/or with low values P 5 101:3 exp , (3c) a a 8200 of RH (i.e., Stull 2000). The Stull (2000) methodology was also only valid at sea level. with H being the elevation (m). Additional uncertainties are the result in the choice of If we assume that the thermodynamic wet-bulb tem- which value of the psychrometric constant is used, par- perature [from Eq. (1)] and the wet-bulb temperature Tw* ticularly when (i) a constant value is chosen [e.g., 6.53 3 2 2 2 (empirical, measured by a thermometer) are approxi- 10 4 8C 1 (Tejeda-Martı´nez 1994), 6.67 3 10 4 (Schurer 2 2 2 2 mately the same, then Eq. (1) takes the following form: 1981), and 5.68 3 10 4 8C 1 to 6.42 3 10 4 8C 1 when Unauthenticated | Downloaded 09/23/21 02:57 PM UTC AUGUST 2013 S A D E G H I E T A L . 1759 ~ Tw # 08 and 08,Tw , 308C(Simoes-Moreira 1999)] and TABLE 1. Coefficients for calculating saturated vapor pressure of (ii) theoretical estimations of g outlined in Eq. (1) are pure water as a function of temperature after Buck (1981). ~ being questioned empirically (Simoes-Moreira 1999; Temp interval (8C) ab c Loescher et al. 2009). As such, these studies have em- 0 to 50 0.611 17.368 238.88 pirically shown that g is independent of Ta and Pa 240 to 0 0.611 17.966 247.15 (Simoes-Moreira~ 1999; Loescher et al. 2009), and they have demonstrated its value is strongly dependent on T (Simoes-Moreira^ 1999) and the wet-bulb depression w 2 8 # , 8 8 # # 8 (Loescher et al. 2009). Tw* ranges, that is, 30 Tw* 0 C and 0 Tw* 40 C, The objective of this study is to derive a direct solution and in preliminary analyses, Eq. (6) closely followed the shape of a second-order polynomial (Fig. 1). Not sur- for calculating the Tw at any desired elevation while maintaining the high levels of accuracy needed for most prisingly, this relationship was more curvilinear for applications.