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Direct Calculation of Thermodynamic Wet-Bulb as a Function of 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 (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

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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 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 . 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 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

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~ 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. This will be achieved by first finding a di- positive Ta values. Note that the intersection of each curve with the horizontal axis represents T *. Based on rect solution for calculating Tw* from Eq. (4) and then w exploring its mathematical derivations to enhance the this, an equivalent form for Eq. (6) can heuristically be considered as accuracy in Tw estimates. ! bT * 2H a w 2 e 2 : g T 2 T * 2. Methodology exp 1 a 101 3 exp ( a w ) Tw* c 8200 Mathematical derivations ffi 2 1 1 5 l(Tw*) fTw* c 0, In this study, we are interested in the range of T used a (7) for most basic and applied research (2178C # Ta # ~ 408C) and assumed the Simoes-Moreira’s (1999) values where l, u, and c are empirical coefficients, and the for g are accurate. The saturated vapor pressure in Eq. advantage of using Eq. (7) can be rearranged to explic- (4) is estimated by the Magnus equation (Murray 1967): itly calculate Tw* as pffiffiffiffi bT 2u 1 D e (T ) 5 a exp a , (5a) ffi s a 1 Tw* , (8a) Ta c 2l ! bT * where e (T *) 5 a exp w , (5b) s w T * 1 c w D5f2 2 4lc. (8b) where a, b, and c are empirical coefficients that depend It is worth noting that Eq. (7) is inherently an ascendant on the interval by which temperature is being measured function (the derivative at any arbitrary point is always (Table 1). Buck (1981) reported that Eq. (5a) led to positive), making an acceptable value of T * always the a maximum 5%–6% deviation from ‘‘truth’’ and re- w larger root of the second-order polynomial (i.e., Figs. 1a–h) sulted in a positive divergence (when conditions were and therefore,pffiffiffiffi the positive sign should always be inserted 08 # T # 508C) and negative divergence (when condi- a before D in Eq. (8a). tions were 2408 # T # 08C), when compared with the a Because the solutions for l, u, and c are infinite given Wexler (1976) derivation of e (T ). Similarly, e (T *)can s a s w any suite of environmental conditions [Eq. (7)], we be estimated by Eq. (5b). Substitution of P from Eq. (3c) a evaluated them at three different logical conditions of and es(Tw*) from Eq. (5b) into Eq. (4) results in Tw*. For the positive Tw* region (08 # Tw* # 408C), these ! fixed conditions were Tw* 5 0, Tw* 5 Ta/2, and Tw* 5 Ta. bT * 2H # a exp w 2 e 2 101:3g exp (T 2 T *) 5 0. This is a logical selection because Tw* is always Ta, and T * 1 c a 8200 a w w in other words, the ambient Ta is the last value to be 2 8 # (6) substituted into Eq. (7). For negative values ( 17 Tw* , 08C), the selection of Tw* 5 Ta could not be ad- dressed mathematically because values of Tw are always To find a solution to estimate Tw* [Eq. (6)], we nu- ,Ta and do not necessarily cover the whole negative merically applied different combinations of Tw*, Ta, ea, range of Eq. (7). Applying the logic from Stull (2000, his and H. In each combination, Ta, H, and ea were held Fig. 1), the wet-bulb depressions (Ta 2 Tw*) did not exceed constant and only Tw* was altered. We used two discrete 68C when Tw* was negative and when the RH was .20%.

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FIG. 1. Variation of F(Tw*) in Eq. (6) with (top) negative range of the wet bulb where for line a, H 5 1700 m, ea 5 0.13 kPa, and Ta 52158C; line b, H 5 0m,ea 5 0.08 kPa, and Ta 52118C; line c, H 5 660 m, ea 5 0.23 kPa, and Ta 5288C; and line d, H 5 3000 m, ea 5 0.3 kPa, and Ta 5 248C; and (bottom) positive range of the wet bulb where line e, H 5 4500 m, ea 5 0.6 kPa, and Ta 5 38C; line f, H 5 1300 m, ea 5 1kPa,andTa 5 138C; line g, H 5 100 m, ea 5 3kPa,andTa 5 98C; and line h, H 5 700 m, ea 5 2.4 kPa, and Ta 5 408C.

  This similarity of T and T * provides the rationale to use bT bT a w 2a exp a 2 2 exp a 1 1 the same set of fixed conditions when Tw* is negative as T 1 c T 1 2c l 5 a a , (9b) when Tw* is positive. Thus, the mathematical derivations 2 Ta yielded

c 5 a 2 gP T 2 e , (9a) 5 1 a a a f z(Ta) gPa , (9c)

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FIG. 2. Relationship between l (black line) and z (dotted line) as a function of temperature 2 207 3 when 2178 # Ta # 408C, l 5 0.0014 exp(0.027Ta)withR 5 0.9998, and z 52[3 3 10 (Ta) ] 2 205 2 205 2 [1 3 10 (Ta) ] 1 [2 3 10 (Ta)] 1 0.0444 with R 5 0.9999. Note that Ta in the denominator of the l function has the exponent 2 [Eq. (9b)], whereas its exponent value in z is 1 [Eq. (10)], making the shapes of the two functions quite different. Hence, we choose different regression equations. where the function j is given by solution found in Eq. (1). For this purpose, a series of numerical, iterative computations were carried out to bT bT calculate a highly accurate and theoretical solution for 4a exp a 2 a exp a 2 3a T 1 2c T 1 c Tw at various combinations of the input data as follows: z 5 a a . (10) (i) Ta between 2178 and 408C (by an equal increment of Ta 18C) and es(Ta) between 0.01 and 7.2 kPa (by an equal increment of 0.05 kPa), and (ii) elevation from 0 (P at To determine the dependency of T on l and j, a series a a sea level 5 101.325 kPa) to 4500 m (P 5 58.5 kPa) in of numerical operations were again carried out. For this a 500-m increments. After excluding results leading to a purpose, the values of T were varied from 2178 to 408C a RH . 100% and RH , 5%, 29 515 remaining combi- by an increment of 0.18C (Fig. 2). For both parameters, nations were analyzed. For the purpose of a direct the statistical relationships were nonlinear, as presented comparison, the analytically approximated values of T * below and described in the Fig. 2 caption. The proposed w were also calculated and plotted against T obtained analytical solution therefore includes the derivation of l w by the numerical analysis (Fig. 3). We found a good and j from these statistical relationships, calculating c and agreement between T and T *, such that the absolute u by Eqs. (9a) and (9c), and finally T * byEq.(8a),where w w w maximum difference between the numerical and ana- j j8 0:027T lytical solutions was approximately 1.3 C. However, in l 5 0:0014e a , (11) order to further improve the accuracy, a regression of 5 2 3 27 3 2 25 2 Tw with Tw* was also derived for both the positive and z ( 3 10 )Ta (10 )Ta negative ranges of Tw* (see the regression equations in 1 3 25 1 : 3 22 (2 10 )Ta 4 44 10 . (12) Fig. 3 caption). Using this modification, the maximum difference between Tw* and Tw decreasedtolessthan 3. Results j0.65j8C. Prior to the use of the regression equations in Fig. 3, Validation the sign of Tw* has to be determined; as such, when Ta , The accuracy of the analytical solution derived here 08C, Tw* will always be negative. In support of these for calculating Tw* was tested by comparing it with the analyses, the m, n, and q coefficients in Eq. (2), and r and

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5 : 2 2 F(0) 0 611 ea gPaTa , (14a) 5 2 F(Ta) es(Ta) ea . (14b)

Because RH , 1andea , es(Ta), F(Ta) from Eq. (14b) will always be positive. This means that Tw* will be positive only when F(0) , 0 [since F(0) 3 F(Ta) is neg- ative], or

0:611 2 e . a Ta . (15) gPa

We further examined the differences between Tw and the modified Tw* (Fig. 3) as not only a function of Ta but also differences in RH and elevation as well (Fig. 4).

Interestingly, when Ta , 08C, the maximum difference was only j0.17j8C. However, when Ta $ 08C, the differ- ence ranged between 60.658C with the largest observed differences ($j0.5j8C) occurring when conditions were

(i) Ta $ 398Cand31%, RH , 65%, (ii) Ta $ 358C, 5% , RH , 8%, and when (iii) 278,Ta , 348CandRH. 98%. This means that the actual temperature and/or the relative humidity values have to be very high to cause a difference larger than j0.5j8C 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* 2 Tw were relatively small and its general pattern across the range of Ta and RH remained FIG. 3. The quantity Tw* versus Tw when (a) Tw* , 08C, Tw 5 2 similar (Fig. 4). 1.011(Tw*) 2 0.0419 with R 5 0.9999 and (b) Tw* $ 08C, Tw 5 2 The variation of Tw* 2 T with H, RH, and T was also 1.0301(Tw*) 2 0.213 with R 5 0.9995. w a evaluated (Fig. 5). We found that (i) an elevation change s in Eq. (3a), and t and u in Eq. (3b) were statistically from the sea level to H 5 2000 m does not significantly 2 estimated and were 100.9254 (for m), 0.155 77 (for n), affect the difference of Tw* Tw, although a slight in- 0.621 94 (for q), 2 502 535.259 (for r), 2385.764 24 (for s), crease was observable (cf. Fig. 5a). For H . 3000 m, 2839683.144(fort), and 212.563 84 (for u). Neverthe- differences increase at a higher rate when compared to less, when the Ta is positive (Ta $ 08C), the positive sign those sea level conditions. These results indicate that 2 . of Tw* is not guaranteed. For example, assume that the Ta larger differences in Tw* Tw are expected when H is 28C, the elevation is H 5 3500 m, and ea is 0.16 kPa 3000 m, and (ii) as shown in Fig. 5b, four distinct be- 2 (RH ’ 23%); the Tw* calculated by the iterative solution haviors are exhibited for the Tw* Tw as RH increases. (Eq. 4) is 24.528C(Ta $ 0andTw* , 0). However, in the The differences decrease as RH increases from 5% to , , same example, if Ta is 148C, then Tw* would be 1.68C 20%, increases between 20% RH 50%, decreases , , (Ta $ 0andTw* . 0). To find out whether Tw* is positive between 50% RH 85%, and finally remain ap- or negative, we define the F(T) function from Eq. (4) as proximately constant as RH varies between 85% and 100%. In addition, the largest differences were apparent bT when RH is near to 5% or about 50%, and (iii) Fig. 5c F(T) 5 a exp 2 e 2 gP (T 2 T). (13) T 1 c a a a demonstrates that the proposed analytical method over- estimates Tw (relative to Tw*Tw*)when218C # Ta # 258C Notice again that the root of this function represents the regardless of the value of the RH and H.WhenTa . 88C, Tw* [F(Tw*) 5 0]. Considering the graphical derivation of the range of Tw* 2 Tw increased as Ta increases, with the Figs. 1e–h, Tw* can only be positive if the root of F(T) rate being considerably higher when Ta $ 268C. The two falls between F(0) and F(Ta)orinotherwords,F(0) 3 distinct sections in the range of Tw* 2 Tw when 08C , F(Ta) , 0. Substituting these values into Eq. (13) yields Ta # 88C result from the fact that positive dry-bulb

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FIG. 4. Difference between Tw Eq. (1) and Tw* as a function of Ta and RH at elevations ranging from sea level to 4500 m. The y axis is % RH.

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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 (Simoes-Moreira~ 1999). Moreover, since the equation for calculating the atmospheric pres- sure 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 tempera- ture for both positive and negative ranges of the air tem- perature can be simulated by a second-order equation. The suggested technique seems to converge with the other

known approach when 2178 # Ta , 08C, so that the maximum difference in predicting Tw did not exceed j0.17j8C. When Ta $ 08C, Tw* 2 Tw were within 60.658C. 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 positivevaluesspannedalarger range and (ii) the relationship between wet-bulb temper-

ature and Ta is more curvilinear when Ta $ 08C. There- fore, the comparative use of the three environmental conditions leads to larger uncertainties in estimating the second-order polynomial coefficients (i.e., l, u,andc). 2 This study also proved again that values of 5.68 3 10 4 to 24 6.42 3 10 when Tw , 08 and 08 # Tw # 308Cconverged with other estimates of the thermodynamic psychrometric 2 FIG. 5. Variation of Tw* Tw versus (top) H, (middle) RH, and constant. The proposed method can provide more de- (bottom) Ta. tailed understanding of hydrostatic properties of air con- taining water vapor, particularly in the estimation of the temperatures can provide negative T values (i.e., the w wet-bulb temperature without the use of alternative and upper section). Finally, when T , 08C, the T * 2 T a w w computationally cumbersome iterative approaches. differences decrease as Ta decreases and both maximum and minimum values of Tw* 2 Tw became smaller (Fig. 5c). The mean absolute error (MAE) and the root-mean- Acknowledgments. The authors wish to thank Prof. squared errors (RMSE) for the proposed analytical so- R. Stull for his response to our inquiry, and also Dr. Th. lution were less than 0.158 and 0.28C, respectively. When Bellinger for suggesting useful references that greatly improved the quality of this work. HWL wishes to thank Ta was positive, MAE and RMSE were 0.158 and 0.218C, respectively, but were 0.078 and 0.088C, respectively, the National Science Foundation under the Grant EF- 102980. Any opinions, findings, and conclusions or rec- when Ta was ,08C. ommendations expressed in this material are those of the 4. Conclusions authors and do not necessarily reflect the views of the National Science Foundation. The authors are thankful The wet-bulb temperature is an important psychro- for the thoughtful comments from three anonymous metric parameter with implications in environmental, reviewers. meteorological, and agricultural basic research and ap- plied applications. Historically, the theoretical equation REFERENCES for the Tw has no direct solution, and to find and accurate estimation typically relied upon a trial and error process Alves, I., J. C. Fontes, and L. S. Pereira, 2000: Evapotranspiration that requires significant CPU time. On the contrary, the estimation from infrared surface temperature. II: The surface

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