# Annex 3. Background on physical parameters used in evapotranspiration computations

Latent Heat of Vaporization (l)1

1 Reference: Harrison (1963)

l = 2.501 - (2.361 x 10-3) T (3-1)

where:

l latent heat of vaporization [MJ kg-1]
T air temperature [°C]

The value of the latent heat varies only slightly over normal temperature ranges. A single value may be taken (for T = 20 °C): l = 2.45 MJ kg-1.

Atmospheric Pressure (P)2

2 Reference: Burman et al. (1987) (3-2)

where:

P atmospheric pressure at elevation z [kPa]
Po atmospheric pressure at sea level = 101.3 [kPa]
z elevation [m]
zo elevation at reference level [m]
g gravitational acceleration = 9.807 [m s-2]
R specific gas constant == 287 [J kg-1 K-1]
a1 constant lapse rate moist air = 0.0065 [K m-1]
TKo reference temperature [K] at elevation zo given by

TKo = 273.16 + T (3-3)

where:

T mean air temperature for the time period of calculation [°C]

When assuming Po = 101.3 [kPa] at zo = 0, and TKo = 293 [K] for T = 20 [°C], equation (3-3) becomes: (3-4)

Atmospheric Density (r)3

3 Reference: Smith et al. (1991) (3-5)

where:

r atmospheric density [kg m-3]
R specific gas constant = 287 [J kg-1 K-1]
TKv virtual temperature [K] (3-6)

where:

TK absolute temperature [K]: TK = 273.16 + T [°C]
ea actual vapour pressure [kPa]

For average conditions (ea in the range 1 - 5 [kPa] and P between 80 - 100 [kPa]), equation (3-6) may be substituted by:

TKv » 1.01 (T + 273) (3-7)

T is set equal to mean daily temperature for 24-hour calculation time steps.

Saturation Vapour Pressure (es)4

4 Reference: Tetens (1930) (3-8)

where:

e°(T) saturation vapour pressure function [kPa]
T air temperature [°C]

Slope Vapour Pressure Curve (D)5

5 References: Tetens (1930), Murray (1967) (3-9)

where:

D slope vapour pressure curve [kPa C-1]
T air temperature [°C]
e°(T) saturation vapour pressure at temperature T [kPa]

In 24-hour calculations, D is calculated using mean daily air temperature. In hourly calculations T refers to the hourly mean, Thr.

Psychrometric Constant (g)6

6 Reference: Brunt (1952) (3-10)

where:

g psychrometric constant [kPa C-1]
cp specific heat of moist air = 1.013 [kJ kg-1 °C-1]
P atmospheric pressure [kPa]: equations 2 or 4
e ratio molecular weight of water vapour/dry air = 0.622
l latent heat of vaporization [MJ kg-1]

Dew Point Temperature (Tdew)7

7 Reference: Bosen (1958); Jensen et al. (1990)

When it is not observed, Tdew can be computed from ea by: (3-11)

where:

Tdew dew point temperature [°C]
ea actual vapour pressure [kPa]

For the case of measurements with the Assmann psychrometer, Tdew can be calculated from (3-12)

Short Wave Radiation on a Clear-Sky Day (Rso)8

8 Reference: Allen (1996)

The calculation of Rso is required for computing net long wave radiation and for checking calibration of pyranometers and integrity of Rso data. Q good approximation for Rso for daily and hourly periods is:

Rso = (0.75 + 2 x 10-5 z)Ra (3-13)

where:

z station elevation [m]
Ra extraterrestrial radiation [MJ m-2 d-1]

Equation (3-13) is valid for station elevations less than 6000 m having low air turbidity. The equation was developed by linearizing Beer's radiation extinction law as a function of station elevation and assuming that the average angle of the sun above the horizon is about 50°.

For areas of high turbidity caused by pollution or airborne dust or for regions where the sun angle is significantly less than 50° so that the path length of radiation through the atmosphere is increased, an adaption of Beer's law can be employed where P is used to represent atmospheric mass: (3-14)

where:

Kt turbidity coefficient [], 0 < Kt £ 1.0 where Kt = 1.0 for clean air and Kt = 1.0 for extremely trubid, dusty or polluted air.

P atmospheric pressure [kPa]

f angle of the sun above the horizon [rad]

Ra extraterrestrial radiation [MJ m-2 d-1]

For hourly or shorter periods f is calculated as:

sin f = sin j sin d + cos j cos d cos w (3-15)

where:

j latitude [rad]
d solar declination [rad] (Equation 24 in Chapter 3)
w solar time angle at midpoint of hourly or shorter period [rad] (Equation (31) in chapter 3)

For 24-hour periods, the mean daily sun angle, weighted according to Ra, can be approximated as: (3-16)

where:

f 24 average f during the daylight period, weighted according to Ra [rad]
j latitude [rad]
J day in the year []

The f 24 variable is used in Equation (3-14) or (3-18) to represent the average sun angle during daylight hours and has been weighted to represent integrated 24-hour transmission effects on 24-hour Rso by the atmosphere. f 24 in Equation (3-16) should be limited to ³ 0.

In some situations, the estimation for Rso can be improved by modifying Equation (3-14) to consider the effects of water vapour on short wave absorption, so that:

Rso = (KB + KD) Ra (3-17)

where:

KB the clearness index for direct beam radiation []
KD the corresponding index for diffuse beam radiation []
Ra extraterrestrial radiation [MJ m-2 d-1] (3-18)

where:

Kt turbidity coefficient [], 0 < Kt £ 1.0 where Kt = 1.0 for clean air and Kt = 1.0 for extremely trubid, dusty or polluted air.

P atmospheric pressure [kPa]

f angle of the sun above the horizon [rad]

W precipitable water in the atmosphere [mm]

W = 0.14 ea P + 2.1 (3-19)

where:

W precipitable water in the atmosphere [mm]
ea actual vapour pressure [kPa]
P atmospheric pressure [kPa]

The diffuse radiation index is estimated from KB:

KD = 0.35 - 0.33 KB for KB ³ 0.15
KD = 0.18 + 0.82 KB for KB < 0.15 (3-20)

As with Equation (3-14), the f 24 variable from Equation (16) is used for f in Equation (3-18) for 24-hour estimates of Rso.

Ordinarily, Rso computed using Equations (3-13), (3-14) or (3-16) should plot as an upper envelope of measured Rs and is useful for checking calibration of instruments. This is illustrated in Annex 5.