A Computer System of Estimating Reference Crop Evapotranspiration
Using the Modified Penman’s (FAO - 24) Method
Menelaos Theocharis 1
1 Department
of Crop Production Technological Educational Institute of Epirus, 47100 Arta, Greece theoxar@teiep.gr
Abstract:
The rational designing of irrigation systems requires the calculation of the evapotranspiration in the best possible way.
Reference crop evapotranspiration (ETr) is the rate at which water, if available, would be removed from the soil and
plant surface of a specific crop, arbitrarily called a reference crop. The classical Penman method is a combination
equation that considers both the energy and aerodynamic aspects of the ETr process. Food and Agricultural
Organization (FAO) modified the original Penman’s method in 1977. This method uses mean daily climatic data, with
an adjustment for day and night time weather conditions. The modified Penman’s (FAO-24) method is considered to be
one of the most accurate methods for the computation of evapotranspiration and must be used when all the required
climatological data are available. The required calculating procedure when the above modified Penman’s (FAO-24)
method is used, is very complex and calls for a lot of calculations. In this paper we present a computer system of
estimating reference crop evapotranspiration using the modified Penman’s (FAO-24) method. The system facilitates
irrigation network designers in calculating reference crop evapotranspiration. It is simple in use, do not demand
special informatics knowledge, and it is approachable to a wide range of researchers. System’s architecture includes a
user-friendly interface for the data input, a computational module and the output subsystem, where the user is given a
detailed table of results. The computer system provides a quick and accurate solution to the problem of estimating
reference crop evapotranspiration. In addition, it could be incorporated as subroutine of integrated software dealing
with hydraulic calculations.
Key words:
Computer system, Crop evapotranspiration, Irrigation, Penman’s method.
1. INTRODUCTION
The rational designing of irrigation systems requires the calculation of the evapotranspiration in
the best possible way. Reference crop evapotranspiration (ETr) is the rate at which water, if
available, would be removed from the soil and plant surface of a specific crop, arbitrarily called a
reference crop. The classical Penman’s method (Penman 1963) is a combination equation that
considers both the energy and aerodynamic aspects of the ETr process. Food and Agricultural
Organization (FAO) modified the original Penman’s method in 1977. This method uses mean daily
climatic data, with an adjustment for day and night time weather conditions. The modified
Penman’s (FAO-24) method is considered to be one of the most accurate methods for the
computation of evapotranspiration and must be used when all the required climatological data are
available.The required calculating procedure when the above modified Penman’s (FAO-24) method
is used, is very complex and calls for a lot of calculations. In this paper we present a computer
system of estimating reference crop evapotranspiration using the modified Penman’s (FAO-24)
method. The system facilitates irrigation network designers in calculating reference crop
evapotranspiration. It is simple in use, do not demand special informatics knowledge, and it is
approachable to a wide range of researchers. System’s architecture includes a user-friendly interface
for the data input, a computational module and the output subsystem, where the user is given a
detailed table of results. The computer system provides a quick and accurate solution to the problem
of estimating reference crop evapotranspiration. In addition, it could be incorporated as subroutine
of integrated software dealing with hydraulic calculations.
2. THE MODIFIED PENMAN’S (FAO-24) METHOD
Food and Agricultural Organization (FAO) modified the original Penman equation (Penman
1963) in 1977. The modifications included a revised wind function, an adjustment factor to account
for local conditions and the assumption that the daily average ground heat flux is zero. This method
uses mean daily climatic data, with an adjustment for day and night time weather conditions. The
modified equation used in this method (Doorenbos and Pruitt 1977; Jennifer, 2001) is:
ETr 
C
ρw
 (R n  G)

 (1  W)f(u)(e a  e d )
W
λ


(1)
where ETr is the reference evapotranspiration, mm/day; W is a temperature and the altitude of
the area related weighting factor; Rn is the net solar radiation, MJm-2d-1; G is the soil heat flux in
MJm-2d-1; λ is the latent heat of evaporation, MJ kg-1; ρw = 1000 kgm-3, is the density of water;
f(u) is the wind related function, kg hPa-1m-2d-1; ea is the saturation vapour pressure at mean air
temperature, hPa; ed is the mean actual vapour pressure of the air, hPa; and C is an adjustment
factor to account for day and night weather conditions.
2.1 Weighting Factor (W)
This is a weighting factor for the effect of solar radiation, the wind and humidity on ETr. It is
given by the equation:
W
Δ
Δγ
(2)
-1
where Δ is the rate of change of saturation vapour pressure with temperature, hPa 0C , and γ is
-1
the psychometric constant, hPa 0C , which are calculated from the equations provided by Brunt
(Brunt 1952), and Bosen (Bosen 1960) respectively:

4098
Ta  237.3
2
 17.27 T 
-1
a 
6.11 exp 
hPa 0C
 T  237.3 
 a

(3)
and
5.2586

0.0065 
1 
h
273  Ta 

γ
1.5156  0.0014307 Ts
hPa 0C
-1
(4)
where Ta and Ts are the daily mean air and water temperature, 0C; and h is the altitude of the area, m.
2.2 Humidity (ea-ed)
Humidity is expressed as the difference between the mean saturation water vapour pressure (ea)
and the mean actual vapour pressure (ed). From Tetens (Tetens 1930), saturated air vapour pressure
as a function of temperature, ea, can be computed as follows:
 17.27 T 
a 
e a  6.11 exp 
hPa
 T  237.3 
 a

(5)
The actual vapour pressure is calculated using saturation vapour pressure and the mean relative humidity:
e d  e a Rh mean
hPa
(6)
Therefore:
 17.27 T 
a 
e a  e d  (1 - Rh mean )6.11 exp 
 T  237.3 
 a

hPa
(7)
2.3 Wind Factor ( f(u) )
Wind Factor expressing the effect of wind on ETr using the modified Penman’s (FAO-24) method, is
calculated (Doorenbos and Pruit 1997) from:
f (u)  0.27(1  0.86u 2 ) kg hPa-1m-2d-1
(8)
Wind speed is measured at many different heights above the ground surface. The Penman’s
equation requires wind speed at a height of 2.00 m. It can be adjusted to a height of 2.00 m by:
2
u2  uz 
z
0, 2
ms-1
(9)
where: u2 and uz is the wind speed corresponded at height of 2.00 m and z m, in ms-1; and z is the
height of wind measurement, m.
2.4 Net Radiation (Rn)
Net Radiation is the difference between incoming and outgoing solar radiation. It can be
measured, but such data are seldom available. Instead, it is calculated from solar radiation on
sunshine hours, temperature, and humidity data (Koutsoyiannis and Xanthopoulos 1997; Papazafiriou
1999). The FAO-24 Penman’s approach calculates Rn as follows:
The net radiation is the algebraic sum of the net shortwave radiation (Rns) and net longwave
radiation (Rnl). As Rnl constitutes a loss, this relationship is:
R n  R ns  R nl
MJm-2d-1
(10)
Net shortwave radiation (Rns), can be calculated by correcting the incoming solar radiation for
surface albedo (α) of the crop surface by:
R ns  1- α R s MJm-2d-1
(11)
where: α = 0.05 for water surface; α = 0.15 for bare soil surfaces; and α = 0.23 for green vegetated
surfaces. For most crops
= 0.25.
Solar radiation (Rs), is calculated using the amount of radiation received at the top of the
atmosphere (Ra). This value is calculated (Duffie and Beckman 1980,1991) by:
R a  3.6

2 I s d r 
180
ω s sin φ sin δ 
sin ω s cosφ cosδ 


15 
π

MJm-2d-1
(12)
-2
where: I s  1.367 kWm is the solar constant; dr is the relative distance from the sun to the earth;
φ is the station latitude, degrees; δ is the declination of the sun, degrees; ωs is the sunset hour angle,
degrees; The values of dr , δ and ωs are calculated respectively from:
 360

d r  1  0.034 cos
J  2.86479 


 365

0
(13)
 360

δ  23.45 sin 
( 284  J ) 
 365

and
0
0
(14)
ωs  arccos ( tan φ tanδ) degrees
(15)
In the above relations, J is the Julian day of the year considered given from:
J  30,5M  14,6 (integer)
(16)
where M =1,2,3,…,12 is the number of the month considered.
After that solar radiation (Rs), is obtained with the Angstrom formulation:
n

R s   0.25  0.50 R a MJm-2d-1
N

(17)
where n and N are correspondently the actual and the maximum possible sunshine hours , hd-1; and
Ν
2
ωs
15
hd-1
(18)
Net longwave radiation (Rnl), can be determined from available temperature, the vapour pressure
and the n/N ratio data by the equation:
R nl  ε n f L σ(Ta  273) 4
MJm-2d-1
(19)
where σ = 4.903*10-9 , MJ K-4 m-2 day-1, is the Stefan-Boltzmann constant ; εn is the net emissivity
factor which is calculated by e n  0.34 - 0.044 e d ; and fL is the cloudiness adjustment factor which is
calculated by f L  0.1  0.9
n
.
N
Therefore Rnl can be determined by:

n
R nl  σ(Ta  273) 4 0.34 - 0.044 e d 0.1  0.9 

N 


MJm-2d-1
(20)
2.5 Soil Heat Flux (G)
Soil heat flux is calculated from the relation provided by Jensen et al. (1990):
G  cs d s
Tn 1  Tn 1
Δt
or G  2 c s d s
Tn  Tn 1
Δt
MJ m-2d-1
(21)
where Tn+1, Tn and Tn-1 are respectively the mean temperature on next month n+1, month considered
-1
n, and on previous month n-1, 0C; cs is the soil heat capacity = 2.1MJ m-3 0C ; Δt is the length
period ; ds is the estimated effective soil depth and ds = 0,18 m for daily temperature fluctuations and
ds = 0,03 m for monthly temperature fluctuations (Jensen et. al. 1990).
Therefore G can be determined by: G  0,38 Td  Td1 MJ m-2d-1 for daily temperature fluctuations
and by: G  0,14 Tn  Tn 1 or G  0,07 Tn 1  Tn 1 MJ m-2d-1 for monthly temperature fluctuations.
Note: The soil heat flux is generally very small and it is assumed to be G = 0.01Rn during the
day and G = 0.02Rn during the night. According to FAO-24 modifications it is assumed that the
daily average ground heat flux is zero.
2.6 Adjustment Factor (C)
The Penman equation assumes that the most common conditions occur when radiation is
medium to high, maximum relative humidity is medium to high and moderate daytime wind about
double the night time wind. These conditions are not always met. The Penman equation has to be
corrected in such situations using an adjustment factor. The adjustment factor as developed by Allen
et al. (1989) and obtained from Jensen et al. (1990) is given by:
C  0.892 - 0.0781u d  0.00219 u d R s  0.000402 Rh max R s  0.000196
ud
u d Rh max 
un
0.0000198
ud
un
u d Rh max R s  0.00000236 u 2d Rh max
u
0.0000000292  d

 un
u
R s  0.0000086  d

 un
2

 u Rh

 d max

(22)

2
 u 2 Rh 2 R   0.0000161 Rh
max
s
max R s
 d

where ud and un are correspondently the daytime and night time wind speed, ms-1;
λ  2.501 0.002361Ts MJ kg-1 ;
(23)
and
R s 
Rs
ρwλ
mmd-1
(24)
Note: The modified Penman’s (FAO 24) method requires day and night wind speed. If the wind
data to evaluate the correction factor are not available, it is adequate to estimate that the ud/un value
is 1.5 during the convective season and 1.85 during the frontal season.
2.7 Reference Crop Evapotranspiration (ETr)
After combining the above equations, the modified Penman’s (FAO-24) equation is given by:
ETr 
C
ρw
 Rn

 W λ  (1  W)f(u)(e a  e d )


(25)
where all the parameters used, are as above have been described.
3. ESTIMATING REFERENCE CROP EVAPOTRANSPIRATION ACCORDING TO THE
MODIFIED PENMAN’S METHOD (FAO-24) USING PERSONAL COMPUTER
The required calculating procedure when the above modified Penman’s (FAO-24) method is used, is
very complex and calls for a lot of calculations. For this reason, we found necessary to develop a simple
computer system that enables researchers and students to easily calculate reference crop evapotranspiration.
This program, being accessible by those who need it, could be embedded as subroutine in larger
computer systems dealing with hydraulic calculations.
3.1 Organising the Input Parameters
The first step is to organise the data for the problem on the spreadsheet. In Figure 1, the input
parameters for the altitude of the area, h in m, the height of wind speed measurement, z in m , the
wind speed corresponded at height z, u z in m/s, the number of the month considered, M, the station
latitude, φ in degrees, the daily mean air temperature, Ta in 0C, the daily mean water temperature,
Ts in 0C, the actual possible sunshine hours, n in h/d, the ratio of daytime and nighttime wind speed
ud/un, the mean relative humidity, Rhmean in %, and the maximum relative humidity, Rhmax in %, were
entered in cells e15, i15, m15, e18, i18, m18, e21, i21, m21, e24 and i24 respectively.
Figure 1. Spreadsheet of data and calculations of ETr using the modified Penman’s
(FAO-24) method.
3.2 Solving the Problem
In order to solve the problem the following macro command is created:
Sub Crop_Evapotranspiration ()
‘Macro recorded in 24/1/2009 by M. Theocharis
Range("f3") = "Estimating Reference Crop Evapotranspiration "
Range("f4") = "Using the Modified Penman's Method (FAO 24)"
Range("h12") = "Input Parameters "
h = Range("e15").Value
z = Range("i15").Value
uz = Range("m15").Value
M = Range("e18").Value
φ = Range("i18").Value
Ta = Range("m18").Value
Ts = Range("e21").Value
n = Range("i21").Value
Ru = Range("m21").Value
Rhmean = Range("e24").Value
Rhmax = Range("i24").Value
Δ = (4098/((Ta+237.3)^2)*6.11*exp(17.27*Ta/(Ta+237.3)))
γ = ((1-0.0065*h/(273+Ta))^5.2586)/(1.5156-0.0014307*Ts)
W = Δ/(Δ+γ)
Range("e30") = W
ea = 6.11*Exp(17.27*Ta/(Ta+237.3))
ed = ea*Rhmean/100
Range("i30") = ea-ed
Fu = (0.27*(1+0.86*uz*(2/z)^0.2))
Range("m30") = Fu
J = Int(30.5*M-14.6)
dr =1+0.034*Cos((4*Atn(1)/180)*(360*Int(30.5*M-14.6)/365-2.86479))
δ = 23.45*Sin((4*Atn(1)/180)*(360*(284+Int(30.5*M-14.6))/365))
cosωs = -Tan(4*Atn(1)*φ/180)*Tan(4*Atn(1)*δ/180)
ωs = (180/(4*Atn(1)))*(Atn(-cosωs/Sqr(-cosωs*cosωs+1))+2*Atn(1))
Ra = 1/15*(3.6*2*1.367*dr)*(ωs*Sin(4*Atn(1)*φ/180)*Sin(4*Atn(1)*δ/180)+(180/(4*Atn(1)))*
Sin(4*Atn(1)*ωs/180)*Cos(4*Atn(1)*φ/180)*Cos(4*Atn(1)*δ/180))
Ν = (2/15)*ωs
Rs = (0.25+0.5*n/Ν)*Ra
Rns = (1-0.25)*Rs
Range("e33") = Rns
Rnl = 4.903*10^-9*((Ta+273)^4)*(0.34-0.0442719*ed^0.5)*(0.1+0.9*n/Ν)
Range("i33") = Rnl
Rn = Rns-Rnl
Range("m33") = Rn
λ = 2.501-0.002361*Ts
Range("e36") = λ
C = 0.892-0.0781*(uz* (2/z)^0.2)+0.00219*(uz*(2/z)^0.2)* Rs/λ+0.000402* Rhmax* Rs/λ+
0.000196* Ru*(uz* (2/z)^0.2)* Rhmax+0.0000198* Ru*(uz* (2/z)^0.2)* Rhmax *Rs/λ+
0.00000236* (uz*(2/z)^ 0.2)^2*Rhmax*Rs/λ-0.0000086*Ru^2*(uz*(2/z)^0.2)*Rhmax0.0000000292* Ru*(uz*(2/z)^0.2)^2* Rhmax^2*Rs/λ-0.0000161*Rhmax*(Rs/λ)^2
Range("i36") = C
ETr = C*(W*Rn /λ+(1-W)*(0.27*(1+0.86*uz*(2/z)^0.2))*(ea-ed))
Range("m38") = Round(ETr,3)&" mm/d"
End Sub
After that we click the “Solutions” button and the solution of the problem results.The described
procedure is shown in Figure 1.
4. APPLICATION
Below the reference crop evapotranspiration in Arta, Greece area for the month July are
estimated, using the above described computer system, Figure 1. For the input parameters:
h = 150.00 m; z =10.00 m ; u z = 3.30 m/s; M=7 ; φ = 39.13 0 ; Ta = 25.30 0C ; Ts =15.00 0C ;
n =11.26 hd-1; ud/un = 1.50 ; Rhmean = 64.90 % ; and Rhmax = 79.90 % ; it emerges that: W = 0.7445 ;
ea-ed = 11.323 hPa ; f(u) = 0,825 kg hPa-1m-2d-1 ; Rns= 19.489 MJm-2d-1; Rnl = 4.275 MJm-2d-1 ;
Rn =15.214 MJm-2d-1; λ= 2.466 MJ kg-1; C = 1.063 and finally the reference crop evapotranspiration
ETr =7.421 mm/d.
5. CONCLUSIONS
Solving problems of estimating reference crop evapotranspiration is of prime importance for the
study of irrigation networks.
The required calculating procedure when the modified Penman’s (FAO-24) method is used is
very complex and calls for a lot and very laborious calculations. This is why FAO developed
various tables where the values of the calculating parameters are provided.
But these tables are very unpractical in use and include significant inaccuracies in values
determination. Besides, it is impossible to use them when trying to solve the problems using a
computer.
The computer system presented in this paper is simple in use, does not demand special
informatics knowledge and is approachable to a wide range of researchers. In addition, it could be
incorporated as subroutine of integrated software dealing with irrigation calculations.
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