Fig. 1  Vegetated earthmounds and surrounding grasslands, Pantanal of Mato Grosso, Brazil.



Victor M. Ponce

Professor Emeritus of Civil and Environmental Engineering

San Diego State University, San Diego, California

05 March 2024

ABSTRACT.  In this article we show how to calculate online reference crop potential evapotranspiration by the Penman-Monteith method. The latter is a combination method based on a modification of the classical evaporation method developed by Penman in the 1940s. The Penman-Monteith method has been endorsed by the United Nations Food and Agriculture Organization (FAO) for general use in the calculation of reference crop potential evapotranspiration. The online calculator ONLINE_PENMAN_MONTEITH was developed in 2020 by the Visualab at San Diego State University.


The calculation of reference crop evapotranspiration is a preeminent analytical tool used in the design of irrigation projects. The FAO Penman-Monteith method has been recommended as the method of choice for calculating reference crop evapotranspiration (ETo) (Irrigation and Drainage Paper 56 - Chapter 2, FAO, 1998). In this article we present an online calculator for the Penman-Monteith method. This calculator was developed in 2020 by the Visualab, a computational laboratory of the Department of Civil, Environmental, and Construction Engineering, San Diego State University, San Diego, California.


The Penman-Monteith method is a modification of the Penman (1948) method. The original Penman method is a combination method to calculate evaporation (Ponce, 2014a: Combination Methods) in which the total evaporation rate is calculated by weighing the evaporation rate due to net radiation and the evaporation rate due to mass transfer, as follows :

             Δ En  +  γ Ea
E  =   _________________ 
                  Δ  +  γ

in which E = total evaporation rate; En = evaporation rate due to net radiation; Ea = evaporation rate due to mass transfer; Δ = saturation vapor pressure gradient, a function of air temperature; and γ = psychrometric constant, which may be shown to vary slightly with temperature. The mass-transfer evaporation rate Ea is calculated with an empirical mass-transfer formula (Ponce, 2014a: Mass-Transfer Approach).

In the Penman-Monteith method, the mass-transfer evaporation rate Ea is calculated based on physical principles. The original form of the Penman-Monteith equation, in dimensionally consistent units, is the following (Ponce, 2014b):

                 ΔH  +  [ ρa cp (es  -  ea) / ra ]
ρλE  =   _______________________________ 
                                Δ  +  γ *

in which

  • ρλE = total evaporative energy flux, in cal/(cm2-s);

  • ρ = density of water, in gr/cm3;

  • λ = heat of vaporization, in cal/gr;

  • E = evaporation rate, in cm/s;

  • Δ = saturation vapor pressure gradient, in mb/°C;

  • H = energy flux supplied externally, by net radiation, in cal/(cm2-s);

  • ρa = density of moist air, in gr/cm3;

  • cp = specific heat of moist air, in cal/(gr-°C);

  • (es  -  ea) = vapor pressure deficit, in mb;

  • ra = external (aerodynamic) resistance, in s/cm; and

  • γ * = modified psychrometric constant, in mb/°C, equal to:

    γ *  =   γ   ( 1  +  _____ )

    in which:

  • γ = psychrometric constant, in mb/°C, which varies slightly with temperature (Ponce, 2014c), and

  • rs = internal (stomatal or surface) resistance, in s/cm.

The quantity ra-1 is the external conductance, in cm3 of air per cm2 of surface per second (cm/s).

In evaporation rate units, Eq. 2 is expressed as follows:

               Δ En  +  [ ρa cp (es  -  ea) / (ra ρ λ ) ]
E  =   _________________________________________ 
                                    Δ  +  γ *

in which

  • E = total evaporation rate, in cm/s;

  • En = evaporation rate due to net radiation, in cm/s;

  • ρ = density of water, in gr/cm3;

  • λ = heat of vaporization, in cal/gr;


  • Δ, γ *, ρa, cp, (es  -  ea), and ra  are in the same units as in Eq. 2.

Equation 4 is the equation of the Penman-Monteith method used in the online calculator presented here.


We run the calculator ONLINE_PENMAN_MONTEITH with the following input data:

  • Month [Select one]:   

  •    April
  • Air temperature Ta (C°):   

  •    20
  • Net radiation Qn (cal cm-2 d-1):   

  •    550
  • Relative humidity φ (%):   

  •    70
  • Wind velocity v2 (km d-1):   

  •    200
  • Atmospheric pressure pa (mb):   

  •    1013.25

Output from the calculator is shown below.

The reference crop potential evapotranspiration for the month of April is:  19.32 cm. ANSWER.


Food and Agriculture Organization of the United Nations (FAO). 1998. Crop evapotranspiration: Guidelines for computing crop water requirements, FAO Irrigation and Drainage Paper 56.

Penman, H. L. 1948. "Natural Evaporation from Open Water, Bare Soil and Grass," Proceedings of the Royal Society, London, Vol. 193, 120-145.

Ponce, V. M. 2014a. Engineering Hydrology: Principles and Practices.

Ponce, V. M. 2014b. The Penman-Monteith method.

Ponce, V. M. 2014c. The psychrometric constant.