The original Penman model is a combination method 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 (Ponce, 1989):
in which E = evaporation rate due to mass transfer;
Δ = saturation vapor pressure gradient, varying with air temperature; and γ = psychrometric constant, varying slightly with temperature. In Eq. 1, the mass-transfer
evaporation rate is calculated with an empirical mass-transfer formula.
_{a}
Unlike the original Penman model (Eq. 1), in the Penman-Monteith model the mass-transfer
evaporation rate
in which
- ρλ
*E*= total evaporative energy flux, in cal cm^{-2}s^{-1}; - Δ = saturation vapor pressure gradient, in mb °C
^{-1}; *H*= energy flux supplied externally, by net radiation, in cal cm^{-2}s^{-1};- ρ
_{a}= density of moist air, in gr cm^{-3}; *c*= specific heat of moist air, in cal gr_{p}^{-1}°C^{-1};- (
*e*-_{s}*e*) = vapor pressure deficit, in mb;_{a} *r*= external (aerodynamic) resistance, in s cm_{a}^{-1}; and- γ* = modified psychrometric constant, in mb °C
^{-1}, equal to:*r*_{s} γ* = γ ( 1 +^{_____})
*r*_{a}(3) in which - γ = psychrometric constant, in mb °C
^{-1}, varying slightly with temperature, and *r*= internal (stomatal or surface) resistance, in s cm_{s}^{-1}.
The quantity ^{-1} is the external conductance, in cm^{3} of air per cm^{2} of
surface per second (cm s^{-1}).
In evaporation rate units, Eq. 2 is expressed as follows:
in which *E*= total evaporation rate, in cm s^{-1};*E*= evaporation rate due to net radiation, in cm s_{n}^{-1};- ρ = density of water, in gr cm
^{-3}; - λ = heat of vaporization of water, in cal gr
^{-1};and - Δ, γ*, ρ
_{a},*c*, (_{p}*e*-_{s}*e*), and_{a}*r*are in the same units as in Eq. 2._{a}
Equation 4 is the Penman-Monteith model of evaporation.
The density of dry air at sea level is: ρ
in which
For instance, at
ρ
The specific heat of moist air, in the range 0°C ≤
^{-1} °C^{-1}
Converting to calories:
^{-1} °C^{-1}) (0.239 cal/J) = 0.2402 cal gr^{-1} °C^{-1}.
In evaporation units of cm d
in which
*E*= total evaporation rate (cm d^{-1});*E*= evaporation rate due to net radiation (cm d_{n}^{-1}); and- Δ, γ*, ρ
,_{a}*c*, (_{p}*e*-_{s}*e*),_{a}*r*, ρ, and λ are in the same units as Eqs. 2 and 4._{a}
Equation 6 can be conveniently expressed in Penman form (Eq. 1) as follows:
in which ^{-1}:
Comparing Eqs. 6 and 7, the evaporation rate due to mass transfer is obtained:
Simplifying Eq. 8:
in which
In Eq. 10, the units of ρ c, ρ, λ, and γ are the same as in Eqs. 2 and 4.
_{p}
The psychrometric constant γ, in mb °C
in which ^{-1} °C^{-1}; p = atmospheric pressure, in mb;
λ = heat of vaporization of water, in cal gr^{-1}; and r = ratio of the molecular weight of water vapor to dry air: _{MW}r = 0.622._{MW}Substituting Eq. 11 in Eq. 10:
in which the constant
Replacing
in which the constant
At
in which
*E*= evaporation rate due to mass transfer, in cm d_{a}^{-1};- (
*e*-_{s}*e*) = vapor pressure deficit, in mb;_{a} *r*= external (aerodynamic) resistance, in s cm_{a}^{-1}; and*r*= internal (stomatal) resistance, in s cm_{s}^{-1}.
The external (or aerodynamic) resistance The external resistance for evaporation from open water can be estimated as follows:
in which
*r*= external resistance, in s m_{a}^{-1};*z*= height at which meteorological variables are measured, in m;_{m}*z*_{o}= aerodynamic roughness of the surface, in m; and*v*_{2}= wind speed, in m s^{-1}, measured at 2-m height.
The external resistance ^{-1})
for the reference crop (clipped grass 0.12-m high),
for measurements of wind speed (m s^{-1}),
temperature and humidity at a standardized height of 2 m is:
For instance, for
The internal (stomatal or surface) resistance L, i.e., the projected area of vegetation per unit ground area. An empirical relation is:
in which ^{-1} and L is in m s^{-1}.
The leaf-area index
h is in m, varying in the range _{c}h ≤
0.15._{c}
From Eq. 18, the stomatal resistance of the reference crop (clipped grass 0.12-m high) is:
^{-1} = 0.694 s cm^{-1}
h is in m, varying in the range
_{c}h ≤ 0.5._{c}From Eq. 18, the stomatal resistance of an alfalfa crop, with
^{-1} = 0.541 s cm^{-1}
Ponce, V. M. 1989. |

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