1. PENMAN MODEL 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 = total evaporation rate; E_{n} = evaporation rate due to net radiation; E_{a} = 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 masstransfer evaporation rate E_{a} is calculated with an empirical masstransfer formula. 2. PENMANMONTEITH MODEL In the PenmanMonteith method, the masstransfer evaporation rate E_{a} is calculated based on physical principles. The original form of the PenmanMonteith equation, in dimensionally consistent units, is:
in which
The quantity r_{a}^{1} is the external conductance, in cm^{3} of air per cm^{2} of surface per second (cm/s).
In evaporation rate units, Eq. 2 is expressed as follows:
in which
Equation 4 is the PenmanMonteith model of evaporation. 3. PHYSICAL CONSTANTS The density of dry air at 0°C and sealevel atmospheric pressure is: ρ_{ad} = 1.2929 kg/m^{3}. The density of moist air ρ_{a} may be approximated as follows:
in which T = air temperature, in °C. For instance, at T = 20°C and sea level (standard atmospheric pressure): ρ_{a} = 0.0012046 gr/cm^{3}. The specific heat of moist air, in the range 0°C ≤ T ≤ 40°C, is: c_{p} = 1.005 J/(gr°C) Converting to calories: c_{p} = (1.005 J/(gr°C) (0.239 cal/J) = 0.2402 cal/(gr°C). 4. DAILY EVAPORATION RATE In evaporation units of cm/d, Eq. 4 is expressed as follows:
in which
Equation 6 can be conveniently expressed in Penman form (Eq. 1) as follows:
in which E_{a} = evaporation rate due to mass transfer, in cm/d. 5. EVAPORATION RATE DUE TO MASS TRANSFER Comparing Eqs. 6 and 7, the evaporation rate due to mass transfer is obtained:
Simplifying Eq. 8:
in which K = a constant varying with air temperature and atmospheric pressure,
in units of
In Eq. 10, the units of ρ_{a}, c_{p}, ρ, λ, and γ are the same as in Eqs. 2 and 4. The psychrometric constant γ, in mb/°C, is:
in which c_{p} = specific heat of moist air,
in cal/(gr°C); p = atmospheric pressure, in mb;
Substituting Eq. 11 in Eq. 10:
in which the constant K remains in units of s/(dmb). Replacing r_{MW} = 0.622 into Eq. 12:
in which the constant K remains in units of s/(dmb).
At T = 20°C and standard atmospheric pressure (sea level):
ρ_{a} = 0.0012046
in which:
6. EXTERNAL RESISTANCE The external, or aerodynamic resistance r_{a} varies with the surface roughness (water, soil, or vegetation), being inversely proportional to wind speed (Eq, 15). In other words, the external conductance and, thus, the evaporation rate, increases with wind speed, as originally postulated by Dalton (Ponce, 1989). The external resistance for evaporation from open water can be estimated as follows:
in which:
The external resistance r_{a} (s/m) for the reference crop (clipped grass 0.12m high), for measurements of wind speed (m/s), temperature and humidity at a standardized height of 2 m is:
For instance, for v_{2} = 200 km/d = (200000 m) / (86400 s) = 2.31 m/s, the external or aerodynamic resistance of the reference crop is:
7. INTERNAL RESISTANCE The internal, or stomatal or surface resistance r_{s} is inversely proportional to the leafarea index L. An empirical relation for surface resistance is (Maidment, 1993):
in which r_{s} is in s/m. The leafarea index L is empirically related to crop height h_{c}. The leafarea index for clipped grass is:
in which h_{c} = crop height, in m,
varying in the range From Eqs. 18 and 19, the surface resistance of the reference crop (clipped grass 0.12m high) is: r_{s}^{rc} = 200 / (24 × 0.12) = 69.4 s/m. The leafarea index for alfalfa is:
in which
crop height h_{c} is in m, varying in the range
From Eq. 20, for h_{c} = 0.3 m, the leafarea index for alfalfa is: L = 3.69. From Eq. 18, the surface resistance for alfalfa is: r_{s} = 200 / 3.69 = 54.2 s/m.
8. ILLUSTRATIVE EXAMPLE Calculate the evaporation rate of the reference crop (clipped grass) by the PenmanMonteith method for the month of April, for the following atmospheric conditions: air temperature T_{a} = 20°C; net radiation Q_{n} = 550 cal/(cm^{2}d); wind speed v_{2} = 200 km/d; and relative humidity φ = 70%. Assume standard atmospheric pressure. Solution.
ONLINE CALCULATION.
Using ONLINE PENMAN MONTEITH, the answer
is: Daily reference crop
REFERENCES Maidment, D. R. 1993. Handbook of Hydrology. McGrawHill. Ponce, V. M. 1989. Engineering Hydrology: Principles and Practices. Prentice Hall, Englewood Cliffs, New Jersey. Ponce, V. M. 2014. Engineering Hydrology: Principles and Practices. Online text.

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