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 mass-transfer evaporation rate E_a is calculated with an empirical mass-transfer formula.

2.  PENMAN-MONTEITH MODEL

In the Penman-Monteith method, the mass-transfer evaporation rate E_a is calculated based on physical principles. The original form of the Penman-Monteith equation, in dimensionally consistent units, is:

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);

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

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

• (e_s  -  e_a) = vapor pressure deficit, in mb;

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

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

  rs γ *  =   γ   ( 1  +  _____ )                              ra

  (3)

  in which:

• γ = psychrometric constant, in mb/°C, which varies slightly with temperature, and

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

The quantity r_a^-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:

in which

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

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

• ρ = density of water, in gr/cm³;

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

  and

• Δ, γ *, ρ_a, c_p, (e_s  -  e_a), and r_a  are in the same units as in Eq. 2.

Equation 4 is the Penman-Monteith model of evaporation.

3.  PHYSICAL CONSTANTS

The density of dry air at 0°C and sea-level atmospheric pressure is: ρ_ad = 1.2929 kg/m³. 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/cm³.

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

• E = total evaporation rate (cm/d);

• E_n = evaporation rate due to net radiation (cm/d); and

• Δ, γ *, ρ_a, c_p, (e_s  -  e_a), r_a, ρ, and λ are in the same units as Eqs. 2 and 4.

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 s/(d-mb), expressed as follows:

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; λ = heat of vaporization, in cal/gr; and r_MW = ratio of the molecular weight of water vapor to dry air: r_MW = 0.622.

Substituting Eq. 11 in Eq. 10:

in which the constant K remains in units of s/(d-mb).

Replacing r_MW = 0.622 into Eq. 12:

in which the constant K remains in units of s/(d-mb).

At T = 20°C and standard atmospheric pressure (sea level): ρ_a = 0.0012046 gr/cm³, ρ = 0.99821 gr/cm³, and p = 1013.25 mb. Thus, the constant K in Eq. 13 reduces to: K = 0.064 s/(d-mb), and Eq. 9 reduces to:

in which:

• E_a = evaporation rate due to mass transfer, in cm/d;

• (e_s  -  e_a) = vapor pressure deficit, in mb;

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

• r_s = internal (stomatal) resistance, in s/cm.

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:

• r_a = external resistance, in s/m;

• z_m = height at which meteorological variables are measured, in m;

• z_o = aerodynamic roughness of the surface, in m; and

• v₂ = wind speed, in m/s, measured at 2-m height.

The external resistance r_a (s/m) for the reference crop (clipped grass 0.12-m high), for measurements of wind speed (m/s), temperature and humidity at a standardized height of 2 m is:

For instance, for v₂ = 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 leaf-area index L. An empirical relation for surface resistance is (Maidment, 1993):

in which r_s  is in s/m.

The leaf-area index L is empirically related to crop height h_c. The leaf-area index for clipped grass is:

in which h_c = crop height, in m, varying in the range 0.05 ≤ h_c ≤ 0.15.

From Eqs. 18 and 19, the surface resistance of the reference crop (clipped grass 0.12-m high) is: r_s^rc = 200 / (24 × 0.12) = 69.4 s/m.

The leaf-area index for alfalfa is:

in which crop height h_c is in m, varying in the range 0.1 ≤ h_c ≤ 0.5.

From Eq. 20, for h_c = 0.3 m, the leaf-area 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 Penman-Monteith method for the month of April, for the following atmospheric conditions: air temperature T_a = 20°C; net radiation Q_n = 550 cal/(cm²-d); wind speed v₂ = 200 km/d; and relative humidity φ = 70%. Assume standard atmospheric pressure.

Solution.

• The saturation vapor pressure gradient is (Ponce, 2014: Section 2.2):

  Δ = (0.00815 T_a  +  0.8912)⁷ = 1.447 mb/°C.

• The net radiation in evaporation rate units is (Ponce, 2014: Section 2.2):

  E_n = Q_n / ( ρ λ ) = 550 / (0.99821 × 586) = 0.94 cm/d

• Using Eq. 16, the external resistance of the reference crop is:

  r_a^rc = (208 × 86400) / (200 × 1000) = 90 s/m = 0.9 s/cm

• The internal resistance of the reference crop is:

  r_s^rc = 200 / (24 x 0.12) = 69.4 s/m = 0.694 s/cm

• From Eq. 11, the psychrometric constant is:

  γ = (0.2402 × 1013.25 ) / (586 × 0.622) = 0.668

• From Eq. 3, the modified psychrometric constant is:

  γ * = 0.668 [ 1 + (0.694 / 0.9 ) ] = 1.18

• The vapor pressure deficit is (Ponce, 2014: Section 2.2):

  (e_s - e_a) ≅ (e_o - e_a) = e_o [ 1 - (φ / 100) ] = 23.37 [ 1 - (70 / 100)] = 7.01 mb.

• From Eq. 14, the mass-transfer evaporation rate is:

  E_a = ( 0.064 × 7.01 ) / ( 0.9 + 0.694 ) = 0.2815 cm/d.

• From Eq. 7, the total evaporation rate is:

  E = [ ( 1.447 × 0.94 ) + ( 1.18 × 0.2815 ) ]  /  ( 1.447 + 1.18 ) = 0.644 cm/d.

• The evaporation rate for the month of April is: E = 30 × 0.644 = 19.32 cm.

ONLINE CALCULATION. Using ONLINE PENMAN MONTEITH, the answer is: Daily reference crop PET = 0.644 cm/d; monthly reference crop PET (April) = 19.32 cm. These results are the same as the hand calculation shown above.

REFERENCES

Maidment, D. R. 1993. Handbook of Hydrology. McGraw-Hill.

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.