PENMAN-MONTEITH METHOD

• The enhanced physical basis and considerable flexibility of the combination models (such as Penman's) have resulted in their extensive use in the calculation of both free surface water evaporation and potential evapotranspiration.

• An advanced combination model which is gaining wide acceptance is the Penman-Monteith equation.

• Unlike the original Penman model, in the Penman-Monteith equation 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:

  ρλE = [ΔH + ρa cp (es - ea) (ra-1)] / (Δ + γ*)

  (2-50)

  in which:

  • ρλE = total evaporative energy flux (cal cm^-2 s^-1);

  • Δ = saturation vapor pressure gradient with temperature [mb (^oC)^-1];

  • H = energy flux supplied externally, by net radiation (cal cm^-2 s^-1);

  • ρ_a = density of moist air (gr cm^-3);

  • c_p = specific heat of moist air [cal gr^-1 (^oC ^-1)];

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

  • r_a = external (aerodynamic) resistance (s cm^-1); and

  • γ* = modified psychrometric constant [mb (^oC^-1)], equal to

    γ* = γ [1 + (rs /ra)]

    (2-51)

    in which

  • r_s = internal (stomatal or surface) resistance (s cm^-1).

• The quantity r_a^-1 is the external conductance, in cm³ of air per cm² of surface per second (cm s^-1)

• In evaporation rate units, Eq. 2-50 can be expressed as follows:

  E = [ΔEn + ρa cp ρ-1λ-1 (es - ea) (ra-1)] / (Δ + γ*)

  (2-52)

  in which

  • E = evaporation rate (cm s^-1);

  • E_n = evaporation rate due to net radiation (cm s^-1);

  • ρ = density of water (gr cm^-3);

  • λ = heat of vaporization of water (cal gr^-1);

    and

  • Δ, ρ_a, c_p, (e_s - e_a), and r_a are in the same units as Eq. 2-50.

• The density of dry air at sea level is: ρ_ad = 1.2929 kg/m³.

  The density of moist air can be approximated as:

  ρ_a = ρ_ad [273/(273 + T)], where T is the air temperature (^oC).

  For instance, at 20^oC and standard atmospheric pressure:

  ρ_a = 0.0012 gr cm^-3.

  The specific heat of moist air is:

  c_p = [1.013 J gr^-1 (^oC ^-1) ]/4.186 J/cal) = 0.242 cal gr^-1 (^oC ^-1);

  At 20^oC:

  ρ = 0.998 gr cm^-3; and

  λ = 586 cal gr^-1.

• Eq. 2-52 reduces to:

  E = [ΔEn + 86400 (0.497 x 10-6) (es - ea) (ra-1)] / (Δ + γ*)

  (2-53)

  in which

  • E = evaporation rate (cm d^-1);

  • E_n = evaporation rate due to net radiation (cm d^-1); and

  • Δ, γ*, (e_s - e_a), and r_a are in the same units as Eq. 2-50.

• Equation 2-53 can be conveniently expressed in Penman form (Eq. 2-36) as follows:

  E = [ΔEn + γ*Ea] / (Δ + γ*)

  (2-54)

  in which

  Ea = 86400 K (es - ea) (ra + rs)-1

  (2-55)

  with

  E_a = mass-transfer evaporation rate (cm d^-1);

  and (e_s - e_a), r_a and r_s are in the same units as Eq. 2-50 and 2-51, and

  K = ρa cp ρ-1 λ-1 γ-1

  (2-56)

  is a constant expressed per unit of vapor pressure deficit (mb^-1), varying with temperature and atmospheric pressure.

• The psychrometric constant γ is equal to:

  γ = 0.389 (cal gr^{-1 o}C^-1) * [atmospheric pressure (mb)] / [heat of vaporization at the air temperature λ (cal/gr)]    {mb (^oC^-1)}

• At 20^oC and standard atmospheric pressure,

  γ = 0.389 [1013.2]/[586] = 0.673 mb (^oC^-1), and:

  K = [(0.497 X 10^-6) γ^-1] mb^-1 = 0.738 X 10^-6 mb^-1.

• Taking K' = 86400 K, Eq. 2-55 reduces to:

  Ea = K' (es - ea) (ra + rs)-1

  (2-57)

• Thus, at 20^oC and standard atmospheric pressure, K' = 0.0638 mb^-1 s d^-1.

• The external (or aerodynamic) resistance r_a varies with the surface roughness (water, soil, or vegetation) and is inversely proportional to wind speed.

• In other words, the external conductance (and thus, the evaporation rate) increases with wind speed, as postulated by Dalton (Eq. 2-27).

• The external resistance for evaporation from open water can be estimated as follows:

  ra = 4.72 [ln (zm/zo)]2 / (1 + 0.536 v2)

  (2-58)

  in which

  • r_a is given in s m^-1;

  • z_m = height at which meteorological variables are measured (m),

  • z_o = aerodynamic roughness of the surface (m), and

  • v₂ = wind speed (m s^-1), measured at the 2-m height.

• The external resistance r_a (s m^-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:

  rarc = 208 / v2

  (2-59)

• The internal (stomatal or surface) resistance is inversely proportional to the leaf-area index L, i.e., the projected area of vegetation per unit ground area:

  rs = 200 / L

  (2-60)

  in which r_s is given in s m^-1, and

  L = leaf-area index, which is empirically related to crop height.

• For clipped grass, L = 24 h_c (m s^-1),

  with height in the range 0.05 < h_c < 0.15 m.

• For alfalfa, L = 5.5 + 1.5 ln h_c,

  with 0.1 < h_c < 0.5 m.

• From Eq. 2-60, the stomatal resistance of the reference crop (clipped grass 0.12-m high) is:

• r_s^rc = 69 s m^-1.

• Likewise, for an alfalfa crop with h_c = 0.3 m: r_s = 54 s m^-1.

Example 2-8. Calculate the evaporation rate of the reference crop by the Penman-Monteith method for the same atmospheric conditions as Example 2.5 in the textbook. Assume standard atmospheric pressure.

• From Eq. 2-37, the saturation vapor pressure gradient is: Δ = 1.45 mb (^oC^-1).

• From Example 2-5, the net radiation in evaporation rate units is E_n = 0.94 cm/d.

• From Eq. 2-59, the aerodynamic resistance is: r_a^rc = (208 X 86400)/(200 X 1000) = 89.9 s m^-1 = 0.9 s cm^-1.

• The stomatal resistance of the reference crop is: r_s^rc = 69 s m^-1 = 0.69 s cm^-1.

• From Eq. 2-51, the modified psychrometric constant is:

  γ* = 0.673 [1 + (0.69/0.90)] = 1.189.

• The vapor pressure deficit is:

  (e_s - e_a) ≅ (e_o - e_a) = e_o [1 - (RH/100)] = 23.37 [1 - (70/100)] = 7 mb.

• From Eq. 2-56, K = 0.738 X 10^-6 mb^-1, and thus, K' = 0.0638 mb^-1 s d^-1.

• From Eq. 2-57, the mass-transfer evaporation rate is: E_a = 0.28 cm d^-1.

• From Eq. 2-54, the evaporation rate is: E = 0.64 cm d^-1.