 The Penman-Monteith Method Victor M. Ponce11 March 2014

 Abstract. The Penman-Monteith combination method for the calculation of evaporation is reviewed and clarified. Unlike the original Penman model, in the Penman-Monteith model the mass-transfer evaporation rate is calculated based on physical principles. An illustrative example is worked out to clarify the computational procedure. An online calculation using ONLINE PENMAN-MONTEITH gives the same answer.

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):

 Δ En  +  γ Ea E  =   __________________                      Δ  +  γ (1)

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.

2.  PENMAN-MONTEITH MODEL

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:

 Δ H  +  [ ρa cp (es  -  ea) / ra ] ρλE  =   ________________________________                                  Δ  +  γ * (2)

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:

 rs γ *  =   γ   ( 1  +  _____ )                              ra (3)

in which:

• γ = psychrometric constant, in mb/°C, which varies slightly with temperature, 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 * (4)

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;

and

• Δ, γ *, ρa, cp, (es  -  ea), and ra  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: ρad = 1.2929 kg/m3. The density of moist air ρa may be approximated as follows:

 273 ρa  ≅   ρad  ( __________ )                        273 + T (5)

in which T = air temperature, in °C.

For instance, at T = 20°C and sea level (standard atmospheric pressure):

ρa = 0.0012046 gr/cm3.

The specific heat of moist air, in the range 0°C ≤ T ≤ 40°C, is:

cp = 1.005 J/(gr-°C)

Converting to calories:

cp = (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:

 Δ En  +  [ 86400 ρa cp (es  -  ea) / (ra ρ λ ) ] E  =   _________________________________________________                                             Δ  +  γ * (6)

in which

• E = total evaporation rate (cm/d);

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

• Δ, γ *, ρa, cp, (es  -  ea), ra, 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:

 Δ En  +  γ * Ea E  =   ___________________                     Δ  +  γ * (7)

in which Ea = 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:

 86400 ρa cp (es  -  ea) Ea  =   ___________________________                       ρ λ γ  (ra  +  rs) (8)

Simplifying Eq. 8:

 K  (es  -  ea) Ea  =   __________________                      ra  +  rs (9)

in which K = a constant varying with air temperature and atmospheric pressure, in units of s/(d-mb), expressed as follows:

 86400 ρa cp K  =   _________________                     ρ λ γ (10)

In Eq. 10, the units of ρa, cp, ρ, λ, and γ are the same as in Eqs. 2 and 4.

The psychrometric constant γ, in mb/°C, is:

 cp  p γ  =  __________               λ rMW (11)

in which cp = specific heat of moist air, in cal/(gr-°C); p = atmospheric pressure, in mb; λ = heat of vaporization, in cal/gr; and rMW = ratio of the molecular weight of water vapor to dry air: rMW = 0.622.

Substituting Eq. 11 in Eq. 10:

 86400 ρa rMW K  =   __________________                        ρ p (12)

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

Replacing rMW = 0.622 into Eq. 12:

 53740.8 ρa K  =   ________________                      ρ p (13)

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

At T = 20°C and standard atmospheric pressure (sea level): ρa = 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:

 0.064  (es  -  ea) Ea  =   _____________________                         ra  +  rs (14)

in which:

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

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

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

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

6.  EXTERNAL RESISTANCE

The external, or aerodynamic resistance ra 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:

 4.72 [ ln (zm / zo) ] 2 ra  =   ________________________                    1  +  0.536 v2 (15)

in which:

• ra = external resistance, in s/m;

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

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

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

The external resistance ra (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:

 208 rarc  =   _______                  v2 (16)

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:

 208 rarc  =   ________   =  90 s/m                 2.31 (17)

7.  INTERNAL RESISTANCE

The internal, or stomatal or surface resistance rs is inversely proportional to the leaf-area index L. An empirical relation for surface resistance is (Maidment, 1993):

 200 rs  ≅   _______                 L (18)

in which rs  is in s/m.

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

 L  =   24 hc (19)

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

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

The leaf-area index for alfalfa is:

 L  =   5.5 + 1.5 ln (hc) (20)

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

From Eq. 20, for hc = 0.3 m, the leaf-area index for alfalfa is: L = 3.69. From Eq. 18, the surface resistance for alfalfa is: rs = 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 Ta = 20°C; net radiation Qn = 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 Ta  +  0.8912)7 = 1.447 mb/°C.

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

En = Qn / ( ρ λ ) = 550 / (0.99821 × 586) = 0.94 cm/d

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

rarc = (208 × 86400) / (200 × 1000) = 90 s/m = 0.9 s/cm

• The internal resistance of the reference crop is:

rsrc = 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):

(es - ea) ≅ (eo - ea) = eo [ 1 - (φ / 100) ] = 23.37 [ 1 - (70 / 100)] = 7.01 mb.

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

Ea = ( 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.

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