Penman equation

Numerous variations of the Penman equation are used to estimate evaporation from water, and land. Specifically the Penman–Monteith equation refines weather based potential evapotranspiration (PET) estimates of vegetated land areas.[1] It is widely regarded as one of the most accurate models, in terms of estimates.

The original equation was developed by Howard Penman at the Rothamsted Experimental Station, Harpenden, UK.

The equation for evaporation given by Penman is:

${\displaystyle E_{\mathrm {mass} }={\frac {mR_{n}+\rho _{a}c_{p}\left(\delta e\right)g_{a}}{\lambda _{v}\left(m+\gamma \right)}}}$

where:

m = Slope of the saturation vapor pressure curve (Pa K⁻¹)

R_n = Net irradiance (W m⁻²)

ρ_a = density of air (kg m⁻³)

c_p = heat capacity of air (J kg⁻¹ K⁻¹)

δe = vapor pressure deficit (Pa)

g_a = momentum surface aerodynamic conductance (m s⁻¹)

λ_v = latent heat of vaporization (J kg⁻¹)

γ = psychrometric constant (Pa K⁻¹)

which (if the SI units in parentheses are used) will give the evaporation E_mass in units of kg/(m²·s), kilograms of water evaporated every second for each square meter of area.

Remove λ to obviate that this is fundamentally an energy balance. Replace λ_v with L to get familiar precipitation units ET_vol, where L_v=λ_vρ_water. This has units of m/s, or more commonly mm/day, because it is flux m³/s per m²=m/s.

This equation assumes a daily time step so that net heat exchange with the ground is insignificant, and a unit area surrounded by similar open water or vegetation so that net heat & vapor exchange with the surrounding area cancels out. Some times people replace R_n with and A for total net available energy when a situation warrants account of additional heat fluxes.

Temperature, wind speed, relative humidity impact the values of m, g, c_p, ρ, and δe.

In 1993, W.Jim Shuttleworth modified and adapted the Penman equation to use SI, which made calculating evaporation simpler.[2] The resultant equation is:

${\displaystyle E_{\mathrm {mass} }={\frac {mR_{n}+\gamma *6.43\left(1+0.536*U_{2}\right)\delta e}{\lambda _{v}\left(m+\gamma \right)}}}$

where:

E_mass = Evaporation rate (mm day⁻¹)

m = Slope of the saturation vapor pressure curve (kPa K⁻¹)

R_n = Net irradiance (MJ m⁻² day⁻¹)

γ = psychrometric constant = ${\displaystyle {\frac {0.0016286*P_{kPa}}{\lambda _{v}}}}$ (kPa K⁻¹)

U₂ = wind speed (m s⁻¹)

δe = vapor pressure deficit (kPa)

λ_v = latent heat of vaporization (MJ kg⁻¹)

Note: this formula implicitly includes the division of the numerator by the density of water (1000 kg m⁻³) to obtain evaporation in units of mm d⁻¹

δe = (e_s - e_a) = (1 – relative humidity) e_s

e_s = saturated vapor pressure of air, as is found inside plant stoma.

e_a = vapor pressure of free flowing air.

e_s, mmHg = exp(21.07-5336/T_a), approximation by Merva, 1975[3]

Therefore ${\displaystyle m=\Delta ={\frac {de_{s}}{dT_{a}}}={\frac {5336}{T_{a}^{2}}}e^{\left(21.07-{\frac {5336}{T_{a}}}\right)}}$, mmHg/K

T_a = air temperature in kelvins

• Pan evaporation
• Evapotranspiration
• Thornthwaite model
• Blaney–Criddle equation
• Penman–Monteith equation