Penman equation
Encyclopedia
The Penman equation describes evaporation
(E) from an open water surface, and was developed by Howard Penman in 1948. Penman's equation requires daily mean temperature
, wind speed
, relative humidity
, and solar radiation to predict E. Simpler Hydrometeorological equations continue to be used where obtaining such data is impractical, to give comparable results within specific contexts, e.g. humid vs arid climates.
from water, and land. Specifically the Penman-Monteith
equation refines weather based potential evapotranspiration (PET) estimates of vegetated land areas. 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:
where:
which (if the SI units in parentheses are used) will give the evaporation Emass 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 λv with L to get familiar precipitation units ETvol, where Lv=λvρwater. 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 Rn 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, cp, ρ, and δe.
, which made calculating evaporation simpler. The resultant equation is:
where:
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
Therefore
, mmHg/K
Evaporation
Evaporation is a type of vaporization of a liquid that occurs only on the surface of a liquid. The other type of vaporization is boiling, which, instead, occurs on the entire mass of the liquid....
(E) from an open water surface, and was developed by Howard Penman in 1948. Penman's equation requires daily mean temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
, wind speed
Wind speed
Wind speed, or wind velocity, is a fundamental atmospheric rate.Wind speed affects weather forecasting, aircraft and maritime operations, construction projects, growth and metabolism rate of many plant species, and countless other implications....
, relative humidity
Relative humidity
Relative humidity is a term used to describe the amount of water vapor in a mixture of air and water vapor. It is defined as the partial pressure of water vapor in the air-water mixture, given as a percentage of the saturated vapor pressure under those conditions...
, and solar radiation to predict E. Simpler Hydrometeorological equations continue to be used where obtaining such data is impractical, to give comparable results within specific contexts, e.g. humid vs arid climates.
Details
Numerous variations of the Penman equation are used to estimate evaporationEvaporation
Evaporation is a type of vaporization of a liquid that occurs only on the surface of a liquid. The other type of vaporization is boiling, which, instead, occurs on the entire mass of the liquid....
from water, and land. Specifically the Penman-Monteith
Penman-Monteith
Like the Penman equation, the Penman–Monteith equation predicts net evapotranspiration, requiring as input daily mean temperature, wind speed, relative humidity, and solar radiation...
equation refines weather based potential evapotranspiration (PET) estimates of vegetated land areas. 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
Rothamsted Experimental Station
The Rothamsted Experimental Station, one of the oldest agricultural research institutions in the world, is located at Harpenden in Hertfordshire, England. It is now known as Rothamsted Research...
, Harpenden, UK.
The equation for evaporation given by Penman is:
where:
- m = Slope of the saturation vapor pressureVapor pressureVapor pressure or equilibrium vapor pressure is the pressure of a vapor in thermodynamic equilibrium with its condensed phases in a closed system. All liquids have a tendency to evaporate, and some solids can sublimate into a gaseous form...
curve (Pa K-1) - Rn = Net irradianceIrradianceIrradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...
(W m-2) - ρa = densityDensityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of air (kg m-3) - cp = heat capacityHeat capacityHeat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...
of air (J kg-1 K-1) - ga = momentum surface aerodynamic conductance (m s-1)
- δe = vapor pressureVapor pressureVapor pressure or equilibrium vapor pressure is the pressure of a vapor in thermodynamic equilibrium with its condensed phases in a closed system. All liquids have a tendency to evaporate, and some solids can sublimate into a gaseous form...
deficit (Pa) - λv = latent heat of vaporization (J kg-1)
- γ = psychrometric constantPsychrometric constantThe psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. This lets one interpolate actual vapor pressure from paired dry and wet thermometer bulb temperature readings....
(Pa K-1)
which (if the SI units in parentheses are used) will give the evaporation Emass 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 λv with L to get familiar precipitation units ETvol, where Lv=λvρwater. 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 Rn with and A for total net available energy when a situation warrants account of additional heat fluxes.
temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
, wind speed
Wind speed
Wind speed, or wind velocity, is a fundamental atmospheric rate.Wind speed affects weather forecasting, aircraft and maritime operations, construction projects, growth and metabolism rate of many plant species, and countless other implications....
, relative humidity
Relative humidity
Relative humidity is a term used to describe the amount of water vapor in a mixture of air and water vapor. It is defined as the partial pressure of water vapor in the air-water mixture, given as a percentage of the saturated vapor pressure under those conditions...
impact the values of m, g, cp, ρ, and δe.
Shuttleworth (1993)
In 1993, W.Jim Shuttleworth modified and adapted the Penman equation to use SISi
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
, which made calculating evaporation simpler. The resultant equation is:
where:
- Emass = Evaporation rate (mm day-1)
- m = Slope of the saturation vapor pressureVapor pressureVapor pressure or equilibrium vapor pressure is the pressure of a vapor in thermodynamic equilibrium with its condensed phases in a closed system. All liquids have a tendency to evaporate, and some solids can sublimate into a gaseous form...
curve (kPa K-1) - Rn = Net irradianceIrradianceIrradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...
(MJ m-2 day-1) - γ = psychrometric constantPsychrometric constantThe psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. This lets one interpolate actual vapor pressure from paired dry and wet thermometer bulb temperature readings....
=
(kPa K-1) - U2 = wind speed (m s-1)
- δe = vapor pressureVapor pressureVapor pressure or equilibrium vapor pressure is the pressure of a vapor in thermodynamic equilibrium with its condensed phases in a closed system. All liquids have a tendency to evaporate, and some solids can sublimate into a gaseous form...
deficit (kPa) - λv = 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
Some useful relationships
- δe = (es - ea) = (1-relative humidityRelative humidityRelative humidity is a term used to describe the amount of water vapor in a mixture of air and water vapor. It is defined as the partial pressure of water vapor in the air-water mixture, given as a percentage of the saturated vapor pressure under those conditions...
)es - es = saturated vapor pressure of air, as is found inside plant stoma.
- ea = vapor pressure of free flowing air.
- es, mmHg = exp(21.07-5336/Ta), approximation by Merva, 1975
Therefore
, mmHg/K
- Ta = air temperature in kelvins
See also
- Pan evaporationPan evaporationPan evaporation is a measurement that combines or integrates the effects of several climate elements: temperature, humidity, rain fall, drought dispersion, solar radiation, and wind. Evaporation is greatest on hot, windy, dry days; and is greatly reduced when air is cool, calm, and humid...
- EvapotranspirationEvapotranspirationEvapotranspiration is a term used to describe the sum of evaporation and plant transpiration from the Earth's land surface to atmosphere. Evaporation accounts for the movement of water to the air from sources such as the soil, canopy interception, and waterbodies...
- Thornthwaite model
- Blaney-Criddle equationBlaney-Criddle equationThe Blaney–Criddle equation is a method for estimating reference crop evapotranspiration.- Usage :...
- Penman-MonteithPenman-MonteithLike the Penman equation, the Penman–Monteith equation predicts net evapotranspiration, requiring as input daily mean temperature, wind speed, relative humidity, and solar radiation...