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2013-10-21T15:53:38Z

78.53.30.87: /* Representation theory */

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 [[Evapotranspiration#Hydrometeorological_equations|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 [[evaporation]] from water, and land. Specifically the [[Penman-Monteith]] equation refines weather based [[Evapotranspiration#Potential_evapotranspiration|potential evapotranspiration]] (PET) estimates of vegetated land areas.<ref>{{cite book |last=Allen |first=R.G. |coauthors=Pereira, L.S.; Raes, D.; Smith, M. |title=Crop Evapotranspiration—Guidelines for Computing Crop Water Requirements |url=http://www.fao.org/docrep/X0490E/x0490e00.HTM |accessdate=2007-10-08 |series=FAO Irrigation and drainage paper 56 |year=1998 |publisher=Food and Agriculture Organization of the United Nations |location=Rome, Italy |isbn=92-5-104219-5 }}</ref> It is widely regarded as one of the most accurate models, in terms of estimates.{{Citation needed|date=February 2007}}

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

The equation for evaporation given by Penman is:
:<math>E_{mass}=\frac{m R_n + \rho_a c_p \left(\delta e \right) g_a }{\lambda_v \left(m + \gamma \right) }
</math>
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)
:''g''a = momentum surface aerodynamic conductance (m s-1)
:δ''e'' = [[vapor pressure]] deficit (Pa)
:''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''v''ρ''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 δ''e''.

==Shuttleworth (1993)==
In 1993, W.Jim Shuttleworth modified and adapted the Penman equation to use [[SI]], which made calculating evaporation simpler.<ref>Shuttleworth, J., Putting the vap' into evaporation http://www.hydrol-earth-syst-sci.net/11/210/2007/hess-11-210-2007.pdf</ref> The resultant equation is:

:<math>E_{mass}=\frac{m R_n + \gamma * 6.43\left(1+0.536 * U_2 \right)\delta e}{\lambda_v \left(m + \gamma \right) }
</math>

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]] = <math>\frac{0.0016286 * P_{kPa}} {\lambda_v}</math> (kPa K-1)
:''U''2 = wind speed (m s-1)
:δ''e'' = [[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

==Some useful relationships==
:δ''e'' = (es - ea) = (1-[[relative humidity]])es
:''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<ref>Merva, G.E. 1975. Physio-engineering Principles. AVI Publishing Company, Westport, CT.</ref>
Therefore <math>m= \Delta =\frac{d e_s}{d T_a} = \frac{5336}{T_a^2} e^{\left(21.07 - \frac{5336}{T_a}\right)}</math>, mmHg/K
:''T''a = air temperature in kelvins

==See also==
*[[Pan evaporation]]
*[[Evapotranspiration]]
*[[Thornthwaite model]]
*[[Blaney-Criddle equation]]
*[[Penman-Monteith]]

==Notes==
{{Reflist}}

==References==
{{refbegin}}
* Jarvis, P.G. (1976) The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field. Phil. Trans. R. Soc. Lond. B. 273, 593-610.
* Neitsch, S.L.; J.G. Arnold; J.R. Kliniry; J.R. Wolliams. 2005. Soil and Water Assessment Tool Theoretical Document; Version 2005. Grassland, Soil and Water Research Laboratory; Agricultural Research Service. and Blackland Research Center; Texas Agricultural Experiment Station. Temple, Texas. http://www.brc.tamus.edu/swat/downloads/doc/swat2005/SWAT%202005%20theory%20final.pdf
* Penman, H.L. (1948): ''Natural evaporation from open water, bare soil and grass.'' Proc. Roy. Soc. London A(194), S. 120-145.
{{refend}}

{{DEFAULTSORT:Penman Equation}}
[[Category:Agronomy]]
[[Category:Equations]]
[[Category:Hydrology]]

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