1. Derivation of Penman Equation for estimating water evaporation from an open pan
The Penman equation for estimating evaporation from an open pan of water was essentially derived
from an energy balance that focuses on net radiation input (including solar and long-wave radiation)
and convective heat exchange between the water and the atmosphere. Heat exchanged with the
environment through the pan is ignored. This energy balance can be written as follows:
K + L - H = Eλvρw EQ. 1
where: K= short wave net radiation input (cal/cm2
day)
L = long wave net radiation input (cal/cm2
day)
H = convective heat exchange from the water to the atmosphere (cal/cm2
day)
E= quantity of water evaporated (cm/day)
λv = heat of vaporization (cal/gm)
ρw= density of water (g/cm3
)
Radiation inputs can be measured or estimated in a variety of ways. The derivation of the Penman
equation centers around the connective heat transfer from the water body to the atmosphere, H. If the
water surface temperature (Ts) is known, H can be calculated from the following equation:
H = KH va (Ts –Ta) EQ. 2
Where: KH = convective heat transfer coefficient
va = velocity of air
Ts = temperature of the water surface
Ta = temperature of the atmosphere
However, it is relatively rare to have surface water temperature data. The need for water temperature
data can be circumvented by using an approximation of the slope of the relationship between
temperature and saturated water vapor pressure at atmospheric temperature, Δ. The slope of the
relationship between water vapor pressure and temperature can be approximated as
Δ= e*s – e*a EQ 3
Ts – Ta
Where:
Δ= slope of the relationship between water vapor pressure and temperature at as specific temperature
e*s = saturation vapor pressure at Ts
e*a = saturation vapor pressure at Ta
For a given temperature Δ can be calculated from the following equation:
Δ = 2508.3 • exp 17.3 T
EQ 3-b
(T +237.3)2
(T +237.3) (same as Eq. 7-6 in Dingman)
1

2. Equation (3) can be re-arranged as follows:
Ts – Ta = e*s – e*a EQ 4
Δ
And the right-hand side can be substituted into equation (2) to give
H = KH va {e*s – e*a} EQ 5
Δ
If we add and subtract atmospheric water vapor pressure, ea , within the brackets of EQ 5, we obtain
H = KH va {e*s – e*a + ea –ea } EQ 6
Δ
And we can rearrange EQ 6 to obtain:
H = KH va {e*s –ea } – KH va {e*a –ea } EQ 7
Δ Δ
According to the mass transfer approach to evaporation,
E = KE va {e*s –ea} EQ 8
And this equation can be rewritten as
{e*s –ea } = E N EQ 9
KE va
We can now substitute E N for {e*s – ea } in EQ 7 to obtain
KE va
H = KH va E – KH va {e*a –ea } EQ 10
Δ KE va Δ
We can now substitute this expression for H back into the original energy balance (EQ 1) to obtain:
Eλvρw = (K+L) – KH va E + KH va {e*a –ea} EQ 11
Δ KE va Δ
2

3. Equation 11 can be solved for evaporation, E, in three steps:
Eλvρw + KH E = (K+L) + KH va {e*a –ea} EQ 12
Δ KE Δ
E{λvρw+ KH } = (K+L) + KH va {e*a –ea} EQ 13
Δ KE Δ
E = (K+L) + KH va {e*a –ea}/Δ EQ 14
{λvρw+ KH / Δ KE}
If we multiply the numerator and denominator of the right hand side of Equation 14 by Δ, we obtain:
E = Δ (K+L) + KH va {e*a –ea} EQ 15
{Δ λvρw+ KH / KE}
This equation can be simplified further by recognizing the mathematical similarity between the
convective heat exchange coefficient KH and convective mass transfer coefficient KE:
KE = Dwv 0.622 ρa EQ 16
DM P ρw 6.25 {ln [(zm - zd)/zo]}2
KH = DH ca ρa EQ 17
DM 6.25 { ln [(zm - zd)/zo]}2
Where Dwv = diffusivity of water vapor in the atmosphere
ρa = density of air
DM = diffusivity of momentum in the atmosphere
P = atmospheric pressure
ρw = density of water
zm = height at which wind velocity is measured
zd = zero plane displacement = 0.7 vegetation height
zo = roughness height = 0.1 vegetation height
DH = diffusivity of heat in the atmosphere
ca = heat capacity of the atmosphere
In general, diffusivity is the proportionality constant between a rate of diffusion and the
appropriate concentration gradient. For instance, the diffusivity of water vapor in the atmosphere Dwv
is the proportionality constant between the rate of water vapor diffusion, and the water vapor
concentration gradient in the atmosphere.
Fwv = Dwv (d ea/ dz) EQ 18
Where Fwv = vertical flux (rate of upward movement) of water vapor
3

4. (d ea/ dz) = change in atmospheric water vapor pressure (ea ) with elevation (z).
Diffusivity of momentum and heat are similar to the diffusivity of water vapor except that the
gradient for momentum flux is velocity and the gradient for heat flux is temperature. The ratios of Dwv
/DM and DH /DM are approximately 1 under stable atmospheric conditions (air temperature declines
with elevation at the same rate as the temperature of rising air would decline due to reduced
atmospheric pressure). To simplify the calculation of KE and KH, it often assumed that atmospheric
stability predominates.
If we assume the diffusivity ratios are 1, then the ratio of KH /KE becomes:
KH = ca P ρw EQ 19
KE 0.622
This is very similar to the psychrometric constant, γ, which has been defined as
γ = ca P EQ 20
0.622 λv
and so the ratio of KH /KE can be rewritten as
KH = γλvρw EQ 21
KE
Substituting γλvρw for KH /KE in equation 14 gives
E = Δ(K+L) + KH va {e*a –ea} EQ 22
{Δ λvρw+ γλvρw }
which can be rearranged and simplified as:
E = Δ(K+L) + KH va {e*a –ea} EQ 23
λvρw {Δ+ γ}
We can also rearrange equation 21 to solve for KH and substitute γλvρw KE for KH to give
E = Δ(K+L) + γλvρw KE va {e*a –ea} EQ 24
λvρw {Δ+ γ}
Dividing all terms by γλv gives
E = Δ(K+L)/λvρw + γKE va {e*a –ea} EQ 25
{Δ+ γ}
Which is the now familiar form of the Penman Equation.
4

5. Monteith’s modification of the Penman equation to account for transpiration:
To account for the difference between pan evaporation and leaf transpiration it is useful to represent
the transport processes in terms of atmospheric conductance, which has been defined as:
Cat = va EQ 26
6.25 {ln [(zm - zd)/zo]}2
Both KH and KE can be expressed as functions of Cat:
KE = Dwv 0.622 ρa = Dwv 0.622 ρa Cat EQ 27
DM P ρw 6.25 {ln [(zm - zd)/zo]}2
DM P ρw va
KH = DH ca ρa = DH ca ρa Cat EQ 28
DM 6.25 {ln [(zm - zd)/zo]}2
DM va
Convective heat transfer from the vegetative canopy can be expressed as:
H = KH va (Ts –Ta) = DH ca ρa Cat (Ts –Ta) EQ. 29
DM va
Which can be simplified if we assume DH /DM =1.
Water vapor transport from the canopy is a two step process, as previously discussed, that involves
canopy conductance (Ccan) as well as atmospheric conductance (Cat). Since conductance is the
reciprocal of resistance, and the overall resistance to water vapor transport will be the sum of canopy
and atmospheric resistance, the overall conductance (Ct) is:
Ct = 1 = Cat EQ 30
1/Cat +1/Ccan (1+ Cat/Ccan)
Thus vapor transport from the vegetative canopy can be expressed as:
ET = KE va {e*s –ea } = Dwv 0.622 ρa Cat {e*s –ea} EQ 31
DM P ρw (1+ Cat/Ccan)
Which can be simplified by assuming Dwv /DM =1, and rewritten using γ = ca P/ 0.622 λv
and solved for {e*s – ea}as follows:
5

6. {e*s –ea}= ET γ λv ρw (1+ Cat/Ccan) EQ 32
ca ρa Cat
The right hand side of EQ 32 can then be substituted into Equation 7 and following the same steps
from Equation 10 to Equation 25 in the derivation of the Penman equation leads to the following
expression:
ET = Δ(K+L) + KH va {e*a – ea} EQ 33
λvρw {Δ+ γ (1+Cat/Can )}
For KH va we can substitute ca ρa Cat
ET = Δ(K+L) + ca ρa Cat {e*a – ea} EQ 34
λvρw {Δ+ γ (1+Cat/Can )}
Also note that relative humidity, Wa = ea/e*a, and, therefore, ea = Wa · e*a. Consequently, the term
{e*a – ea} = {e*a – Wa · e*a }. By factoring out e*a the we can also show
{e*a – ea} = {e*a – Wa · e*a }= {1– Wa}e*a.
Substituting the right hand expression into EQ 34 gives
ET = Δ(K+L) + ca ρa Cat {1– Wa}e*a. EQ 35
λvρw {Δ+ γ (1+Cat/Can )}
which expresses the Penman-Monteith equation in terms of atmospheric conductance and canopy
conductance. Data required to estimate ET include air temperature, net radiation input, wind speed,
vegetation height, relative humidity. Other parameters (e.g., Δ, ca , ρa) can be obtained from air
temperature. Like the Penman equation, it can be used to estimate instantaneous ET if instantaneous
measures of radiation, wind, relative humidity and air temperature are used. Using daily or monthly
average values of temperature, wind speed and relative humidity reduces the accuracy of the equation
because it would not capture some of the interactions between the parameters. ET can be very
different if high winds occur at night, when there is little radiation and relative humidity is high, as
opposed to during the day. In estimating water use by corn in the Midwest, the Illinois State Water
Survey uses a model (Ceres Maize) that uses the Penman-Monteith Equation with hourly
measurements of atmospheric data. Water use by corn is then used to estimate soil moisture
availability, which is then used as an indicator of drought severity.
References
Monteith, J.L and M.H. Unsworth. 1990. Principles of Environmental Physics. Edward Arnold, London.
Smugge, T.J and J.C. Andre. 1991. Land Surface Evaporation: Measurement and Parameterization.
Springer-Verlag.
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7. {e*s –ea}= ET γ λv ρw (1+ Cat/Ccan) EQ 32
ca ρa Cat
The right hand side of EQ 32 can then be substituted into Equation 7 and following the same steps
from Equation 10 to Equation 25 in the derivation of the Penman equation leads to the following
expression:
ET = Δ(K+L) + KH va {e*a – ea} EQ 33
λvρw {Δ+ γ (1+Cat/Can )}
For KH va we can substitute ca ρa Cat
ET = Δ(K+L) + ca ρa Cat {e*a – ea} EQ 34
λvρw {Δ+ γ (1+Cat/Can )}
Also note that relative humidity, Wa = ea/e*a, and, therefore, ea = Wa · e*a. Consequently, the term
{e*a – ea} = {e*a – Wa · e*a }. By factoring out e*a the we can also show
{e*a – ea} = {e*a – Wa · e*a }= {1– Wa}e*a.
Substituting the right hand expression into EQ 34 gives
ET = Δ(K+L) + ca ρa Cat {1– Wa}e*a. EQ 35
λvρw {Δ+ γ (1+Cat/Can )}
which expresses the Penman-Monteith equation in terms of atmospheric conductance and canopy
conductance. Data required to estimate ET include air temperature, net radiation input, wind speed,
vegetation height, relative humidity. Other parameters (e.g., Δ, ca , ρa) can be obtained from air
temperature. Like the Penman equation, it can be used to estimate instantaneous ET if instantaneous
measures of radiation, wind, relative humidity and air temperature are used. Using daily or monthly
average values of temperature, wind speed and relative humidity reduces the accuracy of the equation
because it would not capture some of the interactions between the parameters. ET can be very
different if high winds occur at night, when there is little radiation and relative humidity is high, as
opposed to during the day. In estimating water use by corn in the Midwest, the Illinois State Water
Survey uses a model (Ceres Maize) that uses the Penman-Monteith Equation with hourly
measurements of atmospheric data. Water use by corn is then used to estimate soil moisture
availability, which is then used as an indicator of drought severity.
References
Monteith, J.L and M.H. Unsworth. 1990. Principles of Environmental Physics. Edward Arnold, London.
Smugge, T.J and J.C. Andre. 1991. Land Surface Evaporation: Measurement and Parameterization.
Springer-Verlag.
6