Copyright
– Regents of the University of California

Created
January 25, 2001 - Last Update June 2, 2002

These
notes contain two methods for determining the Penman-Monteith equation. The first is the flux gradient and the
second is a psychrometric derivation.
The flux gradient method is simpler to derive, but the
psychrometric
method is more conceptually understandable.
In the end, a discussion of weather effects on evaporation from
wet
surfaces and other equations are discussed.

Evaporation
rate from a wet surface is determined using the flux gradient approach
for
estimating sensible and latent heat flux density. Recall that sensible
heat
flux density is estimated as

_{}
(1)

where
*T* is the air temperature, *T _{o}*
is the surface
temperature, and

_{}
(2)

where
_{} is the resistance to
sensible heat flux and *T _{o}*
is the surface temperature. The
variable

_{}
(3)

Assuming
that *T _{w}* »

_{}
(4)

Substituting
for *e _{s}*(

_{}
(5)

However,
recall that *H* is a function of *T *- *T _{o}*. Rearranging
Eq. 1, we can express

_{}
(6)

Substituting
this expression into Eq. 4, we get

_{}
(7)

Rearranging
terms, we get

_{}
(8)

For
a non-adiabatic process, where *Q _{n}*
is the net source of external energy flux,

_{}
(9)

or

_{}
(10)

Moving
*l**E* to one side of the
equation, we get

_{}
(11)

but

_{}
(12)

so *l**E *is

_{}
(13)

This
is the Penman-Monteith equation for evaporation from a wet surface. Note that the left-hand term in the
numerator is the diabatic contribution and the right-hand term is the
adiabatic
contribution to evaporation from a wet surface.

*l**E*** for an amphistomatous leaf
(stomata on one surface)**

Recall
that the resistances to latent and sensible heat transfer from a wet
surface
are different and a correction for the psychrometric constant is

_{}
(14)

Ignoring
the small transfer through the cuticle, the resistance to vapor
transfer for a
leaf can be separated into the stomatal resistance to transfer from the
cell
surfaces, inside the leaf, to the stomata openings (*r _{s}*)
and the aerodynamic resistance from the leaf surface
through the boundary layer (

_{}
(15)

Then
Eq. 13 can be written as

_{}
(16)

Equation
16 is the Penman-Monteith equation for a leaf.

*l**E*** from a canopy**

For
a canopy, the resistance to vapor transfer from the canopy to the
ambient air
above can be separated into aerodynamic resistance from a fictitious
level in
the canopy to the air above (*r _{w}*)
and the canopy resistance (

_{}
(17)

If an
independent measure of evaporation is available, the evaporation is not
reduced
by water stress, soil evaporation is negligible, and the aerodynamic
resistance
is known, then the Penman-Monteith equation:

_{}
(18)

provides
a method to evaluate differences in plant stomatal effect on
evaporation from
canopies.

The
difference in the enthalpy (total energy content) of the ambient air
and air at
a wet surface occurs as a result of energy transfer to the surface that
affects
the sensible and latent heat content of the air. If the rate of
adiabatic and
diabatic heat transfer is known, then psychrometric relationships can
be used
to estimate the latent heat flux density.
Using a psychrometric chart (Fig. 1), the rate of energy supply
to
increase the vapor pressure from *e* at
point A to *e _{s}*(

Figure
1. Adiabatic process to raise vapor pressure from *e* to *e _{s}*(

Based on the psychrometric
relationships shown in Fig. 2, it is clear that

*e _{s}*(

and

*e _{s}*(

where
_{} accounts for the
difference in resistance to sensible and latent heat flux.

Figure
2. Energy partitioning in a diabatic process

Because

_{}
(21)

we
know that

_{}
(22)

and
by rearranging, we get

_{}
(23)

Since
the energy required to increase vapor pressure from *e*
to *e _{s}*(

_{}
(24)

By
substitution, the latent heat flux density due to adiabatic heat
transfer (*l**E _{a}*) is

_{}
(25)

For a diabatic process [e.g., increasing
vapor
pressure from *e _{s}*(

Figure
3. Enthalpy change due to heating and evaporation from a wet surface.

Part of the external energy supply
contributes to
evaporation and the remainder increases temperature.
The amount of energy needed to increase the temperature from *T _{w}* to

_{}
(26)

Dividing
the numerator and denominator by (*T _{o}*
–

_{}
(27)

If
we simplify the ratio, we get the fraction of total external energy
supply
going to vaporization _{}. Therefore, the rate
of diabatic contribution to latent heat flux density (*l**E _{d}*) is

_{}
(28)

* *

The
sum of the diabatic and adiabatic contributions provides an estimate of
the
latent heat flux density

_{}
(29)

Clearly,
this is identical to the Penman-Monteith equation for a leaf (Eq. 13),
which
was derived using flux gradient concepts.
The same steps (Eqs. 14-17) are used to arrive at the
Penman-Monteith
equation for a canopy (Eq. 18). Remember
that _{} is simply an estimate
for *T - T _{w}* where