The goal of this post will be to derive the Penman-Monteith equation with isothermal radiation balance. We begin with an energy balance:

(1)

Assuming that we can set to 0. We then develop equations for latent heat transfer using the Ohm’s law analog for latent heat transfer and the substitution , we get:

(2)

We can also describe sensible heat transfer using the Ohm’s law analog for sensible heat transfer:

(3)

Equations 1, 2, and 3 are the three equations used to derive the Penman-Monteith equation (isothermal radiation will be added later). In order to eliminate from the equation, we can use a Taylor Series linearization of the vapor pressure gradient in the latent heat transfer equation, and solve for :

(4)

where and . Plugging back into Eq. 2 and solving for , we get

(5)

This equation for can then be plugged into the equation for sensible heat transfer, and the resulting equation for H can be plugged into the energy balance and rearranged to get the final Penman-Monteith equation.

(6)

(7)

And the final Penman-Monteith equation becomes:

(8)