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

(1)   \begin{equation*}  \begin{aligned} \Delta U & = R_n - \lambda E - H - G \\ 0 & = R_n - \lambda E - H - G,  \end{aligned} \end{equation*}

Assuming that we can set \Delta U to 0. We then develop equations for latent heat transfer using the Ohm’s law analog for latent heat transfer and the substitution \gamma = \frac{C_P P}{\lambda \frac{MW_W}{MW_A}}, we get:

(2)   \begin{equation*}  \begin{aligned} \lambda E & = (e_s(T_s)-e_a) \frac{\rho_a MW_W}{P MW_a} \lambda G_W\\ \lambda E & = (e_s(T_s) - e_a) \frac{C_p \rho_a}{\gamma} G_W \end{aligned} \end{equation*}

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

(3)   \begin{equation*}  H = (T_s - T_a) \rho_a C_p G_H \end{equation*}

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 T_s - T_a from the equation, we can use a Taylor Series linearization of the vapor pressure gradient in the latent heat transfer equation, and solve for T_s - T_a:

(4)   \begin{equation*}  \begin{aligned} e_s(T_s) & = e_s(T_a) + \frac{de_s}{dT} (T_s - T_a) \\ e_s(T_s) - e_a & = e_s(T_a) - e_a + \frac{de_s}{dT} (T_s - T_a) \\ e_s(T_s) - e_a & = D + s (T_s - T_a) \end{aligned} \end{equation*}

where s = \frac{de_s}{dT} and D = e_s(T_a) - e_a. Plugging back into Eq. 2 and solving for \Delta T = T_s - T_a, we get

(5)   \begin{equation*}  \begin{aligned} \lambda E & = [ D + s (T_s - T_a)] \frac{C_p \rho_a}{\gamma} G_W \\ \Delta T & = (\lambda E \frac{\gamma}{C_p \rho_a G_W} - D) \frac{1}{s} \end{aligned} \end{equation*}

This equation for \Delta T 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)   \begin{equation*} H = \left[(\lambda E \frac{\gamma}{C_p \rho_a G_W} - D) \frac{1}{s} \right] \rho_a C_p G_H \end{equation*}

(7)   \begin{equation*}  \begin{aligned} 0 & = R_n - G - \lambda E - \left[(\lambda E \frac{\gamma}{C_p \rho_a G_W} - D) \frac{1}{s} \right] \rho_a C_p G_H \\ \lambda E \left[s + (\frac{\gamma}{C_p \rho_a G_W}) \rho_a C_p G_H \right] & = s(R_n - G)+ D  \rho_a C_p G_H \end{aligned} \end{equation*}

And the final Penman-Monteith equation becomes:

(8)   \begin{equation*}  \lambda E   & = \frac{s(R_n - G)+ D  \rho_a C_p G_H}{s + \gamma \frac{G_H}{G_W}} \end{equation*}