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The enhanced physical basis and considerable flexibility of the combination models (such as Penman's) have resulted in their extensive use in the calculation of both free surface water evaporation and potential evapotranspiration.

An advanced combination model which is gaining wide acceptance is the Penman-Monteith equation.

Unlike the original Penman model, in the Penman-Monteith equation the mass-transfer evaporation rate ($E_a$) is calculated based on physical principles.

The original form of the Penman-Monteith equation, in dimensionally consistent units, is:

\rho \lambda E = \frac {\Delta H + \rho_a c_p r_a^{-1}(e_s - e_a)}{\Delta + \gamma^*}
\hspace{0.99in} (2-50)
\end{displaymath} (1)

in which:

$\rho\lambda$E = total evaporative energy flux (cal cm$^{-2}$ s$^{-1}$);

$\Delta$ = saturation vapor pressure gradient with temperature [mb ($^o$C)$^{-1}$];

H = energy flux supplied externally, by net radiation (cal cm$^{-2}$ s$^{-1}$);

$\rho_a$ = density of moist air (gr cm$^{-3}$);

$c_p$ = specific heat of moist air [cal gr$^{-1}$($^o$C)$^{-1}$];

($e_s$ - $e_a$) = vapor pressure deficit (mb);

$r_a$ = external (aerodynamic) resistance (s cm$^{-1}$); and

$\gamma*$ = modified psychrometric constant [mb ($^o$C)$^{-1}$], equal to

\gamma* = \gamma{\hspace{0.05in} [1 + \frac {r_s}{r_a}] \hspace{2.0in} (2-51)
\end{displaymath} (2)

in which

$r_s$ = internal (stomatal or surface) resistance (s cm$^{-1}$).

The quantity r$_a^{-1}$ is the external conductance, in cm$^3$ of air per cm$^2$ of surface per second (cm s$^{-1}$).

In evaporation rate units, Eq. 2-50 can be expressed as follows:

E = \frac {\Delta E_n + \rho_a c_p \rho^{-1} \lambda^{-1} r_a^{-1} (e_s - e_a)}{\Delta + \gamma^*}
\hspace{0.99in} (2-52)
\end{displaymath} (3)

in which

$E$ = evaporation rate (cm s$^{-1}$);

$E_n$ = evaporation rate due to net radiation (cm s$^{-1}$);

$\rho$ = density of water (gr cm$^{-3}$);

$\lambda$ = heat of vaporization of water (cal gr$^{-1}$);

$\Delta$, $\rho_a$, $c_p$, ($e_s$ - $e_a$), and $r_a$ are in the same units as Eq. 2-50.

For instance, at 20$^o$C and standard atmospheric pressure:

$\rho_a$ = 0.00118 gr cm$^{-3}$;

$c_p$ = 0.242 cal gr$^{-1}$($^o$C)$^{-1}$;

$\rho$= 0.998 gr cm$^{-3}$; and

$\lambda$ = 586 cal gr$^{-1}$.

Eq. 2-52 reduces to:

E = \frac {\Delta E_n + 86400 \hspace{0.05in}(0.49 \times 10...
...r_a^{-1}(e_s - e_a)}{\Delta + \gamma^*}
\hspace{0.99in} (2-53)
\end{displaymath} (4)

in which

E = evaporation rate (cm d$^{-1}$);

$E_n$ = evaporation rate due to net radiation (cm d$^{-1}$);

$\Delta$, $\gamma*$, ($e_s - e_a$), and $r_a$ are in the same units as Eq. 2-50.

Equation 2-53 can be conveniently expressed in Penman form (Eq. 2-36) as follows:

E = \frac {\Delta E_n + \gamma^* E_a}{\Delta + \gamma^*}
\hspace{0.99in} (2-54)
\end{displaymath} (5)

in which

E_a = 86400\hspace{0.05in} K \hspace{0.05in} \frac {e_s - e_a}{r_a + r_s}
\hspace{0.99in} (2-55)
\end{displaymath} (6)

with $E_a$ = mass-transfer evaporation rate (cm d$^{-1}$); and ($e_s - e_a$), $r_a$ and $r_s$ are in the same units as Eq. 2-50 and 2-51, and

K = \frac {\rho_a c_p}{\rho \lambda \gamma}
\hspace{0.99in} (2-56)
\end{displaymath} (7)

is a constant expressed per unit of vapor pressure deficit (mb$^{-1}$), varying with temperature and atmospheric pressure.

At 20$^o$C and standard atmospheric pressure, $\gamma$ = 0.667 mb ($^o$C)$^{-1}$, and:

$K$ = [(0.49 $\times$ 10$^{-7}$) $\gamma^{-1}$] mb$^{-1}$ = 0.735 $\times$ 10$^{-7}$ mb$^{-1}$.

Taking $K'$ = 86400 $K$, Eq. 2-55 reduces to:

E_a = K' \hspace{0.05in} \frac {e_s - e_a}{r_a + r_s}
\hspace{0.99in} (2-57)
\end{displaymath} (8)

Thus, at 20$^o$C and standard atmospheric pressure, $K'$ = 0.00635 mb$^{-1}$ s d$^{-1}$.

The external (or aerodynamic) resistance $r_a$ varies with the surface roughness (water, soil, or vegetation) and is inversely proportional to wind speed.

In other words, the external conductance (and thus, the evaporation rate) increases with wind speed, as postulated by Dalton (Eq. 2-27).

The external resistance for evaporation from open water can be estimated as follows:

r_a = 4.72 \hspace{0.05in} \frac {[ln (z_m /z_o)]^2}{1 + 0.536 v_2} \hspace{0.99in}
\end{displaymath} (9)

in which

$r_a$ is given in s m$^{-1}$;

$z_m$ = height at which meteorological variables are measured (m),

$z_o$ = aerodynamic roughness of the surface (m), and

$v_2$ = wind speed (m s$^{-1}$), measured at the 2-m height.

The external resistance $r_a$ (s m$^{-1}$) for the reference crop (clipped grass 0.12-m high), for measurements of wind speed (m s$^{-1}$), temperature and humidity at a standardized height of 2 m is:

r_a^{rc} = \frac {208}{v_2} \hspace{0.99in} (2-59)
\end{displaymath} (10)

The internal (stomatal or surface) resistance is inversely proportional to the leaf-area index L, i.e., the projected area of vegetation per unit ground area:

r_s = \frac {200}{L} \hspace{0.99in} (2-60)
\end{displaymath} (11)

in which $r_s$ is given in s m$^{-1}$, and

L = leaf-area index, which is empirically related to crop height.

For clipped grass, L = 24 $h_c$ (m s$^{-1}$), with height in the range 0.05 $\le$ $h_c$ $\le$ 0.15 m. For alfalfa, L = 5.5 + 1.5 ln $h_c$, with 0.1 $\le$ $h_c$ $\le$ 0.5 m.

From Eq. 2-60, the stomatal resistance of the reference crop (clipped grass 0.12-m high) is: $r_s^{rc}$ = 69 s m$^{-1}$.

Likewise, for an alfalfa crop with $h_c$ = 0.3 m: $r_s$ = 54 s m$^{-1}$.

Example 2-8 . Calculate the evaporation rate of the reference crop by the Penman-Monteith method for the same atmospheric conditions as Example 2.5 in the textbook. Assume standard atmospheric pressure.

From Eq. 2-37, the saturation vapor pressure gradient is: $\Delta$ = 1.45 mb ($^o$C)$^{-1}$.

From Example 2-5, the net radiation in evaporation rate units is $E_n$ = 0.94 cm/d.

From Eq. 2-59, the aerodynamic resistance is:
$r_a^{rc}$ = (208 $\times$ 86400)/(200 $\times$ 1000) = 89.9 s m$^{-1}$ = 0.9 s cm$^{-1}$.

The stomatal resistance of the reference crop is:
$r_s^{rc}$ = 69 s m$^{-1}$ = 0.69 s cm$^{-1}$.

From Eq. 2-51, the modified psychrometric constant is:
$\gamma_*$ = 0.667 [1 + (0.69/0.90)] = 1.178.

The vapor pressure deficit is:
($e_s - e_a$) $\cong$ ($e_o - e_a$) = $e_o$ [1 - (RH/100)] = 23.37 [1 - (70/100)] = 7 mb.

From Eq. 2-56, K = 0.735 $\times$ 10$^{-7}$ mb$^{-1}$, and thus, K' = 0.00635 mb$^{-1}$ s d$^{-1}$.

From Eq. 2-57, the mass-transfer evaporation rate is: $E_a$ = 0.028 cm d$^{-1}$.

From Eq. 2-54, the evaporation rate is: E = 0.53 cm d$^{-1}$.

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Victor Miguel Ponce 2003-03-02