Free water evaporation by Penman method, June

Free-water evaporation is obtained from Penman with $r = 0.05$ for a water surface.

Penman equation (potential evapotranspiration):

$$PET = \frac{A\,H_n + E_a\,\gamma}{A + \gamma}$$

with

$$H_n = H_a(1-r)\left(a + b\frac{n}{N}\right) - \sigma T_a^4\,(0.56 - 0.092\sqrt{e_a})\left(0.1 + 0.9\frac{n}{N}\right)$$ $$E_a = 0.35\left(1 + \frac{u_2}{160}\right)(e_w - e_a), \qquad a = 0.29\cos\varphi,\ b = 0.52$$

where $H_n$ is the net radiation in mm of evaporable water per day, $E_a$ the aerodynamic term in mm/day, $u_2$ the wind speed at 2 m in km/day, and $e_w, e_a$ in mm of Hg.

Data and constants used

• $T = 30\,^\circ$C $\Rightarrow T_a = 303$ K; RH $= 60\%$; $n = 12$ h; $N = 14.2$ h; $u_2 = 240$ km/day
• $H_a = 16.9$ mm/day; $r = 0.05$; $\varphi = 33.00^\circ$; $e_w = 31.82$ mm Hg; $A = 1.83$ mm Hg/$^\circ$C; $\gamma = 0.49$ mm Hg/$^\circ$C; $\sigma = 2.01\times10^{-9}$ mm/day/K$^4$
• Assumed values (not given): $e_w$ at $30^\circ$C $= 31.82$ mm Hg and slope $A = 1.83$ mm Hg/$^\circ$C; for $33^\circ$N in June, $H_a \approx 16.9$ mm/day and possible sunshine $N \approx 14.2$ h (standard tables / solar geometry); wind $10\ \text{km/h} = 240$ km/day; $\gamma = 0.49$.

Step 1: Actual vapour pressure

$$e_a = \frac{RH}{100}\,e_w = 0.60\times31.82 = 19.09\ \text{mm Hg}$$

Step 2: Net radiation $H_n$

$a = 0.29\cos(33.00^\circ) = 0.2432$, and $n/N = 0.8451$

$$\begin{aligned} H_a(1-r)\left(a + b\tfrac{n}{N}\right) &= 16.9\times0.950\times(0.2432 + 0.52\times0.8451) = 10.960\\ \sigma T_a^4 &= 2.01\times10^{-9}\times(303)^4 = 16.942\\ \text{Back radiation} &= 16.942\times(0.56 - 0.092\sqrt{19.09})\times(0.1 + 0.9\times0.8451) = 2.304\\ H_n &= 10.960 - 2.304 = 8.656\ \text{mm/day} \end{aligned}$$

Step 3: Aerodynamic term

$$E_a = 0.35\left(1 + \frac{240}{160}\right)(31.82 - 19.09) = 11.137\ \text{mm/day}$$

Step 4: PET

$$PET = \frac{1.83\times8.656 + 11.137\times0.49}{1.83 + 0.49} = \frac{15.841 + 5.457}{2.320} = 9.18\ \text{mm/day}$$

Answer: Free water surface evaporation = 9.2 mm/day, so about 275 mm in June (30 days).