Penman-Monteith Evapotranspiration Formula
Calculate reference evapotranspiration using the FAO Penman-Monteith equation.
Essential for irrigation planning and water management.
The Formula
The FAO Penman-Monteith equation is the international standard for calculating reference evapotranspiration (ET₀). Evapotranspiration is the combined water loss from soil evaporation and plant transpiration.
This formula was adopted by the United Nations Food and Agriculture Organization (FAO) in 1998 as the sole recommended method. It accounts for radiation, temperature, humidity, and wind — the four key drivers of water loss from the land surface.
Accurate ET₀ estimates are critical for irrigation scheduling, water resource management, and drought assessment. Overwatering wastes resources; underwatering kills crops. This formula helps find the balance.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| ET₀ | Reference evapotranspiration | mm/day |
| R_n | Net radiation at the crop surface | MJ/(m²·day) |
| G | Soil heat flux density | MJ/(m²·day) |
| T | Mean daily air temperature at 2 m height | °C |
| u₂ | Wind speed at 2 m height | m/s |
| e_s | Saturation vapor pressure | kPa |
| e_a | Actual vapor pressure | kPa |
| Δ | Slope of the saturation vapor pressure curve | kPa/°C |
| γ | Psychrometric constant | kPa/°C |
Example 1
A summer day with: T = 28°C, R_n = 18 MJ/(m²·day), G = 0.5 MJ/(m²·day), u₂ = 2.0 m/s, e_s = 3.78 kPa, e_a = 2.27 kPa, Δ = 0.22 kPa/°C, γ = 0.066 kPa/°C
Numerator radiation term: 0.408 × 0.22 × (18 − 0.5) = 0.408 × 0.22 × 17.5 = 1.571
Numerator wind term: 0.066 × (900/301) × 2.0 × (3.78 − 2.27) = 0.066 × 2.99 × 2.0 × 1.51 = 0.597
Denominator: 0.22 + 0.066 × (1 + 0.34 × 2.0) = 0.22 + 0.066 × 1.68 = 0.331
ET₀ = (1.571 + 0.597) / 0.331 = 2.168 / 0.331
ET₀ ≈ 6.5 mm/day — typical for a warm, moderately dry summer day
Example 2
A mild spring day with: T = 15°C, R_n = 10 MJ/(m²·day), G = 0.2 MJ/(m²·day), u₂ = 1.5 m/s, e_s = 1.70 kPa, e_a = 1.19 kPa, Δ = 0.11 kPa/°C, γ = 0.066 kPa/°C
Radiation term: 0.408 × 0.11 × (10 − 0.2) = 0.408 × 0.11 × 9.8 = 0.440
Wind term: 0.066 × (900/288) × 1.5 × (1.70 − 1.19) = 0.066 × 3.125 × 1.5 × 0.51 = 0.158
Denominator: 0.11 + 0.066 × (1 + 0.34 × 1.5) = 0.11 + 0.066 × 1.51 = 0.210
ET₀ = (0.440 + 0.158) / 0.210 = 0.598 / 0.210
ET₀ ≈ 2.8 mm/day — much lower water demand in cooler conditions
When to Use It
- Irrigation scheduling — determining how much water crops need
- Water resource planning for reservoirs and river basins
- Drought monitoring and early warning systems
- Climate change impact studies on water availability
- Agricultural research and crop water requirement estimation