Engineering & Fluid Mechanics
Penman Combination Equation Formula: Reservoir & Open-Water Evaporation (mm/day)
Combine net solar radiation energy balance E_rad and aerodynamic wind-vapor deficit drying power E_aero without needing surface water temperature via Howard L. Penman’s 1948 equation E_0 = (Delta * E_rad + gamma * E_aero) / (Delta + gamma).
How Howard Penman’s 1948 combination trick eliminated unmeasured water-surface skin temperature!
Before Howard Latimer Penman (1948, Proceedings of the Royal Society A), hydrologists faced a dilemma: Dalton bulk-aerodynamic mass-transfer equations required hard-to-measure water-surface skin temperature T_s, while Bowen-ratio energy balances also required T_s! Penman linearized the saturation vapor-pressure curve (slope Delta = de_s / dT in kPa/°C) to algebraically eliminate surface temperature—weighting radiative evaporation equivalent E_rad (mm/day) and aerodynamic drying power E_aero (mm/day) by Delta and the psychrometric constant gamma (≈ 0.066 kPa/°C at sea level): E_0 = (Delta * E_rad + gamma * E_aero) / (Delta + gamma)!
Notice the physical weights: on warm humid days (24.5 °C -> Delta = 0.185 kPa/°C, gamma = 0.066 kPa/°C), radiation carries Delta / (Delta + gamma) = 73.71% of the weight! With E_rad = 6.20 mm/day and E_aero = 3.80 mm/day, open-water evaporation E_0 = 5.569 mm/day—climbing to 6.542 mm/day when dry desert winds boost E_aero to 7.50 mm/day!
| Hydrometeorological Regime (Delta=0.185, gamma=0.066 kPa/°C) | Radiation Weight Delta/(Delta+gamma) | Penman Evaporation E_0 (mm/day) |
|---|---|---|
| Moderate Wind: E_rad = 6.2, E_aero = 3.8 mm/d | 0.7371 (73.71% Energy-Driven) | 5.569 mm/day |
| Dry Advective Wind: E_rad = 6.2, E_aero = 7.5 mm/d | 0.7371 | 6.542 mm/day |
| Invalid Psychrometric Constant gamma <= 0 | Error Guard | "Invalid Penman Hydrometeorology Inputs" |
Penman Open-Water Combination Evaporation Rate E_0 (mm/day) = (Delta*E_rad + gamma*E_aero)/(Delta + gamma)
Computes daily open-water evaporation in mm/day from saturation vapor-pressure slope Delta (B2, kPa/°C), psychrometric constant gamma (B3, kPa/°C), radiative equivalent E_rad (B4, mm/d), and aerodynamic drying power E_aero (B5, mm/d).
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B5 < 0), "Invalid Penman Hydrometeorology Inputs", ROUND((B2 * B4 + B3 * B5) / (B2 + B3), 3))Priestley-Taylor Energy-Term Share = Delta / (Delta + gamma)
Evaluates the dimensionless fraction of Penman weighting governed by net radiation energy supply.
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B5 < 0), "Invalid Penman Hydrometeorology Inputs", ROUND(B2 / (B2 + B3), 4))Try the example
Use the original values or change them to test your understanding.