---
title: "Penman equation and its variables"
subject: "Engineering Hydrology"
chapter: 3
key: "semester_5/engineering_hydrology:penman-equation-variables"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/penman-equation-variables
site: BCT Engineering
---

# Penman equation and its variables

Engineering Hydrology · Chapter 3: Hydrological Losses

- Asked in: 2072 Chaitra, 4 marks
- Video: [MAQ-22806 Combination methods - Penman equation](https://www.youtube.com/watch?v=m3Yp4wt4_E4) — Arnold Moene
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/penman-equation-variables
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-3

## Question

Write down the Penman equation and explain all variables and constants involved in it.

## Short answer

PET = (A·Hn + Ea·γ) / (A + γ). Hn is net radiation (mm/day), Ea is aerodynamic term (mm/day), A is slope of saturation vapor pressure curve (mm Hg/°C), γ is psychrometric constant (0.49 mm Hg/°C). Hn depends on extraterrestrial radiation Ha, albedo r, sunshine ratio n/N, air temperature Ta (K), and actual vapor pressure ea; Ea depends on wind speed u₂ (km/day) and vapor pressure deficit (ew − ea).

## Answer

The Penman equation for potential evapotranspiration (or open-water evaporation) is

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

$$
H_n = H_a(1-r)\left(0.29\cos\varphi + 0.52\frac{n}{N}\right) - \sigma T_a^4(0.56 - 0.092\sqrt{e_a})\left(0.1 + 0.9\frac{n}{N}\right)
$$

$$
E_a = 0.35\left(1 + \frac{u_2}{160}\right)(e_w - e_a)
$$

| Symbol | Meaning | Unit |
|---|---|---|
| $PET$ | Potential evapotranspiration | mm/day |
| $A$ | Slope of the saturation vapour pressure-temperature curve at the air temperature | mm Hg/°C |
| $\gamma$ | Psychrometric constant (0.49) | mm Hg/°C |
| $H_n$ | Net radiation in evaporation units | mm of water/day |
| $E_a$ | Drying power of air (aerodynamic term) | mm/day |
| $H_a$ | Extraterrestrial (top of atmosphere) solar radiation | mm/day |
| $r$ | Reflection coefficient (albedo): 0.05 water, 0.15-0.25 green crop | - |
| $a$ ($=0.29\cos\varphi$), $b$ (0.52) | Constants of the radiation relation; $\varphi$ = latitude | - |
| $n, N$ | Actual and possible sunshine hours | h |
| $\sigma$ | Stefan-Boltzmann constant ($2.01\times10^{-9}$) | mm/day/K⁴ |
| $T_a$ | Mean air temperature | K |
| $e_w, e_a$ | Saturation and actual vapour pressure ($e_a = RH\times e_w$) | mm Hg |
| $u_2$ | Wind speed at 2 m above ground | km/day |

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does A represent?
   A. Slope of saturation vapour pressure-temperature curve at air temperature
   B. Area of the basin
   C. Albedo
   D. Wind coefficient
   Answer: A. A is the slope of the saturation vapour pressure curve, in mm Hg/°C. (Key point: Meaning of A)

2. Ea in the Penman equation is
   A. Actual evaporation
   B. Drying power of air (aerodynamic term)
   C. Net radiation
   D. Extraterrestrial radiation
   Answer: B. Ea = 0.35(1 + u2/160)(ew - ea) is the drying power of air. (Key point: Ea)

3. Ha is
   A. Heat flux in soil
   B. Extraterrestrial solar radiation at the top of atmosphere
   C. Back radiation
   D. Net radiation
   Answer: B. Ha is extraterrestrial radiation; Hn is net radiation after losses. (Key point: Ha vs Hn)

4. Typical r values are
   A. 1.0 for water
   B. 0.5 for water
   C. 0.05 for green crop
   D. 0.05 for water, 0.15-0.25 for green crop
   Answer: D. Water reflects little (0.05); green crop 0.15-0.25. (Key point: Reflection coefficient)

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