---
title: "Derive PET expression by Penman's method"
subject: "Engineering Hydrology"
chapter: 3
key: "semester_5/engineering_hydrology:derive-penman-pet"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/derive-penman-pet
site: BCT Engineering
---

# Derive PET expression by Penman's method

Engineering Hydrology · Chapter 3: Hydrological Losses

- Asked in: 2080 Baisakh, 6 marks
- Video: [[WAPORCV] Unit 1.2.1 The physical basis and derivation of Penman Monteith equation](https://www.youtube.com/watch?v=yJUuqO-82uc) — IHE Delft - Water Accounting
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/derive-penman-pet
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-3

## Question

Derive the expression for potential evapotranspiration using Penman's method.

## Short answer

Penman's equation PET = (A H_n + γ E_a)/(A + γ) is derived by combining energy balance (H_n = E + H_a), Bowen ratio (H_a/E = γ (T_s – T_a)/(e_w – e_a)), and aerodynamic equations, eliminating surface temperature using the slope A of saturation vapour pressure.

## Answer

**Penman's method** combines the energy-balance method and the aerodynamic (mass-transfer) method to compute evaporation (or PET) from standard weather data.

#### Basic relations
Let $e_w$ = saturation vapour pressure at the surface temperature $T_s$, $e_s$ = saturation vapour pressure at the air temperature $T_a$, and $e_a$ = actual vapour pressure of the air.

1. Energy balance (heat stored in the ground is ignored): net radiation $H_n$ is used for evaporation $E$ and for heating the air $H_a$ (all in mm of water/day):
$$
H_n = E + H_a
$$
2. Bowen's ratio for sensible heat:
$$
\frac{H_a}{E} = \gamma\,\frac{T_s - T_a}{e_w - e_a}
$$
3. Aerodynamic (Dalton type) equations:
$$
E = f(u)(e_w - e_a), \qquad E_a = f(u)(e_s - e_a)
$$
$E_a$ is the evaporation if the surface were at air temperature.
4. Slope of the saturation vapour pressure curve:
$$
A = \frac{e_w - e_s}{T_s - T_a}
$$

#### Derivation
From (3): $e_w - e_a = E/f(u)$ and $e_w - e_s = (E - E_a)/f(u)$.

From (4): $T_s - T_a = \dfrac{e_w - e_s}{A} = \dfrac{E-E_a}{A f(u)}$.

Put these in the Bowen ratio:
$$
H_a = \gamma E\,\frac{(E-E_a)/(A f(u))}{E/f(u)} = \frac{\gamma\,(E - E_a)}{A}
$$

Substitute in $H_n = E + H_a$:
$$
H_n = E + \frac{\gamma(E - E_a)}{A}
\;\Rightarrow\; A H_n = (A+\gamma)E - \gamma E_a
$$
$$
E = PET = \frac{A H_n + E_a\gamma}{A+\gamma}
$$

This is **Penman's equation**. The unknown surface temperature has been eliminated, so only standard weather data are needed.

#### Terms
- $H_n = H_o(1-r)(0.18+0.55n/N) - \sigma T_a^4(0.56-0.092\sqrt{e_a})(0.1+0.9n/N)$ mm/day, where $H_o$ is the solar radiation reaching the top of the atmosphere (not the same as the heating of air $H_a$ above), $r$ = reflection coefficient, $n/N$ = ratio of actual to possible sunshine hours and $\sigma$ = Stefan-Boltzmann constant.
- $E_a = 0.35\left(1+\dfrac{u_2}{160}\right)(e_s - e_a)$ mm/day, with $u_2$ = wind speed at 2 m height (km/day) and vapour pressures in mm Hg.
- $A$ = slope of the saturation vapour pressure curve at $T_a$ (mm Hg/°C); $\gamma = 0.49$ mm Hg/°C.

When $A$ is large (hot weather) the radiation term dominates; when $A$ is small (cold weather) the aerodynamic term dominates.

## Check yourself

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

1. What does Penman's method combine?
   A. Energy balance and aerodynamic (mass-transfer) methods
   B. Darcy and Manning
   C. Pan and lysimeter data
   D. Water balance and lysimeter
   Answer: A. It combines the energy budget and Dalton-type aerodynamic equation. (Key point: Basis of Penman)

2. What is the energy balance used?
   A. H_n = E - H_a
   B. H_n = E x H_a
   C. H_n = E + H_a (net radiation goes to evaporation and heating air)
   D. H_n = H_a only
   Answer: C. Ground heat is ignored, so net radiation splits into latent (E) and sensible (H_a) heat. (Key point: Energy balance)

3. What is the purpose of the slope A in the derivation?
   A. To find humidity
   B. To compute radiation
   C. To measure wind
   D. To relate (e_w - e_s) to (T_s - T_a) and eliminate the unknown surface temperature
   Answer: D. A = (e_w - e_s)/(T_s - T_a) lets the surface temperature be removed, so only standard weather data are needed. (Key point: Elimination of Ts)

4. Which result follows from substitution?
   A. E = A H_n - gamma E_a
   B. E = (H_n + E_a)/2
   C. E = (A+gamma)/(A H_n)
   D. E = (A H_n + gamma E_a)/(A + gamma)
   Answer: D. Substituting Ha = gamma(E - Ea)/A into Hn = E + Ha gives the Penman equation. (Key point: Result)

5. When does the aerodynamic term dominate?
   A. When A is small (cold weather)
   B. Always
   C. When A is large (hot weather)
   D. When wind is zero
   Answer: A. Weights are A/(A+gamma) for radiation and gamma/(A+gamma) for aerodynamic, so small A favours the aerodynamic term. (Key point: Role of A)

## Related in Engineering Hydrology

- [PET versus AET, Penman's method](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Apet-vs-aet-penman) — Engineering Hydrology, chapter 3
- [PET by Penman method for May](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Apenman-may-20c) — Engineering Hydrology, chapter 3
- [PET by Penman method, Kathmandu in March](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Apenman-kathmandu-march) — Engineering Hydrology, chapter 3, asked 2×
- [PET by Penman method, Kathmandu in February](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Apenman-kathmandu-february) — Engineering Hydrology, chapter 3, asked 2×
- [PET by Penman formula, Dharan in April](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Apenman-dharan-april) — Engineering Hydrology, chapter 3
- [PET by Penman formula, Simara in April](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Apenman-simara-april) — Engineering Hydrology, chapter 3

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