## Penman’s equation for lake evaporation

Penman published an important paper in 1948 (1), which allows calculation of the rate of evaporation E_{o} from a water surface like that of a lake, too large to be much affected by the additional evaporation that occurs at the edge. The derivation of his formula is uncomplicated and interesting for its ingenuity, notably at the point indicated by an asterisk (*) in what follows. It is founded on six basic equations -

1. The __Dalton equation (__Note 4.E__)__

E_{o} = k.u (e_{s} - e) |
kg/(m^{2}.s) |

where k is a factor related to surface roughness (in units of 0.01 s^{2}/m^{2}), u is the wind speed (m/s), e_{s} is the saturation water vapour pressure at screen temperature (hPa), e is the vapour pressure of the air (hPa).

2. The definition of the '__diffusion resistance'__ r_{a} between water and air -

r_{a} = r.c/ (K_{s}.L.k.u) |
s/m |

where r is the density of the air (kg/m^{3}), c is its specific heat (J/kg.K), K_{s} is the pyschrometric constant in Regnault's equation (hPa/K - see Section 6.3), L is the latent heat of evaporation (J/kg).

3. The definition of the ‘__saturation deficit__’ S -

4. The ‘__psychrometric slope__’ D, the tangent to the saturation vapour pressure/temperature curve, defined as follows -

D = (e_{s} - e_{w}) / (T - T_{s}) |
hPa /K |

where e_{w} is the saturation water vapour pressure at the water surface temperature T_{s}, and T is the air temperature (°C).

5. The __convective heat flux__ H from water surface to air -

H = r.c (T_{s} - T) / r_{a} |
W/m^{2} |

6. The __energy balance__ at an insulated ground surface (see Section 5.1) -

R_{n} = L. E_{o} + H |
W/m^{2} |

where R_{n} is the net radiation inflow.

Then those equations can be combined:

- First 1 & 2 can be combined to produce

E_{o} = r.c (e_{w} - e) / Ks . r_{a} |
kg/(m^{2}.s) |

(*)8. Then split the term (e_{w} - e), using equations 3 & 4, as follows -

e_{w} - e = (e_{w} - e_{s}) + (e_{s} - e) = D (T_{s} - T) + S |
hPa |

9. Combine equations 7 & 8 -

E_{o} = r.c [D(T_{s} -T) + S] / K_{s}.L.r_{a} |
kg/(m^{2}.s) |

10. Rearrange 9 thus -

r.c.D(T_{s} -T) = K_{s}.L.E_{o}. r_{a} - r.c.S |
J.kg/( m^{4}.s^{2}.K) |

11. Combine 5 & 10 -

H = [K_{s}.L. E_{o} - r.c.S/ r_{a}] / D |
W/m^{2} |

12. Then join 6 & 11 -

L. E_{o} = (D.R_{n} + r.c.S/ r_{a}) / (D + K_{s}) |
W/m^{2} |

This is *Penman’s formula*.

References

(1) Penman, H.L. 1948: Natural evaporation from open water, bare soil and grass. *Proc. Roy. Soc. A, ***193**, 120-45.