Penman’s equation for lake evaporation

E. Linacre

11/'97

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/(m².s)

where k is a factor related to surface roughness (in units of 0.01 s²/m²), 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/m³), 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 -

S = e_s - e

hPa

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/m²

6. The energy balance at an insulated ground surface (see Section 5.1) -

R_n = L. E_o + H

W/m²

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/(m².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/(m².s)

10. Rearrange 9 thus -

r.c.D(T_s -T) = K_s.L.E_o. r_a - r.c.S

J.kg/( m⁴.s².K)

11. Combine 5 & 10 -

H = [K_s.L. E_o - r.c.S/ r_a] / D

W/m²

12. Then join 6 & 11 -

L. E_o = (D.R_n + r.c.S/ r_a) / (D + K_s)

W/m²

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.