## 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 Eo 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)

 Eo = k.u (es - 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), es 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' ra between water and air -

 ra = r.c/ (Ks.L.k.u) s/m

where r is the density of the air (kg/m3), c is its specific heat (J/kg.K), Ks 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 = es - e hPa

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

 D = (es - ew) / (T - Ts) hPa /K

where ew is the saturation water vapour pressure at the water surface temperature Ts, and T is the air temperature (°C).

5. The convective heat flux H from water surface to air -

 H = r.c (Ts - T) / ra W/m2

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

 Rn = L. Eo + H W/m2

where Rn is the net radiation inflow.

Then those equations can be combined:

1. First 1 & 2 can be combined to produce

 Eo = r.c (ew - e) / Ks . ra kg/(m2.s)

(*)8. Then split the term (ew - e), using equations 3 & 4, as follows -

 ew - e = (ew - es) + (es - e) = D (Ts - T) + S hPa

9. Combine equations 7 & 8 -

 Eo = r.c [D(Ts -T) + S] / Ks.L.ra kg/(m2.s)

10. Rearrange 9 thus -

 r.c.D(Ts -T) = Ks.L.Eo. ra - r.c.S J.kg/( m4.s2.K)

11. Combine 5 & 10 -

 H = [Ks.L. Eo - r.c.S/ ra] / D W/m2

12. Then join 6 & 11 -

 L. Eo = (D.Rn + r.c.S/ ra) / (D + Ks) 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.