Evaporation Calculator — Methods, Coefficients, and Calibration

Method 2 · radiation-driven

Priestley-Taylor (1972)

Formula:

E = α · (Δ / (Δ + γ)) · (R_n − G) / λ,  α = 1.26

Source: Priestley & Taylor (1972), Monthly Weather Review 100, "On the Assessment of Surface Heat Flux and Evaporation Using Large-Scale Parameters." Derived from the Penman combination equation by dropping the aerodynamic term and replacing it with an empirical multiplier α applied to the equilibrium evaporation rate Δ/(Δ+γ)·R_n/λ.

Why α = 1.26

Priestley & Taylor's original analysis of saturated land surfaces, ocean surfaces, and reservoirs found that observed evaporation consistently exceeded the equilibrium rate by a factor of ~1.26 when the surface was wet and not water-stressed. The coefficient absorbs the contribution of large-scale aerodynamic transport — wind, the humidity deficit, boundary-layer entrainment — into a single empirical scaling. Stewart & Rouse (1977) and Eichinger et al. (1996) confirmed 1.26 ± 0.05 for open water bodies; the calculator uses 1.26 as the canonical value.

When to use

Best for open water (lakes, reservoirs, tailings ponds) when you have confidence in radiation but weak wind data. The textbook standard in most hydrology curricula for lake evaporation precisely because it is less sensitive to the most error-prone input — wind. Slightly under- predicts on small, sheltered ponds where advected heat from the surroundings matters; slightly over-predicts at very high wind speeds.

Sensitivity

Linear in R_n. ±10% on R_n → ±10% on E. Insensitive to wind and humidity (those enter only through R_n's longwave term). This is the reason Priestley-Taylor lake monitoring programmes can rely on satellite radiation products without ground stations.

Method 4 · field-engineering standard

Aerodynamic Mass-Transfer (Harbeck 1962)

Formula:

E = N(A) · u_2 · (e_w − e_a)
N(A) = 0.291 · A^(−0.05),  clamped to [0.12, 0.30] mm/day per m/s per hPa

Source: Harbeck, G.E. Jr. (1962), U.S. Geological Survey Professional Paper 272-E, "A Practical Field Technique for Measuring Reservoir Evaporation Utilizing Mass-Transfer Theory."

Why N is area-dependent

Earlier Dalton-style mass-transfer models used a fixed coefficient, typically calibrated against the ~9 km² Lake Hefner (Oklahoma) evaporation study of 1950–51 (~0.113 mm/day per m/s per hPa). Harbeck's contribution was to fit measurements across 19 reservoirs spanning four orders of magnitude in surface area and show that smaller water bodies have a higher mass-transfer coefficient — the boundary layer over a small pond is less developed, so vapour escapes more efficiently. The power-law fit N = 0.291 · A^(−0.05) (A in m²) collapses the cross-reservoir scatter to within ±15% on annual evaporation.

The calculator clamps N to [0.12, 0.30]: below 0.12 you have entered Harbeck's large-lake asymptote (good); above 0.30 you have left the regression's domain (sub-square-meter puddles do not behave like reservoirs — droplet physics takes over). Outside this range, prefer a method that does not rely on N.

Why this matters in practice

A 10-acre (40 000 m²) reservoir gives N ≈ 0.166 — 47% higher than the Lake Hefner value of 0.113. A 1-acre pond gives N ≈ 0.183, 62% higher. For typical industrial sites (< 10 ha), pinning N to the Lake Hefner value understates evaporation by 30–80%. Pre-Harbeck calculators that hard- coded 0.113 (or a single locally fitted N) are systematically wrong on small reservoirs in this direction — a fact the calculator's method-comparison table makes immediately visible.

Adjustments layered on top

The calculator multiplies the Harbeck base rate by four modifiers that earlier mass-transfer implementations either bury in a constant or omit:

• Pressure factor 1013.25 / P(elev) — corrects vapour-pressure deficit for site elevation (barometric formula).
• Sun exposure — 1.05 / 1.00 / 0.85 for full / partial / heavy shade.
• Surface features — ×1.20 when sprays, fountains, or aerators are present (turbulent surface area).
• Fetch factor — Lake-Hefner-style logarithmic reduction for short upwind distances (no boundary-layer equilibrium below ~50 m fetch).
• Salinity — linear water-activity reduction (1 − salinity %).

When to use

When wind and water-temperature data are reliable and you want a method that does not depend on radiation estimates. Strongly preferred for short-period studies (hours to days). The historical method of record in USBR and USGS reservoir-evaporation reports for the western United States.

Shared physics — the cross-cutting closures

Saturation vapour pressure (Magnus / Tetens form)

e_s(T) = 6.112 · exp(17.67 · T / (T + 243.5))    [hPa]
e_s(T) = 0.6108 · exp(17.27 · T / (T + 237.3))   [kPa, FAO-56]

Tetens / Magnus form; accurate to < 0.3% over −30 °C to +50 °C. FAO-56 (Allen et al. 1998) uses the kPa variant; the calculator's aerodynamic block uses the hPa variant for compatibility with the Harbeck literature.

Slope Δ and psychrometric constant γ

Δ = 4098 · e_s(T) / (T + 237.3)²    [kPa/°C]
γ = 0.000665 · P                    [kPa/°C], P in kPa

Closed-form expressions from FAO-56 §3. P is atmospheric pressure from the barometric formula at site elevation.

Extraterrestrial radiation R_a — closed form

δ  = 0.409 · sin(2π/365 · J − 1.39)         solar declination [rad]
ω_s = arccos(−tan φ · tan δ)                sunset hour angle [rad]
d_r = 1 + 0.033 · cos(2π/365 · J)           inverse Earth-Sun distance²
R_a = (24·60/π) · 0.0820 · d_r · (ω_s · sin φ · sin δ
       + cos φ · cos δ · sin ω_s)            [MJ/m²/day]
N   = (24/π) · ω_s                          daylight hours

FAO-56 Annex 3, equations (21)–(35). The calculator clamps −tan φ · tan δ to [−1, 1] so polar-day / polar-night conditions do not produce NaN. When latitude is unknown, the calculator falls back to φ = 35°.

Net radiation R_n at the water surface

R_s   = R_a · (0.25 + 0.50 · n/N)         shortwave from sunshine hours
R_ns  = (1 − albedo) · R_s                absorbed shortwave
R_nl  = σ · T_K^4 · (0.34 − 0.14·√e_a) · (1.35·R_s/(0.75·R_a) − 0.35)
R_n   = max(0, R_ns − R_nl)               net radiation

FAO-56 §3, equations (38)–(40). σ = 4.903 × 10⁻⁹ MJ/K⁴/m²/day. Coefficients 0.25 / 0.50 are the FAO-56 Angstrom calibration; 0.34 / 0.14 are the longwave humidity correction; 1.35 / 0.35 are the cloudiness adjustment. Default open-water albedo is 0.08; the calculator overrides this to 0.70 (snow / ice) when monthly T_a < 0 °C — without that override, frozen-month evaporation is over-predicted by an order of magnitude.

Water-column heat storage G (Edinger-Brady)

G = ρ · c_p · depth · ΔT_w / Δt    [MJ/m²/day]
ρ = 1000 kg/m³, c_p = 4186 J/(kg·K), Δt = 30 · 86 400 s

Edinger, Brady & Geyer (1974). For shallow ponds (≤3 m) the storage term is < 5% of R_n and the calculator zeros it. For deep reservoirs (>3 m) in the 12-month loop, the calculator computes G from the month-over-month change in surface water temperature and caps it at 20% of R_n to guard against noisy T_w month-step inputs. This term is what makes spring evaporation lower than expected (water is absorbing heat) and autumn evaporation higher than expected (water is releasing stored heat) on deep lakes.

Wind profile normalisation

u(2) = u(z) · ln(2 / z₀) / ln(z / z₀),  z₀ = 0.001 m

Logarithmic wind profile with open-water roughness length z₀ = 1 mm. Weather services typically report wind at 10 m; FAO-56 requires it at 2 m for the Penman-Monteith term. This conversion reduces the wind speed by roughly 25% (10 m → 2 m).

Fetch factor for short upwind distances

f = max(0.7, 1.0 − 0.06 · ln(fetch / 50))   for fetch > 50 m

Lake-Hefner-style boundary-layer correction applied to the aerodynamic and empirical methods. A 5 m fetch (small pond, large wind exposure) gets the full Harbeck N; longer fetches reduce it slightly as the boundary layer equilibrates. The 0.7 floor matches Harbeck (1962) Table 6.

Salinity correction

f_salinity = 1 − salinity_pct / 100

Linear water-activity reduction. Pure-water evaporation is multiplied by this factor; sea water (~3.5%) gives a ~3% reduction; tailings pond (10–15% TDS) gives a 10–15% reduction. The model is valid for dilute brines (< ~20%); above that, activity drops non-linearly and a more sophisticated Pitzer-equation model is warranted.

Net loss = E − P

Evaporation is the gross water leaving the surface. Net basin loss is evaporation minus rainfall onto the same surface. When the calculator's "Net water loss" toggle is on, monthly rainfall_sum from Open-Meteo's climate normals is subtracted from evaporation month-by-month before the annual mean. This is the correct quantity for makeup-water sizing and cost cards on covered- pond business cases. For permit work the regulator may ask for the gross figure — toggle accordingly.

Known limitations — disclosures for permit submissions

1. Daily mean inputs — all methods compute a daily evaporation rate from daily mean meteorological inputs. Sub-daily extremes (a high-wind afternoon) are not resolved. For high-resolution operational planning, use a sub-hourly Penman-Monteith model with eddy-covariance forcing.
2. Stratified deep lakes — the heat-storage term G captures bulk seasonal lag but does not resolve thermocline dynamics. For lakes > 20 m deep, prefer a coupled hydrodynamic model (DYRESM, GLM).
3. Brackish to brine conditions — the salinity correction is linear; above ~20% TDS use a Pitzer-equation activity model.
4. Sub-zero open water — the calculator clamps T_w to 0 °C. Sublimation from ice is a different regime (Stefan flow + Knudsen layer) and is not modelled. Cold-climate ice-on basins should switch to a sublimation model in winter months.
5. Sheltered ponds with strong advection — small ponds surrounded by tall buildings or vegetation receive advected sensible heat ("oasis effect") that none of these five methods captures. Apply a site-specific upward multiplier or use a Bowen-ratio energy budget.
6. Spray ponds and aerated surfaces — the ×1.20 "surface features" modifier is a heuristic. For high-rate spray cooling, use a manufacturer-specific spray-cooling model (Berman 1961, Porter et al. 1977).
7. Snow / ice albedo override — the calculator switches to albedo 0.70 below 0 °C T_a. Partial ice cover (mixed open water + ice) is not interpolated; the model returns either the open-water or the ice-bound estimate, not a weighted average.

For each of these, the engineer's responsibility in a regulatory filing is to disclose the assumption, cite this methodology page, and bracket the result with the calculator's uncertainty band (Gaussian propagation; ±10 to ±25% typical for annual evaporation).