Q. how to calculate evaporation rate

Q: How do I calculate evaporation from measured mass loss?

Weigh the reservoir over time. Use E_m = \dfrac{\\Delta m}{A\,\\Delta t}. Convert to depth with E_h = \dfrac{\\Delta m}{\rho_w A}. Use \rho_w \approx 1000\ \text{kg m}^{-3}.

Q: How do temperature, humidity, and wind enter the calculation?

Temperature sets saturation vapor pressure e_s. Humidity gives e_a. Wind affects aerodynamic resistance and k. Example for e_s: e_s = 0.6108 \exp\!\left(\dfrac{17.27\,T}{T + 237.3}\right) with T in °C.

Q: How can I convert evaporation depth to mass or volume?

For areand depth h use m = \rho_w\,h\,A. Volume is V = h\,A. For h in meters and \rho_w in kg m^{-3} you get kg directly.

Q: What are typical evaporation rates and coefficients?

Freshwater open-water evaporation often ranges 2 to 10 \text{mm day}^{-1} . Mass transfer coefficients depend on wind and scale, e.g. k_m \sim 10^{-3} \text{ to } 10^{-1} \text{ m s}^{-1} in practical situations. Values vary widely by conditions.

Answer

Quick answer and formulas. Evaporation rate can be computed from energy balance, from mass transfer, or by combining both (Penman). Common forms are:

Energy-limited form (latent heat method):
\[
E \;=\; \frac{R_n – G}{L_v}
\]
where \(E\) is evaporation (kg m^{-2} s^{-1} or mm s^{-1}), \(R_n\) is net radiation (W m^{-2}), \(G\) is ground heat flux (W m^{-2}), and \(L_v\) is latent heat of vaporization (\(\approx 2.45\times 10^6\) J kg^{-1} at 20 °C). To convert kg m^{-2} to mm use 1 kg m^{-2} = 1 mm of water.

Mass-transfer (aerodynamic) form:
\[
E \;=\; k\,(e_s – e_a)
\]
where \(k\) is a transfer coefficient (kg m^{-2} s^{-1} Pa^{-1}), \(e_s\) is saturation vapor pressure at the evaporating surface, and \(e_a\) is ambient vapor pressure.

Penman combination (energy + aerodynamic):
\[
E \;=\; \frac{\Delta}{\Delta + \gamma}\,\frac{R_n – G}{L_v} \;+\; \frac{\gamma}{\Delta + \gamma}\,f(u)\,(e_s – e_a)
\]
where \(\Delta\) is the slope of the saturation vapor pressure curve (Pa K^{-1}), \(\gamma\) is the psychrometric constant (Pa K^{-1}), and \(f(u)\) is a wind function (e.g. proportional to wind speed).

FAO Penman–Monteith reference evapotranspiration (widely used, units mm day^{-1}):
\[
ET_0 \;=\; \frac{0.408\,\Delta\,(R_n – G) \;+\; \gamma\,\dfrac{900}{T + 273}\,u_2\,(e_s – e_a)}{\Delta \;+\; \gamma\,(1 + 0.34\,u_2)}
\]
where \(T\) is mean air temperature (°C), \(u_2\) is wind speed at 2 m (m s^{-1}), and the other symbols are as above.

Use the form that fits your available data: energy form if you have radiation and heat flux, mass-transfer if you have humidity and wind, Penman or Penman–Monteith to combine both for best accuracy.

Detailed Explanation

Overview. Evaporation rate is the mass or depth of water lost from a free water surface per unit area per unit time. Typical units are kg m^{-2} s^{-1} for mass flux, or mm day^{-1} for depth rate. Below are step‑by‑step methods: a physically based mass‑transfer method (bulk aerodynamic) for practical calculations, and the underlying relations you need to compute each term.

Step 1. Choose your target output and collect inputs. Decide whether you want the evaporation as mass flux \( \)kg m^{-2} s^{-1}\( \) or as a depth rate \( \)mm day^{-1}\( \). Required observations are:

– Water surface temperature \(T_s\) in °C.

– Air temperature \(T_a\) in °C.

– Relative humidity \(RH\) in percent (at the air temperature).

– Wind speed \(U\) at a reference height (m s^{-1}).

– Atmospheric pressure \(P\) (Pa). If unknown, use standard pressure \(P=101325\) Pa.

Step 2. Compute saturation vapor pressure above water. Use the Tetens formula (accurate for ordinary ranges):

\[ e_{s}(T) = 0.6108 \,\exp\!\left(\frac{17.27\,T}{T + 237.3}\right) \quad\text{(kPa)}, \]

where \(T\) is in °C. Compute \(e_{s}(T_s)\) for the water surface and \(e_{s}(T_a)\) for the air temperature, if needed.

Step 3. Compute actual vapor pressure of the air. Use the air relative humidity to get the vapor pressure in kPa:

\[ e_{a} = \frac{RH}{100}\; e_{s}(T_a). \]

Step 4. Convert vapor pressures to specific humidities. The bulk aerodynamic formula uses specific humidity \(q\) (kg water vapor per kg moist air). Compute saturated specific humidity at the surface \(q_s\) and actual specific humidity of the air \(q_a\) by

\[ q = 0.622\;\frac{e}{P – 0.378\,e}, \]

\[ q_{s} = 0.622\;\frac{e_{s}(T_s)}{P – 0.378\,e_{s}(T_s)}, \qquad
q_{a} = 0.622\;\frac{e_{a}}{P – 0.378\,e_{a}}. \]

Step 5. Compute air density. The bulk formula needs dry air density \( \rho_a \) (kg m^{-3}). Use the ideal gas relation for moist air approximation with the dry air gas constant. A simple and adequate formula is

\[ \rho_a = \frac{P}{R_d\;T_K}, \]

where \(R_d = 287.05\) J kg^{-1} K^{-1} and \(T_K = T_a + 273.15\) is air temperature in kelvin. For more accuracy with moisture included you may correct for vapor, but this form is usually sufficient.

Step 6. Use the bulk aerodynamic (mass transfer) formula. The mass flux of evaporated water (kg m^{-2} s^{-1}) is given by

\[ E_m = \rho_a\; C_e\; U\; (q_s – q_a), \]

where \(C_e\) is the exchange coefficient (dimensionless), and \(U\) is wind speed at the reference height (m s^{-1}). Typical values for \(C_e\) over open water are about \(1.0\times 10^{-3}\) to \(1.5\times 10^{-3}\). Choose a value consistent with your site and measurement height. The difference \(q_s – q_a\) is the driving humidity deficit.

Step 7. Convert mass flux to depth rate. One kilogram of water per square meter equals 1 mm of water depth. To convert \(E_m\) (kg m^{-2} s^{-1}) to mm day^{-1} use

\[ E_{\text{mm/day}} = E_m \times 86400 \quad\text{(mm day}^{-1}\text{)}, \]

because there are 86400 seconds in a day. If you prefer mm hr^{-1} multiply by 3600 instead.

Step 8. Optional: alternative physical expression for comparison. If you prefer to work with vapor densities rather than specific humidity, use

\[ E_m = k_m \left(\rho_{v,s} – \rho_{v,a}\right), \]

where \(k_m\) is a mass transfer coefficient (m s^{-1}), \( \rho_{v} \) are vapor densities (kg m^{-3}), and

\[ \rho_v = \frac{e\,M_v}{R_u\,T_K}, \]

with \(M_v = 0.018016\) kg mol^{-1}, \(R_u = 8.314462618\) J mol^{-1} K^{-1}, and \(T_K\) in kelvin. This form is equivalent to the bulk aerodynamic formula if \(k_m\) is related to \(\rho_a C_e U\).

Step 9. Example workflow summary (no numbers). Compute \(e_s(T_s)\), compute \(e_a\) from RH and \(T_a\). Convert \(e\) values to \(q_s\) and \(q_a\). Compute air density \( \rho_a \). Choose \(C_e\) and use observed wind \(U\). Evaluate \(E_m = \rho_a C_e U (q_s – q_a)\). Convert to mm day^{-1} by multiplying by 86400. That result is the evaporation rate.

Notes and cautions. The accuracy depends strongly on the choice of exchange coefficient \(C_e\), the representativeness of wind speed, and whether surface temperature differs from air temperature. For hydrology and meteorology, the Penman or Penman–Monteith equations combine radiative and aerodynamic terms and are preferred when radiation and energy budget data are available. The bulk aerodynamic method above is widely used for open water and evaporative engineering when meteorological inputs are available.

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Chemistry FAQs

What is evaporation rate and what units are used?

Evaporation rate is mass or depth of water lost per areper time. Common forms: mass flux \(E_m = \dfrac{\\Delta m}{A\,\\Delta t}\) in kg m^{-2} s^{-1} and depth rate \(E_h = \dfrac{\\Delta h}{\\Delta t}\) in mm day^{-1}.

How do I calculate evaporation from measured mass loss?

Weigh the reservoir over time. Use \(E_m = \dfrac{\\Delta m}{A\,\\Delta t}\). Convert to depth with \(E_h = \dfrac{\\Delta m}{\rho_w A}\). Use \(\rho_w \approx 1000\ \text{kg m}^{-3}\).

How does the mass transfer coefficient method work?

Use concentration or vapor-pressure driving term. For mass flow: \(\dot{m} = k_m (C_s - C_a)\). Using partial pressures: \(\dot{m} = k_p (p_s - p_a)\). Determine k experimentally or from correlations.

What is the Penman or Penman-Monteith approach?

Penman-Monteith combines energy balance and aerodynamic transfer. One form is \[ E = \dfrac{\Delta (R_n - G) + \rho c_{p} \dfrac{(e_s - e_a)}{r_a}}{\Delta + \gamma \left(1 + \dfrac{r_s}{r_a}\right)} \] Use standard meteorological inputs for each symbol.

How do temperature, humidity, and wind enter the calculation?

Temperature sets saturation vapor pressure \(e_s\). Humidity gives \(e_a\). Wind affects aerodynamic resistance and k. Example for \(e_s\): \(e_s = 0.6108 \exp\!\left(\dfrac{17.27\,T}{T + 237.3}\right)\) with T in °C.

How can I convert evaporation depth to mass or volume?

For areand depth h use \(m = \rho_w\,h\,A\). Volume is \(V = h\,A\). For h in meters and \(\rho_w\) in kg m^{-3} you get kg directly.

How do I determine the transfer coefficient experimentally?

Measure steady evaporation flux and the vapor concentration or pressure difference. Plot flux versus driving difference; slope gives k. Ensure controlled temperature and wind to isolate k.

What are typical evaporation rates and coefficients?

Freshwater open-water evaporation often ranges 2 to 10 \( \text{mm day}^{-1} \). Mass transfer coefficients depend on wind and scale, e.g. \( k_m \sim 10^{-3} \text{ to } 10^{-1} \text{ m s}^{-1} \) in practical situations. Values vary widely by conditions.

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