Evaporation from a Water Surface: Theory and Experiment

 

 

D.G. Kröger^I; G.R. Branfield^II

^IProfessor Emeritus. Department of Mechanical and Mechatronic Engineering. University of Stellenbosch Private Bag X1. 7602 Matieland. E-mail: dgk@sun.ac.za
^IIPost-graduate student. Department of Mechanical and Mechatronic Engineering. University of Stellenbosch. Private Bag X1. 7602 Matieland

 

 

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ABSTRACT

Evaporation from a horizontal water surface that is exposed to the natural environment is analysed. An approximate equation is deduced that predicts the turbulent convection mass transfer or evaporation rate for cases where the water surface temperature is measurably higher than that of the ambient air. An empirical equation is recommended in the case where the temperature difference is relatively small and for application at night. Evaporation rates are measured and the results are found to be in good agreement with predicted values.

Additional keywords: Mass transfer

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Nomenclature

Roman

cMolar concentration of species [mol/m³]

C_fFriction coefficient

c_pSpecific heat [J/kg K]

DDiffusion coefficient [m²/s]

gGravitational acceleration [m/s²]

kThermal conductivity [W/m K]

hHeat transfer coefficient [W/m² K]

I Solar irradiation [W/m²]

mMass flux [kg/sm²]

pPressure [N/m²]

qHeat flux [W/m²]

TTemperature [°C or K]

tTime [s]

v Speed [m/s]

wHumidity ratio [kg vapour/kg dry air]

z Coordinate

Dimensionless numbers

Le Lewis number, k/(ρc_pD)

Pr Prandtl number, μc_p/k

Ra Rayleigh number,

Sc Schmidt number, μ/(ρD)

Greek

αThermal diffusivity, k/(ρc_p)₉ [m²/s] or solar absorptivity

δ Concentration or partial density layer thickness, [m]

θ_ZZenith angle, [°]

μ Dynamic viscosity, [kg/ms]

ρ Density, [kg/m³]

φRelative humidity

Subscripts

aAir

avAir-vapour mixture

aviAir-vapour mixture at initial condition

avoAir-vapour mixture at z = 0

cConcentration

DMass diffusion

enEnergy

expExperimental

hHorizontal

i Initial condition

mUniform mass flux

oAt z = 0

qUniform heat flux

TTemperature

tTime

uUnstable condition

v Vapour

wWater or wind

z Zenith

 

1. Introduction

In 1802 Dalton's classical paper entitled "Experimental essays on the constitution of mixed gases; on the force of steam or vapour from water and other liquids at different temperatures, both in a Torricellian vacuum and in air; on evaporation and on the expansion of gas by heat", was published. In this paper Dalton stated that the rate of evaporation from a water surface is proportional to the difference in vapour pressure at the surface of the water and that in the surrounding air, and furthermore that the wind speed affects this proportionality¹. Subsequently numerous researchers investigated the problem of evaporation on the basis of Dalton's model. A recent critical and comprehensive review of many of the empirical equations employed to predict evaporation rates from water surfaces is presented by Sartori². Other references are listed by Bansal and Xi³ and Tang et al.⁴. It follows from these publications that there was essentially no further more detailed theoretical modelling of the process of evaporation of water from a horizontal surface into the natural environment subsequent to Dalton's publication. Much uncertainty exists and significant discrepancies occur between empirical equations that predict rates of evaporation under different conditions.

 

2. Analysis

In the following analysis an approximate equation is deduced that predicts the convective mass transfer or evaporation rate per unit area from a horizontal water surface exposed to the natural environment. Initially the transfer rate due to natural convection only is deduced and the equation is then extended to make provision for windy (forced convection) conditions.

Consider a stationary semi-infinite fluid (binary mixture consisting of air and water vapour) in which the concentration c_vi of the species of interest (water vapour) is initially uniform.

Beginning with the time t = 0, the concentration at the z = 0 boundary or surface is maintained at a greater level c_vo as shown in figure 1(a).

 

 

Water vapour will diffuse into the semi-infinite medium to form a concentration boundary layer, the thickness of which increases with time.

The mathematical equation of time dependent diffusion in a binary mixture, expressed in terms of the molar concentration c is as follows:

The diffusion flux is driven solely by the concentration gradient strictly in an isothermal and isobaric medium. Nevertheless, equation 1 is a good approximation in many non-isothermal systems, where temperature differences are relatively small. If changes in Kelvin temperature are small the diffusion coefficient D can be assumed to be constant.

Equation 1 is analogous to the time-dependent equation for heat conduction into a semi-infinite solid body i.e.

If the temperature of a semi-infinite solid is initially uniform at T_i and a sudden increase in temperature to T_o occurs at z = 0 as shown in figure 1(b), Schneider⁵ shows that the temperature gradient at z = 0 is given by

An effective heat transfer coefficient can be expressed in terms of this heat flux i.e.

Similarly, by solving equation 2 for the case where the semi-infinite solid at an initial uniform temperature T_i is suddenly exposed to a constant surface heat flux q_q, the latter can, according to Holman⁶, be expressed in terms of an effective surface temperature T_oq as

Although equation 2 is applicable to a solid, it is a good approximation when applied to a thin layer of gas or vapour near a solid surface.

Due to the analogy between mass and heat transfer the solution of equation 1 gives the following relations corresponding to equations 3 to 8 respectively:

If the initial concentration at z = 0 is suddenly increased to

If vapour is generated uniformly at a rate m_vm at z = 0, this mass flux can be expressed in terms of an effective concentration c_vom to give analogous to equation 6

These latter equations are applicable in the region of early developing concentration distribution in a semi-infinite region of air exposed to a water or wet surface. According to Merker⁷ for a Rayleigh number Ra > 1101, unstable conditions prevail with the result that water vapour is transported upwards away from the wetted surface by means of "thermals"as shown in figure 2.

 

 

The generation of such thermals is periodic in time, and both spatial frequency and rate of production are found to increase with an increase in heating rate.

For an analysis of the initial developing vapour concentration distribution near the suddenly wetted surface at z = 0, consider figure 1(a).

The approximate magnitude of the curvature of the concentration profile is the same as the change in slope dc_v /∂z across the relatively small concentration layer thickness or height δ_D i.e.

The approximate magnitude of the term on the right-hand side of equation 1 can be deduced by arguing that the average concentration of the δ_D-thick region increases from the initial value c_vi by a value of (c_vo - c_vi)/2 during the time interval of length t.

It is stressed that these equations are only applicable to the first phase of the heat or mass transfer process (growth of concentration layer) and do not include the second phase during which thermals exist (breakdown of concentration layer). No simple analytical approach is possible during this latter phase, although the mean mass transfer coefficient during the breakdown of the concentration layer will probably not differ much from the first phase. This would mean that the mean mass transfer coefficient over the cycle of conduction or concentration layer growth and breakdown is of approximately the same value as that obtained during the first phase of the cycle.

By following a procedure similar to the above, the analogous problem of heat transfer during natural convection above a heated horizontal surface for a constant surface temperature of T_o can be analysed to find according to Kröger⁸

Note the similarity between equations 23, 24, 25 and 26 respectively. These equations are applicable to natural convection mass and heat transfer respectively.

In the absence of winds, effective values of in equations 23 and 24 and ρ_i in equation 25 and 26 respectively, change with time (t > t_u).

During windy periods (forced convection) evaporation rates generally increase with increasing wind speed. According to the Reynolds-Colburn analogy and the analogy between mass and heat transfer, the following relations exist⁶

In general the rate of mass transfer or evaporation from a horizontal wetted surface at a uniform concentration c_vo is thus

For relatively small temperature differences, the concentrations in equation 29 can be replaced by the partial vapour pressures i.e. c=p_v/R_vT_oi where T_oi = (T_o + T_oi)/2and R_v =461.52 J/KgK. Furthermore, for air-water vapour mixtures Sc ≈ 0.6.

Substitute these values into equation 29 and find

Similarly, if the vapour is generated uniformly at z=0 find the rate of evaporation according to equations 24, 27 and 28 i.e.

Since the thermal conductivity of water is not negligible, it is not possible to achieve a truly uniform heat flux situation. The value of the dimensionless mass transfer coefficient as given by equation 24 may thus be less than 0.243, i.e. it will be some value between 0.243 and 0.155 as given by equation 23. Burger and Kröger⁹ report the results of experiments conducted during analogous heat transfer tests between a low thermal conductivity horizontal surface and the environment. They obtain a value of 0.2106 instead of the theoretical value of 0.243 given in equation 26 and the analogous equation 24. They furthermore obtain a value of C_f= 0.0052 based on a wind speed measured 1 m above the test surface. With these values equation 26 applied over a wetted surface can be extended to become

When density differences are very small and conditions near the surface are relatively stable or at night when T_oq < T_i and the heat flux is uniform they recommend

In cases where T_oq < T_i and h_q according to equation 33 is larger than the value of h_q obtained according to equation 32, the former is applicable. If the above values (0.2106 and C_f= 0.0052) are substituted into equation 31 find

Equation 37 is found to be in good agreement with equations recommended by Tang et al.⁴. This expression is applicable at night and during the day when the value for m_vom is found to be larger that that given by equation 34.

The density of the ambient air is given by

Sartori² lists many empirical correlations that predict the rate of evaporation. The values given by these equations may differ significantly over a range as shown by the shaded area in figure 3. In part this may be due to errors in the determination of the water surface temperature, as well as different vapour concentration distributions above water surfaces having different areas. In figure 3 the rate of evaporation is shown as a function of wetted surface temperature T_o and ambient air temperatures T_i= T_o - 5, a relative humidity of φ = 45 per cent and a wind speed of v_w = 3 m/s at about 1m above the wetted surface. Sartori² recommends the equations proposed by the WMO¹⁰ and McMillan¹¹ as shown in figure 3. Equation 34 is also shown in figure 3 for an ambient pressure of p = 10⁵ N/m².

 

 

3. Experimental Apparatus and Procedure

To experimentally determine the rate of evaporation from a water surface exposed to the natural environment, the use of an evaporation pan as shown in figure 4 was employed. It consisted of a 50 mm thick horizontal polystyrene plate having an effective upper surface area of approximately 0.97 m², which was painted with a waterproof matt black paint. A 3 mm high bead of silicon sealant was run along the perimeter of the pan, with the purpose of containing a 1 - 2 mm deep layer of water. Five type-T thermocouples were embedded flush in the surface of the plate with the purpose of measuring the water temperature. Four of the five were positioned in the corners of the pan, 150 mm from adjacent sides, while the fifth was placed in the centre. The pan was surrounded by a dry stony surface.

 

 

The wind speed, ambient air- and dew-point temperatures were measured with the aid of a weather station at a height of 1 m above the ground⁸. A Kipp and Zonen pyranometer was used to measure the total incident solar radiation on the surface, while diffuse solar radiation readings were measured by shielding the pyranometer from direct sunlight for a period long enough for stable measurements to be taken. All tests were conducted on clear sunny days.

The evaporation rate from the water surface was measured by adding consecutive quantities of water (500 ml) to the evaporation pan at a temperature similar to that of the remaining water, and recording the period of time taken for each to evaporate. With the average temperature of the water known during the particular period, the mass flowrate or evaporation rate could be determined. This evaporation rate will be denoted by m_exp.

By applying an energy balance to the surface of a film of water on an insulated base that is exposed to the natural environment as shown schematically in figure 4, it is possible to obtain an expression for the rate of evaporation per unit surface area i.e.

where I_hα_w represents the solar radiation absorbed by the surface, ε_wσ(T_oq⁴- T_sky⁴) is the sky radiation while h_q(T_oq- T_i) is the convective heat transfer rate. The absorptivity of solar radiation at the surface of the water is given by Holman⁶ and can be approximated by the following equation

where θ_Z is the zenith angle measured in degrees.

The surface emits radiation to a sky temperature T_sky, which can be calculated according to

Berdahl and Fromberg¹² express the emissivity of the sky ε_sky during the day as

where T_dp is the dew-point temperature measured in degrees Celsius.

The heat transfer coefficient h_q is given by equation 32 while equation 33 is applicable when T_oq > T¡ or when the value of h_q according to equation 33 during the day is larger than the value obtained according to equation 32. The heat transfer through the polystyrene plate is negligible.

According to Holman⁶ the intensity of solar radiation in clear water at a distance z from the surface is given by

An analysis was performed on equation 34 to determine the sensitivity of the expression to an error of ±1°C in measured water temperature. It was found that equation 34 is very sensitive to water temperature early in the morning and towards evening, while a difference in the predicted evaporation rate of approximately 10% was found during the day. These findings suggest that extreme care should be taken when measuring the water surface temperature T_oq.

Tests were conducted on the 13^th and 14^th of April 2005 at the University of Stellenbosch Solar Energy Laboratory (33.93° S, 18.85° E), at an altitude of 100 m above sea level with an ambient pressure of 100990 Pa. The associated weather data is given in figures 5 and 6 and the results are shown in figures 7 and 8.

 

 

 

 

 

 

 

 

Figure 7 shows a comparison between the experimental and theoretical evaporation rates between approximately 8 h and 15 h (14^th April). Note that no wind was present until roughly 10 h; figure 7 shows almost 'stunted' evaporation rates until this point which may be due to the accumulation of moist air above the water surface, which lead to an increase in ρ_vi and thus reduced evaporation. Figure 8 shows evaporation rates over a period of 24 hours. Note that a negative evaporation rate, or condensation in the form of dew is found to occur during the night when the water surface temperature T_oq is less than the dew-point temperature T_dp.

Table 1 compares the experimentally measured quantity of water evaporated between 8.319 h and 14.752 h, with the values predicted by equations 34 and 45. The error margin is calculated with respect to the experimentally measured quantity.

 

 

Results from above show that equation 34 predicts the measured evaporation rate most accurately, while equation 45 is also within reasonable accuracy.

Both equations predict condensation on the water surface during night-time operation, while the magnitudes of the predicted rates seem to be of the same order.

 

4. Conclusion

Horizontal natural water or wetted surfaces (surface area of approximately one square metre) having a relatively low thermal conductivity and exposed to solar radiation, transfer both mass and heat to the natural environment. Under these conditions the rate of evaporation at the wetted surface will be determined primarily by the heat flux due to solar radiation and can thus be evaluated according to equation 34 for cases where ρ_avi > ρ_avo, the free stream vapour concentration c_vi is uniform and the wind speed is less than 4 m/s at a height of 1 m above the surface⁸. Although equation 34 is an approximation it is based on a sound theoretical approach. Equation 37 is recommended for use during the night-time and when it gives a value of m_vom that is larger than that given by equation 34.

 

References

1. Dalton J, Experimental essays on the constitution of mixed gases on the force of steam or vapour from water and other liquids at different temperatures both in a Torricellian vacuum and in air; on evaporation and on the expansion of gases by heat, Memoirs and Proceedings of the Manchester Literary and Philosophical Society 1802, 5(11), 5-11.         [ Links ]

2. Sartori, E, A critical review on equations employed for the calculation of the evaporation rate from free water surfaces, Solar Energy, 2000, 68(1), 77-89.         [ Links ]

3. Bansal PK and XG, Aunified empirical correlation for evaporation of water at low air velocities, International Communications in Heat Transfer, 1998, 25(2), 183- 190.         [ Links ]

4. Tang Y, Etzion I A and Meir IA, Estimates of clear night sky emissivity in the Negev Highlands, Israel, Energy Conversion and Management, 2004, 45, 1831 - 1843.         [ Links ]

5. Schneider PJ, Conduction Heat Transfer, Addison-Wesley, Reading, Massachusetts, 1955.

6. Holman JP, Heat Transfer, McGraw Hill, New York, 1986.

7. Merker GP, Konvektiewe Wärmeübertragung, Springer-Verlag, Berlin, 1987.

8. Kröger DG, Convection heat transfer between a horizontal surface and the natural environment, R&D Journal of the South African Institution of Mechanical Engineering, November 2002, 18(3), 49-53.         [ Links ]

9. Burger M and Kröger DG, Experimental convection heat transfer coefficient between a horizontal surface and the natural environment, (submitted for publication).

10. WMO-World Meteorological Organization, Measurement and estimation of evaporation and evapotranspiration, In Technical Report, 83, Working Group on Evaporation Measurement, Geneva, 1966, 92.

11. McMillan W, Heat dispersal - Lake Trawsfynyd cooling studies, Symposium on Freshwater Biology and Electrical Power Generation, Part 1, 1971, 41 -80.

12. Berdahl P and Fromberg R, The thermal radiance of clear skies, Solar Energy, 1982, 32(5), 299-314.         [ Links ]

 

 

Received 1 February 2007
Revised form 26 September 2007
Accepted 8 October 2007