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The methods for calculating evapotranspiration from meteorological data require various climatological and physical parameters. Some of the data are measured directly in weather stations. Other parameters are related to commonly measured data and can be derived with the help of a direct or empirical relationship. This chapter discusses the source, measurement and computation of all data required for the calculation of the reference evapotranspiration by means of the FAO Penman-Monteith method. Different examples illustrate the various calculation procedures. Appropriate procedures for estimating missing data are also provided.Meteorological data can be expressed in several units. Conversion factors between various units and standard S. I. units are given in Annex 1. Climatic parameters, calculated by means of the equations presented in this chapter are tabulated and displayed for different meteorological conditions in Annex 2. Only the standardized relationships are presented in this chapter. The background of certain relationships and more information about certain procedures are given in Annex 3. Annexes 4, 5 and 6 list procedures for the statistical analysis, assessment, correction and completion of partial or missing weather data.Meteorological factors determining ET Solar radiation Air temperature Air humidity Wind speed
Several relationships are available to express climatic parameters. The effect of the principal weather parameters on evapotranspiration can be assessed with the help of these equations. Some of the relationships require parameters which express a specific characteristic of the atmosphere. Before studying the four principal weather parameters, some atmospheric parameters will be discussed.Atmospheric pressure (P)The atmospheric pressure, P, is the pressure exerted by the weight of the earth's atmosphere. Evaporation at high altitudes is promoted due to low atmospheric pressure as expressed in the psychrometric constant. The effect is, however, small and in the calculation procedures, the average value for a location is sufficient. A simplification of the ideal gas law, assuming 20C for a standard atmosphere, can be employed to calculate P: (7)whereP atmospheric pressure [kPa],z elevation above sea level [m],Values for atmospheric pressure as a function of altitude are given in Annex 2 (Table 2.1).Latent heat of vaporization (l)The latent heat of vaporization, l, expresses the energy required to change a unit mass of water from liquid to water vapour in a constant pressure and constant temperature process. The value of the latent heat varies as a function of temperature. At a high temperature, less energy will be required than at lower temperatures. As l varies only slightly over normal temperature ranges a single value of 2.45 MJ kg-1 is taken in the simplification of the FAO Penman-Monteith equation. This is the latent heat for an air temperature of about 20C.Psychrometric constant (g)The psychrometric constant, g, is given by: (8)whereg psychrometric constant [kPa C-1],P atmospheric pressure [kPa],l latent heat of vaporization, 2.45 [MJ kg-1],cp specific heat at constant pressure, 1.013 10-3 [MJ kg-1 C-1],e ratio molecular weight of water vapour/dry air = 0.622.The specific heat at constant pressure is the amount of energy required to increase the temperature of a unit mass of air by one degree at constant pressure. Its value depends on the composition of the air, i.e., on its humidity. For average atmospheric conditions a value cp = 1.013 10-3 MJ kg-1 C-1 can be used. As an average atmospheric pressure is used for each location (Equation 7), the psychrometric constant is kept constant for each location. Values for the psychrometric constant as a function of altitude are given in Annex 2 (Table 2.2).EXAMPLE 2. Determination of atmospheric parameters.Determine the atmospheric pressure and the psychrometric constant at an elevation of 1800 m.With:z =1800mFrom Eq. 7:P = 101.3 [(293 - (0.0065) 1800)/293]5.26 =81.8kPaFrom Eq. 8:g = 0.665 10-3 (81.8) =0.054kPa C-1The average value of the atmospheric pressure is 81.8 kPa.The psychrometric constant, g, is 0.054 kPa/C.Air temperatureAgrometeorology is concerned with the air temperature near the level of the crop canopy. In traditional and modem automatic weather stations the air temperature is measured inside shelters (Stevenson screens or ventilated radiation shields) placed in line with World Meteorological Organization (WMO) standards at 2 m above the ground. The shelters are designed to protect the instruments from direct exposure to solar heating. The louvered construction allows free air movement around the instruments. Air temperature is measured with thermometers, thermistors or thermocouples mounted in the shelter. Minimum and maximum thermometers record the minimum and maximum air temperature over a 24-hour period. Thermographs plot the instantaneous temperature over a day or week. Electronic weather stations often sample air temperature each minute and report hourly averages in addition to 24-hour maximum and minimum values.Due to the non-linearity of humidity data required in the FAO Penman-Monteith equation, the vapour pressure for a certain period should be computed as the mean between the vapour pressure at the daily maximum and minimum air temperatures of that period. The daily maximum air temperature (Tmax) and daily minimum air temperature (Tmin) are, respectively, the maximum and minimum air temperature observed during the 24-hour period, beginning at midnight. Tmax and Tmin for longer periods such as weeks, 10-day's or months are obtained by dividing the sum of the respective daily values by the number of days in the period. The mean daily air temperature (Tmean) is only employed in the FAO Penman-Monteith equation to calculate the slope of the saturation vapour pressure curves (D) and the impact of mean air density (Pa) as the effect of temperature variations on the value of the climatic parameter is small in these cases. For standardization, Tmean for 24-hour periods is defined as the mean of the daily maximum (Tmax) and minimum temperatures (Tmin) rather than as the average of hourly temperature measurements. (9)The temperature is given in degrees Celsius (C) or Fahrenheit (F). The conversion table is given in Annex 1. In some calculation procedures, temperature is required in Kelvin (K), which can be obtained by adding 273.16 to the temperature expressed in degrees Celsius (in practice K = C + 273.16). The Kelvin and Celsius scale have the same scale interval.Air humidity Concepts Measurement Calculation procedures
ConceptsThe water content of the air can be expressed in several ways. In agrometeorology, vapour pressure, dewpoint temperature and relative humidity are common expressions to indicate air humidity.Vapour pressureWater vapour is a gas and its pressure contributes to the total atmospheric pressure. The amount of water in the air is related directly to the partial pressure exerted by the water vapour in the air and is therefore a direct measure of the air water content.In standard S. I. units, pressure is no longer expressed in centimetre of water, millimetre of mercury, bars, atmosphere, etc., but in pascals (Pa). Conversion factors between various units and Pa are given in Annex 1. As a pascal refers to a relatively small force (1 newton) applied on a relatively large surface (1 m2), multiples of the basic unit are often used. In this handbook, vapour pressure is expressed in kilopascals (kPa = 1000 Pa).When air is enclosed above an evaporating water surface, an equilibrium is reached between the water molecules escaping and returning to the water reservoir. At that moment, the air is said to be saturated since it cannot store any extra water molecules. The corresponding pressure is called the saturation vapour pressure (e(T)). The number of water molecules that can be stored in the air depends on the temperature (T). The higher the air temperature, the higher the storage capacity, the higher its saturation vapour pressure (Figure 11).As can be seen from Figure 11, the slope of the curve changes exponentially with temperature. At low temperatures, the slope is small and varies only slightly as the temperature rises. At high temperatures, the slope is large and small changes in T result in large changes in slope. The slope of the saturation vapour pressure curve, D, is an important parameter in describing vaporization and is required in the equations for calculating ETo from climatic data.FIGURE 11. Saturation vapour pressure shown as a function of temperature: e(T) curveFIGURE 12. Variation of the relative humidity over 24 hours for a constant actual vapour pressure of 2.4 kPaThe actual vapour pressure (ea) is the vapour pressure exerted by the water in the air. When the air is not saturated, the actual vapour pressure will be lower than the saturation vapour pressure. The difference between the saturation and actual vapour pressure is called the vapour pressure deficit or saturation deficit and is an accurate indicator of the actual evaporative capacity of the air.Dewpoint temperatureThe dewpoint temperature is the temperature to which the air needs to be cooled to make the air saturated. The actual vapour pressure of the air is the saturation vapour pressure at the dewpoint temperature, The drier the air, the larger the difference between the air temperature and dewpoint temperature.Relative humidityThe relative humidity (RH) expresses the degree of saturation of the air as a ratio of the actual (ea) to the saturation (e(T)) vapour pressure at the same temperature (T): (10)Relative humidity is the ratio between the amount of water the ambient air actually holds and the amount it could hold at the same temperature. It is dimensionless and is commonly given as a percentage. Although the actual vapour pressure might be relatively constant throughout the day, the relative humidity fluctuates between a maximum near sunrise and a minimum around early afternoon (Figure 12). The variation of the relative humidity is the result of the fact that the saturation vapour pressure is determined by the air temperature. As the temperature changes during the day, the relative humidity also changes substantially.MeasurementIt is not possible to directly measure the actual vapour pressure. The vapour pressure is commonly derived from relative humidity or dewpoint temperature.Relative humidity is measured directly with hygrometers. The measurement is based on the nature of some material such as hair, which changes its length in response to changes in air humidity, or using a capacitance plate, where the electric capacitance changes with RH. Vapour pressure can be measured indirectly with psychrometers which measure the temperature difference between two thermometers, the so-called dry and wet bulb thermometers. The dry bulb thermometer measures the temperature of the air. The bulb of the wet bulb thermometer is covered with a constantly saturated wick. Evaporation of water from the wick, requiring energy, lowers the temperature of the thermometer. The drier the air, the larger the evaporative cooling and the larger the temperature drop. The difference between the dry and wet bulb temperatures is called the wet bulb depression and is a measure of the air humidity.The dewpoint temperature is measured with dewpoint meters. The underlying principle of some types of apparatus is the cooling of the ambient air until dew formation occurs. The corresponding temperature is the dewpoint temperature.Relative humidity and dewpoint temperature data are notoriously plagued by measurement errors. Measurement error is common for both older hygrothermograph types of instruments and for the more modem electronic instruments. These instruments are described in Annex 5. Great care should be made to assess the accuracy and integrity of RH and dewpoint data. The user is encouraged to always compare computed dewpoint temperatures to daily minimum air temperatures, as described at the end of this chapter and in Annexes 5 and 6. Frequently, it is better to utilize a dewpoint temperature that is predicted from daily minimum air temperature, rather than to use unreliable relative humidity measurements. The user is encouraged to utilize good judgement in this area.Calculation proceduresMean saturation vapour pressure (es)As saturation vapour pressure is related to air temperature, it can be calculated from the air temperature. The relationship is expressed by: (11)wheree(T) saturation vapour pressure at the air temperature T [kPa],T air temperature [C],exp[..] 2.7183 (base of natural logarithm) raised to the power [..].Values of saturation vapour pressure as a function of air temperature are given in Annex 2 (Table 2.3). Due to the non-linearity of the above equation, the mean saturation vapour pressure for a day, week, decade or month should be computed as the mean between the saturation vapour pressure at the mean daily maximum and minimum air temperatures for that period: (12)Using mean air temperature instead of daily minimum and maximum temperatures results in lower estimates for the mean saturation vapour pressure. The corresponding vapour pressure deficit (a parameter expressing the evaporating power of the atmosphere) will also be smaller and the result will be some underestimation of the reference crop evapotranspiration. Therefore, the mean saturation vapour pressure should be calculated as the mean between the saturation vapour pressure at both the daily maximum and minimum air temperature.EXAMPLE 3. Determination of mean saturation vapour pressureThe daily maximum and minimum air temperature are respectively 24.5 and 15C.Determine the saturation vapour pressure for that day.From Eq. 11e(Tmax) = 0.6108 exp[17.27(24.5)/(24.5 + 237.3)]3.075kPaFrom Eq. 11e(Tmin) = 0.6108 exp[17.27(15)/(15 + 237.3)]1.705kPaFrom Eq. 12es = (3.075 + 1.705)/22.39kPaNote that for temperature 19.75C (which is Tmean). e(T) =2.30kPaThe mean saturation vapour pressure is 2.39 kPa.Slope of saturation vapour pressure curve (D )For the calculation of evapotranspiration, the slope of the relationship between saturation vapour pressure and temperature, D, is required. The slope of the curve (Figure 11) at a given temperature is given by. (13)whereD slope of saturation vapour pressure curve at air temperature T [kPa C-1],T air temperature [C],exp[..] 2.7183 (base of natural logarithm) raised to the power [..].Values of slope D for different air temperatures are given in Annex 2 (Table 2.4). In the FAO Penman-Monteith equation, where D occurs in the numerator and denominator, the slope of the vapour pressure curve is calculated using mean air temperature (Equation 9).Actual vapour pressure (ea) derived from dewpoint temperatureAs the dewpoint temperature is the temperature to which the air needs to be cooled to make the air saturated, the actual vapour pressure (ea) is the saturation vapour pressure at the dewpoint temperature (Tdew) [C], or: (14)Actual vapour pressure (ea) derived from psychrometric dataThe actual vapour pressure can be determined from the difference between the dry and wet bulb temperatures, the so-called wet bulb depression. The relationship is expressed by the following equation:ea = e (Twet) - g psy (Tdry - Twet) (15)whereea actual vapour pressure [kPa],e(Twet) saturation vapour pressure at wet bulb temperature [kPa],g psy psychrometric constant of the instrument [kPa C-1],Tdry-Twet wet bulb depression, with Tdry the dry bulb and Twet the wet bulb temperature [C].The psychrometric constant of the instrument is given by:g psy = apsy P (16)where apsy is a coefficient depending on the type of ventilation of the wet bulb [C-1], and P is the atmospheric pressure [kPa]. The coefficient apsy depends mainly on the design of the psychrometer and rate of ventilation around the wet bulb. The following values are used:apsy =0.000662for ventilated (Asmann type) psychrometers, with an air movement of some 5 m/s,0.000800for natural ventilated psychrometers (about 1 m/s),0.001200for non-ventilated psychrometers installed indoors.EXAMPLE 4. Determination of actual vapour pressure from psychrometric readingsDetermine the vapour pressure from the readings of an aspirated psychrometer in a location at an elevation of 1200 m. The temperatures measured by the dry and wet bulb thermometers are 25.6 and 19.5C respectively.From Eq. 7 (Table 2.1), at:z=1200mThen:P=87.9kPaFrom Eq. 11 (Table 2.3), forTwet =19.5CThen:e(Twet) =2.267kPaVentilated psychrometerapsy =0.000662C-1From Eq. 15:ea = 2.267 - 0.000662 (87.9) (25.6 - 19.5) =1.91kPaThe actual vapour pressure is 1.91 kPa.Actual vapour pressure (ea) derived from relative humidity dataThe actual vapour pressure can also be calculated from the relative humidity. Depending on the availability of the humidity data, different equations should be used. For RHmax and RHmin: (17)whereea actual vapour pressure [kPa],e(Tmin) saturation vapour pressure at daily minimum temperature [kPa],e(Tmax) saturation vapour pressure at daily maximum temperature [kPa],RHmax maximum relative humidity [%],RHmin minimum relative humidity [%].For periods of a week, ten days or a month, RHmax and RHmin are obtained by dividing the sum of the daily values by the number of days in that period. For RHmax:When using equipment where errors in estimating RHmin can be large, or when RH data integrity are in doubt, then one should use only RHmax: (18) For RHmean:In the absence of RHmax and RHmin, another equation can be used to estimate ea: (19)where RHmean is the mean relative humidity, defined as the average between RHmax and RHmin. However, Equation 19 is less desirable than are Equations 17 or 18.EXAMPLE 5. Determination of actual vapour pressure from relative humidityGiven the following daily minimum and maximum air temperature and the corresponding relative humidity data:Tmin = 18C and RHmax = 82%Tmax = 25C and RHmin = 54%Determine the actual vapour pressure.From Eq. 11 (Table 2.3), at:Tmin =18CThen:e(Tmin) =2.064kPaFrom Eq. 11 (Table 2.3), at:Tmax =25CThen:e(Tmax) =3.168kPaFrom Eq. 17:ea = [2.064 (82/100) + 3.168 (54/100)] =1.70kPaNote that when using Eq. 19:ea =1.78kPaVapour pressure deficit (es - ea)The vapour pressure deficit is the difference between the saturation (es) and actual vapour pressure (ea) for a given time period. For time periods such as a week, ten days or a month es is computed from Equation 12 using the Tmax and Tmin averaged over the time period and similarly the ea is computed with one of the equations 4 to 19, using average measurements over the period. As stated above, using mean air temperature and not Tmax and Tmin in Equation 12 results in a lower estimate of es, thus in a lower vapour pressure deficit and hence an underestimation of the ETo (see Box 7). When desired, es and ea for long time periods cal also be calculated as averages of values computed for each day of the period.EXAMPLE 6. Determination of vapour pressure deficitDetermine the vapour pressure deficit with the data of the previous example (Example 5).From Example 5:e(Tmin) =2.064kPae(Tmax) =3.168kPaea =1.70kPaes - ea = (2.064 + 3.168)/2-1.70 =0.91kPaThe vapour pressure deficit is 0.91 kPa.BOX 7. Calculation sheet for vapour pressure deficit (es - ea)Saturation vapour pressure: es (Eq. 11 or Table 2.3)TmaxCkPaTminCkPasaturation vapour pressure es = [e(Tmax) + e(Tmin)]/2 Eq. 12kPaActual vapour pressure: ea1. ea derived from dewpoint temperature (Eq. 14 or Table 2.3)TdewCkPaOR 2. ea derived from maximum and minimum relative humidityRHmax%kPaRHmin%kPaea = [e(Tmin) RHmax/100 + e(Tmax) RHmin/100]/2 Eq. 17kPaOR 3. ea derived from maximum relative humidity (errors in RHmin)RHmax%ea = e(Tmin) RHmax/100 Eq. 18kPaOR 4. ea derived from mean relative humidity (less recommended)RHmean%ea = es (RHmean)/100 Eq. 19kPaVapour pressure deficit: (es - ea)kPaRadiation Concepts Units Measurement Calculation procedures
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