5 Hygrometry The objective of atmospheric humidity measurements is to determine the amount of water vapor present in the atmosphere by weight, by volume, by partial pressure, or by a fraction (percentage) of the saturation (equilibrium) vapor pressure with respect to a plane surface of pure water. The measurement of atmospheric humidity in the field has been and continues to be troublesome. It is especially difficult for automatic weather stations where low cost, low power consumption, and reliability are common constraints. 5.1 Water Vapor Pressure Pure water vapor in equilibrium with a plane surface of pure water exerts a pressure designated efs. This pressure is a function of the temperature of the vapor and liquid phases and can be obtained by integration of the Clausius-Clapeyron equation, assuming linear dependence of the latent heat of vaporization on temperature, L = L0[l+a(T-T0)], , , \U(T-TQ , (T\ tt(r^r0)Y| (5.1) where T0 = 273.15 K, L0 = 2.5008 x 106 J kg"1, the latent heat of water vapor at T0, Rv = 461.51 Jkg^K"1, the gas constant for water vapor, efs0 = 611.21 Pa, the equilibrium water vapor pressure at T = T0, and a = —9.477 x 10~4K_1 = average rate of change coefficient for the latent heat of water vapor with respect to temperature. 86 Hygrometry 87 Table 5-1 Coefficients for the empirical equation 5.2 for equilibrium vapor pressure over a plane surface of pure water and over ice. Coefficient Water Ice C3 c4 c5 <% c7 Since water vapor is not a perfect gas, the above equation is not an exact fit. The vapor pressure as a function of temperature has been determined by numerous experiments. Wexler (1976, 1977) fitted an empirical equation to the experimental vapor pressure data, -2991.272 0.0 -6017.0128 -5865.3696 18.87643854 22.241 033 -0.028354721 0.013749042 0.17838301 x 10" -4 -0.34031775 x 10~4 -0.841 50417 x 10' -9 0.26967687 x 10"7 0.444125 43 x 10" -12 0.0 2.858487 0.6918651 í Cl C2 (|| + y + c3 + c*T + c5 T2 +c6 T3 +c7 T4 +c8lnTj (5.2) (T in K, e^ in Pa) and the coefficients for vapor pressure over water and over ice are given in table 5-1. Both eqns. 5.1 and 5.2 are cumbersome; an equation that is readily inverted but with sufficient accuracy would be preferable. Buck (1981) developed an equation that is easy to use and sufficiently precise over the temperature range —30 to 50°C, és = 6.1121 expf 17,502M (5.3) where Tis in degrees Celsius and e's is in units of hPa (or mb). Equations 5.1, 5.2, and 5.3 are contrasted in table 5-2 and fig. 5-1. The error in eqn. 5.1 is tolerable but eqn. 5.3 is preferred because it is easier to invert to obtain the dew-point temperature given the ambient vapor pressure. As noted above, the term equilibrium vapor pressure is more accurate but the term saturation vapor pressure is commonly used. We will use the term saturation vapor pressure. Contrary to Dalton's1 law of partial pressures, the total air pressure does have a small affect on the saturation vapor pressure; this is called the enhancement effect (Buck, 1981); see table 5-3. Saturation vapor pressure in a mixture of dry air and water vapor (moist air) is the saturation vapor pressure of pure water vapor times the enhancement factor: es = efs{T)f{T,p). There is a pressure effect and a weak temperature dependence. For p > 800 hPa, we can use / = 1.004. The enhancement factor has been incorporated into the following equations for the vapor pressure, and so eqn. 5.3 becomes / 17.5027 \ , x es = 6.1365 exp--------------- (5.4) FV240.97 + 77 v J and the equilibrium vapor pressure over an ice surface is Table 5-2 Comparison of the theoretical equation (5.1) for water vapor pressure with the expression obtained from experimental results (5.2) and the more convenient approximation (5.3). Temperature <°C) Experimental results, eqn. 5.2 (hPa) Error in Buck approx. eqn. 5.3 (hPa) Error in eqn. 5.1 (hPa) 0 30 50 6.1121 42.4520 123.4476 0.0 -0.0169 0.2447 0.0 -0.0789 -0.5831 Table 5-3 Enhancement factor for various temperatures and pressures. T(°C) Enhancement factor f (dimensionless) p = 1000 hPa p = 500 hPa p = 250 hPa -50 -40 -30 -20 -10 0 10 20 30 40 50 1.0058 1.0029 1.0014 1.0052 1.0026 1.0013 1.0047 1.0024 1.0012 1.0044 1.0022 1.0012 1.0041 1.0022 1.0012 1.00395 1.00219 1.00132 1.00388 1.00229 1.004 1.00251 1.00426 1.00284 1.00467 1.00323 1.00519 -50 -40 -30 -20 -10 0 10 20 30 40 50 Temperature (C) Fig. 5-1 Error in the Buck approximation. Hygrometry 89 (22 452T \ ------:--------™ (5.5) 272.55 +T/ v ' where, as before, T is in units of °C for both of the above equations. Water vapor saturation pressure varies over two orders of magnitude in the normal temperature range; see fig. 5-2. On the basis of this figure, one would expect the accuracy of almost any humidity instrument to decrease with decreasing temperature. Figure 5.2 can be used to illustrate several humidity relationships. Let point A represent ambient temperature and vapor pressure. Then the saturation vapor pressure is es (point B). If the air parcel were cooled, at constant pressure, until condensation just starts to occur, the new air temperature would be the dew-point temperature 7d and the ambient vapor pressure would be unchanged and would now be equal to the saturation vapor pressure at Td (point D). Starting at point A again, a thermal bulb covered with water would be cooled by evaporation and the vapor pressure in the immediate vicinity would increase, due to the increased evaporation rate of water molecules, until the temperature of the wet bulb becomes the wet-bulb temperature Tw and the new vapor pressure would be the saturation vapor pressure at Tw, esw (point C). 5.2 Definitions There are many variables commonly encountered in the study of humidity. Absolute humidity, dV9 is the ratio of the mass of water vapor mv to the total volume of moist air V in units of kgm-3. Dew-point temperature, Td) is the temperature at which ambient water vapor condenses. The frost-point temperature, Tp is the temperature at which ambient water -30 -20 -10 0 10 20 30 40 50 Temperature (C) Fig. 5-2 Saturation vapor pressure as a function of temperature. Inset shows saturation vapor pressure with respect to water (top curve) and with respect to ice (bottom curve) for T < 0°C. 90 Meteorological Measurement Systems vapor freezes. The dew- and frost-point temperatures can be obtained from the ambient vapor pressure by inverting eqns. 5.4 and 5.5: Td = 240.97 ln(e/6.1365)/(17.502 - ln(e/6.1365)) Tf = 272.55 ln(e/6.1359)/(22.452 - ln(e/6.1359)) Mixing ratio, w, is the ratio of the mass of water vapor mv to the mass of dry air m^ Relative humidity, U, is defined as the ratio, expressed as a percentage, of the actual vapor pressure e to the saturation vapor pressure es at the air temperature T: U = 100e/es (5.7) This definition always uses saturation vapor pressure with respect to a plane surface of pure water, even for temperatures below freezing. Some of the earliest humidity sensors, and still the most common, are the class of sorption sensors which, as will be shown later, generate an output proportional to relative humidity. Specific humidity, q, also known as the mass concentration, is the ratio of the mass of water vapor mv to the mass of moist air, mv + md. Temperature or dry-bulb temperature is the ambient air temperature T as measured, for example, by the dry-bulb thermometer of a Psychrometer. Vapor pressure, e, is the partial pressure of water vapor expressed in hPa. Virtual temperature, Tv, is the temperature that dry air would have if the dry air had the same density as moist air at the same pressure. Tv > T: T T =----------------------- 1--(1-0.622) Wet-bulb temperature, Tw, is the temperature indicated by the wet bulb of a Psychrometer, that is, the temperature of a sensor covered with pure water that is evaporating freely into an ambient air stream. The following relations are useful approximations that are sufficiently accurate for most meteorological applications. The temperature is in degrees Celsius. w = 0.622e/(p - e) e = wp/(0.622 + w) q — w/(l + w); when e <ší p, w ^ q ^ 0.622 e/p dv = 0.2167e/(t + 273.15) kgm"3. The formulae for mixing ratio w and specific humidity q are dimensionless; from the definition of these variables, the units are kg/kg. Frequently, w and q are multiplied by 1000 because it is easier to write 15.2 than 0.0152, and then the assigned units are g/kg. The constant 0.622 in the expression for w is the ratio of the gas constant for dry air to the gas constant for water vapor. EXAMPLE Given p = 1000 hPa, T = 35.00°C and e = 24.85 hPa, find the relative humidity and the dew-point temperature. Compute the saturation vapor pressure using eqn. 5.4: (5.6)