**UFBA**

# Livro Instrumentação - 1798 05

(Parte **1** de 5)

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 e's. 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 =

Hygrometry where T0 = 273.15K, L0 = 2.5008 x 106Jkg a, the latent heat of water vapor at TQ, Rv = 461.51Jkg~1K~1, the gas constant for water vapor, e'sQ = 611.21 Pa, the equi- librium water vapor pressure at T = T0, and a — —9.477 x 10~4 KT1 = average rate of change coefficient for the latent heat of water vapor with respect to tempera- ture.

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

Ci C

io-4 io-7

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,

(T in K, e's 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,

where Tis in degrees Celsius and e 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 Barton'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 = e's(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 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)

Table 5-3 Enhancement factor for various temperatures and pressures.

Enhancement factor f (dimensionless) p = 1000hPa p = 500hPa p = 250 hPa

Fig. 5-1 Error in the Buck approximation.

Hygrometry 89 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 pres- sure 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

Tj 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).

There are many variables commonly encountered in the study of humidity.

Absolute humidity, dv, is the ratio of the mass of water vapor mv to the total volume of moist air V in units of kg m3 .

Dew-point temperature, Tj, is the temperature at which ambient water vapor condenses. The frost-point temperature, Tf, is the temperature at which ambient water

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:

Mixing ratio, w, is the ratio of the mass of water vapor mv to the mass of dry air md. 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:

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 mea- sured, 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:

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 eva- porating 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.

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.

Given p = 1000hPa, T = 35.00°C and e = 24.85hPa, find the relative humidity and the dew-point temperature. Compute the saturation vapor pressure using eqn. 5.4:

Hygrometry 91

Instruments that respond directly to relative humidity and those that indicate the dew-point temperature are prevalent. Conversion of error expressed in relative humidity to error in dew-point temperature is a nonlinear process, as shown in figs. 5.3 and 5.4.

In fig. 5-3, an error of ATd — 0.20°C is converted to an equivalent error in relative humidity. This situation would arise if the user wished to calibrate a relative humid- ity sensor using an instrument that measured the dew-point temperature. The dewpoint instrument error would be expressed in terms of the dew-point temperature and the user would need to know the equivalent error in percent RH. This is a function of relative humidity and of temperature, so there is a family of curves for temperatures from —30°C to 50°C. Examine the relationships shown in fig. 5-2 to understand the role of temperature in the error conversion.

Given air temperature T — 20°C, es = 23.47hPa (using eqn. 5.4). If relative humidity U = 80%, then e = 18.77hPa and Td = 16.44°C. If the dew-point sensor has an error of -t-0.20°C, the dew-point temperature indicated by the sensor

Fig. 5-3 Conversion of a 0.2°C error in dew-point to relative humidity.

Use eqn. 5.7 to obtain relative humidity:

Then we can obtain the dew-point temperature by inverting eqn. 5.4 (to obtain 5.6):

U=10*24.85/56.48=4.0%

92 Meteorological Measurement Systems

Fig. 5-4 Conversion of a 2% error in relative humidity to the equivalent error in dew-point temperature for T = -10, 0, 10, 20, 30, and 40°C.

will be Tj = 16.64°C. If a line were drawn, on fig. 5-3, vertically from the x-axis value of 80% to the curve for T = 20°C and then horizontally to the left, it would intersect the y-axis at a relative humidity error of about 1% (absolute value). Verify this using eqn. 5.4 to convert Td - 16.64°C to e = 18.53hPa; thus the calculated U — 78.95%, which means a relative humidity error of —1.05%.

If the sensor error had been -20°C, the indicated Td = 16.24°C; then e= 19-OlhPa and the calculated U = 80.9% for a relative humidity error of

+0.9%. The y-axis of fig. 5-3 represents absolute error, implying symmetry for positive and negative dew-point temperature errors. However, the error is not exactly symmetrical because the slope of the vapor pressure curve, fig. 5-2, is not constant.

A sorption sensor, as will be seen later, generates an output proportional to relative humidity. If such a sensor made an error in relative humidity, that error could be converted to the equivalent dew-point temperature error using the curves in fig. 5-4, which are plotted for air temperatures from — 10°C to 40°C. Again, the error conversion is a function of temperature.

Given air temperature T = 20°C, then the saturation vapor pressure is es = 23.47 hPa. If the actual relative humididy is U = 80%, then vapor pressure e = 18.7 hPa and dew-point temperature Td = 16.44°C. But if the sensor has a +2% error in RH, it reports 17 = 82%. This would be equivalent to e=19.24hPa and

Td = 16.83°C. If we drew a vertical line on fig. 5-4 from the x-axis value

Td = 16.83°C to the 20°C isotherm and from there horizontally to the y-axis, the value found there would be +0.39°C. In this case, a 2% error in RH is equivalent to a dew-point error of 0.39°C.

Hygrometry 93 5.3 Methods for Measuring Humidity

Wexler (1970) defined six classes of hygrometric methods based on physical prin- ciples: removal of water vapor from moist air, addition of water vapor to moist air, equilibrium sorption of water vapor, attainment of vapor—liquid or vapor—solid equilibrium, measurement of physical properties of moist air, and by chemical reactions.

5.3.1 Removal of Water Vapor from Moist Air

Separation or removal of water vapor from moist air can be accomplished by using a desiccant to absorb water vapor, by freezing out water vapor, or by separation of moist air constituents using a semipermeable membrane. These are standard laboratory techniques that operate on a sample of moist air. After removal of the water vapor the mass of the water vapor and the remaining air sample are determined in a variety of ways and then the humidity can be calculated.

5.3.2 Addition of Water Vapor to Air

Humidity can be determined by measuring the amount of water vapor that must be added to a sample of moist air to achieve complete saturation. This is a laboratory technique, but there is a variation of this method that is suitable for field measurements .

Psychrometry is a method of adding water vapor to moist air where complete saturation is not achieved. The humidity is determined from the cooling of a wet bulb relative to the ambient air temperature. The psychrometer comprises two temperature sensors exposed to the ambient air flow. One sensor, called the dry bulb, measures the ambient air temperature. The other sensor, called a wet bulb, is covered with a wick moistened with water and measures a lower temperature, caused by evaporation of water into the ambient air stream. The wick can be moistened intermittently by dipping into water or continuously by capillary flow through the wick material. Forced ventilation is normally required for optimum performance; natural ventilation may be adequate only when the temperature sensor and wick are very small and/or the ambient wind speed is sufficiently high.

A functional model of a psychrometer is illustrated in fig. 5-5 which shows two separate sensors: a dry-bulb thermometer and a wet-bulb thermometer. The dry-bulb function is simple but the wet-bulb function is more complex. Two primary inputs are shown, temperature (dry-bulb) and vapor pressure, and two secondary inputs, pressure and wind speed. Wet-bulb temperature is only weakly dependent upon pressure and wind speed. The sources of error in a psychrometer have been well documented and are readily controlled, as noted below.

Sensitivity, accuracy, and matching of the temperature sensors. A psychrometer is less sensitive to the absolute error in the temperature sensors than to the relative error, or matching error, between wet- and dry-bulb sensors. Ventilation rate. Typically, the ventilation rate should be at least 3 m/s to maximize the heat transfer by convection and evaporation and to minimize

94 Meteorological Measurement Systems

Fig. 5-5 Functional model of a psychrometer. Raw outputs y and y2 become, after calibration, estimates of T and Tw, respectively.

heat transfer by conduction and radiation. The minimum ventilation rate needed is a function of the sensor's thermal mass. Sensors made from small- diameter thermocouple wire with a fine cloth wick have been successfully used without forced ventilation (Stigter and Welgraven, 1976.) Radiation incident on the temperature sensors. The sensors must be shielded from direct and reflected solar radiation and from long-wave or earth radiation. This is a major source of error in the field that is not usually a factor in the laboratory. Size, shape, material, and wetting of the wick. Specially prepared psychrometer wick, available from instrument vendors, should always be used and not ordinary cotton cloth. Most commercial cotton cloth contains hydrophobic or

anti-wetting chemicals that will eventually impede wetting of the wick. When used in continuously operating psychrometers, even standard psychrometer wicking should be boiled in a solution of lye and detergent, then boiled in distilled water and flushed with distilled water before use. Relative positions of the wet- and dry-bulb sensors. The air must not flow from the cooled wet bulb to the dry bulb.

(Parte **1** de 5)