Psychrometrics uses thermodynamic properties to analyze conditions and processes involving moist air. This chapter discusses perfect gas relations and their use in common heating, cooling, and humidity control problems. Formulas developed by Herrmann et al. (2009) may be used where greater precision is required.
Herrmann et al. (2009), Hyland and Wexler (1983a, 1983b), and Nelson and Sauer (2002) developed formulas for thermodynamic properties of moist air and water modeled as real gases. However, perfect gas relations can be substituted in most air-conditioning problems. Kuehn et al. (1998) showed that errors are less than 0.7% in calculating humidity ratio, enthalpy, and specific volume of saturated air at standard atmospheric pressure for a temperature range of −50 to 50°C. Furthermore, these errors decrease with decreasing pressure.
1. COMPOSITION OF DRY AND MOIST AIR
Atmospheric air contains many gaseous components as well as water vapor and miscellaneous contaminants (e.g., smoke, pollen, and gaseous pollutants not normally present in free air far from pollution sources).
Dry air is atmospheric air with all water vapor and contaminants removed. Its composition is relatively constant, but small variations in the amounts of individual components occur with time, geographic location, and altitude. Harrison (1965) lists the approximate percentage composition of dry air by volume as: nitrogen, 78.084; oxygen, 20.9476; argon, 0.934; neon, 0.001818; helium, 0.000524; methane, 0.00015; sulfur dioxide, 0 to 0.0001; hydrogen, 0.00005; and minor components such as krypton, xenon, and ozone, 0.0002. Harrison (1965) and Hyland and Wexler (1983a) used a value 0.0314 (circa 1955) for carbon dioxide. Carbon dioxide reached 0.0379 in 2005, is currently increasing by 0.00019 percent per year and is projected to reach 0.0438 in 2036 (Gatley et al. 2008; Keeling and Whorf 2005a, 2005b). Increases in carbon dioxide are offset by decreases in oxygen; consequently, the oxygen percentage in 2036 is projected to be 20.9352. Using the projected changes, the relative molecular mass for dry air for at least the first half of the 21st century is 28.966, based on the carbon-12 scale. The gas constant for dry air using the current Mohr and Taylor (2005) value for the universal gas constant is
Moist air is a binary (two-component) mixture of dry air and water vapor. The amount of water vapor varies from zero (dry air) to a maximum that depends on temperature and pressure. Saturation is a state of neutral equilibrium between moist air and the condensed water phase (liquid or solid); unless otherwise stated, it assumes a flat interface surface between moist air and the condensed phase. Saturation conditions change when the interface radius is very small (e.g., with ultrafine water droplets). The relative molecular mass of water is 18.015 268 on the carbon-12 scale. The gas constant for water vapor is
2. U.S. STANDARD ATMOSPHERE
The temperature and barometric pressure of atmospheric air vary considerably with altitude as well as with local geographic and weather conditions. The standard atmosphere gives a standard of reference for estimating properties at various altitudes. At sea level, standard temperature is 15°C; standard barometric pressure is 101.325 kPa. Temperature is assumed to decrease linearly with increasing altitude throughout the troposphere (lower atmosphere), and to be constant in the lower reaches of the stratosphere. The lower atmosphere is assumed to consist of dry air that behaves as a perfect gas. Gravity is also assumed constant at the standard value, 9.806 65 m/s2. Table 1 summarizes property data for altitudes to 10 000 m.
Pressure values in Table 1 may be calculated from
The equation for temperature as a function of altitude is
where
Z |
= |
altitude, m |
p |
= |
barometric pressure, kPa |
t |
= |
temperature, °C |
Equations (3) and (4) are accurate from −5000 m to 11 000 m. For higher altitudes, comprehensive tables of barometric pressure and other physical properties of the standard atmosphere, in both SI and I-P units, can be found in NASA (1976).
3. THERMODYNAMIC PROPERTIES OF MOIST AIR
Table 2, developed from formulas by Herrmann et al. (2009), shows values of thermodynamic properties of moist air based on the International Temperature Scale of 1990 (ITS-90). This ideal scale differs slightly from practical temperature scales used for physical measurements. For example, the standard boiling point for water (at 101.325 kPa) occurs at 99.97°C on this scale rather than at the traditional 100°C. Most measurements are currently based on ITS-90 (Preston-Thomas 1990).
The following properties are shown in Table 2:
t |
= |
Celsius temperature, based on the ITS-90 and expressed relative to absolute temperature T in kelvins (K) by the following relation:
|
|
|
|
|
|
 |
|
|
|
Ws |
= |
humidity ratio at saturation; gaseous phase (moist air) exists in equilibrium with condensed phase (liquid or solid) at given temperature and pressure (standard atmospheric pressure). At given values of temperature and pressure, humidity ratio W can have any value from zero to Ws.
|
vda |
= |
specific volume of dry air, m3/kgda.
|
vas |
= |
vs – vda, difference between specific volume of moist air at saturation and that of dry air, m3/kgda, at same pressure and temperature.
|
vs |
= |
specific volume of moist air at saturation, m3/kgda.
|
hda |
= |
specific enthalpy of dry air, kJ/kgda. In Table 2, hda is assigned a value of 0 at 0°C and standard atmospheric pressure.
|
has |
= |
hs – hda, difference between specific enthalpy of moist air at saturation and that of dry air, kJ/kgda, at same pressure and temperature.
|
hs |
= |
specific enthalpy of moist air at saturation, kJ/kgda.
|
sda |
= |
specific entropy of dry air, kJ/(kgda · K). In Table 2, sda is assigned a value of 0 at 0°C and standard atmospheric pressure.
|
ss |
= |
specific entropy of moist air at saturation kJ/(kgda · K).
|
4. THERMODYNAMIC PROPERTIES OF WATER AT SATURATION
Table 3 shows thermodynamic properties of water at saturation for temperatures from −60 to 160°C, calculated by the formulations described by IAPWS (2007, 2009, 2011, 2014). Symbols in the table follow standard steam table nomenclature. These properties are based on ITS-90. The internal energy and entropy of saturated liquid water are both assigned the value zero at the triple point, 0.01°C. Between the triple-point and critical-point temperatures of water, both saturated liquid and saturated vapor may coexist in equilibrium; below the triple-point temperature, both saturated ice and saturated vapor may coexist in equilibrium.
The water vapor saturation pressure is required to determine a number of moist air properties, principally the saturation humidity ratio. Values may be obtained from Table 3 or calculated from the following formulas (Hyland and Wexler 1983b). The 1983 formulas are within 300 ppm of the latest IAPWS formulations. For higher accuracy, developers of software and others are referred to IAPWS (2007, 2011).
The saturation (sublimation) pressure over ice for the temperature range of −100 to 0°C is given by
where
C1 | = | −5.674 535 9 E+03 |
C2 | = | 6.392 524 7 E+00 |
C3 | = | −9.677 843 0 E−03 |
C4 | = | 6.221 570 1 E−07 |
C5 | = | 2.074 782 5 E−09 |
C6 | = | −9.484 024 0 E−13 |
C7 | = | 4.163 501 9 E+00 |
The saturation pressure over liquid water for the temperature range of 0 to 200°C is given by
where
C8 |
= |
−5.800 220 6 E+03 |
C9 |
= |
1.391 499 3 E+00 |
C10 |
= |
−4.864 023 9 E−02 |
C11 |
= |
4.176 476 8 E−05 |
C12 |
= |
−1.445 209 3 E−08 |
C13 |
= |
6.545 967 3 E+00 |
In both Equations (5) and (6),
pws | = | saturation pressure, Pa |
T | = | absolute temperature, K = °C + 273.15 |
The coefficients of Equations (5) and (6) were derived from the Hyland-Wexler equations. Because of rounding errors in the derivations and in some computers’ calculating precision, results from Equations (5) and (6) may not agree precisely with Table 3 values.
The vapor pressure ps of water in saturated moist air differs negligibly from the saturation vapor pressure pws of pure water at the same temperature. Consequently, ps can be used in equations in place of pws with very little error:
where xws is the mole fraction of water vapor in saturated moist air at temperature t and pressure p, and p is the total barometric pressure of moist air.
Humidity ratio W (or mixing ratio) of a given moist air sample is defined as the ratio of the mass of water vapor to the mass of dry air in the sample:
W equals the mole fraction ratio xw/xda multiplied by the ratio of molecular masses (18.015 268/28.966 = 0.621 945):
Specific humidity γ is the ratio of the mass of water vapor to total mass of the moist air sample:
In terms of the humidity ratio,
Absolute humidity (alternatively, water vapor density) dv is the ratio of the mass of water vapor to total volume of the sample:
Density ρ of a moist air mixture is the ratio of total mass to total volume:
where v is the moist air specific volume, m3/kgda, as defined by Equation (24).
Humidity Parameters Involving Saturation
The following definitions of humidity parameters involve the concept of moist air saturation:
Saturation humidity ratio Ws(t, p) is the humidity ratio of moist air saturated with respect to water (or ice) at the same temperature t and pressure p.
Relative humidity ϕ is the ratio of the actual water vapor partial pressure in moist air at the dew-point pressure and temperature to the reference saturation water vapor partial pressure at the dry-bulb pressure and temperature:
Note that Equations (12) and (22) have been revised so that they cover both the normal range of relative humidity where e(tdb) < p and the extended range (e.g., atmospheric pressure drying kilns) where e(tdb) ≥ p. The definitions in earlier editions applied only to the normal range.
Dew-point temperature td is the temperature of moist air saturated at pressure p, with the same humidity ratio W as that of the given sample of moist air. It is defined as the solution td(p, W) of the following equation:
Thermodynamic wet-bulb temperature t* is the temperature at which water (liquid or solid), by evaporating into moist air at dry-bulb temperature t and humidity ratio W, can bring air to saturation adiabatically at the same temperature t* while total pressure p is constant. This parameter is considered separately in the section on Thermodynamic Wet-Bulb and Dew-Point Temperature.
6. PERFECT GAS RELATIONSHIPS FOR DRY AND MOIST AIR
When moist air is considered a mixture of independent perfect gases (i.e., dry air and water vapor), each is assumed to obey the perfect gas equation of state as follows:
where
pda | = | partial pressure of dry air |
pw | = | partial pressure of water vapor |
V | = | total mixture volume |
nda | = | number of moles of dry air |
nw | = | number of moles of water vapor |
R | = | universal gas constant, 8314.472 J/(kmol · K) |
T | = | absolute temperature, K |
The mixture also obeys the perfect gas equation:
or
where p = pda + pw is the total mixture pressure and n = nda + nw is the total number of moles in the mixture. From Equations (14) to (17), the mole fractions of dry air and water vapor are, respectively,
and
From Equations (8), (18), and (19), the humidity ratio W is
The saturation humidity ratio Ws is
The term pws represents the saturation pressure of water vapor in the absence of air at the given temperature t. This pressure pws is a function only of temperature and differs slightly from the vapor pressure of water in saturated moist air.
The relative humidity ϕ is defined in Equation (12). Using the second equality and eliminating the enhancement factors, which are not applicable using the perfect gas assumption, gives
Substituting Equation (21) for Ws into Equation (13),
Both ϕ and μ are zero for dry air and unity for saturated moist air. At intermediate states, their values differ, substantially at higher temperatures.
The specific volume v of a moist air mixture is expressed in terms of a unit mass of dry air:
where V is the total volume of the mixture, Mda is the total mass of dry air, and nda is the number of moles of dry air. By Equations (14) and (24), with the relation p = pda + pw,
Using Equation (18),
In Equations (25) and (26), v is specific volume, T is absolute temperature, p is total pressure, pw is partial pressure of water vapor, and W is humidity ratio.
In specific units, Equation (26) may be expressed as
where
v | = | specific volume, m3/kgda |
t | = | dry-bulb temperature, °C |
W | = | humidity ratio, kgw/kgda |
p | = | total pressure, kPa |
The enthalpy of a mixture of perfect gases equals the sum of the individual partial enthalpies of the components. Therefore, the specific enthalpy of moist air can be written as follows:
where hda is the specific enthalpy for dry air in kJ/kgda and hg is the specific enthalpy for saturated water vapor in kJ/kgw at the mixture’s temperature. As an approximation,
where t is the dry-bulb temperature in °C. The moist air specific enthalpy in kJ/kgda then becomes
7. THERMODYNAMIC WET-BULB AND DEW-POINT TEMPERATURE
For any state of moist air, a temperature t* exists at which liquid (or solid) water evaporates into the air to bring it to saturation at exactly this same temperature and total pressure (Harrison 1965). During adiabatic saturation, saturated air is expelled at a temperature equal to that of the injected water. In this constant-pressure process,
-
Humidity ratio increases from initial value W to Ws*, corresponding to saturation at temperature t*
-
Enthalpy increases from initial value h to hs*, corresponding to saturation at temperature t*
-
Mass of water added per unit mass of dry air is (Ws* − W), which adds energy to the moist air of amount (Ws* − W)hw*, where hw* denotes specific enthalpy in kJ/kgw of water added at temperature t*
Therefore, if the process is strictly adiabatic, conservation of enthalpy at constant total pressure requires that
Ws*, hw*, and hs* are functions only of temperature t* for a fixed value of pressure. The value of t* that satisfies Equation (31) for given values of h, W, and p is the thermodynamic wet-bulb temperature.
A psychrometer consists of two thermometers; one thermometer’s bulb is covered by a wick that has been thoroughly wetted with water. When the wet bulb is placed in an airstream, water evaporates from the wick, eventually reaching an equilibrium temperature called the wet-bulb temperature. This process is not one of adiabatic saturation, which defines the thermodynamic wet-bulb temperature, but one of simultaneous heat and mass transfer from the wet bulb. The fundamental mechanism of this process is described by the Lewis relation [Equation (40) in Chapter 6]. Fortunately, only small corrections must be applied to wet-bulb thermometer readings to obtain the thermodynamic wet-bulb temperature.
As defined, thermodynamic wet-bulb temperature is a unique property of a given moist air sample independent of measurement techniques.
Equation (31) is exact because it defines the thermodynamic wet-bulb temperature t*. Substituting the approximate perfect gas relation [Equation (30)] for h, the corresponding expression for hs*, and the approximate relation for saturated liquid water
into Equation (31), and solving for the humidity ratio,
where t and t* are in °C. Below freezing, the corresponding equations are
A wet/ice-bulb thermometer is imprecise when determining moisture content at 0°C.
The dew-point temperature td of moist air with humidity ratio W and pressure p was defined as the solution td(p, W) of Ws(p, td). For perfect gases, this reduces to
where pw is the water vapor partial pressure for the moist air sample and pws(td) is the saturation vapor pressure at temperature td. The saturation vapor pressure is obtained from Table 3 or by using Equation (5) or (6). Alternatively, the dew-point temperature can be calculated directly by one of the following equations (Peppers 1988):
Between dew points of 0 and 93°C,
Below 0°C,
where
td | = | dew-point temperature, °C |
α | = | ln pw |
pw | = | water vapor partial pressure, kPa |
C14 | = | 6.54 |
C15 | = | 14.526 |
C16 | = | 0.7389 |
C17 | = | 0.09486 |
C18 | = | 0.4569 |
8. NUMERICAL CALCULATION OF MOIST AIR PROPERTIES
The following are outlines, citing equations and tables already presented, for calculating moist air properties using perfect gas relations. These relations are accurate enough for most engineering calculations in air-conditioning practice, and are readily adapted to either hand or computer calculating methods. For more details, refer to Tables 15 through 18 in Chapter 1 of Olivieri (1996). Graphical procedures are discussed in the section on Psychrometric Charts.
SITUATION 1.
Given: Dry-bulb temperature t, Wet-bulb temperature t*, Pressure p
SITUATION 2.
Given: Dry-bulb temperature t, Dew-point temperature td, Pressure p
SITUATION 3.
Given: Dry-bulb temperature t, Relative humidity ϕ, Pressure p
Moist Air Property Tables for Standard Pressure
Table 2 shows thermodynamic properties for standard atmospheric pressure at temperatures from −60 to 90°C calculated using the ASHRAE RP-1485 (Herrmann et al. 2009) research project numerical model. Properties of intermediate moist air states can be calculated using the degree of saturation μ:
These equations are accurate to about 350°C. At higher temperatures, errors can be significant.
A psychrometric chart graphically represents the thermodynamic properties of moist air.
The choice of coordinates for a psychrometric chart is arbitrary. A chart with coordinates of enthalpy and humidity ratio provides convenient graphical solutions of many moist air problems with a minimum of thermodynamic approximations. ASHRAE developed five such psychrometric charts. Chart 1 is shown as Figure 1; the others may be obtained through ASHRAE.
Charts 1, 2, 3 and 4 are for sea-level pressure (101.325 kPa). Chart 5 is for 750 m altitude (92.634 kPa), Chart 6 is for 1500 m altitude (84.54 kPa), and Chart 7 is for 2250 m altitude (77.058 kPa). All charts use oblique-angle coordinates of enthalpy and humidity ratio, and are consistent with the data of Table 2 and the properties computation methods of Hyland and Wexler (1983a) and ASHRAE research project RP-1485. Palmatier (1963) describes the geometry of chart construction applying specifically to Charts 1 and 4.
The dry-bulb temperature ranges covered by the charts are
Charts 8 to 16 are for 200 to 320°C and cover the same pressures as 1, 5, 6, and 7 plus the additional pressures of 0.2, 0.5,1.0, 2.0, and 5.0 MPa. They were produced by Nelson and Sauer (2002) and are available as a download with Gatley (2013).
Psychrometric properties or charts for other barometric pressures can be derived by interpolation. Sufficiently exact values for most purposes can be derived by methods described in the section on Perfect Gas Relationships for Dry and Moist Air. Constructing charts for altitude conditions has been discussed by Haines (1961), Karig (1946), and Rohsenow (1946).
Comparison of charts 1 and 6 by overlay reveals the following:
-
The dry-bulb lines coincide.
-
Wet-bulb lines for a given temperature originate at the intersections of the corresponding dry-bulb line and the two saturation curves, and they have the same slope.
-
Humidity ratio and enthalpy for a given dry- and wet-bulb temperature increase with altitude, but there is little change in relative humidity.
-
Volume changes rapidly; for a given dry-bulb and humidity ratio, it is practically inversely proportional to barometric pressure.
The following table compares properties at sea level (chart 1) and 1500 m (chart 6):
Figure 1 shows humidity ratio lines (horizontal) for the range from 0 (dry air) to 30 grams moisture per kilogram dry air. Enthalpy lines are oblique lines across the chart precisely parallel to each other.
Dry-bulb temperature lines are straight, not precisely parallel to each other, and inclined slightly from the vertical position. Thermodynamic wet-bulb temperature lines are oblique and in a slightly different direction from enthalpy lines. They are straight but are not precisely parallel to each other.
Relative humidity lines are shown in intervals of 10%. The saturation curve is the line of 100% rh, whereas the horizontal line for W = 0 (dry air) is the line for 0% rh.
Specific volume lines are straight but are not precisely parallel to each other.
A narrow region above the saturation curve has been developed for fog conditions of moist air. This two-phase region represents a mechanical mixture of saturated moist air and liquid water, with the two components in thermal equilibrium. Isothermal lines in the fog region coincide with extensions of thermodynamic wet-bulb temperature lines. If required, the fog region can be further expanded by extending humidity ratio, enthalpy, and thermodynamic wet-bulb temperature lines.
The protractor to the left of the chart shows two scales: one for sensible/total heat ratio, and one for the ratio of enthalpy difference to humidity ratio difference. The protractor is used to establish the direction of a condition line on the psychrometric chart.
Example 1 shows use of the ASHRAE psychrometric chart to determine moist air properties.
Example 1.
Moist air exists at 40°C dry-bulb temperature, 20°C thermodynamic wet-bulb temperature, and 101.325 kPa pressure. Determine the humidity ratio, enthalpy, dew-point temperature, relative humidity, and specific volume.
Solution: Locate state point on chart 1 (Figure 1) at the intersection of 40°C dry-bulb temperature and 20°C thermodynamic wet-bulb temperature lines. Read humidity ratio W = 6.5 gw/kgda.
The enthalpy can be found by using two triangles to draw a line parallel to the nearest enthalpy line (60 kJ/kgda) through the state point to the nearest edge scale. Read h = 56.7 kJ/kgda.
Dew-point temperature can be read at the intersection of W = 6.5 gw/kgda with the saturation curve. Thus, td