On the Question of Calculating Air Humidity Using a Psychrometer
Abstract Generated abstract
This paper develops calculation procedures for determining air humidity from dry-bulb and wet-bulb psychrometer readings. It derives relations for moisture content, water vapor partial pressure, relative humidity, and absolute humidity using moist-air enthalpy balances, with explicit allowance for barometric pressure. The authors show that pressure effects can be substantial and argue that the proposed formulas are preferable to approximate corrections. They also outline a refinement for cases of significant vapor partial pressure, using steam tables to account for deviations of water vapor from ideal-gas behavior.
Full Text
Proceedings
of the Tomsk Polytechnic Institute named after S. M. Kirov,
Holder of the Order of the October Revolution and the Order of the Red Banner of Labor
Volume 205 1972
On the Question of Calculating Air Humidity Using a Psychrometer
G. I. Fuks, V. V. Banyukha
(Submitted by the Department of Theoretical and General Heat Engineering)
If one assumes that water vapor in air is an ideal gas, then its moisture content \(d\) and partial pressure \(P_p\) are related by
\[ d = 0.622\,\frac{P_p}{B - P_p}\, \frac{\mathrm{kg}}{\mathrm{kg\ dry\ air}}, \tag{1} \]
where \(B\) is the barometric pressure.
If the air is saturated with water vapor at the wet-bulb temperature \(t_m\), point \(A\) in Fig. 1, then its moisture content is
\[ d_{nm} = 0.622\,\frac{P_{nm}}{B - P_{nm}}, \tag{2} \]
where \(P_{nm}\) is the pressure of saturated vapor at \(t_m\).
Fig. 1
The enthalpy of the air in this state is
\[ i_{nm} = 1.004 t_m + d_{nm}\left(2500 + 1.9\cdot 6\, t_m\right)\, \frac{\mathrm{kJ}}{\mathrm{kg\ dry\ air}}. \tag{3} \]
The saturation of the air as it passes by the wet-bulb thermometer may be regarded as a process of blowing over a free surface.
The state of the air changes in this case from \(B\) to \(A\) (Fig. 1); the temperature decreases from the dry-bulb temperature \(t_c\) to \(t_m\), the moisture content increases from \(d\) to \(d_{\mathrm{nm}}\), and the enthalpy increases from \(i\) to \(i_{\mathrm{nm}}\). As has been established, the enthalpy and moisture content vary according to the law of an isotherm in the fog region at temperature \(t\).
\[ i=i_{\mathrm{nm}}+\left(d-d_{\mathrm{nm}}\right)4.187\,t_m, \tag{4} \]
where \(4.187\ \mathrm{kJ/(kg\cdot deg)}\) is the heat capacity of liquid \(\mathrm{H_2O}\).
At the dry-air temperature, enthalpy and moisture content are related by
\[ i=1.004\,t_c+d\left(2500+1.926\,t_c\right). \tag{5} \]
The state of the air at the dry-bulb temperature is determined by the intersection of the isotherm \(t_c\) with the isotherm \(t_m\) (point B in Fig. 1). Use of (4), (5), taking (3) into account, gives
\[ d=\frac{d_{\mathrm{nm}}\left(2500-2.261\,t_m\right)-1.004\left(t_c-t_m\right)} {\left(2500-2.261\,t_m\right)+1.926\left(t_c-t_m\right)}. \tag{6} \]
From this value of \(d\), the partial pressure of water vapor in the air is determined from (1):
\[ P_{\mathrm{p}}=\frac{dB}{0.622+d}, \tag{1'} \]
and then the relative humidity
\[ \varphi=\frac{P_{\mathrm{p}}}{P_{\mathrm{nc}}} \tag{7} \]
and the absolute humidity
\[ \gamma_{\mathrm{n}}=\varphi\gamma_{\mathrm{nc}}, \tag{8} \]
where \(P_{\mathrm{cn}}\) and \(\gamma_{\mathrm{nc}}\) are the pressure and density of saturated vapor at the dry-bulb temperature.
It is also possible to compute the partial pressure of the vapor in the air from the relation
\[ \frac{P_{\mathrm{p}}}{B}= \frac{ \dfrac{P_{\mathrm{nm}}}{B}\left(2500-2.261\,t_m\right) -1.614\left(1-\dfrac{P_{\mathrm{nm}}}{B}\right)\left(t_c-t_m\right) }{ \left(2500-2.261\,t_m\right) +0.312\left(1-\dfrac{P_{\mathrm{nm}}}{B}\right)\left(t_c-t_m\right) }. \tag{9} \]
As can be seen from the expressions given, the influence of barometric pressure can be quite significant. Thus, at temperatures \(t_c=30^\circ\mathrm{C}\) and \(t_m=20^\circ\mathrm{C}\) and \(B=500\) torr, the relative humidity will be \(42.2\%\), while at \(B=1000\) torr it will be \(34.5\%\). Accounting for barometric pressure by the relations given is more reliable than by approximate dependencies.
With a significant partial pressure of vapor, it may become necessary to take into account the deviation of the properties of \(\mathrm{H_2O}\) vapor from the ideal-gas state. The moisture content of the air is determined by the relation
\[ d=\frac{\gamma_{\mathrm{p}}}{\gamma_{\mathrm{sv}}}. \tag{a} \]
The density of dry air \(\gamma_{\mathrm{sv}}\) is determined from the equation of state
\[ B-P_{\mathrm{p}}=287.2\,\gamma_{\mathrm{sv}}(273.15+t), \tag{10} \]
where \(287.2\ \mathrm{J/(kg\cdot deg)}\) is the gas constant of dry air.
From (a) and (10) one obtains
\[ d = 287.2\,\frac{\gamma_{\mathrm{p}}(273.15+t)}{B-P_{\mathrm{p}}}\,\frac{\mathrm{kg}}{\mathrm{kg.d.a.}} . \tag{11} \]
If the temperature and relative humidity of the air are known, then the vapor density \(\gamma_{\mathrm{p}}\) is determined from (8). From the table for superheated vapor, using \(t\) and \(V_{\mathrm{p}}=\dfrac{1}{\gamma_{\mathrm{p}}}\), one finds \(P_{\mathrm{p}}\), and then \(d\) can be determined from (11). For air saturated with vapor, the density and pressure are determined from the table for dry vapor.
Determination of the partial vapor pressure \(P_{\mathrm{p}}\) from known values of temperature and moisture content is carried out by successive approximations.
First, from \((1')\) an approximate value \(P'_{\mathrm{p}}\) is determined, and from (11) the corresponding value of \(\gamma'_{\mathrm{p}}\) and \(\left(V'_{\mathrm{p}}=\dfrac{1}{\gamma'_{\mathrm{p}}}\right)\). Using these values, from the tables of superheated vapor we determine the corrected value \(P_{\mathrm{p}}\), and from (11) a more accurate value of \(\gamma_{\mathrm{p}}\). Further refinement is performed analogously, but usually it is unnecessary. For air saturated with water vapor, relation (11) takes the form
\[ d_{\mathrm{n}} = 287.2\,\frac{\gamma_{\mathrm{n}}(273.15+t)}{B-P_{\mathrm{n}}}, \tag{11'} \]
where \(P_{\mathrm{n}}\) and \(\gamma_{\mathrm{n}}\) are taken from the table for dry vapor.
3
The enthalpy of moist air is determined from the relation
\[ i = i_{\mathrm{da}} + d i_{\mathrm{p}} . \tag{12} \]
The enthalpy of dry air \(i_{\mathrm{da}}\) (at \(t_{\mathrm{c}}\)) may be taken from tables according to its temperature. The enthalpy of vapor \(i_{\mathrm{p}}\) depends, in addition, on the partial pressure of the vapor in the air. For saturated air at the wet-bulb temperature,
\[ i_{\mathrm{nw}} = i'_{\mathrm{da}} + d_{\mathrm{nw}} i_{\mathrm{np}}, \tag{13} \]
where \(i'_{\mathrm{da}}\) and \(i_{\mathrm{np}}\) are taken at the wet-bulb temperature, and the quantity \(d_{\mathrm{nw}}\) is determined from \((11')\). If the variation of the heat capacity of liquid \(\mathrm{H_2O}\) with temperature is neglected, then the relation between enthalpy and moisture content at the wet-bulb temperature in the fog region is expressed by
\[ i = i_{\mathrm{nw}} + 4.187(d-d_{\mathrm{nw}})t_{\mathrm{w}} . \tag{14} \]
From (12) and (14), taking (13) into account, one obtains
\[ d = \frac{d_{\mathrm{nw}}(i_{\mathrm{n}}-4.187\,t_{\mathrm{w}})-(i_{\mathrm{da}}-i'_{\mathrm{da}})} {i_{\mathrm{p}}-4.187\,t_{\mathrm{w}}}. \tag{15} \]
For calculation by this relation it is first necessary to find the vapor enthalpy \(i_{\mathrm{p}}\) at the dry-bulb temperature, which, in turn, requires knowledge of the partial vapor pressure \(P_{\mathrm{p}}\) in the air. But the enthalpy of vapor at a given temperature changes only very slightly with changes in its pressure. Therefore, to determine \(P_{\mathrm{p}}\), one may use (6) and \((1')\) or (9), and then, from \(P_{\mathrm{p}}\) and the temperature \(t_{\mathrm{c}}\), using the table for superheated vapor, find the value \(i_{\mathrm{p}}\). A small error will also occur if, in relation (15), instead of \(i_{\mathrm{p}}\) one substitutes the value of the enthalpy of dry vapor at the dry-bulb temperature.
From the value of moisture content calculated from (15), as indicated above, the absolute and relative humidities of the air are determined.