Characteristics of Air

1. Characteristics of Air #

Physical quantities related to air have the following relationships:

%%{init: {"flowchart": {"defaultRenderer": "elk"}} }%% %%{init: { 'flowchart': { 'curve': 'cardinal' } } }%% flowchart LR phi[Relative humidity
ϕ] T[Temperature
T] H[Pressure of dry air
H] c[Speed of sound in dry air
c] rho[Dry air density
ρ] eta[Viscosity of dry air
η] E[Saturation vapor
pressure of air
E] p[Vapor pressure of air
p] rhow[Density of air
containing vapor at pressure p
ρw] cw[Speed of sound in air
containing vapor at pressure p
cw] gamma[Heat capacity ratio
of dry air
γ] gammaw[Heat capacity ratio of air
containing vapor at pressure p
γw] T:::Base o--> rho:::Dryair H:::Dryair o--> rho phi:::Wetair o--> p:::Wetair T o--> E:::Wetair E --> p rho --> rhow:::Wetair p --> rhow H o--> rhow T o--> c:::Dryair p --> cw H o--> cw c --> cw:::Wetair T o--> eta:::Dryair phi o-.-> eta gamma:::Dryair o-.-> cw gammaw:::Wetair o-.-> cw classDef Base fill:#f8f8f7,stroke:#c0c6c9,stroke-width:1px classDef Wetair fill:#eaf4fc,stroke:#84a2d4,stroke-width:1px classDef Dryair fill:#fef4f4,stroke:#c97586,stroke-width:1px %%classDef Depth3 fill:#d6e9ca,stroke:#769164,stroke-width:1px

Solid lines indicate the physical quantities needed to calculate the physical quantity at the tip of the arrow. Dotted lines indicate physical quantities that do not significantly affect the calculation of the physical quantity at the tip of the arrow.

1.1. Density of Dry Air $\rho$ #

The density of dry air is given by the following equation1,

\begin{align} \rho= \frac{1.293}{1+0.00367\cdot T} \cdot \frac{H}{101325}. \end{align}

where

  • Symbols
    • $\rho$ : Density of dry air (kg·m-3)
    • $T$ : Temperature (℃)
    • $H$ : Pressure of dry air (Pa)

1.2. Speed of Sound in Dry Air $c$ #

The speed of sound in dry air is given by the following equation23,

\begin{align} c =20.05\sqrt{T} \end{align}

where

  • Symbols
    • $c$ : Speed of sound in dry air (m·s-1)
    • $T$ : Temperature (K)
  • Note
    • The speed of sound is almost independent of pressure.

1.3. Viscosity of Dry Air $\eta$ #

The viscosity of dry air is given as follows4,

\begin{align} \eta_2=\eta_1\left(\frac{T_1+C}{T_2+C}\right)\cdot \left(\frac{T_2}{T_1}\right)^{3/2} \end{align}

where

  • Symbols

    • $T_1, T_2$ : Temperature (K)
    • $\eta_1$ : Viscosity at temperature $T_1$ (Pa·s)
    • $\eta_2$ : Viscosity at temperature $T_2$ (Pa·s)
    • $C$ : Sutherland’s constant (dimensionless. For air $C=117$)
  • Notes

    • Viscosity is almost independent of pressure.
    • For air, it is derived from the viscosity of dry air at 1 atm, 20 degrees Celsius (about 293K), which is $18.2\times 10^{-6}\mathrm{(Pa\cdot s)}$.
    • Near room temperature, it seems that there is almost no dependence on humidity5.

1.4. Saturated Water Vapor Pressure of Air $E$ #

The saturated water vapor pressure of air $E$ is given by the following Tetens’ equation6,

\begin{align} E = 100\times 6.1078\times 10^{\frac{7.5 \cdot T }{T+237.3}} \end{align}

More accurately, it is calculated by Wagner’s equation for saturated water vapor pressure7,

\begin{align} E = p_c \exp{\left[\frac{T_c}{T}\left(a_1\theta+a_2\theta^{1.5}+a_3\theta^3+a_4\theta^{3.5}+a_5\theta^{4}+a_6\theta^{7.5}\right)\right]} \end{align}

where

  • Symbols

    • $E$ : Saturated water vapor pressure of air (Pa)
    • $T$ : Temperature (℃)
    • $p_c = 22~064~000$ : Critical pressure (Pa)
    • $T_c = 373.946$ : Critical Temperature (℃)
    • $\theta = 1-(T+273.15)/T_c$
    • $a_1 = -7.859 517 83$
    • $a_2 = 1.844 082 59$
    • $a_3 = -11.786 649 7$
    • $a_4 = 22.680 741 1$
    • $a_5 = -15.961 871 9$
    • $a_6 = 1.801 225 02$
  • Note

    • At 100 degrees Celsius, when water boils, it approaches atmospheric pressure of 101325Pa.

1.5. Water Vapor Pressure in Air $p$ #

\begin{align} p=\phi E \end{align}

  • Symbols
    • $p$ : Water vapor pressure in air (Pa)
    • $\phi$ : Relative humidity (dimensionless, real number value from 0 to 1)
    • $E$ : Saturated water vapor pressure of air (Pa)

1.6. Density of Air Containing Water Vapor at Pressure, $\rho_w$ #

The density of air containing water vapor is given by the following equation1,

\begin{align} \rho_w=\rho\left(1-0.378\frac{p}{H}\right) \end{align}

where

  • Symbols
    • $\rho$ : Density of dry air (kg·m-3)
    • $p$ : Water vapor pressure in air (Pa)
    • $H$ : Pressure of dry air (Pa)

1.7. Speed of Sound in Air Containing Water Vapor at Pressure, $c_w$ #

The speed of sound in air containing water vapor is given by the following equation8.

\begin{align} c_w=\dfrac{c}{\sqrt{1-\dfrac{p}{H}\left(\dfrac{\gamma_w}{\gamma}-0.622\right)}} \end{align}

If we approximate $\gamma_w\approx\gamma$,

\begin{align} c_w=c\sqrt{\frac{\rho}{\rho_w}} \end{align}

  • Symbols
    • $c$ : Speed of sound in dry air (m·s-1)
    • $p$ : Water vapor pressure in air (Pa)
    • $H$ : Pressure of dry air (Pa)
    • $\gamma$ : Ratio of specific heats of dry air (ratio of specific heat at constant pressure $C_p$ to specific heat at constant volume $C_v$, $\gamma=C_p/C_v$) (dimensionless)
    • $\gamma_w$ : Ratio of specific heats of air containing water vapor (dimensionless)
  • Note
    • $\gamma$ is almost 1.4 for air and does not depend on temperature or pressure.
    • Even at 100 degrees with 100% relative humidity, $\gamma_w=1.33$, so it seems to be acceptable to approximate $\gamma_w/\gamma \approx 1$9.

2. References #


  1. National Astronomical Observatory of Japan, “Chronological Scientific Tables, Heisei 21 (2009) 82nd Edition” (国立天文台編『理科年表 平成21年 第82冊』), Maruzen Co., Ltd. (2009) p.376 ↩︎ ↩︎

  2. Air - Speed of Sound vs. Temperature”, The Engineering ToolBox ↩︎

  3. Speed of sound - wikipedia ↩︎

  4. National Astronomical Observatory of Japan, “Chronological Scientific Tables, Heisei 21 (2009) 82nd Edition” (国立天文台編『理科年表 平成21年 第82冊』), Maruzen Co., Ltd. (2009) p.378 ↩︎

  5. M. Boukhriss et al., Study of thermophysical properties of a solar desalination system using solar energy, Desalination and Water Treatment 51 (2013) 1290―1295 ↩︎

  6. Monthly Average Relative Humidity - Official Site of Chronological Scientific Tables, ↩︎

  7. W. Wagner and A. Pruß:" The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use", J. Phys. Chem. Ref. Data, Vol. 31, No. 2, 2002, https://doi.org/10.1063/1.1461829 ↩︎

  8. National Astronomical Observatory of Japan, “Chronological Scientific Tables, Heisei 21 (2009) 82nd Edition” (国立天文台編『理科年表 平成21年 第82冊』), Maruzen Co., Ltd. (2009) p420 ↩︎

  9. I B Amarskaja, V S Belousov and P S Filippov, Analytical calculation of adiabatic processes in real gases., Journal of Physics: Conference Series. 754. 112003. 10.1088/1742-6596/754/11/112003. (2016) ↩︎