Model overview
Constants
\(g_0\): Standard gravitational acceleration [m/s²]
\(M_0\): Sea level mean molar mass [kg/mol]
\(N_A\): Avogadro constant [mol⁻¹]
\(P_0\): Sea level atmospheric pressure [Pa]
\(R^{*}\): Universal gas constant [J/(K·mol)]
\(R\): Specific gas constant [J/(K·kg)]
\(S\): Sutherland’s empirical constant in the equation for dynamic viscosity [K]
\(T_i\): Temperature of the ice point at mean sea level [K]
\(T_0\): Sea level temperature [K]
\(t_i\): Celsius temperature of the ice point at mean sea level [°C]
\(t_0\): Celsius sea level temperature [°C]
\(\beta_s\): Sutherland’s empirical constant in the equation for dynamic viscosity [kg/(m·s·K^(1/2))]
\(\kappa\): Adiabatic index [1]
\(\rho_0\): Sea level atmospheric density [kg/m³]
\(\sigma\): Effective collision diameter of an air molecule [m]
\(r\): Nominal Earth’s radius [m]
\(\beta\): Temperature gradient (layer-specific) [K/m]
Variables
\(\omega\): Collision frequency [Hz]
\(\rho\): Density [kg/m³]
\(\mu\): Dynamic viscosity [Pa·s]
\(g\): Gravitational acceleration [m/s²]
\(\nu\): Kinematic viscosity [m²/s]
\(l\): Mean free path [m]
\(\bar{\nu}\): Mean particle speed [m/s]
\(n\): Number density [m⁻³]
\(p\): Pressure [Pa]
\(H_p\): Pressure scale height [m]
\(\gamma\): Specific weight [N/m³]
\(a\): Speed of sound [m/s]
\(T\): Temperature [K]
\(t\): Temperature (Celsius) [°C]
\(\lambda\): Thermal conductivity [W/(m·K)]
Plots and equations
Collision frequency
Equation: \(\omega = 4 \sigma^2 N_A \left( \frac{\pi}{R^{*} M_0} \right)^{1/2} \frac{p}{\sqrt{T}}\)
Density
Equation: \(\rho = \frac{p}{R T}\)
Dynamic viscosity
Equation: \(\mu = \frac{\beta_s T^{3/2}}{T + S}\)
Gravitational acceleration
Equation: \(g = g_0 \left( \frac{r}{r + h} \right)^2\)
Kinematic viscosity
Equation: \(\nu = \frac{\mu}{\rho}\)
Mean free path
Equation: \(l = \frac{1}{\sqrt{2} \pi \sigma^2 n}\)
Mean particle speed
Equation: \(\bar{\nu} = \left( \frac{8}{\pi} R T \right)^{1/2}\)
Number density
Equation: \(n = \frac{N_A p}{R^{*} T}\)
Pressure
Equation:
\(p = p_b \exp \left[ - \frac{g_0}{R T} (H - H_b) \right] \quad \text{for} \quad \beta = 0\)
\(p = p_b \left[ 1 + \frac{\beta}{T_b} (H - H_b) \right]^{-g_0/(\beta R)} \quad \text{for} \quad \beta \neq 0\)
Pressure scale height
Equation: \(H_p = \frac{R T}{g}\)
Specific weight
Equation: \(\gamma = \rho g\)
Speed of sound
Equation: \(a = \sqrt{\kappa R T}\)
Temperature
Equation: \(T = T_b + \beta (H - H_b)\)
Temperature (Celsius)
Equation: \(t = T - T_i\)
Thermal conductivity
Equation: \(\lambda = \frac{2.648151 \cdot 10^{-3} T^{3/2}}{T + (245.4 \cdot 10^{-12/T})}\)