### ISA VARIATION WITH ALTITUDE

Pressure, temperature, density, viscosity and speed of sound variation for the international standard atmosphere (ISA) can be calculated for a range of altitudes from sea level upward. This is done using an exact solution to the hydrostatic equation for a column of air. The air is assumed to be a perfect gas. In the lower region, the troposphere, the atmosphere has a lapse rate (L) of 6.5K/Km . At an altitude of 36089 ft the stratosphere starts and the temperature remains constant at 217K. The hydrostatic equation, perfect gas law and the lapse rate equations are,

$${∂P}/{∂h} = - ρg\text" "P=ρRT\text" and "T=T_0-Lh$$

where the variables used are,

 P -- Pressure (Pa); T -- Temperature (K); g -- Gravitational acceleration (9.8 m/ m/s2); TO -- Standard sea level temperature (288 K); R -- Gas constant for air (287 m2/s2/K); h -- Altitude above sea level (m), L -- Lapse rate (0.0065 K/m) and ρ -- Air density, ( Kg/m3);

Atmospheric Layers and Temperature Variation with Altitude.

Solving the hydrostatic equation with a constant lapse rate gives the resulting pressure variation in the troposphere.

$$P/P_0=(T/T_0)^{g/{LR}}$$

where sea level pressure, Po , is set at the standard 101.3 kPa.

Solving the hydrostatic equation with a constant temperature gives the resulting pressure variation in the stratosphere.

$$P/P_s=e^{{g(h_s-h)}/{RT_s}}$$

where conditions with subscript (s) are values of altitude (hs), pressure (Ps) or temperature (Ts) at the tropopause, the start of the stratosphere; the line dividing the two distinct atmospheric regions.

Once pressure has been calculated at a particular altitude, density is then calculated using the perfect gas law. Viscosity and kinematic viscosity are found by applying the Sutherland law

$$μ=0.1456×10^{-5}{√{T}}/{1+110/T}$$

And finally speed of sound is found based on the temperature,

$$a=√{γRT}$$

where γ is the ratio of specific heats for air (1.4)