## Pk pj

where k = cp/cv is the specific heat ratio for the gas (for an ideal gas, cp = cv + P/M). If the density is eliminated from Eqs. (4-14) and (4-11), the result is

which relates the temperature and pressure at any two points in an isentro-pic ideal gas. If Eq. (4-15) is used to eliminate T from Eq. (4-12), the latter can be integrated to give the pressure as a function of elevation:

which is a nonlinear relationship between pressure and elevation. Equation (4-15) can be used to eliminate P2/P1 from this equation to give an expression for the temperature as a function of elevation under isentropic conditions:

That is, the temperature drops linearly as the elevation increases. D. The Standard Atmosphere

Neither Eq. (4-13) nor Eq. (4-16) would be expected to provide a very good representation of the pressure and temperature in the real atmosphere, which is neither isothermal nor isentropic. Thus, we must resort to the use of observations (i.e., empiricism) to describe the real atmosphere. In fact, atmospheric conditions vary considerably from time to time and from place to place over the earth. However, a reasonable representation of atmospheric conditions "averaged" over the year and over the earth based on observations results in the following:

For 0 < z < 11 km: — = —6.5°C/km = —G , N

where the average temperature at ground level (z = 0) is assumed to be 15°C (288 K). These equations describe what is known as the "standard atmosphere," which represents an average state. Using Eq. (4-18) for the temperature as a function of elevation and incorporating this into Eq. (4-12) gives d-P = PMg (4-19)

where T0 = 288 K and G = 6.5°C/km. Integrating Eq. (4-19) assuming that g is constant gives the pressure as a function of elevation:

G Az

Mg/RG

0 0