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## The Atmosphere

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**The Atmosphere**• Calculating atmospheric properties • Governing gas equations • Solutions for troposphere and stratosphere • The US standard atmosphere • Reading atmosphere tables • Non-standard atmosphere • Altitude definitions**Governing Gas Equations**• Equation of stateP = rRT • True for a “perfect gas” • Here, P= pressure [lb/ft2] = density [slug/ft3] R= gas constant = 1716 [ft lb/slug °R] T= temperature (absolute) [°R]**Governing Gas Equations**• Hydrostatic equation dP = -rgdh • Ordinary differential equation • Geopotential altitude, h • Gravitational acceleration, g varies • below 50,000 ft, differences are small • assume geometric altitude = geopotential altitude**Calculating Atmospheric Properties**• Mathematical relationships for properties in the atmosphere • Based on governing equations and temperature model • Needed for predicting aircraft / rocket performance at different altitudes • Solve the hydrostatic equation (an ordinary differential equation) • Temperature used as boundary conditions • Two regions of interest require two solutions**Troposphere Equations**• Temperature gradient Ts= sea-level temperature hs= sea-level altitude = 0 ft dT/dh= lapse rate = -0.00356 °R/ft • Solve hydrostatic equation for pressure Ps= sea-level pressure = 2116.2 lb/ft2 • Use equation of state for density**Stratosphere Equations**• Isothermal (to 82,000 ft) T36,152 = temperature at 36,152 ft • Solve hydrostatic equation for pressure P36,152= pressure at 36,152 ft = 472.7 lb/ft2 • Use equation of state for density**1) Start with sea-level conditions**2) Check altitude 3) If in troposphere (h 36,152 ft) a) find T at h b) find P at h 4) If in stratosphere (36,152 ft < h < 82,000 ft) a) find T at 36,152 ft b) find P at 36,152 ft c) set T = T36,152 d) use T36,152 and P36,152 to find P at h 5) Find from equation of state Calculating Atmospheric Properties**Calculating Atmospheric Properties**Find temperature, pressure and density at 12,000 ft 1) Start with sea-level conditions Ts = 518.69°R (about 59°F) Ps = 2116 lb/ft2 rs = 0.002377 sl/ft3 2) Check altitude (which atmospheric region?) 3) a) Find T at h**Calculating Atmospheric Properties**3) b) Find P at h 4) Calculate r**The US Standard Atmosphere**• Based on mean annual US temperature • Uses empirical temperature data • Reference of atmospheric conditions • Appendix B, Introduction to Aeronautics • Table B.1 English (USCS) units, Table B.2 SI units • Includes speed of sound, a and viscosity, m**Non-Standard Atmosphere**• Design to extreme (hot) conditions • Phoenix in summer • Denver in summer • Navy hot day • Generally T • e.g., T = +50°R • changes Tsfrom standard 518.69°R • other properties calculated as before**Definitions of Altitude**From US Std. Atmosphere flight data: P= 786.3 lb/ft2 r = 0.001104 sl/ft3 T= 436.8 °R h= 26,000 ft • Geometric altitude • Density altitude • Pressure altitude • Temperature altitude**Martian Atmosphere**• Mostly carbon dioxide – close to perfect gas • Hydrostatic equation and equation of state still valid • Different constants and “sea-level” conditions R= gas constant = 1148 [ft lb/slug °R] or 192 [J/kg K] g= gravitational constant = 12.21 [ft/s2] or 3.72 [m/s2] T0= “sea-level” temperature = 410.4 [°R] or 228 [K] P0= “sea-level” pressure = 16.165 [lb/ft2] or 774 [Pa] r0= “sea-level” density = 3.43×10-5 [slug/ft3] or 0.01768 [kg/m3] dT/dh= lapse rate = -0.0016 [°R/ft] or -0.0030 [K/m]