AE 1350 Lecture #4PREVIOUSLY COVERED TOPICSTOPICS TO BE COVEREDWhy should we study Atmospheric PropertiesWhat is a standard atmosphere?Slide 6Ideal Gas Law or Equation of StateSpeed of SoundTemperature vs. AltitudePressure varies with HeightSlide 11Forces acting on this Column of AirForce BalanceVariation of p with TVariation of p with T in Regions where T varies linearly with heightVariation of p with T when T varies linearly (Continued..)Slide 17Variation of r with T when T varies linearlyVariation of p with altitude h in regions where T is constantVariation of p with altitude h in regions where T is constant (Continued..)Concluding RemarksAE 1350Lecture #4PREVIOUSLY COVERED TOPICS•Preliminary Thoughts on Aerospace Design•Specifications (“Specs”) and Standards•System Integration•Forces acting on an Aircraft•The Nature of Aerodynamic Forces•Lift and Drag CoefficientsTOPICS TO BE COVERED•Why should we study properties of atmosphere?•Ideal Gas Law•Variation of Temperature with Altitude• Variation of Pressure with Altitude•Variation of Density with Altitude•Tables of Standard AtmosphereWhy should we study Atmospheric Properties•Engineers design flight vehicles, turbine engines and rockets that will operate at various altitudes.•They can not design these unless the atmospheric characteristics are not known.•For example, from last lecture, •We can not design a vehicle that will operate satisfactorily and generate the required lift coefficient CL until we know the density of the atmosphere, . SVLCL221What is a standard atmosphere?•Weather conditions vary around the globe, from day to day.•Taking all these variations into design is impractical.•A standard atmosphere is therefore defined, that relates fight tests, wind tunnel tests and general airplane design to a common reference.•This common reference is called a “standard” atmosphere.International Standard AtmosphereStandard Sea Level ConditionsPressure 101325 Pa 2116.7 lbf/ft2Density 1,225 Kg/m30.002378 slug/ft3Temperature 15 oC or 288 K 59 oF or 518.4 oRIdeal Gas Law orEquation of State•Most gases satisfy the following relationship between density, temperature and pressure:• p = RT–p = Pressure (in lb/ft2 or N/m2)–  = “Rho” , density (in slugs/ft3 or kg/m3)–T = Temperature (in Degrees R or degrees K)–R = Gas Constant, varies from one gas to another.–Equals 287.1 J/Kg/K or 1715.7 ft lbf/slug/oR for airSpeed of Sound•From thermodynamics, and compressible flow theory you will study later in your career, sound travels at the following speed:• •where,– a = speed of Sound (m/s or ft/s) = Ratio of Specific Heats = 1.4–R = Gas Constant–T = temperature (in degrees K or degrees R)RTaTemperature vs. AltitudeTemperature, degrees KAltitude, km288.16 K11km216.66K25 km47 km, T= 282.66 K53 km79 km165.66 K90 kmTroposphereStratospherePressure varies with HeightThe bottom layers have to carry more weight than those at the topConsider a Column of Air of Height dhIts area of cross section is ALet dp be the change in pressure between top and the bottomPressure at the top = (p+dp)Pressure at the bottom = p dhForces acting on this Column of AirForce = Pressure times Area = (p+dp)AForce = p AWeight of air= gA dhdhForce BalanceForce = (p+dp)AForce = p AgA dhDownward directed force= Upward force(p+dp)A +  g A dh = pASimplify:dp = -  g dhVariation of p with Tdp = -  g dhUse Ideal Gas Law (also called Equation of State):p =  R T  = p/(RT)dp = - p / (RT) g dhdp/p = - g/(RT) dh Equation 1This equation holds both in regions where temperature varies,and in regions where temperature is constant.Variation of p with T in Regionswhere T varies linearly with heightFrom the previous slide, dp/p = - g/(RT) dh Equation 1Because T is a discontinuous function of h (i.e. has breaks in its shape),we can not integrate the above equation for the entire atmosphere. We will have to do it one region at a time.In the regions (troposphere, stratosphere), T varies with h linearly. Let us assume T = T1 +a (h-h1) The slope ‘a’ is called a Lapse Rate.hh=h1T=T1Variation of p with T when T varies linearly (Continued..)From previous slide, T = T1 +a (h-h1)An infinitesimal change in Temperature dT = a dh Use this in equation 1 : dp/p = - g/(RT) dhWe get: dp/p = -g/(aR)dT/TIntegrate. Use integral of dx/x = log x.Log p = -(g/aR) log T + C Equation 2where C is a constant of integration.Somewhere on the region, let h = h1 , p=p1 and T = T1 Log p1 = -(g/aR) log T1 + C Equation 3Variation of p with T when T varies linearly (Continued..)Subtract equation (3) from Equation (2):log p - log p1 = - g/(aR) [log T - log T1]log (p/p1) = - g / (aR) log ( T/T1)Use m log x = log (xm)aRgTTpp11loglogaRgTTpp11Variation of  with T when T varies linearlyFrom the previous slide, in regions where temperature varieslinearly, we get:aRgTTpp11Using p = RT and p1 = 1RT1, we can show that density varies as:111aRgTTVariation of p with altitude hin regions where T is constantIn some regions, for example between 11 km and 25 km, thetemperature of standard atmosphere is constant.How can we find the variation of p with h in this region?We start again with equation 1. dp/p = - g/(RT) dh Equation 1Integrate: log p = - g/(RT) h + CVariation of p with altitude hin regions where T is constant (Continued..)From the previous slide, in these regions p varies with h as:log p = -g /(RT) h + CAt some height h1, we assume p is known and his given by p1.Log p1 = - g/(RT) h1 + CSubtract the above two relations from one another:log (p/p1) = -g/(RT) (h-h1) Or,  11hhRTgeppConcluding Remarks•Variation of temperature, density and pressure with altitude can be computed for a standard atmosphere.•These properties may be tabulated. •Short programs called applets exist on the world wide web for computing atmospheric properties.•Study worked out examples to be done in the