AE 1350 Lecture #4

PREVIOUSLY 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 Coefficients

TOPICS 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 Atmosphere

Why 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 C L until we know the density of the atmosphere, .

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 Atmosphere Standard Sea Level Conditions Pressure Pa lb f /ft 2 Density1,225 Kg/m slug/ft 3 Temperature15 o C or 288 K59 o F or o R

Ideal Gas Law or Equation of State Most gases satisfy the following relationship between density, temperature and pressure: p =  RT –p = Pressure (in lb/ft 2 or N/m 2 ) –  = “Rho”, density (in slugs/ft 3 or kg/m 3 ) –T = Temperature (in Degrees R or degrees K) –R = Gas Constant, varies from one gas to another. –Equals J/Kg/K or ft lb f /slug/ o R for air

Speed 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)

Temperature vs. Altitude

Pressure varies with Height The bottom layers have to carry more weight than those at the top

Consider a Column of Air of Height dh Its area of cross section is A Let dp be the change in pressure between top and the bottom Pressure at the top = (p+dp) Pressure at the bottom = p dh

Forces acting on this Column of Air Force = Pressure times Area = (p+dp)A Force = p A Weight of air=  g  A dh dh

Force Balance Force = (p+dp)A Force = p A  gA dh Downward directed force= Upward force (p+dp)A +  g A dh = pA Simplify: dp = -  g dh

Variation of p with T dp = -  g dh Use Ideal Gas Law (also called Equation of State): p =  R T  = p/(RT) dp = - p / (RT) g dh dp/p = - g/(RT) dh Equation 1 This equation holds both in regions where temperature varies, and in regions where temperature is constant.

Variation of p with T in Regions where T varies linearly with height From the previous slide, dp/p = - g/(RT) dh Equation 1 Because 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 = T 1 +a (h-h 1 ) The slope ‘a’ is called a Lapse Rate. h h=h 1 T=T 1

Variation of p with T when T varies linearly (Continued..) From previous slide, T = T 1 +a (h-h 1 ) An infinitesimal change in Temperature dT = a dh Use this in equation 1 : dp/p = - g/(RT) dh We get:dp/p = -g/(aR)dT/T Integrate. Use integral of dx/x = log x. Log p = -(g/aR) log T + C Equation 2 where C is a constant of integration. Somewhere on the region, let h = h 1, p=p 1 and T = T 1 Log p 1 = -(g/aR) log T 1 + C Equation 3

Variation of p with T when T varies linearly (Continued..) Subtract equation (3) from Equation (2): log p - log p 1 = - g/(aR) [log T - log T 1 ] log (p/p 1 ) = - g / (aR) log ( T/T 1 ) Use m log x = log (x m )

Variation of  with T when T varies linearly From the previous slide, in regions where temperature varies linearly, we get: Using p =  RT and p 1 =  1 RT 1, we can show that density varies as:

Variation of p with altitude h in regions where T is constant In some regions, for example between 11 km and 25 km, the temperature 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 1 Integrate: log p = - g/(RT) h + C

Variation of p with altitude h in regions where T is constant (Continued..) From the previous slide, in these regions p varies with h as: log p = -g /(RT) h + C At some height h 1, we assume p is known and his given by p 1. Log p 1 = - g/(RT) h 1 + C Subtract the above two relations from one another: log (p/p 1 ) = -g/(RT) (h-h 1 ) Or,

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 class.