AE 1350 Lecture #4

1. AE 1350Lecture #4

2. 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

3. 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

4. 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 CL until we know the density of the atmosphere, r.

5. 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.

6. International Standard Atmosphere Standard Sea Level Conditions Pressure101325 Pa2116.7 lbf/ft2 Density1,225 Kg/m30.002378 slug/ft3 Temperature15 oC or 288 K 59 oF or 518.4 oR

7. Ideal Gas Law orEquation of State • Most gases satisfy the following relationship between density, temperature and pressure: • p = rRT • p = Pressure (in lb/ft2 or N/m2) • r = “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 air

8. 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)g = Ratio of Specific Heats = 1.4 • R = Gas Constant • T = temperature (in degrees K or degrees R)

9. Temperature vs. Altitude

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

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

12. Forces acting on this Column of Air Force = Pressure times Area = (p+dp)A dh Weight of air= r gA dh Force = p A

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

14. Variation of p with T dp = - r g dh Use Ideal Gas Law (also called Equation of State): p = r R T r = 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.

15. Variation of p with T in Regionswhere 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 = T1 +a (h-h1) The slope ‘a’ is called a Lapse Rate. h h=h1 T=T1

16. Variation 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) 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 = h1 , p=p1 and T = T1 Log p1 = -(g/aR) log T1 + C Equation 3

17. Variation 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)

18. Variation of r with T when T varies linearly From the previous slide, in regions where temperature varies linearly, we get: Using p = rRT and p1 = r1RT1, we can show that density varies as:

19. Variation of p with altitude hin 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

20. Variation 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 + C At some height h1, we assume p is known and his given by p1. Log p1 = - g/(RT) h1 + C Subtract the above two relations from one another: log (p/p1) = -g/(RT) (h-h1) Or,

21. 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.