# GT AE 1350 - AE 1350 Lecture #4 (21 pages)

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## AE 1350 Lecture #4

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## AE 1350 Lecture #4

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Pages:
21
School:
Georgia Tech
Course:
Ae 1350 - Intro to Aerospace Engr
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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 L CL 1 For example from last lecture V 2 S 2 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 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 101325 Pa 2116 7 lbf ft2 Density 1 225 Kg m3 0 002378 slug ft3 Temperature 15 oC or 288 K 59 oF or 518 4 oR Ideal Gas Law or Equation 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 air Speed of Sound From thermodynamics and compressible flow theory you will study later in your career sound travels at the following speed a RT 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 90 km 79 km 165 66 K 53 km 47 km T 282 66 K Altitude km 25 km 11km 216 66K Temperature degrees K Stratosphere Troposphere 288 16 K 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 dh Pressure at the bottom p Forces acting on this Column of Air Force Pressure times Area p dp A dh Weight of air g A dh Force p A Force Balance Force p dp A Downward directed force Upward force p dp A g A dh pA gA dh Simplify dp g dh Force p A Variation of p with T dp g dh Use Ideal Gas Law also called Equation of State p RT 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 h Let us assume T T1 a h h1 The slope a is called a Lapse Rate h h1 T T1 Variation of p with T when T varies linearly Continued From previous slide An infinitesimal change in Temperature T T1 a h h1 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 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 log g aR T p log T1 p1 p T p1 T1 g aR Variation of with T when T varies linearly From the previous slide in regions where temperature varies linearly we get p T p1 T1 g aR Using p RT and p1 1RT1 we can show that density varies as T 1 T1 g 1 aR 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 Integrate log p g RT h C Equation 1 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 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 g h h1 p e RT p1 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

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