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HW_6_Soln - ME 309 Fall 2008 Section 1(Merkle Solution...

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ME 309 Fall 2008 Section 1 (Merkle) Solution: Homework # 6 Due Wednesday 10 September 2008 Problem 6-1 The US Standard Atmosphere is defined as having the following variation between sea level and 20,100 m (See Pg 57 in FPM): T = 288.15 + z*(-55.6-15)/11,000 for 0 < z < 11000 (Troposhere) T = 217.55 for 11,000 < z < 20,100 (Stratosphere) where the temperature is in Kelvin and the altitude in m Tabluate the pressure as a function of altitude between sea level and 20,000 m at increments of 500 m . Check your answers with the standard atmosphere which is available on the web. Solution: Known : Temperature lapse rate in troposphere and stratosphere Fluid = air, Molecular weight = 28.97 Find: Pressure as a function of altitude Sketch : Figure 3.3 in text Assumptions : Fluid at rest, velocity = 0 Only force acting is body force due to gravity Air is a perfect gas Analysis : Apply momentum equation for a system k g ˆ ˆ g p ρ ρ - = = Only component of momentum of interest is vertical: Momentum equation becomes: g dz dp ρ - = Replace density by perfect gas: RT pMWTg dz dp - = And replace temperature by lapse rate equation: For troposphere: ) 00642 . 0 15 . 288 ( z R pMWTg dz dp - - = Separate variables: ) 00642 . 0 15 . 288 ( z R MWTgdz p dp - - = Replace group of constants by R MWTg c / = and define a =288.15, b = 0.00642: ) ( bz a cdz p dp - - = For integration, use substitution: u = a – bz which gives: du = -bdz Equation becomes: bu cdu p dp =
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Integrating: b c u p / ln ln = Change back to z and evaluate between limits: b c a bz a p p / 0 - = Stratosphere : RT pMWTg dz dp - = but now, T is a constant Separate variables: dz RT MWTg p dp - = where ddz dz RT MWTg p dp - = - = with d a constant Integrating gives: ( ) const dz p + = ln Inverting: ( ) - - = 11000 exp 11 z RT MWTg p p Numbers: STANDARD ATMOSPHERE TABLE MWT 28.97 g 9.80665 R 8314.4 MWT*g/R 0.034169471 T(sea Level) 288.15 T(11,000) 217.55 Lapse rate (B8- B7)/11000 - 0.006418182 increment 500 Constants a 288.15 b - 0.006418182 c 0.034169471 delta_altitude 500 TROPOSPHERE Altitude, z pressure T density 0 101325 288.15 1.225224 500 95460 284.9409
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