[TASK 2] The U.S. Standard Atmosphere Model Since the gravitational acceleration is not the constant (it is the function

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Task 2 The U S Standard Atmosphere Model Since The Gravitational Acceleration Is Not The Constant It Is The Function 1 (143.68 KiB) Viewed 27 times

[TASK 2] The U.S. Standard Atmosphere Model Since the gravitational acceleration is not the constant (it is the function of geometric altitude ho), we need to somehow incorporate this non-constant gravity when we calculate standard atmospheric properties. Let me introduce a "fictitious" altitude, called geopotential altitude, as defined as: h= Earth hG "Earenthe When we calculate the standard atmospheric properties (temperature, pressure, and density) at a given geometric altitude (hg), we need to, first, convert this into corresponding geopotential altitude (h), applying this equation. The U.S. Standard Atmosphere Model The U.S. Standard Atmosphere Model can be defined by the assumption of two different simple mathematical models: (i) the gradient region (the region of linearly changing, either decreasing or increasing, temperature) and (ii) the isothermal layer (the layer of constant temperature). 100 10S 225.66 K - 4K 10 km -90 80 -79 165.66 K - 4.5 X 10 Km 60 -53 Geopotential Altitude (km) 40 282.66 K Sea-Level Standard Values: P. = 1.01325 x 105 N/m² = 2116.2 lb/ft? -> *** X 10 K/= 1.2250 kg/m = 0.002377 slug/ft3 T. = 288.16 K = 518.690R 25 20 6.5 X 10 Km 288.16K 0 160 216.66K 1 200 240 Temperature, 210 320
Gradient Region Linear Temperature Variation: T = Ti + a(h - h) -R/AR р P. - (6) т Ti Gradient region Geopotential Altitude (km) -go/aR) +11 Base of gradient region T T P1 - T.p.p T1 PP1 For gradient region, one can calculate the temperature, pressure, and density as: T=T, +ach - h) p=p1 (9) p=p1 90 AR 90+ VaR where, T1, P1, P1: Temperature, Pressure, and Density at the base of a given region a: The "lapse rate" that is the slope of the Linear Function of Temperature (note: negative value means linearly decreasing temperature, and positive value means linearly increasing temperature) R: The Gas Constant of Air (287J/kg-K) go: The Gravitational Acceleration at Sea-Level (9.81 m/s2) Isothermal Layer Isothermal layer Pre(80/RTXA-A) P1 T=constant -T.P.P Geopotential Altitude (km) -Base of isothermal layer -R/RTX-) PL T.PIPI For isothermal layer, one can calculate the temperature, pressure, and density as: T=T(constant) p=pie e hoch-) p=pie RT 一發h-ha) where, T1, P1, P1: Temperature, Pressure, and Density at the base of a given layer R: The Gas Constant of Air (287 J/kg-K) 90: The Gravitational Acceleration at Sea-Level (9.81 m/s)
Using MATLAB Live Script with Symbolic Math (you must employ the MATLAB command: syms hG), take user's keyboard input of geometric altitude (hg) in the range of 0 (sea-level) to 105 km, and then calculate the corresponding temperature (T), pressure (p), and density (p) of the given altitude. Below is an example of MATLAB Symbolic Math Live Script Interactive Session (OUTPUT ONLY): [DISPLAY] Input (KEYBOARD TYPE-IN) Geometric Altitude in km: altitude_hG - 75 "output only" style (MATLAB Live Editor) [DISPLAY] Temperature in K: temperature = 187.5869 [DISPLAY] Pressure in Pa: pressure - 2.5807 [DISPLAY] Density in kg/m^3): density = 4.7928e-05