AER 416: Part 3 Standard Atmosphere PDF

AER 416: Part 3 Standard Atmosphere Instructor: Paul Walsh Dept. of Aerospace Engineering, Ryerson University 350 Victoria Street, Toronto, Ontario M5B 2K3 Tel: 416-979-5016 1 The Standard...

AER 416: Part 3 Standard Atmosphere Instructor: Paul Walsh Dept. of Aerospace Engineering, Ryerson University 350 Victoria Street, Toronto, Ontario M5B 2K3 Tel: 416-979-5016 1 The Standard Atmosphere 3.1 preamble For design and performance assessment, engineers use a standard atmosphere for reference. The standard atmosphere reflects the average atmospheric conditions as a function of height above the earth surface. A standard atmosphere starts at sea level, height zero. The standard atmosphere is usually given in the form of a table. The course text shows this table in Appendices A (SI units) and B (Imperial units). This section will show how the table was obtained, and how it can be put to use. 2 The Standard Atmosphere Standard Sea Level Conditions Pressure 101,325 Pa 2116.7 lbf/ft2 Density 1.225 kg/m3 0.002378 slug/ft3 Temperature 15 oC or 288.16 K 59 oF or 518.69 oR 3 The Standard Atmosphere The diagram at left shows the nomenclature for definitions of of height. 'r' = the radius of the earth from its centre to sea level (6356 km). hG = Geometrical altitude, height above sea level. ha = Absolute altitude from the centre of the earth. ha = hG + r 4 The Standard Atmosphere Note that absolute altitude is used in space flight since acceleration due to gravity 'g' varies with ha. Recall from physics, the acceleration due to gravity at Earth's surface, go, G = universal gravitational constant, 6.67x10-11 me = mass of the Earth, 5.972x1024 kg go = acceleration due to gravity at earth's surface, 9.81 m/s2 The value of g above the Earth at height hG, 5 The Standard Atmosphere Relating both g's, so, 6 The Standard Atmosphere 3.2 Hydrostatic Equation Unlike water, the density and pressure of air does not vary linearly with height This is a result of the compressibility of air The analysis of a column of air begins as it did with water, with the hydrostatic equation, 7 The Standard Atmosphere Consider a cube of air with based dimensions of 1m by 1m. Let the height be dhG. The cube is filled with fluid and is stationary. If the cube is not moving then all the forces applied to it are in balance. F = 0 The forces on the vertical surfaces are equal, so must sum to 0. 8 The Standard Atmosphere The force on the lower surface, p(1m)(1m), acts upward. The force on the upper surface (p+dp)(1m)(1m), acts downward. We need to include dp since pressure changes with height. The weight of the fluid within the cube is, = (volume)(density)g. = (1m1mdhG)()g All forces must be in balance, 9 The Standard Atmosphere This is the Hydrostatic Equation. It applies to both compressible and incompressible static fluids. This is a differential equation which must be solved to create a relation of the form p = f(height), or  = f(height). Note also that g is a function of height as well, this complicates matters making the differential equation non-linear. To simplify, use go (a constant) rather than g = f(height). The results with this equation will be slightly different than if we where to solve it with the actual g. So to distinguish this result from the actual result (using g), we will call the new altitude (h) the geopotential altitude. 10 The Standard Atmosphere Note that the geopotential altitude is fictitious since we used go rather than g. Since the g’s are different in these two equations, the heights will differ between the two (h vs. hG). Note that the difference between the two will be small. Relation between the two, 11 The Standard Atmosphere Why are these heights different? Consider a pressure p at height hG Consider a new height, hG + dhG The new pressure is, p + dp, Use this value of dp in dp = -godh find dh You will then see, that dh  dhG Appendix A and B in the text lists both 12 The Standard Atmosphere Relation between the Geopotential and Geometric Height Ultimately we want, p = p(hG), but we can easily calculate p = p(h), we need a relationship between the two heights Start with, From 13 The Standard Atmosphere We have By convention, set h = hg = 0, at sea level Yielding, Note that the difference between the two will be small since r = 6.357x106m at latitude 45o For altitudes less than 65 km, the difference between the two is less than 1%. 14 The Standard Atmosphere 0 – 10 km: Troposphere ~ 10 km: Tropopause 10 - 47 km: Stratosphere ~ 47 km: Stratopause 47 – 80 km: Mesosphere ~ 80 km: Mesopause 80 – 600 km: Thermosphere 600 – 10,000 km: Exosphere 15 The Standard Atmosphere Region Height (km) Temperature (K) Lapse rate (K/km) Troposphere 0 - 11 288.16 - 216.66 - 6.5 Stratosphere 11 - 25 216.66 constant 25 - 47 216.66 - 282.66 +3 Mesosphere 47 - 53 282.66 constant 53 - 79 282.66 - 165.66 - 4.5 Thermosphere 79 - 90 165.66 constant 90 ~ 600 165.66 ~ 2000 +4 Exosphere 600 ~ 10000 2000 constant 16 The Standard Atmosphere Calculation of the pressure and density at a specific h, must be done in stages since the standard atmosphere is broken into regions Ideally, to use this table we need a mathematical expression that relates pressure, temperature and density Start with, 17 The Standard Atmosphere Isothermal regions , T is constant, Start with the fixed values at the base, T1, p1, 1 18 The Standard Atmosphere Gradient regions , T varies linearly with height, The lapse rate a is the rate at which temperature changes with height, can be positive or negative, Temperature at any height can be found from, 19 The Standard Atmosphere Gradient regions , Pressure and density at any height can be found from, 20 The Standard Atmosphere Pressure, Temperature, and Density Altitudes The standard atmosphere allows the definition of three new ‘altitudes’ which are defined by the standard atmosphere table. For example, if the pressure at some elevation is say 61.6 kPa, on the standard atmosphere table this corresponds to a height of 4,000 m. In this case you can say the ‘pressure altitude’ is 4,000 m If the temperature at some elevation is say 265.4 K, on the standard atmosphere table this corresponds to a height of 3500 m. In this case you can say the ‘temperature altitude’ is 3500 m Note that these two circumstances could happen at the same time since the standard atmosphere is just an average. 21 The Standard Atmosphere Pressure, Temperature, and Density Altitudes If the pressure at some elevation is say 61.6 kPa, on the standard atmosphere table this corresponds to a height of 4,000 m. In this case you can say the ‘pressure altitude’ is 4,000 m. The pressure altitude is the altitude indicated in the standard atmosphere for a given pressure value, not necessarily the actual altitude. The same is true for temperature and density altitudes. Example, at a point in the atmosphere, p = 61.6 kPa, T =265.4 K, density by ideal gas law is  = 0.809 kg/m3 , so pressure altitude is 4000 m, temperature altitude is 3500 m, density altitude is 4120 m, all from standard atmosphere tables. Note this kind of nomenclature is not used much. 22