AAE 251 Formulas. Standard Atmosphere. Compiled Fall 2016 by Nicholas D. Turo-Shields, student at Purdue University. Gradient Layer.

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1 AAE 51 Formulas Compiled Fall 016 by Nicholas D. Turo-Shields, student at Purdue University Standard Atmosphere p 0 = Pascals ρ 0 = 1.5 kg m 3 R = 87 J kg K γ = 1.4 for air p = ρrt ; Equation of State r e h = h G r e + h G Gradient ayer T = T 1 + ah h 1 g 0 p T ar = p 1 ρ ρ 1 = T 1 T T 1 [ ] g 0 ar +1 Isothermal ayer p = e [ g0 p 1 ρ = p ρ 1 p 1 RT h h 1] 1

2 Aerodynamics Continuity Equation: ṁ = constant ρ 1 V 1 A 1 = ρ V A Bernoulli s Equation: p 0 = p + 1 ρv ; total stagnation p = static p + dynamic p Aerodynamic Coefficients c l = = N cos α A sin α D = N sin α + A cos α N = cos α + D sin α A = sin α D cos α q S ; c M = M q Sc ; c p = p p q a = γrt ; Mach Number M = v a C 0 Prandtl-Glauert Correction: C c = 1 M Wave Drag 1 Mach Angle: µ = sin 1 Thicker objects shockwaves form at angle greater than µ M c l = 4α M 1 ; c d = 4α M 1 Induced Drag α eff = α α i D i = sin α i α i for small α i = C πear ; α i = C πear elliptical lift distribution e = 1, otherwise, e < 1 dc l dα = dc dα eff = a 0 lift curve slope for an infinite wing dc dα eff = dc dα α i C = a o α α i + C Flaps W V =, V stall = ρ S C ρ S C, max

3 Aircraft Performance Thrust T W = C ; T R = W C ; C max = 1 πear,0 T A, alt = ρ ρ 0 T A, 0 Power P = F v W W CD V = = ; P ρ S C ρ S C 3 R = T R V ; P A = ηp, where η < 1 ρ0 V alt = V 0 ρ ; P ρ0 R, alt = P R, 0 ρ ; P A, alt = ρ P A, 0 ρ 0 R.C. = P A P R W = v sin θ Range and Endurance W0, E prop = ηc C 3 / R prop = η c C ln W 1 E jet = 1 C W0 8 ln, R jet = c t W 1 ρ S 1 W 0 = total gross weight, W 1 = ending weight Takeoff W 1 / 1 W 1 / 0, max C 1/ c t V = F m t, s = V m F W V O = 1.. For the average, V = 0.7 V O. ρ S C, max s O = anding 1. W gρ S C, max T [D + µr W ] avg. W V T = 1.3. For the average, V = 0.7 V T. ρ S C, max s T = 1.3 W gρ S C, max [D + µw ] avg W 1 / 0 W 1 / 1, max [ [ ] C 3 / ] C 1 / = 3, 0 πar 3 4 4, 0 =, 0 πar 1 4 4, 0 µ r ranges from 0.0 smooth surface to 0.10 grass µ 0.40 paved surface; surface friction plus brakes. 3

4 Propulsion Air-Breathing φ = β α, where β is the pitch angle of the propeller and α is the angle of attack ω = πn, where n is number of revolutions per second the propeller spins. tan φ = V, where r is the radius at a given point of the propeller ωr J = V ; Advance Ratio, where d is the diameter of the propeller nd T turbojet = ṁ air + ṁ fuel V e ṁ air V + p e p A e, where typically ṁ fuel = 0.05 ṁ air T fan = ṁ F V e V + p e p A e T propeller = ηp V, where P is the power of the engine driving the shaft β = ṁf ṁ c ; Bypass Ratio; ṁ F is the fan air mass flow rate and ṁ c is the core air mass flow rate Rocket T = ṁv e + p e p A e I sp Ṫ W = T g 0 ṁ, where g 0 is acceleration at sea-level on EARTH If nozzle is designed such that p e = p, then V e = g 0 I sp [ ] mf Assuming constant V e, ln = V V g g 0 I sp = m 0 = m f e 0 Isp Rocket Equation m 0 m 0 = initial rocket mass = m payload + m inert + m propellant ; m f = final rocket mass = m payload + m inert Inert Mass Fraction: f inert = m inert m prop + m inert ; Propellant Mass Fraction = PMF = ζ = m prop m 0 m prop = m paye V g 0 Isp 11 finert ; m 1 f inert e V inert = m pay f inert e V g 0 Isp 1 ; m g 0 Isp 1 f inert e V 0 = m pay e V g 0 Isp 1 finert g 0 Isp 1 f inert e V g 0 Isp 4

5 Orbital Mechanics T1 T = a3 1 a 3 r = GM + mˆr r = r = p 1 + e cos θ When M m, GM + m GM = µ P E = GMm r = µm r 0 Specific Energy: ε = P E + KE ε = v m µ Vis-Viva Equation r / 5 m1 r SOI = D, where m 1 < M, and D is the distance between celestial bodies M Conic Sections p width of each conic at the focus latus rectum a length of the chord that passes through the two foci major axis c distance between foci e = c a = c a = r a r p r a + r p p = a1 e r p = a1 e periapsis r a = a1 + e apoapsis r a + r p = a; r a r p = c ε = µ a Elliptical Orbits T = πa3/ µ Circular Orbits e = 0; r = constant = a = b = p T = πr3/ µ µ v c = r 5

6 Parabolic Orbits e = 1; r p = p 1 + e = p ; r a = p 1 e = µ v esc = r = v c Hyperbolic Orbits δ is the turn angle or bend angle v is the hyperbolic excess speed δ sin = a c = 1 e ε = v v = Ground Tracks v µ r = v v esc Posigrade Orbit: 0 i < 90 Polar Orbit: i = 90 Retrograde Orbit: 90 < i < 180 = T 360 = T ω; T is the period of orbit, R is the rotational period, and ω is the rotational rate R Effects of aunch Sites cos i = sin β cos λ, where i is orbit inclination, β is azimuth, and λ is latitude of the launch site V Earth s Rotation = ω E r E cos λ sin β Plane Changes V = V i sin i osses During aunch V effective = V ideal + V drag + V gravity + V steering dvdrag D V drag = dt = dt m dt V gravity = g sin θ dt 6