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breguetnoteseps - UNIFIED ENGINEERING Lecture Outlines Fall...

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UNIFIED ENGINEERING Fall 2003 Lecture Outlines Ian A. Waitz UNIFIED LECTURE #2: THE BREGUET RANGE EQUATION I. Learning Goals At the end of this lecture you will: A. Be able to answer the question “How far can an airplane fly , and why ?”; B. Be able to answer the question “How do the disciplines of structures & materials, aerodynamics and propulsion jointly set the performance of aircraft, and what are the important performance parameters?”; C. Be able to use empirical evidence to estimate the performance of aircraft and thus begin to develop intuition regarding important aerodynamic, structural and propulsion system performance parameters; D. Have had your first exposure to active learning in Unified Engineering 1
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L II. Question: How far can an airplane (or a duck, for that matter) fly? OR: What is the farthest that an airplane can fly on earth, and why ? We will begin by developing a mathematical model of the physical system. Like most models, this one will have many approximations and assumptions that underlie it. It is important for you to understand these approximations and assumptions so that you understand the limits of applicability of the model and the estimates derived from it. L D W Figure 1.1 Force balance for an aircraft in steady level flight. 2
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For steady, level flight, T = D, L = W or L WL == D D = T L D The weight of the aircraft changes in response to the fuel that is burned (rate at which weight changes equals negative fuel mass flow rate times gravitational constant) dW =− mg ˙ dt f Now we will define an overall propulsion system efficiency: what you get = propulsive power overall efficiency = what you pay for fuel power propulsive power = thrust flight velocity = Tu o (J/s) fuel power = fuel mass flow rate fuel energy per unit mass = m h (J/s) ˙ f Thus Tu o = overall mh ˙ f η We can now write the expression for the change in weight of the vehicle in terms of important aerodynamic (L/D) and propulsion system ( η overall ) parameters: dW ˙ = W Wu 0 = Wu 0 = dt f L T h L Tu 0 h L η ˙ f ˙ f g D overall D g D mh We can rewrite and integrate 0 dW = ud t ln W = constant tu 0 W hL η h L η g D overall g D overall applying the initial conditions, at t = 0 W = W initial const. = ln W initial L h W t ∴= η overall l n D gu 0 W initial 3
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the time the aircraft has flown corresponds to the amount of fuel burned, therefore L h W final l n D overall gu 0 W initial η t final = then multiplying by the flight velocity we arrive at the Breguet Range Equation which applies for situations where overall efficiency, L/D, and flight velocity are constant over the flight.
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breguetnoteseps - UNIFIED ENGINEERING Lecture Outlines Fall...

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