UNIFIED LECTURE #2: THE BREGUET RANGE EQUATION

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

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 L T T D W W Figure 1.1 Force balance for an aircraft in steady level flight. 2

For steady, level flight, T = D, L = W or L W = L = D = T L D 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 = m f g dt Now we will define an overall propulsion system efficiency: what you get overall efficiency = what you pay for = propulsive power fuel power propulsive power = thrust flight velocity = Tu o (J/s) fuel power = fuel mass flow rate fuel energy per unit mass = m h f (J/s) Thus η overall = Tu o mh f 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: We can rewrite and integrate dw W Wu = m g = 0 Wu = = 0 f dt L T h L Tu 0 h L η f f g overall D m g g D m h D dw udt = 0 W h L η g overall D ln W = constant tu 0 h L η g overall D applying the initial conditions, at t = 0 W = W initial const. = ln W initial L h W t = η overall ln D gu 0 W initial 3

the time the aircraft has flown corresponds to the amount of fuel burned, therefore L h W t η ln final final = overall D gu0 W initial 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. Range = h L η overall ln W initial g D W final Fluids Propulsion Structures + (Aero) Materials Note that this expression is sometimes rewritten in terms of an alternate measure of efficiency, the specific fuel consumption or SFC. SFC is defined as the mass flow rate of fuel per unit of thrust (lbm/s/lbf or kg/s/n). In the following expression, V is the flight velocity and g is the acceleration of gravity. ( ) VL D ln g SFC W initial Range = W final

Thus we see that the answer to the question How far can an airplane fly? depends on: 1. How much energy is contained in the fuel it carries; 2. How aerodynamically efficient it is (the ratio of the production of lift to the production of drag). During the fluids lectures you will learn how to develop and use models to estimate lift and drag. 3. How efficiently energy from the fuel/oxidizer is turned into useful work (thrust times distance traveled) which is used to oppose the drag force. Thermodynamics helps us describe and estimate the efficiency of various energy conversion processes, and propulsion lets us describe how to use this energy to propel a vehicle;. How light weight the structure is relative to the amount of fuel and payload it can carry. The materials and structures lectures you will teach you how to estimate the performance of aerospace structures. 5

Below are some data to allow you to make estimates for various aircraft and birds. MJ/Kg $/Kg $/MJ Comments Prime Beef.0 20 5 Beef.0 2 Whole Milk 2. 0.90 0.32 600 cal/quart Honey 1 0.29 Sugar 1 0.07 0 cal/ounce Cheese 6 0.0 Bacon 29 0.1 Corn Flakes 3.50 0.23 0 cal/ounce Peanut Butter 27 0. 10 cal/ounce Butter 32.50 0.1 Vegetable Oil 36 2 0.06 20 cal/ounce Kerosene 2 0.0 0.0 0.2 kg/liter Diesel Oil 2 0.0 0.0 0.5 kg/liter Gasoline Natural Gas 2 5 0.0 0.2 0.0 0.005 0.75 kg/liter 0. kg/m 3 Figure 1.2 Heating values for various fuels (from The Simple Science of Flight, by H. Tennekes) 0.1 F=0 F=25 1 cabbage white F= F= human-powered airplane sailplane w [meters / second] albatross budgie ultralight pheasant Fokker Friendship Boeing 77 1 0 V [meters / second] The Great Gliding Diagram. Airspeed, V, is plotted on the horizontal axis. Rate of descent, w, is plotted downward along the vertical axis. The diagonals are lines of constant finesse. The horizontal line represents the practical soaring limit, 1 meter / second. Figure 1.3 Gliding performance as a function of L/D (where L/D=F, from The Simple Science of Flight, by H. Tennekes) 6

25 F2-00 F2-000/6000 20 S360 L/Dmax F27 B707-0B/300 A320-0/200 A3-300 RJ200/ER B777 B757-200 B77-00 B77-0/200/300 F0 BAC111-200/00 B767-200/ER ATR72 S30A DHC-300 MD11 DC-30 L11-500 B767-300/ER B707-300B ATR2 BAE-ATP B737-0/200 A300-600 L11-1/0/200 D32 DC9-30 B737-300 DC- MD0 & DC9-0 EMB120 B737-500/600 DC-0 B737-00 BAE16-0/200/RJ70 B727-200/231A BAE-16-300 SA227 J1 J31 5 Turboprops Regional Jets Large Aircraft Data Unavailable For: EMB-15 CV-0 BAE RJ5 Beech 1900 CV-50 CV-600 FH-227 Nihon YS-11 SA-226 DHC--0 L-1 DHC-7 0 1955 1960 1965 1970 1975 190 195 1990 1995 2000 Year Figure 1. Aerodynamic data for commercial aircraft: L/D for cruise (Babikian, R., The Historical Fuel Efficiency Characteristics of Regional Aircraft From Technological, Operational, and Cost Perspectives, SM Thesis, MIT, June 2001) W S b (N) (m 2 ) (m) A F House Sparrow 0.2 0.009 0.23 6 Swift 0.36 0.016 0.2 11 Common Tern 1.2 0.056 0.3 12 12 Kestrel (Sparrow Hawk) 1. 0.06 0.7 9 9 Carrion Crow 5.5 0.12 0.7 5 5 Common Buzzard.0 0.22 1.25 7 Peregrine Falcon.1 0.13 1.06 9 Herring Gull 12 0.21 1.3 11 Heron 1 0.36 1.73 9 White Stork 3 0.50 2.00 Wandering Albatross 5 0.62 3.0 19 20 Hang Glider 00 7 Parawing 00 25 2.6 Powered Parawing 1700 35 2.7 Ultralight (microlight) 2000 7 Sailplanes Standard Class Open Class 3500 5500.5 16.3 25 21 3 0 60 Fokker F-50 Boeing 77 19 x 36 x 5 70 511 29 60 12 7 16 Figure 1.5 Weight and geometry for aircraft and birds (where L/D=F, from The Simple Science of Flight, by H. Tennekes) 7

0 70 C-7A 60 W E / W TO [%] 50 0 C-12A C-123 757 767 UV -1 C-9A A3 C-21A C-10 C-130 C-20A C-11B Concorde L-11 C-17 KC-A C-5A 77 C-5B KC-135A 30 0 0 200 300 00 500 600 700 00 900 W TO [1,000 lbs] Weight Fractions of Cargo and Passenger Aircraft Figure 1.5 Weight fractions for transport aircraft in terms of empty weight over max take-off weight (Mattingly, Heiser & Daley, Aircraft Engine Design, 197) OEW/ MTOW 0.70 0.60 0.50 0.0 0.30 B727-200/231A DHC7 CV600 BAE16-0/RJ70 BAC111-200 F2-00 SA226 BAE16-200 BAC111-00 MD0 & DC9-0 DC9- L11-1/0/200 FH227 DC- F2-000/6000 F27 B767-200/ER B737-0/200 B77-200/300 DC9-50 DC9-0 L1A-0/1C DC-0 CV0 DC9-30 L11-500 DC-30 B77-0 B707-300B S360 DHC-0 J31 SA227 ATR2 B1900 F0 B737-300 B757-200 A300-600 A3-300 B767-300/ER EMB120 S30A D32 DHC-300 ATR72 J1 BAE-ATP RJ200/ER B737-500/600 BAE-16-300 A320-0/200 RJ5 B737-00 B77-00 MD11 B777 EMB15 0.20 0. Turboprops Regional Jets Large Aircraft 0.00 1955 1960 1965 1970 1975 190 195 1990 1995 2000 Year Figure 1.6 Structural efficiency data for commercial aircraft: Operating empty weight over maximum take-off weight (Babikian, R., The Historical Fuel Efficiency Characteristics of Regional Aircraft From Technological, Operational, and Cost Perspectives, SM Thesis, MIT, June 2001)

Takeoff weight W (tons) S (m 2 ) b (m) Sea-level thrust T (tons) Fuel Cruising consumption speed (liters/hour) V (km/hour) Range (km) Seats Boeing 77-00 395 530 65 x 25.7 12300 900 12200 21 Boeing 77-300 37 511 60 x 23. 13600 900 500 00 Boeing 77-200 352 511 60 x 21.3 13900 900 9500 37 Douglas Dc--30 256 36 50 3 x 23.1 00 900 9900 2 Airbus A3 139 219 2 x 22.7 5500 60 600 200 Boeing 737-300 57 5 29 2 x 9.1 2700 00 200 12 Fokker F-0 3 9 2 2 x 6.7 200 720 100 1 Fokker F-2 33 79 25 2 x.5 2500 60 1700 0 Figure 1.7 Aircraft performance (from The Simple Science of Flight, by H. Tennekes) For aircraft engines it is often convenient to break the overall efficiency into two parts: thermal efficiency and propulsive efficiency where the subscripts e and o refer to exit and inlet: mu 2 2 e e m u o rate of production of propellant k.e. o 2 2 η thermal = = fuel power m h propulsive power Tu η = = o prop rate of production of propellant k.e. 2 2 mu o e e m u o 2 2 such that η overall = η thermal η prop During the first semester thermodynamics lectures we will focus largely on thermal efficiency. In next semester s propulsion lectures we will combine thermodynamics with fluid mechanics to obtain estimates for propulsive and thus overall efficiency. The data shown in Figure 1.6 will give you a rough idea for the conversion efficiencies of various modern aircraft engines. f 9

0. 0.7 Overall Efficiency 0.1 0.2 0.3 0. 0.5 0.6 0. 0.7 0.6 0.5 0. 0.3 Future Trend SFC Core Thermal Efficiency 0.6 0.5 0. Turbojets Low BPR '777' Engines CF6-0C2 Current High BPR UDF Engine Advanced UDF 0.3 Whittle 0.2 0.3 0. 0.5 0.6 0.7 0. Propulsive x Transmission Efficiency 30 Figure 1. Trends in aircraft engine efficiency (after Pratt & Whitney) B707-300 B720-000 TSFC (mg/ns) 25 20 B727-200/231A BAC111-00 DC9-0 F2-000/6000 BAE16-0/200/RJ70 CV0 DC9- F2-00 DC9-50 B737-300 DC9-30 B737-0/200 BAE-16-300 MD0 & DC9-0 F0 RJ5 D32 EM170 DC-30 L11-500 B767-300/ER EMB15 B737-00 B737-500/600 B77-0 F27 B77-200/300 B757-200 RJ200/ER RJ700 L11-1/0/200 B77-00 CV600 DC-0 B767-200/ER EMB135 MD11 DC- A300-600 L1A-0/1C J31 A3-300 A320-0/200 B777 SA226 DHC7 B1900 SA227 CV50 S360 EMB120 J1 ATR2 ATR72 D32 DHC-0 BAE-ATP DHC-300 DHC-00 S30A Turboprops 5 Regional Jets Large Jets New Regional Jet Engines New Turboprop Engines 0 1955 1960 1965 1970 1975 190 195 1990 1995 2000 2005 Year Figure 1.9 Engine efficiency for commercial aircraft: specific fuel consumption (Babikian, R., The Historical Fuel Efficiency Characteristics of Regional Aircraft From Technological, Operational, and Cost Perspectives, SM Thesis, MIT, June 2001)

The accuracy of the range equation in predicting performance for commercial transport aircraft is quite good. The Department of Transportation collects and reports a variety of operational and financial data for the U.S. fleet in something called DOT Form 1. Operational data for fuel burned and payload (passengers and cargo) carried was extracted from Form 1 and combined with the technological data shown in Figures 1., 1.6 and 1.9 to estimate range. In Figure 1. these estimates are compared to the actual stage length flown (range) as reported in Form 1. The difference between the actual stage length flown and the estimated stage length is shown in Figure 1.11. Figure 1.11 shows that the percent deviation between the Breguet range equation estimates and the actual stage lengths flown is a function of the stage length. For long-haul flights, the assumptions of constant velocity, L/D, and SFC are good. However, for short-haul flights, taxiing, climbing, descending, etc. are a relatively large fraction of the overall flight time, so the steady-state cruise assumptions of the range equation are less valid. (16 short- and long-haul aircraft) Calculated Stage Length (miles) 6000 5000 000 3000 2000 00 0 0 00 2000 3000 000 5000 6000 Actual Stage Length Flown (miles) Figure 1. Performance of Breguet range equation for estimating commercial aircraft operations (J. J. Lee, MIT Masters Thesis, 2000) 11

160 10 120 0 0 60 0 20 0 0 00 2000 3000 000 5000 6000 Stage Length Figure 1.11 Deviation (%) of Breguet range equation estimates from actual stage length flown is a function of the stage length (J. J. Lee, MIT Masters Thesis, 2000) III. Questions: A. How far can a duck fly? B. Why don t we fly on hydrogen-powered airplanes? Fuel properties are listed below. Fuel Density (kg/m 3 ) Heating Value (kj/kg) Jet-A 00.0 5000 H 2 (gaseous, S.T.P.) 0.02 120900 H 2 (liquid, 1 atm) 70. 120900 C. Why is the maximum range for an aircraft on earth approximately 25,000mi? (Voyager: 311kg of fuel is 72% of maximum take-off weight, flight speed 16.1km/hr = 9 days to circle the earth) D. What are the assumptions and approximations that underlie the Breguet Range Equation? 12