How Cool Is a Flying Aircraft Carrier?
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Aerodynamics Extra Credit Project: How cool is a flying aircraft carrier? Dr. Jonathan Poggie Purdue University AAE 334, Fall 2016 Figure 1: Image from the movie The Avengers (Marvel Figure 2: Image from the movie Captain America: The Studios, 2012). Winter Soldier (Marvel Studios, 2014). 1 Problem assume that these are similar to conventional jet engines. The challenge for this extra credit project was to an- In the Marvel Avengers movies, the S.H.I.E.L.D. Helicar- alyze the operation of the Helicarrier. The assignment rier is a kind of flying aircraft carrier (Fig. 1). It can sail asked for at least the following analysis: on the ocean like a conventional ship, but it can also lift out of the water and fly. Lift for the Helicarrier is pro- • Plot of power required versus altitude and forward vided by four fans,1 and there are two arrays of engines in speed the rear. Stills from the movies, for example Fig. 2, seem • Estimate of the maximum force produced by the fans to show a total of 20 engines. and their maximum rotation rate The means of forward propulsion is not made com- pletely clear in the movies. Small-scale quadcopters fly • Consideration of an alternative design forward by tilting to an angle-of-attack to generate for- ward thrust. There is no clear provision for tilting the fans The assignment also encouraged consideration of the on the Helicarrier, and it is not likely that it can tolerate maximum forward speed and the lift provided by the body a significant angle-of-attack. Thus, forward propulsion of the vehicle. must be provided primarily by the rear engines. We will Comparing the size of the aircraft on deck to that of the Helicarrier, it appears to be comparable to the largest ex- 1These are rotating blades, not comic book enthusiasts. isting vessels, such as the United States Nimitz-class air- 1 Aircraft Scaling by Weight Helicopter Scaling by Weight 103 105 1013 102 1012 104 1011 1010 102 3 ) 9 10 2 10 m ( a 8 1 e r 10 10 A g 2 n 7 10 i 10 W Blade Radius (m) Engine Power (W) Cruise Speed (m/s) 101 106 101 105 104 100 100 103 100 103 104 105 106 107 108 109 103 104 105 106 107 108 109 Maximum Weight (N) Maximum Weight (N) Figure 3: Scaling up from existing large, subsonic air- Figure 4: Scaling up from existing helicopters. craft. Data for winged flying objects [6], from insects to large craft carriers. To approximate a ship the size of a Nimitz- airplanes, indicate that flying speed scales approximately class carrier [5, p. 809], we will take the total mass of the as the one-sixth power of vehicle weight, and that wing vehicle to be M = 1:0 × 108 kg and the vehicle length area scales as the two-thirds power of weight. This rela- to be ` = 330 m. Comparing the size of the fans to the tionship can be derived by assuming a constant lift coeffi- length of the ship in Figs. 1–2, we will take their radius to cient, and using the facts that lift equals weight in steady be R = 25 m. flight, mass scales as length cubed, and wing area scales A Nimitz-class aircraft carrier is powered by two West- as length squared. inghouse A4W pressurized water reactors [7], which pro- Figure 3 illustrates this scaling. The solid lines repre- duce steam used to generate approximately 1:0 × 108 W sent the theory and the symbols the properties of a variety electrical power and another 1:0 × 108 W of shaft power. of aircraft, including the Boeing B747-400, Airbus A380- These figures provide a bound on the power available to 800, and the Antonov An-225 at the largest scale. If we support flight. can extrapolate3 by three orders of magnitude with our correlation, a vehicle with a mass of 1:0 × 108 kg would cruise at roughly 600 m/s (around Mach 2), and the wing 2 Scaling area would be about 3 × 104 m2. The largest helicopters are an order of magnitude It is worth beginning by comparing the Helicarrier to ex- smaller than the largest aircraft. The Mil Mi-26 is a isting large flying vehicles.2 For example, it is larger than very large helicopter, with a rotor diameter of 32 m, to- the Saturn V rocket, which had a mass of 3:0 × 106 kg. tal engine power of 17 MW, maximum take-off mass of Very large existing aircraft include the Boeing B747-400 5:6 × 104 kg, and a cruise speed of 70 m/s. A four (about 4:0 × 105 kg maximum mass at takeoff), the Air- lift-fan system equivalent to four Mi-26s would be about bus A380-800 (5:8 × 105 kg), and the Antonov An-225 2 × 105 kg, five-hundred times smaller than an aircraft (6:4 × 105 kg). We see that an aircraft carrier is more carrier. than one-hundred times heavier than the largest existing We can also make scaling estimates for helicopters. As airplanes. we will see in the next section, the engine power of heli- 2The aircraft data presented here were obtained from Wikipedia, 3In accordance with the order-of-magnitude accuracy of these calcu- Jane’s All the World’s Aircraft [3, 4], and Tennekes [6]. lations, all the numbers will be rounded to one significant digit. 2 Power to Keep Helicarrier Aloft Power to Keep Helicarrier Aloft 0.25 1.0 Drag power z = 0.0 km Fan power z = 5.0 km Total power z = 10.0 km 0.20 0.8 z = 15.0 km z = 20.0 km 0.15 0.6 Power (TW) 0.10 Power (TW) 0.4 0.05 0.2 0.00 0.0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 Speed (m/s) Speed (m/s) Figure 5: Power required to keep Helicarrier aloft at sea Figure 6: Power required to keep Helicarrier aloft at vari- level (1 TW = 1012 W). ous altitudes. copters scales as weight to the two-thirds power, and the We now consider the combined effects of the lift from blade diameter scales as weight to the one-fourth power. the fans and the forward motion for different altitudes and Figure 4 illustrates this scaling, along with data from sev- speeds. As an upper bound on performance, we gener- 4 eral existing helicopters, such as the Bell 206B-L4, Mil ously estimate the lift and drag coefficients of the hull to Mi-26, and Sikorsky S-67 Blackhawk. For a helicopter be CL = 0:6 and CD = 0:3. The total power to keep the extrapolated to 1:0 × 108 kg, the engine would need to helicarrier aloft is given by: 12 produce 2×10 W and have a rotor blade radius of 90 m. s [(W − L)=4]3 For comparison, this power consumption corresponds P = 4 + DV (2) to about ten thousand Westinghouse A4W nuclear reac- 2ρA tors, and about twice the total capacity for electrical power 1 2 generation of the entire United States [8]. where the lift is L = CLS 2 ρV and the drag is D = 1 2 CDS 2 ρV . Here the reference area is taken as S ≈ 0:25 `2. The speed at which the full weight of the vehicle 3 Estimates of Performance is supported by the lift due to forward motion is: s From actuator disk theory [2, pp. 671–672], the minimum 2W Vs = (3) power required to hover with the four lift fans can be es- CLρS timated as: s This is a form of the stall speed. Below this speed, the (W=4)3 P = 4 (1) fans must provide part of the lift that supports the vehicle. 2ρA Figure 5 shows the minimum power required for sea where W = Mg is the total weight of the vehicle, the level conditions for speeds up to Vs. At low speed, the acceleration of gravity is g = 9:8 m/s, ρ = 1:2 kg/m3 is fans provide most the required power (blue line), and their the sea-level density of air, and A = πR2 is the area of contribution drops to zero as the speed approaches Vs.A each fan. Each of the four fans is assumed to support one constant drag coefficient has been assumed, so the drag 4 quarter of the total weight. The minimum power required A large commercial airplane at cruise typically has about CL = 0:6 11 to hover is about P = 2 × 10 W. and CD = 0:04; a blunt object has CD ≈ 0:5. 3 power (red line) increases as the cube of speed. The sum is not possible to provide sufficient power abord the craft, of these two components gives the total power (black line, nor can the fans develop sufficient thrust to lift the vehicle. Eq. 2), which has a minimum at relatively high speed. In considering a redesign, we have to examine the value The total power required to stay aloft at altitudes be- that the product gives to the customer. What does a Heli- tween sea level and 20 km is shown in Figure 6. The den- carrier do for S.H.I.E.L.D.? sity and temperature as a function of altitude were taken A conventional aircraft carrier is a device for efficiently from a correlation given in Ref. [1]. The power required positioning around 100 aircraft and their supporting in- increases substantially with altitude because the efficiency frastructure anywhere on the ocean.