PART ONE Parameters for Performance Calculations

Transcription

1 PART ONE Parameters for Performance Calculations As an amateur designer/builder of homebuilt aircraft, you are chief aerodynamicist, structural engineer, dynamicist, mechanic, artist and draftsman all rolled into one. This role may require a program of selfstudy to gain the necessary new skills or a request for help from a team of experts because it is difficult to know where to begin and what the next step should be. Whether you are modifying an existing aircraft or beginning the design from scratch, you need to be able to relate the aerodynamic, structural and inertial loads to the physical design parameters of the airplane. While pursuing the design details, you should keep in mind the overall performance goals and have a feel for the effect of each of the basic design parameters. By first defining the mission, you will put constraints on the size, weight and sophistication of the airplane, deciding which is more important speed, fuel economy, fun or ease of construction. Assuming the mission is well defined, we will replace the complicated dragproducing parts of the airplane configuration whether conventional, canard, biplane, tandem, tractor or pusher with an equivalent hypothetical system: a flat plate having the same parasite drag and an elliptical wing having the same induced drag. Then, as sketched in Figure 1, seven basic design parameters can be used to give meaningful estimates of airplane performance. These parameters are: weight, W; drag area, A D ; effective span, b e ; engine power, P e ; propeller diameter, D p ; wing area, 8, and maximum lift coefficient, CL, max' An accurate description of each of these parameters eventually will require a detailed aerodynamic and structural analysis of the airplane. Meanwhile, we must be sure that the airplane will be safely controllable, maneuverable and stable in all phases of flight. We will use a building-block approach so that the solutions to small, "bite-sized" problems can be combined to answer overall questions relating to performance and flying qualities. Details of these types of calculations will be covered in subsequent articles. ( References 1-4 can be used to Figure 1. The Equivalent Airplane

2 2 AIRPLANE DESIGN get started by those who cannot wait. ) Meanwhile, you should be able to make simplified performance calculations with the seven basic parameters as outlined below and summarized in the BASIC computer program. Rate of climb versus velocity is the fundamental measure of performance for an airplane. It is equal to the excess power available the difference between the power available, 13 av, and the power required for level flight, P, divided by the gross weight, W. In the following, we will dissect the expressions for power available and power required into smaller and smaller pieces until we arrive at the design parameters and operating conditions. Power Required for Level Flight The basic equations describing the power required terms are summarized in Figure 2 and described below. P av req R C = w The power required for level flight is equal to the drag, D, times the velocity, V. The drag can be defined as the drag coefficient, C D, times the dynamic pressure, q, times the wing area, S, where the dynamic pressure is equal to one-half the air density, p, times the velocity squared. The drag coefficient is the sum of the zerolift and the induced-drag coefficients, C D, 0 and C D, i. Assuming that the induced drag of the airplane obeys the so-called parabolic drag polar, we let the induced drag coefficient be equal to the lift coefficient, C L, squared divided by it times the effective aspect ratio, ear. The lift-coefficient is defined by the lift, L, divided by the dynamic pressure and wing area, where the lift is equal to the weight, W, for straight and level flight. For banked turns, the lift required is increased by dividing the weight by the cosine of the bank angle, BA. And, finally, the effective aspect ratio is equal to the geometric aspect ratio span, b, squared divided by the wing area multiplied by Oswald's airplane efficiency factor, e. The P q = DV D = C dqs q= fp V2 C d = C d,0+ Cd,i C2 C d, e AR it C L qs L= W cos(ba) v e AR = e ( 8 ) ( 9 ) Figure 2. Performance Equations Figure 3. Effective Span in a Turn

3 PART ONE efficiency factor is related to the geometry of the configuration and is one of the most difficult quantities to calculate accurately. These relations can be more conveniently written in equation form as shown in Figure 2. By rearranging expressions (2-9), we can rewrite the power required for level flight as P req 1 P A d 2( Wl b ei) V3 + ( 10) p V where the drag area, A d, is defined to be equal to the zero-lift drag coefficient times the wing area, and the effective span, b e', is equal to the square root of the airplane efficiency factor, e, times the geometric span, b, times the cosine of the bank angle, BA. This geometric projection is shown in Figure 3. Increasing the bank angle decreases the effective span, b e' In equation form, A d Cd, 0 S (11) b e = \Feb cos( BA) (12) The drag area is calculated by adding the contributions from each of the airplane components, such as the fuselage, wing or horizontal tail. The expression for pr,,ver required ( 10) shows that at high speeds, the drag area is the dominant parameter, but at low speeds, the effective span loading, Wlbei, is more important. In between, there is a speed at which the power required for level flight is minimal. Tho nondimensional curve for power required for level flight is shown in Figure 4. By using logarithmic scales we can make performance calculations with simplified graphical methods. These are described in detail in Reference 1. Stall Speed The relation for power required in the equation ( 10) assumes that the parabolic drag polar holds for all lift coefficients and speed. We know, however, that as the lift coefficient increases, there will come a point where the air can no longer remain attached to the upper surface of the wing, and the wing stalls. The maxim= lift coefficient, C L, max, is a parameter that is dependent on the airfoil section and the wing configuration. It can be increased by A d = A d ( + A d ( -4- A d ( + ( 13) The drag analysis is one of the most important tasks facing the aerodynamicist. We will devote future space to looking at this subject. Meanwhile, the aerodynamicist in each of us can recognize that drag cleanup is important for improvements in fuel economy, speed and climb rate. The sink rate, R/S, that occurs in gliding flight is related to this power by dividing by the weight, W. That is, RIS = P wreq Figure 4. Nondimensional Power Required for Level Flight vs. Nondimensional Airspeed (logarithmic scales).

4 4 AIRPLANE DESIGN Stnearntube Boundary Freestreern Velocity, V Propellor Disk Area, Ap Streorntube Area, A3.4 Slipstream Velocity, V3 Pressure Distribution, P2 P, Pressure Jump Across Propeller Disk Figure 5. Actuator Disk Propeller Theory the addition of high-lift devices such as flaps and leading edge slots. The minimum speed can be found by solving for the velocity that gives the dynamic pressure required to lift the plane operating with the maximum lift coefficient. From equations ( 4), ( 7), and ( 8), we find that W V stall 15) \p C L, ma, S COS( BA) At speeds lower than this speed, the curve for power-required must be cut off. Power Available The power available is equal to the thrust, T, times the velocity, V, which is equivalently equal to the propeller efficiency, 7/, times the shaft power of the engine, Pe. P a = TV=77.13 (16) This is simply a rearrangement of the the definition for the propeller efficiency P avail TV 1= P epe (17) and is equal to the "useful thrust power" divided by the shaft power input. Three additional propeller definitions are given below: advance ratio, J, thrust coefficient, C T, and power coefficient, C p. or, CT= C p J= n D V pn P P P T 2D 4 P Pe prlp3dp5 V= npdpj T=pnp2Dp4CT P e = pnp3dp5cp where the propeller rotational speed, np, (in revolutions per second), air density, p, and propeller diameter, D e, are used to nondimensionalize the velocity, thrust and power. For a constant-speed propeller, the governor changes the blade-setting angle at different airspeeds in order to keep the same engine rpm. If we solve ( 20) for np,

5 PART ONE we have so that f P e n = P tpdp5cp V { e p D p2 1/3 1/3 Cp1/3 (21) (22) A propeller curve that gives 71 versus J/ C p li3 can then be used to find the power available as a function of airspeed. This will be described in the section on the simplified, actuator-disk theory. For a fixed-pitch propeller, the propeller rotational speed will change as the airspeed changes, so the engine torque versus rpm as determined from dynamometer testing is satisfied. As a first approximation, the torque, Q e, remains constant with engine speed. Then, P e = 27 n e Q, (23) Substituting into equation ( 19) and defining the gear ratio, GR, by the engine speed divided by the propeller rotational Figure 6. Propeller Efficiency vs. Nondimensional Speed (logarithmic scales) speed, GR = np then we can solve for rotational speed 27r GRQ n = P 1 pdp5cp the propeller Substituting into ( 18a), ( 20a), and ( 16), we have and P avail = 27-GRQ, v=,\ (26) p D p (27 GR Q e)311/2 P D 1, 5 \IC p (27) Therefore, a propeller curve for y/icp versus Ji\jCp will allow us to calculate power available versus airspeed ( assuming that the engine torque is constant ). This will be discussed further at a later time. Actuator Disk Propeller Theory The actuator disk propeller theory considers the flow of air through the propeller disk plane as shown in Figure 5. The thrust, T, is equal to the static pressure jump, p 2 p i, across the propeller plane times the disk area, A p ( equal to 4 D P 2 ). It is also equal to the mass flux in the streamtube times the difference in velocity downstream and upstream of the propeller, V 3 V. The mass flux is equal to the area of the propeller disk plane times the air density, p, times the velocity at the propeller plane, V p. If the total pressure ( static plus dynamic pressure ) is constant upstream of the propeller disk, then Bernoulli's equation says that Poo-i--P 172=p i-f- Aj p V p 2 (28)