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Transcript
AA241X April 2007 Propulsion for Small Electric Aircraft This section describes a few of the basic considerations needed to estimate the performance of small electric aircraft propulsion systems. An efficient aircraft requires a careful matching of motor, propeller, and airframe -- a matching that is sometimes difficult when the design is highly constrained (as in AA241X). Computer programs such as some of the applets here and specialized programs such as MotoCalc (www.motocalc.com) provide useful ways to estimate system performance, but can sometime be too helpful, eliminating the need to understand why systems behave as they do. The following notes are intended to introduce you to some of the physics underlying such calculations. Enter data in the applets below and click "Compute" to update. Drag the mouse on the plot to zoom in; shift-click to auto-scale. Electric Motor Performance Electric motors are characterized by their weight, design voltage, and maximum power output, but also by several other parameters that determine how the efficiency and power vary with RPM. If gearboxes are available, the optimal RPM of the propeller and motor can be made to match, but for direct drive systems, the matching of these characteristics is especially important. One can compute many useful electric motor characteristics from the 3 basic equations shown below. In this analysis, the motor performance is characterized by 3 basic constants: Kv relates the motor speed (RPM) to the voltage applied to the windings. This characterizes how the motor's back emf changes with speed. For many motors, this relation is roughly linear so we can write: RPM = Kv (V - I R) Where V is the applied voltage, I is the current, and R is the armature resistance. Kv is often specified for a particular motor, or it can be measured by testing the motor and measuring the unloaded RPM and current at several voltages. R is also often published or can be measured directly on the motor. The motor torque is related to the applied current (since this sets the strength of the magnetic fields generated by the motor windings). The expression is: Q = Kt (I - I0) Where Q is the torque and I0 is the no-load current at the specified voltage. I0 is also sometimes specified in motor catalogs or can be measured directly on a given motor. Noting that Q RPM is proportional to output power and that (V-IR)*(I-I0) is the input power minus losses, we see there is a direct relationship between Kt and Kv. With Kt in RPM/volt and Kv in in-oz (as in MotoCalc), the expression is: Kv Kt = 1352. One could do this in many unit systems, but the idea is that only one of these motor parameters (usually Kv) needs to be specified. The net result is that when the motor is characterized by Kv, I0, and R, one may compute the output power, input power (I V), torque, and efficiency. From the equations above. With proper unit conversions, this is what is done in the applet in these notes. Kt = 1352./Kv; I = (volts - RPM/Kv)/R; Q = Kt*(I-I0); // Q in in oz if Kt defined as in Motocalc QinNm = Q*.00706; // Conversion to N m Pin = volts*I; // watts Pout = QinNm*RPM*2.*pi/60.; // watts eta = Pout/Pin; // Motor Efficiency Propeller Performance The analysis of propellers ranges from very simple momentum arguments to very sophisticated computer programs that can tax even modern supercomputers. In the following discussion, we consider a few approaches to propeller analysis. Maximum Efficiency The first step in the study of propeller performance is an inviscid momentum analysis with lots of simplifying assumptions. The thrust produced by an annular region of the propeller at radius r may be estimated by considering the rate of momentum change through the propeller at this station: dT = ρ (U+u) (2π r dr) 2u From a vortex model perspective, the blades are generating lift due to their circulation: dT = Nblades ρ Γ ( ω r) dr The induced angle generates some induced drag on the blades so that at each station, r, the incremental torque is given by: dQ = Nblades ρ Γ (U+u) r dr Using the second expression to compute the blade circulation, we have: Γ = (U+u) (4π) u / (ω Nblades) If the blades generate a uniform (constant with radius, r) increase in velocity through the propeller plane (u) then the blade circulation is also constant and we can integrate the expressions above to find: T = 2 ρ (U+u) u A Q = 2 ρ (U+u) 2 u A/ω where: A = π R2, the disk area. This means that ω Q = (U+u) T So the efficiency is: η = Pout / Pin = T U / ω Q = U / (U+u). Now the expression for T can be used to relate u to the other parameters. Solving the quadratic: 0 = u2 + Uu - T/2ρA 2u/U = sqrt( 1 + 2T/ρAU2) - 1 Note that when the propeller is lightly loaded (2T/ρAU2 << 1) the above simplifies to: u/U = T/2ρAU2 So: η = 1 / (1 + T/2ρAU2) This expression illustrates a few things, even though we have ignored many features such as nonuniform u, viscous drag, propeller swirl, and others. First, the basic momentum efficiency limits propeller performance at low speeds. For example, if we need to generate a thrust of just 1/2 oz at 10 ft/sec with a 3 in propeller, the inflow velocity ratio is u/U = .76 and the efficiency is no more than 57%. Other effects will reduce this further. If the units above bothered you, then good. Propeller performance is generally computed in dimensionless terms, with the following (rather odd) nondimensionalization most common: • CT = T/ ρ n2 D4 • C Q = Torque / ρ n2 D5 • C P = Power_In / ρ n3 D5 • J = advance ratio = U / n D where n = revolutions per second, and D = diameter. So in dimensionless terms: 2u/U = sqrt( 1 + 8C T/πJ2) - 1 or: η = 2 / (1+sqrt( 1 + 8CT/πJ2)). Propeller Thrust Propeller thrust is specified in the above expressions and often we are interested in the available thrust from a specific propeller. This can be done from a more firstprinciples analysis, or based on some measured data. One additional parameter that is useful in describing a propeller geometry is the pitch. If a propeller were designed so that its sections would follow streamlines, generating no lift at a certain speed (the pitch speed), then we would define the distance that it travelled forward in each revolution as the pitch: pitch = Ups / (ω/2π) = Ups / n The advance ratio at the pitch speed is given by: Jps = Ups / n D = pitch / D. So if we have a propeller with a pitch/D of 50/82, it will generate no thrust at J = .61. We can also estimate the torque at this condition, which is due to viscous drag. From the expressions at the link below, the result is: dQ = Nblades ρ/2 ω sqrt(Ups2+ ω2r2) Cd c r2 dr which can be easily integrated with an assumed distribution of chord and Cd. At the static condition, the angle of attack distribution is not so simple and the assumption of constant inflow, u, is not so good. There are a few choices here: one can model the actual incidence, chord, and Cd distributions and iterate to find the inflow distribution, thrust, and torque. One can assume the inflow is constant anyway (probably OK for rough performance estimates). Or one can measure the static thrust and infer the average inflow. MotoCalc appears to use some approximate curves of CT and CQ vs. J and modify these based on measured static thrust -- fine for rough performance estimates, but not for propeller design. Additional details on these calculations and the theory behind vortex-momentum theory of propeller analysis is provided at this link. Combined Motor and Propeller Performance Combining the motor and propeller analysis allows us to compute the variation of thrust, motor current, and efficiency of the propulsion system as shown in the applet below. In this case, the program iterates on RPM at each speed to find the propeller speed at which required torque is equal to that provided by the motor. It may be useful to fit this data in order to take the next step: matching propulsion system characteristics to your airframe characteristics.