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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Blade element momentum theory ： ウィキペディア英語版 Blade element momentum theory Blade element momentum theory is a theory that combines both blade element theory and momentum theory. It is used to calculate the local forces on a propeller or wind-turbine blade. Blade element theory is combined with momentum theory to alleviate some of the difficulties in calculating the induced velocities at the rotor.〔Blade element theory〕 This article emphasizes application of BEM to ground-based wind turbines, but the principles apply as well to propellers. Whereas the streamtube area is reduced by a propeller, it is expanded by a wind turbine. For either application, a highly simplified but useful approximation is the Rankine-Froude "momentum" or "actuator disk" model (1865,1889). Herein, we study the "Betz Limit" on the efficiency of a ground-based wind turbine. A development came in the form of Froude's Blade Element Momentum theory (1878), later refined by Glauert (1926). Betz (1921) provided an approximate correction to momentum "Rankine-Froude Actuator-Disk" theory to account for the sudden rotation imparted to the flow by the actuator disk (NACA TN 83, "The Theory of the Screw Propeller" and NACA TM 491, "Propeller Problems"). In Blade Element Momentum theory, angular momentum is included in the model, meaning that the wake (the air after interaction with the rotor) has angular momentum. That is, the air begins to rotate about the z-axis immediately upon interaction with the rotor (see diagram below). Angular momentum must be taken into account since the rotor, which is the device that extracts the energy from the wind, is rotating as a result of the interaction with the wind. The following provides a background section on the Rankine-Froude model, followed by the Blade Element Momentum theory. ==Rankine-Froude Model== The "Betz limit," not yet taking advantage of Betz' contribution to account for rotational flow with emphasis on propellers, applies the Rankine-Froude "actuator disk" theory to obtain the maximum efficiency of a stationary wind turbine. The following analysis is restricted to axial motion of the air: In our streamtube we have fluid flowing from left to right, and an actuator disk that represents the rotor. We will assume that the rotor is infinitessimally thin.〔Wind Energy Handbook: Burton, Jenkins〕 From above, we can see that at the start of the streamtube, fluid flow is normal to the actuator disk. The fluid interacts with the rotor, thus transferring energy from the fluid to the rotor. The fluid then continues to flow downstream. Thus we can break our system/streamtube into two sections: pre-acuator disk, and post-actuator disk. Before interaction with the rotor, the total energy in the fluid is constant. Furthermore, after interacting with the rotor, the total energy in the fluid is constant. Bernoulli's equation describes the different forms of energy that are present in fluid flow where the net energy is constant i.e. when a fluid is not transferring any energy to some other entity such as a rotor. The energy consists of static pressure, gravitational potential energy, and kinetic energy. Mathematically, we have the following expression: :$$ \frac\rho v^2 + P + \rho g h = \mathrm where $\backslash rho$ is the density of the fluid, $v$ is the velocity of the fluid along a streamline, $P$ is the static pressure energy, $g$ is the acceleration due to gravity, and $h$ is the height above the ground. For the purposes of this analysis, we will assume that gravitational potential energy is unchanging during fluid flow from left to right such that we have the following: :$$ \frac\rho v^2 + P = \mathrm Thus, if we have two points on a streamline, point 1 and point 2, and at point 1 the velocity of the fluid along the streamline is $v\_1$ and the pressure at 1 is $P\_1$, and at point 2 the velocity of the fluid along the streamline is $v\_2$ and the pressure at 2 is $P\_2$, and no energy has been extracted from the fluid between points 1 and 2, then we have the following expression: :$$ \frac\rho v_1^2 + P_1 = \frac\rho v_2^2 + P_2 Now let us return to our initial diagram. Consider pre-actuator flow. Far upstream, the fluid velocity is $v\_$; the fluid then expands as it approaches the rotor.〔http://cdn.intechopen.com/pdfs/16241/InTech-Aerodynamics_of_wind_turbines.pdf〕 In accordance with mass conservation, the mass flow rate must be constant. The mass flow rate, $\backslash dot$, through a surface of area $A$ is given by the following expression: :$$ \dot = \rho Av where $\backslash rho$ is the density and $v$ is the velocity of the fluid along a streamline. Thus, if mass flow rate is constant, increases in area must result in decreases in fluid velocity along a streamline. This means the kinetic energy of the fluid is decreasing. If the flow is expanding but not transferring energy, then Bernoulli applies. Thus the reduction in kinetic energy is countered by an increase in static pressure energy. Why a streamtube expands as it approaches an object is not explained in this document. So we have the following situation pre-rotor: far upstream, fluid pressure is the same as atmospheric, $P\_$; just before interaction with the rotor, fluid pressure has increased and so kinetic energy has decreased. This can be described mathematically using Bernoulli's equation: :$$ \frac\rho v_^2 + P_ = \frac\rho \left(v_(1 - a)\right)^2 + P_ where we have written the fluid velocity at the rotor as $v\_(1\; -\; a)$, where $a$ is the axial induction factor. The pressure of the fluid on the upstream side of the actuator disk is $P\_$. We are treating the rotor as an actuator disk that is infinitely thin. Thus we will assume no change in fluid velocity across the actuator disk. Since energy has been extracted from the fluid, the pressure must have decreased. Now let us consider post-rotor: immediately after interacting with the rotor, the fluid velocity is still $v\_(1\; -\; a)$, but the pressure has dropped to a value $P\_$; far downstream, pressure of the fluid has reached equilibrium with the atmosphere i.e. $P\; \backslash rightarrow\; P\_$ far downstream. Assuming no further energy transfer, we can apply Bernoulli for downstream: :$$ \frac\rho \left(v_(1 - a)\right)^2 + P_ = \frac\rho v_w^2 + P_ Thus we can obtain an expression for pressure difference for and aft the rotor: :$$ P_ - P_ = \frac\rho(v_^2 - v_w^2) If we have a pressure difference across the area of the actuator disc, there is a force acting on the actuator disk, which can be determined from $F\; =\; \backslash Delta\; PA$: :$$ \frac\rho(v_^2 - v_w^2)A_D where $A\_D$ is the area of the actuator disk. If the rotor is the only thing absorbing energy from the fluid, the rate of change in axial momentum of the fluid is the force that is acting on the rotor. The rate of change of axial momentum can be expressed as the difference between the initial and final axial velocities of the fluid, multiplied by the mass flow rate: :$$ F = \frac = \dot(v_ - v_) = \rho A_Dv_D(v_ - v_) = \rho A_D(1 - a)v_(v_ - v_) Thus we can arrive at an expression for the fluid velocity far downstream: :$$ v_w = (1 - 2a)v_ This force is acting at the rotor. The power taken from the fluid is the force acting on the fluid multiplied by the velocity of the fluid at the point of power extraction: :$$ \mathrm_ = Fv_D = 2a(1 - a)^2v_^3\rho A_D 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Blade element momentum theory」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.