Combined.BE.Momentum.Theory

# Combined.BE.Momentum.Theory - IV Combined Blade...

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Unformatted text preview: IV. Combined Blade Element-Momentum Theory In the previous sections we dealt with momentum theory which gave estimates for power and induced velocity in terms of the thrust. We also studied blade element theory, which allowed us to incorporate features such as number of blades, airfoil section drag and lift characteristics, taper and twist, etc. The latter, however, assumed that the flow through the rotor disk is uniform as in the momentum theory. Consider a small annulus segment of the rotor disk, shown. It has a radius r , and width dr, and an area of 2 π rdr. The mass flow through this annulus is 2 π r ρ (V+v)dr. Unlike the previous two theories , here we assume v to be a function of r. According to momentum and energy considerations, the velocity in the far wake for this annulus is V+2v. Using relations given in Handout II, the thrust generated by this annulus is ( 29 dT r V v vdr = + 4 πρ (1) Next, consider the blade elements that this annulus intersects. The thrust generated by these blade elements is, ( 29 ( 29 dT b r c C dr abc r V v r dr l = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅- + ⋅ 1 2 1 2 2 2 ρ ρ θ Ω Ω Ω (2) These two approaches give two estimates for the thrust generated by the annulus of width ‘dr’. Equating these two, we get a quadratic equation for the inflow λ = (V+v)/ Ω R: R V R bc where R r a a c c Ω = = =- - + λ π σ θ σ λ λ σ λ , 8 8 2 (3) Note that the chord c and the solidity σ may vary with r/R for tapered rotors. may vary with r/R for tapered rotors....
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