# turbine, Appunti di Macchine. Università de L'Aquila

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It can be increased to reduce flare

Constant specific mass flow

The free vortex and constant nozzle angle designs are only two possibilities. Another criterion is to satisfy the RE and obtain a constant mass flow per unit area at all radii: the axial and whirl components are chosen so that

Even if the axial velocity is constant throughout the annulus before and after the rows of a free vortex turbine, there is a streamline shift. This is due to the presence of a radial velocity component arising from an increase of the density from root to tip, in contrast with the main hypothesis of the RE. In fact, the pressure at the exit of the nozzle increases from root to tip, to balance the centripetal acceleration; on the other hand, if the flow leaves the rotor axially, it is characterized by uniform velocity, pressure, and density.

ρ2 Ca2=const . ∀ r

Even if there is no evidence that this leads to inefficiency, one can assume a design with constant mass flow per unit area, in order to have zero radial velocity component. In this case, it is useful to write the RE equation in terms of the velocity magnitude and swirl angle

Ca dCa dr

C w dC w dr

C w

2

r = dh0 dr

T ds dr

C cos α d C cos α dr

C sinα d C sinα dr

C 2sin2α r

= dh0 dr

T ds dr

C dC dr

C 2sin2 α r

= dh0 dr

T ds dr

C=Cm exp −∫rm r sin2 α

r dr   A

1 ρ dp dr = 1 ρ  ∂ pρ s dρdr =a

2

ρ dρ dr =C

2sin2α r

T ρ γ−1

=const .=B= T m ρm γ−1

γ RT ρ

dρ dr =C

2sin2α r

γ RB ρ γ−2 dρ dr =C

2sin2α r

If entropy is constant along the radius,

For constant specific mass flow, the product of density and axial velocity is constant

ρC cos α= ρmCm cos αm=σ B γ RB ργ

σ 2 dρ dr

=tan 2 α

r γ RB ρ

γ

σ2  γ1  [ ρ γ1− ρm

γ1 ]=∫ rm

r tan2 α r

dr

1 γ1  M xm

2 [ ρρm γ1

−1]=∫r rm tan2α

r dr C

Assuming that the stagnation enthalpy and the entropy are constant at the stator exit, and starting from the flow conditions at mean radius, the flow variable variations along the radius at the stator exit can be evaluated (density, velocity magnitude, and flow angle). Then, a similar procedure can be used to determine the flow conditions at the rotor exit. However, in such a case, the specific work is not constant along the radius:

tan α 3= tan β 3− U

C 3cos α 3

dh03 dr

= d h01−W

dr = dh01 dr

dW dr

=− d dr [U C 2sin α2C 3sin α3  ]

The relative and absolute outlet flow angles are related by the following equation:

unkwons : h03 , C3 , α 3 , β3 , ρ3

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