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for experimentally determined design parameters α, β, γ that are positive, α,β,γ >0, (2.2) where x = x(t) and y = y(t) are the rotation speed of the turbine rotor and the liquid flow rate as in the angular mo- mentum equation (1.6). In addition to the assumption (2.2), we also assume the condition through each point (x1, y1) in the plane, with constant given by (2.8). The family of hyperbolas is centered at the point (x, y) with coordinates x=2j(α−γ) and y=− 4jαγ (2.9) β(α + γ )2 β2(α + γ )2 and the family has asymptotes given by the two lines L1 and L2 passing through (x, y) with respectives slopes α/β and −γ/β, L1:y = βα(x−x)+y and L2:y = −γβ(x−x)+y (2.10) as indicated in Figure 1. The y-intercept of the line L2 is de- noted as y2 and is negative, γ ≤ α. For the special case γ = α, (2.1) reduces to KE(Fluid Flow) − KE(Rotating Turbine) = β2 y2 − α2x2 and in this case one has (2.3) 22 22 KE(Rotating Turbine) = α x and KE(Fluid Flow) = β y if α = γ. (2.4) Hence(inthecaseγ =α)αandβmaybethoughtofaski- netic energy coefficients respectively for the rotating turbine and for the liquid flow. The potential energy of the hydraulic head may be denoted briefly as H, y=− 2jγ <0 2 β2(α + γ ) (2.11) (2.12) PE(Hydraulic Head) = H, (2.5) since Joukowski’s coefficient is positive j > 0. ✻ L: slope = α/β 1 where the rate of change of H is assumed to satisfy Joukowski’s relation H ̇ = −2 j y ̇ (cf. p. 435 of STEPANOFF ❈❖ ❈ can now be written with (2.1), (2.5) and (2.6) as (β y − α x ) (β y + γ x ) + 2 j y = C Figure 1 which is to hold during the operation of a given radial outflow turbine/fluid system, for suitable design parameters α,β,γ, and j. The constant C = C(x1, y1) of integration satisfies C =C(x1,y1)=(βy1 −αx1)(βy1 +γx1)+2jy1 (2.8) where x1 and y1 may be taken to be the values of the rotation speed and flow at any fixed instant during the operation of the system. A routine calculation (cf. pp. 229–235 of T H O M A S [1983]) using (2.2) shows that the equation (2.7) characterizes a family of hyperbolas in the (x, y)-plane, parameterized by the constant C. There is a unique such hyperbola (2.7) passing (βy − αx)(βy + γ x) ≥ 0 for all (x, y) in the wedge region (1.9). (2.14) L2 :slope=−γ/β ̇ ✲ [1957]) for a fixed design parameter j known as Joukowski’s ✠ coefficient. Upon integration of H = −2 j y ̇ , there holds H=−2jy+C (2.6) for a constant C of integration. The energy relation (1.11) for the radial outflow turbine • ✒ • • (2.7) The earlier shaft coefficients λ1 and λ2 in (1.3) are as- sumed to be distinct and nonnegative as in (1.4). Moreover, the dominant shaft coefficient is assumed to be larger than the ratio α/β of the energy coefficients, λ1 > βα. (2.13) The inequalities (1.9) and (2.13) yield y ≥ λ1x ≥ (α/β)x, which with (2.2) implies βy − αx ≥ 0. Similarly there also holds β y + γ x ≥ 0 everywhere in the wedge region (1.9), and these last two inequalities together yield 3 y2<0 (x,y) PDF Image | Transient Characteristics of Radial Outflow Turbine Generators
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