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outflow turbine these design parameters satisfy τ >0 and λ1 >λ2 ≥0. (1.4) Kimmel’s analytical model (1.3)–(1.4) is consistent with ex- perimental studies of pump characteristics by K N A P P [1937] as summarized in Figure 13.2 of STEPANOFF [1957]. The ex- perimental results of KNAPP indicate that the curves Tturb = const. appear to be hyperbolas with asymptotes given by two distinct straight lines corresponding to the equation Tt ur b = 0; these experimental results are well modelled by (1.3)–(1.4). The applied torque function Tgen(x, y) in (1.2) is the neg- ative of the generator torque and is modelled by a suitable specified function of the turbine speed x and flow y. In fact Tgen is typically taken to depend only on the turbine speed x. For example in the case of a linear generator one uses the model The assumption (1.9) is used here. Turning now to conservation of energy, let PE(Hydraulic Head) denote the potential energy of the hydraulic head (= hy- draulic pressure head), and similarly let KE(Rotating Turbine) denote the kinetic energy of the (centrifugal force of the) ro- tating turbine shaft while KE(Fluid Flow) denotes the kinetic energy of the fluid flow. Conservation of energy requires a suitable balance between these three energies. A key difference between radial inflow turbines (analyzed in SK [1998]) and radial outflow turbines is that the tur- bine centrifugal energy KE(Rotating Turbine) reinforces or supplements the kinetic energy of the fluid flow for the in- flow turbine, but KE(Rotating Turbine) opposes and partially cancels the effects of the kinetic energy of the fluid flow for the outflow turbine. That is, conservation of energy for the radial inflow turbine requires that the hydraulic pressure head must balance the sum of the kinetic energies, PE(Hydraulic Head) = KE(Fluid Flow) + KE(Rotating Turbine) for the radial inflow turbine, (1.10) whereas for the radial outflow turbine the kinetic energy of the fluid flow alone must balance the combined sum of the hy- draulic head pressure and the turbine centrifugal energy with PE(Hydraulic Head)= KE(Fluid Flow) - KE(Rotating Turbine) for the radial outflow turbine. (1.11) It is shown in Section 2 that the energy equation (1.11) for the radial outflow turbine characterizes a family of hyperbolas in the phase plane. The turbine torque (1.3) vanishes along the line y = λ1x, and it is shown in Section 3 that this domi- nant zero torque line is an attractor for the wedge region (1.9) for the de-energized radial outflow turbine. In Section 4 it is shown that a liquid hammer occurs for the de-energized radial outflow turbine, and a quantitative estimate is given for the resulting pulse decay. 2 The Energy Equation Within the framework of classical mechanics, the kinetic en- ergy of a turbine/fluid system depends quadratically on the speeds of motion. For the radial outflow turbine the difference of the kinetic energies appearing on the right side of (1.11) may be modelled by a quadratic relation KE(Fluid Flow) − KE(Rotating Turbine) = (βy − αx) (βy + γ x) (2.1) Tgen(x, y) = − [T0 + ν · (x − x0)] (1.5) where the constant x0 is the synchronous speed of the genera- tor, T0 is the “rated” shaft torque for variable speed turbines, and the constant ν is a measure of the rate of change of gener- ator torque with respect to speed. In current practice, the two most important cases for the linear model (1.5) are: and ν ̸= 0, and ν = 0. Combining (1.1)–(1.3) yields the following equation for the balance of angular momentum, dx ε =(y−λ1x)(y−λ2x)−T(x,y) (1.6) dt where the positive constant ε is the turbine inertia I scaled by the torque parameter τ , ε := I/τ > 0, (1.7) and the function T is the (negative of the) generator torque scaled by τ , T (x, y) := −τ−1 · Tgen(x, y). (1.8) The constant ε in (1.7) is small for typical radial flow turbines for common liquids. According to the experimental studies of KNAPP [1937] (see also Fig. 13.2 of STEPANOFF [1957]), the radial outflow turbine is expected to operate in the following wedge region of the phase plane characterized as y≥λ1x for x≥0, y≥0. (1.9) and Fixed speed generator: Variable speed generator: T0 = 0 T0 ̸= 0 2PDF Image | Transient Characteristics of Radial Outflow Turbine Generators
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