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3 The De-Energized Outflow Turbine The discussion in Section 2 shows that the state of the ra- dial outflow turbine/fluid system will move in the phase plane along a fixed hyperbola (2.7) during the operation of the sys- tem (as long as the state of the system is not forced to jump or switch from one hyperbola to another by any externally im- posed change or modification to any of the model properties). The time evolution of the system is described by the motion of the point (x(t), y(t)) along the appropriate hyperbola, sub- ject to the energy equation (2.7) and the angular momentum equation (1.6). The generator torque function T (x, y) = T (x, y, t) ap- pearing in the angular momentum equation (1.6) may depend explicitly on certain switching times which are values of t for which the form of the torque function may abruptly change. We consider the case of a single switching time t0 and take the applied generator torque as under the influence of a modified momentum equation (1.6) which now becomes T (x, y,t) = (3.1) Tgen(x, y) 0 for t ≤ t0 for t > t0, The functions x(t) and y(t) are assumed to be continuous across the disconnect time, so the system must satisfy the ini- tial conditions x=x0 and y=y0 at t=t0, (3.5) where the initial state (x0, y0) satisfies (3.3) and (1.9) with y0 >λ1x0 and x0,y0 ≥0 (3.6) Since the rated state satisfies (3.3), we need not consider the (trivial) possibility y0 = λ1x0. The initial state in (3.5) determines the value of the con- stant C = C(x0, y0) in the energy equation (2.7) repeated here, (βy−αx)(βy+γx)+2jy=C(x0,y0) for t≥t0 (3.7) for a given nonzero function Tgen(x, y) independent of t (for t ≤ t0). The time t0 corresponds to a power failure at which time the generator is de-energized and the previously nonzero generator torque Tgen is suddenly switched to zero. The time t0 is called the disconnect time since the generator may be considered to be disconnected from the turbine at time t0. During the steady-state energized operation prior to the disconnect time, the driven turbine is assumed to operate at a constant equilibrium state with constant C(x0,y0):=(βy0 −αx0)(βy0 +γx0)+2jy0 >0 (3.8) x(t)=x0 and y(t)=y0 for t≤t0 (3.2) where the positivity of (3.8) follows from (2.12), (2.14) and (3.6). The energy relation (3.7)–(3.8) can be solved for y in terms of x, and the resulting expression for y can be inserted into the right side of (3.4). In this way y can be eliminated in terms of x, and the angular momentum equation (3.4) be- comes a single first order (regular) nonlinear differential equa- tion for the turbine speed x(t). This differential equation is autonomous and it can be solved explicitly up to quadrature in the form x(t) dξ = [Q(ξ)−λ ξ][Q(ξ)−λ ξ] for suitable fixed constants x0 and y0 which are the coordi- nates of the energized equilibrium state in the (x, y) phase plane. The equilibrium state (x0, y0) corresponds to a balance of (the generally opposing) shaft and generator torques, so the point (x , y ) in the phase plane satisfies the equilibrium equa- 00 tion (see (1.6)) (y0 −λ1x0)(y0 −λ2x0)=Tgen(x0,y0)>0, (3.3) where the generator torque Tgen(x0, y0) is positive prior to the disconnect time (during the powered or energized operation). The constant equilibrium operating state (x0, y0) for (1.6) is said to be a rated state for the energized turbine, and it lies in the wedge region (1.9). We consider a power failure corresponding to a discon- nection of the generator from the turbine at the time t0 as in (3.1). Following the disconnect time, the system begins to move away from the previous rated equilibrium state (x0, y0) t−t0 (3.9) ε εdx=(y−λ1x)(y−λ2x) fort>t0. dt (3.4) 4 Q(x)= j2 + (αβx)2 + β2C(x0, y0) − j β2 if α=γ. (3.10) x0 1 2 with y(t) = Q(x(t)), where Q(x) is the function given by the positive root y = Q(x) > 0 of the quadratic energy equation (3.7) considered as a function of y. For example, in the case α = γ , there holds The formula for Q(x) in the case α > γ is only slightly more complicated than (3.10) and is omitted here. The remarks leading to (3.9) demonstrate both existence and uniqueness forPDF Image | Transient Characteristics of Radial Outflow Turbine Generators
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