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The 9th of International Symposium on Transport Phenomena and Dynamics of Rotating Machinery Honolulu, Hawaii, February 10-14, 2002 Transient Characteristics of Radial Outflow Turbine Generators Abstract A nonlinear mathematical model of Kimmel is de- scribed for the hydraulic behavior of variable speed ra- dial outflow turbines, including fixed speed radial out- flow turbines as a special case. The phase plane motion of the system is along a certain hyperbola. Following a power failure during operation, a liquid hammer or pressure pulse occurs and decays quickly as the state of the system surges along the hyperbola toward an equilibrium curve which is an attractor for the relevant “wedge region” in the phase plane for the de-energized radial outflow turbine. A quantitative estimate is given for the pulse decay. 1 Introduction The hydraulic behavior of variable speed turbines is governed by the conservation or balance laws of mass, energy and mo- mentum. It is assumed here that the mass flows across both the inlet and the outlet of the turbine are the same, so conserva- tion of mass is automatically satisfied. The term “hydraulic” is used in a generalized sense to refer to any suitable incom- pressible (or nearly incompressible) liquid such as water but also including other liquids such as oil, liquid natural gas and liquid hydrogen. The present work has been initiated in rela- tion to cryogenic pumps for liquid natural gas. The radial inflow turbine is analyzed in SK [1998].1 Ra- dial outflow turbines are less efficient than radial inflow tur- bines and are therefore not much used in practical system design. However, hydraulic systems can contain elements which, though not intended to operate as outflow turbines, can nevertheless behave like radial outflow turbines in some circumstances. For example if a pump P1 is operating in se- ries with another pump P2, and if P1 is suddenly de-energized at time t0 while P2 remains energized, then P1 may act as an outflow turbine for t > t0. 1We denote the reference SMITH AND KIMMEL [1998] as SK [1998]. Applying the same principles described in SK [1998] a mathematical model is presented here for the radial outflow turbine, and it is shown tht the model predicts a liquid ham- mer for the outflow turbine. The mathematical study of the flow of fluids through tur- bines is difficult because typical turbines operate with irregu- larly shaped internal channels with different curved surfaces and cross sections. As in SK [1998], a simplified black box model is used for the flow of incompressible liquids through radial flow turbines. The model involves only the angular speed of the turbine rotor, the liquid flow rate through the tur- bine, and the hydraulic pressure head between the turbine inlet and outlet, all considered as functions of time t. For the variable speed turbine, the angular speed of the turbine rotor is denoted as x = x(t) while the liquid flow rate is denoted as y = y(t). If the (constant) inertia of the turbine rotor is denoted as I , then the time rate of change of angular momentum (i.e. the inertial torque) is I d x /d t and conserva- tion of angular momentum for the turbine takes the form Donald R. Smith Department of Mathematics University of California at San Diego La Jolla, CA 92093 I dx dt = T(x, y), (1.1) where the quantity T (x, y) on the right side of (1.1) repre- sents the total noninertial torque (or moment) and consists of two parts T (x, y) = Tturb(x, y) + Tgen(x, y). (1.2) The first term Tturb(x, y) on the right side of (1.2) is the torque due to the rotating turbine shaft while Tgen(x, y) is the nega- tive of the applied torque due to the generator. The torque due to the rotating turbine shaft is modelled by the analytical constitutive law Tturb(x, y) = τ (y − λ1x)(y − λ2x) (1.3) for experimentally determined design parameters τ, λ1 and λ2 ; see K I M M E L [1997a, 1997b] for a discussion on the prac- tical determination of analogous parameters. For the radial 1PDF Image | Transient Characteristics of Radial Outflow Turbine Generators
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