Transient Characteristics of Radial Outflow Turbine Generators

Transcription

1 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 Donald R. Smith Department of Mathematics University of California at San Diego La Jolla, CA Abstract A nonlinear mathematical model of Kimmel is described for the hydraulic behavior of variable speed radial outflow turbines, including fixed speed radial outflow 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 momentum. It is assumed here that the mass flows across both the inlet and the outlet of the turbine are the same, so conservation of mass is automatically satisfied. The term hydraulic is used in a generalized sense to refer to any suitable incompressible (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 relation to cryogenic pumps for liquid natural gas. The radial inflow turbine is analyzed in SK [1998]. 1 Radial outflow turbines are less efficient than radial inflow turbines 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 P 1 is operating in series with another pump P 2, and if P 1 is suddenly de-energized at time t 0 while P 2 remains energized, then P 1 mayactasan outflow turbine for t > t 0. 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 hammer for the outflow turbine. The mathematical study of the flow of fluids through turbines is difficult because typical turbines operate with irregularly 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 turbine, 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 Idx/ and conservation of angular momentum for the turbine takes the form I dx = T (x, y), (1.1) where the quantity T (x, y) on the right side of (1.1) represents the total noninertial torque (or moment) and consists of two parts T (x, y) = T turb (x, y) + T gen (x, y). (1.2) The first term T turb (x, y) on the right side of (1.2) is the torque due to the rotating turbine shaft while T gen (x, y) is the negative of the applied torque due to the generator. The torque due to the rotating turbine shaft is modelled by the analytical constitutive law T turb (x, y) = τ (y λ 1 x)(y λ 2 x) (1.3) for experimentally determined design parameters τ, λ 1 and λ 2 ; see KIMMEL [1997a, 1997b] for a discussion on the practical determination of analogous parameters. For the radial 1 We denote the reference SMITH AND KIMMEL [1998] as SK [1998]. 1

2 outflow turbine these design parameters satisfy τ>0 and λ 1 >λ 2 0. (1.4) Kimmel s analytical model (1.3) (1.4) is consistent with experimental studies of pump characteristics by KNAPP [1937] as summarized in Figure 13.2 of STEPANOFF [1957]. The experimental results of KNAPP indicate that the curves T turb = const. appear to be hyperbolas with asymptotes given by two distinct straight lines corresponding to the equation T turb = 0; these experimental results are well modelled by (1.3) (1.4). The applied torque function T gen (x, y) in (1.2) is the negative of the generator torque and is modelled by a suitable specified function of the turbine speed x and flow y. In fact T gen is typically taken to depend only on the turbine speed x. For example in the case of a linear generator one uses the model T gen (x, y) = [T 0 + ν (x x 0 )] (1.5) where the constant x 0 is the synchronous speed of the generator, T 0 is the rated shaft torque for variable speed turbines, and the constant ν is a measure of the rate of change of generator torque with respect to speed. In current practice, the two most important cases for the linear model (1.5) are: and Fixed speed generator: T 0 = 0 and ν 0, Variable speed generator: T 0 0 and ν = 0. Combining (1.1) (1.3) yields the following equation for the balance of angular momentum, ɛ dx = (y λ 1 x)(y λ 2 x) T (x, y) (1.6) where the positive constant ɛ is the turbine inertia I scaled by the torque parameter τ, ɛ := I/τ > 0, (1.7) and the function T is the (negative of the) generator torque scaled by τ, T (x, y) := τ 1 T gen (x, y). (1.8) The constant ɛ in (1.7) is small for typical radial flow turbines for common liquids. According to the experimental studies of KNAPP [1937] (see also Fig of STEPANOFF [1957]), the radial outflow turbine is expected to operate in the following wedge region of the phase plane characterized as y λ 1 x for x 0, y 0. (1.9) 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 (= hydraulic pressure head), and similarly let KE(Rotating Turbine) denote the kinetic energy of the (centrifugal force of the) rotating 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 turbine centrifugal energy KE(Rotating Turbine) reinforces or supplements the kinetic energy of the fluid flow for the inflow 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 hydraulic 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 = λ 1 x, and it is shown in Section 3 that this dominant 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 energy 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) 2

3 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 momentum equation (1.6). In addition to the assumption (2.2), we also assume the condition For the special case γ = α, (2.1) reduces to γ α. (2.3) KE(Fluid Flow) KE(Rotating Turbine) = β 2 y 2 α 2 x 2 and in this case one has KE(Rotating Turbine) = α 2 x 2 and KE(Fluid Flow) = β 2 y 2 if α = γ. (2.4) Hence (in the case γ = α) α and β may be thought of as kinetic energy coefficients respectively for the rotating turbine and for the liquid flow. The potential energy of the hydraulic head may be denoted briefly ash, PE(Hydraulic Head) = H, (2.5) where the rate of change of H is assumed to satisfy Joukowski s relation Ḣ = 2jẏ (cf. p. 435 of STEPANOFF [1957]) for a fixed design parameter j known as Joukowski s coefficient. Upon integration of Ḣ = 2jẏ, there holds H = 2 jy + C (2.6) for a constant C of integration. The energy relation (1.11) for the radial outflow turbine can now be written with (2.1), (2.5) and (2.6) as (βy αx)(βy + γ x) + 2 jy = C (2.7) 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(x 1, y 1 ) of integration satisfies C = C(x 1, y 1 ) = (βy 1 αx 1 )(βy 1 + γ x 1 ) + 2 jy 1 (2.8) where x 1 and y 1 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 of THOMAS [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 through each point (x 1, y 1 ) in the plane, with constant given by (2.8). The family of hyperbolas is centered at the point (x, y) with coordinates 2 j (α γ) x = β(α + γ) 2 and y = 4 jαγ β 2 (α + γ) 2 (2.9) and the family has asymptotes given by the two lines L 1 and L 2 passing through (x, y) with respectives slopes α/β and γ/β, L 1 : y = α (x x) + y and β L 2 : y = γ (x x) + y (2.10) β as indicated in Figure 1. The y-intercept of the line L 2 is denoted as y 2 and is negative, y 2 = 2 jγ β 2 (α + γ) < 0 (2.11) since Joukowski s coefficient is positive j > 0. (2.12) L 1 : slope = α/β L 2 : slope = γ/β y 2 < 0 (x, y) Figure 1 The earlier shaft coefficients λ 1 and λ 2 in (1.3) are assumed 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, λ 1 > α β. (2.13) The inequalities (1.9) and (2.13) yield y λ 1 x (α/β)x, which with (2.2) implies βy αx 0. Similarly there also holds βy + γ x 0 everywhere in the wedge region (1.9), and these last two inequalities together yield (βy αx)(βy + γ x) 0 (2.14) for all (x, y) in the wedge region (1.9). 3

4 3 The De-Energized Outflow Turbine The discussion in Section 2 shows that the state of the radial outflow turbine/fluid system will move in the phase plane along a fixed hyperbola (2.7) during the operation of the system (as long as the state of the system is not forced to jump or switch from one hyperbola to another by any externally imposed 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, subject to the energy equation (2.7) and the angular momentum equation (1.6). The generator torque function T (x, y) = T (x, y, t) appearing 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 t 0 and take the applied generator torque as { Tgen (x, y) for t t T (x, y, t) = 0 (3.1) 0 for t > t 0, for a given nonzero function T gen (x, y) independent of t (for t t 0 ). The time t 0 corresponds to a power failure at which time the generator is de-energized and the previously nonzero generator torque T gen is suddenly switched to zero. The time t 0 is called the disconnect time since the generator may be considered to be disconnected from the turbine at time t 0. During the steady-state energized operation prior to the disconnect time, the driven turbine is assumed to operate at a constant equilibrium state x(t) = x 0 and y(t) = y 0 for t t 0 (3.2) for suitable fixed constants x 0 and y 0 which are the coordinates of the energized equilibrium state in the (x, y) phase plane. The equilibrium state (x 0, y 0 ) corresponds to a balance of (the generally opposing) shaft and generator torques, so the point (x 0, y 0 ) in the phase plane satisfies the equilibrium equation (see (1.6)) (y 0 λ 1 x 0 )(y 0 λ 2 x 0 ) = T gen (x 0, y 0 )>0, (3.3) where the generator torque T gen (x 0, y 0 ) is positive prior to the disconnect time (during the powered or energized operation). The constant equilibrium operating state (x 0, y 0 ) for (1.6) is saiobearated state for the energized turbine, and it lies in the wedge region (1.9). We consider a power failure corresponding to a disconnection of the generator from the turbine at the time t 0 as in (3.1). Following the disconnect time, the system begins to move away from the previous rated equilibrium state (x 0, y 0 ) under the influence of a modified momentum equation (1.6) which now becomes ɛ dx = (y λ 1 x)(y λ 2 x) for t > t 0. (3.4) The functions x(t) and y(t) are assumed to be continuous across the disconnect time, so the system must satisfy the initial conditions x = x 0 and y = y 0 at t = t 0, (3.5) where the initial state (x 0, y 0 ) satisfies (3.3) and (1.9) with y 0 >λ 1 x 0 and x 0, y 0 0 (3.6) Since the rated state satisfies (3.3), we need not consider the (trivial) possibility y 0 = λ 1 x 0. The initial state in (3.5) determines the value of the constant C = C(x 0, y 0 ) in the energy equation (2.7) repeated here, (βy αx)(βy + γ x) + 2 jy = C(x 0, y 0 ) for t t 0 (3.7) with constant C(x 0, y 0 ) := (βy 0 αx 0 )(βy 0 + γ x 0 ) + 2 jy 0 > 0 (3.8) 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) becomes a single first order (regular) nonlinear differential equation 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) x 0 dξ [Q(ξ) λ 1 ξ][q(ξ) λ 2 ξ] = t t 0 ɛ with y(t) = Q(x(t)), (3.9) 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 j Q(x) = + (αβx) 2 + β 2 C(x 0, y 0 ) j β 2 if α = γ. (3.10) The formula for Q(x) in the case α>γis only slightly more complicated than (3.10) and is omitted here. The remarks leading to (3.9) demonstrate both existence and uniqueness for 4

5 solution functions for the momentum-energy problem (3.4) (3.8). The resulting implicit integral representation (3.9) for the exact solution must be inverted to provide x = x(t), and this inversion is nontrivial from a practical point of view. Hence an approach based on (3.9) is perhaps not best suited for use in obtaining qualitative and quantitative information on the liquid hammer. We follow an alternative approach based on a combination of phase plane and perturbation analyses. The zero torque line y = λ 1 x in the phase plane is a line of equilibrium points for (3.4), which means that any point (x 1, y 1 ) on this line in the phase plane is a fixed point solution. That is, the constant functions x(t) x 1 and y(t) y 1 for all t (3.11) provide solutions for (3.4) if y 1 = λ 1 x 1. (3.12) In this case not only is the generator torque equal to zero (corresponding to disconnected power) but also there is no load because the turbine shaft torque (y λ 1 x)(y λ 2 x) is also zero. y = λ y 1 x y = λ 2 x (x, y ) (x 0, y 0 ) L 1 : slope = α/β L 2 : slope = γ/β x (x, y) y 2 < 0 Figure 2 The initial state (x 0, y 0 ) lies in the wedge region (3.6) characterized by the dominant shaft coefficient λ 1. It follows with (1.4) and (3.4) that the turbine speed satisfies initially dx > 0, (3.13) so x(t) increases initially with increasing t t 0. Hence the motion is to the right along the energy hyperbola (3.7), toward the line y = λ 1 x in the first quadrant, as illustrated in Figure 2. The conditions (2.2), (2.11) and (2.13) guarantee that the resulting solution state (x(t), y(t)) remains in the wedge region (3.6), and (3.13) continues to hold, for all t t 0. Hence the rotational speed x(t) of the turbine is a monotonic increasing function for t t 0, and we now show that the same is true for the fluid flow rate y(t). Differentiating (with respect to time) the energy equation (3.7), and then using the angular momentum equation (3.4) to replace dx/ in that result, yields the following equation on the balance of linear momentum, ɛ dy = (3.14) (y λ 1 x)(y λ 2 x) [β(α γ)y + 2αγ x] 2 j + 2β 2 y + β(γ α)x for t t 0. Since the solution (x(t), y(t)) remains in the wedge region (3.6), it follows with (1.4) that there holds (y λ 1 x)(y λ 2 x)>0 for t t 0. Similarly the other factors on the right side of (3.14) are also positive. For example (1.9), (2.2), (2.3) and (2.13) imply β(α γ)y + 2αγ x β(α γ)λ 1 x + 2αγ x [α(α γ)+ 2αγ]x = α(α + γ)x > 0 for t > t 0, so all factors in the numerator are positive on the right side of (3.14). A similar calculation shows 2βy + (γ α)x 2βλ 1 x + (γ α)x (α + γ)x 0, so (cf. (2.12)) 2 j + 2β 2 y + β(γ α)x 2 j > 0 (3.15) everywhere in the wedge region (1.9). These considerations with (3.14) imply the stated monotonicity result for y = y(t), dy > 0 for t > t 0. (3.16) A routine calculation shows that the intersection point (x, y ) of the zero torque line y λ 1 x = 0 and the hyperbola (3.7) (3.8) in the first quadrant has coordinates x = x (x 0, x 0 ) = (3.17) j 2 λ (βλ 1 α)(βλ 1 + γ) C(x 0, y 0 ) jλ 1 (βλ 1 α)(βλ 1 + γ) y = y (x 0, y 0 ) = λ 1 x (x 0, y 0 ) with positive constant C(x 0, y 0 ) given by (3.8). Note also that (2.13) guarantees the positivity of the quantity (βλ 1 α)(βλ 1 + γ)= β 2 λ β(γ α)λ 1 αγ > 0 (3.18) since the two roots of the quadratic function f (λ) := β 2 λ 2 + β(γ α)λ αγ are λ = γ/β and λ =+α/β, and f (λ) > 0 5

6 for λ = λ 1 >α/β. It follows from (3.8), (3.17) and (3.18) that the limiting point (x, y ) lies in the first quadrant with y = λ 1 x > 0. (3.19) The function x is monotonic increasing by (3.13), and x(t) is bounded above by x. Hence x(t) has a limiting value, say x,ast, and there holds lim t x(t) = x x.itis easy to show (by contradiction) that there must hold x = x. Indeed if there were to hold x < x, then the segment Ɣ0 of the hyperbola (3.7) (3.8) extending between x = x 0 and x = x < x is uniformly bounded away from the lines y = λ 1 x and y = λ 2 x, and the right side of the differential equation satisfies a bound (y λ 1 x)(y λ 2 x) δ>0 (3.20) for a fixed positive constant δ, uniformly on a domain containing the hyperbolic segment Ɣ0 but excluding the point (x, y ). It would follow then from the differential equation (3.4) and the bound (3.20) that x(t 1 ) = x for some finite t 1 satisfying t 0 < t 1 t 0 + ɛ (x x 0 )/δ and also dx(t)/ > 0att = t 1. This would contradict the fact that x is the limiting value of x(t) as t,so x < x is not possible. A similar argument applies to y(t), and this proves lim t x(t) = x and lim t y(t) = y. Collecting these results, we have proved the following theorem based on the stated assumptions (1.4), (1.7), (2.2), (2.3), (2.12) and (2.13). Theorem 1 The dominant (zero torque) equilibrium line y = λ 1 x is an attractor for the wedge region (3.6) in the first quadrant for the momentum-energy system (3.4) (3.8). For initial states (x 0, y 0 ) in the wedge (3.6), the solution state (x(t), y(t)) satisfies y(t) λ 1 x(t) >0 and x(t), y(t) 0 for all t t 0, (3.21) so the state of the system remains in the wedge region (1.9). The solution moves steadily along the hyperbola (3.7) (3.8) toward the limiting state (3.17) located at the point of intersection of the hyperbola and the attracting line y = λ 1 x. The solution approaches the limiting point (x, y ) asymptotically along the hyperbola (3.7) with lim (x(t), y(t)) = t (x, y ), (3.22) but the state of the system never arrives at this limiting point in any finite time. Even though the solution never arrives (in an exact mathematical sense) at the limiting state (3.17) in any finite time, we show in Section 4 that the system moves quickly to a very close proximity of the limiting state when the scaled inertia ɛ is small. Hence, in a practical sense, the state of the system can be considered to arrive essentially (or effectively) at the equilibrium limiting state in a short time that is proportional to the scaled inertia ɛ. 4 The Liquid Hammer When we wish to emphasize the dependence of the solution on the initial state (x 0, y 0 ) for the problem (3.4) (3.8), we denote the solution functions as x(t) = x(t; x 0, y 0 ) and y(t) = y(t; x 0, y 0 ). (4.1) The following theorem, which gives estimates on the differences x(t; x 0, y 0 ) x and y(t; x 0, y 0 ) y between the solution functions and the coordinates of the corresponding limiting equilibrium point (x, y ) of (3.17), is proved using techniques from perturbation theory (cf. SMITH [1985]). Theorem 2 Assume that the initial state (x 0, y 0 ) lies in the wedge region (3.6), and assume that the system parameters ɛ, α, β, γ, λ 1 and λ 2 satisfy the natural inequalites (1.4), (1.7), (2.2), (2.3), (2.12), and (2.13). Then there are fixed positive constants κ>0, ξ>0 and η>0 not depending on ɛ but depending on the initial state and on the other parameters of the problem excluding ɛ, so that the solution functions (4.1) for the energy-momentum initial value problem (3.4) (3.8) satisfy the estimates x(t; x 0, y 0 ) x ξ y 0 λ 1 x 0 e κ(t t 0)/ɛ y(t; x 0, y 0 ) y η y 0 λ 1 x 0 e κ(t t 0)/ɛ for t t 0 (4.2) where the limiting values x = x (x 0, y 0 ) and y = y (x 0, y 0 ) are given by (3.17), and the quantity y 0 λ 1 x 0 is positive. Proof. Subtract λ 1 times the angular momentum equation (3.4) from the linear momentum equation (3.14) and find ɛ d [ y λ1 x ] = A(t) [ y λ 1 x ] (4.3) with A(t) : = [ ] β(α γ)y(t) + 2αγ x(t) λ 1 2 j + 2β 2 y(t) + β(γ α)x(t) [ y(t) λ2 x(t) ], (4.4) 6

7 where (4.3) can be integrated with the initial conditions of (3.5) to yield y(t) λ 1 x(t) = (y 0 λ 1 x 0 ) e 1 ɛ t t 0 A(s)ds for t t 0. (4.5) We now show that A(t) is uniformly positive-valued for t t 0. First note that the multiplicative factor y λ 2 x on the right side of (4.4) has the value y λ 2 x = (λ 1 λ 2 ) x at the limiting terminal point (x, y ) of (3.17), and this value is positive because λ 1 >λ 2 and x > 0. Similarly this factor has a positive value at the initial point (x 0, y 0 ) since (1.4) implies y λ 2 x y λ 1 x on the wedge (1.9), and the last quantity is positive at the initial point by (3.6). In fact the factor y λ 2 x is uniformly positive along the entire portion Ɣ 0 of the hyperbola (3.7) (3.8) between the initial state (x 0, y 0 ) and the limiting state (x, y ) since this (compact) portion Ɣ 0 of the hyperbola is bounded away from the line y = λ 2 x; cf. Figure 2. Hence there holds min[y λ 2 x] = min Ɣ0 x 0 x x [Q(x) λ 2x] > 0, (4.6) 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 (as in (3.9)). For the other factor on the right side of (4.4), a routine estimation gives β(α γ)y + 2αγ x λ 1 2 j + 2β 2 (4.7) y + β(γ α)x 2 jλ 1 + (α + γ )(βy αx) 2 j + 2β 2 y + β(γ α)x everywhere in the wedge region (1.9). The expression on the right side of (4.7) is uniformly bounded (cf. (3.15)) and uniformly positive everywhere on the (compact) portion Ɣ0 of the energy curve (3.7) (3.8), including at the endpoints (x 0, y 0 ) and (x, y ). These remarks with (4.4) imply the existence of a fixed positive constant κ for which there holds A(t) κ>0 for t t 0 (4.8) where κ will generally depend on the parameters x 0, y 0, α, β, γ, λ 1 and λ 2,butκ does not depend on ɛ. From (4.5) and (4.8) there follows immediately (see the simpler calculation in Section 4 of SK [1998]) 0 < y(t) λ 1 x(t) [y 0 λ 1 x 0 ] e κ (t t 0)/ɛ for t t 0. (4.9) The angular momentum equation (3.4) can be integrated between t and t 1 > t to yield x(t 1 ) x(t) = 1 ɛ t1 t [ y(s) λ1 x(s) ][ y(s) λ 2 x(s) ] ds 0 for t 0 t t 1, which with (4.9) implies 0 x(t 1 ) x(t) µ ɛ y 0 λ 1 x 0 t1 t e κ (s t 0)/ɛ ds (4.10) with µ := max y λ 2 x = max Q(x) λ 2x Ɣ0 x 0 x x where Ɣ0 is the segment of the hyperbola (3.7) (3.8) extending between (x 0, y 0 ) and (x, y ). The integral on the right side of (4.10) can be evaluated, and (4.10) implies 0 x(t 1 ) x(t) (4.11) µ κ y 0 λ 1 x 0 [e ] κ (t t0)/ɛ e κ (t 1 t 0 )/ɛ µ κ y 0 λ 1 x 0 e κ (t t 0)/ɛ for t 1 > t t 0. Passing to the limit t 1 in (4.11) and using (3.22) yields the stated estimate of (4.2) for x(t) with constant ξ = µ/κ. A corresponding estimate for y(t) (as in (4.2)) follows directly from the estimate (4.9) for y λ 1 x and the estimate of (4.2) for x, using y = (y λ 1 x) + λ 1 x. Remaining details are omitted. The estimates of (4.2) imply that, for small ɛ > 0, the functions x(t) and y(t) tend rapidly toward their constant limiting values x and y. For example (4.2) implies for x upon integration ( ) ξ x y0 λ 1 x 0 x(t) ɛ, t 0 κ and an analogous result holds for y(t). Hence for t t 0 the area between the graphs of x and x(t) (and similarly, the area between y and y(t)) is small, of order ɛ, as indicated in Figure 3 for x(t). Quantitative information can be obtained on the magnitudes of the positive constants κ, ξ and η in (4.2) in terms of the data, along the lines of the result (4.3) of SK [1998] for an analogous parameter κ in the case of the radial inflow turbine. x x(t) x x 0 t 0 Figure 3 t 0 + ε t 7

8 Acknowledgement The author wishes to acknowledge Hans E. Kimmel for the contribution of the mathematical model for radial outflow turbines. References [1] Kimmel, H. E. 1997a Speed controlled turbine expanders in hydrocarbon liquefaction processes, The International Journal of Hydrocarbon Engineering 1 (3), (Surrey, England: Palladian Publications Ltd.). [2] Kimmel, H. E. 1997b Hydraulic performance of speed controlled turbines for power recovery in cryogenic and chemical processing, World Pumps (Oxford, England: Elsevier), June 1997 issue, pp [3] Knapp, R. T Complete characteristics of centrifugal pumps and their use in the prediction of transient behavior, Trans. A.S.M.E., Nov. 1937, pp [4] Smith, D. R Singular Perturbation Theory (Cambridge, England: Cambridge University Press). [5] Smith, D. R. and Kimmel, H. E Liquid hammer for the de-energized radial inflow turbine generator, Methods and Applications of Analysis 5 (2), pp [6] Stepanoff, A. J Centrifugal and Axial Flow Pumps (Wiley, 2 nd ed.). [7] Thomas, G. B. Jr Calculus and Analytic Geometry: The Classic Edition (Addison-Wesley). 8