Twin screw pumps are positive displacement machines. Two meshing screws connected by timing gears push the fluid trapped in the screw cavities axially from suction to discharge. Available steady state hydraulic modets predict pump performance and axiat pressure distribution in the chambers in single-and two-phase flow conditions. However, no model is availabte for their rotordynamics behavior. Due to the hetix angte of the screw, the pressure distribution around the rotor is not batanced, giving rise to both static and dynamic lateral forces. The work presented here introduces a starting point for rotordynamic analysis of twin screw pumps. First, we show that the screw rotor's geometry can be represented by axisymmetric beam elements. Second, we extend the steady state hydraulic modet to predict both the static and dynamic lateral forces resulting from the unbalanced pressure fleld. Finatty, hydrautic forces are applied to the rotor to predict static, synchronous, and nonsynchronous responses. Predictions of the dynamic pressure were compared to measurements from the literature and were found to be in good agreement.
In turbomachines, the transfer of energy between the rotor and the fluid does not-in theory-result in lateral forces on the rotor. In positive displacement machines, on the other hand, the transfer of energy between the moving components and the working fluid usually results in unbalanced pressure fields and forces. Muhammed and Childs (2013, "Rotordynamics of a Two-Phase Flow Twin Screw Pump," ASME J. Eng. Gas Turbines Power, 135(6), p. 062502) developed a model to predict the dynamic forces in twin-screw pumps, showing that the helical screw shape generates hydraulic forces that oscillate at multiples of running speed. The work presented here attempts to validate the model of Muhammed and Childs (2013, "Rotordynamics of a Two-Phase Flow Twin Screw Pump," ASME J. Eng. Gas Turbines Power, 135(6), p. 062502) using a clear-casing twin-screw pump. The pump runs in both single and multiphase conditions with exit pressure up to 300 kPa and a flow rate 0.6 l/s. The pump was instrumented with dynamic pressure probes across the axial length of the screw in two perpendicular directions to validate the dynamic model. Two proximity probes measured the dynamic rotor displacement at the outlet to validate the rotordynamics model and the hydrodynamic cyclic forces predicted by Muhammed and Childs (2013, "Rotordynamics of a Two-Phase Flow Twin Screw Pump," ASME J. Eng. Gas Turbines Power, 135(6), p. 062502). The predictions were found to be in good agreement with the measurements. The amplitude of the dynamic pressure measurements in two perpendicular plans supported the main assumptions of the model (constant pressure inside the chambers and linear pressure drop across the screw lands). The predicted rotor orbits at the pump outlet in the middle of the rotor matched the experimental orbits closely. The spectrum of the response showed harmonics of the running speed as predicted by the model. The pump rotor's calculated critical speed was at 24.8 krpm, roughly 14 times the rotor's running speed of 1750 rpm. The measured and observed excitation frequencies extended out to nine times running speed, still well below the first critical speed. However, for longer twin-screw pumps running at higher speed, the coincidence of a higher-harmonic excitation frequency with the lightly damped first critical speed should be considered.
In three 2010 papers, Tsujimoto et al. (2010, “Moment Whirl Due to Leakage Flow in the Back Shroud Clearance of a Rotor,” Int. J. Fluid Mach. Syst., 3(3), pp. 235–244), Song et al. (2010, “Rotordynamic Instability Caused by the Fluid Force Moments on the Backshroud of a Francis Turbine Runner,” Int. J. Fluid Mach. Syst., 3(1), pp. 76–79), and Song et al. (2010, “Rotordynamic Moment on the Backshroud of a Francis Turbine Runner Under Whirling Motion,” ASME J. Fluids Eng., 132, p. 071102) discussed and explained a novel destabilizing mechanism arising in both hydraulic turbines and the back surface of vertical pump impellers. The destabilizing mechanism can be explained via a reaction force-moment model that includes both the customary radial displacement vector of an impeller plus the pitch and yaw degrees of freedom. This coupling between radial displacements and tilt plus the coupling of the shaft support structure can create negative damping. In 1993, Verhoeven et al. (1993, “Rotor Instability of a Single Stage Centrifugal Pump, Supersynchronous Whirling at Almost Twice the Operating Speed, A Case History,” Proceedings of the 1st International Symposium on Pump Noise and Vibration, pp. 457–468) identified negative damping arising from U-shaped wearing-ring seals as causing a super-synchronous instability in a horizontal coke-crusher pump. However, several case studies have been presented of super-synchronously unstable pumps for which (until now) no explanation could be provided. Tsujimoto–Song started with a 2DOF model for a vertically suspended disk via a cantilevered shaft. They used an f = ma model for the lateral displacements of the disk and used flexibility coefficients to account for reaction forces and moments from the back shroud of the impeller. The present work starts with a 4DOF model that includes the disk's displacements and pitch and yaw degrees of freedom. The Guyan reduction is used to create two reduced 2DOF models: model A that retains the displacements and discards the rotations and model B that retains the rotations and discards the displacements. Model A produces a requirement for instability that is inconsistent with Tsujimoto–Song's experience and predictions. However, it is useful in predicting the reaction moments produced by a nominally planar precession of the impeller. The instability requirement of Model B is consistent with Tsujimoto's experience and predictions. A comparison of the predicted reaction moments of model A and Tsujimoto's reaction-moment data supports the instability predictions of model B (and Tsujimoto–Song) that the instability arises due to coupling between the displacement and rotation degrees of freedom in the 4 × 4 damping matrix.
In turbomachines, the transfer of energy between the rotor and the fluid does not — in theory — result in lateral forces on the rotor. In positive displacement machines, on the other hand, the transfer of energy between the moving and stationary components usually results in unbalanced pressure fields and forces. In [1] the authors developed a model to predict the dynamic forces in twin screw pumps, showing that the helical screw shape generates hydraulic forces that oscillate at multiples of running speed. The work presented here attempts to validate the model in [1] using a clear-casing twin screw pump. The pump runs in both single and multiphase conditions with exit pressure up to 300 KPa and a flow rate 0.6 liter per second. The pump was instrumented with dynamic pressure probes across the axial length of the screw in two perpendicular directions to validate the dynamic model. Two proximity probes measured the dynamic rotor displacement at the outlet to validate the rotordynamics model and the hydrodynamic cyclic forces predicted in [1]. The predictions were found in good agreement with the measurements. The amplitude of the dynamic pressure measurements in two perpendicular plans supported the main assumptions of the model (constant pressure inside the chambers and linear pressure drop across the screw lands). The predicted rotor orbits at the pump outlet in the middle of the rotor matched the experimental orbits closely. The spectrum of the response showed harmonics of the running speed as predicted by the model. The pump rotor’s calculated critical speed was at 24.8 krpm, roughly 14 times the rotor’s running speed of 1750 rpm. The measured and observed excitation frequencies extended out to nine times running speed, still well below the 1st critical speed. However, for longer twin-screw pumps running at higher speed, the coincidence of a higher-harmonic excitation frequency with the lightly damped 1st critical speed should be considered.
In three 2010 papers, Tsujimoto, Ma, Song, and Horiguchi [1–3] discussed and explained a novel destabilizing mechanism arising in both hydraulic turbines and the back surface of vertical pump impellers. The destabilizing mechanism can be explained via a reaction force-moment model that includes both the customary radial displacement vector of an impeller plus the pitch and yaw degrees of freedom. This coupling between radial displacements and tilt plus the coupling of the shaft support structure can create negative damping. In 1993, Verhoeven, et al. [4] identified negative damping arising from U-shaped wearing-ring seals as causing a super-synchronous instability in a horizontal coke-crusher pump. However, several case studies have been presented of super-synchronously unstable pumps for which (until now) no explanation could be provided. Tsujimoto-Song started with a 2DOF model for a disk suspended vertically via a cantilevered shaft. They used an f = ma model for the lateral displacements of the disk and used flexibility coefficients to account for reaction forces and moments from the back shroud of the impeller. The present work starts with a 4DOF model that includes the disk’s displacements and pitch and yaw degrees of freedom. Guyan reduction is used to create two reduced 2DOF models, Model A that retains the displacements and discards the rotations and Model B that retains the rotations and discards the displacements. Model A produces a requirement for instability that is inconsistent with Tsujimoto-Song’s experience and predictions. However, it is useful in predicting the reaction moments produced by a nominally planar precession of the impeller. The instability requirement of Model B is consistent with Tsujimoto’s experience and predictions. A comparison of the predicted reaction moments of Model A and Tsujimoto’s reaction-moment data supports the instability predictions of Model B (and Tsujimoto-Song) that the instability arises due to coupling between the displacement and rotation degrees of freedom in the 4 × 4 damping matrix.
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