Rotordynamic Coefficients for Compressible Flow in Tapered Annular Seals

1985 ◽  
Vol 107 (3) ◽  
pp. 318-325 ◽  
Author(s):  
C. C. Nelson
Keyword(s):  
Compressible Flow ◽  
Flow Model ◽  
Space Shuttle ◽  
Governing Equations ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Annular Seal ◽  
Direct Stiffness ◽  
Stiffness And Damping ◽  
Main Engine

Derivation of the governing equations for compressible flow in a tapered annular seal is based on Hirs’ turbulent bulk-flow model. Zeroth and first-order perturbation equations are developed by an expansion in the eccentricity ratio. These equations are numerically integrated to obtain the leakage, and the direct and cross-coupled stiffness and damping coefficients. Seal parameters similar to the Space Shuttle Main Engine High Pressure Oxidizer Turbopump are used to demonstrate output from the analysis procedure. The effects of preswirl and seal taper are shown for three different length-to-diameter ratios. Generally the results indicate that prerotating the fluid significantly increases the cross-coupled stiffness but has little effect on the other coefficients, and increasing the convergent taper increases the direct stiffness while decreasing the direct damping and cross-coupled stiffness.

scholarly journals Analysis for Leakage and Rotordynamic Coefficients of Surface-Roughened Tapered Annular Gas Seals

1984 ◽  
Vol 106 (4) ◽  
pp. 927-934 ◽  
Author(s):  
C. C. Nelson
Keyword(s):  
Flow Model ◽  
Computational Method ◽  
Governing Equations ◽  
Clearance Ratio ◽  
Small Motion ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Honeycomb Seals ◽  
Gas Seals ◽  
Rotor Dynamic

In order to soften the effects of rub, the smooth stators of turbine gas seals are sometimes replaced by a honeycomb surface. This deliberately roughened stator and smooth rotor combination retards the seal leakage and may lead to enhanced rotor stability. However, many factors determine the rotordynamic coefficients and little is known as to the effectiveness of these “honeycomb seals” under various changes in the independent seal parameters. This analysis develops an analytical-computational method to solve for the rotordynamic coefficients of this type of compressible-flow seal. The governing equations for surface-roughened tapered annular gas seals are based on a modified Hirs’s turbulent bulk flow model. A perturbation analysis is employed to develop zeroth and first-order perturbation equations. These equations are numerically integrated to solve for the leakage, pressure, density, and velocity for small motion of the shaft about the centered position. The resulting pressure distribution is then integrated to find the corresponding rotor-dynamic coefficients. Finally, an example case is used to demonstrate the effect of changing from a smooth to a rough stator while varying the seal length, taper, preswirl, and clearance ratio.

Analysis of Eccentric Annular Incompressible Seals: Part 2—Effects of Eccentricity on Rotordynamic Coefficients

1988 ◽  
Vol 110 (2) ◽  
pp. 361-366 ◽  
Author(s):  
C. C. Nelson ◽  
D. T. Nguyen
Keyword(s):  
Flow Model ◽  
Element Model ◽  
Analysis Procedure ◽  
Governing Equations ◽  
Zeroth Order ◽  
Static Displacement ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Flow Variables ◽  
Factor Equation

In Part 1 of this paper, a new analysis procedure is presented which solves for the flow variables of an annular pressure seal in which the rotor has a large static displacement (eccentricity) from the centered position. This part of the paper (Part 2) incorporates the solutions from Part 1 to investigate the effect of eccentricity on the rotordynamic coefficients. The analysis begins with a set of governing equations based on a turbulent bulk-flow model and Moody’s friction factor equation. Perturbations of the flow variables yields a set of zeroth- and first-order equations. After integration of the zeroth-order equations by means of the method described in Part 1, the resulting zeroth-order flow variables are used as input in the solution of the first-order equations. Further integration of the first order pressures yields the eccentric rotordynamic coefficients. The results from this procedure compare very well with available experimental data, and are clearly more accurate than the predictions based on a Finite Element model.

Rotordynamic Coefficients for Partially Tapered Annular Seals: Part II—Compressible Flow

1991 ◽  
Vol 113 (1) ◽  
pp. 53-57
Author(s):  
J. K. Scharrer ◽  
C. C. Nelson
Keyword(s):  
Pressure Distribution ◽  
Compressible Flow ◽  
Order Continuity ◽  
Small Motion ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Component Test ◽  
Annular Seal ◽  
Complex Differential Equations ◽  
Basic Equations

The basic equations are derived for compressible flow in an annular seal with a partially tapered clearance. The flow is assumed to be completely turbulent in the axial and circumferential directions with no separation, and is modeled by Moody’s equation for roughness. Linearized zeroth and first-order perturbation equations are developed for small motion about a centered position by an expansion in the eccentricity ratio. The zeroth-order continuity and momentum equations are solved exactly, yielding the axial and circumferential velocity components and the pressure distribution. The first-order equations are reduced to three ordinary, complex, differential equations in the axial coordinate Z. The equations are integrated to satisfy the boundary conditions and yield the perturbation pressure distribution. This resultant pressure distribution is integrated along and around the seal to yield the force developed by the seal and the corresponding dynamic coefficients. Since no component test data exist for this type of seal, the results of a parametric study on the effect of the taper length/seal length ratio on the seal leakage and rotordynamic coefficients are presented.

scholarly journals Seal-Rotordynamic-Coefficient Test Results for a Model SSME ATD-HPFTP Turbine Interstage Seal With and Without a Swirl Brake

1991 ◽  
Vol 113 (1) ◽  
pp. 198-203 ◽  
Author(s):  
D. W. Childs ◽  
C. Ramsey
Keyword(s):  
Space Shuttle ◽  
Pressure Ratio ◽  
Tangential Velocity ◽  
Model Space ◽  
Design Guidelines ◽  
Test Results ◽  
Rotordynamic Coefficients ◽  
Direct Stiffness ◽  
Favorable Influence ◽  
Main Engine

Test results are presented and compared to theory for a model Space Shuttle Main Engine(SSME) Alternate Turbopump Development(ATD) High-Pressure Fuel Turbopump (HPFTP) with and without swirl brakes. Tests are conducted with supply pressures out to 18.3 bars and speeds out to 16,000 rpm. Seal back pressure is controlled to provide four pressure ratios at all supply pressures. Three inlet guide vanes are used to provide the following three fluid prerotation cases: (a) no pre-rotation, (b) moderate prerotation in the direction of rotation, and (c) high prerotation in the direction of rotation. Test results demonstrate the pronounced favorable influence of the swirl brake in reducing the seal destabilizing forces. Without the swirl brake, the cross-coupled stiffness k increases monotonically with increasing inlet tangential velocity. With the swirl brake, k tends to either be constant or decrease with increasing inlet tangential velocity. Direct damping either increases or remains relatively constant when the swirl brake is introduced. Direct stiffness is relatively unchanged. No measurable differences in leakage were detected for the seal with and without the swirl brake. Comparisons between Scharrer’s (1988) theory and measurements for the seal without a swirl brake indicate that the predictions can be used to provide design guidelines only. Specific predictions for rotordynamic coefficients should be treated cautiously, since systematic differences were observed between theory and experiment due to changes in running speed, supply pressure, and pressure ratio.

Effects of Wall Flexibility on the Rotordynamic Coefficients of Turbulent Cryogenic Annular Seals

1996 ◽  
Vol 118 (3) ◽  
pp. 509-519 ◽  
Author(s):  
B. Venkataraman ◽  
A. B. Palazzolo
Keyword(s):  
Energy Transport ◽  
Space Shuttle ◽  
Element Formulation ◽  
Local Pressure ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Wall Flexibility ◽  
Main Engine ◽  
Symmetric Finite Element ◽  
Annular Seals

A theory for analyzing the effects of elastic deformations of the seal wall on the dynamic characteristics of high pressure cryogenic annular seals under concentric operation is presented. The bulk flow continuity, axial and circumferential momentum, and the energy transport equations are utilized to determine the pressure distribution in the seal. Thermophysical properties of the cryogenic fluid are assumed to be functions of the local pressure and temperature. The wall deformations are obtained using an iso-parametric, axi-symmetric Finite Element formulation of the seal wall. A perturbation analysis is employed to arrive at the first order solution which yields the rotordynamic coefficients. Results obtained for the case of Space Shuttle Main Engine Oxygen Turbopump (SSME-HPOTP) Preburner Seal show a significant impact of seal flexibility on the dynamic coefficients.

Modified Governing Equations for Unsteady Compressible Flow in Ducts

1978 ◽  
Vol 45 (4) ◽  
pp. 723-726 ◽  
Author(s):  
A. Celmin¸sˇ
Keyword(s):  
Compressible Flow ◽  
Power Law ◽  
Unsteady Flows ◽  
Tube Flow ◽  
Axially Symmetric ◽  
Governing Equations ◽  
Flow Equations ◽  
Circular Tubes ◽  
First Order ◽  
General Conservation

Conventional governing equations for unsteady compressible tube flows are reviewed and it is shown that they neglect first-order terms which can have significant magnitudes. The derivation of correct tube flow equations from general conservation laws is demonstrated for the case of axially symmetric straight tubes. The traditionally neglected terms are computed explicitly for unsteady flows with power law profiles through circular tubes.

Measured Static and Rotordynamic Characteristics of a Smooth-Stator/Grooved-Rotor Liquid Annular Seal

2018 ◽  
Vol 140 (10) ◽  
Author(s):  
J. Alex Moreland ◽  
Dara W. Childs ◽  
Joshua T. Bullock
Keyword(s):  
Radial Clearance ◽  
Test Results ◽  
Dynamic Features ◽  
Axial Pressure ◽  
Pressure Drops ◽  
Rotordynamic Coefficients ◽  
Annular Seal ◽  
Direct Stiffness ◽  
Static Eccentricity ◽  
Dynamic Variables

Electric submersible pumps (ESPs) utilize grooved-rotor/smooth-stator (SS/GR) seals to reduce leakage and break up contaminants within the pumped fluid. Additionally, due to their decreased surface area (when compared to a smooth seal), grooved seals decrease the chance of seizure in the case of rotor-stator rubs. Despite their use in industry, the literature does not contain rotordynamic measurements for smooth-stator/circumferentially grooved-rotor liquid annular seals. This paper presents test results consisting of leakage measurements and rotordynamic coefficients for a SS/GR liquid annular sdeal. Both static and dynamic variables are investigated for various imposed preswirl ratios (PSRs), static eccentricity ratios (0–0.8), axial pressure drops (2–8 bars), and running speeds (2–8 krpm). The seals' static and dynamic features are compared to those of a smooth seal with the same length, diameter, and minimum radial clearance. Results show that the grooves reduce leakage at lower speeds (less than 5 krpm) and higher axial pressure drops, but does little at higher speeds. The grooved seal's direct stiffness is generally negative, which would be detrimental to pump rotordynamics. As expected, increasing preswirl increases the magnitude of cross-coupled stiffness and increases the whirl frequency ratio (WFR). When compared to the smooth seal, the grooved seal has smaller effective damping coefficients, indicative of poorer stability characteristics.

Interrelated Rotordynamic Effects of Cylindrical and Conical Whirl of Annular Seal Rotors

1991 ◽  
Vol 113 (3) ◽  
pp. 470-480 ◽  
Author(s):  
E. A. Baskharone ◽  
S. J. Hensel
Keyword(s):  
Finite Element ◽  
Flow Model ◽  
Computational Procedure ◽  
Rotordynamic Coefficients ◽  
Conical Whirl ◽  
Conical Rotor ◽  
Annular Seal ◽  
Rotor Surface ◽  
Flow Variables ◽  
Sample Case

A comprehensive approach for computing the dynamic coefficients of an annular seal is presented. The coefficients are partly those associated with a uniform lateral eccentricity mode of the rotor (known as the cylindrical whirl mode) and with an angular eccentricity (which gives rise to a conical whirl type). The rotor excitation effects in both cases are treated as interrelated by recognizing the fluid-exerted moments resulting from the lateral eccentricity and the net fluid force resulting from the angular eccentricity. In all cases, the rotor is assumed to undergo a whirling motion around the housing centerline. The computational procedure is a finite-element perturbation model in which the zeroth-order undisplaced-rotor flow solution in the clearance gap is obtained through a Petrov-Galerkin approach. Next, the rotor translational and angular eccentricities, considered to be infinitesimally small, are perceived to cause virtual distortions of varied magnitudes in the finite element assembly which occupies the clearance gap. Perturbations in the flow variables including, in particular, the rotor surface pressure, are then obtained by expanding the finite-element equations in terms of the rotor eccentricity components. The fluid-exerted forces and moments are in this case computed by integration over the rotor surface, and the full matrix of rotordynamic coefficients, in the end, obtained. The computational model is verified against a bulk-flow model for a sample case involving a straight annular seal. Choice of this sample model for validation was made on the basis that no other existing model has yet been expanded to account for the mutual interaction between the cylindrical and conical rotor whirl, which is under focus in this study.

Analysis for Rotordynamic Coefficients of Helically-Grooved Turbulent Annular Seals

1987 ◽  
Vol 109 (1) ◽  
pp. 136-143 ◽  
Author(s):  
Chang-Ho Kim ◽  
D. W. Childs
Keyword(s):  
Space Shuttle ◽  
Helix Angle ◽  
Circumferential Velocity ◽  
Surface Pattern ◽  
Surface Shear Stress ◽  
Surface Shear ◽  
Small Motion ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Annular Seals

An analysis for helically-grooved turbulent annular seals is developed to predict leakage and dynamic coefficients, as related to rotordynamics. The grooved surface pattern is formulated as an inhomogeneous directivity in surface shear stress. The governing equations, based on both Hirs’ turbulent lubrication theory and “fine-groove” theory, are expanded in the eccentricity ratio to yield zeroth and first-order perturbation solutions. The zeroth-order equations define the steady-state leakage and the circumferential velocity development due to wall shear for a centered rotor position. The first-order equations define perturbations in the pressure and axial and circumferential velocity fields due to small motion of the rotor about the centered position. Numerical results are presented for proposed grooved seals in the High Pressure Oxygen Turbopump (HPOTP) of the Space Shuttle Main Engine (SSME) and for a water-pump application. The results show that an optimum helix angle exists from a rotordynamic stability viewpoint. Further, a properly designed helically-grooved stator is predicted to have pronounced stability advantages over other currently used seals.

Dynamic Analysis of Turbulent Annular Seals Based On Hirs’ Lubrication Equation

1983 ◽  
Vol 105 (3) ◽  
pp. 429-436 ◽  
Author(s):  
D. W. Childs
Keyword(s):  
High Pressure ◽  
Space Shuttle ◽  
Water Pump ◽  
Centrifugal Pumps ◽  
First Order ◽  
Lubrication Equation ◽  
Inlet Swirl ◽  
Main Engine ◽  
Perturbation Solutions ◽  
Annular Seals

Expressions are derived which define dynamic coefficients for high-pressure annular seals typical of neck-ring and interstage seals employed in multistage centrifugal pumps. Completely developed turbulent flow is assumed in both the circumferential and axial directions, and is modeled in this analysis by Hirs’ turbulent lubrication equations. Linear zeroth and first-order “short-bearing” perturbation solutions are developed by an expansion in the eccentricity ratio. The influence of inlet swirl is accounted for in the development of the circumferential flow field. Comparisons are made between the stiffness, damping, and inertia coefficients derived herein based on Hirs’ model and previously published results based on other models. Finally, numerical results are presented for interstage seals in the Space Shuttle Main Engine High Pressure Fuel Turbopump and a water pump.

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