scholarly journals Rotordynamic Coefficients for Stepped Labyrinth Gas Seals

1989 ◽  
Vol 111 (1) ◽  
pp. 101-107 ◽  
Author(s):  
J. K. Scharrer
Keyword(s):  
Pressure Distribution ◽  
Parametric Study ◽  
Separation Of Variables ◽  
Reaction Force ◽  
Zeroth Order ◽  
Small Motion ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Gas Seals ◽  
Basic Equations

The basic equations are derived for compressible flow in a stepped labyrinth gas seal. The flow is assumed to be completely turbulent in the circumferential direction where the friction factor is determined by the Blasius relation. 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 pressure distribution is found by satisfying the leakage equation while the circumferential velocity distribution is determined by satisfying the momentum equations. The first order equations are solved by a separation of variables solution. Integration of the resultant pressure distribution along and around the seal defines the reaction force developed by the seal and the corresponding dynamic coefficients. The results of this analysis are presented in the form of a parametric study, since there are no known experimental data for the rotordynamic coefficients of stepped labyrinth gas seals. The parametric study investigates the relative rotordynamic stability of convergent, straight and divergent stepped labyrinth gas seals. The results show that, generally, the divergent seal is more stable, rotordynamically, than the straight or convergent seals. The results also show that the teeth-on-stator seals are not always more stable, rotordynamically, then the teeth-on-rotor seals as was shown by experiment by Childs and Scharrer (1986b) for a 15 tooth seal.

Rotordynamic Coefficients for Partially Roughened Pump Annular Seals

1991 ◽  
Vol 113 (2) ◽  
pp. 240-244 ◽  
Author(s):  
J. K. Scharrer ◽  
C. C. Nelson
Keyword(s):  
Pressure Distribution ◽  
Zeroth Order ◽  
Linear Complex ◽  
Order Continuity ◽  
Small Motion ◽  
Axial Coordinate ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Annular Seal ◽  
Basic Equations

The basic equations are derived for incompressible flow in an annular seal with partially roughened surfaces. The flow is assumed to be completely turbulent in the axial and circumferential directions with no separation, and is modeled by Hirs’ turbulent lubrication equations. 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 numerically, yielding the axial and circumferential velocity components and the pressure distribution. The first-order equations are reduced to three linear, complex, ordinary, differential equations in the axial coordinate Z. The equations are integrated to satisfy the boundary conditions and yield the perturbated pressure distribution. This resultant pressure distribution is integrated along and around the seal to yield the force developed by the seal from which the corresponding dynamic coefficients are derived. The results of a parametric study on the effect of the rough length/smooth length ratio on the seal leakage and rotordynamic coefficients are presented.

scholarly journals Rotordynamic Coefficients for Partially Roughened Annular Gas Seals

1991 ◽  
Author(s):  
Joseph K. Scharrer ◽  
Clay C. Nelson
Keyword(s):  
Pressure Distribution ◽  
Order Continuity ◽  
Small Motion ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Annular Seal ◽  
Complex Differential Equations ◽  
Gas Seals ◽  
Basic Equations ◽  
Perturbation Pressure

The basic equations are derived for compressible flow in an annular seal with partially roughened surfaces. 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 test data exist for this type of seal, the results of a parametric study on the effect of the rough length/smooth length ratio on the seal leakage and rotordynamic coefficients is presented.

Theory Versus Experiment for the Rotordynamic Coefficients of Labyrinth Gas Seals: Part I—A Two Control Volume Model

1988 ◽  
Vol 110 (3) ◽  
pp. 270-280 ◽  
Author(s):  
Joseph K. Scharrer
Keyword(s):  
Pressure Distribution ◽  
Control Volume ◽  
Reaction Force ◽  
Labyrinth Seal ◽  
Wall Friction ◽  
Small Motion ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Volume Model ◽  
Gas Seals

The basic equations are derived for a two-control-volume model for compressible flow in a labyrinth seal. The recirculation velocity in the cavity is incorporated into the model for the first time. The flow is assumed to be completely turbulent and isoenergetic. The wall friction factors are determined using the Blasius formula. Jet flow theory is used for the calculation of the recirculation velocity in the cavity. 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 pressure distribution is found by satisfying the leakage equation while the circumferential velocity distribution is determined by satisfying the momentum equations. The first-order equations are solved by a separation of variable solution. Integration of the resultant pressure distribution along and around the seal defines the reaction force developed by the seal and the corresponding dynamic coefficients.

scholarly journals An Iwatsubo-Based Solution for Labyrinth Seals: Comparison to Experimental Results

1986 ◽  
Vol 108 (2) ◽  
pp. 325-331 ◽  
Author(s):  
D. W. Childs ◽  
J. K. Scharrer
Keyword(s):  
Pressure Distribution ◽  
Reaction Force ◽  
Labyrinth Seal ◽  
Momentum Equation ◽  
Test Results ◽  
Labyrinth Seals ◽  
Small Motion ◽  
First Order ◽  
Basic Equations ◽  
Separation Of Variable

The basic equations are derived for compressible flow in a labyrinth seal. The flow is assumed to be completely turbulent in the circumferential direction where the friction factor is determined by the Blasius relation. 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 pressure distribution is found by satisfying the leakage equation while the circumferential velocity distribution is determined by satisfying the momentum equation. The first-order equations are solved by a separation of variable solution. Integration of the resultant pressure distribution along and around the seal defines the reaction force developed by the seal and the corresponding dynamic coefficients. The results of this analysis are compared to published test results.

The Effects of Fixed Rotor Tilt on the Rotordynamic Coefficients of Incompressible Flow Annular Seals

1993 ◽  
Vol 115 (3) ◽  
pp. 336-340 ◽  
Author(s):  
J. K. Scharrer ◽  
N. Rubin ◽  
C. C. Nelson
Keyword(s):  
Pressure Distribution ◽  
Incompressible Flow ◽  
Order Continuity ◽  
Small Motion ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Annular Seal ◽  
Arbitrary Position ◽  
Basic Equations ◽  
Perturbation Pressure

The basic equations are derived for incompressible flow in an annular seal with large rotor tilt. The flow is assumed to be completely turbulent in the axial and circumferential directions with no separation, and is modeled by Moody’s friction factor equation. Linearized zeroth and first-order perturbation equations are developed for small motion about an arbitrary position by an expansion in the eccentricity ratio. The zeroth-order continuity and momentum equations are solved using a Fast Fourier technique, yielding the axial and circumferential velocity components and the pressure distribution. The first-order 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. Results of a parametric study show that the detrimental effects of a tilted rotor are small.

Rotordynamic Coefficients for Partially Tapered Annular Seals: Part I—Incompressible Flow

1991 ◽  
Vol 113 (1) ◽  
pp. 48-52 ◽  
Author(s):  
J. K. Scharrer ◽  
C. C. Nelson
Keyword(s):  
Pressure Distribution ◽  
Incompressible Flow ◽  
Order Continuity ◽  
Small Motion ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Dynamic Coefficients ◽  
Annular Seal ◽  
Complex Differential Equations ◽  
Basic Equations

The basic equations are derived for incompressible 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 Hirs’ turbulent lubrication equations. 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/total length ratio on the seal leakage and rotor-dynamic coefficients are presented.

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.

Calculation of Rotordynamic Coefficients and Leakage for Annular Gas Seals by Means of Finite Difference Techniques

1989 ◽  
Vol 111 (3) ◽  
pp. 545-552 ◽  
Author(s):  
R. Nordmann ◽  
F. J. Dietzen ◽  
H. P. Weiser
Keyword(s):  
Finite Difference ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Zeroth Order ◽  
Navier Stokes Equations ◽  
First Order ◽  
Fluid Forces ◽  
Rotordynamic Coefficients ◽  
Gas Seals ◽  
Finite Difference Techniques

The compressible flow in a seal can be described by the Navier-Stokes equations in connection with a turbulence model (k–ε model) and an energy equation. By introducing a perturbation analysis in these differential equations we obtain zeroth order equations for the centered position and first order equations for small motions of the shaft about the centered position. These equations are solved by a finite difference technique. The zeroth order equations describe the leakage flow. Integrating the pressure solution of the first order equations yields the fluid forces and the rotordynamic coefficients, respectively.

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.

Finite-Length Solutions for Rotordynamic Coefficients of Turbulent Annular Seals

1983 ◽  
Vol 105 (3) ◽  
pp. 437-444 ◽  
Author(s):  
D. W. Childs
Keyword(s):  
Pressure Distribution ◽  
Finite Length ◽  
Reaction Force ◽  
Centrifugal Pumps ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
Dynamic Coefficients ◽  
Continuity Equations ◽  
Complex Differential Equations ◽  
Annular Seals

Expressions are derived which define dynamic coefficients for high-pressure annular seals typical of wear-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 by Hirs’ turbulent lubrication equations. Linear zeroth and first-order perturbation equations are developed by an expansion in the eccentricity ratio. The influence of inlet swirl is accounted for in the development of the circumferential flow. The zeroth-order momentum and continuity equations are solved exactly, while their first-order counterparts are reduced to three ordinary, complex, differential equations in the axial coordinate Z. The equations are integrated to satisfy the boundary conditions and define the pressure distribution due to seal motion. Integration of the pressure distribution defines the reaction force developed by the seal and the corresponding dynamic coefficients. Finite-length solutions for the coefficients are compared to two “short-seal” solutions.

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