Convergent-Tapered Annular Seals: Analysis and Testing for Rotordynamic Coefficients

1985 ◽  
Vol 107 (3) ◽  
pp. 307-316 ◽  
Author(s):  
D. W. Childs ◽  
J. B. Dressman
Keyword(s):  
Dynamic Pressure ◽  
Momentum Equation ◽  
Computational Method ◽  
Centrifugal Pumps ◽  
Zeroth Order ◽  
Taper Angle ◽  
First Order ◽  
Dynamic Coefficients ◽  
Direct Stiffness ◽  
Annular Seals

A combined analytical-computational method is developed to calculate the pressure field and dynamic coefficients for tapered 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 by Hirs’ bulk-flow turbulent-lubrication equations. Linear zeroth- and first-order perturbation equations are developed for the momentum equations and continuity equations. The development of the circumferential velocity field is defined from the zeroth-order circumferential-momentum equation, and a leakage relationship is defined from the zeroth-order axial-momentum equation. A short-bearing approximation is used to derive an analytical expression for the first-order (dynamic) pressure gradient. This expression is integrated numerically to define dynamic coefficients for the seal. Numerical results are presented and compared to previous results for straight and tapered seals. The direct stiffness and leakage increase with increasing taper angle, while the remaining dynamic coefficients decrease. An optimal taper angle is shown to exist with respect to (a) the direct stiffness, and (b) the ratio of direct stiffness to leakage. Stiffness increases on the order of 40-50 percent are predicted. Experimental results are presented for seals with three taper angles which show generally good agreement between theory and prediction.

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.

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.

Understanding Friction Factor Behavior in Liquid Annular Seals With Deliberately Roughened Surfaces

2005 ◽  
Vol 127 (1) ◽  
pp. 213-222 ◽  
Author(s):  
Larry A. Villasmil ◽  
Dara W. Childs ◽  
Hamn-Ching Chen
Keyword(s):  
Friction Factor ◽  
Flow Behavior ◽  
Shear Stresses ◽  
Centrifugal Pumps ◽  
Typical Application ◽  
Number Range ◽  
Dynamic Coefficients ◽  
Seal Design ◽  
Pattern Size ◽  
Annular Seals

Multistage centrifugal pumps and compressors are among the most widely used pieces of rotating machinery in industry. A typical application demands the arrangement of several impellers or wheels mounted on a shaft that spins within a stationary case. Annular seals are the most common sealing devices used in this type of machinery. The annular seal design affects both (i) machinery performance in terms of energy conversion efficiency, and (ii) stability due to the interaction within the rotor and the stator through the fluid flow within the seals. Traditionally, the “bulk-flow” theory due to Hirs (ASME J. Lubrication Technol., pp. 137–146) has been used to estimate annular seals leakage and dynamic coefficients. To predict the flow behavior through the seal, this theory relies on empirical friction factor correlations. While leakage is well predicted, the dynamic coefficients are not. The discrepancy is attributed to the friction factor model. Several experiments have produced seal leakage data indicating that friction factor increases as the seal clearance is increased, contradicting predictions based on Moody’s pipe-friction model. A Computational Fluid Dynamics (CFD) commercial code was used to simulate flat-plate-channel-flow experimental tests of water flowing with deliberately roughened surfaces, showing an increase of friction factor with clearance increase. The higher friction factor characteristics of these deliberately roughened surfaces are governed by their ability to develop a high static pressure in the trailing face of each roughness cavity, while the wall shear stresses on the smooth land play a secondary role. In a certain Reynolds number range, the maximum friction factor observed on a specific roughness pattern size is independent of the actual clearance, which we have referred to as the friction-factor-to-clearance indifference behavior. This phenomenon is found to be related to the roughness cavity size and its length-to-clearance ratio.

Two-dimensional incompressible magneto-hydrodynamic flow across an elliptical solenoid

1958 ◽  
Vol 4 (6) ◽  
pp. 553-584 ◽  
Author(s):  
Nelson H. Kemp ◽  
Harry E. Petschek
Keyword(s):  
Magnetic Field ◽  
Flow Field ◽  
Hall Current ◽  
Dynamic Pressure ◽  
Magnetic Reynolds Number ◽  
Two Dimensional ◽  
Zeroth Order ◽  
First Order ◽  
Ion Slip ◽  
The Magnetic Field

An analysis has been made of the two-dimensional flow of an incompressible constant-conductivity fluid through an elliptically shaped solenoid containing a constant magnetic field directed normal to the flow plane. The effect of both Hall current and ion slip has been included in the generalized Ohm's law used for the fluid. The analysis is based on a perturbation procedure in two parameters, one being the magnetic Reynolds number Rm and the other the ratio S of magnetic force per unit area to dynamic pressure. Calculations have been carried to the first order in each parameter, and closed-form analytic expressions have been obtained for the force and moment on the solenoid, the current density, stream function, magnetic field and other pertinent physical quantities.It was found that, to the zeroth order, there is a force but no moment on the solenoid. To the first order in S, where the flow field is modified but the magnetic field is not, there is a moment and a force, the latter being anti-parallel to the zeroth order force. To the first order in Rm, where the magnetic field is modified but the flow field is not, there is a moment but no force. Thus, to the first order the lift to drag ratio is the same as in the zeroth order. Graphs which illustrate some of the effects of angle of attack, fineness ratio of the ellipse, Hall current and ion slip, on the forces and moments are presented.

scholarly journals Analysis and Testing for Rotordynamic Coefficients of Turbulent Annular Seals With Different, Directionally-Homogeneous Surface-Roughness Treatment for Rotor and Stator Elements

1985 ◽  
Vol 107 (3) ◽  
pp. 296-305 ◽  
Author(s):  
D. W. Childs ◽  
Chang-Ho Kim
Keyword(s):  
Surface Roughness ◽  
Computational Method ◽  
Circumferential Velocity ◽  
Axial Position ◽  
Round Hole ◽  
Centrifugal Pumps ◽  
Transient Pressure ◽  
Homogeneous Surface ◽  
Small Motion ◽  
First Order

A combined analytical-computational method is developed to calculate the transient pressure field and dynamic coefficients for high-pressure annular seal configurations which may be used in interstage and neck-ring seals of multistage centrifugal pumps. The solution procedure applies to constant-clearance or convergent-tapered geometries which may have different (but directionally-homogeneous) surface-roughness treatments on the stator or rotor seal elements. It applies in particular so-called “damper-seals” which employ smooth rotors and deliberately-roughened stator elements to enhance rotor stability. Hirs’ turbulent lubrication equations are modified slightly to account for different surface-roughness conditions on the rotor and stator. A perturbation analysis is employed in the eccentricity ratio to develop zeroth and first order perturbation equations. The zeroth-order equations define both the leakage and the development of circumferential flow due to shear forces at the rotor and stator surfaces. The first-order equations define perturbations in the pressure and axial and circumferential velocity fields due to small relative motion between the seal rotor and stator. The solution applies for small motion about a centered position and does not employ linearization with respect to either the taper angle or the degree of swirl, i.e., the difference between the circumferential velocity at the given axial position and the asymptotic circumferential-velocity solution. Test results for four different surface-roughness confirm the predicted net damping increase for “damper seals.” A round-hole-pattern stator yielded the highest net damping and lowest leakage of all seals tested. The seals are substantially stiffer than predicted, but the theory does an adequate job of predicting net damping.

scholarly journals Perturbation Solution for One-Dimensional Flow to a Constant-Pressure Boundary in a Stress-Sensitive Reservoir

2021 ◽  
Author(s):  
Amarjot Singh Bhullar ◽  
Gospel Ezekiel Stewart ◽  
Robert W. Zimmerman
Keyword(s):  
Approximate Solution ◽  
Pore Pressure ◽  
Constant Pressure ◽  
Initial Permeability ◽  
Order Perturbation ◽  
Zeroth Order ◽  
One Dimensional ◽  
First Order ◽  
Outflow Boundary ◽  
Perturbation Solutions

Abstract Most analyses of fluid flow in porous media are conducted under the assumption that the permeability is constant. In some “stress-sensitive” rock formations, however, the variation of permeability with pore fluid pressure is sufficiently large that it needs to be accounted for in the analysis. Accounting for the variation of permeability with pore pressure renders the pressure diffusion equation nonlinear and not amenable to exact analytical solutions. In this paper, the regular perturbation approach is used to develop an approximate solution to the problem of flow to a linear constant-pressure boundary, in a formation whose permeability varies exponentially with pore pressure. The perturbation parameter αD is defined to be the natural logarithm of the ratio of the initial permeability to the permeability at the outflow boundary. The zeroth-order and first-order perturbation solutions are computed, from which the flux at the outflow boundary is found. An effective permeability is then determined such that, when inserted into the analytical solution for the mathematically linear problem, it yields a flux that is exact to at least first order in αD. When compared to numerical solutions of the problem, the result has 5% accuracy out to values of αD of about 2—a much larger range of accuracy than is usually achieved in similar problems. Finally, an explanation is given of why the change of variables proposed by Kikani and Pedrosa, which leads to highly accurate zeroth-order perturbation solutions in radial flow problems, does not yield an accurate result for one-dimensional flow. Article Highlights Approximate solution for flow to a constant-pressure boundary in a porous medium whose permeability varies exponentially with pressure. The predicted flowrate is accurate to within 5% for a wide range of permeability variations. If permeability at boundary is 30% less than initial permeability, flowrate will be 10% less than predicted by constant-permeability model.

Consistent section-averaged equations of quasi-one-dimensional laminar flow

2010 ◽  
Vol 656 ◽  
pp. 337-341 ◽  
Author(s):  
PAOLO LUCHINI ◽  
FRANÇOIS CHARRU
Keyword(s):  
Equations Of Motion ◽  
Stability Result ◽  
Momentum Equation ◽  
Dissipation Function ◽  
Dimensional Solution ◽  
First Order ◽  
Averaged Equations ◽  
Classical Stability ◽  
Momentum Integral ◽  
Free Falling

Section-averaged equations of motion, widely adopted for slowly varying flows in pipes, channels and thin films, are usually derived from the momentum integral on a heuristic basis, although this formulation is affected by known inconsistencies. We show that starting from the energy rather than the momentum equation makes it become consistent to first order in the slowness parameter, giving the same results that have been provided until today only by a much more laborious two-dimensional solution. The kinetic-energy equation correctly provides the pressure gradient because with a suitable normalization the first-order correction to the dissipation function is identically zero. The momentum equation then correctly provides the wall shear stress. As an example, the classical stability result for a free falling liquid film is recovered straightforwardly.

scholarly journals Influences of Randomly Uncertain Factors on Dynamic Coefficients of an Interlocking Labyrinth Seal-Rotor System

Machines ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 39
Author(s):  
Xin Xiong ◽  
Yanfei Zhou ◽  
Yiqun Wang
Keyword(s):  
Pressure Difference ◽  
Gas Flow ◽  
Labyrinth Seal ◽  
Momentum Equation ◽  
Rotor System ◽  
Rotor Speed ◽  
Bounded Noise ◽  
Dynamic Coefficients ◽  
Noise Perturbation ◽  
Rotor Center

Many randomly uncertain factors inevitably arise when gas flows through a labyrinth seal, and the orbit of the rotor center will not rotate along a steady trajectory, as previously studied. Here, random uncertainty is considered in an interlocking labyrinth seal-rotor system to investigate the fluctuations of dynamic coefficients. The bounded noise excitation is introduced into the momentum equation of the gas flow, and as a result, the orbit of the rotor center is expressed as the combination of an elliptic trajectory with the bounded noise perturbation. Simulation results of the coefficients under randomly uncertain perturbations with various strengths are comparatively investigated with the traditional predictions under ideal conditions, from which the influences of random uncertain factors on dynamic coefficients are analyzed in terms of the rotor speed, pressure difference, and inlet whirl velocity. It is shown that the deviation levels of the dynamic coefficients are directly related to the random perturbations and routinely increase with such perturbation strengths, and the coefficients themselves may exhibit distinct variation patterns against the rotor speed, pressure difference, and inlet whirl velocity.

scholarly journals Pancreatic secretion and pressure biomechanics in pancreatic acinus

2020 ◽  
Vol 19 (1) ◽  
pp. 6-12
Author(s):  
G. Ya. Kostyuk ◽  
O. G. Kostyuk ◽  
M. V Burkov ◽  
I. A. Golubovsky ◽  
M. P. Bulko ◽  
...  
Keyword(s):  
Flow System ◽  
Dynamic Pressure ◽  
Excretory Duct ◽  
The Other ◽  
Duct System ◽  
First Order ◽  
Pancreatic Acinus ◽  
Mercury Column ◽  
The Mathematical Model ◽  
Oddi Sphincter

The article highlights the mechanism of the mathematical model of acinus, the components of the formation of pressure in its cavity and the formation of pancreatic juice. It has been established that the mechanism for creating pressure in the acinus cavity is similar to the intraductal one. In this case, the question remains open about the causes of such high pressure, which is measured in several hundred millimeters of a mercury column, especially since, as histologically established, the pancreas and its ducts do not have muscle structures, and those rudiments of myofibrils, which are noted in some places of the flow system, of course, cannot ensure the development of such pressure. The increase in pressure in the cavity of the acinus is associated with the phenomenon of osmosis in its cells. Since cell membranes have the property of conductivity, as a result of osmosis, water through the membrane first passes from the blood to the cell, then from the cell through the membrane into the acinus cavity. In addition to the mechanism of osmosis through the membrane, in the cells of the acinus epithelium, there is a filtering mechanism through the pores of the layer of connective tissue to the lymph channel. It has now been established that, together with simple osmosis, the phenomenon of electroosmosis takes place in secreting cells and organs of excretion, not only accelerates the transfer of substances, but also increases the pressure on the other side of the membrane against the gradient by almost several first-order units. Thus, the outflow of fluid from the acinus cavity proceeds continuously, but only with a change in the speed of movement, it is determined by the pressure drop in the acinus – tubule – excretory duct system, the opening of the Oddi sphincter and the pulse of the cardiovascular wave, which creates dynamic pressure in the capillary. This whole mechanism, as a result, leads to the filling of the cavity of the acinus and the creation of a certain pressure in it.

scholarly journals Integrating molecular profiles into clinical frameworks through the Molecular Oncology Almanac to prospectively guide precision oncology

Nature Cancer ◽  
2021 ◽  
Author(s):  
Brendan Reardon ◽  
Nathanael D. Moore ◽  
Nicholas S. Moore ◽  
Eric Kofman ◽  
Saud H. AlDubayan ◽  
...  
Keyword(s):  
Point Of Care ◽  
Single Gene ◽  
Knowledge Bases ◽  
Computational Method ◽  
Precision Oncology ◽  
Clinical Interpretation ◽  
First Order ◽  
Molecular Features ◽  
Molecular Oncology ◽  
Molecular Profiles

AbstractTumor molecular profiling of single gene-variant (‘first-order’) genomic alterations informs potential therapeutic approaches. Interactions between such first-order events and global molecular features (for example, mutational signatures) are increasingly associated with clinical outcomes, but these ‘second-order’ alterations are not yet accounted for in clinical interpretation algorithms and knowledge bases. We introduce the Molecular Oncology Almanac (MOAlmanac), a paired clinical interpretation algorithm and knowledge base to enable integrative interpretation of multimodal genomic data for point-of-care decision making and translational-hypothesis generation. We benchmarked MOAlmanac to a first-order interpretation method across multiple retrospective cohorts and observed an increased number of clinical hypotheses from evaluation of molecular features and profile-to-cell line matchmaking. When applied to a prospective precision oncology trial cohort, MOAlmanac nominated a median of two therapies per patient and identified therapeutic strategies administered in 47% of patients. Overall, we present an open-source computational method for integrative clinical interpretation of individualized molecular profiles.

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