Dr R Tiwari, Associate Professor, Dept. of Mechanical Engg., IIT Guwahati,

Transcription

1 6.3 Dynamic Seals Classification of Seals Seals are broadly classified as liquid and gas seals according to the working fluid used in the system. The most common working fluids are water, air, nitrogen, Triflurobromomethane (CBrF 3 ), liquid oxygen, liquid hydrogen etc. In addition, they can be categorized as static and dynamic seals. Static seals are used where the two surfaces do not move relative to one another. Gasket-type seals are static seals (Fig. 1). Dynamic seals are used where sealing takes place between two surfaces having relative movement viz. rotary, reciprocating, and oscillating. The main focus of the present paper is on rotary seals. It has wide variety of applications in high-speed, high-pressure and cryogenic temperature conditions of aviation and space industries such as in turbine stages, turbo-pumps, compressors, gear boxes, etc. Rotary seals can be subdivided into two main categories as clearance seals and contact seals. Clearance seals are circumferential non-contacting seals (Fig. a). In contact seals, the contact is formed by positive pressure, while in the case of clearance seals; they operate with positive clearance (no rubbing contact). The most commonly used material for dynamic seals (especially for rotary seals) are stainless steel, bronze, aluminium, nickel-based alloys, Polytetrafluroethane etc. Fig. (a) shows a typical rotary seal with the clearance exaggerated. Rotary seals based on geometry can be classified as (i) Ungrooved plain seals (or Smooth annular seals): (a) Straight (Fig. b), (b) Tapered (Fig. c) and (c) Stepped (Fig. d). In geometry they are similar to journal bearings but the clearance/radius ratio is as low as two times and as high as ten times (or more) large to avoid rotor/stator contact. (ii) Grooved/Roughened surface seals: (a) Porous surface seals (b) Labyrinth seals (Figs. 3(a-d)), (c) Helically grooved / Screw seals (d) Circular hole or triangular patterns seals and (e) Honeycomb patterns seals (Fig. 4). These seals are used in centrifugal and axial compressors and pumps and in turbines. Different internal surface patterns of seals are shown in Fig. 5. (iii) Contact seals: (a) Brush seals (Fig. 6a) (b) Face seals and (c) Lip seals (Fig. 6b)) Because of rubbing, these seals are used commonly in low speed pumps, or where the working fluid can act as a coolant. Contact seals provide much lower leakage rates than either of non-contact seals (Adams, 1987), however, the latter can operate at very high speed and pressure conditions. (iv) Floating-ring oil seals: The ring whirls or vibrates with the rotor in the lubricating oil, but does not spin. They are used in high-pressure multi-stage centrifugal compressors. 303

2 Compressive load Gasket Hydraulic end thrust High pressure Flow Seal Low pressure Rotor High pressure fluid Fig. 1. Static seal (gasket) Fig. (a). Rotor-seal assembly Flow Seal Rotor Flow Seal Rotor Fig. (b). Straight annular seal Fig. (c). Tapered annular seal (converging) Flow Seal Rotor Groove depth Expanding cavity Chalk vane Fig. (d). Stepped annular seal Fig. 3(a). Labyrinth seal (teeth-on-stator) Stator Labyrinth seal Stator Rotor Flow Rotor Labyrinth Fig. 3(b) Labyrinth seal (teeth-on-rotor) Fig. 3(c) Labyrinth seal (teeth-on-stator and teeth-onrotor) axial flow type 304

3 Labyrinth seal Stator Impellor (Rotor) Shaft Honeycomb housing Leakage Cell depth Cell size Fig. 3(d) Labyrinth seal radial flow type Fig. 4. Honeycomb seal (a) Plain seal Unwrap (b) Plain seal with porous material Unwrap (c) Labyrinth seal (f) Hole pattern roughness seal (d) Helically grooved seal (g) Triangular pattern roughness seal (e) Honeycomb seal Fig. 5. Different internal surface patterns on seals 305

4 Brush Fluid to be sealed Rubber lip Leak flow Metal stiffner Garter spring Fig. 6(a). Brush seal Fig. 6(b). Lip seal 6.3. Theoretical Estimation of Dynamic Coefficients of Seals In this chapter, basic governing equations to obtain dynamic coefficients of smooth annular turbulent seals (smooth seals) are presented. Dynamic coefficients are calculated from the approximate solution of the bulk flow theory for the configuration of the test rig. Effects of rotor speeds, seal dimensions and operation conditions on these dynamic coefficients are also presented and discussed in detail. Basic governing equations and solution In an annular seal, flows are usually turbulent because of high Reynolds numbers at which they operate. Black and his co-workers (Black 1969, Black and Jensen 1970) were the first to attempt to identify and model the rotor dynamics effects of turbulent annular seals using bulk flow models (similar to those of Reynolds lubrication equations). Bulk flow models employ velocity components, uz ( z, θ ) and uθ ( z, θ ), that are averaged over the clearance, where z u and u θ are the velocities in the directions and z and θ are the coordinates as shown Figure (4.1). Black and Jensen used several heuristic assumptions in their model, such as the assumption that u = Rω /, where R is the radius of the seal and ω is the rotor speed. Moreover, their governing equations do not reduce to recognizable turbulent lubrication equations. These issues caused Childs (1983b) to publish a revised version of the bulk flow model and the present section will focus on Childs' model. θ The geometry of the seal annulus which is filled with fluid is sketched in Figure 4.1, and is described by coordinates of the meridian of the gap as given by Z(s) and R(s), 0 < s < L, where the coordinate, s, is measured along that meridian and t is the time. The clearance is denoted by H(s, θ, t) where the unperturbed value of H is δ(s). Equations governing the bulk flow are averaged over the clearance. This leads to a continuity equation of the form (4.1) H 1 H dr + ( Hus ) + ( Huθ ) + us = 0 t s R θ R ds (4.1) 306

5 where us and u θ are velocities averaged over the local clearance. Stator Rotor H(s,θ,t) Co-ordinateθ and velocity u θ - Normal to sketch Z(s) R(s) u s s τ ss τ rs Figure 4.1. Fluid filled annulus between a rotor and a stator for turbulent lubrication analysis The axial and circumferential momentum equations are as follows 1 P τ ss τ sr uθ dr us uθ us us = us ρ s ρh ρh R ds t R θ θ 1 P τ u u u u u u θ s τθ r θ θ θ θ θ s R = us + ρr θ ρh ρh t R θ s R s (4.) (4.3) The approach used by Hirs (1973) is employed to determine the turbulent shear stresses, τ ss and τ θs, applied to the stator by the fluid in the s and θ directions respectively, which takes the following form τ ρu ms+ 1 τθ A u 1 ( / ) ms = = + u u θ s ( Res ) ρu (4.4) ss s s s s θ and stresses, τ sr and τ θr, applied to the rotor by the fluid in the s and θ directions respectively and are obtained as mθ + 1 τ sr τθ r Ar us = = 1 + {( u R) / u } θ ω s ( Res ) ρu ρ( u ωr) s θ where the local meridional Reynolds number is given as mθ (4.5) 307

6 Re = Hu / ν (4.6) s s and constants A s, A r, m s and m θ are chosen to fit the available data on turbulent shear stresses. Childs (1983a) uses typical values of these constant. A s = A s = 0.066; m s = m θ = -0.5 (4.7) In the following subsection, the solution for the governing equation are presented and discussed in details. Approximate dynamic coefficients of seals In the present subsection, the theoretical and computational analysis performed by various researchers has been compiled. Lomakin (1958) was the first to propose a theoretical model of a plain seal, which predicted that the axial pressure drop across the seal caused a radial stiffness, independent of shaft rotation. The Lomakin radial direct stiffness (k d ) is given by P λl / C kd = 4.7 R with λ = / R λ 1.5+ λl / C 0.5 e (1) where P is the pressure drop and R, L and C are the radius, axial length and radial clearance of the seal, respectively. If the direct stiffness were the only effect of the plain seal, then its effect on critical speeds would be easily and accurately predictable. Black s work (1969, 1971) provided the major initial impetus for the extensive research and the state of the art design information developed on this topic over the last 35 years. Black developed the classical theory for turbulent annular seals, considering the axial fluid flow caused by a pressure drop along the seal, the rotational fluid flow as a consequence of the shaft rotation and a relative motion of the seal between the rotor and housing. Black (1969, 1971) and Childs (1983a, b) formulated and extended Lomakin s theory in terms applicable to the rotor dynamic analysis of centrifugal pumps. Black, Childs and others have shown, however, that k d increases with shaft speed (at constant P) and that the seal also produces crosscoupled stiffness (k c ), direct and cross-coupled damping (c d and c c ), and direct inertia coefficients. Moreover, the pressure drop will vary with the speed in most turbomachineries and the rotor dynamic effects are quite complex. Clearances, pressures and velocities are divided into mean components (subscript 0) that would pertain in the absence of whirl, and small linear perturbations (subscript 1) due to the eccentricity, ε, rotating at the whirl frequency, ω: 308

7 H = H + ε H ; P= P + ε P s = s0 + ε ; 1s θ = 0 θ + ε 1 θ u u u u u u (4.8) These expressions are substituted into governing equations ( ) to yield a set of equations for mean flow quantities and a second set of equations for perturbation quantities; terms which are of quadratic or higher order in ε are neglected. Resulting zeroth-order equations define the leakage and the circumferential velocity development and are solved by numerical methods. From the first order equations, the time and θ dependency is eliminated to obtain the pressure distribution solution which is then integrated and along and around the seal clearance to yield reaction force components. From rotor dynamic force components, following rotor dynamic coefficients and constants are obtained (Childs, 1983). k d 1 4 * = a a ( ω T) k ; k = a T k 0 c ( ) 1 1 ω ; c d = a c 1 ; c c = a ( T ) c ω ; m d = a m (4.9) with k = P L R C ; c = k T ; m k T = (4.10) L T = (4.11) V a = 0.5 A E ; E B 1 A 1 a1 = A + E+ ; a = E+ σ 6 σ 6 (4.1) πσ 1+ 7b A = ; B= ; 1 + ξ + σ 1+ 4b 1+ ξ E = (4.13) (1+ ξ + Bσ ) σ = λl / C ; b = R a / R c ; ρvc R a = and µ R c ρrωc = (4.14) µ where k, m and c are the stiffness, mass and damping coefficients, values of corresponding quantities, k, c and m are reference a o a 1 and a are dimensionless coefficients, ω is the speed of the rotor, T is the transit time as given in equation 4.11, L is the length of the seal, V is the average axial stream velocity, ξ is the entrance loss coefficient, ρ is the fluid density, λ is the friction coefficient, R is the radius of the seal, C is the clearance of the seal and P is the difference between pressures at the inlet and the exit of the seal. Subscripts d and c represent the direct and cross-coupled 309

8 terms, respectively. R a is the Reynolds number for the axial flow and R c is the Reynolds number for the circumferential flow for smooth annulus seals. Dimensional coefficients are thus functions of ξ, σ and b. To determine coefficients a 0, a 1 and a coefficients σ and b are required for the frequently occurring value of ξ =0.5. From Childs (1983a), we have [ 1 (1/ ) ] / 4 λ = 0.066R a + b (4.15) To calculate λ the average velocity V is inserted into equation (4.14). The expression for V can be obtained from the fundamental relationship for the pressure difference, +ξ + ρ σ P= ( 1 ) V (4.16) So, the average axial stream velocity can be expressed as P V = (4.17) ρ (1+ ξ + σ ) Since the desired value of λ is also function of V and thereby σ, it is best obtained iteratively. From the σ, the dynamic coefficients can be obtained for different speed ω. Figure 4. shows an algorithm for the solution of dynamic coefficients of seals. 310

9 Start Input: ρ, ξ, P, µ, C, L, R Set N = 0 rad/s σ = 0 and e = 1 σ = σ Calculate V, b, R a 1 Calculateλ = σ C / L, 1/ 4 [ 1 (1/ ) ] λ = 0.066R a + b e = 1 λ λ If e 10-4 Yes No Calculate A, B, E, a 0, a 1, a Calculate k, c, m Calculate k d, k c, c d, c c, m d Set N = N+1 If N > 5301 No End Yes Figure 4.. Flow chart for the theoretical estimation of dynamic coefficients of seals 311

10 Numerical simulation results and discussion In this subsection, numerical results of dynamic coefficients of seals are presented for the rotor speed up to 50,000 rpm. The input data are taken as mentioned in Table 4.1. Table 4.1. Input data for numerical simulation of dynamic coefficients of seals Length of the seal 11,, 33 and 44 mm Radius of the seal mm Clearance of the seal 0. and 0.4 mm Dynamic viscosity of water at 3 o C m /s Entrance loss coefficient 0.5 Inlet pressure 3, 6, 16, 41, 81 bar Seal exit pressure 1 bar Speed of the rotor 1 to 50,000 rpm Seals dynamic coefficients are dependent on speeds, seal dimensions and pressure differences. The stiffness (k d and k c ), damping (c d and c c ) and mass (m d ) coefficients are presented for various speeds (ω), pressure differences ( P) and ratios L/D. Figures 4.3 to 4.15 show the variation of the direct and cross-coupled stiffness and damping and direct inertia coefficients with respect to the speed up to rpm, for different values of clearances (0. and 0.4 mm), L/D ratios (0.5, 0.50, 0.75 and 1.00) and pressure differences (, 5, 15, 40 and 80 bar). The effects of these variables on seal dynamic coefficients are discussed in detail in following sections. Effect of rotational speeds and pressure differences Direct stiffness coefficients increase with increase in the pressure difference (Figure 4.3). At lowpressure differences ( and 5 bars), the direct stiffness coefficient becomes negative as shown in Figure 4.3. The direct stiffness coefficient reaches maximum nearly at 5000 rpm and then slowly declines as shown in Figure 4.3. The cross-coupled stiffness linearly increases with the rotor speed and also increases with the pressure difference (Figure 4.4). The direct damping coefficient increase slightly to the speed, however, it increases with the pressure difference (Figure 4.5). The crosscoupled damping increases linearly with the speed but, insensitive to the pressure difference (Figure 4.6). The direct inertia coefficient increases sharply with the rotor speed and it is almost insensitive to the pressure difference (Figure 4.7). Effect of L/D ratios 31

11 L/D ratio has significant effect on rotor dynamic coefficients of seals. The direct stiffness increases with the increase in L/D ratio. For L/D= 1.00, after reaching a maximum value nearly to 8000 rpm it starts declining and becomes negative with increase in the rotor speed. At L/D=0.5, the direct stiffness coefficient always has positive values (Figure 4.8). The cross-coupled stiffness and the direct and cross-coupled damping coefficients increase with the increase in L/D ratio as shown in Figures Effect of seal clearances Doubling the clearance show a huge drop in the direct stiffness and damping coefficients, while increasing speeds up to 50,000 rpm. The drop in the cross-coupled stiffness and damping and direct inertia coefficients with increase in clearance is also significant (Figures ). Figure 4.3. Direct stiffness coefficients for C=0. mm, L/D=0.5 and P= to 80 bar. Figure 4.4. Cross-coupled stiffness coefficients for C=0. mm, L/D=0.5, P= to 80 bar. 313

12 Figure 4.5. Direct damping coefficients for C=0. mm, L/D=0.5, P= to 80 bar. Figure 4.6. Cross-coupled damping coefficients for C=0. mm, L/D=0.5, P= to 80 bar. Figure 4.7. Direct inertia coefficients for C=0. mm, L/D=0.5, P= to 80 bar. 314

13 Figure 4.8. Direct stiffness coefficients for C=0. mm, P=40 bar, L/D= Figure 4.9. Cross-coupled stiffness coefficients for C=0. mm, P=40 bar, L/D= Figure Direct damping coefficients for C=0. mm, P=40 bar, L/D=

14 Figure Cross-coupled damping coefficients for C=0. mm, P=40 bar, L/D= Figure 4.1. Direct inertia coefficients for C=0. mm, P=40 bar, L/D= Figure Direct and cross-coupled stiffness coefficients for P=40 bar, L/D=0.5, C=0. & 0.4 mm. 316

15 Figure Direct and cross-coupled damping coefficients for P=40 bar, L/D=0.5, C=0. & 0.4 mm. Figure Direct inertia coefficients for P=40 bar, L/D=0.5, C=0. and 0.4 mm. Basic governing equations to obtain dynamic coefficients of the smooth-annular turbulent seals (i.e. smooth seals) are explained briefly. Dynamic coefficients are calculated from the bulk flow theory for a seal dimension and effects of rotor speeds, seal dimensions and operation conditions on dynamic coefficients of seals are presented and discussed in detail Fluid-Film Dynamic Force Equations A model of a typical annual (or clearance) seal is shown in Fig. (a). The geometrical shape of a clearance seal is similar to that of a hydrodynamic bearing; however, they are different in the following aspects. To avoid contact between a rotor and a stator, the ratio of the clearance to the shaft radius in seals is made few times ( to 10 times) larger than that of hydrodynamic bearings. The flow in seals is turbulent and in hydrodynamic bearings it is laminar. Therefore, unlike hydrodynamic bearing, one cannot use the Reynolds equation for analysis of seals. When a rotor vibrates, a reaction 317

16 force of the fluid in the seal acts on the rotor. In case of a small vibration around the equilibrium position, the fluid force can be linearized on the assumption that deflections x and y are small. The general governing equations of fluid-film forces on seals, which has small oscillations relative to the rotor, is given by the following linearized force-displacement model (Childs et al., 1986) fx k k xx xy x c c xx xy x m m xx xy x = f + y k + yx k c yy y yx c m yy y yx myy y () where f x and f y are fluid-film reaction forces on seals in x and y directions. k, c, m represent the stiffness, damping and added-mass coefficients, subscripts: xx and, yy represent the direct and xy and yx represent the cross-coupled terms, respectively. These coefficients vary depending on the equilibrium position of the rotor (i.e. magnitude of the eccentricity), rotational speed, pressure drop, temperature conditions etc. The off-diagonal coefficients in equation () arise due to fluid rotation within the seal and unstable vibrations may appear due to these coefficients. Equation () is applicable to liquid annular seals. But for the gas annular seals, the added-mass terms are negligible. For small motion about a centered position (or with very small eccentricity) the cross-coupled terms are equal and opposite (e.g., k xy = -k yx = k c and c xy = -c yx = c c ) and the diagonal terms are same (e.g., k xx = k yy = k d and c xx = c yy = c d ) (Childs et al., 1986). Considering these relationships and neglecting the crosscoupled added-mass terms, equation () takes the following form f x kd kc x cd cc x md 0 x = f + + y kc k d y cc c d y 0 m d y (3) where subscripts: d and c represent direct and cross-coupled, respectively. The RDPs largely affect the performance of the turbomachineries as they lead to serious synchronous and sub-synchronous vibration problems. Whirl frequency ratio, f = k c /(c d ω ) is a useful non-dimensional parameter for comparing the stability properties of seals. For circular synchronous orbits, it provides a ratio between the destabilizing force component due to k c and the stabilizing force component due to c d. In experimental estimation of RDPs of seals, these coefficients (of equation () and (3)) are determined with the help of measured vibrations data from a seal test rig. The more recent textbooks on rotor dynamics include information on rotor dynamic characteristics of rotary seals. Vance (1988), Childs (1993), Krämer (1993), Rao (000), Adams (001) and Tiwari et al. (005) provide a good introductory treatments of seal dynamics. References: 318

17 Admas, M.L. Jr, 001, Rotating Machinery Vibration, Marcel Dekker, Inc., New York. Changsen, W., 1991, Analysis of Rolling Element Bearings, Mechanical Engineering Publications Ltd., London. Childs, D.W., 1993, Turbomachinery Rotordynamics: Phenomena, Modeling and Analysis, John Wiley & Sons, Inc., New York. El-Sayed H. R., 1980, Wear, 63, Stiffness of deep-groove ball bearing. Eschmann, P., Hasbargen, I. and Weigand, K., (1985), Ball and Roller Bearings, Theory, Design and Application. John Wiley and Sons:New York. Gargiulo E.P., 1980, Machine Design, 5, A simple way to estimate bearing stiffness. Harris, T.A., 001, Rolling Bearing Analysis, Wiley, New York. Hertz, H., (1896), Miscellaneous Papers, Macmillan, London, On the contact of rigid elastic solids and on hardness. Hummel, C., 196, Kristische Drehzahlen als Folge der Nachgiebigkeit des Schmiermittels im Lager, VDI-Forschungsheft, 87. Johnson. T.L., 1991, Contact Mechanics, nd edition, McGraw-Hills, New York. Jones, A. B., 1946, Analysis of Stresses and Deflections, New Departure Engineering Data, Briston. Jones A.B., 1960, Transactions of ASME, Journal of Basic Engineering, , A general theory for elastically constrained ball and radial roller bearings under arbitrary load and speed conditions. Krämer E., 1993, Dynamics of Rotors and Foundations, Springer-Verlag, New York. R. Kashyap and R. Tiwari, 006, Prediction of Heat Generations and Temperature Distributions at Critical Contact Zones of High-Speed Rolling Bearings, Proceedings of 18th National & 7th ISHMT-ASME Heat and Mass Transfer Conference, January 4-6, 006, IIT Guwahati. Lim, T. C. and Singh, R., (1990a), Journal of Sound and Vibration 139 (), Vibration Transmission Through Rolling Element Bearings, Part I: Bearing Stiffness Formulation. Lim, T. C. and Singh, R., (1990b), Journal of Sound and Vibration 139 (), Vibration Transmission Through Rolling Element Bearings, Part II: System Studies. Lim, T. C. and Singh, R., (1991), Journal of Sound and Vibration 151 (1), Vibration Transmission Through Rolling Element Bearings, Part III: Geared Rotor System Studies. Lim, T. C. and Singh, R., (199), Journal of Sound and Vibration 153 (1), Vibration Transmission Through Rolling Element Bearings, Part IV: Statistical Energy Analysis. Lim, T. C. and Singh, R., (1994), Journal of Sound and Vibration 169 (4), Vibration Transmission Through Rolling Element Bearings, Part V: Effect of Distributed Contact Load on Roller Bearing Stiffness Matrix. Newkirk, B.L., 194, Shaft Whipping, General Electric Review, pp Newkirk, B. L. and Taylor, H.D., 195, Shaft Whipping due to Oil Action in Journal Bearing," General Electric Review, Palmgren, A., (1959), Ball and Roller Bearing Engineering, 3rd ed., Burbank. Ragulskis, K. M., Jurkauskas, A. Yu., Atstupenas, V. V., Vitkute, A. Yu., and Kulvec, A. P., (1974), Vibration of Bearings. Vilnyus: Mintis Publishers. Rao, J. S., 000, Vibratory Condition Monitoring of Machines, Narosa Publishing House, New Delhi. Schweitzer, G., Bleuler, H. and Traxler, A., 1994, Active Magnetic Bearing: Basics, Properties and Application of Active Magnetic Bearings. Vdf Hochschulverlag AG an der ETH, Zürich. Smith, D.M., 1969, Journal Bearings in Turbomachinery, Chapman and Hall, London. Stodola, A., 195, Kritische Wellenstörung infolge der Nachgiebigkeit des Ölpolslers im Lager (Critical shaft perturbations as a result of the elasticity of the oil cushion in the bearings), Schweizerische Bauzeitung, Vol. 85, No. 1, May. Stolarski, T. A., (1990), Tribology in Machine Design. Oxford: Heinemann Newnes. Timoshenko, S. and Goodier, J., 1951, Theory of Elasticity, nd edition, McGraw-Hills, New York. Vance, J.M., 1998, Rotordynamics of Turbomachinery, John Wiley & Sons Inc, New York. 319

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