Evaluation of Seal Effects on the Stability of Rotating Fluid Machinery

The stability of typical rotating fluid machinery such as single and multi-stage pumps is evaluated by using the finite element method. The individual contribution of the impellers, bearings and seals to the stability and the dynamic interactions of these fluid elements are examined. Various types of bearings and seals, such as annular smooth, parallel grooved and damper seals, are compared for better rotor stability. The effect of the operating conditions on the stability is also investigated. The results show that rotor stability can be easily improved by replacing the unstable fluid elements.


INTRODUCTION
n recent years, the operating conditions ofrotating fluid machinery tend towards higher speed and higher pres- sure along with the rapid progress made in the technol- ogy in industry and space development.In order to raise the efficiency and prevent leakage flow at high pressure, the clearance of fluid elements such as bearings, non- contacting seals, and impellers has to be designed possibly smaller.However, with higher pressure, higher speed, and smaller clearance, greater fluid forces occur and some- times these forces cause unstable vibration.
To prevent such unstable vibration, firstly the dynamic characteristics of these fluid elements and their individ- ual contribution to rotor stability must be made clear.Work is being done, and many results have been obtained by various researchers, [1], [2], [4]-[8], [10], [11].Secondly, the combined effect of these fluid elements on rotor stability and their interaction must be investigated.Yang   et al. 1985] investigated the effect of annular smooth and taper seals on the stability of the single-stage pump ro- tor system.Diewald et al. [1987] showed a procedure to investigate the effect of annular smooth and grooved seals and impellers on the stability of the Jeffcott ro- tor and the multi-stage pump rotor system.These re- searches show that the rotor stability is strongly affected by the fluid elements, and the contribution of these ele- ments to rotor stability varies according to the operating conditions.
In this paper, continuing the research of Yang et al.
[1985], the stability of typical rotating fluid machinery such as the single-stage and multi-stage pumps, which consist of impellers, bearings, and non-contacting seals, are evaluated by using the finite element method.The in- dividual contribution of the impellers, bearings as well as seals to the stability, and the dynamic interactions of these fluid elements are investigated.The contribution to the rotor stability is evaluated by the logarithmic decrement.For the linear and non-cross-coupled inertia rotor system, the total logarithmic decrement of the rotor system can be represented as the sum of the individual decrements.Therefore, the stability of the rotor system can be eas- ily improved by changing the unstable fluid elements in the design stage.In the investigation, some types of bear- ings and seals such as annular smooth, parallel grooved and damper seals are compared to seek better ones for the rotor stability.The effect of the operating conditions on stability is also studied.

EQUATIONS OF MOTION AND LOGARITHMIC DECREMENT
The analytical models of the single and multi-stage pump are shown in Figures and 2. For general application, the non-symmetrical single-stage pump is taken.These rotor systems consist of bearings, impellers and seals.For the convenience of analysis, the impeller is looked upon as a disk which has the same mass and same moment of inertia as the practical impeller.If the rotor rotates with a steady angular velocity w, the equation of motion of the disk elements in the coordinates illustrated in Figure 2  where [Mt d] and [Mr d] are resPectively the mass matrices for translational and bending motions; [Gd] is gyroscopic matrix; F d is the force vector acting on the disks.
The equation of motion of the journal elements is given by the expression where {qe} {qt, qr }T {x y qbx (/)y }T M e ,[ t],[Me], [Ge], [Ke] are the mass matrices of translational and bending motions, gyroscopic matrix and stiffness matrix, respectively; {Fe} is the fluid force vector of bearings, impellers and seals, and they are expressed as follows: - where [Mt],[Ms] are the inertia coefficient ma- trices; [CB],[Ct],[Cs] are the damping coefficient matrices; [KB], [Kt], [Ks] are the stiffness coefficient matrices.Substituting eq.(3) into eq.( 2), and combin- ing the equations of disk elements and journal elements, the equation motion of the rotor system can be obtained.

FIGURE
Model of single-stage pump.
[M]{/i/} + where {F} is the external force not including the fluid forces of bearings, impellers and seals.To determine the eigenvalues and eigenvectors, the characteristic equation of eq. ( 4) is rewritten in the following form: Letting the generalized solution of eq. ( 6) have the form {z} {}e xt (7) then eq. ( 6) yields where ,k and {} are the eigenvalue and eigenvector, re- spectively.They can be obtained by solving eq. ( 8).
In general, the eigenvalues are conjugate complex, and expressed in the following form: where means th natural mode.The logarithmic decre- ment of the rotor system is defined as In order to investigate the effects of the fluid elements on the rotor stability, the logarithmic decrements, of the fluid elements have to be determined.These logarithmic decrements can be obtained by the eigenvalue and eigen- vector through the following transformation (Kurohashi et al. [1982]. Substituting the ith conjugate eigenvalues .i,i and ith conjugate eigenvectors {i}, {i} into the charac- teristic equation of eq. ( 4), the following equations are obtained. .
Furthermore, expressing the mass matrix, damping ma- trix and stiffness matrix as the sum of symmetric parts M*, C*, K* and unsymmetric parts AM, AC, AK, then the above equations become Premultiplying equations of (12) by {t }T and {i }T, and introducing the following expressions (13) {ri}T the difference of the two equations of eq. ( 12) can be written as follows: (,k2i .2i )m; + (,k2i + ,2i )Ami + (. Because the unsymmetric mass Ami is caused by the impellers and seals, and it is usually much smaller than the symmetric mass m, it is ignored here.Substituting eq. ( 9) into eq.( 14), the real part of the eigenvalues is obtained.Substituting the above expressions into eq.( 16), the log- arithmic decrement expressed as the sum of the individual where n B, ns, and n / are the total numbers ofthe bearings, seals and impellers, respectively.

Stability of Single-Stage Pumps
The calculation is based on the conditions in Table I.In the calculation, the bearing dynamic coefficients and im- peller dynamic coefficients of JSME 1984] and Ohashi and Shoji 1987] are used, while the dynamic coefficients of seals are obtained by the calculating method given in Iwatsubo and Sheng [1989]; [1990].
The loci of eigenvalues of eq. ( 8) for single-stage pump rotor systems with smooth, damper and parallel grooved seals are illustrated in Figure 3. Here, only the two eigen- values in or near to the unstable region (ti > 0) are given.In these figures, o90 is the first eigenfrequency of the ro- tor without seals and bearings.The numbers marked on   the loci are the ratios at rotating speed N to ao.In the present rotating region, the stability of the rotor system is dependent on the eigenvalue of root 1.Therefore, the stability is discussed as to this eigenvalue by means of the logarithmic decrement.
From the point of view of energy, the rotor system is re- leasing energy to the outside so that the system tends to sta- bilize with the lapse of time, if the logarithmic decrement is positive; while, if the logarithmic decrement is negative, the rotor system is absorbing energy from the outside so that the system enlarges the amplitude and tends to un- stablize with the lapse of time (Kurohashi et al. [1982]).
Figure 4 shows the logarithmic decrements of the bear- ings, seals, impellers and the total rotor systems.For this natural mode, the contribution of the impeller to the stabil- ity is very small.The logarithmic decrements of the seals are positive, which means the stable effect on the rotor stability, but the logarithmic decrements of the bearings are almost negative, which means an unstable effect on the rotor stability.
The logarithmic decrements of the rotor systems with smooth, damper and parallel grooved seals are shown in Figure 5.The results show that the logarithmic decrements ofrotor systems with smooth and damper seals are similar, and the stability of these systems are better than that of ro- tor systems with parallel grooved seals.In actual machin- ery, investigation of rotor stability with a specific rotating speeds or in a specific rotating region is usually necessary.The logarithmic decrements of rotor systems with differ- ent seals at specific rotating speed N/oo 1 are shown in Figure 6, where the mode shape of the rotor system is shown, too.These results are drawn in the form of bar graphs for the convenience of clarity.From these results the contribution of every fluid element to rotor stability can be immediately recognized.The rotor stability can be improved by replacing the unstable elements with stable elements.This will be discussed in next section.The influence of preswirl velocity Vt in seals on ro- tor stability is investigated, and the results are shown in Figure 7.This figure shows that positive preswirl velocity exerts an unstable influence on rotor stability; while neg-ative preswirl velocity exerts a stable influence on rotor stability.This result is in agreement with the individual re- search of seals (see Iwatsubo et al. 1989]; Iwatsubo and  Sheng 1989]; 1990]).

Stability of Multi-stage Pumps
The specification of multi-stage pumps is illustrated in Table II.The logarithmic decrements of rotor systems and fluid elements at N w0 19671.6 rpm are studied and shown in Figure 8, where the mode shapes of rotor systems are shown, too.In the investigation, some of seals or bearings are changed in order to improve rotor stability and find the interactions of these fluid elements.
In the case of (a), five parallel grooved seals are used in the rotor system.The total logarithmic decrement shows a negative value because of the unstable effects of the seals and the bearings.If the parallel grooved seal is replaced by a damper seal, the rotor system becomes sta- ble (b).Furthermore, by replacing all the grooved seals with damper seals, the rotor system becomes more stable (c).It is found that after the replacement of the seals, not only the logarithmic decrements of the seals but also those of the bearings are changed.If the circle bearings are re- placed with tilting pad bearings instead of the seals, as shown in (d), the rotor stability can also be improved.In this case, the logarithmic decrements of the seals are also changed.It is considered that the interactive variation of the fluid elements is caused by the variation of the natu- ral mode, because the logarithmic decrement is dependent on the natural mode, which has been demonstrated in the previous section.According to the above discussion, the interaction of other elements must be considered when evaluating the effect of the fluid element on rotor stability.

CONCLUSION
The present analysis supports the following conclusions: 1.In rotating fluid machinery, the investigation of the   Total effect of the individual fluid element on rotor stability is an effective method for the purpose of the evaluation of rotor stability and dynamic design.
rotor systems, c and Aki can be expressed as the sum of the fluid elements.
With 2 tilting pad bearings and S grooved seals

FIGURE 8
FIGURE 8 Logarithmic decrement of multi-stage pump.

TABLE Specification of
Single-Stage Pump