Open access peer-reviewed chapter - ONLINE FIRST

Groove Shape Optimization on Dry Gas Seals

Written By

Masayuki Ochiai and Yuki Sato

Submitted: November 5th, 2021Reviewed: February 7th, 2022Published: May 10th, 2022

DOI: 10.5772/intechopen.103088

IntechOpen
Tribology of Machine Elements – Fundamentals and ApplicationsEdited by Giuseppe Pintaude

From the Edited Volume

Tribology of Machine Elements – Fundamentals and Applications [Working Title]

Prof. Giuseppe Pintaude, Associate Prof. Tiago Cousseau and Associate Prof. Anna Rudawska

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Abstract

In this paper, a topological optimum design for the shape of a groove in a dry gas seal is described. Dry gas seals are widely used in high speed and high pressure rotating machinery such as gas turbines, compressors, and so on because of their high reliability compared to other types of seals. However, recent requirements for reducing emission with further control of leakage are in order. With this background, we propose applying topological optimization to the groove shape in a dry gas seal to reduce its leakage while keeping its stiffness for safe operation. First, the method of topological optimum design as applied to the groove of a dry gas seal is explained via numerical analysis. Next, results of the topological optimization are shown via categorizing an optimum shape map. Finally, the mechanism of reducing the gas leakage with an optimized seal is discussed based on the prediction of the flow field using a CFD analysis.

Keywords

  • mechanical seals
  • dry gas seals
  • groove shape
  • optimization
  • gas lubrication

1. Introduction

Non-contact dry gas seals with a grooved pattern on a seal face can maintain a film thickness of just a few micrometers. Therefore, these seals have better sealing performance when compared to typical labyrinth seals [X]. Dry gas seals are used in many turbomachinery, such as in gas and steam turbines, turbochargers, and compressors. Moreover, they are applied to high-speed operation and under high-pressure differences.

Recently, to reduce energy consumption, more enhancements toward efficient turbomachinery are required. To solve this problem, one effective way is by enhancing the sealing characteristics of seals. Many types of grooved dry gas seals have been developed [1]. Spiral grooved seals are widely used because of their good sealing ability. Lately, a significant amount of research on spiral grooved dry gas seals focused on analytical methods [2, 3, 4, 5, 6, 7, 8], dynamic force characteristics [9, 10, 11], thermal effects considerations [12], and CFD analysis considering the turbulent flow [13] have been performed.

On the other hand, the optimum design of the grooves is one of the effective ways to enhance the seal characteristics. The optimum design methods have been also applied to gas film bearings. Lin and Satomi [14] and Hashimoto and Ochiai [15, 16] applied an optimum design method to spiral groove thrust bearing towards enhancing performance characteristics from variations in groove depth, groove angle, and so on. Moreover, an experimental verification was conducted comparing the novel configuration against a conventional designed spiral groove bearing. However, it was found that the effectiveness of the optimization is limited because these studies have not been changed the groove shapes which were based on a spiral path.

Under this circumstance, Hashimoto and Ochiai [17] proposed a topological optimum design method for a grooved thrust gas bearing. In this method, the groove shape could be changed freely using a cubic spline function. Novel groove shapes were found in this study. The effectiveness and the applicability of the method were verified theoretically and experimentally. Moreover, Hashimoto and Namba [18] found the best groove shapes against various objective functions such as film thickness, friction torque, and dynamic axial stiffness. Also, the effect of the new groove shape on sealing characteristics of FDB(Fluid dynamic bearing) was studied previously and discussed by authors [19].

To date, many researchers have treated spirally grooved shape dry gas seals. On the other hand, recently, the optimum design of groove shape on the dry gas seal was proposed by authors, and comparison of the flow visualization was presented [21]. However, the process of the optimum design has not been mentioned and also it has not been studied for a wide range of operation conditions. Therefore, in this study, the application of the topological optimum design to the dry gas seal instead of the thrust bearings to find an optimum groove shape that enhances the seal leakage restriction and its dynamic stiffness is presented. Moreover, it is important to know the optimum groove shapes under various conditions, therefore, in this study, we tried to make a categorization map of the seal’s optimum shape based on the results of the optimum design calculations under a wide range of operating conditions. Furthermore, CFD analysis is conducted and compared with the experimental flow visualizations for verification, while the rationale for reducing the gas leakage with an optimized seal is presented.

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2. Topological optimization methods

Figure 1 shows the typical structure of a dry gas seal cartridge. It consists of a rotating shaft, a ring with grooves on its face, a stationary ring, support springs, and housing. The gas film is generated by the hydrodynamic effect induced on the grooves of the face. The film thickness is determined by the force balance between the support springs and the hydrodynamic gas film force. The film thickness can be changed by changing the support springs. The seal leakage is a function of the film thickness, the gas pressure differential between the inner side and outer side of the seal chamber, the viscosity of the gas, and the groove shape mounted on the face.

Figure 1.

Components of a non-contacting dry gas seal.

In the design of dry gas seals, it is important to minimize the gas leakage towards enhancing the efficiency of turbomachinery. Simultaneously, enhancing the dynamic stiffness of a gas film is an important factor for its safe operation, at high speed in particular. Because turbomachinery is likely to be exposed to some outer disturbance such as earthquakes, a hard contact of the rotor on the seal surface leads to serious damage to the mechanical system.

Both a low gas leakage and a high gas film stiffness are trade-off relations, being difficult to optimize both parameters at the same time. Therefore, in this study, sufficient stiffness is selected for safety. The whole structure of the dry gas seal with the gas film is modeled as spring and damper as shown later. Therefore, from the calculation of a linear vibration waveform, the minimum film thickness is obtained. Under the conditions presented in Table 1, Ref. [19], the required gas lubricated film stiffness is defined. Because the leakage rate is strongly affected by film thickness, the value is fixed as 5 μm in this optimization as shown in Table 1.

ParameterValues
Stator mass m1.0 kg
Support spring k5.0 × 105 N/m
Steady-state clearance cr5 μm
Assumed disturbance f5 G
Viscosity of the air1.82 × 10−5 Pa·s
Compressibility number
Λ=(6μΛ/Pa)*(r1/hr)2
100–750
Outer side pressure PO0.5–10 MPa
Inner side pressure PI0.1 MPa

Table 1.

Dry gas seal physical parameters.

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3. Optimum design formula

The optimization method in this study is based on Hashimoto and Ochiai’s topological optimum design theory [17]. The outline of the method is as follows. The initial groove geometry is the usual spiral groove shape, and then, cubic spline interpolation functions are applied to the initial geometry with 4 grids. Moving the grids on the same circumferences changes the groove shape. Applying the optimum design method, an optimized seal groove shape is obtained. Simultaneously, the number of grooves N, the seal radius Rs, the groove depth hg, and the groove width ratio αare set as the optimized design valuables in this study. Therefore, the design vector of parameters is

X=φ1φ2φ3φ4NRshgαE1

where, the ϕ1 to ϕ4 mean the angles from an initial spiral groove shape, shown in Figure 2a.

Figure 2.

Geometry of a seal and optimum design variables.

In the optimum design, the objective functions should be defined. Obviously, the most important one is to minimize the leakage q. Therefore, the objective function is set as

f1X=qE2

Moreover, even if a lesser leakage design is available, it is impractical to have a lesser dynamic stiffness simultaneously. Since dry gas seals are usually used under high speed and high-pressure differential conditions, sudden contact on the seal faces may lead to serious accidents. Therefore, the dynamic stiffness Kmust be used as the second objective function,

f2X=KE3

The constraint relationships in this optimization are

giX0i=119E4

where

g1=φ1minφ1,g2=φ1φ1max,g3=φ2minφ2,g4=φ2φ2max,g5=φ3minφ3,g6=φ3φ3max,g7=φ4minφ4,g8=φ4φ4max,g9=NminN,g10=NNmax,g11=RsminRs,g12=RSRsmax,g13=hgminhg,g14=hghgmax,g15=αminα,g16=ααmax,g17=c,g18=k,g19=WE5

The g1to g16indicate the upper and lower limit of the design variables and the g17is to avoid negative damping.

The optimum design problem is formulated as

FindXtominimizef1Xandmaximizef2X
subjected togiX0i=119E6
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4. Calculation method of seal characteristics

The analysis method to calculate the seal characteristics is shown below. During the optimum design calculations, the groove shape should be changed continuously from its original spiral groove shape into other shapes. Therefore, a boundary-fitted coordinate system is adopted as the numerical calculation method [15]. Moreover, a divergence formulation method is implemented. A Reynolds equivalent equation obtained from flow balance as shown in Figure 3 is used to obtain the pressure distributions on the seal face. This is because the geometry has a step over which there is a discontinuous pressure gradient between the groove and the land areas.

Figure 3.

Control volume and flow rates.

The Reynolds equivalent equation [16] is

Q2Iξ+Q1IIIξQ2IIξQ1IVξ+Q2Iη+Q1IIηQ2IIIηQ1IVη=QΓE7

Subscripts 1, 2, and Ito IVindicate the areas in the control volume shown in Figure 3.

where the mass flow rates through the various boundaries are

Qξ=η1η2ρA0pξ+B0pη+D0+E0E8
Qη=ξ1ξ2ρB0pξC0pη+F0+G0E9
QΓ=ξ1ξ2η1η2ρhtJ0dηdξE10

The coefficients of Aoto Joare

Ao=aoh312μJ,Bo=boh312μJ,Co=coh312μJ,Do=rωsh2rη,Eo=ρrωs2h340μrθη,Fo=rωsh2rξ,Go=ρrωs2h340μrθξ,ao=rθξ2+rη2,bo=rθξrθη+rξr2,co=rθξ2+rξ2,Jo=rξrθηrηrθξ,rξ=rξ,rη=rη,rθξ=rθξ,rθη=rθηE11

Assuming a small amplitude vibration of the seal with frequency ωf, the gas film thickness and pressure are expressed as follows.

h=h0+εejωftp=p0+εptejωftE12

where, εis an amplitude of vibration and p0and ptexpress a static component and dynamic pressure components, respectively.

Substituting Eq. (12) into Eq. (7) and neglecting seconds terms of ε, the following equations regarding the 0th field and 1st field εare obtained.

F0p0=Q2I0ξ+Q1III0ξQ2II0ξQ1IV0ξ+Q2I0η+Q1II0ηQ2III0ηQ1IV0ηE13
Ftptp0=Q2Itξ+Q1IIItξQ2IItξQ1IVtξ+Q2Itη+Q1IItηQ2IIItηQ1IVtηE14

Discretizing Eqs. (13) and (14), and then solving the equations numerically, the static and dynamic components of the gas pressure fields are obtained. Finally, the gas leakage rate qis calculated from the static pressure distributions as

q=02πρh312μp0rr=rirdθE15

Moreover, assuming the simple vibration model of a dry gas seal shown in Figure 4, the dynamic stiffness Kis obtained from the calculated dynamic pressure distributions as,

Figure 4.

Simple vibration model of the dry gas seal.

K=k2+ωfc2E16
k=02πriroReptrdrdθE17
c=02πriroImptrdrdθE18
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5. Topological optimization results

Using the method mentioned above, topological optimum calculations were conducted. The calculation conditions are shown in Tables 1 and 2, and Figure 2. As shown in Table 1, the mass of stator m= 1.0 kg, the support spring k= 5.0 × 105 N/m, the steady-state clearance cr= 5 μm are defined in the reference literature [19]. The assumed disturbance of f= 5G is defined from the magnitude of earthquakes. The Compressibility number Λand outer side pressure P0are set under a wide range of operation conditions. Because we would like to find the general optimum shape of the seal groove under various operating conditions, not a limited condition. The parameters for optimum design calculations are set as shown in Table 2. These set values are defined by representative of the dry gas seals. The groove numbers are discrete values. The other optimum design variables of the groove depth, the angle amount, the groove width, the inner and outer radius are continuous values, and their maximum and maximum values are set as concentration conditions.

ParametersValues
Groove number6,8,10,12,14,16,18,20,22,24
Minimum groove depthhg min= 5 μm
Maximum groove depthhg max= 10 μm
Minimum angle amountϕi min= −π (i = 1–4)
Maximum angle amountϕi max= π (i = 1–4)
Minimum groove widthαmin= 0.4
Maximum groove widthαmax= 0.9
Minimum seal radius to outer radius ratioRs min= 0.86
Maximum seal radius to outer radius ratioRs max= 0.95

Table 2.

Parameters for optimum design study.

By solving the above optimum design problem, a multi objective genetic algorithm is used as this in a multi objective optimization [20].

Figure 5 shows the optimization results for operation with a compressibility number Λ= 500 and inlet pressure Po= 2.5 MPa. The vertical axis shows the gas leakage flow rate as the 1st objective function and the horizontal axis shows the dynamic stiffness as the 2nd objective function. In this figure, the red line at the bottom part denotes the Pareto solution curve. Here, the Pareto result means the cloud of optimum solutions from the multi-optimization results. From this result, it is confirmed that there is trade-off relations between the objective functions of gas leakage and the dynamic stiffness. The leakage minimized seal has less dynamic stiffness, whereas the dynamic stiffness maximized seal has inferior sealing characteristic. Therefore, in this study, the allowable dynamic stiffness which means avoiding the contact of the seal surface against the outer disturbance as criteria is set. Because the gas leakage should be reduced as much as possible under a safe operation. Comparing the gas film thickness and the linear impulse response by using the vibration model as shown in Figure 4, we can recognize whether the contact occurs or not by outer disturbance. The critical dynamic stiffness can be defined from the criteria. Moreover, in this study, we defined the allowable dynamic stiffness adopting a safety factor of 3 which means three times of the critical dynamic stiffness. In this manner, the optimized groove shapes as shown in Figure 5 were obtained and the characteristic values of optimized seals are shown in Table 3.

Figure 5.

The case of Pareto optimum solutions (Pi = 2.5 MPa, Λ = 500).

Seal shapesLeakage flow rate q(kg/s)Dynamic stiffness K(MN/m)
Spiral groove24.9 × 10−5177
Maximum stiffness26.9 × 10−5286
Minimum leakage18.8 × 10−528.9
Optimized geometry18.9 × 10−530.5

Table 3.

Characteristic values.

The initial shape of the spiral groove seal labeled (A) does not have the desired characteristics of both low gas leakage and high dynamic stiffness. Comparing the shapes of (A) through (D), from the point of view of minimizing the gas leakage, the shape of the groove is quite different from the initial spiral groove as shown in Figure 5B. The optimized shape has a bending curve in the vicinity of the outer diameter of the seal face. On the other hand, from the viewpoint of maximizing the dynamic stiffness, the shape of the groove, as shown in Figure 5C is similar to the spiral groove shape in Figure 5A. This is because a high positive dynamic pressure is required. It is well known that the spiral groove shape can effectively generate high positive pressure.

Thus, considering an allowable dynamic stiffness, the optimized shape as shown in Figure 5D is similar to the shape that minimizes gas leakage with a bending curve. However, the length of the bending curve is no longer that of the leakage minimized seal. This is due to gas flow around the outer vicinity of the gas seal face. The gas flow from the outer high pressure is retarded by the effect of the curved shape of the grooves. From these results, the most interesting thing is that quite a different shape is obtained for the case reducing gas leakage only. However, the results are valid only for the case of Λ= 500 and inlet pressure Po= 1.0 MPa.

Figure 6 depicts the tendency of change in the shape of the dry gas seal face on the Pareto optimum solution. Orienting the low leakage design, the strong bending shape in the outer vicinity and the wide plane region in the inner side are obtained. This bending shape reduces the leakage to the inner side of the seal by pump-out effect from the inner to the outer circumference side. On the other hand, emphasizing the stiffness design, it is found that the bending tendency goes weak and finally the shape goes to the spiral shape gradually.

Figure 6.

Change in optimum shape tendency of the dry gas seal face.

From the point of view of the actual seal design, a wider range of operations is required. Therefore, the optimum design calculations were conducted over a wide range of conditions Λ= 100–750, inlet pressure Po= 1.0–10.0 MPa. The inlet pressure is the most important operating condition in a dry gas seal design. On the contrary, the range of compressibility numbers encompasses many operating conditions of rotational speed, film thickness, gas viscosity, and size of seal.

Figure 7 depicts the optimized shape map for a wide range of inner static pressure at the outside diameter and compressibility numbers. There are three types of shapes, one is quite similar to the spiral groove shape and applicable to a low inlet pressure range of Po= 1.0–4.0 MPa and low compressibility number range ofΛ=10 to 350. Another shape has a slight bend curve like in Figure 5B in the range of Po=1.0 to 4.0 MPa and Λ=350 to 750. The other is a shape having a strong bending curve like in Figure 5D for an inlet pressure range of Po= 4.0–10.0 MPa and over the whole range of Λ.

Figure 7.

Optimal design map under a wide range of conditions.

From the results, in the case of low inner static pressure conditions and a low compressibility number (Λ< 350), shown in the green color area, it is a required feature of a dry gas seal to enhance its dynamic stiffness. This is due to the ability to generate a dynamic positive pressure on the seal face. Under the conditions of low inlet pressure and high compressibility number, shown in the blue color area, large film thickness, and low viscosity, etc., it is difficult to generate high dynamic pressure on the seal face. Therefore, the grooves are formed to maximize the seal dynamic stiffness.

On the other hand, for a high inlet pressure or a high compressibility number condition, shown in the red area, the allowable film stiffness could be obtained easily as its basic ability. Because the high inlet pressure condition is expected to deliver a hydrostatic effect and the high compressibility number leads to an enhancement of the hydrodynamic effect. Hence, the main object of topological optimization is to reduce gas leakage. However, for a low inner static pressure condition, the hydrostatic effect is not expected. Therefore, the bending curve shape is weak. In other words, it is found that the topological optimization for reducing gas leakage is effective in the case of a high inner static pressure condition.

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6. CFD analysis of visualization of the flow and discussions

In order to consider the mechanism for reducing the gas leakage of the optimized shape, which is the interesting bending shape, a CFD analysis of the gas flow was conducted using commercial software (ANSYS FLUENT) which can solve the Navier-Stokes equation including the flow of outer side area of dry gas seal and considered to be obtained more accurate solution compared to usually used Reynolds equation, which is neglecting the outer side flow of seals. In the past work of Hashimoto[18], a similar bending shape is obtained in the case of maximizing the bearing stiffness on a high-speed air bearing. However, as mentioned in the previous sections, another tendency is obtained in this case. That is, the bending shape is obtained in the case of minimizing the air leakage instead of maximizing the stiffness. Therefore, the reason why this shape is obtained is unclear.

Figure 8 and Table 4 report the CFD calculation model and the specifications respectively. The inner and outer radii are same as our experimental equipment [21], Moreover, the groove depths and seal clearance of the seals are 60 μm and 30 μm respectively because of their mesh size limitations. The seal radius ratio (Rs/Ro) and Groove width ratios are chosen by representative values for each seal. In addition, Table 4 indicates the calculation conditions of CFD analysis. The inlet pressure, it means the outer side of the dry gas seal, is set as 0.11 MPa, and the rotational speed is set as 5000 rpm. These values are the same as the previous experiment. The calculations are conducted under the area of one groove pattern by using a periodic boundary condition. In addition, the calculation does not use the turbulent model and concludes choked flow. Because the Reynolds number of the gas seal flow is approximately Re= 26, where the representative length is the clearance 30 μm, the representative speed is peripheral speed at an outer radius of 20 m/s. This setting reduces calculation costs. Consequently, the calculation area sizes are not identical (Table 5).

Figure 8.

CFD analysis model.

Spiral grooveOptimized groove
Outer radius Ro32.0 mm
Inner radius Ri25.6 mm
Groove depth60 μm
Seal Clearance30 μm
Seal radius ratio Seal radius ratio (Rs/Ro)0.50.4
Groove width ratio0.86 μm0.93 μm
Number of groove1024

Table 4.

Seal specifications of CFD analysis.

Operating conditions
Inlet pressure0.11 MPa
Outlet pressure0.1 MPa
Rotational speed5000 rpm
Air temperature300 K
Dynamic viscosity of air1.85 × 10−5 Pa·s

Table 5.

Calculation conditions of CFD analysis.

Figure 9 shows the predicted (I) pressure distribution and (II) velocity distribution from the CFD analysis on the middle plane of gas film thickness comparing the conventional spiral grooved seal(a) versus the optimized seal(b). In this study, the film thicknesses are common. Therefore, the closing forces are different. In addition, the visualization areas are different in the two seals because the calculation area depends on the groove shape intervals.

Figure 9.

CDF analysis results of pressure and velocity distributions on grooved seals.

From the results in Figure 9(I), high pressure is generated on the outer region of the seal caused by the hydrostatic effect. However, the high-pressure area in the optimized seal is narrow compared with that of the spiral grooved seal. Moreover, the velocities on the flow in the optimized seal are faster than those in the spiral grooved seal. This is due to groove shape in the outer radius vicinity. The groove shape of the spiral groove is formed along the rotational direction. Consequently, outer air is drawn into the seal and the air velocity is fast. On the other hand, with the optimized seal face, the gas flow velocity in the outer vicinity reduces because the inflow is suppressed by the pump effect of the bending shape groove. As mentioned earlier, reducing the gas inlet flow velocity on the optimized seal face leads to reduce the gas overall leakage.

Moreover, comparing both the Reynolds equation and the CFD results of load-carrying capacity and amount of leakage, are shown in Table 6. As shown in the Table, the load-carrying capacity is in very good agreement with both results. On the other hand, the amount of leakage, there is a little difference in both analytical solutions. This is because the amount of leakage is calculated using pressure difference and it is easy to include the numerical error. Besides, the load-carrying capacity is calculated by the integration of pressure distribution. Therefore, it is considered that the numerical error is very small. However, the difference in the amount of leakages is acceptable.

Reynolds Eq.CFD
Load carrying capacity (kg)Spiral0.790.76
Optimized0.760.71
Amount of leakage (10−5 kg/s)Spiral7.878.19
Optimized5.705.81

Table 6.

Comparison of Reynolds equation to CFD.

Finally, the experimental verification of flow visualization is mentioned. The experimental visualization results are picked up from our past research work [21, 22].

The experimental conditions are same as Tables 4 and 5, except the groove depth of 70μm and the seal clearance of 50μm. Here, the main purpose of the verification is to confirm the qualitative flow difference, therefore we think the comparison is meaningful even if the values between the CFD and experimental visualization are not the same. The specific visualization setup and spec are shown in the previous studies.

Figure 10 depicts the experimental visualization results of our previous study [21]. The velocity distributions are shown as color arrows. The outer side gas flows strongly into the spiral groove seal face through the boundary as shown in Figure 10a. On the other hand, in the case of optimized seal, the flows are very weak compared to that of spiral groove seal. The same tendencies are shown in the CFD analysis results, and the applicability of the optimization was verified experimentally.

Figure 10.

Experimental visualization results [21].

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7. Conclusions

In this study, a topological optimization of the groove shape on a dry gas seal is conducted to improve its sealing characteristics. The main conclusions are as follows:

  1. The groove shape of the topological optimum design to minimize gas leakage has a bending curve near the outer radius of the rotating seal face. On the other hand, the optimum groove shape when maximizing the gas film stiffness becomes quite similar to that of a spiral grooved shape.

  2. For the purpose of obtaining a workable solution over a wide range of operating conditions, an allowable gas film stiffness is adopted. As a result, the optimum shape pattern is similar to that of the spiral groove under conditions of low inner static pressure and low compressibility number. For high inner static pressure and high compressibility number conditions, the outer groove shape bends. The tendency of bending becomes stronger with an increase in the inner static pressure at the outside diameter and the compressibility number.

  3. CFD analysis reveals that the inflow velocity in the optimized seal is low compared with that in a conventional spiral groove seal. The newly found outer bending curve shape of the groove leads to suppress the inflow. Moreover, the same tendency is shown in experimental visualization.

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Acknowledgments

We would like to express our sincere gratitude to Professor Hiromu Hashimoto for his appropriate suggestions, Professor Luis San Andres for his polite advice, and all the students who have supported this research.

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Nomenclature

aa parameter related to the inflow angle β used to define a spiral curvature
b1width of groove [m]
b2width of land [m]
cdamping coefficient of gas film [N·s/m]
f(X)objective function [N/m]
gi(X) (i =1∼2n+2)constraint function
hggroove depth [m]
hrgas film thickness [m]
kspring coefficient of gas film
k1spring coefficient of support spring
Nnumber of grooves
nsshaft angular speed [rpm]
p0static component of gas film pressure (absolute pressure) [Pa]
paatmospheric pressure at inside diameter[Pa]
PIinner side pressure [Pa]
POouter side pressure [Pa]
ptdynamic component of gas film pressure [Pa]
qleakage gas mass flow rate [kg/s]
rcoordinate of radial direction [m]
riinside radius of seal [m]
rooutside radius of seal [m]
rsinner radius of the grooves [m]
Rsseal diameter ratio (= rs /ro)
Rrratio between inside radius and outside radius of seal (=r i /r o)
Xvector of variables used in calculations
αgroove width ratio =b1 /(b1+ b2 )
βinflow angle [rad]
Δrequipartition space of r[m]
θcoordinate of circumferential direction [rad]
Θiangle of basic geometry (spiral curvature) at the ith nodal point [rad]
ϕiextent of angle change from basic geometry (spiral curvature) at the ith nodal point [rad]
δϕIextent of angle change during optimization at the ith nodal point [rad]
Λcompressibility number = 6μω s /Pa)*(r1/hr)2
μviscosity of gas [Pa·s]
ρdensity of gas [kg/m3]
ξcoordinates of change based on boundary fitted coordinate system [m]
ηcoordinates of change based on boundary fitted coordinate system [rad]
ωfangular velocity of squeeze motion [rad/s]
ωsangular velocity of shaft rotation [rad/s]
maxmaximum value of state variables
minminimum value of state variables

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Written By

Masayuki Ochiai and Yuki Sato

Submitted: November 5th, 2021Reviewed: February 7th, 2022Published: May 10th, 2022