Stability and tracking analysis of gas face seals under low-parameter conditions considering slip flow

1. Introduction

A spiral groove gas face seal (Fig. 1) is a type of non-contacting face seals that prevents sealed gas from escaping from one compartment to another. A well designed gas face seal should have an appropriate dynamic property to head off excessive leakage and face contact [1, 2]. As depicted in Fig. 1, the dynamic system of a gas face seal is a typical mass-spring-damper system which consists of the nonlinear gas film stiffness and damping, the flexible support, and the mass and the transverse moment of inertia of the stator. Thus, the dynamic property of a gas face seal is about a systematic internal ability to handle ambient disturbances, which typically include axial endplay and angular wobble. The axial endplay of the rotor is caused by the axial endplay of the shaft, and the angular wobble of the mated rings is a result of manufacturing and assembly tolerances such as the runouts of both rings and the imperfection in the stator’s flexible support. When subjected to these disturbances, the stator may become completely unstable. Even in a stable state, an improper tracking property can still either induce excessive leakage or local face contact [3, 4]. It is therefore imperative to study the stability and tracking properties of gas face seals which have played important dynamic roles in other applications [5, 6].

Dynamic research on gas face seals has been performed using both direct numerical methods [7-11] and semi-analytical methods [12-17]. In such research, the stability and tracking properties of gas face seals act as important focal points. Green and Barnsby [8] used the state-space-form method developed by Miller and Green [9] to gauge the influence of sealing parameters on the stability threshold of coned gas face seals. The tracking property of the stator was also discussed by a parametric investigation [10]. Zirkelback and San Andres obtained the axial stiffness and damping coefficients of the gas film in a spiral groove gas face seal using a perturbation method [12], and subsequently presented a corresponding stability analysis [13]. Ruan [14] added angular motions to the dynamics to further develop the perturbation method, and presented a simple stability analysis. Meanwhile, Miller and Green developed a step jump method [15, 16] to provide a stability and tracking analysis of spiral groove gas face seals [17]. Because gas face seals are usually used in high-speed and high-pressure turbomachinery, all of the above works have been performed under high-parameter conditions.

Fig. 1Schematic of a typical spiral groove face seal and equivalent mass-spring-damper representation

With the continuing development of gas face seals, the applications of these seals are expanding to devices such as mixers and agitators, where the speed and pressure represent comparatively low-parameter conditions [18]. Such low-parameter conditions can even exist in high-parameter turbomachinery in scenarios such as debugging and engine warm up. In such applications, the continuum-flow assumption of the gas film breaks down and the rarefaction effect becomes significant. First- [19], second- [20], and 1.5-order [21] slip flow models have been developed to modify the classical continuum-flow Reynolds equation, accounting for rarefaction effects by allowing molecular slipping to occur at the gas-wall interface. Fukui and Kaneko developed a Boltzmann-Reynolds model for an arbitrary Knudsen number $K_n$ based on the linearized Boltzmann equation and provided a database of values in terms of the inverse Knudsen number $D$ [22]. Subsequently, they published a set of polynomials for the curve fitting of the above database [23]. These slip flow models have been used in the analysis of gas face seals. Pecht and Netzel [24] and Ruan [18] gauged the effect of slip flow on the steady-state performance of gas face seals including lift-off speed, leakage rate, frequency-independent gas film stiffness and load carrying capacity. Ruan [25] also considered slip flow in a study on the transient performance of spiral groove gas face seals during the startup and shutdown operations. However, few previous works have researched the stability and tracking properties of gas face seals under low-parameter conditions.

The objective of the present study is to provide an investigation of the effect of slip flow on the stability and tracking of spiral groove gas face seals under low-parameter conditions. In addition, the approach used to analyze stability and tracking properties can still be improved. In direct numerical methods [7-11], the stability and tracking properties are analyzed numerically, but this approach is inefficient for intensive parametric analyses. In the perturbation method [12-14], it is convenient to obtain expressions for the critical stability condition because of the existence of linearized gas film coefficients in the time domain. Yet, the tracking property is still characterized by axial relative displacement or angular relative tilt angle. In the step jump method [17], the transmissibility is presented in the form of an expression that can be used to characterize the tracking property. However, the step response, which acts as the constitutive model of gas film, is easy to analyze in the frequency domain [16] but difficult to analyze in the time domain. Therefore, a balanced approach is proposed that introduces the transmissibility into the perturbation method. Furthermore, because the transmissibility only reflects the tracking ability while ignores the space required for tracking motions, critical ambient-disturbance amplitude, above which the seal will achieve face contact, is provided based on the transmissibility to cover the defect of transmissibility in characterizing the tracking property.

2. Model

2.1. Gas film lubrication and sealing dynamics

A schematic diagram of a typical gas face seal is shown in Fig. 1. A dynamic schematic of the mated rings and the rotor geometry are depicted in Fig. 2. A total of $N$ spiral grooves are processed on the rotor face at a depth of ${\delta }_g$. $\alpha$ is the spiral angle and varies from 0 to 180°. The groove width $W_g$ relative to the land width $W_l$ is measured by $\lambda =W_g/(W_g+W_l)$, and the groove length relative to the dam length is measured by $\beta =(r_g-r_i)/(r_o-r_i)$. Here, $r_o$, $r_i$ and $r_g$ are the dimensionless outer radius, inner radius and boundary of the groove region and dam region, respectively, where $r_o=r_o^{*}/r_i^{*}$, $r_i=1$, and $r_g=r_g^{*}/r_i^{*}$. Here, superscript $*$ represents the dimensional form of a variable in the present study.

The gas pressure distribution $p^{*}$ in the gas face seal is governed by the lubrication equation. Assuming that the gas flow is ideal and isothermal, the compressible Reynolds equation is then given by [8-11, 14-17, 25]:

1

$\nabla \bullet \left[p^{*}{h}^{*3}\nabla p^{*}-6\mu \omega r^{*}{p}^{*}{h}^{*}{\mathbf{i}}_{\theta }\right]-12\mu \frac{\partial \left(p^{*}{h}^{*}\right)}{\partial t^{*}}=0,$

where $\mu$ is the gas viscosity, $r^{*}$ is the radius, ${\mathbf{i}}_{\theta }$ is the unit vector in the $\theta$ direction, $h^{*}$ is the gas film thickness, $\omega$ is the shaft speed, and $t^{*}$ is the time. The dimensionless form of Eq. (1) and its corresponding boundary conditions are given by:

2a

$\frac{1}{r}\frac{\partial }{\partial r}\left(rp{h}^3\frac{\partial p}{\partial r}\right)+\frac{1}{r}\frac{\partial }{\partial \theta }\left(\frac{p{h}^3}{r}\frac{\partial p}{\partial \theta }\right)=\mathrm{\Lambda }\frac{\partial \left(ph\right)}{\partial \theta }+2\mathrm{\Lambda }\frac{\partial \left(ph\right)}{\partial t},$

2b

$p_i\left(r=1\right)=1,\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{p}_o\left(r=r_o\right)=\frac{p_o^{*}}{p_i^{*}},$

where $r=r^{*}/r_i^{*}\text{,}$$h=h^{*}/{\delta }_g\text{,}$$p=p^{*}/p_i^{*}\text{,}$$t=\omega t^{*}$ and the compressibility number $\mathrm{\Lambda }=6\mu r_i^{*2}\omega /p_i^{*}{\delta }_g^2$. Also, $p_i^{*}$ and $p_o^{*}$ are the pressures at the inner and outer radii. Here, a flow factor $Q_p$ is incorporated into Eq. (2a) to modify the compressible Reynolds equation to account for the slip flow [18, 22, 23]:

3a

$\frac{1}{r}\frac{\partial }{\partial r}\left(Q_{p}rp{h}^3\frac{\partial p}{\partial r}\right)+\frac{1}{r}\frac{\partial }{\partial \theta }\left(Q_p\frac{p{h}^3}{r}\frac{\partial p}{\partial \theta }\right)=\mathrm{\Lambda }\frac{\partial \left(ph\right)}{\partial \theta }+2\mathrm{\Lambda }\frac{\partial \left(ph\right)}{\partial t},$

3b

$Q_p=6\left\{\begin{array}{ll}1/6+1.0162/D+1.0653/D^2-2.1354/D^3,& D\ge 5,\\ 0.13852+1.25087/D+0.15653/D^2-0.00969/D^3,& 0.15\le D\le 5,\\ -2.22919+2.10673/D+0.01653/D^2-0.0000694/D^3,& 0.01\le D\le 0.15.\end{array}\right.$

3c

$D=\frac{p_i^{*}{\delta }_g}{\mu \sqrt{2R_{gas}{T}_a}}ph,$

where $R_{gas}$ is the gas constant, and $T_a$ is the ambient temperature.

The dimensionless rotor tilt angle ${\mathbf{\gamma }}_r$, the stator tilt angle ${\mathbf{\gamma }}_s$ and the relative tilt angle ${\mathbf{\gamma }}_{rel}$ can be decomposed into components along the $X$ and $Y$ axes:

4

${\mathbf{\gamma }}_r=\left({\gamma }_r\cos t,{\gamma }_r\sin t\right),{\mathbf{\gamma }}_s=\left({\gamma }_x,{\gamma }_y\right),{\mathbf{\gamma }}_{rel}=\left({\gamma }_r\cos t-{\gamma }_x,{\gamma }_r\sin t-{\gamma }_y\right),$

where ${\mathbf{\gamma }}_r={\mathbf{\gamma }}_r^{*}{r}_i^{*}/{\delta }_g$, ${\mathbf{\gamma }}_s={\mathbf{\gamma }}_s^{*}{r}_i^{*}/{\delta }_g$ and ${\mathbf{\gamma }}_{rel}={\mathrm{\gamma }}_{rel}^{*}{r}_i^{*}/{\delta }_g$. The gas film thickness $h$ at any point ($x$, $y$) on the stator face is given by:

5

$h\left(x,y\right)=c_0+z-z_r+y{\gamma }_x-x{\gamma }_y-{\gamma }_{r}y\cos t+{\gamma }_{r}x\sin t+〈1〉,$

where $c_0$ is the dimensionless central film thickness at equilibrium, $z$ is the dimensionless axial displacement of the stator from $c_0$, $z_r$ is the dimensionless axial endplay of the rotor, and $〈1〉$ indicates a dimensionless groove depth in the groove region. Here, $c_0=c_0^{*}/{\delta }_g$, $z=z^{*}/{\delta }_g$ and $z_r=z_r^{*}/{\delta }_g$. By considering the stiffness and damping properties of the springs and the secondary seal, the dimensionless dynamic equations for the axial and angular modes are given by [9, 14, 17, 25]:

6

$\left\{\begin{array}{l}m\ddot{z}+c_{sz}\dot{z}+k_{sz}z=F_g-F_{closing},\\ I{\ddot{\gamma }}_x+c_{s\gamma }{\dot{\gamma }}_x+k_{s\gamma }{\gamma }_x=M_{gx}+M_{si},\\ I{\ddot{\gamma }}_x+c_{s\gamma }{\dot{\gamma }}_y+k_{s\gamma }{\gamma }_y=M_{gy}.\end{array}\right.$

In Eq. (6), $m$ is the dimensionless mass, and $I$ is the dimensionless transverse moment of inertia of the stator, where $m=m^{*}{\delta }_g{\omega }^2/p_i^{*}{r}_i^{*2}$ and$I=I^{*}{\delta }_g{\omega }^2/p_i^{*}{r}_i^{*4}$. $k_{sz}$ is the dimensionless axial stiffness of the springs and $c_{sz}$ is the dimensionless axial damping of the flexible support, where $k_{sz}=k_{sz}^{*}{\delta }_g/p_i^{*}{r}_i^{*2}$ and $c_{sz}=c_{sz}^{*}{\delta }_g\omega /p_i^{*}{r}_i^{*2}$. According to the work of Green and Etsion [26], the dimensionless angular stiffness $k_{s\gamma }$ and the damping $c_{s\gamma }$ are given by:

7

$k_{s\gamma }=\frac{1}{2}{k}_{sz}{r}_o^2,c_{s\gamma }=\frac{1}{2}{c}_{sz}{r}_o^2,$

assuming that the support forces act at the outer radius. $F_{closing}$ is the total load of the fluid pressure and the spring support load at equilibrium, which is applied to the rear of the stator to balance the gas film load $F_g$ caused by the fluid effect. $M_{gx}$ and $M_{gy}$ are the gas film moments about the $X$ and $Y$ axes. $M_{si}$ is a moment caused by the dimensionless static stator tilt ${\gamma }_{si}$, which is arbitrarily assumed to be oriented about the $X$ axis. The constant moment is given by $M_{si}=k_{s\gamma }{\gamma }_{si}$. $F_g$, $M_{gx}$ and $M_{gy}$ can all be obtained by integrating $p$ over the dimensionless sealing area $A_n$, as follows:

8

$F_g={\int }_{A_n}^{}pdA,M_{gx}={\int }_{A_n}^{}pydA,M_{gy}=-{\int }_{A_n}^{}pxdA.$

Fig. 2Dynamic schematic of a spiral groove gas face seal and the geometry of the rotor