Dynamic Models of Mechanical Seals for Turbomachinery Application

Abstract

One of the primary causes of mechanical face seal failure is rotor vibration. Traditional dynamic seal models often cannot fully explain failure mechanisms. The dynamic models of seals proposed in this paper, including those developed by the authors, are valuable for predicting seal dynamics during operation in specific turbomachinery and for explaining the causes of seal failure. The single-mass dynamic model is suitable for analyzing the dynamics of contact mechanical face seals and simply designed dry gas seals. The two-mass dynamic model is used to investigate the operational dynamics processes of classical dry gas seals under complex loading conditions. The three-mass dynamic model is used to study various complex types of mechanical face seals. This model can determine the normal operating condition range and explain leakage mechanisms in the presence of excessive rotor vibrations.

1. Introduction

In designing various mechanical face seals (Figure 1), it is critically important to consider the balance of the axial forces, bending moments, and thermal balance of the sealing rings. The balance of axial forces determines the nominal sealing gap, which in turn affects seal leakage and friction power. The roughness parameters of the sealing faces must also be taken into account. The balance of bending moments is particularly critical for seals operating under high-pressure drops and those employing a graphite sealing ring, given graphite’s relatively low Young’s modulus (9–17 GPa). A balanced bending moment ensures the sealing faces remain plane-parallel or maintain a specific angular distortion. Thermal balance is necessary for seals operating under conditions of high-pressure drops (more than 10 MPa) and high rotational speeds (more than 15,000 RPM) in turbomachines, as well as when there is a significant temperature drop (more than 100 K).

A dynamic analysis of seal performance is necessary during the design process of turbomachinery rotor face seals and when investigating emergency stoppages in turbomachinery caused by the loss of seal tightness [1]. One of the most common causes of seal failure is increased rotor vibration [2]. Bo Ruan demonstrated the importance of the dynamic investigation of a dry gas seal to determine whether the stator ring can track the displacement of rotor axial vibrations using the small perturbation method [3]. Green and Varney investigated the effect of stator and rotor ring impact on the dynamic performance of a dry gas seal to predict seal failures resulting from rotor axial vibrations [4]. Traditionally, computational analysis of seal dynamics is conducted, followed by frequency tuning if required. This tuning may involve adjusting the mass of the seal rings, the stiffness of elastomer O-rings, or the fluid film’s (lubricating film’s) stiffness in the sealing gap. If the issue does not have a direct and clear solution, it becomes necessary to provide the required damping of the fluid film in the sealing gap. In such cases, two problems are involved in determining the dynamic performance of the seal: computational fluid dynamic analysis of the fluid flow in the gap and the motion of the seal rings. These two problems must be considered together.

Most of the research dedicated to sealing technology, especially dry gas seals, focuses on the gas-dynamic calculations of the fluid flow in the seal gap (lubricating film). These analyses aim to determine the pressure distribution in the gap and the leakage mass flow rate, which depend on the geometric parameters of the seal rings and face grooves. To solve this problem, methods based on the solution of the Reynolds equation are used [5]. In the absence of external disturbances to the fluid flow in the seal gap, laminar flow is established [6]. A more accurate solution to this problem can be achieved by directly solving the Navier–Stokes equation or the Reynolds-averaged Navier–Stokes equation using turbulence models [7,8]. Taking into account slip flow will further improve the accuracy of simulation results [5,9,10,11,12]. Moreover, considering heat transfer between the fluid in the gap and the seal rings has a significant impact on pressure distribution and leakage rates [9,13].

The problem of seal ring motion, or seal dynamics, comes down to accurately determining the dynamic properties of the gas film (stiffness and damping coefficients), the dynamic properties of its elastomeric O-ring, and selecting an appropriate dynamic model for the seal. The fluid film in the seal gap can be represented as stiffness and damping coefficients, which for liquids are determined by the viscosity and geometric parameters of the seal rings [14,15], and for gases also depend on the frequency and amplitude of vibration [16,17]. For a liquid–gas fluid, it is additionally necessary to consider the influence of the cavitation effect [18]. The stiffness and damping coefficients of the gas film are usually determined for the equilibrium gap value [19,20,21,22]. In order to obtain more accurate results of the seal dynamic calculation, nonlinear gas film stiffness and damping dependencies should be taken into account [6,23]. It should be noted that there are few publications focusing on the determination of O-ring stiffness and damping under dynamic loading, despite the fact that the dynamic properties of the elastomeric O-ring can significantly affect the dynamic properties of the seal [24]. A number of researchers have used the experimental data obtained by Green and Etsion [25].

Typically, a simplified single-mass dynamic model is employed to describe the stator seal ring motion [10,26]. The emphasis is on determining the dynamic characteristics of a thin fluid film in a non-contact mechanical face seal. Traditionally, the method of small perturbations method is used to solve dynamic problems. This method allows for the obtaining of analytical solutions with acceptable simplifications. Thus, the influence of the geometrical parameters of the seal on its performance can be determined relatively easily and in a short time [27,28,29,30,31,32]. In particular, this method has been used to explain the operation of a mechanical dry gas seal with a gap value of 2 µm, even when axial rotor vibration is present with frequencies of up to 100 Hz and amplitudes of up to 0.3 mm [16]. To reduce computation time and increase accuracy in solving dynamic problems, Miller and Green proposed the step jump method and the direct numerical frequency response method [33]. The accuracy of determining the dynamic characteristics of the fluid film in the seal gap can also be improved by taking into account the influence of deformation of the seal ring faces and by considering the problem of fluid flow in the seal gap using transient analysis [34,35]. Especially important is thermal deformation, since the deformation of the sealing ring is mainly caused by the influence of temperature [36]. Both high-temperature and low-temperature fluid (150 K) significantly affect the seal opening force and leakage [37]. A further improvement in the accuracy of the dynamic characteristics of the fluid film in the seal gap is achieved by taking into account the influence of roughness, texturing, and manufacturing methods of grooves on the faces of the seal ring [38,39].

The cited papers do not cover the whole range of existing seal designs but primarily focus on improving the shape and form of rotor ring grooves, including a novel intelligent optimization technique [40]. However, the issue of seal failure still exists, and the existing techniques do not completely resolve the problem of reliable seal operation in the presence of a complex spectrum of rotor vibration loads. Partially, the issue arises from reliance on a single-mass dynamic model of the seal. In addition, most published research works have focused mainly on small changes in the seal gap compared to the nominal equilibrium gap value. For large values of oscillation amplitude and frequency (an amplitude greater than 25 µm and a frequency greater than 150 Hz), it is essential to account for the squeezing of the gas film in the seal gap [4,23,41].

Recently, several research works have studied the numerical determination of stiffness and damping in a mechanical face seal in the presence of large amplitudes and frequencies of axial rotor vibrations [23,30,32,42]. Furthermore, fractal models have been used to further investigate the influence of contact between the seal rings and external stator noises on the performances of the seal during dynamic analysis [4,43,44]. However, these works still use a single-mass dynamic model of the seal.

Therefore, analyzing the dynamic performance of dry gas and hydrodynamic seals requires both a joint solution of the computational fluid dynamics problem and the application of more complex dynamic models.

The aim of this research is to develop recommendations for the use of existing dynamic models and those developed by the authors for various types of mechanical face seals in turbomachinery applications under specific operating conditions and designs.

2. Single-Mass Dynamic Model of Mechanical Face Contact Seal

2.1. Model Description

The dynamic model of a mechanical face contact seal, unlike a lubricated seal, has a displacement limiter—the face of the rotor ring. The dynamic model of the mechanical contact seal for the two operation modes (nominal contact and separation modes) is shown in Figure 2. K₁ is the spring stiffness, R₁ is the friction force, and D₁ is the damping of the elastomer O-ring. K₀ and D₀ are the stiffness and damping coefficients of the fluid flow in the seal gap (in the case of the separation operation mode). The stator seal ring (with mass m) is pressed against the face of the rotor seal ring by the difference in the fluid pressure (force F_cls) and spring force. For the nominal contact operation mode, the displacements of the rotor seal ring and the stator seal ring are identical z₀ = z₁.

The stator seal ring will separate from the face of the rotor seal ring when the axial acceleration of the rotor exceeds the acceleration of the “contact force”, which is the ratio of the unbalanced axial contact force to the mass of the seal ring (m). The boundary conditions of the seal separation mode can be determined. The stator seal ring separation mode can be prevented by limiting the frequency (ω) and amplitude (z_0amp) of the axial rotor vibration. To describe the separated stator seal ring axial motion, it is necessary to determine the stiffness dependence of the fluid in the seal gap on the gap value (h). It is also important to consider that the frictional force of the elastomer O-ring changes its sign during movement. There are two possible solutions. If z₁ < 0 during 0.5·π·ω < t < 1.5·π·ω, continuous movement of the stator seal ring occurs. T is time. If z₁ > 0, the seal ring stops when the axial displacement of the rotor equals its amplitude value (z₀ = z_0amp). The stopping time depends on the value of 2·R₁/(K₁ + K₀). The separation of the stator seal ring ends when z₁ and z₀ are equal again (z₁ = z₀). Then, the seal operates normally in contact mode. The motion equation of the stator ring for two operation modes is as follows:

$$\begin{gathered} m{\ddot{z}}_1+F_{cls}+R_0\mathit{sgn}{\dot{z}}_1+D_1{\dot{z}}_1+K_1{z}_1=z_{0amp}\mathit{sin}(\omega \cdot t\cdot 2\cdot \pi ) \\ m{\ddot{z}}_1+F_{cls}+R_0\mathit{sgn}{\dot{z}}_1+(D_1+D_0){\dot{z}}_1+(K_1+K_0)z_1=K_0{z}_0+D_0{\dot{z}}_0, \end{gathered}$$

(1)

where $\mathit{sgn}{\dot{z}}_1$ is the signum function of ${\dot{z}}_1$:

$$\mathit{sgn}{\dot{z}}_1={\displaystyle \frac{d}{d{\dot{z}}_1}}\left|{\dot{z}}_1\right|,{ \dot{z}}_1\ne 0 \mathrm{or} \mathit{sgn}{\dot{z}}_1=\left\{\begin{array}{c}-1, if ({\dot{z}}_1<0)\\ 0, if ({\dot{z}}_1=0)\\ 1, if ({\dot{z}}_1>0)\end{array}\right\}$$

(2)

Separation of the stator seal ring is undesirable during operating mode. It is necessary to determine the boundary conditions in order to prevent this operating mode by limiting the frequency of axial vibration. The value of the frequency must be lower than the following value:

$${\omega }_{max}=\sqrt{p_0^2+{\displaystyle \frac{F_{cls}+R_1\cdot \mathit{sgn}{\dot{z}}_1}{m\cdot z_{0amp}}}},$$

(3)

where $p_0=\sqrt{K_1/m}$—is the seal natural frequency for the contact operation mode.

It should be noted that if the rotor vibration frequency significantly exceeds the natural frequency of the seal during the separation operation mode, the separation process will stop [41].

If the frequency ω does not exceed ω_max, there is another possible case of stator ring separation. The separation happens when the stator seal ring is in a state of suspension:

$${\omega }_{max}=\sqrt{p_0^2+{\displaystyle \frac{F_{cls}-R_1}{m\cdot z_{0amp}}}}$$

(4)

The properties of elastomer O-rings significantly affect the performance of all types of mechanical face seals. Their parameters must be taken into account: geometric parameters such as the cross-sectional diameter and outer diameter, the frictional force, and stiffness and damping coefficients. With high contact pressures of the rubber O-ring against the metal surface and the small displacement of the stator seal ring, there is little or no slippage in the O-ring. The axial movement of the stator seal ring in this case is mainly due to the elasticity of the O-ring. This leads to changes in the seal’s load carrying capacity and, consequently, changes in its dynamic behavior.

For example, there is a dimension by which the outer diameter of mechanical face seal rings must be changed. In this case, it is also necessary to change «inwards» the contact diameter of the elastomer O-ring with the metal surface. In the conventional design (the sliding contact on the outer surface of the O-ring like in Figure 1), the radial contact displacement for an O-ring with a hardness of 90 (Shore A hardness scale) is approximately 5% of the O-ring cross-sectional diameter at a low-pressure drop. As the pressure drop increases, the radial contact displacement decreases to 2% at a pressure drop of 10 MPa. If the sliding contact is on the inner surface of the O-ring, the radial displacement of the contact moves «inwards» of the O-ring and is twice as great. In this case, the frictional force of the O-ring might not be considered in the calculation.

For estimation calculations, it is possible to use the results obtained by Green and Etsion [25]. The mass of the elastic parts (such as springs or bellows) is taken into account by increasing the mass of the stator seal by one-third of the mass of these parts.

2.2. Calculation Results and Discussion

The phenomena of the stator ring separation from the rotor ring in mechanical face seals is encountered in practice by a number of companies. An analysis was conducted on the mechanical face contact seal of the compressor (

Table 1. Initial operating parameters of the studied mechanical face seal.

Parameters	Values
Inside radius (r1, mm)	120
Pressure difference (∆P, MPa)	0.15
Spring force (Fsp, N)	285
Stiffness of the spring (Ksp, kN/m)	30
Elastomer O-ring friction force (Ffr, N)	100
Stator rings mass (m, kg)	1.25

). This seal had sealing problems when the rotor speed was in the range of 725 to 880 rad/s.

In the compressor, due to its design, pressure fluctuations occurred in the pressurized cavity, leading to rotor axial vibrations at a frequency 3–4 times the rotor operating frequency. It was found that the seal had operational problems with excessive rotor axial vibrations. Figure 3 presents the separated gap value (h) for the different axial vibration frequencies (2025, 2325, 2625, and 2775 rad/s). There was a separation of the stator seal ring from the rotor ring, causing a gap value of a few microns, which led to leakage. At frequencies of 2025 rad/s and 2775 rad/s, the seal operated in contact mode.

The issue with seal separation was solved by adjusting the spring and seal ring parameters. The total spring stiffness was increased to 45 kN/m while maintaining the spring force value. The mass of the sealing ring was decreased to 0.8 kg. This helped to move the separation operating range out of the excitation frequency operating range. The advantage of the developed technique is focused on predicting the separation phenomena and finding the most efficient methods to eradicate such phenomena at the design stage.

3. Single-Mass Dynamic Model of Dry Gas Seal

The traditional dynamic model of a non-contact mechanical face seal (or dry gas seal) is a combination of an axially movable stator seal ring, a thin fluid film between the stator seal ring and the rotor seal ring, and a spring-loaded ring with an O-ring to press the stator seal ring axially against the rotor seal ring [16]. A thin fluid film in the case of liquid cavity pressurization is modeled by parallel stiffness and damping (Kelvin–Feugt model) [14,15]. This dynamic model is similar to the model shown in Figure 2b, but in this case there is no force F_cls (the axial forces are balanced). When optimizing the seal parameters, the goal is to achieve maximum damping to ensure minimal changes in the gap value. In the case of significant seal ring vibration amplitudes (significant changes in gap value), the fluid film static stiffness dependence on the gap value must be taken into account. Experience in studying the dynamic performance of seals has shown that the Kelvin–Feugt model is the simplest method for accounting for the nonlinear properties of the fluid film.

For gas, a relaxation damping model (Maxwell–Zener model) is used. This model consists of a spring element and a Kelvin–Feugt model in series. The model is characterized by the fact that the damping force acts indirectly between the stator seal and rotor ring, but through an additional spring element. The single-mass dynamic model of the dry gas seal (Figure 4) consists of stator seal ring mass (m), an inertia-free Maxwell–Zener model (gas film, K_dyn), a spring element (spring, K_sp), and an elastomer O-ring characterized by damping, friction, and stiffness (D₁, R₁, K₁).

There are three types of vibration that can occur in a dry gas seal: axial, angular, and bending. The calculation must take into account the mass and moments of inertia of the stator seal ring, the axial force and torque generated by the gas film in the gap, the properties of the spring and O-ring, and an external excitation load from the rotor ring (force and torque). The axial, angular, and bending displacements of the seal ring, as well as the shape and value changes of the gap, are determined during vibration. In the case of complex vibrations where the stator seal ring makes small periodic displacements (the vibration amplitude is approximately 10% of the nominal gap value), the dynamic performance (stiffness and damping coefficients) of the gas film depends linearly on the changes in gap value and their rate relative to its equilibrium position.

In most practical cases, it is sufficient to consider only the axial vibration. The amplitude of the rotor axial vibration or its runout is considered as the excitation load. In this case, special attention should be paid to the seal assembly process of the seal, while the angular stiffness of the gas film should be high. In general, the axial, angular, and bending vibrations of the stator seal ring influence each other due to cross-links. In this case, the angular displacements of the rotor ring excite the axial vibration of the stator seal ring, and so on. The vibrations of the stator seal are described by a system of equations [1]:

$$\left\{\begin{array}{c}m{\ddot{z}}_1+F_{op}+F_{cls}=0\\ I{\ddot{\alpha }}_1+M_{\alpha }+L_{\alpha }=0\\ I_P{\ddot{\theta }}_1+M_{\theta }+L_{\theta }=0\end{array}\right.,$$

(5)

where m, I, and I_P are the mass and moments of inertia of the stator ring; z₁, α₁, θ₁ are the axial, angular, and bending displacement of the stator ring; F_op, M_α, and M_θ are the opening force and gas-dynamic moments generated by the pressurized gas film in the seal gap; and F_cls, L_α, and L_θ are the closing force and external moments. The influence of the three types of rotor ring vibration, axial, angular, and bending, is studied:

$$\left\{\begin{array}{c}{z}_0=z_{0amp}\mathit{sin}(\omega \cdot t\cdot 2\cdot \pi )\\ {\alpha }_0={\alpha }_{0amp}\mathit{sin}(\omega \cdot t\cdot 2\cdot \pi )\\ {\theta }_0={\theta }_{0amp}\mathit{sin}(\omega \cdot t\cdot 2\cdot \pi )\end{array}\right.,$$

(6)

where z_0amp, α_0amp, and θ_0amp are the axial, angular, and bending amplitude of the rotor tilt and runout.

The calculation results for a single-mass dry gas seal (classical dynamic model) are well described in the literature.

4. Two-Mass Dynamic Model of Dry Gas Seal

4.1. Model Description

If the stator seal is made entirely of graphite, the springs press the spring-loaded ring (m_sp), which in turn presses the stator seal ring via an elastomeric O-ring. In this case, a two-mass dynamic model of the seal must be used (Figure 5).

An analysis of the cross stiffness and damping coefficient values shows that the following types of vibrations can occur in the studied two-mass dynamic model: axial vibrations z₀ → z₁ → z₂; angular vibrations α₀ → α₁ → α₂; and combined axial and bending vibrations z₀ → z₁, θ₁ → z₂. Bending excitation from the rotor (θ₀) is very rare in practice and is therefore not considered. As noted in [1,33], axial and angular vibrations can be analyzed separately, and the ring displacements can be summed.

To determine the stiffness and damping of the seal gas film, the following assumptions are made: the gas flow is laminar, isothermal, and axisymmetric. The influence of the misalignment between sealing surfaces is neglected due to the high angular stiffness of the gas film.

The film thickness, or the gap value along the radius and over time, can be represented in the following dimensionless form:

$$H=1+\epsilon \cdot e^{i\tau },$$

(7)

where $H={h/h}_0$ is the dimensionless gap value; $h$ is the current gap value; $h_0$ is the gap value obtained through static calculation; $\epsilon =\mathsf{\Delta }{h/h}_0$; $\tau =\omega \cdot t$; $i=\sqrt{-1}$; and $\mathsf{\Delta }h$ is the change in the current gap value.

The Reynolds equation in a dimensionless form for the gas flow in the seal gap under the given assumptions is written as follows:

$${\displaystyle \frac{1}{\overline{r}}} {\displaystyle \frac{d}{d\overline{r}}}\left(\overline{r}{H}^3{\displaystyle \frac{d{\overline{p}}^2}{d\overline{r}}}\right)=2\sigma {\displaystyle \frac{d\left(\overline{p}H\right)}{d\tau }},$$

(8)

where $\overline{r}={r/r}_1$ is the current dimensionless radius; $r_1$ and $r_2$ are the inner and the outer radii of the gap; $\overline{p}={p/p}_2$ is the dimensionless pressure; p is the current pressure in the radius $r$; $p_2$ is the pressure at the radius $r_2$; and $\sigma$ is the squeeze parameter:

$$\sigma =12{\displaystyle \frac{\mu \omega }{p_2}}({\displaystyle \frac{r_1}{h_0}}{)}^2.$$

(9)

For small perturbations ($\epsilon \ll 1$), the gas pressure in the gap also follows the harmonic law:

$$\overline{p}={\overline{p}}_0+\epsilon \overline{c}{e}^{i\tau },$$

(10)

where $\overline{p}$ is the dimensionless pressure obtained from static calculation; and $\overline{c}=\mathsf{\Delta }p/\epsilon$ is the ratio of the current change in pressure to the current change in the gap value. The static pressure along the gap radius is well described in the literature. For the dynamic model, the following equation applies:

$${\displaystyle \frac{1}{\overline{r}}}{\displaystyle \frac{d}{d\overline{r}}}\left(\overline{r}{\displaystyle \frac{d\left({\overline{p}}_0\overline{c}\right)}{d\overline{r}}}\right)+{\displaystyle \frac{1}{\overline{r}}}{\displaystyle \frac{d}{d\overline{r}}}\left({\displaystyle \frac{3}{2}}\overline{r}{\displaystyle \frac{d{{\overline{p}}_0}^2}{d\overline{r}}}\right)=i\sigma ({\overline{p}}_0+\overline{c}).$$

(11)

To solve the obtained equation, it is necessary to replace the static pressure ${\overline{p}}_0$ with its average value over the radius:

$${{\stackrel{\_}{p}}_0}_m={\displaystyle \frac{2}{{{\overline{r}}_2}^2-1}}{\int }_1^{{\overline{r}}_2}{\overline{p}}_0\overline{r}d\overline{r}.$$

(12)

After transformation, the second equation takes the form of a Bessel equation:

$${\displaystyle \frac{1}{\overline{r}}}{\displaystyle \frac{d}{d\overline{r}}}\left(\overline{r}{\displaystyle \frac{d\left(\overline{c}\right)}{d\overline{r}}}\right)=i{\displaystyle \frac{\sigma }{{\overline{p}}_0}}({\overline{p}}_0+\overline{c}).$$

(13)

where $\overline{c}=-{{\stackrel{\_}{p}}_0}_m+{\alpha }_1{I}_0\left(i\sqrt{i}k\overline{r}\right)+{\alpha }_2{K}_0\left(i\sqrt{i}k\overline{r}\right)$, $k^2=\sigma /{{\stackrel{\_}{p}}_0}_m$, $I_0\left(i\sqrt{i}k\overline{r}\right)={\sum }_{n=0}^{\mathrm{\infty }}{\displaystyle \frac{1}{(n!{)}^2}}({\displaystyle \frac{i\sqrt{i}k\overline{r}}{2}}{)}^{2n}$, and $K_0\left(i\sqrt{i}k\overline{r}\right)=-I_0\left(i\sqrt{i}k\overline{r}\right)\mathit{ln}({\displaystyle \frac{i\sqrt{i}k\overline{r}}{2}})+{\sum }_{n=1}^{\mathrm{\infty }}{\displaystyle \frac{1}{(n!{)}^2}}\left({\displaystyle \frac{i\sqrt{i}k\overline{r}}{2}}{)}^{2n}\right(1+...{\displaystyle \frac{1}{n}})$. $I_0$ and $K_0$ are the modified Bessel functions.

Using the boundary conditions $\overline{c}=0$ at $\overline{ r}=0$ and $\overline{c}=0$ at $\overline{ r}={\overline{r}}_2$, the second equation takes the following form:

$$\overline{c}={{\stackrel{\_}{p}}_0}_m[{\displaystyle \frac{K_0\left(i\sqrt{i}k\overline{r}\right)\left\{I_0\right(i\sqrt{i}k{\overline{r}}_2)-I_0(i\sqrt{i}k{\overline{r}}_1\left)\right\}-I_0\left(i\sqrt{i}k\overline{r}\right)\left(K_0\right(i\sqrt{i}k{\overline{r}}_2)-K_0(i\sqrt{i}k{\overline{r}}_1)}{K_0\left(i\sqrt{i}k{\overline{r}}_1\right)I_0\left(i\sqrt{i}k{\overline{r}}_2\right)-K_0\left(i\sqrt{i}k{\overline{r}}_2\right)I_0\left(i\sqrt{i}k{\overline{r}}_1\right)}}-1]$$

(14)

The dynamic stiffness or dynamic response of the gas film in the seal gap is determined by the following equation:

$$K_{dyn}=-2\pi p_2{r}_1^2{\int }_1^{{\overline{r}}_2}\overline{c}\overline{r}d\overline{r}.$$

(15)

The dynamic stiffness can also be represented as follows:

$$K_{dyn}=K_{\mathrm{g}as}+i\cdot \omega \cdot D_{\mathrm{g}as},$$

(16)

where $K_{\mathrm{g}as}$ is the gas film stiffness coefficient, and $D_{\mathrm{g}as}$ is the gas film damping coefficient.

The two-mass dynamic model is a special case of the three-mass dynamic model; therefore, the derivation of the motion equation will be carried out for the more complex three-mass model.

The following assumptions are made in the dynamic analysis of the seal using the two-mass model: inertial forces of the gas film are not considered; the nonlinearity of the dynamic stiffness and damping coefficients is not considered (the small perturbation method is used); frictional forces between the housing parts and the stator seal ring are neglected; the influence of the opening force on the rotor seal ring is not considered; and circumferential non-uniform contact between the seal ring surfaces is not considered.

4.2. Calculation Results and Discussion

The operation of the dry gas seal for a NC-6.3 pumping unit was studied and tested using air as the working fluid. The seal design illustrated in Figure 6 was developed by one of the co-authors, Falaleev, and has been previously published in [45]. Permission to include the design in this paper was granted by Falaleev.

This is a two-stage dry gas seal. The primary purpose of the first stage was to handle the pressure drop and seal the working or process gas, while the second stage acted as a barrier to prevent any process gas from entering the pressurized cavity. The two stages also provided redundancy: if the first-stage seal failed or became less effective, the second-stage seal could still maintain the barrier for a short period.

The operational parameters for the model calculation are shown in

Table 2. Initial operating parameters of the studied mechanical face seal.

Parameters	Values
Outside radius, (r2, mm)	94
Inside radius, (r1, mm)	75
Groove radius, (rG, mm)	85
Groove angle, (α, degree)	15
Ridge width/Groove width, (b1/b2)	1
Balance radius (elastomer O-ring), (rB, mm)	78.85
Groove depth, (dG, µm)	7
Equilibrium gap, (h, µm)	2.5
Outside pressure, (P2, MPa)	5.2
Inside pressure, (P1, MPa)	0.1
Outside temperature, (T2, K)	298
Rotational speed, (ω, RPM)	7300
Groove number, (n)	12
Elastomer O-ring stiffness, (K1, kN/mm)	5.8
Elastomer O-ring damping, (D1, N∙s/mm)	4.2
Fluid	air

. The elastomer O-ring material used was Viton 75.

The pressure drop was 5.1 MPa. The of axial rotor vibration frequency ranged from 100 to 200 Hz, with amplitudes ranging from 100 to 300 µm. The amplitude of angular vibrations was up to 1 mrad. The equilibrium seal gap value was 2.5 µm. Figure 7, Figure 8 and Figure 9 show the variation in the gap value of the dry gas seal during the vibration period.

The presence of axial rotor vibration with amplitudes of 100 to 300 µm and a frequency of 100 Hz will not significantly affect the operation of the dry gas seal. However, if the frequency of the axial rotor vibration increases to 200 Hz (Figure 8), with an amplitude of 200 µm, the gap value can reach 1 µm. With an amplitude of 300 µm, the gap value can drop to 0.3 µm. Operational experience suggests that such small gap values can lead to contact between the seal ring surfaces.

Simultaneous axial and angular rotor vibrations (Figure 9) led to contact between the seal rings. Specifically, contact between the seal ring surfaces occurred at a rotor vibration frequency of 200 Hz, with an angular vibration amplitude of 1 mrad combined with an axial vibration amplitude exceeding 200 µm. However, calculations show that at a rotor vibration frequency of 100 Hz, with the same angular vibration amplitude, the gap value varied between 0.8 and 4 microns, meaning the seal could operate without contact.

For the dry gas seal, one seal manufacturer has recommendations for axial vibration conditions [46]. For single drive-shaft compressor units, the vibration distance (peak-to-peak) should not exceed 50 µm. For a gear compressor unit, the vibration distance (peak-to-peak) should not exceed 70 µm. The requirements for axial rotor vibration acceleration are shown in Figure 10 [46].

According to the seal manufacturing company, the recommendations for rotor vibration conditions are strict. However, in practice, a dry gas seal can maintain normal operation with worse rotor vibration conditions. Typically, operating or manufacturing companies measure or provide the axial vibration of pumping units or turbomachinery as amplitude and frequency. However, taking the second derivative of Equation (6) gives the acceleration based on axial vibration distance and frequency:

$${\displaystyle \frac{d^2}{d{t}^2}}{(z}_0)=-z_{0amp}\cdot 4\cdot {\pi }^2\cdot {\omega }^2\cdot \mathit{sin}(\omega \cdot t\cdot 2\cdot \pi ),$$

(17)

where $z_{0amp}$ is the half of the vibration distance (peak-to-peak).

The amplitude and vibration frequency are used as initial conditions in the dynamic models. These models provide the actual gap value, which allows for a more accurate determination of whether the seal operates normally.

5. Three-Mass Dynamic Model of Dry Gas Seal

5.1. Model Description

An analysis of existing and prospective dry gas seal designs shows that the dynamic model presented in Figure 11 is the most suitable when the rotor ring is flexibly mounted on the rotor [47].

The model consists of three bodies (masses). The spring-loaded ring (m_sp) is installed in the seal housing and is pressed against the stator seal ring (m_st) by a set of springs with stiffness K_sp. The spring masses are accounted for by adding one-third of the spring-mass to the spring-loaded ring mass. An O-ring seal is located between the spring-loaded ring and the stator seal ring, represented by a stiffness element (K₂), damping (D₂), and friction (R₂). Between the stator seal ring and the rotor seal ring (m_rt) is an inertia-free elastic–viscous model, or a Maxwell–Zener model (gas film, K_dyn). An additional elastomer O-ring or secondary seal is positioned between the rotor seal ring (m_rt) and the rotor face, represented by a stiffness element (K₀) and damping (D₀). The rotor ring is pressed axially against the rotor face (δ₀ = 0) by an unbalanced opening force (F_op). On the opposite side of the ring, a gap exists between the rotor sleeve and the rotor seal face (δ₁). The rotor seal is mounted on the shaft via either a corrugated damper or an elastomer O-ring. Mutual axial displacement between the rotor ring and the O-ring is modeled by friction (R₀).

In the general case, the rotor face transmits the excitation load of axial and angular vibration with amplitudes z_0amp and α_0amp. The rotor face may also have an additional bending vibration component with an amplitude θ_0amp. However, these bending vibrations are compensated by the deformation of the elastomeric O-ring, preventing the bending excitation load from being transmitted to the rotor ring. It should be noted that this dry gas seal design allows for reliable operation across a wider range of axial and angular vibration amplitudes of the rotor face.

The three-mass model allows for the consideration of the phenomenon of the seal rotor ring separation from the rotor face (δ₀ > 0) and enables the study of its influence on the seal’s operation and performance.

The following assumptions are made in the dynamic analysis of the seal using the three-mass model: only axial vibration is considered; inertial forces of the gas film are not considered; frictional forces between the housing parts and the stator seal ring are neglected; and the circumferential non-uniform contact between the seal ring surfaces is not considered.

The system of differential equations for the motion of the bodies under axial forces is derived as follows:

$$\left\{\begin{array}{c}{m}_{rt}\cdot {\ddot{z}}_1+K_0\left(z_1-z_0\right)+K_{dyn}\left(z_1-z_2\right)+D_0\left({\dot{z}}_1-{\dot{z}}_0\right)+F_{op}-K_0\left(L-d\right)=0\\ m_{st}\cdot {\ddot{z}}_2+K_{dyn}\left(z_2-z_1\right)+K_2\left(z_2-z_3\right)+{D^{\prime }}_2\left({\dot{z}}_2-{\dot{z}}_3\right)=0\\ m_{sp}\cdot {\ddot{z}}_3+K_2\left(z_3-z_2\right)+{D^{\prime }}_2\left({\dot{z}}_3-{\dot{z}}_2\right)+K_{sp}\cdot z_3=0.\end{array}\right.,$$

(18)

where L is the groove depth for the elastomeric O-ring, described by the stiffness $K_0$, and d is the cross-sectional diameter of the O-ring [25]. The combined effect of the frictional force $R_2$ and the damping $D_2$ can be described by the equivalent damping ${D\prime }_2$.

The system of equations is solved using the Laplace transform and reduced to an algebraic system of equations. The dynamic properties of the gas film are described by a simplified dynamic stiffness:

$$K_{dyn}=K_{\mathrm{g}as}+s\cdot D_{\mathrm{g}as},$$

(19)

where $s=i\cdot \omega$—is the complex frequency domain parameter, and $i=\sqrt{-1}$.

The harmonic axial vibration is investigated with $z_0=z_{0amp}\mathit{sin}(\omega \cdot t\cdot 2\cdot \pi )$. The displacement $z_3$ is derived from the third equation of the system (18):

$$z_3={\displaystyle \frac{z_2\cdot a_2}{a_1}},$$

(20)

where $a_1=m_{sp}\cdot s^2+K_2+{D^{\prime }}_2\cdot s+K_{sp}$, $a_2=K_2+{D^{\prime }}_2\cdot s$.

Next, $z_2$ is derived from the third equation of the system (18) and $z_3$ (20). The impulse response $H_{z_1{z}_2}\left(s\right)$ is given by the following equations:

$$z_2={\displaystyle \frac{K_{dyn}\cdot z_1\cdot a_1}{a_1\cdot a_3-a_2^2}},$$

(21)

$$H_{z_1{z}_2}\left(s\right)={\displaystyle \frac{z_2}{z_1}}={\displaystyle \frac{K_{dyn}\cdot a_1}{a_1\cdot a_3-a_2^2}},$$

(22)

where $a_3=m_{st}\cdot s^2+K_{dyn}+{D^{\prime }}_2\cdot s+{\mathrm{K}}_2$.

Similarly, by expressing $z_1$ from the first equation of the system (18) and substituting $z_2$ (21), the impulse response $H_{z_0{z}_1}\left(s\right)$ can be determined:

$$z_1={\displaystyle \frac{(a_1\cdot a_3-a_2^2)\left({z_0\cdot a}_5-F_{op}\prime \right)}{{(a}_4\cdot a_3\cdot a_1-a_2^2)-K_{dyn}^2\cdot a_1}},$$

(23)

$$H_{z_0{z}_1}\left(s\right)={\displaystyle \frac{\mathrm{z}}{z_0}}={\displaystyle \frac{(a_1\cdot a_3-a_2^2)\left(a_5-\frac{F_{op}\prime }{z_0}\right)}{{(a}_4\cdot a_3\cdot a_1-a_2^2)-K_{dyn}^2\cdot a_1}},$$

(24)

where $a_4=m_{rt}\cdot s^2+K_0+D_0\cdot s+K_{dyn}$, $a_5=K_0+D_0\cdot s,$ and $F\prime =F_{op}-R_0\cdot sgn\left(z_1\cdot s-z_0\cdot s\right)+K_0(L-d)$.

$$sgn\left(z_1\cdot s-z_0\cdot s\right)=\left\{\begin{array}{c}+1, if \left(z_1\cdot s-z_0\cdot s\right)>0\\ -1, if \left(z_1\cdot s-z_0\cdot s\right)<0\\ 0, if \left(z_1\cdot s-z_0\cdot s\right)=0\end{array}\right.$$

(25)

Since the components are connected in series, the resulting impulse response is as follows:

$$H_{z_0{z}_2}\left(s\right)=H_{z_1{z}_2}\left(s\right)\cdot H_{z_0{z}_1}\left(s\right)$$

(26)

If needed, the amplitude–frequency and phase–frequency responses can be determined as follows:

$${\mu }_h\left(s\right)=\sqrt{{Re}^2\left(H_{z_0{z}_2}\left(s\right)\right)+{Im}^2\left(H_{z_0{z}_2}\left(s\right)\right)},$$

(27)

$${\psi }_h\left(s\right)=-arctg\left[{\displaystyle \frac{Im\left(H_{z_0{z}_2}\left(s\right)\right)}{Re\left(H_{z_0{z}_2}\left(s\right)\right)}}\right]$$

(28)

The obtained analytical solution allows for the determination of the seal ring positions when the rotor ring moves away from the rotor face. However, this solution assumes that the stator ring follows the rotor ring’s displacements. If the stator ring fails to do so, it is necessary to account for the nonlinearity in the gas film’s stiffness and damping properties. If the stator ring does not follow the rotor ring’s displacements, it is recommended to use a numerical solution of the three-mass model, such as the Runge–Kutta method.

5.2. Calculation Results and Discussion

The phenomenon of rotor seal ring separation was studied using the seal for the N370 natural gas pumping unit as a case study. The seal design illustrated in Figure 12 was developed by one of the co-authors, Falaleev, and has been previously published in [45]. Permission to include the design in this paper was granted by Falaleev.

This phenomenon was confirmed in practice during the operation of a gas pumping unit equipped with a bidirectional dry gas seal with special groove geometry (

Table 3. The initial operating parameters of the studied mechanical face seals.

Parameters	N370	Bidirectional Seal
Outside radius, (r2, mm)	112.5	54.18
Inside radius, (r1, mm)	89.85	41.35
Groove radius, (rG, mm)	101	50.15/46.4
Groove angle, (α, deg.)	15	21/6.5
Ridge-to-groove width ratio, (b1/b2)	1	5.85
Balance radius (elastomer O-ring), (rB, mm)	92.5	42.6
Groove depth, (dG, µm)	7	5/12.5
Equilibrium gap, (h, µm)	2.25	10.3
Stator ring roughness, (RaS, µm)	0.04	0.04
Rotor ring roughness, (RaR, µm)	0.03	0.03
Groove roughness, (RaG, µm)	0.20	0.20
Outside pressure, (P2, MPa)	6.180	0.1186
Inside pressure, (P1, MPa)	0.101	0.101
Outside temperature, (T2, K)	298	394
Rotational speed, (ω, RPM)	5500	30,000
Closing Force, (Fcls, kN)	80.680	0.454
Groove number, (n)	12	8
Fluid	air	natural gas
Elastomer O-ring stiffness, (K2, kN/mm)	5.84	1.250
Spring stiffness, (Ksp, N/mm)	4.60	4.2
Elastomer O-ring equivalent damping, (D’2, N∙s/mm)	4.18	1.486
Stator rings mass, (mst, kg)	0.35	0.0976
Rotor ring mass, (mrt, kg)	3.82	0.7385
Spring-loaded ring mass, (msp, kg)	0.25	0.0768

). The elastomer O-ring material used in the N370 seal was Viton 75.

The dry gas seal for the N370 pumping unit was studied and tested using air as the working fluid. Both the N370 and the bidirectional seal were two-stage dry gas seals. The study focused on the first (primary) stage of the N370 seal and the second (auxiliary) stage of the bidirectional seal.

The static gas film axial stiffness and damping as coefficients functions of the gap value (Figure 13 and Figure 14) were obtained by employing CFD models (steady-state and quasi-transient two-way FSI models with mesh deformation) [42]. The static gas film stiffness was determined using the steady-state CFD model of the seal gap ($K_{\mathrm{g}as}=-d{F}_{op}/dh$). The damping dependence was obtained using the quasi-transient two-way FSI model with mesh deformation ($D_{\mathrm{g}as}=-d{F}_{op}/d\dot{h}$). For the three-mass model calculation, a constant damping value was used, corresponding to the equilibrium gap value of the seal.

The stiffness and damping values of the first stage of the N370 seal are several times greater than those of the bidirectional seal, as different stages of these seals were studied. According to the obtained data, dynamic stability issues are more likely to occur in the second (auxiliary) stage of two-stage dry gas seal types.

Based on the two-mass dynamic model calculation of the N370 seal, the axial vibration with an amplitude of 150 µm and a frequency greater than 2.2 kHz led to seal failure, as the stator seal ring could follow the displacement of the rotor seal ring (Figure 15 and Figure 16).

The three-mass dynamic model of the N370 seal was calculated for an excitation load frequency of 800 Hz and an axial amplitude of 150 µm (with the rotor suspended by active magnetic bearings).

The vibration amplitude of the rotor seal ring exceeded the rotor runout amplitude, causing separation (δ₀ > 0) at t = 0.19 s (Figure 17). However, due to the presence of the second stop (Figure 11 and Figure 12), the amplitude was limited by the gap value δ’. The rotor seal ring impacted the stops twice, with the second impact occurring at t = 0.51 s.

This separation phenomenon negatively affected the seal operation and the gas film properties in the seal gap (Figure 18). Due to the first impact, the gap value changed from 0.85 μm to 5.67 μm, while following the second impact, it varied from 0.25 μm to 8.09 μm.

Considering the additional negative effects of angular and bending vibrations, the dry gas seal experiences periodic impacts between the seal rings. This results in abrasive wear of the seal ring surfaces and contamination of the seal gap, which can cause seal unit failure within a short period (Figure 19). It should be noted that, according to the two-mass model calculations, the dry gas seal is theoretically capable of operating normally up to a vibration frequency of 2 kHz at an axial vibration amplitude of 150 µm (Figure 16). However, this prediction is not accurate in practice.

During the operation of the natural gas pumping unit N370, it was found that at a rotational speed of 5500 RPM (vibration frequency of 92 Hz), the seal operates normally up to a rotor face runout of 0.25 mm.

A similar phenomenon was observed in the second stage of the bidirectional dry gas seal with special groove geometry. The second stage seal operated at a low differential pressure of 17.3 kPa. The gas pumping unit exhibited axial rotor vibration at 500 Hz with an amplitude of 55 µm. The second stage rotor seal ring, due to the small opening force, was pressed against the internal stop (δ₁ = 0) by the external squeezed force of the elastomeric O-ring (compressed by 0.4 mm) and the external pressure forces on the rotor ring’s external face. Calculation results of the three-mass model confirmed the rotor ring separation phenomenon under these excitation loads. The sealing unit was disassembled for defect investigation. (Figure 20).

The investigation results confirmed the conclusions from the calculations. The rotor ring of the second seal stage showed wear on the sealing face at the inner diameter, while the rotor ring of the first seal stage did not. This wear corresponds to the area where the rotor ring was impacted by the rotor stop.

6. Conclusions

Rotor vibration is one of the main causes of mechanical face seal failure. Traditional dynamic models often fall short in fully explaining the failure mechanisms. The dynamic models discussed in this paper are valuable for predicting seal behavior during operation and for investigating or explaining seal failures, particularly in the context of turbomachinery. The advantages and disadvantages of the proposed models largely depend on the design of the studied seal.

(1) The single-mass dynamic model addresses practical issues for contact mechanical face and dry gas seals with simple designs. For the simplest contact mechanical face seal, this model provides the maximum rotor vibration frequency at which separation of the stator seal ring from the rotor occurs.

A single-mass dynamic model for non-contact mechanical face seals of a simple design is proposed, using the Kelvin–Feugt and Maxwell–Zener models to describe the dynamic response of the fluid film and the dynamic properties of the elastomer O-ring. The simplest method to describe the nonlinearity behavior of a liquid film is to use the Kelvin–Feugt model. For gas, a relaxation damping model (Maxwell–Zener) is used, where the damping element is not placed between the ring and the rotor face but through an additional spring element. The importance of including axial, angular, and bending vibrations of the stator seal ring in dynamic analyses is discussed. Recommendations are also provided for considering the properties of the elastomer O-ring and the spring mass.

(2) The two-mass dynamic model can be used to predict the performance of most types of dry gas seals under complex excitation loads. One seal manufacturer recommends limiting the rotor axial vibration or rotor runout to 70 µm peak-to-peak [46]. However, based on the calculations from the two-mass dry gas seal dynamic model, the stator seal ring can track the rotor ring’s axial displacement over a broad range of rotor axial vibration frequencies and amplitudes (up to 250…300 µm with a frequency of up to 100 Hz). Nevertheless, the presence of axial vibration at 200 Hz, particularly when combined with simultaneous axial and angular vibration, will lead to contact between the seal faces.

(3) When the rotor ring is flexibly mounted on the rotor, the three-mass dynamic model of the dry gas seal is the most accurate. This dynamic model can precisely determine the range of normal operating conditions for advanced dry gas seal designs and help identify leakage loss processes under conditions of excessive rotor vibration.

The safe axial rotor vibration frequency determined by the two-mass dry gas seal model is 2.2 kHz with an amplitude of 150 µm (a minimum gas film gap is 1.15 µm). According to the three-mass model calculation results, an axial rotor vibration frequency of 800 Hz with an amplitude of 150 µm already leads to seal ring contact (a minimum gas film gap is 0.25 µm). This discrepancy is due to the separation of the rotor ring from the rotor stop, causing impacts between the rotor ring and the stops. These impacts negatively affect the gap value, increasing it by four times compared to the equilibrium gap. Such operation conditions will eventually lead to periodic impacts between the seal ring faces, resulting in wear, gap contamination, and ultimately, seal unit failure. According to the operational experience of the natural gas pumping unit with the studied seal, it is observed that the seal functions normally at a rotational speed of 5500 RPM with a rotor face runout up to 250 µm.

In a two-stage dry gas seal design, the dynamic properties of the gas film in the gap (stiffness and damping values) are several times higher in the first (primary) stage compared to the second (axillary) stage. As a result, dynamic stability issues are more likely to occur in the second stage of the seal. The lower gas film stiffness and damping in the second seal stage can lead to increased susceptibility to rotor vibrations, potentially affecting the overall performance and operational lifetime of the seal unit. The second stage failure will lead to the first stage failure within a short period. Therefore, dynamic analysis of the second (auxiliary) seal stage is more critical than the analysis of the first stage.