Effect of Laminar, Turbulent and Slip Conditions on the Dynamic Coefficients of a Dry Gas Seal

Abstract

The dynamic coefficients of a dry gas seal affect the dynamic characteristics of rotor-seal systems. Fluid films in a dry gas seal can be laminar, turbulent or with slip conditions, according to various operating conditions and design parameters. They can be defined as laminar or turbulent, depending on the Reynolds number, and as slip or non-slip, depending on the Knudsen number. However, previous research did not consider the effect of laminar, turbulent and slip conditions on the dynamic coefficients of a dry gas seal. We proposed a mathematical perturbation method to calculate the dynamic coefficients of the dry gas seal according to laminar, turbulent, and slip effects. We derived the perturbed equations of the modified Reynolds equation, which includes the effects of laminar, turbulent and slip conditions. The pressure of the modified Reynolds equation was solved using the finite element method and the Newton–Raphson method, and the perturbed pressures with respect to three degrees of freedom were calculated by substituting the calculated pressure into the perturbed equations. We verified the proposed method by comparing the simulated results with prior studies. The dynamic coefficients of a T-grooved dry gas seal were investigated according to laminar, turbulent, and slip conditions in a fluid film with different clearances.

1. Introduction

A dry gas seal is a non-contacting and dry-running mechanical face seal that prevents leakage of gas in various high-speed machines using gas as a working fluid. The dynamic coefficients of the dry gas seal, which are stiffness and damping coefficients, are important parameters to determine the vibration and the stability of a rotor–seal system during operation. Figure 1 shows the mechanical structure of a dry gas seal. It is composed of a rotating seal and a stationary seal. Pressure is developed in a small fluid gap between the rotating and stationary seals in such a way to balance the closing force and the spring force. The closing force is generated by the outer pressure of the working fluid. As Han et al. showed, the fluid in the gap can be laminar, turbulent or in slip conditions depending on the operating condition and design variables [1]. The behavior of a fluid film can be defined as laminar or turbulent, depending on the Reynolds number ($Re=\rho Uh/\mu$), and as slip or non-slip, depending on the Knudsen number ($Kn=\mu \sqrt{0.5\pi RT}/hp$), where μ, ρ, U, h, R, T, and p represent the viscosity coefficient, density, fluid velocity, fluid film thickness, gas constant, temperature, and pressure of the fluid film, respectively. A high rotating velocity and large clearance of the fluid in the grooved area (generally located at the outer part of the rotating seal) increase the Reynolds number. As such, the fluid state may change from laminar to turbulent. Additionally, slip may occur between the fluid and the solid because low pressure and small clearance of the area in the inner part of the seal increase the Knudsen number. Han et al. investigated the pressure and the leakage of dry gas seals for laminar, turbulent, and slip conditions, but they did not investigate the dynamic coefficient of the dry gas seal for these conditions.

Several researchers analyzed the dynamic coefficients of dry gas seals using the physical perturbation. The dynamic coefficients in the physical perturbation were determined by numerically differentiating the opening force of the seal with respect to finite displacements and velocities at steady-state clearance. Chen et al. analyzed the stiffness, opening force, and leakage rate of a compliant foil face gas seal (CFFGS) with different design parameters using the Reynolds equation. They showed that CFFGS has stable and good leakage performance [2]. Lu et al. calculated the gas film stiffness of dry gas seal by considering the slip effect of fluid films and optimized the groove shape to improve the gas film stiffness [3]. Other researchers also investigated dry gas seals using the physical perturbation method which extracts dynamic coefficients [4,5]. However, the physical perturbation method requires more calculation time than the mathematical perturbation method, and its accuracy is dependent on the finite displacements and velocities. Moreover, prior researchers used the Reynolds equation considering the laminar flow or slip condition of the fluid film to calculate the dynamic coefficients but did not consider the turbulent flow.

Some researchers have derived the perturbed Reynolds equation mathematically to study the dynamic coefficients of dry gas seals, which extracts dynamic coefficients from the perturbed pressure and film thickness in the quasi-static equilibrium state of the dry gas seal [6,7,8,9,10,11,12,13,14,15,16]. Chen et al. analyzed the dynamic behavior of the dry gas seal using the mathematically derived perturbed Reynolds equation [6]. The simulated leakage rate was compared with the experimental one. Ruan et al. calculated the stiffness and damping coefficients of a dry gas seal using the perturbed Reynolds equation. Additionally, they analyzed the stability of the seal by constructing a spring–damper–mass system [7]. Liu et al. analyzed the dynamic coefficients of the dry gas seal and showed that the interactions between angular and axial perturbation were negligible [8]. Other researchers also investigated dry gas seals using the mathematical perturbation of the Reynolds equation [9,10,11,12,13,14,15]. However, they did not consider turbulent flow and slip conditions of the fluid film.

In this paper, we proposed a mathematical perturbation method to calculate the dynamic coefficients of the dry gas seal according to laminar, turbulent, and slip effects. We derived the perturbed equation of the modified Reynolds equation, which includes the effects of laminar, turbulent, and slip conditions. The pressure of the modified Reynolds equation was solved using the finite element method and the Newton–Raphson method. Perturbed pressures with respect to three degrees of freedom were calculated by substituting the calculated pressure into the perturbed equations. We verified the proposed method by comparing the simulated results with prior studies. Finally, the dynamic coefficients of a T-grooved dry gas seal were investigated according to laminar, turbulent, and slip conditions in a fluid film with different clearances.

2. Method of Analysis

The conventional compressible Reynolds equation can be derived from the Navier–Stokes equation, assuming Newtonian laminar flow, ideal gas conditions, no body force, no inertial force, and no slippage, and ignoring the pressure gradient along the film thickness. On the other hand, Ng-Pan proposed a turbulent compressible Reynolds equation that includes turbulent effect in the compressible Reynolds equation [16,17], and Fukui–Kaneko proposed a modified Reynolds equation that includes the slip effect [18,19]. Recently, Han et al. [1] defined fluid state coefficients according to fluid flow to consider laminar, turbulent, and slip conditions, and proposed a modified Reynolds equation considering laminar, turbulent, and slip conditions as follows:

$$C_r\frac{h^3}{\mu }\frac{\partial }{\partial r}\left(p\frac{\partial p}{\partial r}\right)+C_{\theta }\frac{h^3}{\mu }\frac{\partial }{r\partial \theta }\left(p\frac{\partial p}{r\partial \theta }\right)=\frac{U_0}{2}\frac{\partial \left(hp\right)}{r\partial \theta }+\frac{\partial \left(hp\right)}{\partial t}$$

(1)

where h, μ, p, U, C_r and C_θ are the clearance, viscosity, pressure, fluid velocity, and fluid state coefficients of radial and circumferential directions considering laminar, turbulent, and slip conditions of the fluid film, respectively.

Table 1. Fluid state coefficients according to laminar, turbulent, and slip conditions [1].

Fluid Condition	Cr	Cθ
Laminar	1/12	1/12
Turbulent	1/Gr	1/Gθ
Slip	qp/12	qp/12

defines the fluid state coefficients according to laminar, turbulent, and slip conditions.

In Table 1, G_r, G_θ, and q_p are radial turbulent coefficient, circumferential turbulent coefficient, and slip coefficient defined in Equations (2)–(4), and

Table 2. Slip coefficients according to 1/Kn [1].

Range of Inverse Kn	c0	c1	c2	c3
5 < 1/Kn ≤ 1000	1.000	6.097	6.391	−12.812
0.15 < 1/Kn ≤ 5	0.831	7.505	0.939	−0.058
1/Kn ≤ 0.15	−13.375	12.640	0.099	0.0004

[1,16,17,18,19,20,21,22,23,24]:

$$G_r=12+0.0043R{e}^{0.96}$$

(2)

$$G_{\theta }=12+0.0136R{e}^{0.9}$$

(3)

$$q_p=c_0+c_1\left(Kn\right)+c_2{\left(Kn\right)}^2+c_3{\left(Kn\right)}^3$$

(4)

where Re and Kn are the Reynolds number and Knudsen number, respectively.

Figure 2 shows the perturbed film thickness h for a three-DOF ($z$, $\theta$_x and $\theta$_y) of a dry gas seal. The perturbed film thicknesses with respect to $z$, $\theta$_x and $\theta$_y are represented by $\mathsf{\Delta }$h, $\mathsf{\Delta }{\theta }_x$, and $\mathsf{\Delta }{\theta }_y$. The polar coordinate (r, ${\theta }^{\prime }$) is introduced to define the film thickness on a dry gas seal, and the fixed angular coordinate ${\theta }^{\prime }$ is defined from the –x axis in a counterclockwise direction. The perturbed film thickness h and pressure p are defined as follows:

$$h=h_0+\mathsf{\Delta }z-r\sin {\theta }^{\prime }\mathsf{\Delta }{\theta }_x+r\cos {\theta }^{\prime }\mathsf{\Delta }{\theta }_y$$

(5)

$$\frac{\partial h}{\partial t}=\mathsf{\Delta }\dot{z}-r\sin {\theta }^{\prime }\mathsf{\Delta }{\dot{\theta }}_x+r\cos {\theta }^{\prime }\mathsf{\Delta }{\dot{\theta }}_y$$

(6)

$$p=p_0+p_z\mathsf{\Delta }z+p_{{\theta }_x}\mathsf{\Delta }{\theta }_x+p_{{\theta }_y}\mathsf{\Delta }{\theta }_y+p_{\dot{z}}\mathsf{\Delta }\dot{z}+p_{{\dot{\theta }}_x}\mathsf{\Delta }{\dot{\theta }}_x+p_{{\dot{\theta }}_y}\mathsf{\Delta }{\dot{\theta }}_y$$

(7)

$$p_z={\left(\frac{\partial P}{\partial z}\right)}_0,p_{{\theta }_x}={\left(\frac{\partial P}{\partial {\theta }_x}\right)}_0,p_{{\theta }_y}={\left(\frac{\partial P}{\partial {\theta }_y}\right)}_0,p_{\dot{z}}={\left(\frac{\partial P}{\partial \dot{z}}\right)}_0,p_{{\dot{\theta }}_x}={\left(\frac{\partial P}{\partial {\dot{\theta }}_x}\right)}_0,p_{{\dot{\theta }}_y}={\left(\frac{\partial P}{\partial {\dot{\theta }}_y}\right)}_0$$

(8)

Here, h₀ is the film thickness in the equilibrium. $p_z,p_{{\theta }_x}$, and $p_{{\theta }_y}$ are the perturbed pressure generated by infinitesimal displacement with respect to $z,{\theta }_x$and ${\theta }_y$, respectively. Additionally, $p_{\dot{z}},p_{{\dot{\theta }}_x}$, and $p_{{\dot{\theta }}_y}$ are the perturbed pressure generated by infinitesimal velocity with respect to $\dot{z},{\dot{\theta }}_x$, and ${\dot{\theta }}_y$, respectively.

Substituting Equations (5)–(7) into (1) yields the perturbed Reynolds equation as follows:

$$\begin{array}{l}\frac{\partial }{\partial r}\left(\begin{array}{l}{C}_r\frac{{\left(h_0+\mathsf{\Delta }z-rsin{\theta }^{\text{'}}\mathsf{\Delta }{\theta }_x+r\cos {\theta }^{\text{'}}\mathsf{\Delta }{\theta }_y\right)}^3}{\mu }\left(\begin{array}{l}{p}_0+p_z\mathsf{\Delta }z+p_{{\theta }_x}\mathsf{\Delta }{\theta }_x+p_{{\theta }_y}\mathsf{\Delta }{\theta }_y\\ +p_{\dot{z}}\mathsf{\Delta }\dot{z}+p_{{\dot{\theta }}_x}\mathsf{\Delta }{\dot{\theta }}_x+p_{{\dot{\theta }}_y}\mathsf{\Delta }{\dot{\theta }}_y\end{array}\right)\\ \frac{\partial \left(p_0+p_z\mathsf{\Delta }z+p_{{\theta }_x}\mathsf{\Delta }{\theta }_x+p_{{\theta }_y}\mathsf{\Delta }{\theta }_y+p_{\dot{z}}\mathsf{\Delta }\dot{z}+p_{{\dot{\theta }}_x}\mathsf{\Delta }{\dot{\theta }}_x+p_{{\dot{\theta }}_y}\mathsf{\Delta }{\dot{\theta }}_y\right)}{\partial r}\end{array}\right)\\ +\\ \frac{\partial }{r\partial \theta }\left(\begin{array}{l}{C}_{\theta }\frac{{\left(h_0+\mathsf{\Delta }z-rsin{\theta }^{\text{'}}\mathsf{\Delta }{\theta }_x+r\cos {\theta }^{\text{'}}\mathsf{\Delta }{\theta }_y\right)}^3}{\mu }\left(\begin{array}{l}{p}_0+p_z\mathsf{\Delta }z+p_{{\theta }_x}\mathsf{\Delta }{\theta }_x+p_{{\theta }_y}\mathsf{\Delta }{\theta }_y\\ +p_{\dot{z}}\mathsf{\Delta }\dot{z}+p_{{\dot{\theta }}_x}\mathsf{\Delta }{\dot{\theta }}_x+p_{{\dot{\theta }}_y}\mathsf{\Delta }{\dot{\theta }}_y\end{array}\right)\\ \frac{\partial \left(p_0+p_z\mathsf{\Delta }z+p_{{\theta }_x}\mathsf{\Delta }{\theta }_x+p_{{\theta }_y}\mathsf{\Delta }{\theta }_y+p_{\dot{z}}\mathsf{\Delta }\dot{z}+p_{{\dot{\theta }}_x}\mathsf{\Delta }{\dot{\theta }}_x+p_{{\dot{\theta }}_y}\mathsf{\Delta }{\dot{\theta }}_y\right)}{r\partial \theta }\end{array}\right)\\ =\\ \frac{r\omega }{2}\frac{\partial }{r\partial \theta }\left\{\left(h_0+\mathsf{\Delta }z-rsin{\theta }^{\text{'}}\mathsf{\Delta }{\theta }_x+r\cos {\theta }^{\text{'}}\mathsf{\Delta }{\theta }_y\right)\left(\begin{array}{l}{p}_0+p_z\mathsf{\Delta }z+p_{{\theta }_x}\mathsf{\Delta }{\theta }_x+p_{{\theta }_y}\mathsf{\Delta }{\theta }_y\\ +p_{\dot{z}}\mathsf{\Delta }\dot{z}+p_{{\dot{\theta }}_x}\mathsf{\Delta }{\dot{\theta }}_x+p_{{\dot{\theta }}_y}\mathsf{\Delta }{\dot{\theta }}_y\end{array}\right)\right\}\\ +\frac{\partial }{\partial t}\left\{\left(h_0+\mathsf{\Delta }z-rsin{\theta }^{\text{'}}\mathsf{\Delta }{\theta }_x+r\cos {\theta }^{\text{'}}\mathsf{\Delta }{\theta }_y\right)\left(\begin{array}{l}{p}_0+p_z\mathsf{\Delta }z+p_{{\theta }_x}\mathsf{\Delta }{\theta }_x+p_{{\theta }_y}\mathsf{\Delta }{\theta }_y\\ +p_{\dot{z}}\mathsf{\Delta }\dot{z}+p_{{\dot{\theta }}_x}\mathsf{\Delta }{\dot{\theta }}_x+p_{{\dot{\theta }}_y}\mathsf{\Delta }{\dot{\theta }}_y\end{array}\right)\right\}\end{array}$$

(9)

By expanding Equation (9) and assuming that higher order terms are negligible, the governing equations for each variable can be expressed as follows:

$$\frac{\partial }{\partial r}\left(C_r\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_0}{\partial r}\right)+\frac{\partial }{r\partial \theta }\left(C_{\theta }\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_0}{r\partial \theta }\right)=\frac{r\omega }{2}\frac{\partial \left(h_0{p}_0\right)}{r\partial \theta }$$

(10)

$$\begin{array}{l}\frac{\partial }{\partial r}\left(C_r\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_z}{\partial r}+C_r\frac{h_0{}^3}{\mu }{p}_z\frac{\partial p_0}{\partial r}\right)\\ +\frac{\partial }{r\partial \theta }\left(C_{\theta }\frac{h_0{}^3}{\mu }{p}_z\frac{\partial p_0}{r\partial \theta }+C_{\theta }\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_z}{r\partial \theta }\right)-\frac{r\omega }{2}\frac{\partial }{r\partial \theta }\left(h_0{p}_z\right)\\ =-\frac{\partial }{\partial r}\left(C_r\frac{3h_0{}^2}{\mu }{p}_0\frac{\partial p_0}{\partial r}\right)-\frac{\partial }{r\partial \theta }\left(C_{\theta }\frac{3h_0{}^2}{\mu }{p}_0\frac{\partial p_0}{r\partial \theta }\right)\\ +\frac{r\omega }{2}\frac{\partial }{r\partial \theta }\left(p_0\right)+h_0{p}_{\dot{z}}\end{array}$$

(11)

$$\begin{array}{l}\frac{\partial }{\partial r}\left(C_r\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_{{\theta }_x}}{\partial r}+C_r\frac{h_0{}^3}{\mu }{p}_{{\theta }_x}\frac{\partial p_0}{\partial r}\right)\\ +\frac{\partial }{r\partial \theta }\left(C_{\theta }\frac{h_0{}^3}{\mu }{p}_{{\theta }_x}\frac{\partial p_0}{r\partial \theta }+C_{\theta }\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_{{\theta }_x}}{r\partial \theta }\right)-\frac{r\omega }{2}\frac{\partial }{r\partial \theta }\left(h_0{p}_{{\theta }_x}\right)\\ =\frac{\partial }{\partial r}\left(C_r\frac{3h_0{}^2}{\mu }r\sin {\theta }^{\prime }{p}_0\frac{\partial p_0}{\partial r}\right)+\frac{\partial }{r\partial \theta }\left(C_{\theta }\frac{3h_0{}^2}{\mu }r\sin {\theta }^{\prime }{p}_0\frac{\partial p_0}{r\partial \theta }\right)\\ +\frac{r\omega }{2}\frac{\partial }{r\partial \theta }\left(-r\sin {\theta }^{\prime }{p}_0\right)+h_0{p}_{{\dot{\theta }}_x}\end{array}$$

(12)

$$\begin{array}{l}\frac{\partial }{\partial r}\left(C_r\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_{{\theta }_y}}{\partial r}+C_r\frac{h_0{}^3}{\mu }{p}_{{\theta }_y}\frac{\partial p_0}{\partial r}\right)\\ +\frac{\partial }{r\partial \theta }\left(C_{\theta }\frac{h_0{}^3}{\mu }{p}_{{\theta }_y}\frac{\partial p_0}{r\partial \theta }+C_{\theta }\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_{{\theta }_y}}{r\partial \theta }\right)-\frac{r\omega }{2}\frac{\partial }{r\partial \theta }\left(h_0{p}_{{\theta }_y}\right)\\ =-\frac{\partial }{\partial r}\left(C_r\frac{3h_0{}^2}{\mu }r\cos {\theta }^{\prime }{p}_0\frac{\partial p_0}{\partial r}\right)-\frac{\partial }{r\partial \theta }\left(C_{\theta }\frac{3h_0{}^2}{\mu }r\cos {\theta }^{\prime }{p}_0\frac{\partial p_0}{r\partial \theta }\right)\\ +\frac{r\omega }{2}\frac{\partial }{r\partial \theta }\left(r\cos {\theta }^{\prime }{p}_0\right)+h_0{p}_{{\dot{\theta }}_y}\end{array}$$

(13)

$$\begin{array}{l}\frac{\partial }{\partial r}\left(C_r\frac{h_0{}^3}{\mu }{p}_{\dot{z}}\frac{\partial p_0}{\partial r}+C_r\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_{\dot{z}}}{\partial r}\right)\\ +\frac{\partial }{r\partial \theta }\left(C_{\theta }\frac{h_0{}^3}{\mu }{p}_{\dot{z}}\frac{\partial p_0}{r\partial \theta }+C_{\theta }\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_{\dot{z}}}{r\partial \theta }\right)-\frac{r\omega }{2}\frac{\partial }{r\partial \theta }\left(h_0{p}_{\dot{z}}\right)\\ =p_0+h_0{p}_z\end{array}$$

(14)

$$\begin{array}{l}\frac{\partial }{\partial r}\left(C_r\frac{h_0{}^3}{\mu }{p}_{{\dot{\theta }}_x}\frac{\partial p_0}{\partial r}+C_r\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_{{\dot{\theta }}_x}}{\partial r}\right)\\ +\frac{\partial }{r\partial \theta }\left(C_{\theta }\frac{h_0{}^3}{\mu }{p}_{{\dot{\theta }}_x}\frac{\partial p_0}{r\partial \theta }+C_{\theta }\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_{{\dot{\theta }}_x}}{r\partial \theta }\right)-\frac{r\omega }{2}\frac{\partial }{r\partial \theta }\left(h_0{p}_{{\dot{\theta }}_x}\right)\\ =-p_{0}r\sin {\theta }^{\prime }+h_0{p}_{{\theta }_x}\end{array}$$

(15)

$$\begin{array}{l}\frac{\partial }{\partial r}\left(C_r\frac{h_0{}^3}{\mu }{p}_{{\dot{\theta }}_y}\frac{\partial p_0}{\partial r}+C_r\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_{{\dot{\theta }}_y}}{\partial r}\right)\\ +\frac{\partial }{r\partial \theta }\left(C_{\theta }\frac{h_0{}^3}{\mu }{p}_{{\dot{\theta }}_y}\frac{\partial p_0}{r\partial \theta }+C_{\theta }\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_{{\dot{\theta }}_y}}{r\partial \theta }\right)-\frac{r\omega }{2}\frac{\partial }{r\partial \theta }\left(h_0{p}_{{\dot{\theta }}_y}\right)\\ =p_{0}r\cos {\theta }^{\prime }+h_0{p}_{{\theta }_y}\end{array}$$

(16)

Equations (10)–(16) are the governing equations of $p_0,p_z,p_{{\theta }_x}, p_{{\theta }_y}, p_{\dot{z}},p_{{\dot{\theta }}_x}$, and $p_{{\dot{\theta }}_y}$. Equation (10) is, in fact, the modified Reynolds equation in Equation (1). A finite element equation of the Reynolds equation can be obtained by multiplying Equation (10) with the weighting function and integrating using Green’s theorem:

$$\begin{array}{l}w{\displaystyle \underset{\mathsf{\Gamma }}{\int }\left(C_r\frac{h_0^3}{\mu }{p}_0\frac{\partial p_0}{\partial r}+C_{\theta }\frac{h_0^3}{\mu }{p}_0\frac{\partial p_0}{r\partial \theta }-\frac{r\omega \left(h_0{p}_0\right)}{2}\right)\cdot nd\mathsf{\Gamma }}\\ -{\displaystyle \underset{A}{\int }\left(C_r\frac{h^3}{\mu }\frac{\partial w}{\partial r}{p}_0\frac{\partial p_0}{\partial r}+C_{\theta }\frac{h^3}{\mu }\frac{\partial w}{r\partial \theta }{p}_0\frac{\partial p_0}{r\partial \theta }-\frac{r\omega h_0}{2}{p}_0\frac{\partial w}{r\partial \theta }\right)dA}=0\end{array}$$

(17)

Here, $\mathsf{\Gamma },w$, and n are the boundary, weighting function, and normal unit vector of the boundary, respectively. The boundary integral terms of Equation (17) are zero because the weighting functions at the boundary are zero. The pressure in a four-node element can be expressed by the nodal pressure P_e and shape function N as follows:

$$p=N^{\mathrm{T}}{P}_e$$

(18)

Similarly, the weighting function can be expressed by an arbitrary vector η_e as follows:

$$w={\mathsf{\eta }}_e^{\mathrm{T}}N$$

(19)

Substituting Equations (18) and (19) into (17) yields the local matrix of the finite-element equation:

$${\displaystyle \underset{A}{\int }{\mathsf{\eta }}_e{}^{\mathrm{T}}\left(\begin{array}{c}{C}_r\frac{h^3}{\mu }\frac{\partial N}{\partial r}{{\rm N}}^{\mathrm{T}}{P}_e\frac{\partial N^{\mathrm{T}}}{\partial r}{P}_e+C_{\theta }\frac{h^3}{\mu }\frac{\partial {\rm N}}{r\partial \theta }{{\rm N}}^{\mathrm{T}}{P}_e\frac{\partial N^{\mathrm{T}}}{r\partial \theta }{P}_e\\ -\frac{r\omega h}{2}\frac{\partial {\rm N}}{r\partial \theta }{N}^{\mathrm{T}}{P}_e\end{array}\right)dA}=0$$

(20)

Equation (20) is non-linear and can be solved by using the Newton–Raphson method as follows [1]:

$$\begin{array}{l}{R}^{\left(n\right)}+{\alpha }_{relax}\frac{\partial R^{\left(n\right)}}{\partial P^{\left(n\right)}}\mathsf{\Delta }{P}^{\left(n\right)}=0\\ R={\displaystyle \underset{A}{\int }\left\{\frac{h^3}{\mu }{{\rm N}}^{T}P\left(C_r\frac{\partial N}{\partial r}\frac{\partial N^T}{\partial r}P+C_{\theta }\frac{\partial {\rm N}}{r\partial \theta }\frac{\partial N^T}{r\partial \theta }P\right)-\frac{r\omega h}{2}\frac{\partial {\rm N}}{r\partial \theta }{N}^{T}P\right\}dA}=0\\ \frac{\partial R}{\partial P}={\displaystyle \underset{A}{\int }\left\{\begin{array}{l}\frac{h^3}{\mu }\left(\begin{array}{l}{N}^{T}P\left(C_r\frac{\partial N}{\partial r}\frac{\partial N^T}{\partial r}+C_{\theta }\frac{\partial N}{r\partial \theta }\frac{\partial N^T}{r\partial \theta }\right)\\ +\left(C_r\frac{\partial N}{\partial r}\frac{\partial N^T}{\partial r}+C_{\theta }\frac{\partial N}{r\partial \theta }\frac{\partial N^T}{r\partial \theta }\right)P{N}^T\\ +\frac{\partial N}{\partial r}\frac{\partial N^T}{\partial r}P{N}^{T}P\frac{\partial C_r}{\partial P}+\frac{\partial N}{r\partial \theta }\frac{\partial N^T}{r\partial \theta }P{N}^{T}P\frac{\partial C_{\theta }}{\partial P}\end{array}\right)\\ -\frac{r\omega h}{2}\frac{\partial {\rm N}}{r\partial \theta }{N}^T\end{array}\right\}dA}\end{array}$$

(21)

where R and ∂R/∂P are Equation (20) and the pressure derivative of Equation (20), respectively, and n is the iteration number. The relaxation factor used to increase the stability of the analysis results, α_relax, was set to 0.5. The local matrices of R and the local matrix of ∂R/∂P were assembled to generate the global matrices of R and ∂R/∂P, respectively. The analysis was repeated until the ratio of the summation of the all components of the global matrix of R to that of the global matrix of ∂R/∂P became less than 10⁻⁴, and in most of the analyses in this paper, they converged within 20 iterations. Equation (22) shows the boundary condition applied to solve the modified Reynolds equation in which internal and external pressures are applied to internal and external boundaries and pressure along the circumferential direction is continuous.

$$p\left(r_i,\theta \right)=p_i,p\left(r_o,\theta \right)=p_e,p\left(r,\theta \right)=p\left(r,\theta +2\pi \right)$$

(22)

where $p_i$ and $p_e$ are internal and external pressures applied to internal and external boundaries.

The finite element equations of the perturbed modified Reynolds equation can be obtained by multiplying the weight function and integrating using Green’s theorem, as in Equation (17). The finite element equations of the left-hand terms are as follows:

$$\begin{array}{l}w{\displaystyle \underset{\mathsf{\Gamma }}{\int }\left(\begin{array}{l}{C}_r\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_i}{\partial r}+C_r\frac{h_0{}^3}{\mu }{p}_i\frac{\partial p_0}{\partial r}+C_{\theta }\frac{h_0{}^3}{\mu }{p}_0\frac{\partial p_i}{r\partial \theta }\\ +C_{\theta }\frac{h_0{}^3}{\mu }{p}_i\frac{\partial p_0}{r\partial \theta }-{\tilde{V}}_{\theta }\frac{\partial \left(h_0{p}_i\right)}{r\partial \theta }\end{array}\right)}\cdot \widehat{n}d\mathsf{\Gamma }\\ -{\displaystyle \underset{A}{\int }\left(\begin{array}{l}{C}_r\frac{h_0{}^3}{\mu }\frac{\partial w}{\partial r}{p}_0\frac{\partial p_i}{\partial r}+C_r\frac{h_0{}^3}{\mu }\frac{\partial w}{\partial r}{p}_i\frac{\partial p_0}{\partial r}+C_{\theta }\frac{h_0{}^3}{\mu }\frac{\partial w}{r\partial \theta }{p}_0\frac{\partial p_i}{r\partial \theta }\\ +C_{\theta }\frac{h_0{}^3}{\mu }\frac{\partial w}{r\partial \theta }{p}_i\frac{\partial p_0}{r\partial \theta }-{\tilde{V}}_{\theta }\frac{\partial w}{r\partial \theta }{h}_0{p}_i\end{array}\right)}dA\end{array}$$

(23)

where $i=\mathrm{z},{\theta }_x, {\theta }_y,\dot{\mathrm{z}},{\dot{ \theta }}_x,{\dot{\theta }}_y$. The boundary integral terms of Equation (23) are zero because the weighting functions at the boundary are zero. Substituting Equations (18) and (19) into (23) yields the left-hand finite element equation of the local matrix:

$${\displaystyle \underset{A}{\int }{\mathsf{\eta }}^{\mathrm{T}}\left(\begin{array}{l}{C}_r\frac{h_0{}^3}{\mu }\frac{\partial N}{\partial r}{{\rm N}}^{\mathrm{T}}{P}_0\frac{\partial N^{\mathrm{T}}}{\partial r}{P}_i+C_r\frac{h_0{}^3}{\mu }\frac{\partial N}{\partial r}{N}^{\mathrm{T}}{P}_i\frac{\partial {{\rm N}}^{\mathrm{T}}}{\partial r}{P}_0\\ +C_{\theta }\frac{h_0{}^3}{\mu }\frac{\partial N}{r\partial \theta }{{\rm N}}^{\mathrm{T}}{P}_0\frac{\partial {{\rm N}}^{\mathrm{T}}}{r\partial \theta }{P}_i+C_{\theta }\frac{h_0{}^3}{\mu }\frac{\partial N}{r\partial \theta }{N}^{\mathrm{T}}{P}_i\frac{\partial {{\rm N}}^{\mathrm{T}}}{r\partial \theta }{P}_0\\ -\frac{r\omega }{2}{h}_0\frac{\partial N}{r\partial \theta }{N}^{\mathrm{T}}{P}_i\end{array}\right)}dA$$

(24)

where $i=z,{\theta }_x,{\theta }_y,\dot{z},{\dot{\theta }}_x,{\dot{\theta }}_y$.

The right-hand finite element equations of the local matrices corresponding to $p_z, p_{{\theta }_x},p_{{\theta }_y}, p_{\dot{z}}, p_{{\dot{\theta }}_x}$, and $p_{{\dot{\theta }}_y}$ can be obtained similarly, as follows:

$${\displaystyle \underset{A}{\int }{\mathsf{\eta }}^{\mathrm{T}}\left(\begin{array}{l}-C_r\frac{3h_0{}^2}{\mu }\frac{\partial N}{\partial r}{{\rm N}}^{\mathrm{T}}{P}_0\frac{\partial {{\rm N}}^{\mathrm{T}}{P}_0}{\partial r}-C_{\theta }\frac{3h_0{}^2}{\mu }\frac{\partial N}{\partial r}{{\rm N}}^{\mathrm{T}}{P}_0\frac{\partial {{\rm N}}^{\mathrm{T}}{P}_0}{r\partial \theta }\\ +\frac{r\omega }{2}\frac{\partial {\rm N}}{r\partial \theta }\left({{\rm N}}^{\mathrm{T}}{P}_0\right)-h_{0}N{{\rm N}}^{\mathrm{T}}{P}_{\dot{z}}\end{array}\right)}dA$$

(25)

$${\displaystyle \underset{A}{\int }{\mathsf{\eta }}^{\mathrm{T}}\left(\begin{array}{l}{C}_r\frac{3h_0{}^2}{\mu }r\sin {\theta }^{\prime }\frac{\partial N}{\partial r}{{\rm N}}^{\mathrm{T}}{P}_0\frac{\partial {{\rm N}}^{\mathrm{T}}{P}_0}{\partial r}+C_{\theta }\frac{3h_0{}^2}{\mu }r\sin {\theta }^{\prime }\frac{\partial N}{r\partial \theta }{{\rm N}}^{\mathrm{T}}{P}_0\frac{\partial {{\rm N}}^{\mathrm{T}}{P}_0}{r\partial \theta }\\ +\frac{r\omega }{2}\frac{\partial N}{r\partial \theta }\left(-r\sin {\theta }^{\prime }{{\rm N}}^{\mathrm{T}}{P}_0\right)-h_{0}N{{\rm N}}^{\mathrm{T}}{P}_{{\dot{\theta }}_x}\end{array}\right)}dA$$

(26)

$${\displaystyle \underset{A}{\int }{\mathsf{\eta }}^{\mathrm{T}}\left(\begin{array}{l}-C_r\frac{3h_0{}^2}{\mu }r\cos {\theta }^{\prime }\frac{\partial N}{\partial r}{{\rm N}}^{\mathrm{T}}{P}_0\frac{\partial {{\rm N}}^{\mathrm{T}}{P}_0}{\partial r}-C_{\theta }\frac{3h_0{}^2}{\mu }r\cos {\theta }^{\prime }\frac{\partial N}{r\partial \theta }{{\rm N}}^{\mathrm{T}}{P}_0\frac{\partial {{\rm N}}^{\mathrm{T}}{P}_0}{r\partial \theta }\\ +\frac{r\omega }{2}\frac{\partial N}{r\partial \theta }\left(r\cos {\theta }^{\prime }{{\rm N}}^{\mathrm{T}}{P}_0\right)-h_{0}N{{\rm N}}^{\mathrm{T}}{P}_{{\dot{\theta }}_y}\end{array}\right)}dA$$

(27)

$$-{\displaystyle \underset{A}{\int }{\mathsf{\eta }}^{\mathrm{T}}\left(N{N}^{\mathrm{T}}{P}_0+h_{0}N{N}^{\mathrm{T}}{P}_z\right)}dA$$

(28)

$$-{\displaystyle \underset{A}{\int }{\mathsf{\eta }}^{\mathrm{T}}\left(-r\sin {\theta }^{\prime }N{{\rm N}}^{\mathrm{T}}{P}_0+h_{0}N{{\rm N}}^{\mathrm{T}}{P}_{{\theta }_x}\right)}dA$$

(29)

$$-{\displaystyle \underset{A}{\int }{\mathsf{\eta }}^{\mathrm{T}}\left(r\cos {\theta }^{\prime }N{{\rm N}}^{\mathrm{T}}{P}_0+h_{0}N{{\rm N}}^{\mathrm{T}}{P}_{{\theta }_y}\right)}dA$$

(30)

Once the calculated pressure from Equation (21) is substituted into the perturbed Equations (24)–(30), the local matrix equation of the perturbed pressure can be written as linear equations with respect to the perturbed pressure, as shown in Equations (31) and (32).

$$\begin{array}{l}\left[k\right]\left\{p_j\right\}=\left[F\right]+\left[A\right]\left[p_k\right]\\ \left[k\right]\left\{p_k\right\}=\left[F\right]+\left[B\right]\left[p_j\right]\end{array}$$

(31)

$$\left[\begin{array}{cc}\left[k\right]& -\left[A\right]\\ -\left[B\right]& \left[k\right]\end{array}\right]\left\{\begin{array}{c}{p}_j\\ p_k\end{array}\right\}=\left[\begin{array}{c}\left[F\right]\\ \left[F\right]\end{array}\right]$$

(32)

where $j=z,{\theta }_x,{\theta }_y$ and k$=\dot{z},{\dot{ \theta }}_x,{\dot{ \theta }}_y$. Subsequently, a global matrix equation can be assembled, and the perturbed pressure can be determined. Equation (33) shows the boundary condition applied to solve the perturbed Reynolds equations in which internal and external perturbed pressures are zero at the internal and external boundaries and perturbed pressure along the circumferential direction is continuous.

$$p_k(r_i,\theta )=0,p_k(r_o,\theta )=0,p_k(r,\theta )=p_k(r,\theta +2\pi )$$

(33)

where k is $z,{\theta }_x,{\theta }_y, \dot{z},{\dot{\theta }}_x$, and ${\dot{\theta }}_y$, respectively.

The stiffness and damping matrices for a dry gas seal can be calculated from the calculated perturbed pressure as follows:

$$K=\left[\begin{array}{ccc}{\mathrm{K}}_{zz}& {\mathrm{K}}_{z{\theta }_x}& {\mathrm{K}}_{z{\theta }_y}\\ {\mathrm{K}}_{{\theta }_{x}z}& {\mathrm{K}}_{{\theta }_x{\theta }_x}& {\mathrm{K}}_{{\theta }_x{\theta }_y}\\ {\mathrm{K}}_{{\theta }_{y}z}& {\mathrm{K}}_{{\theta }_y{\theta }_x}& {\mathrm{K}}_{{\theta }_y{\theta }_y}\end{array}\right]=-{\displaystyle \underset{A}{\iint }\left[\begin{array}{ccc}{p}_z& p_{{\theta }_x}& p_{{\theta }_y}\\ -p_{z}r\sin \theta & -p_{{\theta }_x}r\sin \theta & -p_{{\theta }_y}r\sin \theta \\ p_{z}r\cos \theta & p_{{\theta }_x}r\cos \theta & p_{{\theta }_y}r\cos \theta \end{array}\right]}rdrd\theta$$

(34)

$$C=\left[\begin{array}{ccc}{\mathrm{C}}_{zz}& {\mathrm{C}}_{z{\theta }_x}& {\mathrm{C}}_{z{\theta }_y}\\ {\mathrm{C}}_{{\theta }_{x}z}& {\mathrm{C}}_{{\theta }_x{\theta }_x}& {\mathrm{C}}_{{\theta }_x{\theta }_y}\\ {\mathrm{C}}_{{\theta }_{y}z}& {\mathrm{C}}_{{\theta }_y{\theta }_x}& {\mathrm{C}}_{{\theta }_y{\theta }_y}\end{array}\right]=-{\displaystyle \underset{A}{\iint }\left[\begin{array}{ccc}{p}_{\dot{z}}& p_{{\dot{\theta }}_x}& p_{{\dot{\theta }}_y}\\ -p_{\dot{z}}r\sin \theta & -p_{{\dot{\theta }}_x}r\sin \theta & -p_{{\dot{\theta }}_y}r\sin \theta \\ p_{\dot{z}}r\cos \theta & p_{{\dot{\theta }}_x}r\cos \theta & p_{{\dot{\theta }}_y}r\cos \theta \end{array}\right]}rdrd\theta$$

(35)

3. Numerical Verification

To verify the developed equations and program for the perturbed Reynolds equation, we developed a finite element model of the dry gas seal analyzed by Faria [25]. Faria analyzed a spiral grooved dry gas seal whose external radius, groove radius, internal radius, groove depth, groove angle, and number of grooves were 88.9 mm, 76.4 mm, 71.1 mm, 2.54 μm, 20°, and 12, respectively. The rotating velocity, external pressure, internal pressure, and clearance were 15,000 rpm, 5.05 bar, 1.01 bar, and 2.54 μm, respectively. The developed finite element model consisted of 7920 quadrilateral elements, similar to Faria’s model. The opening force, stiffness coefficient, and damping coefficient to axial direction were calculated and compared with the result obtained by Faria.

Table 3. Comparison of the stiffness and damping coefficients with Faria’s results.

	Opening Force [N]	Stiffness Coefficient (Kzz) [N/m]	Damping Coefficient (Czz) [Ns/m]
Present analysis	4840	1.34 × 109	4.20 × 104
Faria’s result [25]	4837	1.32 × 109	4.23 × 104

shows the simulated results of this study and Faria. Since Faria solved the compressible Reynolds equation including only the effect of laminar flow, we calculated the opening force, stiffness, and damping coefficients according to laminar flow. The opening force, stiffness, and damping coefficients of this study are very close to those of Faria. The opening forces for this study and that of Faria are 4840 N and 4837 N, respectively, and the difference was less than 0.1%. The stiffness coefficient along the z-direction in this study and that of Faria were 1.34 × 10⁹ N/m and 1.32 × 10⁹ N/m, and the differences were less than 1.5%. The damping coefficient along z-direction used of this study and that of Faria are 4.20 × 10⁴ Ns/m and 4.23 × 10⁴ Ns/m, and the difference was less than 1%.

4. Results and Discussion

4.1. Perturbed Pressure of the Fluid Film in a Dry Gas Seal according to Laminar, Turbulent, and Slip Conditions in the Fluid Film

We analyzed the dynamic coefficients of the fluid film on a T-grooved dry gas seal under laminar, turbulent, and slip conditions. The analysis was performed using three different methods depending on the governing equation. First, the L_method, which can only consider laminar flow, was used to solve the laminar compressible Reynolds equation. Second, the LT_method was used to solve the modified Reynolds equation by considering the laminar and turbulent behaviors of the fluid film. Third, the LTS_method was used to solve the modified Reynolds equation by considering the laminar, turbulent and slip conditions of the fluid film. Figure 3a, shows the geometry of the T-grooved dry gas seal. The FE model of the dry gas seal was developed with 180,000 quadrilateral elements, and Figure 4b shows the FE model corresponding to the boxed area of Figure 4a which has one groove-ridge area. We checked the convergence of the stiffness and damping coefficients by increasing the number of finite elements of the FE model. It shows that 180,000 quadrilateral elements of the FE model used in this study were sufficient to guarantee the convergence of the stiffness and damping coefficients.

Table 4. Geometric parameters of the grooved seal.

Parameter	Value
Outer radius, ro [mm]	69.5
Groove radius, rg [mm]	62.4
Internal radius, ri [mm]	55.7
Groove number	18
Ratio of ridge and groove	0.65
Groove depth [mm]	0.01

shows the design parameters of the T-grooved dry gas seal used in the numerical analysis. The internal and external pressures were assumed to be 1 bar and 60 bar. The clearance increased from 2.5 μm to 4 μm with increments of 0.5 μm. The rotating seal was rotated clockwise at 25,000 rpm. All analyses were performed under isothermal and isoviscous conditions. Figure 4 shows the distribution of the perturbed pressure $p_z$ corresponding to the L_method, LT_method, and LTS_method for a T-groove when the clearance was 2.5 μm. Figure 5 shows the pressure distribution of the perturbed pressure $p_{\dot{z}}$ corresponding to the L_method, LT_method, and LTS_method for a T-groove when the clearance was 2.5 μm. Figure 6a shows the Reynolds number along the radial direction at θ = 0° and θ = 15°. The fluid between the stator and the rotor is assumed to change from laminar to turbulent flow when the Reynolds number exceeds 1000 [1,16,17]. Figure 6b shows the Knudsen number along the radius at θ = 0° and θ =15. A slip between the fluid and wall is assumed to occur when the Knudsen number is greater than 0.001 [1,18,19].

Figure 4 shows that the maximum perturbed pressures $p_z$ obtained using the LT_method and LTS_method were greater than that obtained using the L_method. As shown in Figure 6a, the ridge part of the outer plane, where the maximum perturbed pressure occurred, had a large Reynolds number, indicating turbulent flow. Figure 6b also shows that the inner plane, where the perturbed pressure is small, has a large Knudsen number, indicating the occurrence of slippage. Therefore, the maximum perturbed pressure increased when the turbulent effect was considered. Furthermore, there was no difference in the maximum perturbed pressure $p_z$ obtained using the LT_method and LTS_method because turbulent flow plays a dominant role in generating maximum perturbed pressure of the ridge part and because slippage did not occur in the groove and ridge parts of the outer plane, as shown in Figure 6b.

Figure 5 shows that the maximum perturbed pressures $p_{\dot{z}}$ obtained using the LT_method and LTS_method were greater than that obtained using the L_method. As shown in Figure 6a, the internal plane area (with a radius of around 59.8 mm) where the maximum perturbed pressure occurred had a large Reynolds number to generate turbulent flow. Therefore, the maximum perturbed pressure $p_{\dot{z}}$ increased when the turbulent effect was considered. Furthermore, there was no difference in the maximum perturbed pressure $p_{\dot{z}}$ obtained using the LT_method and LTS_method because the turbulent flow plays a dominant role in generating maximum perturbed pressure of the internal plane area and because slippage did not occur in that area as shown in Figure 6b.

4.2. Dynamic Coefficients of the Fluid Film in a Dry Gas Seal according to Clearance of the Fluid Film and External Pressure

Table 5. Stiffness coefficients according to the clearance.

Clearance [μm]	Stiffness Coefficients (K)
	Kzz [×107 N/m]			Difference [%]			Kθxθx [×105 Nm/rad]			Difference [%]
	L	LT	LTS	LT–L	LTS–L	LTS–LT	L	LT	LTS	LT–L	LTS–L	LTS–LT
2.5	74.86	249.56	248.63	233.4	232.1	−0.4	11.48	46.36	46.20	303.9	302.6	−0.3
3.0	63.17	196.07	195.42	210.4	209.4	−0.3	9.31	35.61	35.50	282.6	281.4	−0.3
3.5	55.34	155.19	154.73	180.4	179.6	−0.3	7.91	27.44	27.37	246.8	245.9	−0.3
4.0	49.92	124.35	124.03	149.1	148.5	−0.3	6.97	21.32	21.27	205.7	205.0	−0.2

and

Table 6. Damping coefficients according to the clearance.

Clearance [μm]	Damping Coefficients (C)
	Czz [×104 Ns/m]			Difference [%]			Cθxθx [×10 Nsm/rad]			Difference [%]
	L	LT	LTS	LT–L	LTS–L	LTS–LT	L	LT	LTS	LT–L	LTS–L	LTS–LT
2.5	27.75	34.26	34.19	23.4	23.2	−0.2	53.34	66.57	66.44	24.8	24.6	−0.2
3.0	17.99	23.90	23.86	32.8	32.6	−0.2	34.99	47.35	47.27	35.3	35.1	−0.2
3.5	12.58	18.05	18.02	43.4	43.2	−0.2	24.69	36.30	36.25	47.1	46.8	−0.1
4.0	9.27	14.31	14.28	54.3	54.1	−0.2	18.31	29.12	29.08	59.0	58.8	−0.1

show the stiffness and damping coefficients of a dry gas seal according to the clearance, as calculated by the L_method (L), LT_method (LT), and LTS_method (LTS). In all three analysis methods, the dynamic coefficients decreased as the clearance increased.

The stiffness coefficient of the LT_method was greater than that of the L_method. This can be explained by Figure 7a, which shows the difference of the perturbed pressure $p_z$ between the LT_method and L_method at 2.5 μm clearance. Figure 7c shows the geometry corresponding to Figure 7a,b. As shown in Figure 7a, the increased area of the perturbed pressure ($p_z$) (red and green in the contour plot) is larger than the decreased area of the perturbed pressure (blue and sky-blue in the contour plot), and the maximum value of the increased area of the perturbed pressure (+1.12 × 10¹¹ N/m) is larger than the maximum value of the decreased area of the perturbed pressure (−0.32 × 10¹¹ N/m). Additionally, the damping coefficient of the LT_method was greater than that of the L_method (Table 6). This can be explained by Figure 7b, which shows the difference in the perturbed pressure $p_{\dot{z}}$ between the LT_method and L_method at 2.5 μm clearance. As shown in Figure 7b, the increased area of the perturbed pressure ($p_{\dot{z}}$) (red and green in the contour plot) was larger than the decreased area of the perturbed pressure (blue and sky-blue in the contour plot), and the maximum value of the increased perturbed area of the perturbed pressure (+3.70 × 10⁷ Ns/m) was larger than the maximum value of the decreased area of the perturbed pressure (−1.18 × 10⁷ Ns/m).

5. Conclusions

In this study, we proposed a mathematical perturbation method of the modified Reynolds equation that includes the effects of laminar, turbulent, and slip behaviors of a fluid film. The pressure of the modified Reynolds equation was solved using the finite element method and the Newton–Raphson method, and the perturbed pressures with respect to three degrees of freedom were calculated by substituting the calculated pressure into the perturbed equations. We verified the proposed method by comparing the simulated results of the previous study. The dynamic coefficients of a T-grooved dry gas seal were investigated according to laminar, turbulent, and slip conditions in the fluid film with different clearances. This study shows that the stiffness and damping coefficients decreased as the clearance increased. The turbulent effect increased the stiffness coefficients, and it decreased as the clearance increased. Additionally, the turbulent effect increased the damping coefficients. The slip effect decreased the stiffness and damping coefficients, but the slip effect was smaller than the turbulent effect. This study shows that the laminar, turbulent, and slip conditions of a fluid film on a dry gas seal should be considered to predict the dynamic coefficients of a dry gas seal accurately. This study will make it possible to develop a dynamic model of the rotor-seal system with predicted dynamic coefficients of a dry gas seal, and it will eventually contribute to developing a robust and stable dry gas seal in various operating environments.

6. Future Work

Herein, we studied a non-contact dry-running mechanical face seal system. However, solid contact between rotating and stationary seal faces may occur due to angular misalignment, assembly tolerances, or large closing force. Friction and wear are important sources to degrade operating performance and shorten a sound life of dry gas seals [26,27,28,29,30,31]. Hong et al. proposed a numerical method to investigate the friction characteristics of oil film journal bearings considering elastomeric lubrication and contact force [32]. We hope to extend this study by considering elastomeric lubrication and the contact mechanism.