Lubrication Analysis of a Mechanical Seal Considering the Mixed Lubricant State of Gas and Liquid During External Power Shutdown of a Reactor Cooling Pump

Abstract

We proposed a method to calculate the pressure, opening force, and leakage rate in a mechanical seal under the mixed lubricant state of a gas and liquid for the mechanical seal in a reactor cooling pump (RCP) during external power loss. We calculated the pressure by solving the nonlinear finite element equation composed of the linear Reynolds equation of an incompressible liquid lubricant and the nonlinear Reynolds equation of a compressible gas lubricant using the Newton–Raphson method. In addition, we calculated the temperature distribution by solving the two-dimensional energy equation utilizing the finite element method. Additionally, we included the turbulence effect in the incompressible liquid lubricant and the turbulence and slip effects in the compressible gas lubricant. The accuracy of the developed program was validated by comparing the simulated opening force and leakage rate of both the mechanical seal with the liquid lubricant and the mechanical seal with the gas lubricant with prior research. Our analysis shows that in high-temperature environments, the increase in the gas region at the lubrication surface leads to a decrease in pressure and opening force and an increase in the leakage rate. Conversely, as the outer pressure increases, the gas region decreases, resulting in an increase in pressure, opening force, and leakage rate.

1. Introduction

The reactor cooling pump (RCP) is a device used to remove heat from a reactor by circulating pressurized coolant between the reactor and the steam generator in a pressurized water reactor at a nuclear power plant. To ensure the safe operation of the RCP, a three-stage mechanical seal is typically employed to prevent coolant leakage. Figure 1 shows a mechanical seal of an RCP with a rotating seal connected to the shaft and a stationary seal fixed to the housing. A dodecagonal groove is inscribed at the surface of the stationary seal. The mechanical seal maintains equilibrium among the closing force generated by outer pressure, the spring force, and the opening force produced by the pressure of the fluid film. The dynamics of the fluid film are governed by the Reynolds equation, while the characteristics of the lubricant change with variations in temperature and pressure. In normal operating conditions, the temperature of the coolant is typically maintained at approximately 40 °C, and the mechanical seal is responsible for reducing the pressure by about 176 bar, in which the pressure drops of the first, second, and third mechanical seals are about 74 bar (42%), 74 bar (42%), and 28 bar (16%), respectively. However, in an emergency where the external power is cut off, the coolant temperature may exceed 300 °C, and a single mechanical seal must withstand a pressure difference of 176 bar [1]. Moreover, fluid circulation becomes disrupted, leading to rapid changes in pressure and temperature. These changes significantly affect the performance of the mechanical seal. The coolant may exist in a mixed state of gas and liquid depending on the pressure and temperature at the seal surface, and the operational characteristics of the seal can vary as the ratio of gas to liquid changes. As shown by Hirs, liquid fluid film behavior can be either laminar or turbulent, depending on the operating conditions and design variables [2,3]. As demonstrated by Hahn et al., the behavior of fluid film in the gaseous state can vary among laminar, turbulent, or slip conditions based on the operating conditions and design variables [4]. The state of the fluid can be defined as laminar or turbulent according to the Reynolds number and as slip or non-slip according to the Knudsen number. As the Reynolds number increases, laminar flow changes to turbulent flow; as the Knudsen number increases, slip occurs between the gas and the solid. Therefore, it is essential to develop a method to analyze the mechanical seal with a mixed gas and liquid lubricant as well as laminar, turbulent, and slip conditions.

Many studies have been conducted to analyze the mechanical seal when gas and liquid coexist, but they have assumed that the lubricant is entirely liquid or gas. Several researchers investigated a phase change to gas in a mechanical seal with a liquid lubricant by applying the incompressible Reynolds equation of the liquid. Hughes et al. derived the relationship between the mass flow and pressure gradient of an incompressible lubricant using the continuity equation to analyze the boiling phenomenon at the lubricating surface of a mechanical seal with a liquid lubricant [5,6]. Lau et al. demonstrated that when a phase change occurs in a mechanical seal, the sealed liquid approaches saturation and the stiffness coefficient can become negative, leading to instability, which may result in failure even at low temperatures depending on the balance of the seal [7]. Yasuna et al. presented a vaporization model using an incompressible lubrication equation and an energy equation to predict the dynamic and thermodynamic behaviors of mechanical seals. They explained the abnormal behavior of seals when leakage increases considering phase change and convection effects, and they conducted experimental validation [8]. Etsion et al. utilized the incompressible lubrication equation and showed that phase change and tilting on the lubricating surface affect seal performance [9]. Other researchers have also analyzed the characteristics of mixed lubrication phenomena such as boiling and cavitation at the lubricating surface using the incompressible lubrication equation [10,11,12,13,14,15].

On the other hand, several researchers have investigated the phase change at the lubricating surface when the lubricant is gas using the compressible governing equation. Zhang et al. calculated the mixed-fluid region using enthalpy under two-phase flow and performed numerical predictions of static and dynamic responses for hybrid journal bearings using the compressible lubrication equation and energy equation [16]. Arauz et al. calculated the mixed-fluid region using enthalpy under two-phase flow conditions in ultra-cold fluid damper seals operating in the liquid–vapor region. They assessed seal performance and reliability using the continuity equation, the momentum equation, and the energy equation and showed that the compressibility effects of the vapor mixture significantly impact stiffness and rotational frequency ratios [17,18]. Jin et al. analyzed the stability of a mechanical seal with a spiral groove at high speeds using the two-dimensional compressible lubrication equation, the three-dimensional energy equation, and the heat conduction equation, demonstrating that a state change on the lubricating surface can lead to a sudden pressure change and instability [19,20]. Additionally, they analyzed the effects of a phase change in easily vaporizing fluid on performance, confirming that dynamic viscosity plays a crucial role in seal stability and phase-change phenomena. Other researchers have also analyzed the characteristics of mixed lubrication due to liquefaction on the lubricating surface when the operating fluid is gas using the compressible lubrication equation [21,22,23]. Even though many studies analyzing the mechanical seal when gas and liquid coexist have been published, they have assumed that the lubricant is entirely either liquid or gas without adequately addressing the mixed lubricant conditions experienced in many applications, such as the external power shutdown of an RCP.

In this study, we proposed a method to calculate the pressure, opening force, and leakage rate in a mechanical seal under the mixed lubricant state of gas and liquid of the mechanical seal in an RCP during external power shutdown. The pressure was calculated by solving the nonlinear finite element equation composed of the linear Reynolds equation of incompressible liquid and the nonlinear Reynolds equation of compressible gas using the Newton–Raphson method. The temperature distribution was calculated by solving the two-dimensional energy equation using the finite element method. Additionally, the effects of turbulence for the incompressible liquid and turbulence and slip effects for the compressible gas were included. The reliability of the developed program was confirmed by comparing the simulated opening force and leakage rate with the results of prior researchers. Using the developed method, the characteristics of the mechanical seal in an RCP during external power shutdown were analyzed.

2. Method of Analysis

2.1. Lubrication Analysis in Fluid Film

2.1.1. Finite Element Equation of Incompressible Lubricant

The two-dimensional incompressible Reynolds equation can be derived from the three-dimensional Navier–Stokes equation and the continuity equation. When the Reynolds number in the fluid between two plates exceeds approximately 1000, the flow changes from laminar to turbulent, and the effects of turbulence increase as the Reynolds number increases [2,3]. Hirs proposed the steady-state incompressible Reynolds equation including the turbulence effect, as shown in Equation (1) [2,3].

$${\displaystyle \frac{h^3}{\mu }}{\displaystyle \frac{\partial }{\partial r}}\left(G_r{\displaystyle \frac{\partial p}{\partial r}}\right)+{\displaystyle \frac{h^3}{\mu }}{\displaystyle \frac{\partial }{r\partial \theta }}\left(G_{\theta }{\displaystyle \frac{\partial p}{r\partial \theta }}\right)={\displaystyle \frac{r\omega }{2}}{\displaystyle \frac{\partial h}{r\partial \theta }}$$

(1)

where $h$, $\mu$, $p$, and $\omega$ represent the film thickness, viscosity, pressure, angular velocity, and $G_r$ and $G_{\theta }$ are flow coefficients. The flow coefficients change according to the state of the fluid; in the case of laminar flow, the coefficients are 12 for both the radial and circumferential directions, while those for turbulent flow are 0.0392 ${Re}^{0.75}$ and 0.0687 ${Re}^{0.75}$. The position-dependent film thickness in the fluid film can be expressed as Equation (2).

$$h=h_c+h_g$$

(2)

where $h_c$ and $h_g$ represent the minimum film thickness at the seal face and the groove depth, respectively. Equation (3) explains the boundary conditions for the inner and outer regions of the mechanical seal. The boundary conditions are specified at the inner and outer radii. Furthermore, the periodic boundary condition ensures the continuity of pressure in the circumferential direction.

$$\begin{array}{c}p(r=r_i,\theta )=p_i\\ p(r=r_o,\theta )=p_o\\ p(r,\theta )=p(r,\theta +2\pi )\end{array}$$

(3)

where $p_i$ and $p_o$ represent the inner and outer pressures, respectively.

To formulate the finite element representation of the Reynolds equation, a weighting function is utilized, and partial integration along with Green’s theorem is employed to derive Equation (4).

$${\displaystyle {\int }_{\Omega }\frac{h^3}{\mu }\left(G_r\frac{\partial w}{\partial r}\frac{\partial p}{\partial r}+G_{\theta }\frac{\partial w}{r\partial \theta }\frac{\partial p}{r\partial \theta }\right)d\Omega }-{\displaystyle {\int }_{\Omega }\frac{r\omega h}{2}\frac{\partial w}{r\partial \theta }d\Omega }=0$$

(4)

where Ω and $w$ represent the domain of interest and the weighting function, respectively. The pressure of a 4-node element can be defined through the nodal pressure ${\mathbf{P}}_e$, an arbitrary vector ${\mathsf{\eta }}_e$, and the shape function $\mathbf{N}$ and is expressed as shown in Equations (5) and (6).

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

(5)

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

(6)

Substituting Equations (5) and (6) into Equation (4), we obtain the following linear local finite element equation.

$${\eta }_e^T\left\{{\displaystyle {\int }_{\Omega }\frac{h^3}{\mu }\left(G_r\frac{\partial N}{\partial r}\frac{\partial N^T{P}_e}{\partial r}+G_{\theta }\frac{\partial N}{r\partial \theta }\frac{\partial N^T{P}_e}{r\partial \theta }\right)d\Omega }-{\displaystyle {\int }_{\Omega }\frac{r\omega h}{2}\frac{\partial N}{r\partial \theta }d\Omega }\right\}=0$$

(7)

2.1.2. Finite Element Equation of Compressible Lubricant

The two-dimensional compressible Reynolds equation can be derived with the additional application of the ideal gas assumption. Han et al. proposed a modified Reynolds equation for compressible thin flow as shown in Equation (8) to account for laminar and turbulent flow, as well as slip conditions occurring in thin-film flow [3].

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

(8)

where C_r and C_θ represent the fluid state coefficients in the radial and circumferential directions, respectively, taking into account the laminar, turbulent, and slip conditions of the fluid film as illustrated in

Table 1. Coefficients of the fluid state according to laminar, turbulent, and slip conditions [4].

Fluid Condition	Cr	Cθ
Laminar	1/12	1/12
Turbulent	1/Dr	1/Dθ
Slip	qp/12	qp/12

.

In Table 1, D_r, D_θ, and q_p represent the radial turbulent coefficient, circumferential turbulent coefficient, and slip coefficient, respectively. These coefficients are defined as shown in Equations (9)–(11).

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

(9)

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

(10)

$$q_p=n_0+n_1\left(Kn\right)+n_2{\left(Kn\right)}^2+n_3{\left(Kn\right)}^3$$

(11)

where Re and Kn represent the Reynolds number and Knudsen number, respectively. The constant of the slip coefficient is determined by the Knudsen number, as shown in

Table 2. Slip coefficients according to 1/Kn [4].

Range of Inverse Kn	n0	n1	n2	n3
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

.

To formulate the finite element representation of the compressible Reynolds equation, Equation (8) is multiplied by a weighting function and partial integration along with Green’s theorem is employed to yield the following expression:

$${\displaystyle \underset{\Omega }{\int }\frac{h^3}{\mu }\left(C_r\frac{\partial w}{\partial r}p\frac{\partial p}{\partial r}+C_{\theta }\frac{\partial w}{r\partial \theta }p\frac{\partial p}{r\partial \theta }\right)d\Omega }-{\displaystyle \underset{\Omega }{\int }\left(\frac{r\omega h}{2}p\frac{\partial w}{r\partial \theta }\right)d\Omega }=0$$

(12)

Substituting Equations (5) and (6) into Equation (12), as shown in Equation (7), generates the following nonlinear local finite element equation:

$${\eta }_e^T\left[{\displaystyle \underset{\Omega }{\int }\left\{\begin{array}{l}\frac{h^3}{\mu }\left(C_r\frac{\partial N}{\partial r}{{\rm N}}^{\mathrm{T}}{P}_{\mathrm{e}}\frac{\partial N^{\mathrm{T}}}{\partial r}{P}_{\mathrm{e}}+C_{\theta }\frac{\partial {\rm N}}{r\partial \theta }{{\rm N}}^{\mathrm{T}}{P}_{\mathrm{e}}\frac{\partial N^{\mathrm{T}}}{r\partial \theta }{P}_{\mathrm{e}}\right)\\ -\frac{r\omega h}{2}\frac{\partial {\rm N}}{r\partial \theta }{N}^{\mathrm{T}}{P}_{\mathrm{e}}\end{array}\right\}d\Omega }\right]=0$$

(13)

2.2. Thermal Analysis in Fluid Film

Assuming that the interface between the fluid film and the solid is adiabatic, we can express the governing two-dimensional energy equation for the fluid film as follows [24,25]:

$$\begin{array}{l}\rho C_p\left[\left({\displaystyle \frac{\omega rh}{2}}-{\displaystyle \frac{G_{\theta }{h}^3}{\mu }}{\displaystyle \frac{\partial p}{r\partial \theta }}\right){\displaystyle \frac{\partial T}{r\partial \theta }}+\left(-{\displaystyle \frac{G_r{h}^3}{\mu }}{\displaystyle \frac{\partial p}{\partial r}}\right){\displaystyle \frac{\partial T}{\partial r}}\right]\\ =\left[{\displaystyle \frac{G_{\theta }{h}^3}{\mu }}{\left({\displaystyle \frac{\partial p}{r\partial \theta }}\right)}^2+{\displaystyle \frac{G_r{h}^3}{\mu }}{\left({\displaystyle \frac{\partial p}{\partial r}}\right)}^2\right]+{\displaystyle \frac{\mu {\left(\omega r\right)}^2}{h}}\end{array}$$

(14)

where $C_p$ represents the specific heat capacity of the fluid. Equation (15) describes the boundary conditions for the fluid film presented in Equation (14). The outer boundary conditions consist of the temperatures applied at the inner and outer radii. Additionally, the periodic boundary condition indicates that the temperature remains continuous along the circumference.

$$\begin{array}{c}T(r=r_i,\theta )=T_i\\ T(r=r_o,\theta )=T_o\\ T(r,\theta )=T(r,\theta +2\pi )\end{array}$$

(15)

where T_i and T_o represent the inner and outer temperatures, respectively. To develop the finite element formulation of the energy equation, a weighting function is applied, partial integration is performed, and Green’s theorem is utilized to derive Equation (16).

$$\begin{array}{l}{\displaystyle {\int }_{\Omega }\rho C_p\left[\left(\frac{\omega rh}{2}\frac{\partial w}{r\partial \theta }T-\frac{G_{\theta }{h}^3}{\mu }\frac{\partial p}{r\partial \theta }\frac{\partial w}{r\partial \theta }T\right)-\left(\frac{G_r{h}^3}{\mu }\frac{\partial p}{\partial r}\frac{\partial w}{\partial r}T\right)\right]d\Omega }\\ -{\displaystyle {\int }_{\Omega }\left[\frac{G_{\theta }{h}^3}{\mu }w{\left(\frac{\partial p}{r\partial \theta }\right)}^2+\frac{G_r{h}^3}{\mu }w{\left(\frac{\partial p}{\partial r}\right)}^2\right]d\Omega }-{\displaystyle {\int }_{\Omega }w\frac{\mu {\left(\omega r\right)}^2}{h}d\Omega }=0\end{array}$$

(16)

The temperature of the four-node element can be expressed in terms of the nodal temperature ${\mathbf{T}}_e$, an arbitrary vector ${\mathsf{\eta }}_e$, and the shape function $\mathbf{N}$. Through the substitution of the nodal equation into Equation (16), the linear local finite element equation can be obtained, as shown in Equation (17).

$${\eta }_e^T\left[{\displaystyle {\int }_{\Omega }\begin{array}{l}\rho C_p\left\{\begin{array}{l}\left(\frac{\omega rh}{2}\frac{\partial N}{r\partial \theta }{N}^T{T}_e-\frac{G_{\theta }{h}^3}{\mu }\frac{\partial p}{r\partial \theta }\frac{\partial N}{r\partial \theta }{N}^T{T}_e\right)\\ -\left(\frac{G_r{h}^3}{\mu }\frac{\partial p}{\partial r}\frac{\partial N}{\partial r}{N}^T{T}_e\right)\end{array}\right\}d\Omega \\ -{\eta }_e^T{\displaystyle {\int }_{\Omega }\left[\frac{G_{\theta }{h}^3}{\mu }N{\left(\frac{\partial p}{r\partial \theta }\right)}^2+\frac{G_r{h}^3}{\mu }N{\left(\frac{\partial p}{\partial r}\right)}^2\right]d\Omega }-{\eta }_e^T{\displaystyle {\int }_{\Omega }N\frac{\mu {\left(\omega r\right)}^2}{h}d\Omega }\end{array}}\right]=0$$

(17)

2.3. Numerical Analysis Method

To analyze the pressure at the lubrication interface, the finite element matrix must be constructed considering the fluid flow. To accurately reflect the flow characteristics of the fluid, various flow conditions must be analyzed, and an appropriate mathematical model must be selected. Figure 2 illustrates the analysis method under gas–liquid flow conditions. In the case of element (1) where the temperature and pressure of the four nodes represent a liquid, the incompressible Reynolds equation is used to construct the local finite element matrix as shown in Equation (7). In the case of element (3) where the temperature and pressure of the four nodes represent a gas, the compressible Reynolds equation is used to construct the local finite element matrix as shown in Equation (13). In the case of element (2) where nodes 1 and 2 represent a gas and nodes 3 and 4 represent a liquid, the flow state of the fluid is determined using the average pressure and temperature of the four nodes. If the calculated flow state is liquid, the local element matrix is constructed using the incompressible Reynolds equation; if it is gas, the compressible Reynolds equation is used.

After combining local element matrices to form the global matrix, the global finite element equation becomes nonlinear. This nonlinear system can be solved using the Newton–Raphson method as follows:

$$\begin{array}{l}{R}_{L,G}^{\left(n\right)}+{\displaystyle \frac{\partial R_{L,G}^{\left(n\right)}}{\partial P_{L,G}^{\left(n\right)}}}\Delta P_{L,G}^{\left(n\right)}=0\\ R_l={\displaystyle \underset{\Omega }{\int }\left\{{\displaystyle {\int }_A\frac{h^3}{\mu }\left(G_r\frac{\partial N}{\partial r}\frac{\partial N^{\mathrm{T}}}{\partial r}{P}_{\mathrm{e}}+G_{\theta }\frac{\partial N}{r\partial \theta }\frac{\partial N^{\mathrm{T}}}{r\partial \theta }{P}_{\mathrm{e}}\right)}-{\displaystyle {\int }_{A}h\left(\frac{r\omega }{2}\frac{\partial N}{r\partial \theta }\right)}\right\}d\Omega }=0\\ {\displaystyle \frac{\partial R_l}{\partial P_l}}=k={\displaystyle \underset{\Omega }{\int }\left\{\frac{h^3}{\mu }\left(G_r\frac{\partial N}{\partial r}\frac{\partial N^{\mathrm{T}}}{\partial r}+G_{\theta }\frac{\partial N}{r\partial \theta }\frac{\partial N^{\mathrm{T}}}{r\partial \theta }\right)\right\}}d\Omega \\ R_g={\displaystyle \underset{\Omega }{\int }\left\{\frac{h^3}{\mu }{{\rm N}}^{\mathrm{T}}P\left(C_r\frac{\partial N}{\partial r}\frac{\partial N^{\mathrm{T}}}{\partial r}P+C_{\theta }\frac{\partial {\rm N}}{r\partial \theta }\frac{\partial N^{\mathrm{T}}}{r\partial \theta }P\right)-\frac{r\omega h}{2}\frac{\partial {\rm N}}{r\partial \theta }{N}^{\mathrm{T}}P\right\}d\Omega }=0\\ {\displaystyle \frac{\partial R_g}{\partial P_g}}={\displaystyle \underset{\Omega }{\int }\left\{\begin{array}{l}\frac{h^3}{\mu }\left(\begin{array}{l}{N}^{\mathrm{T}}P\left(C_r\frac{\partial N}{\partial r}\frac{\partial N^{\mathrm{T}}}{\partial r}+C_{\theta }\frac{\partial N}{r\partial \theta }\frac{\partial N^{\mathrm{T}}}{r\partial \theta }\right)\\ +\left(C_r\frac{\partial N}{\partial r}\frac{\partial N^{\mathrm{T}}}{\partial r}+C_{\theta }\frac{\partial N}{r\partial \theta }\frac{\partial N^{\mathrm{T}}}{r\partial \theta }\right)P{N}^{\mathrm{T}}\\ +\frac{\partial N}{\partial r}\frac{\partial N^{\mathrm{T}}}{\partial r}P{N}^{\mathrm{T}}P\frac{\partial C_r}{\partial P}+\frac{\partial N}{r\partial \theta }\frac{\partial N^{\mathrm{T}}}{r\partial \theta }P{N}^{\mathrm{T}}P\frac{\partial C_{\theta }}{\partial P}\end{array}\right)\\ -\frac{r\omega h}{2}\frac{\partial {\rm N}}{r\partial \theta }{N}^{\mathrm{T}}\end{array}\right\}d\Omega }\end{array}$$

(18)

where ${\mathit{R}}_l$, $\partial {\mathit{R}}_l/\partial {\mathit{P}}_l$, ${\mathit{R}}_g$, $\partial {\mathit{R}}_g/\partial {\mathit{P}}_g$, ${\mathit{R}}_{L,G}$, ${\displaystyle \frac{\partial {\mathit{R}}_{L,G}}{\partial {\mathit{P}}_{L,G}}},$ and n correspond to Equation (7), the pressure derivative of Equation (7), Equation (13), the pressure derivative of Equation (13), the global matrix from Equations (7) and (13), the pressure derivative of the global matrix, and the iteration count, respectively. The local matrices for ${\mathit{R}}_{l,g}$ and $\partial {\mathit{R}}_{l,g}/\partial {\mathit{P}}_{l,g}$ were assembled to create their respective global matrices. The analysis was repeated until the sum ratio of all components of the global matrix and the sum ratio of the pressure derivative matrix were less than 1 × 10⁻⁴. In most analyses, convergence was achieved within 15 iterations. The converged pressure enables the calculation of the opening force and the leakage rate, as shown in Equations (19) and (20).

$$F_{open}={\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_{r_i}^{r_o}prdr}d\theta }$$

(19)

$$\begin{array}{l}{Q}_L={\displaystyle {\int }_0^{2\pi }{G}_r\frac{h^3}{\mu }\frac{\partial p}{\partial r}rd\theta }\\ Q_G={\displaystyle {\int }_0^{2\pi }\frac{C_r{h}^3}{\mu \mathrm{RT}}p\frac{\partial p}{\partial r}}rd\theta \times {\displaystyle \frac{1}{\rho }}\end{array}$$

(20)

where $Q_L$ and $Q_G$ represent the leakage rates of the liquid and gas, respectively. The leakage rates can be calculated considering the flow of fluids that vary with temperature and pressure.

In an emergency where the external power supply is cut off, one must take into account how viscosity coefficients and density change with temperature and pressure to accurately assess the lubrication characteristics under high-temperature and high-pressure conditions. The viscosity coefficient is influenced by both temperature and pressure. In the case of liquids, the viscosity tends to decrease as the temperature increases until a phase change takes place, after which it begins to rise. However, at high pressures, the viscosity decreases with increasing pressure at the same temperature without a phase change occurring. Density also varies with temperature and pressure. The density of both liquid and gas increases with increasing pressure and decreases with increasing temperature. In this study, REFPROP (Version 10) was used to determine the viscosity coefficients and densities that vary with temperature and pressure.

Figure 3 illustrates the entire numerical procedure. First, the initial fluid flow was assumed for the design variables and operating conditions for each finite element and the viscosity coefficient and density of the working fluid. Then, thermal analysis was conducted using a finite element program to evaluate the temperature of the fluid film. Subsequently, the Reynolds number and Knudsen number were computed based on the fluid flow, and lubrication analysis was performed under laminar, turbulent, and slip conditions. An additional iterative process was implemented to find the film thickness that balances the forces, utilizing the bisection method as the iterative approach in this study. Minimum and maximum values for the film thickness were established to achieve this, and the analysis was performed multiple times until the calculated film thickness resulted in a difference between the opening and closing forces that was below 1 × 10⁻⁴. All processes were developed using the C++ programming language and were analyzed in conjunction with REFPROP to account for the fluid flow, viscosity coefficient, and density as functions of temperature and pressure.

3. Numerical Verification

3.1. Verification of the Liquid State in the Developed Program Compared to Previous Studies

To confirm the effectiveness of the developed program in liquid conditions, a finite element model of the spiral groove face seal, as analyzed by Bai, was developed [15]. Bai analyzed the effects of operating conditions on the performance and stability of the spiral groove face seal at high speeds. The outer radius, inner radius, groove radius, groove number, groove depth, groove angle, and film thickness of Bai’s model were 37 mm, 27 mm, 32.5 mm, 12, 6 μm, 18°, and 5 μm, respectively. The number of elements in the developed finite element model was 24,000. The inner pressure was set to 0.1 MPa, and the outer pressure was 3 MPa, while the rotational speed was increased from 0 to 25,000 rpm at increments of 5000 rpm to compare the leakage rates. Figure 4 shows the variation of leakage rate with rotational speed. The leakage rate in this study is similar to Bai’s results, with the maximum rates of 147.61 × 10⁻⁸ m³/s and 147.48 × 10⁻⁸ m³/s, respectively, showing a difference of less than approximately 1%.

3.2. Verification of the Gas State in the Developed Program Compared to Previous Studies

To confirm the effectiveness of the developed program under gas conditions, a finite element model of the dry gas seal, as analyzed by Zhang, was created [23]. Zhang analyzed the inner flow characteristics of the spiral grooved seal and compared the results of the analysis with experimental data. The outer radius, inner radius, groove radius, groove-to-ridge ratio, groove number, groove depth, and groove angle of Zhang’s model were 77.78 mm, 58.42 mm, 64 mm, 1 μm, 12, 5 μm, and 15°, respectively. The number of elements in the developed finite element model was 26,400. The film thickness was compared under two conditions of 3 and 6 μm, and the analysis was conducted with outer pressures ranging from 1.4 MPa to 2.7 MPa with an increment of 0.1 MPa at a rotational speed of 20,000 rpm. Figure 5a shows the variation in the opening force with the change in the outer pressure. The opening force in this study is similar to Zhang’s results, with the maximum opening forces of Zhang’s model and this study being 15.64 kN and 15.488 kN, respectively, showing a difference of less than approximately 2%. Figure 5b presents the leakage rate as a function of outer pressure. The leakage rate in this study is also similar to Zhang’s results, with a maximum leakage rate difference of 1.35 × 10⁻⁴ kg·s⁻¹ and 1.39 × 10⁻⁴ kg·s⁻¹, representing less than 3%.

4. Results and Discussion

4.1. Analysis Model

We investigated the lubrication characteristics of a mechanical seal under various operating conditions, including normal and external power shutdown conditions, by applying the proposed gas–liquid coupled analysis method. Figure 6 shows the shape of the stationary seal with a 12-sided groove. The mechanical seal used in the analysis consists of a stationary seal with grooves and a flat-shaped rotating seal, and the characteristics of the lubrication interface between the seals were analyzed. Figure 7 illustrates the finite element model of the 12-sided lubrication surface. The lubrication analysis and thermal analysis of the mechanical seal utilized the same finite element model, which was developed with 31,200 rectangular elements.

Table 3. Geometric parameters of the mechanical seal.

Parameter	Value
Outer radius of mechanical seal, ro [mm]	131
Groove radius of mechanical seal, rg [mm]	128.7
Inner radius of mechanical seal, ri [mm]	125
Radial groove length [mm]	1
Number of grooves [ea]	12
Groove depth [mm]	1

shows the design variables used in the lubrication evaluation of the mechanical seal, while

Table 4. Operating conditions and boundary conditions for pressure and temperature of the mechanical seal.

Parameter	Value
$\mathrm{Rotating} \mathrm{speed}, \omega$ [rpm]	1200
Outer pressure, po [bar]	66–176
Inner pressure, pi [bar]	1
Seal fluid temperature, To [K]	313, 583
Ambient temperature, Ti [K]	313, 583

explains the pressure and temperature constraints along with the operating conditions for the mechanical seal. The pressure boundary conditions were analyzed at 66 bar and 176 bar, including normal and external power shutdown conditions, while the temperature boundary conditions were analyzed at 313 K and 583 K. The temperature of 313 K reflects the typical stable operating condition of the mechanical seal, while the temperature of 583 K represents a scenario where the coolant temperature may rise during emergency situations when the external power is cut off. By establishing these temperature boundary conditions, we can comprehensively evaluate the performance of the mechanical seal under both normal operation and emergency conditions.

4.2. Characteristics of a Mechanical Seal Under Normal and External Power Shutdown Conditions

We analyzed the characteristics of the mechanical seal using the proposed gas–liquid coupled analysis method under normal and external power shutdown conditions. Under normal operating conditions, the inner pressure is 1 bar and the outer pressure is 66 bar. The temperature boundary condition was analyzed at 313 K. In the case of external power shutdown, the inner pressure remained at 1 bar, while the outer pressure rose to 176 bar, and the temperature boundary condition was set to 583 K. Figure 8 shows the pressure distribution, temperature distribution, and fluid flow distribution of the mechanical seal under normal operating conditions. The opening force, maximum pressure, maximum temperature, and leakage rate under normal operating conditions were 24,730 N, 66.03 bar, 328.42 K, and 3.92 × 10⁻⁵ m³/s, respectively. Figure 8a illustrates the pressure distribution on the lubrication surface, where the wedge effect generated by the grooves is weak, resulting in a maximum pressure very similar to the outer pressure. Figure 8b illustrates the temperature distribution on the lubrication surface, where the highest temperature is found at the inner diameter of the seal and a comparatively lower temperature is noted in the groove area. Figure 8c presents the fluid flow distribution, indicating that the operational fluid flow remained in a liquid state across the entire area, with no phase change occurring at the pressure and temperature values of the normal operating conditions. Figure 9 shows the pressure distribution, temperature distribution, and fluid flow of the mechanical seal under external power shutdown conditions. The opening force, maximum pressure, maximum temperature, and leakage rate under these conditions were 61,564 N, 176 bar, 637.3 K, and 19.1 × 10⁻⁵ m³/s, respectively. Figure 9a shows the pressure distribution on the lubrication surface, which is similar to the normal operating condition. The maximum pressure is very similar to the outer pressure. Figure 9b displays the temperature profile on the lubrication surface, indicating that the highest temperature is located at the inner diameter of the seal, while a lower temperature profile is observed in the groove region where there is a greater variation in temperature across the lubrication surface. Figure 9c illustrates the fluid flow distribution, showing liquid near the outer diameter and gas near the inner diameter. Near the outer diameter, the high-temperature operating fluid remains in the liquid state due to high pressure, while the pressure decreases and the temperature increases toward the inner diameter, leading to a phase change into gas. In the groove area, the temperature is lower and the pressure is higher compared to the ridge area, allowing it to remain in the liquid state.

4.3. Analysis of Mechanical Seals Due to Outer Pressure

To analyze the distribution of the opening force, leakage rate, and fluid flow within the mechanical seal according to outer pressure, the analysis was conducted by increasing the outer pressure from 66 to 176 bar with an increment of 10 bar under two temperature boundary conditions at a film thickness of 5 μm. Figure 10 shows the results of the fluid flow distribution on the lubrication surface for two temperature boundary conditions. At 313 K, there is no phase change in the operating fluid on the lubrication surface as the outer pressure increases. However, at 583 K, the fluid exists entirely in the gas phase from 66 bar to approximately 98.45 bar, after which a phase change occurs, resulting in a mixed gas–liquid state. As the pressure increases, the liquid phase increases while the gas phase decreases. At the maximum pressure of 176 bar, the gas and the liquid phases occupy 62% and 38% of the volume, respectively. Figure 11 shows the simulated opening force according to outer pressure. The opening force at 313 K, where the entire lubrication area is liquid, is higher than that at 583 K, where there is a significant gas area. Additionally, the increase in liquid area at 98.45 bar leads to a substantial increase in the opening force. The pressure concentration due to the wedge effect and pumping effect is very small, and the pressure at the lubrication surface is predominantly influenced by the boundary pressure, leading to relatively small differences in the opening force. Figure 12 shows the simulated leakage rate according to the outer pressure. The leakage rate at 313 K is higher than that at 583K across all outer pressure ranges because the operating fluid near the inner diameter exists as liquid at 313 K, while it exists as gas at 583 K.

5. Conclusions

In this paper, we proposed an analysis method to predict the opening force and leakage rate of the operating fluid in a mixed gas–liquid state of a mechanical seal caused by high temperature and high pressure during an external power shutdown of a reactor cooling pump. We considered the fluid flow using the incompressible Reynolds equation and the compressible Reynolds equation together, and we calculated the temperature distribution using a two-dimensional energy equation. Additionally, we accounted for the state of the fluid through coefficients that reflect turbulence and slip effects. We examined the pressure, opening force, and leakage rate of the mechanical seal by employing the finite element method along with the Newton–Raphson technique. To ensure the accuracy of the developed program, we validated it by comparing the simulated opening force and leakage rate with previously reported data. According to our analysis results, the opening force tends to decrease as the gas phase on the lubrication surface increases in a high-temperature environment, while the leakage rate tends to increase. Conversely, when the pressure increases, the gas phase decreases, resulting in increases in both the opening force and leakage rate. This study demonstrates that it is essential to consider the mixed gas–liquid state of the operating fluid to accurately predict the characteristics of mechanical seals for RCPs in high-temperature and high-pressure environments. This research will help predict the characteristics of mechanical seals with mixed gas–liquid lubricants and promote the development of a robust mechanical seal of RCPs for high-temperature and high-pressure environments.