Numerical Simulation of Supercritical Carbon Dioxide Dry Gas Seal

A theoretical model of the dry gas seal (DGS) was established, considering real gas, inertia, turbulence, and choked flow effects. The steady-state performance of supercritical carbon dioxide DGSs under high-pressure and high-speed operating conditions was analyzed. The results show that: the increase in groove depth will result in higher opening force and leakage; the opening force and leakage initially increase and then decrease with the increase in the spiral angle; the opening force and leakage increase with the increase in the inlet pressure; the increasing rotation speed leads to an increase in opening force and leakage.


Introduction
The dry gas seal (DGS) is the recommended sealing solution for the Brayton cycle system utilizing Supercritical carbon dioxide (S-CO2) as the working medium [1,2].This is due to its reliability, costeffectiveness, safety, and minimal leakage in comparison to other seal types [3].However, under supercritical conditions, the influences of the inertia, turbulence, and choked flow effects, ignored in the conventional design, becomes increasingly evident.Furthermore, the properties of CO 2 vary drastically near the critical point, and the effect of temperature on the properties of the lubricating medium may not be ignored.
Numerous scholars have done research on DGSs.In 1995, Brand [4] derived the Reynolds equation by considering the inertia effect based on the theory of sliding lubrication.Under high-speed and highpressure operating conditions, the rapid variation of properties of CO 2 near the critical point led to the flow in the gas film of DGSs transforming from the laminar flow into the turbulent flow.Zakariya et al. [5] utilized a transport model of shear force to solve the Reynolds-averaged Navier-Stokes (RANS) equation accounting for the effect of the flow state.Furthermore, they investigated the effect of the operating conditions and structural parameters on the seal near the critical point of CO 2 .Zhang et al. [6] studied the effect of turbulence on S-CO 2 DGSs.The results revealed that the turbulence effect significantly affected the opening force and leakage of DGSs, where the maximum reduction in leakage reached 78%.Baltadjiev et al. [7] examined the influence of the real gas effect of S-CO 2 on DGSs of the centrifugal compressor and found that the real gas effect near the critical point was prominent.Liu et al. [8] established the property database to investigate the effect of the real gas on the steady-state performance of DGSs.The influence of structure parameters and operating conditions on steady-state performance were obtained.Ma et al. [9] studied the real gas effect of S-CO 2 on the performance of DGSs and found that the choked flow was more likely to occur at the outlet of the end face seals due to abrupt changes in density and viscosity near the critical point.However, limited research has been conducted on the comprehensive performance of S-CO 2 DGSs, considering all four effects.
In this paper, we investigated the steady-state performance of S-CO 2 DGSs, taking into consideration the inertia, turbulence, real gas, and choked flow effects.The different structure parameters and operating conditions were considered into account.

Geometrical models
The principle of DGSs is analogous to that of thrust bearings and is based on the hydrodynamic lubrication theory.DGSs involve the relative motion of two surfaces, where fluid flows into the wedgeshaped gap, generating the fluid dynamic pressure to create the sealing effect.Fig. 1 shows the schematic diagram of the end face structure of the spiral groove DGS.

Reynolds equation
The Reynolds equation was modified to account for the four effects.The Reynolds equation is as follows:

Energy equation
Due to the high velocity of the fluid flow across the seal end faces, the heat carried away by the fluid flow significantly surpasses the conduction heat dissipation.Therefore, the heat conduction between the rotating ring, stationary ring, and fluid can be neglected.All the heat generated by friction loss and compression of the fluid is taken away by the fluid flow.Therefore, the energy governing equation in the adiabatic state is as follows:

Steady-state performance
The Reynolds equation is solved using the finite difference method to determine the pressure and temperature distribution of the gas film between seal surfaces, from which the opening force F o and leakage Q of DGSs can be obtained.The opening force and leakage are as follows: ( )

Calculation methods and steps
Firstly, the operating and structural parameters are set, and the density and viscosity of CO 2 are obtained using the interpolation method.Then, the Reynolds equation is solved to obtain the pressure distribution, and subsequently, the Energy governing equation is solved to obtain the temperature distribution.If the pressure field does not meet the equilibrium condition, the pressure distribution will be iteratively calculated, and the temperature distribution will be updated accordingly until the pressure and temperature fields reach equilibrium conditions.Afterward, the velocity field is calculated, and the occurrence of the choked flow is determined.If the choked flow occurs, the pressure boundary condition is increased.The calculation is repeated until M a ≤ 1.Finally, the steady-state performance parameters of the S-CO 2 DGS are calculated.

Analysis of steady-state performance
The dimensionless pressure and temperature fields of the S-CO 2 DGS are shown in Figs. 2 and 3. From Fig. 2, it can be observed that the pressure decreases slowly along the extension direction of the groove area.However, after the gas flows from groove to dam areas, the pressure p o rapidly decreases to the ambient pressure.This is due to the fluid dynamic pressure generated by the wedge-shaped gap in the groove area, which reduces pressure loss.In contrast, in the dam area, the gas film thickness is smaller than that in the groove area, and there is no wedge-shaped gap to generate the fluid dynamic pressure, resulting in decreases in the pressure.Fig. 3 illustrates that the temperature does not significantly vary in the groove area but undergoes significant changes in the dam area.This is because the pressure decrease in the groove area is sight, while it is rapid in the dam area, leading to decreases in temperature.From Fig. 4 (a), it can be observed that the opening force monotonically increases with the increase of dimensionless groove depth.As seen in Fig. 4 (b), the leakage gradually increases with the increase of Hg, and the increase of leakage becomes faster when Hg exceeds 2.3.These phenomena can be explained by the increase of Hg enhancing the pumping effect.On the one hand, it increases the fluid dynamic pressure.On the other hand, it increases the pumping volume and the leakage.When Hg increases from 1.0 to 2.3, the opening force increases by 5.8%, and the leakage increases by 3.1%, with the opening force increasing more than the leakage.However, after Hg exceeds 2.3, the increase in leakage becomes more significant.For instance, when Hg increases from 2.3 to 3.3, the leakage increases by 10%, while the opening force only increases by 2.6%.It implies that the increase in opening force is less than the increase in leakage.Therefore, the S-CO 2 DGS can achieve better steady-state performance when Hg is around 2.3.Figure 5 (a) depicts that the opening force increases with the increase of β until it reaches the maximum value at the spiral angle β = 45° and then slightly decreases with the further increase of β.This is because, at low values of β, the pumping effect is gradually enhanced with the increase of β, effectively improving the gas film pressure.However, when β increases beyond a certain degree, the shape of the spiral groove becomes close to that of a rectangular groove, and the pumping effect is weakened.Fig. 5 (b) reveals that the leakage is minor when β is small due to the weak pumping effect.When β is small, the pumping volume and leakage are low.The leakage reaches its maximum at spiral angle β = 45°.With the increase of β from 15° to 45°, the increase in the opening force is smaller than the increase in the leakage.It indicates that the negative effect of increasing β on the leakage is greater than the positive effect on the opening force.Therefore, the S-CO 2 DGS exhibits better steady-state performance at a spiral angle of β = 15°.From Fig. 6 (a), it can be observed that the opening force increases linearly with the increase of p i .This is because, as the p i increases, the density of CO 2 increases, leading to an increase in the gas film pressure.Meanwhile, Fig. 6 (b) demonstrates that the leakage increases linearly with the increase of p i .It is because the leakage is influenced by the density and rotation speed.As the density of CO 2 at the outlet increases, the leakage increases.From Fig. 7 (a), it can be obtained that the opening force monotonically increases with the increase of rotation speed, but the rate of increase slows down after reaching 15000 rpm.This is due to the increase in fluid dynamic pressure with the increase of rotation speed, resulting in an increase in the opening force.On the other hand, Fig. 7 (b) shows that the leakage increases with the rotation speed, with the leakage volume increasing by approximately 4.2% from 5000 rpm to 30000 rpm.Considering the comprehensive effect on the steady-state performance, the opening force increases by 1.7% from 5000 rpm to 30000 rpm, while the leakage increases by 4.2%.This implies that the increase in leakage is more significant than the increase in opening force.Therefore, lower speeds are more conducive to maintaining lower leakage when the opening force meets the requirements.

Conclusion
Based on the current operating conditions adopted in this study, conclusions are as follows: (1) When the dimensionless groove depth is in the range of 1.0 to 3.3, increasing Hg results in an increase in the opening force and leakage.However, when Hg exceeds 2. opening force slows down while the leakage significantly increases.The optimal steady-state performance parameters for the dimensionless groove depth can be obtained in 1.0 to 2.3.
(2) For the spiral angle in the scope of 15° to 80°, both the opening force and leakage initially increase with the increase of the spiral angle but then decrease, reaching a maximum at 45°.However, the increase of opening force is minor compared to that of leakage when the spiral angle is from 15° to 45°.It is not conducive to achieving good steady-state performance when the spiral angle is too large.
(3) The increase in inlet pressure results in an increase in opening force and leakage.
(4) The opening force and leakage increase as the rotation speed increases.However, the increase in leakage is more significant than the increase in opening force.Under the condition of meeting the opening force requirements, lower rotation speeds are obtained to be beneficial for maintaining lower leakage in S-CO 2 DGSs.
These conclusions provide insights into the steady-state performance of DGSs under different operating conditions.Furthermore, they provide significant guidance for the future development of S-CO 2 DGSs.

Figure 1 .
Figure 1.Structure of spiral groove DGS

Figure 2 .Figure 3 .
Figure 2. Dimensionless pressure field Figure 3. Temperature field 3.4.Influence factors on steady-state performance 3.4.1.Groove depth.The groove depth is an essential parameter, and its influence on the steady-state performance is significant.The dimensionless groove depths are considered as 1.0, 1.3, 1.6, 2.0, 2.3, 2.6, 3.0, and 3.3, and the variation of steady-state performance parameters with groove depth is shown in Figs. 4 (a) and (b).

Figure 4 .
Figure 4. Effect of groove depth on steady-state performance 3.4.2.Spiral angle.The steady-state performance of DGSs is dramatically influenced by the spiral angle.In this paper, the spiral angles of 15°, 18°, 20°, 25°, 30°, 45°, 50°, 60°, and 70° are investigated, and the steady-state performance parameters are shown in Figs. 5 (a) and (b).Figure5(a) depicts that the opening force increases with the increase of β until it reaches the maximum value at the spiral angle β = 45° and then slightly decreases with the further increase of β.This is because, at low values of β, the pumping effect is gradually enhanced with the increase of β, effectively improving the gas film pressure.However, when β increases beyond a certain degree, the shape of the spiral groove becomes close to that of a rectangular groove, and the pumping effect is weakened.Fig.5 (b) reveals that the leakage is minor when β is small due to the weak pumping effect.When β is small, the pumping volume and leakage are low.The leakage reaches its maximum at spiral angle β = 45°.With the increase of β from 15° to 45°, the increase in the opening force is smaller than the increase in the leakage.It indicates that the negative effect of increasing β on the leakage is greater than the positive effect on the opening force.Therefore, the S-CO 2 DGS exhibits better steady-state performance at a spiral angle of β = 15°.

Figure 6 .
Figure 6.Effect of inlet pressure on steady-state performance 3.4.4.Rotation speed.To get an insight into the effect of rotation speed on the steady-state performance of DGSs, the range of rotation speed from 5000 rpm to 30,000 rpm is considered.The results are shown in Figs.7 (a) and (b), which illustrate the effect of rotation speed on steady-state performance parameters.From Fig.7(a), it can be obtained that the opening force monotonically increases with the increase of rotation speed, but the rate of increase slows down after reaching 15000 rpm.This is due to the increase in fluid dynamic pressure with the increase of rotation speed, resulting in an increase in the opening force.On the other hand, Fig.7 (b)shows that the leakage increases with the rotation speed, with the leakage volume increasing by approximately 4.2% from 5000 rpm to 30000 rpm.Considering the comprehensive effect on the steady-state performance, the opening force increases by 1.7% from 5000 rpm to 30000 rpm, while the leakage increases by 4.2%.This implies that the increase in leakage is more significant than the increase in opening force.Therefore, lower speeds are more conducive to maintaining lower leakage when the opening force meets the requirements.

Figure 7 .
Figure 7. Effect of rotation speed on steady-state performance

Table 1 .
Parameters of the DGS 3, the rate of increase of