1. Introduction
Dry gas seal (DGS) is a major innovation in fluid sealing technology [
1]. As a sealing form that uses dry clean gas as the lubricating medium, its superior sealing performance has been widely used in the petrochemical industry, centrifugal compressors, and energy transportation. It is currently expanding into aerospace engines, high-temperature gas-cooled reactors, and other fields [
2,
3,
4,
5,
6].
The basis for the theoretical evaluation of sealing performance and sealing operation stability is to obtain the air film pressure distribution in the seal end face gap, and the characteristics of the lubricating gas (density, viscosity) have an important impact on the air film pressure distribution. In the field of gas lubrication, the state equation of gas is an important equation for solving lubrication characteristics. For a long time, researchers regarded lubricating gases as ideal gases, and the ideal gas state equation was chosen as the gas equation of state associated with the Reynolds equation. This is because most of the working conditions studied are low-pressure conditions, and the lubricating gas is regarded as an ideal gas. The results can guide the design and application of dry gas seals under low-pressure conditions. However, as the application of dry gas seals develops towards extremely harsh working conditions such as high temperature, high pressure, and high speed, such as in aerospace engines, compressors, high-temperature gas-cooled reactor main fans, etc., the lubricating gas in the sealing gap no longer follows the ideal gas relationship. Under high-pressure conditions, the gas behavior of common nitrogen and air will significantly deviate from ideal gas [
7]. In addition, some special gases, such as carbon dioxide, hydrogen, helium, water vapor, etc., have significantly deviated from the behavior of ideal gases in common pressure ranges. When the pressure increases, the degree of deviation will be more obvious. At this time, if the lubricating gas continues to be regarded as an ideal gas, the calculation results will produce large errors, and the traditional ideal gas equation of state is no longer applicable. Therefore, it is necessary to expand and break through the existing dry gas seal theoretical design system.
Strictly speaking, all gases that really exist in nature are real gases. When the pressure is low, the volume occupied by the molecules themselves and the interaction forces (attraction and repulsion) between molecules can be ignored. At this time, the gas can be regarded as ideal gas. An ideal gas is a scientifically abstract imaginary gas. Studying the characterization theory and methods of real gas effects suitable for dry gas sealing performance and application is the basis for conducting research on the impact of real gas effects on spiral groove dry gas sealing performance. The behavior of real gases that deviates from ideal gases, including deviations from the gas equation of state, deviations from the thermal process of gases, deviations from gas physical parameters, deviations from gas reaction processes, etc., is called the real gas effect. In dry gas seal research, obtaining the air film pressure in the seal end face gap is the basis for evaluating the sealing performance and the stability of the sealing operation.
S Thomas et al. [
8] calculated the compression factor of nitrogen through the van der Waals equation and gave the density–pressure relationship of nitrogen under three different research methods (ideal gas equation of state, van der Waals real gas equation of state, and database experimental data) at 400 K. A comparison chart was made. The results show that under low-pressure conditions, the density of ideal gases is not much different from the density of real gases. As the pressure increases, this difference gradually appears. On the basis of this research, S Thomas et al. [
9] conducted an in-depth study on the thermoelastic flow behavior of conical non-contact mechanical seals by comprehensively considering the real gas effect. The heat transfer rules between the fluid film and the end face of the sealing ring, the force deformation of the sealing ring, and the thermal deformation of the sealing ring were obtained. It was found that sealing ring deformation can significantly change the geometry and sealing performance of the fluid film. H Chen et al. [
10] fitted the NIST database and obtained the real gas state equation of hydrogen at a temperature of 173 K < T < 393 K and a pressure of 0.1 MPa < p < 100 MPa. Then, compared with the ideal gas equation of state, it was found that the real gas will be significantly different from the ideal gas under high-pressure conditions. Z M Fairuz et al. [
11] took the spiral groove dry gas seal as the research object, and the lubricating medium was carbon dioxide. Based on the S-W equation, they used the CFD method to comprehensively study the influence of the real gas effect of carbon dioxide on the steady-state performance of the spiral groove dry gas seal and reported the results. Comparisons were made with calculation models of ideal gas and air. The results show that when the studied working conditions (temperature and pressure) are close to the critical point of carbon dioxide, the real gas effect has a very significant impact on the sealing performance. When far from the critical point, supercritical carbon dioxide can be regarded as an ideal gas. After, Z M Fairuz [
12] performed isothermal simulations using ANSYS Fluent to investigate the effect of real gas effects on the performance of dry gas seals. The seal ring deformation was then analyzed by one-way coupling, and it was found that the thermal deformation had the most significant effect on the dry gas seal. This project provides some new insights into how to design a seal for supercritical CO
2 that reveal new flow physics and seal distortion management. Hong et al. [
13] proposed a new comprehensive analytical method in order to solve the problem of complex heat flow in dry gas seals with two different characteristic lengths. It was applied to the helium compressor of a high temperature gas-cooled reactor, and better results were obtained. Song Pengyun et al. [
14] used the hydrogen compression factor expression fitted by Chen to connect the real gas state equation and the pressure control equation and used the analytical method to analyze the impact of the real gas effect on the spiral groove dry gas sealing performance. Since hydrogen has a compression factor Z > 1, it is more difficult to compress than an ideal gas. Under standard conditions, the leakage rate is smaller than that of an ideal gas. Hu Xiaopeng, Zhang Shuai et al. [
15,
16] used the virial equation to characterize the real gas effect, used carbon dioxide, hydrogen, and nitrogen as lubricating gases, and considered the influence of the slip flow effect to study the steady-state performance of dry gas seals. In order to more comprehensively study the air film lubrication characteristics under the pressure–temperature coupling field in the dry gas seal gap, Chanwen et al. [
17] used the R-K equation to characterize the real gas effect and numerically simulated the effect of the real gas effect on the spiral in the non-isothermal model. The influence rules of the tank dry gas seal opening force and leakage rate were studied. Yan et al. [
18] analyzed the turbulence effect of supercritical carbon dioxide dry gas seals through a combination of theoretical calculations and experimental verification and found that changes in operating conditions and geometric parameters will affect the performance of dry gas seals. Hu Hangling et al. [
19] pointed out that the temperature distribution in the flow field of supercritical CO
2 spiral groove dry gas seals is affected by the inlet pressure change and the pressure distribution of the flow field. Jiang Peng et al. [
20] compared and analyzed the temperature distribution and steady-state performance of CO
2 dry gas seals under an isothermal model and adiabatic model and pointed out that the isothermal model assumption is applicable to small film thickness and low-rotational speed conditions. Ding Junhua et al. [
21] considered the effects of turbulence, inertia, real gas, and obstruction on the flow field and sealing performance of the gas film generated by ultra-high speed and constructed a turbulence calculation model under multiple effects. They explored the influence of different working condition parameters and structural parameters on sealing performance under ultra-high-speed conditions.
Today, there are very few studies that consider the influence of the real gas effects of lubricating gases on the dynamic characteristics of dry gas seals. Shin Y S and Huber et al. [
22,
23] combined the continuity equation, momentum equation, and energy equation to numerically solve the dynamic characteristics of cylindrical gas seals considering the real gas effect of air. The results show that under high-pressure conditions, the real gas effect has a very significant impact on the seal leakage rate. Hu Xiaopeng et al. [
24] described the gas compression factor through the binomial truncation form of the virial equation, used the Pitzer equation to calculate the second virial coefficient, and solved the dynamic air film pressure, dynamic damping, and stiffness of the end face of the T-groove dry gas seal and compare with the ideal gas model. The results show that the deviation of the real gas dynamic characteristic parameters from the ideal gas dynamic characteristic parameters changes with the change in the compression factor.
There is no single equation that can accurately describe the real gas effect of all gases, so it is necessary to choose different real gas equations of state for different medium gases. Considering the actual scope of the engineering applications of dry gas seals, we choose a total of seven medium gases: nitrogen, carbon dioxide, hydrogen, helium, air, water vapor, and methane. Using several typical real gas equations of state in chemical thermodynamics (VDW equation, R-K equation, S-R-K equation, P-R equation, virial equation, etc.), we analyze the different medium gases and obtain the compression factor calculation formula under different real gas equations of state. Then, we study the extent to which the compression factors of various lubricating gases deviate from the ideal gas and screen out the best theoretical expression for the actual gas. These results are helpful for the existing theoretical design system of dry gas seals.
3. The Theoretical Expression for the Real Gas Effect
According to the existing research base, the equation of state of an ideal gas can be expressed as follows:
Considering the compression factor of the lubricating gas, the equation of state of the real gas can be expressed as follows:
where
M is the molar mass of the lubricating gas, numerically equal to the relative molecular mass of the lubricating gas;
Rg is the gas constant with a value of 8.314 J·(mol·K)
−1;
T is the temperature of the lubricating gas;
p is the pressure of the lubricating gas; and
Vm is the molar volume of the lubricating gas.
Comparing Equations (2) and (1), it can be seen that under the same temperature and pressure, the difference between the real gas equation of state and the ideal gas equation of state is reflected by a dimensionless coefficient. This coefficient is called the compression factor and represents how much a real gas deviates from an ideal gas. When Z > 1, it means that the real gas is more difficult to compress than the ideal gas; when Z = 1, it means that the real gas state equation is the ideal gas state equation; when Z < 1, it means that the real gas is easier to compress than the ideal gas. Obviously, the gas compression factor is only related to the pressure and temperature of the gas. The gas compression factor can be calculated using the state equation of real gases, but so far, there is no state equation suitable for various gases and various state regions with high calculation accuracy.
At this stage, researchers have proposed more than 200 equations of state that can be applied to non-ideal gases, which can be roughly divided into two categories. One category considers the structure of matter (such as the size of molecules, intermolecular forces, etc.), the physical meaning of this type of equation is clear and has certain universality, but the parameters in the equation need to be determined through experiments. The other type is the state equation of real gases derived by experimental, empirical, or semi-empirical methods and theoretical methods, such as the VDW equation, R-K equation, S-R-K equation, P-R equation, virial equation, etc., although this type of equation is only for specific cases. The gas has a high calculation accuracy and is not universally applicable, but this type of equation is still used in industrial applications.
The VDW equation (van der Waals equation) was proposed by van der Waals [
25] in 1873 based on previous research. The reason why this equation has received special attention is not because it is more accurate than other equations but because it proposes two physically meaningful correction factors
av and
bv in the volume and pressure terms when correcting the ideal gas equation of state; these two correction factors reveal the fundamental reason for the difference between real gas and ideal gas. The VDW equation cannot quantitatively accurately reflect the relationship between real gas state parameters, and the calculation of gases that are easily liquefied is not very accurate, which limits its application in engineering calculations to a certain extent.
The calculation formula of the lubricating gas compression factor obtained through the VDW equation is as follows:
where
pc and Tc are the critical pressure and critical temperature of the lubricating gas, respectively; R is the gas constant.
The Redlich–Kwong equation [
26] is obtained by modifying the model of the VDW equation. Compared with the VDW equation, the R-K equation has a simpler form and higher computational accuracy, which is very successful for gas–liquid-phase equilibria and mixtures and is applicable to nonpolar or weakly polar gases.
The lubricating gas compression factor obtained through the R-K equation is given below:
In the formula,
a and
b are related to the critical pressure
pc and critical temperature
Tc of the gas, and the specific expressions are as follows:
On the basis of the R-K equation model, Soave comprehensively considered the eccentricity factor and further corrected the physical property parameters to obtain the S-R-K equation (Soave–Redlich–Kwong equation) [
27], which greatly improved the calculation accuracy and application range of the R-K equation.
The calculation formula of the lubricating gas compression factor obtained through the S-R-K equation is as follows:
where
where
Tr is the contrast temperature, and
ϵ is the eccentricity factor.
Peng and Robinson [
28] found that the R-K equation performs poorly in calculating the density of liquids and improved the interaction term in 1976, proposing the P-R equation (Peng–Robinson equation), which is much more accurate than the R-K equation in predicting the volume of liquids and can be calculated for the gas phase, liquid phase, and hydrocarbons.
The lubricating gas compression factor obtained through the P-R equation is given below:
The computational coefficients in the P-R equation have a similar structure to those in the S-R-K equation, with the specific expression as follows:
The virial equation [
29], which is derived using statistical mechanics to analyze the forces between molecules, is a polynomial with respect to pressure and temperature, and its exact expression is given below:
In the formula, B and C are called the second virial coefficient and the third virial coefficient, respectively. The virial coefficient is related to the contrast temperature and the eccentricity factor. B reflects the interaction between two molecules, and C reflects the interaction between three molecules. As the number of molecules increases, the interaction force between molecules decreases, so the contribution of the higher-order terms in Formula (7) to the compression factor decreases in turn. Generally speaking, the quadratic term and the cubic term are sufficient. Extremely high calculation accuracy is obtained, but the calculation accuracy of the virial equation for the compression factor of high-pressure gas is not high, and it is not suitable for the calculation of the compression factor of mixed gas.
For pure substances, the second and third virial coefficients in the virial equation can be calculated using the Pitzer equation [
30,
31] and the Orbey equation [
32,
33], respectively. The specific expressions are as follows, as shown in (8) and (9).
When using the virial equation to describe the real gas behavior of air, Zhang Jingbo et al. [
34] directly gave the third-term virial equation for dry gas. The specific expression is as shown in Equation (10):
The virial coefficient in the above formula satisfies the following expression:
In fact, the state equations mentioned above all have high calculation accuracy within a certain range of conditions or are only applicable to a certain type of lubricating gas. The theoretical expressions describing gas relationships deviate to a certain extent from the real measured results. Obtaining gas relationship expressions by fitting measured data is an effective way to solve the above problems, but it is necessary to accurately obtain a wide range of (pressure, temperature) compression factor expressions which requires regression analysis on tens of thousands or more experimental data, which places high requirements on the selection of data analysis methods and fitting samples.
For hydrogen, Chen et al. [
10] obtained a state equation suitable for real hydrogen gas by fitting the NIST database data. The expression for calculating the lubricating gas compression factor is as follows:
In the formula,
Bc is a constant whose value is equal to
From a mathematical point of view, if pressure and temperature are regarded as independent variables, Equations (3)~(6) belong to cubic equations of one variable and can be written as a general expression; as shown in Equation (14), the Cardano formula can be used to directly solve and obtain the analytical expression of the compression factor. The coefficients in Equation (14) are shown in
Table 1.
5. Conclusions
With the application of a dry gas seal to high temperature, high pressure, and high-speed direction, the lubricating gas in the seal gap no longer follows the traditional ideal gas relationship, so the existing dry gas seal theory design system needs to be expanded.
(1) Considering the actual engineering application range of dry gas seals, we chose seven medium gases, including nitrogen, carbon dioxide, hydrogen, helium, air, water vapor, and methane. Using several typical real gas equations in chemical thermodynamics (VDW equation, R-K equation, S-R-K equation, P-R equation, virial equation, etc.), we analyzed the different medium gases and obtained the compression factor calculation formula under different real gas equations. Then, we studied the degree of deviation of the compression factors of various lubricating gases from the ideal gas and screened out the best theoretical expression for the real gas.
(2) There is no equation of state that can be suitable for various gases and state regions with high calculation accuracy, and we can only select them according to the specific medium gas and working condition range. By comparing the compression factor error maps of various gases with different media and considering the design working condition range, the second-term virial equation was selected to characterize the real gas effect of nitrogen, third-term virial equation for carbon dioxide, Chen equation for hydrogen, third-term virial equation for air, S-R-K equation for helium, R-K equation for methane, and the third-term virial equation for water vapor.
(3) The surface schematic diagrams of the compression factor, density, and dynamic viscosity with temperature and pressure for different medium gases in the range of the calculation working conditions were obtained from the NIST database, and it was found that nitrogen and air have certain similar patterns, and hydrogen and helium have certain similar patterns. Other lubricating medium gases show unique change rules under different temperature and pressure conditions, so for different medium gases, different real gas equations of state should be selected according to the actual working conditions.
(4) At the present stage of the dry gas seal performance research, there is no report on the compression factor–temperature–pressure triple solution, and the subsequent research can start from the perspective of the simultaneous coupling of the pressure control equation, the energy equation, and the real gas equation to obtain the theoretical analytical analysis model of the temperature field of the dry gas seal with the consideration of the real gas effect. In addition, the study of the real gas effect affecting the dynamic behavior of dry gas seals is also a direction for future development.