Theoretical Expression and Screening of Real Gas Effect of Spiral Groove Dry Gas Seal

Abstract

The emergence of dry gas seals has revolutionized the form of fluid sealing. The traditional research and analysis of dry gas seals is carried out by considering the lubricating medium gas as an ideal gas, but at this stage, the sealing application environment is complicated, so it is necessary to consider the real gas effect of the lubricating medium gas to expand and break through the design system of dry gas seals. We choose seven common lubricating media in dry gas seal applications and screen the optimal density expression of the real gas using different real gas equations of state. Then, we study the extent to which the compression factors of different lubricating gases deviate from the ideal gas and analyze the errors of different real gas equations of state. These results can provide an optimal expression to clarify the mechanism by which the real gas effect affects the dry gas seal performance, which helps to grasp the nature of dry gas seals, predict the dry gas seal behavior, and guide the dry gas seal application.

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 CO₂ 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 CO₂ 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 CO₂ 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.

2. Geometric Model

The pump-in spiral groove dry gas seal is the most common form in the practical application of end face gas film sealing engineering at this stage. It has important value in rotating machinery and equipment. Its laboratory device is shown in Figure 1a; the structural assembly diagram of a common dry gas seal is shown in Figure 1b. The high-pressure side of the seal is located at the outer diameter of the sealing ring, and the low-pressure side is located at the inner diameter of the sealing ring. Its working principle is shown in Figure 1c. When the moving ring rotates, the sealed gas will be sucked into the groove along the tangential pump, and when moving along the groove toward the root of the groove, it is hindered by the sealing weir at the root. The gas gradually decelerates and is compressed, forming a force that pushes away the static ring. The specific geometric structure diagram of the end face of the dynamic ring with the spiral groove dry gas seal is shown in Figure 1d. The end face of the dynamic ring has N_g spiral grooves with a depth of h_g evenly opened in the circumferential direction. Due to the geometric particularity of the spiral, the spiral groove on the end face of the moving ring has convergence in the direction of gas flow. When the lubricating gas is pumped into the sealing gap, the dynamic pressure effect causes the dynamic and static rings to push apart to achieve non-contact sealing.

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:

$$p{V}_{\mathrm{m}}=\frac{Mp}{\rho }=R_{\mathrm{g}}T$$

(1)

Considering the compression factor of the lubricating gas, the equation of state of the real gas can be expressed as follows:

$$p{V}_{\mathrm{m}}=\frac{Mp}{\rho }=Z{R}_{\mathrm{g}}T$$

(2)

where M is the molar mass of the lubricating gas, numerically equal to the relative molecular mass of the lubricating gas; R_g 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 V_m 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 a_v and b_v 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:

$$Z^3-\frac{b_{v}p+RT}{RT}{Z}^2+\frac{a_{v}p}{R^2{T}^2}Z-\frac{a_v{b}_{\mathrm{v}}{p}^2}{R^3{T}^3}=0$$

(3)

where

$$\begin{gathered} a_v=\frac{27R^2{T}_{\mathrm{c}}^2}{64p_{\mathrm{c}}} \\ {b}_v=\frac{R{T}_{\mathrm{c}}}{8p_{\mathrm{c}}} \end{gathered}$$

p_c and T_c 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:

$$Z^3-Z^2-\left(\frac{p^2{b}^2}{R^2{T}^2}-\frac{ap}{R^2{T}^{2.5}}+\frac{bp}{RT}\right)Z-\frac{ab{p}^2}{R^3{T}^{3.5}}=0$$

(4)

In the formula, a and b are related to the critical pressure p_c and critical temperature T_c of the gas, and the specific expressions are as follows:

$$\begin{gathered} a=\frac{0.42748R^2{T}_{\mathrm{c}}^{2.5}}{p_{\mathrm{c}}} \\ b=\frac{0.08664R{T}_{\mathrm{c}}}{p_{\mathrm{c}}} \end{gathered}$$

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:

$$Z^3-Z^2-\left(\frac{p^2{b}_{SR}^2}{R^2{T}^2}-\frac{a_{SR}p}{R^2{T}^2}+\frac{b_{SR}p}{RT}\right)Z-\frac{a_{SR}{b}_{SR}{p}^2}{R^3{T}^3}=0$$

(5)

where

$$\begin{gathered} a_{SR}=\frac{0.42748R^2{T}_{\mathrm{c}}^2{\left[1+m_R\left(1-T_{\mathrm{r}}^{0.5}\right)\right]}^2}{p_{\mathrm{c}}} \\ {b}_{SR}=\frac{0.08664R{T}_{\mathrm{c}}}{p_{\mathrm{c}}} \\ {T}_{\mathrm{r}}=\frac{T}{T_{\mathrm{c}}} \\ {m}_R=0.48+1.574\epsilon -0.176{\epsilon }^2 \end{gathered}$$

where T_r 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:

$$Z^3+\left(\frac{b_{p}p}{RT}-1\right)Z^2-\left(\frac{3b_p^2{p}^2}{R^2{T}^2}-\frac{a_{p}p}{R^2{T}^2}+\frac{2b_{p}p}{RT}\right)Z-\frac{b_p{p}^2}{R^2{T}^2}\left(\frac{a_p+8b_p^{2}p}{RT}+4b_p\right)=0$$

(6)

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:

$$\begin{gathered} a_p=\frac{0.45724R^2{T}_{\mathrm{c}}^2{\left[1+k_p\left(1-T_{\mathrm{r}}^{0.5}\right)\right]}^2}{p_{\mathrm{c}}} \\ {b}_p=\frac{0.0778R{T}_{\mathrm{c}}}{p_{\mathrm{c}}} \\ {k}_p=0.3746+1.54226\epsilon -0.26992{\epsilon }^2 \end{gathered}$$

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:

$$Z=\frac{pv}{RT}\approx 1+B\left(\frac{p}{RT}\right)+\left(C-B^2\right){\left(\frac{p}{RT}\right)}^2+\cdots$$

(7)

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).

$$\begin{array}{l}\frac{B{p}_{\mathrm{c}}}{R{T}_{\mathrm{c}}}=0.1445-\frac{0.330}{T_{\mathrm{r}}}-\frac{0.1385}{T_{\mathrm{r}}^2}-\frac{0.0121}{T_{\mathrm{r}}^3}-\frac{0.000607}{T_{\mathrm{r}}^8}\\ +\epsilon \left(0.0637+\frac{0.331}{T_{\mathrm{r}}^2}-\frac{0.423}{T_{\mathrm{r}}^3}-\frac{0.008}{T_{\mathrm{r}}^8}\right)\end{array}$$

(8)

$$C{\left(\frac{p_{\mathrm{c}}}{R{T}_{\mathrm{c}}}\right)}^2=\left[\begin{array}{l}\left(0.01407+\frac{0.02432}{T_{\mathrm{r}}^{2.8}}-\frac{0.00313}{T_{\mathrm{r}}^{10.5}}\right)-\\ \epsilon \left(-0.02676+\frac{0.0177}{T_{\mathrm{r}}^{2.8}}+\frac{0.04}{T_{\mathrm{r}}^3}-\frac{0.003}{T_{\mathrm{r}}^6}-\frac{0.00228}{T_{\mathrm{r}}^{10.5}}\right)\end{array}\right]\cdot \left(\frac{T}{T_{\mathrm{c}}}\right)$$

(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):

$$Z=1+B_{z}p+C_z{p}^2$$

(10)

The virial coefficient in the above formula satisfies the following expression:

$$B_z=\left(\begin{array}{l}\frac{4.2043632}{T}-\frac{804.35292}{T^2}-\\ \frac{2.5274312\times {10}^5}{T^3}+\frac{1.11227568\times {10}^7}{T^4}\end{array}\right)\times {10}^{-6}$$

(11)

$$C_z=\left[\begin{array}{l}\left(\frac{18.22306}{T^2}-\frac{2.761559\times {10}^3}{T^3}+\frac{9.149028\times {10}^5}{T^4}\right)\\ -{\left(\begin{array}{l}\frac{4.2043632}{T}-\frac{804.3529}{T^2}-\\ \frac{2.5274312\times {10}^5}{T^3}+\frac{1.11227568\times {10}^7}{T^4}\end{array}\right)}^2\end{array}\right]\times {10}^{-12}$$

(12)

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:

$$Z=1+\frac{B_{\mathrm{c}}p}{T}$$

(13)

In the formula, B_c is a constant whose value is equal to

$$B_{\mathrm{c}}=1.9155\times {10}^{-6}\mathrm{K}/\mathrm{P}a$$

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. Expression of each coefficient of Equation (14).

Category of Equations	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$
VDW	$-\frac{b_{v}p+RT}{RT}$	$\frac{a_{v}p}{R^2{T}^2}$	$-\frac{a_v{b}_v{p}^2}{R^3{T}^3}$
R-K	−1	$-\left(\frac{p^2{b}^2}{R^2{T}^2}-\frac{ap}{R^2{T}^{2.5}}+\frac{bp}{RT}\right)$	$-\frac{ab{p}^2}{R^3{T}^{3.5}}$
S-R-K	−1	$-\left(\frac{p^2{b}_{SR}^2}{R^2{T}^2}-\frac{a_{SR}p}{R^2{T}^2}+\frac{b_{SR}p}{RT}\right)$	$-\frac{a_{SR}{b}_{SR}{p}^2}{R^3{T}^3}$
P-R	$\left(\frac{b_{p}p}{RT}-1\right)$	$-\left(\frac{3b_p^2{p}^2}{R^2{T}^2}-\frac{a_{p}p}{R^2{T}^2}+\frac{2b_{p}p}{RT}\right)$	$-\frac{b_p{p}^2}{R^2{T}^2}\left(\frac{a_p+8b_p^{2}p}{RT}+4b_p\right)$

.

$$Z^3+a_1{Z}^2+a_{2}Z+a_3=0$$

(14)

4. Error Analysis of Real Gas Compression Factor and Screening of Density Expression

According to the aforementioned basis, lubricating gas can be regarded as an ideal gas under low-pressure conditions. For the research objects of this article, nitrogen, hydrogen, helium, air, methane, and other medium gases, the working range of the medium pressure and inlet temperature of the dry gas seal are set to 4~20 MPa, 273.15~473.15 K; for carbon dioxide, the working range of the medium pressure and inlet temperature of the dry gas seal are set to 4~20 MPa, 343.15~473.15 K; for water vapor, the working range of the medium pressure and inlet temperature of the dry gas seal are set to 1~5 MPa, 543.15~773.15 K.

For lubricating gases, the NIST database is selected as the reference standard to calculate the relative error between the real gas state equation calculation results and the NIST database. The relevant parameters for calculating the lubricating gas compression factor are shown in

Table 2. Lubricating gas compression factor calculation parameters.

	Parameters	$\mathbf{Critical} \mathbf{Pressure} {\mathit{p}}_{\mathit{c}}$ (MPa)	$\mathbf{Critical} \mathbf{Temperature} {\mathit{T}}_{\mathit{c}}$ (K)	Eccentricity Factor $\mathit{\epsilon }$	Molar Mass $\mathit{M}$ (g/mol)
Real Gas
nitrogen		3.3958	126.192	0.0372	28.0135
hydrogen		1.2964	33.145	−0.219	2.0159
carbon dioxide		7.3773	304.1282	0.22394	44.0098
air		3.77	132.45	0.0357	28.9586
helium		0.22746	5.1953	−0.382	4.0026
water vapor		22.064	647.096	0.3443	18.0153
methane		4.5992	190.564	0.01142	16.0428

. Here, we define the compression factor. The relative error expression is as follows:

$$E_Z=\frac{Z_{REAL}-Z_{NIST}}{Z_{NIST}}\times 100\%$$

(15)

4.1. Nitrogen

The 3D surface diagram of the nitrogen compression factor, density, and dynamic viscosity changing with temperature and pressure obtained using the NIST database is shown in Figure 2. It can be seen from Figure 2a that the compression factor of nitrogen gradually increases as the pressure and temperature increase, and the value changes around 1; as the temperature and pressure gradually increase, the compression factor becomes farther and farther away from 1. The larger it is, the more nitrogen cannot be regarded as an ideal gas at this time. From Figure 2b, it can be seen that the density of nitrogen gradually increases with the increase in pressure, and this upward trend is especially obvious at lower temperatures; with the increase in temperature, the density of nitrogen gradually decreases, because according to the equation of state of the gas, the temperature increases, the lubricating gas expands thermally, and the intermolecular interactions are enhanced, and the volume of the gas increases, but the mass remains unchanged, so the density decreases. Figure 2c shows the change diagram of the dynamic viscosity of nitrogen. Its trend is the same as that of the compression factor, which increases with the increase in temperature and pressure.

The nitrogen compression factor error cloud is shown in Figure 3. From the figure, it can be seen that the R-K equation and the S-R-K equation can control the error of the compression factor within 3% in the calculation range, which indicates that both equations can be used to express the density of the real gas of nitrogen in this calculation range. In addition, it can be seen from Figure 3d that although the maximum error value of the second-term virial equation is larger than that of the R-K equation and the S-R-K equation in the range of higher pressures and lower temperatures, the distribution of the compression factor computational error of the second-term virial equation is better than that of the above two equations in the computational range of higher temperatures, so the second-term virial equation can be chosen to characterize the real gas density of nitrogen at higher temperatures.

To determine the working condition parameters of nitrogen at a temperature of 453.15 K and pressure 4–20 MPa for Case1 and pressure 12 MPa and temperature 313.15–473.15 K for Case2, a comprehensive comparison of the R-K equation, S-R-K equation, and second-term virial equation found that the second-term virial equation had the optimal results under this calculation condition, so the second-term virial equation was adopted to represent the real gas equation of state of nitrogen. Based on the real gas equation of state, ideal gas equation of state, and NIST database, we obtained the density comparison change schematic shown in Figure 4. From the figure, it can be seen that the real gas density of nitrogen is slightly less than the ideal gas density, which is because the compression factor of nitrogen obtained under this calculation condition is greater than 1. In addition to this, the deviation between the real gas density of nitrogen and the ideal gas density gradually increases with the increase in pressure and temperature.

4.2. Carbon Dioxide

The 3D surface schematic of the variation in the carbon dioxide compression factor, density, and dynamic viscosity with temperature and pressure, derived using the NIST database, is shown in Figure 5. From Figure 5a, it can be seen that at lower temperatures, the state of matter of carbon dioxide is a supercooled liquid, so it is not informative. At higher temperatures, the compression factor of carbon dioxide increases with increasing temperature and decreases with increasing pressure, and this decreasing trend is less obvious at higher temperatures, but the compression factor of carbon dioxide is always less than 1 within the calculation range, which indicates that carbon dioxide is more easily compressed than an ideal gas. For lubricating gases, the trend of density change with temperature and pressure is consistent, and the detailed explanation of the reason was given in the analysis of nitrogen, and the subsequent study will not be a special explanation of the density change. As can be seen from Figure 5c, the change rule of the dynamic viscosity of carbon dioxide is no longer purely linear or nonlinear, and its change trend is more complicated, increasing with the increase in pressure and showing different laws with the change in temperature at different pressures; the dynamic viscosity of carbon dioxide rises with the increase in temperature when the pressure is lower, and the dynamic viscosity of carbon dioxide decreases firstly and then rises with the increase in temperature at higher pressures, and the extreme value exists.

The carbon dioxide compression factor error cloud is shown in Figure 6. From the figure, it can be seen that within the calculation range, the VDW equation can obtain better calculation results at lower pressures, but as the pressure increases, the error becomes sharply increased. Due to the large difference in the calculation results using different real gas equations of state, if we consider the problem purely from the trend of the distribution, the R-K equation, the S-R-K equation, and the third-term virial equation have a better error distribution, which indicates that all three equations can be used to express the density of the real gas of carbon dioxide within the range of the calculation, so the influence of the working condition parameters should be considered comprehensively in the characterization of the real gas of carbon dioxide.

To determine the carbon dioxide working condition parameters at temperature 453.15 K and pressure 4–20 MPa for Case1 and pressure 12 MPa and temperature 313.15–473.15 K for Case2, we comprehensively compared the real gas equation of state, and chose the third-term virial equation to characterize the carbon dioxide real gas. Based on the real gas equation of state, the ideal gas equation of state, and the NIST database, we obtained the density comparison change schematic shown in Figure 7. As can be seen from the figure, the carbon dioxide real gas density is significantly larger than the ideal gas under the calculated working conditions, indicating that its compression factor in this range deviates from 1 in the opposite direction to that of nitrogen, i.e., the carbon dioxide compression factor is less than 1, which indicates that carbon dioxide is easier to be compressed than the ideal gas. In addition, as the pressure increases and the temperature decreases, the deviation between the two gas states gradually increases, which is especially obvious at lower temperatures, so the carbon dioxide real gas effect significantly affects the performance of the dry gas seal.

4.3. Hydrogen

The 3D surface schematic of the variation in the hydrogen compression factor, density, and dynamic viscosity with temperature and pressure derived using the NIST database is shown in Figure 8. From Figure 8a, it can be seen that the compression factor of hydrogen is always greater than 1 in the calculated range, which indicates that hydrogen is more difficult to compress than an ideal gas. In addition, the compression factor of hydrogen increases with increasing pressure and decreases with increasing temperature. For the dynamic viscosity of hydrogen, which increases with increasing temperature, the change in pressure has little effect and remains essentially unchanged.

The cloud of errors in the theoretical calculation of the hydrogen compression factor is shown in Figure 9. Purely from the numerical point of view, for hydrogen, the compression factor calculation result of the third-term virial equation is much larger than the NIST database, and the maximum error can be up to 50%, which is obviously not applicable to the calculation of the hydrogen compression factor. Additionally, the errors of the VDW, R-K, S-R-K, and Chen equations are all less than 3.11% in the range of the studied working conditions, among which the computational errors corresponding to the R-K, S-R-K, and Chen equations can even be accurate to less than 0.7%, which indicates that these three equations of state can be used for the expression of the real gas density of hydrogen. Compared with the remaining two equations, the Chen equation has a more obvious expression advantage, which is due to the fact that the Chen equation itself is based on the fitting of NIST data, so its calculation results are more compatible with the NIST database. Therefore, the Chen equation was chosen to express the real gas density of hydrogen within the calculation range.

The hydrogen operating parameters were determined to be the following: Case1 at temperature 453.15 K and pressure 4–20 MPa and Case2 at pressure 12 MPa and temperature 313.15–473.15 K. The Chen equation was chosen to characterize the real gas of hydrogen. Based on the real gas equation of state, the ideal gas equation of state, and the NIST database, we obtained the density comparison change schematic shown in Figure 10. It can be seen from the figure that, in the calculation of working conditions, hydrogen real gas density and ideal gas density with the temperature and pressure change rule are similar to nitrogen; that is, the real gas density of hydrogen is less than the ideal gas density, and the compression factor is greater than 1, indicating that hydrogen is more difficult to compress than the ideal gas. In addition, as the pressure rises and the temperature decreases, the deviation between the two gas states gradually increases, which is especially obvious when the temperature is lower, which is similar to the rule of change in carbon dioxide, so the hydrogen real gas effect will also significantly affect the performance of the dry gas seal.

4.4. Air

The 3D surface schematic of the variation in the air compression factor, density, and dynamic viscosity with temperature and pressure derived using the NIST database is shown in Figure 11. From Figure 11a, it can be seen that the compression factor of air gradually increases with increasing pressure and temperature, with values varying above and below 1; as the temperature and pressure gradually increase, the compression factor moves away from 1 by an increasing magnitude. Figure 11c illustrates a plot of the variation in the dynamic viscosity of air, which increases with increasing temperature and pressure, which is nearly the same trend as that of nitrogen.

Since the NIST database specifies that air is regarded as a mixture of nitrogen, oxygen, and argon with molar components of 0.7812, 0.2096, and 0.0092 in that order, the computational error of the air compression factor in the computational range is shown in Figure 12. From the figure, it can be seen that the third-term virial equation for dry gas can provide extremely high computational accuracy, except for some high-pressure and low-temperature conditions, and its computational error can be less than 1%. Although the R-K equation and the S-R-K equation can provide the same high computational accuracy, the third-term virial equation for dry gas is used to characterize the real gas effect of air.

To determine the air condition parameters at temperature 453.15 K and pressure 4–20 MPa for Case1 and pressure 12 MPa and temperature 313.15–473.15 K for Case2, we chose the third-term virial equation for dry gas to characterize the air real gas. Based on the real gas equation of state, the ideal gas equation of state, and the NIST database, we obtained the density comparison change schematic shown in Figure 13. As can be seen from the figure, the air compression factor exhibits a tendency to be greater than the ideal gas compression factor of 1 under the calculated conditions, and the deviation between the air compression factor and 1 becomes more pronounced as the pressure and temperature increase. Since the compression factor of air at lower temperatures will be less than 1 within a certain pressure range, this indicates that air will have different real gas effects under different conditions.

4.5. Helium

The 3D surface schematic of the variation in helium compression factor, density, and dynamic viscosity with temperature and pressure derived using the NIST database is shown in Figure 14. Since both helium and hydrogen are quantum gases with similar atomic structure arrangements, it can be seen from Figure 14a that the compression factor of helium is always greater than 1 within the calculated range, which indicates that helium is more difficult to be compressed than an ideal gas. In addition, the compression factor of helium increases with pressure and decreases with temperature. For the dynamic viscosity of helium, which increases with increasing temperature, the change in pressure has little effect on it and remains essentially unchanged, and these results are similar to those for hydrogen.

The helium compression factor theory calculation error cloud is shown in Figure 15. The S-R-K equation can provide higher computational accuracy within the computational range, and its error with the NIST database can be kept within 1%. Other equations, especially the R-K equation, can also provide smaller errors in most calculation ranges, but all things considered, the S-R-K equation is used in the subsequent calculations to express the real gas density of helium.

The helium operating parameters were determined to be the following: Case1 at temperature 453.15 K and pressure 4–20 MPa and Case2 at pressure 12 MPa and temperature 313.15–473.15 K. The S-R-K equation was chosen to characterize the helium real gas. Based on the real gas equation of state, the ideal gas equation of state, and the NIST database, we obtained the density comparison change schematic shown in Figure 16. Based on the previous analysis, the variation trend of helium with temperature and pressure is exactly the same as that of hydrogen, which is not repeated here.

4.6. Methane

The 3D surface schematic of the variation in the compression factor, density, and dynamic viscosity of methane with temperature and pressure derived from the NIST database is shown in Figure 17. From Figure 17a, it can be seen that when the temperature of methane is lower than 400 K, the compression factor exhibits a trend of decreasing and then slightly increasing with increasing pressure. For the dynamic viscosity of methane, as shown in Figure 17c, it increases with increasing temperature at lower pressure and decreases and then increases with increasing temperature at higher pressure. This is not the same as any of the previous results.

The methane compression factor error cloud is shown in Figure 18. In the calculation range, all the equations except the virial equation can provide high calculation accuracy. From the figure, it can be seen that the maximum error of the calculation using the virial equation will be more than 25%, so for methane, the virial equation cannot be used. Comparing the results of the computational errors of the other equations, it is found that the R-K equation can provide a computational error of less than 3%, and its error distribution is more uniform, so in the subsequent calculations, the R-K equation is used to characterize the real gas density of methane.

The methane operating parameters were determined to be the following: Case1 at temperature 453.15 K and pressure 4–20 MPa and Case2 at pressure 12 MPa and temperature 313.15–473.15 K. The R-K equation was chosen to characterize the methane real gas. A schematic diagram of the density comparison change based on the real gas equation of state, ideal gas equation of state, and NIST database is obtained, as shown in Figure 19. From Figure 19a, it can be seen that the density difference between the two states is not significant because the compression factor of methane varies less around 1 under the Case1 calculation condition, and it can be approximated as an ideal gas. Figure 19b, on the other hand, shows the difference between the real gas density of methane and the ideal gas at different temperatures. The density of methane is greater than that of the ideal gas under the calculation condition because the compression factor of methane is less than 1 in this case, and the lower the temperature is, the more obvious the difference between the two states is.

4.7. Water Vapor

The 3D surface schematic of the variation in the compression factor, density, and dynamic viscosity of water vapor with temperature and pressure derived using the NIST database is shown in Figure 20. From Figure 20a, it can be seen that the compression factor of water vapor does not vary much under most operating conditions, and the value is very close to 1, i.e., water vapor can be considered as an ideal gas under these conditions. For the dynamic viscosity of water vapor, as shown in Figure 20c, it increases with the increase in temperature, and the change in pressure has little effect on it and basically does not change, which is similar to the results of the calculations for hydrogen and helium.

The water vapor compression factor error cloud is shown in Figure 21. In the calculation range, the virial equation has higher accuracy, and the third-term virial equation has a higher calculation accuracy than the second-term virial equation; the calculation error of the third-term virial equation is not more than 3.5%, and its error in most of the working conditions is not more than 1% in most cases. Therefore, the third-term virial equation is used to characterize the real gas density of water vapor within the range of the calculation conditions.

To determine the water vapor condition parameters at temperature 453.15 K and pressure 4–20 MPa for Case1 and pressure 12 MPa and temperature 313.15–473.15 K for Case2, we chose the third-term virial equation to characterize the real gas of water vapor. Based on the real gas equation of state, ideal gas equation of state, and NIST database, we obtained the density comparison change schematic shown in Figure 22. From Figure 22a, it can be seen that the difference in density between the two states is not significant because the compression factor of water vapor changes less around 1 when the pressure is lower under the Case1 calculation conditions, which can be approximated to be an ideal gas, but as the pressure rises, the difference between the two is gradually revealed, and then water vapor cannot be regarded as an ideal gas. Figure 22b shows the difference between the real gas density of water vapor and the ideal gas at different temperatures; the density of water vapor is greater than the density of the ideal gas under the calculated conditions; this is because the compression factor of water vapor in this case is less than 1, and the lower the temperature, the more obvious the difference between the two states, which is consistent with the trend of methane.

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.